Alarm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠙⣧⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡈⢷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣾⠏⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠱⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠁⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠘⠀⠞⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢀⠎⠀⠀⣀⠶⠋⠁⠀⡈⠂⠀⠀⠀⠨⡀⠀⠉⠳⢎⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⠀⠀⠁⠀⣀⢄⣈⠤⠐⠒⠒⠒⠠⠌⣐⢄⡀⠀⠈⢂⠀⢳⠀⡇⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢧⠀⠇⠀⡀⠂⠂⣡⠞⠁⠀⠀⠀⢀⢀⠀⠀⠠⠀⠑⠺⡀⠀⠈⢀⡞⢰⠃⠀⡰⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣄⠟⠁⠀⠀⠠⠶⠀⠘⠚⠀⠀⢰⠀⠀⠀⠘⢦⣀⡈⠀⠀⠀⠈⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⢀⣠⡏⠀⠄⢰⢴⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠈⡆⠉⠉⠉⠉⠉⠉⠀⠋
⠀⠀⠀⠀⠀⡸⠀⠀⠠⡀⠙⣄⠀⠉⠉⢠⠁⠘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⢀⠔⠁⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠉⠑⡴⠁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⠛⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⡆⠈⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢤⠀⡀⠀⡌⠀⠀⠀⠉⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠖⢊⣠⣤⡴⠶
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢒⠀⢀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠁⠀⠀⠓⠀⢀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⡅⢀⠀⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣠⡴⠋⣠⠊⠀⢀⣀⠀⠀⢂⠜⠓⠤⠄⣀⣀⣁⣀⣀⠤⠔⠛⢄⢂⠀⠀⠀⢀⡀⠀⠑⢄⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠁⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠄⢰⡎⠈⢀⠀⢀⠈⠑⣄⠀⣠⠐⢚⣁⣠⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣮⠀⠐⢸⡇⠀⠀⢸⡇⢰⡏⣄⠀⣤⠀⠜⣴⠛⠄⠈⠉⡁⠀⢠⡆⠀⢀⡄⠁⢀⠀⢀⠀⠀⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠸⡇⢠⡀⠈⠇⣼⢁⣛⡀⠘⠀⠾⠁⣾⠀⢘⠛⠁⠀⢸⡇⠀⢸⡇⢠⡟⠀⣶⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠁⠘⣷⡀⢰⣇⡜⠸⣧⣀⡄⣲⡀⠹⠶⠶⠶⠞⠀⢸⡇⠀⢸⡇⢸⠇⢀⣍⣠⣼⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣠⠈⠷⠴⠞⠀⠀⠀⠿⠤⠞⠷⠤⠖⠀⠀⠀⠀⠈⠻⠶⠤⠖⣿⣄⡀⠀⢠⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣠⠶⠞⠛⣛⣛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣤⡴⠶⠶⠶⢮⣍⠛⠂⢀⣤⠶⠛⢛⣉⣉⣙⠛⠶⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠳⢶⣤⡀⠀⢈⣁⡴⠚⠋⠉⠉⠙⠛⠷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⢠⡾⣰⠏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠋⠙⠻⣆⣀⣀⣀⠀⠀⠀⠈⢻⡄⠈⣷
⠀⠀⠀⠀⣾⢡⡟⢠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢿⣷⣶⠞⣩⣭⣭⣉⡛⢶⣄⠀⠀⢻⠀⣿
⠀⠀⠀⠀⣿⢸⡃⣿⠀⣤⡤⠴⠶⢚⣩⣤⡀⠀⠀⠴⠾⠿⠶⠿⠿⠿⢿⣿⣦⡙⣧⠀⠘⠀⣿
⠀⠀⠀⠀⠻⠸⣇⠸⣦⠀⠀⠀⠘⣆⢻⣿⠷⡠⢂⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣭⣙⡻⢶⣤⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢃⡸⠿⣦⣄⣀⣈⡤⢖⣫⠞⢋⡭⠶⠛⠉⢉⡉⣉⣉⠉⠛⠻⢿⣿⣷⣮⣙⢷⣄⠙⣦⡀
⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢁⠴⠯⢍⠝⠉⠠⡆⢠⡆⢸⡇⣿⣽⠀⠀⢠⡄⠈⠻⣿⣿⣷⡝⢷⡈⢷⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⡇⠸⢿⠸⣷⠖⠀⢸⡇⠀⢀⡈⠻⣮⢻⣆⢻⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣰⠶⡆⠃⠀⠁⠀⣾⠀⣷⠀⠀⠘⠁⠀⠏⣿⠀⠙⣧⢻⣆⢻⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡐⠀⠀⡇⢿⣀⡟⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⢀⣴⣧⡆⠀⠸⣇⢻⡄⣧
⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡇⠀⠈⠁⠈⠛⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⢿⠘⣧⢸
⠛⢠⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⢰⠛⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⢸⡆⣿⠘
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠸⠶⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆⢀⡼⠁⠀⡀⠀⠀⠀⠀⠀⠘⣧⠅⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⢸⠀⡿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠀⡟⣻⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠏⡇⠀⠀⠆⢀⠇⠘
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡄⠸⣧⠀⢠⡟⢳⠀⠀⣤⣤⣤⠀⠀⣠⠀⠀⢀⠶⠆⠘⠛⡟⠀⠀⠀⠘
⠀⠀⢀⠘⣯⣉⣉⣉⣉⣉⣉⡈⢿⣆⠹⣧⡈⠳⠞⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠸⣆⠀⠀⠀⠃⡀
⠀⠀⠘⣧⠈⢻⣭⣭⣭⣭⣭⣥⣤⠻⣷⣌⢿⣦⡀⠀⠀⢀⡏⠀⠀⡏⢻⠀⢠⣤⠟⠀⢀⡠⠮⡭⠋
⠀⠀⠀⠈⠳⣄⠙⢷⣶⠶⠶⠚⠛⠶⣾⠿⢷⣯⣻⣶⣤⣈⠀⠀⠀⠻⠞⠀⠀⣀⣠⠖⣋⡴⢊⡕
⠀⠀⠀⠀⠀⠉⠳⠦⠉⠻⠃⠀⠀⠀⠘⣧⣀⣉⠛⠿⢿⣿⣿⣿⣷⣶⣾⣿⣿⠿⠞⣋⣥⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠤⠾⣷⣶⣒⣒⡂⠀⠀⠀⠉⣉⣉⣀⣤⠴⠖⠋⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢃⣄⠀⣜⡇⠀⠰⠆⠀⠀⠀⠀⠀⠰⡄⠀⣿⡀⠖⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡽⠞⠋⠉⠭⢝⠢⡀⠀⣖⢲⡤⣄⡤⠒⠒⠛⢧⣞⣳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⡤⢒⡶⢋⣀⣈⣽⣖⣧⡀⠀⠀⠀⠀⠉⠛⢄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢺⢀⡤⣚⣩⠿⠛⠉⠉⠉⠉⠉⠙⠳⢤⣈⢳⣄⡀⠀⠀⢸⡌⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢋⡽⠋⣀⣴⠖⠛⠉⠉⢹⠛⠒⠤⡀⠈⠻⣟⠛⠲⢤⣀⣡⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢣⡀⢠⠙⣅⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢳⡀⢇⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣇⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠂⠀⠀⠀⠀⠸⡊⢂⠀⠀⠀⠀⠀⠀⣀⣧⠀⢹⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢇⡏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⣜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⢍⣒⠤⠀⠰⠀⠀⠀⣀⡴⠊⠁⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⢤⣈⠉⠛⠒⠛⠋⠉⢁⣀⡤⢞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢱⣀⣤⠮⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠀⠀⠴⣤⣀⡎⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠟⠉⠀⠀⠀⠀⠀⠈⠳⡀⠀⠨⢥⡬⠇⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠛⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⣸⡇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠠⣔⠩⠔⠐⠂⠉⠉⠁⠀⠀⠈⠉⠉⠐⠒⠤⠍⣲⠄⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠄⢊⠗⠊⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠺⣑⠠⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠊⡠⠞⠁⠀⠀⢀⡠⠔⠂⠉⠀⠀⢀⠈⢁⡀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠈⠳⢄⠁⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠁⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠈⠢⣄⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠳⡀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⠄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠃⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⡀⠁⠀⠘⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠘⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⣀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠰⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡹⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣠⠞⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠓⢄⡠⠀⠀⠀⠀⠀⠀⠏⠰⠀⠀⠀⠀⠀⠀⢄⡠⠚⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠠⠄⠀⣀⣀⠠⠄⠊⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠢⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠔⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⠁⠈⠁⠒⠠⠤⠄⣀⣀⣀⡀⠠⠤⠄⠒⠈⠁⠈⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⢷⠾⠷⠾⠷⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢣⣰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠞⣱⠃⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠈⢳⠦⡀⠀⠀⠻⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⣀⠀⠠⡧⠴⢤⠇⣠⣒⠛⢁⠎
⠀⠀⠀⠀⠀⠈⠫⣀⣨⠆⠀⠀⠀⠙⠃⠹⣸⠀⠀⠀⣠⣠⢄⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠚⠉⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠔⠊⠙⠲⢍⠀⣴⣒⠃⠁⢈⣍⡷⠀⡠⠚⠉⠉⠉⢢⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⡇⠀⠀⠀⠀⣨⠇⠈⠈⡇⠀⢸⠀⠀⠀⠧⣀⠀⠀⠀⠀⡇⠀⠘⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠧⠤⠴⡂⢫⡤⠴⠒⢲⡃⣠⣼⠋⣱⠖⣲⠏⢨⡓⠒⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⢀⠠⢀⣀⢀⠀⠀⠀⢀⠀⠀⣘⡆⠈⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠊⠁⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⠁⠀⢀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⠀⠀⠓⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠀⠀⠀⢢⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠏⡢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠄⠀⢠⠀⢀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠉⠂⢄⣀⠀⠀⠀⠆⠀⠀⠀⢘⠁⠐⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠤⠶⠤⣄⣉⣉⣉⣛⡤⠬⠥⠦⣄⡄⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢀⢀⣤⣤⣤⣤⣤⣄⣀⡁⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⣪⠒⠉⠉⠀⣄⠀⠉⠉⠙⠺⢕⢦⠀⠋⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣁⠀⣸⠇⠸⠀⠀⠀⠏⠀⠀⢰⠀⠀⠀⣧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣯⣅⠢⡀⠀⡶⢦⢄⣈⡤⣖⣒⣿⣦⣦⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢠⢫⣾⣿⣿⣿⣿⣯⡵⠃⣀⣈⣷⣾⣇⠾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢆⠻⣛⣽⣿⣽⣿⠿⠿⣿⣿⣿⣯⣭⣙⣳⣾⢿⣿⣿⣿⣿⡽⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠈⢛⣽⠟⣫⣵⠞⠛⠋⠉⢹⠛⠛⢿⣟⠿⣷⣽⣿⣯⣟⡻⢃⠃⠀⠀⣀⣠⣤⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠂⢠⡞⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡝⣿⣿⡅⠀⠛⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡟⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣷⡈⣿⣿⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⣼⢡⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡇⢸⡇⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⢸⠀⡇⠁⠀⠀⠀⠀⠸⡋⡀⠀⠀⠀⠀⠀⠀⡀⣿⢸⣿⢸⡇⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉⠀⠀⢻⠸⡆⣇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⢸⡇⡇⠀⠀⠉⠙⠛⠲⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⢃⣾⡟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠘⣦⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢃⣾⠟⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⠄⠈⠳⣿⣷⣭⣒⠤⠀⠰⠀⠀⢀⣠⡴⢟⣥⣾⣷⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣈⣿⢿⣯⣭⣽⣿⣿⣿⣿⣿⣿⢿⣟⠉⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⡉⠉⠉⠉⠉⠉⠀⠀⠹⣻⡄⠀⠙⣆⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⠋⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠴⠶⠚⠛⠳⠶⠶⣦⣄⡉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⠲⠶⣦⡄⠀⡞⠉⡇⠀⣠⡶⠞⠛⠛⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⠴⠞⠁⠀⠀⠀⢹⡆⣇⠀⡇⣼⠧⣤⣤⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⠒⠳⠿⣤⣀⠀⠉⠛⠶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣤⠾⣋⣤⠴⠶⠚⠛⠛⠛⠛⠻⢦⣄⠀⠈⢳⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠚⠛⠋⣩⡶⢛⣩⠴⠖⠛⠛⠛⠛⠛⠶⣦⣄⠙⢷⣦⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠟⢁⡴⠋⠁⠀⠀⢰⣉⡷⠀⠀⠀⠀⠉⠻⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣾⠋⣰⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠃⡼⠁⠀⢠⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣸⠃⠀⠀⠀⠈⠓⠦⣄⣙⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⢠⡴⡄⠀⠀⠀⠀⠀⠻⣄⡼⣄⠀⠀⠀⠀⠳⠞⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡀⠀⠀⠀⠀⢰⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⣠⠏⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢆⠙⢦⣀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠁⠀⢀⡼⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⠀⠙⠳⠶⣤⣤⣀⣀⣉⣉⣠⣤⣴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢻⡄⠀⠀⠀⠀⠉⠉⠉⠁⣀⣠⣤⡶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠳⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡷⠖⠚⠛⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠳⠶⢤⡞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⡊⠉⡆⠀⠠⡊⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠔⠊⢁⣀⣠⠠⠬⠯⠀⣀⣀⡈⠑⠢⣀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⢫⠷⠒⠉⠁⣀⠀⣤⣤⢄⣤⢀⣀⠈⠉⠒⠴⣹⠂⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⡠⠚⠁⣀⣔⠶⠙⠋⠉⠉⠁⣄⠉⠉⠘⠻⢷⣠⣀⠈⠓⢄⠈⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⡪⠋⠁⢰⡆⠀⠀⠠⡆⠙⡦⠀⡄⠀⣤⠈⠙⢕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠊⠀⠀⠀⢸⡇⢹⠀⠀⠃⠲⠆⡰⠀⢻⠀⠀⠀⠀⠑⣷⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣯⠃⠰⢆⠀⡠⡈⠁⠈⠀⠀⠀⠀⢠⠁⠀⠉⠀⠀⠰⡄⣤⠘⣵⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣾⠃⠀⠀⢸⠸⣇⠇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠮⠄⠀⠀⠘⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣜⣀⣀⣀⣀⣀⣀⡀⠠⡄⠀⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣇⠀⠋⠈⠞⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠋⠈⠇⠛⠀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⢠⠠⠀⢰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠱⣧⠀⣠⠸⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢀⡀⠀⣼⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠹⣧⡀⠀⠀⠀⠀⣩⠁⠀⠀⢀⠀⠀⠀⢰⡂⠀⠀⠀⠀⢁⣼⠏⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢝⢄⠀⠀⣤⠁⠀⠀⠀⡟⡦⠀⠀⠀⢋⣄⠀⠀⡠⡮⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠯⣦⣄⡀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣡⣴⠟⠊⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⢄⡀⠁⠚⠳⠶⠦⠶⠦⠴⠶⠚⠓⠉⠀⣠⡔⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⠔⠁⠀⠉⠑⠒⠤⠤⢀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠑⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠻⢷⣦⣀⣀⣤⣤⣤⣤⣀⣀⣠⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣷⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠘⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣶⣶⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠻⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠋⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠒⠚⠉⠉⠉⠉⠉⠒⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡝⠶⠛⠻⠿⣷⣶⣄⠀⣤⣤⣀⡠⣴⡬⠾⠟⠚⠳⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⣀⣤⡶⠾⢿⣿⠊⠛⣿⠿⢿⠿⠷⣦⣤⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣏⢻⡀⣨⣟⣁⡀⣴⣿⠄⣈⣳⣄⣼⡿⢛⣿⣶⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠛⡿⣿⣿⣦⣿⠿⠖⢀⣀⣁⣀⣀⡀⠐⠿⣿⣥⣿⣿⠋⠛⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⡹⣫⣿⣋⡴⠞⠋⠉⠉⠀⠄⠉⠉⠙⠲⢦⣝⢻⡙⢯⠒⠺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣩⡾⠋⠀⠀⠀⠀⠀⣰⠺⠀⠀⢠⡞⠂⠈⠳⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⠀⠀⠀⠀⠆⡇⠀⠈⠉⠀⢠⠽⠇⠀⠀⠀⡈⢧⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠃⠀⠀⣾⣷⣀⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢸⡋⠑⠈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡟⠀⡀⣀⠉⠁⠙⠦⠦⣀⣀⢠⡏⠀⠀⠀⠀⠀⢀⡀⣆⢹⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⢰⠂⡿⠀⠀⠀⠀⠀⠈⣷⣾⠀⠀⠀⠀⠀⠀⢈⡇⠛⢸⣷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢿⣿⠞⠀⠀⣀⠀⠀⠀⠀⠠⠋⠈⠣⡀⠀⠀⠀⢀⡄⠀⠂⣼⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⠠⠄⠿⠀⠀⠀⡰⠁⠀⠀⠀⠑⣄⡀⠠⠺⣧⣄⡖⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠃⠐⢆⠀⠈⡅⠀⠀⢠⡆⠀⠀⢨⣦⣀⣶⣿⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⣦⡀⠼⣦⣤⣀⣀⣘⡋⢀⣠⣢⣾⡿⢟⠕⡊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣭⣓⣻⣿⣿⣛⣛⣩⣵⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⠛⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⣠⠤⣴⣲⣽⠇⠀⠀⢀⡀⠀⠀⠀⠠⣤⠖⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣠⠤⣒⣺⣭⣷⣚⣻⣭⣥⣤⣤⣤⣤⣟⣙⢻⡆⠀⠀⢠⣴⡶⠶⢦⣄⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⣯⠵⠞⢋⣩⡼⠟⠋⠉⠀⠀⣀⣀⣀⣀⠀⠈⠉⠙⠳⢦⣄⠈⢷⡱⡄⠀⠈⠛⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣠⠞⠋⠁⢀⡠⠴⣒⣯⣭⠭⠶⠦⣬⣭⣙⡒⠦⣀⠀⠈⠳⣦⡻⣜⣆⠀⠀⠈⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠁⢀⡠⣚⡥⠖⠋⠉⠀⠀⢠⡆⠀⠀⡤⢨⣝⠳⣬⡳⣄⠀⠈⢳⣿⣮⢦⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⡴⣫⠞⠉⠀⠀⠻⠂⠀⠀⠀⠀⠀⢈⡁⠶⡟⠀⠈⠻⣎⢣⡀⠀⠹⣏⢷⡳⡀⠀⣸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢏⡼⠃⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⣀⡘⢧⠹⡄⠀⢹⡎⢳⣝⣤⠏⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢣⡞⠁⠲⠞⠋⠀⠀⠀⠀⠀⠀⠀⢰⣏⡾⠁⠀⠀⠀⠀⠉⠁⠈⣧⢹⡀⠀⣿⠀⠙⠁
⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠠⡇⡇⢀⣠⠀⠀⠀⠀⢠⡾⠋⠛⣲⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢈⡇⡹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⣧⠀⠁⠀⠀⠀⠀⠀⠛⠶⠛⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⢇⡇⠀⣼⠁⡶
⠀⠀⠀⠀⠀⡀⠸⣇⠀⠸⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⠀⢠⡟⡼⠀⢠⡟⠀⠃
⠀⠀⠀⠀⠀⠹⠀⢻⡄⠀⢳⣻⡄⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣰⡿⢣⡟⡼⠁⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠳⣽⣦⡀⠀⠀⣤⣶⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣵⢫⠞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡻⣄⠀⠈⠻⣿⢦⣀⠻⠽⠂⠀⠰⠇⠀⠀⠘⠃⣀⣠⠶⣋⠕⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠐⣯⣀⣽⠟⠻⣦⡀⠈⠙⠯⣟⣓⠶⠶⠶⠦⠤⠶⠖⣛⡭⠔⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠷⣤⣀⡀⠈⠉⠉⠉⠛⠛⠋⠉⠀⣀⣠⡴⠞⠉⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠶⠶⠶⢺⣿⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⣀⠔⢊⠩⠔⠒⠃⠈⠘⠒⢄⡈⣧⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠈⠀⠰⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⡆⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⣠⣡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣷⡀⠀⠤⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠁⠀⠀⠀⠈⠢⡀⠀⠀⢀⠤⢤⠞⠉⠁⠀⠀⠉⠲⢤⡌⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⡇⠀⣀⣠⣤⢶⣶⣾⣿⡷⠀⠀⠀⠑⢲⡗⢤⣀⠀⠀⠀⠀⠀⠈⢑⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣆⢹⡘⣿⡽⠟⢶⢲⠲⠶⢶⣿⣶⣤⣬⣱⣦⣄⡀⠀⠀⣔⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⡿⢿⣛⡭⠖⠛⠋⠉⢉⡍⠛⠒⠯⣽⢿⣷⣿⣿⠛⠻⢿⡒⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢻⣅⠀⠀⠀⠙⢦⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡴⠣⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠃⠈⢣⡙⣆⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡧⠄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢑⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⢠⡄⠀⠀⠀⢀⣠⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡇⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢠⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢷⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⣠⡎⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠴⢿⣷⣋⡄⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⣌⣒⣤⠼⣀⣀⣀⠠⠤⢤⠐⣐⣩⣶⠞⣋⣉⠁⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⣮⠟⠉⠛⠶⢿⣶⣾⣾⣿⣿⡻⠟⠛⠒⣰⠁⠿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⡄⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠛⠶⣄⠀⠀⠀⣴⡶⢦⡀⠀⣠⡴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢠⣾⡟⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠂⢀⣠⡴⣞⣩⡤⠶⠒⠚⠛⠛⠓⠒⠶⢤⣄⡈⠳⣤⡀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠚⣩⡵⠛⣩⡤⠶⠚⠛⠉⠉⠉⠉⠙⠓⠲⢤⣍⠓⢦⣝⠲⣄⡈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⠳⣄⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠹⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⡿⠟⣿⠟⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣷⣤⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡀⠹⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢦⣙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⣌⡙⠲⠦⢤⣤⣤⣤⣤⠤⠴⠖⢋⣡⢴⣞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠛⠉⠉⠛⠒⠶⠶⠦⠴⠶⠶⠒⠛⠉⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠟⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠏⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢨⡅⠀⢠⡉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠉⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⣤⣀⣀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣤⣤⣾⣟⣛⡒⠶⣤⡀⣤⣀⣐⡄⣰⠞⠉⣁⣤⣶⣶⣿⣿⣿⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣷⠞⠃⠉⢻⡏⠁⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣟⢦⠈⢳⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠃⣰⠃⣿⣿⣿⣿⣿⡿⢋⣠⡴⢶⣚⣻⣿⣿⣿⣿⣶⣦⣭⣿⡿⣿⣿⣿⣿⢸⡇⠘⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⠀⢿⣿⡿⢛⣿⠟⣫⣴⣾⠿⠟⠛⣛⠛⠛⠛⠿⣿⣿⣿⣷⣤⡉⠛⠿⣼⠇⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣤⠞⠋⣠⠋⣡⣾⠿⢋⡀⠀⠀⠀⣿⠀⠀⠀⠠⠆⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⡿⠁⠀⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡟⢠⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠿⠇⢸⢿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣼⣿⡄⠉⠁⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡆⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⢃⣿⠇⢀⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⡄⠘⣿⣿⣷⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣾⠏⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠻⣄⠈⢿⣿⣿⣦⣀⠀⣤⠀⠀⠀⣠⡀⠀⠀⠐⢂⣤⠞⣱⡿⠋⠀⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⠿⣿⣿⣷⣦⣤⣀⣀⣘⣁⣠⡤⠶⣋⣵⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⣿⠿⢿⣿⣿⣿⣿⣿⣶⣾⠿⠟⠉⠹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠈⠉⠉⠉⠀⠠⠤⠖


