Yura is a mathematician, and his cognition of the world is so absolute as if he have been solving formal problems a hundred of trillions of billions of years. This problem is just that!Consider all non-negative integers from the interval $$$[0, 10^{n})$$$. For convenience we complement all numbers with leading zeros in such way that each number from the given interval consists of exactly $$$n$$$ decimal digits.You are given a set of pairs $$$(u_i, v_i)$$$, where $$$u_i$$$ and $$$v_i$$$ are distinct decimal digits from $$$0$$$ to $$$9$$$.Consider a number $$$x$$$ consisting of $$$n$$$ digits. We will enumerate all digits from left to right and denote them as $$$d_1, d_2, \ldots, d_n$$$. In one operation you can swap digits $$$d_i$$$ and $$$d_{i + 1}$$$ if and only if there is a pair $$$(u_j, v_j)$$$ in the set such that at least one of the following conditions is satisfied:  $$$d_i = u_j$$$ and $$$d_{i + 1} = v_j$$$,  $$$d_i = v_j$$$ and $$$d_{i + 1} = u_j$$$. We will call the numbers $$$x$$$ and $$$y$$$, consisting of $$$n$$$ digits, equivalent if the number $$$x$$$ can be transformed into the number $$$y$$$ using some number of operations described above. In particular, every number is considered equivalent to itself.You are given an integer $$$n$$$ and a set of $$$m$$$ pairs of digits $$$(u_i, v_i)$$$. You have to find the maximum integer $$$k$$$ such that there exists a set of integers $$$x_1, x_2, \ldots, x_k$$$ ($$$0 \le x_i < 10^{n}$$$) such that for each $$$1 \le i < j \le k$$$ the number $$$x_i$$$ is not equivalent to the number $$$x_j$$$.

The first line contains an integer $$$n$$$ ($$$1 \le n \le 50\,000$$$) — the number of digits in considered numbers. The second line contains an integer $$$m$$$ ($$$0 \le m \le 45$$$) — the number of pairs of digits in the set. Each of the following $$$m$$$ lines contains two digits $$$u_i$$$ and $$$v_i$$$, separated with a space ($$$0 \le u_i < v_i \le 9$$$). It's guaranteed that all described pairs are pairwise distinct.

Print one integer — the maximum value $$$k$$$ such that there exists a set of integers $$$x_1, x_2, \ldots, x_k$$$ ($$$0 \le x_i < 10^{n}$$$) such that for each $$$1 \le i < j \le k$$$ the number $$$x_i$$$ is not equivalent to the number $$$x_j$$$. As the answer can be big enough, print the number $$$k$$$ modulo $$$998\,244\,353$$$.

NoteIn the first example we can construct a set that contains all integers from $$$0$$$ to $$$9$$$. It's easy to see that there are no two equivalent numbers in the set.In the second example there exists a unique pair of equivalent numbers: $$$01$$$ and $$$10$$$. We can construct a set that contains all integers from $$$0$$$ to $$$99$$$ despite number $$$1$$$.


Tanechka is shopping in the toy shop. There are exactly $$$n$$$ toys in the shop for sale, the cost of the $$$i$$$-th toy is $$$i$$$ burles. She wants to choose two toys in such a way that their total cost is $$$k$$$ burles. How many ways to do that does she have?Each toy appears in the shop exactly once. Pairs $$$(a, b)$$$ and $$$(b, a)$$$ are considered equal. Pairs $$$(a, b)$$$, where $$$a=b$$$, are not allowed.

The first line of the input contains two integers $$$n$$$, $$$k$$$ ($$$1 \le n, k \le 10^{14}$$$) — the number of toys and the expected total cost of the pair of toys.

Print the number of ways to choose the pair of toys satisfying the condition above. Print 0, if Tanechka can choose no pair of toys in such a way that their total cost is $$$k$$$ burles.

NoteIn the first example Tanechka can choose the pair of toys ($$$1, 4$$$) or the pair of toys ($$$2, 3$$$).In the second example Tanechka can choose only the pair of toys ($$$7, 8$$$).In the third example choosing any pair of toys will lead to the total cost less than $$$20$$$. So the answer is 0.In the fourth example she can choose the following pairs: $$$(1, 1000000000000)$$$, $$$(2, 999999999999)$$$, $$$(3, 999999999998)$$$, ..., $$$(500000000000, 500000000001)$$$. The number of such pairs is exactly $$$500000000000$$$.


Every true king during his life must conquer the world, hold the Codeforces world finals, win pink panda in the shooting gallery and travel all over his kingdom.King Copa has already done the first three things. Now he just needs to travel all over the kingdom. The kingdom is an infinite plane with Cartesian coordinate system on it. Every city is a point on this plane. There are n cities in the kingdom at points with coordinates (x1, 0), (x2, 0), ..., (xn, 0), and there is one city at point (xn + 1, yn + 1). King starts his journey in the city number k. Your task is to find such route for the king, which visits all cities (in any order) and has minimum possible length. It is allowed to visit a city twice. The king can end his journey in any city. Between any pair of cities there is a direct road with length equal to the distance between the corresponding points. No two cities may be located at the same point.

The first line contains two integers n and k (1 ≤ n ≤ 105, 1 ≤ k ≤ n + 1) — amount of cities and index of the starting city. The second line contains n + 1 numbers xi. The third line contains yn + 1. All coordinates are integers and do not exceed 106 by absolute value. No two cities coincide.

Output the minimum possible length of the journey. Your answer must have relative or absolute error less than 10 - 6.


Vasya likes everything infinite. Now he is studying the properties of a sequence s, such that its first element is equal to a (s1 = a), and the difference between any two neighbouring elements is equal to c (si - si - 1 = c). In particular, Vasya wonders if his favourite integer b appears in this sequence, that is, there exists a positive integer i, such that si = b. Of course, you are the person he asks for a help.

The first line of the input contain three integers a, b and c ( - 109 ≤ a, b, c ≤ 109) — the first element of the sequence, Vasya's favorite number and the difference between any two neighbouring elements of the sequence, respectively.

If b appears in the sequence s print "YES" (without quotes), otherwise print "NO" (without quotes).

NoteIn the first sample, the sequence starts from integers 1, 4, 7, so 7 is its element.In the second sample, the favorite integer of Vasya is equal to the first element of the sequence.In the third sample all elements of the sequence are greater than Vasya's favorite integer.In the fourth sample, the sequence starts from 0, 50, 100, and all the following elements are greater than Vasya's favorite integer.


This is an easy version of the problem. The actual problems are different, but the easy version is almost a subtask of the hard version. Note that the constraints and the output format are different.You are given a string $$$s$$$ consisting of $$$n$$$ lowercase Latin letters.You have to color all its characters one of the two colors (each character to exactly one color, the same letters can be colored the same or different colors, i.e. you can choose exactly one color for each index in $$$s$$$).After coloring, you can swap any two neighboring characters of the string that are colored different colors. You can perform such an operation arbitrary (possibly, zero) number of times.The goal is to make the string sorted, i.e. all characters should be in alphabetical order.Your task is to say if it is possible to color the given string so that after coloring it can become sorted by some sequence of swaps. Note that you have to restore only coloring, not the sequence of swaps.

The first line of the input contains one integer $$$n$$$ ($$$1 \le n \le 200$$$) — the length of $$$s$$$. The second line of the input contains the string $$$s$$$ consisting of exactly $$$n$$$ lowercase Latin letters.

If it is impossible to color the given string so that after coloring it can become sorted by some sequence of swaps, print "NO" (without quotes) in the first line. Otherwise, print "YES" in the first line and any correct coloring in the second line (the coloring is the string consisting of $$$n$$$ characters, the $$$i$$$-th character should be '0' if the $$$i$$$-th character is colored the first color and '1' otherwise).


A prime number is a number which has exactly two distinct divisors: one and itself. For example, numbers 2, 7, 3 are prime, and 1, 6, 4 are not.The next prime number after x is the smallest prime number greater than x. For example, the next prime number after 2 is 3, and the next prime number after 3 is 5. Note that there is exactly one next prime number after each number. So 5 is not the next prime number for 2.One cold April morning Panoramix predicted that soon Kakofonix will break free from his straitjacket, and this will be a black day for the residents of the Gallic countryside.Panoramix's prophecy tells that if some day Asterix and Obelix beat exactly x Roman soldiers, where x is a prime number, and next day they beat exactly y Roman soldiers, where y is the next prime number after x, then it's time to wait for Armageddon, for nothing can shut Kakofonix up while he sings his infernal song.Yesterday the Gauls beat n Roman soldiers and it turned out that the number n was prime! Today their victims were a troop of m Romans (m > n). Determine whether the Gauls should wait for the black day after today's victory of Asterix and Obelix?

The first and only input line contains two positive integers — n and m (2 ≤ n < m ≤ 50). It is guaranteed that n is prime. Pretests contain all the cases with restrictions 2 ≤ n < m ≤ 4.

Print YES, if m is the next prime number after n, or NO otherwise.


Sometimes Mister B has free evenings when he doesn't know what to do. Fortunately, Mister B found a new game, where the player can play against aliens.All characters in this game are lowercase English letters. There are two players: Mister B and his competitor.Initially the players have a string s consisting of the first a English letters in alphabetical order (for example, if a = 5, then s equals to "abcde").The players take turns appending letters to string s. Mister B moves first.Mister B must append exactly b letters on each his move. He can arbitrary choose these letters. His opponent adds exactly a letters on each move.Mister B quickly understood that his opponent was just a computer that used a simple algorithm. The computer on each turn considers the suffix of string s of length a and generates a string t of length a such that all letters in the string t are distinct and don't appear in the considered suffix. From multiple variants of t lexicographically minimal is chosen (if a = 4 and the suffix is "bfdd", the computer chooses string t equal to "aceg"). After that the chosen string t is appended to the end of s.Mister B soon found the game boring and came up with the following question: what can be the minimum possible number of different letters in string s on the segment between positions l and r, inclusive. Letters of string s are numerated starting from 1.

First and only line contains four space-separated integers: a, b, l and r (1 ≤ a, b ≤ 12, 1 ≤ l ≤ r ≤ 109) — the numbers of letters each player appends and the bounds of the segment.

Print one integer — the minimum possible number of different letters in the segment from position l to position r, inclusive, in string s.

NoteIn the first sample test one of optimal strategies generate string s = "abababab...", that's why answer is 2.In the second sample test string s = "abcdbcaefg..." can be obtained, chosen segment will look like "bcdbc", that's why answer is 3.In the third sample test string s = "abczzzacad..." can be obtained, chosen, segment will look like "zzz", that's why answer is 1.


