Reimu's Teleportation Practice

In Reimu's world of Gensokyo, there are $N$ gentlemen. Among them, $K$ are Myojin priests who serve the deity Myojin. Two gentlemen are connected by $N-1$ bidirectional roads, and any two gentlemen can reach each other using one or more roads. Over $Q$ days, Reimu decides to practice her ability, teleportation. On day $i$, she starts at gentleman $A_i$ and wants to reach gentleman $B_i$, where Marisa is. Reimu moves between gentlemen using the following actions.

• Pick one of the gentlemen directly connected to the current one by a road and move there.
• If the current gentleman is a Myojin priest, teleport to one of the $K$ Myojin priests, including the current one, chosen with equal probability.

Marisa wants Reimu to finish her practice quickly so they can play, so she wants to tell Reimu which actions to use in what order. At every step, Reimu chooses the action that minimizes the expected number of actions needed to reach $B_i$. Print the expected number of actions to go from $A_i$ to $B_i$.

The first line contains $N$, $K$, and $Q$ separated by spaces. ($2 \le N \le 100000$; $1 \le K \le N$; $1 \le Q \le 500000$) Each of the next $N-1$ lines contains $u_i$ and $v_i$, meaning gentleman $u_i$ and gentleman $v_i$ are connected by a road. ($1 \le u_i < v_i \le N$) It is guaranteed that any two gentlemen can reach each other using one or more roads. The next line contains the numbers of the Myojin priests $p_1, p_2, \cdots, p_K$. ($1 \le p_1 < p_2 < \cdots < p_K \le N$) Each of the next $Q$ lines contains $A_i$ and $B_i$, meaning on day $i$ Reimu goes from gentleman $A_i$ to gentleman $B_i$. ($1 \le A_i, B_i \le N$; $A_i \ne B_i$)

Print $Q$ lines. On line $i$, print the expected number of actions to go from $A_i$ to $B_i$. Your answer must have a relative error of at most $10^{-9}$ against the correct answer.