
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 $(2,1)$ and $(3,2)$, and is symmetric with respect to the line $x-3y=0$.
$(1)$ Find the standard equation of circle $C$;
$(2)$ If circle $C$ intersects the line $l: x+2y+m=0$ at points $A$ and $B$, and $|AB|=\frac{4\sqrt{5}}{5}$, find the value of $m$.

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