Kys Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠉⣉⠉⠉⠛⠛⠛⠳⠋⠁⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⡾⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡙⢷⡀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⡀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢰⣯⣿⣿⣿⡎⣽⣿⣿⣿⣯⣿⣿⣿⡿⣿⣿⣿⣀⣤⣿⠟⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⣿⣿⣿⣿⡝⠿⠿⠛⠛⠛⠻⣿⣿⡇⣸⣿⣿⡿⠋⠁⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣇⣀⣤⣾⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠋⠀⠈⠻⣿⠟⢛⣿⣿⠟⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠌⠛⢿⣿⡿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⣇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡀⠀⠈⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢣⣠⡶⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⠔⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣶⣾⢿⠟⠓⢲⣿⣿⣦⡄⠀⠀⠀⢀⣤⣾⣿⣿⡶⢋⣥⠾⠋
⠀⠀⢀⣤⠶⠶⠶⣤⣤⣶⣿⣿⢟⣷⠟⠲⢶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⢿⡁
⠀⢠⡟⠁⠀⠀⠀⠀⠈⢻⣿⣷⡿⠃⠀⢀⣾⣿⣿⣿⣿⣿⠿⠿⠟⢿⣿⣿⣾⣿⠇⠈⢷⡀
⠀⣾⠃⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢀⡴⠋⠈⠻⢿⠿⠋⠁⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠘⣧
⢀⡿⣆⠀⠀⢶⣀⡷⠞⠁⠙⣦⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠈⠳⡼⠷⢶⡛⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣠⡾⠃⣼⠃⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⡾⠋⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡏⠀⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠉⠛⠋⠀⠀⠀⢸⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠑⢆⣠⣄⣤⠤⣀⡤⠼⣆⡀⢳⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠿⠋

⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠉⠀⠀⢑⠀⠀⠀⠀⠰⣄⠀⠀⡀⠀⠀⠠⡪⠃⠀⠀⠀⠀⡔⠀⠨⠗⠀⢱
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠇⠀⠀⠀⠀⠀⠀⡆⠐⠀⠀⠀⠀⠁⢠⡤⠀⠁⠀⠀⠀⠐⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢠⢸⡇
⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⣏⠀⠀⠀⠀⠀⢸⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈⠝⠀⠀⠀⠀⠀⠘⡞⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⣿⡂⠀⣿⢸⣆⡇⢿⡁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸⡟⣷⡄⣏⠀⢩⡇⡈⣳⠀⠘⠓
⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠢
⠀⠈⠘⢀⡀⠀⢀⠀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠑⠐⠋⠀⠀⠀⠀⠃⠐⠃⠀⠈⠀⠀⠀⠈⡞⢀⠱⠀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡈⠀⠀⠀⠀⠈⠶⠒⠋
⠀⠀⠀⡀⢰⠀⡀⠀⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠥⠘⠀⡆⠀⠑⠀⠀⠀⢠⢀⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠒⠦⠅⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢤⠇

⠀⠀⠀⠀⠀⠀⡠⠐⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠤⢀
⠀⠀⠀⠀⡠⢊⠔⠉⡱⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠤⠤⡀⠀⠀⠀⠝⢆
⠀⠀⢀⠈⢰⠁⡠⠊⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢇⠀⠀⡱
⠀⠀⠨⡀⠀⠉⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠺⡪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀
⠀⠀⠀⠀⡀⠉⠢⠒⠉⠑⢄⢠⡀⠈⠫⡪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⡂⠀⠑⢕⠀⡉⠢⡀
⢀⠔⢈⠠⠤⢈⠐⢄⠀⠀⠀⠀⠈⠓⡆⠈⠻⣮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠈⠢⡀⠀⠑⢌⠢⡨⡀
⡄⠠⠁⠀⠀⠀⠡⠀⡄⠀⠀⠀⠀⠀⠈⠹⡀⠈⠻⣮⡰⠀⠀⠀⠀⠀⡠⢒⣒⠂⢄⠀⠀⠀⠈⠢⣀⣀⣁⠈⡪⠲⣄
⠃⠠⡀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠹⠄⡈⠻⣶⠆⠀⡠⠌⡐⠁⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠘⠒⣎⡢⡀⢑⢄
⠈⠢⠈⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠘⠠⠉⠀⠀⡠⠘⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⡂⣑
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠤⠀⠔⡡⢌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠑⠠⡑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⡀
⠀⠀⠀⡀⠄⢐⣂⡐⢄⠀⠀⠀⠀⠈⠣⡜⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢄
⠀⡔⣡⠒⠉⠉⠀⠈⢢⡑⢄⠀⠀⠀⠀⢈⠳⡈⠢⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠙⢡⠟⢢
⠐⢠⠃⠀⠀⡀⢨⠔⠁⠈⢢⠑⠄⠀⠀⢌⠠⡈⠑⣌⠢⡀⠀⠀⡐⠁⠀⣀⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⢉⠦⡈⠢⡀
⡄⢠⠀⠀⢠⠄⠀⠀⠀⢀⠜⠠⡀⠀⠀⠀⠘⠔⠒⠀⠑⠂⠀⠀⡇⠀⢈⠀⢨⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡜⡠⡀⠑⢌
⠐⢄⠑⢤⠁⠀⠀⢀⠔⢁⢄⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠈⠐⠤⠒⠁⠁
