Olfaction Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠠⢬⠝⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡠⠀⣀⣈⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡄⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡀⢸⡶⢀⠑⢄⡀⠀⠉⠒⠂⢤⠀⠀⠀⠀⠀⠀⣠⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠦⣄⠘⣇⡀⠉⠓⠤⣀⠀⠀⠀⠂⠀⠰⠂⠀⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠓⡀⠱⢬⣾⠃⠀⠈⠑⠦⣄⡀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⠁⠀⠀⠈⠑⠮⡤⡼⢠⠀⠉⠲⣆⡿⠁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢦⠀⠀⠈⢶⡄⠀⠀⠀⠀⠰⠋⠛⠚⠓⠚⠛⣟⢁⠀⠀⠤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡺⡂⠷⠀⡀⠀⠙⠢⠄⠒⠒⠢⣄⠀⠀⠀⡤⢎⡀⣤⠴⠷⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡀⠀⠀⠀⣿⣀⢙⠳⠄⠹⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣎⣴⠋⠁⠀⠀⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣁⣉⡀⠦⣴⠆⣱⠤⠎⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡸⣏⠁⠀⠀⠀⠀⠘⣄⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠮⠑⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⣿⠀⠀⠀⠀⠀⠀⠈⢣⠀⠘⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠡⣜⠉⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⡈⣋⡙⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣥⡽⠀⠹⣆⠀⠀⠀⠀⠀⢤⣷⣶⠂⠀⠀⠀⠐⢄⠉⢦⣄
⢀⣀⣀⠀⢀⢠⡄⠀⠀⢠⢏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⣄⠙⣦⡀⠀⢀⠤⠚⠙⠛⠧⡄⠀⠀⠀⠸⡄⠀⠙⣆
⣬⡽⣩⡷⠿⢞⠃⠀⢀⡘⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠁⠞⢁⣶⣞⢦⣀⣉⣜⡦⠀⠀⠀⠻⡀⠀⠘⣆
⠀⠁⠀⠈⠀⠙⠲⡄⠈⠄⠀⠀⠸⡄⠸⢧⠀⣠⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢘⡿⠀⠉⠀⢻⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠘⢇⡜⢓⣉⠀⣅⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡞⠇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢠⢮⡈⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢣⡔⠃⠀⠀⠉⠀⠀⠀⠀⠀⢀⡔⢂⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢷⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠗⠊⢐⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢧⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸
⠀⠀⠀⠀⣠⡰⡆⠀⢈⠳⣸⡇⠀⡀⠠⠚⡄⠀⠀⠀⠀⢹⣤⣠⣬⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢧⡀⠀⠘⠳⡄⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠈⣱⠔⡿⠩⠁⣹⣄⠘⠍⣢⣜⡹⣋⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠜⣇⠀⢀⢠⡙⠄⠀⠀⠀⠀⠀⠀⠘⠃⣠
⠀⠀⠀⠀⠀⠉⢻⠀⠀⢀⡟⠙⣧⡉⣄⢹⢀⠟⠀⢀⠃⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠖⠙⢆⠈⠀⢧⡀⠀⠀⠀⠀⠀⢀⠄⢰⡏
⠈⣷⠀⠀⠀⠀⠘⣆⢀⣾⠁⠀⠘⣏⢉⡀⠀⠀⠀⣼⡔⠇⢈⣖⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠰⠏⠿⠀⢣⡕⠀⠀⠀⠀⢀⠎⢀⢸⡇
⠀⢹⡆⠀⠀⠀⠀⠈⢫⡥⠤⡀⠀⢹⡏⠙⢦⠊⣠⠟⠛⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⡀⠃⠀⣞⠀⢰⠀⠀⠟⢦⠀⠤⠤⠤⠄⠊⢠
⠀⢸⡇⠀⠀⠀⠀⠿⣶⠗⡀⠀⢠⡟⢿⣀⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠁⠀⠀⡟⠀⢸⠀⠀⢠⠈⠁⢀⠀⠀⠀⠀⣟
⠠⣤⣇⠀⢠⣤⣤⠞⠃⢾⣧⠒⣿⠀⠘⣿⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⢩⠀⠀⠰⠇⠀⢸⠀⠀⢸⡆⠀⢆⠑⠄⠀⠰⠋
⠀⠀⠹⣆⠀⠈⠀⠀⠀⠸⣇⠀⠸⣆⣠⡎⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⡎⠀⠀⣇⠀⠀⠘⠀⠀⠀⡜⠀⠀⠱⡌⠲⠛⣷
⠀⠀⠀⠸⣧⣄⡀⠀⠀⠀⣿⠀⠀⠘⢿⣄⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠟⠋⣶⡘⠃⠀⢺⠀⠀⠀⢱⡄⢆⠀⠀⠀⡴⠁
⠀⠀⠀⠀⢀⡙⠿⣄⠀⢠⣿⠃⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⡄⠀⠀⢹⣦⣀⡹⠀⠀⠀⡆⠙⣮⠳⡀⠀⢱
⠀⠀⠀⠀⠙⠛⠁⢿⣄⣸⡇⠀⠀⠀⠀⢿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠁⠀⠀⢈⠟⠉⠁⠀⠀⠀⡇⠀⠈⢣⠘⡄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⠀⠀⠀⠀⣺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⡌⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣥⠀⠀⣤⠃⠘⢿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢱⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣶⠀⢰⡏⠀⠀⠛⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢾⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣄⠛⠀⠀⠀⠀⠘⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⢤⡤⠄⢀⣀⣀⣠⣄⠀⠀⢠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠀⠀⠀⠀⠀⠀⠘⠃⠙⢫⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠧⡠⠤⠈⠙⡍⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣀⣤⠖⢤⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠉⠿⠻⣟⡉⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠠⠶⠆⠀⠉⠉⠈⠻⠷⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠲⠀⠀⠠⠊⠠⠄⠉⠀⠀⠀⠀⠑⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⣶⣷⠀⠀⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠶⢤⡴⢆⡀⠀⠀⠿⠋⠀⠀⠿⠟⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠺⡿⠚⠛⠷⣄⠁⠘⠛⣠
⠀⠀⠀⢠⠀⠀⡀⠈⠉⢠⣿⠦⠤⠄⣀⠘⠀⠀⠉⣁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⢸⡄⠀⢷⣀⣀⣈⣿⣦⡀⠀⣽⣄⡠⠖⠀⠙⣷⠐⠶⢠⣄⣀⣀⠀⠴⠶⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡟⢹⡿⢿⠋⢤⠒⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⣶⠄⠀⠀⢴⡆⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠘⢿⣆⡀⠂⠁⠀⠀⠉⠻⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠃⡀⢛⡄⠀⠀⠙⠆⠀⠙⠛⠛⢲⡶⠠⠶⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡀⢷⡌⠹⣦⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠙⢦⡼⠃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠿⠤⢶⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠻⣦⣤⠖⠁⠀⠀⠀⠀⠀⠀⢀⣿⠉⡇
⠀⠀⠀⢠⠀⠠⣆⠀⠛⣠⣄⣺⠇⠸⣏⠀⣄⠀⠀⠀⠀⠀⠀⢠⡌⠻⠀⡇
⠀⠀⠀⠘⠀⠀⠈⠉⠉⢉⠛⡿⢧⣴⣿⠀⣿⠀⠀⠀⠀⠀⢰⠆⢹⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠖⠀⠈⠋⣿⡆⠛⡄⠀⠀⠀⢀⠇⢀⠈⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢏⠀⢰⣇⠀⠀⢀⠟⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠈⠀⠈⡟⠀⢀⠿⠀⠀⡌⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠸⠀⠀⢸⡇⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠘⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠛⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⣤⣌⣠⣤⣼⡿⠃⠀⠀⠀⠀⠀⠈⠛⠛⣋⣤⣄⣀⣤⣾⡂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠀⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣤⣴⣤⣶⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⠿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡞⠀⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣼⠃⢀⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⣉⣴⣿⠁⠀⡼
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⡁⣠⢠⣤⣶⠹⣿⣿⣿⣶⣾⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣧⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⢀⣠⣤⣶⣶⣶⣶⣿⣿⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠄⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠁⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠀⠈⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣶⣾⣷⣿⠶⡆⣌⡲⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣽⣿⢿⣿⣿⣇⣿⣿⣽⣴⣦⡠⠢
⠀⠀⠀⠀⢼⣿⣿⣿⣿⠻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⣾⣿⣧⠀⠀⠀⠀⠉⠁⠀⠀⢰⣀
⠀⠀⠀⠀⢺⣿⣷⣆⣿⣷⣾⣳⣭⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⡿⠀⠀⡟⠀⠀⠀⠀⠀⢸⣧⡿⠁⢂
⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⡀⣿⣤⠁⠀⠀⢰⠀⠀⠀⠀⠀⣸⡏⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠛⠻⢿⣿⣿⣿⡟⢡⣿⡁⢻⡇⠀⠀⢠⡗⠀⠀⠀⢠⡟⠃
⠀⠀⠀⠀⠀⢠⠋⣠⣶⣿⣿⣿⣿⣷⣶⣬⣙⣿⣿⣿⣿⡿⠀⡟⣂⠀⢰⡄⠀⠀⢀⣿⡖⠛
⠀⠀⠀⠀⣤⡀⠛⠥⡏⠹⠛⡿⠿⣿⢾⣿⣿⣿⣿⣿⣿⡇⠀⣧⡿⠀⢸⡇⠀⠀⣼⣿⣧⡀⠀⡬
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠁⠁⠀⠑⠀⠚⠉⠙⠻⢻⡟⠁⠀⠉⠃⠀⢸⠁⠀⣰⣻⣿⣿⣿⣾⣷⣦⣦
⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⡆⠀⠀⠀⠃⠀⠠⣣⠟⠋⠁⢉⣩⣉⣯⡭⠖
⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⣛⠀⠀⢠⠂⠀⢀⡼⠃⠰⠒⠛⠉⠡⠞⠏⠊
⠀⠀⠀⠀⢿⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢻⠋⠀⢰⠃⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠈⠈⠛⠿⢿⣶⣤⣄⡀⠀⠀⠀⣠⣟⣿⣇⡯⠀⠀⡞⠀⠀⡞⢣⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣆⠀⠀⢀⣾⣿⢻⠃⠀⢸⠁⠀⣸⠃⣬⠉⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣠⠶⠿⣿⣿⣿⠀⠀⣾⠀⠀⡇⣘⡻⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣿⡅⠀⠸⢹⣏⡏⠀⠀⠅⠀⠀⡇⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⠀⣾⣿⣷⠀⠀⢻⠀⠀⡇⣀⣬⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠇⠀⢸⣿⠿⣿⠀⠀⠸⠀⠀⡇⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⠶⠶⢸⣿⡀⣿⠀⠀⠘⡆⠀⢿⠀⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠓⠒⠻⠿⠇⠸⠀⠀⠀⠣⠀⠸⠈⢿⣿⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⡖⢶⠶⣷⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠾⣶⠀⣤⢫⡁⠻⣿⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠉⣹⡘⣆⡅⠠⠹⡿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠨⢯⡃⣏⣤⠀⠀⣆⢀⠙⠙⠛⢿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⡷⠄⠀⠀⠘⠁⢧⠄⠀⠐⠢⢤⣘⡋⠛⠿⠿⠿⠷⠦⢤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠂⠂⠀⠀⣂⣠⣤⣤⡴⣤⠤⣤⣶⢦⣾⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣵⣾⡛⠋⠀⠠⠛⢛⣿⣿⣿⣿⣿⣷⣟⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⣴⣿⡿⠻⡷⣄⣀⣸⢠⠊⣸⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⣿⡿⣿⡀⢠⣿⣿⡈⣤⡟⢴⣿⣿⢷⣯⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⣿⣿⢿⣿⡿⣎⣽⠴⡄⢉⣿⠀⣿⡝⢉⢸⣯⣿⣻⣿⣿⠿⣿⣯⣛⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢰⣿⣿⣿⣿⣿⡿⣿⢿⣟⣧⣸⡿⣄⣉⠻⠂⠄⠙⠿⠝⠿⠍⠙⢿⣭⣭⡟⣻⣿⠿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢣⣿⣿⣿⣿⣻⡿⣷⠿⢄⠹⣍⠿⠻⣮⣭⣭⣿⣶⣒⣒⣦⣤⣤⣤⣈⣛⣿⣿⣿⣿⣷⣧⣬⡿
⠀⠀⠀⠀⠀⠀⢀⣾⢋⣴⣾⣿⣿⣿⠿⣿⣿⡇⠠⠬⠱⠎⠑⠲⢦⣬⣯⣦⣈⣛⡉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢰⣿⠃⣮⣿⣿⣿⣿⣿⡊⠛⣿⣿⠀⣴⢶⣤⣀⠀⠀⠀⣻⣿⣷⣾⣿⣥⣤⣠⠈⠁⠒⢠⠗⠁⣉⣿⣿⣿⡧
⠀⠀⠀⠀⠀⢸⣦⡀⢿⣿⣿⣿⣿⠿⠿⣿⣿⠋⢀⠉⡈⣿⠿⣷⣶⣤⣶⣿⣿⣿⣿⣵⣶⣿⣯⣥⣒⣬⣶⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠈⢿⣿⣶⣼⣿⣿⣷⣶⣶⡼⠃⠀⠈⠀⠀⣿⡀⠌⠙⢿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⢛⢻⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⢻⣿⡆⠀⠐⠀⠀⠀⠀⡀⠀⢀⡀⠉⠈⠀⢀⣀⣀⣢⣤⣤⣴⣶⣦⣤⣼⣦⣿⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢨⣋⡄⢴⣨⢌⢿⡗⠣⣊⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⣸⢻⢇⠉⠘⣗⠯⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢇⠀⢗⡆⣯⡃⡀⢩⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢄⣛⣀⢏⢗⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⣟⢛⣿⠛⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠈⠉⣹⣿⡉⣽⠟⠛⣿⠟⣻⡟⢉⡀⣠⣿⠧⠾⠛⠛⠛⠑⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠉⠀⠀⠀⢼⣿⣿⠷⠿⠖⠚⠿⠟⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡇⠀⠈⠀⠀⡄⠀⠀⠀⢀⣾⣶⣿⣾⣦⡀⣌⠆
⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⣻⠀⠈⢿⡐⠄⠀⠀⠀⣷⠀⠀⠀⠘⠻⠛⢿⣿⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⠃⠀⠀⣡⣽⠶⣦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣷⣄⡰⡀⠀⠀⣼⡌⣶⠊⢠⠀⠡⠀⠀⠀⠀⠀⠀⠀⢻⣻⡿⠆⢠⡇⠄⠀⠀⠀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣯⡙⣈⠀⠀⠀⣿⡇⣿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠣⣽⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠋⡃⢸⠃⣔⢼⣿⡇⡇⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⠀⢀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣦⣅⡨⠀⠃⢸⣿⣻⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣹⣿⣤⡈⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡏⡁⠀⠀⠰⣿⡧⠀⣧⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠄⠀⠀⣿⣷
