
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
Find the equations of the following lines:
- $(1)$ A line passing through the point $(-1,3)$ and parallel to the line $x-2y+3=0$.
- $(2)$ A line passing through the point $(3,4)$ and perpendicular to the line $3x-y+2=0$.
- $(3)$ A line passing through the point $(1,2)$ with equal intercepts on both axes.

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