
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 $f(x)=-\frac{1}{3}x^{3}+\frac{a}{2}x^{2}-2x (a \in \mathbb{R})$,

(I) If the slope of the tangent line to the function $f(x)$ at point $P(2,f(2))$ is $(-4)$, find the value of $a$;

(II) When $a=3$, find the monotonic intervals of the function $f(x)$;

(III) If there are three distinct tangent lines to the function $y=f(x)$ passing through the point $(0,-\frac{1}{3})$, find the range of the real number $a$.

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