
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 that point \(P(2,1)\) lies on the circle \(C\): \(x^{2}+y^{2}+ax-2y+b=0\), and the symmetric point of \(P\) about the line \(x+y-1=0\) also lies on circle \(C\), determine the coordinates of the center of circle \(C\).

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