Cocoa Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⠀⠐⠐⠈⠉⠉⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⣷⠘⠆⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠠⣧⠀⠀⠀⠀⠀⣠⣤⣤⣤⣹⠦⣀⣠⣥⠽⠶⠶⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠰⢩⡉⢲⠨⣧⠀⣻⠿⠛⠛⠋⠉⠉⠁⠱⣇⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠆⠀⠀⠈⠀⠻⣆⠨⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣽⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠠⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠸⡿⣯⡳⠤⠀⠀⠀⢀⡢⠄⠓⠚⠍⣎⢧⠀⠀⠒⠒⣂⣉⡹⠴⠥⠔⠒⣒⣈⣉⠭⠭⢋⠄⠀⡀
⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠠⣀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡭⠭⠥⠴⢒⠒⠒⠋⠛⠁⢄⠀⠀⠀⣀⠀⠀⢊⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣮⣙⠢⠀⠀⠀⠐⠀⢀⠄⢳⠀⠀⣞⡳⣄⠀⠀⠀⠀⠀⠑⡄⠐⠃⠀⠀⠀⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣦⡀⡤⠀⢀⠀⠄⠈⠀⠀⠫⣷⣬⡓⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠸⠂⠀⠈⠻⣿⡆⠀⣾⠥⢨⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠁⠀⠈⠓⠚⠃⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⠀⠀⠀⡠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⠈⠂⠀⠀⠂⠚⠰⠰⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠳⠬⢒⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠻⠶⣂⢀⠀⠀⠀⠀⠀⠀⣇⠈⢙⣿⠃⠉⣱⠀⠀⠀⠠⣦⠀⣔⡀⠒⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣈⣉⣰⣤⢆⠀⠀⠀⢀⡟⠛⠻⣿⠛⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠘⠻⠀⠀⠀⠸⣿⣷⣾⡟⠿⠻⠿⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠤⣤⣲⣤⣯⣽⣦⣠⣖⣀⡀⠀⠀⠀⠰⠟⠟⠋⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢀⠁⠈⢋⣀
⠀⠀⠀⠀⠻⠟⡶⠶⠦⠤⣤⣤⡄⠬⢦⣵⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⢀⣄⠠⢀⠠⠐⠀⢀⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠚⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠐⠛⠲⢿⣤⣴⣶⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠂⠀⠀⠦⠄⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡄⠉⠉⠉⣔⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠬⠠⠔⢀⡂⢀⡀⠀⠀⢀⡀⠀⠈⣠⠍⠐⠢⣴⡺⠟⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣀⢠⡀⡂⡠⠀⠎⠁⢉⠀⡓⢂⣀⣐⠦⠶⠛⠀⠐⠻⠯⢔⣠⡄⣾⠔⠉⠀⠀⠀⠀⠀⠦⠤⡀
⠀⠘⢇⡀⠀⠑⢌⢡⠄⠢⣓⣠⡵⠶⠚⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢐⡠⠄⠀⠀⠈⢣⡼⣈⠑
⠀⠀⠈⠋⣨⣀⣠⣷⠶⠛⠋⠁⠀⠀⠀⠀⢰⣴⠁⡄⠁⠀⠀⠰⠒⠂⠬⠃⠀⠀⠀⠀⣤⣀⣀⡎⠀⢀⡒⣸⢸⠆⣬⢠⣸⠾⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠈⠀⠀⠀⠁⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠲⢏⡔⡠⢋⣝⠀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠠⡀⠀⠀⠀⠻⢧⣴⣬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠳⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⢮⣝⣒⣤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⢀⢀⡀⠱⣖⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠘⢃⠵⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠬⠻⠧⢄
⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢛⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠁⠤⠄⠤⠤⠶⠒⡛⠿⢿⠚⢒⣶⠾⠀⠀⠀⠀⠠⠤⡂⣄⢦⣂⠆⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢄⣤⣌⣀⡠⠀⠠⠐⣊⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⠔⢀⡐⣓⠭⠖⣋⠑⠤⠀⠀⠉⠠⠔⣲⣿⠆
⠀⠀⠀⠈⠙⠫⡂⢤⣲⡹⠬⢒⣩⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣅⣤⣦⣿⠁⢀⠄⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠠⢤⣍⣨⣤⣴⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣤⣤⣖⣾⣭⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣿⠍⠉⠁⣿⠩⠛⠉⢸⣧⢀⠉⢠⡆⠀⠀⠀⠀⠀⠀⡷⣀⠀⠀⡇⣠⠄⠀⢸⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⡀⣿⠀⢀⣂⢸⣟⠋⡤⢸⡃⠀⠀⠀⠀⠀⢰⡟⡏⠓⠀⡇⡻⠎⠀⢸⠋⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⣿⡛⡛⠛⢹⡏⡉⠋⢹⠀⠀⠀⠀⠀⠀⢸⡏⢛⡛⠛⣿⠛⠛⠛⢱⠙⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⣿⠆⠿⠀⢹⠋⠀⠁⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⠐⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠿⠟⠷⠛⠿⠟⠻⠶⠛⠃⠀⠀⠉⠀⠀⠀⠀⢸⡷⠿⠒⠒⡔⠶⠶⠶⢦⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⣤⢸⡇⢀⡆⠀⡇⠏⠀⠀⢾⡽⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⡞⠀⠀⠀⠀⠀⢀⠄⢸⣧⡤⠤⠤⡣⢤⠬⠥⡸⠤⠬⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠁⠀⠀⣿⣿⣇⢸⣇⣷⡦⠀⡇⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠶⣀⣴⣿⣦⣤⣖⣤⣤⠶⢺⠁⠀⠀⠀⠀⠀⣧⣭⣥⣄⣮⣭⣥⣤⣣⣤⣤⣄⡽
⠀⠀⡄⠢⠤⠠⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠿⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⣷⠅⠀⢀⡇⠀⡀⠀⡇⠀⠀⠀⡃⠀⠀⠀⢀⣀⠄⠒
⠀⠀⣻⣦⣤⠴⠖⢻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣃⣈⣀⣀⢷⣀⣀⣀⡇⠀⠀⠰⠿⣷⣄⢠⣴⡴⢶⢾⠄
⠀⠟⠁⠀⠀⠀⠀⠈⣿⣿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇⠉⠉⠉⣷⠉⠈⠉⡇⠀⠀⠀⠀⠀⠉⠻⠝⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣍⣀⣀⣀⣀⣀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⣇⢀⣀⣈⣁⢀⠀⢀⣹⣀⣀⣀⡇⠀⠀⠀⠀⠀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⣤⣀⣠⣹⣧⡴⡶⠖⠒⠦
⠀⠀⠀⠀⠀⠀⢀⣔⣂⣷⣶⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⢮⣅⢀⣀⠠⠀⠀⠀⠀⠀⠻⣯⣉⣁⠀⠀⢈⢷⣚⣠⣤⣤⣴⡄
⠀⠀⣠⣴⣮⣤⣸⣿⠿⠻⠿⢭⣿⣿⣿⣶⣤⠆⠀⣠⣄⠀⢰⣞⠫⠀⢀⣙⠦⣔⡲⠚⠁⠛⢯⡀⣀⣤⠤⠀⠀⠀⢻⣳⣿⣿⠿⠭⠵⠞⠉⠛⠒⠀⠁
⠀⠄⠙⠻⣿⣿⣶⣿⣿⣷⣶⣾⣯⣅⡀⢈⡭⠟⣊⣭⣶⣢⠄⠙⢷⣯⡉⢀⣀⣈⡷⢦⣖⠾⠉⠈⠓⢮⣀⣠⣤⣄⠈⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣧⠀⣹⠿⣿⣿⣽⣧⣶⠊⠁⠠⠔⠒⠋⠀⠈⢻⣿⣾⣭⡻⠀⢀⣙⣷⢤⣶⣻⣷⣾⣿⣿⡄
⠀⠂⠀⠀⠀⠀⠀⠈⠛⠓⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣧⣾⣿⣿⠿⠿⠛⠛⠋⠉⠉⠁

⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡀⠀⣀⠀⠀⠀⣠⢄⡄⢠⣤⣄⠀⠀⢤⣤⠀⢀⡀⠀⠀⣤⡄⢠⠤⢤⣤⣄⠀⣀⣤⡤⡤⢤⣄⡀⠀⠀⠀⠠
⠟⠀⠀⣀⡤⠤⠤⣤⣄⣀⣀⠀⠉⠉⢩⣄⣙⡷⠋⡀⢠⡏⡞⣹⣟⠀⠈⢳⣦⣞⠉⡾⢸⠁⠘⠛⠉⣰⣋⣹⣿⢿⠉⣀⣿⣯⣿⠀⣿⣹⣿⣷⣤⣀
⣠⣾⣏⠀⠀⠀⠀⠀⠀⣙⣿⣟⠛⣇⣄⠉⠉⠀⠀⠅⣰⣹⡿⣇⠈⠲⣶⣾⠋⢉⡿⢡⠞⠀⠀⠀⣰⢥⣼⣿⠛⠀⣦⣿⣿⣾⣿⡹⣿⣿⣻⣿⠉⠻⣷
⣾⠿⠿⢿⣿⣿⣶⣿⣿⣿⣿⠿⣿⠿⣿⣀⣀⣠⠀⠀⣿⡽⢧⣤⡿⣿⠥⠾⣛⣭⣒⠁⢠⣠⡿⢠⠷⣾⢿⠉⢦⣼⣿⣿⡿⠦⣼⣱⣹⢿⣇⠈⠳⡀⠈
⣿⣤⣤⣴⣾⣿⡛⢻⣭⣤⠽⠷⣟⠓⠾⡻⢿⡿⣄⣀⠘⣿⣿⣯⠟⠲⠶⠛⠑⡋⣁⣀⡈⣉⡀⢸⣻⡟⣦⣷⣿⣿⡟⡼⠇⠤⣾⣉⡇⣼⣿⣦⡀⢹⡆
⢷⡿⠫⠤⣤⣈⣿⠿⠛⠓⠢⣄⣈⣻⣶⠿⢾⣻⢏⡏⠀⠈⠉⢁⣠⠶⠟⠛⣡⣴⠾⠋⢉⠅⣹⢨⠟⣷⠿⠛⣿⢻⣇⣷⣄⣠⣿⣈⢷⣧⣿⢿⣧⠀⢳
⡞⢛⢒⣶⣾⣿⠟⠒⠲⣤⡴⠾⢿⡁⠈⢷⣀⡟⣸⣷⠀⠀⡴⠋⣁⣤⠶⠛⢋⣡⣤⣾⣿⣾⠅⠘⠊⠁⠀⠀⣡⣸⣿⠧⠤⣼⣇⡼⣾⡾⣿⣄⢻⣄
⣀⣽⣿⣿⣿⣧⣤⣤⠞⠛⠳⡀⠀⢳⣄⣨⡟⢠⢏⡟⢀⡾⣁⡾⠋⠀⠀⢠⡾⠟⢛⣿⠏⡿⠀⢀⡀⢠⡀⠀⢸⣿⣿⠦⢤⡾⣼⡿⣿⠁⠈⠪⣵⣿⣶
⣿⠿⣯⣿⢣⣄⣴⣛⣳⣤⣤⣿⣤⡾⠟⢋⣴⣧⠞⠀⡿⣿⡿⠀⠀⣠⡴⠋⠀⣠⡿⣡⠔⠁⣤⣌⣽⠟⠋⠀⠘⣿⣿⡶⣾⢶⡟⣽⠏⢀⣤⠤⠀⠉⠓
⣏⣀⣿⣧⣬⢭⣭⣭⣭⣭⣭⣥⠤⢒⡾⣿⡟⠁⠀⣤⠞⢡⣴⡾⣛⢁⣀⡤⠞⢉⠞⠁⠀⠀⠀⠁⠀⠰⠽⠿⠂⠹⣿⣟⣵⡯⠞⠁⠀⠻⠇
⠉⠉⠉⠁⠀⠈⠙⠿⣭⣙⣛⣛⣯⠥⠞⠉⠀⠀⠀⠷⠶⣥⢶⠿⠿⠿⠭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⢸⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⡓⠤⣀⠀⠀⠉⠀⠀⢀⠴⠒⠋⠁⠀⠀⠀⠀⠤⠐⣒⣩⠔⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠢⢬⣐⠀⠀⠀⢸⢀⣀⡠⠄⠀⠀⣠⠖⠖⠩⠋⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣀⣈⠀⠐⠀⠁⢠⡄⢈⣉⡥⠴⠚⢋⡟⠁⢠⡤⣀⣀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠁⢸⠁⠀⠀⣰⣎⡠⠖⠛⠛⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⢧⠈⠫⡛⠲⣄⣸⡤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠤⠤⠿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡀⠈⢧⢸⠗⢦⠀⠸⠥⠀⣀⣀⡀⣀⣀⣤⢤⠴⠒⠒⢩⠏⠁⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠈⠻⡀⠀⢱⢲⠒⠉⠁⠙⠉⠀⠀⣷⠋⣠⠊⠀⠃⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠎⣿⠊⠉⡗⢀⠑⢄⡈⢸⠀⠀⠀⠀⢀⣀⡴⠥⠾⠁⠐⠒⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠗⡄⠀⠠⠁⠀⠘⣤⠀⢓⠌⠓⠚⠒⠒⠉⠉⢉⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠃⢀⡀⠀⠀⢳⡜⠀⠀⠀⠀⠀⠀⠤⣴⣶⢿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢸⡄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠠⠤⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉⠁⠀⠐⠄⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣠⣴⠀⠀⠀⠀⢀⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⠓⠚⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠠⠤⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠤⠀⢀
⠀⠀⠀⠀⠀⢄⣀⡀⠈⠁⠀⠀⠀⡀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢀⡀⠀⠀⠁⠒⠁⣄⡤⠄⡀⠈
⠀⠀⠀⠀⠀⠀⢆⢀⠠⣄⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠈⠋⠉⡁⠀⠀⣀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠠⠀⠂⠀⠤⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⡆⢸⣏⡝⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠘⡇⡀⠀⠀⠀⠀⣠⠟⠀⣼⠃⡈⣯⢂⠉⠉⠱⠒⢂⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠖⠒⠊⠉⠁⠀⠸⠁⠀⡇⢹⡆⡑⠀⠀⠀⠈⠀⠉⠑⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠚⣿⣛⡻⠟⠋⠀⠙⠀⣠⠀⠀⠀⡆⢸⠇⠳⣤⠤⢀⠈⠀⠈⠑⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⣀⠀⠁⠀⢠⡏⣶⠀⠀⡇⢸⠀⠠⡟⢀⠀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠠⣾⠷⠋⠀⠀⣠⡜⠁⠀⠀⣿⣿⣷⠿⠀⠸⠀⣿⡄⠀⠁⠘⢦⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣿⡴⠃⠀⠀⢀⡴⠋⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠘⣿⣇⠀⠀⠀⠈⢳⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣎⡁⢠⡟⠁⠀⠀⣠⠊⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⢄⣄⡀⠻⣿⡄⠀⠀⠄⡀⢢⡀⠀⠰
⠀⠀⠀⠀⠈⠛⠶⠾⣿⡾⠿⣷⡟⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠟⢁⢈⡈⠁⠀⢐⣋⣡⠰⢿⣷⣄⢠⠀⡇⣾⢓⠦⢭⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠀⠀⠀⣶⠃⠀⠀⠀⢤⣼⠋⠠⠀⣨⠄⠀⡀⣺⠤⣼⠀⢿⣿⣿⣦⣇⣰⣄⣹⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡰⠁⠀⠀⢀⢰⣾⠃⠀⠀⠀⠴⠀⠤⢤⡟⠲⠖⠀⢸⢹⣟⢻⣿⣿⣝⡛⠻⠛⠛⢓⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣠⠁⠀⡄⠠⢄⣾⡏⠀⠀⠄⣴⠂⠀⠀⣼⠛⠓⡏⠀⠈⣼⣏⢀⠿⡄⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡉⠐⠤⠀⣠⣾⣿⠃⠀⠂⠠⠈⠀⠐⢤⣌⠉⢻⠃⠀⠀⢿⣿⣇⢰⣷⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠄⣠⣾⣿⣿⣿⠀⠀⠐⠲⠉⠀⠒⢾⡅⠉⡋⠀⠀⠀⡞⠈⠙⢾⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⠀⢉⠐⢊⠈⠀⠲⠅⣀⡸⠁⠀⡴⡼⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠋⠁⠀⡄⢨⢨⠄⠀⢀⡾⠦⢤⠃⠀⣰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⣴⠂⠢⣼⣃⡒⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠃⠶⠗⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠾⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠥⠶⠤⡀⡀⠀⠀⠀⠀⠀⢻⣤⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠠⢀⡽⠛⠦⡀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣤⡈⠁⣠⣀⡀⠀⠀⠅⣈⠻⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⡻⢻⢻⣿⢟⣾⣿⣿⣿⠿⣧⣦⣄⠙⠓⠦⣉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠃⠈⡾⠇⢼⡏⣿⠋⠀⠀⠈⠙⢟⠀⠀⠀⠈⠑⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡀⠀⠀⠄⠀⠻⠀⠘⠀⠀⠀⠀⠈⠀⢢⠀⠀⠀⠀⠀⠑⣔⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠓⠲⠶⢶⡶⣶⠇⠀⠀⠀⠀⠀⠐⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢊⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠇⠀⠀⠀⠀⠠⠀⠀⠠⠗⣦⠱⡄⠂⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣌⡃⠀⡹⣆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⠀⠀⡞⠀⢠⣠⣞⠀⠀⣣⠁⠐⠈⡿⣷⣄⢄⢀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣰⢃⣴⡿⣏⢩⡀⠀⠦⠄⠰⠀⠇⢨⠙⠳⢷⣝⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⡿⠛⣄⠹⣆⡦⠀⢺⡆⡀⠈⢀⠆⠀⠀⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠘⢆⠙⣿⣆⢸⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠻⣾⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣩⠞⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣶⡀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣤⡻⣇⠀⢀⡤⠴⠦⣤⡀⠀⠀⠀⠀⢀⡠⠴⠶⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢰⢷⣙⣿⠁⣠⡶⠒⠶⠽⣲⣤⡀⣰⡉⠀⠀⠀⠀⠀⢀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠰⠃⠛⠉⢀⡼⠛⣳⣶⢀⡴⠋⢹⠷⠾⣷⣖⣶⡁⠀⣠⠏⠀⠀⢀⡈⠳⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠤⠶⠓⢼⡀⣾⠏⢀⡄⠊⠁⡼⠧⣾⠁⣀⠖⢁⣸⠿⢿⣷⡉⠠⡈⠒⢯⢤⣀⣀⠙⠂⢤⣄⡠⢤⡀
⠀⠀⠀⠀⡼⠀⠀⠀⢀⡗⠀⠀⠛⠋⢹⡟⢀⡤⠊⢀⠴⠛⢹⣇⣤⠊⠀⡀⠾⣷⣹⡄⠰⡄⠘⠃⠀⠉⡟⠓⠚⣛⠟⠊
⠀⠀⠀⢸⠁⣀⠀⠚⢋⠁⠠⠄⡆⢶⠁⠸⠟⣧⣴⠋⠀⡴⠎⠀⣸⠶⣿⠀⢀⣼⣇⢳⡀⠃⠀⠀⡄⠀⢀⣤⠞⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠐⡆⠀⠀⢻⡜⣷⢀⡔⠉⢹⡶⡶⠁⣠⡼⠃⢀⣵⣾⢁⣈⣿⢨⠃⠻⣧⡘⠛⡛⣯
⠀⠀⠀⢰⠀⠀⠀⡤⠐⢧⠤⠄⠀⢳⣜⢿⣄⡤⠋⢀⠿⠛⣯⣤⠞⠈⢁⣨⣟⢉⡝⣸⠀⠸⠈⠹⣍⠁⢸
⠀⠀⠀⠈⣇⠠⠀⠀⠀⢸⡀⠀⠉⠁⢻⣧⠙⠷⣴⡁⢀⠞⢛⣧⣴⠟⠉⣉⣽⢛⣇⡟⠠⠀⠀⣤⢿⡀⠸⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢘⡇⠀⢀⡀⢨⠁⠈⠐⠠⢝⡛⠶⣾⣀⠿⣶⠋⢠⣤⣿⣽⠉⠙⠦⣄⠋⠀⠑⢦⡛⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠴⠀⣦⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠙⠢⠽⣶⢾⣇⣁⣠⣿⡏⠀⠀⠀⠈⠉⠀⠘⠂⠩⣌⡆
⠀⠀⠀⠀⠀⠀⠈⢧⠀⣰⡋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡙⠻⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠁⣣⡞⠁⠀⠀⢀⠀⠀⠀⣀⣽⠃⠀⠀⠀⠀⠀⠙⢦⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⠀⠀⠀⠀⠀⠳⠶⠛⣩⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣄⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⢦⣄⡀⢀⣀⠀⠀⢀⠀⢀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠒⠀⠒⢤⣥⡀⣈⣙⠋⠉⠀⠻⠦⠆⠚⠏⢶⠀⠈⢉⣀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⣽⣾⡯⠶⠷⠚⠛⠙⡾⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠒⡐⠚⣡⠀⣠⡦⣥⣼⣯⣁⣀⠀⠀⠀⠀⢰⡆⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡴⠃⠀⠀⠛⣤⡇⠐⠀⠈⡇⢸⡄⢳⠀⠀⠀⡾⠀⢰⡄⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡄⠀⠺⠀⠀⠀⠀⢠⣾⠁⡇⠀⢾⡃⢸⡀⣼⠀⠀⢸⠃⢀⠜⠁⢦⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⣾⣄⡀⠀⢀⡿⠃⠠⠇⠀⠸⡇⢸⡇⠸⡄⠀⠸⡇⠋⠀⠀⢸⣦⡀⠁⠷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⣠⠔⢻⠀⠉⢰⡿⠀⠀⡀⠀⠀⠀⠃⠀⠇⠀⠙⢄⠸⣇⠀⠀⣠⠊⠈⣧⣀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⣧⠀⣻⠀⠀⠘⠀⠀⠀⢠⠇⠀⠐⠀⠀⠈⢂⠙⡆⠰⠁⠀⠀⡈⠡⠉⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣽⠉⣛⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠀⠀⠀⠘⢤⡆⠀⠀⠀⠀⠁⠀⠳⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡴⠹⠀⢹⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢠⡆⢱⡄⠀⠀⠀⠀⠀⣸⡇⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⣦⣀⣧⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡸⠁⠈⠇⠀⠀⠀⠀⢀⠇⣵⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠁⢸⡟⢢⡄⠀⠃⠀⠀⡌⠀⣸⠀⠀⢰⠇⠀⠀⠀⠹⣆⠀⣴⠃⣀⣼⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⠴⠋⠀⠀⠙⢦⡀⢦⠀⠀⠰⠁⠀⣰⠃⠀⠀⠀⠀⠀⠈⠓⠇⣀⠋⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠹⣞⡀⠀⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢄⣀⠠⠄⠀⢀⣀⣀⣀⡀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡌⠉⣁⣂⡀⠀⢠⣤⣤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣯⣭⣛⠿⢷⣮⣿⡿⣥⡀⠈⠉⠀⠤⣒⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠙⢶⣄⠀⠀⣠⠀⠈⠙⠲⣤⡀⠀⠀⢀⣠⠀⠀⠀⠀⢻⣷⣹⣶⠙⣶⣆⣀⡽⠿⢷⣽⣷⣦⣤⣀⠀⢉⠀⠀⢀
