
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  $C$  be the point of intersection of the tangent lines  $l,\ m$  at  $A(a,\ a^2),\ B(b,\　b^2)\ (a<b)$  on the parabola  $y=x^2$  respectively.
When  $C$  moves on the parabola  $y=\frac 12x^2-x-2$ , find the minimum area bounded by 2 lines  $l,\ m$  and the parabola  $y=x^2$ .

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