
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 $f(x)=|2x-1|+|x+2|$.

(1) Find the minimum value of $f(x)$.

(2) If the inequality $|2b-a|+|b+2a|\geqslant |a|\left(|x+1|+|x-1|\right)\left(a\neq 0\right)$ always holds true, find the range of real number $x$.

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