Reverse-Engineering Overlap from Frequency Counts

In a risk system, three alert types $A$, $B$, $C$ can fire simultaneously. From historical logs you know:

• $P(A) = 0.5$, $P(B) = 0.4$, $P(C) = 0.3$,
• $P(A \cap B) = 0.2$, but $P(A \cap C)$ and $P(B \cap C)$ are unknown,
• $P(\text{none of } A,B,C) = 0.1$,
• $P(\text{exactly one fires}) = 0.7$.

Determine $P(A \cap B \cap C)$ and $P(\text{exactly two fire})$. Show all steps.