
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 $f(x)=|x-1|+|x+1|-1$.
- $(1)$ Find the solution set for $f(x)\leqslant x+1$;
- $(2)$ If the inequality $f(x)\geqslant \frac {|a+1|-|2a-1|}{|a|}$ holds for any real number $a\neq 0$, find the range of real numbers $x$.

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