You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
In the easy version, the $$$a_i$$$ are in the range $$$[0, n]$$$; in the hard version, the $$$a_i$$$ are in the range $$$[-1, n]$$$ and the definition of good permutation is slightly different. You can make hacks only if all versions of the problem are solved.

You are given an integer $$$n$$$ and an array $$$a_1, a_2 \dots, a_n$$$ of integers in the range $$$[0, n]$$$.

A permutation $$$p_1, p_2, \dots, p_n$$$ of $$$[1, 2, \dots, n]$$$ is good if, for each $$$i$$$, the following condition is true:

- the number of values $$$\leq i$$$ in $$$[p_1, p_2, \dots, p_i]$$$ is exactly $$$a_i$$$.

Count the good permutations of $$$[1, 2, \dots, n]$$$, modulo $$$998\,244\,353$$$.

## Input Format
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 a single integer $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) — the length of the array $$$a$$$.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le n$$$), which describe the conditions for a good permutation.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

## Output Format
For each test case, output a single line containing the number of good permutations, modulo $$$998\,244\,353$$$.

## Examples
```input
5
5
1 2 3 4 5
6
0 2 2 2 4 6
6
0 1 3 4 5 5
6
1 2 3 2 4 6
15
0 0 1 1 1 2 3 4 5 6 7 9 11 13 15
```
```output
1
4
0
0
532305727
```

## Note
In the first test case, the only good permutation is $$$[1, 2, 3, 4, 5]$$$.

In the second test case, there are $$$4$$$ good permutations: $$$[2, 1, 5, 6, 3, 4]$$$, $$$[2, 1, 5, 6, 4, 3]$$$, $$$[2, 1, 6, 5, 3, 4]$$$, $$$[2, 1, 6, 5, 4, 3]$$$. For example, $$$[2, 1, 5, 6, 3, 4]$$$ is good because:

- $$$a_1 = 0$$$, and there are $$$0$$$ values $$$\leq 1$$$ in $$$[p_1] = [2]$$$;
- $$$a_2 = 2$$$, and there are $$$2$$$ values $$$\leq 2$$$ in $$$[p_1, p_2] = [2, 1]$$$;
- $$$a_3 = 2$$$, and there are $$$2$$$ values $$$\leq 3$$$ in $$$[p_1, p_2, p_3] = [2, 1, 5]$$$;
- $$$a_4 = 2$$$, and there are $$$2$$$ values $$$\leq 4$$$ in $$$[p_1, p_2, p_3, p_4] = [2, 1, 5, 6]$$$;
- $$$a_5 = 4$$$, and there are $$$4$$$ values $$$\leq 5$$$ in $$$[p_1, p_2, p_3, p_4, p_5] = [2, 1, 5, 6, 3]$$$;
- $$$a_6 = 6$$$, and there are $$$6$$$ values $$$\leq 6$$$ in $$$[p_1, p_2, p_3, p_4, p_5, p_6] = [2, 1, 5, 6, 3, 4]$$$.

In the third test case, there are no good permutations, because there are no permutations with $$$a_6 = 5$$$ values $$$\leq 6$$$ in $$$[p_1, p_2, p_3, p_4, p_5, p_6]$$$.

Now solve the problem and return the code.