
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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