
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

On a board the following six vectors are written:
$$
(1,0,0), \quad (-1,0,0), \quad (0,1,0), \quad (0,-1,0), \quad (0,0,1), \quad (0,0,-1).
$$

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 \( u \).

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