$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the sum of all terms on the line.

Source: Argentina Math Olympiad

Source: Argentina Math Olympiad

Number Theory

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