
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: $x^2+y^2-2x-4y+m=0$

(I) If circle C intersects with line $3x+4y-6=0$ at points M and N, and $|MN|=2 \sqrt{3}$, find the value of $m$.

(II) If line $x-y-1=0$ intersects with circle C at points A and B, determine if there exists a real number $m$ such that the circle with AB as its diameter passes through the origin. If it exists, find the value of $m$; if not, explain the reason.

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