Busiest railway segment (large)

The city state CINERIS has $N$ railway stations. The $N-1$ tracks that join the stations form a tree, so exactly one route connects any two different stations.

Jeongmin, who governs the country, wants to know which segment the most people passed through. $Q$ tickets have been sold so far, and a single ticket reveals only its departure station and its arrival station.

Every resident of CINERIS travels along a shortest route. In a tree the route between two stations is unique, so one ticket passes through each segment of that route exactly once. Use the $Q$ tickets to find the segment that the most people passed through.

The first line contains $N$ and $Q$.

Each of the next $N-1$ lines contains two integers $a$ and $b$, meaning a two way track joins station $a$ and station $b$.

Each of the following $Q$ lines contains the departure station $c$ and the arrival station $d$ of one ticket. The person who bought that ticket left station $c$ and arrived at station $d$.

$1 \le N \le 222222$, $1 \le Q \le 222222$

The tracks form a tree, so every station is connected to every other one. An input with no track at all, that is $N = 1$, is never given. The departure station and the arrival station of a ticket are different.

Print on one line the segment that the most people passed through and how many people passed through it. Print three integers $a$, $b$, $c$ separated by spaces, meaning that $c$ people passed through the segment joining station $a$ and station $b$. Write the two station numbers in increasing order, so that $a < b$.

If several segments are tied for the most people, print the one whose pair $(a, b)$ is smallest in lexicographic order. That is, take the segment with the smaller $a$, and among equal $a$ take the one with the smaller $b$.