Anus Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠂⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠈⡀⠀⠂⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⢻⡷⣄⠀⠀⠀⠀⠀⢀⠘⠀⢋⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠘⣅⢀⠋⠲⠀⣰⡿⠖⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡴⠛⠛⣻⠟⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⣩⠃⡀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣾⠟⠀⢸⠃⠄⠀⠀⠀⠀⠀⠀⠀⠘⠂⠹⣿⣆
⠀⠀⠀⠀⠀⣼⡏⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣬⣻⡄
⠀⠀⠀⠀⠀⣉⠇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⢺⡉⣷
⠀⠀⠀⠀⠀⠙⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡏⢀⠘⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣱⠀⠁⢻
⠀⠀⠀⠀⠠⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡟⠀⠀⠸⡇
⠀⠀⠀⠀⢲⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⡙⡆⠀⠀⡇⠀⠀⠀⢠
⢦⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠢⠄⠀⢠⡇⠙⠀⠄⡇⠀⠀⢀⡎
⢸⡇⠀⠀⡆⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡇⢸⠀⠀⡇⢀⡴⠉⡇
⢸⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢸⡄⢠⠃⠉⠀⠘⠃
⣼⣏⢷⣄⡰⢀⡇⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢸⡇⠘⠠⠃
⠛⠀⢆⠀⢛⠾⡇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢿⢹⡞
⠀⠀⠈⢁⡞⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢺⡀
⠀⠀⠀⠀⠱⢸⡿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢨⠞
⠀⠀⠀⠀⠀⠛⠇⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⢂⡄⠙⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢀⣤⢻
⠀⢾⡀⠀⠀⠀⣼⠀⡄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢨⡇⡟⠀⠀⠀⢠⡀
⠀⢨⡷⢄⡀⠀⠀⡆⢠⠈⣇⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⣧⠤⣷⡆⠀⠀⠀⡼⠁⠀⠀⠀⠀⠐
⠀⠀⢳⠀⠉⠣⠚⠧⠀⣀⢼⢄⠀⠀⠀⠀⠀⢀⡴⢀⡟⠀⢐⡇⠈⠂⢤⣼⠅⠀⠠⠈
⠀⠀⢸⠑⢄⠀⠐⢄⠈⠉⣸⣀⠂⣀⣀⠠⠔⠁⠈⣤⠁⠀⡼⠃⠀⣰⠁⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⢧⡉⠙⢁⣠⠲⠒⠶⠂⣤⡀⢾⠖⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⢻⠶⢎⡩⠓⠭⡴⠾⠤⣆⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠶⠆⠰⠚⠓⠀⠷⠶⠤⠾⠅⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣤⣿⣤⣷⣒⣒⣦⠄

⠀⠀⠀⠀⢀⣤⠖⠒⠙⠓⠒⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠟⠙⠑⢦⡀
⠀⠀⠀⢀⣾⣠⣶⣿⣟⢡⠞⠫⠽⠯⠟⠛⢦⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣶⣖⡛⢲⠟⠀⠁⠀⡈⣱⣤⠿⠄
⠀⠀⢰⡏⠀⡠⠙⠫⢍⡛⢄⡀⠀⠀⠀⠀⢾⠉⠘⠁⠈⠙⣶⡶⠟⠛⠙⢶⣤⠶⡲⠶⣶⣦⠔⠛⠛⠋⢹⣯⠖⠉⠀⠀⠨⠤⣤⠶⠛⠉⠉⠀⢠⣦
⠀⠀⢸⣀⣾⠁⠀⡴⠟⠁⠈⢛⡤⠦⣖⡒⠢⣀⡀⠀⠀⠐⠋⠛⠀⠀⠀⠈⠁⠀⠀⠀⠙⢿⣀⢀⡀⣀⡸⠟⠀⠀⣀⠤⠀⠒⠁⠀⠀⠀⠀⠀⣼⣿
⠀⢰⣿⠿⣯⠀⢸⡃⠀⠀⢠⡿⠀⠀⠀⢈⠓⠒⠚⠭⢍⣉⣩⣶⣶⣄⡤⠤⠒⢂⣤⣴⣖⣉⣉⣉⠟⠛⠉⠉⠉⠉⡇⠀⠀⣠⠳⣄⣀⣀⣤⡾⢻⣿
⠀⡟⣿⠂⠉⣷⣾⣷⣶⣤⣾⣃⠀⠀⠀⢸⡁⠀⠀⠀⠀⢨⠉⠀⠉⠁⠈⢁⣾⠋⠁⠀⠀⠠⠰⣯⡀⠀⠀⠀⠀⠀⢙⣷⠦⣿⣶⣿⣯⣹⣿⡇⠘⡭
⠀⢣⠁⠀⠀⢸⣿⠛⣿⣿⣾⣧⣦⣀⣀⣸⣧⣀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠙⣧⣀⣀⣤⣀⣠⡾⠋⠀⣇⠙⠋⢿⣿⣁⡀⣤⡼
⠀⢠⣷⠆⠀⡞⡇⠈⠁⠠⠐⢻⠉⠛⠉⠉⠈⠻⠿⠿⠷⠟⠳⣬⣒⣲⣮⣤⡿⢦⣤⣂⣰⣧⡼⠞⠁⠀⠉⠉⠉⠀⠀⢀⡴⠟⠛⢻⢠⡟⠛⠻⠿⢾
⠀⣾⡏⠀⠀⢱⡄⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣸⡌⠀⠀⠀⠀⢨
⠀⢻⣇⠀⠀⢸⡇⠀⢁⣴⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡃⢙⠟⣧⣰⡤⠄⣘⣼
⠀⠈⣿⣶⣂⡎⢿⢀⠚⠻⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡽⠛⢻⠀⠇⠉⠉⠙⣿⡇
⠀⣼⡃⠀⠈⡁⢸⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠀⠀⣼⠘⠀⠀⠀⠘⣹⣿
⠀⣿⡅⠀⠀⡇⠀⠀⠀⢀⠇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡙⣾⠓⠀⠀⠚⢿⠟
⠀⢹⣦⡀⡠⠀⡆⠀⠄⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠆⠀⠁⡏⠀⠈⠀⠋⠉⣄
⢠⠏⠁⠉⠃⠀⡇⠀⠚⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠇⠀⠁⠂⠀⠀⠠⠀⣼⣿
⣿⠂⠀⠀⢸⠀⡇⠀⠀⠀⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢠⠀⡸⡇⠀⠀⠀⢘⣿⡋
⢻⡤⢀⠀⣸⠐⡇⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡗⠀⡅⠸⠈⠙⠛⣻⢿⣿
⠈⣳⣾⡿⢇⠀⣿⣦⣤⣤⣾⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣻⠁⠀⡷⡇⠀⠀⠀⣠⡄⣿
⢰⣿⠓⠀⠸⢧⢸⠇⠀⠀⠑⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⡛⠛⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣉⠀⢀⠃⡇⢠⣤⣤⣈⣴⠏
⢸⡄⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⠉⠉⠙⠲⡦⠤⣾⡛⠛⠛⠲⣤⣤⡤⢤⣴⡟⡿⠛⠀⡈⡞⠁⠀⠙⠛⢻⣧
⠸⣤⢆⠀⠀⢀⣿⠱⡀⢀⣀⣰⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⣀⣀⠀⠀⠌⠓⠪⢿⣤⠠⣴⡏⠛⠀⠀⣿⠰⠀⢀⡴⡙⠀⠀⠀⠀⠀⢴⣿
⠀⠹⢧⣯⣽⣼⢟⠳⣜⣯⡋⠁⠀⠃⠸⡇⠀⠀⠀⠀⠀⢸⣗⠀⢸⣯⣿⡻⣿⣷⡆⠀⠀⠈⠓⣌⣱⣶⣤⡴⠾⠦⣴⡧⠶⠃⠄⢐⠀⠀⢰⣸⡏
⠀⠀⠀⣿⢩⠺⠀⠀⠈⠈⡻⠶⢂⠁⡼⠁⠀⠀⠀⠀⠀⢸⡏⠀⢸⠿⠋⠛⢻⣷⣄⣤⠀⢀⡼⢱⠋⣹⠉⠁⠚⠁⠀⠀⠀⠀⢢⣭⣡⣼⣾⠟
⠀⠀⠀⣿⢏⠀⠀⠀⠀⠀⠌⣳⣤⣝⠠⣄⠀⠀⠀⠀⠀⠀⠁⡇⠘⠀⠀⢰⡎⣿⣿⣿⣾⣾⣷⡈⠇⢈⡀⠀⢰⣆⢠⡀⣀⣠⣶⣿⠏⠉
⠀⠀⠀⢿⡋⢢⢄⢀⠀⠀⣶⣿⢯⣙⡟⢮⣇⠀⠀⠀⠀⠀⡆⠛⠀⠀⠀⠈⣷⢿⣿⡇⠀⠀⠀⠙⠲⠶⠶⠶⠟⠛⠻⠷⠶⠾⠟⠁
⠀⠀⠀⠀⠛⠿⣦⣸⣥⣤⡟⢉⣀⣩⠇⠘⠿⠀⠀⠀⠀⣸⠃⠀⠀⠀⣧⠀⢱⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⣿⠀⠘⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡆⠀⣰⠀⣿⢰⠀⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡆⢨⠀⢿⠘⢠⡇⠜⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⣴⠀⢸⠀⣾⠁⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣼⡇⠹⠀⣟⣷⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣿⠆⢠⡟⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣯⡶⠛
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⠠⠄⣀⢀⠔⠒⠒⠒⠁⡠⠔⠚⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠈⠓⠊⠁⠀⠀⠈⠋⠀⠀⠀⠀⠁⢀⣀⠤⠒⠋⠀⠀⣠⠃⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢉⡀⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠋⢉⡁⠀⠀⣀⠼⡁⠉⠸⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡐⠉⠀⠀⣀⠀⢀⣴⢲⡤⠤⠔⠋⢗⠒⠒⠊⠁⠙⠛⠛⠒⠒⠞⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⠄⡤⣔⠉⠉⠁⡜⠀⢣⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠰⡀⢸⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡜⠁⢩⣀⣠⠴⡅⠀⠀⠑⠦⣀⠀⠀⠀⠈⠉⠉⠙⢯⠀⠀⢰⠁⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢰⠁⢠⠇⠀⠘⠀⠈⠢⢄⡀⢀⣀⣈⠉⡉⠁⢖⠢⠀⠈⡆⠀⠘⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⢣⠀⠀⠀⡀⠀⠐⡄⠀⢹⠉⠉⢣⠀⠀⢸⠀⠀⠀⢰⠀⢰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠇⠀⣸⡀⠀⠀⢡⠀⠀⢰⠀⠈⠄⠀⠀⠣⡀⠀⠆⠀⠀⠸⠃⢈⠆⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⢇⢹⠀⠀⠀⡇⠀⢨⠀⠀⢰⠰⡀⠀⣑⡦⠼⢦⣀⠀⠀⢸⠀⠸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡄⠀⣜⠈⠀⠀⡰⡁⠀⠘⠆⠀⠀⠀⠈⡟⠁⠀⢄⠀⠈⠉⢉⠋⠀⠇⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠙⣉⡂⠉⡀⠙⢦⣀⣀⣠⡞⢩⣉⡉⠲⠤⠤⢑⠤⡤⠼⠂⡜⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢇⠀⠘⠤⡉⢺⡁⠀⠘⡄⠜⠁⡀⠀⠀⠈⠒⠤⣀⠀⠉⠀⣠⠞⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⠢⣀⠀⢸⠀⠑⠀⢀⢼⠀⠀⠑⢤⣀⣀⣀⡀⠀⠈⠉⠉⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⣈⡳⡦⠤⣤⠤⠊⢠⠂⠀⠀⠰⡀⠀⠀⠈⠑⠂⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠁⠒⠺⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁



⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠷⣶⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣴⡶⠿⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⢠⣤⣤⣄⡉⠉⠉⠉⠉⠻⢷⣶⣶⠶⠶⠶⣶⣤⣀⣤⣶⠶⠶⠶⢶⡾⠛⠁⠀⠀⠀⠁⣀⣠⣤⡄⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠉⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠁⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢰⣦⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⣾⠇⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠷⠶⠶⠾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⢀⡈⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠿⠋⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣿⡇⠀⠀⠀⢹⣿⢶⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣀⣠⣴⡿⢋⣿⠋⠀⠀⣶⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⣿⠂⠀⠀⠀⢸⡿⠀⠀⠉⠉⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠉⠙⠛⠋⠉⠁⠀⢸⡟⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣿⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡇⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡄⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣾⡇⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣿⡇⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⣿⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⢿⡇⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢰⡿⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢀⣿⠃⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠈⠿⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠇⠀⢠⣾⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣤⣄⢠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⠶⡿⠋⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⣀⣤⣴⡶⠿⠛⠋⠁⠀⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣠⣶⠶⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢸⡟⠀⠀⣶⣤⣤⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⡧⠀⠀⢹⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠸⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣄⠀⠈⠑⠒⠠⠠⠀⠀⠂⠀⠀⢠⠏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠠⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡎⠀⠀⢦⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢸⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⣯⣩⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⣠⠞⣭⣶⠋⢻⡆
⠀⠀⠀⠀⠀⢠⡴⠛⠯⢅⣠⠞⠋⢛⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠈⣻⢣⡞⠁⠈⣧⡼⢷
⠀⠀⠀⠀⣰⢋⡟⠙⠳⣦⠉⠳⣤⠞⠀⠀⠙⣷⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣤⠶⢾⡏⠈⠳⣤⡴⣻⠋⠙⢄⣴⠛⡃⢸⠇
⠀⠀⠀⠀⣿⠀⣧⠀⢰⠉⠳⣄⠈⠳⣄⣠⠞⠁⠀⠀⣿⠁⠉⣷⠞⠉⢳⡞⠉⠀⠙⡇⠀⠘⣧⠀⢀⣹⣦⠴⢋⣴⣇⠀⠀⣸⠏⢰⡗⢻⡄
⠀⠀⠀⢠⡟⠦⣜⣧⣯⠀⢠⠋⠛⢦⣈⠙⠓⠦⠤⣄⣁⡀⠀⣿⠀⠀⢸⣇⢀⣀⣀⣷⡤⠤⠞⠛⠉⣉⣤⣶⠋⠀⠈⠳⣾⠛⣿⢸⣇⣰⡇
⠀⠀⠀⠸⣇⠀⠀⢹⠉⣷⣎⠀⠀⢠⠟⠓⠦⣄⣀⣀⡈⠉⠛⠛⠛⠛⠛⠉⠉⢉⣉⣀⣠⣤⠶⠲⣏⠀⠀⠈⢳⡀⠀⠀⣼⡀⣃⢸⡇⠉⣇
⠀⠀⠀⢠⡿⠳⠶⣾⠀⡟⠛⣆⣀⡞⠀⠀⠀⠀⡟⠉⠉⠉⢻⠟⠛⠛⠻⣟⠉⠁⠀⠘⡇⠀⠀⠀⠸⡀⠀⠀⠀⢻⣶⠏⠉⠉⢿⠈⣧⣴⡇
⠀⠀⠀⢸⡁⠀⢀⡟⢰⡇⣠⡞⠉⢳⣄⠀⣀⣼⠃⠀⠀⠀⣼⠀⠀⠀⠀⢿⠀⠀⠀⠀⢿⡀⠀⠀⢀⣿⠖⠖⠖⠋⠘⣷⣄⡀⠸⡆⡇⠀⢹⡄
⠀⠀⠀⣾⡷⠶⢾⠃⡾⠋⢙⡇⠀⠀⠉⠉⠉⠉⠛⠲⠖⠚⠛⠛⠶⠶⠶⠛⠛⠶⠶⠶⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⣿⠉⠙⠻⢇⢱⣤⣾⠁
⠀⠀⣼⠋⠀⠀⣿⢰⣁⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠘⡀⠁⠈⣷
⠀⠀⢹⡦⠴⢶⢯⡟⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠒⠲⡇⢸⠖⢿⡄
⠀⠀⣞⠀⣀⣼⢸⡷⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣧⢸⡀⣀⡟
⠀⢠⡟⠋⢹⡇⣼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢻⠘⡟⠙⢧
⠀⢸⣅⣀⣼⠀⣿⣤⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣸⡇⢃⠀⣈⡇
⠀⣼⠋⠉⣿⠀⡏⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⣷⢸⠛⠉⢳
⠀⣿⣀⣠⡟⠀⣷⠶⠶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢘⡌⣄⠀⠀⣶
⢰⡏⠀⠀⣧⠀⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⡇⢿⠋⠙⢻
⢸⣆⣠⣤⣿⠀⡿⠲⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠒⠚⠷⢸⠗⠒⠺
⣸⠋⠁⠀⣿⠀⡇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣠⡆⢸⣇⣀⣀
⡯⠀⢀⣤⣿⠀⣿⠖⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠁⠈⠁⠀⠈
⣿⠞⠁⠀⢿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠓⢶⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣤⡄⢰⣄⣀⢠
