
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 unit square, let $Q_1$ be the midpoint of $\overline{BC}$. 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 $\overline{BC}$. Calculate the sum of the areas of the infinite number of triangles $BQ_iP_i$.

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