
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 functions $f(x)= \frac{1}{2}x^{2},g(x)=a\ln x$.

(Ⅰ) If the tangent line of the curve $y=f(x)-g(x)$ at $x=2$ is perpendicular to the line $x+3y-7=0$, find the value of the real number $a$;

(Ⅱ) If there exists a point $x_{0}$ in the interval $[1,e]$ such that the inequality $f''(x_{0})+ \frac{1}{f''(x_{0})} < g(x_{0})-g''(x_{0})$ holds, find the range of values for the real number $a$.

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