
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. Evaluate the sum $\sum_{i=1}^{2023} \left(\frac{(2i-1)!!}{(2i)!!} + \frac{1}{2^i}\right)$ and express it as a fraction in lowest terms. The denominator should be of the form $2^a b$ with $b$ odd. Find $\frac{ab}{10}$.

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