Prove that A,B,C are mutually independent given that A and B, A and C, B and C are all pairwise independent. Stuck on circular logic, please help?

Question

To prove the mutual independence of events

Here in this question, "If A,B and C are random events in a sample space and if A,B and C are pairwise independent and A is independent of (B∪C), then is it true that A,B and C are mutually independent."

However, in this question,

https://stats.stackexchange.com/questions/173192/if-a-and-b-are-independent-events-and-a-and-c-are-independent-events-h

for A to be independent of (B U C), A, B, and C must be mutually independent already? Is this circular reasoning? I am not sure how one statement can follow from the other.

My goal is to prove why A and (B U C) are independent of each other in Question #1, in order to prove that A, B, and C are mutually independent....but Question #2 says that in order for A and (B U C) to be independent of each other, A, B, and C must be mutually independent! so I am not sure where the flow of logic is coming from and where to begin for a proof of mutual independence between A, B, and C, given that A,B, B,C, and A,C are all pairwise independent.