
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 sequence $\{a_n\}$ whose sum of the first $n$ terms is $T_n$, and $T_n=-a_n+ \frac{1}{2}, n\in\mathbb{N}^*$, let $b_n+2=3\log_{\frac{1}{2}}a_n (n\in\mathbb{N}^*)$, and the sequence $\{c_n\}$ satisfies $c_n=a_n\cdot b_n$.  
$(1)$ Find the general formula for the sequence $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms of the sequence $\{c_n\}$, $S_n$.  
$(3)$ If $c_n\leqslant \frac{1}{4}m^2+m+1$ holds for all positive integers $n$, find the range of the real number $m$.

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