⠀⠀⠤⠖⠂⠀⠀⠈⠉⠛⢢⠤⠦⠈⢿⡄⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠓⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⡬⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢄⡀⠀⠀⠀⠀⠀⠀⣠⠂⢠⠬⠭⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠉⠢⡀⠀⠀⠀⠈⠐⠊⠭⢒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⢠⠄⡶⠓⠊⠀⠀⠀⠀⣀⠤⠀⢖⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣐⠑⠁⠀⠀⠀⠈⠀⠡⣒⢤⠀⠀⠀⠈⠢⢘⡥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢉⠱⣈⠆⠀⠀⠀⠀⠀⢀⢀⡦⠃⠀⠀⠀⣀⠤⣘⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠣⠯⠀⠀⠀⠀⠀⠃⠏
⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⡀⢀⡤⠤⠤⠤⠤⡄⠀⠀⢀⡤⠤⡴⠦⠤⠤⠤⡤⠤⠤⠤⢤⣠⢤⡤⢤⠤⢤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠳⢤⡔⠋⣀⠔⠁⡤⡄⠀⠀⡇⢀⢔⠁⡠⢉⢍⡠⣪⠆⠘⡡⡢⢉⢸⠃
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⠬⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⡇⢐⠅⡨⢒⠕⢅⡞⣱⣟⡝⣾⠔⣡⣎⡳
⠀⠀⠀⠀⠀⠈⢻⡻⡄⠀⠀⠀⠀⠀⠫⠂⠀⣦⠶⠃⠁⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡲⠷⠦⣈⢔⠕⣡⣠⠆⢿⡉⣀⠼⢖⢕⠕⡇
⠀⠀⠀⠀⠀⠀⢀⠗⡇⠀⠀⢀⡀⠀⠀⠀⠀⠑⠤⢤⡴⠞⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠻⠇⣀⣠⡧⡢⢊⠝⡺⢊⠔⡡⢀⢔⢡⠗⢁⡇
⠀⠀⠀⠀⠀⣠⣮⠜⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⠀⠀⢠⢄⠀⠀⠀⡸⢊⢈⠔⡡⢊⡰⣡⢊⢔⠕⣱⣡⣾⠁
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠌⠊⠈⠈⢱⠀⠀⠁⠀⠀⠀⡇⠔⡡⢊⢰⡙⠝⠉⡇⡡⠪⣪⢜⢆⣳
⠀⠀⠀⠀⠘⢷⡄⠀⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣢⠀⠀⢀⡠⡰⠀⠀⠀⢤⢶⢲⢷⣾⣴⡡⡠⠙⠒⡲⢋⢔⣟⢕⡑⡰⣹
⠀⠀⠀⠀⢠⠎⠀⠀⠻⣍⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⣉⠔⠁⠀⠀⢸⡦⠃⠁⠀⠀⠀⢹⡔⡡⠊⡠⣳⠛⡹⣏⡱⢈⢼
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠻⢇⣀⠐⠒⣊⢽⠖⡡⠪⣪⣪⡾⢊⢄⢔⡕⡏
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡴⡖⡶⢦⡀⠀⢾⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡹⢝⠍⠐⢡⡊⢎⠝⡠⣫⣖⣻⡼⠿⠗⣧⣤⡶⠶⠶⠶⠶⠄⣀⣀⠐
⠀⠀⠀⠀⢸⠋⠀⠀⠀⢰⢏⢜⡞⢋⣉⣱⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⡗⡧⡯⠪⣪⣪⣎⠮⠴⠯⣐⡛⠻⠿⢿⡷⠖⣲⣖⣛⡛⠓⠋
⠤⠐⢒⣒⠸⠦⠤⠶⠶⠟⣛⣛⠋⠹⠿⠽⢒⣒⠶⢿⣶⡿⠶⠖⣛⢒⠒⠶⢶⡶⣶⡶⠾⣷⢿⣿⣿⣽⣿⣟⠛⢩⣯⣬⣷⠶⠒⠒⠉⠉

⠀⠀⡀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⢀⢀⠀⢀⢀⡤⠆⠀⠐⠒⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣄⡅⠰⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠀⠀⠀⢱⠶⠒⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⢀⣀⡴⠊⠉⠀⠀⠀⢠⠆⠀⣰⠃⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⢠⠃⣀⣀⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣀⠀⠀⠘⠐⣆⠀⠀⠀⠘⢡⠋⠉⠉⠳⡄⣀⠀⠀⠀⢘⣇⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠈⢦⡀⠀⠀⢸⡄⠀⠀⣴⣏⠉⠀⠀⣠⣾⡽⠋⠁⠀⠉⡆⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣑⣄⣱⠀⠀⠀⠙⠦⠤⣺⠏⠑⠒⠊⠉⠀⣄⣀⣀⠤⠚⠁⠀⠀⠀⠉⠐⢒⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⣉⣉⡉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰⣋⠉⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠛⠚⠋⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣤⠼⠓⠙⠒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⠴⠚⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃⠀⠀⠀⠀⠘⢧⡀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢰⠚⠉⠉⠤⠛⠚⠙⠧⡄⠀⠀⠀⡸⢉⠂⠈⢲⡌⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⠇⠀⠀⠠⠀⠤⢤⡶⠃⢠⠒⠧⣍⡛⣄⠀⠘⠁⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣯⠜⠉⠀⠉⠉⠀⠀⢸⠀⢦⣴⣿⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡹⡁⠀⠀⠀⠀⣀⡇⢸⣀⣀⣤⡀⠀⠀⢀⣀⠴⡠⣈⠀⡽⢏⠒⠄⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⡄⠀⣴⠛⢻⠀⡞⠁⠀⣶⠟⠒⣋⠭⠒⠉⠀⠈⡏⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢜⡦⠃⢠⡏⢠⡃⠀⠀⣟⡏⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢉⣶⢰⠃⡜⠉⠑⠒⠿⠀⠀⠀⠀⠀⠀⢠⠞⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⢠⡿⠻⡜⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⠝⢦⡀⡸⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠔⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠒⠒⢻⠉⣩⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⡏⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠰⢾⠶⠇⢀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⢻⡟⢶⠶⣲⡾⣿⣽⠿⠫⢿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣷⣴⡶⠞⠋⠀⠈⠛⠷⠶⣴⣧⠾⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠴⠠⠒⠒⠉⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⣡⠀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠋⠀⠀⠀⢙⡶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢠⠏⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⠀⠈⠑⣄⢀⠀⠀⠀⠀⢇⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⢣⣄⠀⠀⠀⣄⠀⠀⠬⣿⠁⠀⠀⢀⠔⠀⠋⠀⠀⠈⡷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣦⣘⡆⠀⠀⠈⠒⠒⣒⠏⠀⠀⠀⠀⠀⣄⣀⢠⡴⠚⠁⠀⠀⠀⠈⠁⢙⡳⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣏⡁⣠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠋⠘⠉⢀⡠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡤⠶⠛⠈⠓⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠚⠉⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⡰⠃⠀⠀⠀⠀⣈⠳⣄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠐⠛⠋⣉⠗⣤⠀⠀⠀⣘⢉⠇⠈⠳⡄⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠃⢠⠞⠓⠬⡙⣦⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠀⠀⠸⠂⢰⡴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⢠⣇⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠉⢹⠃⡜⠀⠀⣴⡜⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⣵⠂⠀⡞⢀⣇⠀⠀⢏⡏⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢀⡵⢆⠃⣸⠀⠈⠐⠚⠁⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣄⣾⠛⠾⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠹⠘⠓⠄⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⣀⣀⠠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠹⡇⢸⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⢛⣟⣋⣀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⢹⠿⠽⢟⣭⠿⠿⣾⣿⠗⠻⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣿⡶⠾⠛⠁⠀⠀⠈⠙⠛⠶⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠀⠀⠠⠀⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠆⠆⠀⠄⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣄⢶⣾⠻⣿⢻⣿⢿⡶⣾⣭⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢿⣿⣿⡌⣿⡿⣿⡾⣿⣾⣧⣿⣼⡟⣿⣷⣤⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣜⢗⣽⣿⣇⣿⣷⣿⣷⣿⣼⣿⣻⣼⡿⣿⣿⢾⣿⣷⣤⣤⡶⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣹⣯⣏⣿⣻⣧⣿⣿⣿⣿⡿⠿⠛⠛⢷⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣏⣿⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣍⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⠿⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣟⢿⣿⣿⡟⢷⣶⣭⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣾⣿⢈⣿⣾⡏⣻⣹⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⣶⢹⣿⣿⣿⣇⣿⠿⣿⣸⡟⢿⣝⣷⣿⢻⣦⡀⠀⣠⡶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⣿⣡⣿⢙⣷⣾⣿⣧⣿⣿⣿⣿⣿⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠈⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣏⣹⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣛⣛⠿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡉⠉⠛⠛⠛⠶⢭⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣏⣉⣉⣛⠛⠲⠦⣌⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⣶⢹⣿⣿⣿⣟⢉⣉⣉⣙⣛⠓⠲⠭⣗⣿⣦⡀⢀⣠⡶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⣟⣉⣉⣉⣉⣉⣭⣭⣭⣿⣿⣿⣿⣿⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠈⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣯⡽⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣍⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⡼⠷⡄
⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠎⠀⣠⣇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠞⢫⠟⠛⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡏⠀⢀⡿⠒⢲⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠎⠙⠓⣲⠖⠲⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠾⠀⠐⠋⢀⠴⠚⣓⣦⣀⣴⣯⢷⣦⠀⠀⠀⠀⠐⠚⠁⣀⡴⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠒⠒⠚⠒⠒⠒⠒⠂⠙⣋⣿⣠⣷⠖⠛⠂⠀⠀⣀⣤⣤⣭⡥⠴⠋⠁
⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⢻⣀⡼⠻⠄⠀⢀⡞⠁
⠀⠀⠀⢀⡞⠀⠀⣤⣄⣴⣶⠄⠀⠈⢷⣄⠀⢠⡀⠀⠈⠉⠛⠀⠀⠀⠀⣼
⠀⠀⠀⣼⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠻⣧⠈⢻⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⡇⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⢻⠀⣸⠇⠀⢤⣄⣀⣤⣤⡟
⠀⠀⠀⣧⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣾⠀⠋⠀⠀⠀⠠⣤⣀⣤⣿
⠀⠀⠀⠹⣷⣞⣋⣀⡀⠈⢻⡀⠀⠀⢀⣼⠏⠀⠀⠀⠑⣦⡀⠠⢤⠤⣾
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣧⣸⠇⠀⠀⠈⠀⠀⣀⣠⢤⡞⠁⠉⠛⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠖⠒⡟⢿⡁⠰⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡷⠶⠚⠋

⣿⣷⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣠⢾⣧⣤⣀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣷⣦⣤⣄⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣤⣄⣴⣆⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⠂⢀⡺⠿⣻⣾⡧⠶⢛⠷⢾⡅⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠛⢋⣩⠟⠉⠻⠆⠀⠀⠀⠀⠀⠀⣠⡶⠟⠿⠛⠿⢋⠿⣟⡿⠗⠺⣏
⠀⠀⠀⠀⠰⠠⠀⠈⠀⣴⠛⠉⣠⠖⠁⠀⠀⠑⠀⠀⠀⠀⢰⣿⠟⠋⢩⢀⣩⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠙⠀⠀⣐⡋⠵⠞⠋⠋⠀⠀⠀⠈
⠀⠀⠀⠀⢦⡀⢠⠄⢦⣆⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠟⠃⠚⠛⠦⣄⡀⢀⡴⠂⠀⠀⠀⠀⠀⠀⢀⡤⠀⠻⣿⣿⣿⣿⣷⡄⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣶⣶⣤⣄⣴⡟
⠀⠀⠀⠀⢀⠖⠁⠀⠘⢓⣈⢉⠄⢈⡷⢺⡄⠀⠀⠀⠀⠀⠀⢠⠋⠤⢀⡄⠟⢻⡟⣻⣿⠓⣨⣿⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢿⣿⠟⠉⣩⠟⠁⠙⠳⡄
⠀⠀⠀⣠⠃⠲⠀⠀⠼⠟⠉⢡⠞⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⡀⠀⣸⢃⣺⣿⠿⢿⠟⠀⠀⠙⠆⠀⠀⠀⠀⠀⠻⠟⠋⡡⢤⣤⡃
⠀⠀⠀⠈⠛⠀⠘⠁⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣷⣦⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⢿⣿⣦⣾⣧⡄⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⢙⣷⣿⣶⣄⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⠆⠀⠀⠀⠀⢠⡄⠑⣀⠶⢀⣙⣯⡽⢿⡗⠂⣉⣿⡅⠀⠀⠀⠀⢀⠂⠐⢪⣿⠛⣽⡿⢯⠿⢿⣷⠀⣀⣤
⠀⠀⠀⠐⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠋⠀⠀⠀⠀⠀⠛⠻⠤⠤⢾⣼⣿⠂⠘⠿⠅⠀⠈⠛⠃⠀⠀⠀⠀⠘⠏⠈⣍⠁⠀⣾⠿⣋⡶⢛⡛⠾⣹⢏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⡟⠉⠹⡿⠃⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠐⠛⠶⠶⣤⡀⣠⡖⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣷⣶⣾⡶⢤⣄⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⡶⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠔⠊⢉⡁⣈⠿⠛⠉⠙⠧⠀⠀⠀⠀⠀⢀⠖⣥⠈⠈⠹⠟⠁⢀⡠⠂⠀⠀⠙⠆⠀⠀⠀⠀⣀⡀⢀⣴⣿⣿⣷⣮⣷⣈⡽⢂⠈⢧⡄
⠀⠀⠀⠀⣶⣈⠁⢀⣰⠞⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠐⢦⣴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣭⣶⣛⣿⣋⣷⣟⣡⡤⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣻⣄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠝⢽⠭⡢⣱⠑⠀⢤⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⢄⠴⠊⠁⢱⠊⠀⢈⣳⣄⠔⢭⠎⣰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠽⢷⡾⠦⢤⣠⠎⠀⠀⠸⢋⣉⢳⢄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠉⠉⣡⠙⠶⠊⠀⢠⣶⠆⠀⣤⣴⠿⠋⠉⢉⠐⠢⡄⠀⠀⠀⠰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡀⠀⠀⠀⣀⠔⠁⠀⠀⢀⣔⡧⢁⣠⠴⠿⣀⠀⠀⢀⠆⠀⢀⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠈⠉⠉⠉⠀⠀⢀⣠⠶⣉⢎⡔⠁⠀⠀⠀⠈⠳⡄⠘⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢩⠼⠊⢁⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⢤⣈⡬⠄⠉⠁⢇⠉⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁⠀⠸⣄⣂⠀⠀⠀⠀⠀⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⠁⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠿⠆⠀⠀⠀⠀⣇⢀⠎
