
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 every positive integer  $n$ , define  $S_n$  to be the sum
\[
  S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .
\]
As  $n$  approaches infinity, what value does  $S_n$  approach?

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