Manao is working for a construction company. Recently, an order came to build wall bars in a children's park. Manao was commissioned to develop a plan of construction, which will enable the company to save the most money.After reviewing the formal specifications for the wall bars, Manao discovered a number of controversial requirements and decided to treat them to the company's advantage. His resulting design can be described as follows:  Let's introduce some unit of length. The construction center is a pole of height n.  At heights 1, 2, ..., n exactly one horizontal bar sticks out from the pole. Each bar sticks in one of four pre-fixed directions.  A child can move from one bar to another if the distance between them does not exceed h and they stick in the same direction. If a child is on the ground, he can climb onto any of the bars at height between 1 and h. In Manao's construction a child should be able to reach at least one of the bars at heights n - h + 1, n - h + 2, ..., n if he begins at the ground.   The figure to the left shows what a common set of wall bars looks like. The figure to the right shows Manao's construction Manao is wondering how many distinct construction designs that satisfy his requirements exist. As this number can be rather large, print the remainder after dividing it by 1000000009 (109 + 9). Two designs are considered distinct if there is such height i, that the bars on the height i in these designs don't stick out in the same direction.

A single line contains two space-separated integers, n and h (1 ≤ n ≤ 1000, 1 ≤ h ≤ min(n, 30)).

In a single line print the remainder after dividing the number of designs by 1000000009 (109 + 9).

NoteConsider several designs for h = 2. A design with the first bar sticked out in direction d1, the second — in direction d2 and so on (1 ≤ di ≤ 4) is denoted as string d1d2...dn.Design "1231" (the first three bars are sticked out in different directions, the last one — in the same as first). A child can reach neither the bar at height 3 nor the bar at height 4.Design "414141". A child can reach the bar at height 5. To do this, he should first climb at the first bar, then at the third and then at the fifth one. He can also reach bar at height 6 by the route second  →  fourth  →  sixth bars.Design "123333". The child can't reach the upper two bars.Design "323323". The bar at height 6 can be reached by the following route: first  →  third  →  fourth  →  sixth bars.


Katie, Kuro and Shiro are best friends. They have known each other since kindergarten. That's why they often share everything with each other and work together on some very hard problems.Today is Shiro's birthday. She really loves pizza so she wants to invite her friends to the pizza restaurant near her house to celebrate her birthday, including her best friends Katie and Kuro.She has ordered a very big round pizza, in order to serve her many friends. Exactly $$$n$$$ of Shiro's friends are here. That's why she has to divide the pizza into $$$n + 1$$$ slices (Shiro also needs to eat). She wants the slices to be exactly the same size and shape. If not, some of her friends will get mad and go home early, and the party will be over.Shiro is now hungry. She wants to cut the pizza with minimum of straight cuts. A cut is a straight segment, it might have ends inside or outside the pizza. But she is too lazy to pick up the calculator.As usual, she will ask Katie and Kuro for help. But they haven't come yet. Could you help Shiro with this problem?

A single line contains one non-negative integer $$$n$$$ ($$$0 \le n \leq 10^{18}$$$) — the number of Shiro's friends. The circular pizza has to be sliced into $$$n + 1$$$ pieces.

A single integer — the number of straight cuts Shiro needs.

NoteTo cut the round pizza into quarters one has to make two cuts through the center with angle $$$90^{\circ}$$$ between them.To cut the round pizza into five equal parts one has to make five cuts.


Running with barriers on the circle track is very popular in the country where Dasha lives, so no wonder that on her way to classes she saw the following situation:The track is the circle with length L, in distinct points of which there are n barriers. Athlete always run the track in counterclockwise direction if you look on him from above. All barriers are located at integer distance from each other along the track. Her friends the parrot Kefa and the leopard Sasha participated in competitions and each of them ran one lap. Each of the friends started from some integral point on the track. Both friends wrote the distance from their start along the track to each of the n barriers. Thus, each of them wrote n integers in the ascending order, each of them was between 0 and L - 1, inclusively.    Consider an example. Let L = 8, blue points are barriers, and green points are Kefa's start (A) and Sasha's start (B). Then Kefa writes down the sequence [2, 4, 6], and Sasha writes down [1, 5, 7]. There are several tracks in the country, all of them have same length and same number of barriers, but the positions of the barriers can differ among different tracks. Now Dasha is interested if it is possible that Kefa and Sasha ran the same track or they participated on different tracks. Write the program which will check that Kefa's and Sasha's tracks coincide (it means that one can be obtained from the other by changing the start position). Note that they always run the track in one direction — counterclockwise, if you look on a track from above.

The first line contains two integers n and L (1 ≤ n ≤ 50, n ≤ L ≤ 100) — the number of barriers on a track and its length.  The second line contains n distinct integers in the ascending order — the distance from Kefa's start to each barrier in the order of its appearance. All integers are in the range from 0 to L - 1 inclusively. The second line contains n distinct integers in the ascending order — the distance from Sasha's start to each barrier in the order of its overcoming. All integers are in the range from 0 to L - 1 inclusively.

Print "YES" (without quotes), if Kefa and Sasha ran the coinciding tracks (it means that the position of all barriers coincides, if they start running from the same points on the track). Otherwise print "NO" (without quotes).

NoteThe first test is analyzed in the statement.


You are given a regular polygon with $$$n$$$ vertices labeled from $$$1$$$ to $$$n$$$ in counter-clockwise order. The triangulation of a given polygon is a set of triangles such that each vertex of each triangle is a vertex of the initial polygon, there is no pair of triangles such that their intersection has non-zero area, and the total area of all triangles is equal to the area of the given polygon. The weight of a triangulation is the sum of weigths of triangles it consists of, where the weight of a triagle is denoted as the product of labels of its vertices.Calculate the minimum weight among all triangulations of the polygon.

The first line contains single integer $$$n$$$ ($$$3 \le n \le 500$$$) — the number of vertices in the regular polygon.

Print one integer — the minimum weight among all triangulations of the given polygon.

NoteAccording to Wiki: polygon triangulation is the decomposition of a polygonal area (simple polygon) $$$P$$$ into a set of triangles, i. e., finding a set of triangles with pairwise non-intersecting interiors whose union is $$$P$$$.In the first example the polygon is a triangle, so we don't need to cut it further, so the answer is $$$1 \cdot 2 \cdot 3 = 6$$$.In the second example the polygon is a rectangle, so it should be divided into two triangles. It's optimal to cut it using diagonal $$$1-3$$$ so answer is $$$1 \cdot 2 \cdot 3 + 1 \cdot 3 \cdot 4 = 6 + 12 = 18$$$.


Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that 1 ≤ k ≤ 3 pi is a prime The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.

The single line contains an odd number n (3 ≤ n < 109).

In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found. In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.

NoteA prime is an integer strictly larger than one that is divisible only by one and by itself.


The final match of the Berland Football Cup has been held recently. The referee has shown $$$n$$$ yellow cards throughout the match. At the beginning of the match there were $$$a_1$$$ players in the first team and $$$a_2$$$ players in the second team.The rules of sending players off the game are a bit different in Berland football. If a player from the first team receives $$$k_1$$$ yellow cards throughout the match, he can no longer participate in the match — he's sent off. And if a player from the second team receives $$$k_2$$$ yellow cards, he's sent off. After a player leaves the match, he can no longer receive any yellow cards. Each of $$$n$$$ yellow cards was shown to exactly one player. Even if all players from one team (or even from both teams) leave the match, the game still continues.The referee has lost his records on who has received each yellow card. Help him to determine the minimum and the maximum number of players that could have been thrown out of the game.

The first line contains one integer $$$a_1$$$ $$$(1 \le a_1 \le 1\,000)$$$ — the number of players in the first team. The second line contains one integer $$$a_2$$$ $$$(1 \le a_2 \le 1\,000)$$$ — the number of players in the second team. The third line contains one integer $$$k_1$$$ $$$(1 \le k_1 \le 1\,000)$$$ — the maximum number of yellow cards a player from the first team can receive (after receiving that many yellow cards, he leaves the game). The fourth line contains one integer $$$k_2$$$ $$$(1 \le k_2 \le 1\,000)$$$ — the maximum number of yellow cards a player from the second team can receive (after receiving that many yellow cards, he leaves the game). The fifth line contains one integer $$$n$$$ $$$(1 \le n \le a_1 \cdot k_1 + a_2 \cdot k_2)$$$ — the number of yellow cards that have been shown during the match.

Print two integers — the minimum and the maximum number of players that could have been thrown out of the game.

NoteIn the first example it could be possible that no player left the game, so the first number in the output is $$$0$$$. The maximum possible number of players that could have been forced to leave the game is $$$4$$$ — one player from the first team, and three players from the second.In the second example the maximum possible number of yellow cards has been shown $$$(3 \cdot 6 + 1 \cdot 7 = 25)$$$, so in any case all players were sent off.


Uncle Fyodor, Matroskin the Cat and Sharic the Dog live their simple but happy lives in Prostokvashino. Sometimes they receive parcels from Uncle Fyodor’s parents and sometimes from anonymous benefactors, in which case it is hard to determine to which one of them the package has been sent. A photographic rifle is obviously for Sharic who loves hunting and fish is for Matroskin, but for whom was a new video game console meant? Every one of the three friends claimed that the present is for him and nearly quarreled. Uncle Fyodor had an idea how to solve the problem justly: they should suppose that the console was sent to all three of them and play it in turns. Everybody got relieved but then yet another burning problem popped up — who will play first? This time Matroskin came up with a brilliant solution, suggesting the most fair way to find it out: play rock-paper-scissors together. The rules of the game are very simple. On the count of three every player shows a combination with his hand (or paw). The combination corresponds to one of three things: a rock, scissors or paper. Some of the gestures win over some other ones according to well-known rules: the rock breaks the scissors, the scissors cut the paper, and the paper gets wrapped over the stone. Usually there are two players. Yet there are three friends, that’s why they decided to choose the winner like that: If someone shows the gesture that wins over the other two players, then that player wins. Otherwise, another game round is required. Write a program that will determine the winner by the gestures they have shown.

The first input line contains the name of the gesture that Uncle Fyodor showed, the second line shows which gesture Matroskin showed and the third line shows Sharic’s gesture.

Print "F" (without quotes) if Uncle Fyodor wins. Print "M" if Matroskin wins and "S" if Sharic wins. If it is impossible to find the winner, print "?".


You will receive 3 points for solving this problem.Manao is designing the genetic code for a new type of algae to efficiently produce fuel. Specifically, Manao is focusing on a stretch of DNA that encodes one protein. The stretch of DNA is represented by a string containing only the characters 'A', 'T', 'G' and 'C'.Manao has determined that if the stretch of DNA contains a maximal sequence of consecutive identical nucleotides that is of even length, then the protein will be nonfunctional. For example, consider a protein described by DNA string "GTTAAAG". It contains four maximal sequences of consecutive identical nucleotides: "G", "TT", "AAA", and "G". The protein is nonfunctional because sequence "TT" has even length.Manao is trying to obtain a functional protein from the protein he currently has. Manao can insert additional nucleotides into the DNA stretch. Each additional nucleotide is a character from the set {'A', 'T', 'G', 'C'}. Manao wants to determine the minimum number of insertions necessary to make the DNA encode a functional protein.

