Problem Statement

The Republic of AtCoder has N cities and M roads.
The cities are numbered 1 through N, and the roads are numbered 1 through M. Road i connects City A_i and City B_i bidirectionally.
There is a rush hour in the country that peaks at time 0. If you start going through Road i at time t, it will take $C_i+ \left\lfloor \frac{D_i}{t+1} \right\rfloor$ time units to reach the other end. $(\lfloor x\rfloor$ denotes the largest integer not exceeding x.)
Takahashi is planning to depart City 1 at time 0 or some integer time later and head to City N.
Print the earliest time when Takahashi can reach City N if he can stay in each city for an integer number of time units. It can be proved that the answer is an integer under the Constraints of this problem.
If City N is unreachable, print -1 instead.

Input

Input is given from Standard Input in the following format:
N M
A_1 B_1 C_1 D_1
$\vdots$
A_M B_M C_M D_M

Output

Print an integer representing the earliest time when Takahashi can reach City N, or -1 if City N is unreachable.

Constraints

• $2 \leq N \leq 10^5$
• $0 \leq M \leq 10^5$
• $1 \leq A_i,B_i \leq N$
• $0 \leq C_i,D_i \leq 10^9$
• All values in input are integers.

Sample Input 1

2 1
1 2 2 3

Sample Output 1

4

We will first stay in City 1 until time 1. Then, at time 1, we will start going through Road 1, which will take $2+\left\lfloor \frac{3}{1+1} \right\rfloor = 3 time units before reaching City 2 at time 4$.
It is impossible to reach City 2 earlier than time 4.

Sample Input 2

2 3
1 2 2 3
1 2 2 1
1 1 1 1

Sample Output 2

3

There may be multiple roads connecting the same pair of cities, and a road going from a city to the same city.

Sample Input 3

4 2
1 2 3 4
3 4 5 6

Sample Output 3

-1

There may be no path from City 1 to City N.

Sample Input 4

6 9
1 1 0 0
1 3 1 2
1 5 2 3
5 2 16 5
2 6 1 10
3 4 3 4
3 5 3 10
5 6 1 100
4 2 0 110

Sample Output 4

20

见题面 Input 部分

见题面 Output 部分