
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
Peter and Paul play the following game. First, Peter chooses some positive integer  $a$  with the sum of its digits equal to  $2012$ . Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is  $2012$ . On each of his moves Paul chooses a positive integer  $x$  and Peter tells him the sum of the digits of  $|x - a|$ . What is the minimal number of moves in which Paul can determine Peter’s number for sure?

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