
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 \( S = \{1, 2, \cdots, n\} \), and let \( A \) be an arithmetic sequence with a positive common difference, containing at least two terms, all of which are elements of \( S \). Furthermore, adding any other elements from \( S \) to \( A \) must not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \). (An arithmetic sequence with only two terms is also considered an arithmetic sequence).

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