Original proposer: vwxyz

Takahashi always has exactly \(k\) bags.
Therefore, he stops performing the action when:

• he has received \(k\) bags with “miss,” or
• the row has been emptied.

If \(S\) has less than \(k\) occurrences of x, he can receive all bags. If there are \(k\) or more, he can receive up to the \(k\)-th “miss” bag.

Therefore, the answer can be obtained by enumerating \(i\) such that \(S_i\) is x, and comparing the number of such indices \(i\) with \(k\).