
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 $m\in R$, the distance from point $P(x,y)$ where the line $x+my=0$ intersects with the line $mx-y-2m+4=0$ to the line $l:(x-1)\cos θ+(y-2)\sin θ=3$ is maximized at $\_\_\_\_\_\_$.

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