
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 C that passes through points A(-2,0) and B(0,2), and the center of the circle C is on the line $y=x$. Also, the line $l: y=kx+1$ intersects circle C at points P and Q.  
(1) Find the equation of circle C;  
(2) A line $l_1$ passing through point (0,1) is perpendicular to line $l$, and line $l_1$ intersects circle C at points M and N. Find the maximum area of the quadrilateral PMQN.

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