
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 at $A$ with $AB = 80$, $AC = 150$, and $BC = 170$. Let $C_1$ be the inscribed circle. Construct segments $\overline{DE}$ and $\overline{FG}$ such that $D$ and $F$ are on $\overline{AC}$, $E$ and $G$ are on $\overline{BC}$, $\overline{DE}$ is perpendicular to $\overline{AC}$, and $\overline{FG}$ is perpendicular to $\overline{AB}$, both segments being tangent to $C_1$. Let $C_2$ be the inscribed circle of $\triangle BDE$ and $C_3$ the inscribed circle of $\triangle AFG$. Find the squared distance between the centers of $C_2$ and $C_3$, and show it can be expressed as $\sqrt{10m}$. Determine the value of $m$.

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