
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
Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ be sequences of positive numbers. Assume $a_{n+2} = (a_{n+1} + 1)a_n$ and $b_{n+2} = b_{n+1} + b_n$ for all positive integers $n$. Under which of the following condition(s) does the sequence defined by $x_n = a_n b_n$ form a geometric progression?
A) if and only if $a_2 = a_1$ and $b_1 b_2$ is a constant
B) if $a_2 = a_1$
C) if $b_1 = b_2$
D) under no conditions

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