
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 $(c_n)$ and $(d_n)$ be sequences of real numbers such that
\[ (3 + i)^n = c_n + d_ni \] for all integers $n \geq 0$, where $i = \sqrt{-1}$. What is
\[ \sum_{n=0}^\infty \frac{c_n d_n}{7^n} \, ? \]
A) $\frac{1}{8}$
B) $\frac{1}{10}$
C) $\frac{1}{12}$
D) $\frac{1}{14}$
E) $\frac{1}{16}$

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