 Solution

For $p=1/4$, put $|{\psi }_{\mathrm{in}}⟩=\frac{1}{2}|g⟩+\frac{\sqrt{3}}{2}|b⟩$. After the first phase rotation the state isSince $U{R}_{00}^{\pi }{U}^{†}=I-2|{\psi }_{\mathrm{in}}⟩⟨{\psi }_{\mathrm{in}}|$, the requested state isIt is proportional to $|g⟩$ exactly when $e^{i\varphi }=-1$, hence when $\varphi =\pi$. The state is then $-|g⟩$, whose global phase has no physical effect.