
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 $f(x) = - \frac {1}{3}x^{3}+ \frac {1}{2}x^{2}+2ax$.
$(1)$ Discuss the monotonic intervals of $f(x)$;
$(2)$ If $f(x)$ has a monotonically increasing interval on $[1,+\infty)$, determine the range of values for $a$;
$(3)$ When $0 < a < 2$, the minimum value of $f(x)$ on the interval $[1,4]$ is $- \frac {16}{3}$. Find the maximum value of $f(x)$ on this interval.

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