
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 points $M(-2,0)$ and $N(2,0)$, and a moving point $P$ that satisfies the condition $|PM| - |PN| = 2\sqrt{2}$. Denote the trajectory of the moving point $P$ as $W$.

1. Find the equation of $W$.
2. If $A$ and $B$ are two different points on $W$, and $O$ is the coordinate origin, find the minimum value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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