
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 the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is ${S_n}$, and ${a_{n-1}}+{a_{n+1}}=2{a_n}$ for $n≥2$, $n∈{{N}^*}$, and $a_{1}=1$, $S_{5}=15$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If ${b_n}=\left\{{\begin{array}{l}{{a_n},n\;is\;odd}\\{{2^{{b_{n-1}}}},n\;is\;even}\end{array}}\right.$, 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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