
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 $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, given that $a_{1}=1$, and it satisfies $2S_{n}=a_{n}\cdot \left(n+1\right)$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$, where for $n\geqslant 2$, ${b}_{n}=\frac{1}{{a}_{n-1}•{a}_{n}•{a}_{n+1}}$. If for any $n\in N^{*}$, $T_{n} \lt 1$, find the range of values for $b_{1}$.

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