
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 an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, where $S_{5}=25$, and $a_{3}-1$, $a_{4}+1$, $a_{7}+3$ form a geometric sequence. Find:<br/>
$(Ⅰ)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(Ⅱ)$ If ${b_n}={(-1)^n}{a_n}+1$ and $T_n$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, calculate $T_{2n}$.

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