
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 the era of "Internet$+$" today, passwords are closely related to our lives. Simple passwords such as "$123456$" and birthdays are easy to crack, so it is necessary to generate a set of easy-to-remember passwords using a simple method. There is a password generated using the "factorization" method, which is easy to remember. The principle is to factorize a polynomial. For example, for the polynomial $x^{3}+2x^{2}-x-2$, the factorization result is $\left(x-1\right)\left(x+1\right)\left(x+2\right)$. When $x=18$, $x-1=17$, $x+1=19$, $x+2=20$, at this point, a six-digit numerical password $171920$ can be obtained.<br/>$(1)$ According to the above method, when $x=21$, $y=7$, for the polynomial $x^{3}-xy^{2}$, after factoring, what numerical passwords can be formed? (Write down three)<br/>$(2)$ If the perimeter of a right triangle is $30$, the length of the hypotenuse is $13$, and the lengths of the two legs are $x$ and $y$, find a six-digit numerical password obtained by factoring the polynomial $x^{3}y+xy^{3}$ (only one is needed);<br/>$(3)$ If the polynomial $x^{3}+\left(m-3n\right)x^{2}-nx-21$ is factored, using the method in this question, when $x=27$, one of the six-digit numerical passwords obtained is $242834$, find the values of $m$ and $n$.

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