
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
Xiao Ming 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: Since $a=\frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}=2-\sqrt{3}$, therefore $a-2=-\sqrt{3}$, hence $\left(a-2\right)^{2}=3$, $a^{2}-4a+4=3$, therefore $a^{2}-4a=-1$, hence $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)+1=-1$. Please solve the following problems based on Xiao Ming's analysis process: (1) $\frac{1}{\sqrt{3}+\sqrt{2}}=$______, $\frac{1}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_$. (2) Simplify: $\frac{1}{\sqrt{11}+\sqrt{9}}+\frac{1}{\sqrt{13}+\sqrt{11}}+⋯+\frac{1}{\sqrt{121}+\sqrt{119}}$. (3) If $a=\frac{1}{\sqrt{2}-1}$, please find the value of $4a^{2}-8a+1$ according to Xiao Ming's method.

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