Shipboard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠉⡟⢉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡒⠒⠚⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠄⠀⠀⡏⠀⠉⠉⢉⣭⣭⡉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⡐⢠⠁⠀⣷⠀⠀⠀⠸⠿⠿⠇⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣶
⠀⠀⠀⠠⠀⡘⢤⣤⣷⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⠁⠀⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⠀⣾⣿⣿⡷⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⣀⣀⣈⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⢸⣦⣄⣹⠉⠻⡇⠀⠀⣸⣶⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣧⣄⡀⠀⢀⣿⣿⣿⣷⣤⡀⠀⡰⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⣸⣿⣿⣿⣿⣿⣿⡆⢳⣦⣤⡃⠙⢿⢠⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠟⠛⡛⠿⡿⢜⡛⠻⠿⢿⣿⣿⣿⣷⢨⣿⣿⣿⣶⣦⡀⢷⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⡗⠀⠁⠈⣿⣶⣤⣿⠂⠉⠻⣿⠚⠛⢿⡿⣿⣿⣷⣜⣿⡄⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄⡹⣷⣦⣼⡇⠈⠹⣿⣿⣿⣿⡄⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⡙⣿⣿⣿⣿⣆⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠎⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠿⠿⠿⠇⠀⠀⣸⣿⡿⠿⢿⠛⠛⢿⣿⣿⣿⣿⡿⠿⠿⣿⠁⠀⠈⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣧⣴⣤⣭⣤⣄⣀⣸⣀⣀⣈⣛⣉⣉⣠⣦⣤⣤⣥⣤⣤⣤⠴⠒⣽⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠴⠿⢛⡛⠻⠿⠟⠛⠛⠿⠛⢛⡛⠻⠿⠿⠛⡛⠿⠿⠿⠛⣛⠋⠀⠀⠒⠒⠒⠒⠒⠒⢒⡀
⠀⠀⠀⠒⠀⠀⠀⠁⠀⣀⠤⠤⠀⠀⠀⠀⠒⠋⠃⡀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠐⠂⠀⢀⡤⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⡇⠀⡿⠛⢷⣏⠀⠾⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢂⠀⠀⠀⠀⣼⠀⠀⠙⣯⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠄⠀⠙⢦⠀⢠⡇⠀⠀⠀⢨⣿⣄⠀⠀⠀⡟⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣰⠿⠞⢀⠀⠀⠀⠁⢼⠃⠀⠀⠀⡼⠁⠀⠳⠄⢸⡏⠀⠀⡟⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀⠰⣯⠉⠐⠶⢌⡦⡀⠀⠀⠀⠑⣄⣠⠞⠑⠂⠀⠀⠀⠻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⠀⠠⡀⠀⠀⠉⠄⣤⣴⣞⣋⠁⠾⠖⠄⠂⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠈⠳⢄⠀⠀⣼⡍⠻⣿⣿⣷⡆⠠⢍⡂⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢠⣿⢠⠀⠀⠀⠀⠑⢾⡇⢸⠖⠉⢻⣿⡷⠀⠀⠈⣷⢄⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠄⢎⣰⢂⡧⣀⡀⠀⠀⠀⠀⢠⣹⡇⢲⠆⠀⠈⠻⣄⠀⢰⡿⠀⠈⠂⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠊⠠⢡⣿⢇⣙⠃⠀⠀⠀⠀⣼⣿⢇⠙⠀⠀⠀⠀⠀⠳⢾⡟⠀⠀⠀⠈⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢏⡉⠁⠀⣸⡟⠀⣏⠑⠂⡦⠀⣰⣿⢏⠂⠀⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣴⠛⠂⠀⠀⠐⠾⢹⣮⡿⡓⠦⠟⡤⣂⣜⠃⡀⠀⠀⠀⠀⠀⠓⢄⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠃⠀⡻⢾⣏⠀⣽⣯⣭⣤⣈⣀⣊⡴⠿⣀⡗⠼⣋⢀⠄⠀⠀⠀⠀⠑⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⢦⣤⣻⢷⣿⠙⠻⣦⣉⠻⠛⢻⡷⣶⠠⣀⡃⠌⢿⢠⣟⡄⠀⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢠⡆⠀⠀⠉⠛⠻⣿⣷⠀⠀⠈⠛⠻⢦⣝⡃⠀⠀⢸⡗⠀⡀⠸⣧⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⣸⠧⠀⠀⠀⠀⠂⢈⣿⠀⡀⢀⠀⠀⠀⠈⠛⠳⠦⣼⡁⠀⠀⠳⣍⠃⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢰⣿⢦⡆⠀⠀⠀⢈⡏⣽⡇⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣦⣀⠙⣇⠲⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠺⡸⡆⣱⡿⣿⡏⠿⣠⠂⢂⡞⢰⣿⣷⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⣿⣿⡁⠀⠈⠓⠠⣀⠀⠀⢀⡠⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠊⠁⢙⢮⡿⣽⣿⡹⢻⣼⣹⠿⣡⣿⣿⣿⠀⠀⢺⠀⠈⠀⠀⠀⢀⠀⠀⣤⡆⢈⠿⣿⢼⡆⠄⢸⡄⠀⠙⠒⠿⡁
⠀⠀⠀⠀⠀⠰⠆⠤⠤⠤⢾⣿⣻⣿⣟⢾⣧⡿⠂⣱⣿⣿⣿⢰⡆⢸⡆⣸⣶⢀⡄⠘⠁⠀⢼⡇⣾⠀⣿⣾⡿⠇⢸⢁⡀⢠⡄⠀⠈⠓⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣱⠛⠣⠄⢟⡛⣂⢲⣟⠿⣿⣿⡼⠿⠺⡇⢺⡇⠸⡀⠷⠀⠠⢻⣷⣃⣼⠟⠛⠃⣀⣊⠈⣥⡟⡅⠀⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⢶⣴⣶⣾⣧⡔⣶⡟⠋⠉⠉⠿⣽⣳⢾⠇⢛⣀⣤⣶⣤⣶⠶⠶⠚⠛⢻⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠘⠁⢂⠐⣃⢀⣴⠇
⠀⠀⠀⠀⠀⠀⣸⠏⢚⣹⠻⠻⢿⡿⣿⢶⣶⣾⡞⠀⢂⣷⢚⣩⠿⢟⣋⡥⣴⡞⠛⠛⠛⠉⠉⠛⢫⣹⣟⣻⡟⣹⣷⡀⠀⠀⠀⠈⠌⣿⠋
⠀⠀⠀⠀⠀⢠⡏⢀⣘⠭⠉⣀⠀⠀⠿⠿⣿⣛⣿⣚⣏⣻⣵⣲⡌⠛⠈⠁⠀⠀⠐⣈⣁⠐⠈⠒⣲⣿⣿⡿⢡⢿⡿⣷⠀⠀⠀⡀⠞⠁
⠀⠀⠀⠀⠀⠘⣧⠀⢥⣀⠀⠲⠤⠌⠁⣚⠋⢡⣺⣷⢿⣿⠛⠟⠋⡀⠀⠀⠀⠀⡉⠤⠤⠤⢤⣬⣭⣽⡟⣡⡿⣄⡿⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣄⠒⠠⢤⠉⡑⠒⠠⠔⠀⠬⢡⣄⣻⠈⠷⣤⡀⠀⠐⠀⠀⠀⠀⠒⠶⠦⠤⣽⣿⢯⣾⠏⡳⠈
⠀⠀⠀⠀⠀⢀⣟⣿⡉⣑⡀⠒⢬⣭⡤⠉⢁⣋⣀⠀⣭⣾⢿⡶⠄⠐⢋⡀⠀⠄⠐⢐⣲⣖⣶⡿⣳⡿⢃⠜
⠀⠀⠀⠀⠀⠸⠶⠟⠓⠹⠿⣿⣒⠲⠖⠒⠈⠬⣅⣀⣴⠶⠟⠛⠚⠷⢟⣢⣴⣮⣭⡥⢶⣾⣿⣱⡏⢱⣮⡄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠙⠛⠋⠁⠉⠉⠁⠀⠈⠉⠛⠉⠙⠛⠻⠿⠿⠤⠀⠘⠿⢯⣿⣯⣽⠕⠛⠛⠛⠛
⠀⠈
⠀⢸
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⣴⠰⣾⣾⡇⠆⢀⠤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠈⠇⠠⡻⣺⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠀⠐⣠
⠀⠀⠀⠀⠀⠈⠑⠿⠷⡇⠀⢰⠀⠀⠀⠀⠀⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⢴⣤⣼⠄⢸⠖⢿⣶⣠⣤⠆
⠀⠀⠀⠀⢸⣦⣀⡀⠐⠀⠀⠀⠠⣀⣠⡄⠈⠙⣿⣿⣶⡄⠀⠀⠀⠀⠀⣼⣿⣿⣶⣭⣟⣂⣉⣸⣬⣿⣿⠏
⠀⠀⠀⠀⢾⣿⣿⣿⢷⣤⣄⠀⠘⣻⡿⣷⣠⣾⣿⣟⣻⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⢿⣛⣿⠻⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣵⢿⣿⠛⣶⢶⡀⣿⣼⣿⣿⠛⢻⣷⣖⠠⣄⡀⠀⣿⢠⠏⣾⠼⢻⣭⣶⣾⣿⠿⠇
⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣷⡾⠿⣭⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣾⣢⣴⣜⣿⡆⢿⣷⣿⣿⣏⣠⣦
⠀⠀⠀⠀⠛⠻⢿⣿⣿⠿⣷⣿⣄⣸⣿⣿⣿⡿⠛⠈⢉⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣶⡶⠄
⠀⠀⠀⠀⢲⣷⣶⡿⣅⠀⠰⣶⣯⣿⣿⡫⣭⣶⣿⣿⣿⣿⣯⣿⣟⣙⣛⡻⣟⣩⣭⣭⠭⠀⠀⠲⠿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠜⠻⠿⠿⢟⣛⠛⠛⠻⠿⠿⠿⠿⣿⠿⣿⠉⣉⡉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠂⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠚⠉⠀⠀⠆⠀⢈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡤⠀⣀⡀⠤⣿⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⡿⢤⣤⠴⠟⠀⠉⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⡶⢦⣀⡿⢋⣉⣽⣯⣗⣤⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠀⠀⢿⣋⣠⠾⠷⢦⣿⠏⠁⠁⠀⠀⠀⠛⠋⢀⡀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⢹⣟⣠⡴⠚⠋⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⡥⠔⢀⣰⠋⢀⡀⢀⣀⣀⣀⣀⠸⣷⠶⡾⢷⣦⣄⡀⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠢⣀⠀⠀⠀⢸⣿⡶⣠⣴⣻⠃⢀⡼⠟⠉⢛⣽⣿⣿⣳⣶⣾⣷⣿⣿⣿⣿⣶⣽⣿⡅⠀⢾⡇
⠀⠀⠀⠀⠈⠑⢦⡀⠘⣿⣿⡿⠟⠛⣶⠞⠒⠛⠉⠉⠉⠉⠙⢻⣿⠟⠉⠉⠉⠉⠉⠛⢻⣿⡇⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢌⡻⠅⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢀⠀⣼⣿⣧⠀⣸⣿⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⡈⢷⣴⡷⠿⠷⠶⣤⣄⠀⢿⡿⣿⣿⣶⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣷⢿⣄⣠⣤⣀⣻⡟⠿⣮⠀⠀⠀⢹⡟⢻⣦⣀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡼⣟⣿⣿⣿⡿⣷⣤⣌⡛⢦⠀⢸⣇⠀⢫⣿⣷⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣺⣿⢾⣿⢿⡑⠉⣿⣿⣾⣽⣹⣟⡟⠾⢻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣷⣍⣹⣶⣿⣿⣾⣇⣿⠏⣹⣿⠛⣶⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢅⠒⢼⣿⡏⢙⠛⠿⡿⠛⣿⣿⢿⣿⣿⣿⠟⢿⣿⠿⠟⣻⢿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠈⠉⠉⠀⠈⠀⠐⠃⠀⠉⠓

⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠈⠉⠁⠐⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⣻⣷⠖⠚⠻⣶⠤⣄⣀⢸⠛⠛⠛⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠁⠀⠐⠒⠀⠀⡸⠀⠉⠉⠒⠛⠤⠤⢙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠴⠶⠶⠐⠶⠶⠒⠒⠚⢀⣇⣄⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⣠⡴⠛⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⢤⡬⠁⣼⣿⠿⢿⣿⣾⣷⣧⣶⣄⡀⢁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⣲⠞⢿⣿⣆⡀⠉⢻⣷⣆⠀⢡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠃⠀⠈⠙⣧⠈⠉⠉⠓⢥⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⠔⠀⠀⠀⣿⠀⠀⠀⠀⠈⡀⢀⣼⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢿⣷⣶⣀⡄⠀⠸⠋⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⠋⠀⠀⣠⠄⢀⣠⣠⡶⢰⣿⣿⣿⡿⣷⣼⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸
⣿⡑⠶⠶⠲⣄⠀⠀⠀⠀⠀⠂⠀⣀⣴⡿⠴⢶⣿⣾⣿⣿⣿⣿⣿⠁⠂⠀⠀⠈⠉⠐⠻⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠴⠒⠢⢄⣀⠀⡰⠉⠀⢹
⢸⣯⢖⡂⠀⠨⡇⣀⠤⢤⣤⠴⠲⣽⣷⢦⣖⠛⠺⣿⣿⣿⣿⣿⡧⠼⠀⠈⠀⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀⣿
⠀⢻⣅⠀⠀⠀⠉⠉⠉⠀⠈⠆⠀⠀⠛⢾⣭⢷⣴⠿⠿⣿⡿⠟⢿⡷⢾⣷⣴⣤⡄⢀⠇⡇⢀⣄⣷⢶⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣿⣷⣴⣟⣀⣀⣸⣷⣾⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣅⣨⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁
⠀⠀⠀⠈⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣯⣽⣿⢿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣿⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣖⠀⣤⣶⣄⣀⣁⣾⣶⣀⣿⣿⣿⣿⣿⡿⣿⠿⠛⠛⠿⠿⣿⣿⣿⣿⡆⡸⠶⣤⣤⣴⡿⠿⢦⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣮⣭⣭⣿⣷⣆⢤⣟⠿⠿⡿⢻⣷⣶⠦⠴⠶⣾⣿⣿⣿⣿⣯⣽⣿⣟⣉⣽⣿⣽⣷⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢿⣿⣿⣿⣿⣶⣦⡴⢶⣶⣾⠟⠻⢿⣯⣝⣿⣶⣶⣾⣭⣤⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠲⠿⠿⠿⠶⠾⠿⠛⠛⠛⠛⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣸⣀⠀⠀⠀⠀⠀⠠⠦⢤⣿⣀⣀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢀⠀⢹⡇⠀⠰⠤⢤⣿⣧⠀⠀⠀⠀⠀⠐⠲⠾⢿⣷⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠘⠓⢺⡗⠐⠂⠀⡜⢹⠀⠀⠒⠀⠀⡂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠸⣥⠆⠀⠴⠀⠸⢠⠀⢀⣠⣤⣦⣤⣤⣤⣼⣾⣶⣶⣶⣾⣿⡟⠁
⠀⠀⠰⢶⣶⣿⣿⣿⣶⣾⣟⣻⠻⢟⣿⣿⣿⣭⣿⣿⣿⣭⡯⣿⢽⡿⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠸⠛⣭⠍⠉⡉⠙⡁⢰⠶⠧⢶⣾⠿⣿⣿⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠀⠒⠂⠀⠀⠀⠠⠀⠀⠐⠒⠈⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⡛⠋⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠅⠒⠁⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠀⢴⠒⠚⠓⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⢸⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡈⢸⡀
⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢻⠘⠀⠀⠈⠁⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⢸⡇
⠀⣠⠶⠴⣧⡸⣒⣒⠒⠒⠒⠚⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣀⠀⠀⠙⢹⣯⡠⠂⠀⢀⡴⠋⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡟⠀⠀⠀⢸⡇
⠀⡏⡂⠀⠀⠁⠄⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠞⣫⠏⠀⠀⣠⠔⡿⢁⡀⠀⢸⠁⠀⣹⠀⠀⠀⣀⣠⠤⠚⢻⠂⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⡇⡇⠀⠀⡁⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠠⠊⣴⣣⢴⡾⢳⠃⣀⣽⡈⠀⠀⠈⠞⣽⣻⠶⠒⠉⠁⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⡄⠀⠀⡇
⠀⣷⠁⣀⣀⢀⡇⠠⠶⠃⠀⠀⡀⠀⠀⠀⠐⢦⣀⠙⠒⠲⣤⣤⢡⠞⢉⡴⠋⠀⠘⠛⢙⣾⡅⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣇⠀⠀⠀⢀⡇
⠀⠈⠉⠙⣏⣸⠛⠉⠉⠙⠛⠓⣌⠀⠀⠀⠀⠀⠈⠑⠦⣛⠧⢄⡀⣰⠎⠀⠀⠀⢰⠋⠁⢸⠃⠀⠀⢠⠞⠁⢸⢠⠀⠀⠀⢀⠠⠀⢸⡄⠀⡀⠀⣿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠃⠀⠈⠱⣦⣈⢱⣶⠶⢤⣤⣸⣤⣤⣼⣇⣠⣀⣸⣄⣠⣾⣛⣛⣛⣿⡛⣛⡛⢟⣛⠙⡟⠀⢻⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⢽⡞⠃⠀⠀⠛⠉⠓⠀⠇⠰⠦⠔⠆⠰⠯⠝⠆⠐⠩⠷⠍⠈⠄⣯⣥⠇⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣲⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⡀⠀⠉⣠⣼⡃⠀⠀⠀⡈⡇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⠀⠀⠀⠀⠀⠀⠀⠻⣧⣐⣀⣱⣂⣈⣀⣀⣀⣀⣀⣤⣈⣁⣀⣠⣭⣴⠞⣏⡡⡌⣧⠀⠀⠀⠀⢻⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⠀⢠⣈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⢀⣀⡀⠀⠀⢀⡀⠤⠒⠂⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⢛⣉⠤⠒⠊⠁⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⢈⠉⠬⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠓⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠓⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠋
⠀⠀⠐⠊⠀⠀⠀⢀⣤⠾⢿⣿⣯⢒⣤⡆⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠃
⠀⠀⠀⠀⠀⣀⣴⠿⠃⠙⠁⢠⣤⣾⡿⠀⠀⠀⠀⢀⣤⣶⣿⡿⠍⠉
⠀⠀⠀⣠⣾⡿⠛⠒⠀⠀⠈⣹⣿⣿⣅⣤⣤⣶⣾⡿⠿⠛⠛⠚⠂
⢀⣴⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠓
⣿⣟⠋⠁⠀⠀⠠
⠟⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣮⣯⣥⣀⡄⠀⠀⠐⠒⠛⠁⠀⠐⠈⠁⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⡆⠀⣠⣤⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣠⣤⣶⣿⣿⣿⣙⣧⣾⢛⣛⣀⡶⢶⢂⣉⣉⣀⣠⣀⣀⡄⠾⠋⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⢀⡖⣶⠒⠛⠛⠛⠉⠛⢉⣽⠟⠃⡘⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⣻⣿⡟⠁⠀⠀⠀⣀⣤⣶⡾⠀⠀⠀⠰⠶⠦⠀⠀⠀⠀⢠⡆⠀⠀⠁⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⣫⡵⠟⣻⠿⠷⠶⣾⠛⡿⠟⠻⠃⠀⠀⠀⠀⠀⠀⡇⠀⠒⠚⠛⠁⠀⠀⠀⣶⠠⠿⠋
