
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 $\triangle ABC$ be a right triangle with $AB = 72$, $BC = 96$, and $AC = 120$. Let $C_{1}$ be the inscribed circle of $\triangle ABC$. Construct $\overline{DE}$ with $D$ on $\overline{AB}$ and $E$ on $\overline{BC}$, such that $\overline{DE}$ is perpendicular to $\overline{AB}$ and tangent to $C_{1}$. Construct $\overline{FG}$ with $F$ on $\overline{AB}$ and $G$ on $\overline{AC}$ such that $\overline{FG}$ is perpendicular to $\overline{AB}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle BDE$ and $C_{3}$ the inscribed circle of $\triangle AFG$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{5n}$. What is $n$?

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