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Contest: Task: Related: TaskD TaskF

Score : $500$ points

Problem Statement

In the Republic of AtCoder, there are $N$ cities numbered $1$ through $N$ and $M$ railroads numbered $1$ through $M$.

Railroad $i$ connects City $A_i$ and City $B_i$ bidirectionally. At time $0$, $K_i$, and all subsequent multiples of $K_i$, a train departs from each of these cities and head to the other city. The time each of these trains takes to reach the destination is $T_i$.

You are now at City $X$. Find the earliest time you can reach City $Y$ when you start the journey by taking a train that departs City $X$ not earlier than time $0$. If City $Y$ is unreachable, report that fact.
The time it takes to transfer is ignorable. That is, at every city, you can transfer to a train that departs at the exact time your train arrives at that city.

Constraints

$2 \leq N \leq 10^5$
$0 \leq M \leq 10^5$
$1 \leq X,Y \leq N$
$X \neq Y$
$1 \leq A_i,B_i \leq N$
$A_i \neq B_i$
$1 \leq T_i \leq 10^9$
$1 \leq K_i \leq 10^9$
All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $X$ $Y$
$A_1$ $B_1$ $T_1$ $K_1$
$\vdots$
$A_M$ $B_M$ $T_M$ $K_M$

Output

Print the earliest possible time you can reach City $Y$. If City $Y$ is unreachable, print -1 instead.

Sample Input 1

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

Sample Output 1

First, you take Railroad $1$ at time $0$ and go from City $1$ to City $2$. You arrive at City $2$ at time $2$.

Then, you take Railroad $2$ at time $4$ and go from City $2$ to City $3$. You arrive at City $3$ at time $7$.

There is no way to reach City $3$ earlier.

Sample Input 2

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

Sample Output 2

First, you take Railroad $2$ at time $0$ and go from City $3$ to City $2$. You arrive at City $2$ at time $3$.

Then, you take Railroad $1$ at time $3$ and go from City $2$ to City $1$. You arrive at City $1$ at time $5$.

Sample Input 3

3 0 3 1

Sample Output 3

-1

Sample Input 4