⠀⠀⠀⠀⠀⠐⠒⠤⣤⣤⣬⣬⣥⣤⣴⣦⣤⣶⣾⣶⣾⣿⣿⣿⣿⣿⣶⣶⣴⣷⣶⣦⣴⣤⣶⣷⣷⣿⣿⣶⣶⣶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠛⠿⠿⣟⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⡿⠛⠻⠛⠛⠛⠛⠛⠛⢛⣿⣻⣿⣿⣿⣧⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡆


⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⢀⠤⣤⠴⠚⠀⠀⠀⠀⠀⠀⠀⢠⠦⠤⠐⠙
⠀⠀⠀⠀⡰⠶⠟⣙⣣⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣷⡀⠀⣀⣀⠠⢴⣷⠀⠀⠀⠀⠀⣤⣀⣀⡀⠤⣾⡙
⠀⠀⠀⢸⣤⡔⢻⣯⣭⣽⠖⠉⣛⢃⣰⡤⠜⠁⠀⠀⠀⢀⢟⠉⠿⠃⠀⠀⣀⠀⢀
⠀⠀⠀⣽⣿⣿⣿⣿⡀⠁⠀⣿⣿⣿⣅⣀⣤⣠⡶⢲⡆⠀⠀⠑⣄⣼⡶⠦⠙⠀⢸⣷⣀⠀⠀⠀⡀
⠀⢀⣐⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⢿⣿⣋⣉⣁⢀⠸⣿⣄⣈⣱⣿⣿⣀⣀⣤⣤⣬⣭⡿⢓⡋⠉⠁
⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣤⡀
⠀⠰⠶⣶⣶⣿⣿⣿⣻⣿⣿⣫⣥⣴⣬⠩⠁⢍⡛⠛⠛⠉⠛⠉⠉⠉⠉⠁⠉⠉⠉⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠠⢶⠉⠛⠻⣛⣛⡛⠻⢻⣻⣾⣿⣧⣤⣠⣿⣷⣦⣴⣟⣿⣟⣛⣿⣖⣶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⠽⠶⠼⠟⠺⠮⠉⠩⠉⠉⠉⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠇⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⢋⣡⣤⣤⣤⣾⡿⠋⠉⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⣿⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣤⡄⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠀⠀⣰⣿⡟
⠿⢿⣿⣷⣶⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⣼⣿⣿⢿⣿⣷⣶⣶⣶⣾⣿⡿
⠀⠀⠀⠉⠉⠉⠉⠁⠀⢻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠋⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⢾⣿⣷⣦⣤⣤⣤⣶⣿⣿⣿⣷⣦⣤⣄⠀⠀⠰⣿⣿⣶⣦⣤⣤⣴⣶⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠌⠛⠛⠉⢳⠀⠀⠀⡇⠀⠀⠀⠀⣀⣠⢤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠤⠓⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠊⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠀⠀⠀⢀⣀⣂⠀⠀⣴⣶⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣠⣤⣶⣶⠟⠛⢻⡏⠉⢩⣄⣠⡤⠄⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⢛⢀⣸⣤⣬⣼⡿⡿⢿⢿⡁⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠶⠚⠛⡫⠟⠛⠋⠁⠰⠁⣼⡄⡑⡀⢀⣠⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠃⠀⠘⠟⠛⣛⢛⣿⠾⠷⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⣤⣶⣶⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠘⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣛⡃⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠲⣿⠿⠿⢿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠀⠀⡀⠀⠘⠀⠀⠉⠉⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠩⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⡀⠀⠀⠀⢀⡇⠀⠀⠀⣄⡞⢀⣤⣄⣀⣀⣀⡐⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢀⣾⡇⠀⡀⠀⠙⠓⢲⣶⡤⠬⠝⠻⣿⣿⣆⠈⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣧⠃⠀⡇⠀⢈⣯⣿⡇⢰⣇⢀⠀⠀⠺⣿⣿⣦⡀⠀⠀⠀⠉⠉⣡⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⢻⣇⠀⠃⠀⠌⣿⣿⣇⠀⠀⠘⣿⣿⣇⠛⠛⠻⠿⢦⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⢺⡈⠸⣿⡄⢀⠀⠀⣿⣿⣿⢠⠀⠀⢹⣿⣿⣆⠀⠀⡠⣨⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢑⠂⠸⠷⡀⢻⣷⡘⡀⠀⣿⣿⣿⣿⣖⠒⠒⢻⣿⣿⠗⣡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡄⠛⠛⠿⠷⣦⠶⠿⢟⣋⡉⠙⠀⢈⣿⡿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⣷⣖⣚⣛⣶⣿⠿⠟⠋⠁⡀⠄⠐⠉⣀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦⣴⡞⠁⠀⠀⠀⠀⠹⠀⠾⢿⡧⠀⠔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠿⠿⠿⠛⠋⣙⠿⢿⣿⣿⣂⠀⢤⠈⠛⠀⠁⡀⠀⠀⠐⠉⣉⣉⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠱⠚⠛⠛⠁⠻⠷⠆⠀⢠⣤⢶⣶⣶⠖⡒⢲⣿⣿⣿⠿⣿⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠟⠁⠈⠀⠛⠛⠑⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⢠⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠠⠀⠀⠀⠀⠀⢠⠵⠤⢤⡀⠀⠀⠀⢣⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢄⠢⣀⠀⠀⢠⣇⣀⠀⠠⢌⢱⢶⠒⠚⠑⠂⢰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠂⠁⠀⠀⡎⠀⠀⠈⠉⠂⡟⠈⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⠠⠤⠤⠤⠦⠤⠄⢀⡀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡃⣠⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠀⠂⠒⠒⠒⠒⠒⠂⠀⠴⢌⡳⢄⡇⠀⠀⠀⠀⠀⢠⡏⡀⠈⢳⡈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⣈⠀⠠⠤⠤⠤⠤⠄⠀⢀⣇⣰⣝⢮⡑⢄⠀⠀⠀⠀⢸⠇⠤⣉⣺⠈⠁⠃⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢰⠫⡁⢨⢦⣑⢄⠀⠀⢸⠀⠀⠀⡇
⠀⠰⠈⡉⠉⠉⠉⠒⠒⠓⠂⠤⠤⠤⣀⣀⠀⡋⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡷⡀⠙⢶⣅⠋⡏⠀⠀⠸⠀⠀⠀⠇
⠀⠀⠀⠀⠑⠦⡀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠉⠙⠒⠒⠦⠤⣤⣤⡄⠀⠀⠀⣀⣀⡘⢰⠀⡸⢆⠀⠙⢯⡀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠛⢄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠦⠤⢄⣀⠉⠑⠒⠦⠤⣄⡃⠸⢠⡐⠄⠑⢄⠀⠙⢦⡀⠅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠢⠤⢄⣈⠙⠒⠶⢤⣊⣢⠑⢄⡀⠈⠣⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠒⠢⠤⠄⣀⡀⠉⠐⠒⠤⢄⡉⠒⠲⣍⡢⡀⠈⠢⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⣀⠠⠤⠤⠀⠠⠼⠆⠠⠤⠤⠤⠀⣈⠉⠑⠒⠒⠤⠌⣉⠒⠦⢍⣛⠶⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠂⠁⣀⠀⠤⠄⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠬⠍⣀⡒⠀⠀⠬⢁⡒⠢⠭⠖⡻⠦⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠭⣁⡒⠓⠠⢄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣖⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠐⡾⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠸⡆⢸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣈⣁⡤⠶⢶⢖⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠃⢧⠬⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⣤⣤⡖⢐⢊⣉⠥⠞⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠑⠺⣦⠄⣠⡔⠲⡶⠾⠛⣛⣉⣉⣀⡠⠋⠎⡴⠁⠀⠀⠀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⣀⣀⣀⣀⠙⠀⠀⠐⠩⠂⠥⠚⠁⠀⠀⠀⠠⠤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠈⠃⠈⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠴⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡄⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⢹⣿⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⡷⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⡀⠀⣀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⡀⠀⡴⣠⠀⠀⢀⢀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣞⡋⠉⣙⣿⣹⣗⡃⠫⠄⢨⣧⠨⠤⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠑⠢⡀⠐⠀⠀⠀⠀⡀⣸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣴⣦⣶⣴⣿⡇⠀⠀⢀⣀⣀⣸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠚⠒⢲⡟⣺⣗⡃⠁⠩⠭⠿⠼⣧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣷⠀⠀⠀⠀⠀⠀⠀⠐⠄⠙⠛⠓⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⣻⡆⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣿⣶⣶⣆⣴⠆
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠛⠲⠷⣷⡦⡄⣤⣾⠇⣶⣿⡏⠀⠀⠀⠀⢀⣰⣀⡿⣽⠿⠃⠀⠀⠀⠉⠉⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⠻⠿⠿⣿⠀⡀⣤⠆⠉⢀⣽⣇⣾⣦⣤⣤⣤⡤⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣷⣴⣄⣠⣿⣉⣠⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⢀⣀⡀⣀
⠀⠀⠀⠰⢶⣦⣴⣶⠒⣶⡖⣴⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⢿⡿⢰⣿⣶⣯⣭⣬⣾⣿⢲⣤⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⠛⡋⢉⡉⢉⣉⣭⡿⠉⠙⠟⠋⠉⣰⣾⣿⠃⠘⣙⣳⣾⣿⡏⣍⣙⣛⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠲⠚⠋⠁⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠋⠀⠉⠀⠉⠉⠁⠀⠀⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⠛⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠶⣒⣛⣋⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⣿⡗⠋⠉⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⣁⣀⣻⡄⣿⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣯⠭⠷⠿⣷⣿⠀⠀⢀⣀⣠⣤⣤⠿⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣀⣀⣿⡷⠾⠟⣛⣉⣉⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣿⠉⢹⡟⠛⣛⣋⣉⣉⣩⣭⣤⣤⡤⣤⣤⢤⢶⡴⡴⡶⣤⢤⢤⣤⣬⣍⣉⣙⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠴⠾⠟⢛⢛⡿⠤⢸⣿⣏⣯⣿⣼⠧⠷⠾⠾⠗⠓⠛⠛⠛⣛⣛⣛⣛⣛⠛⠛⠛⠚⠿⠿⠿⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣩⢩⠅⠤⠀⠖⠘⢃⣁⣈⣸⡇⠠⢸⣧⣥⣤⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⢺⡟⠉
⠀⠀⠀⠀⠀⠀⣤⡤⠧⠤⠴⠶⠖⣒⣛⣋⣭⣭⣴⣾⡇⠄⢸⣧⣤⡤⢶⠶⣶⠲⡖⢻⡟⣿⠙⡟⢹⡏⣿⢻⠋⣿⢹⡟⣿⣿⢻⣷⣿⡇⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡿⠶⠾⠿⠿⠿⠟⣛⣛⣛⣋⣭⡭⠿⠁⠐⢸⣿⠀⡇⢸⣀⣿⣠⣧⣼⣧⠿⠴⠷⠼⠧⠿⠾⣶⣿⣼⣧⣿⠾⠾⠟⠛⠛⠉⠉⢹⣷⣄
⠀⠀⣠⣤⣶⠛⡛⠛⡉⢉⠉⡉⠉⡉⢠⡄⢠⠀⠠⠄⠰⠀⣸⡟⠛⠋⠉⠉⠁⠀⣀⣀⣤⣤⣤⠶⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⡏⣿⠆
⠀⠘⢧⣤⣤⣤⣥⣬⣥⣬⣤⣥⣤⣤⡤⠤⠶⠶⠶⠒⠛⠋⠉⠻⠶⠶⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠖⠚⠛⠋⠉⠉⠁⣿⣿⠋
⠀⠀⠀⢸⡇⢀⠀⡀⣀⢀⡀⡀⢀⠀⡄⢠⠠⠄⠤⠀⠄⠐⠀⠃⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠶⠶⠚⠛⠉⢉⣁⣀⣀⣤⣤⣶⣷⡶⠶⠒⠛⢛⡿⠁
⠀⠀⠀⠘⣷⣲⣶⣶⣶⠦⡤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠖⠒⣚⣛⣛⣉⣉⣉⣩⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⠉⠉⠀⣄⠘⢿⡿⠃⠀⠀⠀⣼⠃
⢀⣤⣶⣤⣹⡄⠈⠉⠉⠉⠉⠉⠉⠙⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠷⠶⠛⠀⢰⡏
⠀⠀⠀⠈⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠋⢉⣉⣕⡦⣿⣭⡿⠛⠶⣶⡶⢤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠦⣤⣴⠖⠲⠦
⠔⠋⠀⠀⠉⠳⣄⡀⠀⢠⡟⠛⠷⣶⠶⠤⠴⠏⠉⣹⢿⣭⣛⣛⣻⠗⠶⣶⣲⠶⠶⠶⣶⠶⣶⠤⠤⠤⠤⢤⣤⣤⠤⠶⠶⢖⣺⣿⣍⣹⡟⠳⢦⣀
⠀⠀⠀⣀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠈⠙⠲⢤⣤⠾⠃⠀⠀⠈⠉⠙⠲⠦⠤⢬⣽⣶⡶⠏⠀⠈⠙⢦⣤⣀⣹⠀⠉⠳⢦⣄⡻⠛⣲⠀⠙⢧⣀⣀⠙⠛⠛⠉
⢤⣤⠴⠋⠙⠓⠲⠤⠤⣤⣄⣀⣀⣀⡴⠶⠦⣤⣀⡀⠀⠀⠀⠀⣰⠶⠶⢤⣄⣀⠀⠀⠀⡶⠦⣤⣀⠀⠈⠁⠀⠀⣤⢤⣈⠛⠛⠋⠀⠀⣻⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⢀⣠⡤⠤⠦⠤⣄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⠁⠀⣀⣀⣈⣉⡙⠓⠚⠁⠀⠀⠈⠉⠓⠶⠶⠾⠃⠀⠈⠉⣓⣲⣒⡛⠁⠀⠀⠀⠀⠙⢷
⠀⣀⣤⣶⠾⠋⠁⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀⡀⠀⠀⢀⣀⣤⠶⠋⠁⠀⠀⠈⠉⠙⠒⠲⠤⣤⣀⡀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠈⠉⠉⠓⠲⠤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠿⣟⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢻⣿⣮⣯⡒⣹⡿⣿⣟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⠀⠀⢿⣿⣻⣿⣾⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡾⠟⠛⠛⠁⠀⢻⡄⠀⠀⠀⠀⠸⣿⠟⠉⠀⠀⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠃⠘⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡟⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣋⡀⠀⢹⣧⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⢀⠃⣿⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢍⠙⠛⠛⠻⠷⠶⣴⣤⣤⣀⣸⣏⠛⠓⠾⠶⠾⣦⣄⣘⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶
⠀⠀⠀⠀⣠⣤⣶⠿⣇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠉⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢤⡀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠎⠀⠀⠰⠀⢦⠶⣆⢠⢤⠄⠀⡀⠀⠸
⠀⢴⢤⣄⡀⠀⠀⡐⠷⣏⣼⠇⠀⠀⠀⡞⠁⠀⠉⠈⣿⣿⣿⢇⡀⠀⠀⠀⠐⣗⠲⠶⠾⠦⣿⣼⣯⣄⣀⣀⣀⣲⣀⢀
⠀⠈⠦⣬⣙⣷⡾⢤⣄⣼⣗⠀⠀⠀⠙⢿⠒⠶⠶⢦⣿⣯⣿⣟⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠳⡀
⠀⠀⠀⠀⠈⠀⠉⠀⠈⣹⣿⠶⣤⣴⡀⠀⢣⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠶⠶⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡗⠼⠽⢟⠳⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣷⠀⠀⢠⡁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡄⠀⠀⠑⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣴⣾⣟⣋⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧⠀⠀⡄⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣮⣤⡶⠟⢦⠀⠀⠀⠀⠉⠉⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢀⠀⣹⣂⣾⣧⣶⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠙⣿⢹⣯⣿⣧⣤⣶⣾⢿⣿⠏⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠫⠶⢿⣿⣿⣿⣿⣿⣤⣴⣷⣲⣿⣿⣿⣿⣄⣀⠐⠚⠀⠀⡀⠀⣠⣿⣠⣿⣿⣿⣯⣾⣶⡞⠁⠀⠀⠀⠢⣤⠎⠀⠠⠄⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣛⣯⣽⣶⣿⣿⡟⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⠋⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣄⣠⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠶⠚⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⢄⡀⠠⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠙⢶⣤⣀⠀⠉⠓⠲⣄⠀⠀⠀⠀⢠⣾⣶⠄⠀⠀⠀⠀⠀⠀⠒⡋⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠗⠈⠓⢤⡀⢤⣿⡿⠗⠦⠀⠀⢙⣯⠄⠐⠀⠀⠀⠀⠀⣀⣷⡆⢀⣠⠤⠔⠚⠉⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡧⠄⠒⠀⠁⠀⢿⡄⠀⠀⣀⣀⣤⢤⠄⢿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⣿⡲⣄⠀⠀⠀⡛⣷⠀⠈⠉⠁⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⣔⠁⠄⠒
