
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 $f\left(x\right)=e^{x}+\left(k-2\right)e^{-x}$<br/>$(1)$ When $f\left(x\right)$ is an odd function, solve the following two problems:<br/>① Find the value of $k$;<br/>② If the inequality $mf\left(x\right)-f\left(2x\right)-2e^{-2x}-10 \lt 0$ holds for all $x\in \left(1,+\infty \right)$, find the range of real numbers for $m$;<br/>$(2)$ When $f\left(x\right)$ is an even function, let $g\left(x\right)=\log _{2}f\left(x\right)$, then for what value of $n$, does the function $h\left(x\right)=\left[g\left(x\right)-1+n\right]\cdot \left[2n+1-g\left(x\right)\right]+n^{2}-n$ have a zero.

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