
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 $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. Calculate the sum $\sum_{i=1}^{1005} \frac{(2i-1)!!}{(2i)!!}$ and express it as a fraction in lowest terms. The denominator of this fraction has the form $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.

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