
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
Given set \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \), and for any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} = 0 \);
(2) \( x y = x_{1} y_{1} + x_{2} y_{2} + \cdots + x_{6} y_{6} \).

If a non-empty set \( T \subseteq S \) satisfies \( u v \neq 0 \) for any \( u, v \in T \) where \( u \neq v \), determine the maximum number of elements in set \( T \).

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