B-B
You are given a ternary string (it is a string which consists only of characters ‘0’, ‘1’ and ‘2’).
You can swap any two adjacent (consecutive) characters ‘0’ and ‘1’ (i.e. replace “01” with “10” or vice versa) or any two adjacent (consecutive) characters ‘1’ and ‘2’ (i.e. replace “12” with “21” or vice versa).
For example, for string “010210” we can perform the following moves:
“010210” →→ “100210”;
“010210” →→ “001210”;
“010210” →→ “010120”;
“010210” →→ “010201”.
Note than you cannot swap “02” →→ “20” and vice versa. You cannot perform any other operations with the given string excluding described above.
You task is to obtain the minimum possible (lexicographically) string by using these swaps arbitrary number of times (possibly, zero).
String aa is lexicographically less than string bb (if strings aa and bb have the same length) if there exists some position ii (1≤i≤|a|1≤i≤|a|, where |s||s| is the length of the string ss) such that for every j<ij<i holds aj=bjaj=bj, and ai<biai<bi.
Input
The first line of the input contains the string ss consisting only of characters ‘0’, ‘1’ and ‘2’, its length is between 11 and 105105 (inclusive).
Output
Print a single string — the minimum possible (lexicographically) string you can obtain by using the swaps described above arbitrary number of times (possibly, zero).
Examples
Input
100210
Output
001120
Input
11222121
Output
11112222
Input
20
Output
20
C-C
You are given an array aa consisting of nn integer numbers.
You have to color this array in kk colors in such a way that:
Each element of the array should be colored in some color;
For each ii from 11 to kk there should be at least one element colored in the ii-th color in the array;
For each ii from 11 to kk all elements colored in the ii-th color should be distinct.
Obviously, such coloring might be impossible. In this case, print “NO”. Otherwise print “YES” and any coloring (i.e. numbers c1,c2,…cnc1,c2,…cn, where 1≤ci≤k1≤ci≤k and cici is the color of the ii-th element of the given array) satisfying the conditions above. If there are multiple answers, you can print any.
Input
The first line of the input contains two integers nn and kk (1≤k≤n≤50001≤k≤n≤5000) — the length of the array aa and the number of colors, respectively.
The second line of the input contains nn integers a1,a2,…,ana1,a2,…,an (1≤ai≤50001≤ai≤5000) — elements of the array aa.
Output
If there is no answer, print “NO”. Otherwise print “YES” and any coloring (i.e. numbers c1,c2,…cnc1,c2,…cn, where 1≤ci≤k1≤ci≤k and cici is the color of the ii-th element of the given array) satisfying the conditions described in the problem statement. If there are multiple answers, you can print any.
Examples
Input
4 2
1 2 2 3
Output
YES
1 1 2 2
Input
5 2
3 2 1 2 3
Output
YES
2 1 1 2 1
Input
5 2
2 1 1 2 1
Output
NO
Note
In the first example the answer 2 1 2 12 1 2 1 is also acceptable.
In the second example the answer 1 1 1 2 21 1 1 2 2 is also acceptable.
There exist other acceptable answers for both examples.
D-D
A positive integer xx is called a power of two if it can be represented as x=2yx=2y, where yy is a non-negative integer. So, the powers of two are 1,2,4,8,16,…1,2,4,8,16,….
You are given two positive integers nn and kk. Your task is to represent nn as the sum of exactly kk powers of two.
Input
The only line of the input contains two integers nn and kk (1≤n≤1091≤n≤109, 1≤k≤2⋅1051≤k≤2⋅105).
Output
If it is impossible to represent nn as the sum of kk powers of two, print NO.
Otherwise, print YES, and then print kk positive integers b1,b2,…,bkb1,b2,…,bk such that each of bibi is a power of two, and ∑i=1kbi=n∑i=1kbi=n. If there are multiple answers, you may print any of them.
Examples
Input
9 4
Output
YES
1 2 2 4
Input
8 1
Output
YES
8
Input
5 1
Output
NO
Input
3 7
Output
NO
E-E
A restaurant received n orders for the rental. Each rental order reserve the restaurant for a continuous period of time, the i-th order is characterized by two time values — the start time li and the finish time ri (li ≤ ri).
Restaurant management can accept and reject orders. What is the maximal number of orders the restaurant can accept?
No two accepted orders can intersect, i.e. they can’t share even a moment of time. If one order ends in the moment other starts, they can’t be accepted both.
Input
The first line contains integer number n (1 ≤ n ≤ 5·105) — number of orders. The following n lines contain integer values li and ri each (1 ≤ li ≤ ri ≤ 109).
