
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."

Question: In the set \(\{11, 13, 15, 17, 19\}\), which numbers are "good numbers" and which are not? Explain the reason.

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