
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 three points in the plane, and for convenience, let \( A_4 = A_1 \), \( A_5 = A_2 \). For \( n = 1, 2 \), and \( 3 \), suppose that \( B_n \) is the midpoint of \( A_n A_{n+1} \), and suppose that \( C_n \) is the midpoint of \( A_n B_n \). Suppose that \( A_n C_{n+1} \) and \( B_n A_{n+2} \) meet at \( D_n \), and that \( A_n B_{n+1} \) and \( C_n A_{n+2} \) meet at \( E_n \). Calculate the ratio of the area of triangle \( D_1 D_2 D_3 \) to the area of triangle \( E_1 E_2 E_3 \).

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