
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 with $AB = 2$ and $BC = 1$. Let $Q_1$ be the point on $\overline{CD}$ such that $CQ_1 = \frac{2}{3}$ and $DQ_1 = \frac{1}{3}$. For $i=1,2,\dots$, let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. Calculate the sum of the areas of triangles $\triangle DQ_iP_i$ for all $i \geq 1$.

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