Mailing Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⠄⢀⢀⣀⣀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢆⠀⠀⠀⠚⠰⠭⣙⣄⠀⠈⠁⠁⠀⠐⠀⣀⠀⠀⠀⠀⠉⠀⠀⡤⢶⣔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⡓⠀⠀⠂⠒⠂⠨⠥⠴⠤⠀⠀⢀⣀⡀⠀⠀⡀⠀⠀⠀⠠⣇⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⢀⢀⡤⠄⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠄⠐⠂⠀⢀⣉⡤⣤⡈⢆⢀⠔⠁⢸⡉⠀⠆
⠀⠀⠀⠀⠀⢀⠒⠐⠈⡇⠁⠀⠐⠒⠒⠒⠒⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠍⣵⣀⠇⢸⡇⠀⠀⡆⠈⠑⡆
⠀⠀⠀⠀⠀⠈⠸⠙⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠯⠽⠁⠀⡇⠀⢀⠗⠈⠠
⠀⠀⠀⠀⢀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠶⠒⠘⠋⠁⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠧⠐⡂⠲⠤⠐⠂⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠠⠨⠅⢀⡀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⡽⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠓⠦⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠉⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡿⠛⠒⠒⠒⠚⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⡀⢈⡀⠀⢄⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡣⠊⠨⡂⠘⢅⣩⣪⣤⠮⠍⠒⠂⠈⠉⢉⢀⠀⠠⢂⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⣠⡉⠤⠔⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢀⢂⠀⢀⠆⠌⡀⡠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠐⠘⠁⠉⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⡐⡠⠀⠀⢠⡀
⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣀⠠⠄⠀⠤⢀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠘⣦⠁⡐⠠⢀
⠀⠀⠀⠄⢀⠎⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡀⠈⢀⠀⠄⢀⠄⢂⠊⠢⡀⠙⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢳⠀⢀⡰⡠⢳
⠀⠀⠀⠰⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠊⠀⠀⣀⣀⠀⠀⠀⠀⠈⡀⠀⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠸⡆⣪⠎⠀⠀⡄
⠀⠀⠀⠀⠃⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⡆⠀⠀⠀⠀⠀⠡⠀⠀⡄⠀⢠⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠧⠁⠀⠀⠀⠁
⠀⠀⢀⡴⣩⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⠀⢃⠀⠁⠆⠀⠀⡠⠀⠀⠰⢀⠄⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⡴⠋⢀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠢⡀⠀⠀⠀⠀⡠⠂⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⡐⡨⠊
⠞⠂⢥⣄⢤⢷⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠂⢄⠈⠀⠒⠀⠁⢀⠠⠊⢀⠀⠀⠀⠀⢀⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠙⠺⢆⣄⠀⠌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠁⠀⠀⢁⠀⠢⠂⠀⠀⠀⡀⡀⠈⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠈⠉⠒⠠⢠⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠴⢄⡄⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠋⠐⠀⡠⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡁
⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢤⣀⠀⢀⢀⠀⠁⠀⠰⢀⣸⣊⡠⠀⢀⢌⠔⢁⠄⠐⠠⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠄⣄⠀⠁⢁⠊⠀⠀⠈⠁⠐⠂⠴⠁⠊⠐⠐⠡⢊⠄⠄⠀⠀⠀⠀⠀⢒⠀⠆
⠀⠀⠀⡀⠀⠔⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢠⠀⢊⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀
⠀⠀⠀⠀⡄⠑⡡⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⢀⠕⠐⡡⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠠⢊⠔⡁⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⢀⢌⠄⠔⠐⠀⠁⠁⠀⠀⠀⠀⢀⢀⠀⠀⠖⢀⠖⡠⢋⠔⡡⠊⠀⠁
⠀⠀⠀⠀⠀⢁⠀⡡⢊⠔⠁⢀⠔⠁⡀⠀⠀⠀⠠⠊⢈⠄⠐⡀⠀⠁⠀⠀⠀⠀⠀⢀⡀⢤⠤⢀⠖⡠⢉⠐⢀⠔⡁⠐⠡⠊⠀⠁
⠀⠀⠀⠀⢀⢸⡀⠀⠁⠊⠔⠁⢀⠈⠄⡀⠑⠀⠁⠁⠀⠈⢀⡀⢠⡤⢀⠀⡠⢊⠅⢉⠔⡡⠀⠡⠊⠔⠁
⠀⠀⠀⡀⠐⠈⡁⠐⡡⠂⠀⢁⣀⡀⢤⡄⠠⠆⡐⢂⠝⡩⢊⠔⡡⠔⠡⢊⠔⡡⠂⡠⠊
⠀⠀⠀⠂⢀⠌⡠⢊⠀⡩⢋⠔⠁⠔⡠⢊⠔⢀⠔⡡⢊⠔⡡⠊⠠⠈⠐⠁
⠀⠀⡠⢐⠅⡈⢀⠀⢈⠔⠁⠀⡠⢊⠔⢁⠔⠡⠊⠐⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠙⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠋⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣷⣄⠀⠀⣠⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⣿⣿⣿⣿⣾⣿⣿⣿⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠈⠻⢿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣠⣶⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣄⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⣰⠏⢁⣤⠶⢤⣦⠈⢷⡀⠀⠀⠀⠀⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠉⠻⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢈⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⠿⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠸⣧⣀⠙⠒⠛⠙⣛⡋⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢀⣀⠀⠀⠀⠀⠀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠋⣠⣾⣿⣿⣿⣶⣤⣶⣾⣿⣿⣷⣄⡉⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠈⠙





⣿⣿⣿⣻⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣻⣿⣛⣛⣻⣟⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣶⣄⡀⢿⣿⣿⣿⣙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣉⣋⣉⣉⣉⣿⣿⣿⡿⢀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣦⣍⠻⢿⣿⣿⡿⠟⣩⣴⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣦⣍⣩⣴⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠋⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠙⠛⠁

⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⣿⣿⣿⣿
⣿⣿⡇⠉⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠉⢸⣿
⣿⣿⠇⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣦⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠛⢿⣿⣷⣄⣠⣾⣿⡿⠛⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠉⠻⣿⣿⠟⠉⠀⠀⠀⠈⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⢸⣿
⣿⣿⡇⢀⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀⢸⣿
⣿⣿⣷⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿
⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢰⡟⠹⣿⡛⠛⠛⠛⠛⠛⠛⣛⡛⠛⣃⣽⠏⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢠⡿⢠⣄⡈⠻⣷⣄⠀⠀⠀⣈⣥⠶⠛⠉⢴⢂⡿
⠀⠀⠀⠀⠀⠐⠚⠂⠀⠐⠒⠚⠓⠚⠒⠂⠀⠀⠀⣾⠁⠈⠀⠁⠀⠈⠛⢷⡶⠛⠉⠀⠀⠀⢀⡏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣄⠀⢀⣀⣀⣀⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢸⡇⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣄⣀⣼⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⣴⣿⣉⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⢸⡇⣉⣿⣦
⠀⠀⠀⠀⠀⣿⠀⠉⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣶⠾⠋⠁⠈⠙⢷⣶⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⡿

