
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 $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, $a_{2}$ is the geometric mean of $a_{1}$ and $a_{4}$, and $S_{9}=45$. <br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Given ${b}_{n}={a}_{2n-1}•{3}^{{a}_{n}-1}$, find the sum of the first $n$ terms $T_{n}$ of the sequence $\{b_{n}\}$.

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