= Solution
Label the chain <cluster state> from top to bottom by $1,2,3,4$, and let $k,l,m$ be the phase-measurement outcomes on the first three vertices, with raw <computational basis> outcome $s$ on vertex four. Repeated <one-bit teleportation> leaves the last <qubit>, up to <global phase>, in
$$
|\chi\rangle=X^mU(\gamma)X^lU(\beta)X^kU(\alpha)|+\rangle.
$$
This sequential description applies even though the whole <graph state> is prepared first: a later <Controlled-Z gate> acts only on unmeasured <qubits> and commutes with earlier single-<qubit> measurement operators. It can therefore be moved past those measurements for the purpose of calculating each conditional branch.
Take $\alpha=\theta$ and $\beta=0$, so $U(0)=H$. After the second measurement the logical state is
$$
X^lHX^kU(\theta)|+\rangle=X^lZ^kHU(\theta)|+\rangle.
$$
For binary $p,q$, the identities proved above imply the <Pauli-frame propagation along a measurement wire> rule
$$
U(\eta)X^pZ^q\simeq X^qZ^pU((-1)^p\eta),
$$
where $\simeq$ means equality up to <global phase>. Choose the third angle adaptively as $\gamma=(-1)^{l\oplus1}\phi$. Then $(-1)^l\gamma=-\phi$, and
$$
|\chi\rangle\simeq X^{m\oplus k}Z^lU(-\phi)HU(\theta)|+\rangle.
$$
Reading the target circuit in temporal order gives its premeasurement state
$$
|T\rangle=U(\phi)XHU(\theta)H|0\rangle
=e^{-i\phi}ZU(-\phi)HU(\theta)|+\rangle.
$$
Consequently
$$
|\chi\rangle\simeq X^{m\oplus k}Z^{l\oplus1}|T\rangle.
$$
The <Pauli Z gate> changes only phases in the <computational basis>, while the <Pauli X gate> exchanges its two measurement outcomes. Thus the explicit choices are
$$
\boxed{\alpha=\theta,\quad\beta=0,\quad\gamma=(-1)^{l\oplus1}\phi,\quad\delta=k\oplus m,\quad r=s\oplus k\oplus m.}
$$
Indeed, for every earlier outcome branch,
$$
\Pr(r=t\mid k,l,m)=|\langle t|T\rangle|^2\qquad(t=0,1).
$$
Hence averaging over the earlier random outcomes leaves exactly the target <probability distribution>. This implements the desired <measurement-based quantum computation> without physically removing the final <Pauli frame>.
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