
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
Line $l$ passes through point $A(1,1)$ and its slope is greater than the slope of line $y=\frac{1}{2}x+1$ by $\frac{π}{4}$.  
$(1)$ Find the equation of line $l$;  
$(2)$ If line $m$ is parallel to line $l$ and the distance between them is $\sqrt{10}$, find the equation of line $m$.

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