
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 100, 101, and 102. Define a sequence such that $T_{n+1}$ is formed from $T_n = \Delta ABC$ where $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ with sides $AB$, $BC$, and $AC$. $T_{n+1}$ is defined using the segment lengths $AD$, $BE$, and $CF$. Determine the perimeter of the last triangle in this sequence $\left(T_n\right)$ where the triangle inequality holds.
- $\textbf{A)}\ 150.5$  
- $\textbf{B)}\ 151.5$  
- $\textbf{C)}\ 152$  
- $\textbf{D)}\ 152.5$  
- $\textbf{E)}\ 153$

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