
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 F1, a particle A starts moving from F1 along a straight line with a velocity v. It undergoes reflection within the ellipse (regardless of the number of reflections, the velocity remains constant). If the longest time taken for the particle to return to F1 for the first time is 7 times the shortest time, determine the eccentricity e of the ellipse.

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