Conditional Independence Breaks Under Marginalization

A coin is chosen at random: with probability $1/2$ it is fair ($p = 1/2$) and with probability $1/2$ it is biased ($p = 1$, always heads). Let $A$ be the event that the first flip is heads and $B$ the event that the second flip is heads. Show that $A$ and $B$ are conditionally independent given the coin type $C$, but that $A$ and $B$ are not marginally independent. Compute $P(B \mid A)$.