
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
In triangle \( T_{0} \), a triangle was formed using its midlines and named \( T_{1} \). In triangle \( T_{1} \), a triangle was formed using its midlines and named \( T_{2} \). Continuing in this manner, triangle \( T_{10} \) was obtained. Find the ratio of the sum of the areas of all these eleven triangles to the area of the last triangle. If the answer is a fractional number, it should be written as a decimal using a dot, for example, \(« 0.15 »\).

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