Sum for random sequence

A sequence generator start generating independent and identically distributed random variables whose cumulative distribution function is $$F(x)=\{\begin{array}{ll}0& :x<0\\ x^2& :0\le x\le 1\\ 1& :x>1\end{array}$$ The cumulative sum is updated for every new generated number. When the sum is larger than $1$, the generator stops. Let $C$ denotes the count of generated numbers when it stops. For example, the probability for $C=3$ is $7/45$.

Find the expected value of $C$. Give your answer rounded to 10 digits after the decimal point.

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Thanks to baihacker for the idea.