⡠⢮⠿⣳⢊⡱⠀⠀⣃⣤⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠄⠐⠒⠚⠀⠀⠀⣸⠁⠈⡻
⢩⠁⠀⢠⢃⡠⠐⠊⠁⠀⠈⢉⣗⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠒⡶⠀⠒⠈⠁⡃⠀⢀⢃⠀⢰⡁
⢈⠃⠀⡜⠇⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠉⠙⠓⠶⠖⠠⠤⣄⣿⠀⠀⠀⡘⠀⠀⠀⠘⠅⠲⠀⢰⣺⠁⡜⠁
⠸⠀⡠⠤⣞⣀⠔⠊⠉⠚⣄⠀⠀⠉⢐⣒⣶⡖⠛⠛⠛⠛⠛⢋⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣉⣀⣤⠇
⠈⠊⠁⠀⠉⠀⠀⠀⢠⢤⣌⡩⠍⠬⠷⠌⠙⠓⣶⠠⠤⣄⣀⠀⠀⠀⠨⠀⠀⣀⡠⠤⠐⠊⢯⣸⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣡⠤⠦⠴⠖⠒⠂⠈⠀⠀⠀⠀⠀⢉⠃⠒⠖⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⡀⠊⠀⠉⠠⢄⠀⡠
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠂⠀⠀⠄⠀⠀⠀⠀⠂
⠐⠁⢐⠀⠀⠀⠂⠀⠐⡀⢀⣀⡀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠅⢀⠝⠒⠚⠁⠠⡼⠤⣀
⠀⠀⠀⠀⠀⢧⣀⠀⠀⣇⠀⠀⠀⠀⠀⠓⠢⠄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠉⠉⠉⠢⣄
⠀⠀⠀⡀⠀⡀⠀⠁⠀⢀⠕⠀⢀⠒⣠⠤⢀⠖⠀⠀⠀⡬⠥⠶⠶⢶⡒⠉⠉⠈⠲⠐⡤⠤⠤⠤⠀⣀⣀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠈
⢠⠊⠉⢑⠀⠀⠀⠀⠀⠘⡀⠀⢀⡀⠀⠀⡁⠀⠈⠱⠒⠁⠀⠀⠀⠀⡇⠀⠀⠀⣨⠉⠉⠉⠁⠒⠒⠒⠂⢤
⢠⠓⠐⠉⠀⠀⠀⠂⠀⠀⠉⠉⠁⡝⠉⠉⠑⠦⢤⠞⠀⠀⠀⠀⠀⠠⠛⠀⠀⠁⠉⢢⠀⠀⠀⠀⠀⠀⠀⢸
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⡰
⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠤⠔⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠠⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣒⣢⣤⣤⣤⣤⣤⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢦⢤⣆⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣖⣃⣔⣿⣋⣉⡜⣿⡯⣓⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠾⣛⠋⠉⠉⢭⡉⢹⡍⠉⠛⠿⢶⣤⣈⢵⣗⣢⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠞⠉⢵⠔⡀⠀⠀⣇⠀⠀⠀⢧⣹⣿⠀⠀⢸⡷⣦⠈⠙⢿⣆⠈⢩⣶⡦⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⣠⣆⣴⠆⠀⢀⠀⢫⡎⠀⠀⠑⡄⠀⠀⠈⢾⣿⠄⠀⠀⣗⣄⠀⠀⢰⣯⠳⣾⡟⠁⠁⠀⠀⢀⡠⣺⣽⡞
⠀⠀⠠⣾⡷⠉⠀⠀⠀⠘⢀⡄⡇⠀⠀⢠⠇⠀⠀⠀⢸⡿⠀⠀⠀⢸⣾⠀⠀⢯⣯⠀⠈⡹⠓⠂⢶⠖⠉⣿⣟⡿
⠀⠀⢢⠤⠝⠀⠀⠀⠀⠀⠈⡏⠁⠀⣴⣾⠷⠾⠷⠚⣻⠋⠀⠀⠀⢈⡇⠀⠀⣸⡇⠀⠀⡉⠀⠀⠸⠀⠀⢩⠷⣧
⠀⠀⠈⠒⢄⡀⠀⠀⠀⢀⡴⠃⠀⠀⠉⢻⠆⠀⠒⠂⡹⠀⠀⠀⠀⡷⠀⠀⠀⣸⠁⠀⠀⡠⠔⠂⠒⢦⡠⣾⣳⢿⡄
⠀⠀⠀⠀⠀⠉⠓⢤⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⣴⣿⣗⡶⠀⠀⠀⠙⠦⣝⢳⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⣿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠈⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⠄⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣴⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⢏⣉⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⣷⡦⠀⠉⠙⠋⠉⠀⠀⠀⠀⢰⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢘⡏⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⠉⣧⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡰⣤⢀⠼⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡓⠛⠓⣀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⢷⣤⡀⠀⢠⡇
⠀⠀⠀⠀⠀⣰⠎⠐⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠀⠀⢻⠁⠦⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠂⠲⠂⠀⠀⣀⢁⣅⣀⣼⠃
⠀⠀⠀⠀⢰⠃⠜⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢼⡆⢠⠀⣀⢸⡟⠀⠀⠀⠀⢠⡀⠰⢀⡤⣀⣴⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠘⠠⠀⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢷⣿⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠴⢭⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢁⣠⡔⠛⠙⠛⠛⠳⣄⠄⠀⠀⠀⠀⠀⠀⠐⠿⠿⢿⡶⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⣈⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠟⠢⠤⠶⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠃⢸⢛⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣄⡆⠀⠀⠀⠀⠀⠀⠀⠐⢶⠏⠛⣿⡶⠆⢄⠀⠀⠀⠀⠀⠀⢰⣹⣇⢸⣼⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⠀⠀⠀⠀⠀⠘⣷⣤⡆⠀⠀⠀⢱⣶⣾⠀⠀⠀⠀⠀⣠⠠⠼⠶⠶⠿⠧⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⣾⡃⠀⠀⠀⣾⢁⡟⠀⠀⠀⠀⢀⢹⡆⠀⠀⢤⠀⠀⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡤⡇⠀⠀⠀⠀⠀⠀⠀⠙⠃⣦⠄⢀⣿⠟⠁⠀⠀⠀⠀⠀⡀⠏⠀⠠⠌⢧⡰⠟⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒⠒⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣲⣯⠤⠶⠒⣒⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠊⠉⠀⠀⠘⠉⠉⠉⣉⣩⡭⠽⠶⠦
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠭⠝⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣞⣀⣀⠀⠀⢀⣀⣀⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣞⣉⣡⡤⠤⠤⠭⢽⡯⠤⠤⠤⠤⢵⠄
⠀⠀⠀⢀⡤⠒⠒⠢⣄⡇⠀⠀⠐⠓⡒⠒⠠⠤⠄⠒⠊⠙⠒⠒⠲⣖⡭⠖⠲⣄
⠀⠀⢀⡏⠰⠞⠓⠆⠈⡁⠀⠙⣆⢙⠦⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠠⠾⣁
⠀⠀⠘⢄⠠⠤⠒⠈⠀⠈⠲⣄⠈⢿⡀⢰⠁⢠⠏⠉⡽⠉⠉⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠳⠤⠮⠤⠴⠛⠇⠀⠀⠀⢀⣀⡤⠖⠋⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡀⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡤⡾⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠲⢍⡑⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠳⠴⠋⠽⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠦⡀⣱⠚⠀⠰⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⠀⠀⠀⠀⡄⠉⠀⠀⡿⠧⠤⢄⣀⣀⣀⣀⣀⠀⠤⠇⠰⠋⣿⢄⠀⠀⡜⢰⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡿⠂⠀⢸⠲⢄⣀⣿⠒⠒⠒⠒⠒⠂⠀⠀⠈⢉⠁⢄⡴⡇⠀⢑⡄⢙⠎⢆⠀⠀⠠⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣳⠀⠀⠈⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⡇⠀⢸⠀⡎⠀⠎⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⣿⣟⠁⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠪⡀⠀⡇⠀⡼⠀⡇⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡇⢻⣤⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⠇⠀⠇⠈⠁⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡨⠝⠧⠼⠍⠉⡗⡶⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⢣⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠎⡞⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠀⠶⠶⢾⠾⡃⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠺⡄⠀⠀⠀⠀⠀⠑⢢
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡇⠀⠀⣮⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⣰⠇⠀⠀⠀⠀⠀⣴⢶⡢
⠀⠀⡖⠋⠉⠉⢹⣟⡟⢛⣛⡿⠿⢿⣿⣛⣛⣫⣋⠉⠉⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠛⠿⠶⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠆⠀⠀⠀⠀⠀⠀⠤⠶⠶

⠀⠀⠀⠀⠀⢠⡄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀
⠀⠀⠀⠀⢠⡎⠶⣆⠉⢙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠱⣿⣿⣌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣿⣬⣷⡀⠄
⠀⠀⠀⠀⠀⠙⢮⡅⣂⡉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣽⣿⣛⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣟⣫⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣴⣻⣷⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⠳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠦⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠙
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣾⣦⣤⣴⣇⠀⠀⢠⠄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣦⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠼⣦⣀⠀⠀⠀⢀
⠀⠀⠀⢀⣴⠿⠛⠛⣉⡤⠀⢉⠙⠒⢯⠀⠀⠀⠀⠀⠐⠈⠀⠀⡀⠀⢀⡠⠤⠀⠦⡀⠀⠀⠀⠀⠀⠀⡔⠈⡀⠀⣀⠠⠍⠑⠦⢰⡋
⠀⠀⠀⣿⠍⠁⢀⣀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⡈⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠤⠐⠡⠔⠀⡤⠊⠁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠰⠦

⠀⠀⠀⠐⠂⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣴⣦⠙⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢿⠛⠀⢀⢼⣯⡀⠀⠀⠀⠀⠀⠀⠐⠲⡦⠔⠚⠶⠧⣼⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠶⠶⠦⢴⣟
⠀⠀⠀⠀⠀⠊⠀⠲⠿⣬⣋⣙⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠋⠉⢘⠃⠦⠠⡇⠀⠀⠀⠀⠀⣀⣼⠛⠁⠀⡤⠄⢠⡆⠈⠀⠀⠹⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠱⡀⠀⠀⠐⠈⠁⠉⠃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⣄
⠀⠀⠀⠀⠉⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠲⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠁⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣍⠄⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣤⡀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⣎⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣾⣿⡟⠛⠋⠉⠛⠿⢿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⣻⣿⣿⣿⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⢻⣏⢻⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡛⢔⡙⣰⣿⣯⣿⡻⢈⣉⢹⣿⣿⣿⣧⣰⣈⠩⡇⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⢶⣿⠯⣼⣿⣟⣵⣂⣿⣿⣯⠻⣿⣧⣾⡿⡗⡧⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⣻⣿⣞⣽⣿⢏⣿⠛⣿⣿⣗⣠⣿⣧⠘⠳⠿⣶⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣷⢸⣿⣿⣿⣿⣾⠉⠘⣿⣿⣿⣾⣿⡏⢀⣶⣾⡟⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡄⢀⣿⣿⣿⣿⣿⣡⣾⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠦⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠼⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠸⡄⠸⣆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣾⠉⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢠⡇⣰⠇⢠
⠀⠀⠀⠀⠀⠘⡆⢸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⡻⡄⣰⡊⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟⢀⡞
⠀⠀⠀⠀⠀⢀⠇⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⢻⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⡀⢰⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⣇⠈⢷⣄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⢦⡀⠴⣫⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠳⠀⢠
⠀⠀⠀⠀⠈⠓⣦⠙⣷⢹⡄⠀⠀⠀⠀⠀⠀⠀⡏⠉⡳⡝⣦⡧⣀⡽⠀⠀⠀⠀⠀⠀⠀⢾⡁⢀⠀⣏
⠀⠀⠀⠀⠀⢰⢫⡞⢱⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠁⠈
⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣰⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠈⢳⣌⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢠
⠀⠀⠀⠀⠀⠱⡄⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣠⠞
⠀⠀⠀⠀⠀⠀⡿⠘⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢷⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⡐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣉⠛⠲⠦⣌⣢⡀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠾⠳⢦⣀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢚⣋⣉⣀⠀⠀⠀⠀⠀⢶⡬⣷⣄⠀⠀⠀⠀⠹⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⡡⠖⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠒⠒⠒⠶⠤⠈⠳⢄⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⡔⠒⢄⠈⢇⡀⠀⠀⠀⣿⠗⢃⠤⠒⠂⠈⠉⠁⠀⠀⠀⠀⢀⡽⠒⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠣⡀⠈⠃⠀⠉⠙⠛⠉⠁⠀⠉⠙⠒⠒⠤⠀⠀⠀⠀⠀⠴⣥⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⢤⠀⠀⠊⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⡄⡘⠀⠀⠀⠀⡎⢃⣰⣁⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⠎⢻⢡⠀⠀⠀⠀⢈⠎⡠⢤⡧⢀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⢀⡖⠚⠛⠓⠲⢖⠒⠒⠒⢦⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠲⢞⠁⠀⢸⠉⠁⠀⠀⠈⠉⠐⠧⠒⡗⠠⡎
