
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 $f(x) = e^{2x} - 1 - 2x - kx^2$.

$(1)$ When $k=0$, find the monotonic intervals of $f(x)$;

$(2)$ If for $x \geqslant 0$, $f(x) \geqslant 0$ always holds, find the range of $k$;

$(3)$ Compare the size of $\frac{e^{2n} - 1}{e^2 - 1}$ and $\frac{2n^3 + n}{3}$ ($n \in \mathbb{N}^*$), and provide a proof. $\left( 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \right)$

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