You have reached the final stage of a game show, and now face an endless row of doors labelled $\ldots,-3,-2,-1,0,1, 2, 3,\ldots$, going from minus infinity to plus infinity. Your goal is to detect a secret door that has been chosen by the game host (and that is static and will not change throughout the game).
The door with number $0$ is already open and forms the starting point of your search.
In every further step, you first pay 1 Euro to the game host and then open the left or right neighbor of your current door. (In other words: if you open door $n$ in some round, then in the following round you must open door $n-1$ or door $n+1$.)
Once you open the secret door, you may keep the money that is hidden behind it.
Prior to the game and before the game show host chooses the secret door, you have to specify to him the full (deterministic) strategy that you will follow for the entire evening. Only then the host will select his secret door (with number $x\ne0$) and will put an amount of $8.999|x|$ Euros behind it (where the multiplicator is $8.999=9-1/1000$ and hence slightly smaller than $9$).
Is there any deterministic strategy that guarantees you some positive profit in this game?
(Note: This puzzle was inspired by the puzzle One prize, infinitely many choices. In this other puzzle, the secret door moves in the same fashion as the searcher does move in my puzzle.)