
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
Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\frac{(x-a)(x-b)}{x}=\frac{{x}^{2}-(a+b)x+ab}{x}=x+\frac{ab}{x}-\left(a+b\right)$, the solutions to the equation $x+\frac{ab}{x}=a+b$ with respect to $x$ are $x_{1}=a$ and $x_{2}=b$.  
$(1)$ Understanding and application: The solutions to the equation $\frac{{x}^{2}+2}{x}=5+\frac{2}{5}$ are: $x_{1}=$______, $x_{2}=$______;  
$(2)$ Knowledge transfer: If the solutions to the equation $x+\frac{3}{x}=7$ are $x_{1}=a$ and $x_{2}=b$, find the value of $a^{2}+b^{2}$;  
$(3)$ Extension and enhancement: If the solutions to the equation $\frac{6}{x-1}=k-x$ are $x_{1}=t+1$ and $x_{2}=t^{2}+2$, find the value of $k^{2}-4k+4t^{3}$.

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