Sequence of points on a hyperbola (noch nicht übersetzt)

Let H be the hyperbola defined by the equation 12x² + 7xy - 12y² = 625.

Next, define X as the point (7, 1). It can be seen that X is in H.

Now we define a sequence of points in H, {P_i : i ≥ 1}, as:

• P₁ = (13, 61/4).
• P₂ = (-43/6, -4).
• For i > 2, P_i is the unique point in H that is different from P_i-1 and such that line P_iP_i-1 is parallel to line P_i-2X. It can be shown that P_i is well-defined, and that its coordinates are always rational.

You are given that P₃ = (-19/2, -229/24), P₄ = (1267/144, -37/12) and P₇ = (17194218091/143327232, 274748766781/1719926784).

Find P_n for n = 11¹⁴ in the following format:
If P_n = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.

For n = 7, the answer would have been: 806236837.