
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
Let \( S = \left\{ A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1,2, \cdots, 8 \right\} \). For any two elements \( A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \) and \( B = \left(b_{1}, b_{2}, \cdots, b_{8}\right) \) in \( S \), define \( d(A, B) = \sum_{i=1}^{8} \left| a_{i} - b_{i} \right| \), which is called the distance between \( A \) and \( B \). What is the maximum number of elements that can be chosen from \( S \) such that the distance between any two of them is at least \( 5 \)?

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