
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
Given a set \( \mathrm{S} = \{1, 2, 3, \ldots, 199, 200\} \), a subset \(\mathrm{A} = \{a, b, c\}\) of \(\mathrm{S}\) is said to be "nice" if \( a + c = 2b \). How many "nice" subsets does \(\mathrm{S}\) have? (Note: The order of the elements inside the set does not matter. For example, we consider \(\{a, b, c\}\) or \(\{a, c, b\}\) or \(\{c, b, a\}\) to be the same set.)

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