
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 $A_{1} A_{2} A_{3}$ be a triangle. Construct the following points:

- $B_{1}, B_{2}$, and $B_{3}$ are the midpoints of $A_{1} A_{2}, A_{2} A_{3}$, and $A_{3} A_{1}$, respectively.
- $C_{1}, C_{2}$, and $C_{3}$ are the midpoints of $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B_{3}$, respectively.
- $D_{1}$ is the intersection of $\left(A_{1} C_{2}\right)$ and $\left(B_{1} A_{3}\right)$. Similarly, define $D_{2}$ and $D_{3}$ cyclically.
- $E_{1}$ is the intersection of $\left(A_{1} B_{2}\right)$ and $\left(C_{1} A_{3}\right)$. Similarly, define $E_{2}$ and $E_{3}$ cyclically.

Calculate the ratio of the area of $\mathrm{D}_{1} \mathrm{D}_{2} \mathrm{D}_{3}$ to the area of $\mathrm{E}_{1} \mathrm{E}_{2} \mathrm{E}_{3}$.

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