
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 \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\).

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