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡞⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⡽⡜⡸⢰⢩⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡛⡜⡜⢰⢱⢡⢃⠃⡎⠎⡼⣦⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡠⡶⡻⡱⡱⢱⣰⢡⢣⢃⠇⡎⡼⡸⡰⢡⢡⢻⣄
⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠉⠉⠛⠚⠓⠓⠓⠳⠷⠧⠧⠾⣼⣼⣠⣱⢡⢃⢧⢋⢎⡎⡷⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠚⠚⠺⠿⠷⣄⣀⣀⡀
⠀⠀⠀⣀⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⢹⠂
⠀⢠⣎⣽⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⢸⠀⢸
⠀⠸⠈⠳⣼⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡀
⠀⡄⠀⠀⠀⠙⠧⣖⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡳⣄
⠀⡇⠀⠀⠀⠀⠀⠈⠳⣮⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣧⣼⡆
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⡔⢂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣾⠷⠛⠋⡇
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣔⣊⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣺⠝⠚⠉⠀⠀⢘⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⠵⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⡾⠟⠊⠁⠀⠀⠀⠀⠀⠀⠨⠁⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢥⣒⣪⣄⣠⢄⡠⠀⣔⡮⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠃
⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉⠁⠀⠈⠉⠲⢯⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸
⣾⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸
⡇⡢⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⡇⠨⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣧⠙⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣖⡠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢻⣴⡊⠁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣡⠄⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠉⠙⠒⠒⠯⠜⣁⣃⡂⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣎⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠬⣐⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣡⠀⠀⠀⡌
⠀⢀⢄⡀⢄⡠⣀⠤⡀⢔⡤⣂⠤⢂⡠⣀⠤⣀⠤⣉⢙⠓⠒⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡄⠀⡇
⠀⠀⠀⠀⠁⠈⠁⠋⠈⠁⠊⠁⠘⠑⠊⠑⠋⠐⠊⠔⠡⠊⠗⠡⠞⡵⢊⢝⡹⢓⡶⣲⠤⣤⢤⣄⣀⣀⡀⠙⣦⣇⡄⣠⢀⣀⢄⡀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢰⣄⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⡆⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣦⡀⠙⠻⣿⣿⣿⣿⣿⣿⡿⠋⢀⣤⣾⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣦⡀⠈⠻⣿⣿⠟⠋⢀⣴⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⠋⢀⣠⣄⠈⠁⢀⣤⡀⠙⠻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⠟⠉⢀⣴⣿⣿⣿⣷⣾⣿⣿⣿⣦⣄⠈⠻⢿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠘⠁⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠾⠿⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣌⣿⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠙⠷⣤⡀⠀⠒⠒⠒⠀⠒⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠙⠷⣍⡉⢉⣙⣛⣛⣁⣀⣀⣀⡀⠀⣀⣴⠾⠛⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⢶⣌⡉⠉⠉⠉⠉⣉⣵⠾⠋⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣥⣤⣤⣤⣤⣤⣤⣽⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠲⠲⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⢻⡟⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣠⡴⠶⢦⣴⡄⠘⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⢸⣗⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢨⣿⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⢿⣯⡀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠙⠷⠶⠶⠛⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⡤⠶⠟⠁⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⣦⣄⠀⠀⠀⢀⣤⡾⠿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠈⠛⠷⣶⡶⠟⠋⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢄⠢⠀⠀⠀⠀⠤⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠂⠈⠁⠀⠀⠼⠦⠂⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠂⠀⠀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠢⢄⡀⠈⠀⠂⠄⠀⠀⠀⠀⠠⠒⠒⢀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢇⠀⠀⠉⠒⠤⣀⠀⠀⠀⢡⢀⠏⢙⠀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⢦⠀⠀⠀⠀⠀⠉⠐⠢⡤⠃⠀⢸⠀⠀⠈⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠂⠳⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠰⣀⠀⠤⠐⣠⠞⠀⡀⠀⠀⠀⠀⠀⠒⠀⠉⠁⠀⡔⠒⣇⢱⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠐⠠⣀⠀⠀⠀⠈⠉⠀⠚⠉⠍⠀⠀⠀⠀⡨⠄⡀⠀⡰⠃⢨⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⠀⢄⠠⠏⠀⠂⠈⠀⣶⠚⢣⢳⡔⢧⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠀⠀⡇⠀⢀⠀⢸⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⢽⡶⠼⠈⢍⠀⠀⠈⠧⠤⣀⡀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡇⠀⠀⠀⢸⠀⠀⢀⡨⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠉⢹⣆⣠⠤⠶⢶⠾⣍⣁⣀⠀⠀⠈⢀⡯⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠭⠀⠀⠀⢀⣠⣤⠇⠀⠀⠀⠀⠀⣀⠄⠒⣸⠁⠀⠀⠀⠀⠈⠁⠀⠝⡋⠉⠘⠈⠁⢢⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢍⠈⠁⠁⠻⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⠒⠊⠋⠁⠀⠀⠙⠒⠐⠒⠚⢍⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⢍⠁⣤⡐⠘⡎⠂
⠀⠀⠀⠀⢈⠀⠑⠦⡀⠀⠠⠤⠖⠒⠈⠉⠊⠄⡀⠀⠀⠈⠉⠒⠢⢤⢤⣀⣀⠀⠀⠀⠀⠳⡀⣸⠄⠀⠀⠠⠁⠈⢀⠡⢈⠀⠲⠂⡜⣉⠉⠀⡓
⠀⠀⠀⠀⡄⠀⠀⢀⠌⠢⣄⠀⠀⠀⡀⠀⢀⠠⠊⠁⠐⠠⠤⠐⠚⠥⡀⠀⠀⠉⠉⠙⠒⠲⣒⡡⠄⠈⡓⠦⠤⣈⠐⠀⠀⠁⠀⡜⢀
⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠈⠁⠀⠊⠃⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣀⠀⠀⠀⡀⠀⠙⡀⠀⠄⢁⠀⠀⠀⠉⠑⠦⠤⠜⠀⠒
⠀⠀⠀⠈⠀⠀⠀⠠⠤⠄⢀⣀⡀⠀⠀⠀⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⡸⡄⢀⠠⠀⠀⠀⠀⠀⠐⡄⢀⣀⠲⠒⠶⡖⠖⠶⡖⠀⠀⠒⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢂⠀⠀⠀⣶⠒⡈⡆⠈⢀⠀⠔⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠁⠀⠀⢈⣂⡥⠾⣁⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠔⠊⠉⠉⠐⠂⠄⠀⢀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠀⡀⠀⡀⠄⡀⠀⢀⠠⢀⠀⢐⠠⠐⡠⠄⡀⠄⡀⠀⡀⠀⡀⠄⡀⠀⡀⠀⠀⠀⠀⠈⠠⠀⠠⠀⠐⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠐⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣉⡙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣉⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣿⣿⣶⣌⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣁⣴⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣄⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣡⣴⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡙⠻⣿⣿⣿⣿⠟⢋⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢀⣴⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡙⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣙⡛⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠘⢧⣤⣇⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⠴⣒⣫⣥⡴⡾⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⣾⠿⠞⠟⣛⠈⢩⣅⡹⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡃⠀⠀⠒⢙⠀⢡⠴⠁⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⣅⠀⠀⠼⣇⣀⣠⠎⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡖⢚⣭⣥⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠛⣷⢦⠉⢹⡆⠀⠀⠈⠉⠘⡆⠀⠀⢘⣇⠀⠀⠀⠀⢀⣀⣼⠀⢻⣒⡇
⠀⠀⠀⠀⠀⢀⣀⣶⠒⠛⣶⡤⠿⠚⢻⠀⣧⠀⠀⠀⢠⠔⣃⣤⡖⣿⡼⣶⠶⠖⠚⠉⢀⡷⠚⠉⠁
⠀⠀⠀⠀⣀⡿⣅⢹⣀⣄⢿⠒⠚⢻⣘⣧⢿⣒⠒⠶⢿⣉⡽⠻⡏⢻⣳⡼⡤⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠹⣄⠘⣾⡄⣹⡼⠶⣺⠟⠉⠀⠀⠈⠉⠛⠻⣏⡳⣄⢻⣄⢻⡇⣇
⠀⠀⠀⠀⠀⠈⡷⠛⠉⢁⡴⠊⣡⣀⣠⠴⢾⡟⠂⠀⠀⠀⠉⠻⡜⣿⡌⣇⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣦⡀⢘⣯⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣹⣿⢼⣾
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⣹⠉⠀⡟⢲⠘⡇⠀⠀⠀⠀⠀⠀⢸⡧⠶⢚⡟
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⢹⣲⣣⣳⣼⢘⡇⠀⠤⠶⠚⠃⠀⢸⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠉⠉⣉⣭⡶⢻⡇⠀⠀⠀⠀⠀⢠⡟⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⠶⠛⠉⠀⠀⣼⠁⠀⠀⠀⢀⡴⠿⠶⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣠⣤⡶⠛⣉⣀⣤⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⣉⣠⠟⠛⣻⠏⠁⢠⡟⢀⡾⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟⣿⣀⡈⢁⣠⡼⠃⠀⠀⢿⣶⣿⣥⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠉⠉⠉⠉⠀⠀⠀⠀⠘⠛⠉⠛⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠦⣄⡴⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣸⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣈⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⣠⣤⣤⣀⡀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⢸⣿⡿⢿⣿⡆⠀⢀⣾⣿⠿⠟⢻⠇⠀⣿⡿⠿⣿⣿⡆⠀⢸⣿⠿⠿⠿⠋⠀⠀⣿⡿⠿⠿⠟
⠀⠀⠀⠀⢠⣿⣟⣻⣷⠀⠀⢸⣿⣿⣿⣿⣇⠀⢸⣿⡁⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡟⠀⢸⣿⣷⣾⡶⠀⠀⠀⣿⣷⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢸⣿⣭⣤⣿⡿⠀⠀⢻⣷⣦⣴⣾⡆⠀⣿⣷⣶⣾⡿⠃⠀⢸⣿⣭⣥⣤⣄⠀⠀⣿⡏⠉⠁
⠀⠀⠀⠀⠛⠃⠀⠀⠈⠛⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠙⠛⠛⠋⠁⠀⠛⠛⠛⠋⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠛⠃

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⣠⡄⠀⠀⠀⣤⡀⠀⢠⣄⠀⠀⢠⣄⠀⢀⣤⡀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⣼⣿⠟⠛⠿⠇⠀⠀⣿⣇⠀⠀⢀⣿⡇⠀⢸⣿⠀⠀⢸⣿⣴⣿⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡄⠀⢀⣾⣿
⠀⠀⠀⠀⣿⡇⢰⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣄⣾⣿⣿
⠀⠀⠀⠀⢻⣿⣶⣴⣿⡇⠀⠀⣿⡏⠀⠀⠀⣿⠇⠀⢸⣿⠀⠀⢸⣿⠙⢿⣷⠀⠀⠀⣿⣷⣶⣶⣦⠀⠀⣿⡇⠻⣿⡿⢹⣿
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠙⠁⠀⠀⠀⠋⠀⠀⠘⠁⠀⠀⠘⠃⠀⠈⠋⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠉⠁⠀⠉⠀⠈⠋

