
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 $\{a_{n}\}$ that satisfies $2a_{n+1}=a_n-\frac{3}{2}$ and $a_1=\frac{1}{2}$, and a sequence $\{b_{n}\}$ that satisfies $b_{n}+b_{n+2}=2b_{n+1}$ and $b_{1}=2a_{1}$, $b_{3}=4a_{1}-a_{3}$.<br/>$(1)$ Find the general formula for the sequence $\{b_{n}\}$;<br/>$(2)$ Let $c_n=(a_n+\frac{3}{2})•b_n$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $S_{n}$;<br/>$(3)$ Under the conditions of $(2)$, for a real number $m$, there exists a positive integer $n$ such that $S_{n}\leqslant m $ holds true, find the minimum value of $m$.

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