
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 an ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through the point $(2, \sqrt{6})$, with an eccentricity of $\frac{\sqrt{2}}{2}$.

(1) Find the values of $a$ and $b$.
(2) Let $A$ and $B$ be the intersection points of the circle $N: x^2 + (y - 2)^2 = 1$ and the $y$-axis, and let $P$ be any point on the ellipse $M$. Find the maximum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$.

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