Cute cup Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢀⡴⠒⠲⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⡞⠀⠀⠀⠘⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣤⡈⢻⣄⡀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣠⠤⡈⢻⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⢺⠠⣄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⡇⠀⢸⡞⠀⠀⠀⠘⣷⡀⠀⠀⠀⠈⢻⡄⢸⡄⠘⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠈⣇⠀⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢹⡇⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠻⣆⠀⢀⣀⣸⡟⠀⠀⠀⠀⠀⠀⣸⣇⣸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣇⠀⡿⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣹⣇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⣿⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢼⣿⣿⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣠⠿⣃⣼⡇
⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠀⠺⣦⣈⠻⢧⣤⣀⣀⣠⣿⣿⣿⣇⠀⢀⣀⣀⣀⣠⣤⣼⠿⣛⣫⣵⣾⣿⣿⠇
⠀⠀⠀⠀⠀⡏⠀⣠⣴⡶⠶⠶⣿⣟⠿⢶⣦⣬⣭⣭⣽⣿⣿⣿⣿⣿⣭⣭⣿⣷⣶⣿⣿⠿⠟⢛⡩⢴⡿
⠀⠀⠀⠀⠀⢇⠰⣿⠁⠀⠀⠀⠀⠹⣇⠀⠀⠭⠍⣉⣙⡛⠛⠛⠛⠛⠋⣉⣉⣉⠩⠤⠔⠒⠋⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠘⣆⢿⣆⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⣨⣍⠉⠉⠉⠀⢠⣶⡆⠀⠀⠀⠀⠀⣰⡟⠙⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡻⢷⣤⣀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢽⡻⠿⣿⣷⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠁⠀⠀⢠⣟⠀⠀⠀⠀⠀⠲⣄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡿⠟⡩⢂⢹⣆⠀⠀⠀⠘⠛⠿⠷⠾⠟⠋⠀⠀⠀⣠⠟⠹⣷⠀⠀⠀⠀⢀⣼⠃⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣋⡀⠸⢱⠈⡿⣌⠛⠶⠤⣤⣤⣤⣤⣤⣤⣤⠶⠖⠛⠁⣠⣼⠟⠀⠀⣀⣴⠞⢁⠄⣰⠏
⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠁⠀⠉⠉⠣⢸⣠⣃⡈⠛⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⠶⠾⠛⢋⣁⣤⡶⠟⢋⣠⠖⣡⡾⢋⠜
⠀⠀⠀⠀⠰⢾⣍⣀⣀⣤⣤⣀⠀⣀⣠⠜⢱⡀⠀⢢⣕⢦⠀⢀⣀⣀⣀⣀⣤⣤⣴⡶⠶⠟⠛⢉⣠⠤⠒⣩⣴⠟⣩⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⠖⠚⠋⣡⢄⡾⠑⠦⣄⣉⡃⠑⢌⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣈⣡⡤⠶⠛⠉⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠺⢛⡵⠻⠦⣤⣄⣀⣀⣈⣁⣈⣀⣀⣀⣀⣤⣤⠴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠴⠚⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠉⠀⠀⠄⠤⠔⠑⠁⡀⠀⠂⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠠⠀⠀⠀⢀⠀⠐⠒⠂⠀⢸⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠋⠵⠖⢀⠀⡀⠀⠄⠠⠬⠥⠤⠀⣈⠐⠀⠁⠈⠠⣦⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⠿⣿⠶⡶⠶⠤⠤⣬⡭⣭⣭⣤⣤⣥⣴⣷⣶⣾⡷⢿⠖⢉⡠⠔⠚⠉⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠉⣙⣓⣿⣿⠿⡓⠸⠃⢸⠆⣠⠔⠚⠓⢢⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡠⠶⣒⣶⡿⠿⢃⠈⢔⢸⠞⠁⠀⠀⠀⠀⢳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠒⣒⣚⡯⠅⢀⠅⢸⠀⠀⠀⠀⠀⢀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣁⠀⣩⣤⠭⠥⢶⣒⠁⠔⠂⣿⠀⠀⠀⢀⣠⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⣐⣚⣭⠭⠵⢂⠉⠀⣷⠀⡐⠊⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢒⣒⣚⣫⠭⠀⡂⢠⠏⣠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠄⠭⢵⣒⣚⡋⡠⢂⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡃⢀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢥⣼⣿⣿⡋⠐⣠⣋⡀
⠀⠀⠀⣀⡤⠤⠖⠒⠶⠞⠉⢉⣉⡉⠭⢿⣦⣄⡀⠀⠀⠙⠈⢛⠈⣿⣿⣿⣟⣣⣴⣿⢿⣿⣿⣟⣛⣛⠛⠶⠶⠦⣤⣄⡀
⠀⠀⢾⣡⣀⣄⠀⠀⠆⠀⠀⠀⠀⠰⠀⠈⠙⠤⢀⣀⣀⣀⣀⣉⣍⣱⣾⣷⣦⡦⠟⢉⣠⠄⠀⠀⠀⠉⣉⡳⠀⣠⢈⡹⣹⠆
⠀⠀⠀⠙⠺⢭⣛⡒⠲⠶⠤⠤⠥⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣀⣭⣤⣀⣠⣤⣶⣶⣶⡾⠤⠴⠶⠖⢒⣛⣭⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣐⣂⣀⣠⣤⣤⣤⣤⣤⠤⠴⠶⠖⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⡉⠼⠧⡴⠀⠀⠀⠀⠀⠀⠀⠈⢉⣦
⠀⠀⠀⠀⡠⠲⣃⡀⠉⠁⠀⠒⠒⠒⠒⠒⠀⠉⠉⠁⣈⡗⢢
⠀⠀⠀⠀⡿⢐⣂⠠⠭⠉⣉⣁⣀⠀⢀⣀⣉⠉⠭⠤⣐⣒⢽
⠀⠀⠀⠀⢡⠀⠀⠀⠉⠉⠀⠀⠒⠒⠒⠀⠀⠉⠉⠁⠀⠀⡈
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⠊⠉⣁⣀⡀⡀⢀⣀⣉⠉⠑⠒⢄⡀
⠀⠀⠀⠀⠀⢠⠀⠉⠉⠐⠒⠒⠂⠀⠀⠀⠒⠒⠂⢹⠁⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄⠘⡄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⡜⠱⡀⠀⠀⠀⢀⡞⢳⡀⠘⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣴⣶⣶⡀⠀⠀⠀⠀⠇⠀⠉⠁⠒⠒⠠⠤⠤⠤⠄⠒⠒⠉⠁⠀⡸⠤⣀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⡟⠻⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⣤⠀⠀⠀⣠⣄⠀⠀⠀⢀⠃⠀⠈⡆⠀⢰
⠀⠀⠀⠀⠀⠀⢘⠒⠂⠠⠤⠤⠄⠉⠁⠠⠤⠤⠤⠀⠺⡆⠀⠀⠀⠉⠋⠀⠀⠀⠈⠋⠀⠀⠀⢸⣀⠤⠒⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡎⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠉⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠘⢌⣒⠤⢀⣀⣀⢀⣀⣀⡀⠤⣊⡡⠆⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠤⠤⠤⠤⠤⠀⠒⠙⠒⠦⣤⢀⡀⣈⣉⣉⣉⣉⣉⣉⣉⣁⠀⡀⣤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⢸⡗⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⠤⠴⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣇⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠐⢢⢱
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⡸
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡠⣞⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋
⠀⠀⠀⡀⠤⠒⠚⠉⠉⠁⣈⣭⠶⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠯⢤⣉⠉⠉⠉⠒⠲⢤⡀
⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠂⠿⠿⠿⠿⠿⠟⠛⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⣀⡝
⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⠿⣷⣦⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠈⣿⠇⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠘⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠻⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠉⠁⠀⠀⣿⠀⣤⡾⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠚⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⣸⣿⠇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣴⠶⠛⠋⠉⠉⠉⠉⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⠀⠀⣀⣴⣿⠏⠀⠀⠈⠉⠙⢷⣤⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣠⣬⠏⠐⠴⠼⠯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠈⠙⠿⣶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣦
⠀⠀⠀⢂⢤⠈⢏⠉⣧⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢠⠈⢏⠈⢧⣀⣀⣀⣀⣀
⠀⠀⠀⠈⠉⠀⡼⢆⣈⣷⣶⣶⣶⣾⣯⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣜⢆⣈⣳⣉⣉⣘⣒⡯⢱
⠀⠀⠀⠀⠀⠀⡏⣳⣶⣿⣿⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠒⠒⠒⠒⠒⠒⠉⣽⣀
⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣯⡭⣍⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⣍⢦
⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⢀⡀⠰⠆⠀⣿⡀⡸⢸
⠀⠀⠀⣀⣀⣀⣹⣹⣿⣿⣿⣿⣿⣿⣿⣿⡯⠵⠋⠀⠀⠀⠀⢀⣀⣀⣀⣹⣀⣀⠀⠀⠀⠀⠀⢰⢹⠭⠔⠁
⠀⠀⣿⣿⡿⠋⠙⠋⣟⣿⣿⣿⣿⡿⣫⠋⠀⠀⠀⠀⠀⠀⠀⣿⣾⡽⠉⠉⠁⡏⣧⣀⣀⡠⠔⣡⠃
⠀⠀⠛⠛⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠉⠙⠓⠚⠒⠒⠛⠓⠒⠒⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⢶⣶⡤⣄
⠀⠀⠀⠀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠥⠤⠄⣀⣀⡀⠈⠠⠌⠘⣢
⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠻⠶⠶⠶⠶⠶⠶⠶⠛⢻⠛
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⡇⠀⠰⠆⠀⠀⠀⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⠸⠟⠀⠤⠀⠿⠂⡄⡟
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⡘⠀⡄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⡄⠱⡀⠀⢡⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠇⠀⠀⠑⠂⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠊⡠⠴⠦⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢰⠁⠯⠤⠃⠀⠀⢀⠎⢀⠖⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢂⣀⡀⠀⠀⢒⣁⣀⡁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠉⠀⠁⠀⡍⠍⠭⠤⠤⠭⠭⠭⠭⠁⠉⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠠⠤⠤⠀⠐⠒⠒⠒⠒⢒⣒⣒⡒⠒⠒⠀⠤⠤⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡠⠊⢀⡭⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠑⢤⣀⠀⠈⢂⠀⠠⠆⡠⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠢⣀⠀⢉⣐⠖⠀⠀⠙⠀⠐⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠈⠁⠀⣀⠠⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⡾⠨⡄⠀⡈⢑⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠀⠀⠀⠀⠀⠀⠐⠄⣀⣀⣉⡼⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠊⣠⠔⢋⠇⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣞⣀⠠⠚⠀⡴⡀⠢⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢆⠈⠲⢄⣡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠉⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣤⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⣷⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣾⠋⠉⢻⡆⠀⠀⠀⠀⢀⡿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠻⠶⠀⠀⡇⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠰⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢸⡇⠀⠀⠀⣼⠃⠀⣠⠶⠛⠛⢶⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣼⠃⢀⡾⠁⠀⠀⠴⠟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣠⣼⣧⣠⣾⣅⣀⣀⣀⣀⣀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠋⠉⠉⠉⠉⠉⠉⢻⡅⠈⢿⠉⠉⠉⠉⠉⠉⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⢸⡇⢀⣠⣤⠴⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁⠀⡀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡄⠀⠀⠀⣾⣤⠶⠛⠛⠉⠻⣦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣿⣿⣿⣦⡀⠀⠀⢀⣴⣿⣿⣿⡏⣿⣿⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⣿⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⢠⣿⣿⣿⣿⣿⣥⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⣠⡾⠃⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡟⠁⠀⠀⣿⠀⠀⢀⣤⠞⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢰⣧⡴⠞⠋⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣼⢋⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⠈⠻⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠠⠔⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡠⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢀⠀⠂⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠁⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠒⠒⠓⢄⠀⠐⠒⠒⠂⠀⠀⠄⠀⠀⢀
⠀⡀⠄⠀⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠥⠸⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠂⠠⡀
⡌⠀⠀⠀⠀⠂⠀⣠⣤⣤⣤⣀⠀⠀⠤⠖⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⠀⢂
⡗⢌⠀⠀⠀⠀⠘⠿⣿⣿⡿⠟⠀⠰⢾⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⠀⠀⠀⠠⢀
⡇⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠁⠀⢸⠄⠀⠠⠄⡀⠈⠢⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠒⠒⠒⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⡶⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣿⣦⠀⠀⠀⠀⠀⠀⢸
⠃⠀⠀⠀⠀⠀⠀⢿⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⠟⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠂
⡆⠀⠀⠀⠰⣈⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠀⠀⠐⠄⠀⠀⠀⢰⠀⠀⠀⠠⠂⠀⡠⠊
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⢀⣠⣶⣿⣷⣤⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠣⣤⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠴⠋⠛⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢟⡛⠯⠭⠒⠁⠀⡠⠖⠒⠲⢤⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢱⠘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣠⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣴⡶⠟⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⢁⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠀⠙⢌⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠒⢚⡉⠉⠒⠤⡀
