
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 point P(2, 1) is on the circle C: $x^2 + y^2 + ax - 2y + b = 0$, and the symmetric point of P with respect to the line $x + y - 1 = 0$ is also on circle C, find the coordinates of the center of circle C.

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