
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
 $101$  people,  sitting at a round table in any order,  had   $1,2,... , 101$  cards,  respectively.  
A transfer is someone give  one  card  to one of  the two people  adjacent to him.
Find the  smallest positive integer  $k$  such that   there always  can  through  no more than   $ k $   times transfer,  each person  hold cards of the same number, regardless of  the sitting order.

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