Solution
= Solution
Since $F=1-2p(1-p)$ with $0\leq p\leq1$, its maximum occurs at $p=0$ or $p=1$:
$$
\boxed{F_{\max}=1,\qquad |\psi\rangle=|0\rangle\ \text{or}\ |1\rangle}
$$
up to a <global phase>. These are the <eigenstates> of the measured observable, so their ideal <quantum measurement>[measurement] does not disturb them.