
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
Let  $a,\ b$  be real numbers. Suppose that a function  $f(x)$  satisfies  $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$  and has the maximum value  $2\pi$  for  $-\pi \leq x\leq \pi$ .  
Find the minimum value of  $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$ *2010 Chiba University entrance exam*

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