
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
The number  $2019$  is written on a blackboard.  Every minute, if the number  $a$  is written on the board, Evan erases it and replaces it with a number chosen from the set  $$  \left\{ 0, 1, 2, \ldots, \left\lceil 2.01 a \right\rceil \right\}  $$  uniformly at random. Is there an integer  $N$  such that the board reads  $0$  after  $N$  steps with at least  $99\%$  probability?

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