AUSTRIAN-POLISH 1982 Problem 4

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$ . For any positive integer $x_1$ , define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$ . Prove or disprove that the sequence $(x_n)$ is necessarily bounded.