
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where $A$ is the left vertex and $F_2(2,0)$ is the right focus. Point $B(2, -\sqrt{2})$ is on the ellipse $C$.  
(1) Find the equation of ellipse $C$.  
(2) If the line $y=kx (k \neq 0)$ intersects the ellipse $C$ at points $E$ and $F$, and lines $AE$ and $AF$ intersect the y-axis at points $M$ and $N$ respectively, does there exist a point $P$ on the x-axis such that $\angle MPN$ is always a right angle regardless of the changes in the non-zero real number $k$? If it exists, find the coordinates of point $P$; if not, explain why.

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