
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
If there exists a permutation \(a_1, a_2, \cdots, a_n\) of \(1, 2, \cdots, n\) such that \(k + a_k\) (\(k = 1, 2, \cdots, n\)) are all perfect squares, then \(n\) is called a "good number". Determine which numbers in the set \(\{11, 13, 15, 17, 19\}\) are "good numbers" and which are not, providing justification.

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