⠀⠀⠑⠄⠑⠦⠔⠁⠀⠀⠑⢕⢄⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⣮⣐⢄
⠀⠀⠀⠀⠈⠒⠈⠈⠒⠑⠄⢄⠑⢕⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠄⠄⠒⠢⡀⣀⠀⠙⢷⣕⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢒⣀⠑⢝⢦⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠉⠻⣮⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡄⠑⢝⠦⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢀⡈⠛⢶⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠑⢬⣢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢄⡙⠿
⣿⣷⠾⠿⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣾⣷⣶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠋⠈⣿⣀⡀⠀⠀⠀⠀⢏⠈⠀⠲⠎⣧⣄⠊⠉⠑⡤⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⡎⢡⣼⣸⡇⢿⡄⢀⣷⠀⠀⣠⡶⢦⣀
⠀⠀⠀⠀⠀⠀⢸⣫⢋⠽⣟⣘⣽⡉⢻⡀⠀⠀⠀⠀⠀⣞⠠⢻⠖⠿⢰⡔⠚⠁⠀⠐⠀⠀⠉⡀⠀⠆⠀⠀⣿⣿⣟⡝⠚⣽⣿⣯⡿⠀⠀⠛⢿⡿⠋
⠀⠀⠀⠀⠀⠸⠿⣷⣈⣻⣿⣿⣶⣃⣼⠿⢇⠀⠀⠀⠉⠬⠛⣐⣣⠸⠶⡅⠀⠀⠀⠈⠐⣿⠜⡑⠈⠀⠀⠈⠓⣿⡋⢭⣿⠟⣿⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣰⡸⢙⣿⣿⡿⢶⣷⢾⡟⢿⡧⣸⡷⠀⠀⠐⠶⠚⠙⣯⠓⠖⠃⠀⠀⠀⠀⠀⡉⢽⡆⠀⠀⠀⠀⠀⠙⣿⢢⣴⢺⡷⠃⠀⠀⠀⠀⢸⡷⠖
⠀⠀⠀⠀⠘⣿⡀⠻⠇⠹⣾⢻⡾⢡⡸⢩⣿⠁⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠁⠀⠀⠀⠀⠀⠀⠉⢻⡿⠋⠁
⠀⠀⠀⠀⠀⠈⢷⣴⣤⣠⡟⠺⣧⣼⣷⡿⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡤⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⡇⢠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡄⣿⢀
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣮⣽⡄⠀⡠⢧⣨⠦⠀⠀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡟⢻⠁⠀⠂⢞⡱⠔⠀⠀⠀⠀⠂⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠭⠭⣀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⣀⣀⣀⡴⠋⠀⠀⠀⡖⠀⢦⡆⠹⣦⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠰⣿⣶⣿⠆⠀⠀⠀⠀⢺⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⠁⠀⠀⠀⠀⢠⡄⡀⠃⠃⡀⢤⡄
⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⠀⠀⠀⠀⠀⢈⡨⣵⣿⣷⣥⢙⠄⠀⠀⢠⠆⠉⠁⠀⢠⡄
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠉⠙⠋⡟⠋⠀⠀⠀⠀⠘⣄⡀⣀⢀⣠⠃
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⣿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⢻⣿⠊⠁
⠀⠂⣤⣤⣤⠤⡤⠤⣤⢤⠀⢿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⣿⠋⠀⢀⠀⡠⢆⠈⠀⣸⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣤⣞⣁⣱⡄⢠⣖⠀⣿⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣶⣄⣠⣴⣄⣀⣀⣸⠀⣿⠀⠀⢈⡗⢀⣤⡀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠄⠀⠀⠀⠀⠀⢀⢸
⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⣿⠀⠀⠸⠻⣼⠄⠷⡧⣹⡄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠄⢅⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢄⠈⠀⠀⠀⠀⠀⠈⢻⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⢰⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠁⠀⠀⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠿⠿⠿⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⣀⣤⣈⠉⠛⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠀⠀⡴⠋⠉⣿⢷⡄⠀⠀⠉⠻⢷⣄⠀⠀⣀⣀⣀⣀⣿⣁⣀⣀⡀
⠀⠀⢠⡞⠁⠀⠀⠘⣁⠀⠀⠀⢱⡀⠀⢻⣦⠁⠀⠀⠠⡀⣠⡿⠟⢛⣉⣉⣉⣹⡿⠉⠉⠉⠛⠳⣦⡀
⠀⢀⡟⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠻⣧⣄⠀⣠⠟⠋⠀⠀⣟⢉⣩⡿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣄
⠀⢸⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠃⢀⡄⠀⠀⠙⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠋
⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠁⠀⠀⠀⠈⠙⣧⣤⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠐⣺⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣼⢋⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠘⣯
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⡼⠋
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⢳⡀
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠘⠉⡇
⠀⠀⠀⣿⡆⠀⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡖⠃
⠀⠀⢸⣿⡇⠀⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠣⡀
⠀⠀⣸⡟⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸
⠀⠀⣿⠇⠀⠠⠜⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠘⡄
⠀⢠⡟⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠁
⠀⣼⠇⠀⢠⡏⠉⠁
⠀⣿⠐⠀⢸⡀
⢸⣯⠀⣠⠎
⣾⡧⢰⡇
