
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 $\dfrac{a}{b} = \dfrac{1^2}{2^2} + \dfrac{2^2}{3^3} + \dfrac{3^2}{2^4} + \dfrac{4^2}{3^5} + \dfrac{5^2}{2^6} + \dfrac{6^2}{3^7} + \cdots$, where the numerators are squares of their positions, and the denominators alternate between powers of 2 and 3, with exponents increasing by 2 for terms with powers of 2 and by 3 for terms with powers of 3 respectively. Compute $a+b$.

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