
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 $f(x)=e^{2x}-1-2x-kx^{2}$,
(I) Find the monotonic intervals of $f(x)$ when $k=0$;
(II) Find the range of $k$ such that $f(x) \geqslant 0$ holds for all $x \geqslant 0$;
(III) Compare the magnitude relationship between $\frac{e^{2n}-1}{e^{2}-1}$ and $\frac{2n^{3}+n}{3}$ $(n \in \mathbb{N}^*)$ and provide a proof: $(1^{2}+2^{2}+3^{2}+…+n^{2}= \frac{n(n+1)(2n+1)}{6})$.

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