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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 3.0 seconds
Memory limit: 256.0 MB

# Problem
Let us call an edge-weighted tree with $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$ good if the weight of each edge is either $$$1$$$ or $$$-1$$$ and for each vertex $$$i$$$, the product of the edge weights of all edges having $$$i$$$ as one endpoint is $$$-1$$$.

You are given a positive integer $$$n$$$. There are $$$n^{n-2} \cdot 2^{n-1}$$$ distinct$$$^\dagger$$$ edge-weighted trees with $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$ such that each edge is either $$$1$$$ or $$$-1$$$. Your task is to find the sum of $$$d(1,n)^\ddagger$$$ of all such trees that are good. Since the answer can be quite large, you only need to find it modulo $$$998\,244\,353$$$.

$$$^\dagger$$$ Two trees are considered to be distinct if either:

- there exists two vertices such that there is an edge between them in one of the trees, and not in the other.
- there exists two vertices such that there is an edge between them in both trees but the weight of the edge between them in one tree is different from the one in the other tree.

Note that by Cayley's formula, the number of trees on $$$n$$$ labeled vertices is $$$n^{n-2}$$$. Since we have $$$n-1$$$ edges, there are $$$2^{n-1}$$$ possible assignment of weights(weight can either be $$$1$$$ or $$$-1$$$). That is why total number of distinct edge-weighted tree is $$$n^{n-2} \cdot 2^{n-1}$$$.

$$$^\ddagger$$$ $$$d(u,v)$$$ denotes the sum of the weight of all edges on the unique simple path from $$$u$$$ to $$$v$$$.

## Input Format
The first and only line of input contains a single integer $$$n$$$ ($$$1 \leq n \leq 5 \cdot 10^5$$$).

## Output Format
The only line of output should contain a single integer, the required answer, modulo $$$998\,244\,353$$$.

## Examples
```input
2
```
```output
998244352
```
-----
```input
1
```
```output
0
```
-----
```input
4
```
```output
998244343
```
-----
```input
10
```
```output
948359297
```
-----
```input
43434
```
```output
86232114
```

## Note
In the first test case, there is only $$$1$$$ distinct good tree. The value of $$$d(1,2)$$$ for that tree is $$$-1$$$, which is $$$998\,244\,352$$$ under modulo $$$998\,244\,353$$$.

In the second test case, the value of $$$d(1,1)$$$ for any tree is $$$0$$$, so the answer is $$$0$$$.

In the third test case, there are $$$16$$$ distinct good trees. The value of $$$d(1,4)$$$ is:

- $$$-2$$$ for $$$2$$$ trees;
- $$$-1$$$ for $$$8$$$ trees;
- $$$0$$$ for $$$4$$$ trees;
- $$$1$$$ for $$$2$$$ trees.

The sum of $$$d(1,4)$$$ over all trees is $$$2 \cdot (-2) + 8 \cdot (-1) + 4 \cdot (0) + 2 \cdot (1) = -10$$$, which is $$$998\,244\,343$$$ under modulo $$$998\,244\,353$$$.

Now solve the problem and return the code.