
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 $C_1$: $x^2+y^2-2x-4y+m=0$,  
(1) Find the range of values for the real number $m$;  
(2) If the line $l: x+2y-4=0$ intersects the circle $C$ at points $M$ and $N$, and $OM \perp ON$, find the value of $m$.

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