
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
If the product of the polynomials $(x^{2}+ax-2)$ and $(x^{2}+x+3b)$ does not contain the term $x^{2}$, find:<br/>
$(1)$ the value of $10^{a}\cdot 1000^{b}$;<br/>
$(2)$ if $\left(x+2\right)^{3}=x^{3}+mx^{2}+nx+8$, find the value of $\left(a+3b\right)^{m-n}$.

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