Black and white wing Emojis & Symbols

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⡷⡄
⠀⠀⠀⠀⠀⢷⠘⢦
⠀⠀⠀⠀⠀⠘⣆⠈⢳⡀
⠀⠀⠀⠀⢷⡀⠹⣄⠀⠙⣆
⠀⠀⠀⠀⠸⡙⢦⡘⢦⠀⢌⠳⣄
⠀⠀⠀⠀⠀⢣⠀⠙⢮⣳⡀⠑⣬⡳⢤⡀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠙⢿⣆⠘⢿⣶⣽⡦⣀⠀⣤⣀⡀
⠀⠀⠀⠀⢰⣄⠈⢧⡈⠣⣄⡙⢷⣤⡹⣿⡟⠊⠓⣼⡆⠉⡳⡄
⠀⠀⠀⠀⠀⢿⠳⣄⠳⡀⠙⢿⣷⣮⣛⣮⣻⣆⢲⣦⣝⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⢷⡈⠳⢼⣦⡀⠻⣿⣿⣯⡉⢛⣛⢮⣧⠀⠠⣍⠉⢽⣻⣗⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠙⠿⣶⣌⠻⢿⣿⣮⣿⣦⣜⣷⣰⠿⠷⣼⣿⣿⣷⣶⣍⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⢶⣬⣽⣿⣦⣝⡮⣍⣿⣍⠉⢉⣳⣤⣹⣷⣾⣯⡉⠛⠋⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣬⣳⣄⡘⢿⣿⣯⡉⠉⣙⣯⣉⠷⣤⣿⡋⠉⠸⢷⣤⣿⣿⣒⣂⡀⠀⠐⢯⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠛⠿⠦⣉⠻⠿⣶⣤⣙⠻⣓⠛⢩⣿⣷⠶⠾⣿⣶⣿⠿⣭⣄⡤⠀⠀⠙⢧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠠⣬⣽⣶⣦⣬⣍⠽⣋⡙⢶⡾⢗⣶⣦⡴⠛⠳⣿⠟⣿⡶⣶⣂⠀⠈⡇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡙⢿⣿⣁⠀⢐⣿⣿⣁⣀⣿⣿⡿⢶⣶⣶⠟⠿⣧⡾⣻⡿⠃⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠳⢮⣝⡛⠶⠶⠶⠋⢁⣽⠟⠀⠉⠛⣛⢴⣿⡿⢼⣧⡞⡆⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣍⠉⠉⠭⣭⣿⣿⡷⠖⡽⡿⣶⡞⢙⣻⠏⠈⠛⣡⢻⠛⠋⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠬⠿⠤⢶⣾⣿⡿⠋⢿⣿⣶⣶⢋⣹⣀⣤⡆⢀⡄⢳⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠼⣻⣁⡤⣾⡏⠘⣫⡔⠘⣫⠻⢿⡿⣹⣄⠀⠈⡈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠋⢉⡟⢀⣹⣷⠋⣫⣿⠻⢟⡴⠟⡰⢿⠛⣇⣤⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠰⠷⠚⠁⢸⣤⠞⢧⡴⠛⠼⠳⠏⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⢠⡈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠
⠀⠀⠀⠀⠀⢳⣌⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣴⠃
⠀⠀⠀⠀⠀⠀⠙⣷⣝⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠘⢦⣈⠻⣷⣝⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⣵⡿⢋⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠙⢷⣬⣻⢿⣮⣿⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢟⣯⣾⠿⣫⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠛⢷⣿⣿⣿⣾⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⠟⠋⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣭⣟⣻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣛⣯⣥⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢛⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣯⣿⣛⣛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⣭⣽⡿⣿⣿⣿⣿⡿⣿⠁⠀⠀⢹⡿⣿⣿⣿⣿⡿⣿⣭⣝⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⡀⠈⠀⠀⠀⠈⠀⣸⣿⣿⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣼⠋⣿⠉⠋⠀⠀⠀⠀⠀⠘⠋⢻⡏⢻⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠐⠠⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⡘⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢃
⢠⠇⠀⠀⠀⠀⠀⢀⠎⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠱⡀⠀⠀⠀⠀⠀⠸
⢸⠀⢠⠀⡄⢠⢀⠾⣠⠊⡐⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠑⣄⡷⡀⡄⢀⠀⡄
⡏⠀⡌⢠⠁⡦⠌⠀⠘⠺⣤⢣⢃⢂⠀⡀⢸⢀⢆⠆⠀⠀⠀⠀⠀⠀⠰⡰⡀⡇⢠⠀⠐⡐⡜⣤⠗⠃⠀⠡⢴⠀⡄⢡
⡇⠀⡇⢸⠔⠀⠀⠀⠀⠀⠈⠙⠛⠞⠷⠳⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠞⠾⠳⠛⠋⠁⠀⠀⠀⠀⠀⠢⡇⢸
⡇⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸
⡇⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾
⣧⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⣠⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣄
⠀⠀⠀⢀⣼⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣧⡀
⠀⠀⢠⢾⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢈⡷⡄
⠀⠀⡇⠈⡇⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢺⠁⢸
⠀⢠⡗⠀⢿⡀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢀⡿⠀⢺⡄
⢠⡇⠹⡄⠈⢷⣆⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⡾⠁⢠⠏⢸⡄
⣸⡇⠀⠱⣤⠈⠻⣷⡀⢠⡙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⡄⢀⣾⠟⠁⣤⠎⠀⢸⣇
⣇⠘⣦⠈⠻⣶⡄⢻⡏⠙⠻⣄⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣠⠟⢉⢹⡟⢡⣶⠗⠀⣴⠃⣸
⠘⡆⠈⣳⣄⠈⢿⡶⠷⣅⠀⠈⢕⣦⣄⣀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣀⣠⣴⡪⠁⠀⣺⠾⢶⡿⠁⣠⣾⠅⢰⠁
⠀⠸⢦⡌⠙⢷⣬⣿⡄⠀⠙⠢⣄⣹⣟⠉⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠉⣻⣏⣠⠔⠋⠀⢀⣼⣥⡾⠋⢡⡴⠃
⠀⠀⢣⡉⠲⣆⡙⢿⠛⠓⠢⠤⣭⣿⠋⠁⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠈⠙⣿⣭⠤⠔⠚⠛⡿⣋⣰⠖⢉⡜
⠀⠀⠈⠳⢄⣈⠛⠻⣿⣧⣤⣤⣤⣽⣷⠃⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⠐⣾⣯⣤⣤⣤⣼⣿⠟⠛⣁⡠⠞
⠀⠀⠀⠀⠀⠈⠁⠒⢸⠏⠁⢀⣀⠬⣿⣷⡏⢀⡀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠇⠀⢀⡀⢹⣾⣿⠤⣀⡀⠀⠹⡇⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⠁⠀⣠⠖⢛⣿⣿⣀⡄⠀⠘⢄⠀⠀⠀⡠⠃⠀⢠⣀⣿⣿⡛⠢⣄⠀⠉⡗⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣡⢀⡞⠀⡞⢩⠋⠁⠀⠀⠁⠀⠈⠀⠀⠉⠙⡍⢳⠀⢳⣀⣌⡗⠒⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⡍⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡖⠂
⣿⣿⣿⡷⠒⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣭⡉
⣿⣿⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣟⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠟⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠄⠀⠀⠀⠀⠀⠈⠻⠿⠛⢿⣿⣿⣿⠀⠀⠀⢀⣤⣀⣠⣾⣿⣿⣿⣿⣿⣿⠻⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡿⠍⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢂⡀⠀⠀⢠⣀⣸⣿⣿⠀⠀⠸⢿⢿⣿⣿⡿⢿⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡿⢫⠍⢥⠈⢹⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⠁⣤⠩⡭⢿⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢛⡉⠀⣟⢀⡔⠒⣵⡌⢱⣷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⣇⢡⣮⡑⢒⡀⣸⠁⢈⣛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠼⢿⣷⢻⠂⢈⡽⠛⡿⣧⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣼⢿⠛⢾⡁⠀⠟⢾⣿⠷⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣰⣀⡹⠿⣶⣆⡀⢷⡀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢀⡼⠁⣰⣶⠾⢏⣀⣇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣰⠟⣶⣶⣿⢛⣅⣀⣩⡟⠳⢀⡉⠛⠷⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠾⠛⢉⡁⠞⢻⣍⣀⣈⡙⣿⣶⣶⡛⣧
⠀⠀⠀⠀⠀⣰⠏⣸⡟⠛⢣⣿⣿⣏⣿⣿⡶⣤⣁⠂⠀⠀⠤⢄⠉⢳⣦⡀⠀⠀⠀⠀⣠⡾⠋⡡⠤⠀⠄⠔⣈⣤⢴⣿⣿⢼⣿⣿⣎⠙⢻⣧⠙⣧
⠀⠀⠀⠀⢰⠃⢀⣿⠁⣠⠌⠋⠁⠃⠈⣠⣿⠹⣿⢻⣶⣖⠰⣦⣄⣛⠿⣷⡄⠀⢀⣾⠿⣑⣠⣤⡆⣰⣶⡟⣿⠏⣿⣄⠈⠀⠁⠙⠙⣄⠀⣿⡄⠘⣇
⠀⠀⠀⠀⣾⢀⢸⡟⣾⣿⡇⠀⣀⣿⣿⣿⢿⣧⣀⣾⣧⡜⠻⢟⠳⢈⣿⡋⠁⠀⠈⢙⣻⡃⠸⠻⠿⢣⣼⣿⣄⣴⣿⢿⣿⣾⣆⠀⢠⣿⣷⡿⣇⠂⢿
⠀⠀⢀⡴⠇⣸⡇⢸⢻⠛⠿⣾⠋⠹⠛⣅⠘⠋⣸⣟⠏⠹⣆⠀⠀⠘⣟⡉⠀⠀⠀⠈⣹⠇⠀⠀⣀⠞⠹⢻⣧⠈⠃⢨⡙⠏⠉⣷⡿⠛⡟⠟⢸⣿⠘⢦⡀
⠀⠀⡾⠛⠛⠻⡇⠈⠀⠀⢰⣿⠢⣀⣼⣿⣷⣴⣿⣿⣦⣴⡿⣻⣠⡏⠉⠁⠀⠀⠀⠈⠉⢙⣇⣿⣿⣦⣴⣿⣿⣦⣴⣿⣷⣀⠔⣿⡇⠀⠀⠘⢸⡏⠛⠛⢿⡄
⠀⠀⠀⠀⠀⢰⣇⣰⣀⣴⣼⣏⠀⣇⢹⣿⠋⢸⣯⡉⢹⣿⣟⡓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣻⣿⣏⠙⣹⡇⠘⢿⡟⢸⠄⢸⣿⣦⣀⣎⣸⡇
⠀⠀⠀⠀⠀⢸⠛⣿⣿⡏⠻⠏⢰⣿⡄⠃⠐⣾⣿⣿⢻⣤⣶⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣷⣤⡿⢿⣿⣷⠆⠈⢀⣿⡆⠘⠟⢨⣿⣿⠋⡇
⠀⠀⠀⠀⠀⢸⡄⠟⣿⣷⠀⠀⢸⣿⡟⠂⣸⣷⡈⠁⢸⣯⡷⣦⣍⣧⡀⠀⠀⠀⠀⠀⠀⣴⣫⣴⣶⣽⡇⠈⠉⣼⣧⡰⢛⣿⡇⠀⠀⣼⣿⠛⠀⡇
⠀⠀⠀⠀⠀⠈⢧⣄⢸⣿⢷⢴⡏⣿⡿⠑⠆⣿⡿⢦⣮⡛⠻⠊⢳⡈⠁⠀⠀⠀⠀⠀⠈⢁⡾⠁⠻⢛⣹⣤⢿⣿⠳⠊⠿⣿⢳⡧⡾⢿⡏⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡌⣿⠀⠸⣷⡘⠀⠀⢠⡝⠁⢸⣿⣿⣷⢤⣀⣷⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⣾⣿⣿⣿⠈⠻⡄⠀⠀⠉⣾⡇⠀⢸⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢱⠀⠀⣿⣷⣄⡀⢠⠀⠀⢸⡝⢿⡟⢧⠉⠻⠄⠀⠀⠀⠀⠀⠠⠟⠋⣴⢋⣿⢋⣾⠀⠀⡄⠀⣠⣾⣿⡀⠀⡜⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠀⢸⡏⡟⠋⢷⣾⣆⠀⣼⣿⣯⡑⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠚⣽⣿⣷⡀⢠⣿⣾⠋⢹⠛⡇⠀⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠀⠀⢾⣿⡟⣿⣿⠋⣿⢷⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⢻⠘⣿⣻⠹⣿⣷⠀⠀⢠⡇⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣄⣠⡞⠿⠀⢹⠉⠀⡟⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃⢸⡄⠈⡟⠀⣿⢻⣄⣠⣾⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣟⣿⣇⠀⠀⢼⠀⣸⠃⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢧⠀⣧⠀⠀⢸⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⡏⣿⣿⣄⢀⡌⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣳⡀⢠⣽⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢀⣿⣿⡟⣾⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡟⣷⠹⣿⣿⣇⠙⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣴⢸⡿⠃⠠⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢘⠄⠈⢻⡇⣦⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢋⡿⠚⠁⣰⣿⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣾⣇⠈⠻⢹⡝⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡞⠁⢀⣶⠙⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⠋⣾⡄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⣤⠴⠋⣿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⠙⠦⣤⣌⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⣄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠹⣦⡀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡿
⠀⠀⠀⢀⠀⠀⠡⡱⣄⠈⢞⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠀⣢⣾⠁⢀⡄
⠀⠀⠀⠈⢷⡀⠀⠱⡌⠣⡈⢆⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢿⣿⢀⣾⣿⠃⠀⣻⠁
⠀⠀⠀⠀⠀⢻⠢⡀⠘⢆⡈⠢⣱⡈⠳⣄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢞⣵⣯⢟⣵⢿⡾⢁⡀⣴⠃
⠀⠀⠀⠀⠀⠀⠳⡌⠢⣄⠘⡄⠈⠃⢀⠹⡢⢈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⣄⣰⣿⢻⣿⣿⣿⡟⣠⡿⣽⠃
⠀⠀⠀⠀⢀⠀⠀⠈⠢⡈⠳⢜⣄⠀⠈⠳⠑⡄⠙⠌⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⣿⣿⣻⣿⣿⣿⡿⣫⣾⣯⡞⠁⢀
⠀⠀⠀⠀⠈⠳⡄⢀⠀⠈⠢⡀⠙⠢⡘⣦⡀⠈⠂⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⣿⢏⣼⡿⣿⡿⠋⢀⠀⡿
⠀⠀⠀⠀⠀⠀⠐⠄⡈⠒⠤⣱⣄⠀⠐⢚⠢⡈⠢⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣟⡾⢉⣤⣾⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠤⢄⠈⠣⣀⠈⢗⠀⠢⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣯⣼⣿⣿⣿⠟⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠄⣀⠑⢄⠈⠓⢤⣝⡢⣀⠀⠀⠀⠂⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⣷⢛⣩⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡈⠉⠛⠂⠀⠉⠣⢄⡉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣋⣭⣿⣷⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⡁⠚⠦⡀⠙⠳⣀⡈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣫⣽⡿⣫⣿⠿⣛⠉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢈⡉⠙⠙⠒⠠⣤⠈⠙⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣽⣿⠿⣿⣯⣼⣿⣶⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⠟⠃⠐⠤⣈⣙⡩⠋⠀⠀⠀⠑⠤⣀⡀⠀⠀⢀⡀⣀⣶⣿⣿⣿⣿⣿⡷⢿⣿⣿⣭⡿⠛⣫⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣒⠶⠂⠁⣬⣁⣀⠆⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠿⢿⣿⣯⣩⣿⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⣲⠞⠉⢁⣬⣀⠔⠁⢀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣟⣿⣿⣍⣿⡿⠧⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢐⣼⢁⣠⠄⠶⠔⠿⣿⡿⣻⣿⡻⢿⠿⣿⢿⣿⣭⣿⣻⣷⣚⣿⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠐⣸⣁⡜⢻⣞⡝⠋⠉⠺⡗⣻⡟⢻⠺⡿⠧⠽⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠀⠉⠀⠀⠀⠀⠈⠛⠛⠚⠟⠙

