
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 $f(x)=|2x-1|+|x+a|$, where $a\in \mathbb{R}$.<br/>$(1)$ When $a=1$, find the solution to the inequality $f(x)\geqslant 3$;<br/>$(2)$ If there exists $x\in \mathbb{R}$ such that $f(x)\leqslant |a-1|$, determine the range of values for $a$.

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