
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 that the focal length of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $2$, and it passes through the point $(1, \frac{\sqrt{2}}{2})$.

(I) Find the equation of the ellipse $C$;

(II) A line passing through the point $M(2,0)$ intersects the ellipse $C$ at two points $A$ and $B$. $P$ is a point on the ellipse $C$, $O$ is the coordinate origin, and they satisfy $\overrightarrow{OA} + \overrightarrow{OB} = t \overrightarrow{OP}$, where $t \in (\frac{2\sqrt{6}}{3}, 2)$. Find the range of $|AB|$.

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