
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 $ABCD$ be a rectangle where $AD = 2$ and $AB = 1$. Define $Q_1$ as the point dividing $\overline{CD}$ in the ratio 1:2 (from $C$ to $D$). For each $i=1,2,\dots,$ define $P_i$ as the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and $Q_{i+1}$ as the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is 
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_iP_i \, ?\]
A) $\frac{\sqrt{5}}{15}$
B) $\frac{2\sqrt{5}}{15}$
C) $\frac{\sqrt{5}}{10}$
D) $\frac{3\sqrt{5}}{20}$
E) $\frac{2\sqrt{5}}{10}$

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