
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 $f(x) = (m+1)x^2 - (m-1)x + m-1$.  
$(1)$ If the solution set of the inequality $f(x) < 0$ is $R$, find the range of real numbers for $m$;  
$(2)$ When $m < 0$, solve the inequality $f(x) \geq 3x + m - 2$ with respect to $x$;  
$(3)$ If the inequality $f(x) \geq x^2 + 2x$ holds for all $x \in [0,2]$, find the range of real numbers for $m$.

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