
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
For an arithmetic sequence $\{a_{n}\}$ with a common difference not equal to $0$, it satisfies $a_{2}+2a_{4}=20$, and $a_{1}$, $a_{3}$, $a_{9}$ form a geometric sequence. <br/>$(1)$ Find the general formula for $\{a_{n}\}$; <br/>$(2)$ Let ${b_n}={2^{n-1}}{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted by $T_{n}$.

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