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠚⠁⠀⠀⠀⠈⠑⡄⠀⠈⢣⡀
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣍⣉⣓⠒⠒⠒⣒⣪⣭⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣈⣁⠐⠒⠠⠤⠀⣀⠀⢀⣀⣀⣀⠀⢀⡀⠀⠤⠄⠒⠂⢈⣀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠒⠶⠤⠤⠤⠴⠖⠒⠒⠒⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠉⢁⣠⣾⡠⠚⣩⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣉⡠⠔⠒⠉⠀⠀⠹⣏⠠⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠨⡀⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢶⣶⣷⠠⢹⡿⢿⣋⣐⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⣥⣶⣾⣿⣧⣼⡦⠼⣶⣶⣿⣿⣻⢶⣭⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣵⣿⠟⠉⣸⣿⣿⣿⣿⠳⡶⢾⡿⣿⣿⣿⣷⣜⢿⣦⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣿⣿⣷⣿⣿⣿⡿⠟⢁⠌⠀⠱⠀⠙⢦⡉⠛⢿⣿⣿⣿⣧⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢱⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⢦⠀⠈⠛⢿⣿⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠟⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⢢⡉⢻⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢻⡀⠃⠜⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠙⠻⢤⣀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⣸⠿⠃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢙⣒⠦⢤⣄⣉⣉⠙⠛⠒⠒⠶⠶⠶⠾⠶⠶⠶⠶⠶⠒⠚⠖⠒⠋⢉⣉⣀⣤⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠭⡉⠓⠢⠈⠉⡉⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⣛⣉⣉⡭⠄⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢲⣤⣄⣉⠐⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⣒⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⣏⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡿⣿⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⢹⣿⠏⢠⣶⢹⣿⣿⣿⣿⡟⣫⣽⣿⢟⣿⣿⣿⢟⣿⣿⣿⢳⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡼⣿⠀⢸⣿⣿⡟⣛⢿⡿⠀⣟⣿⠁⣾⣿⣿⣿⣿⣷⣿⡟⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⡇⣿⡀⠸⣿⡿⠀⣿⡾⠃⠰⣿⠃⢰⣿⠋⢰⡿⢋⣾⣿⣻⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢿⣿⣷⣄⣙⣵⣄⣙⣿⡇⢸⣿⠀⣶⣷⣄⣿⣧⣘⣿⣿⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠿⣿⣾⣿⣿⣿⣧⡘⢿⣦⣙⣿⣿⣿⣿⢿⣿⣿⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⣿⡀⢿⣾⡏⢹⣿⣻⡋⢻⣿⡏⠉⢻⡉⡇⢺⣿⡟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢿⣿⡇⠺⢿⣿⡈⢫⣿⡁⢨⣿⡇⢰⢿⡿⢂⣿⣿⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣷⣿⣿⣷⣼⣿⣤⣼⣿⣷⣿⣿⣿⣿⣿⡏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣿⣿⣷⠀⢉⢻⣿⡏⠁⣿⡇⠰⣿⠇⣿⣿⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢻⣿⣿⡇⢸⡏⣿⡇⢠⢿⣧⠀⠛⣸⣿⢹⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣟⣿⣇⠸⡇⣿⡃⣤⢸⣿⣧⡀⣿⡏⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣿⡿⢀⣹⣿⠁⣿⡼⡟⠛⣱⣿⢹⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣘⠿⢿⣿⣿⣿⣶⣿⣶⣿⣿⣿⣫⡿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣭⣙⣛⠛⠿⠿⠟⣛⣋⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠶⠦⠶⠦⠤⠤⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤⣀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡰⣯⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⢰⠃⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠓⠢⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠖⠊⠉⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⠒⠤⣀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⣴⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣦⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡠⠚⠋⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡶⠿⢦⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠊⠉⠉⠒⠢⢤⣀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣠⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢄⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠉⠑⠢⠤⢄⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⡠⠴⠒⠊⠁⠀⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠤⠤⠬⠭⠭⠭⠭⠥⠤⠤⠶⠖⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢖⡒⠠⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠒⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⢤⣀⣈⡉⠉⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠉⠉⣉⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠉⠁⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⣠⡞⠁⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⢿⣦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠈⠻⣷⡀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢹⣧⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣸⠏⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣁⣀⡀⡀⠐⠋⠀⣀⣀⣀⣘⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠂⣉⣉⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣈⣉⣉⠑⠒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡴⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⠒⠉⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣀⣀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⣠⠤⠔⠒⠂⢉⣉⣥⣤⣤⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣥⣤⣤⣬⣉⡉⠐⠒⠢⠤⣄
⠀⠀⠀⠀⠸⢄⡀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⠤⠌⣉⣉⣙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⣋⣉⣉⠡⠤⠔⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⢍⣉⣉⣉⠉⠉⠈⠛⠛⠛⠛⠛⠛⠊⠀⢉⡉⣉⣉⡩⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⣒⣚⣫⣭⣭⣭⣽⣷⣶⣶⣶⣾⣿⣭⣭⣭⣽⣓⣒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣭⣶⣞⣯⠿⠛⠉⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡯⠭⣭⣿⣛⡻⢶⣮⣍⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣴⣿⡿⣫⣾⢋⡴⠊⣀⣤⣶⡶⠶⠶⣦⣝⣿⣿⠟⠋⣁⣀⣀⡒⠒⠦⣌⣙⠳⣮⡛⢿⣷⣮⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⢱⣿⠁⣿⠁⣾⣿⣟⠁⠀⠀⠀⠀⠉⢻⡷⠊⠉⠀⠀⠉⠟⣿⣶⣌⠻⣷⡌⣿⣎⣿⣿⣿⡆⢳
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠾⣆⠹⣿⣿⠘⣿⡀⠻⣧⡈⠻⢿⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠘⠀⣀⣴⣿⣿⡿⢠⡿⢃⣿⣿⣿⣿⣿⡇⣸
⠀⠀⠀⣾⣿⡅⠀⠉⠉⠉⣛⢻⣷⣌⠻⢷⣬⣑⡀⠈⠙⠳⠶⢤⣭⣭⣉⣉⣙⠂⠀⣠⣴⠶⠿⠿⣛⠫⠥⠚⢉⣰⣿⣿⣿⣿⡿⢋⣴⡿⢶⣄⡀
⠀⠀⠀⣿⣿⣿⠿⡿⣶⣶⣼⣼⡿⣿⣿⣶⣬⣙⠻⠿⣿⣷⣶⣶⣶⣬⣭⣭⣉⡓⢸⣿⣭⣭⣥⣶⣶⣶⣾⣿⣿⣿⡿⠟⣋⣥⣾⣿⣿⡇⢅⠫⠻⣦⡀
⠀⢠⡾⣿⡿⣿⣧⠀⠪⡻⠿⢿⡃⣿⣿⣿⣿⣿⣿⣶⣶⣬⡭⣉⣛⣛⣛⡛⠻⠿⠿⠿⠿⠛⢛⣛⣛⣛⣋⣭⣭⣶⣾⣿⣿⣿⣿⠟⣿⠢⠑⠅⠀⠈⠻⣦⡀
⢠⡿⡀⠙⣷⡹⣿⣷⣤⡀⢈⣴⡿⡎⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣰⣿⣮⡢⠀⠀⠑⡀⠘⣷
⣾⢡⠂⠀⠈⠿⣮⡻⢿⣿⣿⣷⣦⣧⠹⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⣰⢏⢻⣿⣷⡀⢀⠁⠀⠀⠸
⣿⢸⢄⡀⠀⠀⣼⣿⣶⣬⣙⡛⠿⠿⠷⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠐⠀⠀⠀⣴⠋⠀⠂⣿⣿⡇⠈
⢻⡌⢇⠈⠉⠒⠪⠝⣿⠻⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠍⠃⠀⠀⡠⢾⠃⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀
⠈⢿⡌⢦⡀⠀⠀⠀⠸⣄⠈⠀⠒⠉⠩⢝⡻⣿⣿⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠁⢀⣠⣴⣿⡇⢸⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⢀⣾
⠀⠈⠻⣦⡑⢄⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠱⡘⣿⣿⣿⣷⣾⣭⣿⣛⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠊⠀⢀⡼⠋⠁⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠈⠻⣦⣉⠢⣄⠀⠀⠈⠓⠦⣄⡀⠀⠙⠮⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠄⠁⠀⠀⠉⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠷⣦⣍⡒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣋⠩⠄⠀⠀⠀⠀⠀⠀⢀⠠⢐⣨⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠀⠂⠀⠀⠉⠉⠁⠀⠀⠀⣀⡀⠠⢄⣒⣭⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢶⣤⣤⣄⣀⣀⣀⡀⠀⠤⠤⠤⠄⠀⠀⣀⣐⣒⣂⣭⣥⠶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣽⣽⣯⣯⣗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣮⡾⠛⠹⠻⠟⠏⠛⢷⣵⡱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠗⡄⠀⢀⣀⣤⣤⣴⣧⣿⡋⠀⠀⣠⣽⣮⢄⡀⠀⢙⣿⣼⣦⣤⣤⣀⡀⠀⢠⠺⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣾⡟⠛⠉⠍⠉⠉⠙⢿⡶⢝⠜⢯⡽⠣⡫⢶⡿⠛⠉⠉⢩⠉⠛⢻⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⡿⣯⣢⠸⡈⠀⡇⡰⢭⠃⣀⣡⣎⣀⠘⣭⣧⢘⠀⠁⠇⣔⣽⢿⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⣈⡚⠿⢇⣣⣤⢧⣽⣷⠞⢋⣉⣉⡉⠻⣾⡧⡼⢬⣜⡸⠿⢓⣁⠠⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⡇⠀⠂⠨⠍⣾⣶⣻⣟⡿⢱⣞⣭⠴⠦⣭⣳⡎⢿⣫⣟⣦⢷⠉⠁⠐⠂⢸⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⣵⡿⠻⣷⣄⢉⡨⣉⣷⣿⡽⡡⢎⠻⠬⢣⠎⠱⣜⠥⠟⠹⢬⢯⣾⣾⣩⣅⡉⣠⣾⠟⢿⣮⢕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⣾⠋⠀⠀⣜⢏⠧⠛⣽⠟⣱⠶⡛⡆⠠⡢⣬⡐⢂⣥⣔⠄⢰⢛⡶⣎⠻⣯⠛⠼⡹⣣⡀⠀⠙⣿⡞⣆
⠀⠀⠀⠀⣠⣀⣄⣠⣗⣿⣷⡖⣄⣞⡾⣁⠄⣸⡇⢰⢃⡟⡩⠖⢂⠻⡿⣫⣝⢿⠟⡒⠲⢍⢻⡘⡆⢸⣇⠠⣨⢷⣳⣠⢐⣾⣿⣺⣄⣠⣀⣀
⠀⠀⠀⠀⠙⠉⠋⠙⡯⣿⡿⠅⠋⢿⠷⡋⠊⢹⡇⠸⡌⣧⣑⠦⠌⣴⣷⣝⣫⣾⣦⠥⠴⣊⣼⢡⠇⢸⡏⠑⢉⡾⡽⠙⠬⢿⣿⢽⠋⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⣄⠀⠈⢫⣎⡖⣤⣻⣦⡹⠾⣭⠇⠐⠝⢻⠡⠌⡟⠫⠂⠸⣬⠷⢏⣴⣟⣤⢶⣱⡝⠁⠀⣠⣿⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⡻⣷⣴⣿⠋⣈⣙⣋⡿⡿⣳⡑⣆⣰⢒⡜⢆⡰⢣⡒⣦⡰⢎⣞⢿⢿⣉⣚⣉⠙⣿⣦⣾⢟⡪⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⡇⠀⠄⠐⣀⢿⠻⣽⣫⣷⡸⢯⣛⠲⠖⣛⡽⢇⣾⣝⣫⠟⡾⣀⡀⠠⠀⢸⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠒⢉⢥⣶⡎⡝⠛⡖⣺⡿⣦⣈⣉⣉⣁⡴⢿⡟⢲⢚⢫⢱⣶⡬⡉⠐⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣷⣟⠝⢰⢁⠁⡇⢰⣛⡄⠉⡹⢏⠉⢠⣟⠞⢸⠀⠀⠆⠫⣻⣾⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⢿⣧⣤⣀⣂⣀⣀⣠⣾⠷⣪⢦⣞⣳⡴⣝⠾⣷⣄⣀⣁⣘⣀⣤⣼⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡦⠃⠀⠈⠉⠛⠛⠻⡟⣿⣅⠀⠈⠑⣺⣟⠊⠁⠀⣨⣿⢻⠟⠛⠛⠉⠁⠀⠘⢴⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢟⢷⣤⣸⣴⣦⣇⣤⣾⡻⡱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢭⣻⣻⣿⣟⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡐⠒⢦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠰⡀⠀⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠸⣦⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠉⠙⢦⡀⠹⣄⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⣳⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣄⣀⡤⠒⠻⣄⠈⢯⡁⠀⠀⠀⠀⠀⠀⣀⠴⠤⢄⣀⡀⣀⠀⠀⢀⣀⣀⣭⠶⠚⣿⠚⢉⣀⣀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉⠙⠛⠛⠓⠒⠛⠒⠒⠒⠒⠒⠒⠛⠒⠒⠛⠛⠛⠉⢯⠉⠉⠁⠀⠀⠀⠀⡿⠚⠉⠁⢹⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣼⠀⣠⠖⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡼⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢰⣷⣚⣋⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⡉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⣠⡴⠛⠉⠁⢀⣀⣠⠤⠤⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠉⠉⠉⠉⠒⠠⠀⠀⣹
⠀⠀⢠⡾⠁⢀⠠⠐⠉⠉⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⢻⡛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⢉⡴⠋
⠀⠀⠀⠙⢯⡙⠒⠲⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⠤⣤⣤⣤⡤⠴⠶⠖⠒⠛⠋⠉⠉⠁⣀⣀⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⡟⠀⠀⠀⠈⠳⣄⣤⡤⠴⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡴⠊⠁

⣿⣿⣦⣀⠀⣹⠜⣶⣛⠚⣧⠆⠉⣲⠻⣯⣿⣿⣿⢿⣏⣽⣯⣯⠙⠀⠀⠿⠁⠆⠈⣿⠈⠛⠈⠛⢿⣶⡦⠟⠀⢰⠏⠀⣹
⣿⣿⣿⣿⣿⣷⣤⢼⠉⠀⠈⠛⠆⢸⠓⠿⡳⢿⣿⣿⠻⡿⣵⣿⠦⠏⠉⣸⣁⡀⡀⠆⠀⠀⠀⠀⠈⣿⠁⠀⠦⢀⢠⠤⠛
⣿⣿⡿⠿⠷⣬⣿⡦⢻⠧⡶⠄⠀⠂⢐⡷⢿⡙⠻⠏⠛⠻⠷⠎⠉⠀⣶⠻⢿⡿⡿⡀⠀⠆⡀⠀⢤⠈⡀⠊⠀⣸⣦⣴⠿
