
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
In the Elective 4-5: Special Topics on Inequalities, it is known that the function $f(x)=|2x-a|+|2x-1|$ where $a\in \mathbb{R}$.

$(1)$ When $a=-1$, find the solution set for $f(x)\leqslant 2$.

$(2)$ If the solution set of $f(x)\leqslant |2x+1|$ contains the interval $\left[\frac{1}{2}, 1\right]$, find the range of the real number $a$.

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