⣿⠀⠀⠀⢸⠀⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⣸⣦⡀⠉⠛⠳⣤⡀⠀⠀⠀⠀⠀⣾⠁⠀⢠⠇⣼⠉⠉⣿
⢻⡀⠀⠀⢸⡀⡿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠴⠞⠻⢦⡙⢦⣀⠀⠈⠙⢶⣄⠀⣀⡴⠿⢦⣀⡿⢠⠏⠀⠀⣿
⠸⣧⠀⠀⢸⡿⣁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠙⠲⢤⣀⣹⡏⠙⢧⣀⣠⠟⢀⣮⠀⠀⣰⠇
⠀⠘⠓⠒⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠲⣦⣀⣠⣴⣦⡀⠀⠀⠀⠀⠉⠓⠦⣤⣈⠉⠛⠒⠚⠉⢁⡠⠟⠉⢳⡶⠋
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⣿⢦⡀⠀⠀⠀⠀⠀⠀⠉⡿⠛⠛⠛⠛⢿⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠙⢦⣄⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠓⠒⠾⢧⣤⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠐⠛⠛⠻⢶⣄⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣅⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⣛⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠒⢉⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⣾⣠⣤⣄⣀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣼⣿⣥⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣴⣿⣿⠏⠙⠃⠈⠧⡍⡿⢻⢷⠆⠐⠂⣠⡦⢴⣶⣦⣤⣤⣶⣰⣄⣸⣿⣿⣿⣿⢿⠿⣿⣻⣿⡿⣟⣿⣿⣿⣷⣽⣳⡀
⠀⠀⠀⠀⠾⣿⣿⣷⡀⠀⠀⠀⢀⣀⣀⣀⠈⠀⠀⠰⠉⠃⡈⠁⢸⢋⠹⠟⢿⣿⢻⣿⡿⠻⢧⠜⠗⣤⣯⣿⣿⠙⠫⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠰⣿⣿⡏⠁⡠⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣁⣀⣀⣌⣀⡀⣀⣻⣀⣠⣅⣤⣤⣶⣿⣿⣿⣿⣿⢿⡄⢾⣻⣿⣿⣷⠾⠇
⠀⠀⠀⠀⡇⣿⡋⠀⣶⠄⢠⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡟⢸⣧⠰⣡⢿⣿⠟⢻⠁
⠀⠀⠀⠀⡏⠙⠀⠰⡧⠄⠈⢻⣿⡀⠀⠀⠉⠈⠛⠛⠻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣟⠉⠉⠀⠀⠀⠀⢰⡂⢸⡇⢈⣛⣾⠿⠋
⠀⠀⠀⠀⡜⠀⠀⢸⠳⢄⠀⢸⣿⠇⠀⠀⠀⠐⠀⠀⠀⠀⠉⠛⠛⠁⠈⠉⠉⠉⣧⡌⢀⣠⣄⠀⢢⠀⠀⡌⢀⣾⡅⣲⠶⢶⣶⠄
⠀⠀⠀⠀⡀⠀⠀⢘⣧⠀⠀⠀⣿⡄⡄⠀⠰⢶⡄⠀⠀⢰⠇⠀⠀⠉⠀⠀⠀⣀⣸⣷⡟⠉⠁⠀⣼⠃⠀⡇⠰⣿⠄⣿⠿⠳⣾
⠀⠀⠀⠈⠃⠘⢦⠀⣷⣠⠖⠠⢿⣄⠿⠶⠿⠺⠟⢶⠤⠀⠀⢰⣄⣀⣀⣤⣴⣿⠛⠐⠅⡀⠀⢨⠁⠀⢀⠀⢸⣿⢸⣿⠄⠘⠃
⠀⠀⠀⠀⡀⠀⣼⡆⠘⡃⠀⠀⢈⣿⣧⠀⢀⡀⣠⣄⣀⢀⢀⣰⠅⠘⠋⠉⠉⠙⠀⠀⣀⣀⣤⠞⠀⠀⠈⠀⠸⣧⣸⣿⡬⣠⠤⡀
⠀⠀⠀⠀⢳⡀⠸⣷⠀⠀⠀⠀⠀⠘⣿⣷⡌⢛⡇⠀⢈⠉⡋⠁⢠⡄⠀⠈⠛⣤⣤⣴⠏⠀⠟⠂⠀⠄⠰⡀⠀⢻⣆⣾⣧⡀⠀⠹⣄
⠀⠀⠀⠀⠀⢇⠀⢱⡄⠀⠀⠀⠀⠀⣿⡟⢀⠈⠁⠀⠈⠁⠷⣄⠀⢸⣶⠄⣠⡧⠀⠰⣶⣤⡦⠤⡆⠀⣠⢸⠀⠈⣿⡹⣧⣚⣃⢂⢿⣧
⠀⠀⠀⠀⠀⠈⠗⠀⠅⠀⠀⠂⠀⠀⣿⡟⠀⠀⠆⠀⢰⣦⡀⠀⠀⠀⠟⠀⠋⠀⣤⠀⠀⠀⠀⢀⣤⢤⠟⠀⣹⠂⠘⣧⣽⣯⣍⣁⡄⣿⡄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⢠⣀⡀⠀⠀⣿⣷⠀⠀⠀⠀⠁⠀⠉⠙⠘⠇⠀⢀⣤⣴⠿⣷⣤⣠⣤⣿⡛⠾⠖⢀⡍⠀⣄⠸⣿⣿⣛⠁⡁⢹⣧
⠀⠀⠀⠀⠀⠀⣏⠑⠀⣿⠏⠀⠀⢸⣿⠉⠃⠀⠀⠀⢀⣀⣄⠀⠉⠀⣤⠘⢻⡞⠀⠈⣿⠉⠉⠼⠧⠀⢀⡿⠣⠀⢈⠀⢻⣿⡧⠤⡄⣼⣿⡆
⠀⠀⠀⠀⠀⠀⢹⠀⢘⡷⠶⠂⠀⢸⣿⠙⣧⠀⠀⠋⠉⠙⢿⣶⠶⢶⠀⠀⢸⡆⠀⣀⠙⢦⣀⣀⣀⣤⣾⠇⢀⠀⢀⠆⢠⣿⡧⠙⠃⣿⣿⡇
⠀⠀⠀⠀⠀⢘⠀⠀⢈⡧⠀⠀⠀⣾⣏⠀⠈⠳⣶⣶⠃⠀⢈⣿⣦⣾⣄⠀⠈⠉⣿⠋⠀⣼⠃⠙⠉⠙⢃⣠⡾⠀⠸⠀⢰⣽⠷⠡⣰⣿⣿
⠀⠀⠀⠀⢰⠉⠓⠀⢰⣤⠀⠀⢸⣿⣟⠀⡄⠀⠀⣿⣤⠤⠼⠿⠀⢀⡈⠛⣾⠀⠉⠀⣰⣿⣴⡾⠿⣿⣾⠉⣀⣬⢀⣼⡿⠃⠀⢰⣿⡿⠃
⠀⠀⠀⠀⢸⡀⠀⠀⣼⠟⠁⠀⢸⣿⡿⢤⣄⠀⠀⠉⠀⠀⠀⠀⡛⢻⡇⠀⠘⠓⠀⢿⣿⠋⣉⣴⣾⡿⢿⠿⠿⠾⣻⡥⡉⠐⠀⣿⣿
⠀⠀⠀⠀⣼⠁⠀⠀⢻⡟⠀⠀⡸⣯⣿⣄⠉⠲⢤⣤⣾⠟⠀⠴⠅⣀⠛⠂⠲⣤⣤⡿⠋⣼⣿⣿⣏⢱⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⣷⣀⠄⠀⠈⣿⡇⠈⢡⡛⠿⠟⣷⣦⡀⠀⠀⢦⣀⡄⠀⠉⢀⠀⠐⠛⠁⢄⠀⢿⣿⣿⣯⡈⣀⣀⠀⠀⠲⣀⣴⣼⣿⣿⠋
⠀⠀⠀⠀⠈⡇⠀⠸⣧⠈⠻⣦⣼⣿⣿⣷⣭⡉⡟⣶⣄⠀⠈⠱⢦⡾⠿⠷⡦⠀⠀⠀⡇⠈⢿⡧⠉⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⢷⣀⠀⢹⣧⡀⠙⠻⢾⣿⣿⣿⣿⣿⡿⠿⣶⡀⠀⠄⠀⠀⣄⡀⣀⣠⣴⡇⠀⠈⠿⣧⣤⡤⠀⠑⣌⠉⠛⠛⠋
⠀⠀⠀⠀⢀⡀⢹⣆⡈⢂⣼⣦⣄⡰⣾⣿⣽⣿⣿⣷⣾⣿⡿⢠⠀⠀⢸⡉⠙⠛⠛⣉⣠⣤⣤⣄⡈⢿⣯⣇⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣍⠹⠟⠿⣿⣿⣿⣿⣿⣿⡯⣿⡿⠁⢸⡆⠀⣸⠁⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡈⣤⣾⣿⣿⣿⣿⣟⣽⣿⡇⠀⢸⠁⠀⣿⢀⣾⣿⣿⠻⠯⠀⠉⠛⠛⢳⠉⠀⠂⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣿⣿⣿⣿⣷⣿⣿⣟⠁⠀⠻⠀⣸⠏⣼⣿⡿⢈⠇⠀⠀⠀⠀⠂⣙⣒⣠⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠋⣅⣀⣀⣶⡿⢳⣿⣿⠄⣤⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠛⠛⠃⠀⠋

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡁⣈⢈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠲⢄⣑⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣬⡑⠦⣉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠈⠰⢀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠁⠂⠄⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡤⠖⠂⣉⣉⡙⠻⠿⣶⣤⣈⠑⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢀⡀⠀⢠⡏⣰⣾⣿⣿⣿⣿⣿⣷⣦⢉⡛⠀⠀⠈⠐⠄⠀⠀⠀⠀⠂⠀⠈⠄⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⣧⠹⣿⣿⣿⣿⠙⠿⠿⠃⢼⣿⣷⣤⡀⠀⠀⠑⢦⡠⠒⠁⠘⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣌⠛⢿⣯⣴⣦⣀⣦⣿⣿⣿⡿⣿⣦⠀⠀⠀⠁⠀⠠⡀⢁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⣿⣿⣿⣿⣶⡀⠀⠈⠿⣿⣿⣷⡀⠄⠀⠀⠀⠡⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣿⣿⣿⣿⣿⡏⡐⠀⠀⠘⠿⢿⣿⣷⠀⠀⠀⠲⣀⣀⠅⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⣀⣄⢀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣷⣦⠄⠀⠀⠀⠐⣛⣿⣿⡆⠀⠀⢰⠀⠀⢸⠀⠃⠀⠀⠀⠁
⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢄⠀⠈⡇⢸⣿⣿⣿⣷⣿⡶⠖⠀⠀⠦⠀⠙⣮⣿⡇⠀⠀⠈⠀⠀⡜⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣦⡄⠀⢨⠳⣄⢹⠀⢻⣿⣿⣿⣯⣍⣀⠠⠄⢠⣘⣷⣿⡟⠀⠁⠀⠄⠀⠀⠃⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠆⠀⠉⠀⠀⢙⣿⣭⣭⣙⡟⡓⠠⣴⢻⣿⡿⠁⠀⠀⣤⠃⠄⠘⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠀⡀⢐⠀⠀⠀⣰⣿⣿⡧⠂⠈⢹⣿⣷⣿⠿⠃⠄⠀⠀⡞⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⠃⠠⣡⡟⠀⠀⣰⣿⣿⡿⠃⠀⠀⣾⡿⠟⠁⠀⠀⠀⣠⠊⡴⠋⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠁⠛⠁⠀⢠⣿⣿⠟⠃⠀⢺⡾⠋⠀⠀⠀⠀⡀⢢⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡇⠀⠀⠀⠀⣾⣿⡿⠀⢀⣿⠂⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣇⣀⣼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⠃⡠⠀⠀⠀⡈⠻⡏⠉⠉⢹⠀⠀⠙⠀⠊⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠵⢿⣤⠂⠀⠀⠙⠁⢸⣶⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⡸
⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠂⠀⠀⠀⢹⡇⠃⠀⠀⠀⠈⣿⡇⠀⠾⡧⠠⠀⡧⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⢘⣭⢴⡬⢠⣤⢤⠤⠀⢀⠘⡭⠀⡀⠀⡀⡄⢻⣧⡀⠸⣿⡀⠃⠁⠀⠀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠙⢤⣄⣄⣀⣐⣲⠒⠈⠀⠀⠀⠂⠀⠀⢧⡂⠘⡿⠿⠓⠛⠓⠂⠐⠚⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⠀⠐⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⣀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠐⠒⠒⠒⠒⠒⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⠿⠿⢶⣶⣤⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣴⣾⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠾⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡰⠃⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠻⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⢱⡀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⠀⢀⡏⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⣇⠀⢸⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⢸⠁⠀⢰⣯⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣍⣿⣷⠀⠀⢹⠀⡸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⡇⢾⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⢸⡆⡇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠻⣿⠟⢋⡹⠃⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡤⠤⣀⣤⣄⡠⠤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡇⠀⠀⠀⠀⠀⠀⢿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠟⠛⠉⢹⡇⠀⠀⠀⠀⠀⠀⡤⠉⠛⠻⠿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⢀⣤⣶⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⣀⠴⠒⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢷⣦⣄
⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠻⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣧⠗⡀⠀⢀⣀⣿⣤⣤⣤⣀⣀
⠀⢀⣠⠤⠤⠄⡲⣾⢢⡀⢹⣮⣶⠟⠙⠉⠁⠀⠀⠠⠤⣬⣿⣶⣄
⢠⡏⡴⠋⠀⠀⠒⣼⣷⣴⢻⠃⠁⠀⣀⠀⢠⡀⠉⠉⠙⠛⠛⠛⠻⢿⣶⣄
⠈⠳⣅⠀⠀⠀⢀⣼⠃⢿⢰⣀⣔⢦⣻⣄⣄⠙⠓⠒⠲⠂⠀⠀⠀⠀⠈⠻⢧⡀
⠀⠀⠘⣧⣀⣠⡞⠀⣬⣮⠟⠉⠛⠛⠛⠛⠛⠛⠓⠒⠿⠿⣿⣷⡦⣄⠀⠀⠀⢳
⠀⠀⠀⠈⢹⡿⠰⣤⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢎⣧⠀⠀⠘⡇
⠀⠀⠀⠀⣼⡇⢲⣿⠇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣟⠀⢆⡇
⠀⠀⠀⠀⢹⡙⣶⣾⠐⠉⠀⠉⠙⠳⠦⠀⠀⠀⠀⣤⠴⠶⠒⠒⢄⣷⣿⡇⢀⡞
⠀⠀⠀⠀⠀⠙⠶⡇⠰⣶⣶⢶⣶⡀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⡀⢹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⡀⣇⠀⠈⠣⠼⠿⠀⠀⠀⠀⠀⠀⠀⣽⣯⣙⠿⠃⢰⢿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡘⡰⠉⠁⠀⠀⠀⠀⢠⣾⠟⠋⠉⠉⠉⠙⠦⡀
⠀⠀⠀⠀⠀⠀⡊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠀⠀⢀⣤⡤⠴⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡸⠣⠃⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠔⣡⣤⣤⣀⠂⠀⡤⠀⠐⠶⠇⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣼⣧⡀⠀⠀⠐⠈⠉⠉⣉⠄⠀⠈⢸⠀⠈⠁⠀⠀⠀⠀⠙⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡟⣿⡷⢄⠀⠀⠀⠀⠀⠀⢠⢀⠀⠸⠀⠀⣇⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢰⡏⣾⢸⡿⡇⠀⠙⠢⢄⡀⢀⡠⡇⠸⠄⠀⠀⠀⣿⠀⠀⠀⠴⠥⠤⠔⠒⢧⡀⠀⠀⠀⠀⢀⢤⣀⠀⢀⣀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⣾⠁⠹⡈⠣⣙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠁⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢻⣧⡐⢄⣀⠀⠙⣳⣄
⠀⠀⠀⠀⠀⠀⣻⡝⠂⠉⠛⠷⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢀⡀
⢀⡠⠤⠒⠉⠁⠀⢹⣦⣀⠀⠀⣬⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠒⠠⢄⡀
⠀⡇⠂⠐⡉⡐⠲⢾⣿⠟⢀⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠫⠁⠂⠅⠌⠀⠀⡍⢀⠀⠄
⢐⡆⡈⠠⠀⠄⠁⣾⠁⢴⡿⠛⢓⠒⠒⠒⠒⢂⠀⠃⠀⠀⠀⠀⠀⢅⡁⡐⠐⢐⠠⠁⢂⠐⡀⡂⠄⠐⡄
⠠⠧⢐⠠⡈⠄⠡⠙⠶⠼⠔⠀⠄⠌⠠⡀⠄⢂⢀⠠⠄⠀⠀⠀⠀⠀⠲⠠⠉⠄⡐⠠⠀⠄⠄⠠⠩⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⣦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡄⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⣀⢴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢀⡾⠃⢀⡞⢁⡏⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠛⠉⠘⣆⠀⠀⠀⠀⠀⠀⢻⠹⣄⠀⠹⡆
⠀⠀⠀⠀⠀⣾⠁⢠⠟⢠⢾⠀⠀⠀⢰⠋⠉⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡄⠹⡆⠀⢷
⠀⠀⠀⠀⢰⡇⠀⡞⢠⠇⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠚⠳⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⠀⠈⠛⣆⠀⠀⢸⡇⠀⢻⠀⢸⡆
⠀⠀⠀⠀⣾⠇⠀⠀⡞⠀⣇⠀⠀⣼⠀⠀⠀⠀⣠⠴⠖⠲⢦⡀⠀⠀⢀⠤⠖⠒⢦⡀⠀⠀⠀⢹⠀⠀⢸⠇⠀⢸⠀⠸⡇
⠀⠀⠀⠀⣿⡀⠀⠀⡇⠀⢻⠀⠀⡇⠀⠀⢀⠎⠀⠀⠀⠀⠀⢱⠀⡰⠁⠀⠀⠀⠀⠱⠀⠀⠀⠘⡆⠀⣸⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢻⡇⠀⠀⠃⠀⠈⢇⢸⠃⠀⠀⠀⠀⣠⣶⣷⢂⠀⠀⠀⠁⢠⣾⣿⣃⡀⠀⠀⠀⠀⠀⡇⢀⡏⠀⠀⠸⠀⢠⡇
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⡴⠚⢾⠀⠀⠀⠀⠀⠻⣏⣈⡿⠁⠀⠀⠀⠸⢧⣀⡿⠁⠀⠀⠀⠀⠀⡗⠚⢢⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⣧⠀⠀⢸⠃⠲⣼⠀⠀⠀⠐⢄⠀⠈⠉⣠⡞⠈⠒⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠶⢄⠇⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠘⣇⡀⣹⠀⠀⠀⠀⢀⠘⠉⠉⠸⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⢀⠜⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢀⡀⠀⠀⡟⠀⠀⣹⡆⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⡄⠀⣆⠀⠀⣷
⠀⠀⠀⢠⣿⠀⢀⡼⠃⠀⠀⣿⠁⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠀⢀⡴⢻⡇⢰⡇⠀⠹⣦⡀⠹⣷⣤⣤⠂
⠀⠀⠀⠈⢿⣿⣯⣤⡾⠀⣰⠏⠀⣰⣣⠆⠙⢦⡀⠀⠀⠀⠀⠱⠤⠃⠀⠀⠀⠀⢀⡴⠋⠀⠸⣧⡈⢿⣄⠀⠉⠛⠿⣿⠏
⠀⠀⠀⠠⣤⣶⡾⣏⣥⠾⣃⣠⡴⠛⠁⠀⠀⠀⠙⠲⣤⣀⠀⠙⠂⠀⢀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠛⠶⠿⠻⠷⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠈⠙⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⢭⣽⣷⣦⣄⣠⣶⣾⡛⠶⣦⡀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⣫⠟⡡⢒⡐⠶⢶⣾⣿⣿⣟⣛⡉⠉⢳⡀⠙⢦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢀⣼⣿⣿⡯⢞⠕⢁⣴⠶⠚⠛⠻⠿⠿⠿⣿⣦⡀⢳⡀⠈⠳⣄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠀⠀⠀⢀⡾⠟⠋⢁⠔⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀⢱⣄⠀⠙⣧⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢠⠏⡀⠠⠐⢁⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣆⢰⡸⣧⠥⠤⠔⢶⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣠⡾⠋
