
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, now having 6 rows and $N$ columns, points are numbered in a similar pattern as originally. This includes consecutive numbering from left to right, beginning with the top row. Each point $P_i$ is selected from row $i$ and is renumbered after switching the numbering to a top-to-bottom pattern, beginning with the first column. If after this renumbering it is found that $x_1=y_2$, $x_2=y_1$, $x_3=y_4$, $x_4=y_5$, $x_5=y_6$, and $x_6=y_3$, find the smallest possible value of $N$.

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