
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 $f(x)=|2x+3|-|2x-a|$, where $a\in\mathbb{R}$.

$(1)$ If the solution set of the inequality $f(x)\leqslant -5$ is non-empty, find the range of values for the real number $a$;

$(2)$ If the graph of the function $y=f(x)$ is symmetric about the point $\left(-\dfrac{1}{2},0\right)$, find the value of the real number $a$.

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