
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
Two geometric sequences ${a_n}$ and ${b_n}$, neither of which has a common ratio of 1, have the product of their first $n$ terms denoted as $A_n$ and $B_n$, respectively. If $\frac{a_5}{b_5}=2$, find the value of $\frac{A_9}{B_9}$.

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