⣿⣿⣶⣶⣾⢁⣿⣧⡾⢞⣀⣤⠀⠠⠮⠗⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣨⠄⠼⠘⠀⣡⡉⠀⠀⢂⣠⣄⠀⡀⠉⡿⠁⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣟⣛⢭⠽⠛⠛⠠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠟⡂⢴⢈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⢁⣀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣠⣿⠏⣿⣾⣆⠀⠀⠰⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣾⣟⣀⡀⣠⡀⠀⠀⠀⠠⠴⠦⠄⠀⠀⠀⠀⠦⠀⢀⣻⡬⢹⡛⡿⢹⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠘⠀⡀⠐⠦⠀⠀⠘⣃⡤⡾⠀⠀⠂⠀⠀⠰⣦⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⡦⠲⣀⠀⠁⠀⠀⠀⠀⠀⠰⠞⢫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡶⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠒⠎⢀⠀⢀⣄⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠅⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⣶⣯⣀⣳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡷⠦⢀⠀⠀⠀⠀⢀⣬⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠸⢫⣮⣤⣀⡘⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢤⣴⡇⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠨⣿⠑⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢺⣥⡾⢿⣧⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⢉⣳⣾⠯⣷⣶⠟⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡄⠀⠀⠀⠀⠀⠀⣴⠙⢻⣌⣾⡟⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣴⡶⠆⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢃⣴⣿⣷⣾⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⡀⠛⢠⡤⠿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣦⣣⣤⠒⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣋⣤⠖⠀⢀⠀⠀⠲⢶⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠠⠶⠾⢿⣭⣭⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠓
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⣿⣦⡝⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⣯⣍⠉⠉⠀⢠
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠌⠛⠃⠠⣰⡆
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⡇⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠁⣆⠀⠈⠀⢰⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡟
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡇⠒⣿⠇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣿⣿⠀⠘
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠃⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⡇⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⠃⠀⠀⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⡏⠀⠀⠀⣼⣿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣼⡿⠁⠄⠀⢠⣾⣿
⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠁⠀⠀⢀⣾⣿⡇⠀⠀⠆
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣼⢟⣿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠜⢁⣾⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣶⡾⠋⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⡿⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⠁⠀⡀⠀⠀⠛⠁
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠒⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⢰⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠐⠲⡄
⠞⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢀⣐⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣈⣥⣴⠆⣝⠙⠀⢩⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣴⣿⣿⣟⣪⣕⣶⣬⡁⠀⠀⣀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠎⣆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢿⣿⣿⣿⣿⠿⢿⣤⣶⣄⠶⢄⠺⠉⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣩⠇⣸⡇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠶⢽⡻⢿⣿⣿⣶⣾⣿⣿⣶⣻⣿⣦⣤⣌⡠⠀⡀⠀⠀⠀⢠⣀⣴⣤⡾⢞⣡⠾⢹⡏⣰⣦⠾⢦⣶⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⠂⠭⢙⠛⡛⠻⠿⠷⠿⠿⠿⠿⠿⠷⠾⢛⣚⣛⣩⣭⡭⠴⠚⣋⡩⣂⣽⣿⡏⠀⠀⠀⠈⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡅⠰⠶⣖⡻⣯⡷⢝⣫⣿⣿⣇⡀⠀⠀⠀⡸⢟⡇
⠀⠀⠀⠀⠀⠀⢀⢀⣤⣖⣘⣻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠶⢬⣝⣲⣭⣶⣿⣯⣿⣿⡯⣫⣷⠞⢡⣾⠁
⠀⠀⠀⠀⡠⠂⠵⣗⡽⣌⢽⣷⣾⣧⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠿⣟⣻⣿⣿⣷⣿⣿⣿⡿⠟⣋⣴⣾⣿⣷⣏⢂
⠀⠀⠀⡐⢰⠳⠺⡋⣾⠃⡸⡿⢻⣿⢆⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢁⣚⠻⢿⣛⣾⣿⣿⣟⣋⣥⣶⣿⣿⣥⢪⠁⠻⠀⡇⠃
⠀⠀⠀⡅⠁⠇⢇⡜⠊⠈⠈⠘⠓⠩⠻⣧⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣷⣾⣿⣿⣿⣿⣟⡿⠟⢻⡟⠾⠛⠛⡛⣁⠁⠀⡠⠁⠆
⠀⠀⠀⠱⡈⠆⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⠓⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⠚⡻⣭⣍⢺⣉⣿⣿⡿⢠⠄⠁⠀⠀⡀⠄⠀⠀⠀⠔⢁⠎
⠀⠀⠀⠀⠈⠢⡑⠤⣝⡣⡀⠀⠀⠀⠀⠀⠀⠈⠲⡅⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⣈⣃⣤⣹⢴⣿⠿⠃⠀⢀⠠⠄⠀⠀⠀⢀⠠⠊⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠠⣉⠂⠁⣀⠀⠀⠀⠀⠀⠀⠱⣈⠁⠀⠒⠒⠀⠐⠛⠂⠛⠙⠛⠋⠉⢁⣠⠞⠁⠂⠀⠀⠀⠀⠀⠠⠔⢊⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀⡐⠂⠀⠀⠀⠀⠀⡉⠓⠲⠶⠶⠤⠤⠤⠤⠶⠶⠒⠛⠉⠀⣀⠀⠀⠠⠀⠀⢀⡤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠥⢀⣒⠂⠀⠤⠤⠀⠀⣉⣈⣀⣀⠀⠀⠈⠩⠤⠄⠐⣂⡀⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠐⠒⠂⠀⠀⠐⠒⠒⠀⠈⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠈⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠾⠳⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⣠⡤⠶⠖⠛⠉⢁⣀⡤⠤⣄⠀⠘⢧⡀⠘⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡥⠴⠚⠋⠉⢀⡠⠤⠔⠚⠉⠉⠀⠀⠀⠈⢆⠀⠀⠻⣄⠈⠒⠡⠙⠡⢐⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠁⠀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⡜⡄
⠀⠀⠀⠣⡀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⣠⠏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⢀⠎⠀⠀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡤⠚⠁⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠘⠀⢀⡞⠀⣠⣤⡀⠀⡇⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⣨⣧⠔⠋⠁⣀⡤⠚⠁⠀⠀⠀⣠⠾⡆⠀⠀⠀⢇⢇
⠀⠀⠀⠀⠀⠀⡀⢀⠊⢀⠄⢀⡞⠀⠀⠀⠈⢻⣤⡷⠤⣄⣀⣟⠶⠶⠶⠶⢒⣫⠕⣃⡤⠖⠋⠁⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠈⠢⣕⡂⠤
⠀⠀⠀⠀⠀⠔⠠⠁⠰⠁⢀⡞⡴⠾⣯⠞⠉⠙⣿⢧⣀⣀⣀⣈⣩⠭⠭⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠱⠀⠀⠀⡞⠰⡁⢿⡏⠀⣾⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠉⠳⣤⣬⣿⣡⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠑⠒⠒⠲⠦⠤⠤⠤⢤⠶⢶⣶⣖⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⡠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣠⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣸⠃
⠣⣈⠢⢄⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠶⠚⠋⠓⠚⠋⣡⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣠⠴⠊⠁
⠀⠀⠑⠢⠬⣙⡒⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣬⣿⣿⠥⠤⠶⠒⠚⠋⣉⣁⡤⠴⠖⠋⠉
⣀⣀⣀⣀⣀⣀⣈⣉⣙⣒⣲⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣖⣒⣚⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣛⢏⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠈⠀⠀⠀⠐⠊⠀⠀⠀⠈⠂⠀⠀⠀⣐⠤⠤⠄⠠⠄⠱⣀⡀⠀⠀⢧⡋⠸⢰
⠀⠀⠠⣔⣛⣈⣡⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠆⠶⠐⠀⠄⠀⠈⠃⢠⣶⣶⣴⣦⠀⠍⠃⠀⢀⣀⣢⣔⡃⢀
⠀⠀⠄⠙⠛⠛⠋⠀⠀⠀⠀⠓⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠋⠀⠀⠀⠀⣆⠐⠷⠶⠂
⡀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄
⠀⡀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡢⠀⠁
⠀⠀⢰⣶⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣬⣤⣵⠆
⠀⠀⠄⠉⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠔
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠗⢺⣯⡿⠃
⠀⠀⠀⣴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠄⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠁⠀⡤⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠀⣀⠐⠄⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠡⠞⠶⠁⠀⠀⠀⠀⠀⠀⠴⠲⠆⠀⠀⠀⠀⠈⠀⠀⠳⠞⠉⠀⠁⠀⠀⠀⠒⠒⠐⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠒
⠀⠰⠆⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠐⡀⠀⢠⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⠆⠀⠀⠀⠀⠴⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢶⣭⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠻⢦⣄⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣼⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠶⠦⣤⣌⣉⣉⣙⣛⡛⠓⠒⠒⠛⠛⣛⣋⣉⣉⣩⣤⠴⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣯⣝⣓⡒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⣟⣿⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡟⣟⣿⢙⡖⠒⠒⠂⠉⠉⠉⡛⡛⣽⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣿⠇⠟⠯⠘⠁⠀⠀⠀⠀⠀⠀⠅⠤⠟⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠶⠶⠶⠖⠒⠚⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢤⡀⣤⡘⢯⠻⡀⠀⡀⠒⢢⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠒⠛⠺⠇⠘⣆⢻⡸⠹⠦⣠⠆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣽⣿⣷⣶⣶⣶⠤⠨⠤⠥⠤⠤⠶⢶⡶⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⠢⣿⠐⠀⠀⠀⠀⠀⢸⢿⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣞⣿⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡞⣼⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠏⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⡇⣿⣤⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣶⣶⣦⠤⠤⢤⠤⠤⠴⢶⢶⠒⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⣘⣯⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣤⡄⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⡤⠔⣚⡇⢸⡉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠢⢄
⠀⢀⡜⠓⠤⠭⣙⣛⣃⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠚⣇⡀
⣎⠁⠧⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠼⠈⣳
⡷⢍⣒⣒⠤⠤⠤⠭⣍⣉⣉⣉⣉⣉⣩⠭⠤⠤⠄⣒⣂⡭⢶
⢡⠀⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⠉⠀⠀⠀⡆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣀⡤⠤⠤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⢡⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠉⠀⣀⣀⡠⠤⠤⠤⠤⠤⢄⣀⣀⠈⠉⠒⢤⡀
⠀⢸⠀⠀⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⢹⠃⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⡀⠙⡀
⠀⠘⡆⠀⠀⠀⡰⠳⡄⠀⠀⠀⠀⠀⢠⠟⣆⠀⠈⡷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢾⠦⠤⣀
⠀⠀⡇⠀⠀⠘⠁⠀⠙⠀⣀⣄⣀⠀⠋⠀⠈⠂⠀⢣⠈⠑⠲⠤⢄⣀⣀⣀⣀⣀⢀⣀⣀⣀⠤⠤⠒⠊⠁⡎⠀⠀⠀⠑⣄
⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠓⡄⠀⠘
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⣀⡸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⡦⠀⠀⠀⢠⣾⣴⠀⠀⠀⠀⢸⠀⠀⠀⡸
⠀⠀⠘⡉⠑⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⢺⢷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣇⡠⠔⠚⠁⢀⡼
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⠀⠀⠀⠀⠀⠀⠂⠘⠀⠂⠀⠀⠀⠀⠀⢸⠉⠀⣀⠤⠖⠉
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠒⠛⠤⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡞⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢱⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠈⠒⠤⢍⣒⣲⣤⣤⣤⣤⣤⣒⣂⡭⠤⠒⠉⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠈⠙⠶⣖⠤⠤⠤⢤⢄⣀⣀⣀⣀⣀⡠⣤⠤⠤⠤⣶⠶⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣠⣡⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣦⣤⠀⠀⠀⠀⢀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣄⡆⠀⠀⠀⠀⢠⠂⠀⠀⠀⠁⠉⠃⢀⣂⠸⢿⣧⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠉⠛⠰⣿⠀⠀⢠⢤⣠⣤⣦⣶⣷⠉⠈⠉⡿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠋⠀⠣⢀⠀⠈⠀⠈⣯⣽⢿⣿⣤⠀⢾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⡆⡶⢆⣠⡄⠸⢆⠀⠀⢋⡍⠛⠃⠀⢀⣸⣷⡏⣁⠀⠀⢀
⠀⠀⠀⢀⣠⣄⠰⠾⠛⠛⢻⣿⡁⠠⠋⠈⡠⢶⠎⠁⠁⣴⣇⠀⠀⣠⠋⠉⠟⠃⢉⣴⣾⣿⣶⣤⣄⡀
⠀⢠⣤⡽⠋⠁⠀⢀⠐⢤⣜⣿⣷⣄⠐⢆⠀⠀⠀⠀⠀⠈⠙⣙⣿⠄⠀⢈⣤⣤⣼⣿⣿⡗⠀⠈⠉⠳⣤⡄
