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 $998\,244\,353$.

有 $n$ 人聚集在一起，为即将到来的比赛召开评委会会议。其中第 $i$ 位评委会成员提出了 $a_i$ 道题目，希望与其他成员分享。

首先，评委会决定描述题目的顺序。设该顺序为 $1$ 到 $n$ 的一个排列 $p$（即一个长度为 $n$ 的数组，其中 $1$ 到 $n$ 的每个整数恰好出现一次）。

随后讨论按如下方式进行：

• 若评委会成员 $p_1$ 还有未讲述的题目，则其向其他人讲述一道题目；否则跳过。
• 若评委会成员 $p_2$ 还有未讲述的题目，则其向其他人讲述一道题目；否则跳过。
• …
• 若评委会成员 $p_n$ 还有未讲述的题目，则其向其他人讲述一道题目；否则跳过。
• 若仍有成员尚有题目未讲述，则流程从头开始重复；否则讨论结束。

若排列 $p$ 满足：在上述整个讨论过程中，没有任何一位评委会成员连续讲述两道或更多道自己提出的题目，则称该排列 $p$ 是“好的”（nice）。

请计算“好的”排列的个数。答案可能非常大，请对 $998\,244\,353$ 取模后输出。

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 $2 \cdot 10^5$.

第一行包含一个整数 $t$（$1 \le t \le 10^4$）—— 测试用例的数量。

每个测试用例的第一行包含一个整数 $n$（$2 \le n \le 2 \cdot 10^5$）—— 评审团成员的数量。

第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$1 \le a_i \le 10^9$）—— 第 $i$ 位评审团成员提出的题目数量。

所有测试用例的 $n$ 值之和不超过 $2 \cdot 10^5$。

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

对于每个测试用例，输出一个整数——“好”排列的数量对 $998\,244\,353$ 取模的结果。

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.

示例中第一个测试用例的解释：

共有两种可能的排列：$p = [1, 2]$ 和 $p = [2, 1]$。对于 $p = [1, 2]$，过程如下：

1. 第一位评委讲述一道题目；
2. 第二位评委讲述一道题目；
3. 第一位评委已无剩余题目可讲，因此被跳过；
4. 第二位评委再次讲述一道题目。

因此，第二位评委连续（连续地）讲述了两道题目，该排列不是“优美的”。

对于 $p = [2, 1]$，过程如下：

1. 第二位评委讲述一道题目；
2. 第一位评委讲述一道题目；
3. 第二位评委再次讲述一道题目。

因此，该排列是“优美的”。