
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
$(1)$ For the polynomial in terms of $x$ and $y$: $4x^{2}y^{m+2}+xy^{2}+\left(n-2\right)x^{2}y^{3}+xy-4$, which is a seventh-degree quadrinomial, find the values of $m$ and $n$;<br/>$(2)$ For the polynomial in terms of $x$ and $y$: $\left(5a-2\right)x^{3}+\left(10a+b\right)x^{2}y-x+2y+7$, which does not contain a cubic term, find the value of $5a+b$.

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