
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 witchcraft and wizardry, there are 13 students. Before the divination exam, the teacher seated them around a round table and asked them to guess who would receive a diploma in divination. Each student modestly refrained from naming themselves and their two neighbors, but wrote about everyone else: "None of these ten will receive it!" Of course, all those who passed the exam guessed correctly, and all the other students were wrong. How many wizards received the diploma?

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