⠀⠀⣿⠀⣧⣇⣴⣿⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣄⡙⣿⣿⢸⣿⢿⡆⠀⠀⠀⠈⠙⠒⠒⠚⠁⠈⠉⠛⠛⠉⠉
⠀⠀⠹⡄⢻⣿⣿⡇⠚⠉⠉⠉⠛⠳⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠁⣿⣿⣿⡟⠘⣧
⠀⠀⠀⠹⣶⣝⣿⡇⢠⣶⡶⢶⡶⠀⠀⠀⠀⠀⠀⠐⣿⡛⢿⡿⠂⣿⣿⠋⡔⢠⠇
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠉⠲⠿⠽⠀⠀⠀⠀⠀⠀⠀⠛⠓⠉⠀⠀⣿⣿⣼⣥⡏
⠀⠀⢀⣾⣽⢣⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣾⡿⣡⢟⡼⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⡀
⠀⠰⣿⣟⠁⣾⣿⣾⡇⠠⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⠃
⠀⠀⢻⣿⣄⠻⣿⣿⡛⠊⠀⢀⠄⠀⢀⣤⣶⣦⡤⠠⠀⠀⠀⢠⣿⣿⡏⡼⣿
⠀⠀⠀⠙⠿⣿⣾⣿⠁⠀⢀⠈⠀⠀⠀⠄⠉⠁⠀⠀⠀⠀⣠⣿⣿⡟⠐⣠⠇
⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠸⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣹⣿⠃⣴⠇
⠀⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⠸⠁⣦⠤⣀⣠⠤⠚⠁⠈⢿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⣠⣥⡾⠇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⡟⢫⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢻⣿⣿⡏⢱⡀
⠀⠀⠀⢸⣿⡜⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⢸⠇
⣀⣠⣤⠘⣿⣌⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⡿⢸⣿⣿⡇⡗⢠⣤⣀⡀
⡁⠀⠈⠑⢜⠻⠿⠿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢾⡙⠿⠟⣡⠟⠉⠀⠈⣹⠟⠁⠂⢀
⢱⡀⠀⠀⠀⠙⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡻⠟⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠑⡀
⠀⣇⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠛⠛⠛⠋⠉⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠈⠉⠳⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⢀⣤
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠚⠋⣽
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⠁⣠⣴⠞⠁
⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠙⠳⠦⣤⣀⣀⣀⠀⠀⣀⣀⣀⣠⣤⡶⠚⠉⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡀⠒⠒⠲⡄⠀⠀⠈⠉⣩⠝⠛⠣⣼⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⢀⣄⠀⠐⡀⠀⠀⢰⠁⢠⣦⠀⢹⠀⠀⠀⠀⢠⡶⢴⠇
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠘⠚⠀⢀⠀⠀⠀⠸⣄⠀⠁⣠⢾⠀⠀⠀⠀⠀⠻⣦⣀
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⠤⠤⠖⠁⠀⠀⠀⣀⠈⠉⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⠉⠛⡇
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠧⠐⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣤⡤⠞⠁
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣠⠤⠒⠒⢢⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣈⣳⠦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣗⠒⠒⢲⣪⠇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠻⣄
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⠇⠀⠀⠀⠀⣴⣅⣀⠀⠀⠈⠉⠉⠉⠀⠀⣠⠾⠷⣄⡀⠀⠀⠀⠀⠀⠈⠓⠶⠦⢤⣤
⠀⠘⠷⣤⣀⣀⣀⣀⣠⡤⠶⠚⠉⠀⢾⢤⡄⠀⣠⠞⠁⣿⠈⠙⠛⢶⣶⠦⠤⠶⠚⠁⠀⠀⠈⠉⠛⠶⠶⢤⣤⣤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣞⣥⠞⠁⢀⣠⠿⣆⡀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠰⢿⣤⣤⣄⣠⠟⠳⢤⣹⠇
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢀⡴⢋⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠙⢿⣯⣡⡀
⠀⠀⠀⠀⠠⡾⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⣧⣾⣷⣼⣿⣻⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⢿⣯⣓⣶⡤⢤⣤⣤⣤⡴⠶⠛⢉⣠⣤⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣦⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢠⠔⠉⠡⡶⠚
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿
⠀⠀⠀⢺⡛⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⢫⣴⣷⣙⣿⣏⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠙⠓⡶⠦⣤⣤⣤⣤⡴⠶⠛⣋⢡⣤⡼⠧⢓⡳⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠑⢦⣄⣠⣤⣴⣸⣦⣻⣜⡣⣾⢧⢶⣵⡞⠷⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣜⣻⠹⠟⠛⠚⠒⠓⠛⠉⣙⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠛⠋⠉⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠙⣟⠒⠶⣤⡀
⠀⠀⠀⠀⢠⡶⠋⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠙⣦
⠀⠀⠀⣴⠋⢀⡴⢊⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢧⠀⠀⠀⠀⠀⠀⠘⡗⢤⠀⠈⢷
⠀⠀⣾⠁⢠⡞⢀⡾⠀⠀⠀⢠⠖⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠋⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢻⠈⢷⡀⠘⣇
⠀⣸⠇⠀⡟⢠⢋⡇⠀⠀⢠⡏⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠳⣄⡀⠀⠀⢸⠀⠘⣇⠀⢻
⠀⣿⠀⠘⠀⡏⢸⡃⠀⠀⡾⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢈⣉⣁⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⢻⠀⢸⡆
⢸⣿⠀⠀⢸⠃⠘⡇⠀⢰⠇⠀⠀⢀⡴⠊⠁⠀⠈⠙⡄⠀⣠⠞⠉⠀⠀⠙⡄⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⢸⡇
⠸⣿⠀⠀⠈⠀⠀⢳⠀⢸⠀⠀⠀⠎⠀⢀⣠⣄⠀⠀⠱⠰⠃⢀⣤⣤⠄⠀⠐⠀⠀⠀⢸⡆⠀⡏⠀⠀⢸⠀⢸⡇
⠀⣿⡄⠀⠀⠀⠀⣈⣇⡘⠀⠀⠀⠀⠀⣿⠟⢣⣶⠀⠀⠀⠀⣾⡟⠓⣾⠀⠀⠀⠀⠀⠀⣇⣼⡁⠀⠀⠀⠀⢸⠃
⠀⠸⣷⠀⠀⠀⣸⢁⡀⡇⠀⠀⠀⠄⠀⠙⠲⠞⠃⣠⠠⣀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⡇⣀⢹⡄⠀⠀⠀⣼
⠀⠀⢻⣧⠀⠀⣿⠀⠘⡇⠀⠀⠀⠀⠐⡤⠤⠔⣿⠁⠀⠀⠁⠁⠒⡒⢀⠀⠀⠀⠀⠀⠀⣿⠉⡹⠁⠀⠀⢠⡇
⠀⠀⠀⢻⣆⠀⠈⠳⠴⡇⠀⠀⠀⠀⠌⠄⠂⠀⠘⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡷⠚⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢿⡄⠀⢀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢸⡇⠀⠀⣧⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡄⠀⢸⡇
⢠⡖⠀⠀⣼⠃⠀⠀⣿⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢠⡇⠀⣧⡀⠈⣿⣄⠀⢀
⢾⣧⣤⠞⢁⡴⠀⣸⠇⠀⢸⢈⠳⣄⠀⠀⠀⠀⠀⠀⣇⢀⡇⠀⠀⠀⠀⠀⢀⡴⠋⢿⡄⢸⣇⠀⠘⢿⣦⣬⣿⡷⠋
⠈⢛⣛⣿⡿⠃⣰⠏⠀⣠⡿⠊⠀⠈⠳⢤⡀⠀⠀⠰⢌⡉⠀⠀⠀⠀⣠⠶⠋⠀⠀⠘⢿⣄⣻⣷⣤⣄⣀⣭⠟
⠙⠛⠛⠙⠒⠋⠑⠶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣄⣠⣤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⣀⡤⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⠃
⠰⣏⣴⠶⣤⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡀
⠀⠈⠁⠀⠈⠻⣆⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠈⢧
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠒⢦⡀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠤⣄⣀⣀⣀⣠⡤⠿⡄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⢀⡴⠋⠀⢀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⢸⣃⣀⡀⢀⣄⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣼⠳⠦⠤⠤⢶⠞⠛⠒⢤⡔⠉⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡤⣄⡿⠀⢀⡞⠁⠀⣭⡏⠀⡌⠁⢹⡆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠃⠀⠶⠀⠀⣷⠘⠃⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡟⠀⠀⠀⠀⠘⣇⠀⠀⠀⣷⣄⣀⣀⠜⠁⣠⠀⢸⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠣⣀⠀⢀⡠⠛⡦⠤⠤⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣾⠀⠀⠀⠀⠀⠀⠈⣳⣶⣏⣁⣤⣤⠴⠒⠋⢹⠀⢸⠐
⠀⠀⠀⠄⠀⠀⠀⠈⢧⡀⠘⡗⠢⠭⢥⣤⣀⣀⡠⠤⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣿⠀⠀⠀⠀⠠⠂⠀⠸⣄⣀⡀⠀⢀⣀⠠⠒⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢱⣄⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⢉⡀⠠⠀⠊⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠂⠀⠀⠀⡼⠳⣤⣀⡀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠟⠋⠉⠉⠹⡍⠉⠻⡇⠀⠀⢀⡤⠒⠒⠾⡁⠀⢀⣈⠉⠉⠋⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢷⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⡏⠀⠤⠀⠀⢸⠎⠉⢀⠉⠳⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⡀⠀⠀⠀⣇⠀⠀⠈⠀⠀⡇⠀⣰⠃⠀⠢⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣆⠀⠀⠀⠀⠀⣰⠃⠃⠀⠀⠉⠒⢲⠋⠙⠦⣀⣀⣠⠔⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠀⠀⠀⠈⠑⠒⣲⠖⠊⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠀⠀⠂⠚⠂⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣤⣤⣴⣶⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠹⠉⠉⠛⣿⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠖⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠈⠑⠠⡀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠆⠀⠀⡜⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠰⣯⣀⠀⠀⠱⡄⠀⠀⠀⠀⢸⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⢒⡄⢻⡀⠀⠀⢀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠘⣧⠸⣧⢠⡾⠋⠁⠀⠀⣠⠊⣀⣀⣦⣤⣤⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠟⢄⡀⠀⠀⠈⢧⢻⣸⣿⡀⠀⣠⡞⣁⣠⠇⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⣤⣠⣊⡦⣿⣿⣿⣧⡾⣧⠞⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠦⣀⣀⣈⣿⣿⣿⣿⣿⣿⣯⣶⣮⢤⡍⢋⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠝⡷⠛⢿⣿⣿⣿⣿⡟⠲⠤⣄⣈⡀⢀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠱⠖⢪⠿⠿⠿⢹⠛⠀⠀⠀⠉⠙⣿⣾⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⡶⠒⠚⠂⡇⠀⠈⠣⡑⢦⣄⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣰⠿⠋⠉⠉⠀⠀⠀⢠⡿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠣
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠛⠻⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢆⠀⠐⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⡀⠀⠀⠀⠀⠀⠀⠄⢙⠀⠈⠀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⢀⣼⡷⢌⣄⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣆⣠⣽⣷⡦⢄⡀⠀⠀⠀⠀⡀⠰⠀⠀⢸⠀⣃⠀⠀⠀⠀⢙⣉⣿⣿⠿⣟⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠠⣀⠙⢟⠿⢿⣿⣿⡟⠀⠀⠀⠀⠀⢐⣧⣤⠀⠀⢸⣀⢻⠒⠒⠀⠀⠉⢉⡡⠾⠿⢛⣫⣽⠶
⠀⠀⠀⠀⠀⠀⠠⠄⣘⣻⠢⡈⠻⢦⣙⣿⣿⣿⣷⠀⠀⠀⠀⠀⠘⢻⣿⡀⠃⢠⣿⠏⡆⠀⠀⠀⠀⠀⣲⣾⢟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣌⡻⢷⣤⣽⣿⣟⣻⣿⣶⡄⠀⠀⠀⠃⢨⣿⣷⠴⣿⣿⢀⣿⠀⠀⠀⠀⠿⣛⣳⡤⢠⣴⠤⠀⠀⡀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠫⣙⢿⣦⣽⣿⣿⣿⣿⡿⡈⠀⠀⠀⣼⡇⢸⣿⣿⠀⣽⣿⣿⣿⡆⠀⠀⠲⣿⣿⣽⠋⠑⣠⣴⠶⠋⣀⡤⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠙⢿⣧⡙⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣷⣼⣿⣿⠀⣿⣿⣿⣿⣇⠀⠀⠙⠉⢋⣠⣶⣿⣿⣷⣾⠟⣉⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣄⢀⠙⢿⣆⠻⣿⣿⣿⣟⠿⣶⣤⣰⣿⣿⣿⣿⡝⠀⣿⣿⣿⣿⣿⣀⣤⣶⣾⣿⣿⣿⣿⣿⢟⣴⢟⢃⣔⣥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡐⣦⡙⣿⣦⡛⢿⡛⢁⠠⡄⢸⣿⣿⣿⣻⣿⠀⢸⣿⣿⣿⣿⠈⠁⠹⢿⣿⣿⡿⢋⣴⣿⢿⣧⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⣡⣿⣾⡙⠮⢛⣿⣤⡀⠀⠀⠀⣼⣿⣿⣿⣿⡇⢾⣽⣿⣿⣿⣿⣧⠀⠀⠐⠋⢹⣿⣬⣛⠮⠊⣀⡴⠊
⠀⠀⠀⠀⠀⠙⢶⣼⣿⣿⣾⣿⣿⣿⣦⡈⢿⡉⠄⠀⠀⣰⣿⣿⡿⣫⣿⣧⢶⣿⡽⣿⣟⢿⣿⣆⠀⠀⠈⠋⠛⣿⠿⠁⣀⣄⣠⣶⣶⣾⣧⡂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣧⡀⠁⠠⠀⣤⣽⢟⣫⣾⣿⣿⣷⣛⢿⣿⡌⢿⣤⡙⠿⣣⡀⠀⠀⢉⠤⣶⣿⣿⣟⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠐⠓⢤⡐⠿⣟⡿⠿⠿⠿⠃⠀⠀⣴⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⢻⣿⣿⣿⣾⠷⢦⠀⠀⠚⣻⣿⠿⠿⢟⣯⣭⣴⣦⣵⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣶⣦⣑⣀⠂⠀⠀⠀⠀⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡿⣿⣟⣿⣿⣇⢀⣠⣶⣞⣤⣀⣀⣀⣀⣀⣤⣼⣿⣿⣿⣟⣯⣿⣤
⠀⠀⠀⠀⠀⠀⢤⡄⢻⣟⣻⡟⡿⢿⣿⣿⣿⠟⣿⣿⡏⠌⢹⣿⣿⣿⣿⣼⣿⣿⣿⣿⣟⣿⣧⣤⣴⣯⡿⢟⣿⣌⣉⣹⣿⣿⠿⢿⣿⡿⢻⡿⠻⠒
⠀⠀⠀⠀⠀⠐⢬⣻⣿⣆⣝⠃⣤⣀⠉⠿⠋⠀⢿⣿⣿⣷⣿⣶⣶⢻⣿⣿⣿⣿⣿⣟⣶⣶⣿⣿⣿⣿⣣⣾⣿⡿⠟⠋⠉⠀⣬⣻⡵⠖⠋
