
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 $\log_b \sin x = a$, then the expression for $\log_b (\cos x \cdot \sin x) $ is
A) $\frac{1}{2} \log_b (1 + b^{2a}) + a$  
B) $\frac{1}{2} \log_b (1 - b^{2a}) + a$  
C) $\frac{1}{2} \log_b (1 - b^{2a}) - a$  
D) $\frac{3a}{2}$

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