⣀⣐⣇⠱⣀⡠⠐⢈⣉⡇⠀⠀⠀⠀⢠⠙⠢⢄⡀⠀⠀⠘⠂⠀⠀⠀⠀⣀⡀⠈⠓⠹⠦⠀⢹⡀⢈⡉⠀⠀⠀⠀⠀⠑⡤⣚⣥⠵⢥
⠑⣤⡬⠀⢀⡐⠊⠥⡄⠙⣄⣀⡠⢤⠇⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠉⠁⠈⢳⠶⠤⠤⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠣⣽⣡⣐⣚⣢
⠈⠃⠄⠊⠁⠈⠉⠉⠁⠀⠈⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠣⡑⣤⠚⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣢⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⠠⣠⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣼⠀⠀⠀⠀⠲⠓⠒⠳⠶⠆⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡹⢮⡧⡽⠒⠤⢄⣀⠄⠀⢸⡆⠀⠀⢀⣀⠴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠭⢄⣐⣒⡒⠒⠀⢸⠄⠀⠀⢏⣉⠉⠉⠀⠀⠀⠈⢀⠔⠃

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⣦⡀⠀⠀⣀⣤⣴⡿⠿⠿⢿⡿⠿⠿⠷⣶⣤⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣧⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣀⣤⡾⠋
⠀⠀⣿⡁⠀⠉⠉⠀⠀⠀⠀⠈⠙⠛⠇⠀⠀⠀⠁⠈⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠛⢦
⠀⠰⠋⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢲⡀⣠⠤⠀⡀
⠀⠀⣀⣤⣤⣴⣿⣷⣴⣿⣷⣄⣀⣤⡀⢀⣠⡤⠀⠀⠸⣷⣄⠀⢀⣄⣠⣤⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀⣀⡀⠀⠀⢀⡴⠛⢦⡀⢸⠆⡇⠀⢀⡱⠀⣠⡀
⠀⠋⠉⠀⠁⢳⡿⠗⠊⠉⠙⠛⠉⠉⠋⠩⣿⣧⠀⠀⢀⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠋⠀⠀⠰⢋⣤⣄⣀⠳⣼⣶⣿⣷⣏⣱⠞⠁⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠼⠟⠁⠀⠘⠟⠓⠀⠀⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠉⠻⣿⣿⣿⡿⠤⠶⠶⠶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠀⠀⠈⠉⠉⠙⠲⣦⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠴⠖⠋⠸
⠀⢀⣠⣤⣴⣦⣤⣄⡀⠀⢠⣦⣤⣤⣄⣀
⢰⣿⠟⠉⠀⠀⠈⠙⠻⢷⣤⡉⠉⠉⠉⢙⡇⠀⠀⠀⠀⠀⣀⣀⣴⣦⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠀⣀⣀
⢻⣿⣦⡀⠀⢀⠀⠀⠀⠀⠙⠻⢷⣶⡶⠟⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠶⢾⣟⠁⠀⠀⢀⣠⣴⣶⣶⣟⣿⣿⣶⣶⣤⣽⣆⣀⣀⠘⢆⠀⠀⢀⣤
⠈⠛⠿⣿⣿⣿⣿⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⠉⠁⠐⠋⠁⠀⠀⠙⠂⠀⠻⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠿⠿⠗⠒⢻⣿
⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠉⠉⠩⠥⠴⠂⠀⠀⠐⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⡤⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⣀⡠⠄⢀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⣠
⠀⣶⠆⠀⢀⣁⣠⣤⣼⣿⣾⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣾⣿
⠀⢿⡄⠀⠚⠻⢿⣿⣿⡏⠀⠈⢹⣿⣿⣿⣿⣟⠂⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠆⠀⠀⠀⠈⠉⠉⠉⠁⠻⠟⠉⠉⠁⣤⣶⠀⠀⠘⠿
⠀⠀⠀⠀⠀⣦⡠⠤⠙⠛⠒⢒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣷⣶⣤⣤⡄
⠠⣤⣀⠀⢀⣄⣀⣰⣶⣿⣿⣶⣿⣿⣦⣤⣀⠀⠀⠀⠀⢠⣤⣴⣶⣾⣶⣷⣧⣤⣀⡀⠀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⢴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣤⣴
⠀⣿⣿⠷⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠻⢶⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡶⠶⣶⣿⣿⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿
⠀⠉⠁⠀⠀⠈⠀⠘⢿⣿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣿⣷⠄⠀⠀⠀⠙⠛⠿⠄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠻⣏⠉⠉⠛⠛⠛⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡶⣦⠖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⡛⢯⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⢿⡉⢧⡈⣧⠈⣧⠹⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣴⣾⡿⢿⠹⡆⢻⡄⢳⡈⣧⢘⣆⣼⣶⠿⠛⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⡏⢳⡘⣇⠹⣆⣻⣤⣷⡾⠛⠋⠉⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡌⢷⣼⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠘⢯⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡛⢦⡈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠙⢦⣙⣷⠿⣿⣧⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡿⠋⠀⠘⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠸⣿⡆⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠙⠂⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠇⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⣿⣇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⢷⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢸⡷⠆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣼⣿⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⡟⠀⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣿⡟⢀⠀⠀⣿⣿⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠉⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣸⣿⣤⣴⡶⠾⠷⢶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⠫⠄⠐⠋⠀⠓⠀⠿⢉⡻⣷⡄⠀⠟⠀⠀⠀⢀⣾⡿⠁⠀⠾⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠁⣌⢻⣆⠀⠀⠀⢠⣿⡿⡿⠿⡿⠷⣦⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⡄⣀⣴⣿⠋⠀⠁⠀⠁⠒⢊⡻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣄⣀⣀⣀⣀⣥⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣦⣤⣴⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⢿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣩⣱⠏
⣀⣤⡶⣶⣿⣿⠿⠿⠯⣻⡟⠃⣰⡇⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⣷⣶⣦⣤⣴⣶⣿⣿⣿⡶⠋
⠹⣝⡗⠈⠯⣉⣀⡐⠒⠛⢣⡿⣿⣇⠀⠤⠦⠢⠠⠩⠉⠁⠸⠈⠠⠑⠤⠨⠅⠁⠉⠩⠭⠌⠈⠄⠁⣈⡚⢁⡒⠤⠄⡨⠭⠥⣭⣤⣿⣿⡏
⠀⠈⠻⢤⣀⢈⣡⠥⢤⣶⣤⡉⡸⢌⡄⢀⡀⠀⠀⠀⠀⠄⠢⠐⠒⠐⠐⠄⠀⠀⠀⠀⠀⠢⠔⠄⠐⠂⠂⠒⠒⣂⣠⣴⠶⠶⠶⣬⣿⣿⣿⣄
⠀⠀⠀⠀⠈⠑⠒⠦⢤⣭⣿⣶⠀⣀⢠⡤⠀⠉⠁⠀⠂⠀⠒⠂⠐⠒⠒⡈⠉⠁⠉⠁⠐⠂⠀⣀⣀⣠⡤⠖⢋⣭⠞⠁⠀⠀⠀⠈⠳⣟⠻⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⣿⠷⠿⠿⠾⠶⠒⠚⢿⣿⣿⡟⠛⠛⠛⠉⠉⠉⠛⠿⣯⡦⣲⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢿⣧

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠠⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠳⠤⠤⢬⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠐⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠠⠄⠚⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠠⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠈⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⣰⣷⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠶⠀⠀⠀⣀⠀⠀⠀⠄⠂⠈⠻⠃⠀⠈⠿⠃⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠔⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠⡀⣀⣠⣠⠀⢄⣀⡄
⠀⢠⠆⡁⠀⠉⠐⠀⠀⠨⠄⠀⠀⠒⠀⠀⠀⠳⠄⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⢀
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠊⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠉⢅⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠢⠆⠀⠈⠢⡀
⠀⠀⠈⢢⠀⠀⠀⠀⢄⠀⠀⠈⠉⠈⠁⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠑⠄⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠈⠢⣀⡀⠀⠀⠄⠀⡀⢀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⣀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⢀⢀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠈⠈⠈⠈⠈

⠀⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣠⣤⣤⣤⣤⣄⣤⢀⣀⣄⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣶⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠈⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠧⠤⢤⣄⠀⠀⠴⠖⠒⠂⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⠠⠶⡶⠀⠀⠀⠀⢶⡖⠆⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠐⠃⠤⠈⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠀⠀⠠⠊⠀⠔⠁⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⣀⣀
⠀⢀⣀⠀⠀⠈⠙⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⣀⡄
⠀⣀⣀⠀⠀⠘⠲⡠⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⠁⠀⠀⣀⣠⣠⠤
⠀⠁⠉⠀⠷⣄⡀⠀⠘⢦⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡀⡘⠀⠀⠀⢠⡀⠈⠁
⠀⠀⠤⠤⠀⢀⣀⠘⠓⢄⠅⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣴⠉⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⠀⠀⠐⠗⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢖⢤⡙⣯⡧⠀⠀⠀⠀⠀⠀⢳⠸⠄⢻⣇⡆⡜⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡈⠃⠀⢐⠕⠀⠀⢀⣤⠀⠙⠋⠁
⠀⠐⠛⠻⠀⠰⠖⠦⠀⢀⠁⡑⢃⣳⡄⠀⠀⠀⢀⠀⠈⡆⡦⣎⣯⣳⢠⠔⣶⣢⠀⠀⠀⠀⠸⠃⠀⠉⠁⡀⡈⠋⢀⠀⠘⠋⠁
⠀⠀⢤⢀⠀⠀⠀⠀⠀⠙⠵⠌⡊⢿⣷⠀⠀⠀⢸⡀⠀⣹⣿⡟⣑⠁⢈⡌⠋⠀⠀⠀⠀⡤⢀⣀⠟⠠⡠⠊⠠⣠⠕⠁⠀⠀⠀⣰⡄⠐⠖
⠀⠀⠀⠀⠀⠀⠉⠛⠃⠠⣤⡐⢾⣏⢿⠃⠀⠀⠀⢳⠴⠋⡿⢿⡹⣗⣟⠀⠀⠀⠀⠀⠼⡁⡀⢡⣀⣠⡰⠄⠀⢀⣀⣠⠄⠀⠀⢀⡀⢠⡤
⠀⠀⠀⠀⢠⣤⠀⠀⣀⡀⠈⠋⣠⢀⣌⠚⠀⠀⠰⢛⣶⠃⣴⣦⢻⣾⣾⠀⠀⢀⠤⢤⡓⠻⣶⠭⠐⣃⠤⠦⠀⠀⢈⡀⠀⠐⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠤⠄⢨⣘⠊⢀⣜⣙⣦⣙⣶⡲⣦⣾⢻⢸⠈⡎⡆⣗⢹⣃⣠⣟⡵⣋⢥⣥⡵⠒⠁⠶⠒⠒⠂⠈⠛⠉⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣿⣽⣷⣿⣍⡽⠟⡜⢼⢠⡇⣧⢛⡌⣇⠰⡒⣚⣭⡱⠾⠗⠶⠶⠛⠛⠛⠻⠦⠶⠶⠶⠶⠴⠶⠚
⠀⠀⠀⠀⠀⠀⠀⢤⡄⠤⠖⠲⣶⢿⣿⣿⢿⠖⠈⢑⣊⠀⣁⠘⡟⠋⢺⡾⣾⣾⣿⣟⢒⡋⢙⣋⣘⠚⠿⠴⣤⣤⡀⣀⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠗⣨⠓⡋⣫⠃⠈⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⢾⢮⢉⠄⢆⡙⠔⢀⠈⠁⢠⡄⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⡼⢀⠚⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢦⡋⢈⠠⣆⠑⢂⣄⠀⠈⠃⠤⢆⡄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⠀⠦⠉⠃⠶⠤⠄

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣦⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⢀⣀⣤⣾⣿⣷⡚⠓⠂⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⡤⠊⠁⢈⠂⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣴⡧
⠀⠀⠀⢀⠀⠉⠘⠒⠤⠀⡤⠀⡀⠴⠦⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⠀⠀⠈⠳⡄⠀⣀⡴⠋⠀⠀⠀⢀⠀⠀⠘⢛⣟⠃⠟⠙⠛⠛⢻⣿⣦⣄⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠷⠶⠶⠶⠓⠚⠉⠀⠀⠀⠀⠚⠰⣦⠀⠀⢹⡶⣤⣀⣀⢢⣄⣮⣵⣿⣷⠶⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣠⣀⣾⣿⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢿⣿⣯⡉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⣇⢁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢻⢠⣏⠂⡴⢐⢸⡇⣴⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠋⠀⠙⠊⠋⠉⠉⣇⠘⢺⡗⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡾⠋⠡⠊⡀⠀⢸⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠘⢷⣶⣆⡘⠁⠀⠸⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⢿⣤⣴⣿⣾⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⣰⡄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣩⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠶⠚⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢿⢻⣽⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⣀
⠑⠃⠀⡰⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡞⠋⠁⡢
⠀⠀⣸⡇⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⢰⡞⠂
⠀⠀⢿⡇⠀⡞⠀⢀⡔⠀⠀⢠⡄⠀⢡⠀⣧⠀⢠⠰⡟⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠁⠀⣻⡇⢠⡇⢀⡿⠁⠀⠀⠀⠀⠘⡉⣇⣻⠀⡆⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠀⣿⣇⡼⠁⣼⠇⠀⠀⠀⠀⠀⠀⢹⣿⡟⣸⣏⣴⣿⢃⡀⠀⠀⣀⣀⡤⠘⢿⣼⣸⠟⠶⣶⣶⣦⣤⣀⠐
⠀⢠⣿⡟⢁⣼⠇⠀⠄⠀⠀⠒⠀⠀⠀⠙⠷⠟⠿⠿⠛⠚⣠⣴⣾⣷⣶⣶⣆⣀⣿⡗⠇⢤⣼⡾⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⣀
⠀⣸⠟⡴⠟⢁⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡏⢀⣹⣧⡷⣀⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⣰⠟⣭⣷⣦⣽⣧
⢰⣿⣫⠔⠚⠋⠉⠀⠈⠉⠙⢶⡀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠒⣶⣿⣶⣿⡿⠙⢿⣮⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡖⠔⡄⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠀⣾⣿⣿⠟⠫⠁⠈⠲⡟⢿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠋⠈⠉⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢀⣀⣀⡀⠀⢤⢀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠀⠀⢀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣋⣭⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠙⠻⣭⠭⠿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠐⠋⠙⠓⠶⣤⠀⣠⠆⠀⠀⠀⠀⢀⣺⣼⣶⣶⣶⣄⢰⠂⠀⠀⠀⠀⠀⣤⡤⢴⣦⣤⡀⣠⡄
