
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
An ellipse (C) is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. $P$ is an arbitrary point on the ellipse (C), and the maximum value of $|PF\_1|-|PF\_2|$ is 4, with an eccentricity of $\frac{\sqrt{2}}{2}$.

(I) Find the equation of the ellipse (C);
(II) Given a line $l$ passing through point $M(0,1)$ and intersecting the ellipse (C) at two points $A$ and $B$, find the range of the area of triangle $AOB$.

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