
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
In Italy, they produce clocks in which the hour hand completes one full rotation per day, and the minute hand completes 24 rotations per day. As usual, the minute hand is longer than the hour hand. Consider all positions of the two hands and the zero marking on the Italian clock that coincide with those on a regular clock. How many such positions exist on the Italian clock within a day? (The zero marking on the Italian clock represents 24 hours, while on a regular clock it represents 12 hours.)

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