
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 sequence with the sum of its first $n$ terms denoted as $S_{n}$, given that ${a}_{1}=1,{S}_{n+1}-2{S}_{n}=1$ for $n\in\mathbb{N}^{*}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies ${b_{n}=\frac{4a_{n}}{(a_{n+1}-1)(a_{n+2}-1)}}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is denoted as ${T}_{n}$ for all $n\in\mathbb{N}^{*}$, and it holds that ${m^{2}-\frac{4}{3}m＜T_{n}}$, find the range of values for $m$.

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