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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, {Pi : i ≥ 1}, as:
 P1 = (13, 61/4).
 P2 = (-43/6, -4).
 For i &gt; 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi 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 Pn for n = 1114 in the following format:If Pn = (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.

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