John has just registered an account on Brilliant. Each day, starting from today, he logs in, chooses a multiple choice problem, randomly answers it, and logs out. So if he is incorrect on one day, he loses his problem solving streak.

He stops this procedure once he gets a problem incorrect, or once he gets a streak of 10 days. The expected length of John's streak can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime. Find the last \(3\) digits of \(p+q\).

**Note:** The streak can be 0, in case that John gets the problem he answers on the first day incorrect.

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