
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 a narrow alley of width $w$, a painter leans a ladder of length $b$ against one wall such that it touches the wall at point $X$, a distance $u$ above the ground, making a $60^\circ$ angle with the ground. The same ladder is then leaned against the opposite wall, reaching point $Y$, a distance $v$ above the ground, where it makes a $30^\circ$ angle. What is the width $w$ of the alley expressed in terms of $b$, $u$, and $v$?
A) $b \times \sqrt{3}$
B) $\frac{b}{2}$
C) $\frac{b(1 + \sqrt{3})}{2}$
D) $b$
E) $\frac{b(3 + \sqrt{3})}{2}$

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