
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 $L$ is leaning against one wall such that it makes a $30^\circ$ angle with the ground and reaches up to height $k$, and against the opposite wall making a $60^\circ$ angle with the ground and reaching up to height $h$, determine the width $w$ of the alley.

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