
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
The overall idea is an important mathematical concept, which means that when considering mathematical problems, one should not be limited to their local characteristics, but should focus on the overall structure. Reading material:<br/>If $x=1+\sqrt{2}$,<br/>$(1)$ Find the value of the algebraic expression $x^{2}-2x-3$;<br/>$(2)$ Find the value of the algebraic expression $x-\frac{1}{x}+\sqrt{2}$. Xiao Nan's approach is:<br/>

解:$\left(1\right)\because x=1+\sqrt{2}$,$\therefore x-1=\sqrt{2}$,<br/>$\therefore \left(x-1\right)^{2}=(\sqrt{2})^{2}$,<br/>$\therefore x^{2}-2x+1=2$,<br/>$\therefore x^{2}-2x=1$,<br/>$\therefore x^{2}-2x-3$,<br/>$=(x^{2}-2x)-3$<br/>$=1-3$<br/>$=-2$.</td><td width="284">$(2)$由（1）得,$\because x^{2}-2x-1=0$,<br/>又$\because x\neq 0$,<br/>$\frac{x^2-2x-1}{x}=\frac{0}{x}$,<br/>$x-2-\frac{1}{x}=0$,即$x-\frac{1}{x}=2$,<br/>$x-\frac{1}{x}+\sqrt{2}$<br/>$=(x-\frac{1}{x})+\sqrt{2}$<br/>$=2+\sqrt{2}$.</td></tr></tbody></table>仿照阅读材料的方法解决问题:<br/>$(1)$若$x=2-\sqrt{7}$,则代数式$x^{2}-4x+5$的值为______；<br/>$(2)$若$2x=\sqrt{5}+1$,求代数式$x^{3}-2x^{2}$的值；<br/>$(3)$若$a^{2}=\sqrt{a^2+10}+3$,求代数式$a^{2}+\frac{1}{a^2}$的值.

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