
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
Let $f\left(x\right)=2^{x}+\left(p-1\right)\cdot 2^{-x}$ be an even function with the domain of $R$. <br/>$(1)$ Find the value of $p$; <br/>$(2)$ If $g\left(x\right)=f\left(2x\right)-2k\cdot (2^{x}-2^{-x})$ has a minimum value of $-4$ on $\left[1,+\infty \right)$, find the value of $k$; <br/>$(3)$ If the inequality $f\left(2x\right) \gt m\cdot f\left(x\right)-4$ holds for any real number $x$, find the range of the real number $m$.

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