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠓⢤⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡞⠋
⠀⠀⠐⠦⣜⣿⡇⣀⣠⣤⡀⠀⠀⠀⢻⡄⠀⠈⠙⢦⡈⣷⠀⠀⠀⠀⠀⠐⠀⠀⢿⡀⠀⠀⠀⠀⠠⡀⠀⡠⠿
⠀⠀⠀⢀⣴⣿⣛⣿⣿⣿⣷⡆⡄⣄⢼⢇⠀⢀⡀⢈⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠠⠒⠀⣠⣊⡴⠁
⠀⠀⠀⠘⢻⣻⣿⣿⣿⣿⣿⠷⡽⢽⣼⣾⢧⣳⣡⣼⣇⡈⣿⠀⠀⠀⠐⠁⠀⠀⠀⣿⣤⣶⣾⡿⢉⡹⠟
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⠂⠾⠁⠬⠵⠤⡿⠂⣘⣦⡀⠿⣩⡤⠀⠀⣠⠞⢋⣵⣏⣵⣴⣶⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣷⣒⡠⠴⠤⠤⠤⠽⠷⢶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣏⡤⠄⠀⠀⠀⠀⠉⠠⣿⣿⡿⢿⢻⡻⢿⣟⣿⣟⠛⣻⣿⠃
⠀⠀⠀⠶⠶⡶⣾⣿⣿⣿⠿⠿⣻⠿⠿⢓⣊⣑⣒⣶⡿⢿⠿⣿⡿⠷⠾⡿⠿⣖⠙⠋⠻⢛⣙⣻⣣⠄
⠀⠀⢀⡀⣀⣐⣶⣾⣽⣋⣈⣀⡰⢈⣩⣭⣭⠙⠾⡿⠶⣾⣷⣦⣹⣿⣿⣋⣛⣛⣻⠭⠀⠀⠨⠁
⠀⠀⠈⠉⠛⠋⢀⣤⣤⣬⡍⠛⠋⠁⢀⠠⢽⠟⠛⣛⣋⣠⠝⣁⠰⠶⠭⠤⠴⠄⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡤⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠈⠀⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⠄⠀⠀⣀⣠⣠⣤⡶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣤⣤⣄⣤⣶⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠩⠭⠍⠍⣉⠉⣉⠐⣯⣮⣉⣙⣿⣾⠆⢉⠻⣎⠓⡻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠉⣉⣉⣉⣩⣥⣶⣤⠌⠙⠿⠛⠛⠿⠒⠈⠀⠈⠉⠉
⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠸⠍⢀⣀⣈⡉⠉⠉⠁⠉⠙⠉⠀⠉⠀⠀⠐⠒⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠹⠒⠉⠉⠑⠔⠒⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠒⠒⠒⠲⢄⡀⠀⢀⡠⢘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡞⠉⠉⠉⠓⣆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⡤⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢼⣤⢀⣀⠀⠀⣗⣒⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣛⣻⠷⣾⣛⣛⣭⣿⣷⡾⡧⣤⣴⢾⣛⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⢘⡇⠉⠉⠛⠻⠿⣿⠿⠛⠉⠁⣿⠻⠷⢿⠍⠛⠋⣉⣸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣶⣾⠿⠿⠛⢻⡇⠷⢀⡀⢠⠄⢸⠀⡀⠀⠀⣹⣤⣤⣼⣶⠟⣛⡥⠖⠛⡇⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣽⣿⣷⢶⣿⣷⣾⣷⣷⣌⣁⣼⡦⠶⢚⣯⠿⠁⣁⡤⠞⠋⠁⢀⠰⠆⡿⢹⡽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣮⣯⠐⡆⡭⢙⡛⣿⠉⣻⠾⠗⣏⣡⡷⠞⠋⠁⢀⠄⠠⠅⠈⣀⣴⠗⣫⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣧⢼⠿⡟⠷⣷⣬⡄⣿⣁⣴⣶⠞⠛⢉⠀⡆⣂⠀⠈⢁⣠⣶⡿⢋⡵⣊⠁⣘⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⠴⡞⠛⡇⣿⠴⠛⣧⣌⣠⠋⡛⡟⣿⠁⢸⠀⠘⠘⠀⢏⣸⠴⣚⡽⠛⣡⠞⡉⠂⢐⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣉⠑⣢⠴⠛⢳⡿⠔⠊⣀⠤⠈⣙⠷⣾⢻⣄⣸⡇⢀⡤⠖⣫⠴⠚⣁⡴⠛⠑⡠⢊⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⡁⠂⣠⠴⠋⣁⠤⠂⢁⠠⠒⢉⠠⠐⡉⠄⣂⠽⣸⡏⠝⣛⡥⠖⠋⢀⡤⠚⡁⠀⠀⢊⡴⠋⡠⠃
⠀⠀⠀⠀⢤⢀⠀⣀⡴⠚⢁⠁⠂⣤⠴⠋⡡⠔⠈⣀⠄⠊⡡⠔⠊⠠⠀⢀⣔⣚⣧⠴⠟⠛⠉⠁⣀⡤⠞⢉⠀⢀⠔⢁⡴⠿⠖⠋
⠀⠀⠀⠀⠘⣧⣞⣁⣈⣒⣢⣔⣋⣀⣔⣈⣀⣤⣊⣠⣴⣥⡤⠴⠶⠖⠚⠋⠉⠁⠀⠀⢀⣠⠴⠚⠉⠰⠀⡠⢀⣤⠞⡛⠠
⠀⠀⠀⠀⠀⠙⢢⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠒⠀⠀⡠⢊⣠⢟⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⡠⠐⢁⡴⠿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠉⠉⠀⠀⡔⠉⢻⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⣴⠏⢔⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠈⢧⣶⣾⠎⠀⠀⠀⠀⠀⠐⠈⠠⠔⣲⡞⠑⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡶⣼⣿⡷⠶⠬⣄⣀⡠⣴⣬⣴⢊⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⡿⠿⡒⠾⠛⠙⠋⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣀⠀⠀⠀⠀⠐⠿⠗⠂⠀⠀⢠⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠐⠒⠒⡆⠀⠀⠨⢅⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣴⡆⠀⢄⠀⠀⢠⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠂⠰⠶⠾⠧⠀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣄⢀⡄⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣶⣄⠀⠀⠒⠉⠉⠁⠈⠁⠀⠀⠀⡏⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠁⠀⠀⠒⠋⠀⠀⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠄⢀⡀⡀⠐⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠋
⠀⠀⠀⠀⠀⣠⣀⣀⠰⣶⠶⠟⠃⠂⠂⢀⠉⠈⠀⢈⣇⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠒⠦⠤⣤⣴⣿⣗⣦⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠀⠉⠉⠙⠛⠻⠆⠀⠀⠀⣀⣀⣀⠀⠁⣤⠈⠀⠡⠄⠀⡀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣤⣄⣀⣀⣀⡀⠛⠂⠀⠀⠈⠉⠉⢹⣾⣿⣿⣿⣧⡆⠀⠀⠀⠀⠐⠇⠈⠒⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⢀⠀⠉⠙⠂⠰⠶⠦⠀⣤⣤⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠒⠚⡀⠂⢤⣀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⠀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠊⠀⠀⠀⠀⠜⠃⠀⠘⠃⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⣠⡄⠒⣧⠈⠀⠐⠀⢠⠁⢀⡀⠠⡤⠄⠀⠀⠀⠉⠀⣷⠀⠀⣠⡼⠁⢰
⠀⠀⠀⠀⠀⠀⠞⠓⠈⠀⠀⠈⠀⠀⠈⢠⡄⢺⡀⠀⠸⠇⢠⠟⠀⡰⢋⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡾⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⢸⡇⢳⠂⠀⢰⣆⠁⣠⠞⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣸⣇⣀⢀⠄⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⡀⢀⡄⢉⡄⠀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡷⠦⣼⣷⣶⣶⣦⣤⣼⣷⠾⠿⣿⣠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣤⣿⣿⣿⣿⣿⣤⣾⣷⣶⣿⠟⢫⣿⣿⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶⣿⣿⡟⢻⣿⣭⣶⣾⣿⡏⣿⣷⣝⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠾⢿⡿⠻⢿⣿⣿⡇⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠘⠁⠺⠿⣿⠿⠿⡿⢿⢿⠿⠿⠛⠛⠛⠉⠉⡀⠀⠀⠀⠀⠀⠀⠙⠿⣷⠿⠃⠀⢤⠠⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠚⠁⠀⠋⢀⠀⠈⠂⠘⠋⠀⠀⠂⠒⠎⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⢿⣿⣿⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣤⣬⣥⣤⣤⣤⣼⣌⣤⣿⣡⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⡀⠠⡯⠣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⢸⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⣏⣩⣖⢦⣤⣄⣀⣀⠀⢈⡇⠀⠀⠁⡀⢀⣆⣩⣤⡶⠃
⠀⠀⠀⢀⣀⣊⣛⣿⣷⣴⣶⣶⣦⣼⣿⣿⣿⣿⣓⡻⠶⠾⠶⠾⢒⡚⢳⣿⣿⣭⣉⣙⠝⢁
⠀⠒⠀⠀⠿⡉⠹⠿⠶⠟⡉⠽⢯⣭⠋⡐⠿⠿⠧⠽⠿⠟⢀⣽⣿⡋⠙⣛⡂⠘⠛⠯⠐⠛⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠖⠐⠛⠛⠧⠄⠀⠛⠂⠰⠦⠉⠉⠡⠄⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠒⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⢄⢠⠄⠀⠀⢸⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠤⢠⠂⠀⠁⡬⠁⠀⢨⡀⢣⠀⠀⠈⠀⢎⠳⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣷⣶⣧⢰⣿⡄⠀⢀⠇⠘⡇⠀⠀⠀⠘⡆⠈⢊⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣂⢀⣼⡆⠀⠀⠀⠀⠀⠀⠈⣢⣀⡇⠀⠀⠀⠀⢻⡀⠘⢷⡀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢸⣿⠍⠐⠶⣖⣂⣂⠧⠀⠀⢳⡄⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣻⠀⠀⠀⠀⠀⠀⠁⡆⠀⠈⢿⡄⠀⠀⠹⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣰⡟⠀⠀⠀⠀⢀⠄⢠⡾⣿⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣍⠿⠿⣯⡀⠒⠛⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢘⣀⡴⣿⠇⣀⣤⡬⢤⠀⠤⠽⢷⡟⠐⣌⠢⠀⣀⣀⡀⢨⡇⡰⠃⠈⠉⢉⣉⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢘⣍⣀⣀⣀⣿⠉⠀⠸⠇⢸⠀⠀⠀⢠⠁⠤⢶⡞⣯⢿⣇⣀⣼⣼⣤⣴⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡧⠘⢛⢙⠛⠛⠟⠛⠲⠾⠶⢶⠖⣛⣒⣚⣻⡿⠛⠭⠿⠻⠛⣛⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡬⠤⣤⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢰⣶⣿⣿⡿⠿⠿⠟

⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉
⣭⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣺⡂
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣀⣀⣀⣠⣿⣇⣀⣀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣉⣩⣿⡏⢉⠉⠉⠉⠉⠉⢭⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⣿⣿⣿⣿⣷⣿⣶⣶⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠛⠉⣭⠉⠉⠉⠉⠛⢛⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣿⣇⠀⠀⠀⠤⠴⠄⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠤⢤⠤⣤⣽⡿⠛⠓⠒⠒⠒⠚⠒⠒⠒⠒⠛⣷⠂
⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢀⡤⠔⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⢠⡏⢿⡷⣄⠀⠀⠀⠀⢠⣮⠀⠀⠁⠀⢀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠈⣷⡈⠃⡀⠀⠀⠊⠁⠀⠀⠀⠀⣼⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣿⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣱⣆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣶⣶⣦⣰⣄⡠⢤⣤⣹⣿⣿⡆
⠀⠀⠀⢀⠂⠀⢳⠀⣀⣹⣆⣰⠟⣨⠓⢄⠀⠀⠰⠿⠛⣿⣻⣿⢹⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⡇⠀⠛⣰⣿⣿⣿⣿⣧⣽⣧⣌⠳⣄⠡⢊⢸⣿⣿⣿⣤⣿⡇⠀⠈⠙⠫⢙⡿⣿⣿⣿⣿⣿⡀
⠀⢠⣾⠏⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⣿⣴⣾⣿⣿⣿⣿⣿⠿⠾⢶⡶⠦⣤⣍⣚⢟⡿⣿⣿⣷⡀⠘⣆
⠀⣼⠏⠀⠀⠘⢟⢁⡾⠛⠀⠈⢿⠿⣿⣿⠃⣼⣿⣿⣿⣿⢿⣿⠈⠀⠀⠀⣸⡇⠀⠀⠀⠉⠙⠛⠿⠿⣿⣷⣴⣿⣧
⠀⠏⠀⠓⠠⡶⣾⣮⣤⣤⣤⣦⣼⣷⣦⣽⣾⣿⣿⣿⣷⣶⣾⣿⣶⣶⣶⣴⣿⣿⣷⡶⠶⡤⠴⠶⠶⢶⡶⢿⠿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠉⠉⠁⣹⣿⡇⠀⠀⠀⠠⠙⢀⡜⠁⠲⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠳⠄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⡐⠁⠀⠈⠁⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡤⠤⠤⠤⠤
⡄⢀⡀⠀⠀⠀⣿⠀⢀⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣑⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⣸⡃⠀⠀⠒⠂
⢟⠙⠙⠉⠉⠙⣛⢛⡛⠋⠉⠉⠉⠉⠉⣿⣿⠉⢉⣍⣋⠉⠉⠉⠉⠉⠉⠉⢩⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣿⣿⡟
⠈⠄⠀⠀⠀⠀⢻⣤⣥⣦⣄⠀⢀⣄⢀⣼⣿⣷⢞⣹⣋⣀⣀⣀⣠⣤⣤⣤⣮⣤⣄⣀⡀⢀⣴⣤⣀⡐⣿⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣷⣿⣿⡿⠿⠛⢿⣿⡿⠿⠟⢻⣿⢿⣿⣿⠟⢻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠈⠀⠀⠀⠀⠈⢿⣿⠿⠛⠉⠁⠀⠀⠀⠈⢿⣇⠀⠀⢸⣿⠀⠀⠂⠀⢸⡟⢻⢣⠀⠀⠈⠁⠀⠀⠉⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣷⣆⣸⣿⣤⠀⢀⡀⢸⡇⠘⡘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢠⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣿⣷⣤⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠷⣶⣦⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣯⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠉⠀⠀⠀⠉⠛⠲⠦
⠀⠀⠀⠀⠰⢶⣶⣿⣿⡿⠿⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⡾⢿⡷⠾⠒
⠀⠀⠀⠀⠀⠀⠊⣁⣤⣴⠶⠶⢶⣦⣤⣀⣀⣀⣠⣤⣴⣿⣿⣶⣦⣭⣽⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠙⠳⠤
⠀⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⢄⠖⠉⠳⠄⠀⠀⠙⠀⠀⣦⠴⢶⣦⠶⣑⠟⢙⣇⣴⣤⣄⣠⣺⣷⣂⢄
⠀⠀⠀⠀⠐⠛⠚⠛⠭⠯⢍⣴⣶⣾⣯⣵⣿⣷⣷⢾⣖⣲⣶⣾⣿⡷⣶⣤⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⡃⢰⡗⣰⣦⣄⣀⣴⡄
⠀⠀⠀⢀⢀⣄⣤⣴⡆⠄⠴⣿⣍⣿⣿⣿⣿⠾⡶⢾⣿⣭⣿⢿⣿⣿⣞⣩⣾⡿⠴⠧⢿⣿⣿⣿⣿⣷⣦⣽⣿⣿⡷⣿⣿⣿⠿⠿⠃
⠀⠂⠰⠗⠶⣾⠛⣛⠿⡿⢷⣶⣿⣿⡟⢻⡟⢺⣿⣾⣿⣩⠇⠀⠀⣠⣿⣿⣷⠁⠀⠀⠀⢀⣼⣿⣟⣿⣻⣿⣟⡿⡟⢹⡂⠀⠀⠄⠀⡀
⠐⠂⠲⣦⣨⣨⣥⣼⡚⠻⢿⣿⣟⣻⣿⣽⣿⣿⣿⣽⣿⠟⣲⡿⠿⠿⢿⣿⣿⡶⠶⡶⢶⠽⢿⣿⠉⢉⠉⠉⣻⣿⣿⣿⣿⣿⣯⠷⠀⠻
⠀⠀⣠⣟⣻⣿⣏⣩⣹⣿⣟⣻⣏⢻⢿⣾⣿⣷⣿⠿⠋⣵⣿⣀⢀⡀⣼⢟⡉⠓⡠⠀⠀⠀⠠⣿⡤⢤⣤⣶⣿⣿⣿⢻⣍⠉⠃⢿⡈
⠀⠀⠀⠀⠀⣉⣩⣟⣛⣻⣿⣿⣯⣶⣭⣿⠟⠉⠀⢀⣼⣿⠏⠉⠙⠉⣳⣯⠀⠀⠃⠀⠀⠐⣶⣿⠇⠀⠀⠀⣹⣿⣿⣶⣾⣿⣶⣴⣄⡀
⠀⠀⢐⡿⠻⣿⣾⣿⣿⣿⣿⣾⣿⠿⢋⡤⢀⡴⢁⣾⣿⣿⠶⠦⣤⣶⣿⣿⡠⠰⢶⡶⣶⣱⣿⣿⣧⣀⣄⣶⣿⡏⢹⣿⣿⣿⣿⣿⣷⣦
⠈⠀⠀⣯⠖⠷⢶⣿⣿⣿⠿⠋⢡⣶⠟⣠⡟⣠⣿⣿⡿⠁⠀⠀⠀⠀⢈⣿⡏⠀⠁⠀⠀⠀⠈⢹⣿⡙⡛⠛⠻⡇⠈⣿⣿⣿⣿⣿⣿⡗⠂
⠀⢈⢉⣿⣭⣿⣶⣿⣯⣥⣔⡚⡋⠽⠾⢿⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡯⡆⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⣷⠀⠘⢿⣿⣿⣻⣿⠷⡦⠁
⠀⠈⠀⠉⣟⣿⣣⣿⣵⣿⣿⣽⣷⣲⣶⣿⣿⣿⣿⣿⣳⣶⣤⣤⡀⢀⣿⠿⣷⣿⣷⣶⣤⣶⣶⣿⢻⣶⠒⢦⣕⣿⣇⣀⢌⣿⡡⠘⢧⣀
⢤⣼⣤⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣤⣤⣤⣤⣦
⠀⠈⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣟⣿⣿⣿⣟⣿⡏⠉⠁