⠀⠈⢻⡄⠀⠀⢢⣜⡡⠄⠉⠀⠙⣻⣷⣦⠀⠀⠀⠀⠀⠀⠀⢉⡠⢤⣶⣿⠟⢻⣧⢤⠼⣥⡠⡖⣀⢄⣏⠓
⠀⠀⠀⠈⣙⠂⠼⣉⠉⠀⠀⠀⠀⠙⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠈⠀⠀⠀⠀⠋⡉⡐⠯⠛⠋
⠀⠀⠀⠀⠹⢿⣶⣶⣤⣭⣄⣀⣀⣑⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⣐⣤⣭⠴⠒
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢟⣛⣋⡭⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢹⡀⢀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡗⠁⣀⣠⠔⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡠⠗⠒⠚⠉⠀⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⠛⠉⣁⣀⣀⠤⠔⠋⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⡠⠤⠤⠤⠤⠤⠔⠊⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⠤⠤⠮⠥⠤⠤⠄⠀⠀⣀⠞⠙⣆⠀⠀⠀⠀⠀⠀⣰⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⣹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢸⣤⠀⠀⢠⣲⠋⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⢀⡤⠚⠁⣀⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⣉⡒⠒⠶⠤⠤⠤⠬⠭⠤⠤⠶⠶⠒⢚⣩⣼⠃⠀⠀⠹⡍⠙⠒⠒⠒⠲⠶⠶⠶⠒⠒⠚⠛⠚⠉⠉⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁⠀⢸⠀⠀⠀⠀⢷⠴⠲⠤⠶⢤⡤⠲⢤⡤⠦⢤⠴⠒⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠚⡇⠀⠀⠀⠀⡌⠀⠀⠀⢀⠞⠉⣠⠤⢬⠳⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠁⠀⠀⠀⣠⠞⡏⢳⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸⠵⢋⡼⠀⡿⠀⠂⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣗⢋⣠⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠣⣈⠓⠉⣀⡴⠃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠔⠋⠉⠀⠀⠀⢈⠉⠉⠒⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠲⠃⠈⠗⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⡤⣀⣀⣀⣠⡤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠤⠤⠤⠤⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣘⣃⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⠶⠶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⡏⠀⠀⠘⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⡀⣀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣭⣭⣭⣭⣭⣭⣭⣭⠅




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠜⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠒⠒⣾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⠲⢍⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠛⢦⣬⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⠴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠋⠁⠀⠀⡀⡸
⠀⠀⠀⠀⢲⠀⠀⠀⠒⠂⢄⢀⢬⠢⡐⢄⠀⠀⠀⠀⠀⠈⠛⠲⠤⠤⣤⡤⢊⡵⠛⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠑⢄⡉⠉⠂⢤⠙⣦⣅⣈⡒⣬⣖⡖⣰⣀⡀⠀⠀⣔⣚⣁⠔⠋⠀⠀⣀⡀⠤⣐⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢐⢢⡤⣽⣿⣦⢜⣶⠚⣅⠭⠷⣉⠯⢭⣭⣭⣭⠭⠤⠭⠭⠁⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣾⢻⣛⢯⡿⡯⢽⠵⣪⣞⢴⡒⣤⡟⡳⣧⢸⡄
⠀⠀⠀⠀⠀⠈⢁⣌⠁⢯⢀⣷⠶⡪⢷⣸⣟⣛⣻⣦⣜⣸⡿⠷⠿⠿⠟⠁⠲⠼
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠢⠤⠒⠚⠁⢱⠇⠎⢤⣺⡳⢖⣙⡃⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣅⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠀⠀⢀⣴⣿⣿⣿⣠⣾⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⣥⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣴⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⢀⣠⡴⠶⠶⠆⠀⠀⠀⠈⠙⣿⣿⡿⠛⡉⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢻⣩⣿⣧⠀⠉⠉⣤⢤⣤⣄⠀⠀⠀⠀⠈⠛⠋⢀⡆⠈⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⡁⢸⡟⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣟⠀⠀⡀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠀⠀⠀⠀⠹⠟⢛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⠀⡄⢀⣀⣀⣀⣀⣐⣿⣥⠦⠖⠒⠀⠀⠀⠀⠀⠀⠀⢀⡬⠁⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣟⡛⠃⠀⠚⠛⠛⠋⠉⠉⠛⣦⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡠⠗⣸⣇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⣠⠂⠀⠀⠀⠘⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣟⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⣶⣿⡽⠟⠒⠀⠀⠠⠎⠁⠀⠀⠀⢀⠀⠀⠈⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠛⣿⣻⣿⡿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣦⣀⣠⡔⠊⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠿⠟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⣸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠴⢾⣿⠟⣿⠁⠀⠀⠀⣀⣤⡤⠴⠶⠛⠁⠀⣰⣿⣿⡇⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠘⠉⣰⠋⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠘⣻⣿⡟⠁⠀⢠⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢳⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⠂⠀⠀⣀⡀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠉⠀⠀⠸⡇⠀⠀⠀⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡦⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠊⠑⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢿⣿⣿⣿⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠋⠻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠢⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⡉⠍
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⠀⢀⠀⢀⠀⢀⠀⣤⠀⢘⡇⠀⢀⠀⢀⠀⠀⠀⢠⡄⠀⣉⠀⡀⠀⢀⠀⠀⠀⡁⠖⠀⠁
⣿⡿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠃⠈⠲⢄⠀⣿⠀⢸⡇⠀⢸⡆⢾⠀⠀⠃⢸⡆⠀⣿⠸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠂⠀⠈⠀⠈⠀⠈⠀⠈⠁⠀⠈⠁⠈⠑⠀⠀⠀⠁⠀⠉⠀⠉⠒⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠀⠀⠿⠀⠀⣟⠀⠈⡇⢠⠁⠀⢹⠄⢻⠀⡇⠀⠣⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠛⠀⠀⠛⠀⠀⠛⠈⠃⠀⠘⠀⢻⠀⠓⠀⠂⠀⠃⠀⠀⠰⣶⣿⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠸⣶⣝⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⣀⣿⣿
⠀⢠⣀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠌⠳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡽
⠀⠀⠀⠀⠰⣍⡉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡼⠁⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢷⠀⠀⢀⡤⠖⠒⠊⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡟⠀⣠⠛⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠎⢀⡼⠥⢤⣀⣀⣨⡇
⠀⠀⠀⠀⠀⢠⠖⢋⡩⠛⠃⠀⠀⢸⠀⠉⠙⠒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠯⠤⠯⠤⠒⠒⠋⠉⠀⡗⠒⠓⠢⡀
⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⠀⢰⠋⢙⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⡤⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣁⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣼⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢀⡴⠚⠉⠁⠈⠓⠤⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠉⠙⠢⡌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠀⠀⢀⠄⠀⣀⠞⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡉⠉⠒⠒⠶⠤⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠤⠖⠒⠚⢉⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⣜⣭⣙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣨⣭⢂
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣯⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⠿⠿⠀⠘⠒⣶⡖⣶⣶⡖⣶⡶⠐⠁⠐⠿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠛⠁⢻⣿⠇⠘⠃⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡇
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠻⠄⠀⠀⠀⠀⠀⠰⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⡄⠒⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠒⣰⠃
⠀⠀⠀⠀⠀⠙⢶⡛⠛⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠶⣄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠛⣳⡾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⢣⡀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⣶⠶⠦⠤⠽⠿⠥⠤⠶⠶⡖⠚⠋⠉⣰⠏⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⢠⠀⠀⢠⡀⠀⡀⢱⡀⠀⣰⠏⠈⠁⠀⣿⣟⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢳⣤⢮⣴⣶⣼⣧⣤⠇⠀⡇⢠⠏⠀⠀⣠⠖⢋⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⡿⠃⣀⡴⠋⠁⠀⠈⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠸⣿⣿⣿⣿⡿⠀⣀⡼⠓⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠒⠠⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠼⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢱⠤⡀⠀⠀⢠⠊⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡸⠦⠭⠐⣂⣀⣀⣐⣒⣮⠥⠖⠉⢢⠴⠃⠀⠇⣀⣀⡤⢤⣖⣓⣶⣋⣱⣞⡙⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⠵⠚⠋⠉⠉⠀⠀⠀⠀⠈⠉⠛⠾⣏⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⡘⠀⠀⠀⠀⠀⠀⣠⣤⡖⢲⡄⠀⠀⠸⡄⠀⠀⠀⡼⠀⣠⠤⠒⠒⠋⠉⠉⠑⣦⠠⣄⠀⠈⠻⣤⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⡇⠀⠀⠀⠀⠀⠀⢧⡈⠀⡜⠁⠀⠀⠀⠈⠓⠒⣎⠁⠀⡟⠲⠤⠤⠤⠤⠖⠚⢹⠿⡨⠃⠀⠀⠈⢷⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⡸⣄⠀⠀⠀⠀⠀⣀⣼⠛⠒⠦⡄⠀⠀⢸⡶⢧
⠀⠀⠀⠀⠀⠀⠀⣠⠒⢺⠗⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢀⡎⠁⠀⠯⣙⣶⡶⠾⠯⠥⠞⠀⠀⣠⠏⠀⠀⢸⠷⡞
⠀⠀⠀⠀⠀⠀⢀⠽⡶⠁⠀⠀⠀⠈⠑⠲⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠋⠉⠀⠀⠀⠑⠦⠤⠤⣄⣀⣀⣀⣀⣠⠤⠖⠋⠁⠀⠀⢀⡞⠳⡇
⠀⠀⠀⠀⠀⠀⢧⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢊⣉⠩⠭⢭⣉⡭⠽⣭⣉⠑⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⠒⠁
⠀⠀⠀⠀⠀⠀⢸⡉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡰⠋⣤⠈⠰⠶⠈⢧⢈⣁⡬⢳⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⡝⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠢⠤⣬⣧⡤⣴⠛⢤⣉⠤⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡴⠻⣄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣀⣀⣈⣂⣀⣽⢒⣊⣠⠤⠖⠚⠉⠀⠀⠀⠀⣀⣤⠞⠙⣦⣀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣏⢉⣳⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⡤⢤⡖⠚⣯⡁⢸⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⠶⠚⣇⣈⡽⡟⢛⣶⠛⠛⣻⣟⣛⡿⣟⣋⠿⣉⡹⠧⣤⠞⠑⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣩⠤⢤⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⣡⠖⢦⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠠⡇⠰⠋⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⠀⠙⠦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠛⠉⠛⠛⠛⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠋⠉⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣈⣹⢿⡆⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠷⠾⣿⣯⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠼⠟⠋⠁⢸⡷⠋⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸⣧⠖⠛⢲⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢰⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⡰⠋⢳⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⣧⠀⠀⠀⠀⠀⣰⠗⠠⠤⠴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠟⠓⠶⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⣀⣀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡖⠒⠀⠀⠀⠒⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢾⣉⣙⠛⠒⠒⠒⠒⠒⠛⠋⣉⣳⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⠒⠦⠭⠭⣭⣭⣭⣭⣭⡭⠭⠥⠖⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣶⣦⠤⠤⣤⣄⣀⣀⣀⡠⠤⣴⣶⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⣿⣿⣿⡯⡆⠁⠀⠈⠉⡍⢹⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠛⠛⠶⠥⠤⠤⠤⠤⠤⠴⠧⠖⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣦⢦⢻⣳⢀⣔⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣦⣤⣤⡄⠑⠇⠨⠭⣤⠤⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣺