
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 $401$, $403$, and $405$. For each integer $n \geq 1$, define the sequence of triangles $T_n$ by letting points $D, E$, and $F$ be the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$ respectively, and finding the next triangle $T_{n+1}$ with side lengths $AD, BE$, and $CF$. Find the perimeter of the last triangle in the sequence $\left(T_n\right)$ before it ceases to exist.

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