⠀⠀⢀⠀⠂⠀⣀⡠⠒⣀⠀⣿⡀⠀⠀⠀⢴⣿⡿⣿⠟⠽⠋⠀⠛⠀⠀⠀⠀⡄⠄⠀⣤⠴⢒⣲⠄⠹⠇
⠀⠀⠈⠀⠀⢤⣌⠁⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⠦⠤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⢈⣀⠀⠻⢠⣆⠀⠀⠀⠀⠀⡊⠁⠹⣿⣿⢿⣿⠉⠻⠇⠀⠀⠀⠀⠀⠄⠀⠿⣿⣿⣯⣤⣦
⠀⠀⠀⠀⠀⠈⠉⡠⠆⠘⠇⠀⠀⠀⠀⠀⠉⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣘⠓⠿⠇⠈⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⣠⣶⣤⡄⡴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢶⣴⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣄⢹⡄⡇⡇⡧⠠⣼⠀⠀⠀⠀⠀⠰⣟⢰⢷⣼⢱⠶⣇⠀⠀⠀⠀⠀⠀⢤⠀⣸⡷⢮⣵⣦⣀⣤
⠀⠀⠤⣾⡘⣧⠗⣧⣏⠈⠻⠀⠀⠀⠀⠀⠙⠿⢿⡟⠛⠻⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠥⠇⠰⠗⠂⠈⠋
⠀⠀⠀⠀⠈⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⣄⡀
⠀⠀⠀⠀⢨⣿⣿⠿⠿⠓⢺⡇⠀⠀⠀⠀⢀⠠⠃⢀⠩⠙⠦⠀⡀⠀⠀⠀⠀⢀⣴⡟⢻⢻⣿⣿⣧⣶⣦
⠀⠀⠄⠈⠀⣙⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡴⠀⠀⠉⠀⠀⠀⠀⠸⠆⠀⡀⣸⠯⠿⠆⠈⠉
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⣿⠀⡝⠣⣴⡆⠀⠀⠀⠀⠀⠀⠀⡩⡮⣿⣿⠂⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡉⠺⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⢠⠐⠀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⡂⣄⠀⠀⣸⠀⡈⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⣀⠀⠀⠀⢀⡶⠀⠀⡄
⣀⡄⠈⣿⠀⠀⠀⠀⠀⠀⠤⠀⡰⣄⠀⠸⠇⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⠐⢶⠗⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠰
⠶⠦⠛⠉⠹⣋⢛⣴⣦⣤⡆⠄⣐⠀⢣⠰⠛⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠔⠊⢁⠀⠀⡀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠙⠂⠘⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠮⠡⡀⠀⠀⠈⠉⠉⠻⠂⠀⢠⣄
⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠄⣀⣀⡄⠔⠂⠈⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣀⢀⣀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠁⠉⠑⠖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠰⠿⡓⠡⠆⠂⠤⣥⣖⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠦⡖⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⡼⠉⢝⠲⠀⢀⣶⠠⠔⠀⠁⠀⠀⠆⠈⠁
⠀⠀⢠⢤⡤⣶⢏⡌⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠙⢤⠤⣶⢺⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢄⠚⣶⡄⠀⠀⠀⣆⡁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠙⢤⡆⡀
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠌⠠⠋⠀⠉⠀⠀⣰⢃⡃⠀⠀⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠠⠀⠀⠀⠘
⣤⣤⣦⣜⣀⡛⠀⠈⠈⠀⠐⢀⠊⠀⠀⠀⢰⡾⠞⠁⠎⠀⠀⠐⠀⠀⠀⡆⠐⠀⠈⠐⠁⠒⠀⠉⠀⢀⣠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤
⠈⠁⡉⠀⠐⠃⡤⠀⠤⠠⠞⠁⠀⠀⠀⠀⠛⠁⠈⠂⠀⠀⠀⠀⠀⠀⠍⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣀
⠠⠐⢀⠄⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣠⡀⢄⠀⠠⠁⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢰⡆⡀⠀⠘⠿⠃⠂⠀⢈
⠟⠀⢁⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠈⡴⠀⢀⡀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⡀⠢⠠⠈⠓⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⡂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢡⠆⠀⠀⠀⠀⠀⠘⡂⠀⠀⠠⢾⡿⠈
⠀⠀⠀⠀⡨⠣⠄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢣⠹⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠈⠀⠀⣠
⠀⠀⠀⠘⠀⠀⠂⠀⠄⠀⠀⠀⢀⣰⠆⠀⠀⠀⣀⠀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣀⡔⠁
⠈⠀⠀⠀⠀⠄⠤⠊⠈⠀⢣⡀⠠⠁⡆⣀⠖⠉⠀⠁⠀⠈⢋⡋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠔⠈⠀⢀
⠀⠈⠐⣄⠀⠀⠀⠀⠀⢈⠄⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢰⡵⠂⠉
⠺⠃⠀⠁⠐⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠠⠀⠉⠊⠉⢳⠂⠀⢄⡰⢾⠀⠀⠚
⢂⠒⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⠺⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠁

⢀⠴⠤⢦⡤⠖⠶⣄⠶⠦⣤⡴⠖⠦⣤⠆⠴⣄⡴⠦⢦⣠⠶⠦⣤⡴⠴⢦⡠⠆⠤⣤⠴⠰⢤⡴⠶⢦⣤⠶⠰⣤⡴⠦⢤⣤⠦⠰⣄⠴⠶⢤⡤⠖⠦⣄⠴⠶⢤
⣿⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠀⠀⢨
⣼⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⣦⢀⣀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢰⡖⣦⠀⣀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫
⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣩⢾⠉⢰⣯⣽⢶⣯⡇⣨⡏⠘⣅⣸⠀⢸⡃⣩⢾⠉⠰⣯⣽⢰⣮⣿⣠⡍⢠⣧⣝⢠⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢹⡤⠀⠀⠐⠀⠒⣀⠀⠀⢂⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼
⣿⠀⠀⠀⠃⠀⠈⠉⠉⠀⠈⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⢀⣄⠀⠀⠀⠀⠀⢨
⣸⠗⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠁⠀⠀⠀⠀⠐⢺
⢿⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀⣰
⣾⠉⠀⠀⠃⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠉⠛⠃⠀⠀⠀⠈⠱
⢙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⣸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⠶⠀⠀⠀⠀⠀⠴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣺⡗⢾⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹
⠻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣸⠗⠀⠀⢸⠓⠛⠛⠛⠛⠈⠛⠋⠛⠋⠉⠛⠋⠉⠙⠛⣻⡛⠜⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾
⢿⡀⠀⠀⢸⡇⠑⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⣼⠉⠀⠀⠰⠤⠤⠤⠤⠄⠴⠶⠖⠀⠀⠀⠘⠂⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹
⠻⣤⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠰⠒⠀⠀⠐⠂⢂⠀⡆⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠁⠀⠀⢸⠀⠘⠃⠛⠙⠛⠉⠛⠋⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘
⣙⠖⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⡀⠀⠀⠄⠐⠐⠒⠀⠀⠄⠀⠀⠶⢾
⢿⡀⠀⡤⠀⠀⠀⠀⠀⠰⠂⠶⠶⠶⠶⠲⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠀⠤⠄⠈⠀⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠀⠒⠂⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠠⠀⢇⣨
⣼⠋⣀⠀⠠⠀⠀⠀⠀⠠⠄⠤⠠⠀⠀⠀⠀⠀⢠⠀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⡉⠹
⢻⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠗⣧⣴
⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡁⢈
⣩⠖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⢾
⢿⡀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣇⢨
⣴⠋⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡷⡀⢀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠂⠀⡙⠻
⠻⣄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⠀⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡂⡇⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⢡⣴
⣿⠀⠀⠀⠀⡀⠀⠈⢂⠀⠈⠙⡂⠀⠀⠀⠀⠀⢸⠒⠒⢲⠖⠒⠒⠲⠶⠶⠶⠶⣖⢶⣿⣷⡶⠿⡷⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘
⣹⠖⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢸⡀⠀⠊⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾
⢿⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠄⠀⠀⠀⠼⢦⠦⠀⠀⠤⠤⠴⠆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣬⠋⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠦⠠⠆⠀⡄⠀⠄⠠⠄⢤⠤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⢻⣄⠀⡆⡦⠀⠒⠀⠀⠔⠁⠂⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⣴
⣾⠁⠀⡃⡄⠰⠂⠈⠰⠀⠀⠀⠐⠈⠙⠃⠙⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⢘
⣩⠦⠀⠀⡇⠀⠀⢈⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⢿⠀⠀⢰⡇⠐⠀⣤⠀⠃⢠⣆⠀⠀⠐⠀⠁⣠⣤⠀⢀⠁⠀⠈⢣⡁⠀⠀⠀⠀⢠⣤⠀⠀⢠⡄⠀⠀⣤⠀⠀⢠⡀⠀⢀⡄⠀⠀⣰⠀⠀⠀⣄⠀⠀⠀⠃
⠀⠙⠚⠋⠈⠓⠛⠈⠓⠚⠋⠙⠓⠀⠀⠘⠛⠉⠛⠓⠋⠂⠠⠀⠋⠙⠚⠲⠀⠐⠛⠉⠛⠚⠋⠙⠒⠛⠉⠓⠚⠋⠙⠒⠋⠙⠓⠚⠉⠙⠛⠚⠉⠚⠛⠃⠐⠀⠈

⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠰⢦⣄⣀⣛⣙⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣟⡽⠚⠋⠉⠀⠈⠁⠀⠀⠀⠠⣄⠀⠰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠔⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⢤⡴⠾⠷⣧⣾⣮⣼⣦⣀⣀⣶⠗⢲⣦⣄⣀⣀
⠀⠀⠀⠀⠀⣀⠠⠒⠋⠉⠥⣄⡀⠀⠀⡀⠀⠀⠀⠂⠀⠈⠀⢋⢉⠙⠛⠛⢶⣮⣄⣩⠙⠒⡒⠤⣀
⠀⠀⣠⠔⠋⡶⠀⠀⠀⡄⠀⠀⠁⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠉⡙⠶⢦⣄⣀⡓⠀⠀⠀⠀⠀⠀⣀⡤⠔⣆
⠀⢮⣀⣀⠀⠀⠀⠀⠀⡇⠀⠠⠀⢾⣻⣿⣿⣿⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠉⠙⠳⠦⣤⣤⠔⢫⢵⠆⠀⠸⡀
⠀⠀⢱⡦⣄⣀⠐⠂⠔⠀⠀⠀⠀⠀⠀⠉⠙⠛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⡀⠬⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⡻⣟⡻⡛⠲⠶⠶⠶⠤⢤⠤⠤⠤⠤⠤⠤⠴⣶⡶⣶⣶⠶⠶⠒⠒⠂⠉⠉⠉⠉⠓⠲⢤⣉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢂⣀⣤⡤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠡⠞⠋⠉⠀⠴⠶⠿⠷⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⡛⢭⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣸⣀⣒⠶⠒⠀⠀⠐⢰⣛⠋⠉⠉⠙⡸⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡇⠀⢀⡀⢨⡥⠍⠉⠓⠊⠉⠀⠀⠈⠉⠉⠙⠣⠒⠈⠁⡲
⠀⠀⠀⢰⠁⠀⠁⠉⠀⠀⢄⡀⠀⢳⡀⡟⠲⢦⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⣀⠇⠀⡜⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠙⠢⠴⠧⠔⠚⠉⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢼⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠯⣉⠻⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠛⠦⠞⠩⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⢙⡖⠃⠀⢄⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠀⠉⠉⠉⠿⠄⠀⢀⣀⠀⣀⣀⡀⠀⠀⠤⠀⠚⢻⣦⡀⠀⢀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠀⠀⠀⠦⢀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⠀⠈⢂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢶⣶⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢰⡀⠀⢠⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⢹⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠃⢀⣸⠀⢸⠁⠘⠁⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡆⢺⢘⡏⠈⢧⣄⣀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠉⠁⠀⡾⡸⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢇⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⣀⣂⣼⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠉⠙⢹⠋⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⢀⣀⠤⢤⣾⣤⣄⣀⣀⣀⣀⣏⣀⣀⣬⣦⢤⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣧⠟⠀⠀⠀⠀⠀⠀⠾⠻⠟
⠀⠀⢰⣇⣀⣀⣀⣸⣦⣟⡦⠬⠟⠋⠛⠿⢶⣮⠥⠾⣄⣀⣠⡼⠃
⠀⠀⠀⠉⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠺⠂⠀⠀⠀⠀⠀⠀⠒⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠄⠠⠠⠀⠠⠐⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠆⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⣀⡤⠶⢚⡛⠉⠉⠉⠙⠉⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢦⡀
⠀⠀⠀⠀⢀⡞⠁⣀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡞⣋⣁⣀⣀⣹⣆⣀⡀
⠀⠀⠀⠀⠘⣇⠰⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⢉⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⢀⣀⣬⠶⠞⠋⠉⠋⠀⠀⠀⠘⠉⡟⠙⡳⢦⣄⡀
⠀⡤⣄⣀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⣠⠏⢀⠔⢂⣤⣶⠶⣦⣤⣄⣼⣣⠴⠞⠋⠁⠀⠀⢢⡶⠶⢦⡀⠀⠀⡇⢀⠃⠛⠃⠀⠈⢳
⠀⣧⠀⠀⡍⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⣰⠋⠀⡆⢠⣿⠃⠀⠀⠈⢆⠈⠉⠉⠆⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⣦⠀⢃⠀⠣⡀⠀⠀⣴⡿
⠀⢸⡀⠀⠀⣹⢦⡀⣀⠀⠀⠠⠀⠀⠀⠀⣠⣮⢁⡴⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢆⠀⠀⠐⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⡗⠀⠈⠢⣀⣈⣴⠞⠉
