
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 $(a_n)$ and $(b_n)$ be sequences of real numbers defined as
\[ (1 + 2i)^n = a_n + b_n i \]
for all integers $0 \leq n \leq 7$, where $i = \sqrt{-1}$. What is
\[ \sum_{n=0}^7 \frac{a_n b_n}{10^n} \, ? \]
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{7}{16}$
D) $\frac{1}{8}$
E) $\frac{1}{4}$

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