This is the hard version of the problem. The only difference between the two versions is the constraint on $n$. You can make hacks only if both versions of the problem are solved.

You are given an array of integers $a$ of length $n$.

In one operation, you will perform the following two-step process:

1. Choose an index $i$ such that $1 \le i < |a|$ and $a_i = i$.
2. Remove $a_i$ and $a_{i+1}$ from the array and concatenate the remaining parts.

Find the maximum number of times that you can perform the operation above.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 800$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the array $a$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $800$.

For each test case, output a single integer — the maximum number of times that you can perform the operation.