The input consists of a single line, containing a string s of length n (1 ≤ n ≤ 100). Each character of s will be from the set {'A', 'T', 'G', 'C'}. This problem doesn't have subproblems. You will get 3 points for the correct submission.

The program should print on one line a single integer representing the minimum number of 'A', 'T', 'G', 'C' characters that are required to be inserted into the input string in order to make all runs of identical characters have odd length.

NoteIn the first example, it is sufficient to insert a single nucleotide of any type between the two 'T's in the sequence to restore the functionality of the protein.


The Holmes children are fighting over who amongst them is the cleverest.Mycroft asked Sherlock and Eurus to find value of f(n), where f(1) = 1 and for n ≥ 2, f(n) is the number of distinct ordered positive integer pairs (x, y) that satisfy x + y = n and gcd(x, y) = 1. The integer gcd(a, b) is the greatest common divisor of a and b.Sherlock said that solving this was child's play and asked Mycroft to instead get the value of . Summation is done over all positive integers d that divide n.Eurus was quietly observing all this and finally came up with her problem to astonish both Sherlock and Mycroft.She defined a k-composite function Fk(n) recursively as follows:She wants them to tell the value of Fk(n) modulo 1000000007.

A single line of input contains two space separated integers n (1 ≤ n ≤ 1012) and k (1 ≤ k ≤ 1012) indicating that Eurus asks Sherlock and Mycroft to find the value of Fk(n) modulo 1000000007.

Output a single integer — the value of Fk(n) modulo 1000000007.

NoteIn the first case, there are 6 distinct ordered pairs (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1) satisfying x + y = 7 and gcd(x, y) = 1. Hence, f(7) = 6. So, F1(7) = f(g(7)) = f(f(7) + f(1)) = f(6 + 1) = f(7) = 6.


The Smart Beaver from ABBYY decided to have a day off. But doing nothing the whole day turned out to be too boring, and he decided to play a game with pebbles. Initially, the Beaver has n pebbles. He arranges them in a equal rows, each row has b pebbles (a > 1). Note that the Beaver must use all the pebbles he has, i. e. n = a·b.   10 pebbles are arranged in two rows, each row has 5 pebbles  Once the Smart Beaver has arranged the pebbles, he takes back any of the resulting rows (that is, b pebbles) and discards all other pebbles. Then he arranges all his pebbles again (possibly choosing other values of a and b) and takes back one row, and so on. The game continues until at some point the Beaver ends up with exactly one pebble. The game process can be represented as a finite sequence of integers c1, ..., ck, where:   c1 = n  ci + 1 is the number of pebbles that the Beaver ends up with after the i-th move, that is, the number of pebbles in a row after some arrangement of ci pebbles (1 ≤ i < k). Note that ci > ci + 1.  ck = 1 The result of the game is the sum of numbers ci. You are given n. Find the maximum possible result of the game.

The single line of the input contains a single integer n — the initial number of pebbles the Smart Beaver has. The input limitations for getting 30 points are:    2 ≤ n ≤ 50  The input limitations for getting 100 points are:    2 ≤ n ≤ 109

Print a single number — the maximum possible result of the game.

NoteConsider the first example (c1 = 10). The possible options for the game development are:  Arrange the pebbles in 10 rows, one pebble per row. Then c2 = 1, and the game ends after the first move with the result of 11.  Arrange the pebbles in 5 rows, two pebbles per row. Then c2 = 2, and the game continues. During the second move we have two pebbles which can be arranged in a unique way (remember that you are not allowed to put all the pebbles in the same row!) — 2 rows, one pebble per row. c3 = 1, and the game ends with the result of 13.  Finally, arrange the pebbles in two rows, five pebbles per row. The same logic leads us to c2 = 5, c3 = 1, and the game ends with the result of 16 — the maximum possible result.


Petya is a big fan of mathematics, especially its part related to fractions. Recently he learned that a fraction  is called proper iff its numerator is smaller than its denominator (a < b) and that the fraction is called irreducible if its numerator and its denominator are coprime (they do not have positive common divisors except 1).During his free time, Petya thinks about proper irreducible fractions and converts them to decimals using the calculator. One day he mistakenly pressed addition button ( + ) instead of division button (÷) and got sum of numerator and denominator that was equal to n instead of the expected decimal notation. Petya wanted to restore the original fraction, but soon he realized that it might not be done uniquely. That's why he decided to determine maximum possible proper irreducible fraction  such that sum of its numerator and denominator equals n. Help Petya deal with this problem.

In the only line of input there is an integer n (3 ≤ n ≤ 1000), the sum of numerator and denominator of the fraction.

Output two space-separated positive integers a and b, numerator and denominator of the maximum possible proper irreducible fraction satisfying the given sum.


Eighth-grader Vova is on duty today in the class. After classes, he went into the office to wash the board, and found on it the number n. He asked what is this number and the teacher of mathematics Inna Petrovna answered Vova that n is the answer to the arithmetic task for first-graders. In the textbook, a certain positive integer x was given. The task was to add x to the sum of the digits of the number x written in decimal numeral system.Since the number n on the board was small, Vova quickly guessed which x could be in the textbook. Now he wants to get a program which will search for arbitrary values of the number n for all suitable values of x or determine that such x does not exist. Write such a program for Vova.

The first line contains integer n (1 ≤ n ≤ 109).

In the first line print one integer k — number of different values of x satisfying the condition.  In next k lines print these values in ascending order.

NoteIn the first test case x = 15 there is only one variant: 15 + 1 + 5 = 21.In the second test case there are no such x.


The three friends, Kuro, Shiro, and Katie, met up again! It's time for a party...What the cats do when they unite? Right, they have a party. Since they wanted to have as much fun as possible, they invited all their friends. Now $$$n$$$ cats are at the party, sitting in a circle and eating soup. The rules are simple: anyone having finished their soup leaves the circle.Katie suddenly notices that whenever a cat leaves, the place where she was sitting becomes an empty space, which means the circle is divided into smaller continuous groups of cats sitting next to each other. At the moment Katie observes, there are $$$m$$$ cats who left the circle. This raises a question for Katie: what is the maximum possible number of groups the circle is divided into at the moment?Could you help her with this curiosity?You can see the examples and their descriptions with pictures in the "Note" section.

The only line contains two integers $$$n$$$ and $$$m$$$ ($$$2 \leq n \leq 1000$$$, $$$0 \leq m \leq n$$$) — the initial number of cats at the party and the number of cats who left the circle at the moment Katie observes, respectively.

Print a single integer — the maximum number of groups of cats at the moment Katie observes.

NoteIn the first example, originally there are $$$7$$$ cats sitting as shown below, creating a single group:  At the observed moment, $$$4$$$ cats have left the table. Suppose the cats $$$2$$$, $$$3$$$, $$$5$$$ and $$$7$$$ have left, then there are $$$3$$$ groups remaining. It is possible to show that it is the maximum possible number of groups remaining.  In the second example, there are $$$6$$$ cats sitting as shown below:  At the observed moment, $$$2$$$ cats have left the table. Suppose the cats numbered $$$3$$$ and $$$6$$$ left, then there will be $$$2$$$ groups remaining ($$$\{1, 2\}$$$ and $$$\{4, 5\}$$$). It is impossible to have more than $$$2$$$ groups of cats remaining.  In the third example, no cats have left, so there is $$$1$$$ group consisting of all cats.In the fourth example, all cats have left the circle, so there are $$$0$$$ groups.


Little penguin Polo loves his home village. The village has n houses, indexed by integers from 1 to n. Each house has a plaque containing an integer, the i-th house has a plaque containing integer pi (1 ≤ pi ≤ n).Little penguin Polo loves walking around this village. The walk looks like that. First he stands by a house number x. Then he goes to the house whose number is written on the plaque of house x (that is, to house px), then he goes to the house whose number is written on the plaque of house px (that is, to house ppx), and so on.We know that:  When the penguin starts walking from any house indexed from 1 to k, inclusive, he can walk to house number 1.  When the penguin starts walking from any house indexed from k + 1 to n, inclusive, he definitely cannot walk to house number 1.  When the penguin starts walking from house number 1, he can get back to house number 1 after some non-zero number of walks from a house to a house. You need to find the number of ways you may write the numbers on the houses' plaques so as to fulfill the three above described conditions. Print the remainder after dividing this number by 1000000007 (109 + 7).

The single line contains two space-separated integers n and k (1 ≤ n ≤ 1000, 1 ≤ k ≤ min(8, n)) — the number of the houses and the number k from the statement.

In a single line print a single integer — the answer to the problem modulo 1000000007 (109 + 7).


Sasha and Ira are two best friends. But they aren’t just friends, they are software engineers and experts in artificial intelligence. They are developing an algorithm for two bots playing a two-player game. The game is cooperative and turn based. In each turn, one of the players makes a move (it doesn’t matter which player, it's possible that players turns do not alternate). Algorithm for bots that Sasha and Ira are developing works by keeping track of the state the game is in. Each time either bot makes a move, the state changes. And, since the game is very dynamic, it will never go back to the state it was already in at any point in the past.Sasha and Ira are perfectionists and want their algorithm to have an optimal winning strategy. They have noticed that in the optimal winning strategy, both bots make exactly N moves each. But, in order to find the optimal strategy, their algorithm needs to analyze all possible states of the game (they haven’t learned about alpha-beta pruning yet) and pick the best sequence of moves.They are worried about the efficiency of their algorithm and are wondering what is the total number of states of the game that need to be analyzed?

The first and only line contains integer N.   1 ≤ N ≤ 106

Output should contain a single integer – number of possible states modulo 109 + 7.

NoteStart: Game is in state A.   Turn 1: Either bot can make a move (first bot is red and second bot is blue), so there are two possible states after the first turn – B and C.  Turn 2: In both states B and C, either bot can again make a turn, so the list of possible states is expanded to include D, E, F and G.  Turn 3: Red bot already did N=2 moves when in state D, so it cannot make any more moves there. It can make moves when in state E, F and G, so states I, K and M are added to the list. Similarly, blue bot cannot make a move when in state G, but can when in D, E and F, so states H, J and L are added.  Turn 4: Red bot already did N=2 moves when in states H, I and K, so it can only make moves when in J, L and M, so states P, R and S are added. Blue bot cannot make a move when in states J, L and M, but only when in H, I and K, so states N, O and Q are added. Overall, there are 19 possible states of the game their algorithm needs to analyze.


