Chef and Minimum Cost of His Tree

You are a given a tree of $N$ vertices. The $i$-th vertex has two things assigned to it — a value $A_i$ and a range $[L_i, R_i]$. It is guaranteed that $L_i \leq A_i \leq R_i$. You perform the following operation **exactly once**: - Pick a (possibly empty) subset $S$ of vertices of the tree. $S$ must satisfy the additional condition that no two vertices in $S$ are directly connected by an edge, i