
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
In the sequence ${a_{n}}$ and ${b_{n}}$, where the first term is $2$ for both, $a_{2}=b_{2}=4$, $2a_{n+1}=a_{n}+a_{n+2}$, $b_{n+1}-b_{n} < 2^{n}+\frac{1}{2}$, $b_{n+2}-b_{n} > 3\times2^{n}-1$, and $b_{n}\in Z$, find the sum of the first $n$ terms of the sequence ${\frac{n{b_{n}}}{{a_{n}}}}$.

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