GRiddles Series B: Puzzle 4

Code mode

Let ℓ be a non-negative integer. Ana and Bob play a game. To start, Bob publicly chooses a sequence (a_n)_n=1^∞    containing every positive integer exactly once, and sets his starting passcode to 0. On each turn n = 1, 2, 3, . . . , the following happens in order:

1. Bob updates his passcode by adding a_n to it or subtracting a_n from it. The new passcode must be a positive integer not used as a passcode before. If no valid move is possible, Ana wins.
2. If n > 2026^20262026, Ana gets up to ℓ guesses for Bob’s passcode this turn. She cannot repeat a guess across the entire game. If a guess is correct, she wins.

If the game goes on forever, Bob wins if every positive integer eventually appears as his passcode. Otherwise, Ana wins. Find the smallest value of ℓ for which Ana has a winning strategy.