
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
A jailer is granting partial amnesty by the following procedure: First, he opens all the cells. Next, he locks every second cell (in this prison, all the cells are in a single row). In the third step, he turns the key in every third cell (if a cell is open, it will be locked, and if it is locked, it will be opened). Continuing in this manner, on the $n$-th step, he turns the key in every $n$-th cell. After finishing this process, he releases all prisoners who are in open cells. Indicate the numbers of the cells that contain these lucky prisoners.

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