
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 rectangular array of points with 4 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N$, the second row is numbered $N + 1$ through $2N$, the third row is $2N + 1$ to $3N$, and the fourth row is $3N + 1$ to $4N$. 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 $P_i$. Now renumber the array 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 $x_1 = y_2$, $x_2 = y_4$, $x_3 = y_3$, and $x_4 = y_1$. Find the smallest possible value of $N$.

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