
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 \( S \) be the set \(\{1, 2, \ldots, 2012\}\). A perfectutation is a bijective function \( h \) from \( S \) to itself such that there exists an \( a \in S \) such that \( h(a) \neq a \), and that for any pair of integers \( a \in S \) and \( b \in S \) such that \( h(a) \neq a \), \( h(b) \neq b \), there exists a positive integer \( k \) such that \( h^{k}(a) = b \).

Let \( n \) be the number of ordered pairs of perfectutations \((f, g)\) such that \( f(g(i)) = g(f(i)) \) for all \( i \in S \), but \( f \neq g \). Find the remainder when \( n \) is divided by 2011.

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