
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
A number \( n \) is called a "good number" if there exists a permutation \( a_1, a_2, \ldots, a_n \) of the numbers \( 1, 2, \ldots, n \) such that \( k + a_k \) (for \( k = 1, 2, \ldots, n \)) are all perfect squares. Determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers" and which are not. Provide reasons for your answer.

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