
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 real number \( x \), let \( [x] \) denote the greatest integer that does not exceed \( x \). For a certain integer \( k \), there exist exactly 2008 positive integers \( n_{1}, n_{2}, \cdots, n_{2008} \), such that \( k=\left[\sqrt[3]{n_{1}}\right]=\left[\sqrt[3]{n_{2}}\right]=\cdots=\left[\sqrt[3]{n_{2008}}\right] \), and \( k \) divides \( n_{i} \) (for \( i = 1, 2, \cdots, 2008 \)). What is the value of \( k \)?

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