
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 $f(x)=e^{2x}-4ae^{x}+(4a-2)x$, where $a\geqslant 1$.
- (I) Discuss the monotonicity of $f(x)$;
- (II) If there exists $x$ such that $f(x)+f(-x)=0$, find the range of the real number $a$;
- (III) If $f(x)\geqslant f(-x)$ holds for all $x\geqslant 0$, find the range of the real number $a$.

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