
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 $f(x)=\frac{1}{2}x^{2}-\frac{1}{2}$ and the function $g(x)=a\ln x$ have a common tangent line at the point $(1,0)$, let $F(x)=f(x)-mg(x)$.

(Ⅰ) Find the value of $a$;

(Ⅱ) Find the minimum value of $F(x)$ in the interval $[1,e]$.

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