
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 $f(x)=|x-a|+|2x-1|$ $(a\in\mathbb{R})$.
- (I) When $a=1$, find the solution set for $f(x)\leqslant 2$;
- (II) If the solution set for $f(x)\leqslant |2x+1|$ contains the set $\left[ \frac{1}{2},1\right]$, find the range of the real number $a$.

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