
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 that points A, B, and C are all on the ellipse $M: \frac{x^{2}}{a^{2}} + y^{2} = 1$ with $a > 1$, and the lines AB and AC pass through the left and right foci of the ellipse, $F_1$ and $F_2$ respectively. When $\overrightarrow{AC} \cdot \overrightarrow{F_{1}F_{2}} = 0$, the following holds: $9 \overrightarrow{AF_{1}} \cdot \overrightarrow{AF_{2}} = \overrightarrow{AF_{1}}^{2}$.
(Ⅰ) Find the equation of the ellipse M.
(Ⅱ) Let P be any point on ellipse M, and EF any diameter of the circle $N: x^{2} + (y - 2)^{2} = 1$. Find the maximum value of $\overrightarrow{PE} \cdot \overrightarrow{PF}$.

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