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: 7.0 seconds
Memory limit: 1024.0 MB

# Problem
This is the hard version of the problem. The only difference between the two versions are the allowed characters in $$$s$$$. You can make hacks only if both versions of the problem are solved.

You are given a permutation $$$p$$$ of length $$$n$$$. You are also given a string $$$s$$$ of length $$$n$$$, where each character is either L, R, or ?.

For each $$$i$$$ from $$$1$$$ to $$$n$$$:

- Define $$$l_i$$$ as the largest index $$$j < i$$$ such that $$$p_j > p_i$$$. If there is no such index, $$$l_i := i$$$.
- Define $$$r_i$$$ as the smallest index $$$j > i$$$ such that $$$p_j > p_i$$$. If there is no such index, $$$r_i := i$$$.

Initially, you have an undirected graph with $$$n$$$ vertices (numbered from $$$1$$$ to $$$n$$$) and no edges. Then, for each $$$i$$$ from $$$1$$$ to $$$n$$$, add one edge to the graph:

- If $$$s_i =$$$ L, add the edge $$$(i, l_i)$$$ to the graph.
- If $$$s_i =$$$ R, add the edge $$$(i, r_i)$$$ to the graph.
- If $$$s_i =$$$ ?, either add the edge $$$(i, l_i)$$$ or the edge $$$(i, r_i)$$$ to the graph at your choice.

Find the maximum possible diameter over all connected$$$^{\text{∗}}$$$ graphs that you can form. Output $$$-1$$$ if it is not possible to form any connected graphs.

## Input Format
Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \le t \le 2 \cdot 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 4 \cdot 10^5$$$) — the length of the permutation $$$p$$$.

The second line of each test case contains $$$n$$$ integers $$$p_1,p_2,\ldots, p_n$$$ ($$$1 \le p_i \le n$$$) — the elements of $$$p$$$, which are guaranteed to form a permutation.

The third line of each test case contains a string $$$s$$$ of length $$$n$$$. It is guaranteed that it consists only of the characters L, R, and ?.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$4 \cdot 10^5$$$.

## Output Format
For each test case, output the maximum possible diameter over all connected graphs that you form, or $$$-1$$$ if it is not possible to form any connected graphs.

## Examples
```input
8
5
2 1 4 3 5
R?RL?
2
1 2
LR
3
3 1 2
L?R
7
5 3 1 6 4 2 7
?R?R?R?
5
5 2 1 3 4
?????
6
6 2 3 4 5 1
?LLRLL
8
1 7 5 6 2 8 4 3
?R??????
12
6 10 7 1 8 5 12 2 11 3 4 9
????????????
```
```output
3
-1
-1
4
4
3
5
8
```

## Note
In the first test case, there are two connected graphs (the labels are indices):

The graph on the left has a diameter of $$$2$$$, while the graph on the right has a diameter of $$$3$$$, so the answer is $$$3$$$.

In the second test case, there are no connected graphs, so the answer is $$$-1$$$.

Now solve the problem and return the code.