
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 that $a$ and $b$ are real numbers, and $a \neq 0$, let $f(x) = |x-1| + |x-2|$.  
$(1)$ If $f(x) > 2$, find the range of real numbers $x$.  
$(2)$ If $|a+b| + |a-b| \geqslant |a|f(x)$ holds for all $a$ and $b$ that satisfy the conditions, find the range of real numbers $x$.

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