
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
Find the number of nonempty sets \(\mathcal{F}\) of subsets of the set \(\{1, \ldots, 2014\}\) such that:
(a) For any subsets \(S_{1}, S_{2} \in \mathcal{F}, S_{1} \cap S_{2} \in \mathcal{F}\).
(b) If \(S \in \mathcal{F}, T \subseteq \{1, \ldots, 2014\}\), and \(S \subseteq T\), then \(T \in \mathcal{F}\).

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