
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
Let us call a natural number \( n \) "squarable" if the numbers from 1 to \( n \) can be arranged in such a way that each member of the sequence, when added to its position number, gives a perfect square. For example, the number 5 is squarable because the numbers can be arranged as 32154, where \( 3+1, 2+2, 1+3, 5+4 \) are all perfect squares. Determine which of the numbers 7, 9, 11, 15 are squarable.

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