Dora and C++
Description
Dora has just learned the programming language C++!
However, she has completely misunderstood the meaning of C++. She considers it as two kinds of adding operations on the array $c$ with $n$ elements. Dora has two integers $a$ and $b$. In one operation, she can choose one of the following things to do.
- Choose an integer $i$ such that $1 \leq i \leq n$, and increase $c_i$ by $a$.
- Choose an integer $i$ such that $1 \leq i \leq n$, and increase $c_i$ by $b$.
Note that $a$ and $b$ are constants, and they can be the same.
Let's define a range of array $d$ as $\max(d_i) - \min(d_i)$. For instance, the range of the array $[1, 2, 3, 4]$ is $4 - 1 = 3$, the range of the array $[5, 2, 8, 2, 2, 1]$ is $8 - 1 = 7$, and the range of the array $[3, 3, 3]$ is $3 - 3 = 0$.
After any number of operations (possibly, $0$), Dora calculates the range of the new array. You need to help Dora minimize this value, but since Dora loves exploring all by herself, you only need to tell her the minimized value.
Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of test cases follows.
The first line of each test case contains three integers $n$, $a$, and $b$ ($1 \leq n \leq 10^5$, $1 \leq a, b \leq 10^9$) — the length of the array $c$ and the constant values, respectively.
The second line of each test case contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \leq c_i \leq 10^9$) — the initial elements of the array $c$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
For each test case, output a single integer — the minimum possible range of the array after any number of operations.
Input
Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of test cases follows.
The first line of each test case contains three integers $n$, $a$, and $b$ ($1 \leq n \leq 10^5$, $1 \leq a, b \leq 10^9$) — the length of the array $c$ and the constant values, respectively.
The second line of each test case contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \leq c_i \leq 10^9$) — the initial elements of the array $c$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case, output a single integer — the minimum possible range of the array after any number of operations.
10
4 5 5
1 3 4 4
4 2 3
1 3 4 6
4 7 7
1 1 2 6
3 15 9
1 9 5
3 18 12
1 4 5
7 27 36
33 13 23 12 35 24 41
10 6 9
15 5 6 9 8 2 12 15 3 8
2 1 1000000000
1 1000000000
6 336718728 709848696
552806726 474775724 15129785 371139304 178408298 13106071
6 335734893 671469786
138885253 70095920 456876775 9345665 214704906 375508929
3
0
3
2
3
5
1
0
17
205359241
Note
In the first test case, we can increase $c_1 = 1$ by $a = 5$. The array $c$ will become $[6, 3, 4, 4]$, and the range is $3$. Note that there is more than one way to reach the answer.
In the second test case, we can increase $c_1 = 1$ by $a = 2$ and then increase $c_1 = 3$ by $b = 3$. Also, we can increase $c_2 = 3$ by $b = 3$ and increase $c_3 = 4$ by $a = 2$. The array $c$ will become $[6, 6, 6, 6]$, and the range is $0$.