
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 line $l: y=k(x+2\sqrt{2})$ intersects with the circle $O: x^2+y^2=4$ at points A and B, where O is the origin, and the area of triangle ABO is denoted as $S$.
(Ⅰ) Express $S$ as a function of $k$, $S(k)$, and determine its domain;
(Ⅱ) Find the maximum value of $S$ and the value of $k$ when $S$ reaches its maximum.

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