
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 $C$: $x^{2}+y^{2}-2x-4y+m=0$ in terms of $x$ and $y$.

$(1)$ If equation $C$ represents a circle, find the range of values for the real number $m$.

$(2)$ If circle $C$ intersects line $l$: $x+2y-4=0$ at points $M$ and $N$, and $|MN|= \frac {4}{ \sqrt {5}}$, find the value of $m$.

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