
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 $1007$, $1008$, and $1009$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is defined with side lengths $AD + 1$, $BE + 1$, and $CF + 1$, if such a triangle exists. What is the perimeter of the last triangle in the sequence $\left(T_n\right)$?
A) $\frac{1515}{128}$  
B) $\frac{1515}{256}$  
C) $\frac{1515}{512}$  
D) $\frac{1515}{1024}$  
E) $\frac{753}{128}$

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