
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
Determine all $(A, B)$ number pairs for which there exist sequences $a_{n}$ and $b_{n}$ such that $\lim (a_{n} b_{n}) = A$ and $\lim (a_{n} + b_{n}) = B$. For which values of $A$ and $B$ will it be true that the sequences $a_{n}$ and $b_{n}$ themselves are also convergent?

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