
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 \( R \) be a set of exactly 6 elements. A set \( F \) of subsets of \( R \) is called an \( S \)-family over \( R \) if and only if it satisfies the following three conditions:
(i) For no two sets \( X, Y \) in \( F \) is \( X \subseteq Y \);
(ii) For any three sets \( X, Y, Z \) in \( F \), \( X \cup Y \cup Z \neq R \);
(iii) \(\bigcup_{X \in P} X = R\).
We define \(|F|\) to be the number of elements of \( F \) (i.e., the number of subsets of \( R \) belonging to \( F \) ). Determine, if it exists, \( h = \max |F| \), the maximum being taken over all \( S \)-families over \( R \).

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