
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
Consider two geometric series with the same first term $a$ but different common ratios, $r$ and $s$, where $|r| < 1$ and $|s| < 1$. The first series is $a, ar, ar^2, ar^3, \ldots$ and the second is $a, as, as^2, as^3, \ldots$. Calculate the limit of the sum of the product of the corresponding terms from both series.
A) $\frac{a^2}{1-rs}$
B) $\frac{a^2}{1-r^2s^2}$
C) $\frac{a^2}{(1-r)(1-s)}$
D) $\frac{2a^2}{1-rs}$
E) None of these

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