
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 the set $S = \{ (s_1, s_2, \ldots, s_6) \mid s_i \in \{0, 1\}, i \in \mathbf{N}_{+}, i \leqslant 6 \}$, define $x = (x_1, x_2, \ldots, x_6)$ and $y = (y_1, y_2, \ldots, y_6)$ for all $x, y \in S$, with the operations $x = y$ if and only if $(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_6 - y_6)^2 = 0$, and $x \cdot y = x_1 y_1 + x_2 y_2 + \cdots + x_6 y_6$. For a non-empty subset $T \subseteq S$ that satisfies $\forall u, v \in T, u \neq v$, we have $u \cdot v \neq 0$. Find the maximum number of elements in the set $T$.

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