
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 $x^{2}+y^{2}-2mx-4y+5m=0$ represents the curve of a circle $C$  
$(1)$ Find the range of values for $m$;  
$(2)$ When $m=-2$, find the length of the chord formed by the intersection of circle $C$ and the line $l: 2x-y+1=0$.

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