Make Two Apple Piles Equal With Doubling Moves

You have two piles of apples: the first pile contains a apples and the second pile contains b apples.

In one move you choose one pile to be the receiving pile and move apples into it from the other pile. The number of apples moved must be exactly the number of apples the receiving pile currently contains, so the receiving pile doubles in size. A move is legal only if the giving pile holds at least that many apples.

Determine whether there is a finite sequence of moves after which both piles contain the same number of apples. Return true if it is possible (including when the piles are already equal), and false otherwise.

Function signature

canBalance(a, b) -> boolean

Input

• a — the number of apples in the first pile
• b — the number of apples in the second pile

Output

• true if the two piles can be made equal through some sequence of legal moves, false otherwise.

Constraints

• 1 <= a, b <= 10^9
• The answer must be produced efficiently for values up to the full constraint range.

Examples

1. a = 3, b = 5 → true Move 3 apples from the second pile onto the first: (3, 5) → (6, 2). Then move 2 apples from the first pile onto the second: (6, 2) → (4, 4). The piles are equal.

2. a = 1, b = 2 → false The reachable states cycle: (1, 2) → (2, 1) → (1, 2) → ... — the piles never become equal.

3. a = 2, b = 4 → false (2, 4) → (4, 2) → (2, 4) → ... — the states repeat and equality is never reached.

4. a = 7, b = 7 → true The piles are already equal; zero moves are needed.