
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
Given a unit square $PQRS$ and an equilateral triangle $\triangle{PQT}$ whose vertex $T$ lies within the square. Determine the area of the region $Z$, which includes points inside $PQRS$ but outside $\triangle{PQT}$. Additionally, restrict $Z$ to those points whose distance from side $PQ$ is between $\frac{1}{4}$ and $\frac{1}{2}$.
A) $\frac{4 - \sqrt{3}}{18}$
B) $\frac{4 - \sqrt{3}}{16}$
C) $\frac{3 - \sqrt{3}}{9}$
D) $\frac{\sqrt{3}}{12}$

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