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09/04/2025 16:42:29

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H. Random Gap

Ограничения: время – 5s/10s, память – 8MiB Ввод: input.txt или стандартный ввод Вывод: output.txt или стандартный вывод copy
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The pseudo-random number genegators (or RNGs for short) are widely used in statistical calculations. One of the simplest and ubiquitious is the linear congruence RNG, that calculates the

$n$ -th random number

$R_n$ as

$R_n\ =\ (a*R_{n-1}\ +\ c)\ mod\ m$ , where

$a$ ,

$c$ and

$m$ are constants. For example, the sequence for

$a\ =\ 15$ ,

$c\ =\ 7$ ,

$m\ =\ 100$ and starting value

$R_0\ =\ 1$ is 1, 22, 37, 62, 37, 62, …

Linear RNG is very fast, and can yield surprisinly good sequence of random numbers, given the right choice of constants. There are various measures of RNG quality, and your task is to calculate one of them, namely the longest gap between members of the sequence. More precisely, given the values of

$a$ ,

$c$ ,

$m$ , and

$R_0$ , find such two elements

$R_i\ <\ R_j$ that:

there are no other $R_k$ : $R_i\ ≤\ R_k\ ≤\ R_j$ ,
the difference $R_j\ -\ R_i$ is maximal.

Input

Input file contains integer numbers

$a$

$c$

$m$

$R_0$ .

Output

Output file should contain the single number – the maximal difference found.

Constraints

$0\ ≤\ a,\ c,\ R_0\ ≤\ 10^7$ ,

$1\ ≤\ m\ ≤\ 16\ 000\ 000$ ,

$a*m\ +\ c\ <\ 2^32$ .

Sample Input 1

15 7 100 1

Sample Output 1

Sample Input 2

2 0 127 5

Sample Output 2

Source: A. Klenin, ICPC NEERC Far-Eastern subregional, 2004