{"problem":{"name":"Even XOR","description":{"content":"How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$. *   Every non-empty subset $T$ of $S$ sa","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc146_c"},"statements":[{"statement_type":"Markdown","content":"How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$.\n\n*   Every non-empty subset $T$ of $S$ satisfies at least one of the following:\n    *   The number of elements in $T$ is odd.\n    *   The $\\mathrm{XOR}$ of the elements in $T$ is not zero.\n\nWhat is $\\mathrm{XOR}$?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 the digits in that place 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 $k$ non-negative 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*   $1 \\le N \\le 2 \\times 10^5$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc146_c","tags":[],"sample_group":[["2","15\n\nSets such as $\\lbrace 0,2,3 \\rbrace$, $\\lbrace 1 \\rbrace$, and $\\lbrace \\rbrace$ satisfy the condition.\nOn the other hand, $\\lbrace 0,1,2,3 \\rbrace$ does not.\nThis is because the subset $\\lbrace 0,1,2,3 \\rbrace$ of $\\lbrace 0,1,2,3 \\rbrace$ has an even number of elements, whose bitwise $\\mathrm{XOR}$ is $0$."],["146","743874490"]],"created_at":"2026-03-03 11:01:14"}}