
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,
\[
S_{n}=\left\{\left(a_{1}, a_{2}, \cdots, a_{2^{n}}\right) \mid a_{i}=0,1\right\}.
\]

For \( a, b \in S_{n} \),
\[
a=\left(a_{1}, a_{2}, \cdots, a_{2^{n}}\right), b=\left(b_{1}, b_{2}, \cdots, b_{2^{n}}\right),
\]

define \( d(a, b)=\sum_{i=1}^{2^{n}}\left|a_{i}-b_{i}\right| \).
If for any \( a, b \in A, a \neq b \), we have \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq S_{n} \) is called a "good subset".
Find the maximum value of \( |A| \).

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