
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
Given points A(2, 3), B(4, 1), and $\triangle ABC$ with AB as the base of the isosceles triangle, and point C is on the line $l: x-2y+2=0$.
(1) Find the coordinates of point C and the area of $\triangle ABC$;
(2) If line $l'$ passes through point C and intersects the positive x-axis and y-axis at points P and Q respectively, then:
   ① Find the minimum value of the area of $\triangle POQ$ and the equation of $l'$ at this time;
   ② Find the minimum value of $|PC|\cdot|QC|$ and the equation of $l'$ at this time.

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