A magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his choice. What is the smallest value of $k$ such that the revealed cards can determine the exact positions of all other cards?

Let's think step by step and output the final answer within \boxed{}.
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