
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
$$
S=\left\{A=\left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i}=0 \text{ or } 1, i=1,2, \cdots, 8\right\},
$$
for any two elements \(A\) and \(B\) in \(S\), define the distance between \(A\) and \(B\) as
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|. 
$$

Determine 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