Prime Routing

Time Limit : 2 sec, Memory Limit : 524288 KB

Problem J. Prime Routing

Fox Jiro is one of the staffs of the ACM-ICPC 2018 Asia Yokohama Regional Contest and is responsible for designing the network for the venue of the contest. His network consists of $N$ computers, which are connected by $M$ cables. The $i$-th cable connects the $a_i$-th computer and the $b_i$-th computer, and it carries data in both directions. Your team will use the $S$-th computer in the contest, and a judge server is the $T$-th computer.

He decided to adjust the routing algorithm of the network to maximize the performance of the contestants through the magical power of prime numbers. In this algorithm, a packet (a unit of data carried by the network) is sent from your computer to the judge server passing through the cables a prime number of times if possible. If it is impossible, the contestants cannot benefit by the magical power of prime numbers. To accomplish this target, a packet is allowed to pass through the same cable multiple times.

You decided to write a program to calculate the minimum number of times a packet from $S$ to $T$ needed to pass through the cables. If the number of times a packet passes through the cables cannot be a prime number, print $-1$.


The input consists of a single test case, formatted as follows.

$N$ $M$ $S$ $T$
$a_1$ $b_1$
$a_M$ $b_M$

The first line consists of four integers $N, M, S,$ and $T$ ($2 \leq N \leq 10^5, 1 \leq M \leq 10^5, 1 \leq S, T \leq N, S \ne T$). The $i$-th line of the following $M$ lines consists of two integers $a_i$ and $b_i$ ($1 \leq a_i < b_i \leq N$), which means the $i$-th cables connects the $a_i$-th computer and the $b_i$-th computer in the network. You can assume that the network satisfies the following conditions.

  • The network has no multi-edge, i.e.,$(a_i, b_i) \ne (a_j, b_j)$ for all $i,j$ ($1 \leq i < j \leq M$).
  • The packets from $N$ computers are reachable to $T$ by passing through some number of cables. The number is not necessarily a prime.


If there are ways such that the number of times a packet sent from $S$ to $T$ passes through the cables is a prime number, print the minimum prime number of times in one line. Otherwise, print $-1$.


Sample Input 1

5 5 1 5
1 2
1 4
2 3
3 4
3 5

Output for Sample Input 1


Sample Input 2

5 4 1 5
1 2
2 3
3 4
4 5

Output for Sample Input 2


Sample Input 3

2 1 1 2
1 2

Output for Sample Input 3


Sample Input 4

3 3 1 2
1 2
1 3
2 3

Output for Sample Input 4