Score : $800$ points

Problem Statement

Given is a string $S$ consisting of 0 and 1. Find the number of strings, modulo $998244353$, that can result from applying the following operation on $S$ between $0$ and $K$ times (inclusive):

• Choose a pair of integers $i, j$ $(1\leq i < j\leq |S|)$ such that the $i$-th and $j$-th characters of $S$ are 0 and 1, respectively. Remove the $j$-th character from $S$ and insert it to the immediate left of the $i$-th character.

Constraints

• $1 \leq |S| \leq 300$
• $0 \leq K \leq 10^9$
• $S$ consists of 0 and 1.

Input

Input is given from Standard Input in the following format:

$S$ $K$

Output

Find the number of strings, modulo $998244353$, that can result from applying the operation on $S$ between $0$ and $K$ times (inclusive).

0101 1

4

Four strings, 0101, 0110, 1001, and 1010, can result.

01100110 2

14

1101010010101101110111100011011111011000111101110101010010101010101 20

113434815