
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
For each positive integer \( n \), let \( S_n \) be the set that contains the integers from 1 to \( n \), inclusive; that is, \( S_n = \{1, 2, 3, \ldots, n\} \).

For each positive integer \( n \geq 4 \), let \( f(n) \) be the number of quadruples \((a, b, c, d)\) of distinct integers from \( S_n \) for which \( a - b = c - d \).

(a) Determine the value of \( f(6) \).

(b) Determine the value of \( f(40) \).

(c) Determine two even positive integers \( n < 2022 \) for which 2022 is a divisor of \( f(n) \).

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