
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]


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$S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$, and the sequence $\{b_{n}\}$ satisfies $b_{n}=n-\left(-1\right)^{n}S_{n}$, $a_{1}+b_{1}=3$, $a_{2}-b_{2}=5$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Let the sum of the first $n$ terms of the sequence $\{b_{n}\}$ be $T_{n}$.  
① Find $T_{10}$;  
② If the set $A=\{n|n\leqslant 100$ and $T_{n}\leqslant 100$, $n\in N^{*}\}$, find the sum of all elements in set $A$.

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