
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
Circle $M$ passes through two points $B(0,-2)$ and $C(4,0)$, and the center of the circle lies on the line $x-y=0$.  
$(1)$ Find the equation of circle $M$;  
$(2)$ Find the length of the common chord of circle $M$ and circle $N: (x-3)^{2}+y^{2}=25$.

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