Path-etic Products

You are given a tree with $N$ nodes (numbered $1$ through $N$). A tree is a connected undirected graph without cycles. You have to assign an integer to each node; for each valid $i$, let's denote the integer assigned to node $i$ by $A_i$. This assignment must satisfy the following conditions: - For each valid $i$, $1 \le A_i \le 2 \cdot 10^{18}$. - For each simple path in the tree, let's denote t