Touch-screen Password (noch nicht übersetzt)

A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the user selects from a rectangular grid of spots on the screen. The user enters their sequence by touching the first spot, then tracing a straight line segment to the next spot, and so on until the end of the sequence. The user's finger remains in contact with the screen throughout, and may only move in straight line segments from spot to spot.

Every spot that the finger passes over will be part of the sequence provided it has not already been included. For example, on a $3\times 3$ grid labelled with digits $1$ to $9$ (shown below), tracing $1-9-3-7$ (which crosses the $5$ spot twice) will give the password $1-5-9-6-3-7$.

There are $389488$ different passwords that can be formed on a $3 \times 3$ grid.

Find the number of different passwords that can be formed on a $4 \times 4$ grid.