
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
For the subset \( S \) of the set \(\{1,2, \cdots, 15\}\), if a positive integer \( n \) and \( n+|S| \) are both elements of \( S \), then \( n \) is called a "good number" of \( S \). If a subset \( S \) has at least one "good number", then \( S \) is called a "good set". Suppose 7 is a "good number" of a "good set" \( X \). How many such subsets \( X \) are there?

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