For integer $n > 1$, consider $n$ cube polynomials $P_1(x), ..., P_n(x)$ such that each polynomial has $3$ distinct real roots. Denote $S$ as the set of roots of following equation $P_1(x)P_2(x)P_3(x)... P_n(x) = 0$. It is also known that for each $1 \le i < j \le n, P_i(x)P_j(x) = 0$ has $5$ distinct real roots. 1. Prove that if for each $a, b \in S$, there is exactly one $i \in\{1,2, 3,..., n\}$ such that $P_i(a) = P_i(b) = 0$ then $n = 7$. 2. Prove that if $n > 7$ then $|S| = 2n + 1$.