
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 geometric sequence $\left\{a_n\right\}$ with the first term $a_1= \frac{1}{4}$ and common ratio $q= \frac{1}{4}$, let $b_n=2\log_{\frac{1}{4}}a_n$ $(n\in\mathbb{N}^*)$, and a sequence $\left\{c_n\right\}$ satisfies $c_n=a_nb_n$.

$(1)$ Find the sum of the first $n$ terms $S_n$ of the sequence $\left\{c_n\right\}$;

$(2)$ If the sum of the first $n$ terms $T_n \geqslant \frac{1}{2}m^2+m-1$ for all positive integers $n$ of the sequence $\left\{\frac{8n+4}{(n+1)^2b_n^2}\right\}$ always holds, find the range of the real number $m$.

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