
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 a line $l$ passing through point $P(2,3)$, find the equation of line $l$ under the following conditions:  
$(1)$ The inclination angle of line $l$ is $120^{\circ}$;  
$(2) l$ is perpendicular to the line $x-2y+1=0$;  
$(3) l$ has intercepts on the $x$-axis and $y$-axis whose sum equals $0$.

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