
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 positive integer  $n$  is called *good* if for all positive integers  $a$  which can be written as  $a=n^2 \sum_{i=1}^n {x_i}^2$  where  $x_1, x_2, \ldots ,x_n$  are integers, it is possible to express  $a$  as  $a=\sum_{i=1}^n {y_i}^2$  where  $y_1, y_2, \ldots, y_n$  are integers with none of them is divisible by  $n.$  Find all good numbers.

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