
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  $\triangle{PQR}$  be a right triangle with  $PQ=90$ ,  $PR=120$ , and  $QR=150$ . Let  $C_{1}$  be the inscribed circle. Construct  $\overline{ST}$  with  $S$  on  $\overline{PR}$  and  $T$  on  $\overline{QR}$ , such that  $\overline{ST}$  is perpendicular to  $\overline{PR}$  and tangent to  $C_{1}$ . Construct  $\overline{UV}$  with  $U$  on  $\overline{PQ}$  and  $V$  on  $\overline{QR}$  such that  $\overline{UV}$  is perpendicular to  $\overline{PQ}$  and tangent to  $C_{1}$ . Let  $C_{2}$  be the inscribed circle of  $\triangle{RST}$  and  $C_{3}$  the inscribed circle of  $\triangle{QUV}$ . The distance between the centers of  $C_{2}$  and  $C_{3}$  can be written as  $\sqrt{10n}$ . What is  $n$ ?

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