
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
Let \( A_{1} A_{2} \cdots A_{2002} \) be a regular 2002-sided polygon. Each vertex \( A_{i} \) is associated with a positive integer \( a_{i} \) such that the following condition is satisfied: If \( j_{1}, j_{2}, \cdots, j_{k} \) are positive integers such that \( k<500 \) and \( A_{j_{1}} A_{j_{2}} \cdots A_{j_{k}} \) is a regular \( k \)-sided polygon, then the values of \( a_{j_{1}}, a_{j_{2}}, \cdots, a_{j_{k}} \) are all different. Find the smallest possible value of \( a_{1}+a_{2}+\cdots+a_{2002} \).

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