
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 circle equation $(x^{2}+y^{2}-2x-4y+m=0)$.

(1) Find the range of $m$;

(2) If the circle intersects with the line $(x+2y-4=0)$ at points $M$ and $N$, and $OM \perp ON$ (where $O$ is the origin), find the value of $m$;

(3) Under the conditions of (2), find the equation of the circle with $MN$ as its diameter.

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