
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
For how long does an object thrown at an initial speed \( v_{0} \) under an angle \(\alpha\) bounce, and what distance does it cover on the horizontal ground, given that the coefficient of restitution is \(\varepsilon\)? Ignore air resistance.

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