
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 $f(x)=|2x-1|$.

$(1)$ If the solution set of the inequality $f\left(x+\frac{1}{2}\right)\geqslant 2m+1$ $(m > 0)$ is $(-\infty,-2]\cup[2,+\infty)$, find the value of the real number $m$;

$(2)$ If the inequality $f(x)\leqslant 2^{y}+ \frac{a}{2^{y}}+|2x+3|$ holds for any real number $x$ and $y\in\mathbb{R}$, determine the minimum value of the positive real number $a$.

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