
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 an arithmetic sequence $\{a_{n}\}$, $a_{1}=2$, and $a_{2}$, $a_{3}+2$, $a_{8}$ form a geometric sequence.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let $b_{n}={2^{{a_n}}}+9$, and let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$. If $S_{n}\geqslant 2022$, find the smallest positive integer value of $n$.

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