
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 $f(x)=\frac{1}{2}x+\frac{5}{2}$ and $g(x)=x^{2}-2ax+4a-3$ where $a\in \mathbb{R}$.<br/>$(1)$ If the range of function $g(x)$ is $\left[0,+\infty \right)$, find the set of possible values for $a$;<br/>$(2)$ If for any $x_{1}\in \left[-1,1\right]$, there exists $x_{2}\in \left[-1,1\right]$ such that $f(x_{1})=g(x_{2})$, find the range of real numbers for $a$.

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