= Solution
The unforced onset is a stationary pattern-forming instability. Under a horizontal translation, the critical <Fourier mode> transforms as $A\mapsto e^{i\phi}A$. A cubic <amplitude equation> without forcing must have the same phase weight: its leading terms are $\widetilde\mu A-c|A|^2A$. Reflection of the unforced spatial pattern conjugates $A$, permitting real coefficients in this stationary problem. They are determined by a <weakly nonlinear expansion> and projection onto the <adjoint eigenfunction>; symmetry alone does not calculate their values or guarantee a nonzero coupling.
Represent the third-harmonic forcing by a complex coefficient $F$ multiplying $e^{3ik_cx}$, whose phase weight is three. The product $F\overline A^{\,2}$ has weight $3-2=1$ and therefore resonates with the critical positive harmonic. Neither a direct third-harmonic term nor $F\overline A$ has the required <wavenumber> balance. A travelling boundary pattern makes
$$
F(T)=\widetilde\epsilon\,e^{3i(\widetilde\Omega T+\delta)}.
$$
A response phase and the sign of its coefficient can be incorporated into $\delta$. The resulting <three-to-one spatially forced amplitude equation> is
$$
\boxed{A_T=\widetilde\mu A-c|A|^2A
+\widetilde\epsilon\,\overline A^{\,2}e^{3i(\widetilde\Omega T+\delta)}.}
$$
This retains the leading resonant forcing term, linear detuning and cubic saturation, while dropping higher powers and nonresonant harmonics. The forcing is weak, the unforced critical <eigenvalue> is near zero, and the amplitude varies on a slow time; very high forcing frequency outside that slow scaling would require a different averaging argument. Reduction to the specific saturating canonical form in part (i) additionally assumes $c>0$ and nonzero forcing.
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