
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 $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation
\[ (3+i)^n = p_n + q_n i\] for all integers $n \geq 0$, where $i = \sqrt{-1}$. Determine the value of
\[ \sum_{n=0}^\infty \frac{p_n q_n}{9^n}.\]
A) $\frac{5}{8}$
B) $\frac{7}{8}$
C) $\frac{3}{4}$
D) $\frac{15}{16}$
E) $\frac{9}{20}$

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