
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
Suppose that a cubic function with respect to  $x$ ,  $f(x)=ax^3+bx^2+cx+d$  satisfies all of 3 conditions:
\[f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1\].

Find  $f(x)$  for which  $I=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx$  is minimized, the find the minimum value.

*2011 Tokyo University entrance exam/Humanities, Problem 1*

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