⡟⢀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣒⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠖⠉⠀⠀⠀⠀⠉⠓⢝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡜⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣤⣍⣉⠒⠁⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠳⠶⣬⣍⣃⡱⠤⢬⣙⢢⡀⠀⠀⠀⢀⣀⡜⣸⣀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡤⠴⠶⢤⣉⠙⠛⠲⠞⢀⡇⣀⠔⢋⡴⠞⣴⡅⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⡇⠀⢰⠁⠀⠀⠀⠀⠙⣧⠀⠀⠀⣼⡷⠁⠀⠘⠛⢛⣋⣤⡀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠸⡄⠀⠀⠀⠀⣠⡟⠀⠀⣸⢻⠃⠀⠀⠀⣼⠋⠁⠀⣸⠆⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠘⣧⣀⠈⠓⠲⠖⠚⠋⠀⣀⡼⠃⣼⠀⠀⠀⠀⠙⠶⢚⣋⣡⡤⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣸⢋⠙⠓⠶⣦⣤⠴⠞⠉⠀⠀⡏⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠈⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⡿⢸⣄⡀⣸⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⣆⣀⣠⠤⠖⠃⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣼⠁⣞⡏⢠⠏⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠴⠒⠋⠉⢱⡈⣧
⠀⠀⠀⠀⠀⢠⡞⡼⣾⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⣤⠴⠖⠋⠙⡄⠻⣄⣀⣀⡀⢈⡇⢹⡄
⠀⠀⠀⠀⢀⡼⡳⡽⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠰⣏⠀⠀⠀⢀⠔⢁⡤⠶⠬⢍⡉⠉⠁⠈⣇
⠀⠀⠀⠀⡼⢱⠱⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⢦⣠⡶⠃⣰⠏⠀⠀⠀⣀⣠⠇⠀⢠⡇
⠀⠀⠀⣸⢃⠃⢁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠸⣇⣤⠴⠖⠋⠉⠀⢀⣴⠟
⠀⠀⢠⠇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠛⠋⠁
⠀⠀⠹⣀⡠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠛⠉⠁
⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣨⠀⠀⠀⠀⠀⠀⠀⠎⠱⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢿⡀⢠⡟⣷⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠸⡇⣼⠁⠸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⡂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉








⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⡄
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⣤⣤⣬⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⣛⣛⡋⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣋⡿⠿⣿⣿⣿⣿⣿⣟⣙⡿⢿⣿⣿⣿⣿⣿⣇⣀
⣿⣿⣿⣿⣿⣿⡇⢿⣿⡿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡔⣯⣼⢛⣤⣧⡤⠤⠤⠤⠤⢤⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠼⠶⢤⡀⡴⢁⡴⠶⠆⢀⠀⠀⠀⠀⢐⡊⠀⠆⢀⠉⠢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⠜⠛⠤⠃⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⡠⠀⠠⢠⠂⢨⡿⠷⣿⠀⢀⠄⠀⠑⠀⠀⠠⠀⢤⣀⣅⠠⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠘⠀⠎⠀⠀⠉⠙⢇⠀⡞⠉⠉⠀⠀⠀⠀⡄⠀⠀⠀⠈⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠙⠋⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠰⡇⠀⠀⠁⠀⠀⢀⡇
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⢈⣶⢤⣴⡀⣴⠟⠀⠀⠀⠀⢸
⠀⠀⠀⠱⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠳⢾⠁⠀⠀⠀⠀⠈⠻⣽⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠒⠓⢄⠀⠀⡆⠀⡂⢀⣁⠔⠂⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⠊⢁⡸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠆⠀⠀⠀⠀⠀⠘⠑⠂⢰⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣤⣄⠀⠀⠀⢠⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⢸⡀⠀⠀⠀⠀⠀⡄⠀⢰
⠀⠀⠘⣧⡟⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⠀⢰⡩⢭⡝⡆⠀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠀⠀⠃⠀⢸⢠⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠱⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⢰⡆⢒⡎⡕⣴⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⣄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠓⣄⣏⠰⠂⣛⣁⡿⠃⠀⠀⠈⠻⣇⢠⠀⠀⠀⢀⠀⠃⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡖⠀⠊⢯⠗⠈⢲⡆⠀⠁⢸⢳⣎⡶⠉⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⠀⠑⠒⠒⠒⠚⠁
⠀⠀⠀⠌⠠⠀⠀⠀⢄⡀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣽⢂⠀⢈⡏⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢠⠄⠀⠀⠀⢸⡉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠚⠙⠈⠛⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣤⣶⣶⣶⣤⣄⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⣠⢀⡠⡆⠀⠀⠀⠸⠶⠂⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠙⡅⠀⠀⠀⠀⣿⠇⠀⠀⠀⣩⠃
⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⡇⠀⠀⠀⠉⡈⠉⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠈⣉⣀⠀⢀⣈⠋⠀⠀⠀⠀⣀⠀⢠
⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠉⣀⡌⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⢀⡤⣄⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⡿⠃⠀⠀⢠⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⡇⣼⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠉⠀⠀⠀⠀⢀⡷⠋⠇⠀⠀⠀⠀⠀⣇⣀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⡼⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠻⠇⠀⠀⠀⢇⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣶⠤⠤⠤⠶⠀⠀⠀⠠⣶⠤⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣞⣿⡦⠤⠤⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡊⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡐⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠
⠀⠀⠀⠀⣀⢺⣷⣦⣾⠁⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠙⠁⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣴⣹⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⠏⣡⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠴⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠖⠲⠆⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⡴
⠀⠀⠀⠀⢀⣿⢋⣤⡇⠹⡆⠀⠀⠀⠀⠀⠀⢰⢀⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠣⠶⣇
⠀⠀⠀⣶⠀⠁⠀⠀⠙⠖⠁⠀⠀⠀⠀⠀⠀⠸⢈⣤⠁⠛⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠶⢄⠀⠀⠀⣿⠄⠀⠀⠀⠈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠐⠠⠞⠁⠀⠈⠁⠀⠀⠀⢀⣻⣟⡆⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠰⠀⠀⢯⣽⠟⠂⠤⢠⠴⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡀⠀⡀
⠀⠀⡾⠀⠰⠦⠖⡀⠀⠀⠀⠐⠶⠂⠀⢰⡀⠀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠋⠁⠙⠛⣦⡄
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣄⠀⠀⠐⣷⣠⣼⡦⡀⠀⠀⠀⠀⢠⠠⣤⠀⡀⠀⠀⠀⠀⠀⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠀⠀⠀⡈⢻⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⣄⢻⣿⡁⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠐⠦⣀⠀⠀⡠⢁⠀⠀⠀⠤⡄⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡀⠐⠂⠰⠆⠀⠀⠀⠀⠐⢆⠼⠉⠁⠉⠃⣤⡀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⣀⡌⣡⡬⠒⠚⠀⠛⠁⠀⠀⠀⢀⡀⠰⠆⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⠒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⢤⡮⠉⢳⡆
⠀⠀⠀⠀⢀⢰⣧⣤⣾⠁⠀⠀⠀⠘⠄⠓⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⢀⡄⠀⢠
⠀⠀⠀⠀⢻⣾⣿⡟⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠄⠀⠀⢴⣜⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣟⣠⡇⠹⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠⡀⠶⠢⠄⠀⠀⠀⠈⠉⠀⠀⠀⠘⣿⣿⠃⡴
⠀⠀⠀⣴⠈⠁⠈⠉⠳⡴⠃⠀⠀⠀⠀⠰⢠⠄⡠⣄⠀⠀⠀⠀⠀⣤⣦⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⡿⠳⠶⣇
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢠⠦⢐⡋⠀⠀⠀⠀⣴⠿⣷⡇⠀⠀⢀⡠⢤⠶⠀⠀⠀⠠⣆⠶⢄⠀⠀⠀⠀⢿⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣀⠔⠋⠀⠀⠈⠁⠀⠠⠀⠀⠙⠛⠁⠀⠁⠘⢩⣀⠀⠀⠀⠀⠀⠀⠸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉
⠀⠀⢠⠦⠀⣤⣠⠀⠀⠀⠀⠰⠈⠂⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠠⠄⠠⠠⣴⠈⢑⣦⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠛⠁⠀⢰⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠋⠀⠛⠓⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣷⣤⣴⡶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣄⠀⠈⠁⠀⠀⠈⠃⠀⠀⡆⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡍⢈⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⢠⣼⠀⠀⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢰⣶⡄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢂⠀⠈⠀⠀⠀⠀⢴⢭⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠣⣟⣻⠟⠀⠀⠀⠀⠀⠀⠛⠞⠳⠾⡄⠀⠉⠐⠘⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⡀⠀⢸⠃⠀⠀⠀⠀⣼⣻⣦⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⡤⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⢸⡀⠀⠀⡀⠀⢘⣿⠁⠸⠄⠀⠀⠀⠀⠀⠈⠐⠂⠀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⣦⠀⠀⠃⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠘⢦⠀⢬⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡏⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠐⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠂⠀⢄