
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
If we add  $1996$  to  $1997$ , we first add the unit digits  $6$  and  $7$ . Obtaining  $13$ , we write down  $3$  and “carry”  $1$  to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.

Does there exist a positive integer  $k$  such that adding  $1996\cdot k$  to  $1997\cdot k$  no carry arises during the whole calculation?

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