{"problem":{"name":"Range XOR","description":{"content":"Given are integers $L$, $R$, and $V$. Find the number of pairs of integers $(l,r)$ that satisfy both of the conditions below, modulo $998244353$. *   $L \\leq l \\leq r \\leq R$ *   $l \\oplus (l+1) \\opl","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc133_d"},"statements":[{"statement_type":"Markdown","content":"Given are integers $L$, $R$, and $V$. Find the number of pairs of integers $(l,r)$ that satisfy both of the conditions below, modulo $998244353$.\n\n*   $L \\leq l \\leq r \\leq R$\n*   $l \\oplus (l+1) \\oplus \\cdots \\oplus r=V$\n\nHere, $\\oplus$ denotes the bitwise $\\mathrm{XOR}$ operation.\nWhat is bitwise $\\mathrm{XOR}$?The bitwise $\\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \\oplus B$, is defined as follows:\n\n*   When $A \\oplus B$ is written in base two, the digit in the $2^k$'s place ($k \\geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.\n\nFor example, we have $3 \\oplus 5 = 6$ (in base two: $011 \\oplus 101 = 110$).  \nGenerally, the bitwise $\\mathrm{XOR}$ of $k$ integers $p_1, p_2, p_3, \\dots, p_k$ is defined as $(\\dots ((p_1 \\oplus p_2) \\oplus p_3) \\oplus \\dots \\oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \\dots p_k$.\n\n## Constraints\n\n*   $1 \\leq L \\leq R \\leq 10^{18}$\n*   $0 \\leq V \\leq 10^{18}$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$L$ $R$ $V$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc133_d","tags":[],"sample_group":[["1 3 3","2\n\nThe two pairs satisfying the conditions are $(l,r)=(1,2)$ and $(l,r)=(3,3)$."],["10 20 0","6"],["1 1 1","1"],["12345 56789 34567","16950"]],"created_at":"2026-03-03 11:01:14"}}