
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 $T_1$ be a triangle with side lengths $1001, 1002,$ and $1003$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. Additionally, for each triangle in the sequence, the radius of the circumcircle decreases by a factor of $2$. What is the perimeter of the last triangle in the sequence $( T_n )$, where the circumcircle can still be defined?
A) $\frac{1503}{128}$
B) $\frac{1503}{256}$
C) $\frac{1503}{512}$
D) $\frac{1503}{64}$
E) $\frac{1503}{32}$

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