
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 $f(x)= \frac {1}{2}ax^{2}+bx(a\neq 0)$, $g(x)=1+\ln x$.  
(I) If $b=1$, and $F(x)=g(x)-f(x)$ has a monotonically decreasing interval, find the range of values for $a$;  
(II) Suppose the graph of function $g(x)$, denoted as $C_{1}$, intersects with the graph of function $f(x)$, denoted as $C_{2}$, at points $M$ and $N$. A perpendicular line to the $x$-axis through the midpoint $T$ of segment $MN$ intersects $C_{1}$ and $C_{2}$ at points $P$ and $Q$, respectively. Is there a point $T$ such that the tangent line to $C_{1}$ at point $P$ is parallel to the tangent line to $C_{2}$ at point $Q$? If it exists, find the $x$-coordinate of point $T$; if not, explain why.

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