
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 (M) with the equation $\frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1\ (a > b > 0)$ whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2} - y^{2} = 1$, and the length of the major axis of the ellipse is 4.
1. Find the equation of the ellipse (M).
2. If the line $y = \sqrt{2}x + m$ intersects the ellipse (M) at points A and B, and point $P(1, \sqrt{2})$ lies on the ellipse (M), find the maximum area of triangle PAB.

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