
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 point $P(-1,4)$ lies on the line $l_{1}$, and ____.<br/>$(1)$ Choose one of the three conditions and fill in the blank line to find the equation of the line $l_{1}$;<br/>① Line $l_{1}$ is parallel to line $l_{2}$: $2x-y+5=0$;<br/>② Line $l_{1}$ is perpendicular to line $l_{2}$: $x+2y-2=0$;<br/>③ The slope of line $l_{2}$ is $45^{\circ}$, and the slope of line $l_{1}$ is twice the slope of line $l_{2}$.<br/>$(2)$ Under the condition in (1), if the distance between the line $l_{3}$: $4x-2y+m=0$ and line $l_{1}$ is $2\sqrt{5}$, find the value of the real number $m$.

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