
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 $n \geq 1$, 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 as a triangle with side lengths $AD+1$, $BE+1$, and $CF+1$. Determine the perimeter of the last triangle in the sequence $\left(T_n\right)$ before triangle inequalities no longer hold.
A) 3015
B) 3018
C) 3021
D) 3024
E) 3027

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