
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 $f(x)=mx^{2}-2x+1$,

$(1)$ If $m=1$ and $x \in [-2,1]$, find the range of the function $f(x)$;

$(2)$ Solve the equation $f(m)=0$ with respect to $m$;

$(3)$ When $x \in [1,2]$, $f(x) \geq 0$ always holds, find the range of the real number $m$.

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