Let the sample space $S$ be partitioned into three events $A$, $B$, and $C$, whose probabilities depend on a real parameter $p$ such that $P(A) = p$, $P(B) = 2p^2$, and $P(C) = 1 - p - 2p^2$. An event $E$ has conditional probabilities $P(E|A) = 0.6$, $P(E|B) = 0.1$, and $P(E|C) = 0.2$. What is the maximum possible value of $P(E)$ over all mathematically valid values of $p$?