
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 $f(x)=\begin{cases} x^{2}-x+3, & x\leqslant 1, \\ x+\dfrac{2}{x}, & x > 1. \end{cases}$ Let $a\in\mathbb{R}$, determine the range of values for $a$ such that the inequality $f(x)\geqslant \left|\dfrac{x}{2}+a\right|$ holds for all $x\in\mathbb{R}$.

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