
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 $f(x)=a\ln x-ax+1$, where $a\in \mathbb{R}$.  
$(1)$ If the line passing through the point $(0,0)$ is tangent to the graph of the function $f(x)$ at the point $(2,f(2))$, find the value of the real number $a$.  
$(2)$ Let $g(x)=f(x)+\frac{1}{2}x^2-1$. If the function $g(x)$ is strictly decreasing on the interval $\left(\frac{3}{2},4\right)$, find the range of values for the real number $a$.  
$(3)$ For the function $g(x)$ in $(2)$, if $g(x)$ has two distinct critical points $x_{1}$ and $x_{2}$, and the inequality $g(x_{1})+g(x_{2}) < \lambda (x_{1}+x_{2})$ always holds, find the range of values for the real number $\lambda$.

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