
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
When solving some more complex one-variable quadratic equations, the "root shrinking method" can be used to simplify the calculations. The "root shrinking method" refers to first transforming the one-variable quadratic equation into a new one-variable quadratic equation with simpler coefficients, solving the new equation, and then shrinking the two roots of the new equation by the same factor to obtain the two roots of the original equation. Given: The two roots of the one-variable quadratic equation about $x$, $ax^{2}+bx+c=0$ $(a\neq 0)$ are $x_{1}=\alpha$, $x_{2}=\beta$. Find the two roots of the one-variable quadratic equation about $x$, $p^{2}ax^{2}+pbx+c=0$ $(ap\neq 0)$. 

**Solution:**  
Since $p^{2}ax^{2}+pbx+c=0$ $(ap\neq 0)$, therefore $a(px)^{2}+b\cdot px+c=0$. Let $px=t$,  
we get the new equation $at^{2}+bt+c=0$.  
Since the roots of the new equation are $t_{1}=\alpha$, $t_{2}=\beta$, therefore $px=\alpha$, $px=\beta$,  
thus the roots of the original equation are ${x}_{1}=\frac{\alpha}{p}$, ${x}_{2}=\frac{\beta}{p}$.  
This method of solving one-variable quadratic equations is called the "root shrinking method".  
For example, using the root shrinking method to solve the equation $49x^{2}+35x-24=0$.  
**Solution:**  
Since $49=7^{2}$, $35=5\times 7$, therefore $(7x)^{2}+5\times 7x-24=0$. Let $7x=t$,  
we get the new equation $t^{2}+5t-24=0$.  
Solving the new equation gives $t_{1}=3$, $t_{2}=-8$, therefore $7x=3$, $7x=-8$,  
thus the roots of the original equation are ${x}_{1}=\frac{3}{7}$, ${x}_{2}=-\frac{8}{7}$.  
Please use the above information to solve the following problems and write out the specific steps:  
$(1)$ Solve the equation using the root shrinking method: $36x^{2}-6x-1=0$;  
$(2)$ Solve the equation using the root shrinking method: $3x^{2}-160x+1600=0$.

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