
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
When calculating $\left(x+a\right)\left(x+b\right)$, person A mistook $b$ for $6$ and obtained the result: $x^{2}+8x+12$; person B mistakenly took $a$ as $-a$ and got the result: $x^{2}+x-6$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ Under the conditions in $(1)$, calculate the result of $\left(x+a\right)\left(x+b\right)$.

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