
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 the circle $O: x^2 + y^2 = 4$ and the line $l: y = kx - 4$.

(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 $k=1$, let $P$ be a point moving along line $l$, and draw two tangents from $P$ to circle $O$, with points of tangency at $C$ and $D$. Does line $CD$ pass through a fixed point? If yes, find the coordinates of this fixed point; if no, explain the reason.

(3) Let $EF$ and $GH$ be two perpendicular chords of the circle $O: x^2 + y^2 = 4$, with the foot of the perpendicular between them being $M(1, \sqrt{2})$. Determine the maximum possible area of quadrilateral $EGFH$.

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