
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
We say a positive integer  $ n$  is *good* if there exists a permutation  $ a_1, a_2, \ldots, a_n$  of  $ 1, 2, \ldots, n$  such that  $ k \plus{} a_k$  is perfect square for all  $ 1\le k\le n$ . Determine all the good numbers in the set  $ \{11, 13, 15, 17, 19\}$ .

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