
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
【Thinking Guidance】  
The teacher posed the following question in a math class: Given $x^2+\sqrt{3}x+x=-1$ ($x\neq0$), find the value of $x^2+\frac{1}{x^2}$.  
Wenwen formed the following approach through observation, analysis, and thinking: first divide both sides of the equation by $x$ to obtain the value of $x+\frac{1}{x}$, then use the formula for the sum of squares to find $x^2+\frac{1}{x^2}$.  
【Analogous Exploration】  
Following Wenwen's approach, solve the following problems:  
$(1)$ Given $x^2-\sqrt{5}x-x-1=0$ ($x\neq0$), find the value of $x^2+\frac{1}{x^2}$.  
$(2)$ Given $x^{2}+1=3x$ ($x\neq 0$), find the value of $\sqrt{x}+\frac{1}{\sqrt{x}}$.

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