
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 vectors $\overrightarrow{p}=(a_{n},2^{n})$ and $\overrightarrow{q}=(2^{n+1},-a_{n+1})$, where $n\in\mathbb{N}^{*}$, and vectors $\overrightarrow{p}$ and $\overrightarrow{q}$ are perpendicular, with $a_{1}=1$  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=\log_{2}a_{n}+1$, find the sum of the first $n$ terms $S_{n}$ of the sequence $\{a_{n}\cdot b_{n}\}$.

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