
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
Consider two sequences $\{a_n\}$ and $\{b_n\}$ where the first terms are both $2$, given with $a_2=b_2=4$, the relationship $2a_{n+1}=a_n+a_{n+2}$ for $\{a_n\}$, and inequalities $b_{n+1}-b_{n} < 2^n+ \frac{1}{2}$,  $b_{n+2}-b_{n} > 3 \cdot 2^n - 1$ for $\{b_n\}$, with $b_n \in \mathbb{Z}$. Find the sum of the first $n$ terms of the sequence $\left\{ \frac{nb_n}{a_n} \right\}$.

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