⠀⠀⠀⠘⠛⠛⠛⠻⢻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣭⣿⣿⣿⣏⣾⣿⣿⢛⣤⣴⣋⣹⠾⠛⠿⠮⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⠺⠛⠳⠛⠛⠻⠟⠻⠟⠿⠿⠿⠎⠹⠤⠌⠽⠉⠛⠉⠛⠉⠁⠈⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠐⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠰⠦⠐⡷⠛⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⣠⣀⣀⡰⠛⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣃⡞⠉⠉⠉⣹⣤⣤⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢰⡟⠰⠧⣤⣤⣄⣙⠀⢀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⠃⠀⠀⡈⠀⡻⠋⠉⠉⠉⠉⠩⠿⠶⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡝⣶⣄⣱⣶⠿⠒⣶⣤⣄⡀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢿⠖⣰⠿⡐⡯⣈⣇⣀⣾⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⣿⡿⣿⣶⣶⣽⣥⠬⣭⡥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠴⣖⣦⠀⠒⠀⡤⡄⢉⠙⠉⠀⠲⠬⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⣿⠈⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢒⣛⡓⣒⠒⠒⠛⠒⠒⠶⠶⠾⠶⠶⠶⠶⠶⡀⠔⠊⠀⣷⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣿⣿⣿⣿⣿⣿⣿⡟⠿⡷⠶⣶⣾⣷⣶⣶⣶⣶⠿⠶⠶⠶⠼⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠿⠿⡿⠿⠿⣿⡿⢋⣴⣿⣿⣿⡿⠿⠿⠛⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⣒⣚⣛⣛⣒⣒⣒⣒⣒⣻⣟⠒⠒⢲⣶⠿⠛⠛⠛⠒⠒⠒⠒⠀⣼⡇⠀⠀⠀⠈⠙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⢛⣛⣓⣒⣒⣒⣒⣒⣒⣲⠈⠓⢲⡶⡶⠶⠶⠶⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⠶⣛⢅⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡏⠀⠀⠀⣠⠾⣫⣵⡿⢣⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣻⡇⠀⠀⠈⠛⢛⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⢀⣴⢟⣵⣾⣿⡟⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⢀⡴⢋⣵⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⢋⡴⣋⣴⠿⠿⠿⠿⠋⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢳⣶⠀⠀⢀⣛⣻⣿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣡⠥⢶⣒⡚⣛⠃⠀⠀⣹⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣽⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠿⢛⣫⣵⣶⡿⠟⣋⣱⠃⢠⣆⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣿⣥⠾⠟⠋⠉⠁⢸⡟⢀⡤⢖⣫⣵⣾⡿⣟⣫⣵⣾⣿⢋⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⡟⠀⠀⠀⠀⠀⢀⡿⣛⣥⣾⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⣾⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣭⠿⠧⠤⠤⠤⠤⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⠇⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣭⣴⣶⣿⠛⣛⡛⢛⣟⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠏⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣿⣧⣼⢟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⠏⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠋⠠⠴⠿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠽⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠤⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⡛⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠛⠓⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⢺⡇⢰⠀⠐⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⢸⡇⠀⠀⠒⠒⢺⠛⠲⣶⡖⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠠⠀⢸⡇⠀⠀⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠈⠀⠸⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠂⠀⡀⠀⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠂⠐⠾⠶⠶⠾⠿⠿⠿⡇⠀⠐⠛⠀⠰⠶⠖⢸⡇⠀⠀⠉⠁⣀⣠⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⡤⣤⢴⡧⡄⠀⢀⢀⡀⠀⢀⣈⣧⣤⣶⣞⣫⣿⡟
⠀⢀⡀⠀⠀⠀⠀⡀⢀⣜⣶⣷⣴⣬⣭⣿⣷⣿⣮⣷⣶⣶⣷⣾⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣣⣤⣤⣄⢀⡀⢀⡀⠀⠀⠀⡀
⠶⠚⣛⣳⠾⣿⣾⡿⣶⣿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⢿⠻⠟⠛⠿⢿⣿⠿⠹⢋⣠⢠⣤⠙⣿⣿⣿⣿⣿⡿⠯⣭⣥⣤⡄⣡⣤⣤⠄
⠒⠂⠁⠀⠀⠀⠒⠛⠋⠠⠤⠄⠀⢤⡤⠄⣀⣀⡀⢀⣠⡄⠀⢐⣖⣤⣄⣠⣤⣶⣶⣿⠋⠀⢙⣻⣿⠷⠶⣤⣀⣀⣭⠍⠉⢴⣤⡤
⠀⠀⠀⠉⠉⠉⠁⠈⠛⠛⠛⠃⠀⠀⠀⠀⠈⠉⠀⠀⠒⠲⠖⠒⠁⠀⠙⠛⠟⠛⠋⠀⠀⠀⠀⠉⠻⣿⣷⣷⠦⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢾⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⣿⢤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⢻⡆⠠⠈
⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠶⠼⣧⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠁⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉⠁⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣛⣻⣿⡿⣿⡾⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡵⢌⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠲⠤⠶⣼⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡀
⠀⠀⠀⠀⠀⠰⠧⠐⠒⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢐⣃⣹⣿⣠⡤⠬⣽⡿⠶⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⢀⣿⣄⣠⣴⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣻⡷⣧⣙⠤⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠾⢿⣛⣿⣿⣿⣿⣍⡉⣴⣜⣯⡾⠞⠓⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⠀⢻⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠛⠻⣿⣍⢙⡿⢿⣿⡇⢠⣿⡿⣼⢀
⠀⠀⠀⠀⠀⠀⣨⠿⣿⣿⣿⣯⣮⡀⠛⢛⣤⢀⠈⠛⠲⢿⡟⠓⢿⣇⠳⣄⡀
⠀⠀⠀⠀⢠⣶⣿⠆⠠⠀⠘⠉⠉⠉⠒⠀⠀⠈⠠⠂⠀⠠⠄⠀⢘⠻⠿⢮⣉⠁⢤⣴⣶⣤⡤
⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⣽⣄⣈⠙⠳⠦⠀⠀⢀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠐⠀⠈⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣝⣿⣭⣉⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⢀⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⡠⠖⠲⣿⠒⠐⠂⠀⠀⠀⠡⡰⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⣹⡏⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣄⣀⡀⠀⠀⠂⠒⠛⣿⠒⠂⠀⠂⠀⠀⠀⠀⠠⠄⠴⣾⡦⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⣿⢀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠙⡏⢀⠀⠀⠀⠀⠀⠀⠀⣻⠈⠙⠉⠉⢹⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠚⠁
⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣶⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⡿⠉⠀⠄
⠠⣤⠄⠈⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡠⠤⠤⠤⠴⠶⠤⠤⠤
⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣿⣿⣿⣿⣵⣦⣤⣤⣤⣄⣀⣈⣈⣀⣠⣤⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⣿⠏⢩⠝⣋⡥
⣿⣯⣛⡛⠿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣮⣷⣦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡟⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⠈⠀⠈⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣄⠀⠀⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⣿⢻⣏⢙⣁⠀⠈⠈⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠵⠓⠛⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡠⠀⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⣤⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⣴⣺⡥⢿⣤⣉⣰⣄⣠⣨⣦⣼⣅⣀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡴⠞⣛⣩⡤⠷⠞⢻⠚⠛⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⢿⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣢⠘⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡄⣘⡿⣿⣿⣿⣿⣯⣝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠞⠁⣹⣁⡟⣻⣔⣻⣿⣿⣿⣷⣮⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣦⣤⢿⣻⣮⣷⣿⣿⣿⣿⣿⣿⣿⡏⠈
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠛⠉⠉⠙⢿⡿⠋⠄
⠀⡂⠀⠀⠀⠀⢤⠤⣤⣤⠤⠶⠦⢤⣤⣬⣿⢿⠿⢁⣤⡀⠀⢀⢢⣼⡇⠀⡄⠖⠠⣤⠄⠀⠀⠀⠠⠶⠦⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠡⢶⣾⢂⣶⢶⣦⣤⡤⠀⠐⢺⣽⡿⠈⣽⣾⣿⣇⣀⠻⢿⣿⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢶⣶⡬⢙⣠⣴⠚⠿⠃⠀⠈⣛⣯⡌⠻⣿⣦⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠷⠐⠀⠀⠀⠀⠀⢀⣀⢼⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢀⡀⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣙⠶⠟⠊⠉⠀⠀⠀⠀⣀⣧⡄⣰⡼⢤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⢀⣶⣤⠤⠔⠚⠉⠁⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠐⠈⠉⠀⠀⠀⠠⣶⠀⢸⠀⢀⣀⣠⣤⠤⢤⠘⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⡆⠶⠒⠂⠀⠀⠀⠀⢀⠀⠀⣰⣿⠀⠾⠛⠉⠉⠀⠀⠀⠀⣤⠀⠹⢦⡀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢀⣆⡠⢾⡶⠛⠛⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⡀⠈⠛⠻⠛⠃⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⠭⠀⠒⢈⣽⠷⠶⠛⠀⠀⠀⠀⠀⢀⣀⣄⣤⣷⠿⠷⠆⢀⣀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠦⠶⢒⡃⠈⠀⠀⠀⢿⠀⢀⣠⠂⣀⠠⠖⠚⢻⡉⢻⣧⣤⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠀⢠⠀⠘⠀⠀⠀⠀⠀⠸⡆⠀⢧⡾⠁⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣧⠀⣈⣥⣴⣶⣷⣿⣿⣿⣛⡹⠿⢿⣿⣿⣷⣾⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠯⠼⢿⣶⣶⣿⣿⠭⠍⣉⣨⣽⣿⣿⣿⣿⣿⣿⣿⠟⣯⣷⣶⣶⣤⡀⢹⣿⣿⣿⣷⠆
⠀⠀⠐⢿⣿⣦⣤⣀⠀⠈⢿⣷⠇⣰⣶⠶⣾⣿⡿⢿⡿⣭⣭⣭⣛⠛⠛⠛⣙⣧⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣯
⠀⠀⠀⠐⠋⠉⠙⠛⠿⠷⣶⣾⠟⡋⠙⠋⠻⣍⡁⣤⣠⠤⠀⣤⣴⣶⣤⣾⠿⣿⣿⣿⣿⣿⣿⠏⠘⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠷⠈⠛⠀⢲⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠙⡷⢄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠷⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡓⠲⠦⣤⣇⣀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠊⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠴⢦⢤⠀⠀⠀⠀⠈⠉⠙⠛⢆⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠀⠉⣟⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⣼⣦⡠⠀⠀⣀⠀⠀⢴⡀⠀⢀⠀⠀⢸⠈⠉⠒⠻⢦⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⠦⠤⣤⣀⣀⣿⣿⡇⠈⠀⠀⣼⠀⠀⠀⠀⠀⠀⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡀⠀⠀⠀⠉⠉⠙⠒⠒⢦⣼⠂⠐⣄⣀⣠⠀⢀⡆⣠⢸
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠐⠒⠚⠿⠿⣤⣀⣁⣛⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢶⣟⣳⡶⠤⣤⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⠀⠈⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⣄⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣧⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠃⠀⠀⠛⠀⣴⡖⣤⢠⣼⡏⠀⠒⠂⣶⡶⣶⣶⣿⠒⠚⠿⠀⣼⣰⢿⡆⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠀⣰⢨⣽⣏⠙⠒⠒⠒⠿⠧⠿⢬⣿⡙⠛⠒⠒⠻⠷⠾⠿⠿⣤⣤⣄⣀⣿⣃⡈⡇⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⡏⠀⠀⠀⠀⢱⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣷⣿⣹⡇⠀⠀⠀⠀⠀⠀⢰⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣿⣸⠇⠀⠀⢀⣠⣼⡗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣀⡾⠵⠚⢻⣿⠉⢿⡏⠃⠀⣰⢀⣀⠆⣀⢼⡇⠀⢀⠄⢀⣠⡄⡀⣠⠂⣀⡐⠀⣠⣾⣹⣿⣿⣷⣶⣿⣯⣰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⢶⣾⢿⡦⣾⣟⣴⣄⣩⣿⡇⠰⣿⠆⣠⡼⠋⣾⣿⠁⠀⢹⣏⣼⣿⣶⣾⣿⡿⢶⣞⠉⠀⠀⢹⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣼⣿⣺⣗⣿⣟⡹⣿⣿⣿⣇⡀⣿⠞⣿⡇⢠⣿⣿⣶⣶⣿⣿⣿⠣⡿⡿⡿⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣤⣼⣷⣼⣿⣿⣿⣿⣿⠿⠿⠻⣴⣾⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣾⣿⣿⣿⣛⣿⡟⢻⣿⠋⢻⣿⠛⢻⣿⡟⠻⣿⣏⡉⣿⣿⣼⠿⠿⢶⣾⣿⣿⣏⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠉⠻⠛⠿⠿⡿⠿⢿⣿⡿⣿⣿⠿⣿⣯⠿⣭⣭⠿⣶⣤⡤⢤⢶⢾⡟⠻⣿⣿⠿⠿⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠁⠐⠂⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠂⠒⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠰⡧⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢢⡄⠀⠈⠀⢀⡇⠀⣀⠠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢍⣭⡝⢛⣒⢦⣈⣀⡀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠛⠀⠄⢠⡄⢈⣑⣠⣴⠎
⠀⠀⠀⠀⠀⠠⠤⠶⣤⣤⣤⡀⣀⣀⣤⣀⠘⣿⣿⣾⣿⣶⣬⣭⣽⣏⣉⣾⣀⣐⣃⣀⣶⣤⣤⣽⡶⠶⠛⢻⡟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⢐⣶⣶⣶⣮⣿⣟⣻⣿⣿⣿⣿⠿⣿⣿⠫⣿⣿⣧⠖⢦⣭⣭⣭⠨⢭⣥⣶⣢⣤⣿⣟⠛⢻⣶⣶⠏⢀⣒⠂⠀⠀⠁
⠀⠀⠉⠀⠀⠚⠤⠀⠙⠛⢛⡛⠡⠐⠺⠿⠶⠦⣀⣬⠛⠛⠿⠿⣒⡾⠛⠉⠠⣾⣿⣥⢂⠀⢩⡍⠀⣉⠉⠒⠂⠉⠉⠀⠀⠘⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠲⠂⠘⠛⠛⠻⠤⠀⠀⠐⠒⠀⢲⡀⠉⠉⠩⠤⠄⠀⠀⠉⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣭⣭⣽⣤⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣴⣶⠶⢾⣷⠶⢶⣶⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣄⣀⣨⣿⣯⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣉⣉⣉⣛⣉⣹⣿⣹⣉⣋⣈⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣟⣿⡟⣻⠋⠉⠛⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣿⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠋⢻⢻⣿⣿⡛⢋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡉⢛⣿⣧⣿⢰⡇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⠿⠿⠛⠻⣿⠿⢿⣿⣿⠿⠿⠿⠿⠾⠿⠿⠟⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡦⢼⣿⣿⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤⣤⣤⣤⣬⣧⣴⣷⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠻⡇⢠⣿⣿⢸⡟⠀⡦⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣦⣼⣦⣤⣦⣴⣴⣷⣼⣿⣿⣿⣶⣦⣷⣦⣤⣬⣤⣤⣴⡶⢦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠹⠻⡍⣿⣿⣿⣿⣿⣿⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣤⣤⣽⣦⣤⣬⣿⣿⣿⣾⣿⣿⣿⣦⣤⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⡿⠿⠿⠛⠛⢿⠟⠛⣿⣟⣿⣿⣿⣿⣿⣿⠿⡿⢿⠿⠿⠛⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠿⣶⣶⣷⣶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⠾⠿⣿⢿⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣠⣄⣍⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣭⣩⣍⣉⣉⣁⣤⣀⣀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠋⠙⠛⠛⢿⠿⠿⠻⠟⠿⣿⣿⣿⣿⡿⠿⠛⢿⡿⠿⠿⠟⠿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣬⡆⢀⡿⢠⢲⣿⣿⣿⣮⢷⡀⣦⣬⠒⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣧⡾⣠⣶⣯⣿⣿⣿⣿⡌⢿⡄⠀⣿⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠸⣿⣷⠟⠉⠰⣿⣿⣿⠟⣷⠈⢿⠀⢻⣄⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⠿⣿⣿⢟⣀⣴⣾⣿⣿⣿⣦⣭⣆⠈⢷⣼⣿⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⢷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣟⣛⣛⣛⣛⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣻⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣉⣭⣭⣭⣭⣭⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡂
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣉⣽⣉⣙⣋⣽⣉⣷⣈⣉⣛⣉⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣧⣤⣥⣤⣤⣬⣿⣿⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⣴⠆⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⣾⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠋⠉⠀⠂⡀⠉⠀⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠠⠄⢀⣔⡏⠀⢰⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢿⣇⣀⢸⡇⢰⡶⣄⣀⠐⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠈⠙⡛⣸⠃⠀⠉⠙⠻⡾⠧⣤⣛⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⣤⠄⠀⠀⢨⣵⡏⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠚⠂⠲⠀⡜⣨⡿⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠛⠛⠿⠿⣿⣿⣾⣶⣿⣥⣶⣿⣿⡛⠛⠻⠿⣿⣆⡉⢰⡦⢿⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢩⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠯⠿⣿⠃⠀⠀⠀⠀⠀⠉⠙⠳⢾⡧⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠚⣇⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣠⣾⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣰⣃⢋⢌⣼⣦⣗⣴⣷⣧⣾⣿⠋⠀⠐⠇⠴⠃⠀⠀⠀⠀⠀⢰⣽⠃⠀⣀⣀⠐⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣬⣅⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣤⡀⠀⡀⣀⢀⣼⡿⢀⣴⠟⠉⠓⢢⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣉⡉⠛⢻⡿⣿⣿⣷⣮⣿⠋⠰⢟⡿⠿⡿⣿⡿⣿⣷⣦⣤⣤⣽⣿⣿⣷⣿⣧⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠷⢤⢄⣉⠛⠃⠀⠀⢈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣞⣛⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣀⣸⠃⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡅⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡤⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢴⢞⣛⣩⣥⣶⣤⣿⣿⣿⣿⣿⡿⣿⢟⣩⣾⣏⠀⠀⠉⠿⢿⢿⡿⠿⠿⢿⡅⠾⢿⠀⠀⠂⠀⢀⣤⡞
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣐⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣛⠛⠿⣿⡅⠀⠀⠀⠀⢠⠷⣍⠢⣶⣿⣧⣀⢤⣀⣀⣰⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠐⠓⠶⢚⣋⣩⣭⣭⣭⣏⣉⡛⠿⣦⣝⠂⠀⢐⣕⣾⣶⣮⣷⣮⣽⣿⣿⣿⣿⣿⣟⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⣁⣐⣨⣽⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢷⣄⠀⠉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣭⣭⣤⣶⣾⣿⣿⣿⣿⢿⡏⠁⢾⣆⠙⠶⣤⣈⣛⠿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢈⠙⢿⣿⣧⡍⠙⠀⠶⢿⣷⣴⣤⣉⣉⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠻⠶⠦⠉⠀⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⡖⠈⠁⠠⠔⢋⠺⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠻⣕⠢⢄⡀⢸⠖⠘⡟⡃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠑⢦⡈⣿⠀⣀⠁⠀⠀⠀⠀⠀⣤⠐⠖⠀⠀⢰⠏⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠉⠣⣀⣽⠛⣄⠠⢀⠀⢸⣇⠠⣴⠁⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄⠀⠀⢀⠞⠁⠀⢀⠀⠀⠀⠀⠀⢈⡟⠀⠈⠑⢦⡀⣿⠉⠄⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣶⣗⠒⠒⢰⡅⢀⡀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠈⠻⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⢸⡏⠙⢦⣾⠀⠀⠉⢳⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢸⡷⠀⠀⡏⠓⢦⣀⣿⡟⢦⣈⠉⠒⢤⣀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⡗⡀⢠⠀⠀⠈⡟⣿⠇⠀⠘⠷⢤⡀⠈⢳⣤⡰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠻⣿⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢦⣼⡃⠙⢦⡰⠄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢏⡛⠲⢦⣄⣼⠃⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⠀⠙⠦⣀⠹⣄⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠙⠲⣄⡼⢣⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢳⢾⣞⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣠⠞⠠⠀⠀⡐⠁⠀⠀⣠⣏⣀⣀⡀⡠⠃⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢹⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣖⠲⠦⢼⣅⣀⠀⠔⢹⡏⠫⣍⠉⠛⠶⢤⣀⣴⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⠈⠙⠲⣤⣀⠉⠻⢶⣿⣾⠀⠈⠻⠲⣄⠀⡝⢻⣦⡀⢀⢣⠀⠀⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⠀⠀⠀⠈⣉⠛⢶⣿⣿⡟⠀⠀⠀⠀⠈⠻⠷⣾⡆⠈⠻⡀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⠇⠀⠀⠀⢻⢀⣾⡿⠟⣷⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣄⠈⢠⡎⠀⢸⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⡘⠀⠀⣸⠃⠀⠀⠀⢀⣾⣿⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣻⢦⣵⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠐⠌⠀⠀⡟⠈⠀⠀⠀⣾⣛⡺⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠾⣿⡀⠀⠀⠀⠀⠀⢀⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠠⠀⢸⡇⠂⠀⠀⡼⣽⢯⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠀⠀⠀⠈⠑⠦⣀⠀⠀⣠⠟⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠛⠲⢤⡿⠀⠀⢀⣼⣱⣏⠏⠀⣸⢃⠁⠀⠀⠀⠀⢀⠂⠀⠀⡀⢿⡾⠀⠀⠀⠀⠀⠀⠀⠙⠿⢇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠄⠃⣼⢣⠟⡞⠀⣰⢏⠳⠀⠰⠂⠀⢀⣎⣀⣀⣐⣐⣿⠳⠃⠀⠆⠀⠀⡀⠀⠀⠀⠀⠙⠒⢄⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡞⣁⠀⣀⣼⣯⣯⣼⣥⡴⠏⣀⣠⠴⡟⠚⠙⠍⣿⣿⠉⣉⣽⣟⣛⣷⣟⣒⢲⣶⣤⣄⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠾⢽⣿⡯⢍⣉⣠⢇⡾⢰⢁⣾⡷⠊⠉⢙⣳⣦⣤⣞⣛⣯⣽⡟⠛⠛⡛⠓⠛⠾⢯⣿⢿⡄⠈⠙⢦⣼⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠐⠮⠽⣛⣾⢿⠿⡷⡶⢿⣿⣗⣛⠿⢽⡛⣯⣁⠀⠈⠙⠞⠋⣩⣭⣭⣭⣉⣓⢞⣵⡿⢿⡆⠀⢈⠆⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢑⠂⠬⢉⣉⣉⣭⢉⣭⢁⡷⠠⠟⠚⠋⣡⡤⠶⠐⢒⡻⢭⣁⣀⣀⣈⣉⣋⣾⡯⠱⡎⢁⣠⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⡀⡉⠐⠒⠶⠦⠄⢠⣀⢴⠆⡐⠛⠋⣉⠀⠄⠀⠀⠨⢛⣋⠠⠤⣤⣭⣿⣾⠷⣤⠋
⠀⠀⠀⠀⠀⠀⣠⣤⠤⠚⠚⠛⢷⣄⣉⣱⣶⣶⣶⠦⠤⣲⣶⣂⣙⣁⣀⣠⣤⣖⡚⢛⣋⣛⣛⠟⠿⠿⠿⣿⠶⠗⢒⣋⣀⣀⠄
⠀⠀⠀⠀⠀⠉⠓⠈⠒⠂⠀⠀⠡⠟⠛⠋⠉⠓⢂⣠⣀⣬⡟⠛⠉⠁⠀⠉⣭⠽⠿⠯⠤⠀⠀⢈⡉⠤⠤⠤⠤⢉⣉⠀⠀⠠⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⣟⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⢸⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⠃⠀⠀⠎⠀⠀⠳⣾⣷⠷⠟⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢤⣴⠄⠾⢿⢿⠤⡤⣼⣿⡦⠤⡄⡀⢀⣀⣀⣀⠓⡲⠶⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡇⠀⠀⢶⠶⠀⠀⣹⢿⡇⠀⠀⠀⠩⠉⠀⠉⢹⢿⠑⠂⠰⠔⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢧⣸⡀⠀⠀⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⣿⡀⠀⠀⠂⢼
⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣠⣧⣤⠀⠀⠈⢀⢸⢧⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⠀⢠⠀⠐⢸⠈⠳⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⣿⡆⠀⠀⠀⢸⠀⠀⠑⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⣈⡵⡀⠂⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣯⣤⣄⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢾⢿⠏⣤⢂⠀⣎⠜
⠀⠀⠀⠀⠀⠀⠀⣰⠋⣸⠻⡛⠟⡿⣧⡈⣇⠀⠀⠁⠀⠀⠁⣄⠀⡀⣀⢀⠂⠀⡆⠀⠀⢸⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣷⣟⣓⠳⠶⠤⢄⣔⠙⡺⠦⠤⠤⠄⠀⠬⠑⠀⠞⠐⠛⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⢻⣯⢙⠟⠛⠛⠿⠷⣷⣞⠛⠂⠂⠂⠀⠠⠤⠄⠰⠄⠐⠂⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⡛⠛⠛⣻⣦⠸⠆⠰⠀⢤⠙⠿⠿⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣿⣷⣿⣾⢿⣦⣀⣀⡀⠀⠀⠰⣶⠆⠀⠀⢰⣦⠀⠀⠀⠻⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⠠⠀⣨⣟⣿⣹⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠞⣓⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠠⠍⣭⢙⡿⠛⣿⣿⡛⣿⣿⣿⣿⢛⣿⣻⣟⡉⢉⠉⠩⢁⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠚⠑⢟⣿⠋⢿⣟⠊⠙⠩⣶⣟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⣠⠇⠀⣈⠀⣠⠤⣀⠁⣟
⠀⠀⠀⠀⠀⠀⢀⢀⡤⣄⢠⡖⠓⠖⠃⠀⠀⠀⠖⣦⡀⠛⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠚⠃⠀⢠⠀⠀⠀⣰⠋⠲⠢⠔⠃
⠀⠀⠀⠀⢀⣰⠋⠁⠀⠰⠈⠈⠀⠶⠖⠀⠁⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠖⠛⠂⠋⠀⠂⠀⠂⠀⠠⠀⠀⠈⠁⣀⠙⡄
⠀⠀⡆⠀⠁⢉⠀⣀⠀⠀⡀⢈⠠⠴⣿⡦⡴⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⡀⠐⠂⠱
⠀⠀⡷⠀⠐⡊⠘⠹⠇⠀⣀⢄⣰⠤⠀⠀⢀⡠⢤⠶⠊⠉⠈⠉⠁⠗⠈⠁⠈⢉⡶⠄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠆⠀⠀⠁⠀⠀⡀⠈⠁⠀⠀⢀⣠⠞⠉⠀⠀⠀⡀⠃⠀⠀⠀⡠⠀⠀⠀⠛⠀⠀⠀⡀⠀⠐⠯⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⡀⠀⡘⢿⣀⠀⡟⠋⠀⢀⠞⠁⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⡀⠀⠨⠶⡀⠀⠀⠀⠀⢀⡄⠀⣚⠁
⠀⠀⠃⠈⠀⡀⠀⠙⠁⠀⠴⠃⠀⠀⠀⠀⠀⠀⣭⠀⠀⣀⣼⣀⡀⢀⣠⠦⡤⠀⠀⢀⢀⠀⠀⠀⠀⢠⠀⠈⢧⡀⠀⢰⠈⣇⡿⠁⡀⠀⠀⠀⠠
⠀⠀⣰⠳⠴⠁⠀⠀⡠⢊⠂⠀⠀⠀⠀⠀⠠⠚⠋⢠⢈⡀⣉⣉⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠨⠀⠀⠀⠘⡁⠄
⠀⠀⡇⠀⠀⠀⠀⡼⠃⠀⠀⠐⠂⠀⠀⢀⠀⢠⡀⠟⠀⠀⣯⣀⣤⠐⢻⣏⣹⠄⠀⢀⠁⠀⠀⡀⠀⠐⠀⡄⠀⠀⠟⠀⠂⠄⠀⢀
⠀⠀⡏⠀⠀⠀⣰⠁⠀⠀⠀⢀⣼⠀⠀⠋⠀⢸⠅⣀⠂⠉⠁⠀⠙⠉⠀⠀⠀⢀⠤⠰⠤⡶⠀⠀⣀⠄⢀⣅⠀⠀⠄⠀⠀⠐⠀⣈⠁
⠀⠀⠇⠀⠀⠠⠆⠀⠀⠀⠀⢀⣶⠀⠀⢀⣴⣨⡆⠶⠖⠒⣟⠘⣧⠔⠘⡟⠳⡆⠀⠀⠀⠀⣠⣤⠤⡤⠀⠈⠠⠀⢀⠀⠐⠀⣀⢀⢨⡇
⠀⠀⠎⠁⠀⢸⠀⠀⠀⠀⠀⠰⠋⠀⡴⠛⠀⢨⡦⢀⠤⠐⠛⠛⢻⠤⠘⠓⠚⠟⢒⢶⡶⢧⡁⠀⣴⡃⠀⠀⠀⠀⠈⠁⠀⡈⠾⣿⠆⠁
⠀⠀⠇⠀⠀⠘⠀⡀⠀⠀⠐⠀⠠⠊⠀⠘⠦⢜⣧⠼⠀⠀⣀⣀⠀⠀⠀⡀⠀⣦⠘⠚⠓⠺⢃⡴⢯⡥⠸⡇⢄⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⣠⠀⠠⡄⠀⠀⠜⠀⠠⠂⠀⠀⠂⡗⢘⡋⢨⣯⣬⡤⠀⢘⣿⣈⡉⠀⢀⣀⣈⠀⢀⢀⡄⠀⠁⠀⠀⠀⠀⢰⡿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠩⠀⠀⠐⢀⠆⠠⠀⠀⠀⠀⠠⠤⣿⠜⠇⠀⠀⠀⠐⠀⠀⠀⠈⣀⠌⠸⢿⡿⣀⣽⡇⠄⣸⠀⠆⠀⠀⡀⢸⣃⡀⠠⠄
⠀⠀⠀⠀⠀⠘⢳⠃⠐⠖⠦⣤⣀⠀⠘⠁⠀⠀⣿⡠⢠⣤⣠⢼⣄⠀⣠⡀⠐⣇⠀⠀⠀⠏⠁⠀⢀⣀⣽⡴⢦⣀⠴⠋⣸⠏
⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠦⣄⠉⠙⠲⡦⢤⠶⠿⡷⣶⣷⡿⠘⣇⣸⣁⣍⣷⢻⣇⣰⣤⣷⢻⢹⢿⠿⣿⣿⢛⡟⠁⢀⠟⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⢦⡀⠀⠀⣬⡈⠉⢩⡿⠧⠤⡽⠟⠻⠛⠿⠭⠯⠭⠭⠝⠛⢉⣓⢠⠖⠛⠊⠚⠀⣠⢋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣔⠡⢰⡆⠀⠀⠀⠀⠃⠀⠋⠉⠟⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠔⠒⠁⠀⠀⠻⣧⡀⠀⠀⠀⠀⡴⠃⠀⠇⢀⡀⢀⣈⡅
⠀⠀⠀⠀⠀⠁⡤⠈⠃⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⢠⡀⠈⠱⠀⠀⢳⡀⠀⠀⣀⠀⠈⠀⠀⠤⣀⡄⢀⠀⢠⠞⠁⣠⠀⠠⠋⠀⠀⠀⠂
⠀⠀⠀⢀⠠⠀⠹⠀⠀⠀⠀⡄⠀⠄⠀⠈⠉⠢⠀⠀⠙⠀⠀⠐⠄⠀⠃⠀⠀⠀⡀⠀⠆⠀⠀⠠⢰⡯⠄⠁⠀⠂⠀⣀⢠⡜
⠀⠀⠄⠀⠆⠀⠀⠀⠁⠀⡠⠄⠀⠀⠀⠂⢀⠀⠁⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⣀⡤⢔⡉⠁⠀⠠⠈⢧⣐⠈
⠀⠀⠊⠁⢀⠀⠀⠀⠀⠈⢀⡀⠄⠄⠀⠀⠀⠈⠁⠐⠢⠤⠀⠀⣤⣀⣀⡀⢤⠄⠀⠂⠈⠁⠀⣁⣄⠀⠀⣐⠀⠀⠁
⠀⠀⢠⠀⠁⠀⠀⠀⠆⠒⢀⡜⠰⣎⡐⠃⠀⠀⠀⠀⠐⠂⠁⠄⠀⠀⢠⡄⠉⢀⡀⠀⠀⠀⠈⠋⡉⢲⡿⠉⠟
⠀⠀⠃⠠⠇⠀⠀⢀⡆⠀⡿⠀⠀⠉⠁⠀⠀⠀⠠⠀⠐⠒⢶⡀⠀⠄⠈⢠⠴⠛⢧⡠⢤⡀⠈⡀⢏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠾⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠫
⠀⠀⠤⠂⠀⢠⠀⢠⢪⡵⠈⠁⠀⠀⠀⠌⠁⠂⠀⠰⠀⠀⠉⠀⠀⠀⠀⠎⠐⠋⠀⡀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⠶⠀⠀⡀
⠀⠀⠀⠂⠄⠔⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠡⢤⣀⠁⠀⠠⠓⢸⡦⢀⠒⠠⠀⠠⠈⡀⠀⠀⠀⠀⠀⠀⠀⠋⠁⡀⠃⠀⠀⠤⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠢⢀⣠⠀⠀⠀⠀⠰⠀⠀⡙⠂⠴⢀⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⠁⠃⠀⠀⠐⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⢀⠀⠘⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⠈⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣤⡤⠼⠗⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⢀⠀⠀⣀⢀⣀⣼⠦⠻⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠴⠖⠒⠛⡟⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡍⣀⡀⠀⠀⠶⢺⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣀⣠⣤⠶⠖⣶⠏⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠉⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢹⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢷⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⠀⠂⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠀⢀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡠⢀⣴⠿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠤⣂⣠⡾⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠀⠀⢀⣰⣤⡶⢞⡁⡀⢈⡟⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣃⣤⣤⠴⠆⢻⡏⠄⠋⢀⣀⣠⣄⣼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢢⡤⠟⠞⢹⡃⠀⡠⠔⠈⣠⣴⣶⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⣀⠸⠗⢈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣨⣥⣴⣾⣿⣿⣿⣿⡿⠟⠙⣿⢿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡭⠛⠟⠛⠛⠛⢛⣛⣉⠻⠞⣻⣗⡜⢼⣟⣿⣍⠛⢿⣶⣾⣿⡛⠿⢿⣤⣴⣬⣶⣶⣋⠁
