
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 $2a_{5}=a_{2}+14$, $S_{9}=72$. Find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b}_{n}=\left\{\begin{array}{l}{a}_{n},n\text{ is odd}\\{2}^{n},n\text{ is even}\end{array}\right.$, find the sum of the first $2n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{2n}$.

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