
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 mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with its vertex on the circle, whose beam forms an angle of \(15^\circ\), is centered at an angle of \(37.5^\circ\) away from the center. The mouse will die in the dark. What fraction of the total area of the cage can keep the mouse alive?

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