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠶⠖⠋⢲⣶⠚⣳⣶⢿⣦⡾⢿⣿⣋⣀⣴⣿⡿⠿⣿⡿⠿⢛⣡⡘⠋⢘⠛⠶⠿⠿⢿⣿⣶⣿⣷⣬⣽⣿⣿⣿⡿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠥⠤⠶⠾⠛⠋⠹⣿⣿⣿⠿⠂⠒⠻⣭⣿⠿⠿⠿⢓⡠⠀⢾⣿⣿⠇⠀⢨⡛⠻⢶⣶⠀⠀⠛⢛⣉⣩⣍⡙⠛⠛⠃⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢀⣙⠫⠥⠤⠄⠈⠉⠋⢡⣤⣤⣤⣉⡁⠀⠐⢛⣫⡵⠀⠀⠈⠻⣷⣾⣯⡭⠉⠉⠛⠿⣿⡿⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠖⠂

⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⡃⠀⠀⢠⡚⠀⠀⠀⢰⣷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠍⡀⠄⠀⠀⣾⠀⠀⠚⠉⠓⢦⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢⣿⠀⠀⠐⠀⠀⠀⠸⣿⣿⣆⢀⣀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⣠⣿⣿⣿⣷⡄⠘⣟⠙⠲⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡾⠈⣇⠀⠀⠀⠀⠈⠉⣹⠙⣏⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⡀⠀⠀⠀⠀⠀⣿⢺⣿⠀⠁⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣾⠁⠀⣿⠁⠀⠀⠀⠀⢠⣿⣾⣼⡇⢢⡤⢈⠉⠁⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢰⣿⣴⣾⡟⠀⠀⠀⠀⠀⢸⣿⣿⣼⣇⠀⢷⡀⠀⠐⢢⣠⡤⠲⠛⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⣥⡤⠤⠤⣿⣿⣿⠿⢿⡏⢻⣇⠀⠁⣹⡏⠈⠉⣻⣀⣾⣧⣶⣾⣿⠛⠁⠀⠀⠀⠀⠀⢀⣤⢿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣤⣶⣶⣼⣿⣿⣹⣯⣧⣿⣷⣶⣿⣿⣿⣿⡿⡿⣿⠁⠀⠀⣀⡀⠀⠀⠴⢚⣶⣟⣖⣶⣦⡄
⠀⢀⢀⡀⠺⠗⠚⣛⡒⠶⠟⠿⣿⣿⠟⢻⣿⡟⠋⣿⡿⢟⣭⡿⠋⠈⢷⣶⣼⣿⣴⡾⠠⠂⣉⡁⢸⡟⠉⠘⣥⣙⣛⣉⠑⠒⠂
⠀⣬⣤⠭⢀⣭⠥⠤⣤⡉⠩⠥⢶⣃⣒⠛⣩⣤⠾⠚⣋⣀⣩⣤⡴⠦⣬⣽⣿⣽⣶⠖⠒⣲⣿⡷⣶⣾⡧⠐⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠬⠝⠛⠛⠉⠭⠩⠦⠤⢉⣍⠉⠉⢉⣉⣤⠴⠶⠿⣟⣠⡶⠞⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠤⠖⠛⠉⠁

⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠠⠒⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⢴⡤⠴⠓⠀⠀⠀⠀⢸⠀⠀⣀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠐⠛⠉⠀⠀⠀⢀⡀⡀⠀⣈⣺⠼⠓⠚⠉⠙⣦⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⢀⡰⠍⠋⠁⠀⢈⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠉⠀⠀⠀⠀⠀⠀⢺⡇⠀⢈⣤⣤⣶⠶⠓⠒⠚⡂⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠿⠑⠛⠉⠁⠀⠀⠀⠀⠠⡄⣀⣠⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⣄⠀⠛⢧⣀⣀⣀⡀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⡯⠴⠾⣟⡋⠉⠩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠈⠉⠋⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡬⠉⢒⠈⠁⣥⠟⠷⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⠾⡾⠿⢿⣀⣀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠲⠶⠖⢒⡫⠈⠀⠀⠀⢸⡆⠀⡀⡴⢀⡄⠴⠒⠋⢹⡅⠹⣿⣤⣶⣾⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠀⠀⢤⠀⠘⠀⠀⠀⠀⠀⠀⣷⠀⠀⡴⠛⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣫⣠⢴⣶⣶⣿⣿⣿⡿⢻⠿⠿⣿⣿⣿⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣯⣼⣯⣤⣥⣾⣷⣖⡚⠻⢉⣑⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣤⣤⣬⡁⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣶⣶⣤⣀⡀⠀⠀⠙⣶⣶⠒⢀⡤⣤⣶⣶⣶⣿⣿⣟⣛⣛⣛⠻⠿⠿⠟⠻⣂⣽⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣏
⠀⠀⠀⠈⠽⠟⠛⠿⢷⣶⣤⣬⣭⣶⠿⠿⠖⢸⡋⠋⠀⡉⣈⡉⠉⢉⣩⣄⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠃⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠁⠀⠀⠀⠓⠂⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠈⠻⠛⠿⠛⢫⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⡿⠟⠃⢰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣤⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡛⠿⠚⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣿⠋⢹⣷⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠦⣾⣤⣬⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠶⢿⡀⠀⠀⠀⠀⠉⠁⠁⠉⠈⠙⠉⠛
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣀⣤⣴⣾⣷⣿⣿⣿⣯⣭⣧⡀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⠯⣥⣬⣿⣿⣛⣿⣟⣛⣿⣀⣀⡀⠀⠀⠂⠄⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⢆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠒⠶⠶⢶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡚⢀⡀⢀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⡠⢔⡲⠚⣿⣿⠁⣉⣀⣈⣁⣿⣧⣤⠤⠤⠄⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠶⠿⠶⠒⠛⠛⢻⣿⠋⠉⠉⢀⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣾⣿⣷⣲⣿⣿⣿⢦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⡦⣉⠲⢆⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡌⣿⡿⠿⢟⣛⣛⣻⣿⣌⠛⠋⢑⣤⠻⣯⣴⣾⠷⠶⠐⠖⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⠉⠻⣦⣻⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣽⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⡏⠙⠿⣿⣧⡄⢛⢹⢿⣿⣿⡏⠀⢰⣿⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣖⣋⠀⠛⠻⢇⡈⠀⠀⠙⠿⢦⣶⣿⡿⠵⣾⣿⣆⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣛⣣⠀⠉⠉⠙⣿⠿⠿⠿⢶⣄⡀⠉⠛⠐⠃⣀⠀⠀⠀⠉⠀⠀⠈⣿⣿⣦⣍⡛⠶⠦⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠉⠈⠀⠋⠓⠀⠀⢤⣉⠛⠛⢿⣦⣀⠀⠳⠿⠿⠿⠿⠗⠂
⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣶⣦⣬⣤⣀⡀⠉⠛⠻⠶⠀⠀⠀⢠⣠⣶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠿⠟⠛⠛⠓⠢⠀⠉⠛⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣩⣷⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⢧⡒⠚⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠈⠢⡀⠀⣼⠀⠀⠀⢻⣦⡀⠀⠀⡶⠀⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠿⠢⠄⠀⠈⠢⡏⠀⠀⠀⡾⠈⠑⢄⢰⡇⠀⢹⠇⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⢰⡏⠁⠸⠺⣆⡄⠀⠀⠈⣦⣤⢞⠑⠀⠀⠀⠙⠅⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠐⠠⡀⠀⠡⡻⡶⣿⣭⣀⣋⠟⢪⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⡇⠀⠀⠈⠢⡀⢸⠿⣈⠻⣿⣿⡏⠑⠪⢀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⡠⢀⢺⡃⡃⠀⠀⠀⠀⠻⣤⠉⡄⠘⠻⣇⠀⠀⣼⠃⠢⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠓⢳⢯⣾⣙⡶⠀⠀⠀⠀⣿⡿⠔⠃⠀⠀⠀⠱⣤⣿⠀⠀⠈⢂⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⡈⠀⠀⣼⡇⢹⣣⠒⣀⣀⣰⡿⢕⠀⠄⠀⡀⠀⠀⠀⠙⢄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⣶⠛⡃⢀⡈⠛⢻⣿⣿⣷⠜⣻⣦⡴⣁⣠⡀⠀⠀⠀⠓⠄⡀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣾⣁⣈⠿⢿⣅⢽⣿⣵⠦⣤⣬⣴⣛⠠⠟⢻⣅⣴⡀⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡟⠻⠳⣾⣿⣿⣏⡽⡗⢮⣥⣍⢿⡿⠑⠺⣅⡹⠷⢻⣦⣬⣇⣧⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡷⣄⣖⢀⣿⠀⠀⠈⠉⢿⣿⢱⠀⠀⠃⠉⠛⠧⣄⡀⣿⠉⡃⢬⠻⡚⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠠⢃⡥⢙⣮⣯⡀⠀⠀⠀⣶⢻⡎⣇⡜⠈⠀⠀⠀⠈⠉⠙⣤⣅⠀⠻⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡖⣼⠃⣾⣿⡟⢷⣰⠀⣰⢅⣾⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠙⣶⣿⠀⠈⠲⠠⣀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠞⠈⠿⣿⣯⣿⡿⢳⣿⣾⢯⣾⣿⣿⠂⠐⡇⠀⢀⠀⡀⢀⠀⢤⡇⢨⢿⣿⣷⣠⠰⡆⠀⠙⠺⢏⡀
⠀⠀⠀⠀⠀⠃⠐⠒⠲⡿⣿⣿⡿⣿⣿⠏⣡⣿⣿⣿⣿⢠⣿⢠⣷⢠⠇⡙⢂⢰⣧⣾⢀⣿⣿⠾⢣⠍⡄⣸⡄⠀⡍⠀⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣑⣶⠏⣘⣒⡿⠭⢬⣾⣟⣻⡟⣿⠈⢉⠈⠋⣘⣀⣇⣤⣯⣿⣷⣾⣭⣍⣤⡬⠄⣺⠯⠀⠰⠀⢀⡼
⠀⠀⠀⠀⠀⢠⣿⠿⡿⣿⣿⣾⣧⣤⣀⣬⡍⠐⣹⡥⢶⣶⡿⠟⣛⣩⣭⣤⡭⠟⣿⢿⢿⣿⣿⡿⣧⠀⠀⠈⠀⠵⣴⡾⠁
⠀⠀⠀⠀⠀⣾⠀⠾⢻⡂⠀⠉⢿⣿⡿⢿⣤⣾⣿⣿⣭⠳⠎⠛⡉⠁⠤⠐⠂⠐⠠⣼⣿⣿⡿⣸⣿⣷⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⣷⡈⢉⡁⠀⠭⠉⠒⠚⠛⢾⣿⣿⣿⡿⠟⢇⡀⠀⠀⠀⡈⠉⠭⣠⣠⣿⣿⡟⣵⣇⡹⠛⠀⠒⠈
⠀⠀⠀⠀⠀⢹⣄⠑⠢⠬⢙⣒⡀⡐⠀⡟⡡⣿⡎⣿⡤⠠⠀⠂⠀⠀⠀⠒⠒⠶⣾⣿⣯⡾⣛⠎⠁
⠀⠀⠀⠀⠀⡮⣿⣄⣵⣐⡒⢴⡆⠠⢬⣥⠌⠹⡾⣿⣖⣂⣈⡥⠀⠐⡀⢩⣭⣷⡿⣷⡟⡤⠁
⠀⠀⠀⠀⠘⠛⢫⣥⣭⡿⠿⣭⣭⣤⡤⠶⠶⠾⠛⠉⢌⡭⢭⣽⣾⣿⣶⠛⠻⣿⣿⣧⡼⣷⣀⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠛⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢸⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⣰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⢀⣼⣵⠟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃⡴⢀⡴⠟⠋⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣵⠟⠁⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠟⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⠏⣸⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⢞⡏⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡞⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⡞⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢀⡞⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⡞⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢠⡞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⢠⠟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠟⢉⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠘⠉⠁⣸⠁
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⡀⣰⠋⠀⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⣠⣾⣫⣤⠆⠀⠀⣀⣀⣴⠿⠁⠀⠀⠀⠀⠀⠈⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⣠⣾⣿⠿⠿⠶⠚⠛⠉⣩⠿⠷⠶⠶⠶⠤⠤⠶⣶⣞⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠈⠉⠁⠐⠒⠒⠒⠶⢶⣾⣷⣶⣶⠶⠗⠊⢦⠉⣽⠘⠧⠄⠉⠉⠀⠀⠀⠒⠒⣖⠒⠶⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠀⣭⡽⠿⢷⣶⣶⣤⣬⣍⣛⣒⣒⣒⣒⣒⣒⣒⣀⣉⣤⠴⠋⢀⡴⠀⢀⣠⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣤⠆⠀⠈⠙⢿⣶⣄⠉⠉⠽⠛⠛⠻⣿⣿⣿⣶⣶⡶⠶⠛⠁⣀⣥⡶⠞⠛⠉⠉⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠄⣀⠀⠀⠉⠻⣿⣶⠶⠖⠀⠀⣸⠏⠛⠿⠷⠶⠶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⢰⣤⡆⢰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡏⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣀⣾⠃⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠏⠙⢿⠦⠄⠀⠀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠧⣉⡒⠂⢦⣿⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠻⠷⠤⣤⣿⣀⢿⠻⠶⣤⣈⣀⢠⡄⠀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⡏⠀⠀⠀⠀⠙⣟⠓⠲⢬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⢤⠂⠀⠀⢀⣠⠆⠀⠀⢀⢾⣿⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠘⣅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠢⠎⠀⠖⣘⡿⠀⢀⣤⣾⣿⣋⠀⠀⢀⠀⡀⠀⠀⣿⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡟⠛⠻⠿⢿⣿⣾⣾⣶⣾⣧⣳⣿⣿⣟⠛⠻⠿⢿⣿⣀⠉⢰⣦⢿⡄⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢈⡁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣧⣄⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠉⠁⠒⢾⡀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡿⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃⠀⠀⠒⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⡴⡴⢁⢆⣤⣠⣎⣠⣤⢄⣴⣿⡿⠁⠀⣴⠀⣠⠂⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠊⢾⢶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣄⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⠃⢀⣹⠾⠷⣤⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣿⣾⣾⣶⣿⣙⣿⣿⡿⢛⡿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠈⠐⢶⣼⣿⣿⣠⣟⣁⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⣤⣬⣟⠙⠛⠿⢿⣿⡟⠈⠉⢠⣯⣬⣛⣺⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠓⠲⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⡞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠀⠀⠀⢁⣠⣴⣾⣯⣤⡝⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠟⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀⣼⠇⠀⢀⣀⣶⣾⣿⣿⣿⣿⣿⣿⡇⣼⠉⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⡶⠶⠄⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢾⣛⣩⣥⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣋⣤⣿⣿⡄⠀⠀⠙⠟⢎⢻⡛⠛⠛⢻⡇⠸⠿⠄⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠘⠶⣤⣀⣘⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣏⣛⡛⠿⢿⡅⠀⠀⠀⠀⠀⠴⠳⣌⠢⣶⣿⣷⣄⡠⣄⣀⣀⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠒⠶⠒⣛⣉⣭⣭⣭⣭⣍⣉⠛⢿⣦⣍⠳⠀⠀⣢⣒⣷⣶⣼⣿⣶⣭⣿⣿⣿⣿⣿⣿⣛⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠀⢐⣂⣭⣽⣿⣿⣿⣿⣿⣿⣿⣤⠈⠛⢿⣦⡀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣛⣋⣉⣤⣤⣿⣿⣿⣿⣿⣿⣟⠃⢰⣦⠙⠧⣄⡈⠑⠾⣷⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠉⠁⠛⢿⣿⣿⣏⡞⠻⠀⣬⣿⣶⣄⣄⡉⠛⠛⠶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠭⢽⢿⣷⣤⣄⠐⠌⠙⠻⠿⠿⠟⠛⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠈⠀⠀⠈⠻⣿⡟⠻⠿⣿⣿⣿⡿⢿⣿⠃⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⠀⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⠯⠿⠛⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⡿⢿⡿⠏⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠎⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠿⠛⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷⣦⣤⣶⡚⠡⣄⡀⢀⣀⡀⠠⣤⠀⠪⢥
