 Solution

Very near one planet, that planet dominates both the particle's free-precession coefficient and its forcing coefficient, with the leading terms arranged approximately as $\dot{z}\simeq iA(z-z_p)-z/\tau$. Since $|A|\tau \gg 1$ close to the planet, the late-time forced solution approachesThe particle's orbit therefore tends toward the planet's eccentricity and apsidal direction. Extremely close to the planet, close encounters, co-orbital dynamics, and individual mean-motion resonances invalidate the orbit-averaged linear approximation.