
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 centered at the origin with its major axis along the x-axis, one of its vertices is the point $(0, \sqrt{5})$, and its eccentricity is $\frac{\sqrt{6}}{6}$. The left and right foci are denoted as $F_1$ and $F_2$, respectively.  
(1) Find the equation of the ellipse;  
(2) If point $M$ is on the ellipse, find the maximum area of $\triangle MF_1F_2$;  
(3) Investigate whether there exists a point $P$ on the ellipse such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. 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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