
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
Consider the set \( S = \{1, 2, \cdots, 15\} \). Let \( A = \{a_1, a_2, a_3\} \) be a subset of \( S \), with \( (a_1, a_2, a_3) \) satisfying:
\[
1 \leq a_1 < a_2 < a_3 \leq 15, \quad a_3 - a_2 \leq 6.
\]
Determine the number of subsets \( A \) that satisfy these conditions.

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