
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
The right endpoint of the major axis of the ellipse is A, the upper vertex is M, O is the center of the ellipse, F is the right focus of the ellipse, and $\overrightarrow {MF} \cdot \overrightarrow {FA} = \sqrt {2}-1$, the eccentricity is $\frac { \sqrt {2}}{2}$.
(1) Find the standard equation of the ellipse;
(2) Line l intersects the ellipse at points P and Q. Determine whether there exists a line l such that point F is exactly the centroid of △PQM? If it exists, find the equation of line l; if not, explain the reason.

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