
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
Find the smallest positive integer  $m$  satisfying the following proposition: 

There are  $999$  grids on a circle. Fill a real number in each grid, such that for any grid  $A$  and any positive integer  $k\leq m$ , at least one of the following two propositions will be true:  $\bullet$  The difference between the numbers in the grid  $A$  and the  $k$ -th grid after  $A$  in clockwise direction is  $k$ ; $\bullet$  The difference between the numbers in the grid  $A$  and the  $k$ -th grid after  $A$  in anticlockwise direction is  $k$ .

Then, there must exist a grid  $S$  with the real number  $x$  in it on the circle, such that at least one of the following two propositions will be true:  $\bullet$  For any positive integer  $k<999$ , the number in the  $k$ -th grid after  $S$  in clockwise direction is  $x+k$ ;  $\bullet$  For any positive integer  $k<999$ , the number in the  $k$ -th grid after  $S$  in anticlockwise direction is  $x+k$ .

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