
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 $f(x)=|ax-1|-(a-1)|x|$.  
$(1)$ When $a=2$, find the solution set of the inequality $f(x) > 2$.  
$(2)$ If $x\in (1,2)$, and $f(x) < a+1$ always holds, find the range of real numbers $a$.

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