
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
For  $ a\in\mathbb{R}$ , let  $ M(a)$  be the maximum value of the function  $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$ .

Evaluate  $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$ .

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