CF1569C.Jury Meeting

$n$ people gathered to hold a jury meeting of the upcoming competition, the $i$ -th member of the jury came up with $a_i$ tasks, which they want to share with each other.

First, the jury decides on the order which they will follow while describing the tasks. Let that be a permutation $p$ of numbers from $1$ to $n$ (an array of size $n$ where each integer from $1$ to $n$ occurs exactly once).

Then the discussion goes as follows:

• If a jury member $p_1$ has some tasks left to tell, then they tell one task to others. Otherwise, they are skipped.
• If a jury member $p_2$ has some tasks left to tell, then they tell one task to others. Otherwise, they are skipped.
• ...
• If a jury member $p_n$ has some tasks left to tell, then they tell one task to others. Otherwise, they are skipped.
• If there are still members with tasks left, then the process repeats from the start. Otherwise, the discussion ends.

A permutation $p$ is nice if none of the jury members tell two or more of their own tasks in a row.

Count the number of nice permutations. The answer may be really large, so print it modulo $998\,244\,353$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first line of the test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ) — number of jury members.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^9$ ) — the number of problems that the $i$ -th member of the jury came up with.

The sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print one integer — the number of nice permutations, taken modulo $998\,244\,353$ .

Explanation of the first test case from the example:

There are two possible permutations, $p = [1, 2]$ and $p = [2, 1]$ . For $p = [1, 2]$ , the process is the following:

1. the first jury member tells a task;
2. the second jury member tells a task;
3. the first jury member doesn't have any tasks left to tell, so they are skipped;
4. the second jury member tells a task.

So, the second jury member has told two tasks in a row (in succession), so the permutation is not nice.

For $p = [2, 1]$ , the process is the following:

1. the second jury member tells a task;
2. the first jury member tells a task;
3. the second jury member tells a task.

So, this permutation is nice.