
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
Consider an angular sector on a plane with its vertex at the origin and an angle of 30 degrees. Place a ball (a material point of zero diameter) on one of the sides of the angle (ray) and launch it inside the sector with a constant speed at an angle of 110 degrees to the considered ray. The ball will fly within the sector, elastically bouncing off its sides, preserving the magnitude of its speed: the angle of incidence is equal to the angle of reflection. How many times will it hit the sides of the sector before flying off to infinity?

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