Welcome to the Grand Banquet Dilemma! Picture a gathering of folks connected by a web of relationships. This web of relationships is given as a connected tree with $N$ vertices depicting people and $N-1$ edges representing relationships (for each edge the corresponding two people know each other).

You're on a mission to throw an unforgettable party, but there's a twist – you've got two banquets, and you want to ensure that no two people who know each other end up in the same banquet.

Now, your task involves answering $Q$ queries, each presenting a potential new relationship between two individuals, $X$ and $Y$. The burning question: If this new connection happens, can you still invite everyone to the party using the two banquets? If the answer is "$No$", you must figure out the minimum number of relationships to remove so that everyone gets an invite and the number of ways to do so.

NOTE: the queries are independent of each other. Adding a new relationship doesn't permanently change the relationship web.

Input format

• The first line contains a single integer $T$, which denotes the number of test cases.
• For each test case:
  • The first line contains $N$ denoting the number of people.
  • The following $N-1$ lines contain 2 space-separated integers, $u$ & $v$ indicating that there is a relationship between person person $u$ & person $v$ 
  • The next line contains $Q$ denoting the number of queries.
  • The next $Q$ lines contain 2 unique space-separated integers, $X$ and $Y$. It's guaranteed that $X$ and $Y$ initially don't have a relationship.

Output format

For each test case, answer all the $Q$  queries. For each query print $\text{Yes}$ if it's possible to invite everyone to the party using the 2 banquets, after adding the current query's relationship to the initial existing $N-1$ relations, ensuring that no two people who know each other end up in the same banquet, otherwise print $No$. If the answer is $No$, in the next line, print 2 integers, the minimum number of relationships to remove so that everyone gets an invite, and the number of ways to do so.

Constraints

$1 \leq T \leq 10^5 \\ 3 \leq N \leq10^6 \\ 1≤u,v≤N\\ \\ 1 \leq Q \leq 10^6\\ 1≤X , Y≤N\  \\ \text{The sum of all values of N over all test cases doesn't exceed }10^6 \\ \text{The sum of all values of Q over all test cases doesn't exceed }10^6$