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.

An array $b$ of $m$ non-negative integers is said to be good if all the elements of $b$ can be made equal to $0$ using the following operation some (possibly, zero) times:

• Select two distinct indices $l$ and $r$ ($1 \leq l \color{red}{<} r \leq m$) and subtract $1$ from all $b_i$ such that $l \leq i \leq r$.

You are given two positive integers $n$, $k$ and a prime number $p$.

Over all $(k+1)^n$ arrays of length $n$ such that $0 \leq a_i \leq k$ for all $1 \leq i \leq n$, count the number of good arrays.

Since the number might be too large, you are only required to find it modulo $p$.

Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^3$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains three positive integers $n$, $k$ and $p$ ($3 \leq n \leq 3000$, $1 \leq k \leq n$, $10^8 < p < 10^9$) — the length of the array $a$, the upper bound on the elements of $a$ and modulus $p$.

It is guaranteed that the sum of $n^2$ over all test cases does not exceed $10^7$, and $p$ is prime.

For each test case, on a new line, output the number of good arrays modulo $p$.