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⣧⠀⢠⣾⠷⠶⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣈⠀⠀⡁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⠶⣾⡟⠀⠈⢻⡶⠾⠛⠈⠁⠀⠀⠀⠀⠀⠀⠈⢸⡿⠏⢿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⣴⠟⠛⠿⣦⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢾⣿⡷⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣾⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠈⠻⣶⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠿⠃⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⣧⠶⠶⣴⠟⢷⣄⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⡿⣤⣠⢿⡄⣼⠿⠛⠛⠛⠛⠛⠿⠻⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡺⣣⠞⠁
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⢊⢴⠃
⠀⠀⢝⠪⡫⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠕⢁⠆
⠀⠀⠈⠑⢌⠑⢝⢵⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⠪⠑⠋⠙⠢⡀⠀⡠⠂
⠀⠀⠀⠀⠀⠉⠢⡑⠫⡪⡉⠑⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠂⠁⠀⠀⠀⠀⠀⠈⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠛⠑⢁⡼⠓⠒⠢⠍⣒⠂⠤⣀⠀⡎⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣴⠏⠀⠀⠀⠀⠀⠀⠈⠁⢢⠱⣴⡘⢤⣖⣒⠶⡄⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⠀⠀⠀⠀⠀⢀⣀⣀⡸⢀⠏⠉⠢⢙⢿⡙⣮⢢⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⠀⠀⠀⡬⠟⣽⣿⢃⡊⠀⠀⠀⠀⠑⠽⣹⢺⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢄⢠⠾⡻⢓⡁⠈⠈⠪⡐⡀⠀⠀⠀⢀⢌⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢁⠑⢱⣇⠲⣧⠀⠀⠈⣚⣀⡠⢐⢵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⡰⠀⠉⠓⠖⣉⠠⠀⣉⡒⣬⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡔⢕⠄⣀⢀⡀⢀⣙⣄⡁⠢⢅⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣴⡿⠝⠒⠛⠒⢄⣟⣉⠽⡖⠉⠐⠶⣌⡐⠨⣑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣡⣾⡻⠁⠀⠀⠀⠀⠀⠀⠙⣄⠀⠷⡀⠀⠀⠀⠉⠳⣢⢍⣒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⢼⠘⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⢷⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⢀⣴⣶⠞⠭⠰⡖⠨⠝⢶⣶⣄⠀⠀⠀⠀⢰⠶⠽⣯⡀⠶⣀⣿⡧⢶⡄⠀⠀⠀⠀⠀⢀⣾⠟⢫⣽⣿⣍⠻⣷⡀
⠀⠈⠋⢸⠟⠲⢦⣤⠶⢺⢯⠉⠋⠀⠀⠀⠀⠈⠋⢁⣼⡼⣶⠇⣧⣉⠙⠀⠀⠀⠀⠀⣴⡾⠟⢰⣯⣻⣿⣽⡆⠹⢷⣦⡀
⠀⢀⢀⡾⠇⠀⠀⠀⠀⠐⠿⣆⢀⠀⠀⠀⠀⣠⢊⠉⣁⣼⣟⣦⡈⢈⠳⡄⠀⠀⠀⢼⣏⣠⣼⣆⣸⣿⣿⣇⣠⣧⣄⣹⣧
⠀⠈⣮⡄⠀⡀⠀⠀⠀⡀⠀⣼⡞⠀⠀⠀⠀⠀⡋⢰⠏⣠⣿⡀⠻⡆⢹⢀⠀⠀⠀⠸⣯⣾⣿⠋⡋⣿⣿⢙⠉⣿⣷⢿⠇
⠀⠠⢹⣵⣿⣿⣧⢰⣿⣿⣶⣽⡁⠀⠀⠀⠀⠛⠓⢾⡾⠉⡀⠉⢶⠷⠛⠛⠀⠀⠀⠀⠈⠛⠛⠻⢿⣎⣱⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠟⣯⠈⠽⢿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⢿⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢼⣿⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣷⢾
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣈⣉⣅⣰⡶⡦⠀⠀⠀⠀⠀⢐⣓⣿⣧⣿⣾
⠀⠀⠀⠀⠀⠀⣼⣿⡼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣩⡹⠋⠻⠶⠖⠀⠀⠀⠀⠀⠸⠇⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠸⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠏⠀⢈⡈⠀⠈⡙⠀⠘⠋⠀⡄⣀⢀⣀⠀⠘⢀⠈⠠⠇⠀⠔⠈⡐⠀⣈⠉⡀⠀⢰⡄⠀⢺⣦⠾⠛⠊⠓⠘⣢⠀⢦⠀⠈
⢄⠀⠚⠙⣢⣀⡀⠀⠀⠀⠀⡀⠀⠈⠙⡈⠀⠀⠀⠀⢀⠄⢚⠘⠃⣀⠀⠈⢳⢄⡉⢦⣴⠀⠀⠀⠀⠀⠀⠰⢤⣬⡤⢄
⠀⠀⠀⠀⢠⠿⡦⢀⠈⢐⡁⢁⣻⡆⠀⠁⠀⠈⠀⢶⣶⠀⢨⣍⠀⠀⠀⡀⠀⠠⠐⣊⡋⠃⢀⠐⠂⠀⠈⡉⡄⡸⢻⣶⣴
⠀⢐⠈⠁⠀⠀⠀⠀⠀⠙⠉⣂⠁⠐⣃⣄⠲⠀⠀⠀⠊⢠⣾⡶⣤⣠⣶⣁⡌⢁⠈⠉⠀⠛⠀⠀⠀⢀⣦⣡⣈⠑⠈⠉⠘
⠦⢤⣶⣄⠀⠀⠀⠐⢒⢰⡇⠀⠚⠃⠉⠹⣇⠀⢆⡸⢿⠞⠢⣧⠀⠻⠿⠟⠃⠀⠀⠀⢀⡠⠀⠰⠋⢸⣿⠏⠋⢀⣦⠀⠘
⠈⠻⣿⠾⠀⢀⡀⠠⠿⠋⠡⠤⠌⡸⠯⢤⡞⢁⠤⠀⠠⢶⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢾⣛⣿⡟⠀⠀⠙⣩⡀
⣴⡊⣡⣶⣦⡍⣿⠀⢤⠀⢠⢦⣀⠀⠀⠈⠁⠋⠈⠓⡤⠈⠱⠄⣀⠀⡀⠠⠀⠉⣘⠀⢰⡧⡀⣫⢟⠉⠀⠤⣅⠀⠿⠃⠈
⢠⡴⢀⣤⢸⠇⢙⣀⣠⠀⠀⡀⠙⠷⠂⠀⣀⣀⡀⠈⠀⢄⠀⣀⠄⣀⡄⢀⣀⡠⠃⣠⡿⣗⣻⣶⡿⠇⠐⢷⠞⢈⣤⠀⢀
⠀⠑⠙⠁⠀⠀⢄⠀⢆⠀⢀⣄⡰⠆⢀⣀⠒⠒⠺⡆⠁⢈⠀⢷⡒⠁⠀⠙⠋⢠⣴⣿⡇⣿⡏⢻⠀⠀⢀⠸⠁⠈⠉⠀⡹
⠀⠀⠀⠀⠀⠀⢈⣄⢸⡧⢄⠘⡏⡡⠌⣁⡂⠠⡀⠉⠀⠀⠰⢻⡟⢀⠒⣀⣠⡍⣭⡀⠁⠻⠉⠛⡄⠀⠠⣼⣷⡔⣠⠠⠁
⠂⠘⠈⠂⢀⠀⠀⡈⡎⠉⢀⡷⠀⠀⠀⠉⢁⠔⠁⠒⠖⢀⠄⡀⠀⠀⠀⠁⠈⠀⢀⡉⠰⢀⡀⠀⠀⠀⠀⠀⠘⠈⣁⠀⠈
⡄⠀⠈⢿⡟⠛⠉⡽⡆⠐⢬⡀⠀⠀⣠⣴⡆⡈⠄⠠⠖⢊⣭⣴⠦⢀⡀⠀⠀⠀⠘⡟⢀⣀⡄⠀⠲⠂⠀⠀⠀⠔⠀⠑⣦
⡅⠀⠒⠁⠀⣀⡀⢀⡀⠠⣎⣯⣴⣮⡒⢬⣠⣇⡀⢴⠆⠀⣏⣀⣠⣴⣙⣍⠄⢡⣉⡻⠉⠥⢶⠃⠈⠀⠀⠀⠁⠀⢀⡤⠈
⠀⣟⠤⠘⢃⠈⢁⠀⢀⣤⠹⢅⣤⣿⣿⡦⠹⡟⠋⠀⠀⢀⢿⠿⢹⣏⠉⡖⠟⢻⣿⡆⠀⠀⠻⢀⡼⢡⣦⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣛⣋⠡⠤⠄⠀⠰⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠀⠀⠀⠀⠐⠒⠂⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠠⠀⠀⠑⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⢣⠀⠀⠀⠀⠀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⡄⠀⠀⠀⢹⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⡀⠀⢤⣤⣀⣹⣛⠿⠿⡿⠇⠀⠀⠀⣸⣿⣶⣤⣼⣦
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣾⣄⠀⣄⠐⢤⡀⠀⡀⠰⠀⠀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣆⠀⠀⣸⣿⣿⡟⠉
⠀⠀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⡇⣏⠀⠀⠁⢀⣇⠀⣰⢰⣧⣿⣈⣏⣿⠿⠿⠛⠛⠻⠿⢿⢀⣴⡟⢸⣿
⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⣀⡈⢻⠋⠛⠛⠛⠿⢿⣿⣿⣿⣷⣦⡤⡀⣀⣼⣿⣿⡆⢸⣿
⠀⠀⠠⡿⠿⢻⣿⣿⣿⣿⣋⠉⠟⠿⠛⠛⠉⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣀⣽⣮⣙⡟⢃⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⠁⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛

⣷⣶⠿⠾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⢠⠋⠀⢀⠀⠀⠀⠀⠙⡄⠀⠀⡇
⠀⠀⢠⠇⠀⢠⠆⠀⠀⠀⠀⢀⡇⠀⢀⡇
⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⢀⡼⠁⠀⣼⣁
⠀⠀⣿⠀⠀⠘⢧⣄⣠⡴⠛⠀⢀⣼⡁⠈⠙⠢⣄⡀
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠙⠲⢤⣀⠀⠉⠓⠤⣀
⠀⠀⠀⠈⠳⠦⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⡜⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢌⠞⠠⢂⢄⠌⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠎⡠⢁⢊⠎⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡱⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣿⡏⣽⣿⣿⡿⢿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠕⢡⡾⠛⠃⠙⠁⠛⠃⠈⠁⠁⠋⢹⡿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢑⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⢠⣿⢀⣤⣶⣤⠀⠀⠀⠀⢶⡶⠄⠀⣷⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠁⢸⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢀⢸⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⡾⠛⠁⠤⠤⠴⠥⠤⠀⠒⣲⡄⠀⠀⢸⣧⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⠁⠀⠀⣀⡠⠤⠖⠒⠉⠉⣉⡿⠀⠀⢸⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠋⠁⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⢉⣹⠏⠀⠀⣿⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⣋⠕⠁⠀⠀⠀⣿⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⠒⡗⠒⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡆⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣧⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⡀⣀⢀⢀⣈⠄⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⡗⠋⢁⡴⣊⣽⣥⠴⢧⡅⠂⣀⣤⠶⠖⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠢⣘⣰⣷⣶⣿⣻⣧⣯⣼⣿⣱⣷⣾⢯⣰⠶⣿⣶⠾⠒⡀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢥⣯⡴⣻⡟⣹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣵⢧⠬⢾⣡⠗⡃
⠀⠀⠀⠀⠀⠀⠀⠂⠠⢈⣮⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣶⢻⣞⡴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠟⡯⠬
⠀⠀⠀⠀⠀⠀⡰⢀⡜⢁⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⠖⡷⢊⠂
⠀⠀⠀⠀⠀⠀⠀⢘⡰⢊⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣥⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠴⣠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣷⣿⣷⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⣽⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣯⣽⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡯⢝⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⠟⠋⠉⠉⠸⠖⣿⣿⠯⠶⢿⣿⣿⣟⠓⡋⠠⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠊⠁⢀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠸⣄⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠉⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⣠⡾⠻⡁⢳⣤⣶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⣀⣴⠟⠉⢀⣠⠿⠛⠁⠀⠹⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠈⢧⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⢀⣤⠾⠻⣄⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢀⡾⣟⠳⣦⣀⡴⢻⡟⠁
⠀⠀⠀⣠⠖⠋⠉⠉⠙⠛⠶⣦⣄⠈⠳⣦⡀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⣾⢦⣀⡿⢮⡆⠀⠉⠛⠋
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠘⣮⠻⡉⠓⠚⠁
⠀⣸⠃⠀⠀⠀⢠⡴⠶⢦⣄⠀⠀⠀⠈⠳⣤⡼⠛⠁⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠞⣻⠇
⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠴⠃
⠀⣿⠀⠀⠀⢸⡿⡄⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢀⣤⣾⠋
⠀⢹⣆⠀⠀⠈⣇⠹⣄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⣧⠘⣧
⠀⠀⢻⡄⠀⠀⠘⣦⠘⢷⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠉⠳⣄⡀⢀⡽⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠈⠙⠓⠚⠛⠋⠁⠀⠀⠀⢀⣿⡾⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⣀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⣉⡛⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠛⣋⡉⠛⢻
⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠸⡀⢾⣿⣿⣷⣤⡀⠈⠙⠀⠀⠀⠀⠀⠚⠁⠀⣠⣶⣿⣿⡿⠆⡼⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⠀⣾⠟⠒⠛⠻⡇⠀⠀⣸⣆⢿⡿⢿⣿⣿⣶⣤⣀⠀⠀⠀⣀⣠⣴⣿⣿⣿⢟⣿⢃⣾⡀⠀⢀⡿⠛⠓⠚⢿⡆
⠀⠀⠀⠀⢸⢧⣿⣿⣿⣷⡌⡀⣾⠋⢿⠀⠙⠦⣝⣻⣿⣿⡿⠀⣀⡀⢿⣿⣿⣟⣋⡵⠚⠁⠸⠉⣻⠆⡀⣼⣾⣿⣷⣇⣿
⠀⠀⠀⠀⣸⡜⢿⣿⣿⣿⢧⡷⢈⢳⣄⠀⠀⠀⣀⣉⠉⠉⠀⣼⣿⣿⡄⠉⠉⢉⣁⡀⠀⠀⢀⡴⢋⠴⣏⢿⣿⣿⣿⠏⣺⡀
⠀⠀⢀⢎⣬⣷⣦⣭⣭⣴⣿⣿⣷⡕⡈⠛⠶⢿⣿⡏⠃⠀⢰⣿⡏⢻⣷⠀⠀⠋⣿⢿⠷⠞⠋⡐⣵⣾⣿⣷⣬⣭⣥⣾⣯⣜⢆
⠀⠀⣎⠻⡟⢹⣿⣿⠿⣿⣿⣇⣘⢡⣷⠀⠀⡾⣸⣷⣄⡀⠈⠛⠁⠘⠋⠀⣀⣴⣿⠸⡆⠀⢰⡇⣝⣀⣿⣿⡿⣿⣿⣿⠙⡿⢚⡆
⠀⠀⣿⢆⣾⣿⣿⣿⡞⣿⣿⣿⣿⣆⢻⠀⣼⠃⣿⣷⣸⢹⠋⡎⢲⠋⠞⠙⠈⣠⣿⣇⢻⡄⢸⢣⣾⣿⣿⣿⣿⢾⣿⣿⣿⡖⢺⡇
⠀⠀⢹⡜⣿⠋⢿⣿⣧⢿⣿⠿⢈⢋⠆⣸⠇⠀⠉⠳⣽⣻⡆⠆⢸⠄⠄⠐⢈⡰⠋⠀⠀⣿⡈⢎⠃⠸⣿⣿⣿⣿⣿⡏⢻⡟⡼⠁
⠀⠀⠀⠙⢦⣤⣼⣿⣿⠘⣷⣴⡶⠋⠀⠙⢧⣄⠀⠀⠀⠈⠁⠑⠘⠂⠁⠉⠀⠀⠀⢀⣤⠟⠀⠈⠳⣶⣿⢿⡇⣻⣯⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠈⣿⣷⠫⡘⣉⣿⡆⢠⣶⣤⣴⣮⣝⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣭⡶⣤⣴⣦⠀⣾⣿⢟⠕⡞⣻⡏
⠀⠀⠀⠀⠀⠀⣿⡿⣄⣓⣺⣿⡇⣿⢸⡏⡀⠀⠙⣄⠀⠙⢷⣶⣶⣶⣶⠞⠁⢀⡞⠁⠀⡈⣷⢸⡇⣿⡿⣌⣱⣹⣿⡇
⠀⠀⠀⢠⣴⣦⣼⣿⣾⣿⣿⣿⣿⡇⢼⣿⣆⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣯⣤⣶⣤
⠀⣠⠖⠃⢰⡟⠁⢿⢩⡹⣽⣸⠏⠙⠻⣿⡿⢷⡀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠿⣿⡿⠛⠉⢿⢸⢽⣻⣹⠇⠙⣷⠈⠓⢦⡀
⢀⣿⣤⣀⢸⣧⣠⣬⣗⠨⡍⣾⣦⣤⣤⣿⠘⣌⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣏⡜⢹⣧⣤⣤⣼⡦⢷⢵⣯⣤⣤⣿⢀⣠⣼⣇
⣿⡏⠀⠉⠛⠻⢹⡿⢿⠘⠘⢸⣿⣿⠋⠙⠛⠈⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡏⠈⠛⠉⢻⣿⣿⠐⠉⢸⠿⣯⠹⠛⠋⠁⠈⣿
⢸⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣽⠀⠀⠀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡀⣀⡀⠀⠀⠸⣽⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿
⣿⡟⠻⢦⣤⣤⣤⣦⣀⣼⣷⣴⠶⠞⠛⠋⠉⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⠛⠻⠶⢶⣴⣿⣀⣠⢦⣤⣤⣤⠾⠋⣿
⠹⣧⡀⠀⠀⠀⠀⠀⠉⢹⡆⣀⣀⣠⣤⡴⠶⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⣤⣤⣀⣀⡀⣿⠉⠁⠀⠀⠀⠀⠀⣰⡿
⠀⠈⠛⢷⡶⠒⠚⠙⢦⡾⢿⠛⠉⠁⢀⡀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣀⠀⠉⠙⣻⠿⣦⠞⠙⠒⠲⣶⠞⠋
⠀⠀⠀⠈⠉⠛⣿⣶⣤⣤⣾⣧⣾⣟⡿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⣿⣿⣦⣿⣦⣤⣴⣾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣏⣺⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣌⣿⣿⣯⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣉⣭⣍⣁⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡴⠛⠉⠁⠀⠈⠉⠙⠷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⣀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣏⣁⣤⣤⣀⠙⠶⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣤⠿⣿⢯⡀⠀⢙⣧⡀⠈⠳⣶⣲⠖⢲⡖⢦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣧⣀⣘⣷⣝⡲⠿⣿⠇⠀⠀⣼⣁⡴⣋⡤⠞⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠎⠁⣰⠳⡄⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡆⠀⢸⡀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⢠⡇⠀⠀⠀⣰⠏⠹⣄⠀⠀⢀⡀⠀⢳⡀⠀⢻⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃⠀⡾⠀⠀⣾⠛⠁⠀⠀⠈⠓⠚⠉⠉⠛⢦⢳⡀⠀⢻⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⠀⢰⠇⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⣼⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡄⠀⢳⡄⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠀⢠⡏⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢻⡄⠀⢻⡄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠇⠀⡾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡼⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠀⣸⠇⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⢀⡟⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆⠀⢻⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⣼⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⡠⠐⠁⢀⣠⣤⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀⠈⠒⢄
⠀⠀⠀⠀⢀⠊⠀⣠⡴⠝⣩⠴⣚⣉⢙⡍⡑⣦⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠑⢄
⠀⠀⠀⡶⢡⣀⣼⠏⢀⠀⡵⠾⠛⠉⠙⠻⠽⣷⣵⠀⠀⠀⠈⠁⠒⠀⠀⠘⢷⡄⠀⢣
⠀⣘⣽⠀⠀⣼⠃⠀⠀⠂⢃⠀⠀⠀⠀⠀⠀⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢃
⠀⢷⡏⠀⢰⠇⠀⠀⠀⠀⠀⠉⠒⠀⠐⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⡀
⠸⢿⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⡇
⠀⢹⣷⠀⠸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⡘⠀⢐⡇
⠈⡼⣟⣄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⢀⡎⠁
⠀⢥⢿⡘⣦⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⢀⢀⠀⠀⡎⡸
⠀⠐⠈⢧⡅⠳⢄⠊⠈⠁⠀⠄⠀⠀⣀⣀⠀⢀⠀⠄⠒⠉⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⠣⡀
⠀⠀⠈⠁⣙⢶⣤⣉⡖⠦⠤⣄⣀⣀⠀⠀⠁⠀⠀⠀⢀⣠⡍⠂⢲⡶⣄⠀⠘⢯⡳⣄⠀⠈⢢
