
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 \( b>1 \), \( \sin^2 x = 2\sin y \cos y \) for some angle \( y \), and \( \log_b \sin x = a \), find \( \log_b \cos x \).
A) \( \log_b(b^{a-2}) \)  
B) \( 2\log_b \sin y \)  
C) \( \frac{1}{2} \log_b (1 - b^{2a}) \)  
D) \( 2a \log_b b \)  
E) \( \log_b (1-b^{2a}) \)

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