⠀⠀⠀⠀⠀⠀⡀⠸⠩⣛⡿⢷⠟⡋⣱⠀⠐⢢⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣃⠙⠛⠋⠀⡠⠀⢈⣁⣀⡨⣆⡠⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣍⣟⡚⡿⡷⡰⠄⠀⠀⠈⢸⣿⣿⠟⠛⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣷⣟⠿⢯⣴⣭⣥⣴⣶⡟⣩⡄⠉⠉⠀⠈⠁
⠀⠀⠀⠀⠀⢀⢶⣄⣹⣿⣿⡿⢷⣿⢓⡆⠰⠀⣀⠐⠉⣽⢿⣥⣿⣿⣿⠏⠀⠈⠛⠛⠛⠛⠋⠙⣡⣥⣍⣨⣭⡿⣿⣿⣞⡻⣧⡱⠶⠒⠂
⠀⠀⠀⠀⠀⠀⢻⣮⣟⣷⣾⣟⣷⣿⣿⣿⣷⠶⡋⣔⢄⠀⠀⠛⠛⠻⣿⡃⠀⠘⣆⠀⠀⠲⣶⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣷⣿⣯⣍⠐⠂
⠀⠀⠀⠀⠀⠈⠛⠧⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣧⣶⣦⣷⣦⣤⣶⡞⠁⠀⠁⠀⠙⣧⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢀⣴⣾⣿⣿⣶⣄⣠⣤⡤⠄
⠀⠀⢀⣤⣤⣤⣀⡉⠙⠻⣿⣿⣿⣦⣴⣾⣿⣿⣿⣦⣤⣴⣶⣶⣦⣤⣴⣶⣶⣶⣤⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⠿⠋⢁⣤⣶⣶⣶
⠀⢰⣿⣿⣿⣿⣿⣿⣷⣦⡀⠙⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡿⠿⠟⠿⠿⢿⣿⠟⠋⠉⠉⢠⣶⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣄⠉⣉⣁⣀⣈⡉⠉⢉⣠⣤⣤⣤⣄⠉⣠⣴⣶⣶⣦⣤⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⢿⡛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠋⠁⢿⣦⡙⠿⣿⣿⣿⣿⡿
⢿⣿⣿⣿⣿⣿⡿⢸⡷⠀⢠⣌⠉⠉⠻⢷⣝⠿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⠟⢋⣀⣀⣀⡀⠀⠀⠀⠀⣽⣿⠆⣸⣿⣿⣿⣧
⠘⣿⣿⣿⣿⠟⢁⣿⣅⠀⠀⠙⢷⡀⠀⠀⠹⣇⠀⠈⠀⠀⠀⠀⠀⢹⡿⠻⠟⠛⠉⠀⠀⠀⠀⠈⠻⣾⠋⠉⠉⠉⠙⠛⠀⠀⠀⣿⣿⠀⢻⣿⣿⣿⡟
⣰⣿⣿⣿⣧⠀⢻⣿⣿⣷⡀⠀⠈⣿⠀⠀⢀⡿⠀⠀⠀⣦⠀⠀⠀⠘⣧⠀⠀⠀⠀⠲⣆⠀⠀⠀⠀⠸⣇⠀⠰⣦⡀⠀⠀⠀⠀⢻⣿⣧⢨⣿⣿⣟
⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⢸⡆⠀⠘⠃⠀⠀⢀⡿⠀⠀⠀⠀⠘⠷⠀⢀⠀⠀⠹⣧⠀⠀⠀⢀⡿⠀⠀⠈⠙⠳⠶⠶⠞⢻⣿⡇⢸⣿⣿⣿⡆
⠈⠻⣿⣿⡟⠀⣾⣿⣿⠛⠀⢀⣀⣼⠃⠀⠀⠀⠀⢀⣾⡷⠞⠛⠷⢶⣄⠀⢀⡿⠀⠀⣰⡟⠀⠀⠀⣸⣧⣄⣀⣀⣠⣄⠀⠀⠀⠸⣿⣇⠸⣿⣿⣿⠃
⠀⣰⣿⣿⣷⣆⠈⠻⣿⣷⣄⠛⠉⠙⠷⣤⣤⣤⡶⠛⠁⠀⠀⠀⠀⠀⠉⠻⣟⠁⠀⣰⠟⠀⠀⠀⢰⠏⠀⢈⣩⣭⣭⣀⡀⠀⠀⢀⣼⡿⠀⣹⣿⡁
⢰⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠙⢷⣤⡟⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠉⠻⠆⣴⣿⡿⠁⣰⣿⣿⣷⡀
⠸⣿⣿⣿⣿⣿⡿⠀⣿⣿⣏⠀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣤⠞⠋⠉⠉⠉⠙⢷⡀⠉⠛⠶⠶⠞⠛⠉⠀⠀⠀⣸⠇⠀⠀⠀⠀⣿⣿⡇⢰⣿⣿⣿⣿⠃
⠀⠘⢿⣿⣿⣿⡁⠸⣿⣿⣿⣷⡄⠛⠉⠀⠉⠛⠻⠛⠋⠁⢀⣠⡄⠀⠀⠀⢸⣧⣤⣴⣤⣤⡀⢀⣤⡴⠶⢶⡏⠀⠀⠀⢀⣤⣿⣿⡄⢸⣿⣿⣿⠃
⠀⢀⣾⣿⣿⣿⣿⡆⢸⣿⣿⣿⠇⠀⠐⢦⣀⠀⠀⠀⠀⣰⣟⣁⣀⡀⠀⠀⠾⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⣁⣠⣼⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⠇⢸⣿⣿⣏⠀⠀⠀⠀⠙⠛⠶⠶⠶⠛⠋⠉⠉⠙⢷⡄⠀⠀⢸⡇⠀⠀⠀⠋⠀⠀⢀⣴⣾⣿⣷⡟⠁⣰⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠻⣿⣿⣿⡏⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⡇⠀⠀⢀⣤⣶⣶⣿⡿⠛⠋⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣷⣦⡈⠻⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⣿⣤⣤⣤⣿⣦⣴⡿⠛⠛⠛⠋⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⠀⠉⠛⠛⠶⠶⠞⠛⠁⠀⠀⢀⣰⣿⠿⠛⠋⠉⠉⠉⢠⣶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠋⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⣿⠏⠈⠓⠦⣤⣀⣀⣠⣤⠶⠚⠉⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠤⠐⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡷⠠⣿⡇⠸⣶⠖⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣾⣾⣶⣶⢶⣦⣤⡀⣠⣄⠈⣿⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡿⠇⠃⠻⡟⢿⣧⠙⠙⠌⢝⠃⡀⠉⠉⣮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡸⠰⣧⣠⠠⡀⢀⠈⠻⡡⣱⡤⡠⡀⣙⣆⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡇⠀⠘⠙⢧⠻⣦⣿⣶⣷⣼⣿⣽⣿⣿⣿⡷⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠷⡄⠀⠀⢉⣉⣽⣋⣁⠴⠂⠈⠉⠉⠙⢣⠉⢻⣿⠇
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠿⣶⣶⣤⣴⡞⢋⡿⣿⣷⠄⢹⡟⠳⠦⡐⠀⠀⠂⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠄⠙⡟⠙⣧⡀⠈⠉⢀⣠⡟⠀⠀⠀⣿⣦⠡⠐⠀⠀⢣
⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠶⠺⠠⠄⡈⣙⠻⠟⠛⠍⠀⠀⠀⢀⣿⠙⠀⢠⠀⠀⢰⡄
⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢆⠀⠀⠀⠀⠁⠀⠒⠄⡀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣧⣿⡎⣆⢸⣿⣿⣶⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠒⠤⠤⠤⠴⠖⠋⢀⠀⢹⣿⡏⣻⣧⣹⣿⣿⡿⠀⠀⠀⠀⡂⣾⡄⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠆⠹⢿⣝⣿⣿⠛⠛⠁⠀⠀⠀⢸⣇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⢭⣿⣧⣤⡀⠄⢆⠀⠈⠻⠿⢿⣿⣷⠀⠀⠀⠀⠀⠀⢡⣴⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣡⡤⠔⠚⠂⠀⠀⠀⠀⠉⠛⠣⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⣾⠟⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣤⣴⣧⡀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠙⣿⡿⠋⢳⣤⡀⠀⠀⠀⠀⠀⣀⡠⠄⢺⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠤⠀⡀
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠠⢄⡘⠒⠢⡤⠤⢀⡀⠀⡀⣀⣀⠀⢀⣀⡀⢀⡀⠔⠋⢈⣠⠄⠀⠰⠈⡄
⠀⠀⠀⠀⢀⠆⠐⠢⠴⡄⠀⡈⠁⠋⠐⠠⠤⠈⣀⠀⣙⣁⣀⣈⡁⡀⠀⠡⠅⠒⠌⠂⢁⡐⠕⠉⠰⠁
⠀⠀⠀⠀⠈⡆⠀⢠⠀⡏⢈⢟⠀⠔⠦⣀⣀⢦⡀⡀⣄⠀⠀⣄⠀⠀⠀⣈⠠⠤⢲⠒⠁⡃⢰⠀⢢⠃
⠀⠀⠀⠀⠀⢄⠀⣾⠀⢾⡀⠑⠩⠘⢄⠀⠈⡄⠫⡦⡀⠉⠋⠄⣁⡐⠊⡠⠀⠢⠓⢀⢼⠀⡾⠀⡠⠆
⠀⠀⠀⠀⠀⠪⠀⡟⠀⣱⠌⡢⢀⠠⢤⢔⠕⣁⠠⠌⢀⣊⣤⢄⡀⠀⢙⢄⠀⡐⠨⠂⠘⠀⠇⠀⡙⡄
⠀⠀⠀⠀⠀⣱⠀⠃⠀⠽⠁⠒⠥⢐⠀⢀⠮⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠋⠐⠄⠀⠨⠃⢠⠈⠳
⠀⠀⠀⠀⠰⡈⠀⢰⠀⢘⠇⠠⢬⡐⠀⡘⠀⠀⠔⢀⠒⠉⠡⢀⠀⠀⡀⡠⠀⠀⡈⠢⣰⠕⢸⠀⢴
⠀⠀⠀⠀⠀⠎⠀⠀⡇⠀⡃⠀⢠⢏⣂⠃⠀⢸⡀⠘⣜⢄⠣⠑⠩⠍⡒⢷⠀⢸⠪⠂⢠⡁⢸⠀⢄⠇
⠀⠀⠀⠀⠀⠡⡴⠂⠡⠀⠈⣡⠘⡘⢀⡇⠀⠠⢏⠢⢄⡙⠫⠖⠠⢊⠐⠉⢀⢌⠆⠈⢣⡀⢸⠀⡸⡄
⠀⠀⠀⠀⠀⠀⢄⣀⠀⠑⡄⠉⡏⢀⡀⢰⢄⠀⢙⠀⠉⠘⠑⠂⠀⢠⡓⠀⠹⡀⠀⠁⣮⠀⠈⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠃⠤⠠⠄⢄⢐⠂⠈⠀⠀⠁⠁⠁⠖⢀⠉⣑⠒⠮⠀⢤⡄⠾⠁⠉⠀⡠⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠦⠸⠂⠠⣙⠲⠦⠬⠀⠠⠄⡒⠉⢤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠀⠢⠉⠁⠑⠂⠐⠓⠐⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣀⣤⣴⣶⣿⣿⣷⣄⡀⠀⢀⣀⣤⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣠⡴⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣤⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠛⠙⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣤⣤⡾⠋⠉⠁⠀⠀⠀⠀⠀⣸⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠂⣤⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠛⠱⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠌⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣴⡶⠾⣄⡀⠀⡐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⡀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣶⣤⠸⠏⠙⢿⣦⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣾⣿⣿⣿⣿⠇⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠙⠳⠴⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠠⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⡍⠙⠻⣶⣄⡀
⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⡟⠀⠀⢩⣿⣿⣿⣿⣿⣿⣟⠁⠀⠘⢿⣦⡀⠀⠙⠻⣶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠙⢿⣿⡋⠙⢿⣷⣤⡀⠀⠙⠿⣶⡖⠀⠸⠙⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠈⠛⣿⣷⡄⠀⠻⠯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣾⡿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣾⡿⠋⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢠⣾⠟⠃⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⠿⣿⡟⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣷⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡀⠤⠴⠖⠒⠒⠒⠀⠀⣀⣄⡐⠄⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢤⣶⣶⠶⣶⣶⡶⢤⣤⣄⡙⠻⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⢂⣔⡕⣡⣶⡿⢟⣻⣭⣭⣾⣧⣤⣌⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡌⡜⠄⡀⢈⣟⣰⣵⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⡟⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢰⢰⠃⢸⠀⣾⢣⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠸⠘⠀⠉⢰⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡆⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⢿⡿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠸⠄⢀⠀⠘⣷⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢸⢀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠁⠘⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⢣⢧⠏⠀⠃
⠀⠀⠀⠀⠀⠀⠰⡄⠈⢢⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣛⣵⡿⡫⠋⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠐⢌⢮⡂⠉⠂⠀⢤⣀⠈⠩⠀⠀⠠⠤⠀⢐⣊⢥⡮⢞⡵⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠢⡀⠀⠀⠈⠅⠀⣈⣠⣭⣭⣭⠶⣛⣩⠶⠋⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠀⠀⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠖⠛⠶⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⣀⣤⣤⣀⠀⠀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⡴⠟⣻⣿⠟⢉⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⢠⠋⢀⡴⠟⠀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠰⣃⣴⠿⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⡟⠃⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠊⠻⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠀⠀⠀⠙⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⠖⠉⠀⠀⠀⠀⢀⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠟⠁⢀⣀⣀⣀⣀⣀⡿⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⢀⣤⣾⣿⣿⣿⡙⡉⠉⠠⠀⢹⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⣾⡇⠀⠀⠸⠃⠐⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠛⠀⠉⢿⣧⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⢀⡏⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡈⠀⠀⠀⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⡀⠀⢠⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠛⡏⠸⠋⠈⠉⠀⠀⠀⠀⠀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣴⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣄⣠⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠿⠻⠗⢿⣿⠁⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠋⠋⠀⠀⠀⠀⠀⢈⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠧⣒⡄⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⣿⣿⣿⠋⠘⢯⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢈⠁⠋⠀⠀⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠋⠁⠀⣀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⢿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⢇⣀⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠒⠒⠒⠠⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠉⠛⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠙⠀⠀⠀⠌⡛⠦⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠄⠑⢽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠈⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡐⠤⡐⠄⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠑⠨⠑⢌⠢⡀⠙⠿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠘⠂⠁⠀⠈⠲⠀⠘⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⡀⠘⢆⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⢄⠣⠣⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡀⠙⢌⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡐⣌⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⣝⢆⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⡌⠂⠹⡳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠀⠈⠳⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠙⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠈⠢⡈⠓⢝⠢⡀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠑⠮⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠈⠓⢌⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⢀⠈⠑⠠⢀⡁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠂⢀⠲⠬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠈⠋⠐⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠚⠛⠛⠛⠛⠓⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠖⠋⢉⣉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠢⣄⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠊⣀⣴⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣀⠑⢄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡁⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⠠⣄⣠⣾⣤⣼⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣦⣤⣶⣤⣤⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣶⠞⠉⠀⠀⠀⠀⠉⠻⣄⡀⠀⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠀⠉⠻⣧⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣾⠁⠀⠀⢀⣴⠖⠀⠀⠀⢹⡙⠀⠀⠀⠀⢊⡏⠀⠀⣠⣤⠄⠀⠀⠀⠘⣷⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠸⢿⠀⠀⠀⢸⣟⣷⣾⠇⠀⢀⡂⠀⠀⠀⠀⠀⡀⠀⢸⣟⣷⣤⡆⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⡴⠲⢾⠀⠀⠀⠘⢆⠀⠀⠀⠉⠉⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉⠛⠋⠀⠀⠀⣰⠃⠀⠀⢸⡖⠲⡄
