
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
(1) Find the equation of the circle that passes through points $A(5,2)$ and $B(3,2)$, with its center on the line $2x-y-3=0$;

(2) Given a point $A(2,3)$ on the circle, its symmetric point with respect to the line $x+2y=0$ is also on this circle, and the chord intersecting the line $x-y+1=0$ has a length of $2\sqrt{2}$. Find the equation of the circle.

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