Nothing has changed since the last round. Dima and Inna still love each other and want to be together. They've made a deal with Seryozha and now they need to make a deal with the dorm guards...There are four guardposts in Dima's dorm. Each post contains two guards (in Russia they are usually elderly women). You can bribe a guard by a chocolate bar or a box of juice. For each guard you know the minimum price of the chocolate bar she can accept as a gift and the minimum price of the box of juice she can accept as a gift. If a chocolate bar for some guard costs less than the minimum chocolate bar price for this guard is, or if a box of juice for some guard costs less than the minimum box of juice price for this guard is, then the guard doesn't accept such a gift.In order to pass through a guardpost, one needs to bribe both guards.The shop has an unlimited amount of juice and chocolate of any price starting with 1. Dima wants to choose some guardpost, buy one gift for each guard from the guardpost and spend exactly n rubles on it.Help him choose a post through which he can safely sneak Inna or otherwise say that this is impossible. Mind you, Inna would be very sorry to hear that!

The first line of the input contains integer n (1 ≤ n ≤ 105) — the money Dima wants to spend. Then follow four lines describing the guardposts. Each line contains four integers a, b, c, d (1 ≤ a, b, c, d ≤ 105) — the minimum price of the chocolate and the minimum price of the juice for the first guard and the minimum price of the chocolate and the minimum price of the juice for the second guard, correspondingly.

In a single line of the output print three space-separated integers: the number of the guardpost, the cost of the first present and the cost of the second present. If there is no guardpost Dima can sneak Inna through at such conditions, print -1 in a single line.  The guardposts are numbered from 1 to 4 according to the order given in the input. If there are multiple solutions, you can print any of them.

NoteExplanation of the first example.The only way to spend 10 rubles to buy the gifts that won't be less than the minimum prices is to buy two 5 ruble chocolates to both guards from the first guardpost.Explanation of the second example.Dima needs 12 rubles for the first guardpost, 14 for the second one, 16 for the fourth one. So the only guardpost we can sneak through is the third one. So, Dima can buy 4 ruble chocolate for the first guard and 6 ruble juice of the second guard.


Recently, Vladimir got bad mark in algebra again. To avoid such unpleasant events in future he decided to train his arithmetic skills. He wrote four integer numbers a, b, c, d on the blackboard. During each of the next three minutes he took two numbers from the blackboard (not necessarily adjacent) and replaced them with their sum or their product. In the end he got one number. Unfortunately, due to the awful memory he forgot that number, but he remembers four original numbers, sequence of the operations and his surprise because of the very small result. Help Vladimir remember the forgotten number: find the smallest number that can be obtained from the original numbers by the given sequence of operations.

First line contains four integers separated by space: 0 ≤ a, b, c, d ≤ 1000 — the original numbers. Second line contains three signs ('+' or '*' each) separated by space — the sequence of the operations in the order of performing. ('+' stands for addition, '*' — multiplication)

Output one integer number — the minimal result which can be obtained. Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cin (also you may use %I64d).


The circle line of the Roflanpolis subway has $$$n$$$ stations.There are two parallel routes in the subway. The first one visits stations in order $$$1 \to 2 \to \ldots \to n \to 1 \to 2 \to \ldots$$$ (so the next stop after station $$$x$$$ is equal to $$$(x+1)$$$ if $$$x < n$$$ and $$$1$$$ otherwise). The second route visits stations in order $$$n \to (n-1) \to \ldots \to 1 \to n \to (n-1) \to \ldots$$$ (so the next stop after station $$$x$$$ is equal to $$$(x-1)$$$ if $$$x>1$$$ and $$$n$$$ otherwise). All trains depart their stations simultaneously, and it takes exactly $$$1$$$ minute to arrive at the next station.Two toads live in this city, their names are Daniel and Vlad.Daniel is currently in a train of the first route at station $$$a$$$ and will exit the subway when his train reaches station $$$x$$$.Coincidentally, Vlad is currently in a train of the second route at station $$$b$$$ and he will exit the subway when his train reaches station $$$y$$$.Surprisingly, all numbers $$$a,x,b,y$$$ are distinct.Toad Ilya asks you to check if Daniel and Vlad will ever be at the same station at the same time during their journey. In other words, check if there is a moment when their trains stop at the same station. Note that this includes the moments when Daniel or Vlad enter or leave the subway.

The first line contains five space-separated integers $$$n$$$, $$$a$$$, $$$x$$$, $$$b$$$, $$$y$$$ ($$$4 \leq n \leq 100$$$, $$$1 \leq a, x, b, y \leq n$$$, all numbers among $$$a$$$, $$$x$$$, $$$b$$$, $$$y$$$ are distinct) — the number of stations in Roflanpolis, Daniel's start station, Daniel's finish station, Vlad's start station and Vlad's finish station, respectively.

Output "YES" if there is a time moment when Vlad and Daniel are at the same station, and "NO" otherwise. You can print each letter in any case (upper or lower).

NoteIn the first example, Daniel and Vlad start at the stations $$$(1, 3)$$$. One minute later they are at stations $$$(2, 2)$$$. They are at the same station at this moment. Note that Vlad leaves the subway right after that.Consider the second example, let's look at the stations Vlad and Daniel are at. They are:   initially $$$(2, 9)$$$,  after $$$1$$$ minute $$$(3, 8)$$$,  after $$$2$$$ minutes $$$(4, 7)$$$,  after $$$3$$$ minutes $$$(5, 6)$$$,  after $$$4$$$ minutes $$$(6, 5)$$$,  after $$$5$$$ minutes $$$(7, 4)$$$,  after $$$6$$$ minutes $$$(8, 3)$$$,  after $$$7$$$ minutes $$$(9, 2)$$$,  after $$$8$$$ minutes $$$(10, 1)$$$,  after $$$9$$$ minutes $$$(1, 10)$$$. After that, they both leave the subway because they are at their finish stations, so there is no moment when they both are at the same station.


Barney is standing in a bar and starring at a pretty girl. He wants to shoot her with his heart arrow but he needs to know the distance between him and the girl to make his shot accurate.  Barney asked the bar tender Carl about this distance value, but Carl was so busy talking to the customers so he wrote the distance value (it's a real number) on a napkin. The problem is that he wrote it in scientific notation. The scientific notation of some real number x is the notation of form AeB, where A is a real number and B is an integer and x = A × 10B is true. In our case A is between 0 and 9 and B is non-negative.Barney doesn't know anything about scientific notation (as well as anything scientific at all). So he asked you to tell him the distance value in usual decimal representation with minimal number of digits after the decimal point (and no decimal point if it is an integer). See the output format for better understanding.

The first and only line of input contains a single string of form a.deb where a, d and b are integers and e is usual character 'e' (0 ≤ a ≤ 9, 0 ≤ d < 10100, 0 ≤ b ≤ 100) — the scientific notation of the desired distance value. a and b contain no leading zeros and d contains no trailing zeros (but may be equal to 0). Also, b can not be non-zero if a is zero.

Print the only real number x (the desired distance value) in the only line in its decimal notation.  Thus if x is an integer, print it's integer value without decimal part and decimal point and without leading zeroes.  Otherwise print x in a form of p.q such that p is an integer that have no leading zeroes (but may be equal to zero), and q is an integer that have no trailing zeroes (and may not be equal to zero).


A long time ago (probably even in the first book), Nicholas Flamel, a great alchemist and the creator of the Philosopher's Stone, taught Harry Potter three useful spells. The first one allows you to convert a grams of sand into b grams of lead, the second one allows you to convert c grams of lead into d grams of gold and third one allows you to convert e grams of gold into f grams of sand. When Harry told his friends about these spells, Ron Weasley was amazed. After all, if they succeed in turning sand into lead, lead into gold, and then turning part of the gold into sand again and so on, then it will be possible to start with a small amount of sand and get huge amounts of gold! Even an infinite amount of gold! Hermione Granger, by contrast, was skeptical about that idea. She argues that according to the law of conservation of matter getting an infinite amount of matter, even using magic, is impossible. On the contrary, the amount of matter may even decrease during transformation, being converted to magical energy. Though Hermione's theory seems convincing, Ron won't believe her. As far as Ron is concerned, Hermione made up her law of conservation of matter to stop Harry and Ron wasting their time with this nonsense, and to make them go and do homework instead. That's why Ron has already collected a certain amount of sand for the experiments. A quarrel between the friends seems unavoidable...Help Harry to determine which one of his friends is right, and avoid the quarrel after all. To do this you have to figure out whether it is possible to get the amount of gold greater than any preassigned number from some finite amount of sand.

The first line contains 6 integers a, b, c, d, e, f (0 ≤ a, b, c, d, e, f ≤ 1000).

Print "Ron", if it is possible to get an infinitely large amount of gold having a certain finite amount of sand (and not having any gold and lead at all), i.e., Ron is right. Otherwise, print "Hermione".

Ron Hermione Hermione Hermione Ron Hermione Ron

NoteConsider the first sample. Let's start with the 500 grams of sand. Apply the first spell 5 times and turn the sand into 1000 grams of lead. Then apply the second spell 4 times to get 600 grams of gold. Let’s take 400 grams from the resulting amount of gold turn them back into sand. We get 500 grams of sand and 200 grams of gold. If we apply the same operations to 500 grams of sand again, we can get extra 200 grams of gold every time. Thus, you can get 200, 400, 600 etc. grams of gold, i.e., starting with a finite amount of sand (500 grams), you can get the amount of gold which is greater than any preassigned number.In the forth sample it is impossible to get sand, or lead, or gold, applying the spells.In the fifth sample an infinitely large amount of gold can be obtained by using only the second spell, which allows you to receive 1 gram of gold out of nothing. Note that if such a second spell is available, then the first and the third one do not affect the answer at all.The seventh sample is more interesting. We can also start with a zero amount of sand there. With the aid of the third spell you can get sand out of nothing. We get 10000 grams of sand in this manner. Let's get 100 grams of lead using the first spell 100 times. Then make 1 gram of gold from them. We managed to receive 1 gram of gold, starting with a zero amount of sand! Clearly, in this manner you can get an infinitely large amount of gold.


Given the number n, find the smallest positive integer which has exactly n divisors. It is guaranteed that for the given n the answer will not exceed 1018.

The first line of the input contains integer n (1 ≤ n ≤ 1000).

Output the smallest positive integer with exactly n divisors.


Polycarpus has got n candies and m friends (n ≥ m). He wants to make a New Year present with candies to each friend. Polycarpus is planning to present all candies and he wants to do this in the fairest (that is, most equal) manner. He wants to choose such ai, where ai is the number of candies in the i-th friend's present, that the maximum ai differs from the least ai as little as possible.For example, if n is divisible by m, then he is going to present the same number of candies to all his friends, that is, the maximum ai won't differ from the minimum one.

