
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 \(AB = 80\), \(AC = 150\), and \(BC = 170\). Let \(C_{1}\) be the inscribed circle. Construct \(\overline{DE}\) with \(D\) on \(\overline{AC}\) and \(E\) on \(\overline{BC}\), such that \(\overline{DE}\) is perpendicular to \(\overline{AC}\) and tangent to \(C_{1}\). Construct \(\overline{FG}\) with \(F\) on \(\overline{AB}\) and \(G\) on \(\overline{BC}\) such that \(\overline{FG}\) is perpendicular to \(\overline{AB}\) and tangent to \(C_{1}\). Let \(C_{2}\) be the inscribed circle of \(\triangle DEC\) and \(C_{3}\) the inscribed circle of \(\triangle BFG\). The distance between the centers of \(C_{2}\) and \(C_{3}\) can be written as \(\sqrt {10m}\). What is \(m\)?

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