
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 $\{a_{n}\}$ be an arithmetic sequence with a positive common difference, given that $a_{1}=2$ and $a_{3}=a_{2}^{2}-10$.  
$(Ⅰ)$ Find the general formula for $\{a_{n}\}$.  
$(Ⅱ)$ Let $\{b_{n}\}$ be a geometric sequence with a common ratio of $3$, starting with the smallest positive period of the function $y=4\sin ^{2}\pi x$. Determine the sum of the first $n$ terms of the sequence $\{a_{n}-b_{n}\}$ denoted by $S_{n}$.

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