⠀⠀⠀⣠⣮⣀⠀⠀⠀⠀⡈⠷⣮⡀⠀⠀⠀⠀⠀⢙⣶⡿⠛⠁⠀⠀⠀⠀⠘⣿⣯⠻⣿⣀⡔⠛⣲⢶⣈⡩⠿⣮⣿⣍⠉⠐⠂⠀⠀⠳⢄
⠀⠀⠀⠻⣎⡻⣷⣄⠐⠊⠀⠀⠀⠙⠷⣤⣠⣦⣇⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⢿⣧⣼⣁⡀⠛⢣⣔⠉⣻⣿⣿⣶⠶⠖⡠⠄⠊
⠀⠀⠀⠀⠈⠙⢧⣽⣷⣦⡀⠀⠀⣀⣤⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣍⡛⠻⠿⣶⣇⣨⠿⣥⡬⣿⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣗⣮⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣦⣭⣛⣓⣻⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⣤⠤⡀
⠀⠀⠀⠀⠀⠀⡀⢠⣞⡷⣑⢟⡳⢯⢤⢿⣿⣥⣛⡧⢰⣒⡀⠄
⠀⠀⠀⠀⠀⠀⢹⠂⢈⠡⠉⠙⠉⠉⠉⠙⠒⠛⠛⠃⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⡴⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠤⠶⠽⠷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣠⣤⣴⡞⡁⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣻⣿⣿⣿⣎⣷⣿⣦⡀⠀⠀⡀⠐⣀⠍⡄
⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡂⠀⠀⠀⠀⠀⠀⠙⣟⣽⣽⣿⣿⣿⣿⣿⣿⣖⣀⣴⣪⣽⡧
⠀⠀⠀⠀⠀⠰⢧⣤⣤⣬⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢿⡿⣿⡿⢻⡁⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⢖⠿⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠒⠉

⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢹⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⢄⡀
⠐⢤⣤⢠⣄⣀⠐⢷⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣆⠹⢆
⠀⠀⠈⢲⣦⣈⠄⠀⢇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢍⡓⠲⢤⡀⠘⡆⠈⠂
⠀⠀⠀⠀⢀⠨⠗⡲⢿⣵⣂⢑⢂⡤⠤⠤⠤⢤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⢀⣼⣄⠀⢦⣀⠀⠠⠤⠤⠤⠤⣄⣀⠀⡀
⠀⠀⢀⠔⡡⠀⠀⠀⠚⣿⡿⠀⠙⠢⢤⠖⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢒⡤⠛⠋⠁⣄⠉⠳⢀⢃⡀⣀⠤⠤⠖⠖⠚⠉⠁
⠀⢀⣿⢋⡀⠐⠁⠀⠀⢿⣇⢳⢆⡀⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠔⠁⡠⠄⠈⠱⡹⡙⠰⣀⠈⠀⢀⠀⠄
⠀⢸⣿⣿⣁⣀⠄⡈⠠⠀⣝⣧⠛⣵⡆⠩⠄⠁⠀⠀⠀⠀⠀⠀⠘⠋⡶⣊⢁⠠⠔⠚⢳⣌⢤⡈⠒⡤⠀⠈⢆
⠀⠛⠛⠛⠳⠂⠀⠂⠀⣶⡉⢻⣷⣭⡸⠣⠄⠁⠀⠀⠀⠀⠀⠀⣷⢀⣬⡿⢄⡀⠔⠖⠋⠱⠶⠈⡒⠤⠫⣄⠈⠄
⠀⠀⠀⠀⠀⠀⣁⡰⠦⢄⣹⣾⡀⠙⠻⠿⠿⣾⣶⣄⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠚⠿⢶⣬⣥⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠛⠳⢦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠶⠟⠋⢀⣶⣟⣻⣦⠀⠀⠀⠀⠀⠉⠛⣷⣶⡦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⡋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢮⣙⢷⡄
⠀⠀⠀⠀⠀⣠⡶⣫⣵⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⢰⢏⣼⠋⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀
⠀⠀⠀⡰⢡⠞⠁⠀⠀⠀⠀⣿⣿⣷⣌⣳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⣌⢷⡄
⠀⢀⡾⠕⠁⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⢼⣿⣿⣟⣻⡄⠀⠀⢸⢸⡗
⠀⣿⠁⠀⠀⣀⣴⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠁⠀⠀⣾⢸⣇
⠀⡏⠀⠘⣿⣿⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⠀⠀⠀⠀⠀⠀⠈⠃⢿
⢰⠃⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⢷⠀⠀⠀⢀⣶⣆⢆⠸
⣿⣮⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢸⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠘⠦⠿⠿⠋⠉⠀⠀⣴⣻⣿⡿⣸⢸
⢿⣿⡀⠀⠀⠀⠀⣾⣿⣿⣮⣷⠀⠀⠀⠈⠛⠿⢿⣻⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢣⣿⡟
⠈⢧⣛⣦⣐⣶⣤⣜⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⣿
⠀⠈⠻⢻⡝⠿⠋⣙⠿⠛⣷⣶⣄⡀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⣠⣼⡇⣿⠋
⠀⠀⠀⠰⠍⠀⠀⠉⠀⠀⠉⠉⣿⣅⢸⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⡇⠀⠀⣨⣵⠟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠀⢟⡿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⠁⢀⣾⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⢙⡷⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⢧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣧⣤⠀⠀⠄⣿⣿⣶⣿⡇⣠⣴⣶⠷⣟⣋⣉⣩⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⣿⣛⣛⣓⣚⡛⠛⣋⣈⣥⣾⣶⡿⠟⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣬⣽⣿⣿⣿⣯⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠦⠶⢶⠦⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠛⣁⡀⣀⣒⣢⣍⣠⣬⢁⣨⣿⣖⣄⣀⣠⣴⣾⠁⢀
⠀⠀⠀⠀⠀⠀⠙⢷⣦⣽⣋⣹⡟⢉⡟⠉⣥⣋⣁⣞⣡⠏⠋⠁⠀⣴⠋⠈⣛⢦⣛⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠿⠟⠛⠛⣋⣁⡀⣀⣠⣾⠁⠀⠀⢻⣄⣿⣷⣦⢄⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠠⢐⣻⣿⣿⣿⠁⣼⠙⣆⣀⣀⡀⠙⠿⢿⣮⣤⣱⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣬⠶⣫⣿⣿⣿⣿⠇⣸⠇⢀⡏⢻⣷⣜⠒⠀⣄⠈⠉⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣫⡶⣿⣷⡿⣿⣿⣿⣿⣧⣴⣏⠀⠸⣷⡀⠻⠿⢷⣦⡈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢫⣾⣿⣻⣿⣿⣿⡿⢷⠿⢿⣻⡆⠀⣿⡦⠀⡆⠈⢿⣿⣦⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣫⣿⣿⣿⢿⡿⣽⣏⣉⡟⣿⣷⠀⠘⣿⣾⡇⠀⠈⢹⣟⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣱⣯⣿⡟⠠⣿⣛⣿⣿⡀⠀⢻⣿⣆⠀⠀⠀⣿⣷⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣳⢷⣾⣏⣑⣾⢯⣼⣿⣿⣷⡀⠀⠙⣿⣿⡆⢠⠟⣿⡍⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⣿⣯⣟⣻⠿⣄⣽⠷⣴⢿⣿⣿⣟⢻⡄⠀⠈⠹⣷⡜⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣏⣳⡾⣉⣳⣿⣿⣯⠙⠻⢷⣷⣄⠀⠀⠈⢿⣶⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⢦⣟⡶⣻⣿⣿⣿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⢿⣟⣿⡿⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠿⠛⠉⠀⠀⠻⣧⣅⡧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⠿⠶⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠊⢛⣿⣿⠀⠀⠀⠀⠐⢦⣤⡀
⠀⠀⠀⠀⢻⡇⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⡖⠀⠀⡀⠈⣿⣿⣦⠀⠀⠀⢰⣦⣄⣠⣾⠿⠃⠙⢷⣤⣀⣀⣸⣿⠇
⠀⠀⠀⠀⢸⡟⠀⠊⠀⠋⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⠀⠀⠀⣷⠀⢸⡟⡏⡆⠀⠀⠀⠉⠙⠉⢁⠀⣠⣤⣤⡈⠉⠉⠁⠄⠰⠲⣦
⠀⠀⠀⠀⠘⣿⣄⠀⠋⡄⠀⠁⠔⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠇⠀⠀⠀⢹⠀⢸⣏⣧⢡⠀⠀⠀⢀⡔⠀⠀⢀⣽⣿⡿⠃⠀⠀⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⡠⢀⠀⠈⠀⡔⠀⠀⠀⠀⠀⢸⠀⡿⠀⡀⠀⠀⠘⠀⢸⣿⣿⢸⠀⠀⠀⠀⠙⠻⠾⠛⠛⠉⠀⠀⠀⢰⣄⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣧⡈⠋⢠⠀⠈⠤⠂⠀⠀⠀⠀⢸⠀⠁⠀⠁⢠⠀⠀⠀⠘⣿⣽⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠘⣤⠂⠴⣡⠀⠀⠀⠈⡀⢀⡄⠀⠈⠀⠀⡇⢀⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣤⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣄⡀⢶⡄⠉⠆⠀⠀⠀⢳⢸⣧⡄⠀⠀⢰⣇⣸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢚⣶⡤⣶⣿⣧⣀⣠⣽⡗⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣍⠁⢨⠄⠀⠀⢾⡇⢿⣿⣄⠀⣸⣿⣿⣿⠏⠀⠀⠀⣀⡀⠰⠷⣤⣾⠁⣠⣿⣯⣉⣉⣙⣿⣷⡈⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠒⠒⠀⠀⠈⢿⣼⣿⣿⣇⣿⣿⣿⠋⠀⠀⠀⠊⢙⣷⣶⡔⠿⣁⣈⣽⣿⡟⢻⣿⣿⢿⣯⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣄⣀⣠⣤⠄⣀⣀⣀⣤⣉⠛⠿⢿⡿⠛⠁⠀⠀⠉⠻⠶⣿⣽⣏⣻⣟⣛⡛⣿⣾⣯⣭⣽⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣾⡀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠿⠿⠿⠛⠛⠉⠉⠉⠀⠀⠀⢀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢻⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠛⠉⠉⣽⡆⠙⠋⠉⠉⣿⣆⠛⠛⠉⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠐⠂⠀⠀⠀⡠⠀⢠⠆⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠏⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠷⠶⢻⡶⠶⠦⠒⠲⣿⠲⠶⠦⠄⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠂⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⢧⠀⢀⠀⠀⠀⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣠⠴⠆⠘⠿⠶⠂⠰⠾⠿⠶⠶⠒⣇⢲⠉⢷⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⠇⠀⠀⢝⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠏⠀⢀⡴⠊⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡠⠭⣭⣭⠥⠤⠿⠃⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣰⡀⢠⠾⠁⣐⣦⡎⠀⠀⠀⣤⣤⣄⡀⠀⣍⣉⣉⣉⡋⠁⠀⠛⠩⠡⠴⠄⣾⣿⣯
⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡀⠠⣶⣼⣥⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⡇⠐⢦⣀⠤⣈⡈⠛⠓⠬⠿⣯⡉⠉⠛⠛⢓⣶⣶⠶⠶⠿⠿⠿⢯⣄⣠⣶⣾⣷
⠀⠀⢀⡿⢷⣄⢦⣄⣀⣸⣷⣾⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠢⣉⠛⠻⣍⡛⠶⣤⣀⣀⡀⠁⠛⠂⠉⠏⠀⠀⠀⠀⣠⣴⢾⣿⣿⣿⣿⡋
⠀⠀⠀⣀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠂⢄⠈⠛⢶⡛⠉⠉⠛⠛⠷⠶⣤⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⡾⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁⠢⢀⠙⠷⣤⣴⣶⣾⣿⣭⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⡿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣦⢰⡇⢰⡆⣴⠛⣦⢰⡞⢷⡄⣾⢻⡆⣺⠀⠀⢰⣿⡆⠛⣿⠓⢺⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢸⣧⣼⡇⣿⠀⣿⢸⡇⠈⠈⣿⢸⡇⣹⠀⠀⣸⠟⣇⠀⣿⠀⢸⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣿⢸⡇⢸⡇⢿⣀⣿⢸⣇⣼⠇⣿⣸⡇⣸⣄⣀⣿⠛⣿⠀⣿⠀⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠁⠀⠉⠀⠀⠈⠁⠀⠈⠉⠀⠈⠉⠁⠁⠀⠈⠀⠉⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠉⠉⡁⠀⠈⠉⠉⠭⠹⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠠⠁⢠⡟⠙⠦⠒⠒⠒⠶⠦⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠁⡐⡽⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠄⢀⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⡜⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⡀⠀⠌⡜⠀⠀⣴⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠩⠄⢒⠒⠒⠂⠀⠀⡀⠀⢀⢀⣀⠎⡜⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠉⢹⠁⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠂⠀⠀⠀⠀⠈⠉⠉⠉⠰⠋⠉⢹⢁⡞⠀⢀⡾⣉⡀⠉⠐⠒⠒⠲⢤⡄⣀⣸⣀⣴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⢃⡞⠀⠀⡾⠁⠀⠉⠙⠒⠂⠀⠚⠁⠀⢀⠝⢋⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⣀⠠⠀⠀⢴⣠⡀⠠⣡⡞⠀⠀⣼⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢊⠅⠊⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠊⠀⠁⠀⠒⠀⢤⡇⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠘⠛⠀⠐⠂⠀⠀⠀⠀⠤⠤⠊⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠞⡄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡼⠀⣾⠶⠶⠦⣤⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣼⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⢸⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠄⠉⠀⢀⣀⠀⠀⠀⠐⠀⠄⠀⠀⣼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠚⢀⢠⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠩⣾⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠤⠤⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠶⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠉⣀⣄⣀⣀⠙⠒⠠⠄⠉⠉⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠾⢏⣉⣉⠤⢠⣤⣓⠶⣤⣉⠙⠛⠉⠓⠀⠀⢀⣠⠴⠚⠫⢑⣈⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢯⣝⣳⠦⣤⣉⠉⠛⠻⠾⣍⣁⠈⢉⣀⡠⠤⠖⠒⠋⣉⣤⡤⠖⣋⣩⡤⡖⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠠⠝⡹⡟⠓⢮⣙⣓⠲⠤⣌⣉⢉⡏⣀⣁⡤⠶⣚⡋⠭⠀⢶⢺⠁⠁⠀⣀⡤⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣢⣤⣃⡐⠂⠳⠉⣿⠓⠦⣬⣍⡉⣉⠭⡖⢉⠋⢠⠀⠀⢈⣨⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠂⠬⢭⣛⡳⢦⣌⡀⠀⠘⠇⢻⠀⠒⠁⣀⣠⣤⠶⢏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢥⣌⣓⠦⣄⣀⠀⠀⠉⠓⠲⢭⣓⡲⠦⠼⠴⠾⣟⣛⣭⠭⠶⡚⠋⢳⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣀⡘⠀⣌⠙⢦⣍⣙⠓⠶⣤⣀⠐⠤⠉⠳⡖⡊⠉⠉⢀⣀⣤⠤⣖⣿⢿⡶⠒⠋⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠞⠉⠻⣦⣄⡈⠀⠙⠉⠙⡓⢾⢿⣗⠢⠤⠥⠴⣒⣺⣋⣥⢶⠟⠋⡉⡜⠙⠋⣀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡼⡜⢆⠀⠠⡉⡛⠿⢶⣤⣒⡀⠈⠛⠉⠙⢲⡞⠛⠉⠁⠟⠀⣊⣀⣠⠤⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠈⠧⡳⡀⠘⣎⠂⠒⣀⣈⣉⣛⡦⣄⣀⣸⢀⣀⣤⠶⠞⢋⣉⣸⡌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠈⠣⡤⢱⡝⢆⠊⠟⠛⠉⠉⢁⣀⣀⣠⠭⠿⠛⢛⣓⣻⣭⡭⢥⡶⠖⠚⢻⡀⠀⠀⠀⠀⠀⡠⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠉⠷⣀⠀⠀⢳⡈⠙⣎⢑⣒⣒⣚⣯⡭⠽⠖⠞⠻⠋⠹⠩⠃⠀⠀⠀⠊⢀⣠⣴⠾⠓⠀⠀⠀⠀⠎⠀⠀⣠⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠸⠦⣹⠂⠀⠀⠙⢆⢸⠙⠃⠘⢁⣊⣀⣭⣤⣤⣤⣶⠶⢾⣿⠿⠽⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠙⠟⢳⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠚⠋⠀⠀⠀⠀⠈⠿⠾⠿⠟⠛⠉⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⣉⣳⣀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣖⣩⣏⣙⣦⠀⠀⠪⢨⣛⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⠏⠀⠀⠀⠀⠼⠃⠀⠀⠀⠈⠓⢶⣀⠀⠀⠇
⠀⠀⠀⢀⠐⢄⠀⠠⣤⠠⢥⣀⣀⣀⡼⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠑⠶⣶⢁⠂⠀⠀⠀⠀⠀⠓⠨⠁⠀⡠
⠀⠀⠀⠉⠑⠀⡈⠑⠀⠉⠌⠀⠧⣄⡴⠀⠉⠉⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢠⠏⠀⠀⠀⢰⣣⠃⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠲⢦⣉⠂⠀⠀⠀⠀⠀⠈⠋⠀⠀⠄⢸⣄⠀⠀⠀⠀⠄⣀⣤⣄⡰⡏⠀⠀⠀⢠⢒⠏⠀⠀⠬⡁⠀⢢⠐⡄⠀⠀⠀⠀⠀⢀⣀⡠⡆
⠀⠀⠀⠀⢾⠂⠀⠉⠃⠐⠠⠀⠀⠀⠉⠀⠐⠀⡀⠀⠀⠀⠠⠂⠀⠀⠈⡙⠁⠛⠶⢶⣤⡋⠎⠀⠀⠀⣄⠈⢦⠀⠇⡇
⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⣨⡋⠀⠀⠀⠀⠉⠓⠚⠂⠃⠀⠀⠀⠀⠙⠓⠒⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠃⠀⠀⢶⣤⣀⣀⣴⠏⠀⠀⠀⠀⢠⠟⠀⠀⣠⣤⢶⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄⢄⢄
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⠿⣦⣄⣀⠀⣴⡎⠀⠀⢪⠞⡵⠚⠿⣯⣧⠀⠀⠀⠀⠰⠀⠀⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⣴⣄⢘⢀⠀⠈⠁⣠⠦⠀⠀⠀⠸⣤⣠⣀⡀⡰
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠈⠀⠀⠀⠀⣴⠄⠀⠀⠉
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠄
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠂⠀⢦⣄⣠⣔⣺⡼⣷⣄⠀⠀⠀⠀⠀⠀⣠⠤⠂⢩⣿⡍⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠇⠀⠀⣄⠀⠀⠄⢀⡿⢟⣯⠾⠛⣫⠼⢆⡀⠀⠀⡀⢺⣿⣦⡀⠈⠙⢯⣌⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢗⣋⣽⡔⢛⣯⣴⣿⣭⣉⣙⡿⠆⠁⠀⠀⠙⢯⡻⡯⠈⠀⠈⠛⠒⠤⣉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠰⠄⠀⠀⠀⠀⠉⠻⠻⠟⠊⠉⠛⠂⠓⠛⠋⠹⠁⠀⠀⠀⠀⠀⠀⠙⠂⠈
⠀⠀⠀⠀⠂⠀⠀⢐⡎⠀⠀⠀⡄⢀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⡟⠒⠂⠀⣸⠎⠀⠀⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⣠⠄⠉⠍⠉⣇⠀⠈⠲⠗⡉⠁⣀⣀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠠⡀⠀⠀⠠⠄⢨⠇⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠓⠀⠀⠀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⢄⣀⣤⣤⠤⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠹⢦⡘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⡀⠠⠄
