
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 positive integers  $n_1, n_2, \dots , n_8$  with the following property:

For every integer  $k$ ,  $-1985 \leq k \leq 1985$ , there are eight integers  $a_1, a_2, \dots, a_8$ , 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