
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 unit square. Let $R_1$ be the point dividing $\overline{CD}$ in the ratio 1:3 (from $C$ to $D$). For $i=1,2,\dots,$ let $S_i$ be the intersection of $\overline{AR_i}$ and $\overline{BD}$, and let $R_{i+1}$ be the point on $\overline{CD}$ such that $\overline{R_iR_{i+1}}$ is a third the length of $\overline{R_iD}$. Find $\sum_{i=1}^{\infty} \text{Area of } \triangle D R_i S_i \,?$.

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