⠀⠀⠀⢠⣀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈⠀⡖⠀⠀⢰⣦⡤⢄⡀⠀⣠⢟⠦⡀⠀⢄⠀⢀⠀⢀⡴⠀⠀⢀⡀⠀⠠⠀⠐⡇⠀⠀⠠⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⢈⠀⠀⠀⢀⣷⠄⠀⠈⠙⠣⠀⠈⢀⡴⠁⠀⠈⢢⣄⣧⣤⠆⠈⠀⠀⠀⢋⠀⠀⠀⠐⠔⣀⡀⠴⠤⠖⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⢳⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡁⠀⠈⠑⠂⠄⠀⠀⠀⠀⣀⡤⠤⠒⠉⠁⠀⠀⠀⢀⣀⠼⠇⣈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠢⢍⡓⠢⢄⡀⠀⠀⠀⢠⠉⠁⠀⠀⠀⠀⠀⠀⢂⣉⢥⡴⠾⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡍⠙⠖⠦⡁⠀⠀⢼⣤⠴⠖⢈⣁⣀⡤⠊⠉⠃⠘⠀⠁⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠲⣤⣄⡀⠈⠀⠀⢸⡇⢠⡤⠖⠚⠋⠁⡼⠁⢀⣴⣊⣤⣴⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢿⠿⣶⣄⡀⠐⠀⢸⠀⠀⠀⣠⡿⠤⠚⠛⠉⠉⠀⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠸⣆⠈⢮⡙⠓⡤⣼⠶⠖⠋⠁⠀⠀⠀⢀⡀⠠⠤⠤⠶⢛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡄⠀⢹⡄⣿⠓⣄⠀⠸⠄⠀⣀⡀⣀⣀⣀⣤⡤⡴⠶⠒⠊⣹⠏⠀⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠙⣇⠀⠈⠷⢶⠒⠛⠉⠹⠉⠁⠀⢸⡞⢁⠤⠂⠰⠃⠐⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣞⣻⠖⠋⢳⢀⠈⢦⡀⠇⢸⠀⢀⠀⠀⠀⠀⣠⣏⡴⠿⠥⠴⠓⠒⠒⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣱⡦⡀⠀⢀⠛⠀⠰⢏⡀⢹⢦⠙⠦⠼⠴⠖⠒⠒⠘⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠃⠀⠞⠀⠀⠀⠈⠳⣇⡏⠀⠀⠀⠀⠀⠀⣀⣈⣭⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠚⠩⢤⣄⣀⡐⠂⠨⠉⠑⢒⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣴⡉⠧⠤⣶⣶⠮⠟⠶⠄⠉⢈⣁⡠⠔⢚⣉⡭⠴⡓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢙⢳⡿⢟⡲⠤⣌⣙⠒⠲⠚⣋⣡⠤⢖⡛⢩⢴⠚⠋⢁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⣤⣆⣂⠙⠹⡗⠪⣭⣉⣉⢽⠂⠏⠠⠀⣈⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢎⡀⠀⠐⠻⠽⣒⡦⣄⣈⢘⣀⣉⣤⣴⢶⣛⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠹⢓⠺⣝⡲⠤⣄⣀⠩⢑⠺⣭⠭⡝⠛⠋⣉⣩⣤⣾⣡⠵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⠛⢶⣅⣈⠉⠛⠓⡻⣿⣖⠤⠬⠤⣶⣞⣫⢷⠿⠩⠝⠛⣀⣸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠫⡧⡀⢪⣍⣛⠶⠮⣄⡉⠁⠉⡿⠉⠁⣘⣀⡤⡤⣖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⣝⣄⠳⡤⠵⠒⠒⠋⠓⢶⣷⣶⡿⣭⣔⣒⣺⣸⣦⠀⠀⠀⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⡀⠀⢣⡙⢮⣒⣀⣖⣶⣬⡽⠿⠳⠒⠚⠋⠉⢉⠙⣁⣤⡼⠆⠀⠀⠠⠋⠀⣠⣽
⠀⠀⠀⠀⠀⠀⠀⣡⠘⢓⣿⠀⠀⠑⣬⣉⣁⣁⣬⣬⣤⠶⠶⠖⠻⠿⠛⠛⠋⠋⠉⠀⣀⠀⠀⠙⠻⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠦⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣂⣛⣦⠀⠀⢜⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⠀⠀⠀⠀⢀⣴⣾⣿⢿⣿⣦⠀⠀⠀⠀⠀⢠⠶⠿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡄
⠀⠀⠀⡯⣝⣫⢿⣿⣿⣿⡿⠀⠀⠀⠀⣬⣟⣿⣾⣿⣻⣽⡇⠀⠀⠀⢀⡞⢶⣶⢼⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠯⣓⢺⠿⠟⠋⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠉⠓⠦⢬⡿⠛⠁⠀⠀⠀⠀⠺⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣀⣀⣀⣀⣤⣶⡀⠀⠀⠀⠀⠀⠀⢀⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢴⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡚⢻⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⢟⡁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⣿⢿⣷⡹⣷⠀⠀⠀⠀⠀⠀⠋⢿⣷⠘⣿⡆
⠀⠀⠀⢰⣯⣽⣯⣽⣶⡿⠁⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣷⡝⣧⢻⡇⠀⠀⠀⠀⡘⠀⢸⣿⣶⠿⠃
⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠻⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠈⠐⠠⠋⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣻⣿⡏⣿⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⡤⠠⣄⠀⠀⠀⠀⠀⠀⢀⡠⣦⡄⣀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢿⣿⣟⣿⠀⠀⠀⠀⠀⠀⢰⣯⣶⣒⣒⣦⣵⠀⠀⠀⠀⠀⣰⣮⢿⡎⣿⣼⡄⠀⠀⠀⠀⢨⣟⢿⣿⢿⣻⡅
⠀⠀⠀⠀⠀⣿⣿⡷⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⡸⢏⣿⣧⡶⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣾⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⢦⣳⡄⠀⠀⠀⠀⢀⣴⢦⡰⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣷⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢀⣀⣔⣶⣿⡟⣷⡽⠿⠀⠀⠀⢰⣿⠱⠂⣿⡝⣷⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⡃
⠀⠀⠰⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠹⣿⣷⣾⣿⡿⡿⠃⠀⠀⠀⠀⠘⣿⡿⣸⣿⡿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠋⠛⠁

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

⡛⢛⡛⠛⠛⠳⠮⣭⣁⠚⠣⢤⡐⢣⡷⣿⣧⢄⣤⣼⣯⣀⡧⢸⡆⠀⣠⠾⢳⠀⠀⢀⣀⠀⠀⣀⡴⠋⢀⡤⢞
⠿⣿⣭⠗⢂⣀⠀⠀⠙⠻⠦⠄⣉⡁⠐⢧⠀⠀⠈⣿⠋⢹⡇⡼⠀⠀⠛⠀⣠⣴⣿⣿⡿⠿⣷⣿⣻⣾⣿⠟⠁
⣆⡟⢿⣿⣯⣭⡵⢦⠀⠀⠀⠀⢿⢸⣽⣏⠉⠒⢻⠟⣷⢟⣴⡵⠀⠀⣠⣾⡟⠋⠀⢀⣀⠀⠈⣿⣧⣿⣡⣀⣰
⣿⣧⡆⠸⢿⣷⣶⣛⣛⣦⠀⠀⠈⣇⣽⣟⠓⢺⢿⠖⣡⡾⠙⠀⢀⣼⣿⣶⣒⡾⠿⣟⣿⠿⣦⣿⢧⡟⠈⠉⠉
⣷⢿⣧⣆⣸⠉⡿⣿⣿⣤⠷⡄⠀⠹⣾⣟⣻⡟⣷⠋⠋⠀⠀⢠⡿⢿⢧⣏⣴⡷⢒⡛⠁⢀⣿⢫⠏⠀⠀⠀⡀
⣿⠿⣿⣿⢿⣶⡀⡟⢹⣿⣶⣻⡀⠀⠘⠛⣿⠿⠋⠀⠀⠀⠀⣿⠛⣧⡾⠥⠞⢹⣡⠟⣠⠞⣵⠟⢀⣤⠴⣉⢀
⣳⡴⠛⢯⡻⣿⣿⣧⣾⣸⠿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣄⡽⠆⣀⣤⢖⡵⠞⢁⣾⠋⠓⠶⠻⣿
⢙⣷⠖⠋⢳⣹⡿⣿⠿⠿⣶⣟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠰⣟⣿⣿⣦⣤⣀
⡉⣻⣤⠤⠾⢧⣿⡘⡇⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⣿⣿⣿⣯
⡹⣀⣷⡦⠔⠛⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣏
⣿⣿⣯⡷⡶⠚⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣼⠟⠀⠀⢉⣉
⠻⣾⡻⣷⣾⣟⣻⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣷⡦⢀⣤⡴⠊⠉⠉
⠀⠈⠛⠮⣽⣯⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣤⣤⣴⣤
⣤⣿⠷⠦⠤⣌⠙⠛⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣲⣿⣟⣻⡴⣟⣻⢿⣗⡤⠀⠀⠀⠀⢠⣎⣴⣿⠛⠋⠛⣻⣿
⣉⣀⣤⠖⣡⠏⠀⠀⠀⠀⢴⣾⣷⠄⣰⡇⣰⣟⡇⣹⡞⣷⢿⣯⡽⣻⣞⣻⣵⡄⠀⠀⠘⢧⣿⡿⣶⣶⡾⠟⠛
⣿⣿⣿⣿⠋⠀⠀⢠⣄⣠⣄⠀⠀⢰⣿⣄⢿⣩⠿⣩⡿⣿⣼⣋⣶⣧⣹⢺⡾⣏⣄⠀⠀⠀⠉⠻⣾⠏⠒⠒⣶
⠿⠋⡠⠋⠀⠀⣴⣤⡽⠛⠉⠀⠀⠘⣿⣿⣆⠙⢦⣿⡾⣧⢶⣏⣽⣏⣽⣏⣿⣻⣽⡗⢦⠀⠀⢠⣟⣉⣹⣿⣿
⠚⢩⡶⢴⣖⠂⡀⠀⣀⡄⠀⣀⠀⣾⣿⢿⣿⡷⠄⠙⠲⢿⣴⣷⡴⢾⡦⣽⣧⣽⣾⣿⠀⡇⢠⣋⣀⣼⣿⣿⣿
⢞⣿⠷⠾⣯⡽⢿⠳⡬⠛⠛⠁⠀⣾⣿⣷⣝⠻⠶⣄⠀⠀⠀⠈⠙⢿⣶⡿⠾⣿⠏⢁⣾⡇⣌⣭⣿⡿⢿⣼⡏
⣛⣿⣟⣋⣻⡶⢾⣷⣞⣡⠦⢀⠀⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⢘⣉⡽⢈⣩⠟⢻⣇⣸⣿⣶
⣹⣯⣉⣻⣧⣤⣾⣥⣬⣻⣟⣛⣡⣿⣿⣟⣉⣉⣛⣛⣻⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣀⣘⣓⣂⣉⣉⣉⣉⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡴⠀⠀⠀⢀⡀⠀⠀⠀⢰
⠀⠀⠀⠆⡀⠈⠀⠀⠀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣤⣀⠼⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⡙⠷⠂⠀⠈
⠀⠀⠀⣤⡉⠂⢄⠒⠠⠱⠲⠤⠤⠞⠤⢄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠉⢻⠃⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊
⠀⠀⠀⠀⡐⠠⢄⠀⠀⠂⠈⠙⠢⣮⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡼⠁⠀⠀⠀⢿⠎⠀⢀⡤⠄⡂⢀
⠀⠀⠀⠀⣍⠒⠠⢤⢀⠀⠀⠀⠀⠄⠀⡈⠀⠛⠢⠀⠀⠠⠈⠒⠹⢲⠎⣅⣀⡀⢀⠎⠎⠀⠀⠈⠀⡀⠐⡌⡆⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠉⠛⢿⠌⠀⠀⠀⠳⢄⣰⣀⠤⠃⠀⠀⠀⢄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⡀⠀⢀⣀⠀⠀⢀⡾⠀⠀⠀⠀⢠⡟⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣿⣁⡀⠀⠀⢀⡎⠀⠠⣪⢟⡭⣿⣿⣆⠀⠀⠀⠀⡀⠘⠐⠊
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⡟⠀⢀⣄⠁⠊⠀⠀⠛⢉⣄⠀⠀⠀⢇⡀⡀⠀⠀⠄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠃⠂⠀⠉⠀⠀⠀⣤⠀⠈⠉⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠰⣄⢀⣄⣀⠼⣦⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⢁⣀
⠀⠀⠀⠉⠑⠒⠀⠀⠀⠀⠒⠒⠀⠀⠀⠠⠀⠀⠀⡀⢀⠄⢀⡽⢞⣫⠝⢛⣡⣷⣀⠀⠀⠀⠰⣿⣦⣀⠈⠙⠯⣌⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⣋⣷⡦⣻⣽⣾⡯⣭⣭⣹⠿⠀⠀⠀⠈⠻⣝⠿⠁⠀⠈⠉⠒⠠⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠤⠐⠀⠀⠀⠀⠀⠉⠛⠓⠉⠉⠉⠁⠉⠙⠊⠁⠁⠂⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⢆⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡎⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⡀⠀⢏⡉⠁⣀⠜⠁
⠀⠀⠐⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡂⠀⠀⠀⠰⠀⢀⡉⠓⠀⠀⠀⠀⠀⠐⠒⠢
⠀⠀⠀⠀⠠⠇⠀⡀⠀⠀⠇⡀⠀⠈⠂⠚⠀⠀⠀⠀⠀⢯⠀⢀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢷⣄⠡⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⣐⢁⠀⠀⠀⠙⣣⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⡆
⠀⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠛⠯⠉⠑⠂⢊⠕⠉⠢⣄⠑⡔⣀⠀⠚⠁⠀⠐⠂⠈⠀⢀⡀⠁⢀⣴⣁⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠻⠿⠉⠀⠀⠀⠀⠈⠒⠀⠄⠈⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠐⠀⠐⣶⡈⠉⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠛⣻⡜⡂⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢠⣤⣤⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢶⠤⣉⣻⡐⢠⣤⣾⡿⠿⠿⠟⠛⢿⡛⠛⠉⠉⠉⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠉⠁⠚⠨⣷⡸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠩⡒⠦⣤⣘⣷⢹⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⢺⣿⣳⣤⡀⠀⠀⢤⣄⠺⠿⠟⣣⢷⡀⠠⠤⠶⠒⣒⣦⣼⠤⠤⠔⣒⣒⣛⡿⢃⠀⢀
⠀⠀⠀⠀⠈⠻⢿⣷⣤⠠⣂⠉⠀⠀⠀⠀⠨⠿⠿⡯⠭⠭⠽⠖⠒⠶⠛⠋⢍⠀⠀⠀⣀⠀⠀⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣮⣟⠦⠀⠀⠀⠂⠈⡤⢳⠀⢰⣞⢦⣄⠄⠀⠀⡀⠙⣦⠐⠁⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⢠⠄⢀⠄⠀⠈⠀⠈⠻⣷⣍⠂⠀⠀⡐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣿⠂⢸⠟⢠⣿⣄⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢀⠀⠀⠐⢆⠐⠀⢀⡍⠀⠀⢁⠔⠀⠀⠀⠘⠯⣽⠆⠀⠀⠀⢀⠀⢴⡦⠄⠀⠠⠆⠀⠀⠀⡀⠐⠻⢶
⠀⠀⠌⠙⢙⣿⣭⠇⠀⣿⡿⠿⢧⣠⡆⠀⠀⢄⠀⢀⢀⣸⡦⠀⡀⠸⠷⣶⣿⠿⢠⡿⠀⠀⠀⢰⡿⠉⠈⠉⢰
⠀⠀⠢⠄⣭⡵⢖⡃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⣷⠇⠀⠁⠀⠀⠛⠃⠀⠈⠛⠀⠤⠂⢸⡿⠷⡆⠀⢾
⠀⠀⠀⠀⠚⠋⠁⠀⢀⠈⠁⠀⠀⠀⣀⣴⣀⣀⡀⠀⠉⠉⠀⠀⠈⢀⡀⢤⡤⡄⠀⠲⠲⠶⠂⠸⣧⣤⣄⠀⠤
⠀⠀⠀⠀⣤⣥⣀⠂⠤⠂⠀⠀⠀⠞⠻⠮⠭⣯⣄⠀⠀⠀⠀⠀⠀⠒⠂⠘⠊⠙⠙⠛⠛⢋⡁⢀⣀⣀
⠀⠀⠀⠀⠸⣾⡽⣿⣏⣷⢦⡀⠀⠈⣁⣀⡒⠠⢿⠀⠀⠔⠋⠀⠑⢛⣿⡇⠀⠀⠀⠀⠠⠬⡥⠍⠉⢩⣥⢾⠃
⠀⠀⠀⠀⠀⠙⠛⠓⠻⠮⡾⠁⠀⠉⠒⠢⢈⣑⣶⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠐⢶⣤⠒⠄⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡪⢋⠻⣿⢄⡒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠶⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢤⡀⠀⠀⠀⠀⡎⢠⠋⢠⠃⠀⠀⢡⠘⢆⠃⠀⠀⠀⠀⠀⠀⡸⣤⣰⣀⣀⣠⡼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⣆⣊⠜⣣⠄⠀⠀⡼⠺⡤⠜⠠⠤⠥⠐⠂⠒⢳⡀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢠⠊⢤⠀⠀⠉⢳⠉⠀⠀⠀⠘⡃⠒⠰⠖⠒⠒⠒⠒⠐⠈⢁⠃⠀⠀⠀⠐⠯⠿⠶⡶⠶⠶⢶⠟⠋⡏
⠀⠀⠀⠀⠀⠀⠨⠀⠀⡀⠐⠆⡐⠀⠀⠀⠀⠀⠇⠀⢠⡄⠀⠀⠀⠠⠦⠀⡜⠀⠀⠀⠀⠀⠘⢄⠈⠁⠀⠀⠈⠀⡰
⠀⠀⠀⠀⢀⡔⡸⡦⣄⣀⠠⠔⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⡴⢀⠢⡀
⠀⠀⠀⠀⠀⠙⠺⢼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠑⠒⠒⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⡜⠁⠂⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠙⣿⡿⠿⠀⠀⡸⠁⠠⠆⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣀⡠⠏⢢⣀⠠⡀⠀⠀⠀⠀⠀⢀⠜⠂⠀⠒⢅⠀⠀⠀⠀⠢⣀⢄⣈⣡⣴⡾
⠀⠀⠀⠀⠀⠀⠐⢸⠒⠂⡙⠢⡀⠀⠀⠀⠀⠀⠸⣀⣄⠵⠘⠉⠀⠀⣸⡄⠀⠀⠀⠠⢁⠀⠲⠀⣀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢏⡉⠉⣉⣃⠀⠀⠀⠀⠀⢸⠁⠀⢀⡠⠔⠊⠁⠀⠈⡆⠀⠀⢸⡍⢧⠶⣠⡄⡠⠃⠀⠀⠀⡠⢔⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⠤⢈⡆⠀⠀⠀⠀⠀⢗⠉⠀⠀⠀⣄⠤⠒⠀⢡⠀⠀⣸⠻⡀⢶⡹⠋⠀⠀⠀⠀⠾⠄⣠⣏⣨⡋⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⠲⠇⠀⠀⠀⠀⠀⠘⠤⠐⠊⠁⠀⣀⠠⠔⠊⠀⢀⡿⣎⣱⠎⠀⠀⠀⠀⠀⢸⠤⠐⣆⡸⠷⢛⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⡀⠀⠈⠢⠔⠊⠁⠀⠀⠀⠀⠀⢸⢛⡽⠃⠀⠀⠀⠀⠀⠀⠘⣤⠋⡬⠓⠶⠻⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡀⢮⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠂⡀⣪⡸⠋⠶⠲⠤⠤⣄⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⡤⢦⡄⠀⠀⠀⠀⠈⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠙⠁⠀⠀⠀⠀⠀⠀⣀⣤⢆⠔⠒⠲⡠⣄
⠀⠀⠀⠀⠀⠀⣿⣇⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⠀⠀⢀⡀⠀⠀⠀⠀⡴⠍⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠂⠀⠻⣷⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣾⡇⠀⠀⠏⢈⡷⠀⠀⢸⠃⠀⠈⢡⠄⠛⢄⠉⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⣼⡟⠀⠀⣴⣳⢀⡴⠀⣠⡶⠀⢸⣿⠓⠒⠒⠒⠋⠀⠀⠀⢰⢄⠀⠀⢣⡀⠀⢀⠇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢀⣿⠃⢠⡄⢈⢇⣾⠁⢰⡿⠁⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡀⠀⠀⠉⠈⠁⠀⠀⢠⠏
⠀⠀⠀⠸⢤⣀⡀⠀⣀⣼⡿⠸⣿⣀⣼⣿⡌⢸⡏⠀⢼⡇⢀⠜⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣠⡀⣀⣀⡠⠞⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⠉⠀⠉⠀⠀⠁⠀⠀⠁⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣦⡀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣩⣭⣭⣭⣍⠛⠿⠿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣯⣽⣭⠜⣉⣤⣴⣶⣿⣿⣿⣿⠟⣋⣵⠾⠛⠛⠿⠿⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣯⡟⣡⡾⢋⣤⣾⣋⣿⡉⣳⢦⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣿⣿⢛⢉⡼⠋⣠⠞⢤⣼⣅⣀⣈⣿⠒⢳⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢋⣴⢃⡾⠁⡴⠓⠢⣴⡧⣀⣀⣈⣿⠛⠻⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢟⣵⡿⠇⣼⠃⣸⠷⠦⣴⣧⣀⣀⡀⢹⠟⠃⠲⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣱⣿⣿⡿⢰⡏⢀⡿⠤⢤⣼⣅⣀⠀⠉⡿⠓⠲⣴⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⣾⣿⣿⠟⠁⢸⡇⢸⣧⣄⣠⣏⠀⠀⠉⣿⠷⠶⢤⣼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⣿⠿⠋⠁⠀⠀⢸⡇⢸⣄⣀⣸⡋⠉⠁⠺⡿⠤⢤⣤⠏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠸⣇⢸⣉⠈⢹⠋⠉⠓⡾⠧⣤⣠⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⠿⠛⠲⣶⣦⣄⣠⠋⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⡖⠛⣦⠴⣧⣀⠙⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠳⣤⣧⣤⠞⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠁⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣩⣍⢳⡴⢛⡛⢦⠀⠀⠀⢀⣄⣰⠏⣰⢸⣇⣰⠋⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡘⢁⡏⡘⠁⢋⠍⣘⠓⠒⠚⠋⣸⡏⠐⢣⣿⣉⠁⢠⣴⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⣧⣬⡴⠁⠾⠶⢚⡻⡟⡛⠟⢫⠙⠀⡴⢋⠉⣹⢀⡟⣩⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡿⠟⡁⢀⠀⠄⠸⢡⠀⠷⡀⠬⣰⠀⣀⣉⠀⣡⣘⣃⠱⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠛⠚⠓⠛⢺⡟⠷⠾⠿⠛⠛⠀⠉⣤⣤⣶⠌⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠴⠖⠛⠉⠳⠦⡭⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡏⠁⠀⠀⢸⡇⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡇⣀⣀⣀⣈⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⡏⠉⠉⠀⢠⡏⠀⠀⠀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠇⠀⠀⠀⢸⠁⠀⠀⠀⢿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡹⣤⠤⠤⢴⡧⣤⣶⠶⢾⡦⠶⠶⠦⣾⢿⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢍⠙⠷⠀⠀⠀⠀⠉⠁⠐⢬⣄⣀⣀⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠒⠀⠐⠂⠤⠄⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡲⢦⣤⠴⣦⠤⠄⠠⠄⠐⠶⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣥⣍⡂⢈⣷⡤⠓⢂⣠⠄⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡪⣭⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢠⠏⣿⠀⠀⠉⠀⠀⠈⠣⡾⡑⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⡴⢀⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡟⠀⡴⠗⠐⠃⡀⢀⠐⡀⠀⠀⠀⠀⠀⠀⡴⠿⠛⠂⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠐⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡿⠁⢰⡃⣈⠀⠈⢀⠼⡆⠙⠈⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡀⢠⣄⠀⠀⢻⡷⠛
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠀⢰⡿⠁⠀⠘⡇⢠⡀⠀⣰⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠁⣀⣸⠦⣴⡆⠀⢘⠃⠈⠂
