26202번 - Fun with Stones 다국어

Alice and Bob will play a game with $3$ piles of stones. They take turns and, on each turn, a player must choose a pile that still has stones and remove a positive number of stones from it. Whoever removes the last stone from the last pile that still had stones wins. Alice makes the first move.

The $i$-th pile will have a random and uniformly distributed number of stones in the range $[L_i , R_i ]$. What is the probability that Alice wins given that they both play optimally?

The input consists of a line with $6$ integers, respectively, $L_1$, $R_1$, $L_2$, $R_2$, $L_3$, $R_3$. For each $i$, $1 ≤ L_i ≤ R_i ≤ 10^9$.

Print an integer representing the probability that Alice wins modulo $10^9 + 7$.

It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not\equiv 0 \pmod{10^9 + 7}$, that is, we are interested in the integer $p × q^{-1} \pmod{10^9 + 7}$.