
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
Determine the pair of constant numbers  $ a,\ b,\ c$  such that for a quadratic function  $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ , the following equation is identity with respect to  $ x$ .
\[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\]
.

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