NZ-NZMO 2022 Problem 5

The sequence $x_{1},x_{2},x_{3},\ldots$ is defined by $x_{1} = 2022$ and $x_{n + 1} = 7x_{n} + 5$ for all positive integers $n$ . Determine the maximum positive integer $m$ such that

$${\frac{x_{n}(x_{n}-1)(x_{n}-2)\ldots(x_{n}-m+1)}{m!}}$$

is never a multiple of 7 for any positive integer $n$ .