
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 power function $f(x)=({m}^{2}-3m+3)•{x}^{2-{m}^{2}}$ is monotonically increasing on $R$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Let $g(x)=kf^{2}(x)+\left(k-3\right)f(x)+1$. If the zero point of $g(x)$ is at least one on the right side of the origin, find the range of real number $k$;  
$(3)$ If ${h}_{1}(x)=|{f}^{2}(x)-3|$, $h_{2}(x)=|h_{1}(x)-3|$, $h_{3}(x)=|h_{2}(x)-3|$, and $h_{3}(x)=h_{1}(x)$, find the range of $x$ that satisfies the condition.

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