A \textit{divisibility chain} is a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that $a_k$ divides $a_{k+1}$ for all $1 \le k < n$. Compute the number of divisibility chains of the form $(a, b, a^2, c, a^3, 360^9)$.

Let's think step by step and output the final answer within \boxed{}.
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