
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 square  $ABCD$  is divided into  $(n - 1)^2$  congruent squares, with sides parallel to the sides of the given square. Consider the grid of all  $n^2$  corners obtained in this manner. Determine all integers  $n$  for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly  $n$  points of the grid.

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