
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
Let ellipse M be defined by the equation $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defined by $x^{2}+y^{2}=4$.  
(1) Find the equation of ellipse M;  
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and there is a point $P(1, \sqrt {2})$ on ellipse M, find the maximum area of triangle PAB.

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