What is the sum of all \(x\cdot y\) where \((x,y)\) is an integer solution to \[xy-2x+y=7?\]

Suppose \(x,y\) are real numbers such that \(x+y=20\). What is the maximum possible value of \(x\cdot y\)?

Define a sequence \(S_n\) as follows: Given a starting value \(S_0\), for all \(n\geq 1\), define \(S_n\) as \(S_{n-1}^2\) modulo 10 taken to be a number from 0 to 9 inclusive. For example, if the starting value \(S_0=2\), then our sequence goes \(2,4,6,6,6,\ldots\).

Suppose a random integer is picked as the starting value \(S_0\). What is the probability that \(S_{2017}\) is not divisible by 3?

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