The single line of the input contains a pair of space-separated positive integers n, m (1 ≤ n, m ≤ 100;n ≥ m) — the number of candies and the number of Polycarpus's friends.

Print the required sequence a1, a2, ..., am, where ai is the number of candies in the i-th friend's present. All numbers ai must be positive integers, total up to n, the maximum one should differ from the minimum one by the smallest possible value.

NotePrint ai in any order, separate the numbers by spaces.


Vadim is really keen on travelling. Recently he heard about kayaking activity near his town and became very excited about it, so he joined a party of kayakers.Now the party is ready to start its journey, but firstly they have to choose kayaks. There are 2·n people in the group (including Vadim), and they have exactly n - 1 tandem kayaks (each of which, obviously, can carry two people) and 2 single kayaks. i-th person's weight is wi, and weight is an important matter in kayaking — if the difference between the weights of two people that sit in the same tandem kayak is too large, then it can crash. And, of course, people want to distribute their seats in kayaks in order to minimize the chances that kayaks will crash.Formally, the instability of a single kayak is always 0, and the instability of a tandem kayak is the absolute difference between weights of the people that are in this kayak. Instability of the whole journey is the total instability of all kayaks.Help the party to determine minimum possible total instability!

The first line contains one number n (2 ≤ n ≤ 50). The second line contains 2·n integer numbers w1, w2, ..., w2n, where wi is weight of person i (1 ≤ wi ≤ 1000).

Print minimum possible total instability.


A sequence of non-negative integers a1, a2, ..., an of length n is called a wool sequence if and only if there exists two integers l and r (1 ≤ l ≤ r ≤ n) such that . In other words each wool sequence contains a subsequence of consecutive elements with xor equal to 0.The expression  means applying the operation of a bitwise xor to numbers x and y. The given operation exists in all modern programming languages, for example, in languages C++ and Java it is marked as "^", in Pascal — as "xor".In this problem you are asked to compute the number of sequences made of n integers from 0 to 2m - 1 that are not a wool sequence. You should print this number modulo 1000000009 (109 + 9).

The only line of input contains two space-separated integers n and m (1 ≤ n, m ≤ 105).

Print the required number of sequences modulo 1000000009 (109 + 9) on the only line of output.

NoteSequences of length 3 made of integers 0, 1, 2 and 3 that are not a wool sequence are (1, 3, 1), (1, 2, 1), (2, 1, 2), (2, 3, 2), (3, 1, 3) and (3, 2, 3).


Vasya has got an undirected graph consisting of $$$n$$$ vertices and $$$m$$$ edges. This graph doesn't contain any self-loops or multiple edges. Self-loop is an edge connecting a vertex to itself. Multiple edges are a pair of edges such that they connect the same pair of vertices. Since the graph is undirected, the pair of edges $$$(1, 2)$$$ and $$$(2, 1)$$$ is considered to be multiple edges. Isolated vertex of the graph is a vertex such that there is no edge connecting this vertex to any other vertex.Vasya wants to know the minimum and maximum possible number of isolated vertices in an undirected graph consisting of $$$n$$$ vertices and $$$m$$$ edges.

The only line contains two integers $$$n$$$ and $$$m~(1 \le n \le 10^5, 0 \le m \le \frac{n (n - 1)}{2})$$$. It is guaranteed that there exists a graph without any self-loops or multiple edges with such number of vertices and edges.

In the only line print two numbers $$$min$$$ and $$$max$$$ — the minimum and maximum number of isolated vertices, respectively.

NoteIn the first example it is possible to construct a graph with $$$0$$$ isolated vertices: for example, it should contain edges $$$(1, 2)$$$ and $$$(3, 4)$$$. To get one isolated vertex, we may construct a graph with edges $$$(1, 2)$$$ and $$$(1, 3)$$$. In the second example the graph will always contain exactly one isolated vertex.


What are you doing at the end of the world? Are you busy? Will you save us?Nephren is playing a game with little leprechauns.She gives them an infinite array of strings, f0... ∞.f0 is "What are you doing at the end of the world? Are you busy? Will you save us?".She wants to let more people know about it, so she defines fi =  "What are you doing while sending "fi - 1"? Are you busy? Will you send "fi - 1"?" for all i ≥ 1.For example, f1 is"What are you doing while sending "What are you doing at the end of the world? Are you busy? Will you save us?"? Are you busy? Will you send "What are you doing at the end of the world? Are you busy? Will you save us?"?". Note that the quotes in the very beginning and in the very end are for clarity and are not a part of f1.It can be seen that the characters in fi are letters, question marks, (possibly) quotation marks and spaces.Nephren will ask the little leprechauns q times. Each time she will let them find the k-th character of fn. The characters are indexed starting from 1. If fn consists of less than k characters, output '.' (without quotes).Can you answer her queries?

The first line contains one integer q (1 ≤ q ≤ 10) — the number of Nephren's questions. Each of the next q lines describes Nephren's question and contains two integers n and k (0 ≤ n ≤ 105, 1 ≤ k ≤ 1018).

One line containing q characters. The i-th character in it should be the answer for the i-th query.

NoteFor the first two examples, refer to f0 and f1 given in the legend.


To quickly hire highly skilled specialists one of the new IT City companies made an unprecedented move. Every employee was granted a car, and an employee can choose one of four different car makes.The parking lot before the office consists of one line of (2n - 2) parking spaces. Unfortunately the total number of cars is greater than the parking lot capacity. Furthermore even amount of cars of each make is greater than the amount of parking spaces! That's why there are no free spaces on the parking lot ever.Looking on the straight line of cars the company CEO thought that parking lot would be more beautiful if it contained exactly n successive cars of the same make. Help the CEO determine the number of ways to fill the parking lot this way.

The only line of the input contains one integer n (3 ≤ n ≤ 30) — the amount of successive cars of the same make.

Output one integer — the number of ways to fill the parking lot by cars of four makes using the described way.

NoteLet's denote car makes in the following way: A — Aston Martin, B —   Bentley, M — Mercedes-Maybach, Z — Zaporozhets. For n = 3 there are the following appropriate ways to fill the parking lot: AAAB AAAM AAAZ ABBB AMMM AZZZ BBBA BBBM BBBZ BAAA BMMM BZZZ MMMA MMMB MMMZ MAAA MBBB MZZZ ZZZA ZZZB ZZZM ZAAA ZBBB ZMMMOriginally it was planned to grant sport cars of Ferrari, Lamborghini, Maserati and Bugatti makes but this idea was renounced because it is impossible to drive these cars having small road clearance on the worn-down roads of IT City.


Manao is trying to open a rather challenging lock. The lock has n buttons on it and to open it, you should press the buttons in a certain order to open the lock. When you push some button, it either stays pressed into the lock (that means that you've guessed correctly and pushed the button that goes next in the sequence), or all pressed buttons return to the initial position. When all buttons are pressed into the lock at once, the lock opens.Consider an example with three buttons. Let's say that the opening sequence is: {2, 3, 1}. If you first press buttons 1 or 3, the buttons unpress immediately. If you first press button 2, it stays pressed. If you press 1 after 2, all buttons unpress. If you press 3 after 2, buttons 3 and 2 stay pressed. As soon as you've got two pressed buttons, you only need to press button 1 to open the lock.Manao doesn't know the opening sequence. But he is really smart and he is going to act in the optimal way. Calculate the number of times he's got to push a button in order to open the lock in the worst-case scenario.

A single line contains integer n (1 ≤ n ≤ 2000) — the number of buttons the lock has.

In a single line print the number of times Manao has to push a button in the worst-case scenario.

NoteConsider the first test sample. Manao can fail his first push and push the wrong button. In this case he will already be able to guess the right one with his second push. And his third push will push the second right button. Thus, in the worst-case scenario he will only need 3 pushes.


A magic number is a number formed by concatenation of numbers 1, 14 and 144. We can use each of these numbers any number of times. Therefore 14144, 141414 and 1411 are magic numbers but 1444, 514 and 414 are not.You're given a number. Determine if it is a magic number or not.

The first line of input contains an integer n, (1 ≤ n ≤ 109). This number doesn't contain leading zeros.

Print "YES" if n is a magic number or print "NO" if it's not.


Nicholas has an array a that contains n distinct integers from 1 to n. In other words, Nicholas has a permutation of size n.Nicholas want the minimum element (integer 1) and the maximum element (integer n) to be as far as possible from each other. He wants to perform exactly one swap in order to maximize the distance between the minimum and the maximum elements. The distance between two elements is considered to be equal to the absolute difference between their positions.

The first line of the input contains a single integer n (2 ≤ n ≤ 100) — the size of the permutation. The second line of the input contains n distinct integers a1, a2, ..., an (1 ≤ ai ≤ n), where ai is equal to the element at the i-th position.

Print a single integer — the maximum possible distance between the minimum and the maximum elements Nicholas can achieve by performing exactly one swap.

NoteIn the first sample, one may obtain the optimal answer by swapping elements 1 and 2.In the second sample, the minimum and the maximum elements will be located in the opposite ends of the array if we swap 7 and 2.In the third sample, the distance between the minimum and the maximum elements is already maximum possible, so we just perform some unnecessary swap, for example, one can swap 5 and 2.


John Doe has recently found a "Free Market" in his city — that is the place where you can exchange some of your possessions for other things for free. John knows that his city has n items in total (each item is unique). You can bring any number of items to the market and exchange them for any other one. Note that each item is one of a kind and that means that you cannot exchange set {a, b} for set {v, a}. However, you can always exchange set x for any set y, unless there is item p, such that p occurs in x and p occurs in y.For each item, John knows its value ci. John's sense of justice doesn't let him exchange a set of items x for a set of items y, if s(x) + d < s(y) (s(x) is the total price of items in the set x). During one day John can exchange only one set of items for something else. Initially, he has no items. John wants to get a set of items with the maximum total price. Find the cost of such set and the minimum number of days John can get it in.

The first line contains two space-separated integers n, d (1 ≤ n ≤ 50, 1 ≤ d ≤ 104) — the number of items on the market and John's sense of justice value, correspondingly. The second line contains n space-separated integers ci (1 ≤ ci ≤ 104).

Print two space-separated integers: the maximum possible price in the set of items John can get and the minimum number of days needed to get such set.

NoteIn the first sample John can act like this:   Take the first item (1 - 0 ≤ 2).  Exchange the first item for the second one (3 - 1 ≤ 2).  Take the first item (1 - 0 ≤ 2).


