C L.'s periodic sequences

The sequence $x_0, x_1, x_2, \ldots$ is defined by the recurrence relation

$x_{n+1} = a x_n + b x_{n-1} + c x_{n-2} + d x_{n-3}, n\ge 3.$

For fixed integers $a, b, c, d$, it turns out that regardless of the initial values $x_0, x_1, x_2, x_3$, the sequence is eventually periodic. Find the sum of all possible periods.

This problem is posed by C L.

Details and assumptions

As an explicit example, $3$ is a possible period since the recurrence relation $x_{n+1} = x_{n-2}, n\ge 3$ is eventually periodic regardless of the starting values $x_0, x_1, x_2, x_3$.