Shrink-Reverse
Description
You are given a binary string $s$ of length $n$ (a string consisting of $n$ characters, and each character is either 0 or 1).
Let's look at $s$ as at a binary representation of some integer, and name that integer as the value of string $s$. For example, the value of 000 is $0$, the value of 01101 is $13$, "100000" is $32$ and so on.
You can perform at most $k$ operations on $s$. Each operation should have one of the two following types:
- SWAP: choose two indices $i < j$ in $s$ and swap $s_i$ with $s_j$;
- SHRINK-REVERSE: delete all leading zeroes from $s$ and reverse $s$.
What is the minimum value of $s$ you can achieve by performing at most $k$ operations on $s$?
The first line contains two integers $n$ and $k$ ($2 \le n \le 5 \cdot 10^5$; $1 \le k \le n$) — the length of the string $s$ and the maximum number of operations.
The second line contains the string $s$ of length $n$ consisting of characters 0 and/or 1.
Additional constraint on the input: $s$ contains at least one 1.
Print a single integer — the minimum value of $s$ you can achieve using no more than $k$ operations. Since the answer may be too large, print it modulo $10^{9} + 7$.
Note that you need to minimize the original value, not the remainder.
Input
The first line contains two integers $n$ and $k$ ($2 \le n \le 5 \cdot 10^5$; $1 \le k \le n$) — the length of the string $s$ and the maximum number of operations.
The second line contains the string $s$ of length $n$ consisting of characters 0 and/or 1.
Additional constraint on the input: $s$ contains at least one 1.
Output
Print a single integer — the minimum value of $s$ you can achieve using no more than $k$ operations. Since the answer may be too large, print it modulo $10^{9} + 7$.
Note that you need to minimize the original value, not the remainder.
8 2
10010010
8 2
01101000
30 30
111111111111111111111111111111
14 1
10110001111100
7
7
73741816
3197
Note
In the first example, one of the optimal strategies is the following:
- 10010010 $\xrightarrow{\texttt{SWAP}}$ 00010110;
- 00010110 $\xrightarrow{\texttt{SWAP}}$ 00000111.
In the second example, one of the optimal strategies is the following:
- 01101000 $\xrightarrow{\texttt{SHRINK}}$ 1101000 $\xrightarrow{\texttt{REVERSE}}$ 0001011;
- 0001011 $\xrightarrow{\texttt{SWAP}}$ 0000111.