
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 that the function $f(x)= \frac{1}{2}ax^{2}-2ax+\ln x$ has two distinct extreme points $x_{1}$, $x_{2}$ and $x_{1} \cdot x_{2} > \frac{1}{2}$,  
(Ⅰ) Determine the range of the real number $a$;  
(Ⅱ) Let the range of $a$ from part (Ⅰ) be $M$. If there exists $x_{0} \in \left[1+ \frac{\sqrt{2}}{2},2\right]$, such that for any $a \in M$, the inequality $f(x_{0})+\ln (a+1) > m(a^{2}-1)-(a+1)+2\ln 2$ always holds, find the range of the real number $m$.

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