
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
Given $ABCD$ be a square with side length 2, define $Q_1$ as the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$ (diagonal of the square), and let $Q_{i+1}$ be the midpoint between $P_i$ and $D$ on $\overline{CD}$. Calculate the sum of the areas of the infinite number of triangles $DQ_i P_i$.

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