
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 square where each side is 2 units. Let \( Q_1 \) be the midpoint of \( \overline{CD} \). For \( i = 1, 2, \ldots \), define \( P_i \) as the intersection of \( \overline{AQ_i} \) and \( \overline{BD} \), and define \( Q_{i+1} \) as the perpendicular projection of \( P_i \) onto \( \overline{CD} \). Determine the sum of areas of all triangles \( \triangle DQ_iP_i \) as \( i \) approaches infinity.
A) \( \frac{1}{36} \)
B) \( \frac{1}{18} \)
C) \( \frac{1}{9} \)
D) \( \frac{1}{6} \)
E) \( \frac{1}{3} \)

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