
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 distinct positive integers \( a, b < 2012 \), define \( f(a, b) \) to be the number of integers \( k \) with \( 1 \leq k < 2012 \) such that the remainder when \( a k \) is divided by 2012 is greater than that of \( b k \) divided by 2012. Let \( S \) be the minimum value of \( f(a, b) \), where \( a \) and \( b \) range over all pairs of distinct positive integers less than 2012. Determine \( S \).

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