
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  $ f(x)$  be a polynomial and  $ C$  be a real number.
Find the  $ f(x)$  and  $ C$  such that  $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$ .

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