
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 a circle $x^2+y^2-2x-4y+m=0$.
1. If the circle intersects with the line $x+2y-4=0$ at points M and N, and $OM \perp ON$ (O is the origin), find the value of $m$;
2. Under the condition of (1), find the equation of the circle with MN as its diameter.

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