
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 an alley of width $w$, a ladder of length $l = 10 \text{ meters}$ leans against two opposite walls. At one wall the top part of the ladder reaches a height of $4 \text{ meters}$ and forms a $30^\circ$ angle with the ground. At the opposing wall, the ladder reaches a height of $3 \text{ meters}$ and forms a $120^\circ$ angle with the ground. Calculate the width $w$ of the alley.
A) $10(\sqrt{3} + 1)\text{ meters}$
B) $5\sqrt{3}\text{ meters}$
C) $10(\sqrt{3} - 1)\text{ meters}$
D) $5(\sqrt{3} + 1)\text{ meters}$
E) $15\text{ meters}$

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