Examination Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢀⣔⣉⠑⣤
⠀⠀⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡸⢫⢽⢯⢺
⠀⠀⠀⠀⠀⠀⡃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢾⠀⢠⡊⠔⢢⣀⣼⢊⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢢
⠀⠀⠀⠀⡠⠞⢿⡖⡶⢅⣠⢄⠀⣀⡀⠀⠀⠀⠀⠈⡶⢡⣆⡸⢒⡈⢆⣿⣁⡷⡤⣤⡆⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⣦⡀
⠀⠀⠀⠀⣇⠈⠻⠷⠄⢧⠘⠚⣾⡿⢃⢖⠒⡄⠀⠀⠐⣺⡏⠃⠘⡗⠈⢯⠍⠇⡤⠄⢰⠀⢀⠀⠀⠀⠀⠀⢠⠵⠒⠟⠇⣼⠋⠑⡄
⠀⠀⠀⢀⠿⠳⣤⣤⡄⢀⣵⣾⣏⡷⠬⠅⡊⠤⠤⠄⠒⠻⣁⡾⢿⣷⡒⡶⢲⡂⣷⡤⠼⠀⠀⠉⢢⡄⠀⢠⠃⡄⢃⠙⠀⢫⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠂⢹⡉⠙⡞⢩⣷⠾⠧⡔⣡⡆⢲⡼⠦⠟⠋⠒⣎⣁⡀⠿⠹⡁⠉⡳⣇⢀⠤⠤⣠⣿⣤⣖⣛⠶⠻⢴⠶⠦⠴⢤⡀⠸⣿
⠀⠀⠀⢳⡶⠖⢞⠧⢠⢫⡟⠁⢰⡟⠱⠠⢵⠆⠀⡔⠀⢺⠀⠏⠁⠙⣌⣠⠿⡌⢝⠂⣹⡴⣖⠛⠫⣇⣠⣓⡶⣶⠃⠀⠀⠀⠀⠀⢠⢿⡄
⠀⠀⠀⠀⣿⠆⣌⠀⡖⣾⠀⠀⠀⠉⠑⠛⠼⢴⣼⣣⣲⣉⣬⣎⣼⢣⣿⣤⣦⣴⣮⠼⠽⠿⠏⠀⠀⣨⢿⡉⠀⠈⡏⠩⠯⢋⠉⠉⠉⠂⠰⡀
⠀⠀⠀⢠⠏⠀⠈⠙⣹⠇⠈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⢻⢾⣷⠀⠀⠲⠀⠸⡀⠀⠐⠀⠁⠀⠱⡀
⠀⠀⠀⢸⣦⠤⢤⣿⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⢀⣀⡀⣀⢤⢤⠤⠖⠊⢁⠁⣋⠷⡋⣼⡇⠀⡆⠔⠂⡾⠒⣄⡀⠀⠀⠀⣘⠄
⠀⠀⢀⠎⢺⡄⣾⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠀⢂⠀⠀⠈⠀⠂⡀⣘⣺⠥⣣⣿⠀⠙⠀⢠⣿⡄⢹⠍⢹⠫⢹⠇
⠀⠀⠈⢢⡃⢰⣖⣿⣾⡟⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⠐⠀⢥⠞⣺⢴⣿⠀⠐⠂⣸⢿⣣⠤⠆⠀⠀⡞
⠀⠀⠀⡄⠉⠏⣿⣟⡟⠀⢰⠀⠀⢩⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠉⢎⠌⠀⢀⠂⡚⣡⢦⣫⡏⠀⢂⢀⡿⠀⣿⡀⠘⡀⠀⠱⡀
⠀⠀⢀⠓⠺⢱⣮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⡇⢸⠆⠀⠊⡰⠡⠓⠪⣟⠃⠐⠀⢸⡇⠀⢹⣟⠋⢿⠀⢀⡷
⠀⠀⣄⣀⣯⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⢡⠏⠀⠀⠀⣁⢋⣙⣾⡟⢀⡂⠛⣿⠁⠀⠸⣿⡴⠼⣾⠿⢷
⠀⠰⠈⠹⠒⠻⣿⠟⠒⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠋⠀⠀⠀⠐⢛⣊⢿⠎⠀⠨⣤⢨⡏⠀⠀⠀⣿⡇⠀⢃⣀⣠⠆
⠀⠑⢢⠇⢙⣟⣺⠀⠰⣥⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⠀⠀⠀⠀⡩⣥⣶⠊⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⣿⠉⠈⡄
⠰⠀⠄⠀⣾⣩⣇⡀⠠⡎⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⢠⠑⠂⣯⢦⡀⠀⠀⢀⢿⡆⠦⢁
⠦⠼⠶⡖⠲⠍⠉⠁⠀⢈⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢐⡷⠁⠀⠀⠀⠀⢆⡀⢸⡇⠀⡘⣕⢮⡀⠹⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⠁⠀⠀⠀⠀⠘⠀⠀⣿⠗⠚⠉⠀⠈⠙⠛⣿⡈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣠⡥⠴⠒⠉⢈⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡏⠀⠀⠀⠀⠀⡌⠙⢸⡏⠀⠀⠀⠀⠀⠀⠀⢹⡇⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠬⡇⠀⠀⠀⠀⠰⢠⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠤⠇⠀⠀⠀⣀⡄⠀⢰⣷⣤⣤⡤⠤⣤⣄⣀⣀⣀⣿⡎⠁⢃⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⡄⡀⡀⠀⣀⠀⣀⠀⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢸⡗⠂⣽⡇⣰⠿⣆⣿⣧⣾⡇⠀⠀⣿⠀⢠⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠈⠉⠈⠈⠁⠀⠀⣿⠀⡿⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⢰⡟⠛⢻⠀⣿⢸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠸⠧⠤⠾⠀⣿⡿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⢀⣤⣤⣤⠀⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡇⠀⣸⠀⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⠁⠈⠉⠉⠉⢠⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡟⠋⢻⠸⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠷⠶⠾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠁⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡈⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⣆⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠈⠉⠩⠍⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠘⠃⠒⠺⠿⠀⠫⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠺⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠑⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⢀⠀⠀⠀⠀⠀⠈⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡐⠁⢀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠⢄⠀⠠⣄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠆⠀⠀⠀⠀⠀⠠⠊⠉⡆⢔⠋⠀⠀⠈⠂⠃⠑⢄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠣⠃⡁⠤⠀⠀⠀⠀⠀⠀⠀⠅⠀⣠⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠛⠛⠒⢄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣄⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠹⡄⢀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⠄⠀⠀⣈⠥⠊
⠀⠀⠀⠸⣤⠐⢰⡆⡰⠁⠀⠀⠙⢲⣶⡖⠸⣶⡞⠀⠀⠀⠀⠀⢈⣣⣉⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠏⠀⠀⠙⡅⠀⠀⠀⢀⡌⠉⡥⣁⢩⠩⠀⠀⠀⠀⠀⠈⠁⠌⣍⢀⢀⠔⠉⣀⣉⠀⠐⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠣⠀⣑⡂⠉⣓⣀⠈⢉⣁⠀⠀⠀⠀⠸⣏⠁⠀⢁⣀⣀⡀⠀⠉⢀⡠⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣆⢰⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣁⡂⠀⠐⠀⠙⠓⠊⠀⠛⠂⠈⠉⠉⠉⠉⠈⠉⠉⠀⠀⠘⣩⡟
⠀⠀⠀⠀⢀⣀⡀⠀⣀⢀⣀⠀⠀⢀⣀⡼⡅⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠻⠇
⠀⠀⠀⠀⠜⡟⠠⢿⠃⠀⠀⠀⠀⠀⠛⣦⠁⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠒⠛⠛⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠆
⠀⠀⢀⢀⣧⣴⣴⣦⣶⣶⣶⣶⣶⣶⣾⡍⣿⣦⡆⢸⣤⣀⣀⣀⣤⣴⣾⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⠀⡻⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠲⠀⠀⠐⠐⠐⠐⠀⠀⠂⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠂⠁⠘⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⠶⠒⠛⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⡀⠉⠀⠀⠀⠀⠈⠳⣄⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢸⡆⠀⠀⠠⣄⠀⠀⠀⠈⢳⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡤⠻⣦⡀⠀⠈⠻⣦⡀⠀⠀⢳⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣶⣿⣿⣶⣴⠆⠈⢳⡄⠀⠘⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⡿⠙⠛⢋⠙⣦⠀⠈⣧⠀⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣴⣷⣶⡄⠌⠀⠘⣇⠀⣿⠀⢸⠀⢀⠞⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠁⠀⠀⠀⣿⠀⣿⠀⢸⡆⠈⠧⡴⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣦⡤⠀⠀⣿⠀⢿⠀⡨⠿⢄⡀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢇⠀⣠⣿⡇⠀⢰⡁⡀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠏⢷⡀⠈⠙⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⠃⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠙⠲⡶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠀⣀⠚⣴⣾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢹⠀⠀⣀⠀⠀⠀⢀⣀⣴⣶⡡⠀⣀⢀⡠⢞⣡⠟⠁⠈⠻⣄⠀⠀⠀⠀⠀⢀⠞⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣦⣦⠞⠋⢀⣀⣀⠀⢸⣧⣘⢓⡒⣧⢸⠖⠞⢡⡀⠀⠀⠀⠙⢦⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠸⣙⣦⣀⣀⢸⡟⢷⠻⡏⠹⡋⠉⢣⠞⢣⡀⠈⢷⡀⠀⠀⠀⠈⢷⡀⠀⡟⠻⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣀⣴⣛⣯⡓⠬⠉⣹⣀⠈⣴⠻⢄⣙⠂⠈⠹⡦⣻⢄⠀⠙⢦⡀⠀⠀⠈⢷⠀⣇⠘⠀⠀⢸
⠀⠀⠀⠀⠀⣠⡴⠋⠁⡠⠊⠀⠛⠹⡟⠻⠁⠈⠉⠉⠂⣹⠏⣠⠞⣹⠷⡀⠈⠃⠎⠀⠀⠀⠙⣦⠀⠀⠈⠀⣿⣘⠀⢀⡞
⠀⠀⣀⡴⠞⠁⠀⠀⠚⠒⠒⠂⣀⣀⣀⣀⣈⣉⣉⣀⣌⣥⡞⢁⡜⠁⠀⠙⡄⠀⠀⠀⠀⣠⠞⠀⠀⢀⡼⠁⣿⣿⣴⣿⠇
⠠⠾⠧⠶⠶⠶⠞⠛⠛⠛⣋⣉⣉⣭⣭⣭⣭⢤⣤⡤⠤⢾⣷⡏⠀⠀⠀⠀⠸⣆⠀⠀⡰⠃⠀⢀⣴⡟⠀⠀⢿⣟⠁⣿
⠤⠶⠶⠖⠲⣿⡟⠉⠉⠉⠁⠀⢸⣿⡁⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⡄⣰⢁⣠⡶⠋⣸⣧⠀⠀⠀⣟⠀⣿
⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⣿⡇⠀⠀⣾⣧⠀⠀⠀⠀⢀⣾⠉⠉⠻⡿⠶⠛⠉⢻⣧⠀⢠⡏⢠⡏
⠀⠀⠀⠀⢤⠈⣷⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢿⡟⢀⣀⡤⠄⠘⠛⣶⢦⣤⠇⠀⠀⠀⠀⠙⣧⢸⡇⢸⡇
⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⣠⣶⡟⣻⣿⡷⠤⡠⠔⠁⢀⣠⠄⠀⠀⠀⠀⠀⣹⢸⡇⢸⡇
⠀⠀⠀⠀⢸⠀⣷⣀⣀⣀⡀⠀⢠⠤⠤⠤⠤⠰⣿⡷⠊⠀⠙⣷⡎⣁⠿⠦⣀⣴⡏⢡⠞⡀⠀⠀⠀⠀⣰⠋⢹⠋⣽⡇
⠀⠀⠀⠀⢸⡆⢿⠤⠤⠤⢸⡃⣻⡴⣆⠀⠀⠘⡏⡇⠀⠀⠀⣠⡿⡅⢶⠀⠈⡀⣵⡎⢀⠃⠀⡇⠀⢸⣃⣀⣸⣃⡾⠁
⠀⠀⠀⠀⢠⡇⢸⠀⣬⠋⠀⠉⠉⠀⠈⢷⡀⠀⡃⡇⠀⣰⠞⠉⠋⠀⡄⠰⠀⠻⢿⣧⣼⠀⠀⣿⠀⢸⡍⠉⣿⣿⡇
⠀⠀⠀⠀⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⡇⣿⣼⠁⠀⠀⠀⠘⠂⠀⣀⡤⠒⣽⣿⣧⣀⣿⣆⣼⠿⣿⢁⣿⠃
⠀⠀⠀⠀⠀⣷⢸⢸⣇⣴⠶⡶⠶⠶⢶⣶⣿⠀⡇⡷⢿⡀⠀⠀⠀⠈⠙⣇⠀⠀⠀⢸⡏⠀⠉⢹⡏⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⣿⢸⢸⡟⡋⠉⠉⠛⠛⠉⢻⢹⡇⡇⡇⠀⠙⢦⡀⠀⠀⠀⠈⢆⠀⠀⢸⡇⠀⠀⢸⡃⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⣿⣾⣷⣧⠿⠤⠤⠤⠤⠤⠬⠦⠿⠿⣿⠷⡄⠀⠙⢦⣀⠀⠀⠈⢷⣄⣸⣧⣀⣀⣼⣁⣀⣀⣿⠸⡿
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠙⠳⣄⠀⠈⢣⣿⣷⡠⠤⣷⠠⠴⠀⣿⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⣶⠀⣸⢸⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⡟⠳⠄⣿⠀⠀⢠⡇⠀⡇
