
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 6 rows and $M$ columns, points are numbered consecutively from left to right beginning with the top row. Thus, the top row is numbered 1 through $M$, the second row is numbered $M + 1$ through $2M$, and so forth. Six points, $P_1, P_2, P_3, P_4, P_5, P_6$, 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 $x_1 = y_2$, $x_2 = y_1$, $x_3 = y_4$, $x_4 = y_3$, $x_5 = y_6$, and $x_6 = y_5$. Find the smallest possible value of $M$.

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