
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 $$f(x)= \frac {1}{2}x^{2}+(a-3)x+\ln x$$.
(Ⅰ) If the function $f(x)$ is monotonic on its domain, find the minimum value of the real number $a$;
(Ⅱ) The equation $$f(x)=( \frac {1}{2}-a)x^{2}+(a-2)x+2\ln x$$ has two distinct real solutions, find the range of the real number $a$;
(Ⅲ) On the graph of the function $f(x)$, is there any two distinct points $A(x_1, y_1)$, $B(x_2, y_2)$, where the midpoint of the line segment $AB$ has an x-coordinate of $x_0$, and $f′(x_0) = \frac {y_1-y_2}{x_1-x_2}$ holds? If it exists, find the value of $x_0$; if not, explain why.

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