
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
Let the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) have its left and right foci as $F_1$ and $F_2$, respectively. A point $M$ is on the hyperbola $C$, and $|MF_1| - |MF_2| = 2\sqrt{2}$. It is known that the eccentricity of the hyperbola $C$ is $\sqrt{2}$.  
(Ⅰ) Find the equation of the hyperbola $C$;  
(Ⅱ) Through a moving point $P$ on the hyperbola $C$, draw two tangents to the circle $E: x^2 + (y-4)^2 = 1$, with the tangency points being $A$ and $B$, respectively. Find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$.

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