
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
A line \( l \) passes through point \( P(3, 2) \) and intersects the positive halves of the x-axis and y-axis at \( A \) and \( B \), respectively.
(1) If \( P \) is the midpoint of \( AB \), find the equation of \( l \).
(2) If the product of the lengths \( |PA| \cdot |PB| \) is minimized, find the equation of line \( l \).
(3) If the area \( S \) of triangle \( AOB \) is minimized, find the equation of line \( l \).

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