
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
In mathematics, we define the operation rule of a 2x2 determinant $|\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}|$ as $|\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{2}\\{3}&{4}\end{array}|=1\times 4-2\times 3=-2$.  
$(1)|\begin{array}{l}{2}&{-3}\\{4}&{5}\end{array}|=$______；  
$(2)$ There are two polynomials $M$ and $N$, where $M=m^{2}-2mn$. Mingming mistakenly calculated $|\begin{array}{l}{M}&{N}\\{1}&{3}\end{array}|$ by incorrectly using the operation rule of a 2x2 determinant as $|\begin{array}{l}{a}&{b}\\{c}&{d}\end{array}|=ad+bc$, and obtained the result $6m^{2}-7mn$.  
① Find the polynomial $N$;  
② Find the correct result of $|\begin{array}{l}{M}&{N}\\{1}&{3}\end{array}|$.

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