
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 a circle with a point $A(2,3)$ on it, the point symmetric to $A$ with respect to the line $x+2y=0$ is also on the circle. Additionally, the chord formed by the intersection of the circle and 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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