
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 defined by:
\[ (3 + i)^n = c_n + d_ni \]
for all integers \(n \geq 0\), where \(i = \sqrt{-1}\). Calculate:
\[ \sum_{n=0}^\infty \frac{c_n d_n}{9^n} \]
A) \(\frac{1}{3}\)
B) \(\frac{6}{11}\)
C) \(\frac{5}{9}\)
D) \(\frac{8}{17}\)
E) \(\frac{11}{15}\)

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