
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
Read the following materials and answer the questions:<br/>Material 1: We define a new operation: we call expressions of the form $\left|\begin{array}{c}a&b\\ c&d\end{array}\right|$ "determinants". The operation of determinants is: $\left|\begin{array}{c}a&b\\ c&d\end{array}\right|=ad-bc$. For example: $\left|\begin{array}{c}2&3\\ 5&6\end{array}\right|=2×6-3×5=12-15=-3$; $\left|\begin{array}{c}x&3\\ x&4\end{array}\right|=4x-3x=x$.<br/>Material 2: When exploring $(x-y)^{3}=$?, we can expand it using polynomial multiplication: $(x-y)^{3}=(x-y)(x-y)(x-y)=(x^{2}-2xy+y^{2})(x-y)=x^{3}-3x^{2}y+3xy^{2}-y^{3}$, we call this formula "the difference of cubes formula". Using the same method, we obtain "the sum of cubes formula" as: $(x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}$. These two formulas are commonly used in processes such as factorization and simplification.<br/>$(1)$ Calculate: $\left|\begin{array}{c}5&4\\ 8&9\end{array}\right|=$______; $a^{3}-3a^{2}+3a-1=\_\_\_\_\_\_$.<br/>$(2)$ Given $x+y=3$, $xy=1$, find the value of $x^{3}+y^{3}$.<br/>$(3)$ Given $m=x-1$, $n=x+2$, $mn=5$, find the value of $\left|\begin{array}{c}m&3{m}^{2}+{n}^{2}\\ n&{m}^{2}+3{n}^{2}\end{array}\right|+\left|\begin{array}{c}m+n&-2n\\ n&m-n\end{array}\right|$.

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