
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 sequence $a_{1}, a_{2}, \ldots$ consists of integers and has infinitely many positive and infinitely many negative elements. Also, for every $n$, the sequence $a_{1}, a_{2}, \ldots, a_{n}$ gives pairwise distinct remainders when divided by $n$. How many times does the number $2008$ appear in the sequence?

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