
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 the center of a circle lies on the line $y=x$, and the circle passes through the point $(-2,2)$ with a radius of $2\sqrt{2}$, and the equation of line $l$ is $\left(m+1\right)x+\left(2m-1\right)y-3m=0$.  
$(1)$ Find the standard equation of this circle.  
$(2)$ If line $l$ passes through a fixed point $A$, points $B$ and $C$ lie on this circle, and $AB\perp AC$, find the range of values for $|BC|$.

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