
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 O passes through point A (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. Determine the equation of circle O.
2. If EF and GH are two perpendicular chords of circle O with foot of the perpendicular being N (1, $\frac{\sqrt{2}}{2}$), find the maximum area of quadrilateral EGFH.
3. Given the line $l: y=\frac{1}{2}x-2$, and P is a moving point on line l. Tangents PC and PD are drawn from P to circle O, with C and D being the points of tangency. Investigate whether the line CD passes through a fixed point; if it does, find the point; if not, provide a reason.

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