
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
Let  $S$  be the set of  $10$ -tuples of non-negative integers that have sum  $2019$ . For any tuple in  $S$ , if one of the numbers in the tuple is  $\geq 9$ , then we can subtract  $9$  from it, and add  $1$  to the remaining numbers in the tuple. Call thus one operation. If for  $A,B\in S$  we can get from  $A$  to  $B$  in finitely many operations, then denote  $A\rightarrow B$ .

(1) Find the smallest integer  $k$ , such that if the minimum number in  $A,B\in S$  respectively are both  $\geq k$ , then  $A\rightarrow B$  implies  $B\rightarrow A$ .

(2) For the  $k$  obtained in (1), how many tuples can we pick from  $S$ , such that any two of these tuples  $A,B$  that are distinct,  $A\not\rightarrow B$ .

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