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢫⡁⠈⡲⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠰⡊⠀⣩⠁
⠀⠀⠀⠀⠀⠀⢀⢱⡖⢒⡀⠸⣉⣗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣏⡹⠀⣐⠒⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣤⣴⣇⣟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⣷⣠⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣟⠿⣺⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠠⣑⢊⡲⠆⢠⠀⢠⠠⠶⣊⢐⡡⠘⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡌⠁⠭⣗⠠⠀⠠⢐⡧⠥⠉⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⡳⠶⢤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⠛⠁
⠀⠀⠀⠈⢍⣉⣿⣿⣿⣿⣿⣟⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣟⣿⣿⣿⣿⣿⣟⣉⠍
⠀⠀⠀⠀⠀⠠⠀⠀⠭⠿⠿⣮⢿⣯⡛⢶⣯⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣵⠞⣫⣿⢫⡿⠿⠯⠅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠤⢔⠞⣻⣷⡢⢙⣷⣯⣻⣶⣿⣷⣤⣤⣴⡆⣦⣤⣤⣴⣿⣷⡾⣫⣷⣟⠡⣲⣟⡛⠗⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣉⡩⣿⣷⣅⡍⡺⣿⣿⣿⣿⣿⣿⠏⠀⠈⢿⣿⣿⣿⣿⡿⡿⡊⣁⣵⣿⡿⣌⣑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠟⠽⠿⠟⠚⠒⠛⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠛⠃⠚⠚⠿⠟⠝⠏⠂
⠀⠀⠈⡲⢦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⠶⡊
⠀⠀⠀⠠⠭⢽⣿⢿⣿⣟⢾⣿⡻⠯⣭⣽⣿⣦⡀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⣨⣾⣿⣭⡭⠿⣻⡿⢟⣿⡿⢿⣿⠭⠥
⠀⠀⠀⠀⠐⠒⠚⠓⣛⡿⣧⡛⢛⢶⣽⣙⣷⣿⣿⣷⣶⣶⣾⣿⠆⢾⣿⣶⣶⣶⣿⣿⢶⣟⣩⣵⠞⢛⣣⣿⣿⡓⠚⠒⠒
⠀⠀⠀⠀⠀⠈⠌⠍⠐⢺⣿⣿⣕⢙⠋⡻⡿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣯⣿⢿⡿⡿⡋⢛⢑⣽⣿⣿⡂⠨⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠟⠻⠿⠽⠮⠞⠔⠵⠾⢿⢿⡿⠃⠀⠀⠀⠀⠻⣿⣿⡿⠶⠕⠜⠌⠮⠨⠉⠛⠋⠂⠉

⣿⣿⠟⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠆⠻⠀⠀⠀⠀⠀⠀⢢⠗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⣛⠀⢐⡃⡐⠀⠐⠐
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠐⠲⡐⠖⠒⠀⠐⠦⠔⠀⠒⡫⠀⠈⠀⠠⡀⠸⠒⠂⠰⠄⠒⠀⠐⠒⠖⡰⠂
⠀⠀⠀⠐⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⡅⣀⢀⡄⡤⠤⠐⠂⠀⠀⠀⠀⠂⠢⠤⡔⣄⢀⡀⡰⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠈⠀⠀⠐⠒⢦⠄⡴⠒⠂⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠂⠐⠉⠠⠂⠒⢀⢁⣆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠠⡀⠀⠀⠁⣀⣈⡁⠈⠀⠄⢈⣑⡀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠘⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠀⣀⠄⠀⢰⡁⠄⠀⠀⠀⠀⠁⠈⠁⢀⠄⡀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⡄⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄⡼⡃⠀⣜⠆⠄⠀⠀⠔⠂⠑⠒⠒⡋⣠⢷⠀⡾⡆⢙⠒⠒⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣀⠀⢠⠀⠀⠄⠒⢤⣈⠉⢇⡤⣞⡴⢋⡼⠀⢧⡹⣦⡲⢤⠚⠁⣁⡤⠂⠠⠀⡀⡄⠀⣠⠄
⠀⠀⠀⠀⡀⢠⠀⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⢠⡀⢄⠀⠀⠀⠲⣤⡠⠐⠀⡠⠀⠈⠈⠈⡖⣋⣔⠊⠀⠀⠈⠑⣠⣙⢂⠃⠁⠁⠀⢄⠐⠂⢄⡤⠖
⠀⠀⠀⠊⠀⠁⠀⠀⠀⠀⠁⠹⠀⠜⠀⠀⠀⠀⠀⠁⠀⠃⠈⠀⠀⠀⠀⠙⠛⠖⠧⠺⠘⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠃⠖⠾⠲⠛⠋
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⢱⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⡱⣦⢤⢾⣀⡽⠀⠀⠀⡶⢄⠀⠀⠀⠀⡽⣇⠀⠀⠀⢀⣼⣧⠀⠀⠀⠀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡴⣿⢸⢱⡌⡤⠀⠀⡤⢷⠈⠳⡀⠀⠀⣏⣾⡁⠀⠀⢸⣏⡷⠀⠀⢠⠞⢀⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡣⡇⢸⡎⡛⠁⠀⠀⣧⠀⢣⡀⠙⡄⠀⢾⣿⣇⡥⣄⣹⣿⡇⠀⢠⠃⢀⠜⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢱⣧⢰⡼⠀⠀⠀⠸⡌⠳⡀⠑⢠⡌⢀⡈⠚⠻⠃⠹⠟⠚⣁⡠⢁⠄⠋⣠⠎⢸⠇
⠀⠀⠀⠀⠀⣠⣮⠉⠉⢆⠀⠀⠀⠘⠉⢉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⢬⣁⡤⠆⠠⡀⢹⠦⣀⣁⣒⡝⡲⢆⠳⣄⠀⠀⣴⠛⣱⡖⢫⣒⣀⣀⠴⡏
⠀⠀⠀⠀⣤⣬⠙⠟⣦⣀⡀⠠⣀⣠⣖⠛⠁⣤⡄⠀⠀⠀⠀⠀⠀⠐⠁⡤⣻⡀⢸⡟⢆⡀⣈⠳⡦⠬⠭⢟⡓⠈⡆⠈⡧⣾⠁⢸⠁⢚⡻⠭⠤⢴⠞⠁
⠀⠀⠀⠀⠠⠙⠲⣵⣉⡝⠁⠀⠈⢯⣹⡬⠞⠁⠀⡀⠀⠀⠀⠀⠀⠘⡋⣠⢹⠀⠈⡌⣦⣈⣀⠀⠓⠶⡒⣈⠙⠡⢚⡘⠃⠘⢦⢲⠍⠋⣑⢒⠶⠚
⠀⠀⠀⡖⠣⠤⣠⣈⣹⠀⠀⠀⠀⢀⣯⣀⣡⠤⠚⠆⠀⠀⠀⠀⠀⠛⠋⢅⡾⠀⠀⠸⣄⠈⠉⠠⣄⠀⠀⠓⠴⠲⣤⠵⠀⠐⠯⣬⠖⠤⠊
⠀⠀⢠⠇⠀⠀⠈⠉⣩⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⢸⠀⠀⠀⣚⣃⣠⡤⠚⠀⠀⠀⠀⣨⣥⡖⠲⠾⠂⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⢘⠚⠴⠤⢦⣍⡟⠀⠀⠀⠀⠀⣟⢣⠴⡴⠜⠚⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⠞⠁⠤⢨⠁⢰⢲⠀⠲⢯⠀⢍⠢⡀⠙⢦⡀
⠀⠀⠸⡀⠀⠐⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⠁⠔⢃⡀⠢⠸⡉⢊⠀⣘⠈⡩⠰⠲⣠⠓⠀⠙⣦
⠀⠀⠀⠁⣐⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⢓⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠂⠹⣀⠃⠐⢄⢈⠉⠀⠈⡋⢀⠶⠐⣇⡼⠐⢆⣀⡄
⠀⠀⠀⠰⣿⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣿⣷⠀⠀⠀⠀⠀⢀⡠⠐⢀⡔⠁⠉⠀⠀⢀⠀⡈⠉⠀⠀⠀⠈⢉⠀⠄⠈⠀⠈⠀⠡⣄⠐⠤⣀
⠀⠀⠀⠀⢿⣦⠸⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁⣼⠇⠀⠀⡤⠖⢊⣁⣤⠖⢋⡴⠃⣠⠆⢀⠾⠤⠇⠀⠀⠀⠀⠀⠈⢦⠼⢄⠈⣦⠈⠢⣈⠳⢦⣄⡉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠉⠉⠉⠯⠤⠒⠯⠤⠊⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠙⠢⠼⠗⠢⠭⠏⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣿⣿⣼⣿⣶⣾⣿

⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⢿⣆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣠⣿⠁
⠀⢀⡈⢿⣧⡸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⡿⢃⡄
⠀⣀⠻⣦⡻⣷⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⣾⣟⣵⡟⠁
⠀⠘⢻⣿⣿⣿⣟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣿⣿⣿⣿⠾⠋
⠀⢀⣙⢻⣿⣿⣿⣿⣯⣿⢿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣿⣿⣿⣿⣿⣿⡿⢋⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠘⠻⠿⣿⣿⣿⡿⠿⠛⠘⢿⣿⣷⡦⢰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡧⢤⣾⣿⡿⠋⠿⢿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠙⠛⢋⣡⣴⣾⣿⡿⣛⡝⢹⢟⣭⣫⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣼⣿⣟⢷⣾⣿⣿⡟⢹⣻⣿⣿⣿⣿⣿⡟⠛⠒
⠀⠀⠀⠀⠉⣽⡿⢟⣵⢾⣿⡿⠟⣬⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣸⣿⢟⢿⣯⣿⣿⣭⢾⣿⣿⣿⣮⣻⢿⣿⡉
⠀⠀⠀⠀⠀⠀⠐⣩⡾⢟⣫⣴⣾⣿⣵⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⣿⢿⣦⢷⢿⣻⣾⣿⣿⣿⣿⣿⢿⣝⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣩⢾⡿⡿⣻⢯⠞⢋⣿⣿⣿⡿⣧⠀⠀⠀⣴⣿⣶⣿⣷⣝⣷⣿⣿⢿⣿⡿⣝⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠧⡵⣴⣕⡼⣿⣿⣿⣦⡏⠀⢸⣼⣿⣟⣿⣿⣿⣿⣿⢿⠏⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠿⣿⣿⣏⣿⠀⣼⣿⣿⣿⡟⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⢀⠤⣀⣀⠀⠐⠂⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⣀⠀⠤⡀
⠀⠈⠒⢄⡂⠨⢭⣟⣲⡶⠲⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡄⢶⠆⣺⣿⡭⠭⠄⢐⡤⠂
⠀⠀⠀⠀⠉⠈⠙⢚⡿⠋⢆⡑⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠈⣢⡐⠈⢻⡏⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢍⠀⠠⠎⠑⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠋⠑⠂⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⡌⠄⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⢃⠔⡡⠡⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠒⢓⠀⠀⠀⠀⡛⠊⠂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⣿⣿⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⣿⣿⢾⡽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⣸⣿⣿⣿⣟⣿⣿⣧⠀⠀⠀⢠⡄⣰⠀⠀⠀⢀⣆⢠⡄⠀⠀⠀⣴⣿⣿⢻⣿⣿⣼⣇⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣾⣿⣿⢻⡿⣿⣽⣿⣿⣷⣄⢀⣾⣷⣿⡇⠀⠀⢸⣿⣾⣧⡀⣠⣾⣿⣿⣋⣿⣿⡟⣿⣿⣷⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢿⣯⣿⣿⣿⡜⢷⣽⣿⣷⣟⣿⣿⣷⣿⣿⣿⣥⠄⢠⣬⣿⣽⣿⣾⣿⣿⣿⣽⣿⣯⡶⢣⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⣿⣿⣿⣿⣾⣿⣭⡿⣿⣿⣶⣯⣿⣿⡻⠏⠀⠀⠻⣟⣟⣿⣽⣷⣿⣿⢿⣭⣿⣧⣿⣿⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣿⣽⢿⣿⣿⣹⠇⠀⠀⠀⠀⠿⣋⣿⣟⠿⣼⣷⣾⣿⣿⣇⣿⣿⣿⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠈⢫⡟⢹⣿⣿⣿⣿⣿⢧⣿⢻⡇⣿⡿⣿⢻⡿⡄⠀⠀⠀⠀⠸⢿⢿⣿⣻⡟⣿⢿⣿⡽⣿⣿⣿⣿⣿⡏⢻⡹⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣿⢿⣿⢿⣿⠿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⡿⣷⣿⣼⣿⣷⣿⣿⣿⢿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⣿⡇⣿⡞⣿⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⣿⢸⡏⣿⣿⡿⢸⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⢇⠸⣿⠁⢹⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⠇⢋⣿⢃⢸⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⡸⢽⣧⠹⢰⡌⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⢸⢃⣾⢸⢣⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣷⡜⣿⣧⣼⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣠⣾⣿⣠⣾⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢧⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⢻⣧⢿⣷⡹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢣⣿⢏⣾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣾⣿⡻⣆⢻⣷⡈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⣠⣿⠋⡼⢫⣶⡜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢸⣿⡎⣳⣝⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣷⢚⣵⣿⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⢸⣿⡇⢿⣿⡄⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣾⣿⠹⣿⣿⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣿⣿⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⣀⣀⣀⠤⠀⠀⠀⠶⠛⢁⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⣯⣴⣿⡛⠹⣋⣁⡀⠀⢠⡴⠞⠉⠠⢐⡿⡟⠀⡜⠀⠀⠠⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⣉⡋⠀⣀⡄⠀⢠⠂⢀⡞⢰⠁⡸⢀⣾⣀⠁⠸⠘⠁⡠
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⣴⣦⡾⣠⣿⠁⡏⣰⢁⡾⢿⣧⢰⠀⣠⠀⠇⠀⠀⠾
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣨⣿⣫⣿⣿⣳⣼⢯⣏⣼⢁⡾⣿⢹⣦⣿⣆⠸⠂⠀⠀⢸⠀⣄⢀
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣿⡿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣦⣦⢀⢈⠀⣿⢈⣠⠀⠀⠠
⠀⠀⣠⣴⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⣿⡎⡏⠀⠀⡀⠈⠀⠀⠀⢀
⢀⣾⡿⠟⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⢠⠐⠀⣸⡆⣆⣰⠀⡀⢀
⠉⠁⠀⠀⣴⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡿⣰⣿⣿⣤⣷⡌
⠀⠀⢀⣾⡿⠋⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿
⠀⠀⠘⠉⠀⠀⢀⣾⣿⠟⠡⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠏⣿⣿⢸⣿⡇⣿⡿⢸⣿⠇⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⢰⣿⣿⠏⠀⣼⣿⣿⠏⣿⣿⡿⢃⣿⣿⠏⢸⣿⡟⠀⣿⡏⠀⠿⠁⠈⠁⠀⠉⠀⠙⠟⠸⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⠋⠀⣸⣿⣿⠃⢸⣿⡿⠁⢸⣿⡟⠀⢸⣿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⢸⣿⢹⣿⢻⣿⡿⣿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⢠⣿⡿⠃⢠⣿⡿⠁⠀⣿⡿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠈⣿⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⣼⡿⠁⠀⢸⡿⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠀⡈⠀⡀⡀
⣀⣀⠀⢀⣀⢀⠀⢀⣀⣀⠀⠀⠀⡀⠛⡁⡀⣀⡈⠀⠀⣀⡀⣀⣀⡀⠀⡀⡀⠀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣁⠀⠀⣠⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣡⠔⠋⠁⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡺⣟⣵⠿⠋⠀⠀⠀⠀⣠⠋⠀⠀⣀⣠⣤⣤⢤⣶⡤⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣹⣿⣾⣿⣇⣵⠊⠀⢀⡴⠚⣁⠴⠚⠋⠉⠀⣀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠞⠁⠀⠀⢉⣿⣿⣿⡿⠋⠄⢺⣧⠔⠋⠀⠀⠀⠀⠀⠀⢫⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣰⢋⠀⠀⠀⡠⢋⣿⡿⢋⠄⣀⡴⠋⢀⡠⠂⠁⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣸⣷⣯⠂⠀⠊⠀⠋⣯⣾⣯⢞⣵⣢⣶⡿⠁⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⣿⡿⠁⠀⠀⠀⠀⢠⣿⣟⣵⣿⣿⠿⠋⠀⠀⣰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣿⠃⠀⣻⠆⡀⠀⠀⠀⠀⢸⢷⣿⣿⣿⠏⠀⠀⢠⣞⠃⠀⠀⣀⣀⣤⣤⣤⠤⢶⣦⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣤⣡⡾⠁⠀⠀⠀⠀⣱⣿⣿⡿⠋⠀⠀⠈⠠⠞⠛⠋⣉⠉⠁⠀⠀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⢼⣿⣿⠏⣴⣣⠄⠀⠠⠚⠉⣩⣿⡾⢃⣠⣤⣦⠶⠒⠋⠁⠀⠀⠀⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣁⠘⠿⣡⣾⠟⢉⢀⡀⠀⠀⣴⣫⣵⣾⣿⠿⠛⠋⠀⠀⢀⣀⣠⠷⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠈⠝⢡⣾⣿⡏⠀⠀⠀⢠⠾⠛⠉⣀⡀⠠⠶⠤⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣫⡀⠀⢉⣟⣩⣶⡆⠀⢀⣷⡶⠖⠒⠀⠀⠀⠠⠴⠯⠤⠤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⢿⣟⡃⢈⠸⢛⣉⠀⠀⢉⣼⣦⣤⣦⣤⠤⠤⠂⠀⠀⢠⣤⣬⠿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢚⣯⡆⠀⣼⠟⢋⠠⠾⠟⠛⡛⠉⠁⠀⠀⠀⣀⣀⠤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠰⠿⠧⠤⢶⣾⡶⠆⢀⣠⣖⠀⠀⠀⠀⠴⢟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣚⣉⠀⠀⣠⣅⠀⠐⠚⠿⠿⠛⠁⣀⡀⠤⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠿⠅⠒⢛⣉⠠⢤⣬⣄⣀⣀⠀⠀⠾⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⣀⣀⣀⡠⡤⠲⠗⠀⣾⢓⣈⣉⡍⢥⣤⣄⣚⡛⠛⢉⣁⣀⣠⠤⠄⠚⠋
⠀⠀⠀⠀⠀⣠⠚⠁⠼⠿⣤⠶⠇⠐⠶⠀⣀⣾⢟⣀⣬⡛⢉⣀⣬⣭⡉⠉⢀⡀⠝
⠀⠀⠀⠀⠰⠁⠀⠀⠐⠒⠰⣇⡀⠀⢤⠋⢃⠐⣤⣌⠉⣛⡀⠐⡍⣉⡙⠓⠀⠙⠦⡀
⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⢉⢠⣤⣰⡄⣤⣧⡉⠉⣀⡈⢛⠛⠛⠦⠙⢟⠉⠛⠓⠚⠳
⠀⠀⠀⠀⠀⠙⢦⡀⢀⣲⠺⠁⠤⠴⠌⠁⠀⠑⢄⣤⣬⡛⠻⢯⡲⢦⠤⢄⡳
⠀⠀⠀⠀⠀⠀⠀⠙⠶⢮⡳⠦⠺⠦⢀⣠⣽⢛⡛⠋⠉⡉⠱⡦⢄⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠼⠃⠚⠉⠒⠤⠽⠏⠉⠉⠁⠛⠀⠈⠒⠧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢠⠖⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠀⠂⠀⠹⠀⠀⠀⢸⠀⠈⠀⠰⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠱⢛⡀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⣈⠳
⠀⠀⠀⠀⠀⠀⠀⡠⣿⣷⢀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⢀⣰⣿⡷⠀⠀⠀⠀⠀⠀⣤⣶⡿⠃⣤⣶⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣦⣤⠛⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡟⠻⡄⠀⠀⠀⠀⠀⡿⠿⢷⣾⡟⢻⡿⠳⣄⣠⡀⠀⠀⠀⣀⣄⡴⠺⣿⠟⣿⣶⠿⠿⡇
⠀⠀⠀⠀⠀⣀⠜⠁⡀⠙⣿⠀⠀⢠⠀⠀⠀⠀⠀⠰⠀⠀⣻⠟⠀⣄⣈⢦⣀⡀⠀⠀⢸⣶⣦⣡⡁⢸⡛⠶⠸⠷⢉⠆⠀⢈⠱⠿⠀⠞⣻⡁⢩⣍⣶⣾⠃
⠀⠀⠀⠈⣽⣿⣾⣿⣧⣤⢻⣄⠀⢨⠶⠛⠀⠊⠙⠖⠀⢰⡿⣱⣶⡿⡿⣿⣿⡄⠀⠀⠈⠹⣿⣿⣷⣾⠿⣦⢒⣤⡌⠀⠀⠈⣤⣴⣢⣶⢿⣷⣿⣿⡿⠁
⠀⠀⠀⠰⢻⣿⠈⡅⣉⣽⣿⡵⠦⠀⢀⠰⠀⠙⠀⠀⠸⢻⣷⣯⣦⣄⣇⠻⣿⠹⠀⠀⠀⠀⠉⠿⡈⠉⣧⡀⣿⠋⠉⠀⠀⠈⠉⢻⡇⣠⡎⠉⣹⠏⠀⡀
⠀⠀⠀⠀⣼⢁⣿⣿⢿⣻⡾⢛⣀⠤⠒⠀⠀⠀⠈⠐⠠⣌⠛⢿⣏⡟⣿⣆⢹⡀⠀⠀⠀⠰⣄⡀⠙⢄⣿⡟⣿⢢⡜⠀⠀⠘⣤⢚⡟⣿⣧⠐⠉⣀⡰
⠀⠀⠀⢀⣿⣾⡏⢸⣿⠇⡕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⣷⡀⢿⣿⣇⠀⠀⠀⠀⠻⢿⣦⣼⣿⣷⡀⣰⠀⠀⠀⠀⢰⡌⣰⣿⡿⣧⣼⠿⠃
⠀⠀⠀⡼⠻⡏⢸⣿⠛⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡹⣷⣀⡟⠻⠆⠀⠀⠀⠀⠈⠹⣿⣏⠃⠹⡟⠀⠀⠀⠀⠘⡛⠁⢋⣷⡟⠉
⠀⠀⠀⠀⠀⣿⣿⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠈⢿⡶⣀⣠⠀⠀⠀⠀⢠⣀⡠⣾⠏⠁
⠀⠀⠀⠀⠀⠛⡏⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠙⡈⠿⠀⠀⠀⠀⠸⠏⡠⠁⠿
⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢱⢸⠀⠀⠀⠀⢠⢠⠁⠆
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⠀⠀⠀⠀⢸⣸⡎
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⣠⠤⠀⠠⣠⣤⣬⣧⣐⣀⡀⠀⠀⣠⣐⣠⣮⣤⣬⡠⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠰⠋⣾⢿⣿⠿⣏⣿⡻⠃⠀⠀⠛⢿⣝⣿⢿⣻⢿⡎⠳⡌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠘⠀⣰⣿⣿⢿⡼⣯⠃⠀⠀⠀⠀⠀⠀⢫⡿⣯⢿⣿⣷⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠠⠂⠀⠐⠁
⠀⠀⠀⠀⠘⢠⡃⠿⠻⣸⣾⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣼⡸⢿⢧⢢⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠖⠶⠶⣶⣶⣶⣳⣾⠯
⠀⠀⠀⠀⠀⠈⠃⢠⡤⣿⡿⢦⡛⠀⠀⠀⠀⠀⠀⠀⠀⢣⠼⣿⡷⣠⠀⠟⠁⠀⠀⠀⠀⠀⠀⣿⡏⢻⣓⢲⣶⣯⣈⠿⢿⣿⣻⣵⣶⠄⠊⠁⣠⣂
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡸⡄⢈⣧⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⡘⣱⣿⡂⠀⠀⠀⠀⠀⠀⠀⢠⣿⢩⣾⣟⠏⠉⢉⣱⣾⣿⣿⣿⡿⠟⠂⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣎⠹⣿⣟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢁⡅⢸⠗⠀⠀⠀⠀⠀⠀⢀⣾⣳⣿⠿⣱⣶⣶⣿⣿⣭⣭⣿⡿⠗⠢⡤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⡿⠦⠀⠀⠀⠀⠀⠀⠞⢿⣷⣿⣿⠃⠀⠀⠀⢀⠀⢀⣴⣟⣿⣿⠯⣤⡾⠿⠿⢿⣿⣛⣛⣛⣛⣉⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠃⠀⠀⠀⠠⣴⡾⣽⣿⠛⣫⣽⣶⣿⣿⣿⣿⣿⡟⠁⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠈⠱⣦⣿⣿⣿⣶⣾⣿⣛⠻⠟⠁⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⡿⠟⢿⠦⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡏⠀⠀⠀⠀⠀⢹⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏⠋⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣠⣾⣿⣿⣿⣿⣿⢿⣿⠏⠿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⠹⣿⡿⣿⣿⣿⣿⣿⣷⣄

⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿
⠛⠛⠻⢂⣤⣤⡞⠻⠛⢧⣤⣤⣜⠻⠛⠛⣠⣤⣤⡿⠿⠿⢧⣤⣤⡼⠿⠻⢿⣤⣤⣴⠟⠛⠻⣤⣤⣤⡸⠛⠛⢓⣤⣤⣤⠿⠛
⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣯⠀⠀⠀⣿⣿⣿⡁⠀⠀⢺⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷
⣶⣶⣾⠏⠉⠉⣹⣶⣶⡎⠉⠉⠁⣶⣶⣶⠈⠉⠉⣷⣾⣶⡏⠉⠉⢹⣾⣶⣾⠉⠉⠉⣾⣾⣶⠏⠉⠉⢹⣶⣶⡏⠛⠉⠙⣶⣶
⠿⠿⠿⣆⣀⣀⡼⠿⠿⢇⣀⣀⣸⠿⠿⠿⣄⣀⣀⡿⠿⠿⣇⣀⣀⣸⠿⠿⢿⣀⣀⣰⠿⠿⠿⣃⣀⣀⡸⠿⠿⢧⣀⣀⣐⡿⠿
⠀⠀⠀⣾⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠁⠀⠀⢸⣿⣿⡇⠀⠀⢨⣿⣿⣿⠀⠀⠀⣽⣿⣿⡇⠀⠀⢸⣿⣿⣿
⣶⣶⣶⡙⠋⠛⢳⣤⣶⡜⠋⠛⢻⣶⣶⡶⠟⠛⠛⠦⣤⠤⣜⠛⠛⢡⣦⣶⣼⢛⠙⠛⣶⣶⣶⠟⠛⠛⢳⣦⣴⡎⠛⠛⢻⣶⣶
⣿⣿⣿⠄⠀⣀⢸⣿⣿⡇⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⡏⠉⠀⠈⠈⠁⠉⠉⠻⠥⠄⡀⢸⣿⣿⡇⠀⠀⠘⣿⣿
⠀⠀⠀⣾⣿⣿⡇⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⠀⢰⣿⣿⣯⠁
⣤⣤⣤⠿⠟⠟⢧⣤⢄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣤⣼⠛⠛⠷⣤⣤
⣿⣿⣿⠆⠀⠀⠨⣫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⢀⣿⣿
⠉⠉⠉⣿⣶⣾⡉⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢵⣲⣾⡯⢀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣶⣷⣾⠈⠉
⣀⣀⣀⡿⠿⠿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣀⣈⣁⣼⠿⡿⢧⣀⣀⣘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣄⣀
⣿⣿⣿⡂⠀⠀⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⣿⣿⣿⡗⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⣿⣿
⠛⠛⠛⣷⣤⣦⡾⣻⠟⣽⣯⣿⣿⠁⡛⠓⣴⣤⣤⡟⠛⠛⢷⣦⣤⡞⠛⠛⢻⣦⣦⣦⠐⠐⠀⣀⠀⡀⠀⠀⠈⠢⣵⣤⣴⠛⠛
⠀⠀⠀⢿⣿⣿⡚⠫⠀⣹⣿⣿⣧⠀⠀⢀⣿⣿⣿⠃⠀⠀⣸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⡀⢀⣿⣷⣷⡄⠠⠀⢄⣹⢿⣿⠂
⣿⣿⣿⡆⠁⠀⣼⣿⣿⡇⠀⠀⢸⣿⣿⣿⠁⠀⠀⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⠀⠀⠨⣿⣿⣿⡅⠀⠈⢸⣷⣷⡇⠀⠀⢘⣿⣿
⠿⠿⠟⣣⣄⣠⡞⠻⠿⢣⣄⣤⣸⠿⠿⠿⣆⣀⣠⠻⠟⠻⢃⣄⣀⡼⠿⠻⠿⣤⣠⣴⠿⠟⠻⣤⣤⣤⡼⠿⠿⢳⣀⣄⣌⠻⠿
⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⣶⣶⣶⡍⠉⠉⢱⣶⣶⡎⠉⠉⠙⣶⣶⣶⠉⠉⠉⣷⣶⣶⡏⠉⠉⢱⣶⣶⣾⠉⠉⠉⣶⣶⣶⡉⠉⠉⢱⣶⣶⡎⠉⠉⠩⣶⣶
⠿⠿⠿⢅⣀⡀⠼⠿⠿⣇⣀⣀⡰⠿⠿⠿⢀⣀⣀⡿⠿⠿⣇⣀⣀⣸⠿⠿⢿⣀⣀⡀⠿⠿⢿⣀⣀⣀⡸⠿⠿⢧⣀⣀⣈⡿⠿
⠀⠀⠀⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠁⠀⠀⢸⡿⣿⡏⠀⠀⢸⣿⣿⣿⠂⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣯
⣶⣶⣦⠟⠙⠛⢳⣤⣴⡾⠛⠛⢫⣤⣴⣤⠟⠛⠛⣷⣤⣴⣫⣛⣛⣱⣦⣶⣤⠛⠛⠛⣶⣴⣦⡙⠛⠛⢣⣴⣦⡾⠋⠛⢱⣦⣤
⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡟⠉⠉⢹⣿⣿⣿⠀⠈⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⠘⣿⣿

⠀⠀⠀⠀⢀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀
⠀⠀⠀⢠⡏⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡄
⠀⠀⡠⣟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡻⢄
⠀⢰⠃⢸⠂⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠐⡇⠘⡆
⠀⢨⡇⠸⡦⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢴⠇⢸⡇
⢰⠃⢳⠀⢹⣆⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣰⠏⠀⡞⠘
⢸⡇⠈⢦⡀⠹⣷⡀⢀⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡀⢀⣾⠏⢀⡴⠁⢸
⡇⠱⣄⠘⢷⣄⢺⡟⠛⢧⡀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⢀⡼⠓⢻⡟⣡⡾⠃⣠⠎
⠘⡄⢙⣦⠈⢻⡦⠷⡇⠀⠉⣦⣄⡀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⣠⣔⠉⠀⢼⠾⢴⡟⠁⣴⡋⢠
⠀⢱⣄⡙⢿⣆⣳⣄⠈⠓⢤⣘⢿⠏⠁⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠊⠹⡿⣃⡤⠚⠁⢀⣞⣰⡿⢋⣠⡎
⠀⠈⣌⠳⣤⡙⢿⡿⠦⠤⣐⣻⣿⠓⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠚⢿⣟⣃⠤⠶⢿⡿⢋⣤⠞⣡
⠀⠀⠘⢇⡈⠛⠾⣷⣦⣀⣀⣩⣿⡶⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⢶⣿⣍⣀⣀⣴⣾⠷⠛⢁⡸⠃
⠀⠀⠀⠀⠉⠓⠠⢽⡟⠉⠉⢀⣻⣷⡴⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⢦⣾⣟⡉⠉⠉⢻⡯⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠦⢴⠋⠁⣨⠽⢿⣾⡁⡀⠈⢣⡀⠀⠀⢀⡜⠀⢀⢈⣷⡿⠯⣅⠈⠙⡦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠞⡁⢠⠋⣹⠛⠗⠂⠀⠘⠀⠀⠃⠀⠐⠺⠛⣏⠙⡄⢈⠳⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣠⢀⣠⣴⠟⡀⠀⠀⠀⠀⠀⠀⢠⡆⠠⡆⠀⣠⠀⠀⠀⡠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡤⣴⢂⣒⡿⠿⢛⣫⠵⠾⣃⠞⠀⠀⠀⢀⡠⢒⣀⣜⣡⠞⣁⠴⠛⠂⢠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣔⡶⡹⠋⢉⣲⣿⣛⣒⣚⣩⠭⠴⠶⢋⡴⠂⠀⣀⡀⣫⢿⣿⣋⠽⠒⣉⠡⠤⠖⢻⡽⠂
⠀⠀⠀⡀⢀⣶⡟⡛⠟⣀⣠⡴⢷⣶⣶⣒⣒⣒⠒⠲⣚⣛⡭⠆⠀⠀⠸⢳⣁⠾⣇⣊⣀⠀⠤⠄⣠⠀⠁
⠀⠀⠘⢰⣿⢟⡼⢠⣿⣝⠯⣒⠯⣔⣂⡬⢍⣉⣉⣉⡭⠗⠀⠀⠀⠀⠂⠉⣿⢸⡏⠉⢶⣶⠆⠉⠁
⠀⠀⠀⣼⠳⢎⢸⣿⣿⡿⣷⣦⣝⣶⡯⢽⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢁⠆⣘⣟⣥⠭⣽⠃
⠀⠀⡼⠅⠂⠁⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⡿⢒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⣘⡧⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣭⣬⣭⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⡎⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢳
⠀⡦⣀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣀⢶
⠀⢷⠈⢇⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⡸⠁⡞
⢠⣜⡆⢌⢆⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡰⡁⣰⣣
⠀⢣⠙⢆⠪⡳⣄⠀⠈⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠁⠀⣠⢞⠄⡴⢋⡜
⢠⣀⠳⡄⠑⢌⡠⡽⠦⣄⡀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣠⠴⢯⢄⡡⠊⢠⠞⣀
⠀⠳⡉⠛⠦⡐⠍⠙⠪⣒⠭⢓⣦⣄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣠⣴⡚⠭⣒⠕⢋⠩⢂⠴⠛⢉⠞
⠀⠀⠈⠢⣄⠈⠓⠿⣭⣉⡉⠛⡷⠊⠳⡀⠀⢹⠀⠀⠀⠀⠀⠀⡏⠀⢠⠞⠑⣾⠛⢍⣉⣭⠿⠚⠁⣠⠔⠁
⠀⠀⠀⠳⣞⠛⠒⠿⠯⢍⣈⣋⡺⣗⡂⠹⡄⠸⡆⠀⠀⠀⠀⢰⠃⢠⠏⢐⣺⢟⣙⣁⡭⠽⠷⠒⠛⣳⠞
⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⡴⢋⡦⢂⡿⣥⡜⢆⢱⡀⠀⠀⢀⠏⡴⢣⣬⢿⠐⢀⠙⢦⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠕⢹⢡⠾⢠⡎⡟⡹⠛⠳⠀⠠⠟⠛⡏⢻⢱⡌⠷⡌⡏⠪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢧⠊⢧⢳⠃⠀⠀⠀⠀⠀⠀⠘⡞⡼⠑⡼⠈⠃




⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠿⣌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⠿⠗⣤⡀
⠀⠀⠀⠀⠀⠀⡔⠑⢁⣴⣤⠘⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢃⣤⣤⡉⠙⢦
⠀⠀⠀⠀⠀⣮⣤⣿⣸⣿⢿⣆⢌⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡥⣼⡿⣿⣇⣷⣤⣧
⠀⠀⠀⠀⠀⢠⡁⣸⣧⣠⣶⣟⢱⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡎⣻⣶⣅⣼⣅⢸
⠀⠀⠀⠀⠀⠸⣿⣿⠛⢸⡿⢉⣌⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠛⣥⡙⢿⡇⠟⣿⣿
⠀⠀⠀⠀⠀⠀⡟⠛⢳⣾⣄⢨⣵⣷⡌⣿⣿⣷⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⣾⣿⣿⣡⣾⣮⣀⣸⣷⡟⠛⣿
⠀⠀⠀⠀⠀⠀⢻⣴⠟⣿⢿⡿⡟⠛⢉⣋⡻⣿⣃⠈⢣⠀⠀⠀⠀⠀⢀⠜⠀⣚⣿⢛⣙⡋⡛⡻⣿⡿⡿⠿⣤⡏
⠀⠀⠀⠀⠀⠀⠈⡿⣦⣿⣀⢀⣝⣷⣿⡟⢣⣿⣿⡄⡀⢣⠀⠀⠀⠀⡎⢀⢸⣿⣿⡘⢻⣿⣾⣯⠀⣀⣷⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠸⡈⢿⣿⣿⣿⡿⣿⣯⡙⠿⠷⠟⢻⣿⡇⠀⠀⢸⣿⠏⠻⠾⠿⢍⣽⣿⢿⣿⣿⣿⡿⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣿⣿⣿⡿⡍⠉⠀⢲⣤⣀⠀⢿⠃⠀⠀⠸⡟⠀⣀⣤⡖⠀⠙⡁⢿⣿⣿⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⡟⠈⢈⠁⠙⣦⣠⡴⣿⣿⣆⣚⡄⠀⠀⢰⣓⣸⣿⡿⣦⣄⡴⠋⠈⠑⠉⣫⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⣆⠀⠰⢠⣼⣷⣿⣯⠻⠉⡏⠁⠀⠀⠈⢹⠈⢟⣿⣿⣾⣡⡤⠎⠀⡰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⣦⢿⣶⣍⣿⢿⡟⢻⢶⠇⠀⠀⠀⠀⠸⡴⡟⢻⡿⣷⣩⣶⣷⣾⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⡻⣿⣿⣧⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⣿⣿⢟⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣷⣟⠻⠯⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠿⠛⣻⡞⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡿⣷⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣾⢿⣃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢻⣠⢹⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣵⡇⣄⠟⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠛⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⠻⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠝⢦⡘⣷
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠢⡙⢾⣷⣄
⠀⠘⢿⡓⠦⣄⡀⠀⠀⠀⠈⢧⡈⢢⡙⢿⣷⣄
⠀⠀⠀⠙⢦⡐⠭⣙⠲⠤⣄⣀⠙⢦⡈⠢⢌⠻⢟⡶⣤⣄⣀⡀
⠀⠀⠀⢰⡦⢬⣳⣦⣉⠒⠤⣉⠉⠒⠻⠶⢤⣉⠒⠬⢁⡚⠉⠍⠛⠛⠛⣻⠛⠛⢛⡛⠛⠛⠶⠶⢦⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠙⠦⣄⠨⢉⠛⠲⠦⣍⣐⠒⠤⠄⣈⠙⠒⠦⠤⢉⣀⡀⠠⠤⠽⢦⣀⠀⠐⢯⡉⠉⠐⠒⠀⠉⠙⠻⠛⢤⡀
⠀⠀⠀⠀⠠⣤⠬⣽⣦⣬⣁⠒⠤⢈⣉⡓⠲⠦⠤⣍⣬⣀⡐⣇⠐⠀⠤⠄⠸⣉⠙⠓⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤
⠀⠀⠀⠀⠀⠈⠳⢦⣐⠈⠍⣙⠓⠶⠦⢬⣍⣀⣒⣂⣀⠀⡀⠈⢙⡗⠂⣀⠀⣈⣱⠒⠛⠓⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠈⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣓⣶⣬⣍⣀⣐⡒⠂⠠⠭⠭⠭⠭⠉⠀⣉⣒⣤⢤⣤⡠⠽⠋⠁⠀⠀⠀⠀⢀⣧⣀⣀⠀⠈⠲⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⠀⠤⠉⢉⣉⣉⣙⣋⣉⣉⠉⠩⠅⠀⢀⣀⣳⠶⠀⠀⣀⠀⠀⠀⣀⣫⣀⡀⠉⠀⠀⠀⢣⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⢚⠋⠍⢰⣁⠂⠉⣁⠀⠀⠀⢎⡀⣀⣈⠉⠁⠀⠀⠀⢸⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢥⣄⣒⣒⣒⡒⣒⣒⣂⣉⣉⣤⠤⠖⠊⢉⡉⠈⣣⠴⠖⠁⠀⣏⢉⣉⠀⠀⢀⠄⠀⠈⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠛⠛⠉⠉⠉⠤⠄⠀⠒⠐⣈⣠⠔⣋⠅⢒⣈⠤⠖⠺⡉⢁⡀⠀⢸⣀⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⢒⡖⣛⣛⡩⢽⠟⠑⠈⣠⠖⢋⡠⢐⣠⠤⢺⢋⣥⠀⠀⣌⠀⡀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠶⢚⡟⡡⠊⣡⡔⢋⠔⢂⣼⠃⢀⠀⡀⠈⢻⢁⡄⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣤⡴⢃⠔⢁⡴⢃⠌⣩⠓⢾⠁⠀⠈⢹⣀⡞⠀⢠⠄⠀⠀⠈⠙⠓⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡼⠑⣡⣴⢋⡴⢃⡴⢁⡜⣹⠆⢰⡀⠀⠉⢇⣠⠞⢆⠀⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⣱⠣⣊⡴⣻⡡⢋⣴⢃⢊⡼⡉⣹⠀⢨⢱⢸⠈⢻⠓⢤⡜⠢⣴⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⠀⠷⠶⠋⢸⡤⠞⡇⢠⡇⠀⣿⡘⣸⢱⢸⣇⣸⡇⢰⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋⠻⡼⠁⠳⠛⠧⠞⠀⠁⠑⠉⠛⠁

⠓⠖⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠖
⠀⠒⠉⠀⠀⠀⡀⠀⠀⠄⠈⠂⠀⠦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠆⠀⠊⠀⠄⠀⠀⡀⠀⠀⠈⠓⠂
⠀⠈⢋⡥⢒⠁⢀⠀⠀⠀⠁⡀⠀⠐⠀⠄⠀⠀⠉⠃⠀⠀⠋⠁⠀⠀⠄⠐⠀⠀⡀⠁⠀⠀⢀⠀⢑⠢⣍⠃
⠀⠀⠀⠈⠉⠒⠓⣂⡀⠀⠀⠀⠐⠂⠀⠄⠠⠀⠀⠂⠀⠀⠂⠀⠠⠀⠄⠀⠒⠀⠀⠀⠀⣀⡒⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣶⣦⣴⣲⠄⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠂⠀⠀⠲⣄⠀⠀⠀⠀⠀⠐⡟⣕⢿⢻⣿⠇⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠐⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠠⠊⠀⢀⣤⣶⠛⣻⣿⠶⣦⣄⡀⠑⠄⠀⠀⠀⠀⢹⢣⢷⡼⣿⣷⠀⠀⠀⠀⡠⠊⢀⣠⣴⠶⡿⣿⠓⣶⣤⡀⠀⠑⠄
⠀⠀⠔⠁⢀⠶⢿⠁⣿⡰⢿⣌⣼⡏⢃⣟⣦⡀⠁⡀⠀⣴⣮⣯⣿⣍⡹⣷⡵⠀⢀⠌⢀⣴⣿⡉⢹⣧⣀⡿⢆⣿⠈⣿⠶⡀⠈⠢⡀
⠀⠀⠔⢠⣿⣧⣼⠶⢫⠃⡺⠉⣼⢳⠏⣇⣰⠻⣆⠐⡀⠧⡊⠐⠙⢁⠘⠟⠁⢀⠂⣰⠟⣆⣸⠹⡎⣧⠉⢷⠈⡜⡶⣧⣼⣿⡄⠢⠈
⠀⠀⠂⡸⣿⣯⠋⣆⡾⢷⢡⡿⢉⡦⢪⠙⢠⣆⣩⠀⠁⢰⣯⡢⡴⡿⣿⣽⣏⠏⠀⣏⢰⡀⠋⡕⢤⡉⢿⡊⡾⢷⣰⠉⣽⣟⠇⠐
⠀⠀⠀⠣⢻⡙⢶⡌⠱⡌⣸⡀⣬⠁⡜⣤⢃⠙⣹⠠⠂⢸⡝⢿⣦⡫⣵⢸⠀⠐⠄⣏⠋⡘⣠⢣⠈⣥⢀⣇⣡⠎⢡⡶⢋⡟⡜
⠀⠀⠀⠀⠈⣷⢑⢝⠶⣈⠋⠳⠛⣦⠁⣀⡿⣴⠏⠀⠀⢸⡨⡗⢿⢿⣮⡸⡀⠀⠀⠹⣦⠿⣀⠈⣴⠛⠞⠙⣁⠶⡫⡊⡾⠁⠠
⠀⠀⠀⠃⠀⠈⠈⣷⣅⠈⠻⣄⣸⠀⣰⠉⢀⠁⢠⠀⢠⢾⠷⢾⣿⡴⢝⠿⡀⠀⠀⡆⠈⡀⠉⣆⢀⣇⣠⠟⠁⣨⣾⠁⠁⠀⠘
⠀⠀⠀⠀⠀⠘⡄⠈⠉⡙⠦⣌⢳⡄⠁⣷⣾⠀⠀⠀⠈⠉⠉⣩⡯⠤⢄⠉⠀⠀⠀⠀⠀⣷⣾⠈⢠⡞⣁⠴⢋⠉⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠘⣦⣄⠈⢷⣄⡈⠻⣦⡿⡈⠀⠀⠀⠀⢠⣾⣞⣥⢾⠷⣏⠶⠀⠀⠀⠀⢁⢿⣤⠟⢁⣠⡾⠁⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢜⠟⢦⠈⠎⢙⢗⠾⣧⡀⠘⣦⣀⠀⣮⣿⠐⣿⡙⣮⣭⠆⠀⢀⣴⠃⢀⣼⠷⡲⡛⡱⠁⡴⠻⠣
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠈⠃⢷⡀⢣⡁⠘⣷⣄⢁⠀⠀⢰⣿⣏⢟⢁⠀⣿⡆⠀⠀⡘⣠⣾⠋⢈⡜⢀⡾⠘⠁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢈⢣⢠⡀⢹⢦⣿⡦⡘⡆⠙⠆⠀⠀⠑⠽⠷⡬⠽⠟⠀⠀⠰⠋⢰⢃⢴⣿⡴⡏⢀⡄⡜⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⡉⠙⠈⠣⠀⠀⠀⠀⡠⣲⠎⣑⣢⣀⡀⠀⠀⠀⠌⠁⠋⢉⡜⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠟⢆⠈⢦⡁⠀⠀⣸⣿⠗⣾⠩⡿⣇⠊⠀⠀⢌⡴⠁⡠⠻⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠳⡄⠓⠆⠀⢛⣿⡨⣿⡫⢟⢟⠁⠀⠐⠎⢠⠞⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣆⠀⠀⠘⣿⢧⣹⣉⡴⣣⠇⠀⠀⣰⠏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢡⠘⠶⠀⢀⠈⢙⣋⢷⡚⠁⠀⠀⠾⠃⡄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠃⢀⠀⠯⢟⢻⢷⣶⠁⠀⠀⠀⠀⠘⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⡞⣿⢸⣿⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢯⣿⣸⢿⡀⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣃⡿⠛⣿⠶⣾⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣰⢠⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠻⣆⠀⠀⠀⠀⢠⣿⡄⣇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡘⣇⠙⣧⡀⠀⠀⣈⣧⠻⣜⣆⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⠷⣜⢆⠈⠻⢦⡀⢿⠹⣆⠈⠂⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⢷⡈⠛⠇⠀⢀⠙⢿⣷⢬⠑⢀⡞⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡙⠦⣀⠀⡞⠀⠀⠈⠛⢦⣼⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠻⠿⣖⠀⣇⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠶⠦⠄⠘⣦⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣯⣟⡋⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⠀⣠⠋⠀⣀⣀⣀⡀
⢀⣀⣀⣤⣀⠠⣶⣶⣟⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⡏⣰⠋⢀⡴⠋⠉⠉⠉⡙⢷⣄
⠈⠉⠛⠳⢯⣿⣮⣝⡿⣿⣮⣝⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢽⣛⡒⠂⠀⠀⠀⠀⢀⡤⠞⠛⠛⠛⠟⠀⠀⠀⠀⠘⠗⠀⢽⡿
⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣬⣝⡛⠦⡍⠛⠷⠶⠤⠤⠤⠴⠶⠂⠭⣭⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣾⣶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠯⠞⠷⢶⣴⣴⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠛⠉⢳⣄
⢶⣶⣶⣶⡒⠒⠒⠂⠤⠤⢄⣀⡀⠀⠀⠀⠀⠋⡸⠃⠒⠬⣯⡭⢤⡀⠀⠀⠀⢀⠀⠄⠐⠒⠂⠀⠀⠀⠙⣀⣠
⠀⣨⣙⠓⠒⠒⠂⠐⡤⠀⢖⡠⡌⠳⡄⠀⠘⣸⣁⡄⠐⠂⠘⣦⡲⡁⠀⠀⡞⠁⢾⣭⠋⠀⣾⠁⠈⠉⣩⢅⡤
⠀⣈⡙⠒⠀⠄⠀⠘⡀⢀⣬⠉⢀⡶⠁⠀⢼⣸⣷⡀⠀⢀⠘⡿⢓⣻⡆⠀⠙⢦⠀⠫⣤⠠⠄⠂⠀⢀⣈⣁⣀
⠀⠑⠂⠁⠀⠀⢠⠒⠩⠄⣱⠂⠈⢄⠀⠀⠘⢏⡻⣿⡏⠁⠀⢻⣶⣽⣟⠀⢀⠄⢀⣫⡄⠒⡉⠤⠤⠈⠁⠒⠉
⠀⠀⠴⣒⠓⢒⣀⠴⠤⠀⢁⠺⠤⠈⠢⣄⡔⢻⠿⡉⠁⠀⠀⣀⡜⠉⠻⡴⠋⣀⡜⠰⠊⠠⡔⢁⣀⡀⠭⠷⠂
⠀⠀⠀⠰⠞⠍⠁⠀⠸⣏⠀⣄⡀⠹⣆⡜⢀⡏⠁⠈⠙⠿⠉⠈⠀⠀⠀⢳⡞⠋⠀⠰⠤⠄⠒⠀⠈⠭⠿⠂
⠀⠀⠀⠀⣠⠮⢀⠀⠀⢢⡄⢀⡖⢠⣼⠃⣼⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢷⣿⡛⣅⢢⣑⠦⣔⡪⣶⣄
⠀⠀⠀⠀⠑⠛⢁⣴⠤⢀⡠⢾⣿⣷⡇⠧⣿⡸⠰⣄⠰⢀⠀⠀⡄⣶⡇⠘⠈⢳⣿⡟⢄⡈⢑⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠇⢚⣵⠀⢈⣿⡄⠀⣩⣾⡿⠟⠻⢿⠭⡿⠶⠻⣿⣄⡔⠉⠉⣻⣜⡆⣉⡙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⡴⢏⡟⠀⢄⣼⣿⡅⠀⠀⠀⠀⢇⠄⠀⣿⣿⡇⢠⠎⠉⢿⣿⡌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⢀⣼⣿⢸⡇⡆⠀⠀⠀⠈⡠⠀⢹⣿⣷⣤⣦⡀⠘⢯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠏⣠⣾⣿⡇⢸⠁⢷⡀⠀⠂⠬⠂⠀⠈⣿⣿⣿⣿⣷⣄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡏⠹⣿⡇⢸⡄⠸⡷⣄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⣿⣇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠿⢁⣧⢹⡇⢸⣇⢀⢿⣮⡰⠁⠀⠀⠀⣸⣿⣿⠇⣽⣯⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡆⢻⣯⠘⡇⢸⡟⡌⡎⢿⡇⠀⠀⠀⠀⣿⣿⡿⣆⠈⢹⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⡆⢡⣾⣿⠈⣷⢱⣸⡌⠁⠀⠀⠀⢀⣿⣿⠃⣿⡆⠸⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⢸⣿⣿⡄⢿⡆⢻⣇⠚⠀⠀⠀⢸⣿⣇⠸⣿⢁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣙⢿⣷⠸⣿⡌⢿⣧⠀⠀⠀⠈⣿⣿⠟⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⢸⣿⡁⢻⡵⠈⢯⢀⠀⠄⣸⣿⣅⠘⢣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⢀⣿⡿⣧⠈⢷⡁⠈⢧⡃⠐⠋⣹⣿⣷⠀⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⢒⣬⠂⣌⢙⣄⢿⡆⠘⢷⡀⠀⠙⠶⠶⠟⣽⣿⡰⢇⣈⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢇⢐⣘⣇⣻⢿⣿⣿⡄⠈⢷⡀⠀⠀⠀⠛⠽⢾⣷⢠⠆⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡾⠋⡼⣿⣿⣷⡀⡀⠓⠸⣞⡷⠊⡉⠦⠱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠲⢜⣇⣿⠋⢋⠻⢝⡀⠀⣉⣐⠋⠨⠱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠐⠉⠋⣣⡭⠰⠶⠔⠈⠤⠌⣙⡲⠁

⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⢠⣶⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣦
⢀⡜⣿⣞⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣾⡿⣤
⢸⣿⡽⣿⣮⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣾⣿⣽⣿
⣦⣻⣿⣮⡿⣷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣫⣾⣿⣳
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣵⣿
⢻⣝⣿⣿⣿⣿⣿⣿⣿⢫⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣮⢻⣿⣿⣿⣿⣿⣿⣟⣵
⠀⠻⣿⣿⣿⣿⣿⣿⣏⣜⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⢟⣌⣿⣿⣿⣿⣿⣾⡿⠃
⠀⠘⢷⣿⣿⣿⣿⣿⢎⣻⢿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢠⣾⢟⣿⣿⣿⣿⣾⢿⣋⢾⣿⣿⣿⣿⣽⠟
⠀⠀⠀⠉⣿⣿⣿⣿⢞⣿⣿⣿⣿⣿⡿⠿⣛⣻⣧⠀⢠⣿⣑⡻⠿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠈⠙⠛⢳⣞⣿⣿⣿⣿⢯⣷⠟⠛⠛⠿⠀⠸⠟⠛⠛⠷⣯⢿⣿⣿⣿⣟⣶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣣⣻⣿⣿⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣫⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠉⣿⣭⣻⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣻⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣰⡽⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣹⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠯⣿⠻⣿⡇⠀⠀⠀⠀⠀⠀⣾⡿⢛⡯⠿⠟⠉

⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⢤⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⡤⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠈⠻⡿⢦⣔⢭⣛⣯⠓⢦⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⡴⠚⡽⣛⡭⣢⡴⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠲⣿⣦⡈⠑⡾⣿⡷⣦⣘⠿⣿⡓⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢞⣿⠿⣃⣤⢾⣿⠷⠊⢁⣴⣯⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠓⢮⣵⣿⣦⣌⡙⠲⢷⣄⠀⠈⠉⠙⠓⢦⣄⠀⠀⣠⡴⠚⠋⠉⠁⠀⣠⡾⠖⢋⣡⣴⣿⣾⡵⠚⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣿⠶⣤⣰⣿⣦⣍⡛⠒⠮⠷⣦⡀⠀⢰⠋⠙⠂⠰⠋⠙⡆⠀⢀⣴⠾⠵⠒⢛⣩⣴⣿⣆⡥⠶⣿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠈⠹⢮⣍⡓⠶⠬⢬⣽⣦⡘⣆⠀⠀⠀⠀⣰⢃⣴⣯⡭⠭⠶⢚⣩⡵⠏⢁⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣟⡛⠒⠶⣿⣟⣛⠿⠿⠿⣿⣾⣆⠀⠀⣰⣷⣿⠿⠿⠿⢛⣻⣿⠶⠒⢛⣻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣛⠒⠶⠿⢿⣭⣭⠿⠿⣿⡿⠀⠀⣿⡿⠿⠯⣭⣭⡿⠿⠶⠒⣛⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣗⣒⣛⣛⣛⣿⣻⠟⠁⠀⠀⠈⠻⣟⣿⣛⣛⣛⣒⣻⣯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⢀⠈⠫⡑⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢊⠝⠁⡀
⠀⠀⠀⠀⠀⠀⠙⢷⡚⠷⢄⡀⢙⢖⠢⢤⡄⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⢠⡤⠔⣺⡋⢀⡠⠾⢓⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠓⢟⠶⣄⣀⠙⠻⣟⠢⣵⣤⡉⠘⠻⣕⠢⢼⢢⡀⠀⠀⠀⠀⠀⠀⢀⡔⡯⠔⣪⠟⠋⢉⣤⣮⠕⣻⠟⠋⣁⣠⠶⡳⠞
⠀⠀⠀⠀⠀⠀⠀⠲⣤⠵⣦⣈⠉⠛⣞⡓⢭⣲⣬⣙⠒⠺⣆⠈⢸⡇⠀⠀⠀⠀⠀⠀⢸⡅⠁⢰⠗⠒⣉⣡⣖⡭⢚⣳⠛⠉⣁⣴⠮⢤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡈⠉⠒⢤⡭⠶⢽⣦⣌⣉⣂⢻⡄⢸⠃⠀⠀⠀⠀⠀⠀⠘⡇⢠⡏⣐⣉⣡⣴⡯⠶⢩⡴⠒⠉⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣙⠓⠒⠤⢭⣶⡤⠭⢲⡶⠤⠌⣷⡜⢦⡀⠀⠀⠀⠀⢀⡴⢣⣾⠁⠠⢶⡖⠭⢤⣶⡭⠤⠒⠚⣋⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠶⠶⠤⠤⠽⠶⠦⣽⣿⣭⣭⣽⣦⡙⢦⠀⠀⡴⢋⣴⣯⣭⣭⣿⣯⠴⠶⠯⠥⠤⠶⢶⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣰⣖⣊⡭⢥⣴⣊⡽⣫⣖⡩⡿⢮⠇⢸⡵⢿⢍⣲⣝⢯⣑⣶⡬⢭⣙⣲⣆⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠡⠟⠩⠟⠿⠋⡧⠋⠀⠀⠑⢼⠑⠿⠻⠍⠻⠌⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠐⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⠀⠀⠀⢄⢀⣀⠀⢀⡀⠀⠀⠄⠈⠁⠈⠁⠄⠀⠀⣀⠀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⢀⢀⢀⠀⠀⠀⠀⠀⠠⠀⢀⢀⢀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⢐⡒⠆⠴⢒⠀⠀⠀⠈
⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠠⠈⠀⠒⠀⠀⠄⠂⠀⠀⠀⠀⠐⠤⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡀⠀⡠⠀⠀⠀⠄⠠⠤⠜⠸⡄⡰⡌⠢⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠲⢤⢉⢣⢰⡵⢃⠇⢰⡹⢴⡦⡞⢉⡴⠂⠀⠀⠄⠀⠄
⠀⠀⠠⠀⠆⠀⠀⠀⢄⠀⡄⠀⠀⠀⠢⠠⡀⠀⠉⠒⠠⣤⣀⣀⢀⠠⠭⠒⠀⠀⠀⠀⠦⠭⢠⢀⣠⣀⣬⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⢦⢤⠞⡸⠃⠀⢰⢦⠀⠀⠀⣿⡄⠀⠀⣰⣧⠀⠀⠀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⢹⢸⣱⡴⠀⢰⠚⡆⠑⡀⠀⣧⣷⠀⡀⣾⡞⠀⢀⠎⢰⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⡓⠀⠀⢸⠣⡈⠢⡈⠀⠹⠻⠎⠳⠟⠇⠀⢁⠐⢁⠜⡆
⠀⠀⠀⠀⣤⡈⠑⠀⠀⢀⠊⢁⣤⡀⠀⠀⠀⠀⠀⠀⠹⢼⣁⡔⠂⠈⡧⢌⣐⣺⠲⠟⠢⡀⢀⡴⠻⠖⣕⣀⡡⢾⠁
⠀⠀⠀⠶⢎⢋⢶⡦⠀⢤⡶⠋⣡⠶⠀⠀⠀⠀⠀⣈⡔⡽⠀⠿⡐⠤⠌⣗⢒⡚⢥⠘⡄⡽⢿⢀⠃⡬⢓⠒⣺⠁
⠀⠀⢠⢄⣈⠙⡞⠀⠀⠀⢓⠏⣁⡠⣤⠀⠀⠀⠀⠤⠞⣹⠀⠸⡘⠓⠂⡀⠉⠲⣀⣌⣐⠄⠨⣃⣡⡀⠖⠉
⠀⠀⡆⠀⠈⠉⣇⠀⠀⠀⢘⡉⠁⠀⢸⡀⠀⢐⣂⠤⠔⠁⠀⠀⣈⢵⡂⠽⠄⠀⠀⠀⣀⢄⡀
⠀⠀⠉⠚⠒⠎⠃⠀⠀⠀⠘⠵⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠒⠊⡰⡉⠈⡻⡀⠒⠢⡙⠢⡀
⠀⠀⠡⡠⢶⠁⠀⠀⠀⠀⠀⠀⣶⢄⠈⠀⠀⠀⠀⠀⠀⠘⢀⠠⡐⠄⡉⠠⠔⠀⠤⠀⡅⠆⡀⢀⠙
⠀⠀⠰⣷⠿⠃⠀⠀⠀⠀⠀⠀⠻⣾⠆⠀⠀⠀⢀⠀⢀⡉⠁⠀⠉⠀⢈⠊⠀⠀⠈⢂⠀⠈⠁⠈⠩⣁⠠⢀
⠀⠀⠀⠻⠌⣧⠀⠀⠀⠀⠀⣰⡡⠟⠀⠀⣖⣈⣤⠖⣋⠔⣡⡞⡠⠓⠊⠀⠀⠀⠀⠘⠒⠣⡘⡦⡑⢬⡓⢦⣍⣑⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠄⠀⠈⠁⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⣠⠔⠚⠹⣃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣹⠉⠑⠢⣀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠞⢁⡠⠊⠐⢇⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⢤⡝⠈⠓⢄⠈⢳⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡴⡏⢠⠎⢠⠊⢔⠼⠦⡄⠀⠀⠈⠓⠒⠢⣄⠀⢀⡤⠒⠉⠉⠀⠀⡀⢠⠞⠮⡢⠱⡀⢱⡀⢻⢆⡀⠀⠣⡀
⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⣨⡗⣏⣰⠇⢠⠃⠪⢢⡛⡳⠴⢆⡤⠖⠒⠈⠁⠈⠉⠑⠲⢤⡴⠶⠏⢳⡀⡁⢱⡀⢹⣀⡗⢫⡀⠹⡆⠀⠱
⠀⠀⠀⠀⢠⠃⢠⠔⠋⣴⠇⣠⠏⡹⢒⡿⣤⣇⢠⠁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡆⢰⡀⡷⡼⢿⠛⡌⠳⡄⠹⣄⠙⢦⡀⠱⡀
⠀⠀⠀⢀⠃⢠⠞⢀⡼⠃⡰⠃⡴⢃⡾⠤⡏⢸⠋⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡏⠹⡀⡓⠬⢧⡘⣆⠙⣄⠙⣦⠀⠣⡀⠡
⠀⠀⠀⠂⣠⠋⢠⠞⢀⡜⣁⠜⠓⠉⠀⠀⠙⠻⢤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠴⠋⠁⠀⠀⠉⠚⠦⣈⣦⡈⢧⡀⠹⡀⠠
⠀⢀⣢⠜⠳⠖⠛⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠒⠛⠦⣑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠀⠀⠀⠀⠀⣠⣤⣤⡤⢤⣤⠤⠶⣦⣤⣤⣤⣤⠴⠖⠀⠀⠀⠀⢀⣠⢶⡚⡟⣻⣿⣿⣯⡷⠚
⠀⠀⠀⠀⠀⠀⠉⠻⣭⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⠀⠀⣸⣁⣦⠶⣟⠻⡍⣿⣜⣛⣒⣒⡒⠛⠀⠀⠀⠀⠀⢺⡫⡝⢏⣯⣽⣛⣛⣓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡭⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⢮⣗⠿⠍⠛⠳⠾⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠧⣟⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈

⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⠴⡶⠶⡶⠶⠖
⠀⠀⠀⠀⠀⣀⣴⢏⣾⣷⣿⠿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢀⡼⣳⣾⣿⡿⠿⢭⡭⠝⠋⠉
⠀⠀⠀⠀⠀⣿⣴⡵⣝⢶⡭⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⣭⣶⣟⠾⠭⠍⠛⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢋⣤⠆⠀⠀⠀⠀⠀⡘⠇⣾⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣴⠾⢋⣡⠄⠀⠀⠀⠀⢸⣟⡸⢏⣾
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡄⠀⠀⠀⠓⢿⣥⠾⢛⣡⠀⠀⠀⠀⣼⡇⣻⠕⢟⣥⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠾⣛⣛⣛⣛⣛⣛⣿⡿⢿⠟⠋⠀⠀⠀⠀⠀⠀⣿⣚⣋⡁⠀⠀⠀⣠⡾⢓⡖⣨⡭⠔⠀⠀⠀⠀⢋⣽⣿⣿⣿⡿⢛⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⢘⣻⡷⢾⣭⠭⠍⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⢴⠿⠭⠁⠀⠀⠀⠀⢓⡚⣃⡶⠖⠀⠀⠀⠀⠀⠀⠘⢿⣻⠿⠏⣩⣾⠿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛

⠀⠀⠀⠀⠀⠀⣴⢟⡷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⢏⣾⢏⣲⣯⡟⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠞⣯⡿⢍⣽⡯⠭⠭⠭⠭⠭⠭⠭⠉⠀⠀⠀⣠⣾⢿⡿⢟⣛⣻⣛⣿⠿⠿⠿⡿⠛⠋
⠀⠀⠀⠀⠀⠉⠛⢬⠻⢏⣁⠴⣿⡳⠿⠭⠛⠓⠒⠂⠀⠀⠈⣯⣵⣿⠋⣤⣭⡭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⠿⠯⠻⠯⠝⠷⠯⠍⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠻⢿⡾⣟⣴⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⣠⡟⣰⣿⠿⠏⣼⣟⣛⣛⣛⠿⠿⠛⠁⠀⠀⠈⠽⣿⣿⣿⣿⣯⢭⣭⣽⣿⣛⡛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠐⣾⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣩⢡⡝⠷⢍⡑⠿⠶⠮⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⣭⠟⠳⠖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣠⣤⠴⠶⠒⠄⠀⠀⠀⠀⠀⠓⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡴⢌⣇⠀⠀⠈⠃⠀⠀⠀⠀⠀⣤⠀⣰⡶⠋⠀⠀⡬⠙⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢐⣿⣀⣤⠛⠢⡭⡳⢀⡀⠠⠀⠀⡀⠀⠀⡁⣰⣛⠁⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⡟⠂⠁
⠀⠀⠈⠁⠀⠀⠀⠹⡭⢏⣀⠈⠒⠟⠛⠁⠀⠀⠀⢨⠃⢰⠠⣆⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⣤⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠂⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡁⣞⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡠⡤⠖⢙⠒⠂⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠷⢆⠀⢀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠐⠀⠀⠀⢀⣀⠄⠀⢀⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠛⠄⡀
⠀⠀⢀⡈⠄⠁⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡈⠑⠀⠀⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢿⡿⣠⣄⠙⠆⠀⠀⠀⠀⠀⠀⠠⠀⣸⡛⠇
⠀⠀⠀⠀⡈⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠪⠂⠈⠞⢱⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⣼⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⡀⠀⠀⠀⠀⠀⢠⣿⢁⠀⠠⠆⠁⢫⣄⠀⠐⣀⡀⠀⠀⠀⠀⠀⠹⣇⠀⠀⣀⡈⢂
⠀⠀⠀⠀⢄⠀⠀⠹⢿⣄⠐⠀⠀⢀⠀⠀⡞⠹⠈⠀⠀⠀⠀⠞⡽⣿⠺⠜⠃⠀⠀⠀⠀⠀⠀⠛⣧⡄⠙⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠻⢷⣀⠀⠘⠁⠀⠃⠀⠀⠀⠀⡁⠂⠘⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠐⠈⢻⣧⣤⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢄⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⣙⣕⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣮⡑⣄
⠀⠀⠠⠤⠔⠺⠀⠀⠉⢧⡄⠘⣆⠀⠐⢄⠀⠀⠀⠀⠐⠀⢰⣿⢂⡇⠄⠀⠀⠀⠀⠀⠐⠠⠽⠅⠀⠘⡆
⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠁⠮⢑⠀⠀⠁⠒⠀⠀⠀⠀⠉⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶
⠀⠀⠀⠉⠂⠞⠐⠁⠀⠂⠀⠀⡀⠀⠀⠳⠤⠀⠀⠀⠀⠀⠈⠉⢿⣇⡑⡀⠀⠀⠀⠀⠀⠠⠀⠰⣆⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠻⠶⠶⠃⠀⠀⠀⠈⠀⠀⠾⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣴⡶⠊⠀⠀⠀⠀⠀⠈⠑⣷⣤⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣻⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⡛⢻⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⡛⠁⠐⢟⣄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⣬⠛⠄⠉⣳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣺⣿⡇⡤⣀⢈⢈⠃⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠈⡈⡀⣐⢄⢼⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡼⡿⣻⢀⣾⣿⣷⣿⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠸⣿⣿⣿⣧⠸⡟⣟⣮⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⢱⠃⢏⠘⡁⣿⠘⠃⣾⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣇⠙⡃⣿⣙⠀⡿⠹⡜⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡰⢀⡾⢸⡯⠹⡿⡆⢿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⣇⢿⠋⣿⠆⣧⠘⣄⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⡼⡷⡌⢡⠀⡴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⣆⠰⡉⣀⣿⡇⣿⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠇⡠⣇⣾⣸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⡇⡇⣿⣄⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣷⠉⢿⣿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡇⡏⠉⣾⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠘⡄⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣰⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡆⢷⣌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⡟⢸⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⣎⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣼⢹⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡾⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠰⣄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⢀⡤
⠀⠀⠀⠀⠀⠹⢦⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣰⠇
⠀⠀⠀⢠⠀⠀⢡⡳⣄⠘⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣾⠇⢠⣴⡏⠀⣴
⠀⠀⠀⠈⢳⣄⠀⠹⡈⠣⡘⢎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣴⣿⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠫⡳⣄⠈⠢⡈⠢⣧⠈⢷⡀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⣻⣿⣿⣿⣿⣷⢋⣴⣡⠟
⠀⠀⠀⠀⠀⠀⠙⢌⠳⢄⠘⡄⠀⠀⣤⢫⠂⣽⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣶⣼⣿⣽⣿⣻⡿⢵⡿⣻⠏
⠀⠀⠀⠀⢠⣄⠀⠈⠳⢄⠑⠻⣄⠠⣌⠃⠱⡄⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣽⣿⣿⣷⣿⣿⣻⣻⣷⣿⡿⠁⢀⣤
⠀⠀⠀⠀⠀⠙⢅⠢⢄⡀⠱⡀⠈⠣⡹⣗⠀⣀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣩⣿⣵⣿⣻⠿⢋⣠⣶⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⠁⠚⢦⠀⠘⣬⡀⢀⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣻⣯⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠉⠢⡀⠑⢦⡈⠢⣀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⠋⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⡁⠒⠬⠶⣄⠈⠻⢞⡓⠢⠀⠀⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣋⣿⡿⢿⣿⣯⣶⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠂⠤⢄⠀⠠⣤⡍⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⡿⠟⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⡐⠒⠶⠷⠦⢀⣉⠓⠦⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣷⣾⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡲⠤⠠⢄⠙⠳⠖⠖⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣵⣿⣿⣛⣿⠿⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⣤⠄⠊⣹⠋⠀⡄⠀⠀⠂⠈⢁⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣾⣟⣛⣿⡭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠠⢤⡶⠋⠋⣙⣀⡴⠃⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣶⣝⣿⣿⣯⣥⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠐⣲⠋⣉⠁⠶⠴⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣹⡿⣿⡟⢿⡧⠼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠾⠄⣰⢀⡜⢿⣞⡟⠋⠓⢿⠞⣷⢻⡟⢿⠯⣽⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠘⠉⠀⠀⠀⠀⠙⠛⠒⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠶⠶⣶⣾⣿⡿⣷⡯⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠙⠙⠳⠤⠴⢾⡿⡙⠻⢷⣄⠀⠀⠀⠀⠀⠠⠀⠀⢈⡀⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠑⢄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⡀⠀⡀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠒⠺⠶⢦⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣀⡀⠀⠓⢈⠁⠀⡀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣷⣷⣌⣀⣒⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⡾⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⠿⠟⠋⠁⠈⣹⡿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠤⠒⠋⠉⠀⠀⢀⣀⣤⠴⠞⠋⠀⠀⠉⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⢰⠂⠀⡀⠀⠀⠀⢀⣢⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠐⠄⡊⠐⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠉⠀⠀⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣠⢞⠃⢒⠢⡀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠤⣄⠀⢰⣽⠰⣶⡆⣿⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠐⢦⣌⠙⢦⠛⢷⣶⡶⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⣈⣷⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣼⠏⠀⠀⠀⠀⢀⣠⡴⠶⠛⠉⢹⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣼⠇⠀⠀⢀⣤⠾⠛⠁⠀⠀⢀⣀⣏⣼⠁⠀⠀⣀⣠⣶⠁⠀⠀⠀⠀⠀⣸⡏
⣿⡀⠀⣴⠟⠁⠀⢀⣤⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⣿⠀⠀⠀⠀⠀⢠⡿
⠘⢷⣾⣧⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⣰⡿⠁⣀⣀⣠⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣼⣿⠶⠟⠛⠉⠉⠉⠀⠈⠉⠉⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣛⣋⣽⣇⣀⣀⣠⣤⣴
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠤⣄⣀⠈⠙⢿⡉⠉⠉⠉⠉⢀⣾
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠴⠋⣀⣀⠉⠙⠛⢾⡁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠉⠉⠁⣀⡤⠞⠀⠀⢘⣿⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⠀⠲⡞⠉⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠉⠙⢧⡀⠐⠋⠀⢀⡜⠀⠀⣀⡴⠋⣽⠏
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⣀⠴⣋⠠⠔⠊⠁⣠⡾⠃
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠄⠐⠒⠚⠓⠚⠋⠉⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣴⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⣿⠋⠉⠙⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠻⣧⠀⣴⡟⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣠⡿⠀⢿⣄⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⣿⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣶⠶⠶⠶⠟⠛⠋⠁⠀⢈⣻⣧⣤⡿⠀⠸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠶⠶⢧⣤⣀⣀⣤⣴⠞⠻⡿⠛⠋⠉⠁⠀⠀⣀⣤⣿⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⠛⠛⠛⠉⢁⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠊⡉⠀⡤⠀⠀⠀⣲⣦⣴⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⢁⠀⠈⢠⣬⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡶⠖⠄⣀⡐⠁⠀⠁⡖⢡⣎⢄⡰⠃⠀⠂⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣡⡶⣷⡞⠁⠉⠡⣮⢔⡫⣕⡠⣻⠟⠁⠀⠀⠈⠀⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠙⢋⣀⣤⣴⣿⣷⣾⣭⡵⠋⠀⠀⠀⠀⠀⠀⢄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡡⠶⠀⠳⠬⣿⣿⣯⣿⣿⣿⡏⠉⡀⠀⠀⠀⠀⠀⢐⣵⠛⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢿⣶⣶⠿⠶⠾⡶⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⢀⠐⡶⠛⠀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⠤⠞⠍⠿⠠⣷⠾⡄⡀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠞⠛⠛⠛⣦⡴⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⣼⣯⣵⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠔⠮⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢀⠠⣪⠾⠋⠉⠃
⠀⠀⠀⣀⢔⡪⢖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣦⣶⠋⠁
⢀⣴⣿⢟⠟⣨⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⠿⣿⠿⠟⠉
⠈⠻⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣟⠷⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⣯⣿⣿⣿⣿⣿⣶⣶⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣝⢷⣦
⠀⠀⠀⠀⢀⣴⢟⣿⣿⣿⣿⣿⢼⢿⣼⣽⣿⣿⡛⣿⣷⣤⡀⠀⠀⠀⢀⣴⣿⢿⢟⣯⡿⣿⣿⣼⢹⣼⣿⣿⣿⣿⣿⣻⣦⡀
⠀⠀⠀⠀⢨⢟⣿⣿⣿⣿⣿⠛⠟⣼⣏⢻⢱⣿⣷⡇⣏⢻⡇⠀⠀⠀⣼⡏⣽⢸⡾⣿⡟⣿⣹⣿⡞⡟⠹⣿⣿⣿⣿⣿⡛⡄
⠀⠀⠀⠀⣾⣾⣿⣿⡿⢿⢹⣰⣇⣟⢿⣿⣾⣿⣦⣽⣿⣾⡿⠀⠀⠀⢿⣷⣷⣿⣴⣽⣷⣿⡿⢻⣄⡷⣸⢙⠟⣿⣿⣿⣿⣗
⠀⠀⠀⣼⣿⣿⢿⣿⢳⣹⡜⣿⡿⣿⣼⣄⡏⣏⢹⣽⣟⠳⣆⠀⠀⠀⣔⠋⣿⣏⡇⣹⢹⣆⣟⣾⡿⣷⡯⣼⣰⢻⣿⡏⣿⣿⠷
⠀⠀⠀⢸⡿⣿⣿⢹⡈⠻⡟⡟⠷⢸⠙⢳⡟⣿⣬⢿⡀⠡⠈⠀⠀⠀⢠⠌⢀⡿⣧⣿⠟⣿⠉⣹⢰⠟⡟⣿⢻⢰⡏⣷⣟⣿⣇
⠀⠀⠀⣻⣿⣻⢹⡟⢷⠂⡇⣧⢹⠈⡇⠈⣇⣿⠙⡸⡇⠀⠃⠀⠀⠀⠘⠀⢼⢣⠋⣿⢰⠇⢠⠇⢸⢨⡇⠇⢸⡟⣿⢹⡏⣿⣞
⠀⠀⠀⢉⡿⣿⡿⣷⣸⡀⡆⢽⠘⣆⠘⡄⡏⠃⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠈⢩⠇⡏⣠⠏⣸⡃⣇⢸⣠⡿⣾⣿⢿⡉
⠀⠀⠀⠈⠀⣟⢻⣿⣿⣧⡇⣿⡘⠃⣿⣇⢻⡄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡏⢠⢿⠃⢙⣸⡇⣷⡼⣿⢷⣇⣇⢀⠁
⠀⠀⠀⠀⢠⠩⡋⢹⡇⡇⢱⢣⣿⠀⣻⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⣸⠀⢸⡇⠻⠃⠀⣾⡇⢹⠉⡄
⠀⠀⠀⠀⠈⠀⠃⢸⣷⡇⠰⣶⠿⢧⡽⠁⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢻⣴⠿⣧⣷⠀⢸⢹⠃⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢘⠁⠸⠈⠸⡀⠡⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⡈⠈⠸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡆⠀⡇⠀⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢰⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠃⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣧⡈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠇⢠⣇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡻⣵⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢺⡚⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠈⠙⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠀⢀⡠⠀⠒⢒⡨⠭⠭⠭⠭⠭⢅⣐⠒⠠⢀⠀⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡠⠁⣠⠆⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⢦⡐⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠰⠀⡇⠀⠀⠀⡃⠀⡇⠀⠀⠀⠀⠰⡄⠈⣄
⠀⠀⠀⠀⠀⠘⣡⣴⡏⠀⢠⠀⢠⠀⢸⠀⠘⠀⠀⠀⠀⠈⠙⠳⠦⠬⣐⣒⣒⣒⣒⣒⡨⠥⠶⠖⠋⠀⠀⠀⠀⠃⠀⡇⠀⡀⠀⡄⠀⢻⣦⣌⠂
⠀⠀⠀⣠⢠⣶⣿⣿⠀⢠⣏⠀⡜⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢃⠀⣻⡄⠀⢿⣿⣶⡄⣀
⠀⠀⠈⠁⡜⠉⣿⠃⠰⣿⡇⢀⠃⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡀⢸⣿⠆⠘⣿⠈⢧⠈⠁
⠀⠀⠀⢰⠄⠀⣧⠀⠀⢿⠇⣼⠀⢸⠁⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡐⠉⠃⠀⠀⠈⠁⢆⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⡆⠀⣧⢸⡟⠀⠀⣿⠀⠠⡆
⠀⠀⠀⠟⠀⢠⡏⠀⠀⣸⣸⡿⠀⡆⠀⢰⠇⢀⠀⠈⠒⠶⠆⠒⠁⠀⠀⣶⠀⠀⣷⠀⠀⠉⠒⠤⠖⠒⠁⠀⡀⠸⡆⠀⢱⠀⣿⣆⣇⠀⠀⢹⡄⠀⢻
⠀⠀⠰⠀⢀⣾⠃⠀⢠⡿⠃⢻⣰⠆⢠⡏⠀⡞⠀⣠⠀⠀⡀⠀⠀⡀⠀⡟⠀⠀⢻⠀⠀⠀⠀⢀⠀⠀⡄⠀⢳⠀⢻⡀⠸⣆⡟⠘⢿⡄⠀⠸⣷⡀⠀⡄
⠀⠀⣴⠀⣿⣿⠀⠀⣾⡇⠀⣼⠟⢀⣿⢀⡜⠀⢰⡏⠀⣾⠀⣴⢀⣧⣿⠁⠀⠀⠈⣿⣼⡀⣦⠈⣶⠀⢻⡄⠀⢧⠀⣷⡀⢻⣇⠀⢸⣷⠀⠀⣿⣿⠀⣦
⠀⢀⠇⠜⢸⡟⠀⣼⣿⠀⢰⣿⠰⠹⣇⣾⠁⠐⣿⡇⢰⣏⣸⣧⣾⣿⠏⠀⠀⠀⠀⠹⣿⣷⣽⣆⣿⡆⢸⣿⠄⠘⣧⣸⠇⢂⣿⡇⠀⣿⣧⠀⢻⡇⢣⠹
⠀⢨⠞⠀⢘⡇⠀⠘⢻⢀⣿⣿⠀⢰⡟⣧⣾⠀⣹⣤⠻⡏⣹⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⡉⡏⢻⠏⣰⣏⠀⣷⣼⢻⡄⠈⣿⣷⠀⡟⠁⠀⢸⡃⠀⠳⡄
⠀⠈⢰⠀⢸⣿⠀⢀⣸⡌⠈⣿⠀⣸⡇⢰⢿⢠⣿⡇⢰⡣⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠼⡆⢹⡿⡀⡿⡄⢸⡇⠸⡿⠁⢻⣇⠀⠀⣾⡇⠀⡆⠁
⠀⠀⠀⢀⣿⡿⠀⢠⣿⡇⠀⢸⠀⢿⡇⡈⢸⡇⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢼⡇⠁⢸⡟⢠⡇⠀⢸⣿⡆⠀⢿⣷⡄
⠀⠀⠀⢸⠟⠇⠀⢨⢿⡇⠀⢻⣿⠈⣷⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣷⠨⣾⠁⣿⡇⠀⣸⣿⡀⠀⢸⠻⠇
⠀⠀⠀⢸⠀⡆⠀⠸⠈⢷⠀⢸⣿⠀⣿⡞⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢻⣯⠀⣿⠇⠀⡟⠁⡇⠀⢰⠀⡀
⠀⠀⠀⣸⠀⡇⢀⡇⢀⣿⡀⢸⠘⡄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣷⢸⠃⡇⢀⣷⠀⢸⠀⢸⠀⣇
⠀⠀⠘⠘⠀⠃⣼⠀⢀⣿⡇⢸⠸⡇⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢸⠃⡇⢸⣿⠀⢸⣇⠸⠀⠃⠇
⠀⠀⠀⠀⡆⡠⣿⡄⢨⡁⠁⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠉⢸⡀⢸⡿⢤⢰
⠀⠀⠀⠀⠃⠁⢻⡇⢸⣷⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡟⢀⣾⡇⢸⠟⠈
⠀⠀⠀⠀⠸⢠⡀⡃⡟⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢳⢸⢀⡀⠃
⠀⠀⠀⠀⠀⡆⢃⢳⣇⠈⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡙⠁⡸⡆⡜⢡
⠀⠀⠀⠀⠀⠀⠘⡄⢸⡆⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢹⠀⡿⡄⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⣿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠣⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠱⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡄⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢠⣿⠁
⠀⠀⠀⠀⠀⠀⡀⠹⣿⣄⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⣠⣿⠇⢀
⠀⠀⠀⠀⠀⠀⢻⣄⠹⣿⣦⡻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢟⣴⣿⠏⣠⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡘⢿⣷⡽⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢯⣾⡿⢃⣴⡿⠁
⠀⠀⠀⠀⠀⢠⡀⠈⢻⣿⣮⣻⣿⣞⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣳⣿⣟⣵⣿⡟⠁⢀⡄
⠀⠀⠀⠀⠀⠀⢻⣦⣀⠙⢿⣿⣿⣿⣷⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢯⣾⣿⣿⣿⡿⠋⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣹⣿⣿⣿⣿⣮⡻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢟⣵⣿⣿⣿⣿⣏⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣾⣝⢿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣤⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣮⣻⢿⣷⡄⠀⠀⢠⣾⡿⣟⣵⣾⣿⣿⣿⣿⣿⣿⣛⣉⣀⣤⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⡄⢠⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡞⡇⢸⢳⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢛⣉⣭⣿⣿⣿⡿⣿⣿⡿⠃⠘⢼⣿⣿⢿⣿⣿⣿⣭⣉⡛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁⣰⣿⡿⠃⠀⠀⠘⢿⣿⣆⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠘⡝⢦⠹⡀
⠀⣤⣀⠀⠀⠀⠙⣄⢙⢿⣄
⠀⠈⠪⣙⠒⠦⢄⡈⠢⡑⠈⠛⠦⣤⣀⡀
⠀⠀⠲⣚⠳⠦⣔⡀⠍⡉⠓⠤⢀⡀⠉⠉⠉⠉⢝⠉⠉⣉⠟⠛⠒⠦⣤⣤
⠀⠀⠀⣬⣕⣤⣀⡈⠙⠒⠢⠤⠀⣀⡁⣖⠁⠀⠈⣗⠒⠈⠂⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠈⠑⠦⢌⣉⠙⠒⠒⠤⠤⠤⠤⠀⠹⣁⠀⠈⠋⡉⠁⠀⠀⠀⡴⠂⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠐⠮⣉⡙⠛⠓⠒⠒⠒⠒⠒⠒⠊⠉⢹⡭⠀⠀⠀⢀⣰⣉⠓⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠲⣛⡛⠛⠒⠒⠒⠒⠒⠈⠉⣀⠧⠀⣤⢀⡀⢣⠤⠌⠁⠀⠀⢱⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡶⠶⠚⠒⠒⠈⠉⠀⣀⠄⠊⢉⡠⠌⡖⠂⠀⢄⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠿⠭⠝⣟⣁⣠⠖⠁⡠⠔⢚⡥⠀⠀⢤⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣤⠊⢀⡔⠉⡲⢴⠂⠀⢚⡠⠀⡀⠈⠙⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⢴⠋⣀⡔⡑⡨⠂⡦⠈⠡⠔⡣⣠⡀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠒⠹⠬⠚⣧⢾⢀⡆⢰⣀⡇⢰⢸⡅⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⠋⠛⠑⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠙⣆
⠀⠀⠀⠀⣾⡉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⡀⢰⡇⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⣾⠋⠙⠛⠒⠦⢬⣙⡳⠦⣤⣀⡀⠀⠀⢀⡴⠃⠀⠙⢯⣁⠀⠀⢀⣀⣈⡷⠶⠖⠛⠋⠉⠉⠀⠀⠀⢀⠀⠄⠍⡻⣦
⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠉⠛⠒⠭⢭⣻⠗⢺⡀⠀⠑⢄⠀⠙⠻⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠐⣀⠀⢹⡧⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠉⣻⡶⠶⠴⠶⠶⠶⠶⠶⣾⠀⣾⠁⢄⠀⠀⠁⠠⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠀⢈⡷⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⣀⣀⣤⣤⡇⠀⠸⡄⠀⠁⠀⡀⠀⠀⠁⠊⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠋⠁⠀⠀⢀⠔⠁⠀⡀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠠⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡒⠀⢶⠋⠁⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⢤⣌⣙⠓⠾⣧⣤⣄⣀⣀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣾⣿⣇⡬⣿⡍⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠤⢿⣷⣫⣇⡀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣶⣶⣞⡋⢀⣀⣀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣯⠻⣽⠋⠀⠈⣷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⢧⣈⣷⣤⠖⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠸⡇⠀⢹⣄⣤⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠒⠋⢷⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣰⠇⠀⠀⠘⢧⡀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿⣟⣻⣝⣻⣿⣷⣮⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣵⣾⣿⣟⣻⣟⣻⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣻⣿⣿⣿⠿⣝⣿⣿⣿⡟⢬⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⡥⢻⣿⣿⣿⣫⢿⣿⣿⣿⣟⣻⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣿⣿⣿⣏⡽⢿⣿⣧⠻⣿⣿⣿⣿⣿⣿⣯⣿⢟⣼⣿⡿⢯⣽⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣾⣿⣿⣿⣟⣽⣿⣿⣿⣷⣬⣹⣋⠛⢪⣯⣿⣿⣿⣿⣽⡅⡻⣙⣏⣧⣾⣿⣿⣿⣭⣻⣿⣿⣿⣷⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⡿⢿⣟⣴⣿⠻⣿⣟⣿⠟⣿⣾⣿⣿⣿⣿⣿⣿⣷⣽⠻⣿⣻⣿⡟⣿⣦⣿⡿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣻⣷⣿⣏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⣿⣾⣟⣻⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠐⠀⠀⠀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠐
⠀⠀⠄⠀⠀⠀⠂⠀⡄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⠀⠂⠀⠀⠀⡄
⠀⠀⠁⠀⡄⢔⠐⡇⢰⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢲⠀⡕⢀⠆⡄
⠀⠀⠀⣀⢀⠨⢧⡸⣎⣧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣏⡾⣠⠯⢀⢀⡄
⠀⠀⠀⢈⠒⢢⣌⢳⣽⡙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣹⣵⢃⣤⠗⢊
⠀⠀⠀⠢⠳⢠⠙⢷⣽⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣵⠞⢡⠴⠣⠒
⠀⠀⠀⠁⢀⡎⠐⢤⣇⠱⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣷⠔⠈⣆⠀⠁
⠀⠠⠀⠧⡀⣤⠓⢶⡿⣆⢏⠀⠀⡘⢦⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠞⠀⠐⢉⢇⡾⣷⠖⢣⡄⡠⠇⠠
⠀⠀⠀⠠⠬⣌⣙⠲⣿⡌⢃⠁⠆⡘⠢⣛⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡫⠚⡀⠖⢁⠋⣼⡷⢚⣉⣬⠤
⠀⠀⠀⠀⠀⡌⢉⣽⣟⣿⠦⠑⠀⣍⡓⢾⣿⡄⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⣸⣿⠖⣋⡅⠐⠡⢾⣟⣿⣍⢁
⠀⠀⠀⠀⠁⡠⠤⠤⠼⠿⣷⣌⠀⠄⢭⣻⣿⡦⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠠⣾⣿⣫⡅⠀⢪⣴⡿⠷⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢒⡛⢛⣛⣷⣦⡀⠡⠯⠽⢿⡦⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠠⣾⠿⠭⠯⠁⣠⣶⣟⣛⠛⣓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠙⠛⣻⣦⡄⠐⢛⢿⣦⡝⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⣥⣾⠟⠓⠂⣤⣾⡛⠛⠁⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠴⠿⡿⢧⣤⡜⠻⠦⣭⠢⢄⣀⠀⠀⠀⠀⠀⢀⣀⠤⢪⡵⠞⠛⣤⣤⠿⡿⠷⠭⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠙⠛⠛⠛⠿⠓⠛⠶⠒⠀⠐⠲⠖⠓⠻⠟⠛⠛⠛⠉⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡿⠟⠩⠚⡁
⠀⠀⠀⠲⣶⣶⣛⣛⣛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣫⣾⠵⠓⠉
⠀⠀⠀⠀⠘⠟⣝⠿⣿⣿⣾⢿⣧⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣴⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠘⣿⣬⣉⣭⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢣⣍⣅⣂⢻⡿⣿⢿⣿⣿⣟⣿⣿⣿⢿⣽⣷⢟⡉⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⢢⡂⣛⢾⣿⣿⣟⣾⣿⣿⣣⠿⠻⣽⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠐⠒⠂⢙⣿⠿⡿⡿⣿⣻⠬⠚⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠂⠀⠈⠀⠉⠉⠉⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣶⡆⢀⣀⣀⡈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣯⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠸⢯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⢛⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠟⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠾⠟⣿⣿⣿⣷⠀⠀⢰⣿⡿⣿⣿⠿⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢾⣾⣿⣿⣿⡁⠀⠀⠀⢹⣿⣿⣾⣷⠦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⠁⢠⠂⠀⢢⠀⠙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠈⠀⠀⠀⠀⠀⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠒⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⣴⣖⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈⠉⠑⠢⠤⡤⠤⠤⢄⠠⠀⠀⠀⠒⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠤⠤⠤⣄⠀⢤⠄⠈⠐⠤⣀⣀⡀⠀⠀⠀⠀⠉⠓⠲⢶⡒⠀⠉⠉⠁⠀⠐⠂⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁⠢⡈⠻⣌⠉⠉⠉⠙⠒⠦⣄⡀⠈⠀⠀⡀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⠀⠀⢰⡀⠀⠀⠈⠒⢄⠀⠀⠠⡀⠈⠠⠈⠫⡍⠉⠀⠂⠄⡀⠉⠢⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⣆⠀⠀⠀⠈⢦⠀⠀⠀⠈⠢⠀⠈⠢⡤⠤⠀⠂⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢄⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠈⠆⠀⠘⣦⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡄⠘⢆⠘⠂⠀⠙⢆⠀⠀⠀⠀⠘⣄⠀⢩⠻⠦⢀⠀⠀⠈⠲⠤⣀⠀⠀⠀⠀⠀⠀⠙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠐⢤⣤⠀⠀⠀⠀⠣⠀⠀⠀⠀⠃⠙⢬⠀⠀⠀⠈⢰⡦⠄⡀⠢⠉⠁⠒⠀⠠⠤⠤⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠀⠈⠇⠐⡀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠳⣦⣄⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢣⡀⠀⠠⡀⠀⠀⠡⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢘⣴⠀⠀⡀⠀⠀⠀⢱⡀⠀⠘⠏⠀⠠⢌⣢⣀⠀⠐
⠀⠀⡀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠈⠈⢀⡘⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢁⠀⠈⠉
⠀⡎⠀⠀⠀⠁⠀⠀⠀⠑⡈⢀⠔⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈⡓⠢⠄⠀⠀⠀⠀⠆
⠀⠣⢄⣠⠞⠀⠀⢸⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠧⡀⠀⠀⡇⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢄⠀⠀⠀⠧⠀⠀⠀⠀⠘⠢⡀⡷⠀⠀⠰
⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠈⠦⠀⠈⠀⠱⠀⠀⠆⠀⠀⠈⠀⠀⠑⠒
⠀⠑⠰⠀⠀⠁⢄⡀⠈⢀⠀⠀⠠⠀⠀⠀⠆⠈⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⡀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⢨⡀⠀⠛⠋⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠠⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠐⠊⠁⠈⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⣧⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣿⡆⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣄⣀⠀⠀⠀⠀⢹⣏⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣣⡀⠀⠀⠀⡠⠀⠀⣀⣀⣠⣬⠿⣟⠻⠶⣶⣤⣴⣿⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⣄⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠉⠙⠳⠦⣤⣀⣀⣀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠤⠁⢀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠉⠉⠙⠻⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠐⢒⠀⠃⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠒⡲⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋
⠀⠲⣤⣈⠻⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⢋⣠⡴⠂
⠀⠀⡈⠻⣿⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣾⡿⠋⡀
⠀⠀⠙⠶⣮⣝⣻⣿⣿⣿⣿⣿⣿⡻⠿⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⣻⣿⣿⣿⣿⣿⣿⣛⣭⡶⠞⠁
⠀⠀⠀⠀⢈⡙⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣉
⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣶⠶⠒⢒⣀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⡰⠟⠀⣀⣐⣒⠲⢶⣾⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣟⣋⣥⡤⠖⢀⠈⠀⠀⠀⠀⠀⠈⢀⠐⠦⣤⣍⣛⣿⣿⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⣿⣿⣿⣶⣟⣡⠔⠀⠀⠀⠀⠀⠐⢤⣙⣷⣾⣿⣿⣿⣿⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠉⠉
⠀⠀⠀⠀⠀⣤⠀⣀⢠⡤⣤⠀⣄⡀⢠⠀⣀⡤⣀⠀⡠⣀⠀⠀⠀⣤⡤⣀⠀⣤⡤⣄⢀⠤⡀⢠⡄⢀⣠⢄⡀⢠⡀⠀⡄
⠀⠀⠀⠀⠀⢹⣦⢻⡼⠀⣿⠀⡏⠳⣬⠰⣿⠀⣬⢌⠳⣦⠀⠀⠀⢸⠀⢸⡇⢸⡗⢀⢘⠷⣆⢸⡇⢾⡇⢠⡥⢸⠹⣦⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠉⠀⠉⠀⠉⠉⠉⠈⠉⠉⠀⠀⠀⠉⠁⠉⠀⠉⠁⠉⠈⠙⠁⠈⠁⠈⠙⠉⠁⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢠⣶⣆
⠀⠀⠀⠀⠀⠀⣾⠀⢿⡀⠀⠀⡾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⡀⠀⣇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿
⠀⠀⠀⠀⠀⣠⡈⢷⠀⠈⢻⣄⢻⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣼⠃⢠⡟⣀
⠀⠀⠀⠀⠀⣿⠙⣾⣧⡀⠀⠙⢿⣷⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⡾⠁⢀⣾⡾⢻
⠀⠀⠀⠀⠀⢻⡆⠀⠙⢿⣄⠀⠀⠙⢷⡄⠀⣸⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣼⠏⠀⢀⡾⠋⢀⡿
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠈⠻⢦⣀⠀⠉⠳⣿⣆⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⡴⢿⠁⢀⡴⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⢸⡷⠾⣿⣦⣀⠀⠀⠉⠓⠦⣄⣿⡈⠳⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠔⠉⣰⣯⠔⠁⠀⢀⣴⡿⣶
⠀⠀⠀⠀⠀⠈⢻⣄⠀⠉⠙⠓⠶⢤⣄⣀⡀⣾⠻⢆⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⡀⠐⠈⣼⣃⣠⠴⠚⠋⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠈⠉⠙⢧⡀⠙⠲⠄⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡧⠶⠶⠶⣤⣤⡞⠉⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡞⠛⠛⠓⠒⠒⠒⠒⠒⠛⠳⢦⣤⣀⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣤⣀⣀⠈⠻⣶⠚⠛⠛⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣀⣀⣀⣠⣤⡤⠤⢠⣼⢧⣤⠀⠀⠀⣿⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠀⠈⠑⡄⠘⣧⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡿⡤⠤⠤⠤⠤⠤⠺⣧⣀⣀⣀⠀⠀⠘⢷⣄⣀⣠⡟⠀⠀⠀⠀⠒⠾⣿⣿⠀⠀⢸⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⠤⠄⢾⡉⠀⠀⠀⠀⠀⠙⢿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠉⠁⠀⠀⢀⣨⠷⣄⣀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⢶⡖⠋⠁⠀⣀⠌⠋⢙⠛⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣖⠋⢁⣠⠖⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⡋⠀⠀⠀⠀⠀⢻⡖⠀⠀⣀⣤⣞⣀⣤⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⣏⡡⠀⠀⢀⡔⠀⠀⠀⠀⣾⢷⠾⡏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⡿⠛⡁⠄⢀⡴⠃⢀⠌⠀⡌⢰⣿⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⠟⠉⠄⠀⢀⡴⠋⠀⡰⠃⢀⠞⢠⡏⣿⠘⡇⢿⡀⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣞⡁⢀⣀⣠⡶⠋⠀⣠⠞⠀⣠⠏⢀⡿⠀⠹⢦⡿⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⣼⠁⣀⣴⠃⢀⣴⡏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⢿⠶⠟⠛⠷⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⠛⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⠻⠙⠇⠙

⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼
⠀⠀⢿⣆⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣼⡏
⠀⠠⣌⢿⣧⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⣾⠏⣠
⠀⢤⣝⣷⣝⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⣿⣿⣣⠄
⠀⠀⢽⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡥
⠀⠐⢶⣿⣿⣿⣿⣿⣿⡿⣭⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣽⣿⣿⣿⣿⣿⣿⣥⠶
⠀⠠⣤⣿⣿⣿⣿⣿⣿⡯⣿⣿⣏⣀⣹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡋⣘⣿⣿⡯⢽⣿⣿⣿⣿⣿⣯⣤⠄
⠀⠀⠴⠾⠛⣋⣭⣵⣶⡶⡾⢻⡿⣳⢚⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⣿⡿⣏⣶⣾⣿⠻⡶⣶⣿⣿⣿⣿⠿⠦⠤
⠀⠀⠀⠀⢚⣿⡿⢟⣵⣾⣷⡾⣸⣿⣿⣿⣿⣻⣿⡀⠀⠀⠀⠀⢸⣿⡿⣷⣾⣿⣿⣻⣶⣿⡿⣽⡿⣿⣟⠃
⠀⠀⠀⠀⠈⠉⢚⣫⠾⣛⣡⣾⣿⣽⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⣾⠿⣼⣾⣿⢿⣵⣿⣿⣿⣿⣿⣟⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢋⢿⡿⡿⣻⣯⠞⢫⣿⣿⣿⣷⡄⠀⠀⣰⣿⣤⣿⣮⣛⣾⣿⣿⣿⣿⢯⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠧⠵⢜⢮⢞⣿⣿⣷⣼⠀⢘⣵⣿⣏⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⢾⠹⠄⢸⣿⣿⠿⠟⠻⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⡤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣚⣉⣝⣉⣉⣡⣶⣴⣿⣶⣀⣀⡤⠛⣷⣖⣀⣀⠀⢐⣀⣴⣟⣀⣀⠤⠀⠀⠀⠀⠲⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡭⣾⣟⣭⣯⣉⣻⣿⡝⢻⣿⣟⣛⣩⣽⣭⣿⣿⣿⣿⣿⣿⠿⠏⡁⠀⠀⠂⠀⠀⠀⣀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣵⣾⣿⠟⠻⣷⣦⣍⠻⣿⣯⣥⣽⣿⣿⡟⠻⣿⣟⠢⠤⣀⠙⠿⣷⣶⣤⣶⣾⣷⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢠⢎⣾⣿⣿⣽⣿⡳⠌⣻⣶⣅⣪⡴⢿⣿⣿⢿⣌⣱⡈⠙⢿⣶⣶⣶⣶⣿⣿⡛⠛⠉⠠⠤⢤⡤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⢮⡙⢦⣜⣿⣭⣭⣟⠀⠻⣿⠣⡙⢿⡿⣟⢿⣿⣿⣮⣉⡒⠠⠀⠀⠀⠀⠀⠀⣀⣬⡿⠋
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠛⣷⣷⣤⣉⣿⣿⠿⠿⣿⣶⣿⣷⣌⠃⠙⢮⡓⢬⡛⢿⡻⠿⣿⣶⣶⣶⣿⡿⠿⠛⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⣽⣆⢻⣿⣿⡿⣷⣁⠈⢿⣿⣿⣿⣿⣶⣾⡿⠦⣍⠀⠈⠒⢄⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢳⣽⣯⣿⣿⣿⣷⡈⣿⣿⢿⣇⠙⢿⣿⣿⣿⣿⣷⣤⣉⠒⠤⣤⠉⠑⢤⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⣿⣿⣿⣷⣾⠻⣦⣀⡑⢬⣻⣿⣮⡙⠙⠿⣟⠷⠶⣶⣤⡶⠿⠟⠀⠀⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⡿⣜⣿⡏⢇⠐⠹⣿⣿⣶⣄⠈⢻⡿⢶⣶⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣟⢻⣿⣍⣙⡏⠻⠏⢷⣙⣸⣿⣮⡓⠀⢼⣷⣷⣌⠢⡀⣽⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣷⡬⢿⡉⠁⠀⠀⠀⠙⠛⠉⠙⠿⣶⡾⠇⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⡐⢉⠇⠀⠀⠀⠀⡼⠀⣸⢇⡀
⠀⠀⠀⢹⣿⣿⢿⣿⣿⣿⣈⣛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⡿⠀⠀⠀⠀⣰⠁⢠⣟⡜⡇
⠀⠀⠀⠘⣿⣿⣿⡛⣿⣯⠛⠛⠋⠀⠀⠀⠀⣀⠠⡤⢢⠀⠀⠀⠀⠠⣀⣴⠆⠀⢀⡴⣣⣿⣻⡇⠀⠀⠀⣼⣿⣴⣿⠋⣸⠇
⠀⠀⠀⠀⢹⣿⣿⣿⣶⣾⡇⠀⠀⠀⣀⣄⣴⢾⣿⡿⠁⢀⣴⣿⣿⣛⣽⡟⠀⣠⢾⣿⣔⣿⢿⡁⠀⢀⣼⣿⣿⡿⠁⣰⣟⣆
⠀⠀⠀⠀⠀⣿⢿⡟⣧⡁⠀⠀⢀⣾⣿⣿⣿⣫⣽⠃⣰⣻⣿⣿⣿⢿⣿⠅⣸⣿⣿⣿⣿⣿⡋⠀⢀⣾⣿⠟⣽⣧⣾⠟⣡⡏
⠀⠀⠀⠀⠀⠈⠈⢷⡈⣧⠀⠀⣿⣟⢿⣿⡿⣖⡏⠀⣿⣟⣿⣿⣿⡿⠃⠀⣿⣿⣿⣿⣿⣾⡞⠀⣾⣿⣷⢊⢽⣿⣁⡴⣋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⢹⡟⣸⣿⡻⣟⠁⠀⠹⣧⣿⣿⠶⣶⠀⠀⢻⣿⣿⣿⣿⣵⡋⠀⣿⣿⣿⠗⢉⣿⣋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣿⡦⠄⠀⠀⢠⣟⣿⣿⣗⠂⠀⠀⢀⣿⣿⣿⣟⣽⠄⠀⢹⣿⣿⣏⣠⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣿⣯⣝⠀⠀⠀⢱⣿⣿⣕⠀⠀⠀⢰⣿⣿⡿⢿⣿⠶⠀⠠⢮⣿⣿⣦⡾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡳⠄⠀⠀⠀⠈⠙⠙⠊⠀⠀⠀⠀⠻⣧⠻⠆⠉⠀⠀⠀⠘⣿⣶⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡛⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣽
⠀⠀⠀⠀⠀⠀⣆⠀⢳⡘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣰⠃⢀⡆
⠀⠀⠀⠀⠀⠀⢻⡳⣄⣧⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⣇⡴⣻⠃
⠀⠀⠀⠀⠀⠀⢀⠙⣌⠻⢧⡀⢈⢦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢎⠀⣠⠿⢋⡼⢁
⠀⠀⠀⠀⠀⠀⠸⣷⢼⣆⠀⠙⠦⣕⢭⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⢕⡥⠞⠁⢀⣾⢴⡿
⠀⠀⠀⠀⠀⠀⠀⡈⠢⣍⠳⢦⣕⠮⣙⢿⣧⡙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣡⣿⢟⡩⢖⣥⠶⢋⡥⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣗⠮⣧⣀⠨⣙⠺⢿⣧⡙⠳⣮⣝⢦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣭⡶⠛⣡⣿⠿⢚⡩⢀⣤⡯⢖⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⡉⠙⠒⠯⢷⣿⣙⠺⣿⣿⡄⢳⠀⠀⠀⠀⠀⢰⠃⣼⣷⡿⠚⣹⣷⠯⠗⠚⠉⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣗⠶⠬⠵⣾⣯⣑⠻⣿⣇⣿⠀⠀⠀⠀⠀⢸⣇⣿⡿⢓⣩⣿⡶⠯⠴⢖⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⡒⠲⠶⠿⢦⡬⠿⣿⣜⣆⠀⠀⠀⢀⣞⣼⡿⠯⣤⠾⠷⠖⠒⣺⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣓⣒⣛⣛⣿⣋⡽⢿⣷⡄⣴⣿⠿⣭⣻⣟⣛⣓⣒⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠭⢥⠤⠚⣫⠾⣵⠝⠀⠘⢵⡽⢮⡓⠢⢤⠭⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠐⠈⠀⠀⠀⠀⠀⠈⠐⠙⠂⠈⠁

⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠸⡏⠻⣦⣿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣿⣄⢻⠙⢷⣄⠀⠀⠀⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⢧⡀⠈⠳⣄⠀⡇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⡍⠳⢄⡉⠢⣄⠈⠳⣿⡀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣀⠉⠒⢬⣇⠀⠀⠙⢦⡙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠮⠉⠒⠢⢷⡤⠀⠀⠀⣷⡑⢌⠳⣄⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠲⠦⠤⠤⠷⢤⠀⠀⠸⣜⣣⠀⠈⢧⠀⠀⠀⣀⡞⠁⢀⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡒⠒⠒⠒⣚⣇⡀⠀⠙⢷⡅⠀⠈⠻⠛⠋⠁⠀⠀⠸⠟⠀⢀⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣉⣉⡠⠔⢣⡤⠀⠀⠙⠲⠦⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠤⠂⢁⡤⢚⠖⡠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢾⣁⡤⠳⠚⠁⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⢷⠶⠶⣠⣄⠀⠀⠀⣀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣋⡴⠋⠀⢼⣵⣧⣾⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣴⡾⠋⢀⠔⢁⠜⣡⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⢞⡿⠊⠀⡰⠋⣠⠊⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠀⣠⠶⠋⣁⡎⢀⣠⣞⢀⡴⠁⣴⣧⠞
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠙⠉⠁⢉⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠦⢤⡀
⠀⠀⠀⠀⠀⠀⡰⢛⣴⠶⢶⣦⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣴⣶⠶⣶⡜⢢⡀
⠀⠀⠀⠀⢀⡞⡰⣛⡿⢿⠷⣿⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣿⡿⢿⠿⣟⢢⠙⣄
⠀⠀⠀⢀⡞⣼⣿⣿⣿⣿⡯⢾⡇⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⢼⣿⣿⣿⢿⣷⡘⡄
⠀⠀⠀⡼⣀⣵⣿⢴⣧⣾⣷⣯⢷⠙⢷⡀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡰⠟⣸⢯⣷⣿⣾⣷⣼⣧⣁⢹⡀
⠀⠀⢀⣷⣿⠿⢛⠘⣱⡉⡝⣻⣹⡳⣄⣘⠓⠒⣋⣨⣷⠀⠀⢼⣅⣘⡓⠒⢋⣠⣴⣻⣹⣛⡉⣦⠛⡘⠿⢿⣿⡇
⠀⠀⢸⡙⢱⢀⣾⣷⣿⣿⣿⢿⣿⣟⣿⢿⢟⢻⢿⡏⠁⠀⠀⠀⠉⣿⢿⢛⢿⢿⣻⣽⡿⢿⣿⣿⣷⣿⡄⣤⠙⣿
⠀⠀⢸⣹⣿⣿⡿⡟⢋⣄⣴⣈⢙⣏⣿⣽⣽⣴⠿⠃⠀⠀⠀⠀⠀⢻⢶⣬⣽⣹⣙⡛⣏⣰⣄⣌⠛⠿⣿⣻⣿⣿
⠀⠀⢸⡼⠏⡏⢠⣧⣾⣿⠿⡿⢻⠏⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⣟⢻⡿⣿⣿⣦⣇⠉⡙⠟⡟
⠀⠀⠀⣟⣤⣷⣾⣿⠿⢱⣴⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣮⠻⣿⣿⣾⣧⢼⠇
⠀⠀⠀⠹⡽⣿⡟⠉⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⢩⢿⡿⡞
⠀⠀⠀⠀⢳⡁⣿⣸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣸⠀⡸⠁
⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣍⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣿⣿⣿⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡽⡍⠲⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠊⡝⣞⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⡄⠱⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡼⢁⠞⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡘⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡎⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡿⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣿⣛⣛⣻⣿⣻

⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠃
⠀⠠⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠱
⠈⠀⣄⠀⠠⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⡔⠈
⠴⡀⠀⠀⠀⢀⠀⠀⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠁⠀⡀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⢑⠀⠑⢄⢈⢦⡐⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⣠⢎⢀⠔⠁⢀
⠉⠢⠀⠀⠀⠑⢄⠌⢿⡄⠙⣆⠈⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⡞⠁⣼⠏⢄⠔⠁⠀⠀⠠⠊
⠠⢄⠀⢀⠘⢄⠀⠳⡄⠙⠄⠈⡁⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡉⠀⠜⠁⡴⠃⢀⠜⢀⠀⢀⠤
⠀⠀⠑⢄⠙⠦⣝⣦⠙⢂⡀⣥⠛⢀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢀⠘⢣⡅⣀⠚⢡⣞⡥⠞⢁⠐⢁
⠀⠰⠲⠄⠙⠲⠬⢭⠛⠂⢁⡌⠑⠼⣜⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣼⠔⠉⣠⠁⠚⢫⠭⠴⠚⠁⠴⠢
⠀⠀⠀⠀⣈⠐⠲⠤⠤⠲⠀⠯⠉⠙⣳⡈⠙⡳⢄⠀⠀⠀⠀⠀⢀⠴⡛⠉⣰⡛⠉⠩⠇⠰⠢⠤⠤⠒⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢲⣂⣒⡣⠤⢀⡏⣐⠟⢢⣄⠀⢀⣤⠚⠖⡈⣇⠠⠤⣓⣂⣲⠚⠒⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡩⠥⠚⣠⠞⣡⢔⣊⠈⠋⠤⠀⠠⠜⠏⢈⣒⢤⡙⢦⠘⠢⠬⡉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠡⠠⡄⠁⠘⠀⠘⠀⠀⠀⠀⠀⠘⠀⢘⠀⢁⡤⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢲⡀⠀⡴⠊⠁⠈⣧⠀⢀⣠⠔⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡟⣠⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⣧⠎⠀⠀⠀⠀⣿⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠐⠋⠸⠵⣶⠄⠀⠀⠀⡼⠀⠀⠀⢸⠃⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠛⠒⠺⢅⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⢠⣄⡀⠀⢠⠞⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠳⣍⢳⠏⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⣠⠤⠒⠚⠋⠙⢲⣀⣠⠤⠶⠓⠒⠦⡄
⠀⠀⠀⢠⢞⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⣿⡤⠤⠤⢄
⠀⠀⠀⠙⠉⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⣀⣠⠴⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠁⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠣⢤⣠⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⣀⡀⠀⠀⣀⣀⣠⠴⠚⠙⢆⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣽⣿⣿⡿⣶⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⢿⡛⣧
⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⡿⢿⣿⣿⣿⣞⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣝⡷⡝⢆⠈⢸⣇
⠀⠀⠀⠸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧⣧⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣵⣜⣦⣴⣾⣿⡄
⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣄⣀⠀⠀⢠⣯⣿⣿⣿⣿⣿⣿⣿⡿⢽⠯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣿⡭⣶⢿⣿⣿⣶⣿⣿⣿⡟⠛⠙⣿⣿⣿⣧⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⢿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣂⣤⣂⡘
⠀⠀⠀⠀⠐⠒⠶⠶⣦⣤⣄⣠⣤⣤⡸⠻⢿⣿⢿⡋⣛⣋⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⢼⣻⡗⢻⣾⡻⡗⠌⠪⠻⣿⣿⣿⡿⠿⠏⠈⠈⠻⠚⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣧⠴⣿⣟⢛⡷⠃⠀⢹⡏⠫⣁⣤⡖⠒⠀⠈⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠾⠷⠇⣀⣼⣧⣶⡼⠋⠁

⢰⡦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢴
⠀⢷⣿⣷⣶⣮⣤⣤⣤⣄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣠⣤⣤⣴⣵⣶⣾⣿⡾
⠘⠷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⡾
⠀⠠⣬⣽⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣥⠄
⠀⠢⡄⠿⠿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⠿⠿⢠⠔
⠀⠀⢀⣠⣤⣤⣶⣶⠾⠿⠿⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠒⢒⠿⠿⠷⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠉⠙⠫⠭⠤⠀⣀⣉⣡⠤⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣄⣉⣀⡐⠤⠭⠉⠋⠉
⠀⠀⠀⠀⠀⠒⠚⠛⠛⠋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢰⠟⠉⠙⠒⠒⠚⢁⡼⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⣮⡟⠀⠀⠀⠀⢀⡤⠒⠋⢁⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⣰⡏
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣩⡽⠃⠀⠀⢀⠔⠊⣉⣠⣤⣴⣿⠟⠁⠀⠀⢠⠖⠁⠀⠀⠀⠃⠈⡇⠀⠀⠀⢀⡤⠒⠊⡧⢀⡼⠛⣹⠇
⠀⠀⠀⢀⡿⠀⠀⢀⣠⡴⠞⠁⠀⠀⣴⠋⠀⢠⣒⣦⣭⠟⠁⠀⠀⠀⣴⠃⢠⠀⣤⠘⢷⡤⠞⠁⠀⠀⢰⠏⠀⢰⡫⠾⠠⠖⣿⠃
⠀⠀⠀⠼⠧⠶⠚⠋⠁⠀⠀⠀⠀⣼⡃⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠘⠋⠚⠛⠶⠚⠒⠋⠀⠀⠀⠀⠀⢸⡶⢇⣚⠓⣲⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠁⠈⠉⣿⣟⡀⠀⢀⣀⠤⠔⠒⠊⠉⠈⢁⣠⣤⠞
⠀⠀⠀⠀⠀⡠⠉⢀⠀⠄⣀⠿⣯⠀⢰⠀⠀⠀⠀⠀⠈⠀⣠⡴⣁⠀⠀⣤⠤⠊⠀⠀⠀⢈⣫⡀⠀⠘⡄⠀⠀⠀⠀⠀⠠⢤⠄⠛
⠀⠀⠠⡀⠀⠀⢰⢛⡤⣓⣬⡟⠀⠀⠸⣄⠀⠀⠀⠀⣀⠐⠶⣞⡁⠀⠀⠹⣄⣄⠀⢠⠀⡰⣝⡿⠀⠀⠘⡄⢠⣤⣬⣑⡖⠊
⠀⠀⠀⠈⠲⢤⣈⡾⠷⠛⠋⠀⠀⠀⠀⠙⠿⣔⣦⡲⣮⠽⠯⠁⠀⠀⠀⠀⠈⠉⠛⠚⠳⣿⣾⡟⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄
⠀⠀⠀⠀⡔⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠀⣨⡇⠀⠀⡠⠄⠀⠀⠒⠂⠈⠀⢠⢾⡅⠀⠀⣀⣀⣀⣀⣀⡠⠂⠀⢳
⠀⠀⠀⣸⠁⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⣶⠅⠀⢸⠇⠀⠀⠀⠀⢀⠲⢶⣿⠋⠀⠰⠋⠀⠀⠀⠀⠀⠀⣒⣲⠋
⠀⠀⢀⡟⢀⣰⣄⣠⡑⢄⠁⣄⣀⠀⠀⠀⣴⠃⠀⢀⠠⣬⠽⠃⠀⠀⣾⠀⣠⣀⡀⣀⣦⠽⠋⠁⠀⠀⢠⡀⠀⠀⠀⠀⢦⡤⠞⠁
⠀⠀⠈⠱⠛⠉⠀⠁⠉⠉⠙⠚⠉⠁⢠⡾⠟⢷⣼⡿⠗⠚⠀⠀⠀⠠⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡤⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡆⠀⠀⠀⠉⠑⢒⣦⡤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⢤
⠀⠀⠀⣠⠞⠋⠉⠁⠀⢀⣠⠴⡿⠀⠀⢀⡤⠤⠤⠤⠀⠠⠀⢂⡜⠀⠀⠀⣷⠀⠀⣀⣤⡑⠒⣭⠽⣋⣀⠀⠀⠀⣠⠔⠁⠠⠄⢶⠋
⠀⠀⢰⡇⠀⠀⠀⣀⠀⠠⣴⡞⠁⠀⠀⡟⠀⠀⠀⠀⠀⠠⢖⣫⠟⠀⠀⠀⢹⡄⣴⢆⣶⣜⡫⣭⡶⠞⠉⠀⢀⠔⠁⠀⠀⠤⠴⣟
⠀⠀⠘⣧⣶⡶⡾⠏⠋⠀⠁⠀⠀⠀⠀⢧⢠⡀⣀⠀⣀⣋⡩⠞⠁⠀⠀⠀⠀⢷⣼⡾⠿⠞⠛⠉⠉⠀⠀⣀⣡⣀⡀⢀⣐⣲⡶⠚
⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠓⠛⠉