{"problem":{"name":"Unfair Nim","description":{"content":"There are $N$ piles of stones. The $i$\\-th pile has $A_i$ stones. Aoki and Takahashi are about to use them to play the following game: *   Starting with Aoki, the two players alternately do the follo","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc172_f"},"statements":[{"statement_type":"Markdown","content":"There are $N$ piles of stones. The $i$\\-th pile has $A_i$ stones.\nAoki and Takahashi are about to use them to play the following game:\n\n*   Starting with Aoki, the two players alternately do the following operation:\n    *   Operation: Choose one pile of stones, and remove one or more stones from it.\n*   When a player is unable to do the operation, he loses, and the other player wins.\n\nWhen the two players play optimally, there are two possibilities in this game: the player who moves first always wins, or the player who moves second always wins, only depending on the initial number of stones in each pile.\nIn such a situation, Takahashi, the second player to act, is trying to guarantee his win by moving at least zero and at most $(A_1 - 1)$ stones from the $1$\\-st pile to the $2$\\-nd pile before the game begins.\nIf this is possible, print the minimum number of stones to move to guarantee his victory; otherwise, print `-1` instead.\n\n## Constraints\n\n*   $2 \\leq N \\leq 300$\n*   $1 \\leq A_i \\leq 10^{12}$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$A_1$ $\\ldots$ $A_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc172_f","tags":[],"sample_group":[["2\n5 3","1\n\nWithout moving stones, if Aoki first removes $2$ stones from the $1$\\-st pile, Takahashi cannot win in any way.\nIf Takahashi moves $1$ stone from the $1$\\-st pile to the $2$\\-nd before the game begins so that both piles have $4$ stones, Takahashi can always win by properly choosing his actions."],["2\n3 5","\\-1\n\nIt is not allowed to move stones from the $2$\\-nd pile to the $1$\\-st."],["3\n1 1 2","\\-1\n\nIt is not allowed to move all stones from the $1$\\-st pile."],["8\n10 9 8 7 6 5 4 3","3"],["3\n4294967297 8589934593 12884901890","1\n\nWatch out for overflows."]],"created_at":"2026-03-03 11:01:14"}}