
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 an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and it satisfies $a_{3}=8$ and $S_{5}=2a_{7}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b}_{n}={(-1)}^{n}{a}_{n}+{2}^{n+1}$, find the sum of the first $2n$ terms $T_{2n}$ of the sequence $\{b_{n}\}$.

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