
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
Bětka was playing with gears arranged as shown in the diagram. When she turned one gear, all the other gears also turned. Eventually, she was satisfied with a gear system where the first gear had 32 teeth and the second gear had 24 teeth. When the third gear made exactly eight rotations, the second gear made five rotations and a part of a sixth, and the first gear made four rotations and a part of a fifth.

Determine how many teeth the third gear had.

(E. Novotná)

Hint: How many times do the teeth of the first gear interlock with the teeth of the second gear when the first gear turns four times?

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