= Solution
Introduce the <slow time> $T=\varepsilon t$ and treat $t,T$ independently in the <method of multiple scales>. Write
$$
y=y_0(t,T)+\varepsilon y_1(t,T)+\cdots,\qquad
y_0=R(T)\cos\phi,\quad \phi=t+\theta(T).
$$
At the next order,
$$
(\partial_t^2+1)y_1=
f(R\cos\phi,-R\sin\phi)+2R_T\sin\phi+2R\theta_T\cos\phi.
$$
For fixed $R$ define the <period average> by
$$
\langle h\rangle=\frac1{2\pi}\int_0^{2\pi}
h(\phi;R)\,d\phi.
$$
Here $f$ inside the averages is evaluated at the leading <position> and <velocity> $R\cos\phi,-R\sin\phi$. <Orthogonality> to both fundamental harmonics is the <solvability condition in the method of multiple scales>: it removes the resonant forcing that would otherwise generate a <secular term>. Since the squared sine and cosine averages are $1/2$, the <amplitude-phase equations> are
$$
\boxed{R_T=-\langle f\sin\phi\rangle,\qquad
\theta_T=-\frac1R\langle f\cos\phi\rangle}.
$$
Equivalently $\dot R=-\varepsilon\langle f\sin(t+\theta)\rangle$ and $\dot\theta=-\varepsilon\langle f\cos(t+\theta)\rangle/R$ at first order. These equations describe a bounded, weakly perturbed oscillation over $t=O(\varepsilon^{-1})$ while the <amplitude> remains in a range where the expansion is ordered. At zero <amplitude> the <oscillation phase> coordinate is singular; Cartesian harmonic coefficients or an equilibrium analysis should replace it.
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