
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 $C$: $x^{2}+y^{2}-2x-4y+m=0$,  
$(1)$ If the equation $C$ represents a circle, determine the range of the real number $m$;  
$(2)$ When the equation represents a circle, this circle intersects with the line $l$: $x+2y-4=0$ at points $M$ and $N$, with $|MN|= \frac{4}{5}\sqrt{5}$, find the value of $m$.

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