
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
Define $n!!$ as in the original problem. When $\sum_{i=0}^{1004} \frac{(2i)!!}{(2i+1)!!}$ is expressed as a fraction in lowest terms, determine the form of the denominator and find the value of $\frac{cd}{10}$, where the denominator is $2^c d$ with $d$ odd.

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