
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. Determine all sets \( \mathcal{F} \) consisting of certain subsets of \(\{1,2, \ldots, n\}\) such that for every fixed, nonempty \( X \subseteq \{1,2, \ldots, n\} \), the number of sets \( A \in \mathcal{F} \) for which the size of \( A \cap X \) is even is equal to the number of sets \( A \in \mathcal{F} \) for which the size of \( A \cap X \) is odd.

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