
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 lies on the line y = x. Moreover, a line l: y = kx + 1 intersects the circle C at points P and Q.

(I) Find the equation of circle C.
(II) If $\overrightarrow{OP} \cdot \overrightarrow{OQ} = -2$, find the value of the real number k.
(III) Draw a line $l_1$ through point (0, 1) perpendicular to line l, and suppose $l_1$ intersects circle C at points M and N. Determine the maximum value of the area of quadrilateral PMQN.

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