⠀⠀⠱⣄⡞⠁⠈⡇⠀⠑⡄⠀⠀⢀⡀⡿⢁⡾⠋⠀⠀⠀⠀⢃⠘⣿⣆⠀⠀⠀⠀⠘⣤⠶⡶⣤⣀⡀⠀⠈⢄⠻⢶⣦⣶⠶⣠⡤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠁⣀⣥⠞⢃⠌⠀⠀⠀⠀⠀⠀⠀⠣⢌⠻⢷⣶⣤⣤⡾⠋⠀⠳⣄⣈⣉⣉⣹⠟⠛⣯⣀⣠⡼⠃⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⢠⠏⠀⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣀⡀⠠⣤⢯⠀⠈⠀⠙⠛⠛⢛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢁⡞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣖⡶⠿⠟⠛⠛⠋⢉⠉⠉⠠⠀⠐⠀⠀⠸⣄⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⠏⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⠋⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡜⠁
⠀⠀⠀⠀⢸⢱⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠘⡀⠀⠀⠀⢀⡜⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⡠⢂⣴⣿⠇
⠀⠀⠀⠀⡾⡘⢇⠀⠀⠀⣠⣞⠁⠀⠀⠀⠀⠀⣇⠀⠀⢀⡞⢄⡈⠛⠁⠀⠀⠀⠀⠸⡄⠀⣠⢤⣶⠞⣩⠞⠁⠀⣠⣴⡿⠋⠁
⠀⠀⠀⣸⢗⢔⣌⣳⡶⣴⣿⡎⠀⠀⠀⠀⠀⠀⡸⡆⠀⣸⣤⣤⢬⣢⠀⠀⠀⠀⠀⢰⠇⠀⠓⠛⠛⠛⣋⣤⣴⡯⠟⠋⠀⠀⠀⣤⣤⡤⣀⡀
⠀⣴⠞⢯⣑⠠⡈⣻⠇⣏⠟⠁⠀⠀⠀⠀⠀⢠⣿⡟⠀⠹⣤⣒⣂⣿⠃⠀⠀⠀⢀⣞⣀⡠⠤⠖⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠳⢭⡢⡀
⠀⠉⠺⠦⣬⣙⣻⠏⢸⠃⠀⠀⠀⠀⠀⠀⠀⠘⣽⠃⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠦⢤⡼⠛⠛⢲⡈⢳⣷
⠀⠀⠀⠀⠀⠀⠙⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⢀⣀⣠⠴⡮⠥⠤⠤⠴⠶⠖⠛⠁⠀⠀⢸⡇⢸⣿
⠀⠀⢤⠶⠛⠁⢴⣾⠀⠀⠀⡠⠄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⢖⣚⢩⡅⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⢾⠛⠓⠺⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⡀⢀⠀⠀⠀⣸⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢫⠡⣄⡎⣭⠺⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⠘⣇⢈⣹⠀⠀⢠⠏⠋⡀⠳⢄⠀⠀⠀⠀⣠⠞⠙⠒⠀⠀⠘⠇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣠⠋⠸⡄⠆⣠⠊⠀⠀⠀⠀⠈⠀⠀⠀⡰⢧⣦⠀⠀⠀⠀⠀⠀⠀⣤⣴⠚⠙⡄⠀⠀⠀⠄⠀⢠⡟⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣇⠀⢰⢃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠁⠐⠀⠀⠀⠀⠀⠀⠿⣭⡀⢀⡇⠰⠀⢀⠀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣶⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠉⠀⠀⠂⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠃⢿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢦⠀⠀⠀⠀⠀⠡⠄⢠⠀⣐⣈⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⠀⠀⠀⠀⠀⠠⠄⠀⠐⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣏⣤⣤⣴⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠛⣩⣼⡿⢋⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⠛⡻⠏⠉⠀⠈⢀⣿⣿⡿⠿⠧⢤⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣗⣚⣿⣀⠀⠀⠀⠀⠐⣫⡉⠁⡀⠀⠀⠀⢙⠩⢍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠤⠤⠿⠟⠉⢿⡄⠀⠀⢀⣠⣞⢲⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⣿⣆⠀⡾⠟⠉⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣭⣿⣿⣿⣏⣀⡀⢠⣤⠀⣼⣿⣻⣷⡀⢶⣅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣻⣿⣿⣿⣿⣽⣿⠀⢻⠀⢹⡇⠀⢻⡁⢸⡇⢿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣷⣿⡄⢸⡀⢸⡇⠀⣸⠁⣿⡅⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⢸⡇⠀⣿⠀⣿⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⡧⢸⡄⢸⡇⠀⣿⠀⣿⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢸⡇⠸⣇⠀⣿⢀⣿⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣿⠀⣿⠀⣿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡗⠀⣿⠀⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠞⣹⢟⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⠀⢹⠀⣿⠰⡟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠀⠈⠀⠛⢸⡷⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣭⣿⣛⡟⠚⠀⠴⢀⢭⣥⡶⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣿⡂⢸⡗⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣽⣿⣿⣿⣿⣧⣤⣤⣤⣽⣷⡾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠋⠡⠀⠒⠒⠒⠤⠉⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢚⣤⢞⣧⠀⠲⠀⠀⠰⠀⣸⡯⡀⡄⠂⡴⠂⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠳⠼⠁⠈⠻⣏⡙⠛⠒⠒⠒⠒⠚⠛⢁⠞⣁⠀⢃⣠⡇⠀⠦⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⡀⠀⠀⠀⢀⠀⠀⡼⡏⠐⠀⠀⠀⠀⠀⠀⡿⡍⠛⢿⣯⠿⠧⠤⡀⢈
⠀⠀⠀⠀⠀⠠⠐⡞⠉⠐⠳⠄⠱⣿⣍⣷⣴⠃⠈⠉⠒⠒⠒⠒⠂⠉⡈⢷⣤⠋⠉⢆⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠜⠉⠐⢦⣠⡄⢹⡷⠛⠉⠉⠓⠲⠤⠤⠤⠤⠤⠤⠴⠶⢟⠉⠳⢎⡑⠌⢉⣁
⠀⠀⠀⠀⠀⠀⡑⠀⠀⠸⢷⡷⠋⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⠃⠀⠁⠀⠁
⠀⠀⠀⠀⠠⠴⠟⠒⠒⢨⡟⠀⠀⢠⠁⠤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⣧⠀⣰⢒⠆
⠀⠀⠀⠀⠀⠀⠈⠂⠀⣸⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠅⠈⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠔⠠⡏⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠢⣿⠀⠘⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⣿⠊⠁⡄
⠀⠀⠀⠀⠀⠀⢈⠦⠤⣹⣇⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⣀⠴⠋⣼⠛⡓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⣀⣠⡤⠶⠚⠁⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠜⢏⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⠋⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠲⠤⠜⢙⣿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠛⡁⠈⡁⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠈⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣶⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣄⡀⠀⡤⠶⠶⢤⠀⢸⡟⠛⠛⠀⠀⠀⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠁⠀⢳⡀⠀⡟⠀⣼⠃⠀⠀⢀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠉⠒⣤⣤⡾⢷⡾⠦⢠⡟⠀⠀⠀⠘⡀⢀⣽⠾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⡰⣿⣁⣀⡀⢀⣈⡍⡀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠀⣿⠁⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠿⣮⣝⣛⣛⣛⣭⣷⠸⣄⣀⣀⡀⢰⠏⠀⢸⠀⠀⢯⠉⠉⠉⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⣧⣭⣭⡛⢡⠟⠿⠿⠿⠿⠿⠃⠭⣤⣤⣬⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢭⣉⡓⠲⠦⣤⣀⡑⢄⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠉⠛⠳⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣶⣄⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠛⠋⠉⠉⠈⠙⠶⣄⠀⠀⠀⠀⠀⠈⢧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⣉⠥⠤⠤⠤⣄⠀⠀⠀⠀⠀⠙⢗⠺⠿⠦⠀⠀⠀⠀⠀⢳⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢁⡤⠚⠉⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠗⠒⠒⠲⠦⠄⠉⠲⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⠖⠦⡄⠘⣇⠀⠀⠀⠀⠰⣿⡷⠋⣠⠤⠒⠒⠙⠛⠁⠀⠀⠀⠀⠀⢀⣹⠦⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠸⡀⠀⠘⡄⠈⠙⠒⠶⠖⠚⠉⠀⠚⠓⠶⠤⢤⡀⠀⠀⠀⠀⠀⠀⣴⣋⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠋⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠋⠀⠀⠀⠀⢀⡀⣼⠀⠀⠀⠀⢠⠰⠋⢸⠄⠀⠀⠀⠀⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⢾⣡⠇⠀⠀⠀⠀⠘⢆⡷⠚⠒⣄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣤⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⠀⠀⣀⡴⠋⠈⣇⣿⣀⠀⢀⣀⣀⡜⢀⠞⢙⣞⠒⢢⡄
⠀⠀⣠⢄⠀⠀⠀⠀⠀⣀⣾⠉⠉⠈⠉⠉⠻⣍⠉⠉⠉⠳⣤⠴⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⠙⠻⣄⠀⠀⣇⣀⣀⣀⡀⠀⠀⢀⠘⠒⢉⣿⡑⢫
⣠⢄⣳⡈⢣⣀⠔⠂⣁⣀⣸⠀⠀⠀⠀⠀⢠⠈⠙⠢⢄⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣄⠀⠈⠓⠚⠳⠀⠈⣇⠀⣆⡇⠀⠀⠀⠀⠀⠀⢱⠶⣻⣥⠑⠓⣄
⠑⢦⣤⡁⠀⢀⠰⠯⠥⣄⠈⢧⠀⣀⣠⠤⡞⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠚⠉⠁⠈⠙⡶⠦⠤⠤⠴⠖⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢣⢻⢣⣔⣒⣛⡤
⠀⠫⠤⠴⠚⠉⠑⠒⠒⠁⠀⠀⠉⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡶⢯⠱⣅⡤⠋⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⠉⠒⠒⠒⠒⠒⡖⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⡼⡀⠀⠀⠀⠒⢻⠉⠉⠉⡟⠛⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡞⢦⡷⡼⠓⠲⠤⣄⣘⠀⠀⠀⣇⠀⠀⠀⣀⣀⠤⠧⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠭⣍⣐⣒⣒⣒⣒⣒⣒⣺⡄⠀⠀⢸⠉⠉⠉⠉⠀⠀⠀⠀⠉⢀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠸⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⣀⡀⠀⠀⠀⠀⠑⠉⠁⡝⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠲⣀⠀⣀⠀⠀⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⡍⠀⡱⠒⠀⠀⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣚⣛⡛⠓⠦⣄⠀⣠⠔⠋⠁⠀⢹⠀⢺⠀⢀⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠈⠉⠒⣬⣿⣧⠀⠀⠀⢀⡿⠀⡤⠃⠈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠒⠒⠒⠒⠛⠛⠂⣀⠉⠉⢇⣀⣀⡠⠔⠛⠀⠸⠒⣋⣉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⢲⠤⠤⡤⠔⠒⠉⣀⠀⠀⠈⠑⠀⠀⠀⠀⠀⣤⣈⣉⠡⠔⣇⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠇⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠋⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢖⣂⣉⡦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣓⠭⡎⠑⠢⠄⡈⠢⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠉⢡⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡖⠒⠁⠀⠀⠀⠈⠒⢄⠀⠀⠠⣴⡒⣉⠀⠀⠀⠀⠀⠉⠐⠊⠁⢠⣤⣶⡏⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡈⠀⠀⠀⠀⠀⣠⣳⠀⣰⠕⢱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣦⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⣂⠴⡖⠚⢂⠖⠙⠋⣠⠔⠉⠉⠉⠉⢉⠿⠧⠄⠀⠀⠀⢰⠠⡼⠔⣣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣈⠗⠀⠀⠀⢀⡏⠙⠒⢦⣀⣮⠗⠀⠀⠈⠉⠁⠑⠤⣭⣭⡇⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠓⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣣⢣⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⠖⠊

⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠀⠀⠀⠀⠀⠤⠤⠠⠄⠀⠀⠀⠀⠠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⠻⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠⡄⠀⠰⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⠷⣄⣀⣀⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⣤⠤⠀⠐⠺⠓⠲⡲⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠈⢉⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠐⠋⡁⠠⠶⠒⠀⠀⠈⠛⠋⠓⠦⠀⢉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⢀⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠳⠀⠀⠀⠀⠁⡀⡀⠠⡀⡀⡀⠀⠀⠀⠀⠀⢠⡖⢀⠿⠀⠀⢀⡞⠑⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⠀⣤⡀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠂⣀⠀⠋⡄⠁⣠⢞⠀⠠⡊⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣅⠀⠉⠚⠛⡋⠘⠉⢉⠠⠴⠞⠁⠀⢂⡰⠋⠀⠀⠠⢊⠔⣩
⠀⠀⠀⠈⠄⠤⠠⠄⠠⠀⠀⠠⠀⠄⠠⠀⠠⠀⠤⠠⠄⠀⠄⠠⠄⠀⠀⠉⠉⠝⠩⠀⠀⠀⢄⠀⢀⡠⠋⠀⠄⠠⠂⠔⡡⢊⠤
⠀⠀⠀⠠⠅⠀⠁⠈⠁⠈⠀⠁⠀⠁⠈⠁⠈⠀⠉⢈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⠛⠋⠀⠀⠀⡀⠂⠀⢈⠔⡡⢚
⠀⠀⠀⢀⡧⠀⠀⠄⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡐⠂⣐⢲⠞⠀⡀⢀⠔⡁⢂⠄⠀⠀⡠⢊⠔⣩
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠓⠚⠒⠓⠚⠒⠓⠒⠐⠚⠐⠃⠚⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠤⠤⠤⠤⠤⠄⢄⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠐⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠄⠠⠔⠈
⠀⠀⠀⠀⢄⠀⣠⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⡉⠳⡉⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠄⡀⠐⠀⡄
⠀⠀⠀⠀⠈⠑⢄⡀⠀⠂⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠴⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⣉⠉⠛⠛⠓⠒⠒⠛⠉⠉⠁⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠤⢄⡀⢸⠵⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡤⠄⠼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣣⠀⠀⡜⠉⠈⢦
