
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
Mary and Pat play the following number game. Mary picks an initial integer greater than  $2017$ . She then multiplies this number by  $2017$  and adds  $2$  to the result. Pat will add  $2019$  to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by  $2017$  and add  $2$  to the result when it is her turn. Pat will always add  $2019$  to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by  $2018$ . Mary wants to prevent Pat from winning the game. 
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

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