
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 power function $f(x)=(m^2-3m+3)x^{m^2-\frac{3}{2}m+\frac{1}{2}}$ is an increasing function on its domain.<br/>$(1)$ Find the analytical expression of the function $f(x)$;<br/>$(2)$ If the function $g(x)=x+a\sqrt[3]{f(x)}$, $x\in \left[1,9\right]$, does there exist a real number $a$ such that the minimum value of $g(x)$ is $0$? If yes, find the value of $a$; if not, explain the reason.

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