
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
Denote by  $C,\ l$  the graphs of the cubic function  $C: y=x^3-3x^2+2x$ , the line  $l: y=ax$ .

(1) Find the range of  $a$  such that  $C$  and  $l$  have intersection point other than the origin.

(2) Denote  $S(a)$  by the area bounded by  $C$  and  $l$ . If  $a$  move in the range found in (1), then find the value of  $a$  for which  $S(a)$  is minimized.

50 points

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