Restore Modulo

### 题意简述 给定 $n$ 和 $a_1, a_2, a_3, \dots, a_n$，求**最大**的 $m$ 使 $a_1 \sim a_n$ 在模 $m$ 意义下是**等差数列**。 **注意第一项也要小于 $m$。** 若 $m$ 无解输出 `-1`，若 $m$ 可以无限大输出 `0`，否则输出一行两个整数：最大的 $m$ 和对应的公差 $c$，保证 $0 \leq c \lt m$。 **本题有多组数据。** ### 数据范围 $1 \leq \Sigma n \leq 10^5, 1 \leq t \leq 10^5, 1 \leq a_i \leq 10 ^ 9$。

For the first place at the competition, Alex won many arrays of integers and was assured that these arrays are very expensive. After the award ceremony Alex decided to sell them. There is a rule in arrays pawnshop: you can sell array only if it can be compressed to a generator. This generator takes four non-negative numbers $ n $ , $ m $ , $ c $ , $ s $ . $ n $ and $ m $ must be positive, $ s $ non-negative and for $ c $ it must be true that $ 0 \leq c < m $ . The array $ a $ of length $ n $ is created according to the following rules: - $ a_1 = s \bmod m $ , here $ x \bmod y $ denotes remainder of the division of $ x $ by $ y $ ; - $ a_i = (a_{i-1} + c) \bmod m $ for all $ i $ such that $ 1 < i \le n $ . For example, if $ n = 5 $ , $ m = 7 $ , $ c = 4 $ , and $ s = 10 $ , then $ a = [3, 0, 4, 1, 5] $ . Price of such an array is the value of $ m $ in this generator. Alex has a question: how much money he can get for each of the arrays. Please, help him to understand for every array whether there exist four numbers $ n $ , $ m $ , $ c $ , $ s $ that generate this array. If yes, then maximize $ m $ .

The first line contains a single integer $ t $ ( $ 1 \leq t \leq 10^5 $ ) — the number of arrays. The first line of array description contains a single integer $ n $ ( $ 1 \leq n \leq 10^5 $ ) — the size of this array. The second line of array description contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 0 \leq a_i \leq 10^9 $ ) — elements of the array. It is guaranteed that the sum of array sizes does not exceed $ 10^5 $ .

For every array print: - $ -1 $ , if there are no such four numbers that generate this array; - $ 0 $ , if $ m $ can be arbitrary large; - the maximum value $ m $ and any appropriate $ c $ ( $ 0 \leq c < m $ ) in other cases.

6
6
1 9 17 6 14 3
3
4 2 2
3
7 3 4
3
2 2 4
5
0 1000000000 0 1000000000 0
2
1 1

19 8
-1
-1
-1
2000000000 1000000000
0