The King of Zugzwangtria is dead in a war. Since the king had no successors, his existing marvellous brothers Dratox and Keraze are claiming for the throne. Dratox being the elder has the right on the throne but his younger brother Keraze refuses and threatens to war.

The wise ministress Freixola suggests both the players to play a game, whoever wins would get the throne. Both agree to this! The game is as follows:

First Dratox picks up a natural number from $1$ to $N$, call it ‘$a$’.

Then Keraze choses a real number ‘$b$’ of the form $\frac{a^{1+\sqrt{i}}}{i^{\sqrt{a}}}$ (where $i$ is some integer from $1$ to $N$)

Then Freixola tries to pass the golden stone through the magical ring. If the stone can possibly pass through the ring then Dratox wins, else Keraze wins.

The golden stone is a rigid regular tetrahedron of side length ‘$a$’ and the magical ring is a rigid circle of radius ‘$b$’ (negligible width).

Let $X$ be the total number of $a$’s for which Dratox has a guaranteed win.

Enter ${X^{X}~mod~{10^{9}+7},~N=10^{123456789}}$