
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 line $l_{1}: x-2y+3=0$ and line $l_{2}: 2x+3y-8=0$, their intersection point is $M$,

$(1)$ Find the equation of line $l$ that passes through point $M$ and is parallel to line $l_{3}: x+3y+1=0$.

$(2)$ Find the equation of line $l$ that passes through point $M$ and has a distance of $1$ from point $P(0,4)$;

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