
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 $m$ and $n$ are real numbers, $m \neq 0$, and $f(x) = |2x-1| + |x-2|$.
$(1)$ If $f(x) > 2$, find the range of real numbers for $x$;
$(2)$ If $|m+n| + |m-n| \geq |m|f(x)$ holds for all $m$ and $n$ satisfying the conditions, find the range of real numbers for $x$.

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