
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 a line $l$ that passes through point $P(1,4)$, intersecting the positive semi-axis of the $x$-axis and $y$-axis at points $A$ and $B$ respectively, with $O$ as the origin of the coordinate system.

$(1)$ Find the equation of line $l$ when the sum $|OA|+|OB|$ is minimum.

$(2)$ Find the equation of line $l$ when the product $|PA|\cdot|PB|$ is minimum.

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