
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 $\triangle{XYZ}$ be a right triangle with $XY = 80$, $XZ = 150$, and $YZ = 170$. Let $C_{1}$ be the inscribed circle. Construct $\overline{MN}$ with $M$ on $\overline{XZ}$ and $N$ on $\overline{YZ}$ such that $\overline{MN}$ is perpendicular to $\overline{XZ}$ and tangent to $C_{1}$. Construct $\overline{AB}$ with $A$ on $\overline{XY}$ and $B$ on $\overline{YZ}$ such that $\overline{AB}$ is perpendicular to $\overline{XY}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{MZN}$ and $C_{3}$ the inscribed circle of $\triangle{YAB}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{m\cdot p}$. What is $p$?

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