Roller Skate Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡓⠦⠶⢦⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠐⠒⠶⠤⠬⠉⣅⢚⡓⠋⠵⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣅⠦⠀⡠⠆⣠⠀⠀⠇⠀⠀⠀⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⣀⣈⠻⣝⣃⣄⠀⠀⢂⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣻⢮⡛⢭⣍⣧⡀⠀⠀⠀⠈⠢⠄⡀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢯⡻⢛⣝⣱⠮⠕⡤⠁⠢⢄⠀⠀⠀⠈⠑⢄⡸⣇⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡻⠒⠽⢟⣯⣏⠹⠝⠂⠀⠀⠀⠀⠉⠢⡀⠀⢀⠜⠕⠙⠈⠣⠭⢉⣋⡙⠓⠲⠤⢶⣒⣀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⢯⡅⠉⣗⡴⢆⣈⠉⠃⠀⠢⡀⠀⠀⠀⠀⠀⢈⡶⡫⠊⠀⢺⠠⡀⠀⠀⣘⣀⠉⠉⢒⠒⠂⠤⠌⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠈⢙⢧⡶⢧⠿⢦⣄⡀⠀⠀⠀⠈⢢⠀⠀⠀⡰⡫⠊⠀⠀⠀⣮⠭⠭⢭⢯⡗⠊⠀⠀⠈⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⡉⠈⠉⠂⠀⠈⡷⠃⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠑⣄⠜⠜⠀⠀⠀⠀⢠⣧⡲⣤⠼⠛⠾⠓⠀⠀⠈⠢⣄⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⠙⠛⠒⠒⠒⠚⢷⡄⠀⠐⣭⠶⣄⠀⠉⠳⣄⡀⢀⠎⠎⠀⠀⡀⣠⣶⡻⣾⣿⣦⢝⡻⠟⡃⠤⣀⠀⠀⠀⠉⠢⢄⡀⠀⣧
⠀⠀⠀⡘⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢷⡄⠀⠸⣧⠈⠳⣦⡀⠈⣫⡮⠼⠒⠈⠁⢠⣼⢷⣯⢮⡲⣟⣿⡭⡁⠚⠀⠀⠑⠢⣀⠀⠀⠀⠈⡖⡹⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢷⡀⠀⢹⡂⣀⠴⠛⠉⣀⠠⠤⠐⣲⢾⠩⠿⠄⣈⡟⠛⠷⡌⠣⡄⠀⠀⠀⠀⠀⠈⠑⢄⡠⢎⠜⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣉⠤⠒⠉⠙⡀⢀⡴⠚⠠⣿⣄⣨⡻⣖⣉⠓⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡠⣊⠕⠁⠀⠀⡄
⠀⠀⠀⠘⣌⣀⠒⠀⠤⠤⠤⠤⠤⠤⠤⠀⠒⢂⡉⠥⠒⠉⠀⠀⠀⢀⡠⠚⠃⠄⣀⡀⢶⣝⠇⠑⠁⠙⠳⣦⣀⠀⠀⠀⠈⢆⠀⢀⢜⠔⠁⠀⠀⠀⠀⡃
⠀⠀⡠⠊⠉⠐⠨⢍⣀⣈⠠⠒⢋⠵⠒⠭⣁⡌⠀⠀⠀⠀⢀⠤⠚⠙⠲⠤⣤⣤⣤⣤⣼⣯⠀⠀⢀⢠⣀⠀⠙⢷⣄⡀⠀⠀⠳⢣⠂⠀⠀⠀⠀⢀⠴⡇
⠀⠸⡀⠀⠀⠀⠀⠀⡘⠀⢀⡜⠁⠀⡠⠔⠚⠛⠢⡀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡉⠳⣄⡀⠈⠻⣦⣀⢃⣃⣀⡤⠤⠐⠊⠁⠀⡅
⠀⠀⠈⠓⠤⢀⣀⡠⠧⠤⡞⠠⢤⠊⠀⡠⠒⢲⠀⢸⠀⠈⠉⠐⠒⠒⠀⠀⠀⠒⠒⠀⠈⠉⠻⡆⠀⠘⣧⠀⠈⢻⣦⠤⠐⠋⠉⠁⢀⣀⣀⠤⠄⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠆⠀⣌⠀⡠⠊⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⠆⠂⣉⠄⢤⠒⠐⠂⠈⠩⡁⠀⡀⠤⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣈⣁⠠⠔⠁⠀⢦⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠓⣈⠤⠒⠉⠀⠀⠈⠢⡀⠀⠀⠰⣀⠊⠁⢐⠤⠐⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠯⣒⠠⠤⠉⣉⣉⣉⣉⢉⡡⠤⠄⠲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⡅⠀⠠⠋⢠⠖⠉⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠉⠐⢢⠔⠊⢁⠔⠋⠉⠑⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡇⠀⢇⣀⡠⠊⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠄⠀⢰⠃⠀⣠⠚⠉⢀⡀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡠⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠔⠃⠒⡇⠈⠲⠁⢀⠞⠁⢘⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡃⠀⠠⡀⠔⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡄⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣶⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣷⣴⣾⣧⣌⠹⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⢻⣿⡇⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡛⠻⣿⣿⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣟⢂⠀⠀⢩⣭⣿⣧⣤⣤⣤⣤⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⣠⣾⢀⣯⣿⠈⠀⠀⠛⣚⣦⡉⠉⠛⠛⠓⡻
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⢀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢃⠆⣼⢻⡰⠀⣰⣤⢰⣷⠰⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⡥⢠⡾⠙⠷⠃⣸⣟⡉⣸⢿⢀⠃⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⣀⡤⠖⢾⠟⡱⣂⡴⠏⠁⠀⠀⣴⠏⠋⣵⢻⣜⡌⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠉⣵⠲⠟⠁⠀⣀⣴⢞⡁⣉⡽⠏⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣻⢀⢋⣵⠏⠁⠀⠀⠀⢱⠄⠀⠀⠀⢸
⠀⠀⠀⠂⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣄⣀⠀⢠⠏⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⢚
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣤⣀⣀⣀⣠⡀⢀⠄⠫⢙⣛⡿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣋⣤⣶⣶⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠉⠁⠀⠀⠈⠉⣷⠈⠄⠀⠀⠈⢹⠋⣨⠟⠻⢷⣦⣤⣤⣤⣴⣖⣯⢥⠖⠫⣝⡟⢁⡴⠒⠓
⠀⠀⠀⠀⣮⣿⣶⣦⣧⣤⣤⣤⣴⠒⠻⣷⣺⡃⠀⠀⠀⠀⠁⠸⢰⡇⠈⠂⠢⣼⣫⠽⠛⣫⣅⡀⠀⠀⠀⠈⢆⣼⠰⠿
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⡌⠖⠈⠁⠀⣸⡄⠀⠀⠀⠀⠀⠙⠻⣤⣀⠀⡀⡏⢠⠞⣁⠈⢱⠀⠀⠀⠀⠀⠙⠦⣀⠤
⠀⠀⠀⢸⠠⠀⢠⠴⠒⠘⣿⣿⣏⣡⡴⠾⣩⡻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢼⡈⠛⠁⣸
⠀⠀⠀⢘⠀⠀⠸⣀⠔⠊⠅⢀⣀⣧⠀⠀⢎⡔⠣⢉⢗⡲⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠉⠀⠀⢀⠻⣀⡴⠞⢏⡭⢅⡘⠒⠒⠘⠬⠥⢜⣺⠑⠂⠈⠑⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⢱⠤⢄⣉⠔⠁⢀⠀⠘⢱⠛⠎⠽⡑⣲⠤⣀⡀⠀⠀⠀⠐⠤⠀⣹⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠑⠒⠊⠢⠤⠰⣑⠏⠛⠀⠉⠳⣬⣤⣤⠤⢖⡽
⠀⠀⠀⢰⡀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢈⣷⢶⠒⢏⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠠
⠀⢀⡠⠤⠷⣤⣅⣀⣀⣉⣉⣙⡓⠒⠶⠤⠭⠭⠥⡤⠤⠖⣹⢏⣁⠀⡈⣧⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⡰⠁⠠⢓⣤⠊⠉⠀⠐⠒⠠⠍⠭⢍⣛⣛⡒⠲⠶⢖⠖⢫⣤⣎⠹⡉⠒⠃⡠⠃⠀⠀⠀⠀⠂
⠯⣉⣱⡎⢧⠘⠮⠖⣩⣏⠢⣦⡄⢒⣦⢁⠠⣈⠭⣥⡀⢀⢹⡌⢧⠈⠉⠉
⠀⠀⠀⠀⠀⠑⠒⠚⠁⠈⠢⠤⠴⠋⠘⢦⣉⣁⠴⠉⢆⠑⠛⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠯⣱⣄⡀⠤⠄⠐⠒⠒⠀⠉⠉⠙⠐⢄
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⡆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⡈⠁⢱⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠠⠰⠟⠙⣦⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠿⠈⣶⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡻⠂⢩⣷⠠⢀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠂⠉⠉⠒⠠⡀
⠀⠀⠀⠀⠀⢱⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠂⠤⠤⢄⡄⠀⡀⠀⣀⢀⣀⣀⣀⣀⠤⠤⣀⡅
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⣶⣶⣦⣦⡶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⢠⠔⠉⠉⠉⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠟⠋⠉⢉⠙⢏⠁⠀⠑⡀
⠀⠀⠀⠀⢀⠃⠈⢀⣤⡀⠡⠘⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡎⠐⠁⡤⡄⠐⠀⡆⡠⠔⠁
⠀⠀⠀⠀⠀⢆⠠⠀⠉⠀⠌⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢄⠉⢁⠐⢀⠋
⠀⠀⠀⠀⠀⠈⠑⠤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠛⠒⠒⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠧⣤⠤⡀⠀⠀⠀⠀⠈⢆⠉⠉⠉⠁⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡼⢻⡿⠒⠷⠤⠃⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⢂⢴⢸⣏⣳⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣿⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠞⣠⠏⣼⣾⣇⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⠤⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠏⢠⣏⡰⢣⡿⠤⣧⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡳⠠⠐⣸⡗⡦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠁⣠⠟⢘⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡴⡋⠀⢾⣩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠋⠡⣨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠛⢄⡴⡄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣾⠿⠋⠑⡤⢦⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⢶⠿⠟⠛⠉⡁⠀⣤⠶⡀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠈⡴⢦⠀⢟⡹⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠾⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣒⣒⠲⢤⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⠩⠅⠒⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⢶⠤⠍⣑⡒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⢒⡩⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⡉⠒⠒⠀⠭⠭⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠵⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠲⠤⢼⠇
⠀⠀⠀⠀⠀⢟⠓⠤⡀⠀⠀⠀⠀⢀⡠⢶⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣤⡀⠀⢸
⠀⠀⡰⠉⠑⠚⠑⠢⡬⠽⠋⢉⡠⠟⠓⠚⠓⠛⢶⡒⠦⠤⣤⠠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⣀⠀⠤⣤⠤⠚⠚⠓⠛⠶⢎
⠀⢼⠀⠀⠀⠀⠀⠀⠑⢦⡴⠁⢀⡤⠖⣒⠲⢤⡀⠈⢦⠤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢤⠞⠁⣀⠴⢒⡒⠢⢄⠀⠑⡄
⠀⠈⠓⢄⡀⠀⠀⠀⠀⡿⠀⢠⢃⠞⣁⣤⣈⠲⡘⡄⠀⣦⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡎⠀⡴⠡⠊⣡⣄⡙⢦⢱⠀⠘⡀
⠀⠀⠀⠀⠈⠑⠒⠰⠊⡇⠀⢠⢸⡀⢿⣥⡿⢀⡇⡇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢇⠰⣯⣼⠇⢸⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢦⡑⠤⠤⠤⢊⠔⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⢌⠲⠤⠤⠔⣃⠎⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠈⠑⠒⠊⠁⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠒⠒⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⡨⠭⠭⡲⡄
⠀⠀⠀⠀⣠⣲⣤⣶⣾⣿⣷⣶⣶⣶⣎⠀⠀⠀⠀⠀⠀⠀⢸⠲⡄
⠀⠀⠀⠀⡟⠿⣿⣿⣿⣿⠿⠿⠿⠿⢿⠒⠒⠒⠢⣄⠀⠀⠈⡆⣇
⠀⠀⠀⠀⡷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣤⣄⣨⣇⣠⣤⣷⢿⠁
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⡾⢟⠉⢹⠉⠀⠸⡘⣆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣀⠈⣇⣠⣴⡿⢟
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣫⣿⠟⠹⠁⠀⢸⡸⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡈⠁⠀⠀⢇⠀⢈⣶⣿⠂
⠀⠀⠀⡜⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣⢐⣿⣿⠾⡟⠋⠹⡈⢆
⠀⠀⢰⠁⠀⠙⣄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⡌⠚⠂⠀⠱⡀⢀⣳⡾⣇
⠀⠀⣾⠀⠀⠀⠘⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⡀⣾⣿⡶⣷⠛⠉⢣⠘⣆⡀
⠀⢠⣿⠀⠀⠀⠀⠘⡈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢃⠋⠉⠀⠈⣆⠀⣀⣷⡟⢣⡀
⠀⢸⣿⡆⠀⠀⠀⠀⢣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⡄⠀⣶⣶⡾⢿⠋⠁⠘⢦⣹⣶⡀
⠀⢸⣿⣿⣦⣀⠀⠀⠈⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡰⡄⠫⠍⠀⣀⠱⣄⣠⡾⠛⠣⣹⣶⣄⢀
⠀⠘⣿⣿⣿⣿⣿⣶⣤⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢆⠀⠊⢻⠟⠋⠙⢦⣀⣴⠟⠓⢤⣽⠗⠦⡀
⠀⠀⢹⠿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠳⣌⠉⠀⢠⣶⣾⠟⠉⠒⣤⣴⠋⠉⢲⡼⠤⣀⡀
⠀⠀⠈⣧⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠦⣴⡛⠏⠀⣼⢿⡏⠀⣧⢖⠉⠀⠀⠀⠈⠉⠒⠶⢤⣀
⠀⠀⠀⣿⣷⣦⣄⡈⠑⠒⢭⣝⡻⠿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠽⠯⢅⣜⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠈⠛⣿⣿⣿⣿⣶⣾⠙⣿⣷⣦⣍⡻⢿⣿⣿⣿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡆
⠀⠀⠀⠀⠀⢹⡿⠻⠿⢿⣿⢶⣾⣿⣿⣿⣿⣶⣍⠻⢿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⠀⠀⢰⣿⢃⡀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣇⣀⣀⢸⠇⠀⠉⠻⣿⣿⣿⣃⡈⠻⣶⣬⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣭⡚
⠀⢀⡤⠴⠶⢿⠟⠛⠉⠙⠦⣀⣠⣤⠼⠛⣋⣩⣽⢢⠉⢻⣉⠍⠙⡶⠲⠯⠭⣍⣛⣛⡛⠛⠛⠛⠛⠛⠛⣛⣛⣋⣭⣭⣵⣶⠞⠿⡛⠋⠉⠑⢄
⢠⠋⠀⢀⡠⢄⡙⢦⠀⠀⠀⠹⣭⣴⣶⡟⣏⠉⠀⣸⠀⣸⣤⣆⣠⠏⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⢻⡟⠙⢿⣿⣿⣿⣿⠏⠠⢄⠈⢆⠀⠀⠈
⣇⠀⢀⡏⠀⣴⡝⡌⡖⠃⢀⣠⡇⠀⠻⣿⣮⣙⣚⣥⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣀⣠⣬⣷⣶⣶⣾⡀⠀⠀⢉⣻⣿⠿⠸⣗⡀⠇⠨⡆
