
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 $f\left(x\right)={\left(m-1\right)}^{2}{x}^{{m}^{2}-4m+2}$ which is monotonically increasing on $(0,+\infty)$, and the function $g(x)=2^{x}-k$.

(1) Find the value of the real number $m$;

(2) When $x\in(1,2]$, let the range of values of $ƒ(x)$ and $g(x)$ be sets $A$ and $B$, respectively. If $A\cup B=A$, find the range of the real number $k$.

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