⠀⠀⠀⠀⣶⡄⠀⠀⣴⡄⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⣶⣶⣶⣤⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢠⣤⣤⣤⡀
⠀⠀⠀⠀⣿⣿⣦⡀⣿⡇⠀⠀⢰⣿⠟⠋⠙⢻⣿⡆⠀⠀⠀⣿⡟⠻⣿⡇⠀⠀⠀⣾⣿⠟⠛⠻⣿⣷⠀⠀⠀⢸⣿⠿⣿⣿
⠀⠀⠀⠀⣿⡟⢿⣷⣿⡇⠀⠀⢸⣿⡀⠀⠀⢀⣿⡇⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⣿⣇⠀⢠⣶⣼⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠃⠀⠻⣿⠇⠀⠀⠈⢿⣿⣷⣾⣿⠏⠀⠀⠀⠀⣿⠏⠉⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣿⡇⠀⠀⠀⢸⡿⠙⢿⣷
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠘⠃⠀⠈⠋

⠀⠀⠀⠀⣠⣶⣾⣶⣄⠀⠀⣶⣶⣶⣶⣶⡆⠀⣶⣆⠀⠀⢀⣶⠄⢠⣶⣄⠀⢠⣶⠆⠀⣤⡄⠀⠀⣠⣤⠀⢠⣤⣦⣤⣤⣤
⠀⠀⠀⠀⣿⣿⣭⣽⡟⠀⠀⠙⠛⣿⡟⠛⠁⠀⢻⣿⡄⢀⣾⡟⠀⠈⢻⣿⣶⣿⠏⠀⠀⢿⣿⣆⣴⣿⠇⠀⠘⠛⠛⣿⣿⠋
⠀⠀⠀⠀⣬⡛⠻⢿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣿⣾⡟⠀⠀⠀⣠⣿⢿⣿⣄⠀⠀⠀⠻⣿⣿⠁⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⢰⣿⠏⠈⢻⣿⠂⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⡿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠛⢦⣄⣠⡴⠛⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠉⠁⢀⣨⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠚⠁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣼
⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣤⣤⣤⣀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶
⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⢀⣀⡀⠙⠁⣀⣀⠀⠀⠀⠀⠀⣾⣿⡶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣠⡀⠉⠉⠀⠀⠀⠀⣼⡏⠹⣷⡀⠉⠙⠛⠿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⢰⡿⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣦⣤⣀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣦⣤⣀
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀
⠛⠛⠀⠀⠘⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣤⣀⡀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣷⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⢸⣿⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣹⣿⣿⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⢸⡇⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣿⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣼⠏
⠀⢸⡇⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢰⡟
⠀⢸⡇⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⠂⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠉⠛⠻⠶⣦⣤⣀⡈⢻⣆⢀⣿⠁⠀⠀⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠛⢿⡿⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⠃⠀⠀⠘⠛
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⣦⣤⣀⣀⣀⣀⣠⣴⡾⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⡇⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⢸⡇⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠻⠗
⠀⢸⣧⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠄⠢⡄⠀⠀⠠⣀⡤⠒⠂⢄⣠⠄⠀⠀⣠⠄⠐⠦⣄⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀
⠀⢸⠁⠈⠹⣷⡀⠀⠀⠀⠀⠀⣿⢳⣤⠇⠀⠀⣰⡏⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⠘⣷⠀⠀⢸⣇⡀⢀⡀⠀⠀⠈⣿⠀⠀⠉
⠀⠘⣗⡶⠀⣿⣷⠀⠀⠀⠀⠀⣿⡼⠋⣷⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡿⠀⠀⢸⡏⠀⠘⠁⠀⠀⠀⣿⠖⢦⠄
⠀⠀⠉⠀⢸⠏⣿⣇⠀⠀⠀⠀⠿⠀⠀⠛⠀⠀⠈⠻⢦⣀⣀⠤⠂⠄⠀⠀⠾⠆⠀⠴⠛⠁⠀⠀⠸⠇⠀⠠⠄⠀⠀⠠⠿
⠀⠀⠀⢀⡏⠀⠸⣿⡀
⠀⠀⠀⣼⠥⠤⠤⢽⣷⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠉⠀⠀⠘⡏
⠀⠀⢠⠇⠀⠀⠀⠀⢿⣇⠀⠀⠸⣇⠀⠀⢲⠀⠀⢀⡧⢠⠶⠼⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠐⢆⠀⠀⠀⠀⡅⠀⠐
⠀⢠⣿⠀⠀⠀⠀⠀⠸⠿⣧⠀⠀⠈⣱⡆⠸⠀⠀⢸⣧⠀⠀⠀⢇⠀⠀⠀⢠⡀⠀⠀⣦⠐⠁⠀⠀⢸⡇⠀⠘⠦⠀⠀⠀⠃⠀⠬⠃
⠀⠚⠚⠀⠀⠀⠀⠀⠈⠛⠉⠳
⠀⠀⡀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⢠⡦⠀⢠⢤⡆⠀⠀⠀⠀⣀⠄⠀⠐⠦⣄⠀⠀⠀⠰⣶⠖⠒⢦⡀⠀⠀⣠⠖⠀⠉⠑⠢⣀
⠀⠠⠗⠀⠀⠀⠀⠰⣿⡄⠀⠀⠀⠀⢈⣿⡄⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⣠⣀⣸⠇⠀⣼⡏⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢸⣯⠻⣄⢸⡇⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⠃⠉⠁⠀⠀⢹⣇⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣀⣇⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⢠⣿⠀⠈⠿⠇⠀⠀⠀⠈⠓⢄⣀⠀⠠⠘⠁⠀⠀⠰⠿⠄⠀⠸⠁⠀⠀⠈⠂⠀⠀⠐⠺⠥

