
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
For a positive integer \( k \), we write
\[
(1+x)(1+2x)(1+3x) \cdots (1+kx) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k,
\]
where \( a_0, a_1, \cdots, a_k \) are the coefficients of the polynomial. Find the smallest possible value of \( k \) if \( a_0 + a_1 + a_2 + \cdots + a_{k-1} \) is divisible by 2005.

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