
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 and define \(\mathrm{S}_{\mathrm{n}} = \left\{\left(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{2^{n}}\right) \mid \mathrm{a}_{\mathrm{i}} \in\{0,1\}, 1 \leq \mathrm{i} \leq 2^{\mathrm{n}}\right\}\). For \( \mathrm{a}, \mathrm{b} \in \mathrm{S}_{\mathrm{n}} \) where \( a = \left(a_{1}, a_{2}, \ldots, a_{2^{n}}\right) \) and \( b = \left(b_{1}, b_{2}, \ldots, b_{2^{n}}\right) \), define \( d(a, b) = \sum_{i=1}^{2^{n}} \left|a_{i} - b_{i}\right| \). If for any \(a, b \in A\), \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq \mathrm{S}_{\mathrm{n}} \) is called a "good subset". Find the maximum possible value of \( |A| \).

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