= Solution
The free precession rate $A(a)$ tends to zero far inside the inner planet, diverges on approaching $a_1$, diverges on both sides of $a_2$, and tends to zero far outside the outer planet. Between $a_1$ and $a_2$ it diverges at both ends and has at least one minimum. A horizontal line $A=g_1$ therefore crosses once inside $a_1$ and once outside $a_2$, plus zero, one tangent, or two times between the planets. There are consequently
$$
\boxed{2, 3, \text{or }4}
$$
locations of the corresponding <secular resonance>. This argument uses the smooth <Laplace-Lagrange secular theory> away from the immediate neighborhoods of the planets and from <mean-motion resonances>.
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