⠀⠘⣿⡏⠉⠹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣶⠶⠒⠀⠀⠀⠀⠐⠉⡲⠂⠀⠀⠈⣿⠀⠀⠈⢿⡄⠀⠀⠀⢸⠋
⠀⠀⣿⡇⠀⢀⣷⠇⠀⠀⠀⠀⠀⠀⢸⣂⣀⠀⠀⠀⠀⠀⣴⡟⠀⢠⣦⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⡀⠀⢠⠃
⠀⠀⣿⡇⠚⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣦⠀⠀⠀⠰⣿⡀⠀⢿⠻⠐⠄⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠹⣧⠠⠎
⠀⠀⡿⡇⠀⠀⠣⣦⣀⢰⡀⠀⠀⣦⡀⠀⢀⡟⠀⠀⠀⠀⠛⢧⣀⠀⡀⡠⠆⠀⠀⠘⢿⣦⣀⣀⡀⢀⣿⡀⠀⠀⠀⠀⢻⡇
⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⡴⣤⡤⡤⡴⠦⣄⣤⠄⠀⢶⢀⡆⠀⠀⢶⣲⡆⠀⠀⠀⣴⠶⠀⠀⠀⠀⡒⢦⣄⠀⠀⠀⠰⡆⡄⠀⠀⠦⠀⠐⠶⠤⠤⢤⣲⠆
⠀⠀⠁⠀⡶⠀⠀⠀⣰⠃⠀⠀⠸⣦⡇⠀⠀⠀⠹⣿⣄⠀⣠⠄⠀⠀⠀⠀⣰⠇⠸⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⠐⠂⠀⠀⣰⣿⠃
⠀⠀⠀⣾⡇⠀⠀⢰⡏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢟⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡰⡿⠃
⠀⠀⠘⣿⡆⠀⠀⢿⣧⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⡼⠣⣿⣄⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⣀⡴⣿⡇⠀⠀⠀⠀⠀⣰⡼⠁
⠀⠀⠀⠹⢵⣄⣀⣈⡿⣦⣀⣀⣠⣟⡇⠀⠠⢴⠄⣘⡁⠀⠙⣿⢦⡀⠀⠀⣤⣄⠀⠀⠀⠉⠉⠀⡿⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠁⠉⠈⠁⠈⠀⠀⠀⠚⠓⠁⠀⠀⠘⠛⠛⠀⠀⠉⠋⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠙⠚⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⣠⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣤⡀
⠀⠀⠀⠀⠀⣿⠉⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⣀⣀⣀⡀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠋⠉⠉⠻⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣾⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⠿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣻⣿⣛⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠉⠀⠀⠀⠀⠀⠠⠐⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡀⠀⠄⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⠡⠂⠀⡀⣤⠀⠁⠰⡄⡈⠙⠂⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠶⠁⢀⠊⠄⣷⣢⣄⡀⠘⢿⠀⣖⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⢀⡇⠀⠊⠁⣀⠤⠈⠛⢿⣶⡌⢩⢋⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠨⡄⢿⠁⠠⠘⠁⠀⠀⠠⠒⠚⠻⣆⡌⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⢄⣀⠀⠀⠀⠀⢠⠖⠁⠀⡾⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⡐⠂⠀⡰⠀⢠⡟⠀⢀⠳⡄⠃⠈⠁⠺⠁⠀⠀⢐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣧⠃⠀⠀⢔⠂⠀⠸⠀⠀⠄⠀⠙⢆⡀⢀⠃⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠂⠀⠋⠀⠀⠀⠀⠐⠀⠂⠀⠈⢿⡏⠀⠀⠀⠀⢘⢸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⡔⠀⠀⠀⠸⢂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠠⡰⠊⡌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⢰⠀⠀⠀⠀⡃⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡇⠀⡆⠀⠀⠀⡸⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⡉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢱⠀⡅⠀⠀⢰⠁⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢰⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⢄⠡⡀⠆⢁⠀⠀⣇⠎⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣀⢀⢣⡎⢀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠛⠒⠳⠄⢬⣼⡀⢸⡿⠀⣠⠴⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠌⠐⡄⣀⣄⡀⠀⠠⠀⠀⠀⠠⠚⠉⠁⠀⣢⣤⣈⣙⣷⣌⣴⢸⠁⠶⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⣀⡤⠖⠉⠀⢀⡀⠠⠼⠏⣿⣬⡂⠀⠀⣀⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢐⡓⠂⠀⠀⠐⠒⠀⠐⠒⠚⠣⠤⠤⠀⠀⠀⠁⢀⡐⢀⠞⠇⡿⡿⢦⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠁⠀⠀⡏⠀⠀⠍⡉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢘⣶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⡐⠆⠨⠐⠀⠀⠙⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠒⢲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⢾⡿⠀⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣀⣀⡤⠛⠁⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣒⣒⣒⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠂⠤⣝⣦⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣭⣭⠭⣭⣭⣤⠄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠂⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠞⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢀⣄⠀⠂⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣀⣴⡆⠀⠸⢤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⣾⠙⠆⠀⠄⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠘⠙⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⣹⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⢀⡀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣧⢠⣀⣀⢀⣀⣀⣀⢀⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠌⠙⠛⠘⠛⠛⠋⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢉⠉⠉⡡⠀⡄⠈⠈⠉⠹⠡⣈⡙⣇⠀⠀⠀⠀⠃⠒⠠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠒⣀⣤⡤
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠙⠲⠤⠀⠀⠀⠠⢍⣉⣒⣀⣀⣀⠀⠀⠀⠀⡈⡀⠀⢐⠀⠀⠀⠀⢸⠇⠈⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠐⠒⠀⠤⠤⠀⠈⡉⠙⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⠠⠄⠒⠒⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐⠐⢄⠀⠀⣀⠔⢱⠧⠅
⠀⠀⠀⠀⠆⠀⢀⠀⡀⢰⠁⡀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣤⠄⠀⠀⠠⠈⠒⠀⠀⠀⠈⠁⠀⠀⠀⢰⠶⣟⠁⣃⠈⣦⠊⠁⠀⡞⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠇⠀⠨⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣾⢷⡜⠀⢹⠀⠀⢠⣇⡀⠀⠃
⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠛⠁⠀⢸⠀⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⡄⢈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠒⠲⠂⠂⠉⠉⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡀⠠⠦⠄⢖⠐⠂⠄⠒⠂⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢁⣉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢩⡃⠀⠆⠄⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠄⠤⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⢸⡏⠓⡞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠮⣭⣝⣛⣳⠶⣦⣤⣬⣇⣀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣄⣀⡀⠉⠉⠙⠛⠶⠦⣭⣍⣛⡛⠻⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣿⠛⣷⣶⠦⣤⣼⣛⡃⠉⠉⠛⠓⠶⢦⣭⣉⣛⠲⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠉⠉⠁⣠⠿⠽⠶⠏⢸⡏⢹⡟⠻⣶⣦⣤⣄⣀⣈⠉⠙⢿⠉⣿⣤⣤⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇⠸⣇⣰⣿⣿⣀⡼⢻⣿⡛⠻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠤⠤⣤⡤⠶⠶⠚⠋⠁⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⢸⣇⣴⡟⠡⣿⠈⠉⢀⡾⠻⠿⠟⠁⠈⠉⠛⠓⠶⠶⠶⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣋⣥⣤⣀⡀⠀⠀⠀⠀⢀⣼⡏⢹⡇⢸⡿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠀⣿⠻⠥⣄⣉⠉⠛⢳⣶⣾⣏⣽⡇⢸⡇⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄⣀⣬⠿⠻⠿⣤⣤⣈⡉⠙⢻⡷⢿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠙⣷⣍⣩⡿⢳⣶⣤⣀⣀⣉⣽⣿⠿⢦⣤⣈⡉⠛⠻⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠉⠉⠀⣸⣿⣦⣬⡿⠻⣷⣬⣄⣀⠀⠉⠙⠛⠳⢶⣤⣌⣉⢛⣿⣦⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠛⠛⠒⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣿⠀⢨⡏⢻⣷⣶⢤⣄⣀⡀⠈⠉⣿⢁⣿⠦⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣸⡿⠷⠶⠿⣏⣿⠛⣶⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⢶⡏⢸⣏⠀⠀⢀⡿⠛⠛⠁⠀⠉⠛⠲⠶⠶⠶⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣿⠁⢸⡇⢸⡟⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣤⠴⠞⠛⠉⠀⠀⣿⠀⢸⡇⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⢼⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠶⠖⠛⠛⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣸⣇⣼⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠢⠐⢆⠀⢀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣦⣤⣤⡄⢀⣀⣀⣀⠈⠲⣶⣶⠀⣼⣷⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⢿⣿⣿⣿⠿⢇⢪⠴⠤⠷⢤⡀⣹⠉⠀⣹⠛⠻⡟⠋⡡⠀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠟⠻⢄⢀⣴⣿⢀⣴⠿⠀⠀⠀⠀⠀⠁⠀⠙⠣⠀⠑⠄⠁⠼⠃⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣥⣤⣤⣶⡿⡛⢡⣾⠁⠀⠀⣶⣶⣿⣿⡀⠀⠀⠘⣦⢀⡀⠀⠊⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠂⢤⣶⣷⣿⣿⣿⣿⣣⠿⠛⠛⠂⠀⠀⠛⠛⢿⣿⡷⠤⠀⠐⠁⠸⠧⡠⠴⠋⠉⠀⣄⠀⢤⠆
⠀⠀⠀⠀⠀⢰⠌⠒⣼⣿⣿⣛⣍⢳⣰⠁⠀⠐⠟⢻⡄⠀⠀⠀⠈⠋⠀⠀⢀⡖⠀⣀⠀⠦⢆⡈⠐⡿⠉⢀⢀⣀
⠀⠀⠀⠀⠀⠠⠶⠸⣿⣿⣿⣿⡟⢈⣿⡀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⣤⠟⣏⠰⣿⣿⡇⢰⣄⣥⠀⠀⠰⠚⠃⠁
⠀⠀⠀⠀⠀⠠⣤⣤⣘⢯⠟⠋⡀⢼⣟⢿⣟⢿⡋⠀⠀⣰⡄⠀⠀⠀⠈⣷⣿⠆⣈⣇⡴⡎⢶⣴⡄⠠⡦⠘⠓⢶
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⠕⢤⣿⣾⢿⣌⡟⣬⠁⠀⢾⣿⣿⡀⠀⠀⠀⠘⡄⠑⣾⣿⡿⣷⢠⠌⠃⠷⠞⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠈⢹⣦⡘⠶⡆⠻⣯⣿⠟⢻⠆⡀⠀⠀⠟⠛⠃⠀⠀⠀⠀⠐⠀⠘⠏⡄⢻⣶⠀⣤⡸⠃⠘⣫⡄
⠀⠀⠀⠀⠀⠀⠀⣈⠻⡁⠐⣪⡿⡞⣿⣟⠟⠀⠛⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⠁⠝⢶⠀⣀⢿⠒⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠷⠿⢙⡿⢞⠚⡿⠌⢠⣿⣀⣴⡶⠖⢢⣿⣿⡄⠀⠀⠀⠈⠢⣧⠔⠂⠐⡲⣿⣿⣷⣤⣆
⠀⠀⠀⠀⢀⠀⣴⣤⣤⣴⢿⡇⢻⡷⠖⠀⣞⠛⡯⣋⡁⠰⣾⣿⣿⣷⡄⠀⠀⠀⠘⢿⡁⡄⢀⡐⣻⣷⣿⡿⠇
⠀⠀⢀⡴⠋⠛⡆⢈⣿⣥⢶⠾⣟⡙⠀⠸⣿⢶⠴⣿⠂⢀⢘⡉⣿⣿⣿⡄⠀⠀⠀⠈⠁⠦⠍⣁⠹⡿⠋⣤⣶⠒
⠀⠀⣎⠀⠀⣠⣿⣿⣿⣟⣻⠤⠿⠃⠀⠁⡭⠘⢆⠹⠃⣈⣉⠉⢳⣾⣗⣶⡄⠀⠀⠀⠰⣶⠄⡝⠰⢷⠀⠿⣇
⠀⠀⣼⠂⠀⣿⣿⣿⣿⡟⠣⣱⠶⣶⠚⣴⡆⠙⢌⣢⠀⢩⡌⠰⣰⢿⣶⠾⢛⢄⠀⠀⠀⠈⠠⣧⡁⣈⠀⢘⡿⠷⠄
⠀⠀⢹⡆⠀⠈⠿⢿⡋⢀⣠⣤⡀⠘⡇⠬⠶⠋⠀⢸⢡⡸⠇⠼⠷⠭⢿⢎⡐⠋⠁⠀⠀⣀⡀⠀⢀⡁⣤⠞⠁
⠀⠀⠀⠹⠦⠖⣀⣿⡀⠘⣿⣿⠋⢀⡾⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠓⢀⣀⣀⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠈⠓⢦⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⡀⠀⠀⠀⠀⠀⠈⢓⡴⠋⠙⢯⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠙⢦⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢙⡆
⠀⠀⠀⠀⣠⠞⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠾⡀
⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡜⠃⠀⠙⣆
⠀⠀⠀⡇⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡴⠋⢀⣠⠔⠋⢹
⠀⠀⢠⡇⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⢎⡠⠖⠉⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠈⠓⢦⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡼⠗⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡥⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡤⠖⠒⠤⣠⡴⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠠⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⣸
⠀⠀⠀⠸⣤⣚⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣙⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣾⣋⣁⣀⣀⣈⣙⣷⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⠀⠀⣴⠛⢉⣭⣍⣉⠙⣦⠀⠀⠀⠀⠀⠀⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣿⠀⠀⠀⠀⠀⢰⡇⠀⡏⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⣀⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠿⣤⡀⠀⠀⠀⠀⢷⣄⠙⠒⠚⠛⢒⡛⠀⠀⠀⠀⢀⣤⠿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢶⣄⠀⠀⣠⡶⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣿⣟⣻⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣥⣤⣠⣸⣿⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣙⣛⣛⠛⡛⠛⠻⣿⡆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣛⣛⠛⢿⣿⠀⣿⠇
⠀⠀⢠⣾⡿⠟⠛⠋⠉⣿⡇⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣥⣼⣿⣉⣉⠀⢸⢿⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡤⣾⣿⣭⣭⠀⢸⡿⢸⣿
⠀⠀⠀⢻⣧⠀⠀⠘⠉⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⠓⠒⠒⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣼⢿⣽⣿⣿⠏⢸⣿
⠀⠀⠀⠸⣿⡀⠀⠀⠄⣿⡇⠀⠀⠀⠚⠛⠛⠒⠒⠒⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⣠⠤⣤⣤⣀⡀⠀⣀⣀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠉⣿⡇⠀⠀⠀⠛⠛⠒⠲⠶⠖⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠦⠴⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⢤⣄⠀⣀⣀⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠉⠈⠀⠘⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠛⠋⠉⠉


