
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
Vukasin,  Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements:
Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.( $n$  is perfect if  $\sigma\left(n\right)=2n$ ). The sum of the digits of the third number equals to the number of it's positive divisors".
Dimitrije:"Everyone of those three numbers has at most two digits equal to  $1$  in their decimal representation".
Dusan:"If we add  $11$  to exactly one of them, then we have a perfect square of an integer"
Stefan:"Everyone of them has exactly one prime divisor less than  $10$ ".
Filip:"The three numbers are square free".
Professor found the right answer. Which numbers did he mention?

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