
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
Consider the polynomial  $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ , with degree less than or equal to 2.

When  $ f$  varies with subject to the constrain  $ f(0) \equal{} 0,\ f(2) \equal{} 2$ , find the minimum value of  $ S\equal{}\int_0^2 |f'(x)|\ dx$ .

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