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Consider two events $A$ and $B$. If $P(A) \neq 0$ and $P(B) \neq 0$, which of the following statements must be true?

$A$ and $B$ cannot be both mutually exclusive and independent.

$A$ and $B$ are mutually exclusive, but not independent.

$A$ and $B$ are independent, but not mutually exclusive.

$A$ and $B$ are both mutually exclusive and independent.