Output
Print the maximal number of orders that can be accepted.
Examples
Input
2
7 11
4 7
Output
1
Input
5
1 2
2 3
3 4
4 5
5 6
Output
3
Input
6
4 8
1 5
4 7
2 5
1 3
6 8
Output
2
F-F
Array of integers is unimodal, if:
it is strictly increasing in the beginning;
after that it is constant;
after that it is strictly decreasing.
The first block (increasing) and the last block (decreasing) may be absent. It is allowed that both of this blocks are absent.
For example, the following three arrays are unimodal: [5, 7, 11, 11, 2, 1], [4, 4, 2], [7], but the following three are not unimodal: [5, 5, 6, 6, 1], [1, 2, 1, 2], [4, 5, 5, 6].
Write a program that checks if an array is unimodal.
Input
The first line contains integer n (1 ≤ n ≤ 100) — the number of elements in the array.
The second line contains n integers a1, a2, …, an (1 ≤ ai ≤ 1 000) — the elements of the array.
Output
Print “YES” if the given array is unimodal. Otherwise, print “NO”.
You can output each letter in any case (upper or lower).
Examples
Input
6
1 5 5 5 4 2
Output
YES
Input
5
10 20 30 20 10
Output
YES
Input
4
1 2 1 2
Output
NO
Input
7
3 3 3 3 3 3 3
Output
YES
Note
In the first example the array is unimodal, because it is strictly increasing in the beginning (from position 1 to position 2, inclusively), that it is constant (from position 2 to position 4, inclusively) and then it is strictly decreasing (from position 4 to position 6, inclusively).
G-G
Greatest common divisor GCD(a, b) of two positive integers a and b is equal to the biggest integer d such that both integers a and b are divisible by d. There are many efficient algorithms to find greatest common divisor GCD(a, b), for example, Euclid algorithm.
Formally, find the biggest integer d, such that all integers a, a + 1, a + 2, …, b are divisible by d. To make the problem even more complicated we allow a and b to be up to googol, 10100 — such number do not fit even in 64-bit integer type!
Input
The only line of the input contains two integers a and b (1 ≤ a ≤ b ≤ 10100).
Output
Output one integer — greatest common divisor of all integers from a to b inclusive.
Examples
Input
1 2
Output
1
Input
61803398874989484820458683436563811772030917980576 61803398874989484820458683436563811772030917980576
Output
61803398874989484820458683436563811772030917980576
H-H
You are given a 4x4 grid. You play a game — there is a sequence of tiles, each of them is either 2x1 or 1x2. Your task is to consequently place all tiles from the given sequence in the grid. When tile is placed, each cell which is located in fully occupied row or column is deleted (cells are deleted at the same time independently). You can place tile in the grid at any position, the only condition is that tiles (and tile parts) should not overlap. Your goal is to proceed all given figures and avoid crossing at any time.
Input
The only line contains a string ss consisting of zeroes and ones (1≤|s|≤10001≤|s|≤1000). Zero describes vertical tile, one describes horizontal tile.
Output
Output |s||s| lines — for each tile you should output two positive integers r,cr,c, not exceeding 44, representing numbers of smallest row and column intersecting with it.
If there exist multiple solutions, print any of them.
Example
Input
010
Output
1 1
1 2
1 4
Note
Following image illustrates the example after placing all three tiles:
Then the first row is deleted:
I-J
You are given three integers k, pa and pb.
You will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability pa / (pa + pb), add ‘a’ to the end of the sequence. Otherwise (with probability pb / (pa + pb)), add ‘b’ to the end of the sequence.
You stop once there are at least k subsequences that form ‘ab’. Determine the expected number of times ‘ab’ is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and . Print the value of .
Input
The first line will contain three integers integer k, pa, pb (1 ≤ k ≤ 1 000, 1 ≤ pa, pb ≤ 1 000 000).
Output
Print a single integer, the answer to the problem.
Examples
Input
1 1 1
Output
2
Input
3 1 4
Output
370000006
Note
The first sample, we will keep appending to our sequence until we get the subsequence ‘ab’ at least once. For instance, we get the sequence ‘ab’ with probability 1/4, ‘bbab’ with probability 1/16, and ‘aab’ with probability 1/8. Note, it’s impossible for us to end with a sequence like ‘aabab’, since we would have stopped our algorithm once we had the prefix ‘aab’.
The expected amount of times that ‘ab’ will occur across all valid sequences is 2.
For the second sample, the answer is equal to 341/100.