
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with a moving point P on it. The sum of the distances from P to the left and right foci $F_1$ and $F_2$ is $2\sqrt{2}$, and the maximum distance from P to one focus of the ellipse is $\sqrt{2} + 1$.

(I) Find the equation of the ellipse;

(II) A line $l$ passing through the right focus $F_2$ intersects the ellipse at points A and B.

① If there exists a point M (0, $\frac{1}{2}$) on the y-axis such that $|MA| = |MB|$, find the value of the slope $k$ of line $l$;

② Is there such a line $l$ that makes the maximum value of $S_{\triangle ABO}$ equal to $\frac{\sqrt{2}}{2}$ (where O is the origin)? If it exists, find the equation of line $l$; if not, explain why.

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