
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 equation of circle C can be represented as $x^2+y^2-2x-4y+m=0$, where $m\in \mathbb{R}$.
(1) If $m=1$, find the length of the chord cut by the line $x+y-1=0$ on circle C.
(2) If circle C intersects line $l: x+2y-4=0$ at points M and N, and $OM \perp ON$ ($O$ is the origin), find the value of $m$.

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