
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
Observe the multiplication of the following polynomials and answer the questions:<br/>①$(x+3)(x+4)=x^{2}+(3+4)x+3\times 4=x^{2}+7x+12$;<br/>②$(x+3)(x-4)=x^{2}+(3+(-4))x+3\times (-4)=x^{2}-x-12$.<br/>③$(x-3)(x+4)=x^{2}+((-3)+4)x+(-3)\times 4=x+x-12$;<br/>④$(x-3)(x-4)=x^{2}+((-3)+(-4))x+(-3)\times (-4)=x^{2}-7x+12$.<br/>$(1)$ Calculate $(x+2)(x+3)=\_\_\_\_\_\_$. Based on the pattern you discovered, guess $(x+a)(x+b)=$______;<br/>$(2)$ If $(x-2)(x+m)=x^{2}+nx-2$, find the value of $n$.

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