
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 $Q$ lie on $\overline{XY}$ and $\overline{XZ}$ respectively. What is the minimum possible value of $YP+PQ+QZ$?
A) $\sqrt{200 + 96\sqrt{3}}$
B) $\sqrt{208 + 96\sqrt{3}}$
C) $\sqrt{212 + 96\sqrt{3}}$
D) $\sqrt{224 + 96\sqrt{3}}$

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