⠀⠀⠀⠀⠀⢸⢻⡆⠘⣷⡰⣿⣿⣖⣽⡆⢀⡿⠀⡿⢸⠃⠀⠀⠀⠀⠀⠰⠶⣿⣧⡀⠈⣶⣄⠿⠀⡀⢸⠇⣶⣿⡗
⠀⠀⠀⠀⡀⠸⠀⢿⣄⠘⢧⣓⣚⣙⣚⣡⠞⣡⠞⢡⣾⢠⠖⣒⣶⠀⠀⠀⠘⣿⡿⣣⠀⣿⣿⡆⠉⣉⣛⣴⢿⡟⠁
⠀⢺⡟⠛⢿⣈⣛⣛⣛⣷⣤⣀⣈⢁⣈⣴⣾⣥⠶⣩⣤⣼⣿⡿⠿⠀⠀⢰⣷⣿⡓⠻⣧⣿⡟⢻⣏⣉⣉⣧⣾⡇
⠀⠈⠛⠛⠛⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢻⣿⣾⣿⠏⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡴⠚⠋⠙⠙⠛⠛⠳⠶⣬⡄⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⠀⢀⠔⣻⢠⡤⡄⣴⢢⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠸⠶⢻⢸⣄⡟⢿⡼⢰⣫⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢸⡇⠀⠀⠀⠀⣸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⡇⠀⠹⡁⠀⠀⠂⠉⠀⠀⠀⢰⠀⠀⢸⢹⠀⠀⠰⠶⠋⣀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣇⠀⠀⠱⣄⠀⠀⠀⠀⢀⡰⠃⠀⢠⠿⠋⠀⠀⠀⠀⠀⣿⠀⣀⠀⡀⢀⢸⣇⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠙⠒⠒⠒⠉⢀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⠀⡀⡇⢸⡏⢆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠿⣶⣶⠶⠖⣲⠛⢻⡉⠳⣄⠀⠀⠀⠀⠀⠀⠀⣇⡀⠘⠛⠀⢧⡘⠃⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⢲⠏⠋⠉⠉⠁⠀⠀⢳⣶⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⣀⠀⠀⡞⠓⠒⠒⠒⠒⠒⠊⠉⢷⠀⣀⠽⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣌⠀⠈⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠙⢲⡾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢹⡶⠚⠉⠀⢳⡀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣼⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢷⡀⠀⠀⠀⠱⣼⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣰⢇⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣸⣷⣴⠶⣶⢴⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⣿⣾⡻⠚⠲⠶⠶⠶⣶⣤⣤⣤⣤⡴⠖⣛⣩⣍⣂⣦⣽⣿⣿⡀⠀⢀⠇
⠀⠀⠀⣤⣤⣤⣤⣤⣼⣤⠯⢭⣭⡭⣽⣶⣶⣾⣿⣿⡿⢿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠯⠷⠶⠾⠶⠶⠶⠶⠶⣶
⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣼
⠀⠀⠀⠉⠉⠉⢻⡍⠁⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢹⡄⢨⠃⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⢠⢾⣯⣷⣼⡇⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⡀⠈⠉⠀⣠⣴⣲⢿⡄⠀⠀⠀⠊⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⡖⣾⡏⡿⣿⣽⣽⡇⠀⠀⠀⠘⣟⡮⡷⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⡤⣆⢄⠀⠀⠀⠐⠢⢄⠀⡐
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣇⠀⠀⠀⢻⣻⣟⡧⣏⡄⣽⡇⠀⠀⠀⠀⢹⡾⣿⠟⠀⠀⠀⠉⠒⠂⠀⢌⠀⠁⠀⣿⣹⡳⣻⣹⣯⠷⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠀⢹⣿⠿⠿⠛⠋⢀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠁⠀⠀⠸⣧⣳⣷⠷⠟⠚
⣶⠀⠀⢁⠀⠀⠠⠄⠀⠀⠀⢠⣿⣿⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⢄⣀⠀⠀⢸⣿⣿⣷⠀⠀⠀⢀⣀⣀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⡟⡗
⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠈⠀⣘⢻⠃⠈⣿⣷⡿⠀⢤⣾⣿⣿⣿⣗⡀⠀⢀⣤⠶⣴⢄⠀⠀⠀⠘⣿⣯⡇⣷
⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣤⣴⡶⢧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⡂⠍⢀⡞⠀⠀⠁⠁⠀⠀⠀⢻⣿⠿⠛⠁⠀⠐⣟⢙⢖⣿⠗⠀⠀⠀⠀⠀⠙⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⡞⢭⣿⡆⠀⠀⠀⠀⠀⠀⡸⠉⠷⣦⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠈⡿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠀⠀⠀⢠⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⡄⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠻⠯⣷⣿⠿
⢠⠀⠀⠀⠀⠀⣀⡀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣴⣯⣷⣄⠀⢀⡠⢞⠷⣗⠀⠀⠀⢀⣬⣍⡛⠃⠀⠀⠀⠠⣾⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣆
⢹⠀⠀⢀⣴⣾⣿⣷⣌⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡘⢿⣅⡷⠋⠁⠀⠀⣼⣿⣿⣿⣷⣶⠀⠀⠈⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⡀⠀⠀⠀⢸⣁⣉
⢸⠀⠀⠙⢿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠃⠀⠉⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⢀⣄⠀⠀⠉⠻⣷⠟⠀⠀⢀⠀⠀⡀⠀⠀⠀⢸
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⢀⣌⡛⠁⠀⠀⢰⣾⣿⣿⠀⢰⣏⡿⣿⣿⣿⠏⢰⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢷⣜⣥⣶⡀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠈⣯⣿⣿⠄⠀⠉⠙⠛⠛⠁⠀⠀⠙⢿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣤⢿⣤⠀⠀⠀⠀⣀⣀
⡇⠀⠀⠀⠀⠀⣠⣾⢯⠣⡱⣝⢿⡶⢋⣢⠀⠀⠀⠈⠛⠿⣿⣿⠟⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣤⡤⡶⣶⡀⢸⠧⠆⣸⠀⠀⢸⣿⣿⣿
⠋⢀⠀⠀⠀⠸⣿⢮⡳⡑⣝⢮⣢⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣤⣶⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⡟⡼⢱⢣⣇⠈⠑⠚⠃⠀⠀⠈⠿⠿⠛
⢠⣾⣿⣦⣄⠀⠈⠳⣽⣌⣮⠿⠋⣀⣄⣤⡀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡄⠀⣀⣴⢦⠀⠀⠀⠀⠙⠻⠟⠛⠋⠉⠀⠀⠀⠀⠀⣤⣤⣶⣶⣧
⠈⢿⣿⣿⣿⠏⠀⠀⠙⠟⠁⠀⠀⢿⣿⣼⡇⠀⠀⡀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡜⣏⡧⡪⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠘⠿⠟⠛⠉⠀⠈⢢⠞⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣢⣤⡀⠀⠀⠀⠀⠀⢀⡟⠳⠒⣢⣤⠀⠀⠀⠀⣤⣶⡞⣿⢸⡇⠀⠀⣾⣿⣿⣷⣆⠀⢀⣶⣿⣷⡄⠀⠁⠀⠀⠀⢀⣠⣴⣶⣦⡄⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⢀
⡽⣿⡵⠀⠀⠀⠀⠀⣸⠐⠉⠐⢠⡇⠀⠀⠀⢰⣷⣿⣿⣿⣿⣷⠀⠀⠙⠛⠿⣿⠏⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣛⡄⠀⠀⠄⠀⠀⠀⠀⢀⣴⣿
⡾⠋⠀⠀⠀⠀⠀⢰⠷⣦⣄⣂⡜⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣻
⠀⢀⣀⣀⣀⣀⠀⠈⠐⠲⠇⠈⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⣠⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⡀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢶⡗⢸⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠷⠛⠛⠃⠀⣴⡞⠉⠄⠀⠀⢳⡞⢳⠀⠀⠀⣶⣾⣿⣿⣿⠀⠀⠰⠀⠀⠐⣾⣿⣻⣷⣦⣄⡀
⠀⠀⠛⠚⠓⠛⠀⢰⣶⣾⣿⣷⡄⠀⣀⣲⢤⣄⣀⣀⣾⣷⣄⠀⠀⠀⠈⡀⠈⠀⠈⠀⠘⣏⣀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣷⣻⣷⣳⣻⢯⡟⣄
⠀⠀⢀⣀⣀⡀⠀⠈⣿⣿⠿⠿⠿⠀⢹⣏⠣⡈⡟⠻⣿⣿⣿⡇⠀⠀⠀⢱⣦⣠⠤⠖⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠛⠷⣿⣣⣯⢧⣿⢻⡗
⣿⣿⣿⡟⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠙⠍⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠷⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣭⡙
⠿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡳⢤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⠀⠀⡾⢲⡖⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣠⡿⡇⠀⠀⠀⠀⠀⠈⠻⢿⣿⡁⠀⠀⠀⠀⠀⠿⠼⠥⠿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡄

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠑

⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠀⠀⠒⠀⠄⠄⠀⠠⠀⡄⠀⠐⠄⢠⠆⠀⠂⠀⢠⠒⠒⠒⠀⠀⠔⠒⠒⠀⠀⢠⠄⠂⠀⠀⠐
⠀⢠⢸⠀⠀⡤⡤⢃⠠⠀⢀⡇⠅⠀⠀⠰⠅⠀⠀⢸⠸⢸⠀⢠⠀⠇⠰⠅⠀⢀⠀⠐⡇⠀⡴⠂⠙⠋
⠀⠸⢸⠀⠀⠁⠀⢶⢨⠀⢸⡇⡇⠀⠀⠀⢧⠀⠀⢸⡀⡚⠀⢸⠀⢸⢸⠕⠀⠈⠀⢆⠣⡀⠈⠁⠀⠂⡀
⠀⢨⢸⠀⠀⠆⠒⢫⢸⠀⢸⡇⡇⠀⢠⠀⠘⠀⠀⢸⡃⡇⠀⠈⠀⠸⣐⢈⠀⢰⠀⠀⠑⠈⠀⣍⠂
⠀⢐⠨⠀⢠⠀⠀⠔⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣇⠃⠀⡏⡄⠀⣷⢸⠀⠀⠀⣥⠊⠈⠉⠀⢀⡠⠃
⠀⠨⠴⢡⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢾⠭⠋⠊⠉⠋⠱⠭⠝⠙⠊⠩⠬⠟⠙⠊⠁⠋⠛⠉
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠴⠟⠩⠀⠀⢠⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠄⠂⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠈⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡤⢀⡤⠖⠤⢄⣰⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣠⠤⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠴⢄⠄⠄⢆⡀⠀⠀⠠⣑⠶⠠⠊⠼⠄
⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⢹⣀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡨⠇⠈⠀⠀⠀⠀⠀⡰⣤⢀
⠀⠀⠀⠀⠀⠀⠀⢉⣒⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⡎⠁⠀⢀⡠⠴⠋⡀⡀⠀⠀⠐⠀⠄⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠈⠁⠢⣤⠀⠀⠀⢸⣇⡠⠚⠉⠀⠀⠀⠀⢀⠉⠐⠂⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⠄⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡿⠁⣠⡀⣠⡄⠹⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⡆⣰⣷⣴⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠦⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⡇⣹⣿⣿⣿⠿⠿⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠀⠈⢁⠀⠀⣀⡀⠀⢈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠙⠄⠀⠀⠈⠀⣄⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣶⣶⠀⢸⡏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⣼⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⠿⠣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠓⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣋⣉⡀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠻⠿⠟⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢹⣿⣿⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⢀⣠⣿⣿⢁⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉

⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠀⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢃⡀
⠀⠀⠀⠈⢇⠀⠀⠀⠀⣠⢾⠭⡍⠂⠀⡇⠀⠀⠀⠀⠀⣠⣲⡍⠉⠉⣉⣹⡶⣄⠀⠚⣁⠉⣡⡄
⠀⠀⠀⠀⠈⡆⠀⠐⣽⠁⢇⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⣱⣼⠟⣻⣻⣹⠍⠀⢹⢣⢈⣉⠰⠦
⠀⠀⠀⠀⠀⢱⠀⠀⢻⣀⣀⣁⠴⡂⠀⡇⠀⠀⠀⠀⣴⡿⠙⠤⠔⠏⠙⢽⡂⠸⢼⡈⠁⠘⠛
⠀⠀⠀⠀⠀⠈⡆⠀⢩⡄⠱⡀⢘⡃⢰⣛⣲⠀⠀⠀⠈⡆⢰⡉⠋⠷⡄⠀⢯⣳⣤⡈⠉⢲
⠀⠀⠀⠀⠀⠀⢹⠀⠰⠎⢸⡥⢈⡥⠀⢸⢸⠀⣠⠤⠤⢝⣄⠈⠉⣒⠵⠒⡟⠋⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⡇⣀⡀⢮⣝⣟⡛⡄⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠄⠒⠒⠒⠀⠉⠉⠉⢙⠞⠉⠀⠀⡜⠀⠈⠁⡦⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡼⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠐⠒⠧⠤⠤⠐⠚⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡖⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣵⣶⣶⣶⣶⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣮⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣿⣿⣷⡂⠈⠀⢰⡇⠀⠀⢠⣬⣇⣀⠀⠀⢠⢠⠀⠸⣿⢿⣿⣿⣿⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣿⢹⠘⠁⠀⠃⠸⡇⢸⣷⠋⢢⣿⢩⣁⣰⣷⣾⠀⠀⠉⣸⣿⣿⡟⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠂⢀⠀⠀⠂⣼⡇⣼⡇⠁⠀⢸⠂⠙⠉⠉⠉⠀⠀⡀⠁⠀⠈⣧⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣴⣷⣄⠛⠑⡶⠶⠼⣧⣿⠛⢆⠀⢸⠀⠀⠀⠀⠀⠉⢨⠗⠒⠀⠉⠉⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢹⣼⡀⠀⠀⣿⣿⡶⢾⡞⠉⢣⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠱⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢻⣿⣿⣆⠀⣾⣷⠒⠰⡇⠀⠀⢣⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣸⣿⣿⣿⠐⣿⣧⣄⣀⣡⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡋⠉⢻⣿⣿⣆⡷⣦⠄⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄⠀⠹⣿⣿⣿⣿⢰⣶⣶⠉⢆⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣘⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣿⣿⡇⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⡿⢏⠁⢄⣹⣿⠟⠁
⣿⣿⣿⡿⠿⠀⠆⣰⣾⠟⠁
⣿⣯⡘⠃⠙⣡⣿⠟⠁
⣿⡿⣳⣤⣿⠟⠁
⠉⢿⡯⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⣠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣈⠛⠛⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣟⠛⠛⠛⢻⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠚⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⡇⠀⠲⠶⠶⠶⠶⠶⠶⠶⠀⠀⣿⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣷⣦⣶⣶⣾⡇⠀⢴⣶⣶⣶⣶⣶⣶⣦⠀⢠⣿⣤⣤⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⣏⣉⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⠀⠈⣿⣿⡏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣔⡆⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣡⠤⣄⠀⠀⠀⠀⣠⠤⣅⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⡀⠙⢀⡿⠀⠀⠀⠀⢿⠀⠛⢈⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠛⠉⠀⠟⠛⠛⠛⠀⠉⠛⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⢸⠀⠀⠀⠀⢸⣧⠤⠤⠤⠤⠤⣾
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣆⣀⣀⣀⡼⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⣠⠟
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠛⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠟⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠄⢀⡀⠀⠀⠀⢦⠟⠀⠀⠀⢳⡄⠀⣠⣤⠰⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⠈⠀⠐⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢴⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⢠⠀⠀⠀⠀⠀⠀⠲⡄⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⠀⠈⠉⠑⢄⠀⠀⠀⠀⠀⠈⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠊⠁⠈⠐⠄⠀⢀⠀⠀⠀⠀⠀⠄⠀⣑⣤⠀⠀⣄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠀⠩⢦⠈⠆⠊⠀⠀⡰⠊⢱⡀⡜⠁⠀⠈⠀⠘⠑⢄⠸
⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⣈⠢⠀⠀⠀⠀⡀⠀⠀⡁⢠⣼⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⢀⣉⣀⢤⠀⢠⠀⠘⠀⢀⠀⠈⠁⠈⠛⠓⠂⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢰⠤⢠⠁⡃⠈⡀⠀⢀⢤⡀⠉⠀⠀⠠⠄⠀⠠⠐⠁
⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⡸⡇⠃⠀⢱⢀⠈⠒⠘⣄⡠⢐⠖⠓
⠀⠀⢸⠀⠀⠀⡀⠀⡴⠃⠀⡘⠀⠀⡐⡐⠀⡇⠀⠀⠀⠳⠢⣀⠀⠤⠂⠁
⠀⠀⠈⣷⠋⠹⢧⠌⠀⠀⠀⠀⣿⣿⢡⠹⣿⠁⠀⠀⠀⠀⢐⠕⢠⡀⠀⠀⢀⠤⠠⡄⠀⡀⠄
⠀⠀⠀⠇⡄⠀⣸⠋⠁⠀⠀⠼⡇⠰⣓⠤⣄⠁⠠⢤⠠⠀⠀⠀⠐⢦⡠⠔⠁⠤⠂⠇⠈⠄⠤⠂
⠀⠀⠀⠀⠁⠠⠇⠀⠀⠀⠀⠀⠑⠒⠀⠒⢲⠶⠂⣒⣒⣀⣠⣄⡰⠈⣷⣰⣤⣤⣤⣾⣮⣤⡤⠞⠋⢠⡴⢶
⠀⠀⠀⠐⠒⠈⠒⠒⠂⠀⠀⠀⠀⠀⡤⠀⠐⢲⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢴⡇
⠀⠀⠀⢸⣶⠒⣶⠖⠂⠀⠀⠀⢻⠿⡹⠀⠀⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢘⠉
⠀⠀⠀⠂⠂⠘⠉⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠧⠴⠴⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣬⣖⢲⣬⡭⠀⣆⡀⠀⣀⣀⣥⣴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠀⢰⡆⠄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠉⠓⠤⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠿⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠈⢆⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠂⠈⠉⠉⠭⣁⡐⠆⠀⠀⠀⠉⠏⠀⠀⠀⠀⠘⠓⠒⠒⠿⠇⠘⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢴⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⠀⠈⠑⠋⠱⡀⠀⠀⠀⠀⠀⠀⠑⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⡠⠒⠒⠂⢄⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡁⠀⠐⡢⡀⠱⡀⡰⠁⠀⠀⣠⠄⠄⠀⣠⠖⠈⠑⠠⢈⡗⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡁⢀⠀⠈⠁⠀⠁⠀⠀⠀⢰⠁⠀⠘⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠇
⣠⡤⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣁⠰⠀⠁⠀⠀⠀⢀⠆⠀⠀⠘⠄⠴⢳⣤⡔
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⡦⢠⠐⠆⠀⠴⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⡂
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠸⠤⢀⣇⣠⠀⠀⠀⠀⢀⠴⢄⠈⠀⠀⠀⠰⠀⠀⠀⠂⠁
⢰⡄⠀⠀⠀⠀⠀⠀⢠⠴⠊⠀⠀⠀⠀⠀⡸⢏⠈⠀⡀⡖⣄⠀⠚⠀⢇⡠⠄⡲⠂⠓
⠀⣇⠀⠀⠀⠀⠀⡠⠊⠀⢀⠋⠀⠀⠀⡐⠁⠀⠀⠀⠀⠘⢯⢂⠀⢀⡀⠔⠊
⠀⢳⣶⠒⢺⣇⡐⠁⠀⠀⠈⢘⣷⣶⠁⢳⣶⡃⠀⠀⠀⠀⠀⠀⡩⢄⡀⠀⠀⠀⠀⣀⢄⣀⡀⠀⣀
⠀⢸⠹⠀⠀⢸⠶⠒⠂⠒⠀⣼⠉⢹⠮⣀⡍⠣⠀⠀⠀⢀⠄⠂⠀⠤⣏⠡⣀⡴⠉⣠⣈⡅⠀⡋⠀⡠⠂
⠀⠀⡄⠀⢀⡏⠀⠀⠀⠀⠀⠸⠄⣈⣒⣘⣛⣒⣀⣈⣉⣀⠀⠀⠀⠀⡬⢇⡁⣀⣈⣀⣀⣨⠴⠂⠁⢀⡤⠄⠀⣀⣀
⠀⠀⣇⡀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠓⠒⠚⠑⠚⠓⠚⠛⠛⠋⠉⠛⠛⠛⠛⠒⠒⠘⢉⡽
⠀⠀⡇⠀⠀⠀⠈⠉⠉⢉⣉⣉⣉⣁⣇⠀⠀⢩⠉⡏⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⢻⣧
⠀⢀⠾⠟⢩⣿⠏⠀⠀⠀⠀⠀⠻⢃⡜⠀⠀⠈⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠
⠀⠌⢠⠀⡌⠈⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⢤⠾⠼⠤⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠈⢀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣑⡶⢶⣒⣚⠀⢰⠠⠀⠀⠀⠀⣱⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⠀⢦⣆⡰
⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠘⠛⠿⣀⡼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣡⠴⠚⠚⠛⠛⠛⠳⠶⣧⣌⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢀⡤⢿⢀⡤⡀⣰⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠸⡴⢿⢸⡇⡵⢿⣸⢇⣖⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡂⠀⠀⠀⠀⢸⠈⠛⠁⠈⠉⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣷⠀⠐⡞⠀⠀⠰⠐⠆⠀⠀⠀⣸⠀⠀⢸⢣⠀⠀⢠⣤⠟⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣿⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⣯⠞⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⡠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀⠀⠈⠓⠤⢄⠤⠤⠚⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠘⡜⠀⡟⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣶⣤⣤⣤⣤⣤⣤⠴⣞⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡤⠃⣇⠀⡷⠈⠢⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠯⣈⣸⠿⠗⠶⠶⠃⠀⠹⣆⣨⠷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢠⡯⣄⣀⣀⣀⣀⡠⠤⢿⡗⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠎⠑⠢⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠴⠊⠁⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⠦⣄⣠⡞⠢⠤⠤⠤⠤⠤⠤⠤⠤⠒⠚⣇⣀⣤⠜⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠘⣆⢸⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣼⠓⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠊⢻⡄⠀⣀⣀⢈⡿⢯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣀⣴⣇⣞⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⢟⡛⢭⣽⣾⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣤⠿⣷⣦⣄⣒⣛⣋⣉⣽⣿⣿⣿⣿⣿⣭⣥⣴⠶⠾⠶⠞⠓⠺⡿⣄⡀⣠⠃⠀⣀⣀⣀⣀
⠀⠀⠀⡶⠒⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠒⠛⠛⠛⠛⠙⠛⠛⠉⠛⠉⠋⠉⠉⠙⠙⠉⠉⠋⠉⠉⠉⠙⠋⢻⡇
⠀⠀⠀⠷⠤⠤⣤⡤⠤⠤⣤⠤⠤⢤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⡶⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢰⠃⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠚⠛⠛⠓⠲⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣉⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣴⡾⣿⣿⣿⣷⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢰⣿⣿⣗⡖⠒⠒⠒⠒⣾⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠀⢀⢀⡇⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢸⣿⠿⠏⡿⠀⠀⠤⠀⣿⠀⣰⣄⡴⢋⠉⣿⠟⡞⠀⢠⣯⣽⠁⠀⠈⠋⠌⣿⣿⣿⡿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢿⡘⠃⠀⠀⠀⠀⢀⣽⠀⢻⣿⢀⠀⠙⣿⡀⠳⡶⠾⠿⠿⡀⠀⠀⠀⠼⠛⠻⢿⡇⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⣀⣄⠀⡀⢀⡰⣿⢠⣿⣧⠀⠀⠀⢩⠁⠈⠀⠀⠀⠀⢇⣀⣀⡄⠀⠀⠀⣀⣧⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⣤⣾⣷⣄⡛⠃⢐⡶⠶⠖⢿⣼⣯⠛⠳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠒⠂⠉⠁⠀⠀⠈⢦⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠉⣥⣜⠀⠀⠀⣿⣿⣿⣶⣶⣷⠞⠉⢣⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢶⣿⣿⣷⡄⠀⣻⣹⣉⣁⡀⣿⠀⠀⠀⢧⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⢿⠷⠀⠂⣹⡄⠀⠀⠈⢧⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇⣿⠿⠷⠢⠤⠤⢧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡋⠁⠙⢿⣿⣿⣇⣿⢷⡶⠄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠈⢻⣿⣿⣿⣿⡇⠰⣶⣶⡖⠘⡄⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠹⣿⣿⣿⣿⣾⣿⣿⣧⣤⣼⣦⣤⣤⣶⣿⣿⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉






⣿⣿⣷⣾⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⢶⠾⡷⢿⡿⠿⣾⡿⠿⠿⠶⠶⠶⠶⠾⡶⡷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⢳
⠀⠀⠐⠐⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠒⠄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠐⢠⡔⠘
⠀⠀⠀⠀⠀⠀⠀⢐⠂⡉⢲⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⢸⣟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⡠⢂⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢭⠀⡁
⠀⠀⠀⠀⠀⠀⠄⣤⣤⣤⣀⡄⠀⠀⠀⠀⣄⡠⡉⣲⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡒⠒⠢⠀⡄⣇⠀⠂⠐⠊⡱⠄
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⣘⠆⠀⠀⢠⡔⣾⠇⢐⡀⠀⠀⠀⠀⠀⡰⠚⡥⠀⠀⠁⠀⠀⠰⣄⠁⠀⠚⢃⣀⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠟⠉⠃⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠛⣀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠉
⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠐⠀⠠⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⢄⣤⢠⣄⠠⡄⠀⢐⡇⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢬⡵⠄⠚⠀⠔⠰⠊⠑⡅⠀⠁⢈⠔⠁⠀⡁⠂⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠈⠛⠀⡠⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⢀⠎⣁⢀⠴⠀⠀⠀⠈⡐⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠈⡁⣨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⢠⢰⡎⢀⣽⠄
⠀⠀⠀⠀⢀⠤⡕⠀⠀⠉⠀⣀⣄⠀⠀⠀⠀⠊⠒⠊⠒⠓⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⣙⡀⠀⠀⠘⠀⠀⣔
⠀⠀⠀⠀⠳⠤⠄⠩⠿⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠠⠐⠤⠤⠤⠀⠀⠠⠀⢮⣀⡀⠉⠳⠆⠀⠀⢄⣀⠤⢤⠄
⠀⠀⠀⠀⠀⠰⠚⠓⠲⠶⠛⠚⠲⠆⠀⠀⠀⠀⠄⠒⠒⢄⠀⠀⠀⠀⠀⠀⠂⢄⣀⠀⢰⠶⠶⠶⣤⠶⠶⠶⡆⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣀⡀
⣾⣿⣿⣡⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣷⡄⡀⠀⠀⠀⢈⡟⠛⠿⣿⣷⣦⡀
⣿⣿⡏⠁⠀⡾⢯⠄⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠸⣾⠾⠿⠛⠛⢻⡻⠿⣶⣤⣀
⢹⣿⣧⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠋⢀⡿⠛⢿⣷⣦⣀⣀
⢨⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉⠠⠀⡼⠋⠈⠙⣿⣿⣶⣄
⠈⠻⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⣠⣀⠀⠀⠀⣤⣤⣀⠀⠀⣠⣤⣤⣀⣤⣤⣤⣤⡄⠀⣇⠀⠀⠀⠸⣿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣽⠟⠀⠈⠻⣿⣿⡿⠋⠀⠀⢀⣹⣿⣇⠀⠀⢹⣿⣿⣆⣸⣿⣿⡏⠘⢻⣿⣜⠿⠇⠀⠙⠦⣀⡀⠀⢻⣇⡠⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣀⡀⢀⣴⣿⣿⣧⡀⠀⢠⣿⣿⣿⣿⣆⠀⢼⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⣿⣿⡆⠀⠀⠀⠈⠉⠉⠉⠛
⠀⠀⠀⠀⣴⡖⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠓⠿⠿⠯⠻⠿⠿⠶⠿⠿⠧⠘⠿⠿⠶⠻⠿⠷⠻⠟⠻⠿⠷⠞⠛⠻⠿⠟⠁⠀⠀⣀⣀
⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⡾⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢡⣮⠿⠁⢁⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠈⠳⠴⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠈⠃



⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠴⠒⠒⠂⠐⠒⡆⡰⡂⠒⠒⣆⢴⠒⠒⠒⠲⡀⢠⡖⠒⠒⢲⠀⢀⢴⠂⠈⠉⠑⠀⠀⠰⡖⠀⠐⡆⠀⠀⢀⡴⠒⠈⠉⠉⠉⢹
⠐⡅⣿⠀⠀⣠⣀⣠⠿⡆⡇⠀⠀⣿⢸⠀⠀⠀⠀⢱⡃⡇⠀⠀⢸⠀⡇⡚⠀⢀⡀⠀⡇⢠⢚⡇⠀⠀⡇⠀⡔⣹⠀⢀⣴⣖⣚⡻⠛
⠀⡂⣍⠀⠀⠏⠚⠧⢄⡃⡇⠀⠀⣿⢸⡀⠀⠀⠀⠘⡇⡇⠀⠀⢸⢰⢂⡇⠀⢸⡇⠀⢣⢸⡅⡇⠀⠀⠇⠀⡀⢺⠀⠀⠓⠤⠤⢀⣀
⠀⡇⣿⠀⠀⢀⣀⣀⣸⠀⡇⠀⠀⣿⢸⡃⠀⠀⠀⠀⢹⡅⠀⠀⢸⢘⢠⡇⠀⢸⣇⠀⢸⢸⢚⡇⠀⠀⡇⠀⠳⡈⠢⢄⣀⣀⡀⠀⠈
⠀⡄⣹⠀⠀⡟⠛⠛⢹⠀⡇⡀⣀⣿⣸⣇⠀⠀⣆⠀⠈⠃⠀⠀⢸⣏⢸⠁⠀⢀⡀⠀⠈⣿⡥⡇⠀⠀⡇⢀⡀⠈⠁⠚⠚⣤⣩
⠈⠆⣽⠀⠀⡇⠀⢀⡸⠞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⡄⠀⠀⢸⡗⣽⠀⠀⡾⢗⠀⠀⣿⡖⡇⠀⠀⠉⠀⣧⢾⠉⠉⠉⠁⠀⠀⣠
⠀⡇⡸⠤⢤⡧⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⢠⣖⢶⠞⣧⢳⡶⡶⢫⡸⠶⢲⠾⣧⢓⠖⡲⢲⡶⢻⡼⡶⢒⢶⢖⡶⠟⠁
⠀⠉⢉⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣭⡶⠛⠉⠉⢯⡁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠈⠉⠉⠀⠀⠁⠉⠉
⠀⣲⠄⠀⠀⠤⠤⠤⠤⠤⠀⠤⠴⠃⠶⠛⠫⣴⠃⠀⡠⢀⣿
⢠⠁⠄⠐⠒⣒⠒⠉⠉⠉⠉⠁⠉⠉⢉⣀⣀⣘⣧⠔⠂⠈⢀⠁⠐⠄
⠈⠔⡠⠀⠀⠀⠀⠠⠄⠀⠶⠶⠶⢶⣤⠀⠀⠄⣀⡀⠖⡈⠛
⠀⡐⠇⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⡤⠴⠦⠠⠅⠀⠀⠘⢉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠆⠀⠀⠀⠀⠀⠠⠀⣀⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠠⢀⠀⠀⠀⠀⠀⡀⢀⠀⣀⣤⢤⣉⠀⠁⡆⠀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠐⠃⠢⠀⠀⠈⢉⡩⠍⠚⠉⠀⠀⠈⠉⠛⠿⠬⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⣀⣀⣘⣷
⠀⠀⠀⡇⣀⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣀⣀⡠⠠⠤⣀⠄⠠⠠⣀⠤⠔⢒⣲⠤⢠⣀⠀⠀⠀⠨⡏⣇
⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠺⢧⡤⠴⠤⠜⡀⡀⠀⠓⣓⣛⣚⡛⠷⠶⠯⠵⠿
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣢⠈⠀⠈⠀⠁⠀⠉⠑⠐⠠⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⡏⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸
⠀⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠔⡇⠲⢴⠒⠤⠤⣠⣀⡀⠀⠀⡏⢐⠄
⠀⠀⠴⢋⠑⡶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉⠀⠁⠀⣀⡥⠟⠀⠀⠀⠁⠐⠂⠒⠨⠝⣚⣷⡧⡄
⠀⠀⠀⠀⡘⠀⠀⠀⠉⠓⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡞⠉⠀⠀⠀⢀⣤⣾⣋⠀⠀⠀⠀⠀⠀⠞⠢⢄⠀⠰⠁⠀⠉⠁
⠀⠀⠀⠀⠁⠸⠩⠛⠛⠉⠙⠫⠭⣙⢒⠤⣀⡀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡴⠟⠛⠂⠤⣀⠉⠉⠓⠲⠤⣌⢤⠤⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠓⠢⠄⣀⠀⢼⡷⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠠⠊
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠀⠀⠀⠐⠄⢠⠀⠀⠀⡄⠀⠐⠄⢀⡆⠀⠢⠀⠠⡔⠒⠒⠂⠀⢠⠒⠒⡄⠀⢀⡤⠐⠀⠀⠀⠂
