
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 power function $f(x) = (m^2 - 2m - 2)x^m$ is monotonically increasing on $R$, and the function $g(x) = x^{-1} + k$.  
$(1)$ Find the value of $m$;  
$(2)$ Let $f(x)$ and $g(x)$ have value ranges on the interval $(1,2)$ as sets $A$ and $B$ respectively. If $x \in A$ is a necessary condition for $x \in B$, find the range of real number $k$.

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