
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
Let $\{a_{n}\}$ be a geometric sequence with the sum of its first $n$ terms denoted as $S_{n}$. It is known that $a_{n+1}=S_{n}+1$, where $n\in \mathbb{N}^{*}$. <br/>$(1)$ Find the general term formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Let ${b_n}={({-1})^n}({{a_n}+n})$, 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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