
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 $C$ passing through the point $P(1,1)$, and it is symmetric to the circle $M$: $(x+2)^{2}+(y+2)^{2}=r^{2}(r > 0)$ about the line $x+y+2=0$.
(1) Find the equation of circle $C$;
(2) Let $Q$ be a moving point on circle $C$, find the minimum value of $\overrightarrow{PQ} \cdot \overrightarrow{MQ}$.

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