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E - Sequence Sum

Score : $500$ points

Problem Statement

Let us denote by $f(x, m)$ the remainder of the Euclidean division of $x$ by $m$.

Let $A$ be the sequence that is defined by the initial value $A_1=X$ and the recurrence relation $A_{n+1} = f(A_n^2, M)$. Find $\displaystyle{\sum_{i=1}^N A_i}$.

Constraints

• $1 \leq N \leq 10^{10}$
• $0 \leq X < M \leq 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $X$ $M$

Output

Print $\displaystyle{\sum_{i=1}^N A_i}$.

6 2 1001

1369

The sequence $A$ begins $2,4,16,256,471,620,\ldots$ Therefore, the answer is $2+4+16+256+471+620=1369$.

1000 2 16

6

The sequence $A$ begins $2,4,0,0,\ldots$ Therefore, the answer is $6$.

10000000000 10 99959

492443256176507