
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 that the line $l_{1}$ passes through the point $A(2,3)$.  
$(1)$ If $l_{1}$ is perpendicular to the line $l_{2}: x+2y+4=0$, find the equation of $l_{1}$.  
$(2)$ If the intercepts of $l_{1}$ on the two coordinate axes are equal, find the equation of $l_{1}$.

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