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢳⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⠉⠀⠀⠀⠉⠑⠦⡀⢔⠉⠉⣉⣩⠆⠀⣠⠖⠋⠁⠀⠀⠈⠙⠲⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠉⢹⠁⠀⠀⠸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣠⠴⠊⣁⣠⠤⠤⠼⠦⠤⠤⢤⣀⡉⠓⢤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⡤⠖⣻⠿⠓⠉⠀⣀⣀⡤⠤⠤⠤⣀⣀⡀⠈⠑⠺⢿⡓⠢⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⡤⠚⠁⣀⠤⠒⠉⠁⠀⠀⠀⡀⣀⠀⠀⠉⠑⠲⢄⡀⠉⠲⣄⠉⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⠞⠁⠀⠀⢀⣤⠀⠀⠀⣏⣹⠃⠀⠀⣀⠀⠀⠙⢦⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠁⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠙⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠎⠀⠀⣰⡔⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡞⠀⠀⠀⠙⠳⠾⣀⢰⣦⣄⠀⠀⢰⠇⠀⠀⣀⣤⣶⠟⠻⠷⠀⠀⠘⡆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⢀⡤⡄⠀⠀⠈⠉⠛⠿⣿⣦⣼⣤⣴⡾⠟⠉⠀⠀⠀⠀⢠⡀⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠈⠓⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣘⠇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢣⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣆⠀⠀⢠⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢧⠀⠀⠘⠛⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠳⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣴⡀⠀⠀⠀⢙⣳⠀⠀⠀⡰⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠷⠏⠀⠀⠀⠈⠁⢀⡠⠊⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠈⠙⠲⠤⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣹⠶⢤⣀⡀⠈⠉⠉⠉⠉⠉⠀⢀⣠⠤⠾⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡜⠁⠀⠀⠀⠈⠉⠉⠓⠒⠊⠉⠉⠁⠀⠀⠀⠙⣌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡴⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⢦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⣰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⣄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢈⡷⠟⠚⠋⠉⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡖⠟⠋⠙⠒⡿⣾⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢫⠂⠁⠀⠀⠀⢀⣠⠤⠾⠂⣀⣀⣀⣀⣀⣀⠘⠧⢤⣄⡀⠀⠀⠀⠈⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⣀⣴⠚⣉⡤⠖⠚⣉⠩⠤⠄⠤⠤⠉⣉⠓⠲⢤⣉⠓⣦⡀⠀⠀⠀⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⢑⡴⠋⡡⠴⡈⠁⠀⠀⢰⠀⡀⠀⠀⠀⠀⢑⠢⢈⠛⢤⡋⠳⣄⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠁⡴⠋⡠⠊⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢆⠈⢣⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡞⢡⠚⠠⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⡈⢧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠃⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⡠⢄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⠆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⠀⠀⡀⢀⡔⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡁⠢⣀⡐⠀⠀⠀⠀⠀⠙⠤⠁⠀⠀⠀⡀⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢉⡷⢤⣉⡐⠀⠀⠠⠤⠤⠄⠀⠐⢂⣁⠤⢾⣁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣡⠎⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⠻⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⢀⠁⠀⠠⠄⠄⠀⠀⢈⣁⡐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠞⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⡈⣿⣶⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣭⠿⠿⠇⡇⠀⠀⡀⡐⢌⠠⡆⠁⡀⣤⡡⠆⠐⢀⡤⡂⠁⠀⠀⠀⠀⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⡀⠃⠀⠀⠁⠥⢙⠙⠀⠀⠙⣯⠀⠀⠀⢆⠀⠀⠀⡀⠄⡀⠄⠀⠄⠴⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⣈⠔⠊⠁⠀⣀⣀⡀⣸⡀⠀⠀⠀⠈⠐⠀⠨⣃⠁⠈⢀⡠⠀⢁⡀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⢀⠀⠐⣾⠇⠒⠘⠃⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠃⠂⠒⠻⠁⠈⠂⠤⢈⡛⡘
⠀⠀⠀⠀⠀⠠⠐⠀⡁⠔⠊⠁⡔⡏⠁⠀⠀⣀⢤⣶⣶⣿⣷⣾⣾⣾⣿⣶⣶⣤⣄⡀⠀⠠⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠊⠁⠀⢀⣴⡾⢋⣩⡽⠅⠔⠒⢘⣯⠈⠁⠃⠙⠛⣿⡿⠻⢦⣄⡀⠂⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⡴⠟⠙⠷⠀⠀⠀⠠⢑⢂⠄⠀⠀⠀⢀⣼⠿⣄⠉⠻⣦⠀⠔⠈⡄⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⣰⡟⠁⠀⠀⠄⢡⠀⠀⠜⠙⠁⠀⠀⠀⠀⡀⠀⠀⠈⢳⡄⠙⣷⡅⡀⠀⡀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⡿⠁⢠⠏⠙⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠾⠋⢄⠈⢡
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⣼⡇⢀⡏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⣇⢸⡆
⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⣿⠀⢸⢤⠄⠸⠀⠀⠀⠀⠀⠰⡠⣭⢦⣼⠿⠿⠾⠟⠟⠛⠚⠛⣿⠻⠶⢿⠈⠇
⠀⠀⠀⠀⠀⢃⠀⠈⠀⠀⣿⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢘⣏⣗⡈
⠀⠀⠀⠀⠀⠈⠀⠀⢃⠀⢹⣇⠘⣄⡠⡀⢟⡄⠒⠀⠀⠀⠆⣸⠃⠀⠀⠀⡠⠀⠀⠀⠴⠄⣀⡐⡞⢀⣹⠽⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠻⣦⢺⣏⠀⠀⠀⠀⡄⠀⢰⢀⡟⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⢉⡟⠀⣀⣡⠃⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡘⠷⣄⢀⣄⠀⠀⠈⠋⣴⡄⠀⠀⠀⠈⣀⠀⠀⠀⠄⠟⠁⣠⡿⣳⠂⡠⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠢⡀⠀⠙⢧⡝⠻⣧⣀⣀⠀⠀⢴⡄⢀⣀⢀⢤⡹⢆⠨⠋⠡⣢⣼⡯⢪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣀⠠⠈⢳⠦⣤⣭⣛⣒⠛⠓⠚⠉⣉⠁⠬⣀⣠⡴⠫⠟⢋⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⡼⠛⠂⠥⢈⠀⠪⠉⡩⠛⠻⡻⠓⢂⠊⡉⠠⠅⠐⣩⠴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡜⠁⠀⠀⠀⠀⠈⠀⠐⠂⠀⠤⠤⠀⠠⠀⠀⠒⠈⠁⠈⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⡀⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⠠⠀⡠⠀⠔⠂⠀⠀⠀⠀⠊⠁⠀⠀⠀⠂⠁⠌⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⠉⠈⠑⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣶⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣀⣀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⣀⣸⣇⣀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣯⣤⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣤⣽⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠋⣴⣾⠿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠻⣷⣦⠙⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠲⠄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠆⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠓⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣁⡀⠀⠀⠠⠄⠀⠀⢀⣈⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠿⢿⣾⣶⣶⣾⡿⠿⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠉⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡀⢤⣙⣷⣴⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣦⣼⣿⣿⣿⣦⠀⢤⣶⠇⠰⣴⣿⡃⠀⠉⠻⣥⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣩⡿⠁⢀⣽⠟⠁⡀⠤⠛⡒⠒⡂⠩⢿⣿⣶⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣸⣿⣯⡶⢿⣥⢖⡩⠖⠋⠉⢉⠉⠉⠩⠓⠦⣶⣍⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⢉⣴⠏⡴⠋⢠⣤⠀⠹⠼⠅⠀⢣⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⡾⢀⡠⡌⠋⠀⢀⠀⠀⠀⠀⠀⢨⠂⠁⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣾⠁⠈⠗⠛⠐⣶⣌⣦⣀⠀⠀⠀⠈⠠⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠰⣦⠀⠀⠀⠈⠻⣼⠀⠀⠀⠀⠐⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢁⡀⠀⠀⠀⠀⠈⣷⡀⠀⠠⣷⢀⠀⡞⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⣿⣿⣿⣤⠘⠟⠀⢠⠀⠀⡀⠻⢿⢦⠀⠀⢀⡼⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣿⣿⣷⣄⡀⠘⠀⠘⠿⠆⠀⠉⣀⠴⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣾⣿⣿⣿⢿⣿⣷⣶⣶⠖⠒⠚⢋⣡⡴⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠙⠙⢣⡿⢿⣿⣿⣿⣿⣾⣿⣿⣭⣤⣌⣹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⡠⠊⠁⡀⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠈⠢
⠀⠀⠀⠀⠀⠌⠀⠔⠁⠰⡡⠔⠒⠓⠒⠠⢄⠸⠤⠤⠤⠤⠤⠤⠀⡠⠄⠀⠀⠠⢔⣄⡠⡈⠡⡀⠑
⠀⠀⠀⠀⠀⠀⠊⠀⢀⠎⠀⠀⠀⠀⢀⠔⢈⡀⠤⠀⠤⠄⠠⠄⣈⠂⢄⠀⠀⠀⠀⠈⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡠⢚⠕⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠐⠱⡢⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠔⢁⠜⠁⠀⢀⠔⠂⠁⠀⠪⠇⠈⠉⠒⠤⡀⠀⠈⢣⠑⠠⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠔⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢢⠀⠀⠱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠆⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢡⠀⠀⢡⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢘⣀⠀⠀⠀⠀⠀⠀⠧⢇⠀⠀⠀⡀⢀⡀⣀⡄⠀⠘⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⠀⠀⢘⠚⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠈⠈⠉⠊⠇⠀⢀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⡄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠑⠄⡀⠀⠀⠠⡄⠀⠀⣀⠤⠊⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⣄⡀⠀⠀⠉⠀⠈⠁⠀⠉⠀⠀⢀⠤⡊⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊⠀⠈⠑⠂⠠⠄⠀⠀⠤⠄⠒⠈⠀⠀⠘⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠂⣄⠄⠐⠒⠀⠀⠈⠉⠉⠉⠯⢳⡄⠀⠀⠘⡄⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠐⠁⠀⢀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠻⣄⠀⠀⠃⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠊⠀⠀⠀⠉⠀⠀⠀⠠⢄⠀⠀⠑⠀⢀⣈⢢⡀⢀⡀⠘
⠀⠀⠀⠀⠻⠭⣁⣂⣀⠀⠀⣀⣷⠞⠉⠉⠉⠉⣱⢦⠀⠀⢒⠂⠊⠀⡠⠂⠁⠀⠀⠈⠙⢦⡁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠔⣪⠾⠓⠊⠭⠭⠍⠑⠒⠾⣄⠀⡀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⡱⠊⠐⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢝⢮⡢⡀⠀⠀⡸
⠀⠀⠀⠀⠤⠤⠔⠒⠉⠀⠀⠁⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⡙⣮⡢⡴⠃⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠏⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠟⠀⠀⠐⠘⡍⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢁⢡⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠄⠀⠀⠉⠛⠀⠀⠒⠺⠃⠀⠀⠀⠀⠀⠀⠐⢈⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡕⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣡⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠲⢭⣒⠂⠤⠤⠤⠐⣒⡩⠔⢫⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠀⠈⠛⠉⠉⠉⠉⠁⠀⠁⠉⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠩⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠔⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠜⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⠀⠐⢂⣾⣷⠚⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⣰⣿⣿⣿⣿⢟⡿⠁⠀⠀⠀⠀⠀⠇⠀⠸⠄⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠲⡙⢿⣿⣿⣿⣧
⠀⠀⣰⣿⣿⣿⡿⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢻⣿⣿⣿⣧
⠀⢰⣿⣿⣿⡿⠁⠈⠀⠀⣦⠀⡤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⢻⣿⣿⣿⣆
⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⠀⢧⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡁⠀⠀⠀⠀⢿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢢⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⡓⢘⠇⠀⠀⠰⢾⡷⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢱⡄⠀⠀⢸⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣼⣿⣿⣿
⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⢬⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡇⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⢻⣿⣿⣿⣧⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠂⣴⣿⣿⣿⡿⠁
⠀⠀⠀⢻⣿⣿⣿⣷⣝⣦⡀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⡠⢁⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⢐⠐⡆⠀⠀⠀⣠⢎⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢿⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡒⢠⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠄⢐⡇⠀⠀⠀⠀⢸⣿⡇⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄
⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⣼⣿⡿⠁⠀⠂⠀⠠⡄⠀⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠒⠀⠈⢿⣿⣆
⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⡇⢸⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⡇⠸⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⣿⣧
⠀⣿⣿⡇⠀⡀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⣇⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⠀⢸⣿⣿
⠀⣿⣿⡇⠀⠁⠀⠓⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠀⢸⣿⣿
⠀⢿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⢸⡏⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣸⣿⡟
