
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
In the coordinate plane, denote by  $ S(a)$  the area of the region bounded by the line passing through the point  $ (1,\ 2)$  with the slope  $ a$  and the parabola  $ y\equal{}x^2$ . When  $ a$  varies in the range of  $ 0\leq a\leq 6$ , find the value of  $ a$  such that  $ S(a)$  is minimized.

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