
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 foci $F_1$ and $F_2$, a point $M\left(\frac{2\sqrt{6}}{3}, \frac{\sqrt{3}}{3}\right)$ on the ellipse satisfies $\overrightarrow{MF_1} \cdot \overrightarrow{MF_2} = 0$.  
(i) Find the equation of the ellipse.  
(ii) If the line $l : y = kx + \sqrt{2}$ intersects the ellipse at two distinct points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} > 1$ ($O$ is the origin), find the range of the real number $k$.

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