
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 left focus of an ellipse is $F_1$, a small ball $A$ starts moving in a straight line from $F_1$ with velocity $v$. It rebounds off the ellipse wall (ignoring the radius of the ball and assuming the speed remains constant regardless of the number of rebounds). If the longest time it takes for the ball to return to $F_1$ for the first time is five times the shortest time, determine the eccentricity of the ellipse.

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