
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
Find eight strictly positive integers \( n_{1}, n_{2}, \ldots, n_{8} \) with the following property:

For every integer \( k \) such that \(-2007 \leq k \leq 2007\), there exist eight integers \( \alpha_{1}, \alpha_{2}, \ldots, \alpha_{8} \), each belonging to the set \(\{-1,0,1\}\), such that

\[ k = \alpha_{1} n_{1} + \alpha_{2} n_{2} + \ldots + \alpha_{8} n_{8} \]

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