
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 two elements $A = \left(a_{1}, a_{2}, \cdots, a_{8}\right)$ and $B = \left(b_{1}, b_{2}, \cdots, b_{8}\right)$ in $S$, denote $d(A, B) = \sum_{i=1}^{8}\left|a_{i}-b_{i}\right|$, and refer to it as the distance between $A$ and $B$. How many elements can be selected 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