
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 $f(x)=-x^{3}+x^{2}$ $(x\in\mathbb{R})$, and $g(x)$ satisfies $g'(x)= \frac {a}{x}$ $(a\in\mathbb{R},x > 0)$, and $g(e)=a$, where $e$ is the base of the natural logarithm.
- (I) Given $h(x)=e^{1-x}f(x)$, find the equation of the tangent line to $h(x)$ at the point $(1,h(1))$;
- (II) If there exists $x\in[1,e]$ such that $g(x)\geqslant -x^{2}+(a+2)x$ holds, find the range of values for $a$;
- (III) Let the function $F(x)= \begin{cases} f(x), & x < 1 \\ g(x), & x\geqslant 1\end{cases}$, and $O$ be the origin. If for any point $P$ on the graph of $y=F(x)$ when $x\leqslant -1$, there always exists a point $Q$ on the curve $y=F(x)$ $(x\in\mathbb{R})$ such that $\overrightarrow{OP}\cdot \overrightarrow{OQ} < 0$, and the midpoint of $PQ$ is on the $y$-axis, find the range of values for $a$.

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