Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2 \pmod{n}$. Compute the number of positive integers $k$ such that $k \mid f(k)$.

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