= Solution
The basic <one-bit teleportation> primitive uses the input as <qubit> 1 and a fresh $|+\rangle$ <quantum ancilla> as <qubit> 2. Apply the <Controlled-Z gate> $E_{12}$ and measure <qubit> 1 in the <equatorial qubit measurement> basis with angle $\alpha$. If $s\in\{0,1\}$ is the result, the normalized <post-measurement state> on <qubit> 2 is
$$
\boxed{X_2^sJ(\alpha)_2|\psi\rangle.}
$$
Both results have probability one half. For example, writing $|\psi\rangle=a|0\rangle+b|1\rangle$, the unnormalized branch is $(a|+\rangle+(-1)^se^{i\alpha}b|-\rangle)/\sqrt2=X^sJ(\alpha)|\psi\rangle/\sqrt2$. Thus $J(\alpha)$ is applied deterministically as a logical gate with known <Pauli frame> $X^s$; a permitted conditional <Pauli X gate> would remove this byproduct physically.
The literal resource restriction does not need an unstated direct <Pauli X gate>. A <Pauli Z gate> can be enacted by applying $E$ to the data and a fixed $|1\rangle$ <quantum ancilla>. Two consecutive angle-zero <one-bit teleportations>, with results $p,q$, map an arbitrary current state $|\chi\rangle$ to $X^qH X^pH|\chi\rangle=X^qZ^p|\chi\rangle$. Apply the available $Z^p$ to remove the latter factor, up to an irrelevant <global phase>. This implements a heralded $X^q$: if $q=0$, the input is unchanged; if $q=1$, the required <Pauli X gate> has been applied.
If the original result is $s=1$ and a physically corrected output is required, repeat this <heralded Pauli X correction using controlled-Z and measurements> until $q=1$. Each attempt succeeds with probability one half independent of the input, so it terminates with probability one, using two attempts on average. Only $E$, fixed ancillary <quantum states> and single-<qubit> <quantum measurements> are used. This exact physical correction has no finite worst-case measurement bound; the standard finite deterministic realization is the logical <Pauli frame> version, which suffices for the later output-simulation parts.
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