A function $f(x)$ satisfies $f(x) = f\left(\frac{c}{x}\right)$ for some real number $c > 1$ and all positive numbers $x$.  

If $\int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3$, evaluate $\int_1^c \frac{f(x)}{x} \, dx$.

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