
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
Mr. Stříbrný organized an exhibition. He displayed 120 rings laid out on tables along the walls of a hall, forming one large circle. The tour began at the entrance in a designated direction. Every third ring in the sequence was gold, every fourth ring was antique, and every tenth ring had a diamond. Any ring that did not have any of these three properties was a fake.

How many gold rings were also antique and had a diamond? How many fakes did Mr. Stříbrný exhibit?

Hint: According to which rules were the rings with various combinations of the three mentioned properties arranged?

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