
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 a school of wizardry, there are 13 students. Before a clairvoyance exam, the teacher seated them around a round table and asked each to guess who would receive the diploma of clairvoyance. Each student humbly did not mention themselves or their two neighbors and wrote: "None of these ten will receive it!" Of course, all those who passed the exam guessed correctly, and all the other students guessed incorrectly. How many wizards received the diploma?

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