
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 $f\left(x\right)=ax^{2}+bx+2$, and the solution set of the inequality $f\left(x\right)\leqslant 0$ is $\left[1,2\right]$. $g\left(x\right)$ is an even function with a domain of $R$, and when $x\geqslant 0$, $g\left(x\right)=f\left(x\right)$.  
$(1)$ Find the analytical expression of $g\left(x\right)$;  
$(2)$ If $g\left(m\right) \gt 2$, find the range of real number $m$.

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