
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 the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at \((0,0)\) and the line segment from \((-1,0)\) to \((1,0)\). David stands at the point \((-1,0)\) and shines a flashlight into the room at an angle of \(46^{\circ}\) above the horizontal. How many times does the light beam reflect off the walls before coming back to David at \((-1,0)\) for the first time?

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