Problem:
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}.

Sample Input 1

6 2 1001

Sample Output 1

1369

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

Sample Input 2

1000 2 16

Sample Output 2

6

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

Sample Input 3

10000000000 10 99959

Sample Output 3

492443256176507

Solution in Pascal (FPC 3.0.4) (judge verdict: Accepted)

program sequence_sum;
var
  n, x, m, num: int64;
  i, k, offset, period, rem: integer;
  a: array[0..100000] of int64;
  b: array[0..100000] of integer;
begin
  read(n, x, m);
  if n = 1 then
  begin
    writeln(x);
    halt(0);
  end;
  a[1] := x;
  b[integer(x)] := 1;
  for i := 2 to integer(m) + 1 do
  begin
    x := (x * x) mod m;
    a[i] := a[i - 1] + x;
    if i = n then
    begin
      writeln(a[i]);
      halt(0);
    end;
    k := integer(x);
    if b[k] > 0 then
    begin
      offset := b[k] - 1;
      period := i - offset - 1;
      break;
    end;
    b[k] := i;
  end;
  num := (n - offset) div int64(period);
  rem := integer(n - offset - num * period);
  writeln((a[offset + period] - a[offset]) * num + a[offset + rem]);
end.