⠀⢰⠨⠀⠀⣤⡠⢞⢸⠀⠀⡇⡄⠀⠀⠘⡇⡇⠀⢠⠀⢂⠁⢠⠀⢸⠀⢼⠀⠀⡇⠠⢫⠀⢠⠂⠄⠛⠁
⠀⢸⢈⠀⠀⠁⠀⢸⢸⠀⠀⡇⡇⠀⠀⠀⢡⡇⠀⢸⠸⢸⠀⢸⡇⠘⡀⣿⠀⠀⡇⠰⠈⡄⠀⠉⠁⠐⢤
⠀⢸⢰⡄⠀⠦⠐⢻⢸⠀⢀⣇⣇⠀⢠⡀⠈⠇⠀⢸⣗⣼⠀⠈⠁⠀⣧⣾⠀⠀⡇⠀⠁⠂⠁⢌⣒
⠀⢸⢨⠇⠀⡀⢀⠸⠊⠁⠀⠀⠀⠀⠉⠁⢀⠀⠀⢸⡇⡇⠀⡘⡆⠀⢻⡺⡀⠀⠉⢹⡔⠉⠉⠉⠀⢀⡜
⠀⠘⠴⢃⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠼⠭⠋⠓⠭⠛⠱⠬⠨⠋⠂⠍⠍⠽⠋⠧⠅⠔⠋⠛⠁
⠀⠀⣄⠀⠀⢀⣀⠀⠀⠀⠠⠤⡴⠷⡫⠁⠀⠀⡅
⠀⠈⠁⠀⠐⢒⣂⡀⠀⠀⠀⠀⠀⣀⣸⠤⠂⠈⡁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠄⠀⢁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠉⠉⠁⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠔⠤⣄⣀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠁⣀⠠⠄⠂⠀⠀⠀⠀⠀⠉⠙⠂⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠤⣆⠀⠀⠤⠀⠀⠀⠠⢥⠶⠄⠄⠠⠧
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡦⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⣇⡀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⢉⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠷⠈⠀⠀⠁⠀⠂⠠⢮⡄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⣒⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⢀⡤⢞⡁⠀⠀⠀⠀⠂⠄⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠀⠩⠐⣠⢄⡀⠀⠀⠀⣿⣀⠔⠊⠉⠀⠀⠀⠈⠁⠒⠂⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣐⣦⠤⠄
⠀⠀⠀⠀⠀⢤⡟⠉
⠀⠀⠀⠀⠀⣖⠀⡀⠀⠀⠀⠴⠒
⠀⠀⠀⠀⠀⠾⢿⣿⣧⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠈⠙⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣧⣼⠿⣤⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠂⣿⠏⠁⢹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡥⢴⣿⡥⠀⢬⡀⠤⠤⠤⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠗⠂⠨⠇⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣦⣤⣤⣤⣤⣤⣤⣤⣀⣁⡉⢳⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣀⣠⣰⣿⡻⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠾⠿⠿⠿⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⠀⠀⠃⠈⡁⠀⠀⣠⠀⠠⠀⠚⠘⠀⠀⠀⠀⠀⠀⡁⢄⡀⠀⢀⠈⠀⠀⠀⠀⡀⢫⠏⠀⢀⠄⠀⠀⠤⠐⠀⠁⠂⢊⠀⠀⡀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠄⢀⠊⠀⠀⢲⠀⠘⠈⠐⠠⢭⠄⠈⠀⢀⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⠉⠀⠀⠐⠁⠀⠀⠘⠀⢠⡤⠀⠁
⠀⠁⠂⠐⠀⠀⠀⠀⠑⠄⠠⠂⠀⠘⡀⠀⠀⠀⠀⠠⠐⡀⡀⠈⠀⠀⠸⠁⣀⣁⠀⠀⠀⡆⠀⠁⡄⠄⠀⠀⠀⠀⠀⡀⠀⠘⠀⠁
⠈⡠⡀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⡰⡀⠀⡗⠒⢩⡀⠜⠀⠀⠇⠠⠀⢀⠀⡈⢀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⠈⠀⠘⠂⠁⠀⠀⠀⠐⣄⡁⠉
⠐⠒⠉⠀⠀⠀⠀⠐⠀⠀⢀⠀⠠⡀⠀⠀⠈⠱⠎⡄⠫⢀⠟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠈⠀⠀⠀⠁⠀⠀⡤⠂⠎⠂⠀⠀⠀⠠
⣀⠀⠢⠀⠠⠀⠀⠀⠀⠀⢠⠃⣁⠃⠈⠀⠀⠁⢀⢡⠘⠱⠂⠐⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠀⠀⠐⠀⠀⠀⠒⣫⠗⠒⠁⠀⠀⠀⢄⢲⣃⠄
⠴⣀⠙⠀⠀⠀⠀⠀⠀⠀⠃⠠⠆⠊⠂⠀⢀⢀⣀⡀⠀⠂⠀⣀⡤⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠈⡡⢬⠀⠀⠠⡀⠵⠤
⠐⠃⠐⠂⡠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⡖⠈⠁⠀⡉⠓⢦⠀⠘⢒⡄⢀⠄⠀⠀⠁⠀⢈⠁⠀⠒⠄⠀⠘⡌⠄⠐⠀⠠⠨⠀⠀⠋⠉⠔⠀⠀⡀⠁
⠀⠀⢤⡨⢀⠀⠀⠀⠀⠀⠠⠈⣥⠀⣸⠀⠀⡣⠱⠁⠆⠀⠷⠀⠡⡑⠃⠀⠐⠑⠄⠀⠀⢀⠀⢀⠀⠀⠄⠈⢀⡀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠈⢁
⠐⠂⠀⠈⠁⠈⠀⠀⠀⠀⢲⠀⠛⠀⠛⠉⠈⠀⠀⠈⡀⣀⡖⠀⢀⣤⣴⡤⠀⠀⠀⠀⠀⠂⠓⡀⠀⠀⢂⢠⠥⠢⡈⡈⠀⠰⠀⠀⠃⠀⠀⢀
⠀⠠⠀⠀⢠⣀⠀⠤⠤⠒⠚⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠙⠀⣠⠸⣄⠜⠀⠀⠀⠀⠋⠀⠀⠠⠅⠀⠀⠤⣁⢡⢐⠒⠀⠀⠀⠤⠀⢠⠀⢠⠌
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⠀⠀⠉⠀⠀⠀⠶⠆⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⢂⠀⠤⢀⡀⠀⠀⠀⠁⠒⠈⠀⢀⡀⠀⠰⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠂⠡⡀⠀⠁⠀⡀⠀⠀⠄⢠⠤⠔⠉⠀⠀⠀⠈⡂⢀⠑⡀⠀⠀⡀⠶⠆⢀⠄⠈⡉⠒⠂⠠⠄⠀⢲⠘⡅⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⠀⠀⠈⠐⠈⢃⠀⠁⠀⠶⠀⠀⠀⢀⠁⠈⠁⠀⡈⠀⢸⠁⠈⢆⠈⠀⠈⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠋⠀⠀⠀⠁⠂⡀⠀⠘⠑⠥⣊⡖⢤⣄⠀⠀⠁⠐⠈⠄⢀⠀⠈⢁⡀⠀⢠⠁⠀⠀⠀⠀⠠⠆⠀⠙⠂⠁
⠁⠀⠀⠢⠄⠤⠀⠁⠀⠀⣀⢤⠐⠄⠀⠀⠢⠀⠐⢁⠀⡀⠀⠀⠀⠈⠙⠂⠄⠀⠀⠠⠒⠀⠤⠨⠀⠀⠀⠉⠈⠀⠀⢀⡆⠀⠀⠠⡀⠀⠀⡄⠀⠦⢀
⡄⢀⠀⠀⠠⠀⠠⠐⢈⠁⠀⠀⠀⢊⠀⠠⠀⠀⠀⠠⠈⣑⠠⠀⠀⣄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠒⠀⠠⠀⡓⠄⠈⡇⠀⢆⠀⠙⠀⠀⠁⠐⠃
⠉⠀⠁⠀⠀⠄⠀⠀⢎⠄⢊⠀⠀⠈⠄⠀⠣⡈⠀⠀⠀⢠⠂⠀⢰⠇⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⣰⠄⡀⠀⠀⠠⠀⠀⠁⡀⡄⠀⠁⠀⠀⠀⠄⠒
⡀⠀⣄⠀⠈⠄⠀⠀⢀⠀⠄⠂⠀⠀⠀⠀⠀⠀⢄⠔⠀⠉⠀⠀⣸⠃⠈⠀⠀⠀⠀⠀⠀⠄⠀⠠⡈⠂⠀⠁⠀⠐⠀⠈⠄⠈⠀⠀⠐⡀⠀⠀⠁
⠁⠓⠘⠀⠀⠀⠀⠈⠀⠀⣀⠀⠤⠀⠀⠦⠂⠀⠠⠀⠀⣀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠔⠀⠀⠸⡀⢠⠀⠀⠀⡂⠀⠀⠄⠀⠀⢀⠀⠳⠀⡀
⠀⡀⠁⠀⠀⠀⠀⠀⠃⠘⠃⠀⠉⠀⠀⠀⠀⠠⢠⠜⣃⠠⠣⠀⠁⠀⠀⠀⠈⠀⠠⠀⠐⡀⣌⠀⠂⠀⠄⠀⡘⠀⠀⠀⠐⠀⠀⢉⡁⣁⠀⠠⠉⡄⠁
⠀⠀⠀⠀⠠⠀⠐⠒⠂⠄⠀⢲⠧⡀⠀⠒⢀⡰⠁⠀⢈⡟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠠⠈⠈⢃⠒⠀⡄⠃⠀⠀⡷⡈⠘⣴⠂⠀⠡⣄⠀⠂⠘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠻⢥⣜⠀⠀⠈⠀⢓⡰⠊⡆⠐⠀⠊⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠁⢀⣀⠀⠁⢁⣀⡀
⠀⠓⠁⠀⣀⠀⠀⠠⠁⠀⠀⠀⠚⠈⠂⠀⠁⠀⠈⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢠⢦⡙⠋⠄⠀⣀⠀⠽⠀⠀⠀⠀⠰⢎⡠⠂
⠁⠰⢄⡀⠐⢄⢉⡀⠀⠀⠀⠀⢀⡀⠀⠠⠂⢀⡀⠀⠀⠾⠑⠀⠑⠈⠀⠀⠀⠁⠀⠀⠀⢁⠂⠁⠊⠀⠀⠀⠀⠐⠊⠀⢀⡀⡄⠀⠠⡆⡦⠀⠀⢅
⠀⢀⡀⠙⠰⠀⠠⡐⠀⠄⡁⠔⠀⠀⠀⠀⠘⠒⠃⠁⠀⠂⠁⠄⣤⠀⠀⠀⠀⡔⡤⠀⠀⡏⠝⠀⠀⣀⠀⠄⠰⠄⠂⢈⠡⠀⢠⠀⠀⣀⡀⠀⠤⠀⠱
⢷⠀⠂⠀⠀⠀⠀⠀⠄⠈⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⢈⠄⠀⠀⠂⠭⠀⢠⠀⠈⠀⠐⠂⠀⠀⠈⠉⢠⡄⠠⠐⠀⠉⡀⢀⡄⢀⠂⢸⠀⡀⠀⠀⢋
⢀⣃⡀⣰⣀⡀⠀⠀⢀⡐⡀⣀⣀⢂⡄⠀⡈⣀⠀⣀⣈⣀⣀⣆⠀⢀⣂⡂⣀⣠⣀⣀⣀⡐⠀⠀⣀⣀⣠⡀⠀⣀⣈⣁⣀⣄⠒⠈⢈⣇⣀⠀⣀⢸⡀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣤⠄⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠾⠂⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠈⠦⠱⠄⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡌⠑⠦⣄⡀⣀⣀⡀⠀⡸
⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠁⠁⡀⠜⠁
⠀⠀⠀⢂⢃⠂⠀⠀⠀⠀⠀⠀⢰⡲⠀⠀⠀⠀⠀⠈⠁⠁⠢⢀⠀⠀⠀⠀⢀⡾⠇⠀⠈⠀⠈
⠀⠀⢀⠀⠀⠎⠄⠀⠀⠀⠀⠠⠻⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠑⡀
⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠣⡀⠀⡀⠀⠀⠠⠀⠤⠀⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠤⠤⠤⠤⠿⠿⠿⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡀⠈⠀⠠⠀⠀⠀⢶⣆⠁⠀⠠⠀⠀⠈⠀⢆
⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡤⠤⠤⣄⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣼⡇⠀⢸⣧⣤⣤⡄⠀⠀⣿⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⢻⡇⠀⢸⡟⠛⠛⠁⠀⠀⣿⠀⣿⠀⠀⣴⠟⠉⠙⠷⣦⣤⣈⣿⡛⠛⠶⠶⠶⠶⠚⣷⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⣤⣾⠇⠀⠀⠀⠀⠀⣿⠀⢻⣄⣼⠃⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⣇⣼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠏⠁⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠉⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⡤⠸⣇⠀⠀⠀⠀⠀⢦⣄⣀⣤⠆⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠹⣦⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣴⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⢈⣿⠛⠛⠳⠖⠛⠛⢹⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠶⢦⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠙⠳⣤⡾⠋⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⣶⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⢀⣀⣴⠞⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⡴⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣚⣛⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠛⠲⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠶⠶⠦⣤⣥⠴⠶⠶⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⢀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣤⠞⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣤⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠙⢦⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠻⡄⠀⠈⢧
⠀⠀⠀⠀⠀⡸⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠈⠇
⠀⠀⠀⠀⢠⡇⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡇
⠀⠀⠀⠀⠘⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡶⠶⢶⠀⢸⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡁⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⢀⣤⠤⠶⢤⠈⢧⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⡼⠁⣠⠤⠶⢤⡀
⠀⠀⠀⢸⡤⢤⣄⣸⡇⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢸⣧⠤⣤⣀⡇
