
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]


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If a function has the same domain and range as $\left[m,n\right]$, then the function is called a same-domain function on $\left[m,n\right]$, and $\left[m,n\right]$ is called the same-domain interval. It is known that the function $f(x)=\sqrt{a}x^2-2\sqrt{a}x+b+1$ is a same-domain function on the interval $\left[1,3\right]$.<br/>$(1)$ The analytical expression of the function $f(x)$ is ______; <br/>$(2)$ If the function $g(x)=k-\sqrt{{f}(x)-\frac{3}{2}}(k≥0)$ has a same-domain interval when $x\leqslant 0$, then the range of the real number $k$ is ______.

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