 Solution

Choose an integer $k$ large enough thatPrepare an ancillary qubit in $\sqrt{1-c^2}|0⟩+c|1⟩$ and declare only $|\xi ⟩|1⟩$ to be good. The initial good amplitude ofis $pc=\sin {\theta }^{\prime }$. Applying $k$ amplitude amplification iterations gives good amplitudeThe final state is therefore $|\xi ⟩|1⟩$ up to a global phase. Discarding the ancilla prepares $|\xi ⟩$ exactly; this is exact amplitude amplification.