
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
Elective 4-5: Special Topics on Inequalities

Given the function $f(x)=|2x-1|$.

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

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

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