
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 $c$ and $d$ be relatively prime positive integers such that $\dfrac cd=\dfrac1{2^2}+\dfrac2{3^3}+\dfrac3{2^4}+\dfrac4{3^5}+\dfrac5{2^6}+\dfrac6{3^7}+\cdots$, where the numerators increase by 1, and the denominators alternate between powers of $2$ and $3$, with exponents starting from $2$ and $3$ respectively, and increasing by $1$ for each subsequent term. Compute $c+d$.

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