
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, set  $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$ . For any two elements  $a=(a_1,a_2,\cdots,a_{2^n})$  and  $b=(b_1,b_2,\cdots,b_{2^n})$  of  $S_n$ , define 
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call  $A \subseteq S_n$  a  $\textsl{Good Subset}$  if  $d(a,b) \geq 2^{n-1}$  holds for any two distinct elements  $a$  and  $b$  of  $A$ . How many elements can the  $\textsl{Good Subset}$  of  $S_n$  at most have?

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