
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]


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Two identical gears \(A\) and \(B\) are placed flat on a horizontal surface, with \(A\) placed on top of \(B\) and both fully overlapping. This makes their projections on the horizontal surface completely coincide. Then, four pairs of overlapping teeth are arbitrarily removed. If \(A\) and \(B\) originally each have 14 teeth, can gear \(A\) be rotated around the common axis to a suitable position so that the projections of the two gears form a complete gear projection on the horizontal surface? What if \(A\) and \(B\) each originally have 13 teeth? Please provide a proof for your conclusion.

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