⠀⠀⠀⠀⢸⠠⣄⢸⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⡀⠔⠊⠀⡠⠂⠀⠀⠠⣄⠐⢕⠢⣀⣀⣀⣀⡤⠚⡁⠀⠀⠀⢸⢁⡤⢹
⠀⠀⠀⠀⢸⠀⢞⢹⡄⠀⠀⠀⠀⠉⠒⠶⠆⠀⠀⠀⠀⢸⠀⢠⠀⠀⡀⢈⡇⠀⠙⠲⠶⠶⠶⠶⠊⠁⠀⠀⠀⢸⠛⡆⣸
⠀⠀⠀⠀⠀⠣⣀⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣷⣤⣷⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠛⠁⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠰⣄⠀⠀⠐⠒⠂⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠷⣦⣤⣀⣀⣤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢷⣄⡙⠋⠉⠻⣷⣤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣤⣤⣶⠟⠋⢀⣠⣴⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣤⠉⠛⠷⣦⣄⣀⠀⠉⠛⠛⠉⠛⠛⠋⠙⠋⢁⣀⣤⣶⠾⠛⠉⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⢸⡿⢶⣶⣦⣀⣉⠙⠛⠿⠶⠶⣶⣶⠶⠶⠾⠛⠛⠉⢁⣤⣤⣴⡿⣷⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣼⡇⢰⣿⠉⠛⠛⢷⣶⣶⣦⣠⣤⣤⣠⣤⣄⣤⣶⣦⡾⠏⠉⣿⡆⣿⡆⠰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠁⢀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⣿⡄⢸⡇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⣿⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠘⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢻⡇⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣸⡟⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠘⣷⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⣴⠟⠃⣰⡟⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⢹⣧⠀⠀⣿⡆⠀⠀⣠⡾⠛⠿⣿⡏⠙⠛⠋⠉⠛⢉⣉⣴⠾⢋⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣗⠀⠀⠹⣷⣾⠋⠀⠀⠀⣿⡇⠀⠀⠘⠿⠶⠶⠶⠶⠿⠛⠋⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⠾⠋⣿⡄⠀⠀⣼⡟⠀⠀⢶⣄⠀⣀⣀⣠⣤⣤⣴⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡧⠀⠀⣿⡀⠀⠀⠈⠙⣿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡴⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠛⠦⣄⠀⠉⠙⠢⠴⣋⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠒⠚⠛⠛⣇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣷⣶⣶⣾⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠋⠁⠀⠈⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠉⠙⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡏⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠋⠉⠙⠻⢦⣤⡴⠶⠶⠶⢦⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣿⠋⠁⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢹⡿⠶⠤⣤⣤⠶⠶⣤⣄⣀⣠⣤⠶⠶⢶⣶⠞⠋⠁⢸⠇⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣾⠋⠉⠁⠀⢀⣴⠟⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠀⠀⠀⠀⠀⢻⣤⡴⠶⠶⠶⠾⢷⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⠄⢸⡷⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠋⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠉⠙⠿⣟⠛⢻⡿⠉⠁⠀⠀⣾⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢿⣄⣠⡴⠶⠶⢦⣄⡀⠀⠀⠙⣷⡟⠀⠘⢦⣄⠀⢿⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠈⠙⢷⡄⠀⠈⢻⡄⠀⠀⠉⠳⢾⡧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠆⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣷⣄⣀⡀⠀⣸⡇⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣶⡟⠁⠀⠀⠛⠲⠶⢦⣤⣤⣤⣤⡴⠟⠀⠈⠉⠉⠉⣹⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⡤⠶⢦⣴⠟⠉⠷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⣴⠟⠉⠛⠶⠶⣤⣤⣤⣴⡟⠁⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣇⠀⠀⠀⣀⣤⣤⣄⣠⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⣀⣠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣿⣴⣶⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠾⠛⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠂⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠓⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡤⠶⠖⠲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠻⠋⠀⠉⠉⠉⠉⠉⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠍⢤⣭⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠉⠑⠒⠉⠐⠣⠄⠂⠐⠠⠘⠉⠁⠒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠼⠗⠀⠀⠀⠾⠓⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣀⠀⢀⠀⣀⠀⠄⠁⠀⠀⠠⠤⠐⠀⣀⡀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠁⠂⠁⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢈⠄⠀⠠⠀⠀⠐⠀⡁⢀⠀⠀⠀⢀⠘⠀⠁⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠂⠀⠀⠄⠠⠠⠀⠐⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢⢀⣀⢀⠀⠀⠀⠄⠀⠀⠐⠉⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠫⠁⠉⡥⠎⠉⠀⠀⠀⠠⠀⠒⠂⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠀⠀⠀⠀⢀⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠃⠀⠘⠤⢦⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣱⠒⠒⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠠⣄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠙⠉⢿⣦⣴⡿⠉⢋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣧⣼⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⢰⡟⢻⡆⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁




⡶⣶⢶⢶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠈⠉⠁⠉⠀⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠆⢤⣤⣤⣤⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⣄⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⢄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⡀⠈⠃⠀⡠⠋⠀⠀⢀⣄⠀⣀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠆⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⡎⠀⢈
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠟⠀⢀⣤⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠀⠀⠸
⠀⠀⠀⠶⠶⠲⠖⠂⠀⠀⠀⣀⠄⠁⠀⠀⠈⡅⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⣼⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⣴⢞⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠖⠁⠀⡀⢀⣠⠞⠁⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠁⠴⠀⠀⠠⠀⠀⠀⠄⠐⠒⠊⠁⠀⢀⡀⣤⠔⠉⠀⠰⠀⠀⡇⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠴⠆⠴⠂⠀⠀⢈⠀⠀⠀⢂⠀⠀⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡀⢀⣄⠀⠀⢀⣀⠴⠂⠀⠐⢲⠟⠋⠉⠠⠀⡔⠀⠀⠀⠀⣈⣀⠀⠀⡂⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⢀⣈⡀⢀⣀⡤⠞⠁⢀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠁⠀⠐⡀⡕⠀⠀⠃⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⠆⠁⠀⠀⠀⠀⠀⠀⢠⡌⠑⠀⣀⣠⠀⠀⠀⠀⠀⢇⠀⠀⠀⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠘⠂⠄⣀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⢀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠢⠄⠐⠐⠃⠠⣀⠀⠀⠀⠀⠸⠀⠐⠆⠀⠈⠣⡀⠀⠀⠚⠣⡄⠘⠆⠀⠀⠀⠀⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠏⠀⠀⠈⢂⠀⡄⠀⠀⣆⠀⠀⠀⠀⠂⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠆⠁⠀⠆⠀⢀⡀⠀⡠⠀⠀⠐⠊⠁⠳⠀⠀⠂⠀⠀⠘⢀⡴⠁⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠴⢀⠀⠀⠀⡴⠋⢠⠂⢀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢀⡰⠁⠀⠀⡀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠁⠀⢀⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠔⢠⠂⠈⠀⠀⠠⠘⠉⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠁⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠄⠀⠀⠀⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠡⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⣴⠒⠀⠀⠀⠬⠶⠶⠖⠶⠖⠲⠾⠖⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⣧⢀⡀⠀⠀⠁⠀⠀⠹⠡⠶⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠂⠒⠀⠉⠁⠀⠀⠀⠀⠤⠴⠶⠶⠆
⠀⠀⠀⡁⢠⣦⣤⣤⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⡿⠛⠛⠓⠛
⠀⠀⠀⣤⡠⣤⢤⣤⢤⡤⣤⡄⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠂
⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⣤⣀⣀⣠⡄⢀⣀⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠦⠲⠶⠖⠆
⠀⠀⠀⠉⠉⠉⠉⠁⠈⠈⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⡌
⠀⠀⠀⣠⣀⣄⣠⡄⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠉⠁
⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠛⠛⠛⠛⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠛⠛⠛⠿⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠿⢷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠿⢷⣶⣄⡀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠁⠀⠀⠀⠀⡠⠴⠒⠒⠲⢤⡀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⢀⡠⠴⠒⠒⠲⢤⡀⠀⠀⠀⠀⠻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢠⠊⠀⢀⣠⣤⡀⠀⠈⢦⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣠⠋⠀⢀⣤⣤⡄⠀⠈⢦⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⠀⠀⠀⠈⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⣸⡏⠀⢀⡇⠀⢰⣿⣿⣏⠀⠀⠀⠈⡇⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣇⠀⠸⣿⣿⣿⣿⡿⠀⢀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⡇⠀⠸⣿⣿⣿⣿⡟⠀⢀⠇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠘⢦⡀⠈⠛⠛⠋⠀⣠⠞⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠘⢆⡀⠈⠉⠋⠉⠀⣠⠞⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠉⠒⠂⠄⠒⠊⠁⠀⣠⣾⠏⠀⠀⠤⠔⠲⠤⡀⠀⠈⢿⣦⡀⠀⠉⠒⠒⠀⠒⠊⠁⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣀⠀⠀⠀⠀⢀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡀⠈⠙⠛⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠰⠿⢶⣤⣤⡶⠾⠆⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠉⢀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢁⣀⣀⣀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠉⠉⠉⠉⢙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠴⠝⠒⠤⡀
