
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 a positive geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}$, and $a_{1}=1$, $2a_{1}+a_{2}=a_{3}$, the sequence $\{b_{n}\}$ satisfies ${2}^{b_{n}}=4{a}_{n}({S}_{n}+1)$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{{b}_{n}{b}_{n+1}}\}$, where a positive number $m$ always satisfies $m\leqslant T_{n}$, find the range of values for $m$.

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