= Solution
Label the unknown input $1$, Alice's resource photon $2$, and Bob's resource photon $3$. First distribute photon $3$ through the channel while keeping photon $2$ with Alice; at $p=1$ this leaves the perfect shared <Bell state> $|\Phi^+\rangle_{23}=(|00\rangle+|11\rangle)/\sqrt2$. When the input arrives, the three-qubit state is
$$
|\Omega_0\rangle=\frac{1}{\sqrt2}
\left(\alpha|000\rangle+\alpha|011\rangle+\beta|100\rangle+\beta|111\rangle\right).
$$
Alice applies $\operatorname{CNOT}_{1\to2}$, giving
$$
|\Omega_1\rangle=\frac{1}{\sqrt2}
\left(\alpha|000\rangle+\alpha|011\rangle+\beta|110\rangle+\beta|101\rangle\right).
$$
She next applies $H_1$, producing
$$
|\Omega_2\rangle=\frac12\big(
|00\rangle_{12}|\psi\rangle_3+
|01\rangle_{12}X|\psi\rangle_3+
|10\rangle_{12}Z|\psi\rangle_3+
|11\rangle_{12}XZ|\psi\rangle_3\big).
$$
These two gates followed by computational measurement implement a perfect <Bell state> analyzer. For outcome $(m,n)$ the normalized conditional state is $|mn\rangle_{12}X^nZ^m|\psi\rangle_3$, with probability $1/4$. Alice sends the \b[two classical bits] $m,n$; Bob applies $Z^mX^n$, recovering $|\psi\rangle$ exactly. Averaging before reception of the bits gives Bob $I/2$, independent of the input, so this protocol provides no faster-than-light communication.
For $p=1$, directly sending the original <qubit> is already perfect. <Quantum teleportation> therefore has no fidelity advantage in this ideal case and consumes a shared entangled pair plus classical communication. Its operational advantage is that entanglement can be distributed beforehand: once the unknown input appears, only classical information need travel, and the original input photon need not itself propagate to Bob. The Bell measurement consumes the input; it does not create a second copy.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2015-teleportation-and-dense-coding-readout.png]
{title=<Quantum teleportation> corrections and restricted <superdense coding> readout}
{description=Original teleportation circuit with the distributed resource, two measurement bits and conditional corrections, beside the four Bell-state computational-basis outcome probabilities. The right panel describes direct measurement without a two-qubit Bell decoder, not measurement after an unrestricted decoding circuit.}
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