
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 $f(x)=x^{3}-ax^{2}+2 (a\in R)$, where $f′(x)$ is the derivative of $f(x)$.
(I) Find the interval where the function $f(x)$ is monotonically decreasing.
(II) If for all real numbers $x$, $f′(x)\geqslant |x|- \frac {3}{4}$ holds, find the range of values that $a$ can take.
(III) When $a=0$, does there exist two points $A(x\_1,y\_1)$ and $B(x\_2,y\_2) (x\_1\neq x\_2)$ on the curve $y=f(x)$ such that the tangents to the curve at points $A$ and $B$ intersect at the same point on the line $x=2$? If there does, find the maximum value of the y-coordinate of the intersection point. If not, explain the reason.

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