
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
【Full score: 15 points】  
A line $l$ passes through point $P(2,1)$ and intersects the positive x-axis at point $A$ and the positive y-axis at point $B$.  
(1) Find the minimum value of $u = |OA| + |OB|$ and write the equation of line $l$ when $u$ takes its minimum value;  
(2) Find the minimum value of $v = |PA| \cdot |PB|$ and write the equation of line $l$ when $v$ takes its minimum value.

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