
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
An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let points A and B correspond to its two foci. If a stationary ball is placed at point A and then sent along a straight line, it bounces off the elliptical wall and returns to point A. Calculate the maximum possible distance the ball has traveled.

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