
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 has the optical property that a ray of light emanating from one focus of the ellipse, after being reflected by the ellipse, will pass through the other focus. Now, there is a horizontally placed elliptical billiard table, with points A and B as its foci, the major axis is 2a long, and the focal distance is 2c. A small ball (the radius of the ball is negligible) placed at point A starts moving in a straight line from point A, and after being reflected by the wall of the ellipse, it returns to point A for the first time. The distance traveled by the small ball is __________.

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