--description--

Let $C(x,y)$ be a circle passing through the points ($x$, $y$), ($x$, $y + 1$), ($x + 1$, $y$) and ($x + 1$, $y + 1$).

For positive integers $m$ and $n$, let $E(m,n)$ be a configuration which consists of the $m·n$ circles: { $C(x,y)$: $0 ≤ x < m$, $0 ≤ y < n$, $x$ and $y$ are integers }

An Eulerian cycle on $E(m,n)$ is a closed path that passes through each arc exactly once. Many such paths are possible on $E(m,n)$, but we are only interested in those which are not self-crossing: A non-crossing path just touches itself at lattice points, but it never crosses itself.

The image below shows $E(3,3)$ and an example of an Eulerian non-crossing path.

Let $L(m,n)$ be the number of Eulerian non-crossing paths on $E(m,n)$. For example, $L(1,2) = 2$, $L(2,2) = 37$ and $L(3,3) = 104290$.

Find $L(6,10)\bmod {10}^{10}$.

--hints--

eulerianCycles() should return 6567944538.

assert.strictEqual(eulerianCycles(), 6567944538);

--seed--

--seed-contents--

function eulerianCycles() {

  return true;
}

eulerianCycles();

--solutions--

// solution required