
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
Let  $k$  be a positive integer. Find the smallest positive integer  $n$  for which there exists  $k$  nonzero vectors  $v_1,v_2,…,v_k$  in  $\mathbb{R}^n$  such that for every pair  $i,j$  of indices with  $|i-j|>1$  the vectors  $v_i$  and  $v_j$  are orthogonal.

*Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.*

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