
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
In the Cartesian coordinate system xOy, there is an ellipse C: $$\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) whose eccentricity $e = \frac {\sqrt {2}}{2}$, and the point P(2, 1) lies on the ellipse C.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) If points A and B both lie on ellipse C, and the midpoint M of AB lies on the line segment OP (excluding the endpoints). Find the maximum area of triangle AOB.

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