
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
Find the standard equation of a circle based on the following conditions:<br/>$(1)$ The center of the circle lies on the line $y=x$, the radius is $2$, and the circle is tangent to the line $y=6$;<br/>$(2)$ The circle passes through the three points $A\left(4,3\right)$, $B\left(5,2\right)$, $C\left(1,0\right)$.

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