⢏⠁⣀⢷⠛⣿⢃⡇⢸⠀⠀⣀⡇⠀⠀⠈⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠈⠙⢿⡿⠿⠟⠛⣶⢛⣋⣥⣴⣶⠛⠛⠁⡇⣼⣏⣤⣤
⠘⣏⢀⣴⣿⣿⢪⡀⣾⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠒⠒⢢⡀⢳⠖⠀⠀⠸⡛⠉⠁⠙⢿⣷⣶⣾⣿⣿⣿⡿⠋
⠀⠈⠻⢷⣿⣿⡦⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢷⣦⢿⣷⢓⠀⡗⠋⠀⠤⠇⠀⠀⠀⠀⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⢦⣞⣠⠏⠀⣇⣠⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣷⢀⣼⡿⠟⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣀⣠⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠗⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⢾⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠿⠀⠀⢠⠤⣤⣤⢀⡿⢿⡿⣿⣦⣀⡀⠀⢀⣽⣿⣿⡿⠿⠛⢻⣷⣶⣦⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⡠⠔⢒⣚⣛⣁⡾⣷⣄⠉⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠈⠉⣿⣳⣶⣦⣤⣀⠀⠘⢿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠒⠿⢿⣿⣿⣿⣧⠀⠈⠋⢀⣿⣿⠟⠁⠀⠀⣤⣍⣉⡉⠉⠉⢉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢒⣦⣤⣀⠀⠈⠙⢿⡏⠀⠀⠀⢸⡜⣿⣄⣀⣀⣴⣿⣿⣿⣿⠀⠀⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣍⡲⢦⣬⣉⠛⠿⣷⣦⣼⠀⣾⢶⠀⢼⣿⣾⣿⣿⣿⡿⠋⠉⢙⣿⣧⣤⣠⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢂⣱⣬⡲⣄⡙⠳⣯⣿⣿⣦⣌⡽⠛⠶⠶⠖⠊⠀⠙⠿⠿⣿⣿⣁⣀⠀⠀⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠒⠛⠛⠿⠿⠿⣿⣿⣦⠙⢿⣿⡿⢋⣠⣶⣾⣿⠀⣴⣿⣿⣶⡄⠈⣿⣿⣿⠇⠰⢷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣶⣶⣶⣤⣤⣤⣤⣀⣀⠙⠿⠷⠚⠁⠈⠉⠉⠉⠉⠁⠀⠈⠉⠀⠀⠀⣰⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣶⡷⠒⣶⣿⣿⣿⡿⠿⣶⣶⣒⡒⠒⠒⠉⠛⠚⢋⣉⣉⣩⣭⣙⠳⣶⡶⠠⡀
⠀⠀⠀⠀⠀⠀⢸⣍⡚⡩⠋⢠⠾⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠶⠲⠚⠛⠛⠛⠛⠛⠛⠛⠆⠈⠙⠓⣻⣦
⠀⠀⠀⠀⠀⠀⠸⣿⣥⡀⠉⠤⡰⣶⡞⢲⣦⣤⠈⡁⢠⢴⣶⠒⣶⣤⡄⢈⠀⣤⣴⡖⢲⣶⢀⠀⠁⢤⣭⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣷⣾⣿⠟⠁⠀⠙⠷⣾⣧⣿⡿⠋⠀⠈⠻⠶⣿⣼⣿⠟⠁⠀⠙⠿⢷⣾⣷⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠏⠉⠙⢦⠀⠤⠔⠒⠂⠰⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠄⠀⠨⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠂⠄⠀⠠⠖⠁⠀⠠⠤⠄⠀⠀⠀⠀⡓⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠏⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡄⠀⠀⠀⠀⠀⠰⠀⣬⠀⠐⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⢉⡀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣱⡀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠂⠤⠤⡤⠴⣊⠀⠐⠛⠂⠀⠈⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠂⠀⠐⡀⠀⠀⠀⠀⠈⢀⠀⠀⠐⢦⠤⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠈⠀⠀⢀⣠⣤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠒⠂⠃⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠈⠉⠀⠀⠿⠐⠒⠲⠖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠎⢀⣈⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠐⠂⠀⠊⠀⠀⠙⣍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠤⠒⠫⢀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠐⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢄⠁⢀⡥⠖⠛⡆⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠤⠴⠶⡏
⠀⠈⠀⠁⠀⠀⠀⣠⠜⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⡀
⠀⠀⢀⠀⠀⢠⠞⠁⠀⠀⠀⠀⡰⠁⠀⠉⠑⠲⠤⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢳
⠀⢀⠁⢁⡾⠁⠀⠀⠀⠄⣀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⢀⠞⠀⠀⣀⡤⠖⠋⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡀⠘⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⣤⠔⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣀⣀⣀⣀⡠⠴⠫⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠀⠩⠛⠢⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠠⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠁⠀⠀⠀⠀⠀⠈⢠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⠿⣟⣯⠥⢆⣀⣤⣶⣾⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡎⠿⣛⣯⠵⢚⣫⣭⣶⣿⣿⠿⠛⢋⣩⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣟⣫⠵⢒⣨⣥⣶⣿⡆⠿⠛⣋⣡⣴⣶⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⣫⣴⣿⣿⣿⠟⣛⣩⣤⢺⣿⡿⠿⢛⣋⣥⣴⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣛⣭⠥⢶⣾⣿⣿⣿⣿⣿⣿⣿⠿⢛⡈⣥⣶⣾⣿⡿⠟⣛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣥⣏⡴⠶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠟⢘⣋⣭⣴⣾⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢵⣏⣹⢓⡆⢿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣎⣿⠿⠛⣋⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣹⠟⣼⣿⡘⣿⣿⣿⣿⣿⣿⠟⣛⣋⡉⣤⣶⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⣼⠛⣐⢿⣿⣷⣌⠻⣿⣿⣿⣿⣿⡿⠿⠃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣷⡟⢉⢸⣿⣆⢻⣿⣿⡷⣠⣬⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠛⣧⡾⠛⡁⢴⣿⣇⢻⣿⡎⢿⡿⣱⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣯⣨⠿⠏⣡⣴⣿⣿⡌⣿⣿⡜⣿⡿⠘⠃⠿⠛⣋⣭⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡴⠛⢁⣴⣿⣿⣿⣿⣿⣧⠸⠿⣃⣥⡶⣞⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣾⠁⠰⠿⠿⠿⠿⠟⣛⣩⣥⡶⠟⢫⣷⣿⡟⢰⡶⣯⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡷⠶⠶⣶⠿⠛⢿⣯⡛⠀⠀⠈⠻⣿⣧⣈⣾⣿⣿⡟⡩⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠺⣷⣿⣿⣿⠷⢯⡀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⡐⣇⠞⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⢡⠖⢦⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣈⠓⢋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⣽⣷⣿⣿⣿⣿⣿⣿⣿⣏⣉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⢹⢸⣿⡿⠛⠛⠛⠛⠛⣛⣻⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣏⡏⠀⠀⢀⣠⣤⣤⣤⠀⣰⡶⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⢀⠀⠠⠴⠾⠿⠋⣰⣿⣆⠈⠙⠧⠤⠤⠤⣤⣤⣤⣤⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠤⠒⢶⣦⣄⣀⣒⠒⠋⠛⠋⠙⣿⣷⣄⠀⢛⣻⣿⡯⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠶⢿⣽⣿⣿⣷⣦⡀⠀⠈⢿⣿⠂⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠙⠛⠾⣿⣿⣿⣿⣿⠀⠀⠀⠉⢀⣾⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢶⡦⣤⣀⡀⠀⠀⠉⠻⣿⡟⠀⠀⠀⠀⢸⣇⢴⠞⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠫⣤⣈⡉⠛⠾⢿⣷⣦⣄⣀⣿⠁⣤⣤⡄⠀⢘⣿⣞⢃⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢪⣀⠘⠳⣦⣝⡻⣷⣦⣄⡉⠛⠿⣻⣇⡀⠙⢉⣃⡴⠙⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠐⠒⢉⣁⣤⣿⣷⣬⡻⣦⡀⠙⢿⣾⣿⣿⣿⣦⡾⠋⠉⢛⣛⡋⠁⢀⣀⣀⠉⠛⠛⠯⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⣿⣿⣿⣦⠈⢻⣿⣿⠟⢋⣠⣴⣿⣿⣿⠇⢠⣾⣿⣿⣿⣶⠄⠀⣺⠯
⠀⠀⠀⠀⠀⠀⠀⠈⢶⣷⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⡀⠉⠻⣿⣷⠴⠋⠁⠐⠛⠛⠛⠛⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⠽⣯⣭⣿⣛⣛⣛⣛⣛⣛⣛⣂⣀⣥⣤⣤⠤⠤⠤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠤⣴⡶⠾⣁⡀
⠀⠀⠀⠀⠀⠀⡔⠉⢀⠠⣤⠔⠁⠀⠉⠉⠉⠉⠉⠉⠈⠓⠤⠽⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢀⡈⠓⢄
⠀⠀⠀⠀⠀⢸⠀⢠⣉⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⡍⣄⠈⢇
⠀⠀⠀⠀⠀⠘⡆⠈⣬⠤⣈⡠⠴⠒⠊⣻⠻⡒⠢⣤⢄⣉⣠⡤⠖⠚⡟⢻⠒⢦⡤⢈⣁⣠⡤⠖⢚⠛⡏⠑⠒⠤⣈⣀⢼⡬⠀⣘
⠀⠀⠀⠀⠀⠀⠘⢦⡈⠐⠓⠀⠋⠀⣠⠃⠀⠑⢄⡀⠐⠒⠐⠋⢀⡴⠁⠈⠳⣄⠁⠚⠂⠘⠁⣀⠞⠀⠈⠦⡀⠙⠀⠚⠂⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⡄⣀⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠀⠈⠙⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⣀⣀⣤⣤⠴⠶⠤⢀⣀⠀⣤⡄⠈⣛⣛⣛⣻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢳⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⢻⡏⣿⠀⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡴⠶⠛⠛⠛⢛⣛⣛⣛⣿⣷⣿⡀⠉⣩⠿⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⢺⣍⣉⣉⣉⣉⣉⡇⠼⠃⣠⡤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠉⢉⣭⠿⢿⡟⠀⠀⠀⣠⡶⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣠⣤⣧⠀⠀⡼⠋⢠⡾⠃⠈⣹⠟⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⡀⠀⢰⣟⣻⡆⣸⠃⠈⣡⡾⠛⣦⣀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠛⠀⠉⠟⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⢷⡈⠁⠶⣤⣉⣉⣴⠟⢀⡾⠋⢠⡾⠃⠉⣿⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⢦⣄⡀⠉⠉⠀⠀⢿⣁⡴⠋⠀⠀⠀⠉⠀⠀⠟⠉⠙⡿⠶⢦⣤⡀⠀⢠⣤⣤⡴⠾⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠛⠳⠶⢦⣄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠴⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠲⢶⡌⢁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⣄⣀⣤⣤⣌⣻⣤⣤⡀⠀⠉⠉⠉⣹⣿⣿⣛⠛⠛⠛⢻⣯⣭⣍⡀⠀⠀⠀⢀⣰⡇⣸⠇
⠀⠀⠀⠀⠀⠀⣴⠋⣡⣤⣌⣙⣧⣠⣿⢋⣥⣤⣄⠰⣆⢠⡞⢉⣤⣤⣉⢻⣟⣿⠟⣻⣤⣌⡙⠗⠀⠀⣉⣡⡴⠏
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠴⢛⡟⢹⣿⠉⣿⠳⠶⢻⡟⣻⣿⠛⣿⠷⠾⣻⠏⣿⡟⢻⡿⠷⠈⣷⢀⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢷⣌⠛⠶⠛⠁⠾⠙⣧⡘⠳⠶⠋⣠⡟⠹⣦⡙⠳⠾⠋⢰⠟⢻⣄⠛⠶⠞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠉⠛⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⡿⢛⣓⣂⣤⡔⠁⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠻⠿⢿⣿⣿⡁⠀⠀⠀⠀⠈⢿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠁⠀⠀⠀⠉⡙⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠈⣄⢀⣀⣠⣤⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⢿⣲⣒⣋⠓⠷⢺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣊⣹⡤⣮⢬⡿⠵⣿⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢖⡄⡁⠢⠤⡴⠱⢻⣟⡛⣲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢾⡼⡶⠟⣫⢛⡷⣴⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⣬⣋⢚⣛⡴⠿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠐⠂⠐⢺⡫⢻⠛⣻⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠚⠦⢥⣭⣼⣿⡄
⠀⠀⠀⠀⠀⢠⢣⡖⠒⣔⡀⢰⣿⠻⣬⣉⣉⣿⣿⣿⡿⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠃
⠀⠀⠀⠀⠀⠘⡘⣯⡘⢀⣷⣿⣿⠁⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⢸⣯⣿⠻⢶⣶⣦⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠑⠘⠛⠵⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠻⢤⡋⡛⢿⣿⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣀⠀⠐⠄⠀⠹⣄⣹⣿⣿⡀⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢢⠀⠀⣀⣴⡟⠛⣯⣿⣷⣲⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡻⣷⣄⠀⣿⣿⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠚⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢶⣤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⡄
⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣶⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡲⠃
⠀⠀⠈⠀⠀⠀⠟⢛⣦⣄⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠩⢗⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣅⠐
⠀⠀⠈⠀⠀⠀⡀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⡀⢈⢀⡠⢄⡀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠠⠀⠀⠀⠘⠋⢀⡒⠂⠘⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠐⡀⢙⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠠⠰
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠉⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠁⣰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡤⠅
⠀⠀⠀⠀⠀⢺⡠⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠾⣯⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⡟
⠀⠀⠀⢠⠀⠀⠣⣃⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠬⠀⠈⠙⠲⠀⠀⠀⠀⠀⠀⠀⠴⠞⠉⠁
⠀⠀⠀⠀⢀⠄⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠂⠈
⠀⠀⠰⢂⡑⠠⠐⣂⠤⠀⢠⠀⠀⠀⠀⠀⣀⠁⠦⠴⠨⠠⠐⠀⠄⠈⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠈⠆⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⢦⣸⠇⢸⡇⣤⢀⡶⠀⣤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢿⣸⠂⠀⠀⣿⠀⠸⣧⢿⣼⠃⣼⢧⣼⠃⣸⡟⠓⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣿⣁⡤⠈⢿⠛⠶⠤⣿⠀⠀⠛⠈⠋⠐⠋⢰⠃⢸⡋⠙⠂⣰⣏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠹⣆⠈⢿⣄⡤⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡼⠹⣇⠀⣠⡿⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⠙⠳⣽⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢴⡋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⡞⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⠀⣠⠖⢲⠒⠀⠀⠒⠠⢄⠀⠀⠀⠀⡀⠂⡀⠁⢀⡲⠀⠀⠀⢒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠲⣦⠀⠀⠀⠀⠀⡜⠁⠀⠀⣀⢤⠀⠀⠀⠀⠈⠂⢄⣠⠁⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠁⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡙⠶⠞⠀⠀⠀⠀⢠⠇⠀⠘⢄⠈⠀⠀⠀⠀⠀⠀⠀⠠⠜⠀⠖⠁⠒⡇⠀⠋⠘⠀⠀⠀⠀⠀⠀⠀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢻⡶⢦⠄⠀⠀⠀⠞⠀⠰⠂⣄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠁⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠃⠀⠀⠀⠀⠂⠰⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠁⠒⠻⠂⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢉⡀⠠⠶⡴⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢃⡀⠀⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢯⠙⠛⠛⠛⠻⠄⠀⠀⠀⠸⠄⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠶⠠⠄⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣠⡴⢾⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⢤⠉⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⡀⠀⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣆⠀⠀⣶⠛⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⢈⠐⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⡀⠘⠀⡁⠘⠋
⢸⡀⢀⠸⡏⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⣤⠚⢁⠄⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⡔⠀⠀⠀⠀⢎⠁⢀⣨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠸⠷⠋⠀⠙⠛⠁⠀⠀⠀⠀⠀⣠⣀⠒⠄⠀⡄⢈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠴⡎⠁⠈⠋⠐⠚⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢊⠀⠐⢄⣰⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⠤⠒⠀⣐⠓⠖⠁⡄⠀⡀⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⠠⠤⠐⠒⠉⠉⠒⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⡠⠔⠉⢀⠈⡀⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡤⠓⠒⠒⠀⠈⡡⠘⠶⠚⢣⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⢄⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡠⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣆⣀⣀⣀⢀⣀⠞
⠀⠀⠉⠀⠒⠒⠒⠀⢀⡤⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⢀⠄⠒⢀⡉⠀⣠⠒⢢⡀⠘⠔⠀⢠⣄⢀⡇⣀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⣄
⠀⠀⠀⢀⠁⠀⠀⠂⢧⡾⠋⠉⠀⠈⠉⠰⡄⠀⢀⡜⠋⠐⢶⡴⠀⠒⢦⣍⢉⠉⠉⠉⠁⠀⠀⣀⢠⡤⠀⠀⠄⠠⣤⡄⢠⡤⠤⣤⠐⢊⡡⠂⡹⠧⠀⢀⣤⣀
⠀⠀⠀⢸⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠈⠀⢤⡈⢁⣼⠁⠘⠋⠑⠦⡄⠄⢘⡇⠩⠀⠀⠀⠀⢠⠀⣿⠀⠀⠀⠀⣠⠒⣇⢠⠊⠁⡠⠂⠀⠀⠈⠑
⠀⠀⠀⠀⠣⡀⠀⠐⡇⠀⠀⠀⣀⠚⢦⠀⠀⡀⠀⠠⠀⠀⠈⠙⠋⠀⡠⠒⠉⠁⠈⠉⠢⡈⢁⢰⠀⠀⠐⠀⠀⣀⠋⡀⠐⠀⠈⠀⠀⢈⡁⠀⠾⠁⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠂⠤⢇⠀⠀⢀⠉⠀⠁⠀⠀⠳⡄⠀⠀⠀⠀⠄⢀⠄⠁⠀⠀⢀⡀⠀⠀⠁⡈⠂⠦⣄⣀⡀⠔⠁⠀⠀⠤⣀⠀⠀⢀⠸⠃⠸⠀⠀⠰⣮⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⠀⠀⠀⠐⠀⠀⠘⠦⠀⠀⠀⠀⢘⠀⠀⢠⣘⠋⣷⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢢⠀⠀⠀⠈⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⢋⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣷⣿⣦⣄⠀⠀⡠⠐⠓⠲⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⣿⣭⣶⣶⣾⣿⠀⠀⠀⠀⠀⢹⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠤⠉⠛⠛⠿⠧⡄⠀⠀⠀⠀⠀⢻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣲⣶⣦⡄⠀⠀⠀⡠⠤⠤⣤⣤⣸⣦⣠⠀⠀⣼⠷⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣿⣟⣫⣭⣴⣶⠋⠀⠀⠀⠀⢻⣿⣉⣯⡽⠯⢥⢉⣙⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⡛⠿⠿⢿⣿⡃⠀⠀⠀⠀⠀⢻⣷⣣⠽⢔⡵⠢⡿⠛⡿⢭⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⣿⣿⠿⢆⣅⡔⢖⠞⠴⢫⣯⢍⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣠⣿⣿⣦⠈⠋⠰⣿⠾⣟⠏⣡⣛⠟⣰⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⢿⢲⣶⣚⠓⠲⢾⣧⠀⠀⠀⠀⠻⠊⣲⣏⣚⡉⡵⠿⢷⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⢱⣄⣴⣅⣝⡻⣿⣗⣦⠀⠀⠀⠀⠈⠀⠁⠘⠺⠭⣭⣵⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡅⢙⢀⣬⣤⢊⠜⣿⣓⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣶⣡⡶⢖⢛⡵⢿⢏⣿⣶⣦⣤⣤⣤⣤⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡶⢫⡿⠣⠟⢋⣽⣾⡙⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⣀⡠⠤⠖⠦⡛⣟⣿⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠱⠿⣒⣘⣛⣼⣧⠀⠙⣦⡀
⠀⠀⠀⠀⠼⣩⡥⢆⡄⠀⢸⣾⢿⡒⠒⢻⣿⣿⣿⣿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⢤⣺⠝⠁
⠀⠀⠀⠀⠀⢿⡂⡆⠉⣦⣿⣿⠚⠟⠛⡿⢿⣿⠿⠏⠀⠈⠉⢙⣟⣷⣶⣤⣤⣀⣀⣀⣀⣀⣠⣼⣿⠏⠉
⠀⠀⠀⠀⠀⠘⠿⠴⣳⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣻⡇⡹⢹⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⡀⠈⠳⡔⠀⣿⣿⡏⠉⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣶⣀⠀⠑⡄⠀⣹⣾⣿⢿⣷⣀⡠⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣼⠀⠀⢰⣾⣿⠀⠘⠿⠛⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠑⣂⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⣤⣾⡿⣷⠴⠾⢤⠈⠛⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⡇⠀⠀⠸⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣉⢈⣩⡅⡁⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠘⠛⡃⡇⠀⠀⣠⡷⠴⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠘⠟⢓⢻⡀⠀⠀⢠⡤⠔⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡆⠻⠛⣊⢷⡀⠀⠀⣡⡄⠋⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⣿⣿⣿⣿⣶⣆⠻⠟⣊⡳⣄⠐⢁⣰⠎⠁⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡜⣿⣿⣿⣿⣧⣶⠜⠛⣉⣬⡲⣌⡁⠴⠛⢁⣰⠿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣾⣷⡎⢉⠩⣴⠾⠒⡄⠛⢁⡶⠋⣫⣷⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣟⡛⠛⠻⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡚⣫⣵⣶⣤⠀⠾⠍⠁⠀⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠃⢈⠙⣌⣙⣓⣶⣴⣬⡉⠛⢿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣧⠀⠀⠀⠈⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⠀⠀⠈⡏⠉⠀⠀⠈⠉⠓⠶⣬⣝⡻⢿⣿⣿⣸⣿⣿⠿⠟⡇⠖⠒⠖⢛⡶⠚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⢀⠄⡰⠃⠀⠀⠀⠀⠀⠀⣰⠋⠁⠄⠉⠓⠆⠠⢰⠒⠂⠉⠇⠀⠀⣿⢦⣠⣤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡀⠀⢘⣯⣀⣀⣀⡶⠶⠦⠖⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠚⣁⣠⡞⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⡶⠶⠟⠛⠛⠛⠻⠿⣶⣶⠾⠻⠷⣦⡀
⠀⠀⠀⢀⣾⠟⠻⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣶⣶⣿⣧⣤⣄⣤⣤⣤⣄⡀
⠀⠀⠀⢸⡏⠀⠀⠀⣠⣴⠾⢿⣿⣶⣤⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⡉⠉⣹⡷
⠀⠀⣀⣿⡇⣠⣶⠿⠋⠀⠀⣾⠏⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⢈⣷⢀⣿⠃
⢠⣾⠏⠙⠛⠋⠁⠀⠀⠀⢸⣿⠀⠀⣠⡶⠟⢩⡵⠶⠶⠶⠦⢭⣽⣿⣶⣦⣄⣸⡏⡞⠃
⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠶⠟⠉⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣷⣆
⢿⣇⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠙⢻⣿⣶⡀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣷⣶⣶⣤⣀⣼⣿
⠀⣴⣿⣿⣿⠃⠀⠘⣿⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⠿⠿⠿⠿⣿⡟⠋
⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⢀⣠⡶⠟⠻⣦⣄
⠘⢿⣶⣦⣀⠀⠀⠀⣸⡏⠉⠛⣿⡆⢠⡾⠛⢷⣄⠀⣿⠃⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⢙⣿
⠀⢰⣿⠈⣻⡆⠀⠀⣿⣧⣀⠀⢸⣧⠹⣦⣤⣼⠟⢠⣿⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣤⣾⣿⠿⢷⣦⣀
⠀⣿⠿⠶⠟⠁⠀⠀⠀⢸⡿⣿⣆⠙⢷⣤⣌⣠⣴⣿⣿⣀⣾⠋⣄⣀⣠⣴⠆⣴⣿⠿⠿⠷⠶⢶⣭⣙⠻⠶⢶⣦⣤⣤⣄⣀⣀
⠀⣿⡆⠀⠀⠀⢰⠀⢠⣿⣧⣼⠏⠀⢸⡏⠛⣛⣋⡁⠈⢿⣿⣤⣭⣩⣭⣴⣾⠿⠷⠶⣦⣤⣀⡀⠈⠙⠻⢶⣤⣤⣤⡾⠟⠋⢹⣿⣦⡀
⠀⠘⢿⣦⣤⣴⠟⠀⢸⣇⠈⠀⠀⠀⣼⠁⣼⡿⠛⢿⣦⠈⠻⣯⡉⠉⠉⠉⠀⣀⣤⣤⣤⣬⣉⡛⠳⢶⣤⣤⣄⣀⣤⣤⣴⠾⠛⠁⠈⣷
⠀⠐⣿⣿⡉⠀⠀⠀⠈⣿⣆⡀⢀⣴⠏⠀⢻⣧⣀⡀⠙⣷⣄⠈⠛⠷⣤⣄⣸⡿⠋⠉⠉⠉⠙⠛⢷⣦⣀⠈⠉⠉⠉⠀⠀⢀⣀⣤⣴⣿⡇
⠀⠀⠘⢿⣿⣦⣄⣀⠀⣼⣿⠿⠛⠁⢀⣀⣀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠾⠟⠛⠛⠉⠁⠀⠀⣷
⠀⠀⣠⣴⡏⠁⠉⠙⢛⣻⣿⣷⣄⡀⠸⣯⡿⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣾⣷⣶⣶⣤⣄⠀⢰⡖⢶⡄⠀⠀⢀⣀⣀⣠⣤⣴⡾⠿
⠀⣰⣿⣿⠁⢀⣀⢰⣿⣯⣿⣼⠋⠛⠷⠶⢶⣶⣦⣤⣶⣶⣿⣿⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣮⣛⣋⣠⣶⡟⠛⠙⠻⣿⣿⣤⣀
⠀⣿⣾⣿⠀⣿⣙⡷⠉⣿⢿⡟⠀⠀⠀⣰⣾⣿⡍⠉⣿⣷⣦⠀⠀⠀⣴⣾⣿⡋⠉⣿⣿⣷⡄⠀⠈⠙⣿⣿⡍⠉⣿⣦⡀⠀⠀⠙⣿⣽⣷
⠀⢹⣟⢿⣧⣬⣩⣥⣼⣿⣿⡇⣰⡶⣶⡻⠿⠿⠛⠛⠛⠋⠁⣠⡶⣦⠉⠛⠻⠷⠾⠿⠛⠋⣴⡾⣶⡀⠉⠉⠛⠛⠛⠛⠁⣰⡶⣶⡜⣿⣿
⠀⠀⠙⠿⣽⣿⣿⣿⣿⣟⣿⣧⡙⠷⠟⣀⣠⣤⣶⢶⣶⣦⣄⣹⠷⠟⣀⣤⣴⣶⠶⣶⣦⣄⡙⠻⠛⢀⣤⣴⡶⢻⣶⣦⣄⡙⠷⠟⢃⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠹⣯⣻⢿⣿⣿⢿⣿⡿⠁⠀⠻⣿⡿⣿⣿⣿⡿⣯⡿⠃⠀⠙⢿⣿⣿⣿⡿⣿⣿⠟⠁⠀⠻⣾⣻⢿⣿⣿⢿⣫⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣿⣟⣋⣁⣀⣀⣀⣀⣈⣛⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣈⣙⣻⣿⣿⣛⣉⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⣬⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⡿⠂⣠⣄⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⡿⢋⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢟⣛⣛⠿⣿⣿⢟⣛⣛⠿⣿⣿⢟⣛⣛⠿⣿⣿⢟⣛⣛⠻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠻⣷⠀⢰⣿⠛⠻⣷⠀⢰⣿⠋⠻⣷⡀⢰⣿⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣾⠟⠀⠘⢿⣦⣾⠟⠀⠘⢿⣶⣾⠟⠀⠘⢿⣦⣾⠟