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⡠⠒⡞⠁⠀⠚⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠷⡀⠑⣴⠋⠙⠣⡴⣠⣆⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣜⣄⠘⣏⠉⠉⡏⠥⡄⣀⣀⣀⠤⠴⠊⠁⠀⠀⠔⠁⠀⠉⢢⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣾⡄⠈⠒⠰⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣓⣢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⢳⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡢⢄⠠⡂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⡎⠢⡀⢳⡝⡳⠂⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡮⣹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡶⢦⡇⠀⣸⠀⢀⠤⢦⣤⣤⡴⠶⠞⠛⠳⣋⣽⢆⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠒⠢⣤⠭⠎⣀⠀⢸⠁⠀⠀⠀⠀⠀⢐⠉⠇⣤⣇⣉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠊⠈⠀⢣⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢮⡉⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡤⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡔⠒⡶⢼⠒⠙⣰⣖⡖⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡈⢶⡊⠑⠂⠋⢹⠤⠤⡘⠐⠯⢝⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠪⠽⢽⣹⡞⠀⠀⠰⢲⠋⡇⠀⠉⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠕⠒⠒⠒⠚⠪⠽⠬⠃⠀⠀⠀⠁⢎⡑⠤⠤⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣁⡠⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠳⢶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣀⣠⣤⠀⢀⡀⠈⡁⠐⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⢀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⡀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡠⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠑⠤⡀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⡴⠆⢀⡔⠁⡠⠂⢀⡾⠀⢔⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠠⠄⠀⠀⠘⠓⠄⠀⠈⠻⠖⠀⠀⠀⠰⠖⠂⠁⠀⠠⠀⠉⠀⠈⠠⠖⠋⠀⠔⠡⡊
⢰⠂⠀⠀⠀⠀⠠⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡓⠀⣀⠀⠀⠐⠛⠃⠀⠀⠀⠀⣠⠇⠀⠀⠀⠔⠠⠊⠆
⢰⠄⠠⠀⠠⠀⠠⠀⠠⠀⠥⠀⠁⠀⠀⠀⠀⠀⡀⠀⡀⠂⡀⠂⡐⠂⠐⠀⠀⠀⠄⢠⡤⠜⠁⠀⠀⢀⠀⠀⠀⢐⡅
⢸⡤⠀⠁⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡐⢂⠐⠀⠀⠀⡀⠀⠠⠀⠠⠌⡼⠋⢐⠁⡠⠀⠀⠀⠀⢀⠀⠐⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠂⠒⠒⠀⠀⠀⠀⠀⠈⠩⠛⠶⠰⠥⠮⠆⠴⠤⠬⠤⠔⢵⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠚⠀⠃⠂⠐⠀⠐

⠠⠱⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⡶⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠋⠃⠀⠠
⠀⠂⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠄⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢧
⠀⠀⠀⠀⠀⢣⣥⡆⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠈⠀⠲⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⡃⡔⠀⠀⢸⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠄⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠢⢞⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠘⠁⢐⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢁⡀⢀⠀⠀⠀⡀⠀⠐⠋⠘⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠊⠉⠑⠀⠀⠈⠃⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠖⢰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠰⠀⢀⡂⠀⠀⠀⣀⣀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠉⠀⢀⠛⡔⠀⠁⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣶⣌⣠⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠘⠃⢋⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠂⠀⠀⢠⢷⡾⠋⠁⠄⠸⡷⠆⠀⠐⠃⠀⠰⠆⠈⠁⠀⣄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿⠀⠀⠀⡄⠀⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡀⠘⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠇⠀⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⠀⠀⠀⣼⠇⠀⠀⡟⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡁⠀⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠈⠅⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣠⣤⣀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠛⢙⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⣱⣾⣿⣿⣿⣿⣶⣶⡮⢻⣷⡄⠀⡀⠀⣠⣾⡿⢋⢀⣴⣿⣯⢻⡆⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣴⣿⣿⣿⣿⣿⣿⡿⠋⣴⣾⣿⣷⣆⣤⣾⣿⢿⣷⣴⣿⣿⣿⣿⣿⠁⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣤⣤⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠾⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣈⠙⠛⠋⠄⡀⠀⠈⣿⣿⣿⣿⠿⣻⣧⡀⠀⠀⠈⠉⢉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣅⡀⠀⠀⠀⣠⠄⡠⢜⣿⣿⣿⢏⡾⢿⣿⡿⣦⣄⣀⠀⠀⠀⠀⠀⢤⣙⢻⡿⣋⣥⠶⠞⢛⠟⠛⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢉⠁⠘⢿⣶⣿⣿⣧⢊⣴⠟⠋⠀⠀⢈⣠⣽⣿⣷⣶⣍⠛⠿⣶⣤⣀⣀⣠⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣷⡘⠿⣧⠐⣶⣤⣤⣿⡿⠃⠀⢀⣠⣶⣿⠟⢛⣿⣷⣄⡉⠻⢦⡀⠈⢻⣿⣙⣿⣿⡿⠏⠀⠀⠀⠈⠀⢠⣿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣿⣦⡀⢉⣙⠂⠈⠷⣿⣿⣿⠏⠀⣠⣴⣿⠟⣩⠴⣟⡛⠶⣌⡙⠻⣷⣦⡈⢷⣤⣿⣿⠿⠏⢰⣧⢀⡂⠀⠀⢀⣿⡟⣸⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡾⢿⣭⣁⠈⢉⣙⠀⣿⣿⣿⡶⠛⠋⣀⣴⣿⡿⢋⣵⣿⡋⠳⢬⡙⠷⣶⣤⣤⡘⢿⣿⣾⠿⠋⠀⡀⠲⣆⠛⢸⠁⠀⠀⣼⣿⣱⣿⣿⣿
⠀⠀⠀⢀⣴⣿⠉⠻⠆⠙⠻⠷⠈⣻⣿⣿⡿⠋⠀⣠⣾⣿⠟⣫⠾⣿⣿⣌⠻⣶⣤⣈⠢⣌⠻⣿⣿⠿⠋⠁⠀⢀⠀⢿⣆⠙⢷⡈⠂⡠⠀⣿⣿⡿⠛⢹⣿
⠀⠀⢠⣾⣿⣿⣷⡄⢶⣶⣦⣀⣼⣿⣿⠟⠀⢀⣴⣿⣋⣡⣾⠻⠶⣦⣤⣈⣀⠈⠛⠻⢷⣾⡿⠋⠁⠀⠀⠀⣄⠘⢷⡄⠹⣷⡀⠻⠆⠱⠘⠋⢁⣀⣴⣶⣿
⢀⣴⣿⣿⣤⣤⣤⣤⣀⣘⣿⣿⡿⠋⠠⢂⣴⣿⡟⠉⠉⠛⠻⢷⣦⣀⠉⡉⢛⣿⣮⡿⠟⠉⠀⠀⠀⠀⠘⢦⡌⠳⡄⠙⠆⠈⡀⣀⣤⣴⣶⣿⣿⣿⠏⠉⠛
⣼⣿⡇⠀⠉⠁⠺⢿⣿⣿⣿⣿⣄⣀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠌⢛⣿⣷⣽⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣒⣠⣴⣾⣿⠋⠻⠿⠋⠀⠉⠁
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⡟⠀⠀⠀⠀⠀⠀⠐⠦⢭⣛⣿⣧⣀⡀⡀⠤⠀⠀⠀⣀⣤⣶⣿⣿⣿⡟⢿⠏⠉⠁⠀⠤⠂
⢻⣿⣿⣀⣀⣀⣀⡠⠄⢀⣀⣤⣴⣿⣿⣿⣿⠀⠀⢠⠐⠂⢀⣀⡈⠲⣮⣽⣿⣏⣠⣤⣤⣴⣶⣿⣿⢿⣿⠋⠈⠉
⠀⢹⣿⣯⣭⣭⣴⣶⣾⣿⣿⣿⣿⡿⠋⡉⢻⣦⣤⣶⣦⣬⣴⣭⣝⣻⣾⣿⣿⣿⣿⣿⡿⢿⡿⠛⠋
⠀⠀⠙⠛⠿⠿⢿⣿⠛⠛⠋⠀⠉⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⡿⠿⠟⠙⠛⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣈⣉⣹⣝⣫⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⠟⣂⣀⠀⠀⢀⡴⠋⢁⣀⣀⠀⠀⠀⠱⣄⡘⢀⣠⡴⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠙⠳⣦⣼⣖⠛⠋⠉⠁⠀⠀⢀⡀⢻⣴⣞⣋⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⣴⣴⣾⡟⣿⡍⢷⡄⠀⢀⠴⠚⠉⣉⠿⣏⠉⠉⠙⠻⡀
⣿⣿⣿⣿⣏⣿⣶⣿⣭⣿⣿⣾⣿⣿⣧⣿⣿⣬⣷⠀⠀⠀⢀⡾⠛⠀⢹⡆⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣏⣀⣀⡀⡈⣇⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠛⠿⢛⠟⠛⠻⠟⠛⠛⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠗⠒⠢⠤⢀⡎
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢀⡔
⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⠃⠀⠀⠀⠀⢀⡤⠊
⣿⣿⣿⣿⣿⣏⣈⣦⣴⠟⠁⢀⣀⣤⠄⠚⠉
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠻⠛⢿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠄
⣿⣿⣿⣿⣿⣿⣿⡋⢟⣿⣦⡀⠀⣀⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠞⠑
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠰⣿⠀⠀⠀⠈⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣼⣿⡄⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⣄⣀⣀⡀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠿⠿⠿⣿⣿⣏⡊⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠻⢻⣿⣿⣿⣿⣶⣤⣀⠀⠀⠑⡄
⠀⠀⠀⢀⣴⣿⡿⠟⠋⢉⣠⡤⠶⠶⠶⠶⢮⣭⡛⠷⣤⡀⠀⠀⠀⠀⣠⣴⢶⣟⣿⡿⢿⣿⣛⠛⠿⣿⣷⣦⡀⠘⡄
⠀⠀⣰⣿⠟⢁⣠⡴⠒⠉⢁⣀⣀⣀⣀⠀⠀⠀⠉⠓⢄⡙⠢⠀⠀⢋⢵⣞⣉⣀⣀⣀⠀⠀⠈⠙⠶⣄⠉⠻⣿⣄⠃
⠀⣼⡿⠃⣴⡟⢉⣴⣶⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣶⣤⡀⠀⠈⠳⣆⠘⢿⣇
⢸⡿⢁⣾⣯⣴⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⣿⣿⣿⣷⣄⠀⠹⣆⡼⣿⡄
⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⣄⣿⡁⢸⣇
⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣴⣾⣿⣿⣷⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⣿
⢻⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠈⢿⣿⣿⣿⣿⣿⣿⡀⣴⣿⣿⣿⣿⣿⣿⣿⠇⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡟
⠈⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢫⣾⣟⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠈⢿⣌⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⢢⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⡟
⠀⠀⠀⠻⣷⣈⠙⢿⣿⣿⣿⣿⣣⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⢁⣴⠟
⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣿⣿⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠛⢉⣤⡾⠋⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢻⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠿⢉⣽⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣌⡇⡙⠻⢿⣿⣿⣷⣾⣶⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⠋⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⣦⣿⣷⣤⣄⡀⠀⠉⠙⠛⠿⠉⠿⣿⣿⠿⠿⠿⠛⠛⣽⢃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠰⣇⠐⠿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣰⣦⣭⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣤⣍⣛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣻⣿⣿⣷⣶⣶⣶⣭⣭⣭⣭⣭⣛⣛⣛⣋⣛⣫⣭⣥⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⢉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡷⢶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⣿⣯⣼⣕⠓⠊⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡦⠤⠤⣄⡀⠀⠀⢄
⠀⠀⠀⠈⠛⢿⣗⡛⢿⠿⢿⣿⣷⣦⢾⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⢠⣄⣠⡀⠀⠀⠈⣼⡇⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⢤⡄⠀⠀⣠⠈⠒⠲⠊⣀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⢭⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢆⠈⠈⠀⡻⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢹⠀⠘⠤⠀⠀⣠⠤⠾⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠙⡶⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⠀⡀⠀⡴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⣴⠟⠀⠀⠀⠀⠈⠹⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠖⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠀⠀⠀⠀⠈⠐⠐⠀⠀⠀⠀⠠⢮⣾⣯⣟⣫⠗⡦⢀⢀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠳⣦⣤⣤⣀⣀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠎⠀⠀⠀⡀⠌⠑⠂⠀⠀⠀⠀⠀⢀⡀⠊⠄⡟⢿⡟⣛⢿⡟⠛⠛⠻⠿⡧⣤⣬⡞⠁⠀⠀⠀⠀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⡀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠯⠉⠙⠋⡝⠛⠓⠋⠉⠀⠈⠁⠙⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠐⠄⠈⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⡿⣿⠟⠛⡟
⠀⠀⠀⠀⠢⠄⠨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡓⡋⠣⠆
⣶⣶⡶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠸⢼⠇⠀⠀⠀⠀⢠⣶⣦⡀⠀⢰⠀⠀⠁⠀⠀⠀⢘⣈⠁⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠀⠀⠀⢸⡇⠈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹
⠀⠀⢠⠀⠀⠀⠀⠀⠚⡀⠁⠀⢠⡥⢿⣄⠀⢻⣶⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⢸⡀⠀⠀⠀⠀⠠⠁⡄⠀⠈⠀⠈⠙⣧⢸⡆⠀⠤⣀⢒⣾⣿⣀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀⣀⣀
⡿⠋⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⢀⣿⡥⣾⡿
⠃⠀⠀⣸⠏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣧⠀⢀⡀⠀⠀⠰⣶⣦⣼⡄⠀⠀⠀⠀⠈⢧⠀⠈⢿⡇⠀⠃
⠀⠀⠀⢫⠾⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡏⠁⠐⠂⠀⠀⠀⠀⠈⠐⠀⠀⠀⡀⠈⢇⠀⠀⠈⢀
⠀⡠⠀⠀⠀⠀⠉⠀⣀⣀⣤⣶⠛⠛⡐⠂⠀⠀⢤⣀⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡼⠶⣶⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⠏⠉⢛⡷⣾⣛⠉⠉⠀⠀⠀⠈⠈
⠀⠀⢤⡀⢀⠠⠖⠉⠀⢀⠈⠉⠀⠀⡠⠀⢤⡀⠀⠀⠀⠁⠀⠀⠱⡍⠢⢭⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠐⠓⠀⠀⠐⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⠀⡶⣧⡇⠀⠀⠀⠉⠉⠛⠛⢷⡴⠞⠁⠀⠀⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣀⡘
⢙⣀⢀⠀⠀⠀⢀⡦⠀⠀⠀⢀⣀⣀⡦⠀⠀⢀⠀⠀⠀⠀⢀⠎⠲⣀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠐⣾⠐⠉⠳⠒
⠋⡏⣸⠧⠤⠖⠋⢀⣶⢟⡺⢯⣍⣭⡄⠀⠀⠠⠀⠀⠀⢠⡟⣗⠀⠈⠑⠂⠀⠀⠀⢻⠀⡔⠻⠆⡠⠚⠐⠟
⢀⡁⠀⠀⠀⠀⠀⢸⡟⢸⣋⣐⡤⠄⠀⠀⠀⠀⠳⡀⠀⠈⡤⠚⣀⠀⠀⠀⠀⠀⠀⢈⣾⣱⡁⢀⡷⠤⣴⠋⢰
⠘⠀⠀⠀⠀⠀⠀⣠⡿⢛⠿⡿⣿⢯⣧⣠⡀⣰⠀⡗⠀⠀⣹⡉⠱⠆⠀⠀⠀⠀⠀⢸⣬⡉⠙⣿⡀⠠⣟⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠏⠐⣾⠚⠇⠈⣷⡦⣤⣴⡇⠘⠀⠀⠀⢹⡖⠆⢛⠀⠀⠀⢀⡜⡀⠠⠿⣤⣯⣤⣟⣁⣠⣀
⢶⡄⠀⠀⠀⢀⡀⠀⠀⢩⠀⠀⠀⣿⣼⠛⢿⡸⡆⠀⠀⠀⠘⣷⠬⠋⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠉⠈⠉⠉⠉
⠉⠀⢀⡴⡞⢯⠉⠳⠄⠈⠀⠀⢀⣿⡇⠀⠰⣇⠁⠀⠀⠀⠀⠿⡂⠴⣢
⠀⠀⢸⠀⠀⠀⠀⣄⡗⠀⠀⠠⠞⡿⠁⠠⠴⠋⠀⠀⠀⠀⠀⠈⠣⠿⠟⠀⠀⠀⠀⠀⢀⡔⠛⠛⠐⠒⢠⡀
⣀⣀⣰⣛⣆⣀⣂⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣐⣒⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣸⣀⣘⣛⣀⣀⣀⣘⣄
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠳⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣀⣤⣾⣿⢻⡙⡄⢣⠸⡌⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⣾⢿⢻⡱⡈⢆⢳⣸⣬⡶⠿⠛⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡌⣎⣧⣵⠾⠛⠋⠁⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠰⣚⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣷⡀⠀⠀⠀⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣍⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡑⢌⣳⡿⣿⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠋⠀⠘⣿⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢠⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⣷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⢾⠍⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣽⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢘⣿⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢀⣾⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣼⡟⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣿⡇⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠊⠀⠀⣼⡟⡀⠀⣹⡇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⡇⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣿⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡷⡛⠩⠉⠝⢛⡷⢦⣄⠀⡠⠀⠀⠀⣸⡟⠀⢠⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢻⣧⡀⠀⠀⠀⠀⠀⠁⡙⣧⠀⠀⢀⣼⡿⡷⠶⠦⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠠⢼⣇⣤⡾⠋⠀⠀⠀⠘⠨⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⣶⣿⠛⠉⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢸⠇⠇⣆⡎⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⠓⠂⠀⠀⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢢⣿⠀⠀⣾⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢁⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⢻⣿⣿⡿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠡⢰⡀⠀⠀⠙⢻⣿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢲⢀⠄⠘⠀⠀⠀⠁⣸⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠛⠈⠇⢈⠀⠀⠀⠀⠀⠈⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠸⠀⠔⠀⠐⣠⣶⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠀⠀⢘⠠⠀⠀⣒⣂⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠡⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠁⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣀⣐⣈⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡿⠿⠛⢿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠚⠋⠉⠀⠀⣀⠀⠀⠉⠀⠀⣠⢠⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⣤⢶⡆⠀⠃⠢⠀⠀⠀⣁⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣮⢡⡟⣳⢸⡏⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡟⢻⠃⡿⢸⡇⣻⠈⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠁

⣶⢶⢶⢶⢿⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⡷⢶⡶⢶⣶

⠀⠀⠀⢀⢠⠄⠐⠶⠀⠈⢈⡠⠊⠁⠀⠐⠃⠸⠃⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢟⠏⠉⠀⠈
⠀⠀⢰⢁⡾⠀⠀⠀⣀⡼⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⡔
⠀⠀⠈⠠⢤⣀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢠⠄⠁⠀⠶⠘⢲⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⡤⠀⠀⠀⠶⠁⢀⡴⡁⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢶⣶⢶⢲⠒⡆
⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠐⡷⣮⡏⠿⠘⢰⠃⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣤⣴⡖⡖⡆⠀⠀⠀⠚⠳⠧⠆⠎
⠀⠀⠀⠾⠀⢠⠆⠀⠀⠰⣄⠀⢸⣴⡇⠀⠀⠀⠀⠤⠔⠀⠀⠀⢲⣾⡏⡇⠀⡇
⠀⠀⡀⠀⠰⠈⠀⠀⠀⠐⢂⢼⣿⣷⠯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⣰⡾⣠⠀⠀⠀⠀⡤⣡⠻⡟⠻⡕⠀⠀⠀⠀⢠⢿⣷⠀⣠⡤⣶⣶⡶⠀⠀⠀⣄⠒⣄⠀⢠⠀⠀⡆⠂
⢠⠀⠀⢐⣿⠁⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠩⠬⠱⡰⠉⠁⠀⠀⠀⠀⠈⠁⠘⢀
⢸⡄⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡽⣆⠁⠀⠀⠀⠑⣃⠀⠀⠀⠀⠀⠀⣀⡝⠀⠄⠀⣤
⠀⢳⡄⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢸⣿⠁⠝⠀⠠⠄⠀⠄⡑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃
⠇⣺⡄⣾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⡄⠚⠃⠀⠈⠠⠢⠁⡀⠒⠬⣆⠀⠄⠀⠀⠀⠶⠄⠀⠁⠈
⣇⣟⣼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⠀⠸⠾⠛⠋⠈⠙⠀⣆⠀⠀⠀⠀⠀⡀
⠘⠋⠁⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣺⡇⠀⠀⠈⠢
⣤⠶⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠈⣆⠘⡇⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⡀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠨⡇⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡯⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢆⣰⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⢀⠀⢀⠄⠀⠀⢸⠀⠀⠀⠀⠀⢰⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠴⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠁⠁⠀⠉⠀⠀⣀⡀⢀⣀⠀⣠⠤⠖⠒⠒⠚⠂
⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⣀⠀⠀⠀⣠⣤⠀⠀⣠⡄⢀⣠⡀⠀⠀⡤⠠⠄⠀⠠⠀⢀⣠⣄⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠸⠀⠉⡜⠀⠀⠙⠿⠃⠐⠿⠀⠐⠛⢋⣤⣤⣤⣶⣶⣤⣤⣄⣈⠙⠉⠉⠁⠀⠻⢿⣿⣿⡿⠂
⠀⢠⠠⠊⠀⠴⢶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠁⣀⣀⠀⠀⠀⠀⡀⠄⠀⢸⢰⢺⡇
⠀⡄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⠿⣿⣿⣿⣿⣿⡿⡛⠛⠛⠛⠛⢻⣿⠛⠁⠀⢤⢠⣿⠇⠀⠀⠈⠈⠈⠁
⠀⠈⠑⠀⠀⣾⣖⠀⠀⢠⣤⠀⠀⠉⠉⠀⠀⠀⢡⠄⠀⣀⠀⠀⢠⣶⣆⠀⠈⠻⠀⠀⠀⠈⠛⠀⠀⠀⠀⣶⢠⡄⡆
⠀⢰⣿⣷⡄⠙⠇⠀⠀⠘⠋⢠⠀⠀⢠⠀⠀⠀⠉⠀⠀⢉⠀⠀⠀⠉⠁⢠⣴⡶⠿⡇⠀⠀⠀⠀⣍⡀⠀⠛⠘⠃⠃
⠀⠘⣿⠀⠀⠀⠀⠘⠃⠀⡀⢸⡇⠀⠈⠉⠡⢠⢀⣤⢠⠉⠉⠀⡤⠀⢀⠘⠻⢷⣦⡇⠀⠀⠀⢸⣿⠄⠀⢠⢀⣀⡄
⠀⠀⣿⠀⠀⠀⠀⢸⣦⠀⣿⢸⣷⠀⠑⠀⠀⠀⠄⠀⠀⠀⢀⠠⠈⠀⠈⣀⣀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⢸⠸⠯⡇
⠀⠂⠻⠤⢝⣤⡄⠀⣿⣆⣿⣿⡿⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⢿⡇⠀⠐⠄⠀⠀⢰⡀
⠀⠘⢷⣲⣼⡽⠀⠀⠈⠿⣿⠟⢁⣀⡈⠉⠀⠀⡆⢰⢶⡶⠂⠀⠀⡄⠈⠀⠀⠀⣰⣿⣿⣿⣦⠀⣳⠀⠀⢸⣶⣴⡇
⠀⠀⠀⠉⣿⣇⠀⠐⠀⢀⣴⣾⣿⡟⠓⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠰⣿⡄⠀⠈⠈⠉⠁
⠀⠠⡀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⠃⠀⠀⢰⣿⣷⡄⠐⠐⠄⠀⠄⠀⠀⠀⠀⣀⣸⣿⣿⣿⣥⣄⢼⠃⠀⢀⢤⣤⡆
⠀⠀⠀⠀⠘⣿⠿⡿⢛⡉⠿⠛⠁⠀⠄⠀⠀⠉⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⢿⠿⣿⢻⡆⢀⠀⠰⠛⠹⠇
⠀⠀⠀⣶⣤⠈⠓⠁⠋⡀⠀⠀⣴⠀⠘⡶⠀⠸⠏⢹⡏⠀⠀⣿⠘⣿⡄⠙⢛⣱⢇⡇⣿⢸⣇⡀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⣀⡄⢄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠄⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⢠⠠⠴⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠂
⠀⠀⢠⠁⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢁⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⠀⠀⠀⠀⠉⠠⠄⠀⠀⠈⠀⠀⠐⠀⠀⠀⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠒⠁⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⡀⠂⠛⠛⠛⠀⠛⠓⠈⠁
⠀⠀⠀⠀⠁⠀⠰⠄⠀⠀⠀⠀⠀⠀⠔⠂⢴⠄⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⢦⠐⠀⠀⠀⠈⠉⠉⠛⠹⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠠⠀⠀⠀⠀⠀⠓⠒⠒⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠥⢤⣤⣤⣤⠬⠒
⠀⠀⠀⠀⠀⠀⣀⣁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠚⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠜⠁⠙⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠂⠀⠀⠐⢐⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠀⡀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠠⠄
⠀⠐⠃⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⢠⡾⣧⠀⠀⠀⠀⠀⠀⠀⡀⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠀⠀⠀⠀⠀⠀⠀⠵⠾⠿⠧⠄⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠀⠠⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠃⠀⢀⡠⠖⠑⠒⠿⠦⡀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠋⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣈⣐⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠠⣤⣀⣀⡤⠘⢹⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠽⠵⠿⠃⠀⠀⠀⠀⢘⡒⢀⣀⡀⠀⠀⠀⠀⠀⠉⠉⢁⠀⢀⠀⠀⠀⠠⣀⠀⣀⣘⢃⣁⠤⠂⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⣀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠉⠉⠆
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⡀⢈⠡⠆⠀⠀⠀⠠⣄⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⡀⠀⠷⠀⠸⠀⠀⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠩⡟⢋⡠⠄⠀⢀⠀⠀⠀⣴⢄⡄⢀
⠀⢸⡀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⣳⠒⠼⡃⠀⠈⠀⠒⠂⠀⠴⠆⠉⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠸⠿⠛⠿⠂⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝
⣻⣶⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⠀⠀⠀⠀⠑⠓⠀⠀⠀⠀⢸
⠛⡻⠀⠀⠀⠀⣀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣵⣠⣶⣶⡶⢶⣖⠒⠀⠀⠀⠀⠀⠐⠛⠀⠺⠿⠿⠀⠀⠀⠀⠀⡏⡀⠀⠠⣧⣷⣾⣷⣾⢿⠇
⠀⠀⠰⠶⡀⠳⣿⣜⢿⣛⠛⢛⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⣼⣿⠿⣟⡆
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣳⠫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⣄⡀⠀⠀⠀⠀⠘⣯⣾⡿⠁
⡀⠀⠀⠀⠀⠀⠣⡘⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣠⡴⠾⠟⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⠆⠀⠀⠀⢰⣶⣶
⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⣠⠀⣄⠄⠄⠀⠀⠀⡟⠉⠀⠀⠘⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠠⢤⣀⣄⣠⠎⣽⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠑⢤⣀⢹⣷⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⣀⣀⣀⣺⠩⠁⣀⣬⣤⣤⣒⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠛⢷⣿⡟⠢⠀⠀⠀⡀⢀⣿⣿⣶⡎⣀⠀⢈⣀⣠⣤⣴⡿⣽⣿⢾⡷⣷⡷⠟⢿⣿⣿⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⢀⣤⣾⣿⣿⣷⣿⣶⣿⣿⣛⣿⢿⣟⠿⣯⡭⠾⠃⠀⠀⠈⠙⢻⡿⠟⠉
⣤⣄⣤⣴⣆⠀⠀⠀⠀⠀⠰⠝⠿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠐⠀⣠⠬⠟⠚⡻⠟⠀⠀⠀⢀⡠⠔⠋⠁⠀⢀⣀
⠉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⣽⣿⣀⡀⡠⠄⠒⠀⠠⠽⠖⣚⣋⣉⠠⠄⠀⠶⠿⡿⠿⠂⠀⠀⠀⠛⠛⠃
⠀⠹⣿⠏⠀⠀⠰⠶⠶⠀⠀⠀⠀⣠⠒⠛⣿⣿⠋⠐⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠬⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠟⠛⡉⠉⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⢀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⡿⠂⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣶⣶⣭⣵⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠟⠯⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⢀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠃⢈⣿⡷⠂
⠿⠟⠛⠒⠤⠀⠀⢸⡁⡠⠄⠀⠀⢠⣶⣴⣤⣷⠄⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢺⠟⠛⠁⠀⢺⠟⠁⠀⠀⢰⠤⠤⠴
⠀⠐⠂⠀⠀⠀⠀⠀⠑⠚⠃⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⡀⢠⣶⣾⣿⣯⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⢀⣠⢄⣀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠁⠀⠿⣷⣷⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