
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
We say that a number  $n \ge 2$  has the property  $(P)$  if, in its prime factorization, at least one of the factors has an exponent   $3$ .

a) Determine the smallest number  $N$  with the property that, no matter how we choose  $N$  consecutive natural numbers, at least one of them has the property  $(P).$ b) Determine the smallest  $15$  consecutive numbers  $a_1, a_2, \ldots, a_{15}$  that do not have the property  $(P),$  such that the sum of the numbers  $5 a_1, 5 a_2, \ldots, 5 a_{15}$  is a number with the property  $(P).$ 

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