Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.

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