
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
Given that the sequence $\{a_{n}\}$ is a geometric sequence with positive terms, and $a_{1}=2$, $a_{5}=32$; the sequence $\{b_{n}\}$ satisfies: for any $n∈N^{*}$, $a_{1}b_{1}+a_{2}b_{2}+…+a_{n}b_{n}=(n-1)⋅2^{n+1}+2$.

1. Find the general term formula for the sequence $\{a_{n}\}$;
2. Find the general term formula for the sequence $\{b_{n}\}$;
3. By inserting $k$ terms of $(-1)^{k}b_{k}$ ($k∈N^{*}$) between any two adjacent terms $a_{k}$ and $a_{k+1}$ in the sequence $\{a_{n}\}$, a new sequence $\{c_{n}\}$ is obtained. Find the sum of the first $2016$ terms of the sequence $\{c_{n}\}$.

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