
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
It is known that circle C passes through point P(1, 1) and is symmetrical to circle M: $(x+2)^2+(y+2)^2=r^2 \ (r>0)$ with respect to the line $x+y+2=0$.
(1) Find the equation of circle C;
(2) Draw two different lines through point P that intersect circle C at points A and B, respectively, and the inclination angles of lines PA and PB are complementary. Let O be the origin. Determine whether line OP is parallel to line AB and explain your reasoning.

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