⡟⠉⢻⡟⠿⠿⠿⠿⢿⠻⠟⠻⠿⠿⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠸⠀⠀⠀⠀⢀⣀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢆⠀⠀⠀⢀⣀⣀⣠⢤⣤⣆⣀⣠⣤⣶⣶⠿⠛⠛⠉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⡠⠴⢒⣪⣭⣥⣤⣤⣴⣿⣿⣀⣜⠉⠉⠙⠺⢤⣀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⣀⠄⠊⣉⡥⠴⣿⣭⣭⣤⣴⣇⣤⣤⣄⣀⡏⠉⠓⠲⣇⡀⠈⠓⣾⡀⠈⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢤⠀⠀⠀⠀⠀⠀⣰⠟⢊⡹⢖⢩⣿⠧⠚⠋⣉⣀⣀⣤⢼⣷⣨⣿⣨⠉⠓⠦⣼⡀⠉⠳⣄⠇⠙⢦⣤⡚⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠆⠀⢀⣀⠁⣾⣫⣶⣁⠤⣾⣇⢤⣾⡟⢉⣀⣾⢿⠧⠤⣤⣾⣁⡙⠑⣦⣤⡎⠙⣶⣿⠟⣳⡔⠉⠻⣿⣦⣈⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢻⡼⢿⣿⡾⡡⢖⢉⠠⣴⣾⢉⡀⢠⣶⡞⠀⠈⠁⣾⠿⠋⠲⢿⠋⣹⣦⣌⠹⢎⢀⣽⣄⢀⠹⡏⠀⠀⠈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⢰⡷⢢⠔⡋⢴⣅⠿⣉⠄⠊⢁⡀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⢟⠛⡻⡄⠈⢿⠻⠻⣇⣀⣻⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⢼⡄⠀⣾⡃⠊⡤⢚⡹⠂⣽⡆⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⣿⣿⢹⣿⡟⢇⠀⢿⡿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢼⣷⣷⢦⡰⢏⢴⣾⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⢿⣈⣹⡉⠘⡖⠒⣗⣦⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⢿⣣⣻⡔⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⢁⡼⡏⠉⣇⢈⣷⠒⢻⠁⢸⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠷⣼⠇⠘⢤⣡⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡍⠀⣷⢚⣿⣉⣻⢿⣾⣄⢸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⣀⣩⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠉⠤⢼⣽⠤⢾⣿⣿⠈⣹⠿⢿⣧⡄⠩⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⣁⣬⠃⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠹⠛⣛⠁⣺⣿⣿⠀⣼⢀⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠷⠈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⣿⣤⡏⣠⣇⢀⠇⢠⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⠛⢊⢥⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢻⠉⣻⠂⡻⠛⡼⢤⣼⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣤⡾⠱⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡎⢈⠃⢰⣿⣴⠃⣼⡇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠿⣿⣾⣿⠛⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡟⢻⠯⢠⣯⣨⠋⢸⠛⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⣤⠏⣤⣠⣤⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⡀⢸⣯⣤⢂⣠⡉⡻⠇⡴⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠁⢿⣘⣷⣾⡇⡶⠀⢠⡆⠀⣀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠾⠟⠀⠔⢩⡟⢁⠾⣛⣿⣀⠞⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠉⠉⠉⠀⠘⡛⠀⠢⠘⠊⡷⢀⣠⣀⠀⠀⢉⣠⠀⢰⣊⣤⣤⣤⠞⡁⠐⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢆⣀⣵⠘⢡⣾⠾⣡⡄⡀⢿⣃⠈⢩⣭⢾⡿⠰⣾⠇⠂⢁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⢔⡐⠉⣭⠀⠙⢜⢋⡀⢈⣶⠟⠩⡌⠖⠈⠁⡀⢀⣀⣤⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⣡⠾⣌⠛⠁⠀⠂⠈⠙⠦⠀⠈⠉⠁
⣿⡛⢙⢙⣟⠟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⠁
⡟⣝⠙⠙⠉⠚⢋⡇
⣷⠀⡀⠀⠀⠀⡸⡇

⢀⡆⢠⢠⠏⢾⢦⠪⣃⡜⠁⠖⠒⠋⠀⢠⢠⠇⠠⠃⡷⠼⢂⡛⠀⠞⠠⠆⠴⢀⠀⠄⣈⣀⠠⠤⠠⡷⠈
⢈⡀⡉⣜⠦⣼⠾⠀⠁⡀⠰⠐⠐⠬⣄⢏⠆⠶⢤⠛⠁⠀⠇⠈⡆⠀⢷⠘⠛⠛⠃⢠⡀⠲⠄⠠⠤⠄
⠀⡇⠀⡟⢀⣸⠀⡾⢶⠶⠆⠀⣇⠄⢀⠀⠀⣄⠀⡀⠂⢱⠀⢜⠀⠶⠀⠖⠰⠀⠆⣸⠃⠂⠀⠀⣀⠀⡜
⠘⣀⣘⣁⠀⢰⣀⡇⣸⢀⡄⡴⠀⠀⠒⠒⠒⠀⢰⠉⢀⣁⣀⢈⠉⠀⢰⠀⡉⠉⠈⡁⣀⠀⠃⠈⠉⢠⠅
⡄⡂⢀⢀⠀⠆⠨⠤⠅⢈⠁⠖⠛⠠⠂⠆⠰⠐⠂⠀⣉⣈⡠⠀⠀⠹⣙⡆⢀⡤⡄⠀⠠⠀⠠⠖⠒⠀⠰
⠀⡁⢨⢤⡥⠀⠆⠄⠄⢸⠄⠂⠐⠀⠒⠖⠂⠀⠀⠁⢁⠀⠀⠀⠋⠀⢈⠀⡀⢀⠄⠀⠀⠆⠂⠁⠁⠒⠂
⠐⡇⡀⣸⢀⡄⠀⠈⠓⠀⠀⡘⠀⠠⠠⠆⠄⠀⣀⢀⡘⠀⠈⡁⡀⡉⠡⠀⠀⡄⠰⠃⠀⠆⠀⠱⠀⠤⠤
⣀⡀⠀⢀⠀⠠⢤⡘⠀⠂⠈⠁⠰⠒⠀⢀⡀⠈⠀⠠⠤⠤⠀⠀⠀⠠⠶⠂⠰⠁⠀⣀⣀⠀⣀⠀⠠⠠
⠀⠄⠀⡘⠀⠠⠀⠄⠀⠐⠒⠀⠀⠈⠀⣬⠁⠀⠀⠔⠠⠠⢀⣠⡄⠠⠀⠂⢀⡀⠠⠠⠠⠀⡉⠁⠀⢀⠇
⡄⠀⠈⠁⠠⠖⠂⠀⠀⠀⠈⠀⠸⡀⠀⠀⠀⠉⠓⠀⠀⠀⡈⢀⠀⠀⣀⠀⢀⢉⠀⠀⡀⠀⠀⣬⠄⠉⠰
⢠⠀⠀⠀⠄⠰⠀⠀⠊⡁⠀⠀⠀⠉⠀⢀⡀⠁⠘⠀⠀⡄⠁⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⢀⠀⠄⠀⠰
⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠐⠲⠀⠀⢀⠁⠄⠀⠀⠀⡤⠁⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠁⠀⠀⡤⠄⠀⢀
⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⠀⠄
⠀⠀⠀⢀⡀⠀⠑⠀⠈⠄⠀⠀⠀⠠⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠄⠀⠀⠠⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠟⣛⣛⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⡤⠶⠶⠶⠖⠛⠛⠛⠉⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣶⡶⠿⢿⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⢿⣧⡀
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⠛⠁⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⢷⣄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⢉⣿⣶⣤⣀
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⣿⢻⡄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠾⡿⠋⠁⠈⣹⣿⡷⠶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⡶⣿⡄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⠻⢾⣅⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⠿⠋
⠀⠀⢀⣠⣤⡴⠶⠞⠋⠀⠈⠉⠉⠙⠛⠋⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠻⣶⣤⣀⡀
⢀⣴⠟⠛⠷⠶⠶⠶⠶⠶⣶⡶⠶⠶⣦⣤⣤⣤⣶⠶⠶⠶⢶⣶⡶⠶⠶⠶⠶⣶⣶⣶⠶⠶⢶⣶⣶⠖⠚⠛⣿⠿⠛⠋⠉⠙⠛⢷⣄
⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠈⠻⣦⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢿
⣿⡄⠀⠀⠀⣠⡶⣦⡀⠀⢀⣿⠀⠀⠸⣿⠀⠀⠀⠀⣠⣄⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢀⣤⣤⡀⠀⣹⡇⢿⡇⠀⠀⢀⣴⠞⣶⠀⣼
⠘⣷⡀⠀⠀⢿⣇⣸⠇⢀⣾⠏⠀⠀⠀⠻⣧⣄⠀⠺⣧⡿⢁⣼⠏⠀⠀⠀⠻⣧⣄⡀⠻⣧⣼⢇⣴⠟⠁⠈⠻⢶⣤⣈⣻⣶⣿⠾⠋
⠀⠈⠛⠷⢶⣤⣭⡵⠶⠛⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⢹⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠣⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⢒⠤⠖⠲⡄⠇
⠀⠀⠀⠀⠀⣖⡿⢸⠀⠀⠀⠀⠀⠀⠀⢠⣌⡀⠀⠀⠈⣀⠀⠀⠠⠄⡈⠉⠉⢉⠀⠔⠀⠀⠀⠀⢈⣠⣼⣷⣀
⠀⠀⣠⠂⢲⣍⡐⢹⣇⠀⠀⠀⠀⠀⠀⠈⣷⣉⣌⣁⣉⣎⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⡟
⠀⠀⡇⠀⠀⢯⠀⡆⣿⢷⠀⠀⠀⠀⠀⠀⢆⣈⡙⢂⡐⢛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄
⠀⠀⠈⢢⠖⠹⢶⡀⠹⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠘⢄⡀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⣀⠈⢄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡲⠾⠡⣴⠰⢀⡄⠀⠀⠀⠀⡁⢯⠀⣱⢈⠆
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡬⠄⡩⠩⠌⠦⢤⣼⡁⠀⠀⠀⠑⠀⠩⠅⠊⠀⠀⠀⣠⣶⠟⠻
⠀⠀⠀⠀⠀⢠⡴⣤⡀⡢⡺⠅⠈⠀⠀⠀⠀⠀⠀⠴⠃⠐⠃⠈⠣⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀
⡐⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠐⠈⢀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢏⠄⠡⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⢠⡇⢠⡖⢀⣠⠄⠀⠀⡸⠿⠿⠛⠐⡄⠀⠀⠀⠀⠀⠏⠤⠠⡂⠇
⣄⠀⠀⠀⠀⠀⢸⡁⢘⠣⣊⠁⠲⠤⠀⠀⠀⠱⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢀⠤
⠛⠇⠀⠀⠀⠀⠀⠡⠂⢤⣈⣙⠋⠀⠀⠀⠀⠀⠱⡔⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡼⠛⠟⢢⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣀⣠⢄⣠⠐⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣈⠀⠀⠀⡀⡀⡀⠸⠀⠀⢀⠰⠂
⠽⠉⠴⢫⠡⠴⢶⡒⢢⠄⠀⠀⠐⠠⠄⠁⠄⡀⠀⠀⠀⠛⠀⠐⠉⠁⡤⠀⠀⠁⠀⡀⠶⢰⠀⠀⢠⠀⠀⠀⠈
⠖⠃⢊⡆⠀⠃⠀⠈⠀⠀⠀⢰⠃⢀⠀⡰⠠⡄⠀⠀⠈⠓⠈⠀⠀⠈⠀⠀⠉⠀⣤⣀⣿⣾⣾⣦⣀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⢣⡰⡄⡇⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠉⠉⠉⠀⠻⠥⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⠂⠀⢰⢂⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠀⣼⡆⠀⣤⡄⠀⣤⡀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣧⢹⡇⠀⠀⢀⣤⣤⠂⣈⣛⢹⡟⣦⣤⡲⠖⠂⠠⢛⣾⣿⡉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠌⣿⣠⠊⣡⠔⢩⡿⠟⢉⠫⠿⢿⣳⡗⣲⡂⣂⡚⠞⠋⠁⠚⢩⡕⠿⣦⣀⠂
⠀⠀⠀⠀⠀⠀⣠⣶⢋⣟⢁⡜⢁⡞⢷⡶⢊⠀⠐⢸⡇⢶⡰⠀⠀⠉⠀⠀⠋⠀⠆⠉⠀⡼⢚⡦⠑⠢
⠀⠀⠀⠀⠀⢶⡆⠂⢸⠑⣿⡆⡟⢿⠃⡔⠁⣶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣆⡄⡀
⠀⠀⠀⠀⠀⠈⢻⣦⠸⡄⢰⣴⣇⠸⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡴⡛
⠀⠀⠀⠀⠀⠀⠀⠻⣦⢣⡈⢦⠹⡄⠳⣼⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠇
⠀⠀⠀⠀⠀⠀⡈⠀⢀⠔⠿⡾⢳⡛⠟⠙⢶⠦⣀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢻⡗
⠀⠀⠀⠀⠀⢸⣇⠔⠁⠀⠀⠨⣿⣿⣷⣟⣶⣶⢾⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣓⡄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠂⢀⡌⠂⢜⠿⢯⣃⣴⢫⣉⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣰⡱⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢠⡁⠞⢩⡦⠍⣿⣭⡿⣵⠆⢠⠄⣀⡀⠀⠀⠴⢘⣁⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠩⠿⠚⠛⠋⠀⠘⡋⠐⡈⠹⣡⣒⡀⠆⣨⠟⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠃⣨⠴⠂⢘⡋⠙⡋⠜⠁⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠸⠂⠈⢁⣴⠟
⣶⢶⢶⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡯⠉⠉⠉⢹

⣷⣦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣠⣴
⣿⣿⣿⣦⡀⠀⢸⡇⠀⢀⣤⣾⣿⣿⣿⣿⣷⣄⠀⠀⢸⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣀⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠁⠙⢦⣸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠈⠑⢄⢸⢀⡴⠋⠈⠻⣿⣿⣿⡿⠋⠈⠳⣄⣿⣠⡞⠃⠉⠻⣿⣿⣿⠟⠁⠙⢷⡄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⢉⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⠀⣠⣾⣿⣿⣷⣆⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣷⣀⠀⠀⠀⣿⠀⠀⠀⣀⣴⣿⣿⣿⣦⡀
⡇⢀⣤⠞⢿⣿⣿⣿⣿⡿⠟⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⡿⠳⣄⠀⢸⠀⢀⡔⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⡞⢿⣿⣿⣿⣿⣿⠿⢦⡀
⣷⠟⠁⠀⠀⠙⢿⡿⠋⠀⠀⠀⠙⢾⡶⠋⠀⠀⠈⠻⣿⡿⠋⠀⠀⠈⠳⣾⠖⠁⠀⠀⠈⠻⣿⠟⠁⠀⠀⠈⠳⣿⠞⠃⠀⠀⠙⢿⣿⠟⠁⠀⠀⠙⢶
⣿⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣾⣷⣄⡀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⣿⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴
⣿⣿⣿⣦⡀⠀⠀⡇⠀⠀⣠⣾⣿⣿⣿⣿⣷⣆⡀⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠀⠙⢦⣀⣇⡴⠋⠀⠙⢿⣿⣿⡿⠋⠀⠙⢦⢸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠀⠙⢆⣿⣠⡾⠋⠈⠻⣿⣿⣿⠟⠉⠈⠳⣄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⡇⢀⡴⠛⢿⣿⣿⣿⣿⡿⠛⢦⡀⢸⡇⢀⡴⠛⢿⣿⣿⣿⣿⠿⠛⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⢸⠀⣠⠔⢿⣿⣿⣿⣿⣿⠟⢦⡀
⣷⠋⠀⠀⠀⠉⢿⡿⠋⠀⠀⠀⠙⢾⡷⠋⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠙⢾⡶⠋⠀⠀⠀⠹⢿⠟⠁⠀⠀⠈⠳⣾⠞⠁⠀⠀⠘⢻⣿⠟⠁⠀⠀⠙⢦
⣿⣷⣄⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣾
⣿⣿⣿⡷⣄⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣦⡀⠀⢸⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⡇⠀⠀⣠⣾⣿⣿
⣿⠟⠁⠀⠈⠳⣼⣇⠴⠉⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠈⠻⣿⣿⣿⠟⠁⠈⠳⣄⣇⣠⠞⠁⠙⢿⣿
⡇⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⢀⣽⣧⡀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣀⣿⣅⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣨⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣧⡄⠀⠀⢸⡇⠀⠀⣠⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⣿⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀
⡇⣠⠖⠙⠻⣿⣿⣿⣿⠟⠉⠣⣄⢸⠇⣠⠞⠙⢿⣿⣿⣿⣿⠟⠙⢦⡀⢸⠀⣠⠞⠛⢿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⠞⠻⣿⣿⣿⣿⣿⠟⠢⣄

⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⢸⠂⠙⠀⠀⠀⠀⠀⠄⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢀⠤⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠐⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢁⣤⡖⠂⠉⠉
