Statement:

Given a tree of around $$$10^5$$$ nodes and around $$$10^5$$$ queries. We are also given an integer $$$k$$$. Each query has $$$k$$$ vertices $$$a_1, a_2, ..., a_k$$$. For each query, find a node $$$x$$$ such that $$$dist(x, a_1) + dist(x, a_2) + ... + dist(x, a_k)$$$ is maximum, then print that value. Here $$$dist(u, v)$$$ is the number of edge on the simple path from $$$x$$$ to $$$y$$$.

I have been thinking about this problem for a while, but have not been able to solve it, even for small $$$k$$$ like $$$2$$$ or $$$3$$$. Any idea on how to solve this problem? I appreciate all ideas and any sources of similar problems.