
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 prison has  $2004$  cells, numbered  $1$  through  $2004$ . A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on  $n^{\text{th}}$  trip, turning the key in every  $n^{\text{th}}$  cell, and he finished his mission after  $2004$  trips. How many prisoners were released?

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