
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 1】  
To solve the equation $(x^{2})^{2}-5x^{2}+4=0$, we can treat $x^{2}$ as a whole, then let $y=x^{2}$, so the original equation can be transformed into $y^{2}-5y+4=0$. After calculation, the solutions to the original equation are $x_{1}=1$, $x_{2}=-1$, $x_{3}=2$, $x_{4}=-2$.  
We call the above method of solving problems the substitution method.  
【Reading Material 2】  
Given real numbers $m$, $n$ satisfy $m^{2}-m-1=0$, $n^{2}-n-1=0$, and $m\neq n$, obviously $m$, $n$ are two distinct real roots of the equation $x^{2}-x-1=0$. By Vieta's theorem, we know $m+n=1$, $mn=-1$.  
Based on the above materials, solve the following problems:  
$(1)$ Direct application:  
To solve the equation $x^{4}-x^{2}-6=0$, let $y=$______, the original equation can be transformed into ______.  
After calculation, the solutions to the original equation are ______.  
$(2)$ Indirect application:  
Given real numbers $a$, $b$ satisfy $a^{4}-3a^{2}+1=0$, $b^{4}-3b^{2}+1=0$, and $a\neq b$, find the value of $a^{4}+b^{4}$.

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