
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
There are 10 individuals on a mysterious island, comprising both elves and wizards. Each person has a treasure chest containing a distinct number of gold coins ranging from 1 to 10. When asked about the number of gold coins in their chest, every wizard lies, while every elf tells the truth. If the reported numbers of gold coins are all within the range of 1 to 10 and the total sum of their reported gold coins is 36, then the island must have at least $\qquad$ wizards.

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