You are given an integer $$$n$$$ and an array $$$a$$$ of length $$$n-1$$$.
For a permutation$$$^{\\text{∗}}$$$ $$$p$$$ of length $$$n$$$, define
$$$$$$ v_i=\\min\\left(\\max_{1\\le j\\le i}p_j, \\max_{i+1\\le j\\le n}p_j\\right), $$$$$$
where $$$1\\le i\\le n-1$$$.
In other words, cut the permutation between positions $$$i$$$ and $$$i+1$$$, take the maximum element on each side, and let $$$v_i$$$ be the smaller of these two values.
Your task is to count the number of permutations $$$p$$$ of length $$$n$$$ such that $$$v_i=a_i$$$ for each $$$1\\le i\\le n-1$$$.
Output the answer modulo $$$998\\,244\\,353$$$.
$$$^{\\text{∗}}$$$A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 10^4$$$). The description of the test cases follows.
The first line of each test case contains one integer $$$n$$$ ($$$2\\le n\\le 10^6$$$) — the length of $$$p$$$.
The second line contains $$$n-1$$$ integers $$$a_1,a_2,\\ldots,a_{n-1}$$$ ($$$1\\le a_i\\le n$$$) — the elements of $$$a$$$.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^6$$$.
For each test case, output one integer — the number of suitable permutations, modulo $$$998\\,244\\,353$$$.
112132 231 142 3 253 3 4 22231 243 3 354 4 4 442 1 263 3 5 5 5
220020241208
In the first test case, both permutations $$$[1,2]$$$ and $$$[2,1]$$$ satisfy $$$v_1=1$$$.
In the second test case, the only suitable permutations are $$$[2,1,3]$$$ and $$$[3,1,2]$$$.
In the third test case, no suitable permutation exists.