
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 that circle $C$ passes through points $A(-1,5)$, $B(5,5)$, and $D(5,-1)$.<br/>$(1)$ Find the standard equation of circle $C$;<br/>$(2)$ Find the standard equation of circle $M$ that is externally tangent to circle $C$ at point $B$ and has a radius of $\sqrt{2}$.

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