= Solution
For $f(x)=\sin x$, the period average $\langle\sin(A\cos\psi)\sin^2\psi\rangle$ vanishes: a half-period shift reverses the first factor and leaves the second unchanged. This is the <zero first-order amplitude drift for odd damping>. The <averaged amplitude for position-dependent damping> therefore has $A_T=0$, while its first-order phase equation again has $\theta_T=0$. With the <initial conditions>,
$$
\boxed{x(t;\varepsilon)=\cos t+O(\varepsilon),\qquad 0\leq t\leq C/\varepsilon.}
$$
The first-order forcing has no fundamental resonant component, so it produces bounded corrections rather than a secular amplitude change on this time scale. This statement concerns the leading approximation on $t=O(\varepsilon^{-1})$.
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