
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
Given a continuous function  $ f(x)$  such that  $ \int_0^{2\pi} f(x)\ dx \equal{} 0$ .

Let  $ S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x$ , find constant numbers  $ A_0,\ A_1$  and  $ B_1$  for which  $ \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx$  is minimized.

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