
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 $P(1,2)$ and intersects the positive half of the $x$-axis at point $A$ and the positive half of the $y$-axis at point $B$.
$(1)$ If $\overrightarrow{AP} = 3\overrightarrow{PB}$, find the equation of line $l$;
$(2)$ Find the equation of line $l$ when the area of triangle $AOB$ (where $O$ is the origin) is $4$.

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