Sonic And Circles

Sonic has circles ${A_0, A_1, A_2, ... , A_N}$. First circle $A_0$ has centre at origin and has radius equal to $1$. Subsequent circles touch the previous circle and both halves of the hyperbola ${x ^ 2 - y ^ 2 = 1}$ and always lie in the quadrants $1$ and $2$. Sonic is interested to find the radius of the $Nth$ circle modulo $10 ^ 9 + 7$. Help him find the same as he is not so good with geometry.