
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 \( n \) be a positive integer. Define \( S_n = \{(a_1, a_2, \cdots, a_{2^n}) \mid a_i = 0 \text{ or } 1, \text{ for } 1 \leqslant i \leq slant 2^n \} \). For any two elements \( a = (a_1, a_2, \cdots, a_{2^n}) \) and \( b = (b_1, b_2, \cdots, b_{2^n}) \) in \( S_n \), define
$$
d(a, b) = \sum_{i=1}^{2^n} |a_i - b_i|.
$$

If \( A \subseteq S_n \) satisfies \( d(a, b) \geq 2^{n-1} \) for any two distinct elements \( a \) and \( b \) in \( A \), then \( A \) is called a good subset. Find the maximum number of elements in a good subset of \( S_n \).

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