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G - Sequence in mod P

Score : $600$ points

Problem Statement

There is a sequence $X=(X_0, X_1, \ldots)$ defined by the following recurrence relation.

$X_i = \left\{ \begin{array}{ll} S & (i = 0)\\ (A X_{i-1}+B) \bmod P & (i \geq 1) \end{array} \right.$

Determine whether there is an $i$ such that $X_i=G$. If it exists, find the smallest such $i$.
Here, $x \bmod y$ denotes the remainder when $x$ is divided by $y$ (the least non-negative residue).

You are given $T$ test cases for each input file.

Constraints

• $1 \leq T \leq 100$
• $2 \leq P \leq 10^9$
• $P$ is a prime.
• $0\leq A,B,S,G < P$
• All values in the input are integers.

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Input

The input is given from Standard Input in the following format:

$T$

$\mathrm{case}_1$

$\mathrm{case}_2$

$\vdots$

$\mathrm{case}_T$

Each test case is in the following format:

``` $P$ $A$ $B$ $S$ $G$ ```

Output

Print $T$ lines.
The $t$-th line should contain the smallest $i$ such that $X_i=G$ for $\mathrm{case}_t$, or -1 if there is no such $i$.

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3
5 2 1 1 0
5 2 2 3 0
11 1 1 0 10

3
-1
10

For the first test case, we have $X=(1,3,2,0,\ldots)$, so the smallest $i$ such that $X_i=0$ is $3$.
For the second test case, we have $X=(3,3,3,3,\ldots)$, so there is no $i$ such that $X_i=0$.