
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
$(1)$ Calculate: $\left(x+1\right)\left(x+2\right)=\_\_\_\_\_\_.$  
$(x+1)\left(x-2\right)=\_\_\_\_\_\_.$  
$(x-1)\left(x+2\right)=\_\_\_\_\_\_.$  
$(x-1)\left(x-2\right)=\_\_\_\_\_\_.$  
$(2)\left(x+a\right)\left(x+b\right)=x^{2}+( \_\_\_\_\_\_)x+( \_\_\_\_\_\_).$  
$(3)$ Write down the answers directly for the following expressions:  
$(x+99)\left(x+1\right)=\_\_\_\_\_\_.$  
$(x-2)\left(x+50\right)=\_\_\_\_\_\_.$

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