Let random variables \( \{ X_r : r \geq 1 \} \) be independent and uniformly distributed on \( [0, 1] \). Let \( 0 < x < 1 \) and define a random variable
\[
N = \min \{ n \geq 1 : X_1 + X_2 + \cdots + X_n > x \}.

\]

Find the mean and variance of \( N \).

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