
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 $f(x)=\frac{2}{3}{x^3}-a{x^2}+x+1$, where $a\in R$.  
$(1)$ If $f\left(x\right)$ is monotonically increasing in $(1,+\infty)$, find the range of real number $a$.  
$(2)$ When $a=\frac{3}{2}$, find the maximum and minimum values of the function $f\left(x\right)$ on the interval $[0,2]$.

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