
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
Given that $\{a\_n\}$ and $\{b\_n\}$ are arithmetic sequences with common differences $d\_1$ and $d\_2$, respectively, and $A\_n = a\_n + b\_n$, $B\_n = a\_n b\_n$. If $A\_1 = 1$, $A\_2 = 3$, then find $A\_n$; if $\{B\_n\}$ is an arithmetic sequence, then find $d\_1 d\_2$.

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