
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 $\{a_{n}\}$ is an increasing arithmetic sequence, $a_{1}=2$, ${a}_{2}^{2}=a_{4}+8$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If ${b}_{n}={a}_{n}+{2}^{{a}_{n}}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $S_{n}$.

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