
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 the point $P(1,1)$ and is symmetric to the circle $M$: $(x+2)^2+(y+2)^2=r^2 (r>0)$ about the line $x+y+2=0$.

(I) Find the equation of circle $C$.

(II) Let $Q$ be a moving point on circle $C$, find the minimum value of $\overrightarrow{PQ} \bullet \overrightarrow{MQ}$.

(III) Two distinct lines are drawn through point $P$, intersecting circle $C$ at points $A$ and $B$ respectively. The slopes of lines $PA$ and $PB$ are complementary. Let $O$ be the coordinate origin. Determine the positional relationship between lines $OP$ and $AB$, and explain your reasoning.

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