
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
In a set \( X \) of 17 elements, a family of \( N \) distinct non-empty subsets is selected such that each element of set \( X \) is contained in exactly two subsets from this family. What is the maximum value of \( N \)? Find the number of all possible distinct types of such families for the maximum \( N \). Two families of subsets have different types if they cannot be obtained from each other by permuting the elements of \( X \).

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