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: 3.0 seconds
Memory limit: 512.0 MB

# Problem
This is the hard version of the problem. The differences between the two versions are the constraints on $$$n, m, b_0$$$ and the time limit. You can make hacks only if both versions are solved.

Little R has counted many sets before, and now she decides to count arrays.

Little R thinks an array $$$b_0, \ldots, b_n$$$ consisting of non-negative integers is continuous if and only if, for each $$$i$$$ such that $$$1 \leq i \leq n$$$, $$$\lvert b_i - b_{i-1} \rvert = 1$$$ is satisfied. She likes continuity, so she only wants to generate continuous arrays.

If Little R is given $$$b_0$$$ and $$$a_1, \ldots, a_n$$$, she will try to generate a non-negative continuous array $$$b$$$, which has no similarity with $$$a$$$. More formally, for all $$$1 \leq i \leq n$$$, $$$a_i \neq b_i$$$ holds.

However, Little R does not have any array $$$a$$$. Instead, she gives you $$$n$$$, $$$m$$$ and $$$b_0$$$. She wants to count the different integer arrays $$$a_1, \ldots, a_n$$$ satisfying:

- $$$1 \leq a_i \leq m$$$;
- At least one non-negative continuous array $$$b_0, \ldots, b_n$$$ can be generated.

Note that $$$b_i \geq 0$$$, but the $$$b_i$$$ can be arbitrarily large.

Since the actual answer may be enormous, please just tell her the answer modulo $$$998\,244\,353$$$.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t\ (1 \leq t \leq 10^4)$$$. The description of the test cases follows.

The first and only line of each test case contains three integers $$$n$$$, $$$m$$$, and $$$b_0$$$ ($$$1 \leq n \leq 2 \cdot 10^6$$$, $$$1 \leq m \leq 2 \cdot 10^6$$$, $$$0 \leq b_0 \leq 2\cdot 10^6$$$) — the length of the array $$$a_1, \ldots, a_n$$$, the maximum possible element in $$$a_1, \ldots, a_n$$$, and the initial element of the array $$$b_0, \ldots, b_n$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceeds $$$10^7$$$.

## Output Format
For each test case, output a single line containing an integer: the number of different arrays $$$a_1, \ldots, a_n$$$ satisfying the conditions, modulo $$$998\,244\,353$$$.

## Examples
```input
6
3 2 1
5 5 3
13 4 1
100 6 7
100 11 3
1000 424 132
```
```output
6
3120
59982228
943484039
644081522
501350342
```

## Note
In the first test case, for example, when $$$a = [1, 2, 1]$$$, we can set $$$b = [1, 0, 1, 0]$$$. When $$$a = [1, 1, 2]$$$, we can set $$$b = [1, 2, 3, 4]$$$. In total, there are $$$6$$$ valid choices of $$$a_1, \ldots, a_n$$$: in fact, it can be proved that only $$$a = [2, 1, 1]$$$ and $$$a = [2, 1, 2]$$$ make it impossible to construct a non-negative continuous $$$b_0, \ldots, b_n$$$, so the answer is $$$2^3 - 2 = 6$$$.

Now solve the problem and return the code.