
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 two lines $l_{1}: 3x+4y-2=0$ and $l_{2}: 2x+y+2=0$ intersect at point $P$, find:  
$(1)$ The equation of the line passing through point $P$ and the origin;  
$(2)$ The equation of the line $l$ that passes through point $P$ and is perpendicular to line $l_{3}: x-2y-1=0$.

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