⠀⠠⠀⠠⠀⠁⠀⠉⠁⠀⠀⢐⡿⠀⠐⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢃⣼⠶⢋⣌⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⣠⡾⠋⠀⠀⠆⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡤⢠⣆⠀⠈⠀⠀⠀⠁⠀⠀⢀⣼⠃⠀⠀⠀⠐⠖⠛⡟⠛⠏⢁⠀⠀⠊⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣟⠀⠔⠀⠀⠀⠀⠀⠀⠈⠂⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⣨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡈⠥⣀⠀⠀⠀⠀⠀⢈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠙⣧⡈⡀⠀⠀⡀⠀⠀⢠⡀⠀⠛⠀⢀⠀⠀⠀⢀⠀⠀⠀⢈⡓⠀⠀⠀⠔⠁
⢀⠀⠀⠀⠀⠐⠂⠘⠀⠉⠀⠄⢀⡷⢄⠘⣦⡀⠨⢽⣦⡀⠱⢄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉
⢁⠤⠴⠥⢤⣀⡠⠄⠀⠀⠀⠀⣨⠁⠁⢨⠁⠑⠀⠀⠹⡝⠿⡸⣇⠀⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣠⠛⠀⠉⡇⠀⠀⣧⠀⠘⡜⡆⢡
⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠀⠀⡴⠂⠀⠀⡰⠂⠀⡀⠊⠃⢀⠂⣴⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢤⠄⢁⡠⠞⠁⢀⠤⡂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠑⢲⠆⠀⠀⠈⠔⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠋⠉⠀⠈⢀⣀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠇⠀⢠⠃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢤⣾⢁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠄⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠐
⡤⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠚⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣉⣷⡄⠉⠂
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⡴⢾⣿⠋⠉⠀⠠⠀⢆
⠀⠀⡀⠀⠀⠨⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠒⠉⢁⠪⠕⠁⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⢀⡀⣀⡄
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⣀⣠⡤⢴⠠⠊⣀⠀⠠⠄⠀⠀⠀⡆⣇⠀⢳⣁⡠⠂⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡠⢖⣪⡷⠋⢈⡤⠞⠏⠀⠒⠒⠒⠲⠲⠾⠓⠛⠘⠋⠁⠀⠀⠈⢀⣼⣤⠏
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠌⠉⠐⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠤⠀⡀⠀⠀⠀⠀⠠⠆⣸⠁⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠠⠀⠈⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢀⠄⠉⠀⠤⠄⠀⠀⠈⠀⢀⣀⣀⣉⠀⠒⠀⠀⡀⢀⡀⠀⢤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠁⠀⠄⡠⠀⠁⠠⠠⠆⠊⠀⠀⠀⠀⠀⠁⠀⠀⠁⠒⢄⣀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠂⠐⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠉⠀⠘⠠
⠀⠀⠀⣀⣠⣤⣤⡤⡜⠉⠀⠀⠁⠘
⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠉⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⠉⠛⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠉⠉⠓⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠓⠲⢤⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠶⢤⣈⠹⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣛⠉⠳⣄⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⢿⡛⠙⢯⡀⠀⠈⠓⢦⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠁⠀⠀⠙⢦⡀⢻⡄⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣶⣿⣿⠃
⠀⠀⠀⢀⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⣠⡿⠃⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⢀⣼⠏⠀⠀⠀⠀⠀⠀⢨⣿⣿⠁⢀⣴⡿⠿⠛⠻⠿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣿⠿⠟⠛⠻⠿⣿⣭⡉⠉
⣾⣇⡀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣴⡟⠁⠀⣀⣤⣤⣀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⣀⣤⣄⡀⠀⠙⣿⡄
⠘⢿⣿⣶⣤⣀⣀⣀⣼⣿⡿⢰⡟⠀⠀⣾⠋⠀⠈⠙⣧⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡾⠉⠀⠈⠻⣆⠀⠸⣿
⠀⠀⠉⠛⠿⢿⣿⣿⣿⡿⠁⢸⡇⠀⠀⢷⠀⠀⠀⢀⡿⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇⠀⠀⠀⢠⡏⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣄⠀⠈⠙⠒⠒⠋⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠓⠒⠚⠋⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠄⠘

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣹⡿⠉⠙⠛⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⢋⣭⣴⣿⣤⣖⣦⣤⣀⣀⡙⢿⣿⣿⣿⣿⣿⣿⣞⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣟⡀⠀⠀⣉⣿⣿⣿⣿⠿⠿⠛⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⠈⣻⣿⣿⡿⢿⣟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣴⣾⢿⡿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡄⠊⠀⢀⣠⣾⡟⠙⠣⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⡠⠊⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠁
⠀⠀⠀⠘⠀⠀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⡀⠀⠰⠷⠀⠀⣼⠃⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⢿⣄⡀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠙⠻⣿⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠉⠉⠉⠉⣹⣿⣏⠉⠉⠉⠉⣹⣿⣍⠉⠉⠉⠉⣹⣿⡍⠉⠉⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠀⠙⠳⠶⠶⠞⠋⠀⠙⠳⠶⠶⠞⠋⠀⠙⠷⠶⠶⠟⠉







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣀⣀⡀⠀⠀⣰⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠉⠙⠃⠀⠀⠃⢠⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⢡⠀⠀⠃⣀⣸⡤⠤⣴⡿⢀⣀⡀⠘⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠑⠢⢄⡀⠉⠀⣠⣼⡤⠖⠋⢁⣀⡤⣾⠿⡧⠴⣶⣏⣥⣦⣄⡀⠀⠄⠙⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣷⣤⠟⠛⢡⣴⣚⡙⣿⠧⢿⢺⠛⢳⡖⠠⠷⣿⣿⠛⠳⠄⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣀⠀⠀⠀⠀⡼⠛⢀⡔⢡⣈⠿⢯⣵⣤⠚⠻⣍⣭⡏⢘⠂⣤⣍⢐⣨⡿⢏⣂⡀⠘⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡼⡁⣠⠿⢃⣴⠙⡷⠾⣑⠤⠛⢩⣤⢰⠾⠿⠈⡉⠂⠝⠛⣼⠂⠙⣀⣲⠆⠳⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⢲⣶⡾⢠⣴⡋⣶⣮⢛⡜⠁⣴⠉⠀⠀⢠⡉⠉⠀⠀⠀⠀⠺⠶⠉⠻⣤⠀⣙⡅⢀⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢻⡇⢀⠟⢿⡞⢛⡞⢿⡎⣾⡇⠉⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢜⢠⣛⠣⢲⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢿⡟⣿⡾⠐⣾⣄⡹⢷⡞⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢁⡐⢀⠗⠚⡵
⠀⠀⠀⠀⣀⣤⣾⠋⠉⠁⠀⠀⠀⠘⣷⣸⣷⢀⡏⢙⡃⢰⠉⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡋⢘⡛⠀⢆
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢿⣷⡄⡇⢀⡧⣼⣅⡹⢷⠜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠸⢈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠔⡏⠻⡷⢮⠉⡷⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣙⢂⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠷⣵⡀⢃⠀⡗⢸⡀⣽⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣾⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣟⢾⡤⢾⠈⣇⣼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠛⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⣹⠀⢇⣈⡄⢰⠀⠇⠶⡦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣄⡀⢰⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⢧⣈⢟⣿⡀⠣⠘⡀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠀⢰⠺⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⢻⣈⣮⡛⣄⢷⡾⢌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⠅⢨⡁⠈⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠀⠈⠝⣍⡛⡀⠢⠖⢠⣛⢿⡄⣰⢆⣀⢀⣿⠁⡴⠘⣻⠎⣷⠟⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠿⠿⣿⡳⠎⣻⢧⣷⣿⡤⣘⢃⣀⢛⠁⣠⣬⠆⠚⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡐⠛⣻⠟⢩⠡⠃⠰⠋⠘⠿⠀⠻⠋⠁
⣤⣤⡤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠛⢪⡇⠈⠑⠂
⡿⠴⠆⠟⠥⢼⡇
⡇⠁⠀⠠⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⣤⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⢸⣧⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⢀⣼⠗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣶⡶⠟⠁
⠀⠀⠀⠀⢠⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⡿⠋⠁⠀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣄⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⡉⠉⢹⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠁⠀⣱⠀⠀⠀⢰⠁⠀⠱
⠀⠀⠀⠀⠀⢀⡠⠔⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡠⠔⠚⠋⠉⠉⠳⠖⠄⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠐⠦⣄⡀⠀⠀⣰⠏⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢂⣀
⠀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⣥⣤⣤⡶⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⣷⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣼⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⡟⠋⠉⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣦⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟⠛⢿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣧⣤⣼⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠠⢄⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠁⡃⠘⠇⠸⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⢸⠀⢶⠀⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠆⢠⣤⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⠀⠘⡀⢀⣤⢓⡢
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡀⢀⣤⠀⠀⠀⠑⡌⠁⢀⢗⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢘⣿⣿⣿⣄⠀⠀⠀⠈⣦⡉⠁⢈⢞⢆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠛⠛⢿⠏⠁⠀⠀⠀⠀⣿⣿⣶⣌⡁⠀⣬⠞⠢⣖⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠈⠑⠂⠤⠌⠋⠠⠋⠉⣁⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠉⠐⠒⠤⢄⡸⣿⣿⡄⠀⠀⠀⠀⠀⢀⠖⠉⠁⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⣎⡁⠒⠠⣄⣉⠛⠷⠄⣀⠀⠀⠀⣸⣀⡀⠤⠐⣪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣯⣴⣶⣶⣶⣾⡿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠻⣿⣿⠁⠀⠀⠀⠀⠀⠈⢻⡿⠛⠉⠉⠙⢛⢻⠁⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣔⣭⡕⡀⠸⡟⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⢪⣭⢢⠀⠙⢆⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠭⠛⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠪⠍⠊⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠒⠁

⣧⣿⢹⢳⣿⡿⣿⢸⡟⣿⢸⡇⣿⡈⣿⣸⡏⣇⢹⠻⡇⠀⣷⠀⠀⠀⠀⠀⠀⣰⠇⠀⣠⢞⡥⢖⣊⡩⠭
⣸⡿⡾⡞⣿⣿⡿⣼⠃⣿⢸⡇⢸⡇⣿⡇⣿⢹⢸⡆⠳⢿⠈⠢⣤⣀⠀⠀⢰⡏⢀⣼⢟⣵⣾⣷⠶⢿⣿
⣯⢣⣧⢇⡿⣸⠇⣿⢠⣿⢸⡇⢸⡇⣿⡇⣿⢸⠸⡅⠀⢸⡆⠀⠈⠙⠦⡄⣸⡇⣼⣿⣿⣿⢕⣪⡭⠵⠶
⠃⣼⢝⡾⢱⡟⣸⠇⢸⡟⢸⡇⣼⠇⣿⡇⣿⢸⢠⡇⠀⣼⡅⠀⠀⠀⠀⢌⢻⣽⣿⣿⣿⡷⢋⠤⣒⣨⣭
⣾⢏⡾⣱⠏⢠⡟⢀⡿⢁⡿⣱⠏⢠⣿⢳⣿⣿⡾⠀⠀⡟⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⣱⡷⠛⠉⠁
⡱⢋⡴⠋⣠⠏⣠⠟⣡⠟⣱⠏⣠⣾⣷⢋⣾⠞⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⡟
