
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 the function $f(x)=|x+3|-m$, where $m > 0$, and the solution set of $f(x-3)\geqslant 0$ is $(-\infty,-2]\cup[2,+\infty)$.

$(1)$ Find the value of $m$;

$(2)$ If $x\in\mathbb{R}$, and $f(x)\geqslant |2x-1|-t^{2}+\frac{3}{2}t+1$ holds, find the range of the real number $t$.

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