⡁⠀⠀⠀⠰⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⣿⣿⣿⡇⠀⠀⠀⠰⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⣿⣿⡿⡇⠀⠀⠀⠀⡷⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠙⠓⠂⠀⠀⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⡒⠀⠒⠒⡶⠊⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⠀⣀⡀⣠⣤⢠⣤⠤⠤⠤⣤⢤⣤⣤⣌⣀⠀⠀⠀⠀⢰⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂⠉⠀⠄⠁⢠⡤⠀⠀⠀⠀⠀⡶⠿⠿⠟⠿⣿⣿⣿⣶⣧⣄⡈⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠦⠀⠀⠀⠀⢠⠀⠰⣖⣀⣠⣤⣤⡶⠂⠀⢀⠀⠉⠉⠙⠃⣭⠉⠻⣿⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢦⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢁⠀⠁⢙⠂⠔⠚⢛⣻⣦⣀⠀⠙⢀⣀⣿⣿⠇⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡙⠒⠤⣀⣀⠀⠀⠀⠛⠁⠈⣉⠑⢀⡀⠀⠀⠶⠿⠿⡶⢶⡯⠟⣁⡴⢳⣾⠃⢀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢶⣦⣮⣍⠁⠀⠀⠀⠀⠠⠶⠿⠷⠤⠰⠿⠒⢛⣍⣡⠀⠀⣿⣿⣰⣿⡟⠉⠉⠻⣖⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⣰⢶⠗⠀⠈⠉⠛⡀⠘⠃⣿⢿⣿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣼⣏⣀⣠⣴⠿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣅⣉⣉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠊⠙⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣟⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠠⠃⠀⠀⠀⠀⠀⠴⣻⣷⣵⣆⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣰⠽⣿⡿⣋⡬⣿⠀⠈⠀⡈⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⡁⢹⣧⣠⣹⣿⣆⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⢀⣩⠾⣿⡾⣥⣾⣷⠤⡀⠀⠐⠀⠘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⣹⣭⣿⡋⠥⣺⠏⣻⣷⣷⣦⣄⣀⡀⠀⣀⠀⡂⢀⣉⣀⣴⣾⣿⣱⢊⣽⣓⣺⣿⣾⡷⠏⠀⠁⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢛⣡⠀⢀⠉⠉⠛⠻⠿⠿⢿⣿⣿⠿⣿⡿⠿⠿⣿⣻⢍⣫⡙⣌⠛⠉⠀⠀⠀⠀⢀⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣥⠬⠋⠀⣴⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠋⠛⠉⠀⠀⢀⣥⢴⢊⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣀⣀⣀⣀⠠⣄⣀⠀⡀⠤⣶⢎⢃⠈⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠂⠀⠂⠁⠁⠀⠀⠀⠘⠓⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠉⡀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣴⢿⠿⠂
⠀⠀⠀⢀⣀⣤⣤⡀⠀⠐⠾⣿⣿⣾⣿⣿⣶⣖⣒⣠⣤⣶⣾⣿⣧⡤⠶⠖⠛⠀⠀⣠⣶⣿⡆
⠀⠀⠀⠈⣿⣿⣿⣿⣶⣤⣀⣀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⣀⣀⣀⣤⣶⣿⣿⣿⡯
⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠟⠟⠛⠋⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢿⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡘⡇⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣶⠛⠉⠉⠉⠉⠉⠉⠉⠛⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠩⣤⣶⣶⣶⣶⣶⣶⣶⣤⠍⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠟⠻⡆⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣠⣌⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡼⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠄


⠀⠀⠀⠠⣤⣄
⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠙⠻⠷⠢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢋⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⠭⢭⣉⣉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢂⣼⠛⠛⢉⣉⣉⣉⣉⣉⢉⡉⠉⠉⠉⠉⠀⠀⠒⠊⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⠜⢰⢀⣠⣤⣄⣀⣀⣀⣀⣀⣠⣀⣀⣠⣍⣉⣉⣀⠐⠒⢯⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⣿⡷⠾⢘⡻⢿⠷⡿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠻⠿⠿⠗⠒⣅⠴⡙⢎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⠻⣿⣟⡐⠠⢶⣦⣤⣀⣀⣒⣶⣶⣦⣤⣤⣤⣤⣤⣴⢀⠀⡞⢄⠛⣘⠴
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡵⡻⢩⠋⢂⣬⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣷⡒⠚⢧⡀
⠀⠀⠀⠀⠀⠀⢠⢏⡞⡝⢁⠏⣰⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣷⡌⡄⣇
⠀⠀⠀⠀⠀⠀⡟⠞⡵⠁⡞⣸⣿⣿⡿⠛⠛⣿⠿⢿⣿⠋⣉⣹⡿⠛⠛⢿⣿⣿⣿⣿⣿⡀⢹
⠀⠀⠀⠀⠀⠈⠧⠚⠀⡼⢠⣿⣿⣿⠃⢰⣶⡿⢠⡄⢻⡈⣿⣿⡁⣾⣶⣿⠟⠛⠻⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣿⣿⣿⠀⣿⣿⣧⠸⣧⢸⡇⢠⣾⡇⣴⣼⣿⠀⠛⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣏⣻⡆⠸⠛⢻⡄⠁⢸⡇⠘⣿⣧⠈⢹⣿⠀⠿⣿⢿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣿⣿⣿⣿⣿⣦⣴⣾⣿⣶⡿⣿⠞⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⣿⣿⣿⣿⣿⣿⣿⣩⣿⢏⡾⣿⢿⡿⣿⢿⠟⡿⣿⣿⣿⣿⣿⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣰⣏⣾⣵⣏⣿⣸⣿⣿⣿⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣏⢿⣿⣿⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⡏⣿⣇⣿⣿⣿⣯⣿⣿⣧⣿⣷⢿⣿⢻⣿⡇⣿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢿⣿⣷⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣼⣿⣧⣿⣾⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢎⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡂⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣩⢆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣈⣉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣁⣀⡭⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣤⡀⠀⠀⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣿⣆⣾⣿⣿⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣾⠿⠿⢿⣿⣿⠟⠻⣿⣿⣟⠛⠛⠳⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠈⠻⣿⣧⠶⣶⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣿⣿⣿⣴⣶⣶⣤⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣷⠻⠄⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠈⠉⠁⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⣠⡆⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⢿⡀⠀⠐⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⢻⡆⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠚⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠟⠛⠛⠛⠿⠶⠶⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠙⠻⣦⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣿⣟⣛⣭⡙⢻⣆⠀⠓⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢿⡏⠉⠛⠛⠿⠷⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠫⢱⡟⠋⠉⠈⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣀⡀⠀⠀⠀⠀⠀⠀⣀⣿⠃⣀⣤⠾⢋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⡿⠉⣿⠀⠀⠀⠀⠀⠀⣼⠿⠟⣋⣥⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠆⠀⠀⠀⠹⣷⡅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⠿⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⢷⣤⡀⠀⠀⠀⠸⠷⠶⠶⠶⠿⠀⠀⣀⣴⢿⣥⣤⣀⡀⠀⠀⠀⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⠀⠀⠀⠈⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋⠁⠀⠀⠀⢉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡶⠀⠀⠀⠀⣿⠀⢀⣀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡄⢰⡆⢾⠋⠉⢹⠄⠀⠀⠀⣾⠙⣦⠀⣸⣃⠀⠀⠀⠀⣿⠀⣟⣽⠀⢰⢫⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢀⣠⡤⠸⣇⣼⡇⠸⣇⣤⠟⠀⠀⠀⠀⠸⣶⠏⠀⢹⡏⠀⠀⠀⠀⣿⠀⠻⠷⠞⠈⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠉⠫⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⢱⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢙⣿⣧⣴⡾⣻⣷⣿⣷⣦⣽⣿⡷⠄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⢟⣽⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⣀⣼⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⣵⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣖⣿⣷⢹⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⢿⣿⣿⡞⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣵⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠛⠁⠀⠀⠀⠀⠉⠻⣷⡹⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢌⠻⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠛⠿⠿⠿⠿⠟⣫⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡅⣷⠐⠛
⠀⠀⠀⠀⣤⣤⣤⣤⣬⣥⣥⣯⣤⣤⣤
⠀⠀⠀⠀⢹⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⡆
⠀⠀⠀⠀⠈⡇⢼⣿⣿⣿⣿⣿⣿⣿⠁⢀⣰⠃
⠀⠀⠀⠀⠀⢿⡸⣿⣿⣿⣿⣿⣿⡿⠞⠋
⢠⣤⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣛⣁⣀⣀⣀⣤⡄
⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⣾⣀⣤⣴⣶⣾⣿⣿⡛⢿⣿⣿⣿⣿⣯⣙⣻⣛⣛⣿⣿⠿⣷⣶⣤⣀⠀⢠⡁⠀⠀⠀⢀
⠀⠀⠀⢰⡝⢿⣟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣽⣿⡷⢞⣴⣧⡶⡋⠀⠀⢍⣆
⠀⠀⠀⠘⠃⠀⠊⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⣭⣴⣻⣿⣿⡿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⢸⣿⣿⢸⡿⠟⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠁⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣃⣀⡤⠖⣉⠔⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⢟⣋⡡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠖⠶⠾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡼⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⠏⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠀⡼⠃⠐⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢹⢷⣇⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣹⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢹⡿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣼⣦⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣬⣍⣛⡻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⢸⡏⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢋⠀⢠⡀⠀⠀⣠
⢰⡆⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣿⣿⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠘⢣⠛⠃⢶⡄⠉
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⢿⣷⣌⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢳⣺⣮⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣮⡙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⣼⢛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡶⢯⣼⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⢿⣵⠆⢲⡎⠁⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⠛⠛⣿⢶⣶⡠⢴⠆⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣼⣦⣤⣬⣛⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣟⣻⣋⣜⣛⣟⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣯⣭⣭⣭⣭⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠊⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢍⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠚⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡠⠔⠊⠉⢉⣉⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⠋⠀⠀⠀⠀⡏⢀⣹⠄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⠃⢠⡴⠶⣄⠀⢣⣀⢁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⣔⠙⠎⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣇⠹⡌⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠛⠀⠚⠉⠁⠀⣀⣀⣀⣀⣠⠤⠴⢋⣰⢿⣶⣖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠊⠉⠀⢸⠟⠉⢹⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠸⣄⡀⠈⠀⠀⠀⠀⠀⠀⠒⠉⣀⡤⠟⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣉⡁⠀⠀⠀⠒⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠭⠭⢥⣤⣄⡀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠐⠊⠀⠘⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣶⣦⣴⣦⣷⡘⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢁⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢋⣾⢿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⡷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣇⢈⡉⠛⢿⣿⠁⠀⠀⢹⣿⣿⣟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⢭⢍⣉⣛⡃⣰⣿⣔⣰⠀⠛⠋⢻⣿⣿⠀⢸⡟⠈⠀⠀⢸⣿⡀⠀⠀⣸⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢰⢰⣶⡦⡇⡤⡤⠤⡤⠀⢻⠁⢸⣿⣿⠀⢸⣦⠀⠀⠀⠋⢸⣿⣷⣿⣿⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠠⣿⠷⠇⣿⣿⣟⣿⠀⢸⡇⢸⣿⣿⠀⠺⠿⣧⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢸⡄⣿⣷⡧⡀⠀⢀⢀⠀⠸⠃⠸⠃⠀⠀⢂⣁⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣋⣤⣤⠂⣭⣸⣍⡉⣀⣞⡇⠀⠀⠠⠀⢨⣭⣭⣭⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠃⠙⠺⠃⠀⠀⠈⡃⠄⠀⠀⠀⢐⣓⢶⣶⠿⣿⢿⣿⠟⠁⠀⠀⠉⠉⠉⠻⢿⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠛⢳⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠬⡧⠀⠉⠀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠖⠀⠂⠂⠀⠀⢛⣛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣛⠐⡓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠒⠲⠆⠀⠆⣭⣭⠀⠀⠀⠀⠀⠑⠀⠀⠀⣭⠬⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠴⠶⠀⠀⠀⠀⣶⢄⡀⠀⠀⢛⢒⣿⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡐⠤⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣇
⠀⠀⣾⡏⠈⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⣿⣿

⠀⠀⠀⠀⠀⣀⡠⠤⠤⠶⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠒⠦⠤⣄⣀
⠀⠀⠀⠀⡾⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆
⠀⠀⠀⠀⡇⠀⠈⠉⠙⠓⠒⠲⠶⠦⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠶⠞⢻⠒⠊⠉⠉⠀⡇⢀⣀⣀
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⡡⠤⡌⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠏⠀⠀⡇⣸
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠛⣆⠀⠀⠀⡜⠀⢀⡜⣠⠃
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢠⣧⠔⣋⠔⠁
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠃⠀⢀⡾⠖⠋⠁
⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠥⢤⣀⡀
⠀⠀⠀⡤⠒⠉⠉⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠉⠓⢄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⡤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠉⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡦⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢠⢀⣀⠴⠊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡌⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠚⠃⠀⠈⠙⠋⠂⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⣟⠶⣤⣶⣾⣿⣿⣿⣿⠶⠴⢞⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⣤⡜⢶⣛⢇⡀⠩⠘⣿⣷⣶⠆⠀⠀⠀⠀⠻⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡶⠋⠐⠾⠿⠿⠋⠶⢶⣤⠀⠳⡄⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠔⢉⣤⣶⣤⢦⣤⣤⣶⣶⣾⣿⠀⠀⡇⠀⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠜⠉⠀⢀⣤⣄⢀⣀⣤⣽⠀⠀⣷⠀⠈⣿⡟⠅⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠻⠿⣿⡏⠛⠁⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⠴⠶⠶⢶⣶⣦⣼⡿⠄⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⡤⠶⠛⠁⢠⣤⣤⣤⣤⢀⠉⡯⠙⢦⣄⡀⠀⠀⠐⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠂⣀⣠⣁⣀⢠⣴⣦⣭⡿⠿⠿⠟⠛⠁⠀⠀⠈⠉⠉⢻⡁⠰⣿⣿⡃⠀⠀⠀⠀⠀⠹⢗⡇
⠀⠀⠀⣴⠞⠋⠁⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠷⠦⠤⠤⠤⠴⠖⠚⠉


⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢴⣄⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠈⣿⡆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⢀⣴⠟⠁⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠸⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠹⣷⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠻⠿⠶⠶⢶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠶⠿⠟⠛⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠸⠿⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠙⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣶⣶⣤⣤⣤⣤⣶⣶⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣍⣉⣋⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣭⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠤⠤⢤⡤⣤⣤⡄⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣵⢿⠹⢭⣅⡀⠀⠀⠈⠉⢒⢲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠈⠣⣀⠀⠀⠀⠀⠀⠈⢸⢼⢿⡾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣌⠁⢠⣆⠀⠀⠀⠠⠄⣀⣀⣀⡀⣘⢡⡀⠀⠀⠤⢤⠤⠄⢰⣮⣽⣿⣳⣁⣠⢖⣺⣟⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⣿⣷⢾⣷⡦⠒⣂⢤⣴⢻⣿⡿⣷⣇⣤⠀⠀⣀⡴⠊⣷⠤⢿⡻⣿⠃⣼⣶⣥⡿⠟⢻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⡿⠑⡞⢿⠀⠹⣯⠽⢋⣩⢏⠼⠻⠿⠟⢳⢶⠴⠶⣎⣹⢡⢻⡑⠊⡈⡇⠸⣿⠉⠁⠀⢀⣾⣿⠃
⠀⠀⣠⣴⣚⡋⠉⠹⢄⠳⠄⢸⣦⣛⠀⠀⠀⠈⠻⣿⠯⠥⠤⠤⠀⠀⠈⠃⢳⠒⠖⠉⠎⠀⢻⠯⠁⠴⢞⠇⣡⠤⣦⣺⡽⣯⣀
⠀⣴⣿⣿⣿⣽⣄⣰⣦⡱⢄⡊⠻⣶⡄⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⡼⣩⣤⣶⣷⣿⠡⠐⡀⠉⢲⢄
⠀⢻⣿⣻⣿⣿⠾⣭⢙⠅⠀⠁⠀⠙⢿⣦⣄⠀⢀⠀⠀⠀⠀⠀⢀⠈⠉⠈⠉⠉⠀⠀⠀⡀⢠⢴⣼⣿⠟⠉⣩⣿⡿⣣⠞⣥⣶⣿⣿⣽⠆
⠀⠀⠙⢯⣝⡻⠿⣿⣽⣮⣶⣶⣦⣶⣤⣽⣿⣷⣾⣧⢸⣡⡀⠀⢸⢀⣄⠠⣠⣠⣲⢇⣾⣥⠷⠋⠁⠀⠀⠀⠠⠤⠞⠛⣒⣻⠿⣛⠟⠉
⠀⠀⠀⠀⠈⠛⢿⣶⣯⣭⣝⣛⡛⠿⠿⠿⢿⣿⣿⣿⣟⡿⡷⠿⠾⠿⠿⠗⠛⠛⢛⣻⢏⣀⡀⠤⠤⢤⠖⣒⣒⡨⣭⣭⣷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠯⢿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣯⣭⣭⣭⣼⣶⣷⣶⠶⠶⠶⠒⢒⣋⣉⠹⠙⣁⣐⣲⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠶⠷⠞⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⣴⡄
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠠⡹⠞⠉⠁⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠣⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⣀⢀⠀⢀⣀⡀⠀⠀⠀⢀⡠⢒⡨⢏⣛⡐⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠯⠭⠒⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠈⠉⠉⠉⠉⠉⠓⡄⢀⢎⠔⡪⠄⠤⠤⠊⠙⢄⣣
⠀⠀⠀⢀⠤⠃⠀⠉⠀⠀⠀⠀⠐⣆⡀⢸⠉⠉⠉⠉⠉⠉⠉⠉⣹⠀⢨⣉⣚⣛⣒⣒⣚⣒⣒⣒⣹⠀⠸⡉⢉⣉⣉⣩⣹⣉⣉⡉⣩⠆
⠀⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⡖⠁⠈⡇⠆⠤⠤⡤⠤⠄⠴⡆⠀⠈⣄⡀⣀⠀⠀⠀⢀⡀⣀⡇⠀⠀⡧⠤⠤⠤⢼⢼⠤⠤⠤⡖
⠀⠀⠀⠀⢉⠉⠉⢉⠀⠀⠉⠉⠉⡇⠀⠀⡇⢀⢤⠼⠿⡟⡁⠀⡇⠀⠀⠇⠈⠉⢉⡉⣉⠉⠁⢈⠁⠀⠀⢸⢠⠐⠄⢸⢸⠐⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⢫⠑⢫⠇⠀⢀⠃⠀⠀⡇⠀⢸⢰⠀⠀⢰⠀⠁⠀⠀⢀⠀⡀⡐⡂⠙⡀⡀⢸⠀⠀⠀⢸⠘⠲⠎⢸⢸⢈⠉⢆⠁
⠀⠀⠀⠀⢸⡀⠀⠀⠑⠃⠀⠀⢸⠀⠀⠀⢡⢈⣚⣒⣂⣒⣁⣴⠀⠀⠀⠸⡘⠒⠛⠃⠃⠃⠃⡀⠀⠀⠀⠀⡆⠐⠐⢸⢸⠀⠀⢸
⠀⠀⠀⠀⠀⡍⠉⠉⠉⠉⠉⠉⠹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⢣⠈⠀⠸⠌⡔⠂⡘
⠀⠀⠀⠀⠀⠱⠤⠄⠀⠀⠤⠤⠃⠀⠀⠀⠈⠧⠤⠀⠀⠀⠤⠃⠀⠀⠀⠀⠱⠤⠄⠀⠀⠤⠴⠁⠀⠀⠀⠀⠘⠤⠤⠀⠀⠬⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠄⠀⠀⣀⣀⠀⢁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⡠⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠔⠉⠁⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢈⠀⠤⡁⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠰⠊⠉⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣌⡒⠂⠤⢄⡀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠪⠐⠠⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⡶⢂⠚⡃⠈⠖⣠⣀⢲⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⣀⣠⣭⣧⣼⣄⣧⣌⣼⣾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⡴⣼⡉⢦⣤⣙⣻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢛⣛⣯⣵⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣟⠀⠘⣇⢸⣿⠯⠍⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⢄⠀⢻⡄⢻⡄⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣑⡂⠿⣄⠛⢆⢠⡻⢿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣷⣌⣉⢡⣼⣿⣷⣄⠀⠀⠀⢀⡴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡶⠶⠚⢭⣿⣿⣭⣭⣛⣛⣛⣛⣺⣛⣛⣼⣓⣂⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣿⣿⣿⡃⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⠿⠿⠛⣛⣿⣿⣿⠖⠈⡂
⠀⠀⠲⢼⣿⣿⣶⣶⣽⠿⣿⠿⣿⣿⣦⣤⣉⣙⣛⠛⠛⠛⠛⠒⠶⠆⠀⠀⠀⠀⠛⢉⣉⣡⣴⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢤⣄⣒⠀⠁⠉⠉⠉⠛⠛⠛⠿⠿⠷⠶⠿⠶⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⢦⣤⣤⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣭⣤⡽⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⠁⠲⣾⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢰⠇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⡏⠀⢠⡟⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡾⠀⢀⡾⠀⠀⣰⠏⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿⠀⣸⠇⠀⣼⠃⠀⣰⠏⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠛⢹⡇⠀⣿⠀⢠⡏⠀⢠⡏⠀⢀⡞⠘⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣩⡴⠞⣋⣁⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡃⠀⢸⠇⠀⠀⠒⠤⣌⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢺⣿⣾⣟⣭⣤⣴⣾⣇⣶⣿⣰⣾⡇⣶⣾⣧⣦⣾⣠⣄⣈⠙⠲⣦⣝⠂⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠖⠀⣀⣴⠟⢸⣛⡷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠙⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⣁⣀⣠⣤⠶⠟⣫⣴⡇⣚⣛⠲⢤⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⢋⣉⣩⣥⣴⡾⠿⣛⣥⠀⠉⠉⠙⢷⣷⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⠃⢹⣟⠷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⣉⣉⣭⣴⣶⠿⢛⡅⠀⠒⢂⣠⢿⢏⠄⣾⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⡭⠔⣀⠀⠀⠀⢿⡛⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⣛⣋⣩⣥⣶⠾⡛⢡⡶⢞⣫⣵⡿⢋⡾⢣⣄
⠀⠀⠀⠀⢀⣴⠏⣡⠞⢫⡴⢋⣡⠔⠀⠀⠈⢿⡷⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⣛⣋⣭⣵⣶⠿⢡⣾⣿⠿⣛⣭⠾⠋⠀⠠⡉⢳⣄
⠀⠀⠀⠀⣼⠃⣸⡏⢰⡏⠠⡟⠁⣴⠉⠀⠀⠈⢻⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⣛⣋⣩⣭⣶⠞⣡⣿⣷⣶⣿⣿⣷⡄⢦⡘⢦⢹⡄⢻⡆
⠀⠀⠀⠀⢹⣆⠸⣧⡈⠷⣄⡻⢦⣌⠳⠤⠀⠀⠀⠙⢿⣖⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣩⣭⣭⣶⣾⣴⣿⣯⢿⣿⠹⣿⡌⣿⠈⡧⢸⠀⢁⣼⠃
⠀⠀⠀⠀⠀⠙⢦⣝⠻⣶⣌⡛⠶⢬⣝⡓⠀⠀⠀⠀⠀⠙⠿⣷⣶⠦⢤⣤⣤⣤⣬⣭⡭⢷⣿⣿⣿⣿⠿⣣⣿⡟⣸⣿⢀⣿⢣⠏⣰⠃⢈⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠰⠿⠷⢦⣍⣛⠷⠶⣤⣉⡉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⢖⣛⣛⣛⣭⣥⡶⠿⢛⣩⣾⠟⠡⠞⠁⢀⣠⣤⣾⣿⡿⠯⠍
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⣿⣷⣶⣦⣬⣅⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣭⣭⣭⣭⠵⠶⠶⣛⣛⣉⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣯⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠶⠶⠶⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣉⣉⣉⣉⣩⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣋⣉⣉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠄⠔⣀⡠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⣀⡨⢄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠⠊⠀⢰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡀⠸⠈⡁⡇⡠⠀⣠⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃⣀⢵⣄⠀⠃⢞⡈⢉⡽⠠⠔⠁⠀⠉⠉⠉⠉⠉⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⣚⠳⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠉⠀⠒⠤⠄⡀⠀⣀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢑⠃⡠⠃⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡀⠑⢌⡠⢁⣳⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⡀⡈⣿⣛⡠⠊⠁⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠐⠲⠖⠀⠉⠀⣰⢿⡵⡂⢠⠆⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⣔⠀⠀⠀⠀⢀⡷⢛⡗⢞⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢄⡀⠅⠋⠘⠤⠊⡶⢯⡶⠛⠹⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠦⣙⣀⢠⠛⢄⠊⢃⠸⢷⠆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣿⠀⠆⡡⢚⠀⣆⠤⠢⠊⡜⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠠⠔⠂⣡⣂⣤⣤⣾⣽⡾⠷⠔⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠛⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⡄⠀⢴⣿⡿⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠐⠒⠀⠙⠛⠛⠣⠤⠘⠛⠶⠶⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣄⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⣤⣄⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⢠⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⣀⣤⣤⣶⣶⣶⣤⣄⡀⣸⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣀⣴⠷⠒⠲⣶⣶⣿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣴⣶⠸⣿⣿⣿⣿⣇