⠀⠀⠀⠀⠸⡄⣴⢀⣴⠏⢀⣶⠟⠁⠀⠀⠀⢧⢤⡤⠀⠺⠆⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡲⠗⢦⠃⠸⠛⠃⠀⠟⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠷⣷⣋⡤⠾⠛⠀⠀⠀⠀⠀⠀⢹⠊⢳⠀⠒⣲⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢆⡴⢁⠘⠁⠀⠀⠀⢈⠀⠀⠀⠀⠀⢁⠀⠀⠠
⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠹⣥⣏⠀⣽⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡀⠈⠀⠀⡀⢀⢩⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⡠⠀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠃⣀⢀⠀⠀⠉⠀⠀⠀⠈⠛⠺⡬⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠁⣤⡄⠤⣤⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠔⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⡧⢘⣀⡀
⠀⠀⠀⠀⠀⡈⠢⠀⠠⢀⣄⣀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⠀⣐⢁⡬⡄⢠⡀
⠀⠀⠀⠀⠀⣇⠀⢠⠟⢠⡤⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠀⡠⠀⡆
⠀⠀⠀⠀⠀⠹⣰⡟⣠⣟⣤⠞⠁⠀⠀⡀⠄⠈⠁⡄⠀⠄⠰⠀⠐⠀⠰⠄⢷⣾⡿⠦⣤⡀⢃⠀⠀⠀⠀⢰⠁⠶⢫⢤⡍⠘
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠊⠁⠁⠀⠀⠐⠀⠂⠀⠒⠈⠓⣈⣿⢟⠙⠂⢀⡄⢣⠈⣦⠀⠀⠀⠀⠒⠀⠈⠀⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠤⣠⡀⢀⡀⠀⠀⠀⠠⠤⠔⢒⣴⡿⠋⢑⠀⠀⢀⠋⠀⣸⠀⡻⠀⠈⠒⠄⠀⠀⠀⠀⠀⠈⢶⣄⢀⣴⠃
⠀⠀⠀⢠⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⠀⠴⠖⠔⠛⠟⢁⠀⠀⠀⠐⠂⠀⢠⡏⣴⠃⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠸⢺⡟⠁⠀⠀⠀⡄⠊
⠀⠀⠀⢿⠀⣷⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠀⢠⡼⠋⣴⡇⠀⠀⠀⠐⡃⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠈⣝⡄
⠀⠀⢠⡼⣄⣸⡷⠐⠛⠁⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⡤⢀⡥⢀⣸⠎⡡⠚⠛⠀⠀⠀⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⢈⣿⠋⡴⢂⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⣆⠀⠗⣉⣥⠊⠑⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠐⠊
⠀⠀⠈⠣⠠⡽⠀⠀⠂⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣉⣤⡶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠐⡄⠉⠠⠀⠠⠂⡀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠙⢉⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠐⢧⣈⣄⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠓⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠘⡀⠀⠀⠀⠃⠠⠀⠀⢠⡦⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠱⣀⠓⠾⠤⠀⠛⠁⣴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢩⠄⠀⠀⡀⠀⠃⠀⢾⠀⠀⡢⠌
⠀⠀⠀⠀⠀⠀⠙⣷⣤⠔⠻⡎⠉⠸⡆⠀⠀⢀⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢦⡭⡅⠀⠀⠀⠀⠢⠀⠰⡄⢬⠀⠀⠈⢖⡻⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣰⣀⠜⠃⠀⠀⠲⠦⣀⣤⠂⣠⠆⠀⠀⠀⠀⠀⠢⠴⠒⢂⣈⠀⠙⠒⠚⠀⠀⠀⠀⠄⢹⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠤⠤⠴⠂⠀⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠐⡀⢂⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠛⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠰⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠨⠀⠀⣤⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠂⠀⠀⡀
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠄⡀⠀⠀⠀⠀⠰⡱⠭⠆⠀⠀⠁
⠀⠀⠐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠁⠠⠀⠀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠃⠀⠀⠀⠀⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡠⢐⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢐⠀⠀⠀⡁⡣⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⢰⣿⡏⢀⠀⠀⠀⠑⠬⠅⠁
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡐⠂⡀⠀⠉⠀⠀⠀⠀⠂⠀⠈⡀⠀⠂⠀⢠⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠒⠚⠛⠛⠛⠒⠒⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⠚⢻⡟⢿⡓⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣷
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢧⣸⠇⠀⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⢸⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠶⠶⠖⠛⠛⠉⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠟
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣼⠉⠁⠉⢹⡇
⠀⠀⠀⢸⣿⠆⢠⠶⠤⣄⣄⠀⢀⠀⡔⠚⠛⡆⠀⠀⠀⢰⠉⡆⢸⠀⠀⠀⢆⢀⡴⣿⠀⠀⠀⣾
⠀⠀⠀⠀⢿⣆⡾⢠⠀⠈⠫⡁⢀⠤⡿⠀⠀⣧⠀⠀⠀⣸⠀⠙⠞⠀⠀⠀⠘⠦⠾⠃⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠁⠀⢧⠀⠀⠈⠓⠚⠁⠀⠀⠘⢦⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡤⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠶⠖⠒⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⣦⡀⠀⠀⢀⠈⠛
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠚⠛⠒⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⢦⣀⣀⡠⠢⣀⠀⢀⡠⠶⣄⣀⣠⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⢐⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⣶⣾⣷⣬⣭⣵⣶⣷⣶⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠋⠁⠀⠀⠀⢀⠤⠒⣒⡙⠹⢢⢀⠜⠨⢤⠂⠑⡄⠀⠀⠘⠿⠶⣤⣤⣀⣀⣀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⣼⣿⣿⣶⣤⠁⣠⣤⣴⡖⣖⠓⢦⠀⠀⠀⠀⠈⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠹⢿⣟⣿⡼⠀⢻⣿⣿⣿⠏⣰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⡆⢨⣡⠄⡗⣠⠈⢩⡁⣰⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠉⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠘⠋⠂⡄⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⢀⣌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⣃⠲⠈
⠀⠀⠀⠀⠀⠈⠒⠠⠒⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣀⣴⣤⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣤⣤⣴⠟⠓⠲⠶⢿⣿⠿⠟⠛⠁⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠛⠋⠁⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⢀⠉⠀⠀⠐⠃⠀⠂⠀⠐⠀⠀⠀⠀⠄⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢘⣉⣶⢻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠏⡀⣦⣾⢟⣽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡝⠉⣴⠇⠸⡆⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠆⢠⣾⣷⣾⡿⣿⣵⣾⡿⣿⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣅⠠⣝⢺⠀⣿⠀⠀⠀⣠⣾⢛⣏⣿⡟⣱⡟⠀⣿⣿⣿⠿⣾⢿⠿⣿⣿⣿⣿⣛⣷⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠐⣿⣭⠀⡏⠀⢠⡾⢋⣴⣟⣟⣽⡾⠟⢁⣠⣉⣵⢾⣟⡟⢷⢾⣧⣙⠛⢿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡇⣼⠶⠏⣠⢁⣠⣿⣴⣷⣿⡿⠟⠁⠀⣰⣡⣾⡿⠗⠲⢬⡻⣷⣶⢿⣿⢿⣯⡷⢄
⠀⠀⠀⠀⠀⡠⠚⢏⣛⣌⣳⣍⣉⣹⣓⣶⢲⣦⡋⢟⣂⣽⣭⡝⠋⢉⠉⠙⢶⣖⣶⡛⠛⢱⡏⠀⠀⠀⠀⠙⣌⠻⢿⣭⣛⣿⢿⣏⣳⡀
⠀⠀⠀⠀⠉⠛⠿⣿⠻⠿⣿⣿⣿⣶⣶⠿⠿⢯⡿⡿⢫⢿⠏⠀⠀⠀⢣⠀⠀⠙⣄⣭⣿⢿⣶⡅⠀⠹⠀⠄⠸⡄⠀⠉⠛⠻⠿⠿⠟⠓
⠀⠀⠀⢀⣤⣴⡶⢿⣿⣿⣶⣶⡿⠋⢀⡈⠐⡀⢻⡁⢸⣸⢠⠀⢆⠀⠈⣷⡀⠀⠘⣿⣿⢨⣬⡅⢀⠀⡀⢰⠀⣧⡄⢤⡀
⠀⠀⣰⢿⡽⣿⣷⣶⣶⣾⣿⡿⣤⠀⢰⠀⠀⡀⠀⢷⣾⣿⣾⠐⣿⡄⠀⠘⣧⠀⠀⢹⣿⣯⣽⡇⢠⠀⣇⢸⠀⣿⣿⣷⣀⡙⠲⣄
⠀⠈⠉⠉⠒⠳⠼⠥⠿⣿⣿⢯⣯⠀⣾⠀⠀⢷⠀⢸⣿⣿⣟⠖⢚⡧⡀⠀⠽⡷⠀⠈⣿⣿⣿⣇⢼⡃⡧⡸⢸⣿⣿⣿⣿⣿⣧⣄⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢻⣿⣗⢀⡯⠀⠀⣯⠀⢸⣿⣿⣿⣧⣀⣿⣅⠀⠘⣷⡄⠀⡏⢷⣿⣿⢸⣷⣷⠇⣽⠀⠙⠻⠿⢿⣿⣯⣶⣧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣽⡟⢻⣿⢀⡀⣷⠀⣾⣿⣿⣿⣟⣯⣻⣧⣀⠀⢹⣯⣀⡇⠈⢿⣿⣿⣿⢟⣼⣿⠕⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⣯⠴⢾⡷⢴⣿⣯⣿⣿⣿⣯⣿⣽⣗⡆⢸⣿⣿⠇⠀⠀⠻⣛⡟⢿⣿⣏⢷⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣽⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣾⣿⠏⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣿⣣⡿⣱⣿⣿⣿⣿⣿⣿⣿⠇⠈⠙⠛⠷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⢿⠞⠹⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⢿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠿⠛⠛⢲⣄⣄⠀⠀⠀⠀⡀⠀⠉⣹⣿⠇⣀⠀⠀⠈⢳⣶⡄
⠀⠀⠀⠀⣿⡀⢀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡞⠀⠀⣆⠀⣿⣷⢇⠀⠀⠘⢷⣷⠶⠟⠁⠀⠙⠳⠶⠶⠾⠛⢁⣀
⠀⠀⠀⠀⣿⡄⠀⠠⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣾⠃⠀⠀⢹⠀⢸⣾⡘⠀⠀⠀⠀⡀⠀⠀⠐⣿⣿⡆⠀⠀⠀⠁⠈⣸⡇
⠀⠀⠀⠀⠘⢿⣶⡀⠃⢀⠀⠁⠆⢀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠸⠀⢸⣧⡇⡇⠀⠀⠀⠷⣤⣤⡾⠿⠋⠁⠀⢠⡀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣧⣆⠁⠊⣀⠀⢊⡄⠀⠀⠀⠀⠈⠘⠁⠸⠀⠀⠀⠀⢸⡇⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠡⢆⡄⠠⢆⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡇⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣦⣈⠰⣦⠉⠔⠀⠀⠈⣆⣿⣤⠀⠀⢰⣇⣼⣿⡇⠀⠀⠀⠀⠀⠀⠐⢲⣦⢴⣾⣯⣁⣠⣿⡂⠁⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣬⣁⠸⠄⠀⠀⣿⡸⣿⣧⡀⢸⣿⣿⡿⠁⠀⠀⣀⡀⠐⠳⣶⡏⣀⣼⣿⣍⣤⣝⣿⣆⠉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠑⠀⠀⠘⢷⣿⣿⣇⣾⣿⡟⠀⢀⣀⠈⢉⣿⣶⣞⣫⣤⣴⣿⣿⢙⣿⣿⣻⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖⣐⠲⠶⠶⠶⣂⠴⠶⠶⠶⣌⠙⠛⠛⠉⠀⠀⠀⠈⠙⠻⢾⣷⣿⣯⣭⣽⣿⣿⠷⠾⠿⠿⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠤⣿⠤⠤⠤⠤⣽⠄⡤⠤⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠈⠀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠂⣿⠇⠀⠐⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡴⣯⢤⣤⡠⢤⣿⢤⣤⡤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⢰⣿⠀⠀⠐⢻⡯⡀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⠶⠆⠛⠷⠖⠰⠖⠿⠶⠖⢲⣔⠏⢻⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠀⢠⡴⠁⠀⠀⠀⠀⠀⠤⣴⣶⣤⣴⠒⢶⣖⡖⠚⠃⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠠⣶⣦⣾⣤⣔⣋⣤⣽⣏⠀⡀⠀⠛⠗⢦⣄⠰⣶⣤⠤⣤⣤⣀⣁⡀⠘⡃⣾⣿⣷⠄⠀⠀⣀⣠
⠀⠀⠲⣦⡀⢀⠀⠀⢠⣄⣸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣙⠶⣽⣶⠤⣌⡉⠉⠛⠂⠀⣤⠨⠿⠿⠛⠉⠉⠉⢉⣻⣶⣿⣿⣿⠆
⠀⠀⠜⠋⠻⠮⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠓⢄⠈⠳⢌⣽⠿⠶⠶⣤⣤⣀⡈⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⢀⡴⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠢⠀⠹⢦⣄⣠⣤⣴⣿⣻⣷⣶⣄⣸⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠑⠢⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣤⣤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣰⢀⡒⣎⠜⣁⠦⣤
⠀⠀⠀⠀⠀⣿⡂⠀⠀⢸⣿⣇⣀⡀⠀⣿⠀⠀⠀⠀⠀⢠⡿⠿⠿⠚⣿⣶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⡿⠿⠿⢿⣿⡾⢷⣦⣌⣔⠁⡸
⠀⠀⠀⠀⠀⣿⡿⠿⠿⢿⣿⡿⠿⠛⠛⣿⡀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣿⣿⣀⣀⣀⣀⡶⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⢸⣿⡇⠀⠀⠉⢹⡇⣘⠋
⠀⠀⠀⠀⠠⣿⣀⣀⣀⣸⣿⣇⣀⣀⣀⣿⠂⠆⠀⠀⠀⢠⡿⠿⠿⠛⣿⣷⡿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠐⣶⣿⠿⠿⠿⢿⡿⠛⠿⠿⢿⢿⡇⢉
⠀⠀⠀⠀⠈⠿⣿⣿⠿⣿⡏⢛⠛⠛⠛⠿⡡⢘⠂⠀⠀⠸⣇⣀⣀⣀⣿⣿⣀⣀⣀⣀⡿⠀⠀⠀⠀⠀⠐⣯⡵⠀⢀⡠⠸⡄⠀⠀⠀⠀⢠⠀⡶
⠀⠀⠀⠀⢠⣾⣿⣿⠿⠛⠀⠀⠀⠀⠠⠄⠳⡇⠀⠀⠀⢀⡿⠛⠛⠛⣿⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠘⠓⠀⠀⠀⠀⠀⣟⣰⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣿⣿⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⣿⣾⠟⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣿⣷⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢰⡟⠛⠋⠙⣿⣿⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠤⠀⢀⣴⣿⣅⡀⣸⠏
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢒⡒⠛⠙⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠿⠿⠤⠤⠤⠼⢇⣤⡴⠓⢄⠀⠀⠀⠁⠀⠀⢀⣼⣿⡟⠙⠻⣿⡏
⠀⠀⠀⠀⠀⣇⡀⠀⢀⡀⣀⣐⠤⢩⡿⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣿⠟⠁⣠⠼⠃⠀⠀⠢⠀⢴⣋⣾⡟⠀⣠⣰⠏⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡶⣾⣿⠏⠙⢾⣻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢤⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣿⡇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠒⡒⠀⠀⠈⡉⠩⠙⣿⡄⠀⠀⠈⠀⠄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢻⣆⠀⠀⠀⠁⠌⠐⡐⡡⠁⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢊⢆⠠⠀⠈⠌⠔⠔⡀⠁⠠⠘⢿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡄⢂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠘⡿⣄⠀⠀⢐⣒⣪⣥⣥⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠛⢿⡌⢆⢤⣤⢬⣭⣿⣶⣶⡶⠿⠽⠕⣎⡵⠛⠛⠉⠉⠉⠉⠉⠀⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢥⣄⡀⠀⠀⢱⡈⢆⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⠚⠼⠈⢹⠐⠪⣷⠈⢜⡄⠀⠀⠀⡀⠁⠠⠀⠂⠀⠘⣽⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢉⡗⡦⢤⣀⡘⠃⣦⣧⠈⡿⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⡠⠤⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠇⢨⠙⢲⣿⢧⠸⣄⠀⠀⢀⣀⣠⡄⢠⣤⢤⣄⢸⠓⠀⣀⣉⣉⠥⠤⠤⠴⠶⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠈⠉⢀⠀⢣⠹⣄⣀⣀⣀⣤⠭⠄⠖⠒⠒⠚⣉⣉⣉⡤⠤⠤⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢻⢷⠀⢷⣯⠭⠤⠤⠶⠒⠒⠚⢋⡩⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠺⠑⠀⢿⣗⣤⡑⡀

⠛⠻⠿⠿⠿⠿⠿⠿⣶⣬⣉⡉⠒⠶⢯⣁⣀⡀⣼⢃⣿⣼⣿⣆⣀⢀⡀⢈⣿⡏⠀⢹⡇⢸⣇⠀⠀⢀⣤⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠁⠀⣀⣶⢟
⣯⣬⣷⠶⢦⣀⠀⠀⠈⠉⠉⠿⣶⣤⡆⠀⠉⠳⠦⡜⠇⣻⠛⠉⠉⠙⠛⣿⣿⠿⠷⣿⡇⢸⡏⠀⠀⣾⣟⠁⠘⠁⢀⣠⣴⣶⣶⣶⣤⣐⠾⣿⣇⣀⣤⣾⣏⡶⠋
⠛⢙⡿⣾⠛⢉⣉⣦⣄⠀⠀⠀⠀⠉⠙⠓⠋⣄⡠⣀⡀⢙⣷⡀⠀⠀⢀⣿⡟⠀⢀⡿⢀⡞⠁⠀⠀⠈⠀⢀⣠⡾⠿⠿⠟⠿⠟⠛⠻⣿⣿⣎⢻⣿⣿⠟⠋
⣆⣸⠛⣿⣿⣿⣏⣀⣈⣷⣤⡀⠀⠀⠀⠀⠀⢹⡇⢻⣀⣿⡍⠙⠚⠻⢿⡿⠛⣷⡿⢁⡾⣴⠃⠀⠀⢀⣰⣿⡟⠿⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡇⣿⢃⣠⠐⣒⣶
⣿⣿⠀⡟⠙⣽⣿⣿⣭⣤⣴⠿⣦⡀⠀⠀⠀⠘⢻⡘⡟⣿⣷⣦⣤⣴⣿⣅⡼⢋⣴⡯⡾⠃⠀⠀⣠⣿⣛⣩⣥⣤⣤⣶⣶⣿⣿⣶⣤⡀⣿⣿⢇⣿⠈⠛⠛⠛⠋
⡿⢿⣿⡇⠀⡏⢹⣿⣿⣿⣧⣤⡴⢿⣆⠀⠀⠀⢸⣇⠿⣿⣆⢀⣨⣿⣹⢋⣴⣿⠿⠋⠀⠀⠀⣼⡟⣿⠽⣿⠥⣶⠛⠛⢿⣤⡿⠋⢻⣷⣿⢏⣾⡿⠀⠀⠀⠀⢸
⣿⡷⣿⣿⣼⡆⢸⠁⢹⣿⣿⣿⣷⣤⡼⣧⡀⠀⠀⠻⣆⣿⡉⣹⣿⡟⣵⢋⠎⠁⠀⠀⠀⢀⣾⡷⣾⣯⣴⣇⣠⣿⠷⠒⡛⠋⠀⠀⣸⣿⢏⡼⠃⠀⠀⠀⠀⢀
⣿⣿⣾⣿⣿⣿⣿⣄⣸⠀⢸⠿⣿⣿⣦⣤⣷⡀⠀⠀⠙⢞⢿⣿⣟⣾⠗⠁⠀⠀⠀⠀⠀⣾⡿⢶⡏⣰⠟⠉⣹⡟⠰⠋⣿⡇⠀⢠⡿⢣⣾⡇⠀⠀⢀⣰⠛⠉
⣿⠋⠉⣿⡽⢯⡻⣿⣿⣤⡏⠀⣸⠯⣻⣷⢬⣧⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⠻⠛⣙⣋⡁⠀⠱⠞⠋⢀⡾⢋⣴⡿⠋⣠⣶⣿⠋⢉⣶⣴⣶
⣸⣶⠞⠉⠹⣮⠻⣿⣿⣿⣿⣶⣯⣠⡟⢿⣦⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⣮⣠⡾⠋⢀⣀⣴⠞⣫⠶⠛⠉⠀⣾⠟⠈⠙⡛⠟⠛⢿⣿
⠙⣷⣤⡤⠞⠻⣧⠹⣿⣿⣿⣿⣿⣿⣧⣼⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⢣⡻⣿⣶⣷⣦⣀⡀⠀⠜
⡶⢿⣏⠀⠀⠀⣸⣇⢻⡏⠻⣧⠀⠉⠙⠛⢷⣌⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠶⣝⣛⡿⣿⣿⣿⣿
⣄⣠⣿⣶⠶⠛⠛⢻⣆⣿⡆⢨⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣨⣿⣿⣿⣿⣷⣶
⡹⡇⠈⣿⣤⣤⠴⠾⢿⣽⣷⣼⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣯⠃
⣷⡝⢾⡟⢿⣄⡀⣀⣼⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⣶⣆⠀⠀⠛⠛⠛⠛
⣯⡻⣮⣷⡿⠻⣟⠉⢉⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠐⠢⠀⠂⠀⠂⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⠿⠿⠛⠁⠀⠀⣀⣠⣤⣤
⠹⣷⣮⠻⣾⣶⣿⣿⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⣶⡦⠀⢠⣀⡴⠛⠉⠁
⠀⠈⠫⢷⣌⠻⢮⣝⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢀⣴⣿⣿⠀⠀⠀⢀⡀
⠤⢤⣀⡀⠉⠛⠺⢿⣷⣼⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣴⣿⡿⢿⡦⠴⣦⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⢻⣭⣿⣿⣋⣹⣧⠼⣏⡙⡟⢷⣿⣲⠶⡄⠀⠀⠀⠀⠀⢠⣎⣤⣾⣿⡟⠋⠉⠈⢙⣛⣿⣿
⠋⠉⠀⠀⣠⠖⠋⣽⠁⠀⠀⠀⠀⠀⢠⣔⠶⡶⣦⠀⢀⣾⠃⢠⣾⣿⡕⠛⣯⡽⣧⣸⣍⡟⠙⣿⠾⣦⠿⢾⢣⡀⠀⠀⠀⠀⠸⣏⣉⣿⣿⣧⣤⣤⣶⣾⣿⠟⣭
⣶⣶⣾⣿⣥⣴⡾⠃⠀⠀⠀⠀⠀⠀⠈⠙⠿⠓⠋⢀⣾⡿⢠⣿⡿⢾⣧⣼⣿⢦⣿⠉⢻⣿⡟⢹⡟⢿⡖⢾⡙⣯⡆⠀⠀⠀⠀⠙⢿⣿⣯⣙⣿⣿⠛⢁⣀⣀
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣶⣀⣐⣶⡄⠀⠀⠀⢸⣿⣿⡄⢿⣦⣾⠻⣦⣼⠟⣿⡷⢾⣄⣹⡿⣷⣤⣟⣻⡟⢿⣀⣦⣄⠀⠀⠀⠀⠀⠘⠻⣿⠛⠉⠉⠉⢻⣿
⣾⡟⠋⠀⡿⠁⠀⠀⢀⣴⣀⣨⣿⠟⠛⠁⠀⠀⠀⢸⣿⣿⣿⣆⠙⢿⣄⣸⣧⣾⣧⣤⣴⡏⢸⣷⡋⢙⣿⠉⢳⡾⢿⣹⣯⠳⣄⠀⠀⠀⠀⣼⠟⠋⠙⣻⣿⣿⣿
⣉⣤⠖⣋⠀⢤⣀⡀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣄⠘⠻⣯⣁⣽⡋⢹⡟⠋⢉⣻⣏⠙⣿⡏⠓⣾⣋⣿⣇⠘⡆⠀⠀⣼⠟⠒⠚⣿⡿⣿⣿⣿
⠉⠀⢾⡿⠋⣻⣧⣌⢣⣀⡀⢀⣴⣖⣀⣰⣦⠀⠸⣿⣿⣛⣿⣿⣿⠛⠓⠤⠀⠙⠛⠷⢾⣿⣿⠛⢻⡟⢷⣿⢿⣶⣿⣿⣿⠟⢀⣿⠀⢸⣥⣤⣴⣾⣫⣾⣿⣿⡟
⠖⢛⣿⣷⠶⢿⣯⣸⣿⣟⠳⣦⡬⠟⠛⠋⠁⠀⢠⣿⣿⢿⣶⣝⠻⠿⣦⣄⡀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣾⣿⣾⣿⣿⡟⠀⣀⣾⡇⢀⣋⣀⣴⣿⣿⢿⣿⣼⡿⢀
⡶⢿⣿⣦⡤⠤⣿⣏⣀⣨⣷⣌⡶⠦⢄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠀⠀⠈⠉⢠⡾⠿⣿⡇⠾⢛⣿⣿⣿⡁⠀⣽⣿⣇⣈
⣤⣼⣿⣷⣤⣤⡼⠟⠋⣿⣿⠟⢲⣖⣯⣥⣠⡄⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣤⣴⠟⢰⣦⡾⠋⣀⣽⣧⣴⣿⣿⡿⠟
⣰⣿⣥⣉⣑⣾⣤⣄⣼⣟⣁⣀⣮⠛⣿⡏⢛⣁⣾⣿⣿⣿⣏⠁⣀⡙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⡟⠀⠸⠛⠂⠉⠉⠉⠉⠈⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠄⡠⣤⣤⠀⠀⠀⠀⠀⠀⠄⠶⠶⠛⠛⠉⠉⡉⢉⡽⢣⣤⠒⢆⠀⢀⡖⢢⣄
⠀⠀⠀⠀⠀⠰⡈⠐⢄⣀⠀⢁⡤⠒⠀⢉⡟⠁⡀⠀⢀⣀⠁⠤⠄⠠⢤⣄⣲⣦⡵⠞⠋⣠⠟⣈⠆⠐⡄⡔⠀⢎⡘⢷⡀
⠀⠀⠀⠀⠀⡆⠈⠑⠂⠂⠤⠉⠀⢀⡴⠋⠀⠀⢡⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢠⡟⠂⠀⢋⠖⠁⠑⢎⠊⠀⠚⣧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠚⠓⢮⠀⢈⠖⠛⠂⠀⢸
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠊⠉⠀⠀⠀⠈⠉⠂⠀⠘⡇
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⡆⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡆⣇
⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⣉⢒⠂⠀⠀⠐⠒⠒⠛⠛⠉⠉⢿⡁⠀⠀⠀⢸⠀⢸⠀⠀⠀⠐⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣠⡤⣶⣯⣶⢿⡷⢿⣿⢿⣺⠟⢋⡩⠭⠝⢛⣓⣢⠉⠛⠒⠂⠘⠀⠘⠂⠒⠒⠋⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⠞⣫⠾⣭⢴⡭⢖⢭⣺⣧⣴⠋⢉⣠⡏⢳⡀⠀⢠⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡟⢸⠁⢰⠁⣰⢁⠎⠀⠀⠀⢉⠉⠀⢀⠔⠉⠉⢧⢳⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠃⡾⢀⡇⠀⢀⡏⢰⠀⠎⢰⡆⠠⣸⡿⢸⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⢷
⠀⡼⠁⠀⠀⠀⠠⠤⠄⠒⠁⣀⠔⠁⡿⣇⡀⠀⠀⠀⠀⣧⡼⠀⠀⢸⠀⡇⠀⠀⠘⡇⢀⡜⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⠀⠔⢈⣀⠀⠀⠀⠈⠉⠉⠀⣂⠜⠁⠈⠙⠓⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⠴⠶⠗⠋⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘
⣼⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠴⠤⠴⢖⣒⠒⠒⠦⣄⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡠
⣿⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⠀⡐⣳⣤⠗⠚⠛⠓⠚⢭⣲⢞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠐
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠜⠀⠀⢰⡯⠗⠚⠉⠚⠿⠭⣙⠢⡀⠑⡙⡆
⠈⠓⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣾⠊⠀⢀⠤⠒⠒⠦⣌⠑⣜⣆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠈⠳⡜⡜⡖⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⢳⡦⠙⠁
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡟⠙⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⠖⠑⠀⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠉⠀⠟⠀⠃⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠋⣤⠄⡆⠀⠀⢀⠀⠓⠠⠀⠛⢱⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⠀⣳⠂⠃⠀⣄⠐⠁⣀⠙⢦⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠻⠟⡀⠛⠀⠀⠀⠉⠀⠀⢉⡶⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠴⢾⡷⠃⠀⡀⡀⠀⠃⣴⠐⠀⠁⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡷⠁⠀⡤⠀⢠⡀⠁⠈⠀⠄⠁⠠⡦⣾⢧⡆⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡴⣧⣄⢠⣶⠄⠀⣿⢀⠀⠀⢀⠀⢤⡄⠀⠀⢾⠇⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡈⠉⡌⠁⠀⠀⢀⠈⢞⠗⢁⣀⠀⡀⢀⡄⠀⠀⠀⠀⢆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠛⠋⠁⠀⠀⠆⠠⠶⡜⢀⡀⢰⠎⠛⠂⠉⠀⡀⠀⢀⠈⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢸⡆⠁⠀⠀⡠⠞⠒⠘⠳⠶⠾⣀⡀⠘⠈⠀⠠⡌⠓⠢⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠂⢀⠊⣠⣾⠏⠶⢸⣦⠐⣮⣀⠈⣾⠆⡈⠠⠴⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠂⠠⠄⠲⣿⠰⠈⣥⣰⣀⡌⠁⠖⠉⠁⠀⠀⠻⠃⡐⢃⡄⡀⠂⡀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠡⣶⠂⢫⡶⠀⠙⣆⢀⠀⠟⡡⣴⡶⠁⣼⠇⣀⡻⠟⠱⠿⣯⣂⡐⠓⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⠆⠴⢫⡍⡑⠂⢈⡉⢈⡑⢤⣎⠴⠚⠓⣿⡐⠓⠻⠤⢴⠐⠃⢬⠍⠁⡳⠂⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣌⠁⠀⠀⠀⢰⣄⠀⠀⠘⣃⡘⠿⠇⠉⠀⠙⡃⣠⣦⠸⠇⠀⣦⡈⢳⠂⢐⠋⢢⣄⠈⣠⣭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠘⣷⠀⠈⡉⠁⢀⣘⣿⠇⠐⠚⡛⠃⠛⠂⠘⠤⠄⠀⠆⠛⣵⡎⢡⡾⢠⡍⣀⡀⠹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⣠⡌⠈⠡⠦⠀⣩⣄⠙⣠⣤⠀⠠⣦⣄⢢⣛⡤⠀⢀⣈⠀⠀⠀⢀⠈⠟⠣⠤⠄⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠇⣛⠋⠈⣿⠀⠀⠈⠉⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢿⡛⠃⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠄⠈⣡⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⠞⠱⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠲⠀⠀⠀⠂⠂⠂⠂⠀⠀⠂⠀⠂⠐⠐⠐⠀⠀⠀⠀⠀⠀⠈⠀⠁⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠀⠒⠒⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠈⣿⡟⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⣀⣠⡤⠔⣛⠛⠤⣌⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡰⠀⢀⡠⠒⠈⠁⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠈⠲⢤⡴⣫⢷⡿⠛⠛⡻⠿⣎⢣⡀⠙⢿⣶⣤⣤⣀⣤⣴⣭⣤⡴⠷⠾⠓⠈⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠐⣒⠚⡿⢫⡾⢓⡻⠃⠀⢙⠗⠄⠘⣷⡙⣄⣠⣿⠦⢌⣀⠀⠉⠢⡀⠀⠁⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠋⢤⡉⠀⣈⡟⠃⢀⣹⠳⣤⢻⠀⠈⠆⠀⢋⠀⢤⣄⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⣸⣗⡐⡬⠀⣀⣽⠛⠂⣀⣯⠱⡿⢸⠀⠀⠸⠀⠈⡆⠀⢃⢀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢤⡹⠄⠀⣤⣏⠁⠠⡜⠢⣽⠇⣾⠀⠀⠀⠀⠀⢹⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟⢦⡔⠀⢠⣮⡀⠐⣶⣉⣻⠏⣰⠙⠀⠀⠀⢰⠀⠈⡀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣟⠲⡉⠀⠶⠄⢀⡽⠦⣾⠏⣰⠃⠀⠃⠀⠀⠸⠀⠀⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣳⠄⠸⠓⢠⠞⣒⠞⣡⠾⡀⠠⠀⠈⡀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⡽⢀⡋⣠⢯⠝⣣⠞⠁⠀⠑⡄⠁⠀⢣⠀⠀⡇⠀⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣷⣬⡴⢛⡴⠚⠁⠀⠀⠀⠀⠈⢆⠀⠀⢆⠀⠀⠀⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠄⠘⢠⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⠟⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣟⣛⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠳⢤⣀⣀⣠⣴⢿⣀⢀⣰⣤⣤⠴⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢄⡀⠀⠈⠉⠁⠀⢢⣭⠟⠁⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠛⢶⣀⠀⢀⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠙⣷⣟⡁⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠞⠁⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡍⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣶⣄⢄⠀⠀⡀⢀⣵⡿⠃⠀⠀⠀⠀⠀⡙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣠⡾⠷⣶⠟⠒⠳⣦⡀⠀⢀⣴⠿⠁⠀⠀⠀⠀⠀⠙⢿⣦⣀⣵⡿⠋⠱⠀⠀⠀⠀⠀⢀⣼⣾⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢉⣁⣀⠀⠤⢤⡄⠀⣎⠛⢿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⡿⢿⣷⡀⠀⠀⠀⠐⠀⣸⡾⢣⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡛⠋⠉⠉⢛⣦⠤⠲⠄⡼⠃⠈⣿⠿⢷⡄⠀⠀⠀⠀⠐⣤⢾⠓⠀⠀⠙⠳⣼⡆⠀⢀⣾⣏⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢡⡆⠀⠀⠀⠀⠀⣻⢧⣤⡴⠞⠻⡳⣼⣀⠀⢻⢷⠶⢤⣤⣾⣝⠅⠀⡐⠄⠀⠀⠘⠟⣶⣟⣿⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⣏⠀⠉⠀⠈⠄⢸⠛⠿⣏⣀⣀⣙⡆⠀⠀⠙⠶⠶⣶⡄⠀⠀⣠⣾⣝⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡞⠀⠀⢸⡄⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡜⠀⠀⠈⠉⠙⣿⠀⠋⢠⠶⠆⠀⠘⢷⣌⣼⣿⣷⠟⠁
⠀⠀⠀⠀⠀⣠⡞⠁⠀⢠⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣾⠃⣴⢶⡄⠀⣠⣄⠀⠀⠻⢟⣿⢿⡄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠞⠁⠈⠳⡌⠁⠈⣀⡀⠀⣀⡀⠈⠻⣆
⢀⣠⡾⠋⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⣾⣿⣷⣿⠘⡉⠋⠀⢹⣷⣤⣼⠃
⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣿⡏⠀⠰⣿⡉⢸⣿⠿⠁
⠙⢷⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣠⡾⠉⠄⢠⡼⠛⣧⣀⣴⠿⢷⠾⠁
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣠⠾⡃⠀⢀⣴⠟⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠘⡄⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⣴⠋⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠀⠀⠀⠀⠀⠀⠀⠑⣠⡞⠑⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣠⡾⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⡾⠃

⠀⠀⠀⡾⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠼⢿⣦
⠀⠀⠀⠉⠳⢤⣆⣰⠐⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣠⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠃⠓⠀⣸⠏
⠀⠀⠀⠀⠀⢀⠽⠋⡱⠛⣻⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⠿⣿⠛⠈
⠀⠀⠀⠀⠀⠀⠤⠀⢰⣲⣶⡦⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡭⣞⠿⠁⠀⣇
⠀⠀⠀⠀⠀⠀⡠⠻⢋⡵⣿⢦⡀⠀⠀⠀⠠⠈⠰⠀⡠⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⡞⢡⢤⡀⠀⠀⠀⠀⢰⠚⠳⡄
⠀⠀⠀⠀⠀⢺⣏⠙⣁⠤⠋⣲⣇⠀⠀⠀⠀⠈⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡼⡆⠀⣀⡇⠀⠀⠀⠀⢈⡀⣠⠁
⠀⠀⠀⠐⡻⢓⢊⡏⡉⠎⠛⠏⠁⠀⠀⠀⠘⡜⠁⠀⢢⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⢯⢯⣰⡿⣦⣶⣲⡥⠚⡇
⠀⠀⢀⣰⠧⠱⠪⢆⡀⠄⣍⢶⡅⠀⠀⠀⠀⠇⠀⠒⠀⠀⠀⠐⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢰⢮⣋⢦⡏⢐⢢⣧⡐⠁
⠀⠀⡊⠃⢚⡐⠳⠈⢱⠤⠣⢭⣁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠈⠁⠈⠀⢄⠠⠀⠒⠀⠀⠈⠙⠤⠜⠛⡹⣝⡫⣿⢟⡷
⠀⠀⢫⠫⠇⡀⣋⠘⢈⠃⠇⡘⢁⠀⠀⠀⠀⠘⡀⠘⡖⠀⠄⡱⠀⠐⠋⠁⠀⢀⡄⠀⠀⠀⢀⣠⣄⡀⠘⠋⣿⣻⣛⣿⡆
⠀⠀⡞⠒⢂⠉⡅⢤⢈⠧⢸⠀⠃⠁⠀⠀⠀⠀⠑⢀⠓⢦⢐⡀⣢⠀⠀⠀⠉⠉⠀⠀⠁⠀⡟⠀⠀⣷⠖⠲⠟⣹⡳⠋⠁⡀
⠀⠐⠙⡧⠬⠂⡑⣄⡼⠐⠞⢀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢈⠁⠉⢂⠀⠀⠀⠁⠁⢈⠂⠀⠀⢳⣀⠀⠘⠦⡤⢔⡵⣟⠁⡰
⠀⠀⠀⠇⢠⣠⣁⠎⡙⠃⡇⠃⠀⠀⠀⠀⠀⠀⠀⢁⠠⠈⠠⢄⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠋⠀⠀⠉
⠀⠀⠀⠀⠈⠀⠁⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⠀⡀⢀⠀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⣿⣿⢛⠿⢛⠛⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠉⠉⠀⢠⡄⠀⠀⠉⠉
⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⢇⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⣇⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠎⠀⢀
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁
⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠘⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⣀⠆
⠂⠀⢐⠘⠙⢢⠄⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢢⠆⠀⢘⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠓⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉
⠐⠄⢄⠀⠀⠀⠀⠀⠀⡬⠔⣃⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠈⡭⠔⣂⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠆⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠊⠀⢀⠀⠄
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠊⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠄⠀⢀⠆
⠀⠀⢈⠘⠙⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢀⠄⠀⢐⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠑⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⡀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠇⡀⠀⠀⢀⠀⣠
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⢉⣉⣤⠴⣲⣿⡄⠀⠀⠀⠀⠀⠰⠲⣇⣯⠿⣟⡾⢁⡆⠀⠀⠀⠀⠠⢔⡇⠂⡀⢠⠃⣉
⠀⠀⠀⠀⠠⠂⠀⡀⠔⠛⠁⣠⣴⢚⣵⠟⠀⠀⠀⢠⠊⣅⢲⣻⡭⠶⣛⢁⣴⡞⠀⠀⠀⠀⠘⠘⣿⣇⡜⢿⠬⡽⠇
⠀⠀⠀⠀⠀⣐⣉⣠⡤⠶⣟⣽⣾⠟⠁⠀⠀⢀⣠⣈⣉⣉⣭⣥⣶⣿⡿⠟⠋⠀⠀⠀⠀⠐⠾⣺⣿⡏⠛⢳⣶⠻
⠀⠀⠀⠈⠙⠛⠛⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⣴⣿⡿⠙⢓⣶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠿⠟⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⡾⠀⢀⠉⡨⠆⡋⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⣀⡒⠐⣦⣀⡀⣄⣀⣈⣉⣽⣇⡖⢿⡧⣻⢿⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠂⠈⣀⣤⠤⠴⡶⠴⠶⠒⡻⠿⢿⡓⢛⣶⡶⠿⠁
⠀⠀⣄⣁⣒⣉⣁⣀⠄⢠⠤⢖⢺⣽⢿⣓⠟⢈⣷⠋⠉⠁⠀⠀⢠⡞⢩⡷
⠀⠀⠀⠉⠛⠿⢟⠥⣂⠠⡟⣎⣳⠼⣛⢥⣴⡟⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢠⣤⣤⠀⣠⠤⠄⣤⣤⡄⢠⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠙⣓⣋⣥⣔⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⢹⢪⣿⠀⢸⡇⠈⡇⣿⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢀⡦⢻⣀⡾⠘⣿⠀⡸⣇⣰⠇⢻⣤⢾⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⢠⠤⠤⢤⣄⡀⢐⠋⠒⠐⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡔⢱⠤⣀⡴⡁⠈⠀⠒⡦⠉⠙⢎⣢⠇⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠆⣰⠚⢍⢍⠙⡉⠉⡝⣧⡷⠉⡝
⠀⡹⠐⠉⠉⠐⣼⡦⠀⠉⢠⠀⢺⢻⣅⠄⠒⢊⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠇⠘⠒⢆⡀⢸⡜⠀⣇⠀⡁⣿⡦⣀⠸
⡎⠀⢀⣤⠒⠪⢀⣁⠤⠀⠄⢁⢁⡼⢏⠙⢀⠇⠉⠀⠀⠀⣀⡀⠀⡀⢀⣀⡀⠀⠀⠸⠴⡆⣜⠷⠎⠢⠼⠴⠤⠿⠔⠓⠃⠘⠃
⠈⠐⠉⠀⠱⠤⠜⠣⢇⣒⣱⡥⠚⠀⠀⠉⠉⠀⠀⢀⡒⠃⠄⢈⡶⡶⠭⣄⠬⣙⡆⠀⠘⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠒⠂⠀⠖⠀⠒⠗⠊⢻
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⡇⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡠⠤⣀
⠀⠀⠀⠀⢀⢠⠔⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠈⠠⢀⡇⣀⣀⣇⣀⣀⡧⠔⠃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠅⠀⠀⢰⣉⡈⠣
⠀⠀⠀⠀⣀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠠⠂⠠⣈⠆⠀⠀⢆⠃
⠀⠀⠀⠀⢘⠢⡀⠑⠤⣀⠀⠀⠀⣠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⢣⠠⡀⠀⠠⠀⣀⠹⢀⠁
⠀⠀⠀⠀⠈⢦⡈⠑⠂⠤⢈⣉⣉⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⠖⠂⠈⠁⠒⢢⡄⠀⠀⠑⠆⡈⣉⢀⣤⠴⡃⢤⡀
⠀⠀⠀⠀⠀⠀⡝⠒⡤⠤⠠⠴⠊⠀⠀⠀⠀⠀⢀⠤⢦⣊⠀⠀⠈⠀⠱⠂⠀⣀⣀⠀⢀⠄⠂⠁⢰⠀⠀⠀⠀⠀⢠⠩⠤⡰⠹⡉
⠀⠀⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠓⠒⠂⠬⢲⡤⡀⠀⠀⠀⢀⠀⡠⠊⡄⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⡀
⠀⠀⠀⠀⢠⠏⡰⠁⠀⠀⠀⠀⠀⣠⠖⢉⠀⠀⠀⠀⢀⠔⠀⠊⠉⠑⠒⡶⡉⠓⡄⠀⡧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠱
⠀⠀⠀⢠⠏⢠⠃⠀⠀⠀⠀⠀⠀⢈⠑⠢⢅⠂⠠⢴⠐⠓⢢⠤⠀⠴⡄⠀⠉⠀⡶⠈⠀⠀⠀⢀⡠⠖⠋⠉⠙⠙⠢⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢸⠄⠄⣠⠀⠀⢼⡠⠄⠤⣇⠠⠂⠉⠁⠀⠀⠀⣠⠋⠀⠀⢀⡤⢤⣀⣀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢐⣀⡜⢁⡤⠄⣻⠀⠊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣧⠈⠦⡀⠀⠀⠀⠀⠀⠈⣆⠀⠸⡀⢀⡦⠔⢋⢆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⢍⠡⠀⠉⣧⡤⠀⡆⠀⠀⠀⠀⠀⠀⠈⠣⣌⡙⢧⣀⣀⣼⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠒⠘⠃⢉⠢⠀⠀⠀⢃⢁⡠⠛⠳⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠠⡈⢆
⠀⠀⠀⠠⠠⣤⣀⠀⡔⠉⠭⡉⠢⡐⡀⠀⠀⣂⣣⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠳⡀
⠀⠀⠀⢠⢄⡘⢄⣉⡉⡠⠀⠁⠀⣸⠼⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡆
