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 $N+M$