
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 the following conditions, find the equation of the circle:
(1) It passes through points A(6, 5) and B(0, 1), and the center C of the circle is on the line $3x+10y+9=0$;
(2) It passes through points P(-2, 4) and Q(3, -1), and the chord intercepted on the x-axis is 6 units long.

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