⡥⢛⡴⢊⡥⢞⣡⢞⣡⠞⢁⣴⣿⣿⣷⡿⠃⠀⠀⠀⢹⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣚⡵⢚⡭⣖⡯⠖⢋⣠⣴⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣵⣾⠿⣚⣥⣴⣿⣿⣿⣟⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣽⣶⣿⣯⣅⣈⠀⠀⠈⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠐⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⡀⠀⠈⠈⠉⠙⠛⠷⣦⡀⠀⠀⠀⠈⠑⢝⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⡛⠻⠶⣦⣄⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢹⣧⣠⡾⠋⠂⠒⠀⠀⠀⠀⠀⠀⢀⢀⣠⠃
⠙⠷⢤⣀⡉⠛⠷⣦⣄⠀⠀⠈⠳⣄⠀⠀⠀⣿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠠⠀⠐
⠀⠀⠀⠈⠉⠓⠲⢬⡙⢿⣦⡀⠀⠘⢧⠀⣠⣿⠂⠀⠀⠀⠀⠀⠀⠀⢰⣾⢏⠠⠂
⠛⠛⠛⠲⠶⢤⣀⠐⠙⢷⣝⢷⡄⠀⢸⣿⣻⣥⣄⣀⠀⠀⠀⠀⠀⡄⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀
⣭⣭⣭⣕⣢⢄⡈⠓⢆⠀⢹⡎⣷⣠⡟⠡⠄⠀⠀⠉⠙⠲⣄⠀⠀⢹⣾⣏⠄⠀⣀⠠⡰⠄⠊⠀⠉⠉⠉
⣶⠿⠳⠶⢮⡑⢯⣢⡈⢳⠀⣿⣿⣷⣦⠶⠤⣔⡢⠀⠀⠀⠈⠳⡄⠸⣿⢏⣴⠪⠔⠁⢀⡠⣶⠿⣛⣛⣛
⣉⣉⡛⠛⠦⡍⢲⣹⡱⡀⡇⣹⣿⣿⣭⣽⣛⢶⣭⡳⢅⠢⠀⠀⢹⢸⡿⠃⠁⢀⣠⡾⣛⠭⣒⠭⠒⣒⣲
⡤⢭⣉⠓⠦⡈⢆⢹⣧⢱⣧⡿⠉⠀⠀⠀⠈⠙⢪⣻⣮⠳⡱⡀⢸⣼⠃⢀⣴⡿⡫⡪⠔⠉⣀⣴⡿⢗⡯
⠤⢭⣐⠭⡢⡙⡼⡄⣿⢸⣿⣷⣖⡒⠲⢄⠀⠀⠀⢹⣟⢇⢻⣧⣾⠃⣴⣻⡫⡪⠊⢀⣴⣾⢟⡵⣪⠵⣚

⣀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⠀⡂⠀⠀⠀⠀⠐⠃⠆⠀⢰⠸⠂⠀⠀⠀⢀
⣿⣿⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠓⠁
⠛⢻⠁⠀⠀⠀⠀⠀⣀⠀⠀⠆
⢰⠈⡄⠀⠀⠀⠀⢀⠈⠘⠁⠀⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⣀
⠀⢣⠐⠀⠀⠀⠀⢸⠀⢸⠘⡼⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡞⠂⠐⠂⡀⠂
⠀⠀⠉⠀⠀⠀⠀⢎⢣⡘⣆⡇⡜⠁⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠤⠈⢍⣤⢀⠀⠀⠀⠀⠧⢔⣬⣴⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢣⢹⣸⣿⣧⠀⠀⠀⠀⠀⣄⢆⢋⡦⠀⠀⠀⠒⠨⠔⡤⡁⠀⠀⠀⠀⣰⣿⣦
⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠉⠀⢇⠀⠀⠀⣠⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡇⠀⠀⠀⠀⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠐⡞⢻⠿⣿⠛⠛⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡴⣤⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠲⠿⠇⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣈⣔⠹⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡻⠃
⠀⠀⠀⠀⠐⠇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠈⠀⠀⠀⠀⡾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡀⠄
⠀⠀⠠⠦⠀⠶⠄⠁⠄⠀⠀⠀⠀⠀⠀⡆⡲⠣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠉⠡
⠀⢀⠚⠿⢂⣤⡀⠿⠂⠀⠀⠀⠀⣳⠉⠃⠁⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⢠⡄⠄⢄⠈⠀⠀⠀⠆⠘
⠀⠈⠫⠤⠄⠤⠶⠤⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡀⣂⠀⠀⠀⠀⡀⠀⠀⡄⣸⠈⡁⡀⠀⠔
⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⢀⠀⠄⠀⠀⠀⠙⢢⣦⣁⣿⣾⣿⣿⣆⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠠⠞⠉⠉⠉⠉⠀⠸⠯⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡧⣭⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⡀
⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣏⡇⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣟⢡⢗⣄
⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⣼⡏⠀⠈⠘⢻⠆

⠀⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠤⠤⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣧⡴⠒⠒⠛⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⡶⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢸⠀⣶⡾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢠⣾⠟⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠉⠀⠀⣿⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⣀⠘⢿⠆⠈⣿⡦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠁⠓⠦⣀⠀⢿⡧⠈⠙⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠠⠀⠌⢈⠒⠤⢀⡈⠛⠃⠀⠉⢓⡦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢀⠂⠀⠀⠄⠠⠁⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⢀⠌⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠉⠉⠁⠐⠒⠒⠂⠀⠤⠀⠤⠄⢠⣀⣀⢀⠀⡐⠀⠂⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠒⠐⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠶⠦⠶⠄⠠⠄⠀⠀⣀⠀⠀⠀⠠⠄⠠⠤⠐⠒⢈⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠛⠛⠻⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠧⠝⠛⠿⢿⣿⠿⠋⠁⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡠⠐⠒⠢⢄⠈⢻⠟⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⢿⠋⢀⠤⠒⠒⠢⣄⠈⢦⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠎⠐⣨⢤⡑⡄⠣⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡇⢀⠋⡔⣩⢬⡑⠈⠂⠀⡇⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢇⠀⢆⠠⡙⠞⡣⠃⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄⠣⡙⠞⡡⢀⠄⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠑⠤⠤⠄⠊⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠈⠒⠤⠤⠔⠊⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⢬⣳⣀⡤⠤⠤⠒⠒⠚⡏⠉⠉⠉⠉⠉⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣶⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⡇⠾⠟⢯
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⠿⠁⢱⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⢼⡇⠙⣶⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⢼⡇⠙⢻⣦⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢉⠒⠤⠀⠘⠛⠀⢉⣒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡗⠠⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣀⡀⠀⠈⠉⠁⠀⠀⠒⠒⠠⠤⣄⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢊⡇
⠀⠀⠀⠀⠀⠀⠘⠒⢶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣈⣉⣉⣉⣁⣉⣉⣉⣁⣠⣤⣤⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠿⠿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⠽⠿⠿⣿⣿⠿⠋⠳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⢁⠄⠒⠢⢄⠙⢿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠿⡿⠉⡠⠐⠐⠠⡈⠳⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⡎⠀⠃⢊⡴⡄⠄⠆⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠁⡌⠐⣡⢦⡑⠘⠀⢹⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⢣⠀⢆⠂⠙⠅⣁⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠰⡀⠨⠫⠀⡰⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡉⠀⢉⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠁⠀⢁⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢣⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣿⣿⣿⣆⣠⣾⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣷⣦⣠⡿
⠀⠀⠀⠈⠹⠿⠛⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣠⠀⠀⠀⠀⢀⡾⠂⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣿⣧⢶⣀⣠⣴⠟⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠻⣆⡘⢿⣿⣿⣿⣿⣿⣯⣅⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠈⠛⠛⠛⠉⣻⣿⣿⣿⣿⣿⡷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣶⠄⣴⣶⣆⢈⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⣿⠁
⠀⠀⠀⠀⢠⣿⠀⣻⣿⣯⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠟⠀⠀⢤⣄⣀⠀⣀⣻⣿⣟⣀⢾⣿⣿⠟⢐⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⢻⣿⣿⡿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠟⠉⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡿⠀⠙⣿⣟
⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠀⠀⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢀⣤⣤⣄⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢸⣿⣿⣿⠸⣿⣏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢁⣾⣿⣿⣿⡄⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣠⣿⣿⣿⣿⣿⣧⠘⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣯⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣌⠛⢿⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⡿⢃⣴⣬⠉⣙⡛⠛⣃⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⠟⣠⣿⡿⢡⣾⡿⢃⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⠋⣴⣿⠟⣠⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣛⣛⣛⣛⣁⣛⣛⣁⣘⣛⣃⣘⣛⣋⣐⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⠄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⣤⣤⣤⡙⠻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠿⢋⣡⣤⣤⣌⡉⢡⣾⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⢿⣿⡆⠹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣸⣿⡿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⢠⣶⠶⢠⣄
⠀⠀⠀⠀⠀⢸⠃⠁⠉⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆⣙⣋⣵⠦⡄⠀⠀⠀⠀⠀⢐⠃⠁⠥⠕⠀⠀⠀⠈⣷⡪⡟⣥⡟⣦
⠀⠀⠀⠀⠀⠞⢦⣦⣴⠶⠺⠀⠀⠀⠀⠀⠀⠀⠘⠱⠿⣿⢿⠞⠧⠿⠄⠀⠀⠀⠀⠈⠓⠚⠛⠜⠁⠀⠀⢠⣿⠃⠁⠛⠸⠿⡧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠃⠀⠀⠀⠈⣾⢱⠄⠰⢠⢰⡧
⠀⠀⠀⠀⠀⠀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠸⠀⠀⠀⠀⠀⠀⠆⢠⠀⠉⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡶⢣⣽⢢⡬⡬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⡤⠀⢸⢀⢀⡎⣈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠖⠐⡤⠐⠃⠉⠂⠀⠀⠀⠀⠀⠠⠤⠆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢼⠘⣼⠀⣨⠃⠀⠀⠀⠀⠀⠰⠏⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠀⠒⠠⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⣦⡸⣆⣏⣰⠁⠀⠀⠀⠀⠀⠠⣜⣣⣡⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣠⣆⣉⣠⣡⠏
⠀⠀⠀⠀⠀⡠⡔⢊⠀⠀⠀⠀⡀⠀⠀⠙⣎⢷⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣿⣿⡁
⠀⠀⠀⠀⠘⠛⠒⠒⣶⠒⠒⠐⠓⠀⠀⠀⠉⠈⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⢻⡀⠀⠀⠀⢸⠋⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡶⣦⣤⡀⠁⠀⠀⠀⢰⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣱⣶⣾⣛⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢉⣔⣻⠏⣿⡖⠀⠀⠀⠈⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡋⡽⠻⣿⢡⠉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⣿⠏⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠖⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠙⠉⣉⡁⠀⠀⠀⠀⠀⡀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠅⠰⠶⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⠀⠈⠀⡀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠚⠷⠶⢀⣀⠀⠿⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⡟⡜⠀⠀⠀⢀⡾⠦⠀⠀⠀⠀⠀⠘⢃⠉⠘⢇⡄⢠⠀⡀⠁⠐
⠀⠀⠀⠀⠀⠀⠈⢷⣂⣀⣈⣍⣥⣄⠐⣿⠀⠀⠀⢀⢤⣸⢤⠚⠈⠁⠀⠀⠀⣼⢣⣑⣁⠀⠀⠀⠀⣄⠈⡀⢹⢰⣾⢸⣄⣇⡀⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⡀⠁⠀⠀⠀⠀⢭⡇⠈⠁⠀⠀⠀⠀⠀⣼⢀⡕⢠⠄⡇⠀⠀⠀⠈⠑⣴⣬⣼⣿⣿⡿⢿⣿⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠈⡁⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠘⠿⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠁

