
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
In $\triangle XYZ$, $\angle XYZ = 30^\circ$, $XY = 12$, and $XZ = 8$. Points $P$ and $Q$ are located on $\overline{XY}$ and $\overline{XZ}$ respectively. What is the minimum possible value of $YP + PQ + QZ$?
A) $\sqrt{200 + 80\sqrt{3}}$
B) $\sqrt{208 + 96\sqrt{3}}$
C) $\sqrt{218 + 100\sqrt{3}}$
D) $\sqrt{225 + 105\sqrt{3}}$

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