
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 8 positive integers \( n_{1}, n_{2}, \cdots, n_{8} \) such that for every integer \( k \) in the range \(-1985 \leqslant k \leqslant 1985\), there exist 8 integers \( a_{1}, a_{2}, \cdots, a_{8} \), each belonging to the set \(\{-1, 0, 1\}\), such that \( k = \sum_{i=1}^{8} a_{i} n_{i} \).

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