Problem:
Score : 500 points

Problem Statement

Snuke is conducting an optical experiment using mirrors and his new invention, the rifle of Mysterious Light.

Three mirrors of length N are set so that they form an equilateral triangle.
Let the vertices of the triangle be a, b and c.

Inside the triangle, the rifle is placed at the point p on segment ab such that ap = X.
(The size of the rifle is negligible.)
Now, the rifle is about to fire a ray of Mysterious Light in the direction of bc.

The ray of Mysterious Light will travel in a straight line, and will be reflected by mirrors, in the same ways as "ordinary" light.
There is one major difference, though: it will be also reflected by its own trajectory as if it is a mirror!
When the ray comes back to the rifle, the ray will be absorbed.

The following image shows the ray's trajectory where N = 5 and X = 2.

It can be shown that the ray eventually comes back to the rifle and is absorbed, regardless of the values of N and X.
Find the total length of the ray's trajectory.

Constraints

2≦N≦10^{12}

1≦X≦N-1

N and X are integers.

Partial Points

300 points will be awarded for passing the test set satisfying N≦1000.

Another 200 points will be awarded for passing the test set without additional constraints.

Input

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

N X

Output

Print the total length of the ray's trajectory.

Sample Input 1

5 2

Sample Output 1

12

Refer to the image in the Problem Statement section.
The total length of the trajectory is 2+3+2+2+1+1+1 = 12.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <cmath>

#define lint long long int
using namespace std;

lint Solve(lint N, lint X) {
    if (!N || !X)
        return 0;
    
    //cout << "Solve(" << N << ", " << X << ") - Type ";
    if (2 * X <= N) {
        //cout << "1" << endl;
        return 3 * X       + Solve(N - X, N - 2 * X);
    }
    else {
        //cout << "2" << endl;
        lint C = X / (N - X);
        return X + (N - X) * (3 * C + 1) - N + Solve(N - (N - X) * C, N - (N - X) * (C + 1));
    }
}

void Stress() {
    static const lint NMAX = 1000000000000LL;
    static const double MAGIC = (3 - sqrt(5)) * 0.5;

    lint N = NMAX;
    lint X = N * MAGIC;

    cout << Solve(N, X) << '\n';
}

void Submit() {
    lint N, X;
    cin >> N >> X;

    cout << Solve(N, X) << '\n';
}

int main() {
    //Stress();
    Submit();

    return 0;
}