
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 $f(x)=a\ln x$ ($a\in \mathbb{R}$).
- (I) If the function $g(x)=2x+f(x)$ has a minimum value of $0$, find the value of $a$;
- (II) Let $h(x)=f(x)+ax^{2}+(a^{2}+2)x$, find the monotonic intervals of the function $h(x)$;
- (III) Suppose the function $y=f(x)$ and the function $u(x)= \frac {x-1}{2x}$ have a common point $P$. If there is exactly one common tangent line passing through point $P$, find the coordinates of point $P$ and the value of the real number $a$.

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