
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
\[
\frac{a}{b} = \frac{1}{2^1} - \frac{1}{3^1} + \frac{2}{2^2} - \frac{2}{3^2} + \frac{3}{2^3} - \frac{3}{3^3} + \cdots
\]
where numerators increase linearly by $1$, denominators alternate between powers of $2$ and $3$ with respective exponents increasing by $1$, and terms alternatively add and subtract. Compute \(a+b\).

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