
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 $f(x)=-\frac{1}{3}x^3+\frac{1}{2}x^2+2ax$, where $x\in\mathbb{R}$.
1. Find the monotonic intervals of $f(x)$ when $a=-1$.
2. Find the range of $a$ such that $f(x)$ has a monotonically increasing interval in $(\frac{2}{3},+\infty)$.
3. When $0<a<2$, the minimum value of $f(x)$ on $[1,4]$ is $-\frac{16}{3}$. Find the maximum value of $f(x)$ on this interval.

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