Reziba has many magic gems. Each magic gem can be split into $$$M$$$ normal gems. The amount of space each magic (and normal) gem takes is $$$1$$$ unit. A normal gem cannot be split.Reziba wants to choose a set of magic gems and split some of them, so the total space occupied by the resulting set of gems is $$$N$$$ units. If a magic gem is chosen and split, it takes $$$M$$$ units of space (since it is split into $$$M$$$ gems); if a magic gem is not split, it takes $$$1$$$ unit.How many different configurations of the resulting set of gems can Reziba have, such that the total amount of space taken is $$$N$$$ units? Print the answer modulo $$$1000000007$$$ ($$$10^9+7$$$). Two configurations are considered different if the number of magic gems Reziba takes to form them differs, or the indices of gems Reziba has to split differ.

The input contains a single line consisting of $$$2$$$ integers $$$N$$$ and $$$M$$$ ($$$1 \le N \le 10^{18}$$$, $$$2 \le M \le 100$$$).

Print one integer, the total number of configurations of the resulting set of gems, given that the total amount of space taken is $$$N$$$ units. Print the answer modulo $$$1000000007$$$ ($$$10^9+7$$$).

NoteIn the first example each magic gem can split into $$$2$$$ normal gems, and we know that the total amount of gems are $$$4$$$.Let $$$1$$$ denote a magic gem, and $$$0$$$ denote a normal gem.The total configurations you can have is:   $$$1 1 1 1$$$ (None of the gems split);  $$$0 0 1 1$$$ (First magic gem splits into $$$2$$$ normal gems);  $$$1 0 0 1$$$ (Second magic gem splits into $$$2$$$ normal gems);  $$$1 1 0 0$$$ (Third magic gem splits into $$$2$$$ normal gems);  $$$0 0 0 0$$$ (First and second magic gems split into total $$$4$$$ normal gems). Hence, answer is $$$5$$$.


Qwerty the Ranger took up a government job and arrived on planet Mars. He should stay in the secret lab and conduct some experiments on bacteria that have funny and abnormal properties. The job isn't difficult, but the salary is high.At the beginning of the first experiment there is a single bacterium in the test tube. Every second each bacterium in the test tube divides itself into k bacteria. After that some abnormal effects create b more bacteria in the test tube. Thus, if at the beginning of some second the test tube had x bacteria, then at the end of the second it will have kx + b bacteria.The experiment showed that after n seconds there were exactly z bacteria and the experiment ended at this point.For the second experiment Qwerty is going to sterilize the test tube and put there t bacteria. He hasn't started the experiment yet but he already wonders, how many seconds he will need to grow at least z bacteria. The ranger thinks that the bacteria will divide by the same rule as in the first experiment. Help Qwerty and find the minimum number of seconds needed to get a tube with at least z bacteria in the second experiment.

The first line contains four space-separated integers k, b, n and t (1 ≤ k, b, n, t ≤ 106) — the parameters of bacterial growth, the time Qwerty needed to grow z bacteria in the first experiment and the initial number of bacteria in the second experiment, correspondingly.

Print a single number — the minimum number of seconds Qwerty needs to grow at least z bacteria in the tube.


A boy Valera registered on site Codeforces as Valera, and wrote his first Codeforces Round #300. He boasted to a friend Arkady about winning as much as x points for his first contest. But Arkady did not believe his friend's words and decided to check whether Valera could have shown such a result.He knows that the contest number 300 was unusual because there were only two problems. The contest lasted for t minutes, the minutes are numbered starting from zero. The first problem had the initial cost of a points, and every minute its cost reduced by da points. The second problem had the initial cost of b points, and every minute this cost reduced by db points. Thus, as soon as the zero minute of the contest is over, the first problem will cost a - da points, and the second problem will cost b - db points. It is guaranteed that at any moment of the contest each problem has a non-negative cost.Arkady asks you to find out whether Valera could have got exactly x points for this contest. You should assume that Valera could have solved any number of the offered problems. You should also assume that for each problem Valera made no more than one attempt, besides, he could have submitted both problems at the same minute of the contest, starting with minute 0 and ending with minute number t - 1. Please note that Valera can't submit a solution exactly t minutes after the start of the contest or later.

The single line of the input contains six integers x, t, a, b, da, db (0 ≤ x ≤ 600; 1 ≤ t, a, b, da, db ≤ 300) — Valera's result, the contest's duration, the initial cost of the first problem, the initial cost of the second problem, the number of points that the first and the second problem lose per minute, correspondingly. It is guaranteed that at each minute of the contest each problem has a non-negative cost, that is, a - i·da ≥ 0 and b - i·db ≥ 0 for all 0 ≤ i ≤ t - 1.

If Valera could have earned exactly x points at a contest, print "YES", otherwise print "NO" (without the quotes).

NoteIn the first sample Valera could have acted like this: he could have submitted the first problem at minute 0 and the second problem — at minute 2. Then the first problem brings him 20 points and the second problem brings him 10 points, that in total gives the required 30 points.


Let's assume that we have a pair of numbers (a, b). We can get a new pair (a + b, b) or (a, a + b) from the given pair in a single step.Let the initial pair of numbers be (1,1). Your task is to find number k, that is, the least number of steps needed to transform (1,1) into the pair where at least one number equals n.

The input contains the only integer n (1 ≤ n ≤ 106).

NoteThe pair (1,1) can be transformed into a pair containing 5 in three moves: (1,1)  →  (1,2)  →  (3,2)  →  (5,2).


The HR manager was disappointed again. The last applicant failed the interview the same way as 24 previous ones. "Do I give such a hard task?" — the HR manager thought. "Just raise number 5 to the power of n and get last two digits of the number. Yes, of course, n can be rather big, and one cannot find the power using a calculator, but we need people who are able to think, not just follow the instructions."Could you pass the interview in the machine vision company in IT City?

The only line of the input contains a single integer n (2 ≤ n ≤ 2·1018) — the power in which you need to raise number 5.

Output the last two digits of 5n without spaces between them.


Permutation p is an ordered set of integers p1,  p2,  ...,  pn, consisting of n distinct positive integers, each of them doesn't exceed n. We'll denote the i-th element of permutation p as pi. We'll call number n the size or the length of permutation p1,  p2,  ...,  pn.Petya decided to introduce the sum operation on the set of permutations of length n. Let's assume that we are given two permutations of length n: a1, a2, ..., an and b1, b2, ..., bn. Petya calls the sum of permutations a and b such permutation c of length n, where ci = ((ai - 1 + bi - 1) mod n) + 1 (1 ≤ i ≤ n).Operation  means taking the remainder after dividing number x by number y.Obviously, not for all permutations a and b exists permutation c that is sum of a and b. That's why Petya got sad and asked you to do the following: given n, count the number of such pairs of permutations a and b of length n, that exists permutation c that is sum of a and b. The pair of permutations x, y (x ≠ y) and the pair of permutations y, x are considered distinct pairs.As the answer can be rather large, print the remainder after dividing it by 1000000007 (109 + 7).

The single line contains integer n (1 ≤ n ≤ 16).

In the single line print a single non-negative integer — the number of such pairs of permutations a and b, that exists permutation c that is sum of a and b, modulo 1000000007 (109 + 7).


Nastya received a gift on New Year — a magic wardrobe. It is magic because in the end of each month the number of dresses in it doubles (i.e. the number of dresses becomes twice as large as it is in the beginning of the month).Unfortunately, right after the doubling the wardrobe eats one of the dresses (if any) with the 50% probability. It happens every month except the last one in the year. Nastya owns x dresses now, so she became interested in the expected number of dresses she will have in one year. Nastya lives in Byteland, so the year lasts for k + 1 months.Nastya is really busy, so she wants you to solve this problem. You are the programmer, after all. Also, you should find the answer modulo 109 + 7, because it is easy to see that it is always integer.

The only line contains two integers x and k (0 ≤ x, k ≤ 1018), where x is the initial number of dresses and k + 1 is the number of months in a year in Byteland.

In the only line print a single integer — the expected number of dresses Nastya will own one year later modulo 109 + 7.

NoteIn the first example a year consists on only one month, so the wardrobe does not eat dresses at all.In the second example after the first month there are 3 dresses with 50% probability and 4 dresses with 50% probability. Thus, in the end of the year there are 6 dresses with 50% probability and 8 dresses with 50% probability. This way the answer for this test is (6 + 8) / 2 = 7.


One day student Vasya was sitting on a lecture and mentioned a string s1s2... sn, consisting of letters "a", "b" and "c" that was written on his desk. As the lecture was boring, Vasya decided to complete the picture by composing a graph G with the following properties:   G has exactly n vertices, numbered from 1 to n.  For all pairs of vertices i and j, where i ≠ j, there is an edge connecting them if and only if characters si and sj are either equal or neighbouring in the alphabet. That is, letters in pairs "a"-"b" and "b"-"c" are neighbouring, while letters "a"-"c" are not. Vasya painted the resulting graph near the string and then erased the string. Next day Vasya's friend Petya came to a lecture and found some graph at his desk. He had heard of Vasya's adventure and now he wants to find out whether it could be the original graph G, painted by Vasya. In order to verify this, Petya needs to know whether there exists a string s, such that if Vasya used this s he would produce the given graph G.

The first line of the input contains two integers n and m  — the number of vertices and edges in the graph found by Petya, respectively. Each of the next m lines contains two integers ui and vi (1 ≤ ui, vi ≤ n, ui ≠ vi) — the edges of the graph G. It is guaranteed, that there are no multiple edges, that is any pair of vertexes appear in this list no more than once.

In the first line print "Yes" (without the quotes), if the string s Petya is interested in really exists and "No" (without the quotes) otherwise. If the string s exists, then print it on the second line of the output. The length of s must be exactly n, it must consist of only letters "a", "b" and "c" only, and the graph built using this string must coincide with G. If there are multiple possible answers, you may print any of them.

NoteIn the first sample you are given a graph made of two vertices with an edge between them. So, these vertices can correspond to both the same and adjacent letters. Any of the following strings "aa", "ab", "ba", "bb", "bc", "cb", "cc" meets the graph's conditions. In the second sample the first vertex is connected to all three other vertices, but these three vertices are not connected with each other. That means that they must correspond to distinct letters that are not adjacent, but that is impossible as there are only two such letters: a and c.


Recently Ivan the Fool decided to become smarter and study the probability theory. He thinks that he understands the subject fairly well, and so he began to behave like he already got PhD in that area.To prove his skills, Ivan decided to demonstrate his friends a concept of random picture. A picture is a field of $$$n$$$ rows and $$$m$$$ columns, where each cell is either black or white. Ivan calls the picture random if for every cell it has at most one adjacent cell of the same color. Two cells are considered adjacent if they share a side.Ivan's brothers spent some time trying to explain that it's not how the randomness usually works. Trying to convince Ivan, they want to count the number of different random (according to Ivan) pictures. Two pictures are considered different if at least one cell on those two picture is colored differently. Since the number of such pictures may be quite large, print it modulo $$$10^9 + 7$$$.

