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: 512.0 MB

# Problem
In the country of Dengkleknesia, there are $$$N$$$ factories numbered from $$$1$$$ to $$$N$$$. Factory $$$i$$$ has an electrical coefficient of $$$A_i$$$. There are also $$$N-1$$$ power lines with the $$$j$$$-th power line connecting factory $$$U_j$$$ and factory $$$V_j$$$. It can be guaranteed that each factory in Dengkleknesia is connected to all other factories in Dengkleknesia through one or more power lines. In other words, the collection of factories forms a tree. Each pair of different factories in Dengkleknesia can use one or more existing power lines to transfer electricity to each other. However, each power line needs to be turned on first so that electricity can pass through it.

Define $$$f(x, y, z)$$$ as the minimum number of power lines that need to be turned on so that factory $$$x$$$ can make electrical transfers to factory $$$y$$$ and factory $$$z$$$. Also define $$$g(x, y, z)$$$ as the number of distinct prime factors of $$$\text{GCD}(A_x, A_y, A_z)$$$.

To measure the electrical efficiency, you must find the sum of $$$f(x, y, z) \times g(x, y, z)$$$ for all combinations of $$$(x, y, z)$$$ such that $$$1 \leq x < y < z \leq N$$$. Because the answer can be very large, you just need to output the answer modulo $$$998\,244\,353$$$.

Note: $$$\text{GCD}(k_1, k_2, k_3)$$$ is the greatest common divisor of $$$k_1$$$, $$$k_2$$$, and $$$k_3$$$, which is the biggest integer that simultaneously divides $$$k_1$$$, $$$k_2$$$, and $$$k_3$$$.

## Input Format
The first line contains a single integer $$$N$$$ ($$$1 \le N \le 2 \cdot 10^5$$$) — the number of factories in Dengkleknesia.

The second line contains $$$N$$$ integers $$$A_1, A_2, \dots, A_N$$$ ($$$1 \leq A_i \leq 2 \cdot 10^5$$$) — the electrical coefficients of the factories.

The $$$j$$$-th of the next $$$N-1$$$ lines contains two integers $$$U_j$$$ and $$$V_j$$$ ($$$1 \le U_j, V_j \le N$$$) — a power line that connects cities $$$U_j$$$ and $$$V_j$$$. The collection of factories forms a tree.

## Output Format
An integer representing the sum of $$$f(x, y, z) \times g(x, y, z)$$$ for all combinations of $$$(x, y, z)$$$ such that $$$1 \leq x < y < z \leq N$$$, modulo $$$998\,244\,353$$$

## Examples
```input
3
1 2 3
1 2
2 3
```
```output
0
```
-----
```input
4
6 14 6 6
1 2
2 3
2 4
```
```output
12
```

## Note
In the first example, the only $$$(x, y, z)$$$ possible is $$$(1, 2, 3)$$$. Because $$$\text{GCD}(A_1, A_2, A_3) = \text{GCD}(1, 2, 3) = 1$$$ has $$$0$$$ distinct prime factors, therefore $$$f(x, y, z) \times g(x, y, z) = 2 \times 0 = 0$$$.

In the second example, all triples $$$(x, y, z)$$$ that satisfy the condition are as follows:

- $$$(1, 2, 3) \rightarrow f(1, 2, 3) \times g(1, 2, 3) = 2 \times 1 = 2$$$
- $$$(1, 2, 4) \rightarrow f(1, 2, 4) \times g(1, 2, 4) = 2 \times 1 = 2$$$
- $$$(1, 3, 4) \rightarrow f(1, 3, 4) \times g(1, 3, 4) = 3 \times 2 = 6$$$
- $$$(2, 3, 4) \rightarrow f(2, 3, 4) \times g(2, 3, 4) = 2 \times 1 = 2$$$

So the electrical efficiency is $$$2 + 2 + 6 + 2 = 12$$$.

Now solve the problem and return the code.