The natural numbers from $1$ up to $300$ are evenly located around a circle. We say that such an ordering is "alternate" if each number is either less than its two neighbors or greater than its two neighbors. We will call a pair of neighboring numbers a "good" pair if, by removing that pair from the circumference, the remaining numbers form an alternate ordering. Determine the least possible number of good pairs in which there can be an alternate ordering of the numbers from $1$ to $300$ inclusive.

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