
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 \( m, n \) be integers such that \( 1 < m \leq n \). Define
\[ 
f(m, n) = \left(1 - \frac{1}{m}\right) \times \left(1 - \frac{1}{m+1}\right) \times \left(1 - \frac{1}{m+2}\right) \times \ldots \times \left(1 - \frac{1}{n}\right).
\]
If 
\[ 
S = f(2, 2008) + f(3, 2008) + f(4, 2008) + \ldots + f(2008, 2008), 
\]
find the value of \( 2S \).

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