
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, and provide a justification for your answer!

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