
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 that the line $l$ passes through the point $P(4,1)$ and intersects the positive $x$-axis and positive $y$-axis at points $A$ and $B$ respectively, where $O$ is the origin.<br/>$(1)$ Find the equation of the line $l$ when the area of triangle $OAB$ is minimized;<br/>$(2)$ Find the equation of the line $l$ when $|OA|+|OB|$ is minimized.

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