The only line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 100\,000$$$), the number of rows and the number of columns of the field.

Print one integer, the number of random pictures modulo $$$10^9 + 7$$$.

NoteThe picture below shows all possible random pictures of size $$$2$$$ by $$$3$$$.


A group of university students wants to get to the top of a mountain to have a picnic there. For that they decided to use a cableway.A cableway is represented by some cablecars, hanged onto some cable stations by a cable. A cable is scrolled cyclically between the first and the last cable stations (the first of them is located at the bottom of the mountain and the last one is located at the top). As the cable moves, the cablecar attached to it move as well.The number of cablecars is divisible by three and they are painted three colors: red, green and blue, in such manner that after each red cablecar goes a green one, after each green cablecar goes a blue one and after each blue cablecar goes a red one. Each cablecar can transport no more than two people, the cablecars arrive with the periodicity of one minute (i. e. every minute) and it takes exactly 30 minutes for a cablecar to get to the top.All students are divided into three groups: r of them like to ascend only in the red cablecars, g of them prefer only the green ones and b of them prefer only the blue ones. A student never gets on a cablecar painted a color that he doesn't like,The first cablecar to arrive (at the moment of time 0) is painted red. Determine the least time it will take all students to ascend to the mountain top.

The first line contains three integers r, g and b (0 ≤ r, g, b ≤ 100). It is guaranteed that r + g + b > 0, it means that the group consists of at least one student.

Print a single number — the minimal time the students need for the whole group to ascend to the top of the mountain.

NoteLet's analyze the first sample.At the moment of time 0 a red cablecar comes and one student from the r group get on it and ascends to the top at the moment of time 30.At the moment of time 1 a green cablecar arrives and two students from the g group get on it; they get to the top at the moment of time 31.At the moment of time 2 comes the blue cablecar and two students from the b group get on it. They ascend to the top at the moment of time 32.At the moment of time 3 a red cablecar arrives but the only student who is left doesn't like red and the cablecar leaves empty.At the moment of time 4 a green cablecar arrives and one student from the g group gets on it. He ascends to top at the moment of time 34.Thus, all the students are on the top, overall the ascension took exactly 34 minutes.


Jon fought bravely to rescue the wildlings who were attacked by the white-walkers at Hardhome. On his arrival, Sam tells him that he wants to go to Oldtown to train at the Citadel to become a maester, so he can return and take the deceased Aemon's place as maester of Castle Black. Jon agrees to Sam's proposal and Sam sets off his journey to the Citadel. However becoming a trainee at the Citadel is not a cakewalk and hence the maesters at the Citadel gave Sam a problem to test his eligibility. Initially Sam has a list with a single element n. Then he has to perform certain operations on this list. In each operation Sam must remove any element x, such that x > 1, from the list and insert at the same position , ,  sequentially. He must continue with these operations until all the elements in the list are either 0 or 1.Now the masters want the total number of 1s in the range l to r (1-indexed). Sam wants to become a maester but unfortunately he cannot solve this problem. Can you help Sam to pass the eligibility test?

The first line contains three integers n, l, r (0 ≤ n < 250, 0 ≤ r - l ≤ 105, r ≥ 1, l ≥ 1) – initial element and the range l to r. It is guaranteed that r is not greater than the length of the final list.

Output the total number of 1s in the range l to r in the final sequence.

NoteConsider first example:Elements on positions from 2-nd to 5-th in list is [1, 1, 1, 1]. The number of ones is 4.For the second example:Elements on positions from 3-rd to 10-th in list is [1, 1, 1, 0, 1, 0, 1, 0]. The number of ones is 5.


A string $$$s$$$ of length $$$n$$$ can be encrypted by the following algorithm:  iterate over all divisors of $$$n$$$ in decreasing order (i.e. from $$$n$$$ to $$$1$$$),  for each divisor $$$d$$$, reverse the substring $$$s[1 \dots d]$$$ (i.e. the substring which starts at position $$$1$$$ and ends at position $$$d$$$). For example, the above algorithm applied to the string $$$s$$$="codeforces" leads to the following changes: "codeforces" $$$\to$$$ "secrofedoc" $$$\to$$$ "orcesfedoc" $$$\to$$$ "rocesfedoc" $$$\to$$$ "rocesfedoc" (obviously, the last reverse operation doesn't change the string because $$$d=1$$$).You are given the encrypted string $$$t$$$. Your task is to decrypt this string, i.e., to find a string $$$s$$$ such that the above algorithm results in string $$$t$$$. It can be proven that this string $$$s$$$ always exists and is unique.

The first line of input consists of a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the string $$$t$$$. The second line of input consists of the string $$$t$$$. The length of $$$t$$$ is $$$n$$$, and it consists only of lowercase Latin letters.

Print a string $$$s$$$ such that the above algorithm results in $$$t$$$.

NoteThe first example is described in the problem statement.


One day Vasya heard a story: "In the city of High Bertown a bus number 62 left from the bus station. It had n grown-ups and m kids..."The latter events happen to be of no importance to us. Vasya is an accountant and he loves counting money. So he wondered what maximum and minimum sum of money these passengers could have paid for the ride.The bus fare equals one berland ruble in High Bertown. However, not everything is that easy — no more than one child can ride for free with each grown-up passenger. That means that a grown-up passenger who rides with his k (k > 0) children, pays overall k rubles: a ticket for himself and (k - 1) tickets for his children. Also, a grown-up can ride without children, in this case he only pays one ruble.We know that in High Bertown children can't ride in a bus unaccompanied by grown-ups.Help Vasya count the minimum and the maximum sum in Berland rubles, that all passengers of this bus could have paid in total.

The input file consists of a single line containing two space-separated numbers n and m (0 ≤ n, m ≤ 105) — the number of the grown-ups and the number of the children in the bus, correspondingly.

If n grown-ups and m children could have ridden in the bus, then print on a single line two space-separated integers — the minimum and the maximum possible total bus fare, correspondingly.  Otherwise, print "Impossible" (without the quotes).

NoteIn the first sample a grown-up rides with two children and pays two rubles.In the second sample there are only children in the bus, so the situation is impossible. In the third sample there are two cases:  Each of the two grown-ups rides with one children and pays one ruble for the tickets. In this case the passengers pay two rubles in total.  One of the grown-ups ride with two children's and pays two rubles, the another one rides alone and pays one ruble for himself. So, they pay three rubles in total.


There are n students who have taken part in an olympiad. Now it's time to award the students.Some of them will receive diplomas, some wiil get certificates, and others won't receive anything. Students with diplomas and certificates are called winners. But there are some rules of counting the number of diplomas and certificates. The number of certificates must be exactly k times greater than the number of diplomas. The number of winners must not be greater than half of the number of all students (i.e. not be greater than half of n). It's possible that there are no winners.You have to identify the maximum possible number of winners, according to these rules. Also for this case you have to calculate the number of students with diplomas, the number of students with certificates and the number of students who are not winners.

The first (and the only) line of input contains two integers n and k (1 ≤ n, k ≤ 1012), where n is the number of students and k is the ratio between the number of certificates and the number of diplomas.

Output three numbers: the number of students with diplomas, the number of students with certificates and the number of students who are not winners in case when the number of winners is maximum possible. It's possible that there are no winners.


Vasya has found a strange device. On the front panel of a device there are: a red button, a blue button and a display showing some positive integer. After clicking the red button, device multiplies the displayed number by two. After clicking the blue button, device subtracts one from the number on the display. If at some point the number stops being positive, the device breaks down. The display can show arbitrarily large numbers. Initially, the display shows number n.Bob wants to get number m on the display. What minimum number of clicks he has to make in order to achieve this result?

The first and the only line of the input contains two distinct integers n and m (1 ≤ n, m ≤ 104), separated by a space .

Print a single number — the minimum number of times one needs to push the button required to get the number m out of number n.

NoteIn the first example you need to push the blue button once, and then push the red button once.In the second example, doubling the number is unnecessary, so we need to push the blue button nine times.


It seems like the year of 2013 came only yesterday. Do you know a curious fact? The year of 2013 is the first year after the old 1987 with only distinct digits.Now you are suggested to solve the following problem: given a year number, find the minimum year number which is strictly larger than the given one and has only distinct digits.

The single line contains integer y (1000 ≤ y ≤ 9000) — the year number.

Print a single integer — the minimum year number that is strictly larger than y and all it's digits are distinct. It is guaranteed that the answer exists.


Comrade Dujikov is busy choosing artists for Timofey's birthday and is recieving calls from Taymyr from Ilia-alpinist.Ilia-alpinist calls every n minutes, i.e. in minutes n, 2n, 3n and so on. Artists come to the comrade every m minutes, i.e. in minutes m, 2m, 3m and so on. The day is z minutes long, i.e. the day consists of minutes 1, 2, ..., z. How many artists should be killed so that there are no artists in the room when Ilia calls? Consider that a call and a talk with an artist take exactly one minute.

The only string contains three integers — n, m and z (1 ≤ n, m, z ≤ 104).

Print single integer — the minimum number of artists that should be killed so that there are no artists in the room when Ilia calls.

NoteTaymyr is a place in the north of Russia.In the first test the artists come each minute, as well as the calls, so we need to kill all of them.In the second test we need to kill artists which come on the second and the fourth minutes.In the third test — only the artist which comes on the sixth minute.


You are given a string $$$s$$$ consisting of lowercase Latin letters. Let the length of $$$s$$$ be $$$|s|$$$. You may perform several operations on this string.In one operation, you can choose some index $$$i$$$ and remove the $$$i$$$-th character of $$$s$$$ ($$$s_i$$$) if at least one of its adjacent characters is the previous letter in the Latin alphabet for $$$s_i$$$. For example, the previous letter for b is a, the previous letter for s is r, the letter a has no previous letters. Note that after each removal the length of the string decreases by one. So, the index $$$i$$$ should satisfy the condition $$$1 \le i \le |s|$$$ during each operation.For the character $$$s_i$$$ adjacent characters are $$$s_{i-1}$$$ and $$$s_{i+1}$$$. The first and the last characters of $$$s$$$ both have only one adjacent character (unless $$$|s| = 1$$$).Consider the following example. Let $$$s=$$$ bacabcab.  During the first move, you can remove the first character $$$s_1=$$$ b because $$$s_2=$$$ a. Then the string becomes $$$s=$$$ acabcab.  During the second move, you can remove the fifth character $$$s_5=$$$ c because $$$s_4=$$$ b. Then the string becomes $$$s=$$$ acabab.  During the third move, you can remove the sixth character $$$s_6=$$$'b' because $$$s_5=$$$ a. Then the string becomes $$$s=$$$ acaba.  During the fourth move, the only character you can remove is $$$s_4=$$$ b, because $$$s_3=$$$ a (or $$$s_5=$$$ a). The string becomes $$$s=$$$ acaa and you cannot do anything with it. Your task is to find the maximum possible number of characters you can remove if you choose the sequence of operations optimally.

