
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
Line $l_{1}$: $x+2y-11=0$ intersects with line $l_{2}$: $2x+y-10=0$ at point $P$. Line $l$ passes through point $P$. <br/>$(1)$ If line $l$ is perpendicular to line $l_{2}$, find the equation of line $l$; <br/>$(2)$ If the intercepts of line $l$ on the coordinate axes are equal, find the equation of line $l$.

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