
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 \( P \) denote the set of all subsets of \(\{1, \ldots, 23\}\). A subset \( S \subseteq P \) is called good if whenever \( A, B \) are sets in \( S \), the set \((A \backslash B) \cup (B \backslash A)\) is also in \( S \). (Here, \( A \backslash B \) denotes the set of all elements in \( A \) that are not in \( B \), and \( B \backslash A \) denotes the set of all elements in \( B \) that are not in \( A \).) What fraction of the good subsets of \( P \) have between 2015 and 3015 elements, inclusive?

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