
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 function $f\left(x\right)=e^{2x}-2x$.  
$(1)$ Find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$.  
$(2)$ Find the extreme value of $f\left(x\right)$.  
$(3)$ If the inequality $f\left(x\right) \gt 2\left(e-1\right)x+m$ holds for any $x\in R$, find the range of real number $m$.

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