
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 $l\_1$: $y=2x$ and $l\_2$: $x+y=6$ intersect at point $P$.
(1) Find the equation of line $m$ that passes through point $P$ and is perpendicular to line $l\_0$: $x-2y=0$.
(2) Find the equation of line $m$ that passes through point $P$ and has a distance of $2$ from the coordinate origin $O$.

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