Let $H$ be the hyperbola defined by the equation $12x^2+7xy-12y^2=625$.

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 $H$, $\{P_i:i\ge 1\}$, as:

• $P_1=(13,61/4)$.
• $P_2=(-43/6,-4)$.
• For $i>2$, $P_i$ is the unique point in $H$ that is different from $P_{i-1}$ and such that line $P_i{P}_{i-1}$ is parallel to line $P_{i-2}X$. It can be shown that $P_i$ is well-defined, and that its coordinates are always rational.

You are given that $P_3=(-19/2,-229/24)$, $P_4=(1267/144,-37/12)$ and $P_7=(17194218091/143327232,274748766781/1719926784)$.

Find $P_n$ for $n={11}^{14}$ 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 $(a+b+c+d)mod1000000007$.

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