
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 $f(x)=(x^{2}-2x)\ln x+ax^{2}+2$.  
$(1)$ When $a=-1$, find the equation of the tangent line to $f(x)$ at the point $(1,f(1))$;  
$(2)$ When $a > 0$, let the function $g(x)=f(x)-x-2$, and suppose that the function $g(x)$ has exactly one zero. If $e^{-2} < x < e$, $g(x)\leqslant m$, find the range of values for $m$.

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