
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 intersection point of the lines $l_{1}$: $2x-3y+4=0$ and $l_{2}$: $x+y-3=0$ is $M$. <br/>$(1)$ Find the equation of the line $l$ passing through point $M$ and perpendicular to line $l_{1}$; <br/>$(2)$ Find the equation of the line ${l'}$ passing through point $M$ and parallel to line $l_{3}$: $x-2y+5=0$.

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