Problem Description

Given the indeterminate equation

If this equation has no integer solution, output $-1$.
If this equation has integer solutions and also has positive integer solutions, output: the number of positive integer solutions, the minimum value of $x$ among all positive integer solutions, the minimum value of $y$ among all positive integer solutions, the maximum value of $x$ among all positive integer solutions, and the maximum value of $y$ among all positive integer solutions.
If the equation has integer solutions but has no positive integer solution, you need to output: the minimum positive integer value of $x$ among all integer solutions, and the minimum positive integer value of $y$ among all integer solutions.

A positive integer solution means a solution where both $x$ and $y$ are positive integers, and $\boldsymbol{0}$ is not a positive integer.
An integer solution means a solution where both $x$ and $y$ are integers.
The minimum positive integer value of $x$ means the minimum value of $x$ among all integer solutions with $x$ being a positive integer, and similarly for $y$.

Input Format

The first line contains a positive integer $T$, representing the number of test cases.

The next $T$ lines each contain three positive integers $a, b, c$ separated by spaces.

Output Format

Output $T$ lines.

If the corresponding query has no integer solution, output a single number $-1$.
If the corresponding query has integer solutions but no positive integer solution, output $2$ numbers separated by spaces, representing the minimum positive integer value of $x$ and the minimum positive integer value of $y$ among integer solutions, in order.
Otherwise, output $5$ numbers separated by spaces, representing the number of positive integer solutions, the minimum $x$, the minimum $y$, the maximum $x$, and the maximum $y$ among positive integer solutions, in order.

The input and output sizes are large, so please use fast I/O methods.

7
2 11 100
3 18 6
192 608 17
19 2 60817
11 45 14
19 19 810
98 76 5432

4 6 2 39 8
2 1
-1
1600 1 18 3199 30399
34 3
-1
2 12 7 50 56

Hint

Constraints

For $100\%$ of the testdata, $1 \le T \le 2 \times {10}^5$, $1 \le a, b, c \le {10}^9$.

Translated by ChatGPT 5