
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 $\frac{a}{b} = \frac{1}{2^2} + \frac{2}{3^3} + \frac{3}{2^4} + \frac{4}{3^5} + \frac{5}{2^6} + \frac{6}{3^7} + \cdots$, where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents increasing by $2$ starting from $2$ and $3$ respectively for each subsequent term. Compute $a+b$.

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