
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 \( S = \{1, 2, \ldots, n\} \). Define \( A \) as an arithmetic sequence with at least two terms and a positive common difference, where all terms are in \( S \). Also, adding any other element of \( S \) to \( A \) does not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \) (a sequence of only two terms is also considered an arithmetic sequence).

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