
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 equation of a circle that satisfies the following conditions:
(1) The circle passes through points M(5, 2) and N(3, 2) and its center lies on the line $y=2x-3$.
(2) The circle passes through the intersection points of the circles defined by $x^2+y^2-x+y-2=0$ and $x^2+y^2=5$ and its center lies on the line $3x+4y-1=0$.

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