
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 $\dfrac{a}{b} = \dfrac{1}{2^1} + \dfrac{1}{3^2} + \dfrac{2}{2^3} + \dfrac{2}{3^4} + \dfrac{3}{2^5} + \dfrac{3}{3^6} + \cdots$, where the numerators start from 1 and increase by 0 each second term, and the denominators alternate between powers of 2 and 3, with exponents also increasing by 1 for each subsequent term. Compute $a+b$.

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