
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 sets $A$ and $B$ are defined such that $A+B=\{x|x=a+b, a\in A, b\in B\}$ and $A\div B=\{x|x=\frac{a}{b}, a\in A, b\in B\}$, and the set $A=\{1,2\}$, calculate the sum of all elements in the set $\left(A+A\right)\div A$.

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