
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
In a square array of points with 4 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. The top row is numbered 1 through $N$, the second row is numbered $N + 1$ through $2N$, and so forth. Four points $P_1, P_2, P_3, P_4$ are selected so that each $P_i$ is in row $i$. Let $x_i$ be the number associated with $P_i$. The array is then renumbered consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_3$, $x_2 = y_1$, $x_3 = y_4$, and $x_4 = y_2$. Find the smallest possible value of $N$.

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