
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
Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that  $k$  is chosen with probability  $2^{-k}$  for every positive integer  $k$ , and then it outputs  $k$  tokens. Let  $N$  be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least  $N$  total tokens from the slot machine before Drew receives at least  $M=2^{2018}$  total tokens, and Drew wins if he receives  $M$  tokens before Ann receives  $N$  tokens. If each person has the same probability of winning, compute the remainder when  $N$  is divided by  $2018$ .

*Proposed by Brandon Wang*

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