Recurrence relation 2

Let \(x_{1},\ldots,x_{42}\) be real numbers such that \(5x_{i+1} −x_{i} −3x_{i}x_{i+1} = 1\) for each \(1 ≤ i ≤ 42\), with \(x_{1} = x_{43}\). Let the product of all possible values for \(x_{1} + x_{2} + \cdots + x_{42}=M\), Number of possible values of \(x_{1} + x_{2} +\cdots + x_{42}=N\) . Find \(N+M\)

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