
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]


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Let \( S = \{1, 2, 3, \ldots, n\} \). Define \( A \) as an arithmetic progression with at least two elements and a positive common difference, with all terms belonging to \( S \). Additionally, adding any other elements of \( S \) to \( A \) should not form an arithmetic progression with the same common difference. Determine the number of such sequences \( A \). (Note that a sequence with only two terms is also considered an arithmetic progression.)

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