⠀⠀⠀⢀⠀⠀⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠂
⠀⠀⠀⢠⡀⠐⠶⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠆⠘⡁
⠀⠀⠀⠀⢁⠂⠀⠡⣔⠄⣀⠀⠀⠀⠀⠀⢠⠄⠀⠁⠀⠘⠠⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠘⠱⠖⠀⠄⠀⠀⠀⠀⠠⠄⡄⠀⠀⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣄⣤⡤⢤⣤⣤⠤⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣈⠉⡉⡉⡉⢉⡩⠉⢉⠡⠀⠀⠀⠀⠂⠛⠬⠁⠀⠈⠰⠉⠈⠂⠀⠀⠀⠀⠈⡈⢠⡄⠁⠁⣠⠈⡀⠁⠀⠀⠀⠀⣠⢠⡐⣛⣅⢀⣄⠀⣠⡀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠁⠀⠀⠁

⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠
⠀⠀⣫⠁⠀⠠⢎⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡧⣶⠄⠂
⠀⠀⢄⣀⡀⠐⠀⠄⡑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠛⠷⠄⢀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠈⠀⠈⠁⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠈⠉⠀⠈⠀⡀⠈⠁⠀⠀⠀⠀⠤⠂⢀⠤⡀⡠⢖⠒⠲⡇⠀⠀⠀⠀⠑⢀⠀⢀⡐⠀⢀⠀⢁⡐⠀⠀⠀⠀⠐⠖⠖⠒⠒⠒⠒⠲⠶⠒
⠀⠀⠙⠚⠋⠓⠞⠉⠓⠋⠙⠖⠃⠀⠀⠀⠀⠑⠈⠙⠀⠉⠁⠈⠉⠐⠁⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠉⠀⠉⠁⠀⠀⠀⠀⠠⠐⠂⠀⠀⠆⠄⠀⠂⠠

⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠂⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲
⠀⠀⡀⠀⠀⠀⠀⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢪⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⡀
⠀⠀⠁⠠⠀⠀⠀⠐⠀⠀⠐⠤⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠐⠂
⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠙⠒⠀⠀⠀⢠⣤⠀⢠⢤⠀⢠⢤⠀⢤⢬⠀⠀⠀⢀⠐⡴⢦⠀⡴⢤⠀⡴⢶⢊⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠈⢀⠐⠀⠀⠀⠠⠢⠇⠥⠲⠀⠗⠰⠁⠴⠢⠀⠀⠀⠈⠁⠉⠙⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠠⣤⡤⢤⡤⢤⣤⠤⣤⡄


⠀⠀⠀⠆⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡀⠐⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⡈⠀⠀⠀⠀⠈⠢⢤⣀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠉⠀⠀⠠⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠝⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⠶⣤⠄⡀
⠀⠀⢃⣀⣀⠀⠀⠀⠀⢈⣘⣀⠐⠀⠀⠀⠀⣰⡔⢲⣠⠒⢦⣰⠒⣴⡀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⢀
⠀⠀⠞⠀⠠⠘⠁⠉⠑⠀⠀⠘⠆⠀⠀⠀⠀⠧⠞⠘⠤⠃⠣⠔⠘⠤⠏⠀⠀⠀⠀⠀⠖⠖⠶⠒⠒⠶⠰⠲⠂⠀⠀⠀⠀⠈⠀⠐⠀⠀⠂⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣠⣴⣤⡀⠀⠀⠀⠀⣠⡤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠿⢛⣛⣒⣢⣤⠋⠀⠀⠀⠈⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠳⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠒⠀⠀⠈⠉⠛⠳⢦⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣠⡤⠴⠶⢖⡳⢾⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⠟⢻⣛⣛⣛⣫⠙⢛⡛⣻⣆
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠙⣢⣠⢜⣅⣙⡫⠶⣿⡓⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⢜⡁⠋⡀⣁⣀⣠⢋⡔⢿⠷⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢹⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⣾⠔⢉⡝⣡⣩⣬⣿⢟⣴⣚⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠛⣭⠒⠴⣏⣼⣋⠾⡿⢿⣂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⣎⣅⢩⢍⡴⠟⠻⣷⡹⣆
⠀⠀⠀⢀⣠⠔⠂⠀⠙⠲⣕⢍⠿⠙⠛⢿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠚⠲⢬⣍⣩⣴⣿⣇
⠀⠀⢠⠋⣤⠶⢔⣦⠀⠀⣿⡾⢶⡘⠛⠛⢻⣿⣿⣿⣿⡿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡟
⠀⠀⢸⠰⣿⠀⡄⠁⢆⣴⣿⡿⠂⠙⠻⢿⣿⣿⡿⠟⠋⠀⠀⠈⠉⠛⣿⢷⣶⣦⣤⣀⣀⣀⣀⠀⢀⣀⣀⣸⣿⣿⠃
⠀⠀⠀⢆⠹⣷⣀⡞⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⠀⣹⢹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠁⠂⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠉⠉⠙⠲⣧⢩⠉⣻⣿⣿⠟⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣀⠀⠈⠢⡀⠀⠈⢧⣀⣿⢿⣿⡆⠀⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢤⡄⠄⠈⠄⢠⣾⠟⠛⣎⣿⡿⠤⣚⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢿⣾⣀⢀⠀⣿⣿⣿⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡛⠓⢊⣰⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⠶⡶⢶⣶⣶⠂⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣷⣾⣏⠀⣤⣤⣿⢹⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠛⠛⡏⢻⡍⠀⠋⠉⢻⣷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⠀⠀⢸⠃⠀⢰⣮⠿⣿⣼⣾⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⡿⠀⠀⠘⠀⢈⠙⠀⠈⢀⡠⢿⣿⢹⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⡿⠃⠀⠀⠀⠀⣼⣾⠀⡄⠘⠌⢸⡇⢿⡿⣟⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣇⠀⠐⢸⣇⠀⠹⣿⡟⡋⣧⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠆⠀⠘⣹⠗⠘⣽⡿⡾⢿⣛⡟⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣙⣿⣬⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠀⠐⡿⣇⠈⣿⣿⣳⠀⢠⣁⡝⢷⣏⡣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣄⣤⣤⣄⣀⣤⣐⡇⠒⡲⠁⠄⣨⠻⠶⣿⢻⣿⣀⠂⠭⡀⠀⣷⣿⣿⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣶⣶⣶⣥⣍⣙⣿⣾⣤⣀⡴⠃⠀⠀⢀⠀⢽⣇⠀⠀⠀⠁⠂⠄⡈⢻⣿⡻⡻⢿⣟⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣗⡂⠠⣤⠄⣠⡈⢿⡄⠀⠀⠀⢀⡀⠉⠹⣿⣷⡌⠀⢘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠛⠿⠿⢿⣧⣶⡿⠿⠿⡿⣿⣿⣿⣿⣷⣤⣮⣙⣿⣷⣶⣄⣢⣹⣁⣀⣀⣿⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣏⠓⣦⣤⣤⣤⣴⣶⡿⠷⠬⣭⣛⡛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢛⣿⣋⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⡿⣧⣿⣭⣟⣿⣿⡇⠀⠀⠀⠈⠙⠛⠛⠳⠶⠖⠒⣿⣿⣷⠿⣴⣷⣤⣀⣴⣿⣽⣿⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⠉⢉⣳⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢿⠁⣽⡷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣩⣿⣷⣿⣿⡇⣿⣻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢳⣹⣿⡿⠇
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣋⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣛⣿⣩⡿⡟⣿⣿⣿⡏⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⢿⣿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢷⣿⣿⡷⠀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡟⠯⠓⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⣀⣀⣀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣦⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣟⢻⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣝⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣿⣿⡿⠷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣋⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡝⠛⠋⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⣠⣴⣿⣿⡿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⡿⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠛⣛⣋⢉⠁⣀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠻⠟⠟⠻⠻⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣷⣬⣙⣿⣿⣿⡄⠀⢀⡀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣠⡟⢾⣷⣙⣿⣿⣿⣿⣿⣿⣄⠻⠃⣰⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣇⢻⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠍⣙⡟⠛⠛⠛⠛⠛⢿⣛⣛⣛⣉⣩⣭⣹⠋⠉⠉⠉⠉⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣶⡿⠋⣀⣀⣀⣀⠀⠀⠀⠉⠉⢁⣬⠁⠀⠁⠀⠀⠀⠀⠀⢸⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢿⣶⡶⢢⠆⠰⡴⢶⣶⡶⢶⠖⠶⡶⣶⣶⡶⢲⠖⠶⡶⣶⣾⡿⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠒⠉⠀⠀⠈⠒⠒⠒⠁⠀⠀⠉⠒⠆⠒⠁⠀⠀⠙⠂⠔⠚⠁







⣀⣀⣀⠀⡀⣀⣀⣀⠀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣗⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣗⢾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⠟⠂⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⡀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⣠⣤⣤⣤⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⠿⢻⠟⠉⠉⠉⠙⠛⠿⠛⢛⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⣾⣷⣿⠟⡡⠚⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠟⠭⠓⠉⠀⠀⠀⠀⠀⠀⠐⠾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠙⠛⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⠀⣀⣠⠶⠾⢿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡫⠄⠀⠀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣧⠲⣄⣀⣠⣴⠞⣁⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢋⠁⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣄
⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⢿⣿⣿⣿⣿⠿⣿
⣿⣷⣾⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣂⣠⣄⡀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠿⠟⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡃⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⠤⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⠹⠆⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣷⣶⣤⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⠇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⡼⢩⠟⣹⣿⣿⠛⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⢀⠏⢀⣿⣿⣿⡆⢠⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠀⠈⠁⡟⢸⡇⣼⡟⠃⠇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢨⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢿⠁⣿⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣪⣭⣵⡲⣄⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⣿⣿⣿⡿⠮⠷⠤⢤⣀⣀⠞⠉⠀⠀⠀⠈⠁⠐⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣋⣤⣴⣶⣶⣶⣶⣾⡟⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠈⠛⠿⢿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡏⣿⠓⠦⠐⠠⠀⠈⠉⠙⠛⠦⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣻⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⣤⣤⡴⠖⠚⣛⢛⣣⠾⣿⡿⠟⡄
⠀⠀⠀⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢮⡽⠶⣾⠿⠷⠒⢒⠋⣉⠲⠶⠶⠾⣷⡄
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣶⢐⡛⢞⡙⠋⠒⢚⠇⠢⣶⣶⠖⣩⣿⣄⡀
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣀⡀⢙⣤⠖⣡⡶⣴⣤⣤⠚⢫⣿⣦⢉⣶⡀
⠀⠀⠀⠀⠀⠀⢣⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢫⠴⣃⠘⠓⢀⠀⠀⢀⣀⡠⢋⡴⠻⣿⣿⠖⠶⣄
⠀⠀⠀⠀⠀⠀⢸⣏⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠟⢀⣴⢃⡌⠛⢉⡟⠡⠚⠁⢲⣾⣶⢎⣤⣼⠷⢤
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⢀⢾⡎⣴⠷⣾⣟⠛⠟⣡⠟⣶⣶⢆⣼⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠽⠉⢙⣶⠌⡡⠞⢧⣶⡟⣡⣚⣛⡛⢛⣯⡱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠙⠈⡳⡦⢋⠘⢉⣩⢍⣴⠿⠿⣿⣦⡙⣦⡀
⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⠻⣛⠿⣿⡿⢿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠗⠋⡟⠜⢡⣟⡒⠶⠦⠼⠿⠃⣸⣿
⠀⣠⠶⠊⢁⠀⠀⠀⠀⠈⢳⡌⢣⢸⠃⣀⣀⣙⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⠳⠶⠶⣿⣿⣿
⣸⢁⣴⡶⠶⢌⡲⡀⠀⠀⢸⣿⡿⢻⣆⠉⠙⠛⠛⣻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⡇⢸⣿⣄⢠⡀⢳⠰⡄⣠⣿⣿⣷⢄⡛⠿⢷⣶⣶⣿⣿⣿⣿⣿⠿⠋⠀⠀⠈⠉⠛⠿⢿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⢺⣿⣿⣿
⢡⠀⣿⣦⣈⠅⢨⠀⣿⣿⣿⣿⡇⠀⠈⠁⠀⠒⠛⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠤⣿⡟⠻⢿⣿⣶⣶⣶⣤⣦⣤⣤⣤⣶⣶⣾⣿⣿⡿⠋
⠈⢧⡘⢯⣳⣤⠾⣰⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⣹⡇⠀⣰⠇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠙⠒⠦⠴⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠞⠋⠉⠉⠉⠛⠶⣇⡍⣙⢉⠻⣿⢟⣯⡿⠟⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣈⣉⠁⠢⣄⠀⠀⠀⠈⢳⡉⠀⢻⣿⣿⣿⣷⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣎⣴⡆⠈⠂⠀⠱⡀⠀⠀⠈⢷⣴⣾⣿⢹⣿⣿⡆⠀⠀⢀⡴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠭⢥⠄⢠⠀⠘⠀⢀⣴⣿⡟⠛⢹⣄⣿⣿⡿⠴⢚⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠻⣿⣦⡞⠀⡀⠀⠀⠷⣿⣿⣿⠀⠀⠈⠻⠿⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠩⣍⡿⠗⣤⠁⣰⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⢍⣉⣁⣴⡿⠟⠋⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠤⠀⠀⠀⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⢸⡟⠁⠀⠀⠉⢉⣛⠛⣿⣿⣿⣿⣯⡀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠉⠁⠀⢀⣈⣩⣿⡟⢠⣿⠉⠛⢸⣿⣿⣷⣶⣶⣶⣾⣿⣿⠟⠉⠀⠀⠼⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠠⣤⣏⣀⠀⠒⠒⠒⠛⠻⣷⣤⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢰⠿⣽⣿⣿⣶⣦⡀⠀⠙⢿⡇⠸⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠙⠛⠿⣿⣿⣿⣿⠀⠀⠈⠁⡰⣿⣿⠋⠀⠀⠀⢰⣦⣤⣤⡀⠀⠀⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠓⣖⣶⣤⣀⠀⠀⠉⢿⡟⠀⠀⠀⠀⣿⡘⣿⣄⣀⣠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣅⡒⢶⣤⣈⡙⠻⢿⣷⣦⣼⠁⢶⣷⡆⠀⣿⣿⣽⣿⣿⣿⣿⠟⠉⠀⢹⣿⣦⣄⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⣥⣜⢦⣄⠙⠳⣮⣻⣿⣷⣤⢈⣵⠿⠦⡤⠤⠖⠁⠈⠻⠿⣿⣿⣿⡇⢀⠀⠀⠘⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠀⠀⠶⠶⠿⢿⣿⣿⣿⣷⣄⠈⢿⣿⣿⡿⠋⣁⣤⣶⣶⡆⢀⣴⣷⣶⣤⡀⠈⣿⣿⣿⣿⠀⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣤⣤⣀⣤⣀⣀⣀⠀⠀⠉⠻⢿⣷⡤⠟⠋⠰⠾⠿⠟⠛⠛⠁⠈⠛⠛⠉⠉⠁⢀⣿⣿⠿⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣍⣛⣿⣛⣛⣛⣛⣛⣛⣓⣀⣈⣡⣤⡤⠤⠤⠤⣠⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣚⡉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣨⠝⡿⠛⢁⣼⣿⣿⣿⠿⠏⠈⠋⠻⠓⠂⠀⢀⣀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣦⡈⠻⢯⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡅⠊⢀⠐⠛⠉⢉⣁⠀⠀⠀⠀⡄⠀⠀⠀⢀⠈⠁⠉⠉⢉⠉⠉⠉⠉⢉⣉⠉⠁⠀⡀⠃⢴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠅⣀⠐⣬⣹⣿⠀⠻⣿⡶⢀⡀⢰⣞⣿⡋⠙⣿⣷⠆⣀⠀⣦⣿⣿⠉⢹⣯⣠⡀⢀⣚⢶⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⢾⡾⠟⠃⠀⠀⠙⠻⠿⢿⡶⠿⠑⠁⠀⠈⠛⠻⢿⢷⡿⠟⠃⠀⠀⠉⠻⠷⡾⣷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠈⠉⣉⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠐⠒⠚⠿⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⢰⣿⡅⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⠉⠛⢷⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠐⠒⠂⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠻⣄⠀⠀⠀⠶⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⢠⠀⢳⡀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⢨⠿⠦⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣤⣄⣀⣀⣀⣀⣀⢠⣤⠔⠋⠁
⠀⠀⠀⠀⢸⠀⡀⢢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⢸⠶⣽⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠄⣸⠀⠈⠙⣦⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⡏⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⡜⠁
⡄⠀⠀⡆⢹⠖⠒⠒⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠀⠀⠀⡇⠀⠀⠀⠈⠉⠲⠦⣤⡀⠀⠶
⢧⡀⠈⠉⢸⠆⠀⠀⠀⠈⠳⢦⠀⠀⠀⠀⠀⠀⣀⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠙⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⡘⠃⠀⠀⠈⠉⠛⠛⠢⠤⣄
⠀⠀⠀⠀⠀⣿⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠙⡄⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⣆⠀⠀⡗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⢧⣄⠐⠒⠈⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⠈⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠳⣤⡄⠄⠒⠠⠄⠘⢷⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠈⣿⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠙⠠⠤⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠛⣦⣤⣤⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⡍⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⠁⠈⠁⠈⠀⠈⠉⠀⠈⠉⠁⠈⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⣤⣤⣤⣤⣤⣀⡀⠴⠖⠒⠒⠶⢦⣤⠀⠖⠒⠒⠖⠢⣄⡀⠰⠶⠒⠶⠶⣤⣀⡄⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠃⠀⠀⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣸⡆⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⢀⡆⠀⡄⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠸⡇⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⢀⠀⠀⣦⠀⠀⠀⢠⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⢿⠀⠀⠀⠘⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢰⡀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⢩⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⢸⠁⠀⠈⠀⠀⠀⠀⠇⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡄⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⢁⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠇⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⠀⠀⠀⠃⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⡄⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⠸⠀⠀⡆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⡇⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⣀⡿⢧⣀⠀⠀⠀⣀⡿⢆⣀⣀⣀⣀⡠⠘⠛⠲⠶⠶⠶⠖⠉⠛⠑⠶⠆⠉⠁⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠶⡄⠐⠂⠀⠀⢲⡆⠐⠀⠀⠒⠒⠒⢶⠀⠐⠂⠐⠒⠀⠒⠀⠒⣤⠔⠂⠀⠀⠀⠀⢠⢰⣧
⣤⣤⣀⣤⣤⣤⣄⣠⣤⣤⣄⣤⣸⣷⣤⣷⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧⣠⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⣄⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⢦⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠳⡀⢀⣀⣀⣀⣀⣀⣉⠉⠉⠒⠒⠃⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢇⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠒⠢⢤⡀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⡶⠦⠤⠬⠷⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢠⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⠦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠼⠷⠒⠒⣲⢒⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⠤⢴⣚⡡⠼⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢣⠀⢀⡠⢤⡀⠀⠀⡗⠚⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⠋⠀⣎⠀⠀⢹⠀⠀⢸⡀⠀⠀⢀⣀⡤⠔⠒⠋⠉⠁⠀⠀⠀⠀⠙⠒⠤⣀⡀
⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⢷⣀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠟⠛⠉⠙⣢⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠁⣀⣠⠤⠖⠚⠉⠁⠸⡄
⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣉⠓⠒⠦⠤⣤⠤⠤⠖⠚⣉⣁⠤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠑⠢⢤⣀⣀⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⡇⠹⣄⠀⠀⠀⣀⠤⢶⡋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠦⢤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉
⠀⠀⡠⠖⠛⣩⠟⠀⠀⠈⠓⠚⠋⠁⠀⠀⠙⠶⠯⠭⠽⠿⠶⠶⠶⠶⠦⠤⠤⢤⣴⣛⠟⠒⠶⠤⠤⠤⠤⠤⠶⠒⠉⠓⠶⣋⠉⠉⠉⠓⠦⣀
⢠⠎⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⡦⣄⠀⠈⢧
⡎⠀⢀⡞⡄⡤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡍⠺⣧⠀⠈
⡇⠀⠸⡇⢧⠳⠤⢃⣠⡤⠖⠚⡏⢱⡶⢤⣀⠀⠘⠦⠴⠁⣀⡤⠒⢻⠗⢶⠤⣀⠀⠀⠹⠤⠞⠀⣀⡠⠤⣤⢤⣄⡀⠀⠀⠀⠳⠴⠃⢀⡟
⠹⣄⠀⠙⠢⢭⣬⠭⠚⠁⠀⡼⠁⠀⢳⡀⠈⠙⠷⢶⡶⠿⠋⠀⢀⠎⠀⠈⢆⠈⠙⠲⠶⣶⠶⠟⠁⠀⣰⠃⠀⠹⣌⠙⠒⠤⠤⡴⠾⠋⠀⢀⡼
⠀⠈⠢⣄⣀⠀⠀⢀⣀⡴⠊⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⣀⡠⠞⠁⠀⠀⠀⠈⠲⢄⣀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠐⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣼⠇⢀⣤⠄
⠀⠀⠀⠀⣠⠖⢻⠛⠲⣄⠀⠀⠀⠀⠀⠀⢁⠂⠁⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠃⠘⠟⢠⠞⣅⡰⠶⠾⠓
⠀⠀⠀⠀⡏⠀⣸⣇⡀⠘⡆⠀⠀⠀⠀⠀⠊⠢⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡠⠚⠉⠰⣦⣌⡛⠷⠄
⠀⠀⠀⠀⢳⣟⢹⡇⢙⡾⠁⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠘⢿⣤⣬⠉⠉
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠿⠃⠘⠿⠟⠀⠻⠿⠃⠀⠀⢰⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠠⠀⢀⣠⣾⢿⢿⢷⣿⣿⡿⡷⣤⣀⡀
⠀⠀⠀⠀⣠⡂⠛⡁⠱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠹⣯⠘⡡⠊⣡⣞⣉⡭⠟⠉
⠀⠀⠀⢸⡅⠐⢠⡁⠡⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠔⠋⠐⠁⠐⠿⠛⠉
⠀⠀⠀⠘⠶⢪⣤⣦⠷⠚⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⢐⣠⣄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠐⡈
⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⣂⣸⣿⣿⠇⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⠔
⠀⠀⠀⠀⠀⡏⠀⢿⠀⠀⠀⠀⠀⠀⠠⠤⠄⡀⠉⠉⠉⢀⠈⡉⢉⡁⠈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢃⠀⠈⡆⠀⠀⠀⠀⠀⠐⠐⠀⠒⠂⠀⠐⠒⠉⠉⠑⠒⠒⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢘⣴⣾⡶⡾⢤⣀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⡟⢻⢻⣿⡟⡉⠛⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠀⠊⠀⠀⠀⠀⣠⠴⠋⠀⠇⠀⠀⠀⠀⠈⠚⠶⠾⠟⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢒⡖⠪⠁⠀⠀⠀⠄⠀⣠⠀⡀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⠀⠀⠀⠀⠀⠀⢸⠀⢚⡟⠀⠰⠚⠓⢾⠀⢐⡗⠀⡉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⡁⠐⠀⢄⠀⠀⠀⠀⠀⠈⠢⠬⠥⠔⠁⠀⠀⠈⠂⠌⠡⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠙⠀⠀⠀⠀⠀⠀⢈⠶⡄
⠀⠀⠀⠀⠀⠀⠌⠙⠁⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆
⠀⠀⠀⠀⠀⢄⠘⠀⠀⠻⠿⠟⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⠀⠾⠿⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⣠⣤⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡠⠠⠋
⠀⠀⠀⠀⠀⢀⣤⡶⣶⡄⢠⣶⡷⢀⣤⡾⢿⣃⣠⡶⠛⢻⠆⠀⠀⠀⢠⢉⠀⠸⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠒⠒⠲⠾⠞⡡⠴
⠀⠀⠀⠀⠰⡻⠋⠀⠘⢿⠽⠁⠻⠟⠋⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠙⠘⠋⣹⠃⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠽⠶⠜⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠛⠓⠚⠀⠛⠛⠃