
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 that the power function $f\left(x\right)=(m^{2}-3m+3)x^{m+1}$ is an even function, and $g\left(x\right)=f\left(x\right)+x+2$.  
$(1)$ Find the analytical expression for $y=f\left(x\right)$;  
$(2)$ If $g\left(x\right)\geqslant kx$ holds for $x\in \left[-1,2\right]$, find the range of values for $k$.

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