⣿⣿⣿⢿⣿⠿⢿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣼⣾⣶⣶⣷⣾⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢙⣿⣦⣀⣠⣶⣿⡋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡾⠛⠁⠈⠙⠉⠀⠉⠻⣦⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣵⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠷⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣴⡄⠀⣦⠀⢠⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⣀⣀⣀⡀⢀⢀⠀⢀⡀⢀⣀⣀⠀⣀⣀⢀⡀⢀⡀⠀⠀⣀⡀⢀⣀⡀⠀⢀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⣿⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢠⣦⣤⣴⣤⣦⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⢹⣟⠉⠉⠉⠉⠙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡧⠘⠿⠟⠟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣉⣉⣉⣉⣹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠛⠃⠘⠛⠛⢻⣿⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠺⠇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⣴⣾⡟⢿⢿⣽⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡜⠃⠀⠀⠀⠀⠀⠛⠙⠉⠋⠀⠀⠀⢲⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠨⢧⠤⠒⠊⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⢿⡺⠃⠀⠀⠀⠀⠀⢄⡀⠒⠀⠀⠀⠘⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣸⠆⠀⠠⠜⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠉⠛⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡇⠄⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⣀⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣆⡠⠰⣶⡆⠀⠈⢧⣀⠀⠉⠀⠐⠂⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠿⣽⠹⣷⣀⣠⣤⣄⠀⣤⣤⣄⣀⣤⣿⣿⣅⡤⣾⣒⡀⠀⠘⡆⠀⠀⠀⠂⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣶⣿⣿⣿⡿⡟⡟⢹⣿⣿⣿⣿⡿⠿⣧⠁⠈⠑⢀⠀⠀⠹⠂⠠⠄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⢨⣯⠈⠷⢲⡤⢘⣻⣿⡋⢽⠻⢿⠉⠉⠀⠀⠂⠈⣀⣤⡖⠀⠀⠀⠀⠀⠀⣸⠛⠋⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⣿⠿⠛⢙⠛⠳⠀⠚⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠛⠿⠶⠦⠤⠀⠀⠃⠀⠀⠀⢀⣴⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠲⠶⠀⠀⠀⠂⠤⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠛⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⠁⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠦⢤⣤⣤⣤⣤⡄⠈⠙⠳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⣦⠄⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆⢀⣀⡀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⠀⠋⠓⠀⠈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠠⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⢀⣀⣀⣀⣀⠀⢀⣀⡀⢀⣀⠀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⠀⠀⠀⠙⠈⠙⠯⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢻⣿⠟⠁⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠐⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣬⣭⣥⣬⣭⣥⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠃⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⣿⠘⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣅⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⠀⠀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⢿⣦⣝⠃⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠘⣫⣴⡿⠛⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⢀⣤⣤⡀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣶⡿⠋⠙⢿⣶⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠙⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠛⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣀⣀⣀⣀⣴⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠛⠉⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣤⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣤⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⢠⡄
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠤⢄
⠌⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳
⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⣁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠞⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠈⠀⣀⠜⠁⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠤⠚⠀⠙⠓⠒⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠠⡅⢀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠠⠀⠓⠘⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡄⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠂⠀⠠⠀⠀⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⡀⠐⠀⠀⠄⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠁⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠘⠠⠀⠀⢀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠔⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡰⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡀⠄⣀⠀⠺⣦⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠠⠐⠀⠀⠠⢐⠠⢴⣄⣉⠿⢚⣅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣤⡀⢙⡶⢔⡋⠑⢷⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⣦⡤⠟⠹⣦⣈⡽⠛⣯⣀⣠⡤⠄⠀⠀⠀⠄⠒⠀⢠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⠀⢶⣄⣩⠟⢫⣅⣈⣳⠶⣿⠉⠻⣦⠴⠋⠉⠀⠀⠀⣠⠄⡀⠰⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣙⡷⢚⡁⠘⣷⣤⣾⠙⢷⣄⡬⠟⠋⢤⡀⠀⢀⣠⡴⠎⠋⠀⡁⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠐⠆⠀⠀⠙⠦⢴⠋⠰⣦⣠⡽⠫⣯⣀⣹⠗⠊⠁⠀⠀⠀⠀⠉⠓⠍⠁⠀⠀⠐⠀⠁⢠⠐
⠀⠀⠀⠀⠈⠀⠀⠈⠐⠀⠀⠈⠙⠒⣏⠈⢻⡦⠖⠉⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡀⠀⠄⠈⢦⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣫⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣭⣭⣉⠹⢤⠤⠴⠶⠖⢶⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡉⠙⣿⣿⣦⣴⣺⣿⢟⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢁⣴⡿⠿⢿⣿⣏⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⣠⠟⠁⠀⢠⣼⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⡼⠃⠀⡜⠓⠦⣄⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⡞⢀⡾⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣴⣶⣿⣿⣿⣷⣶⣬⣉⠒⠦⠼⠃⣼⠁⢠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣷⣦⡀⢠⡏⠀⡏
⠀⠀⠀⠀⣠⠤⣤⠀⢰⡇⠀⣿⣿⣿⣿⡿⠂⢰⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⠃
⠀⠀⠀⠘⠷⠀⢸⠇⢸⡇⠀⢿⣿⣿⣿⣶⣦⣌⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠻⣦⠘⣇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣦⣀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⢿⣿⣿⣶⣤⣄⣀⡀⠀⠈⠉⠛⠻⢿⣿⣿⣅⡛⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠈⠙⢿⣿⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠙⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣿⣿⣟⣚⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣼⡿⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠿⠶⠶⠶⠶⠶⠿⢿⣿⣿⣿⣿⣿⣯⣵⠾⣟⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠶⠖⠛⠋⠉⠛⠶⠶⠶⠦⣤⣤⣭⣭⣭⠴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⢿⣶⣾⠿⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉








⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⠿⠿⠿⠿⠿⠛⠛⣛⡛⠛⡛⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⠿⠟⠛⢛⣛⡉⠉⢩⡭⠁⠠⠴⠆⠀⠐⠒⠀⠀⠙⠉⠁⠈⠉⣁⢀⣵⠟⡅⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⠿⠿⢛⡛⢛⡭⠭⠀⠐⠓⠀⠛⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢌⣵⠟⠁⠀⣁⢿⣿
⠀⠀⠀⠀⢺⣿⣟⡛⢭⡉⠐⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣴⠟⠁⠀⠀⠀⢀⢸⣿
⠀⠀⠀⠀⢸⣿⡍⠛⠷⣦⣜⡃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣴⠟⠁⠀⠀⠀⠀⠀⠘⠸⣿⡇
⠀⠀⠀⠀⠈⣿⡇⠓⠀⠀⠉⠛⢷⣬⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⡇
⠀⠀⠀⠀⠀⣿⣧⠺⠀⠀⠀⠀⠀⠈⠙⠻⣦⣔⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣍⡀⢤⡀⢀⣴⠶⠾⠷⣦⣤⡾⠛⠛⠻⣦⣘⣵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⣦⣾⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⣿⠋⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⣿
⠀⠀⠀⠀⠀⠀⢿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡀⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠠⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⢤⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠸⠁⠀⠀⠀⠀⠈⠻⣦⣄⡀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⢷⣄⠀⠀⠀⠀⠀⢈⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⣴⠏⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝⢿⣄⡀⠀⠀⠸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢀⠀⠀⠀⠀⠀⢀⣼⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠝⠻⣦⡀⠐⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⠃⠀⠀⠀⢀⣾⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⡻⣦⣄⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠻⠀⠀⢀⣾⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣤⡄⠀⠠⠤⠀⠶⣒⣙⣀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡄⢠⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢠⠤⠀⠀⠶⠆⢀⣘⣛⣁⣀⣩⣤⣤⣤⣴⣶⣶⣶⣿⠿⠿⠿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢡⡿⠱⠇⠀⠴⠀⠐⢒⣀⣀⣩⣭⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠛⠻⣿⣧⣌⣉⣉⣉⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠀⣿⣿⣿⣿⣿⣿⣿⣄⣉⣉⣙⠛⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡉⠀⢿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⡿⠀⢉⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡙⠻⣿⣿⣿⣿⣤⣭⠉⣭⣤⣿⣿⣿⣿⣿⠟⢋⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣌⠙⢿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠋⣡⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠻⢿⣿⣿⡿⠟⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡉⠉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡙⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣜⣛⣤⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢟⣽⠿⣽⢿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡀⠀⠀⠀⠀⢀⢺⣏⣿⣿⣾⠹⡟⠾⣸⣿⡇⣻⢂⠀⠀⠀⠀⠀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷
⠀⣿⣶⣝⡻⣿⣿⣿⣿⣿⡿⢟⣭⣾⣿⠀⠀⠀⠀⢸⣷⣦⣝⢿⣮⣟⣻⣿⣿⢟⣣⣶⣿⠀⠀⠀⠀⣿⣮⣝⢿⣿⣿⣿⣿⣿⡿⣛⣽⣿⡿⠿⠃
⠀⣿⣿⣿⣿⣮⠝⠿⠟⢫⣾⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⢎⣝⣛⣍⠾⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣮⠛⠿⠟⣽⣾⣿⣿⣿⡇
⠀⣿⣿⡿⣫⣵⣿⣿⣿⣷⣯⡻⢿⣿⣿⠀⠀⠀⠀⢸⣿⡿⣫⣶⣿⣿⣿⣿⣿⣮⡻⢿⣿⠀⠀⠀⠀⣿⣿⢿⣫⣶⣿⣿⣿⣷⣭⡻⣿⣿⡇
⠀⠛⠵⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠯⠛⠀⠀⠀⠀⠘⠩⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠀⠀⠀⠀⠫⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠋⠉⢠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⢡⠀⠀⠀⠀⢠⣿⣥⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⡏⠀⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠺⠂⠀⠐⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⡟⠀⠀⠀⠀⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢾⣿⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠸⣿⣿⡟⠛⠻⠶⠶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣾⣿⠛⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠹⣿⡟⠆⠀⠀⠀⠀⠘⠉⠋⠙⠛⠛⠶⠶⣦⣤⣌⣀⣀⠀⠀⠀⠀⣴⡞⠋⡉⣤⣈⣿⡟⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢿⣇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠻⠷⠶⣾⣿⠠⠟⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣿⣆⠘⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣿⡿⠁⠀⠀⠀⠀⠀⢀⠔⢀⠄⠀⠀⠀⠀⢀⣀⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⣁⣤⡶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡷⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢀⠎⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⣠⠂⠀⠀⠀⠀⠘⡀⠀⠀⠀⡰⠀⠎⠀⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⡰⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⣀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠈⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣠⡧⢂⣀⣤⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⠿⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠒⠒⠉⢉⠙⣀⣀⣀⣀⣀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠂⠀⠀⠐⠉⡀⠀⠀⠀⠠⡴⢀⠀⠀⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠓⢀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠒⡀⠀⠂⠀⠀⠀⢀⣀⡤⠃
⠀⢀⣠⠴⠒⠒⢈⡉⠭⠀⠀⠈⠋⠉⢻⠀⠁⢀⠀⠀⠠⠆⠀⠀⠐⠁⠀⠀⠀⠀⠁⢤⠒⠂⠉
⡰⠋⠀⠀⠀⠔⠁⠀⢀⡀⠀⡀⠀⠀⠰⣄⠂⠀⠀⠀⠀⠀⠀⠀⢄⠂⠒⠐⡀⠀⠀⠀⠑⣤
⡇⠀⠀⠀⠀⡀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠋⠈⠁⠀⠁⠉⠢⢄⠀⠀⠀⠈⠀⠀⡠⠋
⡇⠀⠀⠀⠀⠇⢀⢀⣀⠀⠠⢄⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⣠⠞⠀⠀⢀⣀⣤⡀
⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠐⠢⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠄⠒⠈⠁⠀⠀⠱⡀
⡇⠀⠀⠀⠀⢠⠀⣤⠤⠀⠐⠂⠈⠁⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⣁⠀⠀⠀⠀⠀⣌⠀⠀⠡
⡇⠀⠀⠀⠀⢸⠀⢀⡀⡀⢀⠤⡀⡀⠀⣴⡉⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠂⠀⠀⠀⠀⠉⡀
⡇⠀⠀⠀⠀⢸⠀⠁⠀⡸⠃⠀⠈⡵⠂⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣀⡤⠔⠊⠉
⢃⠀⠀⠀⠀⢸⠀⢀⡃⠀⠑⠀⠊⠈⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠙⠢⡤⠤⠤⡖⠂⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣄⡇⠀⢻⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢫⠙⠙⠊⠀⣸

⡜⠀⠀⠀⠁⠀⠒⠂⠤⠆⠀⠈⠤⠒⠋⠁⡠⢀⠀⢴⡓⠊⡜⠀⠀⠀⠁⠀⠒⠂⠤⠆⠀⠘⠤⠒⠋⠁⠠⢀⠀⠴⡓
⠀⠪⠃⢀⡴⠦⡀⠀⣠⢊⢢⡀⠀⠀⠀⢌⠀⣐⠁⠀⠀⠉⠀⠪⠃⢀⠴⠦⡀⠀⣠⢊⢢⡀⠀⠀⠀⢌⠀⣐⠁
⠀⠀⠰⡇⠐⡡⠋⠀⠈⠷⠊⠀⠀⠀⠀⠀⠉⠀⠀⢠⠄⠤⡀⡀⠰⡇⠐⡡⠋⠀⠈⠗⠊⠀⠀⠀⠀⠀⠉⠈⠀⢠⠄
⡝⠀⠀⠈⠋⠀⠀⢀⡀⠄⡂⠀⠀⠀⠀⡔⠂⠀⠀⡇⢄⠗⡙⠀⠀⠈⠋⠀⠀⢀⡠⠄⡂⠀⠀⠀⠀⠔⠂⠀⠀⡇⢄
⠀⠀⠀⣠⢚⢦⡀⠑⠐⠊⢀⣴⣿⣦⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⡠⢚⢦⠀⠑⠒⠋⢀⣴⣿⣆⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠈⠫⣁⠔⠁⠀⠠⣄⠀⠹⡿⠗⠁⠀⠀⠀⠀⣴⣷⣄⠀⠀⠈⠫⣁⠔⠁⠀⠠⣄⠀⠹⡿⠟⠁⠀⠀⠀⠀⣴⣷
⢀⢤⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠚⡗⣄⠀⠑⠽⠋⢀⢤⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠚⡗⡄⠀⠑⠽
⠠⢀⠊⢀⡅⠀⠀⠀⡂⢆⠀⠀⠀⠻⡉⢉⠕⠁⠀⠀⠀⠀⠠⢀⠊⢀⡅⠀⠀⠀⡂⢆⠀⠀⠀⠻⡉⢉⠕⠁
⠀⠀⠉⠚⢀⡴⢄⠀⠈⠃⠃⢀⢀⡀⠈⠁⠀⠀⠀⠀⠸⠂⠀⠀⠉⠚⢀⡴⢄⠀⠈⠒⠃⢀⢀⡀⠈⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠚⢔⠐⡳⠀⠀⢀⠀⠂⠀⠀⠺⡐⠂⠤⠄⣀⠀⠀⠀⠀⠀⠐⢔⠐⡳⠀⠀⢀⠀⠂⠀⠀⠺⡐⠂⠤⠄⣀
⢀⠔⣀⠒⡄⠀⠉⠀⠀⡰⢻⠑⡄⠀⠀⠇⠘⠤⠐⠈⠉⠀⢀⠔⣀⠒⡄⠀⠉⠀⠀⡰⢻⠑⡄⠀⠀⠇⠘⠤⠐⠈⠉
⠘⠤⢓⠋⠁⠠⢀⠀⢴⡓⠈⡜⠀⠀⠀⠁⠐⠒⠀⠤⠆⠀⠘⠤⢓⠋⠁⠠⢀⠀⢴⡓⠊⡜⠀⠀⠈⠁⠐⠒⠀⠤⠆
⡀⠀⠀⠀⢌⠀⣐⠁⠀⠀⠉⠀⠪⠃⢀⡴⡦⡀⠀⡠⢊⣣⡀⠀⠀⠀⢌⠀⣀⠁⠀⠈⠉⠀⠊⠃⢀⡴⡦⡀⠀⣠⢊
⠀⠀⠀⠀⠀⠉⠈⠀⣠⠤⠤⡀⡀⠰⡇⠐⡡⠋⠀⠈⠗⠊⠀⠀⠀⠀⠀⠉⠈⠀⢠⠄⠤⡀⡀⠰⡇⠐⡡⠋⠀⠈⠗
⠀⠀⠀⠀⠔⡂⠀⠀⡇⢔⡗⡙⠀⠀⠈⠋⠀⠀⢀⠄⠄⡀⠀⠀⠀⠀⠔⠂⠀⠀⡇⢔⠗⡝⠀⠀⠈⠋⠀⠀⢀⠀⠄
⣴⣿⣦⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⣠⢺⢦⡀⠑⠘⠊⢀⣴⣿⣆⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⣠⢺⢦⡀⠑⠘⠊
⠹⡿⠗⠁⠀⠀⠀⠀⣴⣷⣄⠀⠀⠈⠣⣁⠔⠁⠀⠠⣄⠀⠹⡿⠗⠁⠀⠀⠀⠀⣴⣷⣄⠀⠀⠈⠣⣁⠔⠁⠀⠠⣄
⠀⠀⠀⡠⠚⡗⣄⠀⠑⠽⠋⠀⢤⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡗⣄⠀⠑⠿⠋⢀⣄⡤⢀
⠀⠀⠻⡉⢉⠕⠁⠀⠀⠀⠀⠠⣀⡊⢀⡅⠀⠀⠀⡒⢄⠀⠀⠀⠻⡉⢉⠕⠁⠀⠀⠀⠀⠠⣀⠊⢀⡅⠀⠀⠀⣒⢆
⢀⢀⡀⠈⠁⠀⠀⠀⠀⠸⠆⠀⠀⠉⠚⢀⣴⢄⠀⠈⠃⠃⠀⢀⡀⠈⠁⠀⠀⠀⠀⠸⠆⠀⠀⠉⠚⢀⣴⢄⠀⠈⠃
⠀⠂⠀⠀⠸⡒⠢⠤⢄⣐⠀⠀⠀⠀⠀⠚⢝⠀⡳⠀⠀⢀⠀⠂⠀⠀⠸⡒⠢⠤⠄⣐⠀⠀⠀⠀⠀⠸⢝⠀⡳
⠑⡄⠀⠀⠇⠐⠤⠐⠊⣉⠀⢀⠔⣒⠂⡄⠀⠉⠁⠀⡰⢫⠑⡄⠀⠀⠃⠐⠤⠐⠊⣉⠀⢀⠔⣒⠂⡄⠀⠙⠁⠀⡰
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣛⣿⣛⣿

⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⢸⣿⠿⣿⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣽⣿⣿⣷
⠀⠀⠀⠀⢸⣿⠃⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣷⣄⠀⠀⠀⢀⣴⣾⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠙⢿⣿⣦⣾⡿⠟⠉⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⢃⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣼⣿
⠀⠀⠀⠀⠸⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣻⣿⣿
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣿⣿⣷⠀⠀⠀⢎⠀⣧⢈⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⣿⣿⣇⠀⠘⠀⢩⡾⢃⠴⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢿⡿⠿⢄⠄⡐⠋⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸⣿⣷⣦⡈⠓⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠄⡈⠛⢛⣿⣷⡌⣒⣀⠥⡒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢾⣿⠀⠀⠂⢀⠔⢨⠋⣻⣀⡤⠖⠁⠈⢣⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡛⣽⣏⣀⣘⣇⢘⣲⠑⢂⣀⣸⡇⣰⠊⣁⣘⢻⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡢⠀⠑⣿⣿⣻⡟⣟⢻⠛⠷⡏⣟⢹⣛⣿⣿⣿⡿⡟⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠛⠉⢉⣿⣉⡆⠁⠈⠀⠀⠀⠀⠀⠛⢤⣿⣿⣿⡇⠈⠩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠰⠉⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠔⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⣐⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⢣⣀⡠⠤⠇⠀⠀⢀⡠⠂⠁⠀⢱⠀⠀⠰⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢈⠡⠔⠁⠀⠀⠀⠀⠀⠈⢢⣀⠔⠊⠉⠀⠐⠀⣐⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⡐⢀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⡠⠐⢀⢀⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⣀⠔⠊⠁⣿⣀⣀⡄⠠⡀
⠀⠀⠀⠀⡜⠀⠈⠂⠚⠀⠸⡀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠁⠸⠀⠀⢃⠀⠤⠎⢈⡇⠀⠀⢰
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⡇⢸⠑⠻⢀⣉⡀⡄⣀⣠⠀⢸⠀⠀⠀⠀⠢⠔⠊⠡⠤⠤⠤⠜
⠀⠀⠀⢸⠀⠀⠀⠠⠄⢀⠀⡇⠸⠀⠀⠋⠀⠁⠀⠁⠀⠀⢸⠔⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⣎⠀⡀⣀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡖⠀⠉⢁⠜⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠐⠂⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠀⡄⠈⠉⠉⠂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠐⡀⠇⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⡁⠀⣀⠔⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠊⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⣄⣸

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠀⠀⠀⠄⡐⠁⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠲⡄⠀⠀⣀⢐⣊⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠻⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣈⣉⣉⣉⣉⣁⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⢉⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⡉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⠿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠉⠁








⡿⢿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⡀⠄⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠖⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠐⠂⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠜⠀⠃⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡰⠃⠀⠀⠀⠀⠐
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡠⠁⠀⠀⠀⠀⠀⢢⠜
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡄⠃
⠐⠐⠓⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠠⠊
⠀⠀⠀⠀⠀⠈⠉⠒⠦⡀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢠⠀⢀⠖⠃
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⢐⡤⠂⠉⠀⠀⠉⠈⠓⠒⠢⠤⢀⡀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠢⠤⣐⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⡦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠰⠢⠢⢞⠞⠋⣁⣴⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠢⢂⠖⠔⠔⠐⠀⠂⡠⠲⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠄⠐⡀⠀⢂⠀⠠⢂⠄⠀⠐⠁⢀⡀⡤⠤⡒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡀⠠⠀⠀⢈⠌⡀⠡⠂⠔⠀⢀⡀⡤⠰⠂⢈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡔⠡⠀⠀⠁⠀⠌⠐⡀⠀⠀⠂⠀⣀⠄⠤⢂⠋⠈⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠰⡄⢁⠀⠀⠂⡀⡀⢨⠠⠄⠖⠀⠃⡁⠠⠂
⠀⠀⠀⠀⠀⡀⠈⠔⡡⠉⢈⠄⠀⠐⠀⢀
⠀⠀⠀⠀⠂⠀⠀⡈⠀⠐⠁⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⣟⣻⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⠀⣽⣿⡟⣭⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⣿⣿⡟⠀⣰⣿⡿⠿⢶⣾⡿⠻⣿⣿⣿⣿⡟⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠸⣿⠀⢰⡿⠃⢀⣤⠀⢹⡇⠀⡿⠀⠈⣿⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⢠⡇⠀⡟⠀⣸⠃⠀⠀⣀⣀⣼⣇⠀⠁⠀⠀⠃⠀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⣸⣷⠀⠀⠀⣿⡀⠘⢿⡿⠿⢿⣿⠀⠀⣴⡀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣿⣿⣆⣀⣸⣿⣷⣤⣀⣀⣤⣾⣿⣷⣾⣿⣿⣶⣿⠋⠍⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠻⢿⣿⡿⠟⠿⣿⣿⠛⠉⢙⣟⠉⠉⠀⣀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢰⠄⠈⠏⠀⣤⠀⢸⡇⠀⠾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⣀⣠⣾⠀⠸⠟⠀⢸⣿⣦⠀⠘⣿⣿⠁⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⢿⡇⠀⣿⣿⣿⣧⣄⣀⣴⣿⣏⢀⣀⣴⣿⣿⣦⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣿⣿⣧⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠤⠠⠄⠐⠒⠀⠉⠉⢑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⣇
⣠⡄⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢻
⡇⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⠢⢄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠒⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠻
⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⣘
⠀⠀⢀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⣀⣠⠤⠴⠴⠔⠛⠉
⠀⠀⠘⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⡀⣠⡠⠤⠦⠶⠴⠼⠐⠒⠚⠃⠉⠉⠉
⠀⠀⠀⠣⣞⡥⠠⠤⠤⠤⠤⠤⠴⠶⠒⠒⠐⠓⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣇⣀⣠⠤⠤⠶⠒⠒⠒⠉⠉⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠋
⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠈⠑⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠒⠒⠒⠒⠒⠂⠠⠤⠤⠼⢤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⢲⠚⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠶⠒⠒⠛⠉⠉⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣀⣤⠤⠤⠖⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠴⠂⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡏⠉⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠚⠋⠉⠉⠀⠀⠀⣀⣀⣀⡤⠀⠀⠀⢸⡝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣀⣀⠤⠤⠴⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⣀⠴⠋⠁⡇
⠀⠀⠀⠀⠀⠀⠀⣧⣈⡆⠀⠀⠀⠀⠀⠒⠚⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠴⠒⠒⣚⠴⠊⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢧⠈⠉⠒⠦⢤⣀⡀⠀⠠⠤⠤⠖⠒⠒⠋⠉⠉⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⢀⣠⣀⡠⠚⠙⢦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠏⠀⠈⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⢠⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠑⠦⣤⡤⠏⠀⠀⠀⠉⠓⠲⢤⣀⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⠽⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠴⠋⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠒⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠉⢢⢀⡔⠉⠀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡀⠀⣿⠀⠀⠀⢀⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⠁⠀⣿⡀⠀⠀⠈⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⣠⠞⠁⠙⠦⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠛⠋⠁⠀⠀⠀⡠⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠢⣀⠀⠀⣀⡠⠚⠁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠉⠉⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣠⣤⠤⠤⢤⣄⣀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⡽⢿⡿⠟⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠈⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣁⣀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠘⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣤⣶⣾⣿⡿⠟⠋⣿⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠚⠿⠿⠛⠛⠉⠁⠀⠀⠀⢹⡄⠀⠀⠈⢉⣩⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⣭⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⣿⣿⣷⣦
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠐⠂⠄⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡀⠀⠀⠠⠀⠐⠀⠘⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡔⠁⡠⠐⣂⡀⠀⠱⣴⡀⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠄⠀⠡⠗⠛⠒⠤⣬⣣⡘⠤⣾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⣷⠂⠀⠰⠊⠀⠀⣈⡹⢷⣀⡰⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡸⡀⠀⠀⠀⠀⠊⣠⠤⠀⢹⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⠔⠁⠠⠀⢠⡗⠀⠸⣄⠸⠁⠂⠠⠆⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡴⠁⠀⢰⠌⠀⠈⠀⠀⠁⠈⢷⡀⠀⠜⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⠀⠀⠀⠀⠈⡇⠀⠀⠀⢰⠃⠀⡀⠀⠀⠀⠜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡅⠀⠈⡼⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢀⠀⠀⠀⢀⠁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠰⠁⠀⠀⠀⠀⠀⢱⡀⢀⠀⢸⠀⠸⠀⠀⢀⡏⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢂⠀⠆⢠⠀⠀⡜⣠⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢸⠀⡆⠀⠀⠀⠠⡀⠀⠀⠀⢰⣌⣢⡂⡀⡈⠀⠰⣷⠁⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠦⢐⠘⠀⠀⠀⠀⠀⠀⠀⠠⡠⠤⠀⢈⠉⠒⢾⣆⠀⡇⢠⠊⡉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠈⠓⠛⠂⠀⠈⠀⡀⠀⠀⠀⣀⡠⠖⠛⠃⠒⢙⡿⣿⡰⠈⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠖⠉⠠⠤⠤⠀⠀⠤⠀⠈⠠⠤⣀⣈⠁⡀⠀⠔⠃⠀⠀⡠⠿⣿⣦⡀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠊⠀⠀⡇⠀⢉⡒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⢀⠀⢀⠁⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡐⠄⠀⠀⠀⠈⣓⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠉⠉⠙⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠠⢝⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⣦⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣑⠢⠤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠉⠉⠙⠒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⢿⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡴⠒⠤⠤⢄⣀⡀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⠏⠀⠈⠙⢻⣻⠉⢦
⠀⠀⠀⠀⠀⠀⠀⣴⣷⣀⣤⣤⣤⣤⣤⣠⣃⠀⠠⠤⠀⠀⠈⠉⠐⢲⣾⠥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⠃⠀⠀⠀⢀⣿⡏⠀⣸
⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⢉⠏⠛⣛⠷⠶⢤⣤⣀⡀⢀⣾⠃⠀⠀⠈⠉⠑⠒⠤⢄⣀⠀⠀⣗⣇⣀⠀⠀⢠⣞⠞⠀⣰⠇
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢀⠎⠀⠐⠢⠬⣉⡒⠀⠈⠙⣿⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⡳⠈⠙⠚⠽⡽⠽⠋⠀⣰⡟
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠏⢀⡀⠈⠉⠛⠻⠷⢶⣤⣤⣀⣰⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢄⣈⠉⠓⠤⢄⡀⠀⠀⠈⢉⠏⠛⠛⠶⢦⣤⣀⡀⠈⣿⠃
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠠⢇⣀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠐⠢⢄⡀⠉⠒⠠⢄⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠉⠛⣿⡏
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⢀⣿⠁⠈⠑⠂⠀⠉⠒⠢⠄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁
⠀⠀⠀⠱⢤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠏⠑⠒⠤⢄⣀⠀⠤⠄⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠉⠙⠒⠦⠬⣉⣀⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣦⣤⣤⣀⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⣤⠀⢀⡠⠤⠭⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠖⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠏⢰⠁⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡎⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⡊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠷⠶⠶⠭⠭⠤⠤⠤⠤⠤⠤⠀⠨⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠰⣅⠅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠰⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⢤⡤⡀⢠⢄⢠⠂⣄⠤⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠀⠈⠁⠃⠘⠜⡀⡀⠁⡀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠤⠉⠾⣆⢶⣰⣃⣓⢇⣉⡡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠉⠉⠉⠀⠀⡄⢀⣀⣠⠤⠴⠖⠒⢻⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⡇⢸⠁⠀⠀⠀⠀⠀⠸⣿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠧⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠇⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠈⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠻⣦⣄⠀⠀⠀⠀⣠⣴⠞⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠉⠻⢦⡴⠟⠋⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉







⣿⣿⣶⣾⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⢷⠷⠾⠿⠶⣶⣾⠷⠶⠿⠶⢶⡶⢿

⠀⠀⠀⢠⣀⣀
⠀⠀⠀⢸⡍⠉⠛⠛⠶⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣇⠀⠀⣀⡀⠀⠀⠈⠉⠛⠛⠷⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⢋⠉⠉⣽⠏
⠀⠀⠀⠀⠘⡄⠀⠀⠈⠁⠀⠀⠒⠲⠤⠤⠀⠀⢹⣷⠀⠀⠀⠀⣠⣴⡟⠁⠀⣠⠖⢉⣠⣼⠏
⠀⠀⠀⠀⠀⠱⡀⠈⠘⠓⠂⠀⠀⠤⢤⣀⣀⠀⠈⣿⡄⠀⠀⣰⡟⢻⡇⣠⡾⠁⠈⠉⣽⠇
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠒⠒⠂⠀⠀⠤⠤⠤⢀⠀⠘⣷⡀⠀⣿⠀⢠⣾⠏⠀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣄⣀⠀⠀⢀⣀⣀⡀⠀⠘⣷⡀⢻⣴⣿⣃⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣀⠀⠀⣀⣀⣀⡀⠀⠘⢇⣾⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⠚⣡⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠤⠤⠄⠀⠉⠉⠉⠉⠀⠀⠘⡆⠀⠀⠀⠀⠀⢘⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠹⠀⠀⠀⣀⣴⣿⣿⣿⣿⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡿⢚⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣀⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠢⠤⠤⠤⠴⠚⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠒⠊⠁⠀⠀⠀⢀⣴⣛⣩⣷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣀⣠⣤⣶⡾⠿⣟⣻⠭⠟⠓⠒⠒⣒⡶⠖
⠀⠀⠀⠀⠀⠀⠚⠓⢶⠞⢛⣛⣩⡯⠷⠶⠭⠥⠤⠔⢶⠒⢺⠻⠥⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣽⡯⣟⡿⠀⠀⠀⠀⠀⠀⠘⠤⠟⠀⠀⠀⠀⠀⢉⣉⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡟⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢀⠤⠤⠄⢷⠀⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠸⡆
⠀⢀⣴⡶⡶⢾⣶⠖⠒⠎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⡤⢤⣤
⠀⠞⢻⡀⢡⠀⡾⠃⣸⡶⠿⠷⡲⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣠⡴⠃⠀⠀⠀⣠⠗⢒⡦⠀⠀⠀⠀⠀⠀⠸⠜
⠀⢸⡯⠗⣺⣖⣓⣲⠟⠀⡔⠀⢸⢸⠀⠀⠀⢈⠑⢤⣀⠈⢿⢠⢻⡦⣀⣤⠶⡁⡏⠁⠀⠘⢭⣥⣭⣀⡂
⠀⠸⡇⠀⠀⠀⠀⡞⠀⢸⠁⢀⣮⣇⠀⠀⠀⠀⠢⢲⡈⠲⢼⣿⠏⣧⡀⠀⣱⡽⣇⣤⡤⠤⠤⠤⠴⠶⠶⠖⠒
⠀⠀⣇⠀⠀⠀⠀⡇⠀⠸⣶⠛⢡⠿⡿⠑⠲⣄⠀⠀⠀⠉⠈⢋⡀⢸⡉⠉⠉⠀⠐⠀⠀⠈⠉⠉
⠀⠀⠸⣄⠀⠀⠀⠹⣄⡠⢩⠗⣛⣲⡖⢀⣼⡿⠂⠀⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠈⠓⠦⠄⠀⣼⠓⢳⣤⣒⣶⢛⣾⡟⠀⠀⠀⠀⠀⠀⢀⠃⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠾⢤⡞⠀⠀⠀⣾⡏⠀⠀⠀⠀⣀⣀⣀⣤⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠻⣖⡾⠏⠉⠉⠉⠉⠉⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠙⠦⢄⣀⡤⠟⠁⡴⣄⠀⢤⡔⠚⠉⠀⠀⣠⢞⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⠦⢤⣠⡤⠞⠁⢈⠳⣄⡉⠒⠤⡤⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠙⠲⢤⣀⡴⠋

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠛⠟⠛⠻⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠻⠛⢛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠻⢦⣄⠀⠀⠀⠀⣠⡴⠟⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠉⠻⢦⡴⠟⠉⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣴⣼⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