⠀⢠⣿⣿⣿⣿⣿⡿⢸⡏⠀⠈⠉⠉⠛⠛⠛⠉⠉⠀⠈⢿⠃⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⢻⣿⢠⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢏⣼⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣧⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣆⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣏⣤⣭⣥⣼⣭⣿⣥⣾⣯⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠶⠶⠦⠄⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠊⠉⠀⠀⠀⠀⢀⣀⣠⣤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⡤⠤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠒⠈⠉⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⣿
⠀⠀⢀⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿
⢀⣘⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣆⠀⠠⠄⠀⠤⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠹⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠙⠿⣿⣿⣶⣬⣽
⣿⣿⣿⣇⢹⣧⣤⡤⠀⠀⠀⠀⠀⠀⢠⠀⡾⠀⢀⣀⣀⣀⣄⣀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿
⣿⣿⣿⢿⣟⠛⠉⠀⠀⠀⠀⠀⠀⣀⣘⣒⣛⣤⣿⣭⣭⣿⣿⣶⡶⠿⠿⠷⠆⠁⠀⠀⠴⠶⠾⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⡁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⢸⣷⣤⣴⣷⠀⠇⠳⠢⠎⢡⢰⠉⠦⠴⠉⡆⣸⣷⣤⣾⣿⠀⠰⠹⠖⠴⡍⡄⠀⠀⡘⡡⠤⢼⠱
⠀⢺⣿⣿⣿⣿⠀⠀⡄⠀⣤⢸⢰⢰⡆⢠⡖⡆⣿⣽⣿⣯⢹⠆⢀⢠⠀⢠⡄⡇⠀⠀⠃⣤⠀⣠⠈
⠀⠘⣿⣿⣿⠟⠀⢦⡀⠒⢀⠌⠘⣌⠑⠊⣡⠃⠹⣿⣿⣿⡿⠀⠹⣄⠀⠀⡰⠃⠀⠀⢧⠀⠀⠈⡸
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠉

⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠾⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠋⠉⠉⠉⠛⠛⠻⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⣠⣤⣶⣦⣄⡀⣀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣧⣠⣼⠛⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠹⣿⠋⠁⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⢻⣦⣀⣴⣿⣯⣀⣤⡶⠾⠛⠛⠋⠉⠉⠛⠛⠷⣦⣄
⠀⠀⠀⠀⣾⡏⠉⠛⢿⣦⣀⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢀⣀⣼⡿⠛⠉⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠷⠶⣦⣤⣤⣴⣿⣷⣤⣀⣀⣀⣀⣠⣴⣿⣧⣤⣤⡴⠶⠞⠛⠉⠁⠀⠀⠀⣿⣧⣤⣶⠶⠾⠛⠛⠛⠻⠷⣦⣄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣀⣤⣶⠾⠛⠁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡶⠾⠛⠋⠉⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠷⠶⠶⠾⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠒⠂⠈⠉⣉⣁⣀⡀⠀⠀⠀⠀⠀⣈⣉⣉⠁⡒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣤⣿⣿⣡⣒⡠⢌⡉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⢢⠈⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⠉⠉⠆⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠘⠁⠈⠑⠄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⡀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢡⠀⠸⡀⠀⠀⠀⢀⠔⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⣦⠀⠀⠑⢤⡀⠀⠀⠀⠀⡌⠀⠀⡰⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⡀⢡⠀⠀⡀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢁⡀⠀⠀⠉⠢⣀⠀⡰⢁⡠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠉⠁⠀⠀⠣⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠘⠀⠀⠀⡀⢨⡵⠍⣉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠉⣁⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠘⡴⣊⠵⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠉⠀⠀⠀⠀⢡⠀⢢⠟⠵⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠤⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⣁⠒⠰⠊⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠄⠀⠉⢈⠁⠀⠄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡠⠂⠩⠀⠤⠐⠀⠀⠅⠂⠀⠄⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠘⠠⠀⠂⠀⠠⠀⠐⠁⠠⠂⠀⠀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⠀⢀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⣤⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣠⣤⣤⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⢿⣞⣿⣿⢿⣛⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⡼⣿⢿⣷⣿⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣻⣿⡿⣟⣿⣻⣷⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣾⣿⣿⣯⡻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡸⣿⣿⣿⣿⡿⣺⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣮⣿⣿⣯⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⠿⠿⠿⠿⠿⠿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠀⠀⠒⠂⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⠀⠄⠀⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⠠⡄⢀⠀⠀⢒⠒⠒⠯⣷⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣬⣜⠋⠀⣤⣉⡻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⠋⢘⣿⣿⠄⡀
⠀⠀⠀⠀⢸⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣭⣿⡿⢠⣇⡴⠞⠉⠙⣦
⠀⠀⠀⠀⠘⣿⣮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⣼⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣭⣟⣛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣻⡍⡋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣇⠁⢤⣀⢀⠀⠀⠀⠀⠀⢀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠿⠿⠦⠧⠤⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣅⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢤⡈⢓⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡶⠊⠀⠘⠀⠀⠙⢶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣋⡁⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠛⠿⠭⠭⣾⣒⣒⣒⣶⣶⣶⡾⠷⠿⠛⠋⣩⠷⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⢻⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢙⣋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣏⠒⠤⢀⣀⣀⡀⠀⣀⡀⠤⢒⡏⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣌⣉⣓⡒⠒⠒⠶⠶⠒⠒⢊⣉⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠛⠛⠻⠿⠿⠛⠛⠋⠉⠁

⣿⡿⠿⠿⠻⠿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢄⣀⠀⠀⠐⢴⡟⠆⠀⠰⣾⠧⠀⢰⢦⠀⡀⢂⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠤⠄⠞⠻⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠚⠉⠐⠒⠲⠤⣀⠀⠀⢀⠄⣠⠄
⠀⠀⢸⡅⢸⡁⠀⠀⠀⢸⠀⠒⢒⠢⡀⠀⠀⠀⡠⠺⠷⠶⠀⠀⠀⠀⠀⠀⠐⠛⠺⡗⢦⡀⠐⠁⢀⡀
⢾⠀⢺⠀⢮⢠⠆⠀⠀⠈⠦⠼⠷⠤⡐⠀⢀⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠑⡄⠀⠹⠛⠂
⢻⠀⣻⠄⡿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣷⠶⠸⠀⣰⠀⠀⠀⠸⠿⣆⣸⠀⠀⠀⠸⠀⠠⣤⣄
⣹⠀⢻⠀⠗⣧⠀⠀⠀⠀⠂⢤⣀⠤⠀⢸⠀⠀⠀⠀⡀⠀⢎⡀⢤⠄⣤⡀⠀⠀⠉⠒⠀⠀⠀⡇⠀⠛⠁
⢹⠂⡠⠲⡄⢁⣀⠀⠀⠑⢄⣤⡕⠀⠀⢸⠀⠀⡠⠚⠀⢀⡀⠀⠀⠀⣀⣙⠦⠤⢤⠀⠀⠀⠀⡇⠀⢠⢄
⠈⠁⢈⠟⠓⠚⢤⡠⢦⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠈⠣⠤⣀⣀⣀⣀⣀⠤⠜⠃⠄⠀⠀⠀⢠⠃⠀⠲⠄
⠀⢰⡃⠀⢰⡀⠀⢣⠀⠀⠒⠤⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠩⢂
⠀⠈⣧⠀⠈⣧⠀⠈⡆⠀⠐⢄⣤⡶⡀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠽⠇
⠀⠀⠸⡆⠀⠺⡏⠀⠸⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⡀⠀⠔⠊⠁⠀⠀⠀⣱⠒⠒⠄
⠀⠀⠀⢻⡀⠀⢻⡆⠀⢣⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠑⠂
⠀⠀⠀⠈⢷⠀⠀⢷⠂⠈⢆⠀⠀⡠⠉⠀⡣⠀⡐⡀⡀⠀⢀⠀⠠⠂⠀⠀⢠⡀⠰⣄⣀⣛⡃⢹⠀⠈
⠀⠀⠀⠀⠘⣇⠀⠘⣧⠀⠘⡄⠂⠀⠀⠉⠀⠀⡧⠁⢣⠀⡷⠷⣧⠀⠸⡿⠂⠃⠀⠀⠀⠀⠙⠈⣄
⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⢱⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠄⡀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠐⠄⠀⠑
⠀⠀⠀⠀⠀⠀⢷⠀⠸⠿⢇⣀⠇
⣀⣀⣀⣀⣀⣀⣘⣧⣴⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠈⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣀⣤⣤⣶⠏⠐⠲⠦⢤⣀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠛⠛⠛⠂⠀⠀⠀⠐⠛⠋⠛⠳⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡄⠀⠰⠞⣿⠲⡄⠀⠀⠀⡤⢲⡟⠶⠄⠀⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣇⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⢀⡿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣘⡲⡄⠀⢀⣴⣲⡲⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢳⢳⠀⡐⣌⢁⠀⢀⠈⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡏⣇⢳⠘⣮⣉⣩⠞⠀⠀⡜⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢸⣸⡆⢳⡀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠃⠀⢳⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣷⡿⠓⠀⠉⠁⠚⢷⣶⠀⠀⠀⠀⠀⣷⣾⠗⠂⠀⠀⠒⢿⣶⡇⠀⠀⠀⠀⢺⣾⠶⠒⠀⠀⠒⠾⣷⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠻⠟⠀⠘⠿⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠲
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣬⡄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀⡂
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⢀⡀⠔⠋


⠀⠀⠀⠀⣿⡿⠒⠀⠀⠀⠒⠿⣿⠀⠀⠀⠀⠀⣿⣿⠖⠂⠀⠀⠒⢾⣿⡇⠀⠀⠀⠀⢺⣿⠗⠂⠀⠀⠐⠶⣿⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠥⠿⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠸⠁⢀⠀⠀⠀⡀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠑⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠸⡾⠀⠀⢷⠇
⠀⠀⠀⠀⣀⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢬⠄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢨⡅⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠙⠢⠀⠈⠁⠀⠔⠋


⠀⠀⠀⠀⣷⡷⠓⠀⠉⠀⠒⢾⣿⠂⠀⠀⠀⠀⣷⣶⠖⠂⠀⠀⠒⢶⣶⡇⠀⠀⠀⠀⢲⣶⠶⠒⠀⠀⠒⠶⣶⡆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡞⠀⠘⡿⠃⠀⡀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠉⠁⡀
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⡀⠔⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠀⢐⢒⣒⣒⣠⣀⡀⠀⠀⠀⠀⣄⠐⡂⠒⠒⠒⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣤⣴⣶⠖⠃⠁⠀⠉⠘⠉⠈⠁⠀⠀⠉⠁⠁⠀⠀⠀⠁⠀⠀⠀⠁⠹⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣛⠿⢥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⣐⣤⡵⠆⠀⣀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣙⠛⠛⠿⠶⢶⢶⣷⣤⣤⢤⣥⣬⠦⠤⠵⠶⣶⣶⠶⠳⠛⠋⢉⡁⢰⡞⢣⢋⣠⡤⠄⠀⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢹⢇⠀⠢⢤⣐⢂⣤⠞⣽⣾⣵⠒⠛⠛⠛⠲⣄⠻⣌⢲⣦⡉⢡⡀⠘⠟⠘⠀⣏⠂⠀⠈⠲⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠁⠀⠀⠀⠐⡛⠛⠎⠀⢉⣁⠀⡀⠳⡆⣀⡀⠀⡛⠟⢓⠋⠶⠱⠂⢑⠾⡀⡘⠀⠀⠀⠀⢠⢳⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⣘⠒⠰⠖⣴⣱⣤⠆⠁⠉⠔⡉⠹⢏⣛⠋⠀⠉⢸⣿⠙⢻⠀⡟⠈⢡⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠐⣧⠀⢛⡇⠀⡔⠻⣦⣉⢻⣄⠹⣤⡀⠀⢨⡅⠀⢸⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸⠀⠈⡇⠈⢗⣺⢟⢏⣼⠅⢻⠿⠟⠀⠈⠁⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⠀⢸⠀⠀⡇⠀⠀⠃⢼⡹⠤⠖⣯⣄⡘⣿⢿⣿⠆⠘⠀⠇⠀⠸⠀⠀⠀⠀⢀⠌⠜⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⢨⠀⢸⣶⣄⣷⣶⢶⣀⠀⠀⠀⣴⣶⣻⣏⡁⡇⢰⣯⡞⡿⠃⠀⠰⠃⠠⠀⠐⠁⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⡂⠀⠀⠀⢴⣶⡾⢃⡀⠹⣿⢸⡟⡎⢻⣶⣾⣏⣿⣿⢰⠖⠛⠛⣇⠿⠃⠀⢸⣦⠄⠀⣀⣠⣔⡁
⠀⠀⠠⢄⣰⣦⡭⠶⢖⣶⡤⢛⡳⠀⠀⠀⢧⣿⣔⣿⢷⠄⢘⣿⣵⠁⠀⣭⣟⣈⣹⡌⠿⣏⠋⣸⡿⣇⠀⠀⠸⣓⡿⣛⣉⣙⠲⣮⢓⣲⠤⢤⡀
⠀⣵⣿⡟⣍⡂⠉⢪⡧⠟⡙⠚⠁⢦⡀⠙⢩⢚⣥⣌⢯⠺⠟⢘⣮⡷⠀⠾⠟⠻⡟⣧⣶⣶⡿⠿⠃⠀⠀⡴⠁⠵⠒⠋⠉⠉⠇⡴⢫⣭⣭⣳⡌
⠁⣿⢱⠳⠇⠀⠀⠘⠃⠔⠉⠁⠀⠀⠑⢦⣉⠛⠻⣿⢿⣦⣴⣯⡾⢿⣦⠀⣠⣶⢻⠯⣿⣋⠇⠀⢀⡤⠊⠀⠀⠀⠀⠀⠠⣄⡈⣴⠠⠀⢁⡽⢃
⠀⠪⣅⣓⠶⠄⠀⣀⠀⠀⠀⠀⢀⠁⢄⠀⠉⠓⢄⡻⢿⣿⡿⢿⣧⠀⠀⢦⣤⣽⣾⡿⣿⣿⣴⣸⠋⠀⢃⠆⠟⠁⢤⣀⠀⡀⠀⠀⡐⢊⡥⠖⠋
⠀⠀⠀⠈⠉⡛⠿⣶⣥⣲⣞⣂⣈⣁⣀⣀⣀⣀⣀⣛⡒⠇⠀⢄⢀⣀⡀⠈⠳⠯⠉⠙⣋⣉⣁⣀⣐⣒⣂⣐⣺⣥⣴⡶⠞⢛⣉⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠠⠉⢉⠈⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠛⠛⠒⠒⠖⠛⠛⠛⠋⠉⠙⠛⠿⢿⣿⠿⠿⠟⠋⣉⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢦⣒⣂⠀⠀⠤⠄⢀⢀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠠⠤⠤⠀⠀⢒⣒⡨⠋⠁

⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⢿⠟⢠⣶⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠻⠷⠟⠀⠘⢿⡧⠀⠀⠀⠀⢿⠟⢠⣶⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠻⠿⠟⠀⠘⢿⡧
