Problem:
Score : 300 points

Problem Statement

You are given a string S of length N and another string T of length M.
These strings consist of lowercase English letters.

A string X is called a good string when the following conditions are all met:

Let L be the length of X. L is divisible by both N and M.

Concatenating the 1-st, (\frac{L}{N}+1)-th, (2 \times \frac{L}{N}+1)-th, ..., ((N-1)\times\frac{L}{N}+1)-th characters of X, without changing the order, results in S.

Concatenating the 1-st, (\frac{L}{M}+1)-th, (2 \times \frac{L}{M}+1)-th, ..., ((M-1)\times\frac{L}{M}+1)-th characters of X, without changing the order, results in T.

Determine if there exists a good string. If it exists, find the length of the shortest such string.

Constraints

1 \leq N,M \leq 10^5

S and T consist of lowercase English letters.

|S|=N

|T|=M

Input

Input is given from Standard Input in the following format:

N M
S
T

Output

If a good string does not exist, print -1; if it exists, print the length of the shortest such string.

Sample Input 1

3 2
acp
ae

Sample Output 1

6

For example, the string accept is a good string.
There is no good string shorter than this, so the answer is 6.

Sample Input 2

6 3
abcdef
abc

Sample Output 2

-1

Sample Input 3

15 9
dnsusrayukuaiia
dujrunuma

Sample Output 3

45

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
            var vals = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
            var N = vals[0];
            var M = vals[1];
            var S = Console.ReadLine();
            var T = Console.ReadLine();

            var result = IsGoodString(N, M, S, T);
            Console.WriteLine(result);
        }

        public static long Lcm(int a, int b)
        {
            return (long)a * (long)b / Gcd(a, b);
        }

        // ユークリッドの互除法 
        public static int Gcd(int a, int b)
        {
            if (a < b)
                // 引数を入替えて自分を呼び出す
                return Gcd(b, a);
            while (b != 0)
            {
                var remainder = a % b;
                a = b;
                b = remainder;
            }
            return a;
        }

        static long IsGoodString(int N, int M, string S, string T)
        {
            //L＝最大公約数の倍数を調べればよい
            var gcd = Gcd(N, M);
            var n = (long)N * (long)M / gcd;

            for(long i = gcd; i <= (long)N * (long)M; i+=gcd)
            {
                if (i % N != 0 || i % M != 0) continue;
                var chars = new Dictionary<int, char>();
                foreach(var si in Enumerable.Range(0, S.Length))
                {
                    chars[(int)(1 + i / N * si)] = S[si];
                }
                var conflict = false;
                foreach(var ti in Enumerable.Range(0, T.Length))
                {
                    char ctemp;
                    if (chars.TryGetValue((int)(1 + i / M * ti), out ctemp))
                    {
                        if (ctemp != T[ti])
                        {
                            conflict = true;
                            break;
                        }
                    }
                    else
                    {
                        chars[(int)(1 + i / M * ti)] = T[ti];
                    }
                }
                if (!conflict) return i;
            }
            return -1;
        }
    }

}