Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.

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