= Solution
Alison reaches the station at $t_1=1/(5/13)=13/5$ years. Her return <worldline> is
$$
x_A=1-\frac{12}{13}\left(t-\frac{13}{5}\right)=\frac{17}{5}-\frac{12t}{13},
$$
while Belinda's is $x_B=-5t/13$. Equating them gives
$$
\boxed{T=\frac{221}{35}\,\mathrm{yr}\simeq6.3143\,\mathrm{yr},\qquad
x_{\rm meeting}=-\frac{17}{7}\,\mathrm{ly}.}
$$
This is the elapsed Earth-frame time, measurable by synchronized clocks in that <inertial frame>. A signal announcing the reunion reaches a clock physically on Earth later; its light-travel delay is not part of the requested frame-time comparison.
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