
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 $f(x) = (m+1)x^{2} - (m-1)x + m-1$.  
$(1)$ Solve the inequality with respect to $x$: $f(x) \geq (m+1)x$;  
$(2)$ If the inequality $f(x) \geq 0$ holds for all $x \in [-\frac{1}{2}, \frac{1}{2}]$, find the range of values for $m$.

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