
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 $f\left(x\right)={x}^{-{m}^{2}+2m+3}(m\in Z)$ is an even function and is monotonically increasing on the interval $\left(0,+\infty \right)$.  
$(1)$ Find the analytical expression of the function $f\left(x\right)$;  
$(2)$ Let $g\left(x\right)=\sqrt{f(x)}+2x+c$. If $g\left(x\right) \gt 2$ holds for any $x\in R$, find the range of real number $c$.

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