Crazy function

Algebra Level 5

\[ \begin{cases} f(1) = 0 \\ f(2)=1 \\ f(4)=2 \\ f(8)=3 \\ f(16)=4 \\ f(32)=5 \end{cases} \]

Let \(f(x) \) be a polynomial of degree 5 satisfying the system of equations above. If the coefficient of \(x\) in \(f(x) \) can be expressed as \(\dfrac pq\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

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