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠉⠀⠀⠀⠀⠸⠇⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⡖⠀⡆⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⡑⠐⣅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⢻⣿⣷⡀⠀⠄⠀⠇⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠉⠁⠏⠀⠠⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣥⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⢀⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠠⠄⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠄⠀⠀⡠⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠠⠜⠀⠡⢬⠤⠤⠥⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠄⠾⠃⠀⠀⣠⠌⡀⠔⢂⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣄⢀⣀⣀⣀⣀⡂⠤⢀
⠀⠀⠀⠀⠀⡄⢀⡠⢤⠞⡱⢊⢄⡬⠖⠂⠉⠉⠉⠒⠢⣄⠀⠀⡀⠀⠀⠈⠡⣇⠈⠉⠙⠛⠛⠛⠛⠚⣣
⠀⠀⢀⠀⠀⣧⢈⡴⠃⠌⠀⡵⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠑⣄⠈⠤⢈⠄⠀⠻⢄⠀⠀⠀⠀⠀⠀⢈⠏
⠀⠀⠀⠀⠁⠐⠉⠀⡘⠀⢸⠁⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠹⡄⠈⡀⠀⠀⠀⠈⠣⡀⠀⠤⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢨⡧⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢱⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠉⢀⠇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢧⡀⠀⠀⠀⠋⠀⠀⠀⠀⠁⢠⠞⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⢑⠦⣀⡀⠀⠀⠀⣀⣠⡖⠋⢀⠎⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠈⠙⠛⠛⢚⣋⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣠⣤⡤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠉⠒⠢⠴⠾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠂⠀⠀⢠⠀⣤⢤⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠐⠐⠂⡰⠁⠀⠀⠀⠀⠠⠞⠀⠡⠴⡶⠶⠶⠶⠶⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠢⠤⡾⠃⠀⠀⠀⣠⠃⣀⠴⢂⡂⡉⠉⠈⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⢀⣀⣐⣂⠀⠤⢀
⠀⠀⠀⠀⠀⠀⡈⠀⢀⡠⣠⠞⡵⢊⠀⣪⡥⠂⠒⠒⠒⠂⠶⢄⡀⠌⠂⠀⠀⠀⠀⠘⠣⣷⠈⠙⠛⠻⠿⠿⠿⠷⠮⠶⡑
⠀⠀⠀⠀⠀⢰⡇⠎⣩⠞⢡⠊⠐⣡⠎⠁⠀⠀⡁⠀⠠⠀⢀⠀⠉⠣⡔⠌⢢⡀⠄⣈⠁⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠇
⠀⠀⠀⠇⠰⡀⢓⠜⠁⢀⠇⠀⡼⠁⠀⠁⠀⠀⢂⠀⠀⠀⠀⠐⠀⠀⠸⡆⠀⢣⠊⠀⠈⠀⠈⢦⡀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⢠⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⢻⠀⠘⠀⠀⠀⠀⠀⠀⠑⠤⠌⠁⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⢸⠀⠀⠀⠀⠀⠀⠀⡾⠂⠁⠀⠀⠀⢂⠀⢸⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢇⠀⠀⠀⠀⠀⢻⠿⠀⠀⠀⠀⠀⠀⠀⡜⠐⢰⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠑⣆⠀⠀⠄⠀⠨⠀⠀⠀⠀⠀⠀⢀⡼⡁⢠⣧⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠈⡳⢆⡀⠀⠀⠀⠀⠀⠀⡠⡴⠋⠀⡰⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡲⢌⡀⠈⠙⠻⠯⠼⠾⠾⠚⣩⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠙⠒⡶⠒⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠴⠶⠞⠛⠛⠛⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠾⠿⠛⢻⡟⠛⠿⠷⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⢀⡄⠈⣡⢤⡀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠐⢻⠀⠐⢺⠀⠀⠀⠀⠀⠐⠓⠂⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⢀⣸⡀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠧⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⢠⣿⠋⠒⢩⡇⢰⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠋⠙⣿⡄
⠀⢀⣿⠃⠀⠀⠼⠧⠈⠣⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣥⠀⠀⠀⠘⣿⡀
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠃⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘
⣿⠶⠆⠳⠤⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⠰⠶
⣿⡄⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠃⠀⢠
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⡟
⠀⠈⣿⡄⠀⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣹⠀⠀⠀⢀⣿⠁
⠀⠀⠘⣷⣀⣠⠐⣇⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠁⢤⣀⣿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠐⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠃⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠈⢓⡚⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⢶⣶⣤⣼⣧⣤⣶⡶⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠉⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠀⠀⠀⢀⣀⣀⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣰⡿⢾⣿⣿⣿⣆⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣄⣠⣬⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠭⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡋⣩⣤⡶⠶⠲⠾⠒⠲⠶⠴⣶⣥⣂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⣿⠿⡋⠉⠀⠀⠀⠀⠙⠃⠃⠀⠀⠀⠈⢙⠷⣄⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢠⡿⠋⠙⠴⠚⢀⠈⠀⠀⢲⡀⡇⡆⠀⠀⠀⢀⠋⠀⠀⠙⢄⢜⣷⡀
⠀⠀⠀⠀⠀⠀⢨⣾⠞⠀⠀⠀⠡⢮⡈⢦⠀⠀⣨⢧⠇⡇⠀⠀⢠⠎⠀⠀⠀⠀⠀⢑⣻⣷⡄
⠀⠀⠀⠀⠀⠀⣸⠃⡐⠀⢀⠀⠀⠀⣩⠒⠁⠀⠀⠀⠀⠠⠼⡀⠋⠀⠀⠀⢀⣌⠘⣭⡤⢹⣽⡄
⠀⠀⠀⠀⡀⣿⠏⠉⠃⠠⠶⠦⢤⠄⠀⠀⠀⠀⠀⠀⠤⣬⡁⠀⠀⠀⠰⣪⠕⠊⠀⢀⠀⠀⢻⣷
⠀⠀⠀⠀⠱⣾⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢰⠐⣿⠂⡂⠀⡥⠶⠞⠃⠀⠀⠀⠀⠀⠀⢴⣧⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡂⠰⠐⠆⣿⡇
⠀⠀⠀⠈⢀⣿⠈⠁⠀⠓⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠙⡆⠀⠀⠀⠋⠉⠉⠁⠈⠉⢁⣧⡇
⠀⠀⠀⠀⢩⣾⡆⠀⠀⢠⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢱⣶⢄⡀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠐⡛⢿⣤⠶⡀⠙⢋⡴⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣷⡞⢁⠄⠀⢠⢯⠏
⠀⠀⠀⠀⠀⠘⡴⡷⡜⠃⠀⠈⠀⢀⢜⢝⡕⠀⠀⡄⡄⠀⠀⢦⡀⠀⠀⠀⠉⠀⠲⠊⡠⢮⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⡀⠀⣀⠈⠛⠠⠇⠀⠀⡇⢻⠀⠀⠀⠙⠂⠀⠀⠀⠀⢀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⠅⣐⠂⠀⠀⠀⠈⠁⠈⠀⠀⠀⢠⠺⡀⠀⣀⢔⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣝⡐⢤⣠⣄⣀⠈⠈⠧⠀⢀⢠⣀⡡⣖⣾⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠛⠶⠬⠭⠷⠭⠵⠖⠒⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢴⣶⣶⠿⠯⠭⠽⠯⠭⠿⠷⣶⣦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠽⠒⠉⠁⠀⠀⠀⢰⡆⢸⣶⣷⣦⠀⠀⠈⠉⠛⠻⢗⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡺⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣧⣼⣋⠀⠀⠀⠀⠀⢠⣄⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠘⣿⠀⣿⠀⠀⠀⠀⠀⠈⢽⣿⣿⠈⠉⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠉⠻⡳⣄
⠀⠀⠀⠀⣰⡻⠋⠀⠀⠀⠀⢠⣿⠀⣿⠀⠀⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⣼⡝⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠹⣷⡄
⠀⠀⣼⡏⠀⣤⡆⣠⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣣⠟⠀⠙⣿⡄
⠀⣸⡟⠀⠀⢸⡗⣿⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠀⠀⠸⣷⡀
⢠⢻⠁⠀⠀⠛⠃⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⡟⡇⠀⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢶⡖⠀⣿
⣿⡇⠸⣧⣀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⣷⡄⣿
⣷⡇⠸⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠀⣿
⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠈⣞⡄⠀⠀⠀⢀⣀⡀⢀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣸⡟
⠀⠹⣣⠀⠀⠰⣇⣀⡽⠚⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣧⡄⢠⣿⠁
⠀⠀⢻⣧⠀⢸⡏⠈⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⣿⠃
⠀⠀⠀⠻⡷⡀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠘⢾⢆⠀⠀⠀⠀⠀⠾⠻⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⠃⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠑⢕⣤⡀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣄⢹⣷⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣦⣄⡀⠿⠁⠀⠀⠀⠀⠀⠀⣾⡧⢿⣄⠀⠀⠀⠀⠀⠀⠈⠙⢋⣡⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣄⣀⠀⠀⠀⠀⠹⣦⡼⠏⠀⠀⠀⢀⣀⣠⣶⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣧⣶⣶⣄⠀⢀⣀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠛⢻⡉⠀⠀⢀⣴⣶⣤⣘⣆⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣀⠀⠿⠿⠿⠛⣿⣭⣴⣶⣶⣾⣧⣤⣄⡈⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⠁⢀⣴⠃⠀⠀⢀⣴⠿⠋⠁⣄⠀⠀⢠⡄⠈⠙⠻⣦⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣿⡇⢀⣾⠇⠀⠀⣰⡟⠁⢄⠀⠀⠈⠀⣄⠘⠀⠀⠠⠆⠈⢻⣦⠉⠛⠿⠃
⠀⠀⠀⠀⢸⣿⠀⣸⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⢻⣧
⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⣿⠃⠐⠲⠆⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠁⠀⣿⠀⠀⣧⠀⢰
⠀⠀⠀⠀⢻⣿⠀⢻⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡠⠛⣇⠀⠀⠀⠠⢤⣄⠀⣿⠀⠀⣿⠀⣼
⠀⠀⠀⠀⠸⣿⠀⠘⣷⠀⠀⢹⣧⠈⠉⠀⠀⠀⠊⠀⠀⠘⢧⠀⠀⠀⠀⠀⣸⡟⠀⢀⣿⠀⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠹⡄⠀⠀⢻⣦⡀⠠⠆⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⣰⡟⠁⠀⣼⠇⣸⡏
⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣰⣿⠟⢿⣦⣀⠀⠀⠇⠀⠀⠘⠀⣀⣤⡾⠋⠀⢀⡼⠃⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠈⠛⠻⠷⠶⣶⡶⠶⠟⢿⣿⠀⠀⠀⠉⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⠿⠿⣋⡵⠋⠉⠉⣛⠛⣿⣿⣿⣷⣼⠒⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠁⠀⠈⣿⡟⠛⡉⠀⠀⠀⠀⠄⠀⣿⣿⣿⣿⡟⠀⢨⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠈⠂⣠⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⢸⡸⣿⡿⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡴⣿⣿⡇⠀⠀⢠⣚⣢⡀⠀⠀⢸⣷⣿⠃⠀⢠⣄⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣧⠀⠀⠸⡛⠡⠇⠀⠀⢸⠿⠉⠢⠤⢼⣿⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣿⣿⣿⣷⣤⢤⣠⡤⠤⠤⢖⣞⠈⡄⠁⠄⢸⠋⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⠿⠿⠋⢖⣠⣦⣴⣧⣨⣄⣠⣄⣒⠐⠎⡼⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠿⠶⠛⠛⢿⣤⡀⠀⣀⣀⣀⣈⣻⣿⠗⠧⡠⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⣡⣤⣤⣤⣶⣶⣿⣿⣷⣿⣿⣿⣶⣿⣽⣻⡷⣄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⢛⣛⣛⣋⣙⠛⠿⠿⠿⠿⠶⠶⠶⠦⣤⣄⡈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣇⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⢿⣿⠟⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣛⣿⣿⡇⣾⡷⢰⣿⣟⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣶⣎⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠈⠿⣿⣿⣿⡟⣿⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠈⠙⠛⠷⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⣄⡀⠀⢸⣿⣿⣟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣄⠀⣿⡜⣿⣦⠻⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿⣿⣦⠘⣿⣎⢿⣿⠿⠿⢛⣣⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢿⣿⡆⠈⢿⣷⢆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⠀⠀⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢁⣤⣶⠾⠟⠛⠛⠛⠛⠻⠿⠶⣶⣤⣀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⠛⠿⢶⣦⣄⠀⠀⡟⠉⢹⡇⠀⢀⣤⡶⠿⠛⠛⠛⠻⢶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣀⣠⡾⠃⠀⠀⠀⠈⠹⣷⠀⣧⠀⢸⡇⢠⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⡂⣻⣀⣸⡇⣺⡏⠉⠛⠛⠷⢶⣤⣀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠛⢛⣷⣦⣄⡀⠀⠀⠈⠙⠷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⣁⣤⣴⠶⠾⠛⠛⠛⠛⠛⠛⠻⢷⣤⡀⠀⠀⠈⢻⣆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣴⠶⠿⠛⣋⣥⡶⠟⢋⣁⣤⣴⡶⠶⠶⠶⢶⣦⣤⣀⠀⠙⠻⣷⣄⠀⠀⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⣠⡾⠛⢁⣤⠾⠛⠉⠀⠀⣀⣄⡀⠀⠀⠀⠉⠛⠿⣶⣄⠈⠻⣷⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣰⡿⠋⢀⣴⠟⠁⠀⠀⠀⠀⠸⣧⣤⠗⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠟⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠏⢀⣾⠋⠀⠀⢀⣴⠞⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣾⡏⢀⣾⠁⠀⠀⠀⠻⣤⡀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡘⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇⠀⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⣹⡆⠀⠀⠀⠀⠀⠐⢯⣨⠇⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠸⣿⠀⠿⣬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣏⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⢀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⣇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠳⣄⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢿⡀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣙⣷⠀⠀⠀⢀⣼⠏⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢷⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠋⠁⠀⠀⣠⡾⠃⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠙⠄⠈⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠺⣭⡿⠀⠀⠀⠀⢀⣠⡾⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣄⣀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠟⠉⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣀⣠⣤⣶⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠿⠷⠶⢶⣶⣦⣤⣤⣴⣶⣶⠶⠶⠿⠛⠻⣍⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⢺⡆⢚⡷⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠐⣶⠐⡇⠀⠀⠀⠀⠀⠸⠃⠿⠶⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠟⠀⠗⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⢠⡟⠁⢴⢀⡶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢲⡄⠈⢻⡄
