
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 \(k!!\) be defined as \(k(k-2)(k-4)\cdots 3\cdot 1\) for \(k\) odd and \(k(k-2)(k-4)\cdots 4\cdot 2\) for \(k\) even. Consider the sum:
\[ S = \sum_{i=1}^{1005} \frac{(2i)!!}{(2i+1)!!} \]
When \(S\) is reduced to its simplest fractional form, its denominator is of the form \(2^c d\) where \(d\) is odd. Calculate \(\frac{cd}{10}\).

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