
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 the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and if $S_{n}=a_{n+1}-1$, with $a_{1}=2$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b_n}=\left\{{\begin{array}{l}{{a_n},n\text{ is odd}}\\{\frac{{n+1}}{{n-1}}+\frac{{n-1}}{{n+1}},n\text{ is even}}\end{array}}\right.$, find the sum of the first $10$ terms of the sequence $\{b_{n}\}$, denoted as $T_{10}$.

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