
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
A regular dodecagon \( P_1, P_2, \ldots, P_{12} \) is inscribed in a unit circle with center \( O \). Let \( X \) be the intersection of \( P_1 P_5 \) and \( OP_2 \), and let \( Y \) be the intersection of \( P_1 P_5 \) and \( OP_4 \). Let \( A \) be the area of the region bounded by \( XY \), \( XP_2 \), \( YP_4 \), and the minor arc \( \widehat{P_2 P_4} \). Compute \( \lfloor 120A \rfloor \).

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