
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
Let $x_{1}$ and $x_{2}(x_{1} \lt x_{2})$ be two extreme points of the function $f(x)=\frac{1}{3}{x}^{3}+mx^{2}-3x+1$, and $\frac{{x}_{1}+{x}_{2}}{{x}_{1}{x}_{2}}=\frac{2}{3}$.  
$(1)$ Find the value of $m$;  
$(2)$ Find the range of $f\left(x\right)$ on the interval $\left[0,3\right]$.

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