⠀⠀⠀⠀⠀⠄⠀⠉⠓⠨⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⢀⣀⠐⠀⠀⠀⡁⡀
⠀⠀⠀⠀⠀⢄⡀⠀⠠⠀⠈⠐⠢⡀⠀⠀⠀⢀⠀⠀⠀⣀⠠⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⣀⣠⡐⠃⢠
⠀⠀⠀⠀⠀⠀⠃⠛⢆⠀⠀⢀⠄⠉⠉⠑⠰⠖⠐⠒⠂⠲⠄⠐⠈⠉⠈⠁⠀⠀⠀⠀⠔⢁⢈
⠀⠀⠀⠀⠖⠀⠀⠀⠨⠁⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠅
⠀⠀⠀⢘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠀⠀⠎
⠀⠀⠀⢀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠨⠃⠐⠤
⠀⠀⠀⠰⠀⣀⣠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠰
⠀⠀⠀⢠⠀⠐⠁⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⠒⠛⠉⠉⠂
⠀⠀⠀⠠⠀⠀⠀⠀⣀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⡖⢃⠦⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡉⢀⠒⠀⠈⠀⠀⠀⠀⢀⠔⠒⠂⠀⠀⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠑⠒⠀⠀⠀⠀⠀⠄⠀⠀⠡⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠂⠁⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠑⠀⠀⢶⠀⠀⠀⠱⢄⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡔

⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠈⠙⢆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⢭⣽⣷⣶⣄⣠⣶⣾⡟⠶⣦⡀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⣫⠟⡡⢒⡐⠶⢶⣾⣿⣿⣟⣛⡉⠉⢳⡀⠙⢦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢀⣼⣿⣿⡯⢞⠕⢉⣴⠶⠚⠛⠻⠿⠿⠿⣿⣦⡀⢳⡀⠈⠳⣄⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠀⠀⠀⢀⡾⠟⠋⢁⠔⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⢱⣄⠢⡘⣧⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢠⠏⡀⠠⠐⢁⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣆⢱⣸⣷⠥⠤⠴⢶⡀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣀⣠⡾⠋
⠀⠀⣿⠀⣷⣅⣴⣿⠁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣄⠘⣿⣿⢸⣿⢿⡆⠀⠀⠀⠈⠙⠓⠒⠛⠁⠀⠉⠉⠋⠉⠁
⠀⠀⠹⣄⠻⣿⣿⡇⠊⠁⠈⠉⠛⠳⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠁⣿⣿⣿⡟⠈⣷
⠀⠀⠀⢹⣷⣽⣿⡇⢰⣶⡶⢾⡶⠀⠀⠀⠀⠀⠀⠐⣛⡛⢿⡿⠃⣿⣿⢋⡔⢠⠇
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠉⠒⠾⠟⠀⠀⠀⠀⠀⠀⠀⠛⠓⠉⠀⠀⣿⣿⣾⣥⠏
⠀⠀⢀⣾⣽⢣⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣾⡿⣡⢟⡼⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⡀
⠀⠐⣿⡟⠁⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⠃
⠀⠀⢻⡿⣤⠻⣿⣿⡛⠋⠀⢀⠀⠀⠠⣤⣶⣶⡤⠀⠀⠀⠀⢠⣿⣿⠏⡜⣿
⠀⠀⠀⠙⠺⣿⣮⣿⠁⠀⢀⡋⠀⠀⠐⠄⠉⠁⠀⠀⠀⠀⣠⣿⣿⡏⠀⣠⠇
⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⠔⣿⣹⣿⠃⣴⠇
⠀⠀⠀⠀⠀⠀⢸⢿⡅⠀⠀⠀⠜⠁⣶⠤⠤⠤⠴⠚⠁⠀⢿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⢀⣰⣥⡾⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⡟⢡⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢻⣿⣿⣇⢱⡀
⠀⠀⠀⢸⣿⢲⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⢸⠇
⣀⣤⣔⡘⣿⣄⢸⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⠟⣸⣿⣿⡇⡇⣲⣤⣄⣀⡀
⡀⠁⠀⠙⢮⡛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⣜⠿⢟⣱⠏⠁⠀⠀⡸⠋⠉⠀⠀⡀
⠸⡆⠀⠀⠀⠙⢦⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⠿⠛⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠐⡄
⠀⢧⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠢⢄⠀⠀⠚⠁⠙⠋⠻⣌⠓⠫⠹⠦⠑⡂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠙⡆⣀⠀⠀⣀⣤⣔⠉⠤⠠⡤⠀⢠⠤⢍⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠛⠉⠉⠉⠁⠈⠁⠉⠀⠉⠉⠉⠙⠒⠦⢤⣐⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠆⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣦⣤⣄⣀⣠⣠⣄⣠⣀⡘⠂⠀⠀⠀⠀⢀⠞⠁⣀⡀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⢳⠀⠀⠀⠀⠀⡄⠾⣧⠹⠷⠆⣶⠋
⠀⠀⠀⠀⠀⠁⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢝⠱⠾⠛⠀⠁
⠀⠀⠀⠀⠥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠋⣿⢤⡝⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠈⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠠⠚⡽⠀⠀⠀⠀⣴⡏⣹⠿⠀⣶⡞⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠩⠖⠀⠀⠁⣠⠞⠀⠀⠀⠀⢶⣶⠦⠿⠉⢀⣭⠀⠉⠋⠱⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠚⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠘⢦⣄⣀⣀⠄⠊⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⢆⡀⣀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠿⢿⣤⣀⣀
⠀⠀⠀⠉⠉⠀⠘⠿⣦⣀⣀⣀⣀⣤⣴⠴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⣤⣄⣄⣄⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡟⢻⣿⣿⣿⣿⣷⣤⡔⠒⠛⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣦⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠴⠿⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢫⡾⠛⠓⢒⣶⣶⣶⣶⣾⠟⠻⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⣭⡛⠉⣿⣿⣤⣬⣿⡿⣻⡽⠟⠠⣂⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠢⣹⣿⣿⣿⡿⢛⠁⠐⣩⡖⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⣷⣦⣭⣨⣈⣤⣷⠶⡛⠋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢀⠘⣯⣿⣿⣿⠿⡟⣙⢊⠙⣂⣀⢰⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢄⡁⠟⣿⣯⣽⣦⣸⣯⡸⠿⣿⡏⢰⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣬⣵⣿⣿⣿⣩⡆⠿⣛⣷⢀⠃⡈⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⡕⣆⣿⣿⡟⢰⢒⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣧⡿⣿⣷⠟⠁⡼⢠⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠋⠀⢰⡟⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⡠⠤⢤⡤⠒⠒⠒⠒⠦⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠐⠖⠊⠉⠉⠉⡄⠤⠤⢄
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⣀
⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣀⣀⣀⣀⡀⠀⢀⣀⠀⠀⠀⣀⡠⠤⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆
⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠒⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠒⠊⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⣄
⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⢄⣀⡀⠤⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⡰⠃
⠀⡝⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠐⠘⠒⠶⠶⠆⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠈⠢⣄⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢄⠀⠀⠀⢠
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠸⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⢈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠙⡅
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠱⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡
⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡸
⠀⠀⠀⢩⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠢⢄⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠑⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡔⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⣀⡤⣶⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢀⡠⠞⠁⠀⠀⠙⠂⠈⡀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣽⣤⣵⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠋⢉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣄⣀⣀⣀⣿⣿⣿⣿⣦⡀⠀⢀⣀⣀⣤⣴⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠻⡆
⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠋⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃⠀⠀⠹⣿⣿⣿⣿⣿⠟⠉⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠛


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

⠀⠀⠀⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⢀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⢮⡛⢿⣦⡄⠈⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢱⡈⠻⣿⣆⠘⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣆⠀⠈⣿⡷⣶⣿⡷⠄⠀⠙⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⡇⠀⣿⡿⢋⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢿⣇⣼⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢿⣿⣿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⣿⣿⣿⣿⣖⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⡶⣒⡡⠐⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠿⣿⣿⣿⣶⣾⠿⣋⣠⠔⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣿⣿⣿⣿⣿⣯⠥⣊⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡻⣿⣿⣷⣿⣯⣤⣤⠖⠀⢀⣀⠀⠀⠀⠀⠀⠀⡰⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⡻⣿⣿⣿⣿⣿⣿⣏⠁⡠⢀⠔⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢭⣛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠻⠿⠿⠟⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠤⠒⠤⠒⠀⠀⠐⠢⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠒⠊⠉⠉⠱⠄⠀⠠⡀
⠀⠀⡔⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡄
⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠢⠤⠒⠒⢄⠠⠐⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸
⠱⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠶⠆⠒⠁⠍⠀⠀⠀⠀⠀⠀⠘
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠁⠒⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⢂⠀⠀⠀⠠⠄
⠑⣄⠀⠀⠀⠀⢀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠑⠂⠐⠂⠒⢄⠀⠀⠀⠀⣀⠀⠀⠀⣀⠔⠒⠂⠊⠁⠀⠀⠀⠀⠀⠎
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠲⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢀⠔⠁⠀⠀⠀⠂⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢀
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠘⢤⡀⠀⠀⠘⠀⠀⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⡤⠖
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠀⠀⢰
⠀⢄⠀⠀⠀⠀⠂⠀⠀⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠠⠀⠀⠐⡃
