HongKong 2022-23 IMO Selection Tests

A special calculator contains a red button, which counts the number of even digits of an integer. For instance, when the screen shows $2022$, pressing the red button gives $4$ since all $4$ digits of $2022$ are even. Someone inputs a positive integer $n$ into the calculator and keeps pressing the red button until $0$ is displayed on the screen. If $0$ is displayed after the red button has been pressed four times, find the smallest possible value of $n$.