
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, $a-2=-\sqrt{3}$,  
Thus, $\left(a-2\right)^{2}=3$, $a^{2}-4a+4=3$,  
Therefore, $a^{2}-4a=-1$,  
Hence, $2a^{2}-8a+1=2(a^{2}-4a)+1=2\times \left(-1\right)=-1$.  
According to Xiao Ming's analysis process, please solve the following problems:  
$(1)$ Simplify $\frac{1}{{\sqrt{3}+1}}+\frac{1}{{\sqrt{5}+\sqrt{3}}}+\frac{1}{{\sqrt{7}+\sqrt{5}}}+\cdots+\frac{1}{{\sqrt{121}+\sqrt{119}}}$.  
$(2)$ If $a=\frac{1}{{\sqrt{2}-1}}$, find:  
① the value of $3a^{2}-6a+1$.  
② directly write out the algebraic expression $a^{3}-3a^{2}+a+1=$______; $2{a^2}-5a+\frac{1}{a}+2=$______.

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