
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$, and the maximum distance from a point on $C$ to its focus is $\sqrt{2} + 2$. <br/> $(1)$ Find the equation of $C$; <br/> $(2)$ If the tangent line $l$ of the circle $x^{2} + y^{2} = \frac{4}{3}$ intersects $C$ at points $A$ and $B$, find the maximum value of $|AB|$.

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