
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 functions $f\left(x\right)=x^{2}-ax-2\left(a\in R\right)$ and $g\left(x\right)=-x^{2}+x+a$.  
$(1)$ If $x=-1$ is a real root of the equation $f\left(x\right)=0$, find the range of the function $f\left(x\right)$.  
$(2)$ If for any $x_1∈[\frac{1}{4},1]$, there exists $x_{2}\in \left[1,2\right]$ such that $g(x_{1}) \gt f(x_{2})+3$, find the range of real number $a$.

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