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡔⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠣⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⠤⠰⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠁⢣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⠔⠤⠤⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣠⠔⠻⣦⡀⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠐⠊⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢸⣿⣿⡿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⣤⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢀⡏⠀⠀⢀⣀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⠃⣶⣶⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡼⠀⢸⠇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⡇⠀⢸⠀⠀⠀⢀⣶⣟⠀⠀⠀⠀⠀⠀⠀⡾⡀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢸⠁⠀⢸⡀⠀⠀⠘⠿⣯⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡸⠀⠀⠘⢇⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢸⡉⠋⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⠀⠀⠈⠓⠒⠚⠁⠔⠋⠙⠲⡀⠀⠀⠀⢳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠡⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣟⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⠛⠁⢠⠻⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠸⡀⠀⠀⠀⠀⠀⠀⢀⣯⣀⡙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠓⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢳⡀⠀⠀⠀⠀⠀⢸⡄⠉⠉⠉⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢷⡀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠙⠦⣤⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣄⣀⣀⡀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠡⠔⠒⠒⠒⠒⠲⠬⠍⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⢀⡠⢤⣖⣒⠒⠲⠤⢄⣀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⣁⣀⠤⠤⣀⣀⠀⠚⢕⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡠⡰⠋⣠⠖⣉⣤⣶⣶⣶⣶⣤⣉⠲⣄⠙⢞⢆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡸⡝⠀⡔⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠁⠀⢫⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⢠⢷⠁⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⡄⠀⠈⡆⠀⠈⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⢸⠘⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠀⡧⡄⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠃⡏⢀⢧⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⡀⠀⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡜⠀⢜⡞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣵⣄⠙⠦⣉⠛⠿⠿⠿⠿⠛⠉⠴⠋⠀⢠⠎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠠⡀⠈⠚⠑⠤⣀⡉⠉⠒⠀⠀⠀⠀⢀⡠⠖⠁⢀⠞⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡳⢄⡀⠀⠐⠚⠯⠭⠤⠴⠒⠊⠉⢀⡠⢖⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠑⠒⠢⠤⠤⠀⠀⠀⢀⣀⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠿⣿⣿⣦⠤⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣈⠻⣿⣿⣿⡟⢛⣷⣦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠌⢢⣈⠷⣽⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⢡⣾⣟⢙⡿⣬⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⡄⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡿⠿⠀⠀⠀⠐⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣦⣶⣶⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠛⠉⠉⠛⠛⠓⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣠⣶⣶⣄⢹⡄⠀⠀⠀⠀⣸⠁⣠⣶⣤⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⣿⣬⡇⠀⠀⠀⠀⣿⣼⣿⣿⣿⣷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠻⣿⣿⢟⡟⠀⠀⠀⠀⠀⠸⣟⢿⣿⣿⠏⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣁⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠙⠛⠿⠿⠿⠿⠛⠛⠛⢿⡟⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣤⣤⣀⣀⣤⣤⡤⠴⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢀
⠀⠀⠀⠀⠀⡠⠂⠉⠉⠠⠤⣉⡉⠲⠠⣄⣀⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⠈⢁⡠⠀⠀⡄⢱
⠀⠀⠀⠀⠀⡣⠀⠤⣀⣄⠀⠀⠘⠳⠒⠤⢀⡀⠘⠀⠀⠻⠋⠈⠑⠊⠉⠉⢈⡃⠤⢔⠢⠁⠁⢜⠔⠃⡸
⠀⠀⠀⠀⠀⢇⠀⠐⠀⢸⠒⢺⠆⢀⣴⡀⠀⢀⣄⠉⠀⠉⠉⠉⠉⡁⠀⠀⠀⠀⣀⡰⡤⠔⠹⠀⡤⠀⢸
⠀⠀⠀⠀⠀⡮⠀⢠⡇⢉⡄⠑⢕⡂⢆⠉⠉⢹⠈⡷⡉⠙⠂⢖⡉⠚⠠⠤⠊⣉⠩⣐⠅⠀⡎⢀⡇⠀⣹
⠀⠀⠀⠀⠀⡸⡂⢸⠃⠰⡕⢄⠀⠀⠀⠑⡠⡪⠃⠈⠂⠅⠄⣀⠀⠉⠐⠲⡎⠀⠀⡠⣒⠈⡅⠐⡇⠠⢌
⠀⠀⠀⠀⠀⢂⡀⢸⠀⢐⡯⠠⡈⠐⣊⠛⠊⣔⠊⠉⠉⠈⠤⠭⠉⠢⡀⠀⠊⡦⠮⡀⠁⠀⡢⠀⡇⠴⡸
⠀⠀⠀⠀⠀⡾⠂⠀⡀⠀⣷⡀⠀⠩⠒⠂⠞⠀⠀⡀⠀⣠⠤⢄⠀⠀⠇⠀⠀⡄⠀⠘⢄⠀⡡⡀⡇⠀⡸
⠀⠀⠀⠀⠀⠣⠄⠀⢧⠀⣪⠀⠈⢻⢄⢠⠁⠀⠰⠀⢰⢠⠀⡌⢖⠠⠤⢔⢴⡀⠠⣔⢄⠑⠪⠀⢹⠀⣱
⠀⠀⠀⠀⠀⢧⠀⣀⠸⠀⠰⠤⠀⡇⠳⢺⡀⠀⠻⠆⡀⠳⢕⡬⣂⠉⡡⢐⡡⠇⠀⠏⠢⢁⠓⠀⠹⠈⢜
⠀⠀⠀⠀⠀⠀⡟⠁⠀⢣⠀⠰⢇⠕⠁⠜⡇⠀⠠⢇⠈⠔⣤⢈⠁⠉⠑⢄⣀⢠⠊⠎⠠⣀⠷⠀⠆⠀⡺
⠀⠀⠀⠀⠀⠀⠑⢤⠂⠀⠑⠀⠐⡀⠤⡢⠜⠢⠀⠼⣀⠀⣀⠀⢁⠀⢘⠎⠀⠀⠣⡀⢀⡼⠇⠀⠃⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁⢂⡏⠀⠀⠀⠀⠀⠀⢀⠋⢠⠀⠢⣅⡙⠈⠑⠖⠉⠁⠀⢀⠀⠁⠀⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢈⠓⠐⠉⠐⢦⡉⣓⠒⠒⢒⠐⢨⡁⢐⠔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⠀⠀⡁⡅⠀⠀⠉⠂⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⢁⣀⣀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣤⠞⠋⠉⠉⠈⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠊⠀⣠⠔⠋⠉⠁⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠟⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠛⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⣴⢾⣷⣶⣄⠀⠀⢰⠀⠀⠀⡌⠀⠀⣠⠶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⣹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⡇⠀⠀⠅⠀⠀⠀⠀⢻⣿⣶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠟⠀⠀⢰⠀⠀⠀⠇⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣄⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⣠⣶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁

⠀⢠⠊⡽
⠀⢸⠀⡇
⠀⠘⡆⢧
⠀⠀⠘⢎⠢⠄⢀⣀⣀⠤⠄⣒⣒⣒⡒⠒⠀⠤⢀⣀
⠀⠀⠀⠀⢱⠀⠉⠛⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⣆⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡒⠝⡻⡸
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠋⠉⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⡒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡯⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⣍⣒⡒⠤⠤⢤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⢿⣿⡿⢿⣿⣿⠿⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠛⠳⣦⣴⠾⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣿⠂⠀⠀⠀⠀⠀⢤⣾⡤⠀⠀⠀⠀⠀⢹⡷⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⠿⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⠀⠀⢸⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠈⠻⣦⣀⠀⠀⠀⠀⣀⣤⣄⣀⣀⣀⣀⣤⡾⠋⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠈⠉⢛⣿⠛⠛⠉⠁⠈⠉⠉⠉⣿⠅⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠀⠀⢠⡟⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣬⣷⣄⠀⠀⣤⢿⡷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠃⠀⠀⣬⠿⠙⣧⢸⡁⠙⠶⠀⠀⠀⠙⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣞⡋⠁⠀⠀⢀⣠⣤⣤⣤⠴⠋⠈⠛⠒⠶⠶⠶⣤⣀⣀⡀⠨⣽⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠖⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⢠⣤⡄⠀⠀⢀⣠⣤⣀⣠⡀⣀⣀⠀⠀⣀⣤⣄⡀⢀⣀⣀⣤⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇⢸⣿⣿⠀⣴⣿⡿⠿⢻⣿⣷⣿⠇⠀⢰⣿⡿⢿⣿⢸⣿⡿⠿⢻⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢾⣟⠀⣿⣻⣿⡇⢿⣿⡀⠀⢸⣿⣿⣏⠀⠀⢸⣿⡇⢸⣿⢹⣿⣧⡄⢸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠾⠟⠸⠿⠛⠿⠿⠈⠛⠿⠖⠸⠿⠙⠻⠃⠀⠀⠛⠿⠿⠋⠸⠿⠏⠁⠸⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠛⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⡀⠲⡿⠒⠀⣀⠀⣷⠀⠒⣿⠒⠲⠂⠀⠀⣀⠀⠀⣀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠇⣿⠎⣿⠀⡇⠀⣾⣩⠇⡿⣦⠀⣿⠀⣿⠀⣿⡎⣽⠀⣾⣩⠗⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠀⠿⠀⠿⠜⠷⠔⠻⠄⠴⣥⡿⠖⠻⠔⠻⠤⠻⠀⠻⠴⠻⠤⠔⣥⡼⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠉⠀⠀⠀⠀⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⢀⣀⠀⣀⠤⠄⠈⠀⠀⠐⠀⠀⠀⣠⠟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠙⠓⢤⣀⡀⠀⠀⠀⠀⠐⠀⠠⠄⠀⠄⠠⠤⠀⠀⠀⠁⠀⠀⡀⣠⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⢀⠀⢀⠘⠦⠶⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣠⣞⡉⠉⠉⢣⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢡⠈⣦⠈⠻⡄⠀⢈⡇⢹⡛⠛⣶⠶⠖⣷⣶⡞⠉⠻⡄⠀⡀⠁⠀⠀⣰⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢡⠀⠸⠀⠀⣿⠀⠈⠁⣸⣅⣀⡉⠓⠆⠀⠈⢃⠀⠀⡇⣀⡉⢉⡉⠙⢇⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠢⠐⡀⠀⠒⠻⣤⠤⠺⡋⠀⠈⠉⠲⣅⣀⡴⠃⠤⠞⠉⣈⣁⡉⠀⢀⡜⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⠃⠈⣆⠀⠀⣀⡀⠀⣀⠤⠖⠒⠦⡈⠻⢄⣀⡠⠔⠋⠁⠀⠁⠙⢯⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠋⠉⡀⠉⠉⠀⢀⠀⠀⠀⣹⠀⠀⠠⣀⠐⠒⠛⠁⠀⠀⣤⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠱⢲⠖⠒⠚⡱⠊⠉⠀⠉⠁⠢⠧⠀⠀⠀⣦⠖⠚⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⡷⢄⣀⡀⣀⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠀⠀⢦⣀⡀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢀⡴⢞⠙⠦⠤⠔⠻⡄⠀⠀⠈⢲⠖⠚⠳⢄⣠⠴⣄⡀⢀⣶⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⡅⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠉⠉⠉⢻⡻⣶⣿⣟⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⣀⡤⠴⠏⠀⠀⠀⠀⠀⠈⠒⠲⠶⢤⡀
⠀⠀⠀⠀⣀⣤⣶⣿⠈⠀⠀⠀⠀⠀⠀⠙⢮⡻⠿⣥⣀⠈⠛⠷⣦⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⢀⣾⠏⠇⠋⢿⡀⠀⣀⠣⣀⢄⡠⡀⢀⠉⠀⠀⠙⢷⣄⣲⣼⣧⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎
⠀⠀⢸⡇⠀⠀⠀⠤⣽⣶⣬⣷⠾⢿⣿⣿⣿⣿⣶⣤⣽⣦⡙⢿⣿⡿⢧⣄⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⣼⣷⣄⠀⠤⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⠙⠣⠀⠙⢷⣖⠒⠚⠤⣀⣀⣀⡠⠤⣄⣠⣤⡴⠟⠁
⠀⢸⣿⢯⠛⠛⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣠⡀⠀⣌⣿⠀⠀⠀⠀⠈
⠀⢸⣿⡀⠀⢀⣼⣿⢁⡤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢠⣿⣿
⠀⢸⣿⠻⠶⣾⣿⡟⠁⠀⠀⠈⠙⠻⠂⠀⠀⠀⠀⠶⠶⠚⠛⠙⠣⣽⡟⢸⢟⣿⣷⡀
⠀⠀⣻⢷⣤⡽⣿⠁⣶⣶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⢸⡇⡘⠸⣿⣿⣷
⠀⠀⢸⣆⣘⡆⢹⠀⠀⠑⠴⠿⠇⠀⠀⠀⠀⠀⠀⠀⣶⣦⣉⠿⠋⠈⡟⢳⡰⡻⢻⣿
⠀⠀⠀⠙⢿⣷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢡⠋⢀⣰⢿⡿
⠀⠀⠀⠀⣾⣟⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡸⠀⣉⣵⣿⠁
⠀⠀⠀⢰⣿⠟⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡼⠟⢛⣽⣿⣇
⠀⠀⠀⠸⣷⠀⠠⢿⠂⠁⠀⠀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠐⠉⠋⣽⡟
⠀⠀⠀⠀⢸⡗⠞⠂⠀⠀⠖⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⢀⡾⣿⣿⣆⡀⢀⣼⡿⠁
⠀⠀⠀⠀⠸⣷⡴⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡻⣎⠻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⡾⠊⡀⠀⠀⠀⠀⣠⠔⠋⢹⣿⣿⠶⠅⠀⠨⣻⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⢀⠉⠉⠀⠈⠉⠀⠀⠀⢸⡻⢿⣯⣥⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡒⠀⠠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣶⠿⠋
⠀⠀⢀⡀⠤⠒⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀
⣶⣯⣅⣀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣶⣦⣤⣀⣀

⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠐⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⡀
⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠋⠉⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣵⡄
⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡀
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⣶⣤⣶⣶⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣺⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣰⣿⠃⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⡽⣟⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠮⠙⠉⡙⣿⠄⡀⠏⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢱⣄⢟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣷⡁⢠⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡀⡐⠋⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⡝⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠥⡖⢳⠜⢷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⡃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣝⠫⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢻⣯⠠⢀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢼⣷⣘⣿⢿⣗⣄⠀⠀⠀⠀⠀⠀⠀⢀⠸⢸⣿⣤⡤⣅⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢻⡏⠹⣾⣿⡿⡳⡀⠀⠀⠀⠀⠀⠀⡳⡘⢻⣿⡔⢸⠉⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢩⠀⢻⣿⣷⡘⣿⠀⠀⠀⠀⠀⢀⣡⣝⣻⣿⣿⣬⡆⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢀⡢⢀⠈⣿⣿⣧⣌⡇⠐⠠⠈⠀⠁⠀⢲⢻⣿⣿⣿⠧⠉⠀⠙⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⢿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⡄⡹⡿⠋⠘⠀⠁⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⠜⠫⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⢀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⢢⡄⠀⠀⠀⠈⠙⠛⢹⠀⢀⠠⠔⠊⠀⠙⠦⠄⢀⣀⡤⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠄⠠⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠆⠠⠀⠲⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⢧⠀⠀⠈⠉⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⣦⠤⠦⠤⠀⠀⠈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⢀⡇⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠠⠤⣇⡀⠀⠘⠦⣀⣀⣤⣋⠉⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠈⠱⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⢹⡠⠾⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣠⠋⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣢⠤⠤⣤⠒⠉⢳⠒⠋⠁⠀⠀⡠⠂⠀⠀⠘⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠁⠀⣰⡇⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⣖⠊⠁⠈⠲⠤⠤⠘⢦⣀⡀⠀⠀⣰⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠙⠆⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠻⡍⠀⠀⠀⡗⢤⣀⣀⣀⡠⠖⠋⠉⠓⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⡠⢄⣀⠀⠀⣀⠜⠁⠀⡼⠁⠀⠰⢤⡀⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠋⠀⠀⠈⠙⢯⠴⠒⠲⢼⡁⠀⡀⠀⠀⠃⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢤⠀⠀⠀⠈⣆⣀⣀⡀⠙⢆⠙⠦⣀⣀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢺⠀⠀⠀⢠⠇⠀⠀⡽⣆⠈⠳⣄⣀⠴⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠁⢀⣠⠔⠁⠸⣄⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠓⠲⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠉⠛⢿⣿⣿⣿⠿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠉⠳⠎⠁⠀⠉⠳⡴⠛⠉⠙⠳⣴⠋⠉⠉⠙⠳⠖⠋⠙⠲⡎⠉⠙⠻⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣼⣦⣀⠀⢀⣠⣧⡀⠀⠀⠀⣠⠶⣤⣀⣀⣀⣤⠞⠙⢦⣤⣤⠂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⠉⠛⠛⢉⣀⣙⠓⠚⠛⠁⠄⠀⠉⣉⠉⠀⠐⠒⢄⠀⣺⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣸⠀⠀⢢⠉⠀⠐⠒⢄⠀⠈⠠⠀⠀⠠⠊⠀⠠⠀⢀⠀⣀⢈⠀⢈⡷⠄⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠀⠀⠙⣆⠀⠔⢠⠊⠁⠀⠀⠂⡠⠄⠊⠀⡁⠀⠀⠐⠄⠁⠀⢨⠀⢀⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡀⡠⠀⣂⠉⣁⠀⠪⡀⠀⠈⠀⢀⠠⠀⠀⠀⢆⡐⢁⠄⢺⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣼⠀⠀⠈⠀⠀⢀⠉⢢⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡃⢰⠙⣧⠀⠠⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠙⡆⠠⡀⠀⠀⢀⠀⢸⠀⢢⡈⢤⠂⠀⠀⣀⠤⡀⠠⠓⠄⠀⡞⠑⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡇⠀⠀⡠⠔⠁⢀⡘⢆⡀⢂⠘⠀⠀⡌⠀⡀⠄⠀⠀⠂⠘⣇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⡄⠀⠀⠈⠳⣤⠟⠊⠉⠉⠢⠙⡄⠘⡄⡰⠀⠀⡇⠀⢀⠀⠐⢀⠈⢠⠿⢦⠀⠈⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⢀⠄⢀⠀⠐⠈⠀⠀⠀⠀⠀⡰⢡⠊⡡⠄⣀⠁⠀⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠰⣏⠓⠀⠀⣣⡀⠁⠀⣀⣐⠀⠊⠀⠁⠈⠀⣠⠴⠗⠚⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⢹⠉⠉⠂⠈⣩⠽⠛⠲⠴⠒⠛⠛⠳⢦⡼⠁⠀⠀⠀⠀⠀⠀⠠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⢀⡜⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠊⠀⠀⠀⠀⢻⡀⠀⠀⠀⠠⣤⠼⢦⣀⣀⣠⡴⠳⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⠀⠀⣿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⡠⠲⡄⡄⠀⠠⡄⣇⢰⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣥⣧⣼⡇
⠀⠀⠀⢸⡇⠀⠀⢹⠈⡇⢻⢸⠈⡇⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⣾⡿⠋⠀⠀⣹⣯⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠰⣿⡦⠔⠸⠆⠙⠸⠃⡥⢉⡠⣄⢠⠒⡌⡇⠀⠀⢀⣾⡿⠁⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠛⠙⢿⡷⡄
⠀⠀⠀⠀⠈⠠⣴⠒⡆⢀⡀⢇⠙⡎⣧⢘⡞⢧⡀⠘⢀⣠⣿⠟⠀⣠⣶⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠀⣄⠘⠘⡶⠃⢾⡄⠀⠀⣠⣤⣿⣿⣏⠀⣰⣿⣿⣿⣿⣿⣿⠏⠀⢀⣠⣴⣶⣾⢄⣔⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠿⠄⠈⠁⠀⠁⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣠⣿⡟⠉⠉⠙⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⠀⣼⣿⠋⠀⠀⠀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⠂⠀⠀⠀⠀⢠⣾⠿⠃⠀⣤⣤⣶⣿⣿⣿⣿⡟
⠀⠀⠀⣶⣶⣦⣤⣤⣶⣶⣾⣿⣿⣿⡿⠿⠛⣛⣩⡿⠟⠉⠀⣀⣀⣀⣤⣤⣶⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠈⠻⢿⣿⣭⣽⣿⣿⣿⡽⠷⠶⠶⠛⠋⣉⣤⠶⠿⢛⣿⣿⣿⣻⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⣋⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠉⠀⠐⠂⠀⡀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠁⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠆⠀⢀⣤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠁⠈⠈⠀⣈⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠦⣀⢜⠻⡟⢹⢿⡿⠟⠛⠉⠉⠉⠉⠫⢿⣿⣿⡟⠻⢶⣤⣤⣤⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⠿⣶⣾⣺⣅⣀⣀⣀⣤⣠⣴⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠠⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⠀⠀⠈⠟⡏⠀⠀⠀⠀⠀⢸⣷⡆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⢆⣀⣀⡼⣁⣀⡀⠀⠀⠀⣯⠛⠃⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠐⠉⠈⠙⠄⠀⠀⠘⢦⣀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⠋⠉⠉⠉⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠔⠊⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⠞⣏⠉⠉⠉⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣇⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⠀⠈⠉⠙⠓⠒⠲⠶⠦⠼⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⢷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⢦⣀⣀⣠⠴⠶⠒⠶⠦⢤⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⣄⡀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⠉⠓⠲⢤⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠤⠤⠤⠶⠒⠊⠉


⣾⣾⣷⣾⣶⡷⢾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣷⣿⣶⣿⢿⡶⢿⠶⠶⠾⣷⡶⠶⠶⠿⢿

⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⢀⡴⠋⠁⠀⠀⠀⠈⠙⢦⡀
⢠⠏⠀⠀⣠⠖⠒⠶⣄⠀⠀⢹⡀
⣾⠀⠀⣸⠁⠀⠀⠀⠈⣇⠀⠘⡇
⢹⡀⠀⡿⠀⠀⠀⠀⠀⢿⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⡄
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠈⠳⢤⣤⠤⠴⠶⠶⠶⠤⢤⣀⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡇⣀⣠⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣠⠏⠀⠀⠀⠀⠀⢿⡉⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠉⠳⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠰⣿⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠟⠀⠀⠀⠀⢀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢰⣶⡄⠻⠃⢸
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⣟⠲
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣇⣤⠤⠖⠋⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡏⠀⢀⡞⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⣏⠙⠛⠒⠒⠒⠒⢺⠇⠀⠀⠀⠀⣼⣠⡔⠋⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡼⠉⠀⡇⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡇⠀⠀⠀⠀⣸⠁⠀⠀⢀⡼⠁⠀⠀⠹⣄⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠴⠃⠀⠀⠀⠀⠻⣄⣀⡤⠋⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠖⠛⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠉⠀⠉⠀⠀⢀⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠙⠓⠛⠛⠲⣤⡤⠤⣤⡀⣀⣀⡀⣀⣤⣤⣤⡴⠚⠓⠲⡏⠀⠀⠀⠀⠀⣀⡠⠔⠚⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⡀⠀⠒⠦⣄⣀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠉⠉⠙⠋⠀⠀⠈⠀⠀⠀⠀⢀⣀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠢⠤⠤⠤⣀⣀⣀⣀⣀⠤⠤⠤⠤⠖⠒⠊⠉⠉⡀⠀⠀⢀⣼⣀⣤⠀⠀⠀⡄⠀⠈⣷
⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⢄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⣀⣀⡴⠓⠶⠒⠋⠀⠀⠹⣄⠀⠀⡇⠀⣰⠇
⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⢻⡗⠒⠚⠳⢤⣤⣤⠴⠷⣤⣤⠴⠛⠳⠶⠶⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⡇⠀⢀⡿
⠀⠀⠀⠀⠈⣧⡠⠄⠀⡇⠐⠂⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠂⠀⡇⠐⠻⣆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⣇⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⡇⠀⠀⣹
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⡀⢠⡇⢀⣴⠏
⠀⠀⠀⠀⠀⠸⣆⡠⠀⠘⡆⠀⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢸⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢱⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⡼⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢧⠐⠂⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠁⢠⠃⠀⠊⢿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⠀⠈⢧⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠀⢀⡎⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠳⣄⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠴⠶⠦⣤⣠⣤⣤⣠⠶⠒⠾⠃⠀⠀⢀⠞⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠃⠀⢀⡾⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠴⠃⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⡤⣴⠛⢧⠀⠀⠀⠀⢸⠁⠀⠀⠀⠒⠤⢄⣀⣀⣀⡠⠤⠖⠊⠁⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡆⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡾⠀⠀⠀⠀⢠⡏⠉⠂⠀⠀⠀⠀⢠⠤⠶⠛⠛⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠁⠀⠀⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠚⠃

⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣟⣿⣿⡿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