
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
We define an operation $|\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}|=ad-cb$. For example, $|\begin{array}{l}{2}&{3}\\{4}&{5}\end{array}|=2\times 5-3\times 4=-2$, and $|\begin{array}{l}{x}&{-1}\\{2}&{-4}\end{array}|=-4x+2$. According to this operation rule, solve the following questions:<br/>$(1)$ Calculate $|\begin{array}{l}{-3}&{-2}\\{4}&{5}\end{array}|=$______；<br/>$(2)$ If $|\begin{array}{l}{2-2x}\\{3-5x}\end{array}|=2$, find the value of $x$；<br/>$(3)$ If the values of $|\begin{array}{l}{8mx-1}&{-\frac{8}{3}+2x}\\{\frac{3}{2}}&{-3}\end{array}|$ and $|\begin{array}{l}{6}&{-1}\\{-n}&{x}\end{array}|$ are always equal, find the values of $m$ and $n$.

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