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠻⣿⣿⣿⡿⠟⠓⠈⠁⠀⠀⠁⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣴⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡾⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡿⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣿⠃⠀⠀⢠⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡿⠀⠀⠀⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣺⡇⠀⠀⠀⠁⠀⡜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⣿⡇⠀⠀⢨⠀⠀⠀⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣹⣷⠀⠀⡄⠀⠀⠄⢼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⢹⣿⡆⢀⠁⠀⡀⠠⢼⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⢿⣿⡌⠀⠰⠀⠀⣾⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⠀⠘⣿⣷⡀⡀⠀⠀⠈⡇⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⢿⣷⡇⠀⠀⢠⣿⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢄⣦⠈⢿⣧⠀⢐⣾⣿⡄⠀⠀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠉⠺⠾⣿⣧⠀⠁⣿⣶⡀⠀⢉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢐⣾⣿⣿⣧⠀⢻⣿⣯⣶⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣠⣴⣾⡿⠟⠋⠁⠀⢙⠁⠉⠉⣩⣭⣍⠉⢉⣴⡿⠁⠐⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣶⣶⣾⣷⡾⠿⣦⣄⣬⡉⣿⢿⠿⠿⠿⡿⢶⣾⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣴⣮⣤⣤⣬⣤⣴⣦⣌⣙⣏⣉⡛⠻⣛⣛⡹⢟⣋⣟⣉⠉⣉⠛⣿⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⠿⠿⠿⠛⠛⠛⠛⠋⠉


⡶⠶⠶⠷⠖⠲⠶⠶⠶⣶⣷⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡶⣶⣶⣶⣷⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢁⣾⡀⠀⠀⠁⠓⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⣤⣤⣄⣼⣿⣧⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⠃⠀⠈⢩⣭⣤⣤⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⢿⣍⣛⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣤⣠⣤⣤⣤⣴⣿⣿⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠈⠉⣽⣯⣿⣿⣿⣿⣿⣧⣤⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⣹⠟⠋⠉⠉⢉⣽⣿⣿⠻⠻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠀⠋⠀⠀⠀⠰⠈⢋⣽⠛⠛⠶⢶⣾⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠃⠀⠀⣀⠀⠀⠀⠀⠰⣾⠀⠀⢀⣸⡿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⡀⠄⣍⠀⠐⡅⠤⢸⣯⡃⠀⠸⣿⣿⣿⠈⣛⢦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢋⣿⡏⠘⣹⣿⣽⣶⣵⣿⣷⣨⣿⣷⣿⣾⣿⣿⣿⡆⠟⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⣰⣾⠿⠇⢈⣹⣿⢯⣍⣉⣉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢸⣿⣿⣟⠛⠛⠃⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠐⢿⣀⢀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠘⣇⠀⠀⡘⢸⣿⣿⢿⡻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⡌⠫⡀⠀⠀⠀⠰⠚⠛⠛⠻⠿⣿⠀⢹⠀⣀⠶⣺⣿⢿⣿⣵⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡄⠀⠀⢀⠀⠀⠁⠀⣿⠀⠀⢡⠉⠀⣿⣿⣴⣿⣻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣉⣦⣄⠈⠀⠀⢀⣸⣷⣀⣀⢸⡇⠀⣿⣿⣿⡿⠟⢛⡻⡏⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⣿⣻⣿⣿⣿⣯⣍⣛⣛⠿⠷⠶⢿⢿⣿⣿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣿⠿⣿⣿⣿⡿⠿⠿⠿⢿⣻⣻⣿⣿⣿⣟⣛⣯⣷⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡀⢹⠛⠻⠿⠝⠻⠛⠚⠛⠛⠛⠋⠙⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡶⠶⠶⠶⠖⠲⠷⠶⠶⣶⢶⣶⢶⣶⣶⣶⢶⢶⣶⣶⡶⣶⣶⡶⣶⣶⡶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠶⢿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⡄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠓⠀⠀⠀⠁⠀⠀⠐⠀⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⢀⣀⣄⠀⠀⣐⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⠾⣟⣯⡉⠡⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠓⠚⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⡔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠂⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢀⣰⠀⠙⠀⠀⠈⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣤⠜⠃⣇⠠⠔⠏⠙⠉⠁⠀⠀⡇⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣴⣾⣧⣵⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⡤⢰⡎⣧⣼⢻⣿⣿⣿⣟⣳⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢋⠁⣸⡷⡿⣾⣷⣿⣿⣿⣿⣿⡇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠿⠛⠛⢿⣿⠟⡉⢃
⠀⠀⠀⠀⠀⣠⣄⣠⡤⠤⠤⢤⣤⣸⣿⣿⠋⣠⡀⠀⠀⣬⡇⠀⡀⠆⢤⡤⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠈⠡⠶⢖⣶⠶⣮⣶⡄⠀⢺⣿⡿⠉⣿⣿⡏⣀⡻⢿⣟⡠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠷⠎⠩⠤⠍⠛⠃⠀⠨⠿⠶⠙⢿⣦⣀⠀⠀⠀⠀⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣠⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣀⠀⠀⠀⢸⠀⠀⢠⣄⠀⠀⠀⠀⢸⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⡗⢄⡀⢹⠀⠀⠀⣿⣷⠢⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇⢻⠀⠙⣿⡇⠀⠀⡟⡇⠁⠘⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⢈⣿⣇⠀⢠⡇⡅⠀⠀⢀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⣿⠀⠸⠀⠀⠀⠀⠈⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡟⡇⡆⠀⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣇⣿⠃⠀⠀⠀⠀⠀⠀⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠄⠀⠀⠀⢀⠀⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⣿⣯⢲⠀⠀⠀⠀⠀⢀⠀⣀⣐⢄⢀⣱⡆
⠀⠀⠀⠀⠀⠀⠀⡼⣃⠀⠀⠀⠀⠰⢴⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⣸⣿⠘⡀⠀⠀⠀⠀⣨⢄⠁⠈⠁⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠠⠀⠀⠲⢶⣿⠿⡋⣽⣶⢦⡆⠀⠀⠀⢹⣿⡆⡇⠀⠤⠂⢀⣤⡿⠃
⠀⠀⠀⠀⠀⣰⣿⣧⠴⣋⠀⢀⣤⣤⣬⣿⣿⣷⣿⠒⠒⠚⠓⠓⠂⢾⣿⠿⢧⣠⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠉⠁⠉⠀⠐⠒⠒⠀⠀⠀⢽⣿⠿⢿⣇⡀⠀⠀⢀⣶⣿⣷⣾⣿⡿⠿⢿⣿⠃⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣤⣾⣿⣤⣶⣾⣿⣿⣿⡛⣟⣥⡤⠀⣤⠁⢰⣧⠀⢨
⣿⣿⣿⣶⡬⠑⠤⣄⡑⣠⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⢫⣽⣷⣶⣿⠀⣾⣿⠆⣿⣦⣀⠀⠀⠀⣀⠄
⠛⠛⠛⠦⠉⠀⠀⢰⣥⣉⡙⠓⠶⣿⣯⣯⣽⣿⣿⣯⣴⣿⣿⣿⣷⣦⣿⠿⣿⣿⣿⣿⠂⢿⠀⠀⣿⣿⣿⠻⣿⣿⣶⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣈⠙⠛⠛⠻⢯⣭⡍⠉⠁⠀⠀⠈⠙⠓⠀⠈⠀⠀⠀⠀⠀⠂⢻⣿⣿⠃⠐⢯⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠹⣆⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢻⡄⠈⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡜⣷⠀⠈⣿⡟⡿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠍⣹⡇⠀⠘⡄⠁⡈⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠈⢴⣿⣏⠀⠀⠘⠂⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡇⠀⠀⠀⠀⢸⣇⢀⣸⣿⣿⡆⠀⠀⢰⡀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠐⠢⠤⠶⠚⠋⠁⠀⠀⠁⠀⠀⠀⠀⣸⣗⣤⣾⡿⠻⣧⡀⠀⠀⢳⠀⠀⠀⠀⡑⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⢸⣿⣄⠀⠈⢃⠀⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⢸⣿⠛⣄⠀⠀⣦⣿⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠎⢿⣰⣿⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⡟⠘⡄⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢿⠃⠀⣴⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠀⢠⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠌⠀⠀⣠⡿⠁
⠀⠀⠀⠀⢀⣀⠤⠶⢾⣿⣿⣿⣿⣿⣿⣶⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠒⠒⠲⠧⣬⡭⢭⣿⣽⣓⣂⣀⠀⠠⠀⣤⡎⠀⠀⠀⣴⣏⣀⣄⣼⠛⠃⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⣿⡇⠀⡠⢾⣿⣿⣿⣿⣿⡷⠦⣄⠀⠀⠀⠀⠂⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡥⠊⠀⠀⠈⠙⠛⠿⠿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡇⢰⣿⣧⢲⣶⡄⣤⣹⣄⣀⣀⣀⡔⣁⣀⡀⠠⠤⠄⠀⠀⣀⣀⣠⣤⡾⠁
⠀⠀⣠⠰⢶⡤⣤⣼⣿⣿⡿⠿⠿⠿⠍⠉⠉⠃⠀⠉⠉⠉⣩⣥⡈⢩⣭⡭⢩⣿⢿⣉⣴⣶⣶⠶⠾⠟⢛⡫⢵⣶⠟
⠀⠀⠹⣿⣿⢟⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⢀⠀⠤⠄⠠⠶⠶⠶⠖⠒⠚⢿⠟⠛⠛⠩⠏⠁⠀⠂⠁⢀⣤⣤⡿⠁
⠀⠀⠀⠙⢿⣿⣿⡿⠿⠿⣿⣧⢿⣶⣦⣤⣤⣤⡤⠤⣤⣀⣀⣀⢀⣀⠀⣀⠀⠀⠀⠄⢀⣴⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⣀⣙⣿⣷⣶⣶⣿⣿⡔⣀⣠⡤⣤⣤⠀⠀⠉⢉⣭⣉⣉⡩⠌⣉⣁⣀⣀⣠⣉⣽⡿⠿⠿⠧⢤⣀⣀⣀⣀⣀⡀
⠠⠤⢤⣤⣬⣭⣶⣿⣧⠤⠟⠛⠉⠁⠀⠀⠀⠤⠭⡿⠶⠟⠋⠉⠀⠀⠀⠀⠉⣉⡻⠿⠋⠁⠀⢤⣤⣤⠴⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣒⣒⡢⠤⠤⣤⣤⣤⣤⣤⣴⣦⡤⠄⠒⠈⠀⠀⢀⣠⠴⡲⠻⠛⠻⢭⣴⣀⣄⡀
⠀⠀⠀⠠⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡠⠬⠤⠠⠶⠀⠠⠀⠀⢠⣤⣤⡤⠤⠧⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣤⡤⣀⢀⣀⡀⠀⠀⠀⠰⣷⣶⡶⠤⠟⠦⠡⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣶⠶⠚⠚⠛⠃⠁⠀⠒⠈⠉⣻⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢠⣿⣿⣯⡄⠉⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⢴⣤⠄⣀⡀⠀⠀⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣾⢟⣽⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠟⠛⠋⠉⠈⠛⠛⠛⠒⢓⣶⠴⠀⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⢀⣴⠟⡡⠿⠿⠿⠿⠄⣠⡄⢠⣤⣀⣀⣀⠘⢿⣇⢠⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⣾⠟⠁⣴⣷⣾⣿⠛⠛⠊⠉⠉⠈⠉⠉⠉⠉⢉⣡⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣾⡿⠁⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣍⣉⣛⣴⣆⣰⣤⣄⣀⠀⢻⡇
⠀⠀⠀⠀⠐⠛⠛⣳⣄⢸⣿⠁⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⣤⣷⠶⣶⣤⣀⣂⡀
⠀⠀⠁⠲⢤⣀⠈⠉⠛⠛⠿⠀⠀⠀⢸⣿⣿⡿⡷⣤⣄⣀⣀⣀⡀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠉⣳⣾⣤⣀⠀⠀⣤⡈⠉⠀⠀⠀⠀⠈⢹⡏⠻⠁⠀⠉⢶⣬⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠫⢽⢿⣿⣷⣿⣿⣶⣤⣤⣀⣠⣤⣸⡇⠀⡀⠀⠀⠼⠛⠋⠋⠉⠁⠐⢲⠂⣤⣤⡤⢀⠘⡿⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣯⡉⡿⢿⡟⠻⣿⢿⣿⣿⣾⣶⣤⣦⣀⣀⡀⠀⠀⣤⣦⣾⠀⠉⠀⠀⡀⢸⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠿⠿⠛⠛⢉⠛⠙⠿⠿⠿⠿⠛⠛⠛⠙⠻⠿⡿⠟⢿⣿⣿⠷⠶⠶⣶⡶⣂⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠐⠂⠐⠙⠋⠁⠀⠀⢀⣠⣠⣾⣿⣀⣰⣦⣶⣶⣶⣾⣿⣿⣆⣀⣰⣮⣭⣭⢤⣀⣀⣶⣄⣀⣨⣭⣽⣛⣛⠭⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠿⠿⠿⠿⣻⣵⣶⣮⣝⣻⣿⣿⣿⣿⣿⣿⣷⡆⠩⠍⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⢤⣩⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡇⠀⢧⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸⡀⢣⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⠀⡇⠈⡇⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⡇⠀⢹⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡇⠀⢰⠃⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡇⠀⡼⠀⠀⠈⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⡇⢠⠃⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⣧⡏⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⣦⣥⣶⣶⠿⡿⠭⠿⠟⠛⠒⢲⣼⣿⣧⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⣒⣋⣉⣁⣠⣤⣤⣤⣶⣶⣶⡿⠿⠟⠛⠉⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣉⣁⣠⡤⠤⠤⠖⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠤⠜⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⣿⡓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⣿⣇⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢸⣿⢻⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⣿⢸⡀⠀⠀⠀⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⣿⠸⡇⠀⢀⣼⠟⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⣴⡿⠁⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠊⠉⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠋⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⡠⠋⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⢠⠞⢡⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⢀⡴⢃⡴⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⣠⠊⣠⠎⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡟⢁⡜⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣢⣤⣼⣻⣤⡤⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⠛⠙⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡱⣄⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⢀⣠⠃⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠉⠑⠊⡉⠒⠋⠙⠒⢉⠑⠊⠉⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠁⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠉⠓⠋⠀⠀⠈⠛⠉⠙⠋⠙⠚⠁⠀⠀⠀⠀⠈⠓⠋⠙⠋⠉⠋⠁⠀⠀⠙⠚⠉⠛⠋⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠫⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣔⣺⢏⡶⠤⣤⠤⠀⠀⠀⠀⠀⠀⠀⢸⡬⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠐⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣀⠀⠀⠀⡀⠤⢒⠀⢀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡀⠕⠂⠁⠠⠐⠺⡚⠛⠛⠋⠙⢯⡉⠙⢇⠀⠀⠀⠠⢤⠰⠦⠓⠶⠖⠒⡾⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣮⠧⠤⠤⠴⡏⠐⠄⡀⡇⠀⠀⠀⠀⠀⠙⢧⢜⡀⠀⠀⠀⠈⡆⠈⠈⠀⠀⠀⠈⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⡄⠀⢈⠇⠀⢀⣀⣀⣀⣀⣀⡑⡍⠢⡀⠀⢀⠃⢀⣀⣀⠀⠀⠀⠄⠀⠀⠻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⡀⣀⣀⣧⡤⠾⠖⠎⠒⠒⢶⠒⠒⠒⠛⠋⢢⡘⠲⡚⠛⠓⠚⢛⠺⡉⠉⠻⡀⠀⠀⠈⠪⠢
