
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
A student is solving the problem: given $a=\frac{1}{2+\sqrt{3}}$, find the value of $2a^{2}-8a+1$. He analyzed and solved it as follows: first, rationalize the denominator of $a$ as shown below, $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$, so $a=2-\sqrt{3}$, thus $\left(a-2\right)^{2}=3$, $a^{2}-4a+4=3$, hence $a^{2}-4a=-1$, therefore $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$. Please solve the following problems based on the above analysis: (1) if $a=\frac{1}{\sqrt{2}-1}$, rationalize the denominator of $a$; (2) under the condition in (1), find the value of $a^{2}-2a$; (3) under the condition in (1), find the value of $2a^{3}-4a^{2}-1$.

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