
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: $x^2+y^2=2$, and a line $l: y=kx-2$.
(1) If line $l$ intersects circle O at two distinct points A and B, find the value of $k$ when $\angle AOB = \frac{\pi}{2}$.
(2) If $k=\frac{1}{2}$ and P is a moving point on line $l$, with PC and PD being tangents to the circle O at points C and D, investigate whether line CD passes through a fixed point.
(3) If EF and GH are two mutually perpendicular chords of the circle O: $x^2+y^2=2$ with the foot of the perpendicular at M(1, $\frac{\sqrt{2}}{2}$), find the maximum value of the area of quadrilateral EGFH.

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