| There is a hidden array $a_1, a_2, \ldots, a_n$ of length $n$ whose elements are integers between $-m$ and $m$, inclusive. | |
| You are given an array $b_1, b_2, \ldots, b_n$ of length $n$ and a string $s$ of length $n$ consisting of the characters $\texttt{P}$, $\texttt{S}$, and $\texttt{?}$. | |
| For each $i$ from $1$ to $n$ inclusive, we must have: | |
| * If $s_i = \texttt{P}$, $b_i$ is the sum of $a_1$ through $a_i$. * If $s_i = \texttt{S}$, $b_i$ is the sum of $a_i$ through $a_n$. | |
| Output the number of ways to replace all $\texttt{?}$ in $s$ with either $\texttt{P}$ or $\texttt{S}$ such that there exists an array $a_1, a_2, \ldots, a_n$ with elements not exceeding $m$ by absolute value satisfying the constraints given by the array $b_1, b_2, \ldots, b_n$ and the string $s$. | |
| Since the answer may be large, output it modulo $998\,244\,353$. | |
| The first line contains a single integer $t$ ($1 \leq t \leq 10^3$) — the number of test cases. | |
| The first line of each test case contains two integers $n$ and $m$ ($2 \leq n \leq 2 \cdot 10^3$, $2 \leq m \leq 10^{9}$) — the length of the hidden array $a_1, a_2, \ldots, a_n$ and the maximum absolute value of an element $a_i$. | |
| The second line of each test case contains a string $s$ of length $n$ consisting of characters $\texttt{P}$, $\texttt{S}$, and $\texttt{?}$. | |
| The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($|b_i| \leq m \cdot n$). | |
| The sum of $n$ over all test cases does not exceed $5 \cdot 10^3$. | |
| For each test case, output a single integer — the number of ways to replace all $\texttt{?}$ in $s$ with either $\texttt{P}$ or $\texttt{S}$ that result in the existence of a valid array $a_1, a_2, \ldots, a_n$, modulo $998\,244\,353$. | |
| In the first test case, we can see that the following array satisfies all constraints, thus the answer is $1$: | |
| 1. $\texttt{P}$ — ${[\color{red}{\textbf{1}},3,4,2]}$: sum of $1$. 2. $\texttt{S}$ — ${[1,\color{red}{\textbf{3},4,2}]}$: sum of $9$. 3. $\texttt{P}$ — ${[\color{red}{1,3,\textbf{4}}, |