
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 $l_{1}$: $2x+y+3=0$ and $l_{2}$: $x-2y=0$.  
$(1)$ Find the equation of the line $l_{3}$ which is symmetric to line $l_{1}$ with respect to the $x$-axis, and determine the intersection point $P$ of $l_{2}$ and $l_{3}$.  
$(2)$ Find the equation of the line $m$ passing through point $P$ and having a distance of $2$ from the origin $O(0,0)$.

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