
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
For real constant numbers  $ a,\ b,\ c,\ d,$  consider the function  $ f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$  such that  $ f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x|$  for  $ |x|\leq 1.$ 

Find  $ f(x)$  for which  $ \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx$ 　is minimized.

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