
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
The cells of a square  $2011 \times 2011$  array are labelled with the integers  $1,2,\ldots, 2011^2$ , in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer  $M$  such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least  $M$ .
(Cells with coordinates  $(x,y)$  and  $(x',y')$  are considered to be neighbours if  $x=x'$  and  $y-y'\equiv\pm1\pmod{2011}$ , or if  $y=y'$  and  $x-x'\equiv\pm1\pmod{2011}$ .)

*(Romania) Dan Schwarz*

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