
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 placed such that:
- Its foot at point $A$ rests between two walls.
- At point $B$ against one wall, touching a height $m$ above the ground, the ladder makes a $60^\circ$ angle with the ground. 
- At point $C$ against the opposite wall, touching a height $n$ above the ground, the ladder makes a $70^\circ$ angle with the ground. Determine the width of the alley $w$ in terms of the given parameters.
A) \( ℓ \sin(70^\circ) \)
B) \( ℓ \left(\frac{1}{2} + \cos(70^\circ)\right) \)
C) \( ℓ \sin(60^\circ) \)
D) \( 2ℓ (\cos(70^\circ) - \cos(60^\circ)) \)

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