
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
Complete the following questions:  
$(1)$ If the inequality about $x$, $(1-m)e^{x}+e^{-x}\geqslant 2$, always holds on the interval $\left[-\ln 2,\ln 2\right]$, find the range of real number $m$;  
$(2)$ Given that the vertex of the quadratic function $f\left(x\right)$ is $\left(-2,0\right)$ and it is tangent to the line $y=2x+3$, if the function $g\left(x\right)=\ln \left[f\left(x\right)-kx\right]$ is monotonically increasing on the interval $\left[2,+\infty \right)$, find the range of real number $k$.

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