
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 $O$ passing through point $A(1,1)$, and it is symmetric to the 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 $C$;

$(2)$ If $EF$ and $GH$ are two perpendicular chords of circle $O$ with the foot of the perpendicular as $N(1, \frac{\sqrt{2}}{2})$, find the maximum area of quadrilateral $EGFH$;

$(3)$ Given the line $l: \frac{1}{2}x-2$, and $P$ is a moving point on line $l$. Through $P$, two tangents $PC$ and $PD$ to circle $O$ are drawn, with the points of tangency as $C$ and $D$. Investigate whether the line $CD$ passes through a fixed point. If it does, find the fixed point; if not, please explain why.

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