--description--

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 $H, \{P_i : i ≥ 1\}$, as:

• $P_1 = (13, \frac{61}{4})$.
• $P_2 = (\frac{-43}{6}, -4)$.
• 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 $P_{i - 2}X$. It can be shown that $P_i$ is well-defined, and that its coordinates are always rational.

You are given that $P_3 = (\frac{-19}{2}, \frac{-229}{24})$, $P_4 = (\frac{1267}{144}, \frac{-37}{12})$ and $P_7 = (\frac{17\,194\,218\,091}{143\,327\,232}, \frac{274\,748\,766\,781}{1\,719\,926\,784})$.

Find $P_n$ for $n = {11}^{14}$ in the following format: If $P_n = (\frac{a}{b}, \frac{c}{d})$ where the fractions are in lowest terms and the denominators are positive, then the answer is $(a + b + c + d)\bmod 1\,000\,000\,007$.

For $n = 7$, the answer would have been: $806\,236\,837$.

--hints--

sequenceOfPointsOnHyperbola() should return 92060460.

assert.strictEqual(sequenceOfPointsOnHyperbola(), 92060460);

--seed--

--seed-contents--

function sequenceOfPointsOnHyperbola() {

  return true;
}

sequenceOfPointsOnHyperbola();

--solutions--

// solution required