
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
Let the function $f(x)=|2x-1|$.
$(1)$ If the solution set of the inequality $f(x+1)\leqslant 2m$ ($m > 0$) is $\left[-\frac{3}{2}, \frac{1}{2}\right]$, find the value of the real number $m$;
$(2)$ If the inequality $f(x+1)\leqslant 2018^{y}+ \frac{n}{2018^{y}}+|2x-1|$ always holds for $x$, $y\in\mathbb{R}$, find the minimum value of the real number $n$.

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