I. We Must Be Together No Matter How Far
time limit per test
2 seconds
memory limit per test
1024 megabytes
input
standard input
output
standard output
If the sun is always scorching, and if the rainbow never fades, can you not leave?
— Gloria, "We Must Be Together No Matter How Far"

In the ocean of memories, alicespring is desperately searching for traces of Doris's existence. In this ocean, there are a total of $$$n$$$ islands and $$$n$$$ bidirectional roads connecting the islands. With these roads, any two islands are mutually reachable. Each time alicespring travels along a road, it consumes $$$1$$$ point of stamina, and traveling the same road again will consume stamina again.

Once upon a time, when alicespring and Doris had not yet separated, they had a total of $$$q$$$ appointments. In each appointment, Doris gave alicespring $$$k$$$ points of stamina, while alicespring was on the island numbered $$$s$$$ and agreed to meet Doris on the island numbered $$$x$$$. For each appointment, alicespring wants to know if there exists a way to travel along the roads such that alicespring can start from island $$$s$$$ and arrive at island $$$x$$$ with exactly $$$0$$$ stamina left.

It is guaranteed that there is at most one road between any two islands, and each road connects two different islands.

Input

First, input a line with two integers $$$n,(3\le n \le 10^5), q(1\le q \le 2\times 10^5)$$$, representing the number of islands and the number of appointments, respectively.

Next, input $$$n$$$ lines, each containing two positive integers $$$u_i, v_i$$$, indicating that there is a road between the island numbered $$$u_i$$$ and the island numbered $$$v_i$$$.

Then, input $$$q$$$ lines, each containing three positive integers $$$s, x(1\le s,x \le n, s \neq x), k(1\le k \le 10^9)$$$, indicating that in this appointment, Doris gave alicespring $$$k$$$ points of stamina, while alicespring is on the island numbered $$$s$$$ and agreed to meet Doris on the island numbered $$$x$$$.

Output

For each appointment, output a line with a string. If there exists a way to travel along the roads such that alicespring starts from island $$$s$$$ and arrives at island $$$x$$$ with exactly $$$0$$$ stamina left, output Yes; if such a way does not exist, output No. The output is case-insensitive: for example, YES and yEs both indicate that alicespring can legally reach node $$$x$$$.

Example
Input
5 4
1 2
1 3
2 3
3 4
4 5
1 5 5
1 5 4
1 5 3
1 5 2
Output
Yes
Yes
Yes
No
Note

In the first sample group, for the first appointment, alicespring can sequentially pass through islands numbered $$$1, 3, 4, 5, 4, 5$$$; for the second appointment, alicespring can sequentially pass through islands numbered $$$1, 2, 3, 4, 5$$$; for the third appointment, alicespring can sequentially pass through islands numbered $$$1, 3, 4, 5$$$; for the fourth appointment, it can be proven that no matter what, alicespring cannot legally reach island numbered $$$5$$$.

First sample group illustration