
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 $l$ passes through the intersection points of the lines $x-y+1=0$ and $2x-y=0$, and has a slope of $3$.  
$(1)$ Find the equation of the line $l$;  
$(2)$ Find the equation of a line parallel to line $l$ and at a distance of $\frac{\sqrt{10}}{10}$.

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