
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's call a natural number $n$ squareable if the numbers from 1 to $n$ can be arranged in such a way that each member of the sequence, when added to its index, gives a perfect square. For example, the number 5 is squareable because the numbers can be arranged as 32154, such that $3+1=4$, $2+2=4$, $1+3=4$, and $5+4=9$. Determine which of the numbers $7, 9, 11, 15$ are squareable.

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