
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 $\dfrac{a}{b} = \dfrac{2}{2^2} + \dfrac{4}{3^1} + \dfrac{6}{2^4} + \dfrac{8}{3^3} + \dfrac{10}{2^6} + \dfrac{12}{3^5} + \cdots$ where the numerators increase by 2 each time, and the denominators alternate between powers of $2$ and $3$, starting from $2^2$ and $3^1$ respectively, with exponents increasing by 2 for each successive power of the same base. Compute $a + b$.

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