
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
From vertex \( A \) of the square \( ABCD \) with side length 1, two rays are drawn that intersect the square such that vertex \( C \) lies between the rays. The angle between the rays is \( \beta \). From vertices \( B \) and \( D \), perpendiculars are drawn to the rays. Find the area of the quadrilateral whose vertices are at the bases of these perpendiculars.

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