
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 positive integer  $N$  is *apt* if for each integer  $0 < k < 1009$ , there exists exactly one divisor of  $N$  with a remainder of  $k$  when divided by  $1009$ . For a prime  $p$ , suppose there exists an *apt* positive integer  $N$  where  $\tfrac Np$  is an integer but  $\tfrac N{p^2}$  is not. Find the number of possible remainders when  $p$  is divided by  $1009$ .

*Proposed by Evan Chang*

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