HMMT 2022 T10

On a board the following six vectors are written:

$$(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)$$

Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt{2}}(v+w)$ and $\frac{1}{\sqrt{2}}(v-w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.