
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 $a$ and $b$ be relatively prime positive integers such that $\dfrac ab=\dfrac1{2^1}+\dfrac3{4^2}+\dfrac5{2^3}+\dfrac7{4^4}+\dfrac9{2^5}+\dfrac{11}{4^6}+\cdots$, where the numerators increase by $2$ each time, and the denominators alternate between powers of $2$ and $4$, with exponents increasing by $1$ for each subsequent term. Compute $a+b$.

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