
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 \( S = \{1, 2, \cdots, n\} \). Define \( A \) as an arithmetic sequence with a positive common difference consisting of at least two terms, where all terms are in \( S \). Additionally, adding any other element from \( S \) to \( A \) should not form an extended arithmetic sequence with the same common difference. Determine the number of such sequences \( A \). (Note: sequences with only two terms are also considered arithmetic sequences.)

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