409953: GYM103861 K Vision Test

Prof. Pang has an extraordinary vision. He can see the pixels on a 4K monitor. To test Prof. Pang's vision, Prof. Shou will show Prof. Pang several pixels and let Prof. Pang guess a straight line that contains these pixels. Given $$$k$$$ pixels with coordinates $$$(i, y_i)$$$ ($$$0\le i<k$$$), Prof. Pang must find nonnegative integers $$$a, b$$$ and $$$c$$$ (which represent the line $$$y=\frac{ax+b}{c}$$$) such that $$$y_i=\lfloor \frac{ai+b}{c} \rfloor$$$ for all $$$0\le i<k$$$.

Prof. Shou will ask Prof. Pang multiple questions. They are given as follows: Prof. Shou has a fixed array $$$x_1,\ldots, x_n$$$. For each question, Prof. Shou chooses a range in the array, $$$x_l,\ldots, x_r$$$. Then he defines $$$y_i=x_{l+i}$$$ for $$$0\le i\le r - l$$$ and asks Prof. Pang to answer the question for the $$$r-l+1$$$ pixels $$$(0, y_0), \ldots, (r-l, y_{r-l})$$$.

Please help Prof. Pang answer all the questions. For each question, output the answer with the minimum $$$(c, a, b)$$$ in lexical order.

It is guaranteed that the answer exists when Prof. Pang chooses the whole array $$$x_1, x_2, \dots, x_n$$$. So the answer always exists when Prof. Pang chooses an interval of this array.

The first line contains a single integer $$$T$$$ ($$$1\le T\le 10^5$$$) denoting the number of test cases.

For each test case, the first line contains an integer $$$n$$$ ($$$1\leq n\leq 10^5$$$). The second line contains $$$n$$$ numbers $$$x_1, \ldots , x_{n}$$$ ($$$0\leq x_i\leq 10^9$$$).

The next line contains an integer $$$q$$$ ($$$1\le q\le 10^5$$$) denoting the number of questions.

Each of the following $$$q$$$ lines contains two integers $$$l, r$$$ ($$$1\le l\le r\le n$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases will not exceed $$$10^5$$$ and that the sum of $$$q$$$ over all test cases will not exceed $$$10^5$$$.

In the order of input, output one line with three integers $$$a, b, c$$$ denoting the answer for each question.

3
5
1 1 2 2 2
4
1 5
1 1
3 5
2 3
5
1 2 3 4 6
3
1 5
2 4
3 5
3
0 3 5
1
1 3

1 4 3
0 1 1
0 2 1
1 1 1
5 4 4
1 2 1
3 6 2
5 1 2