
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
For the set \( S = \left\{ \left( a_1, a_2, \cdots, a_5 \right) \mid a_i = 0 \text{ or } 1, i = 1, 2, \cdots, 5 \right\} \), the distance between any two elements \( A = \left( a_1, a_2, \cdots, a_5 \right) \) and \( B = \left( b_1, b_2, \cdots, b_5 \right) \) is defined as: 
\[ d(A, B) = \left| a_1 - b_1 \right| + \cdots + \left| a_5 - b_5 \right| \]
Find a subset \( T \) of \( S \) such that the distance between any two elements in \( T \) is greater than 2. What is the maximum number of elements that subset \( T \) can contain? Provide a proof for your conclusion.


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