
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) Given the line $l: x+2y-3=0$, find the equation of the line parallel to $l$ and at a distance of 1.

(2) Find the equation of the line perpendicular to the line $x- \sqrt{3}y+1=0$ and at a distance of 5 from the origin.

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