
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 \( S = \{1, 2, 3, \ldots, 65\} \). Find the number of 3-element subsets \(\{a_{1}, a_{2}, a_{3}\}\) of \( S \) such that \( a_{i} \leq a_{i+1} - (i+2) \) for \( i = 1, 2 \).

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