
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
At 30 palm trees on different parts of an uninhabited island, a sign is attached.

- On 15 of them it says: "Exactly under 15 signs a treasure is buried."
- On 8 of them it says: "Exactly under 8 signs a treasure is buried."
- On 4 of them it says: "Exactly under 4 signs a treasure is buried."
- On 3 of them it says: "Exactly under 3 signs a treasure is buried."

It is known that only those signs under which there is no treasure are truthful.

Under the smallest number of signs can a treasure be buried?

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