
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
A bag contains 999 balls, marked with the 999 numbers \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1000}\) respectively. Each ball is either red or blue, and the number of red balls is a positive even number. Let \(S\) denote the product of the numbers on all red balls. Find the sum of all possible values of \(S\).

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