
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 point P in the plane that satisfies $|PM| - |PN| = 2\sqrt{2}$, where $M(-2, 0)$, $N(2, 0)$, and $O(0, 0)$,
(1) Find the trajectory S of point P;
(2) The line $y = k(x - 2)$ intersects S at points A and B. Express the area of $\triangle OAB$ as a function of $k$.

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