
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 $\dfrac{a}{b} = \dfrac{1}{2^2} + \dfrac{2}{3^1} + \dfrac{3}{2^4} + \dfrac{4}{3^3} + \dfrac{5}{2^6} + \dfrac{6}{3^5} + \cdots$, where the numerators always increase by $1$, and the denominators alternate between squares of $2$ and odd powers of $3$, with exponents increasing by $2$ for powers of $2$ and by $2$ for powers of $3$. Compute $a+b$.

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