⠀⠀⣤⣤⡄⠙⠃⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⣤⣤⡄⠙⠃⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠉⠋⠁⠀⠀⠀⠀⡠⠊⠈⡘⠀⠈⠒⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⡠⠊⠈⡘⠀⠈⠐⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⢰⠀⠀⠀⠈⠀⠐⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢐⠠⠈⠀⠀⠁⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠈⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠉⠀⠀⠀⠀⠀⠀⡈⠨⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢿⣷⠄⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠄⠀⢠⣿⣿⣿⣿⣿⣿
⠀⢤⣴⣶⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠘⠿⠷⠀⣴⣷⠀⠀⠀⢀⣴⣶⠄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠃⠇⠀⠘⠿⠷⠀⣴⣷
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⢀⡴⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⣠⡶⠚⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⡾⠃⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⣧⡀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⣿⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢠⡾⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠞⣀⣠⣤⣴⣶⣶⣾⣟⣛⣛⣛⣛⣛⣛⣓⡒⠶⢦⣤⣀⡀⠲⣄⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡾⠃⣿⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⣠⡟⠛⢲⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠃⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⢻⠏⠀⣠⠞⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢠⣯⡤⠞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢷⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⣠⠟⢉⡟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠛⠛⠻⣅
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⡏⢀⣞⡴⠟⢛⣋⣉⣿⣶⠀⠀⠀⢀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⣄⡞⢹⣿⣯⣭⣭⠴⢞⡟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣝⣛⡛⠛⠦⣤⣤⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠦⠤⠤⠤⠴⠶⠶⠶⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⡿⠿⠿⣿⣿⣿⣿⡿⣿⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣭⣭⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠓⠿⠷⠀⠀⠀⠐⠛⠗⠈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⣶⣦⠀⠀⠀⠐⣿⠆⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⢙⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⢒⣲⠀⠀⠙⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⣏⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠈⠋⠁⢀⡀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠛⠛⣇⣿⡇⠀⠀⠘⢷⡶⠶⠶⠖⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⣀⣿⡟⠁⠀⠀⠀⠀⠙⠶⠴⠞⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠸⡏⠛⠛⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⢸⣷⣤⣦⣼⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⣤⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣶⡘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⣼⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣽⣶⡗⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⠏⠀⠀⠀⣸⣿⣿⣿⣧⢺⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⠟⠀⠈⠀⠀⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⠀⠀⠈⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡞⠿⢿⣿⣿⣿⠍⠀⠀⠐⠒⠀⠀⠀⢹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⣠⣠⣌⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣼⣿⣿⣿
⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣿
⢸⣿⣿⡿⢿⣤⠆⠀⠀⠀⠀⠀⠀⢿⣿⣷⣼⡿⠿⠛⠛⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠙⠿⠛⠛⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿
⠘⣿⡟⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠙⠃⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⢼⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⢐⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠿⠿⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣴⡟⠉⠁⠀⢀⣀⡠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⣄⣀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⣿⠉⠑⠲⠿⠭⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⣦⠈⣉⣑⣦⣤⣶⣿
⠀⠀⠀⠀⠀⠀⣿⢤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⢻⡟⠉⠉⠁⠀⠸⣿⢀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠓⠲⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣀⣠⠴⣿⢋⣠⣤⣤⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠲⠶⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⣿⠋⠉⠉⠁⠀⠀⣿⠟⠉⠀⢨⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⡀⠀⠀⠀⠀⢠⡏⠀⠀⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢉⣀⢤⣹⠀⠀⠀⠀⢸⠃⢀⡴⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠞⡏⠀⠀⠀⠀⣿⠴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠶⠶⠃⠀⠀⠀⢰⣇⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⢀⡤⠖⠋⢉⣉⣈⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⠓⠦⣄⠀⠈⠙⢷⣄
⠀⠀⠀⣾⡏⠀⠰⣏⠉⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠛⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⡿
⠀⠀⠀⠙⢷⣄⡀⠈⠉⠒⠲⠤⢤⣄⣀⣀⡈⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⢉⣁⣀⣀⡤⠤⠴⠒⠚⢉⣁⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣤⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠛⠉⠉⣉⣉⣁⣤⣤⡴⠶⠖⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢔⠒⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠑⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⡤⠔⣃⡠⠴⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠌⣛⣤⣿⡶⠒⠘⢶⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⡿⠉⣶⠛⢠⡖⢦⡶⢲⣶⠶⣰⠿⣶⣿⣶⣮⣷⡿⠀⠀⠀⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⢳⡄⢸⣤⠘⠇⡀⡟⠠⢸⣞⠽⣿⢯⣿⢽⡅⠀⠀⠀⢀⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠸⣤⣸⢯⡝⠠⣠⢠⣴⣰⡡⠶⣿⣆⣹⣷⢿⣿⠃⠀⠀⡠⠞⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢺⣵⠿⣿⡄⢿⣷⢻⣿⡇⣦⢸⣟⡒⣻⣿⣭⣿⣀⠔⢉⡠⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⢹⡧⠹⣷⡀⠟⡗⣿⠢⣘⣼⣷⢿⢿⢛⡍⣿⢁⡴⠋
⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠼⡏⢹⠋⢋⣄⣿⡧⣾⣰⣼⣶⡿⣿⡿⢻⡄⠸⡇⢸⡟⠤⠤⢤⣄⣀⣀⣀
⣴⣾⣿⣿⣽⣿⠌⠀⠀⠀⠀⢡⡀⠆⠰⡆⠘⡆⠈⡗⠈⡏⠀⢰⠀⢸⣇⠀⢇⠎⠀⠀⠀⠀⠉⠙⠋⠛⢏
⠈⠳⢆⠈⢛⣙⡁⠀⠀⠀⠀⠀⡙⠢⣤⢀⢀⣈⡀⠈⠀⡙⠂⠀⣁⣀⣤⡾⠋⢀⡀⠀⠀⠀⢀⡀⠠⣨⠗
⠀⠀⠀⠈⠒⠤⣀⠀⠈⠁⠀⠚⠀⠀⠐⠂⠒⠒⠒⠐⠂⠐⠖⠀⠐⠒⠒⠒⠒⠂⠂⠈⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠻⠛⠛⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⢀⣀⣤⠴⠶⠶⠖⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢠⡾⠛⠁⣀⣤⠶⠖⠛⠛⠛⠶⣄⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⡄⠀⠀⢿⣅⣀⣠⡄⠀⠀⠀⢈⡇⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠈⠛⠷⣦⣄⣀⣉⠀⢀⣀⣀⣤⠾⠁⠀⢀⣴⡿⠃⠐⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠿⣶⣤⣄⣀⡀⠀⠉⠉⠉⠉⠉⠉⣁⣠⣤⡾⠟⠋⢀⡆⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠈⡇⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⢸⡇⠒⠶⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣀⣄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢿⣿⣷⠆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⠚⠋⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⣖⣂⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠻⠿⠛⠀⠀⠈⠙⠷⢤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣈⠙⠛⠛⠛⠛⠛⠉⠀⠀⣀⣀⣠⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠛⠛⢛⡛⠻⠯⣶⣢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣻⢟⡭⠚⢁⠠⠤⠀⠠⣄⠈⣉⠉⠑⠲⠬⣙⠪⣝⢶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣩⢞⣟⡵⢋⠔⡨⠔⠊⠀⠀⢈⡫⡠⠀⠀⠀⠀⠐⠪⡙⠦⡙⢿⣮⣙⡦⠄⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡴⠋⢠⢯⠞⡀⣠⠊⠀⣠⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠈⢢⠘⢎⢫⣧⠀⣀⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠋⠀⢀⡏⡏⡜⠰⠃⠀⠀⠴⠄⠠⠀⠐⠀⠀⠀⠠⠆⠀⠁⠄⠀⠀⠀⠈⠈⡏⣿⠋⠀⢠⢧
⠀⠀⠀⠀⠀⠀⢠⡟⡰⠁⠀⠀⢸⢸⠁⠃⠃⠀⠀⠀⠀⠈⣄⠀⡀⠘⠇⢀⠆⠰⠀⠇⠀⢸⠀⠀⠀⠀⢸⢸⣂⡴⠛⠒⠧⠤⠀⣀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣾⠸⡆⡆⢀⠀⡄⢂⠀⠁⠈⠓⠠⠦⣴⡅⠊⢀⠘⢀⠁⡎⠀⠀⠀⠀⢸⢸⣿⣶⣤⣄⡀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⢸⠇⠄⠀⠀⠀⠀⢻⡆⢧⠰⡈⢆⢘⣄⡢⣀⠀⠀⠀⠀⡈⢀⡼⢋⠔⡡⢊⠀⠀⠀⢰⢃⡟⣼⣿⡋⠋⠧⠙⢿⡖⠦⠤⠜
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣷⡘⣆⠀⠀⠀⠙⠻⣶⡛⠷⢶⣤⡙⠈⣴⣵⠟⠊⠀⠀⠀⡰⢃⡞⣰⣿⠙⢃⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⣿⠰⠀⠀⠀⠀⠀⠀⡟⣷⣌⢧⡀⠐⢤⡀⠀⠈⠙⠲⢌⠻⢸⠟⠁⠀⠀⠀⡠⠞⣡⢏⣴⡿⠁⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⠲⣄⡈⠑⠢⠤⣄⠀⠐⠘⠀⠀⠀⢀⣠⡴⢋⣵⣾⡿⠃⠀⠀⠈⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡳⠮⣍⣓⠲⠦⠤⠤⠤⠤⠴⠒⣚⣩⡤⠞⣋⡴⠏⢀⢀⠂⠀⢀⠆⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⠙⠟⢷⣤⣈⡍⠛⠛⠛⠛⠛⠛⠋⢉⣠⡴⠞⠋⠀⠀⠡⢂⠄⢀⠌⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⠣⡈⠲⣍⠛⠻⠿⢷⠶⠶⠷⠛⠛⠋⠉⠀⠀⠀⡀⠜⠁⡁⢠⠊⡔⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⢑⠢⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢐⡨⢐⡡⢊⠔⣡⠎⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡐⠀⣀⠈⠁⠐⠂⠒⠂⠐⠀⡀⠤⠒⡡⠒⡡⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠨⢐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡠⢖⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢽⣳⡶⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢖⣊⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠾⢭⣭⣭⣽⣭⣭⣭⠭⠶⠖⠛⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⡼⠃⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⣱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠀⠀⠀⠀⠀⡐
⠀⠀⠀⡀⠀⠢⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣠⣤⣆⠏⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠈
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠀⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡀⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠆
⠀⠀⠀⠈⠂⠈⠉⣉⢀⣠⡤⠆⠂⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠐⠀⠀⠀⠀⢀
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⠴⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠂
⠀⠈⠆⠁⠆⠀⠤⠄⡔⠂⠉⠁⡤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⠅⠀⠀⡇⠀⠀⠒⠂⠙⠋⠘⠃⠁
⠀⠀⠈⠐⠓⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⠀⠔⠀⢐⠀⡇⠀⠀⠁⠀⢠⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠌⠀⠀⠢⠀⠠⢍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠠⠀⠀⠘⠀⠐⠊⠉⠀⠀⠠⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⡇⡁⢇⡡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠁⠀⠉⠁⠘⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠓
⠀⠀⢠⠦⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⢠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠁⠀⡀⠀⠀⠀⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⢆⠀⠾⠡⠄⠆⡱⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠤⠤⠛⠀⣀⣥⠀⠀⠀⠀⠀⠀⠐⠛⠐⠘⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠂⠀⠐⠒⠁⠀⠀⠀⠀⠀⢰⠀⢈⠈⠉⠉⣉⢉⡉⠀⢹⠀⠀⠀⠀⠀⠀⡀⢠⠀⠐⢬⡀⠀⠀⠘⠀⠀⠀⠀⢀⠚⠉⠀⠀⠆⠔⠀⠁⠓⣄
⠀⠀⠀⠀⠥⣬⠈⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠊⠜⠃⠨⠀⠄⠂⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠒⠂⠀⠆⠐⠔⠂⣸
⠀⢰⠀⠀⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⣉⡁⠘⠋⠁⢀⣀⠔⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠃⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢰⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