
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)  $\bigcup_{X \in F} X = R$ 

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