
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
From the identity  $$  \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2}  \log \sin x \, dx + \int_{0}^{\pi \slash 2}  \log \cos x \, dx +\int_{0}^{\pi \slash 2}  \log  2 \, dx,  $$  deduce the value of  $\int_{0}^{\pi \slash 2}  \log \sin x \, dx.$ 

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