
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 C of the circle lies on the line y=x, the line L: y=kx+1 intersects with ⊙C at points P and Q.
(1) Find the equation of ⊙C.
(2) A line L₁ is drawn through point (0, 1) and perpendicular to L, and L₁ intersects ⊙C at points M and N. Find the maximum area of quadrilateral PMQN.

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