
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]


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In an alley of width \( w \), a ladder of length \( l \) is placed with its foot at point \( P \) between two walls. It rests against one wall at point \( Q \), a distance \( m \) above the ground making a \( 60^\circ \) angle with the ground. It also rests against the opposite wall at \( S \), a distance \( n \) above the ground, making a \( 30^\circ \) angle with the ground. Determine the width \( w \) of the alley.
A) \( \frac{l(1 + \sqrt{3})}{2} \)
B) \( l \)
C) \( \frac{l}{2} \)
D) \( l\sqrt{3} \)
E) \( \frac{l}{\sqrt{3}} \)

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