
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
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled  $0, 1,\ldots, 2017$  are marked in a circumference and each couple is placed on a different point. For  $i\geq1$ , let  $s_i=i\ (\textrm{mod}\ 2018)$  and  $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute  $0$ . On the  $i$ -th minute, the couple at point  $s_i$  (if there's any) moves to point  $r_i$ , the couple on point  $r_i$  (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after  $2018^2$  minutes. Determine how many couples remain at the end. 

Note: If  $r_i=s_i$ , the couple on  $s_i$  stays there and does not drop out.

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