
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, the sum of distances from point $P$ on the ellipse to the two foci $F_1$ and $F_2$ is $2\sqrt{2}$, and the eccentricity is $\dfrac{\sqrt{2}}{2}$.

$(1)$ Find the equation of the ellipse;

$(2)$ A line $l$ passing through the right focus $F_2$ intersects the ellipse at points $A$ and $B$,

    $(i)$ If a point $M(0,\dfrac{1}{3})$ on the y-axis satisfies $|MA| = |MB|$, find the value of the slope $k$ of line $l$;

    $(ii)$ Let $O$ be the origin. Is there such a line $l$ that makes the area of $\Delta ABO$ reach its maximum value of $\dfrac{\sqrt{2}}{2}$? If so, find the equation of line $l$; if not, explain why.

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