
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
If $b>1$, $\sin x>0$, $\cos x>0$, and $\sin x = b^{-a}$ where $a > 0$, find the value of $a$ if $\cos x = b^{-2a}$.
A) $\frac{1}{4 \log b} \log (\frac{\sqrt{5} - 1}{2})$
B) $\frac{1}{2 \log b} \log (\sqrt{5} + 1)$
C) $-\frac{1}{2 \log b} \log (\frac{\sqrt{5} - 1}{2})$
D) $-\frac{1}{4 \log b} \log (5 - \sqrt{5})$

Present the answer in LaTex format: \boxed{Your answer}
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