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

$(1)$ Solve the inequality $f(x) \geqslant 3$.

$(2)$ If the inequality $|a+b|+|a-b| \geqslant af(x) (a \neq 0, a, b \in \mathbb{R})$ always holds, find the range of the real number $x$.

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