
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 functions $f(x)=x\ln x-\frac{1}{2}ax^{2}$ and $g(x)=-x+a$ where $a\in \mathbb{R}$.
$(1)$ If the graph of $y=x$ is exactly tangent to the graph of $f(x)$, find the value of the real number $a$.
$(2)$ Let $F(x)=f(x)+g(x)$. If $F(x)$ has two distinct critical points $x_{1}$ and $x_{2}$ where $x_{1} \lt x_{2}$.
$(i)$ Find the range of values for the real number $a$.
$(ii)$ If the inequality $e^{\lambda +1} \lt x_{1}\cdot x_{2}^{\lambda }$ always holds true, find the range of values for the positive number $\lambda$ ($e=2.71828\ldots$ is the base of the natural logarithm).

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