⠀⠀⠀⢸⡄⠀⠀⣹⠃⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠘⣇⠀⠀⢩⡇
⠀⢀⣠⣤⣉⠛⢛⣁⣤⣀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⣀⣤⣈⡛⠛⣉⣤⣄⡀
⢠⠏⠁⠀⠈⠙⠉⠁⠀⠈⢳⡀⠀⣄⠀⠀⠈⠉⢹⡆⠀⠀⠀⠀⢰⡏⠉⠁⠀⢀⣀⠀⢀⡞⠁⠀⠈⠉⠋⠁⠀⠀⠹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠙⠳⠦⡤⢸⡇⠀⠀⠀⠀⢸⡇⠤⠴⠞⠋⠁⠀⢸⠁
⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠂
⠀⠀⠀⠀⠰⡏⠉⠁⠀⠀⠀⠀⣠⢔⡒⠲⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠒⠒⠒⠦⣤⣀⠀⢠⡶⠀⠚⠛
⠀⠀⠀⠀⠀⠹⡀⠀⠀⣀⢄⠞⣱⠉⢙⡆⠀⠀⢿⠀⠀⠀⠀⠀⠀⠰⡗⢾⣯⣤⣤⡤⣶⣾⠏⠻⣧⠀⠠⠶⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠒⡎⠀⠹⢤⠞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣷⣿⠋⠶⢖⣧⣟⣁⠀⠀⡇⡇⡤⠦⠘⣛⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣳⢤⣀⣀⠤⢚⡁⠀⢸⠀⠀⠀⠀⠀⢸⡽⠃⠘⢒⣒⠻⠀⠙⠬⣇⠀⠣⢧⣀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣉⡄⠸⠆⠀⡛⠂⢰⣚⡓⡆⠀⠀⠀⠀⢱⠀⢸⠉⠛⠹⢦⠀⠀⢾⢑⣦⣀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢩⡕⠸⣋⡀⢛⣃⠀⠈⡇⡇⠀⢀⣀⣀⣈⢣⡈⠉⠓⠢⠜⡠⠤⣾⠟⠋⠁⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠋⠀⠚⠉⠀⠀⣇⡧⠰⡎⠀⠀⣨⠟⢻⣒⡶⡞⠉⠀⠀⢹⣤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣽⣁⣸⢃⡴⠇⠈⠙⣶⠓⠦⢽⢀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⢠⠯⠁⠀⠀⢸⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⢏⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣷⣿⣷⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠿⡛⢟⣳⣟⣛⣻⡛⢹⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⣿⢟⡻⠕⠂⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠭⣛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⡿⣿⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⡻⣿⣿⣷⣄⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢎⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣉⣭⣴⣶⣿⣿⣷⣿⢧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡦⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣖⣾⣿⡿⢿⠟⠛⢻⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣖⣾⣿⣿⠿⠟⠛⠋⢉⠘⣇⢸⡄⠺⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⡃⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⠿⠟⠛⠉⣻⠀⠸⣶⡄⢸⣧⠘⣆⠙⠀⣧⠀⣤⣿⣿⣿⣿⣿⣭⡇
⠀⠀⠀⠀⠀⠀⠸⠟⣛⣁⣤⣴⣖⣾⣯⣿⣿⠟⢿⠉⣁⡀⢠⡆⠸⣦⠈⣿⡀⠀⠹⣿⠀⣿⡄⢹⡄⠀⢹⡆⠙⢉⣿⣿⣿⢿⢿⣱
⠀⢀⣀⣤⣴⣖⣾⣿⡿⠾⣿⠛⡏⢹⡇⠀⢹⡆⠀⠀⢹⣇⠈⣧⡀⢿⡄⢹⡇⠈⡀⢻⡇⠸⣷⣈⣿⣤⣾⢿⣿⣿⠿⠟⠛⠉⠉⠁
⠀⠘⣷⣿⣿⣿⣿⣿⠀⣄⠘⡄⢻⡀⣿⠀⡀⢻⠀⡀⠀⣿⡄⢹⣇⠘⣷⣀⣷⣸⣷⣶⣿⣿⣿⠿⠟⠛⠋⢉⣠⣴⣶⠇
⠀⠀⢹⣿⣿⣿⣿⣿⡆⢸⡄⢳⠈⣇⠸⡄⢁⠈⠇⠸⣄⣸⣧⣼⣿⣶⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣖⣸⣿⣿
⠀⠀⠀⣷⣿⣿⣿⣿⣿⡀⠓⠈⣆⣉⣠⣧⣼⣷⣾⣿⣿⠏⠻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣽⣿⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⠿⠟⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣿⣿⠏
⠀⠀⠀⠀⢿⣿⡿⠯⠿⠛⣋⣭⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢿⣿⣿⣮⣗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣪⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣮⣴⣢⡤⣀⡀⣀⣀⣀⣀⣀⡀⢤⢒⣯⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠸⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠰⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⢄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⡀⠀⠀⠀⠀⠀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠎⠁⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠀⠀⢤⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠆⠀⠈⠅⠀⠀⠀⠀⠀⠠⠊⠉⡄⢔⠁⠀⠀⠁⠚⠈⠢⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠘⢢⠰⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠄⡠⠂⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈
⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⢣⡠⢂
⠀⠀⠀⠀⡇⠀⠀⣠⠀⡠⠀⠀⣀⣀⡀⢀⣀⠀⡀⠀⠀⠀⠑⠓⣄⠠⠄⠊⠁
⠀⠀⠀⠀⢹⡏⠀⠿⡔⠀⠀⠀⠀⢸⠿⠇⠉⠿⠇⠀⠀⠀⠀⠈⠞⠒⡄⡀⠀⢀⠤⠂⡄⠠⡠⠄⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠏⠉⠉⠈⠀⠃⠀⢍⠢⠧⠄⠀⠤⠄⠂⠀⠀⠀⢲⡴⠒⠁⠔⠂⠢⢀⠀⠀⠂
⠀⠀⠀⠀⠀⢐⠀⡄⠀⠀⠀⠀⠀⠈⠀⠐⠐⠒⠒⠒⠒⠒⠂⠀⠀⠈⠋⠽⠬⠒⠒⠒⠚⠃⠀⠀⠊⠭⠀⠀⣲
⠀⠀⠀⠀⢠⠋⠈⠁⠀⠂⠀⠐⠂⠀⠀⡀⠀⠀⢓⠂⠀⠒⠂⠐⠒⠂⠀⠐⠂⠀⠀⠀⠀⠐⠀⠀⠒⢲⡄⢰⣿
⠀⠀⠀⠀⠀⠛⠛⠠⡞⠁⠀⠈⠁⠀⡿⡁⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⢡⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠰⠴⠗⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⢈⣀⣀⣀⣀⢀⣀⣀⣀⣀⣝⠙⣋⡉⠀⠘⠀⠀⠀⠀⠈⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠘⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢉⣨⢿⣿⣿⣿⣿⣿⣏⡙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⡞⠁⠀⠒⠙⠛⠀⠙⠈⠉⢿⣷⣤⣿⡻⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠁⠀⠠⡶⣠⡆⠸⠆⠀⠀⠈⠁⢹⣷⡙⢦⣌⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⡄⠙⢯⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠦⠤⠆⠀⠀⣠⡞⠛⠿⠿⠛⠛⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠑⢦⠤⠠⠤⢴⡎⠉⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠈⠦⢤⠤⠚⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⣤⠞⠉⠙⢦⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡫⣅⡀⢹⠀⢈⣶⢼⡿⠛⠿⠿⠿⢿⣿⡽⠛⠛⠾⢿⣦⡀⢀⣠⠞⠛⠛⠚⠓⢒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠡⡛⢲⠟⠁⠀⠙⣲⠞⢼⡇⠀⠀⣀⡀⢺⡟⡔⡠⣠⠄⣀⡿⠛⠉⠀⠀⠀⠀⢀⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠁⠀⠀⣠⠞⠡⠤⠀⠐⠒⠒⠒⠂⠀⢙⡷⠳⠷⠋⠁⠀⠀⠀⠀⠀⢀⡴⢋⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠶⠶⠤⠴⠤⠤⠤⢤⣭⣍⣉⣉⣙⣓⣒⣒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠒⠲⠖⠶⠶⠶⠤⠤⠤⠤⢤⣭⣭⣍⣉⣉⣉⣉⣙⣛⣚⠓⠚⠒⣻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⣀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣄⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡆⣀⣀⣀⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⢀⡾⠿⣿⡿⣿⢿⡿⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣠⣤⣀⣀⠀⠀⢀⣿⡇⠠⠆⠐⠃⡈⠁⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠒⠒⠚⠓⠒⠚⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠓⣿⣿⠏⢈⡉⠉⣀⠉⡟⠒⢿⡗⠚⠓⠚⠛⠓⠚⠛⢟⠒⠒⠒⠒⠒⠒⠚⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠈⠁⠀⠉⢠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣿⣿⣷⣤⠀⠀⢰⣿⡧
⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣾⣿⣿⣿⡿⢅⠀⠈⠉⠀⠀⠀⠀⠀⠉⢀⣀⣀⡀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣟⣃⣚⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣵⣔⢀⣀⣀⣀⣤⣄⣀⣀⣀⣠⡤⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠴⠤⠶⠶⠦⠴⠦⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠶⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠧⠜⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣡⠞⠻⠉⠉⠏⠁⢲⣬⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠰⣠⠄⠆⠀⠀⠉⣹⢍⣑⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠈⢀⠀⠀⢀⠞⠙⢦⠙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⢤⣈⣉⣠⡔⠋⠲⢤⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠑⠶⠊⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣖⡆⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠷⡞⢁⠙⣤⣤⣤⣤⣤⣄⣠⡔⠉⠀⠀⠀⣠⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠧⡴⠀⠣⢔⢯⡟⠉⠉⠉⢼⢟⢁⡉⠉⣦⠤⠚⠁⠀⠀⡠⣻⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠀⢀⣔⣁⠀⠐⠀⠀⠂⠈⡺⠓⠗⠋⠁⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⢹⣍⣉⣐⣒⣒⣒⣒⡒⠛⠬⠭⠭⠭⠭⠭⠤⠄⣀⣀⣀⣀⣠⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠙⠛⠒⠒⠒⠒⠒⠚⠁

⠀⠀⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠑⡄
⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⠀⠇⠀⠀⠀⠀⠄⣿⠈⠄⠁⠀⠀⠀⠀⠀⠠⠰⡀⠀⠀⠀⠀⠀⢠⡐⠂⠀⠬⠆⠀⢀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⢰⠗⠊⠀⠀⠉⠂⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡆⢉⠃⠁⠀⢀⠼⢇⠀⠀⠀⠀⠠⠆⠒
⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠤⠀⡠⠀⠈⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠅⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠀⠀⡀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠃⠀⢺⠉