⠀⠀⠀⠀⢠⡿⠀⠀⢸⠞⢧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⠀⠀⠀⢿⡄
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡇⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⡳⢾⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⢾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣯⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠘⣧⡀⠘⠧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣴⡲⠀⠀⠀⠀⠀⠀⠀⣭⡽⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡽⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠿⠛⠛⠛⡿⠛⠿⠷⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⢠⣤⢀⣤⣥⣄⠀⠀⠀⠀⠈⠉⠛⠻⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⢛⡄⢀⡀⠀⠀⠀⠀⠀⠀⡟⠈⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⢹⡇⢹⡇⠀⠀⠀⠀⠀⠀⡇⢸⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠈⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⣰⣿⣿⡁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠹⣿⣧⡀
⠀⠀⣰⣿⡟⢹⡇⡴⠋⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣉⡽⠀⠈⠙⣿⣷⡀
⠀⢠⣿⡿⠀⢸⡇⢧⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣭⣤⠀⠀⠀⠸⣿⣧
⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⡆
⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⣠⠞⠉⣳⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⡽⠀⠀⣹⣿
⣿⣿⠉⠙⠛⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣻⠀⠀⢻⣿
⣿⣿⡄⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⣾⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠹⣿⣧⠀⠀⢠⠖⢒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⢠⣿⡿
⠀⠀⢻⣿⣆⠀⣸⠖⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⡤⣷⠄⠀⢀⣾⣿⠃
⠀⠀⠀⢻⣿⣶⠻⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢨⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠚⠛⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠛⠃⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⣹⠆⠀⠀⠀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠶⣦⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⣦⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠹⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣦⣤⣤⣄⣀⣀⣇⣀⣤⣤⣤⣶⡾⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠙⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣆⡠⡀⠈⠓⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⠀⠠⠓
⠀⠀⠀⠀⠀⡔⡏⠂⠀⠂⠀⠀⢰⣿⣿⣿⡿⣿⡿⠉⠙⠋⠁⠹⣧⠀⠉⢀
⠀⠀⠀⠀⠐⠈⡲⠇⠀⠀⣀⠀⣿⣿⣟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠑⠮⠄⠀⢠
⠀⠀⠀⠀⠀⠉⠠⣄⠀⡠⠀⠣⣹⣿⣿⡗⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠒⠀⠈⢻⣍⠋⠀⠀⠀⠀⠶⠾⠟⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡼⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣣⡀⣀⡠⠀⠀⠙⠳⡖⠤⠤⣔⠂⠱⠃⠤⢇
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠛⣦⡄⠀⠀⠆⠠⠈⠦⠴⠁⠈⠣⣀⠔⠁⢀⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢌⣻⣆⠀⠠⠂⠀⠀⠀⠈⠉⠑⣑⣄⣀⠁⠤⠤⠀⠒⠀⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⢻⡛⣻⣆⠀⣀⠄⢀⠤⠒⠈⠁⠀⢀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢱⠈⣻⣆⡡⠊⠁⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢘⣾⡿⠋⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠉⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁

⠀⠀⣠⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠚⠒⠛⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠒⠒⠲⠶⠤⢄⡀
⢀⠞⢁⡀⠀⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⢸⠉⢀⠄⠀⠶⠒⠆⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠤⢄⡀⠈⠑⠀⠔⡋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠸⠀⠀⢪⢐⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣴⡿⠻⢿⡿⠻⣇⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠀⠀⠘⠶⡁⠰⠆⠀⠀⠀⠀⠀⠀⠀⡻⠇⢀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡸⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⠀⠀⠀⠀⢡⣠⠀⠁⠀⠀⠀⠀⠰⠀⠉⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠉⢀⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⢰⡖⠆⠠⠀⠄⠚⠀⠈⣁⠴⡀⠀⠀⠀⠀⠀⠀⡀⢠⠊⠀⠀⠈⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⢷⠀⠀⡀⠀⠀⢀⡤⠧⡀⠀⠀⠉⠀⠀⠐⠘⡳⠄⠀⠀⠀⠀⣈⠄
⢰⠀⠀⠀⠀⢘⡣⠤⣀⠀⠁⠘⠀⠀⣇⠀⠀⠀⢨⡀⠀⠀⠀⣴⠀⢠⠄⡀⠳⢀⢀⣖⠀⠊
⢈⠠⠄⠀⠚⠁⠀⠀⠀⠉⠑⠒⠐⣷⠈⠀⠀⠀⠀⣡⠀⠀⠀⠚⠚⠉⠚⠓⠀⠤⣈⠁
⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢳⡀⠀⠀⢀⠖⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠦⣀
⠀⢠⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡣⠀⡪⠀⠀⠀⠀⠂⠀⠁⡀⠀⠤⠴⠚⠛⠀⠂⢀⣉⠡⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠀⢦⡀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠊⠁⠀⡎⠀⠀⠀⠂⠀⠀⠠⠋⠀⠀⣀⠤⠂⠁⠈⠀⠀⠀⠀⠀⡀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⢠⠀⠀⠈⠢⣄⠀⠀⠉⠉⢠⢊⠀⠄⠀⣸⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠓⠢⠤⣤⣄⣀⣀⣀⡀⠀⡠⢤⡄⢒⡩⠥⠒
⠈⠀⣠⠔⠀⠙⢄⠀⠀⠰⠃⠀⠀⠁⢀⠇⢀⠎⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⢛⠙⠁⣠⡔⠒
⠀⡐⠁⠀⠀⠀⠀⠉⢱⠁⠀⠀⠀⠀⢨⠀⡜⠀⠀⠀⠀⠀⢀⠴⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠈⠁⠀⡇
⢸⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠃⠀⠃⠀⠀⢀⣀⡔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⠀⠐⢈⠀⠀⡇⡀
⠈⠁⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣰⠟⠙⢛⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠾⠰⠀⠀⠃
⠀⠂⠀⠀⠀⢠⠏⠀⠀⠀⠆⠀⡜⠀⠀⠀⠐⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠀⠐
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠀⠒⠂⠈⠉⠐⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠠⠀⠀⠀⡎⠀⠀⠀⠀⠀⡜⠀⠀⠀⡀⡐⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘
⠈⡄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠
⠀⠈⠂⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢃⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣊⣀⣇⣀⣀⡰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡴⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⢦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⣄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢈⡷⠟⠚⠋⠉⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡖⠛⠋⠹⢒⡻⣾⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢫⠂⠁⠀⠀⠀⢀⣠⠤⠾⠃⣀⣀⣀⣀⣀⣀⠘⠧⢤⣄⡀⠀⠀⠀⠈⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⣀⡴⠚⣉⡤⠖⠚⣉⠉⠤⠄⠠⠤⠉⣉⠓⠲⢤⣉⠓⣦⡀⠀⠀⠀⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⢁⡴⠋⡡⠴⠈⠁⠀⠀⢰⠀⡐⠀⠀⠀⠀⢱⠢⢈⠛⢤⡋⠳⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠁⡴⠋⡠⠊⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢆⠈⢣⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡞⢡⠚⠤⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⡈⢧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠃⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⡠⢄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⠀⠀⡀⢀⠐⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⠢⢀⡐⠀⠀⠀⠀⠀⠙⠤⠁⠀⠀⠀⡀⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢉⡷⢤⣉⡐⠀⠀⠤⠤⠤⠄⠠⠐⢂⣁⠴⢾⣁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣡⠎⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⠳⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⢒⣒⣚⡉⠉⠹⠟⢉⠁⣐⠒⠒⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⡫⠵⠒⠋⠉⠉⠀⠀⢀⡀⠀⠀⢀⢠⡀⠀⠈⠉⠑⠒⠮⠍⣓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⡯⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡆⠙⢃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠽⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣷⡐⣉⣠⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠈⠲⢍⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠊⠀⠀⠐⢿⡇⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢮⡳⢄
⠀⠀⠀⠀⠀⠀⣠⠞⡱⠋⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡷⠀⠀⠀⠀⠀⠀⠀⠉⢪⡳⡄
⠀⠀⠀⠀⠀⡴⢫⠞⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⢦
⠀⠀⠀⢀⣾⣧⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢻⣦⠈⢮⣧
⠀⠀⢀⡞⡰⠁⠰⣿⢀⣾⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⢣⢳⡀
⠀⠀⡼⣰⠃⠀⠀⣿⠸⣿⡀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠿⠶⠂⠀⠈⢣⢷
⠀⢸⢣⠇⠀⠀⠐⠿⠂⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⡇
⢀⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸
⢸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⣿⢸⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⠀⠀⠇
⣇⣸⠰⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⠃⠀⠀⢠
⡟⢛⢠⣽⠟⢚⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⢀⡷⠀⠘
⣷⢸⠈⠙⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠉⠀⠀⡇
⢸⡨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠘⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸
⠀⢻⡸⡄⠀⠀⣠⠖⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢠⢧⡏
⠀⠈⢷⢱⡀⠀⢻⣦⢄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡇⠀⠀⢀⣾⡞
⠀⠀⠘⣧⢡⠰⣿⡁⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣟⠛⢀⡞⡼⠃
⠀⠀⠀⠈⢷⡷⡈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⢾⡿⠁
⠀⠀⠀⠀⠀⠳⡝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⠞
⠀⠀⠀⠀⠀⠀⠘⢦⡱⣄⠀⠀⠀⠀⢠⣶⠶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡾⠷⠂⠀⠀⠀⠀⠀⠀⣠⢞⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠈⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠛⢶⣄⠀⠀⠀⢀⣠⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⠦⣀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣙⡃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣸⠟⠀⣀⣴⣯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣷⠬⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⢴⣮⠵⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣝⡒⠠⠄⣀⡀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⣀⡀⠤⠄⣒⣫⠵⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠦⠬⢭⣥⣴⣶⣶⠢⣬⣭⡤⠔⠒⠚⠉⠁
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⡿⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⣀⠤⠀⠠⢠⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⠀⢄⡇⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⣸⠁⠈⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠳⣄⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠆⠀⡦⠛⠉⠓⠂⢀⠀⣙⣿⣿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⠈⠛⢿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⢲⠀⠀⣀⠀⠀⠄⢀⡤⠋⠀⠀⠀⠑⢦⠦⠆⠼⠀⢀⣀⢐⠂⠒⠀⡅⠀⠆⠀⠀⠀⠀⠈⠈⢹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠀⡀⣀⠀⠀⠸⠁⡀⠀⠀⠀⠄⠨⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠸⠤⠆⠀⠀⠀⠋⠀⠀⠀⠀⠈⠂⠀⠀⠀⡰⠠⡇⠀⠀⠀⠤⠃⠀⠦⠤⠤⡤⠮⠤⡼⢁⠓⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠐⠀⠢⣄⠀⢆⠀⢦⢨⠄⢀⠄⠀⢤⠔⢀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢌⣡⢾⡿⠿⢾⣦⣈⠆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⠢⢈⠐⠂⠀⠢⠹⠂⠀⠐⠀⣠⠐⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⠁⠐⡄⠀⠙⣯⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠤⠠⠄⠰⠀⠈⡆⠀⠀⠀⠀⠀⠈⠁⠠⠄⠤⠄⠇⣀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡈⠉⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⣀⠀⠠⠄⠀⠇⠀⠀⠀⠀⠀⠤⠀⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡟⣶⣶⣞⣷⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠥⡐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⡠⢤⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠒⠀⠀⢸⠀⡇⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠠⠤⠠⠀⠄⠤⠠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠠⠤⠬⠥⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢘⠀⢇⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⢠⡶⠛⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⡄
⠀⠀⠀⠀⢸⣧⡀⢀⣸⠂⠀⠀⠀⡀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⣀⡟
⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⡀⢀⠀⠀⢀⢰⣿⣿⢋⠀⠀⠀⠀⢈⡻⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⠛⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⠱⠁⠀⠀⠁⠂⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⡀⢸⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⠀⢢⣤⣤⣶⣤⡀⢹⣿⡇⠀⠠⣄⣀⣠⠀⠀⢸⣿⣿⣿⡷⠆⠀⢀⣀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⣿⣿⣿⣿⣿⣞⣻⡿⣄⠀⠙⠛⠋⠀⢀⣾⢿⡟⠉⢀⣠⣾⣿⡿⠀⠀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠻⠟⠛⠉⠁⠀⠁⠐⠛⠑⠂⠀⠀⠀⠻⠿⠄⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣾⣷⣾⣶⣄⣀⣀⣀
⣾⡟⠀⠀⠈⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⢹⣿⣿⣿⣿⣿⣿
⢻⣿⣦⣤⣾⡇⠀⠀⠀⠀⣸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣷⣤⣤⡿⢿⣿⣿⣿⣿⣿
⠀⠈⢻⣿⠋⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡿⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣧⠀⠀⠎⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⢰⡿⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣇⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⣾⠁⠀⣸⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡏⠀⢸⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⢸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠟⠛⠉⠉⠉⠀⡘⣃⡀⠉⠉⠉⠛⠻⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⣱⡃⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⢾⠴⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠁⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠈⠙⢿⣿⣦⡄
⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⣰⣿⡿⠇⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠼⢿⣿⡆
⠀⢠⣿⣿⠁⠀⢸⢿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⠀⠈⣿⣿⡄
⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⣀⣿
⣿⣿⠉⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠉⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠘⣿⣿⡀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⢀⣿⣿⠃
⠀⠀⠸⣿⣷⡖⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢰⣾⣿⠇
⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠆⠀⢀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣶⣦⣤⣀⣀⣀⠀⢨⡅⠀⣀⣀⣀⣤⣴⣶⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢠⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢸⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡞⠀⠀⡴⠀⠀⠀⠀⠀⢀⡀⠸⣧⠀⣴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠀⠀⠷⣿⣠⠎⣀⣀⠀⢰⣿⣶⣒⣲⡆⠀⠁⠀⠉⠻⣿⣿⣥⡶⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣯⣙⡦⠈⣻⠋⠉⢡⠖⢛⣭⣽⣷⣧⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⢻⣿⣿⣿⣟⡽⢟⣛⠿⠿⠿⠿⢿⣷⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⢟⡭⠊⢡⠀⠀⠀⠘⠀⠀⠀⠍⠻⣯⣿⣿⠇⡇⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠒⣿⣣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⣸⣷⠃⠀⢹⡿⣄⠀⠀⠀⠀⠀⠀⢀⣤⠔⠛⠋⣿⡇⢄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡏⣠⡀⠀⠀⠈⠓⢤⡤⠀⡴⠞⠉⠀⠀⢀⣀⢹⣧⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣇⠀⠀⠀⠀⠀⠀⠈⠈⡥⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⢸⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣹⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠆⢻⣿⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣱⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢹⣿⣷⣦⣄⡀⠀⠸⠆⠀⢀⡀⢔⣨⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⠾⠛⠻⠿⢿⣿⣿⡿⠶⠾⠛⠿⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣄
⠀⠀⣼⡟⠁⠀⠀⣠⡴⠚⠉⠀⣠⣄⣠⠴⠶⣶⣶⣶⣦⣀⠀⠀⣤⣤⣤⣴⣆⠀⠀⠀⠀⠀⣠⣤⠖⠒⠒⠒⠠⢠⣤⡀⠉⠓⠦⣄⠀⠀⠈⠻⣷⡄
⠀⢸⡟⠀⠀⢠⡾⠋⠀⠀⠀⢀⠜⠁⠀⠀⢀⣼⣿⣿⣿⡿⠗⠀⠛⠛⣿⠉⠁⠀⠀⠀⠐⠿⣿⣿⣤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢻⣄⠀⠀⠘⣿
⠀⢸⠀⠀⢀⣿⠁⠀⠀⠀⢠⠃⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⠀⣀⣀⡤⠿⠤⠤⠤⣤⣀⣀⠀⠈⠛⢿⣿⣿⣷⣦⣄⡀⢸⣷⡀⠀⠀⠀⢹⣇⠀⠀⢹
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣾⠀⢀⣴⣿⣿⡿⣋⡁⣀⠤⠚⣉⣥⣴⣶⣶⣾⣿⣷⣶⣶⣭⣝⣲⣤⡀⢈⡛⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡿⠀⠀⠈
⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⣿⣶⣿⡿⠟⠁⠀⡻⠋⣡⣶⡿⠟⠛⠉⠁⠀⣴⡄⠀⠈⠉⠛⠻⣿⣿⣿⣿⣏⠀⠉⠻⢿⣿⣿⡟⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⠀⠀⢠⠎⣠⣾⡿⠋⠀⢶⠀⠀⠀⠀⠛⠃⠀⣼⠀⠰⠇⠀⠙⢿⣿⣿⣷⡄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⡿⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡿⠁⠲⠆⠀⠀⠈⢢⡀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠞⠃⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡎⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡆⠀⡇⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣿⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⢹⣿⣿⠀⢠⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⡇⣿⣿⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⣿⣿⠀⢘⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⡇⠀⣧⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⡿⠀⢸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠹⡀⠸⣿⣿⣷⠀⢠⡤⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⠦⠠⢡⣿⣿⠇⢀⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⣿⠏⠀⠈⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⢰⠆⠀⠀⠀⢠⣤⠀⠀⠀⠐⠷⠀⡠⢊⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⡀⠀⠀⠀⠘⠟⠀⢀⡀⠤⢒⣡⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣴⣶⣿⣿⣿⠿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠁⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠒⠒⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠒⠒⠒⠂⠀⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠋⠉⢷⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣤⣀⠀⠐⢧⣀⣀⣀⢠⡤⠞⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢬⡇
⠀⠀⠀⠀⠀⢸⠅⠀⠉⠀⠀⠈⣿⠀⠀⠀⣀⣽⠜⠦⠤⠤⠤⢄⣀⠈⠑⡦⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡤⠞⢁⡠⠖⠋⠁⠀⠀⣀⣀⣀⣀⠀⠀⠙⠋⣰⠋⠙⠦⠼⠁
⠀⠀⠀⠀⠀⢧⣀⠤⠞⣏⠳⠊⢁⣠⡤⠶⠚⠛⠉⣉⣉⠉⠉⠛⠶⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣠⠞⠉⠀⠀⠀⠀⢾⡀⣤⣞⣀⠀⣀⡀⠈⠻⣦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⠎⢠⠞⠁⠀⠀⠀⠃⠀⠀⠀⣁⡉⠉⠉⠁⠈⠐⠃⠀⠈⢳⡄⠈⣧
⠀⠀⠀⠀⠀⢀⡎⢠⡏⢀⡀⠙⠀⠀⠀⠀⠀⢮⠁⡇⠀⠀⡄⠀⠀⠀⢤⠀⠀⢻⡀⠘⣆
⠀⠀⠀⠀⠀⣼⠀⣼⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⡆⢳⠀⠞⠁⠀⠀⢀⣠⣤⠀⠈⣇⠀⠸⡆
⠀⠀⠀⠀⠀⡏⠀⣯⣴⣚⣆⠀⠀⠀⠀⠀⠀⠀⢃⢸⡮⠤⠲⡄⠀⠀⣈⠉⣳⠀⣿⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⣿⢈⣁⠞⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠉⠉⠁⠀⠀⠈⠉⠁⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⢹⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢳⠀⢻⡄⠀⠀⡀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠃⠀⢀⡟⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠻⣄⡀⠀⠀⠶⠀⠀⠰⣿⣛⡷⠀⠀⠛⠀⠀⠀⣠⠟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠞⠉⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢦⣄⡀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⣀⡤⠞⠋⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠈⠁⠀⢤⣄⣀⣀⣀⡀⣀⣠⡤⠶⢿⣉⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⡶⠞⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠆⡑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣠⠔⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠁⠀⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠟⠻⣿⠿⢿⣿⣿⡄⠀⠀⠀⠀⢀⡂⠀⠓⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⢤⡀⠀⠈⢁⠀⠀⠈⢄⠀⠀⠀⠀⢸⣿⣿⠛⡁⠀⠀⠀⠀⣀⠀⠹⣿⣇⠀⠀⠀⢀⠎⠀⠠⠈⠀⠈⠀⠀⢠⠔⠁⠀⡠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⢂⡠⠔⠈⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⠑⠠⡠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠢⠀⠀⠀⠀⡄⠂⠘⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠋⠀⠄⠀⠀⠀⠀⠂⠂⠀⠀⠠⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠊⠠⡀⠀⠈⢢⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⡠⠉⠀⠒⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⢀⠂⠀⠀⠀⡀⠀⠀⡌⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠲⠀⠀⠀⠰⠆⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⡀⡠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣴⣦⡀
⡠⠊⠀⠀⠁⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠁⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿
⠇⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠘⠤⣀⡀⠀⠈⠄⠀⠀⠀⢀⠼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠀⠸⠀⠈⠀⠀⣀⡀⠔⠁
⠀⠀⠐⠄⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠄⠀⠀⠀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠄⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣶⣶⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡀
⠀⠀⠀⠀⠀⣠⣿⠋⠀⣠⣷⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣾⣄⠀⠙⣿⣄
⠀⠀⠀⠀⢠⣿⠃⠀⣼⡿⠁⢀⣶⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⡀⠈⢿⣧⠀⠘⣿⡄
⠀⠀⠀⢀⣿⠇⠀⣸⡿⠀⢠⣿⠋⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⣿⡄⠀⢿⣇⠀⢸⣿⡀
⠀⠀⠀⢸⣿⠀⢀⣿⠃⠀⣿⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⣿⠀⠘⣿⡀⠀⣿⡇
⠀⠀⠀⢸⡟⠀⢸⣿⠀⢸⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⡇⠀⣿⡇⠀⢻⡇
⠀⠀⠀⢸⣷⠀⢸⣿⠀⠘⣿⡄⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢠⣿⠃⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠸⣿⡀⠀⣿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢸⣿⠀⢀⣿⠇
⠀⠀⠀⠀⢻⣧⠀⠸⣿⡄⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢠⣿⠃⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣆⠀⠙⣿⣆⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣠⣿⠋⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋

⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⣠⣾⠟⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⢀⣾⠟⠁⠀⢀⣴⠞⠉⠀⠀⣤⡤⠴⠶⢿⣿⣿⣶⣤⡀⠀⣤⣤⣶⣶⣆⠀⠀⠀⠀⣀⣤⡶⠚⠋⠁⠒⠠⣴⡆⠀⠉⠓⢦⣀⠀⠀⠙⢿⣆
⠀⢸⠇⠀⠀⣰⠟⠁⠀⠀⠀⡔⠉⠀⠀⢀⣼⣿⣿⣿⡿⠛⠀⠙⠉⣿⠉⠀⠀⠀⠀⠘⠻⣿⣷⣦⣄⣀⠀⠀⠀⠱⣄⠀⠀⠀⠙⣧⡀⠀⠈⢿
⠀⠸⠀⠀⢰⡟⠀⠀⠀⠀⡌⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⣀⡠⠤⠴⠛⠒⠒⠶⠦⢤⣄⣀⠈⠙⠿⣿⣿⣿⣦⣄⣀⣿⣧⠀⠀⠀⠘⣷⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢰⣇⢀⣴⣿⣿⠿⣫⣤⡠⠖⣋⣥⣶⣾⡿⠿⠿⠿⠿⣿⣿⣶⣯⣿⣦⣄⣬⡙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⣿⣿⡿⠋⠁⠀⡼⢋⣴⣾⠿⠛⠉⠀⠀⠀⣾⡆⠀⢀⠀⢉⠛⠿⣿⣿⣿⣄⠀⠈⠛⢿⣿⡿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⠎⣴⣿⠟⠁⠀⠹⠂⠀⠀⠀⠉⠁⢀⣿⠀⠚⠁⠀⠈⠻⣿⣿⣷⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡿⠃⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣾⣿⠁⠈⠛⠀⠀⠀⠙⢦⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⠉⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠀⢸⣼⣿⡇⣴⣶⡆⠀⠀⠀⠀⠀⠀⠈⣶⣿⡄⠀⠀⠀⠀⠀⠀⠸⢿⠷⠀⣿⣿⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⠀⢸⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡄⠘⣾⣿⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⢰⣿⣿⠁⢠⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⠀⠹⣿⣿⣧⡀⠋⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⡡⢣⣿⣿⠇⠠⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⢀⣄⠀⠀⠀⢋⡀⠀⠀⠀⢠⡄⠀⢀⠔⣡⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⡈⠁⠀⠀⠀⢸⣿⠀⠀⠀⢀⠡⢂⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣭⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢯⡀⢻⡡⢤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢿⠂⣇⣀⠀⠀⠀⠀⠀⠀⠀⠐⣫⠀⠀⠈⠉⠀⠀⠘⠻⡉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣉⣸⠵⠉⠑⠤⣀⠀⠀⠀⠀⣼⠂⢀⣄⠞⣆⣠⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⠀⠈⠑⠦⣄⠀⢺⣍⢉⡁⢀⣀⠈⢳⣤⣀⡇⠀⠀⣀⠴⠒⠉⢥⢹⠀⠻⢟⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠑⢯⣿⢡⣆⢁⡄⠀⠀⠞⣽⡡⠖⠉⠀⠀⣀⡤⠜⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⣸⡌⠒⠋⢀⣠⢏⡴⠋⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠘⣧⡟⢲⠖⢉⠏⡴⠃⠀⢀⡴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⢀⣤⣋⡠⠞⢠⣶⠏⠀⣜⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⣀⠀⠀⠁⠀⠀⠠⡍⠁⠀⠀⠀⠉⡞⠀⠀⠀⠙⠲⣝⢦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠁⠀⠀⠈⠳⣴⢶⠂⠀⢠⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⡬⢷⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⢿⣀⠀⠀⢀⣶⡻⠆⣸⠇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⠴⠶⢶⡶⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠒⠬⢀⡀⠀⠀⠀⡟⠈⠓⠤⣸⡟⢁⡴⠋⠀⢀⡤⠞⠹⡇
⠀⠀⢀⡴⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠤⣄⣉⡑⠢⢤⣇⡀⠀⠀⠀⠉⠉⢀⡤⠖⠉⠀⠀⠀⡇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⢤⠁⠀⠀⠀⠉⠉⢢⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⣷
⠀⢰⡇⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⡇⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣀⡀⢸
⠀⢸⠇⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠴⠚⠁⠹⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠖⠉
⠀⠈⠓⠒⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠤⠀⠀⠠⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠟
⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⣠⣴⣄⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣧⣴⣿⡆⠹⣿⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⣿⡉⣷⣀⣿⣷⣶⣾⠿⠿⠿⠛⠛⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⣀⣤⡴⢶⡞⠻⡏⠉⣉⠉⠉⡿⠛⢻⣶⣄⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣶⣿⣋⠀⠀⠈⡇⠀⣿⠀⢻⠀⢸⡇⠀⢸⠇⠈⣳⣄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣰⣿⠟⠁⠘⢿⣷⣆⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠟⠀⢰⠏⠹⣦⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⢀⣾⡿⢧⣀⡀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢹⡆⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢠⣿⣿⡁⠀⠈⠉⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⡀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣾⣿⡋⠙⠓⠂⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⢸⣇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⡟⠛⠲⠆⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣷⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⣿⡟⠛⠃⠀⢀⡀⠀⠀⠀⠀⡼⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠸⣧⣀⣤⠞⠋⠀⠀⠀⠀⢰⣣⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠙⣿⣁⣤⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⣄⠀⠘⢻⣿⠃⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠈⢿⣇⠀⡴⠀⠀⣼⡟⠀⠀⠀⠀⠀⡄⠀⠀⢳⡀⠰⣌⠛⣶⡾⠃⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠹⣿⣧⡀⢰⡟⠀⠀⢸⠀⢸⠀⣿⠀⠀⠀⣷⣤⣿⠿⠋⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡈⠉⠛⠻⢷⣤⣤⣿⣤⣤⣤⣿⠶⠾⠛⠋⠉⠀⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⣀⡀⠀⠉⠉⠉⠀⠀⠀⣀⣀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⣠⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠸⠃⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠋⠀⠀⠛⠀⢀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠖⠛⠛⣓⠲⣾⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣷⣿⣿⣷⣶⣤⣍⠛⣦⠸⣧⣤⣼⠇⢾⣅⣠⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢠⠄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣀⣿⣆⣀⣀⣈⡙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡟⠀⡞⠀⣿⣿⣿⣿⣿⣟⣡⡴⠞⣛⣯⣭⣿⣿⣿⣿⣿⣿⣿⣶⣾⣟⠻⢿⣿⣿⡇⣹⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⠟⢋⡽⢋⣡⣶⣿⠿⠛⠋⢉⣍⠉⠉⠙⠛⠿⣿⣿⣿⣷⣄⠈⠛⠶⠟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⡶⠛⠁⣴⠋⣴⣿⠟⠫⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠃⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⡿⠃⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡙⢿⣿⣿⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢸⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠄⠀⠀⠀⠀⠀⠀⠲⠷⢸⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡟⣸⣿⠁⢰⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣧⠀⠹⣿⣿⣧⡀⠆⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⣿⠃⢀⡟⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⣧⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣴⠟⣴⡿⠃⠠⠏⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣿⣿⣿⣦⣌⡁⠀⠀⠀⠿⠀⠀⢀⣠⡴⢟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡻⣿⣿⣿⣿⣿⣶⣶⣶⣶⣛⣫⣵⣶⠿⢿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣱⣿⠋⠙⠛⠻⠿⠿⠿⠿⠟⠛⠉⣁⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢜⢂⢠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⢀⡆⠀⠀⠀⠀⠀⠔⠂⠁⡀⠀⠀⡠⠀⢀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣀⡀⠟⠞⠁⠀⠀⠀⠀⠀⢀⠀⠰⠃⠀⠄⡀⠀⠉⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠄⠀⠈⠁⠀⠀⠇⠀⠠⠀⠀⠆⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⡀⠀⡀⠀⠄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢾⢷⣷⢬⠁⠢⠀⠀⠀⠀⠠⠠⠊⠁⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠂⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⠈⡏⡃⠐⠀⠒⠀⠀⠀⠀⠀⡀⠁⠂⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣀⢈⢀⣸⠟⠁⠁⡠⠆⠐⠒⠀⠀⠦⠅⠀⢀⣘⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⠉⠁⠉⠀⡀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⡀⠀⠀⠋⠀⠀⠀⠀⠁⠀⢀⠂⠁⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠠⠄⠈⠁⢲⠶⠒⠀⠀⠀⢀⢈⠀⠀⠆⠀⠇⢀⠀⢀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡅⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡌⠀⠀⠑⠊⠀⠸⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁
⠐⡁⠀⠊⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠀⠤⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⡛⠉⠉⠐⠀⣑⣀⣧⢀⡁⠘⠉⠻⠻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⡉⠰⠀⠁⡀⠀⠀⠀⢹⡿⣿⢀⣿⠀⠀⠀⠀⢁⣘⠛⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠟⠡⠀⠀⠈⣿⠈⣿⠀⠀⠀⢸⣿⣿⣍⣀⡀⠀⠀⠀⠀⣿⠀⠀⠘⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⡁⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠘⠛⣿⠙⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⣤⢠⡄⢀⠤⣀⠀⠛⠐⠛⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⣠⡆⢲⣄⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⠠⠀⢸⣷⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡼⠏⠀⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠐⠂⠀⢸⡟⢿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣶⡴⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠐⠀⣀⡈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢙⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⡁⣾⡇⢸⣆⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣼⡋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣉⣙⡇⢺⡿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢻⣿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⢸⣿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣀⠙⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠁⢤⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⡄⠀⢠⡖⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠠⣼⣿⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠠⠀⠈⠿⢷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢼⡧⡆⠀⢲⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⠲⠺⣄⠀⡼⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠾⠧⠀⢙⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⡈⢀⠀⠀⠀⠀⠘⠛⣿⠏⠀⠀⠀⣠⡄⣤⠀⠀⠀⠀⣋⠙⣏⠀⠀⠀⠀⠄⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠆⠀⠀⠀⣾⣁⡀⠀⠀⢠⣿⠄⢿⡀⠀⠀⢠⣶⠀⣹⠇⢀⠀⢂⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠠⠆⢛⠛⠁⠀⠀⠈⣿⡀⢸⡇⠀⠀⠀⠉⢀⠉⠆⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣃⠀⠂⠠⠈⢱⠉⠄⠀⠂⢀⣀⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠛⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⡄⣤⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠋⡼⠁⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⢰⠃⣀⡀⢸⠃⠀⣠⠴⠶⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠎⠳⣄⠀⠀⠀⠁⠀⣴⢤⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⣴⣦⠀⠀⠙⠋⠁⠙⠋⣤⣾⠵⢦⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣠⡼⠗⠀⠀⠀⠀⠈⠉⠀⠙⣤⡇⠀⠀⠀⢠⢦⣴⢦⡀⠀⠀⠙⠃⠉⠁⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⣠⠔⠚⠉⠛⠲⣌⠁⢀⡴⠖⠚⠀⠀⠀⣉⣙⣦⠀⢀⡴⠛⠉⠉⠉⠙⢦⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⣷⠀⠛⠋⢳⠀⠀⢰⡇⠈⠁⠀⢾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⠃⠀⠀⢾⠀⠀⠀⢀⣠⡤⠖⠋⠀⠀⣀⣸⠀⠀⣸⣇⣀⣀⣀⡈⢙⡶⠠⣤⣀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠛⢯⠀⠱⣶⠚⠋⠉⠹⣭⠄⣴⣾⣅⣀⠼⠉⢩⠿⠁⢰⣇⡈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡇⢀⣠⠆⠀⣀⣀⡀⠀⠀⠀⠀⣀⢀⣀⣀⡳⠄⠨⠂⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠉⣁⠤⠒⠉⠀⠀⠉⠀⠒⠒⠚⠁⠀⠀⠀⠈⠉⠒⠀⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡠⠊⠁⠀⢠⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⡄⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠎⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡄⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠔⠻⣙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠲⢄⠀⠀⠀⡀⠀⠀⡜⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⢢⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠥⠀⠀⠋⢀⠞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠦⣀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠑⠢⣤⣀⡀⠀⠀⠀⡶⠀⠀⠀⠀⢀⣉⠤⠒⠁⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣤⣀⡤⣄⣀⠈⠉⠙⠓⠒⠒⢒⣒⣈⣭⣤⣄⢀⡀⢴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡞⠀⠀⠀⠀⠉⠉⠛⠓⠚⠛⠋⠉⠉⠀⠀⠀⠈⠉⢳⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠞⠛⠉⠉⠙⠛⠓⠶⣤⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⣀⡀⠀⠀⣠⢤⡀⠀⢀⣀⣤⣼⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣀⡼⠁⠈⠙⢳⡄⡇⢸⡇⣴⣋⠁⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣸⣧⡧⢬⣿⣇⡉⠉⠛⠶⣤⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣤⠞⣃⣤⠤⠶⠶⠞⠻⠷⣦⣀⠀⠙⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠶⠚⢛⣭⠗⣋⡥⠴⠒⠛⠛⠓⠶⢦⣄⠙⢷⣄⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠋⣡⠞⠁⠀⠀⢰⣉⠆⠀⠀⠀⠉⠻⣦⡙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⡟⢡⡞⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢷
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡟⢠⠏⠀⠀⢾⣁⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠁⣿⠀⠀⠀⠀⠈⠙⠲⣬⠶⢤⠀⠀⠀⠀⣠⢦⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿⠀⣇⠸⡽⠀⠀⠀⠀⠀⠘⠲⡛⢦⡀⠀⠀⠈⠉⢀⡿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠳⣄⠀⠀⠀⡼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢳⡈⢧⡀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠓⠋⠀⢠⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠁⠀⠙⠶⣤⣄⣀⡀⠀⠳⣃⣀⣀⣤⠾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠛⣦⠀⠀⠈⠉⠉⠋⠉⠉⢉⣉⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢀⡴⠛⠛⠓⠒⠒⠒⠚⠛⠛⠉⠙⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠟⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠦⠤⣤⡀⠀⠀⣙⠀⠐⠋⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣠⣞⡁⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠉⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⢀⢼⣿⣿⣿⣷⣭⣳⠄⢦⣄⣤⡔⢒⣛⣓⣢⣶⡄
⠀⠀⠀⠀⠀⠐⣿⣶⣿⣯⣓⣦⣠⣠⣄⡴⢊⣩⣽⣾⣷⠀⠀⠀⠀⠀⠀⠈⢸⠃⡎⢼⣿⣿⣿⣛⣩⠤⣤⣤⣿⣤⣙⠻⣿⣿⣿⣿⣷⡀⣆
⠀⠀⠀⠀⢰⢡⣿⣿⣿⠿⣋⣡⣤⣿⣤⣉⡛⢿⣿⣿⣿⡷⡄⠀⠀⠀⠀⠀⠘⠀⠧⠞⣋⠔⣫⣵⠾⠛⠛⣛⠛⠻⢿⣿⣾⣟⢿⣿⡏⡇⢸⢰
⠀⠀⠀⠀⢸⡨⠟⣫⢟⣩⡷⠞⠛⣛⠛⠿⢿⣿⣯⣙⠿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⡿⠋⠀⠃⠀⠀⣿⠀⠀⠰⠉⠻⣿⣧⡈⠛⠁⠈⠈
⠀⠀⠀⠀⠀⢀⢞⣵⠟⠁⠂⠀⣿⠉⠀⠀⠒⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⢳⡟⠐⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠹⣿⣧
⠀⠀⠀⠀⠀⣞⣾⠁⠂⠀⠀⠀⢸⡆⠀⠀⠀⠀⠐⠙⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⢁⡄⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⣿⣿⠀⡄
⠀⠀⠀⠀⢸⣽⡇⣀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⡀⢹⣿⡆⠀⠀⠀⠀⢰⢸⡇⢹⣿⡄⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠘⠃⣿⣿⠀⡇
⠀⠀⠀⠀⢸⣿⡇⠉⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠉⣸⣿⠇⠀⠀⠀⠀⠈⠆⢧⠘⣿⣷⡐⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⡀⡰⣿⠏⣰⠃
⠀⠀⠀⠀⠀⢿⣿⡀⠆⠀⠀⠸⠋⠀⠀⠀⠀⠀⠠⣰⢿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⢿⣷⣄⠠⠀⠀⢀⠀⠀⢀⠀⣠⢞⣽⠏⡰⢃⡜
⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠄⠀⠀⣀⠀⠀⠤⢀⡴⣳⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣦⣤⣭⣤⣤⢶⣻⣵⠟⠁⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣦⣤⣭⣤⣶⣺⣵⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠉⠙⠛⠛⠛⠛⡋⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠙⠛⠛⠛⠉⠉⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣶⡄⣰⣦⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⡀⣰⣦⢠⣶⡄
⠀⠀⢸⣿⡇⣿⣿⠀⣿⡇⢲⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣦⢸⣿⡇⣿⣿⢸⣿⡇
⠀⠀⢀⣉⣤⣌⣁⣤⣉⣠⣿⠃⠀⠀⠀⠀⠀⠀⠿⢛⣉⣉⣉⣉⣉⣤⣶⣷⣤⣉⠛⠿⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⣿⣄⣉⣤⣌⣡⣤⣉⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⢙⣉⣋⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣥⣤⣬⣿⣿⣿⣿⣿⣿⣡⣤⣬⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢛⣉⣉⡙⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠘⠿⠿⠃⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠻⠿⠶⠶⠿⠿⢛⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟
⠀⠀⠀⣠⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠚⠻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠟⠓⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠻⠿⠿⠿⠿⠟⢋⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⣠⣤⣍⡙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣩⣤⣄⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣶⣿⣿⣿⣿⣿⣷⡄
⢦⣄⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣠
⠀⠈⠙⠻⠆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠰⠞⠋⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⣋⣉⣉⣉⣉⣉⣉⣉⣁⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣈⣉⣉⣉⣉⣉⣉⣉⣙⠛⠿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⡇
⠀⠀⠀⠀⠈⠉⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⢀⡀⢀⡀⢀⡀⢀⡀⠀⣀⠀⣀⢀⡀⣀⠀⣀⠀⢀⣀⡀⠀⢀⡀⢀⡀⢀⣀⡀⠀⢀⣀⣀⠀⣀⡀⠀⣀⣀⠀⣀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⣿⣼⣧⣼⢀⣟⣿⠀⣿⣾⡁⢸⡇⣿⢧⣿⢰⡏⢡⣅⠀⢸⡇⢸⡇⢹⣬⠿⠀⢸⡧⠄⢠⣟⣷⠀⣿⣼⡇⣿⠀⠈⢷⠟
⣤⣤⣤⣤⣤⣼⣧⣼⣧⣼⣭⣽⣧⣿⣬⣷⣼⣧⣿⣬⣿⣬⣿⣶⣯⣤⣬⣷⣾⣥⣽⣤⣤⣤⣼⣷⣶⣼⣭⣽⣦⣿⣬⣧⣿⣶⣤⣼⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣵⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣦⠴⣶⡏⠉⠙⢦⡋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢻⣿⢻⠋⣿⣿⡶⣿⣦⡀⢰⠿⡛⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢉⣽⣿⣿⣧⣄⠁⣰⣿⣿⡿⠿⣷⡾⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⠾⠟⣉⡿⣟⣯⠿⣿⠿⠛⢳⣶⠿⢷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⢀⡾⠉⠀⠀⠲⣿⣿⣧⣶⣤⡿⢿⣿⣿⣯⣵⡀⣴⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⡇⠌⢻⣤⣶⠋⢉⣙⣿⣿⡏⣽⣭⣷⣴⣿⣟⣩⣿⣾⣏⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⣷⣄⠙⢿⣿⡄⢤⡍⡽⢿⣄⣉⣁⣭⣩⡷⢤⣻⣟⠻⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠞⠀⠀⢻⡄⢀⠙⢿⣶⣼⣿⣾⡟⢉⣽⠛⢛⣷⡆⣽⣼⠛⠛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⣖⣩⣿⣦⡁⢦⣘⣿⠿⣿⡛⣻⣿⣟⠛⢻⣿⣧⠉⠀⠡⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠋⠁⠈⢷⠀⠻⣯⠀⠈⡿⢿⣿⢧⣤⣾⠟⠉⠙⣁⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣿⣀⡇⠀⣾⣟⢀⣭⡷⣶⡄⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠀⡀⠀⠈⠙⠻⣯⣁⠣⢹⠷⣦⠀⠀⣀⠻⣿⢷⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣷⣆⠀⠀⢀⣹⣿⡷⠶⡥⠤⠲⢾⣷⣾⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⠿⠟⠋⢙⣿⣷⣼⣶⣦⠀⠀⠀⠀⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⠷⠀⣠⢔⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢚⣵⣖⢿⣿⢛⡵⢪⡟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠈⠀⢠⣷⡲⠦⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠚⠂⠐

⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠈⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⡇⢀⡀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⡇⠀⠳⡄⠀⠀⠃⠀⠀⠀⠋⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⡀
⠀⣇⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⣄⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠧⣠⢤⣤⡶⠞⠋⠉⠛⠲⣴⠒⠒⠲⢄⡀
⠀⢿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⢢⣴⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⣤⣤⣴⣦⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣀⡶
⠀⠘⣿⣿⣿⡀⣴⣶⣀⣸⣿⣿⣿⣿⣾⣿⣿⣿⣿⡆⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢦⣴⡇⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠢⣄
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⠀⠀⠀⠀⢴⠀⢸⠂⠑⠦⣀⣐⣦⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠠⢻⠀⠀⢠⡤⣇⣠⠎⠀⠀⠀⠀⠀⠈⡇⠉⠉⠙⡏⢁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢹⡇⠀⠸⣆⡴⠋⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏⢳⡤⢤⣀⡎⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⡰⠒⡆⠈⠧⠤⣽⣽⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠀⠀⡏
⠀⠀⠀⠀⠀⢸⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠘⢦⣀⣤⣀⠀⠀⠘⠈⠓⢋⣀⡀⠀⠀⠀⠀⠀⠀⣄⡴⠧⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠇⠀⠀⠀⠀⡴⠋⠁⠀⠈⠱⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸⠳⣄⠀⠀⠀⠀⠈⠉⠉⠒⠲⠤⣀
⠀⠀⢀⡠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⣸⣇⠀⠀⠀⠀⣠⠇⣠⣀⣀⡀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠑⠒⠛⠻⠞⠳⢤⣈⣁⣉⣀⣠⠤⠖⠛⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠢⣤⣀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⣤⣄⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠒⠒⠢⠤⣴⠃⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠛⠒⠶⠶⠒⠋
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡿
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⠾⢤⡀⠀⢰⠇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣒⠚⠉⠓⢤⣀⡀⠀⣠⠋⠀⠀⠙⢶⡏
⡼⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡠⠊⠉⠛⠁⠀⠀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠈⠙⠒⠒⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⢴⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠦⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠛⠋⠉⠉⠉⠀⣘⣣⡀⠉⠉⠉⠙⠛⠿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⡁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠐⡇⢻⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠛⢿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣷⣄
⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦
⠀⠀⣼⣿⠟⠂⣴⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠐⠻⣿⣧
⠀⣸⣿⡟⠀⠀⠘⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⢻⣿⣇
⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⣿
⣿⣿⠶⠀⢿⡇⠀⠀⠀⠰⠶⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠀⠶⣿
⣿⣿⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣿
⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⢹⣿⣧⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣼⣿⡏
⠀⠀⢻⣿⣦⠄⠸⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠠⣴⣿⡟
⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠀⠀⢀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⢁⣦⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣀⠀⢨⡅⠀⣀⣀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