Let
\[f(x)=\cos x \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cdots \cos \frac{x}{2^{99}} \sin \frac{x}{2^{99}}.\]
Then what is the value of \(-\log_2 f\left(\frac{\pi}{12}\right)?\)

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