
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 function $f\left( x \right)={x}^{3}}+{f}'\left( \frac{2}{3} \right){{x}^{2}}-x+c$,

(I) Find the value of ${f}'\left( \frac{2}{3} \right)$;
(II) Find the monotonic intervals of the function $f\left( x \right)$;
(III) Let the function $g\left( x \right)=\left[ f\left( x \right)-{{x}^{3}} \right]{{e}^{x}}$. If the function $g\left( x \right)$ is monotonically increasing in the interval $\left[-3,2\right]$, find the range of the real number $c$.

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