
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 point $P(2,1)$, it intersects with the positive half of the $x$ and $y$-axes at points $A$ and $B$ respectively, forming a line $l$.

(1) Find the equation of line $l$ when the area of $\triangle AOB$ is minimized.

(2) Find the equation of line $l$ when the product of $|PA|$ and $|PB|$ is minimized.

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