
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 $M: x^{2}-2x+y^{2}+4y-10=0$.  
$(1)$ Find the standard equation of circle $M$, and write down the coordinates of the center and the radius of circle $M$;  
$(2)$ If the line $x+3y+C=0$ intersects circle $M$ at points $A$ and $B$, and $|AB|=2\sqrt{5}$, find the value of $C$.

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