
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 $n!!$ similarly as in the original problem. Consider the sum $\sum_{i=1}^{1010} \frac{(2i)!!}{(2i+1)!!} \cdot \frac{(2i+1)!}{(2i)!}$. When this sum is expressed as a fraction in lowest terms, what is its denominator?

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