
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 and at least two elements, where all elements of \( A \) are in \( S \). Additionally, adding any element from \( S \) to \( A \) should not form a new arithmetic sequence with the same common difference as \( A \). How many such sequences \( A \) exist, considering that sequences of just two elements are also considered arithmetic sequences?

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