
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 satisfying $|PM| - |PN| = 2\sqrt{2}$, with $M(-2,0)$, $N(2,0)$, and $O(0,0)$,
(1) Find the locus S of point P;
(2) A straight line passing through the point $(2,0)$ intersects with S at points A and B. Find the minimum value of the area of triangle $\triangle OAB$.

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