
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
The tower in King Arthur's castle is topped with a roof in the shape of a triangular pyramid, where all the planar angles at the apex are right angles. The three slopes of the roof are painted in different colors. The red slope of the roof is inclined to the horizontal at an angle of $\alpha$, and the blue slope at an angle of $\beta$. Find the probability that a raindrop, falling vertically onto the roof at a random location, lands on the green slope.

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