
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 $f(x)=-\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+2ax$.
(1) Find the range of values of $a$ such that $f(x)$ has a monotonically increasing interval in $(\frac{2}{3}, +\infty)$.
(2) When $a=1$, find the maximum and minimum values of $f(x)$ on $[1, 4]$.

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