
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
Alice and Bob play a game on a Cartesian Coordinate Plane. At the beginning, Alice chooses a lattice point  $ \left(x_{0}, y_{0}\right) $  and places a pudding. Then they plays by turns (B goes first) according to the rules

a. If  $ A $  places a pudding on  $ \left(x,y\right) $  in the last round, then  $ B $  can only place a pudding on one of  $ \left(x+2, y+1\right), \left(x+2, y-1\right), \left(x-2, y+1\right), \left(x-2, y-1\right) $  

b. If  $ B $  places a pudding on  $ \left(x,y\right) $  in the last round, then  $ A $  can only place a pudding on one of  $ \left(x+1, y+2\right), \left(x+1, y-2\right), \left(x-1, y+2\right), \left(x-1, y-2\right) $ Furthermore, if there is already a pudding on  $ \left(a,b\right) $ , then no one can place a pudding on  $ \left(c,d\right) $  where  $ c \equiv a \pmod{n}, d \equiv b \pmod{n} $ .

1. Who has a winning strategy when  $ n = 2018 $ 1. Who has a winning strategy when  $ n = 2019 $ 

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