⠀⠀⠀⣼⠀⢠⢦⠤⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠦⢄⡀⠈⠁⠀⢀⠤⠒⠊⠁⠀⠀⠀⠤⢐⣂⡭⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⡒⠢⢅⡀⠀⢀⣄⡤⠄⢂⣀⡠⠒⠛⠙⠙⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣶⣤⡈⠀⠀⢾⠀⡴⠒⠋⢉⠜⢀⣰⣦⡤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢟⠻⣶⣤⣀⢰⣆⣀⡤⠟⠒⠉⠀⠀⠀⢀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⢲⡸⡧⡈⠂⢜⠉⠉⠁⠀⠀⣀⣀⣠⠬⠭⣽⠓⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠈⠻⡁⠈⠶⡔⠒⠋⠙⠋⠉⢹⠞⣠⠆⠰⠁⠀⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠤⠾⡗⠉⣹⠠⡑⢄⠃⡇⠠⠀⠀⣀⣤⠯⠞⠃⠒⠓⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢉⣟⡶⠀⡐⠀⠈⠱⣄⢻⠆⠉⠉⠉⠉⠀⠠⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠘⠋⠀⠀⠀⠀⠀⠚⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⠂⠈⢉⡀⣾⣤⣶⣿⡶⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣴⣿⠯⠍⠩⢤⢐⣮⡟⣉⣛⣿⣿⣿⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⡠⢿⣦⣌⠉⢻⡿⣦⠙⠣⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡀⡀⠀⢀⡠⠞⠉⣴⠏⠙⣿⣧⠈⠉⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡜⣁⠔⠋⣀⡴⠞⠁⠀⠀⢹⣿⡆⢧⠀⢠⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠗⠚⠉⠁⠀⠀⠀⠀⠀⢈⣿⡇⡘⠀⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⣟⠿⣦⠀⠀⢀⣤⣤⣄⣀⣸⠻⣧⣷⡄⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡏⠀⠉⠁⠀⣸⡟⠙⠿⢓⠄⢀⣿⣿⣿⡆⢿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢿⣿⣻⡟⡇⢸⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡙⣧⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⣸⣿⣽⣷⠣⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⡘⢷⡈⢆⠒⠍⠍⢀⠀⠀⠀⠒⠉⠁⠘⣼⣿⠀⣸⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠻⡎⠆⠀⠣⠔⠊⠉⠀⠀⠀⣠⡼⣿⢏⡇⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣭⡻⠆⠘⠀⠀⠀⠀⠀⢀⣶⣿⣵⢞⣁⣾⣷⡏⣿⢼⡟⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣭⡛⠃⠀⠀⠸⣿⠀⠀⠸⣿⣿⡿⢫⡾⣿⣿⢣⠿⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⡍⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⣷⣾⢟⣼⣿⠃⠈⠸⢸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⡶⠞⠉⠀⠀⠀⠀⠀⠀⢸⣯⣥⡾⠛⠁⠀⠀⢀⢸⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⡄⠀⠀⡀⠀⠀⠀⠀⣸⣿⣏⠁⠀⠀⡰⡠⡘⣸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣘⠿⣷⣶⣶⣶⣿⣿⣧⢀⣷⢇⣇⡇⣷⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠉⠙⠛⠛⢸⡿⠸⣿⡇⠹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣶⣶⣶⣶⠶⠾⠿⠛⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣜⣿⣎⢿⣿⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⡴⣡⣴⣾⣧⡀⠀⠀⠀⠀⠀⠀⢿⣿⣭⣍⣙⡻⢿⣿⣿⣿⣿⣎⣿⣿⡄
⠀⠀⠀⠀⣠⣾⡿⢿⣛⣯⡷⣫⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡷⠬⠛⣻⣿⣿⣯⣿⠃
⠀⠀⠀⠰⣻⣵⣶⣿⡿⢃⣾⣿⣷⢟⣾⡿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠬⠖⢚⣋⣩⡭⣭⣭⣍⢳⣤⡴⠆
⠀⠀⠀⠘⠋⠉⠉⠉⢁⣿⣿⡾⢣⣾⣿⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⡶⣿⠳⡛⢏⢹⡉⣏⣯⡧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢣⣿⣿⣣⣿⣿⠇⠀⠀⠀⠀⠀⡤⠞⠁⢿⣌⣧⢸⡖⣗⣻⣉⠯⠗⢊⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣼⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣦⣤⣭⣭⣤⣤⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣯⣭⣭⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠒⠲⠇⠀⠷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⣀⡀⢀⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡴⢅⡀⠀⠀⠀⠀⢀⡌⠀⠀⠀⢀⣽⠀⠀⠀⣹⠄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠘⡇⠀⠘⢿⡓⠦⣰⠋⠀⢀⠀⠀⢛⠁⠀⠀⠈⢫⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠀⠀⠀⡇⠀⠀⠈⢷⠀⠹⣄⡀⠁⠀⠻⡏⠁⠀⠀⠒⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⣿⠀⠀⠀⢹⣆⠀⠀⠘⣧⠀⠈⢇⠀⠀⡾⠓⠀⠀⠀⢰⣇⠀⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⠀⢸⡄⠀⠀⠀⢿⠀⠀⠀⠹⣦⠀⠈⢆⢲⠷⠀⠀⠀⢀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠈⣧⠀⠀⠀⠸⡇⠀⠀⠀⢹⡆⠀⠈⡄⢀⠀⠀⠀⠀⣙⠁⠀⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⢿⠀⠀⠀⠀⣷⠀⠀⠀⠈⣯⠀⠀⢰⠀⠀⠀⠀⠹⠟⠓⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⡇⠀⠘⣧⠀⠀⠀⢸⡆⠀⠀⠀⢹⣆⠀⠀⢣⠀⠀⠠⣾⠆⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢿⡄⠀⢹⡀⠀⠀⠀⢿⠄⠀⠀⠀⢿⡄⠀⠸⠄⣀⣠⡦⠀⡀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⢧⡀⠈⣷⠀⠀⠀⠸⣇⠀⠀⠀⠘⣧⠀⠀⡄⠀⣭⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⣧⠀⠘⣧⡄⠀⠀⢻⣧⠀⠀⠀⣿⠀⠀⠇⢛⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⡆⠀⠘⣷⡀⠀⠀⠻⣆⠀⠀⢿⠀⠀⡌⠍⠁⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢻⡆⠀⠘⣷⡄⠀⠀⠹⣆⠀⠸⡆⠀⣿⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣦⡄⠈⢻⣄⠀⠀⠹⡄⠀⣿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠙⢦⡄⠀⢱⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⠹⣆⠀⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠄⠀⠙⠆⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠠⠤⢤⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠤⠀⠀⠀⠔⠓⠛⠛⢶⡔⠁⢀⢀⠴⠯⢄⣀⠀⠀⠀⠈⠱⢄⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡉⠀⠀⠀⠀⠀⠒⣿⣏⡁⠄⠐⠆⠉⢯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣺⣄⡀⡤⠐⠢⣤⣾⣟⠉⠀⠀⠤⣀⢀⢿⡆⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡛⠻⠻⡟⠿⠓⡀⠐⢆⠀⠶⠏⢆⠈⣧⠾⠶⠶⣴⡗⠁⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠤⣀⡈⠙⠀⠀⠘⠦⠐⢷⡠⠀⠈⢻⡏⠀⠀⠀⠀⠹⣆⢀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠶⠘⢧⡀⠀⠀⠀⠀⣿⣁⣀⠠⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠀⠢⡀⠢⡀⢢⣀⣼⣄⣀⡴⠀⠀⠀⢀⠀⠈⠻⢲⣦⣤⠾⠃⠀⠀⠀⠀⠀⠀⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⠐⠦⣵⣾⡾⠛⠛⠛⠻⣿⣃⠔⡊⠀⠀⠶⡄⠈⠉⠳⡗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⢉⣟⣾⣿⠀⠀⠀⠀⠀⠘⣿⣡⡀⠄⠀⠐⢉⡈⠁⠀⢱⠲⠀⠀⡈⢠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣀⡀⠤⠤⢿⡄⠀⠀⠀⢀⢰⣟⠻⠶⠆⠐⢂⠉⠁⠀⠀⠀⠐⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠐⢩⢻⣿⣶⣾⣶⡿⣍⠛⠀⠠⢄⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠈⠀⠄⠁⠸⠏⢸⡏⡁⠱⠈⠑⢄⢈⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⡀⠀⢸⡀⠘⡄⣰⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠉⣇⠔⢸⣇⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠁⠀⣰⠏⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣍⣀⣉⣉⣉⠙⠛⠒⠒⢐
⠀⠀⠀⠀⠀⣠⣾⣄⣀⣐⣀⣀⡟⢠⠍⠉⣳⣆⡤⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⠁⠀⠉⠉⠉⠳⢀⠉
⠀⠀⠀⠀⠀⢧⣦⣤⣬⣭⣍⣉⣉⣿⡷⠟⣋⠡⠒⠁⠀⠀⠀⠀⠞⠛⠻⠛⠛⠛⠛⠓⠒⠂⠒⠛⡁
⠀⠀⠀⠀⣠⣸⣻⣯⣛⢦⡴⠟⢋⣡⡴⠊⠀⠀⠀⠀⠀⠀⠀⠈⠷⢿⠿⢿⡻⠿⠛⠛⠛⢉⠉⠟⠁
⠀⠀⠀⠀⠓⠮⢭⣛⡛⠿⢷⠞⡉⠋⠰⠦⠶⠀⠀⠀⠀⠀⠀⠀⠈⢂⣰⠁⣀⠤⠐⡀⢅⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠂⠐⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠤⠄⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⠀⠀⠀⠈⣦⣄⣀⡀⡀
⠀⠀⠀⠀⢀⣠⡤⣼⣿⣶⣷⣄⡀⠀⠤⠄⠙⠓⠀⠀⠀⠀⠀⠀⠀⠆⠛⠻⠗⢀⣀⣤⠤⠤⠀⢰⠃
⠀⠀⠀⠀⠻⣷⣾⣦⠙⠛⢿⢿⣿⡆⢀⣠⠤⠶⣦⠀⠀⠀⠀⠀⠀⠰⣦⠤⣤⣀⣀⣀⣀⠀⠀⠀⣹⠆⠄
⠀⠀⠀⠀⠀⣈⡙⠿⣿⣷⣢⡑⠎⠉⠉⢐⡚⠉⠈⠐⡂⠀⠀⠀⢠⡀⢻⣷⣝⢿⡂⠘⠉⠛⠉⠈⢉
⠀⠀⠀⠀⠀⠉⣅⣀⠈⠛⠻⢿⣷⠂⠁⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⣄⠀⠩⡟⠈⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡶⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠴⠄⠀⠀⠀⠀⠀⠀⠠⠈⣀⣀⣀⣉⡑⠚⣉⡀⠡⠀⠀⠀⠓⠦⠧⠤⢀⠢⣴⠤
⠀⠀⠐⠂⢄⣀⢀⣤⠶⠂
⠀⠀⠀⠀⠐⠈⠻⢤⣢⣥⡤⠀⠉⠀⠀⠀⠀⠒⠀⠈⣄⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂
⠀⠀⠀⠀⠀⠁⠠⠄⠀⠐⠆⠴⠦⠀⠀⠀⠈⢆⡰⠶⠊⠀⠀⠀⠠⠤⠀⠀⠀⡠⠀⠀⠀⠈⢦⣀⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠜⠴⠎⠂⠀⠀⠀⢡⣠⠀⠀⡀
⠀⠀⠀⠀⠀⡀⢀⠄⠒⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣡⣤⡶⠏
⠀⠀⡆⠖⣀⠛⠃⠀⠀⢵⡦⡄⠳⠃⠀⠀⠠⠄⢂⠂⠠⣒⠍⠁⠀⠀⠀⠀⠀⡤⢀⠀⢀⠤⠀⠀⠀⠈⠁
⠀⠀⠉⠉⠙⠦⠐⠚⠘⠃⠒⠁⠀⠀⠀⠀⠠⠆⣤⠶⠖⠥⠒⠖⠢⠀⠀⠀⠀⢣⡀⠀⠈⠑⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⣠⠴⠤⠔⠚⠁⠀⠀⠀⠀⠁⠉⠉⠉⠀⠙
⠀⠀⠄⠒⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠠
⠀⠀⢠⡐⠀⠈⠆⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠆⠀⠀⠀⠅⠀⠀⢀⣀⡀⢀
⠀⠀⠀⡆⠀⠀⢀⡄⠀⠀⠀⣀⣀⡀⢁⠤⠀⠀⠀⣠⠁⠀⠀⠀⠀⠂⢀⣠⢀⠏⠀⠀⠀⠀⠉⠀⠈⠈⠀⠀⠉
⠀⠀⠀⢹⠂⠈⠀⠀⠀⠀⠀⢃⠁⠉⢁⠤⠬⢥⠊⠗⠀⠀⠀⠀⠀⠉⠙⠒⠃
⠀⠀⠀⠈⢧⠀⣀⠤⠃⠀⠀⠈⠄⠠⠤⠤⠰⠒⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠤⠐⠐⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠄⠠⠀⠒⠀⠘⠄⠀⣀⣌⣣⡤⠦⠞⠂
⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠈⠳⣀⣀⣤⠽⡴⠶⠛⠛⣏⠉⠁⢀⠹⣄⣀⡤⠥
⡀⠤⠤⠀⠀⠣⠀⠀⠀⠈⠀⢁⠉⠑⠀⠀⠀⠀⠀⢈⣷⠚⠛⠉⢉⠀⠀⠀⣀⡀⠠⠴⠞⠂⠊⡉⠁⣀⠄⠤⠄
⠐⠀⠒⠊⠀⠀⠀⠀⠀⠐⡖⠂⢦⣶⠄⠂⠁⠀⠀⠀⢧⣀⠤⠶⠒⢃⠉⣩⣀⡀⠤⠀⠚⠃⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠫⠀⠒⠒⠉⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣶⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣉⢯⡇⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢹⣯⠻⣤⠚⠁⣠⡤⢌⣙⣦⣄⡀⠀⢀⠚⠁⠀⠀⠀⠀⠈⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢠⠋⡼⠶⠅⢠⡾⠿⣀⡂⠁⣸⠿⢿⣧⣿⣦⣠⣀⠆⠀⠀⢰⠃⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣠⡏⢠⠀⣠⡶⠋⠀⣸⠟⢁⣿⢁⣤⠏⢁⣸⠟⢻⣿⣶⣤⠾⠇⠠⢄⣰⠖⠀⠈⠳⢤⡀
⠀⠀⠀⠀⢀⠞⠁⠐⠈⠁⡇⠀⠹⣦⡿⢅⣴⠇⠀⢠⠜⠉⣩⣧⣤⠎⢁⡴⠚⠉⣹⣏⢦⠀⢂⠀⠈⣫⠓⠲⢶⣤⣄⣽⣥⣖⡫⠇
⠀⠀⠀⠀⡜⠀⠀⠀⣠⣼⠃⠀⠀⠈⣀⡼⢟⣤⡞⠀⣠⠞⠁⢀⡽⠷⢏⠀⣠⠄⠛⢻⡷⢇⠀⠳⠀⠈⡄⠀⠘⠆⠀⢨⡶⠋
⠀⠀⠀⢰⡇⠠⠆⠈⠁⢂⠀⠑⠸⣇⢻⣀⠜⠉⢱⢾⡇⠀⣰⠋⠀⢠⡿⠚⡷⢀⡤⠾⣷⡸⣧⣈⣀⠀⣰⣀⣀⡼⠒⠁
⠀⠀⠀⠸⠃⠀⠀⠀⠘⢹⠀⠀⠀⠹⡎⢿⣀⠞⠀⢀⡿⢻⣃⣠⣶⠋⠀⢠⡿⢿⣤⣤⣾⢆⡇⠈⡻⣦⣄⣡⠸⣷
⠀⠀⠀⠀⡆⠀⠀⢰⠒⠙⡷⠰⠆⣀⡻⣎⡻⣤⣴⠉⢀⠜⠛⢹⣧⡶⠋⡀⣠⣤⣯⣀⣼⢸⡇⡔⠁⠀⢳⡄⠀⢻
⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⢣⠀⠀⠈⠀⢹⡷⡈⡻⢶⣏⣀⡰⠋⣹⣦⡾⠋⢀⣤⣽⣛⣏⣿⣀⠆⠀⢠⡞⢧⠀⢸⡄
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⠇⠀⣠⡀⢨⡇⠈⠙⠢⠬⣛⠳⠶⣧⣰⠿⣾⠉⣠⡤⣿⣷⠋⠉⠳⢤⡜⠁⠀⠱⢆⡛⡄
⠀⠀⠀⠀⠀⠀⢳⡀⢀⠖⠘⡆⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠉⠒⠨⣽⣶⣿⣏⣁⣠⣿⡟⠀⠀⠀⠀⠉⠁⠀⠘⠀⠩⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⣼⠃⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢍⡛⠻⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⠙⣣⡞⠁⠀⠀⠀⡀⠐⠂⠀⢈⡷⠀⠀⠀⠀⠀⠀⠹⣄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⡿⠀⠀⠀⠀⠀⠳⣤⡶⠞⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣀⣀⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⠀⣉⠛⠶⠀⠲⠦⣤⡤⠶⢀⣴⠂⠐⣉⡤⣀⠀⢀⣤⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠙⠛⠖⠲⠶⠦⣄⣠⡄⢉⣉⣁⣌⣹⡘⣥⣠⡤⠦⠶⠚⠓⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⠖⠒⠀⠤⠔⣶⣮⠟⠚⠉⠉⠉⠀⠀⡟⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⡠⠌⠉⣴⠃⢰⠟⡳⠶⣿⠧⡤⢤⡀⠀⠀⠀⢠⠇⢀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⣀⠞⠀⠀⠀⠉⢶⡇⡌⠀⠀⣸⠀⢿⠀⢷⠀⠀⠀⡼⠀⠈⣷⠀⠈⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠀⠈⣻⠀⠀⠀⠀⢀⣾⠇⣿⠀⠘⣿⠀⣿⠀⣾⠀⠀⢀⡃⢀⡜⠀⠻⡦⣀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⢀⡿⣄⡀⠀⢀⣾⠋⠀⠟⠀⠀⢹⠃⢿⡇⠻⡄⠀⠈⣧⠋⠀⠀⠀⣿⣄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡠⠞⢻⠀⠉⢰⡾⠁⠀⣀⠀⠀⠀⠈⠀⠘⡇⠀⠙⢄⠰⣧⠀⠀⢀⠜⠁⢹⣆⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⣦⠀⠺⠆⠀⢰⠁⠀⠀⠀⡼⠀⠀⠂⠀⠀⠈⢆⠙⡇⠠⠎⠀⠀⢀⠉⠡⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠓⣿⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⠌⠀⠀⡶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⡠⢻⠀⢻⡀⠀⠈⡀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⢀⡇⠱⡀⠀⠀⠀⠀⠀⠀⣰⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⢀⣄⠘⣇⠀⠀⠁⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⢸⠇⠀⡇⠀⠀⠀⠀⠀⢰⠛⣧⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣼⠏⢳⣿⢆⠀⠀⢣⠀⠀⠀⠄⠀⣶⠀⠀⢀⡟⠀⠀⠈⠲⣤⠀⢀⣠⠈⠀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣫⢀⡾⠃⠈⠳⣄⠀⢠⠀⠈⠃⡠⠏⠀⢀⡞⠁⠀⠀⠀⠀⠙⠦⣸⠁⠰⠟⢿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠁⠀⠀⠀⠀⠈⢳⣈⡇⠀⢀⡵⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠈⠀⠙⠨⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣤⣄⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣰⢂⡇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠛⢇⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣆⠀⢠⠀⠁⠀⠠⡆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡑⢀⣫⡽⢿⠶⠂⠀⠀⢀⡇⠐⠂⠚⠃⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⡥⣀⡀⠈⠀⣤⠀⠀⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⢧⡀⠀⠙⠒⢶⠊⠐⠄⡀⠀⠂⠠⠤⣀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠀⠀⠙⢦⡄⠀⠀⠑⢦⠀⠁⠀⠀⠀⠀⠸⠇⠀⠀⠀⢠⠋⠀⠀⠴⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠆⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠉⡄⠀⢀⠂⣀⡀⠀⠁⠀⠀⢀⡣⡌⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠔⠁⠀⠀⠀⠐⢒⠐⠀⠈⣄⠀⠀⠀⠀⠈⠀⠃⣴⠀⡀⢀⠀⠓⠒⠀⠀⢀⠀⠁⠀⢲⣀
⠀⠀⠀⠀⠀⠀⠄⠀⣀⠄⠀⠐⠀⢀⠀⠄⠀⠀⠀⠀⢹⡂⠀⠁⠀⠀⠀⠇⠈⠛⡇⠈⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⣚⣚⢉⣠⡜⠂⠀⠀⠀⠀⢁⠀⠀⠀⠄⠀⠁⠀⠀⠘⠑⢀⣘⡃⣄⣠⠀⠁⠈⠉⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⣉⡡⡬⠐⠙⡀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⢈⡿⢄
⠀⠀⠀⠀⠀⠀⠈⡄⠐⡗⠉⡉⠉⠀⣁⡤⠔⠒⠒⠒⠛⢣⡀⠀⠀⠀⠐⡖⣦⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠚⠉⠀⠀⠀⣤⠀⠀⠀⠉⠉⢀⠤⠀⠘⠃⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠁⠀⠀⠀⢀⠄⢰⠁⠋⢳⡀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠐⢶⠆⠛⣶⠑⠂⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠋⡀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠇⠐⠑⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠓⠾⡏⠐⠂⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠠⡀⠒⠹⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠢⡜⠁⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⢄⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠠⠠⠦⠤⠶⢄⣀⠀⠀⠀⠀⠀⡠⢰⠈⠓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠗⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠈⠁⢀⠀⠀⠀⠀⢀⢤⠀⡆⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣉⣉⡤⢬⣉⡉⠉⢉⣡⣈⡷⠁⠀⠀⠈⠒⠒⠦⠤⠦⠀⢶⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡀⠀⢀⣠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠙⣿⣁⢄⣒⢐⡄⠐⠊⠀⠀⠀⠀⠠⠄⠈⠑⣀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⠉⣭⢄⣀⡀⠀⠀⠀⠀⢠⡄⢠⣅⡬⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠓⠈⠁⠀⠀⠀⠀⢠⡉⢩⠀⠠⠀⠄⢉⡐⡀⠀⠀⠀⠙⡜⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠈⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠘⣬⣆⠀⠔⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡠⠂⠊⡀⠀⠰⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣴⡺⠟
⠀⠀⠀⠀⠀⠀⠀⡅⠒⢢⣀⠙⢁⠀⠐⠺⣶⢤⠚⠗⠁⠀⠀⠀⠤⠄⣪⠀⠐⢒⡩⠂⠀⠀⠀⠀⠀⠀⠠⡤⠄⢀⠐⡀⠠⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠷⠤⠴⠛⠚⠛⠒⠚⠀⠀⠀⠀⠀⠀⢰⠡⣤⣴⠒⣭⠄⠲⠖⢤⠀⠀⠀⠀⠀⠇⡀⠀⠈⠙⠖⠲⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢢⣖⣀⡠⠴⠃⠀⠀⠀⠀⠈⠒⠒⠒⠒⠢⠄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⣁⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠃⠀⠀⠀⢠⠂⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢃⠀⠀⠀⠀⠰⠈⡀⢄⡀⠤⠀⠀⠀⠀⠄⡤⠀⠀⠀⠀⠀⡐⠀⠀⠀⢠⡿⠁⠀⠀⠀⠸⡄⠀⣀⡠⠠⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⠰⠆⠀⠀⠀⢀⢠⣤⡀⠉⠰⠂⣀⠀⢂⣰⠄⠀⠀⠀⠀⢀⡓⢀⡠⠴⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠀⠀⠀⠀⡀⠀⠀⠈⢃⠀⢈⢉⠰⠶⠤⠄⠊⠛⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⣀⡌⠗⠀⠀⠀⠨⠀⢠⣤⠠⠀⠴⠆⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠔⠂⠲⡌⠉⠉⢀⠙⠆⣀⣀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠺⡄⠉⠀⠀⠱⣄⣀⣤⣤⠴⣴⠖⠛⠛⠙⢧⡀⠠⠀⢐⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⢰⡀⠀⠀⠀⢠⡖⠠⠀⢤⢀⡀⠀⠀⠀⠀⠀⠙⡆⣀⣠⣤⠽⡶⠖⠛⠋⠛⠉⠀⠄⠀⢙⡄⢀⡠⠴⠌⠓⠒⠛⠉⢉⡄
⠀⠀⠀⠀⠐⡈⠑⠒⢀⡌⢀⡃⠀⠀⠀⠀⢀⢠⠀⠀⢀⠀⠀⢀⠀⠀⠀⠐⡜⡅⠀⠀⠀⠰⡀⢀⣠⠤⠴⠄⠐⠚⠋⣉⢀⣤⠆⠠⠴⠚⠐⠂⠈
⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡽⡙⠚⠲⢿⠆⠉⠀⠀⠀⠀⠀⠹⣠⠔⠒⠛⠉⣉⣀⢠⡦⠤⠠⠖⠂⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠰⡂⠁⠀⠀⠀⠀⠀⠀⠙⠤⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⢠⡾⠁⠉⠀⠀⣀⡀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠴⠂⠀⠘⠁⠀⠀⠚⠉⢀⡀⡀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣠⡀
⠀⠀⡌⠀⠀⣴⠄⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠘⠁⠀⡤⠀⠀⠀⠀⣨⠏⠀⢀⣆⠀⠚⠁⠀⠀⠀⣤
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⠀⣀⠤⠐⢀⣀⠿⠒⠈⠀⠀⠀⠘⣇
⠀⠀⣇⠀⠀⠀⠀⠺⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠐⠛⠂⠀⠀⠶
⠀⠀⠙⠀⠀⠴⡄⠀⠀⠀⣠⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠈⠂⣀⣀⠀⠐⢶⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⣠⡶⢛⣷⠀⣾⠿⠿⠶⣦⣾⠷⠋⠤⢀⣀⣞⡆⠀⠀⠀⠈⠛⠀⠤⠴⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢡⣿⠃⣠⠶⡀⡶⢲⣰⣿⣸⢧⡟⣿⢈⣜⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠙⣰⡿⢤⠿⣟⡾⠻⠟⣧⣈⡯⠭⡅⠻⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣠⠜⠉⠛⠫⣤⣬⣴⡶⠞⠋⠙⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠴⠀⠐⠛⠃⣀
⠀⠀⠀⠴⣀⠀⠤⠀⠀⠈⠉⠀⠀⣤⡄⠀⠀⠀⠀⠀⣠⠖⠒⣀⠃⠈⠙⠂⠀⠀⠀⠀⠀⢰⣧⠶⠒⠉⠁⠉⠂⠀⠀⠒
⠀⠀⢰⡄⠉⠃⠀⠀⣀⠘⠋⢀⡀⠀⠀⣀⠀⠀⡰⠀⡅⠀⠈⠉⢤⠀⠰⠖⠀⠦⠀⠀⠀⠘⢃⠰⡷⠤⠤⠤⠤⢖⠀⠀⠢
⠀⠀⣼⣿⣦⣈⣉⡈⢙⣁⠀⠋⠉⢀⣰⠁⠀⢘⡇⠀⠲⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠲⠀⠀⠀⠈⠘
⠀⠀⠻⣿⣉⣩⡉⠹⠟⠛⠉⠉⣽⣧⣡⡆⠀⠘⠂⠀⠀⠀⠀⠀⠀⠐⠁⠀⠐⠀⠀⠀⠀⢈⠁⠀⠛⠀⠀⠀⠀⠉⠀⠀⠐
⠀⠀⠰⠿⠿⠗⠒⠶⠶⠶⣾⠛⠛⠉⠉⣠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠤⠀⠐⠀⢠⠠⠀⠀⢸⡔⠂⠀⠀⠀⠲⠆⠀⠀⣄
⠀⠀⠘⣿⠗⢲⡶⠶⢶⡾⢯⠶⠶⠞⠛⢃⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠀⠀⢀⠀⠀⠀⠈⢧⠀⠛⠀⢠⠀⠀⠀⠃
⠀⠀⣾⡟⣲⠶⣥⣤⣤⡤⠬⠴⠖⣒⠉⠍⡄⠀⠀⠀⠈⡁⠀⠀⠠⣤⡄⠀⠀⠘⠁⡤⠀⠀⠈⠲⠄⢀⠀⠀⠤⢀⠤
⠀⠀⠐⠳⠿⠧⠤⠛⢂⣀⣠⣤⣤⠭⠗⠊⠀⠀⠀⠀⢀⠈⢤⡀⠀⠀⠀⠠⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠦⠳⡤⠷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⡾⣤⡀⠀⠀⠀⠀⠀⠀⠀⣘⠡⢲⢎⢳⢢⢱
⠀⠀⠀⠀⠀⡠⢸⡁⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡎⡀⠀⠀⢇⢱⠀⠀⠀⠀⠀⠀⠀⠰⡰⢞⣮⡛⢎⡜
⠀⠀⠀⠀⠀⠑⢤⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣓⣓⣒⣒⣚⣒⠆⠀⠀⠀⠀⠀⠀⠘⠕⠋⠭⠜⠢⠇



⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠤⣀
⠀⠀⠀⠀⠀⣠⣣⣜⣒⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠌⠡⠧⢄⠀⠀⠀⠀⠀⠀⠀⢰⣁⣠⠔⢄⠈⢧
⠀⠀⠀⠀⠀⣷⣦⣴⣳⣦⣴⣟⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⠢⠊⢉⡏
⠀⠀⠀⠀⠀⠋⠛⠋⠀⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠮⣀⠖⠢⣀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒
⠀⠀⠀⠀⠀⠀⠠⡓⢝⢮⣲⠃⠀⠀⠀⠀⠀⠀⠀⢸⠭⠝⠫⠭⡇⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠉⣓⢢
⠀⠀⠀⠀⠀⡠⢖⡝⠢⠓⠃⠀⠀⠀⠀⠀⠀⠀⢰⠭⠕⠭⠭⠪⠭⡆⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⡵⠃
⠀⠀⠀⠀⠘⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠰⠭⠥⠬⠭⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠋⠙⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢀⣀⣠⣤⣤⣤⠶⠶⠶⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⣖⣚⠋⠉⣉⣈⣀⣠⣤⣤⣤⣤⣤⠾⠧⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠾⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣬⣤⣷⣶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠤⢐⣀⣤⡴⠶⠛⢻⡿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡭⠶⢟⣋⠁⠀⠀⠀⠀⠀⢡⡿⠟⠋⠁
⠀⠀⠀⠀⠀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡤⣒⡭⠖⠋⠁⠲⠶⣶⠪⠉⠀⢀⣰⣄
⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠷⣪⣵⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠙⢿⣦⡀⠀⠀⢀⣀⣀⣀⡤⠔⣡⣾⣏⣴⣿⡋⢦⣉
⠀⠀⠀⠀⠉⠻⢿⣶⣦⣭⣭⣷⣾⣿⣿⢿⠟⢻⣿⢷⣤⣀⣼⠆⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠸⢦⠀⠹⣧⡀⣍⣁⣰⠀⠀⠀⠀⠀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣆⠈⠻⢶⣭⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⡉⢀⡀⠨⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠋⠀⠀⠘⣿⣮⣧⣀⠀⠐⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡿⢹⡿⠈⣷⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢚⣫⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡇



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣿⣭⣭⣯⣬⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⡶⢄⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠁⠀⠀⠈⠙⠒⠊⠑⣠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⠀⣀⠀⠀⠀⠉⠀⠀⢠⠔⠀⢋⠻⠛⠙⠻⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠈⠀⠀⠀⠀⠁⠈⠠⠀⠀⠀⡀⡀⠈⠆⠈⠀⠀⠀⣸⡦⣿⣈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⡠⢄⡀⠉⠉⠀⠒⠒⠀⠀⢀⠰⠀⠁⠀⠀⠀⠀⠀⡀⢄⡥⠎⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠁⢀⠄⣢⣞⠁⠀⠀⠀⢀⠀⠤⠁⠀⠀⠠⡀
⠀⠀⠁⠀⠠⡬⠁⠀⠀⠂⠀⠀⠀⢀⡀⠀⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⠵⡟⠁⠀⠀⠀⠢⠀⠀⠁⠀⠀⣀⠆⣀⠴⢆
⠀⠀⢀⠀⠀⠀⠀⠀⠐⢄⠂⠠⠘⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡡⠊⠁⢌⠎⡀⣈⣐⣤⠄⣄⡀⢄⣀⡈⠅⠀⠐⠈⠈⢀
⠀⠠⠁⠀⠀⡀⠀⠀⠀⠀⠘⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣔⡍⠀⠀⠈⢀⠜⠀⠉⠉⢁⣉⠉⠉⠁⠀⠀⠀⠀⠀⠛⠳⠘⣷⣀
⠀⠑⢝⡂⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡴⠋⠀⢐⣓⠧⣄⠈⠀⠀⠀⠔⢠⠀⡄⠐⠦⠀⣤⠀⠀⠀⠠⠄⠀⠠⠼⣿⠷⠄
⠀⠀⠀⠈⠒⠤⡂⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠛⢤⡄⡀⠄⠀⣠⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢢⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⡠⠀⠀⢀⣠⣔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣀⠀⢻⢀⢠⠀⠀⡀⠀⢀⣀⣁⡬⠤⠶⠟⠁⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠧⢼⣼⡤⠴⠶⠬⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠚⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠲⠦⠤⣴⣯⣓⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣹⠋⠙⡿⣿⠙⠲⠦⡤⠖⠒⠉⠁⢀⠀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣨⠞⢽⢷⣄⠀⠁⠤⣔⡢⠭⢴⠞⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠎⠀⡀⣼⣿⠀⢸⡾⠋⠓⡆⠉⠁⠀⢀⣮⡴⢎⠁
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⣰⠃⢸⣠⣿⣷⣦⣄⣀⡤⠶⡾⠟⠉⠀⠀⣻⡆
⠀⠀⠀⠀⢀⡾⠳⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⢎⣾⠉⢣⣿⣁⡀⠀⠀⠈⠉⠑⠆⠀⣠⠔⣡⢰⠁⢃
⠀⠀⠀⠀⣾⣕⣒⠠⠀⠀⠈⠉⠓⣆⢀⡴⢡⡏⢸⣴⡟⠛⠛⠫⠭⢭⣄⣀⡀⠀⢈⡴⢺⠡⣋⡴⠋
⠀⠀⠀⠀⠘⡄⣌⠉⡙⠦⢤⣀⡀⠚⠛⣱⠛⢇⣼⣿⣦⣤⣄⣀⣐⠉⠉⠀⠉⠉⢻⠄⣸⡟⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠛⠓⢄⣙⠻⠉⢻⣧⡀⣼⡏⠀⠀⠀⠀⠀⠉⠉⠛⠒⠷⠤⢼⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠳⠿⠋⠀⡠⠶⣤⡀
⠀⠀⠀⠀⠀⠐⠒⣶⣷⠄⢈⡉⣹⢷⡀⠀⠀⡠⠊⠀⢠⣸⣿⡆
⠀⠀⠀⠀⠑⠶⠥⠾⠇⡈⠁⠀⠉⣂⡼⠁⠘⠅⣀⡀⣋⣠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢖⣀⡴⡄⡳⢄⠀⣶⠟⠋⠉⠛⠉⠙⠛⠛⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⠬⠁⠀⠀⡀⢀⣀⣙⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⣤⡌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡀⠐⠊⠡⣠⡼⢟⠉⢸⠉⢳⠀⠀⠠⠀⠀⠀⠈⢾⣿⠀⢸⣗⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠀⠴⠒⣛⠑⠞⣛⣿⣦⡉⠻⣆⡠⠉⠂⠀⠀⠀⠀⣻⣧⠞⢍⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠔⠁⠀⢤⢘⣿⡏⢁⡄⠀⡄⠀⠀⢀⣤⣀⡞⢻⣿⠀⠾⠿⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢖⣿⣽⠋⡟⠀⠘⢇⠤⠇⠀⠀⠈⠜⢻⣷⢸⣿⠀⢀⢠⣿⡿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡿⠛⠟⠀⠀⠀⠀⠀⠀⠈⡀⢀⠊⠀⠊⣿⢸⣿⠀⠈⣸⣿⢻⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⣿⣿⡇⠀⠠⢸⠿⢮⠹⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣿⠁⠀⠀⢈⣽⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⣿⣿⠃⠀⠀⠀⢬⣿⢿⡿⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⢀⠄⣾⣼⣿⣿⠏⠀⠀⠀⠀⠸⣇⢸⣇⣟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⢣⣆⣼⣿⠟⣿⡿⣠⣀⠀⠀⠀⢻⣇⣾⢻⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⠶⠆⢀⣺⣿⣿⣾⣿⢿⡇⢰⣿⠁⠟⣷⠀⠀⠀⢆⠻⣯⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣬⣭⣷⣶⣶⣽⣿⣿⡋⠀⣈⢧⢹⡏⠀⢁⠀⣧⠀⠀⠀⢧⢛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⣴⣶⣿⣿⡿⠿⣻⠟⣻⢟⡽⠉⠉⠉⠉⢾⠃⠀⢸⣄⣿⡆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢐⣛⠋⠉⣉⠁⠁⠁⣀⣔⣁⣠⠤⠋⠀⠀⠀⠀⣼⣼⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠰⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡖⠂⠀⠀⢰⡇⠀⠀⠀⣴⡄
⠀⠀⠀⠀⣰⣾⣦⣄⡀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠀⠀⠀⣿⡇
⠀⣰⣆⠀⠈⠻⠿⣿⣿⢶⣔⣏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣏⣀⣀⣀⣸⣆⣀⣀⣀⣿⡇
⠀⠈⠻⠿⡦⠶⠖⠛⠛⠻⢿⣿⢷⣦⣄⣀⡞⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⡏⠉⠀⠀⢰⠇⠀⠀⠀⣿⣇
⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⣀⣀⣽⣿⠿⡏⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠈⣿⡞⠋⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣲⣶⠒⠶⠷⠶⣶⡖⣺⢶⠖⠒⢲⡿⣯⠜
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢰⡌⠙⠃⠀⠀⠀⠀⠀⠀⠢⡤⣠⠠⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣶⣤⣀⣠⢏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠿⣟⣻⢿⣿⣿⡱⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣶⣮⣽⣿⣣⣴⡟
⠀⠀⠀⢀⣤⣴⣰⣶⡄⠀⠈⠙⠀⠀⠀⠀⠉⠽⠷
⠀⠀⠀⡌⠈⠹⠀⠹⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀
⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⢤⠠⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⣤⣬⠵
⠀⠀⠀⠈⢶⡄⣷⣠⡾⠀⠀⠀⠀⠀⠀⠀⠃⠄⠄⠰⠀⠀⠈⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⢀⠤⠱⠀⠀⠂⠀⠀⡀⡤⠉⠀⠀⣴⣿⠉⠀⠀⢀⣠⠆⢀⣠⠀⠀⠁⣈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠘⠦⠡⠠⠦⠴⠐⠃⠁⠀⠀⠀⠀⠈⠻⠿⠶⠛⠋⠁⠀⠈⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⣄⢀⣤⢦⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⠆⡀⠈⣧⡀⠀⢀⣄⢀⠀⠄⠻⡿⣶⣇⠇⢭⠇⣳⡾⡽⠃⢠⠀⠀⣇⠀⠈⢠⡎⠀⣀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⣠⠘⠓⠨⡀⠀⠈⢃⠤⣍⠁⠀⣴⣷⡿⠀⣲⠠⣷⣶⡄⠀⢉⡅⠠⠈⠀⠀⣈⠓⠚⠀⣄⠄⠈⠉
⠀⠀⠀⠀⠀⡔⠓⠚⠂⠈⠉⠀⢹⣶⣾⣇⠀⠀⠘⠲⠀⢇⠀⣀⣋⠂⠶⡵⠆⢊⣓⣀⢀⡇⠰⠚⠁⠀⠀⣽⣶⣿⠁⡈⠉⠁⠘⠒⠓⣂
⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⢀⡄⠉⣀⣁⣉⣁⣙⣲⣦⣁⣈⣳⣉⣽⣷⣾⣹⣷⣼⣏⣹⣋⣀⣡⣴⣖⣁⣈⣁⣉⡈⡉⣀⡀⠀⠀⠀⠀⠀⠠
⡤⠠⠀⠀⠀⠠⠀⠄⠀⠀⠀⠤⠀⠠⠀⠀⠄⠠⠀⠀⠀⠄⠀⠠⠠⠠⠄⠀⠀⠀⠠⠠⠄⠄⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄⠤⠀⠀⠀⠄⠀⠄⠀⠠
⡇
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣈⡄⢢⢠⢢⢇⢀⢠⢎⢀⡀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠀⠀⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡟⣠⢀⡄⣠⢀⡄⣠⢀⠄⡄⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠐⠃⠘⣰⠃⠊⡼⠁⠊⠘⢱⠇
⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠰⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⡇
⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠒⠖⠒⠒⠒⠲⠲⢒⣶⠶⠒⠒⣶⠒⣶⢶⣶⢲⡶⣶⠖⢲⡒⠒⠲⣶⡒⠶⠲⠒⠒⠒⠂⠖⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠑⡂⣀⣀⠀⠀⠀⠊⣹⣾⣷⠟⠈⠁⢀⢠⠁⡾⠙⢛⠻⠁⣝⢯⣄⠹⡛⠋⠹⡄⢁⢀⠈⠁⠙⣷⣿⣮⠑⠀⠄⠀⣀⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠘⠜⠦⢉⣁⡘⠃⠀⠀⠙⠠⠞⠁⠀⣬⣿⡦⢰⡢⢠⣾⣥⠆⠈⠑⠠⠜⠀⠀⠀⠻⢀⡉⠱⠏⠈⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⠰⠊⠀⡿⠀⠀⠘⡆⠀⠹⠁⣠⢦⣾⠃⡼⠄⡈⣷⣧⣦⡀⠉⠂⠐⠟⠀⠀⠺⡆⠉⠂⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠈⠁⠐⠀⠀⠁⠀⠀⠀⠛⠉⠸⣵⠗⠰⢷⡅⠈⠙⠃⠀⠀⠀⠀⠀⠐⠀⠉⠐⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠄⢀⡀⣠⣠⡄⠀⠠⠶⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠋⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠿⢿⣷⣾⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣷⣿⠂⣾⣲⣾⣾⡷⠠⣼⣵⣾⡇⠰⣶⠴⢿⡆⢸⣷⣾⣥⠌⠩⡭⢯⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣯⣽⣾⣨⣥⠒⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣴⠶⣖⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠉⠈⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⠤⠥⠄⠀⣀⡀⠐⠀⡠⠀⠀⠀⠀⣀⠂⢀⣡⡤⠤⠤⠤⢤⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠂⠰⠶⠂⠦⠀⠈⠉⠁⠉⠤⠀⠀⠀⠀⠀⠤⠬⠤⠌⠙⠀⠁⠶⠴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠐⠒⠐⠒⠲⠂⠀⠒⠒⠐⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠰⠶⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠶⢶⣶⠀⠀⠀⣗⠒⢒⡖⠀⠀⠐⣷⠖⠲⡇⠀⠀⠘⢲⣤⠴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡏⠀⠀⠃⠀⠀⢀⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡤⡄⠀⢰⡇⠀⠠⣧⠀⠀⢠⡇⠀⢰⣵⠀⠀⣸⢠⠇⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠒⠒⠇⢣⠀⣼⠓⠒⠒⢿⠀⠀⢸⠇⠀⠸⠿⠀⢀⠏⢺⠒⠒⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⡆⠀⠁⡆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠈⢃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⣀⡀⣠⠆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠄⣿⠁⠀⣤⣤⣶⣳⣎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⣆⠀⠐⣤⣷⣾⣿⣆⠀⠈⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠲⣀⣄⠘⠾⡁⠀⣀⠤⢉⢻⣷⣠⣾⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⢀⡼⢿⡄⠛⣉⣑⠛⠿⣋⢸⣿⢏⣽⠛⠋⣻⣯⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⢀⡿⠒⠋⢻⠀⣻⠿⢤⢈⡿⢋⣴⡁⣿⣷⣤⣄⢽⣎⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣥⢀⣞⣤⣤⠀⣸⠀⡍⣩⣼⣿⣇⣘⠇⣽⡌⡏⢀⣬⡞⢩⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢠⣆⠳⣤⣍⡳⣶⣾⣇⡩⠝⠿⣯⣷⡾⣿⣣⣏⡘⣛⣴⢫⣾⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠐⠶⣶⡟⠉⣁⡿⢿⡿⡇⠀⢠⡾⠋⢸⣯⢀⣴⣾⠟⣻⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣑⠺⠿⡚⢻⣿⡿⠴⠶⠂⠲⠾⣦⣲⣾⣾⣿⣷⡶⠞⢛⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠐⠒⡤⣭⣭⣭⣁⣀⣀⣈⣉⣉⣉⣭⣥⣶⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⡄⠀⣇⠀⠀⠈⠙⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⠁⢸⡄⢐⢰⣷⠆⠾⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠸⡏⢸⣼⣿⣧⢽⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⣿⡄⣾⣿⣿⣸⣿⣿⣿⣿⢸⣽⣿⣿⣿⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠧⠧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠿⠿⠿⠿⠙⡂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⣀⣠⣤⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠶⢶⣾⣭⣿⢿⣿⠛⠉⠁⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⡤⣤⣤⣤⣴⡾⠛⣩⡥⠀⠸⠏⠀⢀⣾⡿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣀⣴⠿⠋⣥⠀⢠⣿⣵⣾⣿⣿⣶⣿⡏⠀⢹⣿
⠀⠀⠀⠀⢀⣼⡿⠛⠛⣻⣿⠟⣁⠀⣸⠏⣠⣾⡿⠟⣩⣾⣿⣿⣿⣿⣤⡼⣿⣤⣤⣀
⠀⠀⠀⢠⣿⠋⠀⣠⣾⠋⠀⣼⠇⠀⣿⣾⠟⣡⣴⣿⣿⣿⣿⣿⣿⡟⢁⣀⣀⠙⢿⣿⠷⡄
⠀⠀⠀⣿⣧⣴⣿⢿⡇⠀⠀⡟⢀⣾⡟⢁⣾⣿⡟⣩⣾⡟⣹⣿⡟⣠⣿⣛⣻⣷⠈⣿⣶⡙⣆
⠀⠀⢠⡿⢋⡞⠁⣾⠀⠀⠀⢣⣿⡏⢠⣿⣿⠋⣴⣿⠟⣰⣿⡟⣰⣻⡯⢬⣽⠾⡇⢸⡇⣷⢸⡆
⠀⠀⣿⠁⠀⠀⠀⣉⣀⣤⡀⣾⡿⢀⣿⣿⠁⣼⣿⠏⣰⣿⣿⢠⣟⡿⠤⠤⣾⣛⣿⠸⡇⢻⡄⣷
⠀⠘⣧⠶⠚⠛⠛⠛⠋⠉⠀⣿⡇⢸⣿⠇⣼⣿⠃⣰⣿⣿⡇⣼⢻⡧⠶⢶⣿⣭⣽⠀⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣿⢸⣿⢃⣼⣿⣿⣿⠁⣿⣻⠴⠶⢶⡿⢤⣿⠀⠀⣾⢇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⢣⣾⣿⣿⣿⣿⠀⣟⣳⠶⠶⢾⣟⣻⠇⣸⢀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣷⣿⡿⢛⣽⡿⣿⡆⢿⣽⠖⠒⣿⢤⠏⢰⡏⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣾⠟⠋⠀⠸⣷⠘⣿⣏⣙⣿⠏⣰⣟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠙⣷⡈⠻⠞⣡⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⣰⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⣶⣶⡀⠀⠀⠨⠅⠃⠀⢀⣤⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠂⠐⠁⠀⡀⢀⢀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠠⣶⣷⣤⣤⣾⡿⠆⠀⠀⠀⢀⣾⣿⡿⢷⣦⣤⡀⠀⠀⢠⣼⠿⠷⡂⠀⢠⣴⠿⠛⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣉⢮⣁⠀⠠⠀⠀⣐⣩⣆⠒⡉⠂⠿⠛⠀⠀⠀⠈⠉⠀⠉⠁⠀⠘⠁⠀⠒⠄⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠉⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⣄⣠⡿⠃⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⡀⢠⡄⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡀⠀⣰⠃⠀⠘⠀⠘⠓
⠀⠀⠀⠀⠀⢠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠈⠀⠙⠀⠀⠀⠀⣠⣆⠀⠀⠙⠚⠁⠀⠘⠒⠀⠀⠀⠄
⠀⠀⠀⣤⠀⢸⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣶⡆⠀⠀⠀⠤⣴⣠⡀
⠀⠀⠀⠉⠃⠀⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠰⠶⠀⠴⠀⠆⠐⠖⠀⠇⠰⠖⠐⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢷⣤⣛⡿⣦⣀⠀⠀⠀⠲⢄⡀⠀⠀⠀⠀⠀⠀⠑⢇⠉⢈⡷⠈⠽⠁⡰⠋⠀⢛⠟⠀⣴⢭⠞⠁⠈⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣦⣜⣉⠳⠶⠤⣤⣬⡀⣀⠀⠀⠀⠀⠀⠀⠈⠻⠖⣫⣤⠾⡵⣷⠀⠛⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠛⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⡤⢄⡄⢠⡀
⢀⣀⣀⣀⠠⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣁⠫⣀⠷⠞⢻⡬⠚⣠⠮
⠀⢁⣠⠤⠶⢚⣭⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢀⡀⢰⣉⢧⡽⠶⢛⡉⠔⣪⠔⠂
⠁⠀⣀⣠⠶⢫⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⠀⠒⢠⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠈⣉⠠⠔⣲⡟⠁⠀⠀⢀⠴
⣴⠞⡋⣹⣾⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⣤⢼⠃⠀⣄⣀⢰⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠮⠿⠷⣶⠿⠛⠻⠁⡀⠴⠊⠀⢀
⣴⡾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⢀⡰⡿⢠⡆⠛⠥⠦⠤⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣖⡒⣺⣏
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣹⣶⠷⠂⠛⠓⣶⡶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣼⣿⠟⠑⠿⢥⣤⠘⠏⠁⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣎⡲⠶⠂⠘⠃⠀⠀⠀⡀⢠⡀⣄⠀⡄⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠮⣹⠞⠣⠎⣠⠃⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠡⠒⠊⠉
⠈⠉⢀⣴⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡄⠌⠙⠳⢾⣡⡎⢳⠆⣀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⣀⣤⠶⠆
⠴⢾⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣮⣓⠖⢀⣈⠛⠐⠃⠐⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠉⢀⢀⣀
⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠻⣶⣾⣿⣯⠴⠶⠚⠓⠋⠀⠀⠀⣞⣫⣥⠤⠶⠶⡿⣛⣋⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠿⠿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⡀⠈⢒⣤⠓⠈⣀⠀⣤⠚⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡖⠸⣐⣯⣸⠥⣺⠂⠠
⠀⠀⡄⠀⡀⠉⠈⣵⡗⣀⠰⣿⣃⢛⣺⡝⠀⠀⠀⠀⠀⠀⡀⠤⠀⣱⣆⡙⢎⣉⣤⠽⠊⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⣁⣀⣤⣄⣽⣷⣫⣿⡄⠽⠦⣤⣤⣴⠄⠁⠀⠀⠀⠀⠀⢫⣀⢒⣄⢹⡤⠗⠚⣉⣤⣄⣥⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠁⢀⣀⢈
⡈⡅⢤⠠⣤⠈⣭⠑⣆⡀⠴⠶⠆⠈⠀⠀⠀⠀⢀⣀⣀⣀⡉⢨⣁⠤⢐⣒⣬⣧⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠤⠐⠉⠉
⣄⡵⠖⠛⣧⠝⠃⢠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⣤⣤⣤⣤⠔⠈⠰⢃
⣍⡷⠞⢋⠁⢀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢋⠴⠈⡠⠲⡞
⡠⠴⣞⣭⣾⠯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠆⠽⠖⠛
⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠦⠶⠶⣶⣤⣶⣮⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠠⠀⠀⠆⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣛⠲⢦⣤⣉⡉⠀⠀⠒⠤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠛⢌⣥⣀⣴⠶⠃⣀⡈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣍⠳⣄⣄⠈⠙⠛⠶⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢈⠁⡀⢰⣦⡀⣤⠻⣷⡾⠿⢶⣿⢿⣿⣍⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠲⠆⢤⡀⢀⣀⡀⠀⠀⠀⠀⢀⠀⣀⣠⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⢿⡿⠿⠶⢯⣄⠀⠀⠀⠀⠀⠁⠢⠌⠀⠂⢶⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⡄⠀⠀⠀⠊⠹⣿⣿⣦⣄⣀⣀⠀⠀⠀⠈⠀⠀⠁⠀⠉⠈⡀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣄⣠⠀⡁⠉⠛⢻⣛⣿⣿⣿⣿⠛⠉⠋⢈⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⣿⠿⠿⢿⣟⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢾⣿⣿⢻⡆
⠀⠀⠀⠀⠀⢘⣄⡀⠀⠀⠀⠀⠛⠿⣿⣤⣍⣉⡙⠛⠛⠛⠓⠚⠛⣛⣛⣿⡿⠃⠰⣿⣶⣿⣿⢧⡨⠝⣄
⠀⠀⠀⠀⠀⣸⠧⠭⠕⠂⠀⠤⢀⠀⡀⠈⠉⠉⠉⣿⣿⣻⣿⣟⠉⠉⠉⠁⠀⠀⠀⠘⢿⣿⣿⣦⠑⢄⠈⠳⢤⡀
⠀⠀⠀⠀⠀⢻⣕⠒⠤⠤⢤⢀⣁⣀⣀⣀⣤⣀⣌⣹⣿⣟⣟⠛⠧⠀⠀⠄⠄⠀⠒⠀⠀⠿⢿⣿⣷⡌⠳⣄⠀⠐⠰⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⠤⠤⠤⠤⠤⠴⠞⠛⠻⣿⡛⠲⠲⠦⠶⠤⠀⠀⠠⠀⠤⠬⢽⣿⡇⠀⠨⡙⠢⣄⡈⠀⢯⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠆⠤⠤⠤⠤⠤⠦⠒⠋⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⡌⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠇⠀⠈⠀⠀⠀⠀⠀⢀⠐⠁⠠⠛⠉⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠀⠘⠶⣤⡈⠐⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠀⠀⠀⠈⠙⠳⣄⡀⠀⡠⠃⡀⠊⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⢶⡦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⠀⣠⠟⠛⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠍⠓⢤⡈⠐⢄⠰⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠁⠀⠘⢿⡄⠁⠈⠳⣄⠙⠢⠄⠀⠀⢀⠞⠁⠖⠁⠀⠀⠀⠉⠳⣄⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⢀⠀⠀⠙⢦⡀⠀⠀⠉⠢⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠹⠛⢷⠀⠙⠦⣀⠀⠀⠈⢿⣥⣄⠀⠀⠀⠉⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠔⠠⠀⠀⠸⠖⢳⠀⣨⣷⣄⡀⠂⠀⠈⠳⠤⠀⠀⠀⠀⢂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠰⠁⠈⠁⠓⢂⡀⠀⠠⠾⠁⣀⣸⡙⠲⣄⡀⠀⠀⠀⢤⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠑⠊⡅⣰⠋⣼⢤⡀⠀⠻⠽⠋⢹⠂⠀⠉⠓⠦⣤⣠⠏⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠉⠡⠞⡁⢖⣮⡤⠂⠀⢀⢎⣀⣠⠶⣤⠴⢤⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡏⠺⣈⠅⠽⡏⠈⠋⠀⢀⡴⠋⠛⠒⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢁⠨⢄⠇⠠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡞⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⠀⡇⢀⠀⠀⠀⠉⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠉⠉⠉⠉⠙⠀⠀⠏⠀⢀⡤⠔⠀⣀⣀⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉⡡⠴⠊⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⠙⠳⢦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣛⡙⠁⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⣠⠞⠁⠠⠒⠉⠁⠠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢶⡶⠤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⢤⡀⠀⠘⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠁⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⠚⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡾⠁


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠈⠈⠁⠐⣄⠀⢰⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠂⢀⠀⣀⣀⣴⣀⠀⠀⠀⠈⠁⠀⠉⢴⣿⣇⣸⡿⠲⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣵⣿⣿⣿⣿⣷⡶⠒⠂⠉⢷⡀⠀⠉⠛⠉⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⡄⢙⠤⠤⠀⢼⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⡅⠈⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⡀⠀⢃⡄⢠⣿⡄⠀⠀⠈⣻⢿⣿⣿⡿⠟⠓⠦⣄⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⣤⣀⠔⠁⠀⠈⠈⢇⠀⠀⢀⢘⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡄⠀⠀⠀⡌⢦⣴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣮⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣷⡽⡗⠀⠀⠀⢸⣯⣾⣿⣮⢿⠀⠀⠀⢿⢯⣾⣿⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣝⡿⢿⣽⠇⠀⠀⠀⢸⣯⣿⣿⣯⡿⠀⠀⠀⢸⣿⡻⣿⣫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⢰⣶⣶⣶⣆⣀⠀⠀⠀⣶⣶⣶⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⣿⣿⣷⡀⡀⠀⠀⢀⣾⣿⣿⣿⣆⣀⠀⢀⢀⣾⣿⣿⣿⡆⢀⠀⠀⠀⢸⣾⣷⣿⣾⠜⠀⠀⢰⣾⣿⣿⣾⡼⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣟⣀⣻⣿⣿⣿⣿⣿⣿⣿⣃⣻⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣘⣻⣿⣿⣋⣀⣀⣀⣈⣻⣿⣿⣟⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢲⣶⣶⣶⣶⠀⣶⣶⣶⣶⡆⠐⣿⣿⣿⣿⣿⣿⣿⡿⠀⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⡆⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠷⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠿⠿⠿⠇⠰⠿⠿⠿⠿⠇⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⠀⢻⣿⣿⣿⣿⣿⣿⡇⢠⣤⣤⣤⣤⡄⢠⣦⣤⣤⣤⣤⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣧⣀⣿⣿⣿⣏⣀⣿⣿⣿⣿⣿⣿⣇⣨⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠛⠛⠛⠛⠿⠯⣥⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠒⠛⠛⠋⠋⠙⠓⠲⢤⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⢠⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠈⢧
⠀⠀⢺⣄⡴⠷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠘⡆
⠀⠀⣸⣿⠇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢻
⠀⠀⡿⠙⠒⠒⠊⠀⠀⠀⠀⠀⢰⣟⠉⠻⠆⢠⣾⠛⠛⢷⣆⠀⢰⡿⣇⠀⠀⣿⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢰⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡙⠛⢷⣆⢸⣇⠀⠀⢸⡿⢠⣿⣀⣿⣆⠀⣿⠶⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠷⠶⠾⠋⠀⠻⠶⠶⠟⠁⠾⠃⠀⠈⠿⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠈⣷⣠⣀⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⡇⠀⠈⠹⡄⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣄⣾⠃
⠀⢻⣁⣀⣀⣼⠧⣤⣀⠈⠛⠳⠶⠶⠖⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⣷
⠀⠀⠈⡙⡅⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⠴⠃
⠀⠀⠀⣧⠷⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⠴⠶⠖⠚⠛⠉⠉⠉⠉⠉⠉⠙⠛⠒⠶⠦⢤⣤⣤⣄⣀⣤⣤⡶⠟⠋
⠀⠀⠀⠘⠻⠶⠶⠖⠋
⠀⠄⠤
⠀⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣄⢠⡀⣠⣄⢀⣤⡀⣠⣄⢠⡀⠀⣠⡀⣤⣤⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣼⢿⡇⣿⣾⠈⠃⡇⣿⢸⡇⠀⡿⣇⢸⡇⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⡿⢸⠟⣧⡿⢺⣴⠇⣧⡿⢸⣥⣴⠗⢿⢸⡇⢸⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⠤⠤⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⡽⢇⡠⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡂⠀⠀⠀⠈⠉⠉⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠂⠀⠤⠀⠀⠀⠀⠄⠀⠀⡰⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡞⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢩⠔⠒⠞⡰⠁⢠⣟⠁⠀⠠⠤⠤⣄⢀⣰⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⠁⢠⠇⠀⠉⠁⠒⠐⠊⠀⠠⢙⠥⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡖⠀⠀⠓⠂⠤⠼⠁⠀⡿⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡶⠒⠦⢤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠠⠤⠤⣄⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⠊⠙⠿⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠯⡀⠀⠀⠀⠀⠀⠀⠙⠟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠐⠊⠉⠀⠀⠁⠀⡈⠒⢄⠀⠠⠔⡈⠄⣂⠤⠌⡙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠈⠢⡀⠀⠀⠀⠀⢀⠠⢓⠀⠁⠒⢹⠀⠀⠀⠀⠀⠀⠀⠄⠙⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⡏⠀⠀⠀⠀⠀⠀⠂⢀⠑⠄⠐⣨⡔⠂⠅⠒⠀⠂⠀⠑⠤⡀⠀⠀⢀⠠⠀⣈⣀⡼⠿⠃
⠀⠀⠀⠀⢀⡤⠦⠖⠛⠈⠐⠀⠈⠢⢀⠀⠀⢀⠠⠒⠈⠀⢀⠈⠀⢇⠀⠀⠀⠀⠀⠀⠁⣢⠈⠀⢈⣵⠾⠟⠋⢁⣠⠴⠂
⠀⠀⡏⠁⠀⠠⡀⠀⠀⠀⠀⠀⠈⣰⠄⠁⠠⠔⡾⠁⠂⠁⠀⠀⠀⢀⠁⠢⡀⠀⠠⢒⡠⣤⡤⠖⠋⢀⣀⣴⠟⠋⠁
⠀⠀⠳⣄⠈⠠⢌⠐⢄⠀⠤⢐⡡⢄⡢⠤⠄⡀⠑⢄⠀⠀⠀⠀⠀⡀⠜⠂⠀⣑⣾⣿⠟⢉⣀⣶⠶⠛⠁
⠀⠀⠀⠈⠻⢦⡀⠑⠠⡀⠂⢹⠀⠀⠀⠀⠀⠀⠐⠀⡈⠂⢀⢒⡩⢔⣪⠶⠟⠋⣉⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠐⢄⠑⠤⡀⠀⠀⣀⠄⠂⡉⠀⣠⡵⠚⠋⣀⣤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠉⠢⣌⠀⠨⠔⣊⡤⠒⠋⡀⣠⡶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠑⠔⠋⢁⣢⡾⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣼⠞⠋⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⣀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⢀⣀⢠⣤⣤⣴⣶⣶⣤⣄⠀⢨⡿⢯⡽⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣀⢰⣤⣄⣎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣷⣿⣿⣿⡻⠆⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣖⢳⣶⢢⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡯⣯⠻⢿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣆⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠠⣹⣿⣿⣿⣿⣿⣿⠿⣿⣇⠜⣐⣲⣦⠐⢿⣿⣿⣿⣿⣿⣶⢟⣻⣿⣿⣿⣿⣿⣿⣤⡙⢿⡧
⠀⠀⠀⠀⠀⠐⠹⣿⣿⣿⣿⣿⣧⣾⣭⣿⣟⠝⠇⢱⠇⣿⠿⣆⡿⢫⣶⠉⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠗
⠀⠀⠀⠀⠀⠘⣴⢿⣿⣿⣿⣿⣿⣟⣿⡋⣩⣶⣿⡄⡘⠻⡄⣹⣗⠉⠁⠀⠀⠉⠙⠛⣿⣿⣏⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠘⢿⡟⢿⣿⣿⣿⣶⣡⣾⣷⣦⡀⣽⡶⠂⠠⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡏⠛⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⣉⢙⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⣈⣻⣿⣃
⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠈⣿⣿⣿⣿⡿⣿⣏⣛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⣿⣿⣽⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢽⠉⠦⠙⠻⢿⡿⢉⣝⡟⢿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⢫⡄⡹⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠀⠉⠒⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⡑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

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