You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
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 $$$998\,244\,353$$$.

## Input Format
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 $$$5 \cdot 10^3$$$.

## Output Format
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 $$$998\,244\,353$$$.

## Examples
```input
6
4 10
PSPP
1 9 8 10
4 1000000000
????
1 1 1 4000000000
8 1000000000
?P?SSP?P
-857095623 -1424391899 -851974476 673437144 471253851 -543483033 364945701 -178537332
4 7
PPSS
4 2 1 3
9 20
?????????
1 2 3 4 5 6 7 8 9
3 1000000000
P??
-145463248 -974068460 -1287458396
```
```output
1
0
2
1
14
1
```

## Note
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. $$$\texttt{P}$$$ — $$${[\color{red}{1,3,\textbf{4}},2]}$$$: sum of $$$8$$$.
4. $$$\texttt{P}$$$ — $$${[\color{red}{1,3,4,\textbf{2}}]}$$$: sum of $$$10$$$.

In the second test case, it can be shown that no array $$$a$$$ with all $$$|a_i| \leq m = 10^9$$$ satisfies all constraints.

Now solve the problem and return the code.