
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 $M$: $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity that is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and is internally tangent to the circle $x^{2}+y^{2}=4$.
(1) Find the equation of the ellipse $M$;
(2) Given $A(-2, \sqrt{2}), F$ is the lower focus of the ellipse $M$. Determine if there exists a point $P$ on the ellipse $M$ such that the perimeter of $\triangle AFP$ is maximized. If it exists, find the maximum perimeter value and the area of $\triangle AFP$ in this case; otherwise, explain the reason.

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