
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 ladder of length $a$ is placed such that it touches both walls. The base of the ladder is at point $P$ placed between the walls. Resting against one wall at point $Q$, the distance $k$ above the ground makes a $60^\circ$ angle with the ground. Resting against the opposite wall at $R$, the ladder makes a $45^\circ$ angle with the ground at a height $h$ above the ground. You need to determine the width $w$ of the alley.
A) $\frac{\sqrt{3}a}{2}$   
B) $\left(\frac{\sqrt{3} - \sqrt{2}}{2}\right)a$   
C) $\frac{\sqrt{2}a}{2}$   
D) $\frac{a}{2}$   
E) $\frac{\sqrt{3} + \sqrt{2}}{2}a$

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