⠀⠀⠀⠀⠀⠀⠠⢼⠶⠟⠛⠛⢶⡒⣊⢹⣟⡄⠀⠂⠀⠂⠀⢳⡁⠁⠁⠀⠀⢹⢄⠱⠀⠀⠀⢸⠄⠙⠦⢀⠱⠀⠀⠀⠀⠑⠄⠑⢄
⠀⠀⠀⠀⣄⣀⠀⢀⣷⠀⠀⠀⠀⠈⠲⢍⢲⡔⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠃⢲⠀⠀⠀⡈⠠⠀⠀⠢⡙⢇⠀⠀⠀⠀⠈⢢⠀⠑⢄
⠀⠀⠀⠀⠀⠍⠉⠛⢾⠀⠀⠀⠀⡔⠁⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⢳⡑⢄⠀⠀⠀⠸⠀⠀⠀⠀⢀⣠⣀⣀⡑⡄⠉⠢⡀⠀⠀⠀⠡⡀⠀⠑⢤⡀
⠀⠀⠀⠀⢰⠃⠀⠀⡜⠀⣀⣀⢌⡤⠤⢄⡘⢀⣀⠤⢒⣒⣲⣖⣒⡺⣧⡬⠶⠢⣼⣃⣤⣒⣯⣹⠥⠤⠤⠤⠼⠶⠄⠀⠈⠢⡀⠀⠀⠱⡀⠀⠀⠉⠢⡀
⠀⠀⠀⢀⠆⢀⣀⣼⣖⣉⣥⣽⠤⠤⡤⠾⠿⠉⠈⠉⠍⠀⠀⠀⠂⠀⢢⠐⡠⠋⢱⠀⠐⠠⠀⠐⠀⠐⠀⠀⠀⠈⠈⠀⠀⠀⠈⠢⣄⠀⣴⣠⠤⠤⠤⣈⣢⣀
⠀⠀⢀⠌⠀⠀⠈⢆⠀⠐⠐⢀⠠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢱⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡤⠤⠤⠤⡤⠿⣄⠀⠀⣀⣀⡠⡤⠞⠋
⠀⢀⠎⠀⠀⠀⠀⢸⠀⢀⠔⠁⢀⣀⣀⡀⣠⠁⠀⠀⠀⣀⣀⣠⣤⠦⠤⠀⠠⠥⡌⠀⢀⣠⠤⠄⣤⠒⠒⠒⠒⡠⠮⡀⡇⠀⣠⣊⠤⠴⡖⠋⠋⠠
⠈⠉⠙⠛⠒⠒⢒⣆⣶⢶⣶⠤⣯⠩⠁⢀⠋⠠⠒⡞⠁⢸⠔⠻⢈⣃⡀⢀⢀⣰⣒⣉⣉⣉⣀⣤⣽⣤⣤⣴⣶⣶⣶⡾⠛⠛⠛⠂⠀⠀⠗⠂
⠀⠀⠀⠀⠀⢀⠠⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠿⠿⢿⠿⣿⡿⣿⡿⣋⠠
⠀⠀⠀⠠⠭⠤⠖⠓⠀⠉⠀⠬⠤⠴⡬⡸⠯⠄⠠⣉⣒⠲⠁⠑⠿⠿⠷⠭⠦⠖⢉⡊⠡⠞⠿⠣⠴⠤⠽⠿⠦⠬⠭⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣢⣿⣧⣤⡤⠀⠀⣿⣦⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⣿⣾⠷⠆⠀⣿⣿⣧⡀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⡟⢿⣿⣷⡀⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⢀⠀⠀⠀⠀⠀⡟⠀⠈⢿⣿⣯⠈⠉⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡤⡀⠘⣡⣿⠏⠀⠈⠀⠀⢐⠹⣻⣿⣿⠀⣸⣿⡿⠷⢂⡸⣿⠷⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⢀⣴⣿⡟⠀⢀⣀⣀⣀⣠⣾⣿⣿⣿⡄⡿⠙⣸⣧⣾⣿⠁⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⢀⣴⣿⡿⡽⠛⢻⡏⠛⠛⠛⠿⠿⣿⣿⣿⡷⠐⠚⠻⣿⣿⡗⠚⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⣤⣼⣿⡟⡿⣽⠋⠁⠉⠁⠀⠀⢀⠈⠓⡳⣾⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠡⠶⠿⠿⢿⣿⡟⣾⢳⢇⣤⣤⣤⣠⣄⣀⣶⣮⣽⣿⣿⣿⣿⡀⠀⠀⣿⣿⡇⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠐⠀⢀⢘⣡⡿⠷⢦⣾⢇⡟⡀⠀⠀⢸⡇⠀⠀⠈⡍⠉⠙⢻⡿⢿⣧⡀⠀⠛⣿⣷⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢀⠀⡠⠖⠋⠀⠀⡀⠀⠠⠀⠀⠐⢀⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣽⡇⠀⢠⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡤⠚⠁⠀⠀⠠⠊⢚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⡠⢤⣀⣶⣤⣶⣮⣿⣿⣧⣤⣾⣿⡇⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡴⠂⠀⣠⣴⣦⣶⣿⡶⢿⠿⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡷⠀⡀⢟⣫⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣟⣋⣩⠉⠀⠀⡴⠀⢆⢸⡇⠈⠀⠀⠀⠀⠈⢈⡟⢛⠿⢿⣷⣸⣧⠕⣿⣿⣀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠷⠶⣾⣶⠿⠤⢤⣤⣤⣤⣦⣼⣿⣯⣤⣤⡾⠿⠿⠿⢚⣛⣛⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣴⢒⠀⠺⣿⣿⣿⠏⠉⠛⠻⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠐⢷⢸⡼⣷⣼⣿⣭⡤⠈⣻⣏⣸⣯⣶⠀⢹⣿⣿⣯⣿⣿⡇⠐⢸⡿⠿⢿⣶⣄⡈⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡦⠀⠀⠻⠿⠿⣿⣿⠇⠀⣿⣿⡍⠛⠉⠀⢠⣯⡈⠉⠛⠛⠁⠀⢸⣿⠃⠀⠘⠻⠃⠐⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⣿⠀⠀⣤⣤⣤⣶⣶⠄⠀⠐⠂⠀⢻⣿⣷⡀⠀⠀⠼⠿⠂⣶⠄⢤⣀⣠⣿⡭⠑⠂⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠐⠛⠩⠎⠙⠋⢸⣷⠐⢦⣴⣾⣷⠄⠀⠀⠁⠀⢀⣤⡀⠀⠄⠋⠋⠉⠀⠀⠀⠀⠀⠀⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠛⠛⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠋⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⠙⢻⡄⠀⠀⠀⠀⠀⠁⠉⠻⡿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⡿⣿⣧⠀⠀⠀⠀⠤⠤⢤⣶⣷⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠨⡿⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣾⡗⢤⣀⠀⠀⣀⣠⣗⣀⣀⣀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠀⠀⠉⢣⠀⠙⠦⡈⠉⠙⠛⠉⠙⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⠀⠀⠀⣿⢶⣶⠒⠾⠛⠿⡆⣀⣀⣈⢢⣠⣤⣤⣤⣤⣤⣷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢳⣦⣤⣿⢼⢌⡀⠀⠀⠀⠈⠁⠈⠀⠉⢙⣏⡟⠋⠛⠿⢿⡿⣮⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠻⣾⠉⠻⣦⣄⠀⠀⣀⣀⣠⣤⣬⣧⣤⣤⣤⣤⣴⡧⠀⠛⢦⣈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠛⠻⣶⣌⡈⠀⠁⠘⡟⡉⠉⠙⠻⢿⡇⠱⡄⠀⠈⠳⣄⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⡇⠀⠀⠀⠀⠈⡇⠀⠈⠢⡀⣀⠈⠵⠂⠀⠈
⠀⠀⠀⠀⣤⣤⣄⣠⣤⣤⣤⣤⣦⣄⣀⣣⣄⣀⣠⣄⣀⣀⣀⣴⣦⣦⣿⣤⣦⣤⣤⣤⣤⣦⣴⠖⠋⠁
⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⢀⣀⣀⣤⣤⣄⡀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣀⣠⣴⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣾
⠘⠛⠛⠛⠛⠛⠛⠛⠁⠐⠒⠛⠛⠛⠟⣀⣀⣸⠤⡤⠤⠤⢴⠶⠶⠖⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⠛⠛⠲⠾⠿⠿⢿
⢠⣿⣤⣶⣶⣦⣤⣿⢓⣶⣶⣶⣶⣶⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣬⣿⣿⣯⣤⣤⣤⡌⣢⠃⣾
⢀⣟⢻⣿⣿⣿⣿⣿⠧⢀⣈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣁⠀⢀⣤⣽
⠐⣷⣾⣿⣿⣿⣿⣿⣜⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣻⡿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⣿⣿⠿⠂⠀⠀⠰
⠰⣿⣿⣿⣿⣿⣿⣿⣯⡁⠁⠉⠀⠀⠒⠂⠀⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠀⠐⠀⠀⣀⢰
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⣁⣧⣴⣶⣷⣾
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⡀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠦⠄⢠⣤⣼⣿⡋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣍⣁⠀⠠⣶⣦⣽⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⣛⠛⠛⠿⠷⠶⠶⢶⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣍⣻⣟⣿⣶⣶⡶⠶⠦⢀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣀⠀⠀⠛⠀⠀⠀⣀⣠⣀⣀⣀⠀⠂⠀⢀⡰⡆⠀⠨⢀⠠⠤⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠲⠊⠁
⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⣸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡧⢑⡏⠀⠀⠀⠀⠈⠀⠐⣂⡀⠠⠀⠹⣀
⠀⠀⠀⢸⣛⣻⠶⡶⠦⠶⠆⣆⣰⣴⣶⣤⣤⣼⣄⣜⣁⣀⣠⣦⣤⣤⣤⣷⣿⣴⣶⣷⠾⢷⣦⣴⣶⣾⣿⢷⣿⡿⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠛⠛⠲⣶⣦⣤⣤⣤⣎⣃⣇⣁⣀⣉⣉⣉⣉⣉⣉⣁⣖⣐⣜⣐⣐⣒⣉⣭⣭⣭⣭⠤⠖⣂⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⡛⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠷⠴⠀⠴⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠈⠉⠐⠈⠉⠓⠾⠟⠋⠘⠷⠖⠀⠉⠉⠙⠓⠒⠛⠂⠂⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢠⣀⣀⣀⡀⠀⠒⠙⠋⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⣒⣒⣶⡒⣦⣬⣏⡋⠉⠉⣙⠀⣶⣦⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡷⠒⠲⠼⣆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⢁⣀⣠⣤⣤⣤⣤⣀⡀⢸⡇⠀⠀⠀⣃⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⡾⢸⠠⠄⠶⠾⠷⠤⠤⠭⠾⣿⡒⠶⣤⣏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⣸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠉⣍⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣤⡄⢰⡏⠀⢸⠾⠶⠾⣷⡶⠦⣤⣀⠘⣦⣄⡀⠀⡏⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠋⠉⠀⠉⢻⡟⠀⠀⢸⠛⠒⠛⠛⠛⠀⠀⠈⢹⣿⣭⡛⣳⣷⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣰⣿⣿⣦⣤⣜⢀⣀⣀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣿⣟⣿⣛⣛⣻⣿⠿⠷⣶⣦⣷⣦⣤⣄⣸⣄⣄⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠧⠈⠉⠻⢿⣿⣶⣿⣾⣿⣧⣥⣼⣿⣿⣿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠶⣶⣤⠬⢬⣭⣭⣭⣭⣭⣭⣩⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠲⠶⠤⠤⠬⠾⢿⡶⣶⢾⣷⣍⣰⡶⠄⠳⣯⣉⣍⣩⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣛⣏⣙⣛⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠿⠒⠊⠋⠀⠀⢀⡀⠤⠿⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢴⣶⡖⠒⣆⡀⠀⠰⠖⢢⣿⣏⣑⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⣴⠶⠾⠛⠛⠋⡹⠀⢲⢶⠖⠛⠉⠉⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠌⠞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢁⠔⠅⠀⠆⠀⠀⣀⣤⣠⣤⣀⡀⢀⣀⡤⠶⣶⣶⣤⠤⣀⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠀⣐⣁⢀⣼⣿⣿⣤⣤⣶⣿⢶⠶⠿⠟⠛⠛⠛⣻⠋⠁
⠀⠀⠀⠀⢄⡠⠤⠤⠤⠴⠞⠁⠀⠀⠀⠀⢸⠀⠉⠉⠉⠁⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⣇⣤⡴⠆
⠀⠀⠀⠀⠀⠉⠑⠢⣄⡐⠓⠉⠓⠲⢶⣆⠈⢆⢀⣀⣠⣤⣤⣀⠀⠀⠏⣴⠞⠋⣉⡟⣷⣶⠤⣍⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⣶⣶⣾⣍⣚⣻⣿⣀⣀⣿⢹⣷⣦⡄⠀⣘⣀⣈⣧⣿⣿⣤⣿⣆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣿⣿⣿⣟⣿⠓⣾⢣⡜⣿⣿⣿⣿⣿⣿⣯⣭⣯⣽⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠃⠀⠁⠐⠂⢿⣿⣯⣽⣦⣷⣷⣼⣾⣿⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠽⠻⠿⠿⠟⠻⠻⠿⠿⠿⠟⠿⠟⠋⠙⠻⢛⡙⠋⠩⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣨⠧⠔⠶⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠛⠉⠉⠁⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⡀⠀⣀⣣⡤⠖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡦⠤⠖⠚⡟⠉⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠀⢤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣟⣤⣄⠠⠶⠒⣻⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣠⡤⠶⠶⠛⢻⠏⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣧⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⡀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠘⣠⡿⠋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠴⠚⠓⣾⠟⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⣀⡠⣤⠖⠛⠟⠐⠃⠘⢰⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀⣠⢥⢷⡎⠉⠁⡠⠘⢉⣠⣴⣶⣿⣯⣿⣦⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠀⠐⢈⡧⠐⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣒⣋⣡⣴⣾⣿⣿⣿⣿⠟⠻⡿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⠛⠿⠿⠿⠛⠛⣛⣛⠻⠞⣷⣦⠓⣿⣿⣿⡛⢿⣿⣿⣿⡿⠶⣶⣤⣼⣶⣶⡖⠒
⠀⠀⠀⠀⠐⠾⠃⠀⠀⠀⠲⠖⠂⣰⣶⢚⣳⣾⣷⣠⣿⣿⣚⣀⣴⣿⡿⠿⣿⡿⠿⢋⣤⠘⠉⡛⠶⠾⠿⢿⣿⣷⣶⣶⣭⣿⣟⣿⢿⠿⠋⣀
⠀⠀⠀⠀⠀⠠⣤⣤⠄⠿⠶⠼⠟⠛⠉⢹⣿⡿⠿⠷⠒⠚⣿⠿⠿⠿⠟⣛⡄⠀⣺⣿⡿⠀⠀⣝⡛⢷⣖⠀⠈⣙⣋⣩⣍⣉⠛⠛⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣀⡙⠛⠒⠒⠀⠈⠉⢁⣠⣴⣤⣤⡁⠀⠐⠻⡭⠆⠀⠀⠙⠿⢷⡿⠔⠋⠉⡙⠻⠿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⣤⡀⠀⠀⠀⣿⠒⠂⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣿⣅⠁⠒⠒⢻⠀⠀⠀⠀⠀⣀⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣷⣄⣠⣼⣦⣴⣶⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⢀⣀
⠀⠀⠤⠍⠀⠀⠛⣒⣒⣦⣥⣤⣼⣟⣛⣻⣿⣯⣿⣿⢿⣿⣿⣿⣿⡿⠿⠿⣛⣋⣉⣀⣴⣾⣿⣿⣿⣿⠦⣤⠀⠁⠀⠭⠤⠤⠭⠍
⠀⠀⠀⣒⡚⠛⠛⠛⢿⣿⣋⣭⣭⣹⣿⣿⣿⣿⣿⣇⣼⣿⡶⠓⢙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣁⣀⡀⠠⠤⠤
⠰⠶⠽⠿⠿⣿⣶⣶⣶⣿⡟⠛⠛⠛⠽⠿⠿⢷⣶⣶⡷⠟⢛⣿⣿⣿⣿⣽⣭⡽⠛⠛⠛⢳⣖⣻⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠤
⠀⠀⠒⠶⠦⠤⠤⠀⢀⣈⣙⠒⠶⠶⠶⠂⠀⠉⠿⠿⠷⠦⠸⠿⣿⣿⠉⠀⠠⣤⣤⣄⣤⣭⠭⠭⠭⠼⠿⠿⠆⢀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⠶⠖⠀⣀⣀⣀⣀⡉⠉⠉⣻⣿⠁⠀⠀⣀⣀⣀⡀⠤⠤⠶⠛⠋⣀⣀⣀⣀⣀⣀⣤⣄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠖⠋⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠠⠀⡠⢀⡀⠀⠀⠀⠈⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⠀⠈⠑⠲⣄⡀⠀⠀⠀⣾⣷⠆⠀⠀⠀⠀⠀⠐⢚⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠙⠢⣀⠶⣿⠟⠓⠂⢀⠨⣿⠀⠀⠀⠀⠀⠀⠀⢼⣷⣀⡤⠤⠒⠋⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣆⠀⠁⠀⠀⣸⣇⠀⢀⣤⡴⠖⠶⠚⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⣯⠳⣄⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠅⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⡄⠀⠀⠀⠀⠀⠀⢻⡄⠈⠑⢆⡀⠸⣇⠀⠀⠀⠀⠀⠔⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣿⣧⣦⣾⣷⡀⠀⠀⠈⣷⠀⠀⠀⠙⡆⢻⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⠨⠀⡠⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡔⣯⣖⣏⣆⣴⠂⢨⡇⠘⣧⠀⠀⠀⠠⠀⠀⠀⢹⡄⢀⣪⣤⣤⢾⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢿⣿⣿⣿⣯⡉⢩⡿⠚⣫⡍⢹⡍⠱⡄⠉⣴⠠⠀⠀⠀⢀⡤⠞⣿⠟⣛⣉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣤⣬⣨⣉⣍⡉⣈⣿⣿⣣⣭⣿⣯⣤⣼⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠈⠁⠀⠔⣛⣿⣿⣿⡿⣿⣿⣿⣿⣿⠿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⡿⠆⢐⠤⢤⣶⣶⣶⣿⣿⣥⣵⣿⣮⠝⠿⠻⠷⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣟⣿⣓⣒⡀⣅⣒⣚⣛⡛⠭⣽⣧⣼⣷⣏⣳⣶⣾⠓⢶⡗⢦⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⣍⣩⣭⠟⠿⠛⠈⡁⢘⡶⠶⠾⣒⣀⡽⠋⠥⣬⢭⠭⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠠⠤⠄⠀⠀⠀⠀⠤⠀⠀⠈