
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 the lines $l_{1}$: $(m+2)x+my-6=0$ and $l_{2}$: $mx+y-3=0$, where $m$ is a real number.  
$(1)$ If $l_{1} \perp l_{2}$, find the value of $m$.  
$(2)$ If the point $P(1,2m)$ lies on the line $l_{2}$, a line $l$ passes through point $P$, and the intercept on the $x$-axis is twice the intercept on the $y$-axis, find the equation of line $l$.

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