
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 the lines $l_{1}$: $x+y-4=0$, $l_{2}$: $x-y+2=0$, and $l_{3}$: $ax-y+1-4a=0$. <br/>$(1)$ If there exists a triangle with its sides represented by the lines $l_{1}$, $l_{2}$, and $l_{3}$, find the range of real numbers for $a$; <br/>$(2)$ If a line $l$ passes through the intersection point of $l_{1}$ and $l_{2}$, and the distance from point $M(-1,2)$ to $l$ is $2$, find the equation of line $l$.

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