
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
Given three points \( A, B, C \) forming a triangle with angles \( 30^{\circ}, 45^{\circ}, 105^{\circ} \). Two of these points are selected, and the perpendicular bisector of the segment connecting them is drawn, after which the third point is reflected relative to this perpendicular bisector, resulting in a fourth point \( D \). This procedure continues with the set of the four resulting points: two points are chosen, a perpendicular bisector is drawn, and all points are reflected relative to it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure?

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