
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
The cover of a vertical well, 10 meters deep, periodically opens and closes instantaneously, so the well is in an open state for one second and in a closed state for one second. A stone is thrown from the bottom of the well vertically upwards with an initial velocity \( V \) exactly 0.5 seconds before the next opening of the cover. For which values of the initial velocity \( V \) will the stone freely exit the well and fall back onto the cover of the well? Assume the acceleration due to gravity is \( 10 \ \text{m/s}^2 \).

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