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

If \( A \subseteq S_n \) such that for any two distinct elements \( a, b \in A \), \( d(a, b) \geq 2^{n-1} \), 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