write a go solution for Description:
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^∗ 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<=t<=2*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<=n<=4*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<=p_i<=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*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.

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.. Output only the code with no comments, explanation, or additional text.