⠀⠀⠀⠀⣐⠈⠁⠤⣄⡀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠠⣄⡀⠀⠀⠘
⠀⠀⠀⠀⠈⢉⣀⣀⣀⠁⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠑⠒⠆⠀⠀⠁⠀⠀⠀⠐⢂⠩⠛
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠙⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡸⠋⠉⠉⠻⠋⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠀⣴⣿⣿⣶⣶⣿⣿⣦⠀⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⣴⡆⣿⣿⣿⣿⣿⣿⣿⣿⠆⣦⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣠⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⡍⣿⣿⣿⣿⣧⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⢻⡇⣿⡇⣿⣿⣿⣿⣿⣧⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢸⣿⢸⡇⣿⡇⣿⣿⣿⣿⣿⣿⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⣿⢉⣭⢹⡟⢸⣿⢸⡇⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣸⣿⢸⣇⢸⣿⢸⡇⣿⡇⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣛⣙⣛⣘⣃⣘⣛⣘⣃⣛⣃⣛⣻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠴⠾⢷⡄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠶⠾⣦
⠀⠀⠀⠀⠀⠀⢺⣿⡶⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢿⣷⡦
⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇

⡴⢲⣤⡄⣤⢠⣆⢰⣄⡄⢲⠆⣤⢲⡆⣰⡄⢶⠆⡄⣄⣰⢰⣒⠀⣶⣦⢴⡂⢴⡂⣴⡂⢠⣦⢠⣶⣄⡴⠂⣄
⣓⣾⣉⣓⣋⣛⣙⣘⣈⣃⣘⣀⣛⣘⣁⣋⣓⣘⣀⣃⣘⣃⣘⣒⣀⣋⣓⣘⣂⣰⣛⣛⣂⣚⣙⣘⣙⣊⣳⣂⣃
⣿⣟⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⣹⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣠⡖⠋⠡⠌⠻⠿⠉⠛⠙⠛⣷⣦⣨⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡾⠟⠁⠀⢠⡄⢠⠀⣴⡀⠀⠀⠘⠋⢻⣯⠳⣌⡙⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠈⠱⣏⠀⠈⠀⠀⠀⠀⣠⣾⣿⣧⠈⠻⣗⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⣀⣀⡀⠀⠀⠀⡼⢿⣿⣿⣿⠷⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⢄⣀⠈⠁⢀⣀⡴⠚⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢻⡉⠉⢉⡽⠁⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡨⠧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠦⢷⡎⠀⣄⠘⣶⡤⢤⢤⣤⡤⠤⠤⢤⣤⡴⠯⣀⠀⠀⠀⠀⢀⣟⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⢒⣦⠈⢧⡀⠾⡏⣷⠋⠉⠉⠉⠉⣹⣿⢏⠉⠉⠉⠛⣿⣀⣠⠖⠋⠀⠉⠉⠉⣩⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠯⡩⠟⠁⠀⠀⣨⠟⠙⠛⠀⠀⠀⠤⠄⢹⣃⡏⣞⡔⣀⠤⠟⠉⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⣒⣒⣀⡈⠉⠉⠈⠁⠀⣩⠟⠙⠛⠁⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⡛⠛⠓⠒⠒⠒⠒⠲⠶⠤⠤⠤⢬⢭⣭⣭⣍⣉⣉⣙⣁⣀⣀⡀⠀⠀⠀⠀⠀⢀⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠶⠦⠤⠤⠤⡤⣤⣭⣭⣉⣉⣉⣉⣉⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⡿⣏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⣴⣿⣿⣄⣱⣿⣿⠁⠀⠀⠀⠀⢀⣄⠀⠀⢀⣀⣤⣶⢲⡆⠀⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠛⢿⣿⣿⣥⣆⠀⠀⢸⣿⣿⣤⣶⣮⣿⣿⣷⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠙⣿⣿⣿⡿⠟⢛⣦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣤⠀⠀⢶⣿⣿⣿⣄⠀⠈⠉⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣏⢁⣹⣿⣿⣧⡀⣠⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡳⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢶⡄⠆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠀⣠⣤⡴⠁⠀⠀⢀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⠿⠛⠛⠷⣤⣔⣿⣿⣟⣴⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠀⣠⣶⢾⣿⣿⣿⣬⣉⣥⣿⣿⣿⣿⣿⣿⣿⣿⠋⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⡟⠉⢸⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡉⣿⣿⡔⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠠⣾⠀⣻⠘⡟⢿⠻⡻⠟⠋⠃⠉⠀⠁⠀⠈⠁⠀⠉⢻⣿⣿⣿⠿⠛⢿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⠿⠟⣡⠶⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⢁⣬⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⠅⠀⠘⠳⠟⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣹⢿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⣾⣿⣿⣬⣿⣿⡁⠀⠀⠀⣠⣦⡀⢀⢠⣴⣾⠛⠟⠀⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣙⣿⣿⣿⣧⣄⠀⣿⣿⣿⣿⣿⣿⣿⣥⣄⣀⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣄⣤⣤⣤⣤⣶⣶⣶⣶⡶⠿⠿⠿⠻⠟⠛⠛⠛⠉⣉⣉⣉⡉⠉⠉
⠀⣶⣶⣶⠾⠿⠿⠿⠿⠛⠛⠛⢛⣛⠉⣉⠉⣩⣩⣬⣤⣤⡤⠶⣶⠶⢰⣶⠘⢻⡟⢹⡇⢿⠀⡿⢹⣏⡁
⠀⠀⠀⠀⠀⣴⡟⣷⢸⡇⢸⡇⢸⣿⡄⢿⣧⢸⡇⢸⡇⠈⣟⠀⣿⠀⣾⢹⡇⢸⡇⠈⣷⠸⣷⡇⢸⡏⠁
⠀⠀⠀⠀⠀⢸⡇⢻⡾⣧⢸⡇⣸⣬⣧⢸⡏⢿⡇⠘⣷⠀⣿⠀⢹⡆⣿⠉⠿⠨⠇⠀⠿⠀⠛⠁⠘⣛⣋⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠸⠧⣼⡃⠻⠾⠃⠟⠀⢙⣚⣃⣈⣁⣀⣭⣤⣤⣤⣤⣴⣶⣶⣶⣶⠾⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⣤⣤⣤⣶⣲⣿⡷⠶⠾⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁
⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⣻⣯⣧⣿⡀⠀⠀⠀⢠⣤⣤⣤⣀⣀⣀⣀⠀⠉⠀⢠⣴⣲⢿⡄⠀⠀⠀⠋⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⠀⠀⠀⢰⣲⣿⣿⣿⣿⣿⡿⠀⠀⠀⠘⣏⡺⣟⣷⠀⠀⣤⠀⠀⠀⠀⡼⢤⠀⠀⢀⣀⣤⢴⡤⣀⠀⠀⠠⠲⢤⡀⡐⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠀⠹⣟⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⡞⡿⠏⠀⠀⠀⠉⠀⠀⠴⢉⠈⠀⠸⣯⣿⢯⢯⣿⠽⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠐⠀⠻⠿⠟⠛⠋⢁⣀⠀⠀⠀⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢻⣾⡾⠾⠚⠋⠁
⣿⠀⠀⡢⠀⢀⠐⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡴⣶⠤⢄⣀⠀⢹⣿⡿⣇⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣿⣿⣟
⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠼⣿⣿⣿⠃⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠧⠀⣀⠀⣻⠃⠸⣿⣿⠏⠀⢶⣿⣿⣿⣿⡧⠀⢀⣤⡞⢷⣶⠀⠀⠀⠈⠻⣿⣏⣿
⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠉⢳⣶⣿⣻⡄⠀⠀⠀⠀⠀⠀⢀⣞⡀⠀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⢿⠿⠛⠁⠀⠀⠻⣗⢟⡵⠟⠁⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠐⠄⠀⠠⠀⠀⠀⠀⠀⠈⣿⣓⣯⣷⠀⠀⠀⠀⠀⠀⠼⢀⣋⡿⠒⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣄⡀⠀⠀⠸⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⣽⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠘⠿⣿⣿⡟⠀⠀⠀⠀⣔⢦⣀⠀⠀⠀⠀⠀⠉⠛⠛⠿⡟⠀⠀⡀
⢼⠀⠀⠀⠀⣠⣶⡀⢾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⢠⡶⡝⣝⣿⠄⠀⠀⢰⣷⣦⣭⣀⠀⠀⠀⠸⣿⣯⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠧
⢸⠀⠀⢴⣿⣿⣿⡿⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡦⠙⠶⠛⠁⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠈⠳⣿⣿⣿⡟⠁⠀⠀⠀⠂⠀⠀⠀⢸⠀⠂
⣼⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢿⠿⠋⠀⢀⣀⣀⣄⠀⢀⣾⣿⣿⣿⣿⡟⢀⣴⣦⣀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢀⡀⠀⠀⠀⠈⠀⢠⣶⣦⣄⠀⠀⠈⣿⣯⣿⠀⠘⠵⢯⣿⣿⠟⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡀
⡄⠀⠀⠀⠀⠀⠀⣠⢾⡻⡙⢶⣟⣳⢄⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠝⠁⠀⠀⢀⣀⡀⠀⣟⣛⢻⠃⠀⢠⣤⣶⣶
⡇⠀⠀⠀⠀⣴⣾⡫⡱⣝⢎⢮⡻⣥⠶⠃⠀⠀⠀⠀⠈⠙⠿⠃⠀⠀⠀⠀⠀⢀⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⡟⡽⣽⣣⠰⠧⣹⣾⠀⠀⠀⣿⣿⣿
⠀⣰⣤⣀⠀⠈⠻⡕⢝⢮⣢⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣧⠀⠀⠀⣀⡀⠀⠀⠀⠻⢿⣾⣵⡧⠿⠀⠀⠀⠀⠀⠀⠀⣉⣡⣄
⠸⣿⣿⣿⣷⡆⠀⠘⢿⡷⠋⠀⢰⣿⣟⣿⠀⠀⠁⠀⠀⡔⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⢴⡮⣫⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿
⠀⠈⠛⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⠛⠀⠻⣤⡶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉
⣀⡀⠀⠀⠀⠀⠀⠀⢀⣶⡤⠄⣀⠈⠉⠁⠀⠀⣠⣤⣤⠴⣦⠀⠀⣰⣿⣶⣦⡄⠉⠀⣤⣤⣄⡀⠙⠈⠁⠀⠀⠀⣀⣤⣤⡀⠀⠀⢀
⣻⣽⣢⠀⠀⠀⠀⠀⣼⠁⠀⠨⢹⠃⠀⠀⢀⣾⣽⣿⣾⣷⣿⠀⠀⠻⠿⣿⣿⡟⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⡿⢆⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾
⣿⠟⠁⠀⠀⠀⠀⢰⣧⣀⠠⢈⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠀⠀⠙⠻⠿⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠁⣂⠀⠀⠀⠀⠀⠃⠨⢽⠏⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⣿⡏⢹⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠿⠿⠇⠀⣠⡔⠒⡉⠉⠀⢧⡴⢆⠀⠀⢠⣶⣶⣶⣿⡇⠀⢈⡄⠀⠀⢤⣿⣷⣦⣤⣀
⠀⠛⠓⠚⠒⠓⠀⢰⣶⣶⣶⣦⠀⠀⢠⣤⣀⡀⠀⣠⣶⣄⠀⠀⠀⠘⡑⠀⠐⠀⠀⠘⡗⢈⡀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠀⠀⢀⣞⣞⣞⣞⣞⣟⡿⣄
⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⣿⠇⠘⣿⡳⡬⢽⠛⢿⣿⣻⣗⠀⠀⠀⢱⣤⣀⡠⠶⠒⠉⠁⠀⠀⠀⢻⣿⣿⣿⣿⠂⠀⠀⠀⠺⢿⣾⣟⣞⣟⣾⠿⡶
⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠙⠯⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⣤⡀⠀⠸⠯⠷⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⣯⡹⠃
⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠲⢤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⠀⢰⡖⣶⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣮⡾⡗⠀⠀⠀⠀⠀⠙⠻⣿⣟⠀⠀⠀⠀⠀⠸⠧⠽⠼⠟⠀⠀⠀⠀⠀⠀⠀⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦⡀⠀⢠⣟⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⡇⢠⡟⢉⣿⢶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣤⣤⣤⡄⣠⣤⡤⠀⣤⣤⡄⢠⣤⣤⡤⠀⣿⠀⢸⣧⡟⢀⣾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⣿⡷⠆⠀⠿⢶⡆⠀⢸⡇⠀⠀⣿⠀⢸⡿⢁⣾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠘⠁⠀⠙⠛⠋⠀⠛⠛⠃⠀⠘⠃⠀⠀⣿⢠⡿⠁⣾⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢫⡿⠁⣼⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⢾⣇⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣇⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢰⡟⠉⠉⠉⢉⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢀⠀⡀⣀⡀⠀⣀⣀⣀⠀⠀⢠⠒⠀⣆⣀⠀⢀⣀⠀⣀⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⢀⠿⡁⣔⣺⢸⡇⢹⠀⡇⠀⢈⣙⡆⡏⢸⡇⣗⢚⠰⣗⡚⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⡼⠧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢹⡟⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠓⠞⠁⡶⠶⣦⠀⣀⣀⣀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⡇⠀⣀⣀⡀⠀⡇⢀⣿⣤⣯⣈⣿⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡏⠉⣿⢠⡟⢋⣥⢤⣤⡉⠻⣄⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⠀⣿⡟⠀⠉⠀⠀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠀⠓⠒⠛⣇⠀⠀⠀⠀⠀⠀⠀⢸⠟⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠙⢦⡀⠀⠀⠀⢀⣴⣏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠆⠀⠀⢉⡿⠛⠛⠉⠀⢨⣟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣯⣽⣿⣥⣤⡴⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⡎⠀⡱⠀⢰⢀⢰⠰⠒⡆⡗⢰⢸⠂⡐⡆⡗⢰⠰⣠⢢⠲⠄⢀⡎⡆⢰⢲⢀⠰⡄⡗⣄⠆⢆⢰⠰⣆
⠀⠀⠈⠙⠃⠈⠉⠈⠉⠀⠀⠁⠈⠈⠈⠁⠉⠁⠉⠈⠀⠁⠈⠉⠀⠈⠀⠈⠈⠈⠈⠉⠁⠁⠞⠀⠉⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠘⠋⠙⠇⣲⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⣀⠈⠀⢃⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠂⠁⠲⣄⠀⠀⣀⠘⢀⠄⠀⠀⠀⠀⠀⠒⠖⠀⣁⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡰⠁⠢⢄⡀⠀⠀⢈⠙⠂⡴⠀⠀⠀⠀⠀⠀⠸⠀⢀⢀⡀⠀⠀⢂⠀⠀⠀⠀⠀⢀⠤⢀⠌
⠀⠀⢀⠎⠀⠒⠲⠄⣙⠀⢀⠛⠀⠚⠈⠁⠀⠀⠀⠀⠀⠸⠀⠀⠈⠁⠀⠀⠈⠃⠀⠠⠀⠀⠀⠀⣨
⠀⠀⠀⠈⠁⠐⠂⠄⣀⠐⠛⠂⡄⠀⠰⡁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⠀⠐⣀⡤⠾⠀⠀⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡨⠂⠁⠐⡀⠀⠀⠀⠀⠈⢢⠀⠀⠻⣿⣿⠃⡀⠃⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣠⠂⠡⡀⠀⠀⠀⠀⠀⠙⠢⠤⠀⢉⢡⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠐⠀⠤⠤⠀⠄⠖⠀⠾⣂⡓⠋⠀⠀⠀⠀⣀⠀⢀⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠆⠀⠀⠈⠀⠁⠀⢠⠀⠉⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⢰⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⢸⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠠⢀⠇⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠀⠀⠀⠒⠀⠉⠀⠐⠠⢀⠀⠀⠀⠀⡀⠀⠀⠂⠁⠀⠈⢢⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⠀⠀⠀⢀⡠⠔⠊⠁⠀⠑⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⡀⠀⠀⡀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠰⠤⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠒⠤⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣠⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⠂⠉⠉⠉⠉⠉⡆⡇⠈⠉⠈⠁⠂⣰⠂⠐⡆⠀⡤⠤⢤⢀⡤⢀⣀
⠀⠰⢹⠀⠀⠮⠭⠭⣍⠋⠷⣉⢶⠀⢐⡒⢚⢸⠀⠀⡇⡀⡇⠀⣸⢺⠀⠀⠀⠑⣄⡔⠊⠠⡀⠀⣀⠤⠀⠀⢀
⠀⢫⠘⢄⣀⣀⣀⡀⠀⠑⡄⢐⢸⠀⢸⠀⢸⢸⠀⢀⠁⡇⡇⠀⣿⡘⠀⣈⢣⠀⢹⡈⠣⡀⠈⠉⠀⡀⠼⠃⠃⠀⢨
⠀⠀⠉⠒⠒⠒⢢⣸⠆⣀⠤⠼⠼⠀⠼⠀⣨⣸⠀⠸⣀⢣⠁⢠⡇⡇⠀⣿⡌⠀⡸⠈⡆⡸⠀⢰⠉⠀⡘⠸⠀⠀⡎
⠀⠀⠊⡍⠉⠉⡩⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⢀⡼⢵⠁⠀⠉⠀⡠⠃⢰⢀⠇⠀⡏⠀⢠⠃⡇⠀⡘
⠀⠘⠦⣩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣾⣶⠴⠋⢰⡸⢤⣶⠶⠋⠀⠀⡎⡼⠀⢰⠁⠀⢜⣘⠠⡤⠁
⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⢀⣤⡶⠚⡏⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣓⡲⠊⠀⢀⠸⣅⣈⠆
⠐⠊⠀⠀⠠⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠰⡆⣀⠁⠒⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣄⡀⠀⠉⠁⠀⠤⠐⠆
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠔⠤⣄⣀⡀⢀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠈⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠉⠙⠒⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡖
⠀⠀⣀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⣀⣀⡂⠀⠀⠠⡀⠀⠀⠈⢀⣐⣠⣄⣀⣀⢃⣃
⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠭⢉⠐⠂⠐⠀⠀⠐⠂⠀⠀⠉⠉⠈
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⢼⠤⠠⠀⠀⠀⡀⠀⠀⠀⡘
⠀⠀⠌⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠈⠀⠀⠀⣀⠼⠂⠀⠀⠀⠂⠀⠠⠐⣒⢣⠆⡀
⠀⠀⠀⠀⠁⠀⠀⠉⠒⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⠀⠀⠀⣀⡴⣞⣁⠀⠀⠀⠀⢀⠒⠠⡀⠀⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠁⠈⠑⡒⠤⣀⠀⠀⠀⠀⠀⣿⠀⣀⠴⠋⠁⠀⠀⠀⡈⠁⠒⠢⠦⠤⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠄⣀⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠄⣴⣾⣿⣷⣾⣿⣷⣦⡠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣦⣭⣭⣭⣭⣭⣭⣭⣭⣵⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⢰⣶⣶⠶⣩⣾⣿⠿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀⡀⣫⡘⣿⣿⢸⠷⡵⡷⣿⣿⣿⠰⢊⡹⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⡇⣿⡇⣮⢻⣿⣿⣿⣿⣿⣿⣿⡜⣏⢷⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⢸⢰⣶⣶⠶⣩⣾⣿⣿⣹⡞⣧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡟⣼⢸⣷⣵⡷⣿⣿⣿⣿⣧⢻⡸⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡿⠛⢿⣟⢫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢡⣾⠾⣷⠍⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡘⢿⢿⡿⣣⢋⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡹⡎⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠾⠿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡒⢒⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠀⣀⡜⠁⠀⠈⠁⠀⠈⢣⣀⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡴⠊⠁⢧⣤⣤⣤⣤⣤⣤⣼⠈⠑⢦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠙⣄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡇⠀⡇⠀⠀⠀⠘⡆⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⡇⠀⠀⠀⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⢠⡖⢲⠀⡇⢸⠀⡇⠀⡇⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⠃⢸⠀⡇⢸⠀⡇⠀⡇⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠲⡌⢧⠀⠀⠀⠈⠁⠛⠛⠛⠛⠛⠛⠋⠙⠁⠀⠀⡼⢡⠴⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠙⠒⡇⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸⠋⠒⣾
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢬⣓⣊⡥⣄⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⣠⢬⣑⣚⡥⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠀⠀⠈⣧⠀⠦⣄⣀⢸⠀⠀⠀⠀⡇⣀⣠⠴⠀⣼⠁⠀⠀⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠈⢸⠀⠀⠀⠀⡇⠁⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃



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

⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⡿⢏⠐⢌⣿⡿⠋
⣿⣿⣿⡿⢛⠐⢂⣷⡿⠋
⣿⣧⡘⠀⣉⣼⡿⠋
⣿⡿⣻⣶⡿⠋
⠀⢾⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠶⡶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⣰⡿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠟