
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
In an infinite arithmetic sequence $\{a_{n}\}$ with the first term being $0$, it is known that $a_{2}$, $a_{3}$, and $ $a_{4}+1$ form a geometric sequence.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let ${b}_{n}=\left\{\begin{array}{l}{a}_{n}+1, \text{if } n \text{ is odd}\\{2}^{{a}_{n}}, \text{if } n \text{ is even}\end{array}\right.$, determine the sum of the first $2n$ terms of the sequence $\{b_{n}\}$ denoted by $T_{2n}$.

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