
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 $f(x)=x^{2}-2ax+5$ where $a>1$.<br/>$(1)$ If the domain and range of $f(x)$ are both $\left[1,a\right]$, find the value of the real number $a$;<br/>$(2)$ If $g(x)=\frac{{x}^{2}+x-1}{x+1}$, and for any $x\in \left[0,1\right]$, there exists $x_{0}\in \left[0,1\right]$ such that $f(x_{0})=g\left(x\right)$ holds, find the range of values for the real number $a$.

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