The only line of the input contains one integer $$$|s|$$$ ($$$1 \le |s| \le 100$$$) — the length of $$$s$$$. The second line of the input contains one string $$$s$$$ consisting of $$$|s|$$$ lowercase Latin letters.

Print one integer — the maximum possible number of characters you can remove if you choose the sequence of moves optimally.

NoteThe first example is described in the problem statement. Note that the sequence of moves provided in the statement is not the only, but it can be shown that the maximum possible answer to this test is $$$4$$$.In the second example, you can remove all but one character of $$$s$$$. The only possible answer follows.  During the first move, remove the third character $$$s_3=$$$ d, $$$s$$$ becomes bca.  During the second move, remove the second character $$$s_2=$$$ c, $$$s$$$ becomes ba.  And during the third move, remove the first character $$$s_1=$$$ b, $$$s$$$ becomes a.


Olesya loves numbers consisting of n digits, and Rodion only likes numbers that are divisible by t. Find some number that satisfies both of them.Your task is: given the n and t print an integer strictly larger than zero consisting of n digits that is divisible by t. If such number doesn't exist, print  - 1.

The single line contains two numbers, n and t (1 ≤ n ≤ 100, 2 ≤ t ≤ 10) — the length of the number and the number it should be divisible by.

Print one such positive number without leading zeroes, — the answer to the problem, or  - 1, if such number doesn't exist. If there are multiple possible answers, you are allowed to print any of them.


Each student eagerly awaits the day he would pass the exams successfully. Thus, Vasya was ready to celebrate, but, alas, he didn't pass it. However, many of Vasya's fellow students from the same group were more successful and celebrated after the exam.Some of them celebrated in the BugDonalds restaurant, some of them — in the BeaverKing restaurant, the most successful ones were fast enough to celebrate in both of restaurants. Students which didn't pass the exam didn't celebrate in any of those restaurants and elected to stay home to prepare for their reexamination. However, this quickly bored Vasya and he started checking celebration photos on the Kilogramm. He found out that, in total, BugDonalds was visited by $$$A$$$ students, BeaverKing — by $$$B$$$ students and $$$C$$$ students visited both restaurants. Vasya also knows that there are $$$N$$$ students in his group.Based on this info, Vasya wants to determine either if his data contradicts itself or, if it doesn't, how many students in his group didn't pass the exam. Can you help him so he won't waste his valuable preparation time?

The first line contains four integers — $$$A$$$, $$$B$$$, $$$C$$$ and $$$N$$$ ($$$0 \leq A, B, C, N \leq 100$$$).

If a distribution of $$$N$$$ students exists in which $$$A$$$ students visited BugDonalds, $$$B$$$ — BeaverKing, $$$C$$$ — both of the restaurants and at least one student is left home (it is known that Vasya didn't pass the exam and stayed at home), output one integer — amount of students (including Vasya) who did not pass the exam.  If such a distribution does not exist and Vasya made a mistake while determining the numbers $$$A$$$, $$$B$$$, $$$C$$$ or $$$N$$$ (as in samples 2 and 3), output $$$-1$$$.

NoteThe first sample describes following situation: $$$5$$$ only visited BugDonalds, $$$5$$$ students only visited BeaverKing, $$$5$$$ visited both of them and $$$5$$$ students (including Vasya) didn't pass the exam.In the second sample $$$2$$$ students only visited BugDonalds and $$$2$$$ only visited BeaverKing, but that means all $$$4$$$ students in group passed the exam which contradicts the fact that Vasya didn't pass meaning that this situation is impossible.The third sample describes a situation where $$$2$$$ students visited BugDonalds but the group has only $$$1$$$ which makes it clearly impossible.


Little Dima misbehaved during a math lesson a lot and the nasty teacher Mr. Pickles gave him the following problem as a punishment. Find all integer solutions x (0 < x < 109) of the equation:x = b·s(x)a + c,  where a, b, c are some predetermined constant values and function s(x) determines the sum of all digits in the decimal representation of number x.The teacher gives this problem to Dima for each lesson. He changes only the parameters of the equation: a, b, c. Dima got sick of getting bad marks and he asks you to help him solve this challenging problem.

The first line contains three space-separated integers: a, b, c (1 ≤ a ≤ 5; 1 ≤ b ≤ 10000;  - 10000 ≤ c ≤ 10000).

Print integer n — the number of the solutions that you've found. Next print n integers in the increasing order — the solutions of the given equation. Print only integer solutions that are larger than zero and strictly less than 109.


For some experiments little Petya needs a synchrophasotron. He has already got the device, all that's left is to set the fuel supply. Fuel comes through a system of nodes numbered from 1 to n and connected by pipes. Pipes go from every node with smaller number to every node with greater number. Fuel can only flow through pipes in direction from node with smaller number to node with greater number. Any amount of fuel can enter through the first node and the last node is connected directly to the synchrophasotron. It is known that every pipe has three attributes: the minimum amount of fuel that should go through it, the maximum amount of fuel that can possibly go through it and the cost of pipe activation. If cij units of fuel (cij > 0) flow from node i to node j, it will cost aij + cij2 tugriks (aij is the cost of pipe activation), and if fuel doesn't flow through the pipe, it doesn't cost anything. Only integer number of units of fuel can flow through each pipe.Constraints on the minimal and the maximal fuel capacity of a pipe take place always, not only if it is active. You may assume that the pipe is active if and only if the flow through it is strictly greater than zero.Petya doesn't want the pipe system to be overloaded, so he wants to find the minimal amount of fuel, that, having entered the first node, can reach the synchrophasotron. Besides that he wants to impress the sponsors, so the sum of money needed to be paid for fuel to go through each pipe, must be as big as possible.

First line contains integer n (2 ≤ n ≤ 6), which represents the number of nodes. Each of the next n(n - 1) / 2 lines contains five integers s, f, l, h, a that describe pipes — the first node of the pipe, the second node of the pipe, the minimum and the maximum amount of fuel that can flow through the pipe and the the activation cost, respectively. (1 ≤ s < f ≤ n, 0 ≤ l ≤ h ≤ 5, 0 ≤ a ≤ 6). It is guaranteed that for each pair of nodes with distinct numbers there will be exactly one pipe between them described in the input.

Output in the first line two space-separated numbers: the minimum possible amount of fuel that can flow into the synchrophasotron, and the maximum possible sum that needs to be paid in order for that amount of fuel to reach synchrophasotron. If there is no amount of fuel that can reach synchrophasotron, output "-1 -1". The amount of fuel which will flow into synchrophasotron is not neccessary positive. It could be equal to zero if the minimum constraint of every pipe is equal to zero.

NoteIn the first test, we can either pass 1 or 2 units of fuel from node 1 to node 2. The minimum possible amount is 1, it costs a12 + 12 = 4.In the second test, you can pass at most 2 units from node 1 to node 2, and at you have to pass at least 3 units from node 2 to node 3. It is impossible.In the third test, the minimum possible amount is 2. You can pass each unit of fuel through two different paths: either 1->2->3->4 or 1->3->4. If you use the first path twice, it will cost a12 + 22 + a23 + 22 + a34 + 22=14. If you use the second path twice, it will cost a13 + 22 + a34 + 22=14. However, if you use each path (allowing one unit of fuel go through pipes 1->2, 2->3, 1->3, and two units go through 3->4) it will cost a12 + 12 + a23 + 12 + a13 + 12 + a34 + 22=15 and it is the maximum possible cost.Also note that since no fuel flows from node 1 to node 4, activation cost for that pipe is not added to the answer.


Tomorrow Peter has a Biology exam. He does not like this subject much, but d days ago he learnt that he would have to take this exam. Peter's strict parents made him prepare for the exam immediately, for this purpose he has to study not less than minTimei and not more than maxTimei hours per each i-th day. Moreover, they warned Peter that a day before the exam they would check how he has followed their instructions.So, today is the day when Peter's parents ask him to show the timetable of his preparatory studies. But the boy has counted only the sum of hours sumTime spent him on preparation, and now he wants to know if he can show his parents a timetable sсhedule with d numbers, where each number sсhedulei stands for the time in hours spent by Peter each i-th day on biology studies, and satisfying the limitations imposed by his parents, and at the same time the sum total of all schedulei should equal to sumTime.

The first input line contains two integer numbers d, sumTime (1 ≤ d ≤ 30, 0 ≤ sumTime ≤ 240) — the amount of days, during which Peter studied, and the total amount of hours, spent on preparation. Each of the following d lines contains two integer numbers minTimei, maxTimei (0 ≤ minTimei ≤ maxTimei ≤ 8), separated by a space — minimum and maximum amount of hours that Peter could spent in the i-th day.

In the first line print YES, and in the second line print d numbers (separated by a space), each of the numbers — amount of hours, spent by Peter on preparation in the corresponding day, if he followed his parents' instructions; or print NO in the unique line. If there are many solutions, print any of them.


Valera the Horse is going to the party with friends. He has been following the fashion trends for a while, and he knows that it is very popular to wear all horseshoes of different color. Valera has got four horseshoes left from the last year, but maybe some of them have the same color. In this case he needs to go to the store and buy some few more horseshoes, not to lose face in front of his stylish comrades.Fortunately, the store sells horseshoes of all colors under the sun and Valera has enough money to buy any four of them. However, in order to save the money, he would like to spend as little money as possible, so you need to help Valera and determine what is the minimum number of horseshoes he needs to buy to wear four horseshoes of different colors to a party.

The first line contains four space-separated integers s1, s2, s3, s4 (1 ≤ s1, s2, s3, s4 ≤ 109) — the colors of horseshoes Valera has. Consider all possible colors indexed with integers.

Print a single integer — the minimum number of horseshoes Valera needs to buy.


Polycarp has just invented a new binary protocol for data transmission. He is encoding positive integer decimal number to binary string using following algorithm:  Each digit is represented with number of '1' characters equal to the value of that digit (for 0 it is zero ones).  Digits are written one by one in order corresponding to number and separated by single '0' character. Though Polycarp learnt how to encode the numbers, he has no idea how to decode them back. Help him calculate the decoded number.

The first line contains one integer number n (1 ≤ n ≤ 89) — length of the string s. The second line contains string s — sequence of '0' and '1' characters, number in its encoded format. It is guaranteed that the number corresponding to the string is positive and doesn't exceed 109. The string always starts with '1'.