⠀⠀⠀⠀⠀⠈⠡⠝⠿⡗⢶⣦⣴⣭⣭⣉⣉⣏⣹⡄⠀⠻⢿⣆⠀⠙⢾⡙⣦⡀⠻⣬⠳⣔⠠⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠽⠾⢛⢿⣞⢦⡤⡷⣿⣦⠀⠀⠙⢮⣛⢦⡈⠻⣜⢧⡀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠙⣛⡛⢉⣻⣮⡳⣤⡀⢠⡑⢷⣝⢦⡈⠳⣙⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⣿⣮⣻⣦⣀⣀⡙⠾⠛⠂⠈⠺⡛⠆⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠻⠿⢿⣏⣿⠀⠘⢫⣘⣆⡈⠲⡷⣿⠗⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡿⣦⠀⠀⠙⢷⣹⢦⡈⣪⣕⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣙⣿⡷⢤⣤⣤⣝⢮⡳⣮⠙⢮⡵⣄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠚⠿⢯⣷⡙⢇⡝⣄⠙⢮⡳⣀⢂⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢻⠀⠀⡑⢥⡳⣄⠹⣎⠲⣄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣄⠉⠀⠻⣮⡣⡈⠳⣝⢷⣄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢿⣳⡤⡤⢬⣷⡝⢦⠬⢷⣯⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠽⡿⠻⡿⡟⢾⢯⠅

⠀⠀⣠⣄⣤⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤
⢨⠀⣯⡄⣦⡤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢓⠂
⢘⣠⣶⡉⢱⡶⡀⠀⡄⠲⣆⠀⠀⡠⣼⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠰⠿⠗⠀⠀⠀⠀⠀⢀⡀⢈⠄⠀⠐⢎⠟⡙⢿⡧
⢘⠚⠃⠽⢈⡉⠍⠒⢆⠒⡌⢉⠿⣿⣿⢿⣿⡾⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⡀⢀⣀⡈⣿⣶⣶⡠
⠈⠉⠀⠀⡘⠁⠂⠀⡀⠂⠉⠀⣉⡹⣹⢬⣿⡃⠀⠀⣠⣰⣶⣔⣷⣄⣀⣀⢤⣤⢴⣶⠖⢲⣲⣋⣛⣏⣽⣋⣙⠳⠿⠾⠿⠽⠷
⠈⠀⠁⠈⡒⠚⠃⠀⠃⣬⡄⣐⡟⠐⣄⣀⣡⣿⡀⣸⣿⣿⡟⡛⣛⣛⠛⠛⠻⡿⠩⠭⠉⣥⠀⠀⢀⠠⣄⣀⢀⣤⠰⢦⡤⠀⣤
⠸⣷⣄⣸⢷⣤⡆⠀⣠⣿⡧⣴⡾⢿⣶⣞⣿⣿⣿⣿⣿⣷⣆⣷⡶⠻⢶⢶⣶⣿⣷⠲⡆⢹⣷⠄⠿⠀⠹⠟⠸⠗⠀⠘⠟⠀⠟
⢸⣿⢿⣿⣯⣾⡁⠀⣷⣤⣤⣽⣥⣤⣾⣿⣿⣿⣻⢿⣿⣿⣬⣿⣯⣄⣬⢠⣭⡟⣤⢠⣤⠤⣶⣤⣶⠶⢷⣶⣶⣶⠶⢶⣦⢄⣰
⢸⣿⣧⣛⢋⣟⡃⠀⡟⣟⣿⡿⠛⣟⢻⢭⡎⠉⢿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⠀⣯⣇⡳⠀⢿⣿⣯⡀⢨⣽⣋⠿⠀⣴⠛⢨⣟
⢸⢯⣾⣽⣿⣉⡇⢀⣿⣉⢸⣧⣿⣿⣤⢂⢳⡆⣼⡿⢿⣂⡱⣇⢁⠤⣍⣽⣭⣄⣽⣟⡙⠻⠉⣫⠙⣶⠒⠁⣹⣿⣿⣿⣿⡏⡁
⢸⣷⣧⣼⣏⣹⡏⢈⣿⣼⣿⣿⣿⣿⣿⢾⣧⣲⡿⣿⣾⠽⣷⣻⣯⣢⡿⣿⣿⣹⣯⣿⣯⡼⠛⣟⢋⡛⠛⠛⣛⠛⣛⣋⡿⠷⠶
⠈⠟⠛⠛⠚⠛⡃⢘⠛⠛⠿⠛⠻⠿⠛⠻⠛⠛⠳⠟⢿⡿⣿⢿⡿⡿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣾⣿⣿⣷⣧⣤
⠘⠛⠒⠒⠀⠀⠙⠉⠉⠉⠀⠠⠠⠌⠿⠙⠂⣤⣤⣶⠒⠓⠋⠛⠉⠉⠩⠉⠀⠀⣀⣈⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⠿⠿⠿⠻⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠟⠛⠟⠟⠻⠿⠟⠛⠟⠟⠟⠟⠻⠿⠟⠻⠛⠻⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣄⠀⠀⠀⠀⢶⣤⡀⠀⢀⠀⠀⠀⠀⠀⠰⡖⠀⠀⠀⢀⠔⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠈⢳⣅⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡀⠀⠈⠠⠈⠀⠄⢙⡄⠀⠀⠀⢀⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠹⠼⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠐⢢⣀⣀⣛⢶⣄⣀⠬⠀⠀⠀⠀⠙⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⣀⣀⢠⡀⠀⠀⠘⠄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣉⡝⠛⢓⠘⠂⠀⠀⠀⡰⠉⢿
⠀⠀⠀⠀⠀⠀⣶⣦⣤⠀⠀⠀⠀⠁⠠⡿⠋⠉⠉⠛⠛⠛⠛⠛⠓⠰⠘⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⣿⡆⠀⡠⡄⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⣯⡄⠀⠀⠀⠀⢤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⡡⠊⠀⠑⣤⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠈⠢⡀⢠⠂⢤⣿⡇⠀⠀⠀⠀⢀⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠀⠀⠀⠠⣢⡄⠀⠀⠀⠀⢀⡤
⠀⠀⠀⡜⡱⢣⡆⢀⣀⠀⠀⠀⠎⠀⠀⣾⡏⠒⠀⠀⠀⠀⠛⠿⠛⠁⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠊⠀⣿⡇⠀⠀⠀⣠⠟⠁
⠀⠠⠄⠀⠁⠏⠀⠀⠉⠩⠉⠙⠓⠂⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⣶⠒⠋⢀⠘⠁⠀⠀⠀⠐⠁⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠲⡀⠀⠀⠒⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⣀⣰⣆⣰⣦⣤⣀⡀⠀⠀⠀⠀⠐⠲⣿⡄⠀⢈⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⡠⠀⢀⠇⠀⠀⠀⣿⢇⡀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⣿⡆⠀⠀⠀⠀⠐⡿⡇⠀⠀⠀⢀⠞⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⡠⣷⡇⠀⣠⠄⠀⠀⠉⠃⠹⠏⠹⠇⠽⠃⠘⠓⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢿⢟⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣯⣭⣭⣭⣉⣉⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠄⠂⠈⠱
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠁⢀⣤⠀⠘⠂
⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⣉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠚⠋⠉⠉⠉⠉⠙⠲⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣉⣉⣉⣉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠘⣷⣨⣍⣁⣒⠻⢭⠩⠭⢛⠲⣶⣶⣄⠻⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣪⡶⠛⣧⡈⣷⡆⠀⠉⠻⡆⢻⣶⢄⠑⢌⢿⣾⠇⠙⡆⠀⠨⣳
⠀⠀⠀⠀⠀⢂⠼⠉⠀⠀⠈⠉⠡⠤⠀⠐⠀⡄⣼⡯⡀⠀⠀⠈⢫⠀⠈⠁⠀⠠⢸⡄
⠀⠀⠀⠀⢸⠸⣅⠀⠀⠀⣀⣈⣐⣈⠏⢀⠜⣰⣟⠥⡂⠄⠀⡆⣿⡴⠃⠀⠐⠀⠀⢱⣄
⠀⠀⠀⠀⠈⡆⠀⠉⢁⣲⡬⢵⣂⡠⢐⡡⣾⠞⡣⠑⠂⢀⡰⣱⠁⠐⠀⠀⠄⡠⣠⡳⢦⣍⡒⠤⡀
⠀⠀⠀⢠⡊⢀⣾⢣⢶⢌⣤⣴⣶⣾⢷⣿⣫⠵⠬⠒⠊⢩⡔⣇⠂⣁⣁⣁⣪⣜⠉⠛⠶⡬⣛⠷⣮⣑⠦⣀
⠀⠀⠀⢈⢏⢾⣮⢧⢣⣿⠃⠀⠈⠻⠿⣶⣖⣦⢶⡜⡿⡔⣱⡸⠛⠛⠋⠉⠉⠙⠳⢶⢗⡮⣙⣷⣦⣍⡛⢶⣭⣒⠤⡀
⠀⠀⠀⢃⣾⣿⢣⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⠐⡹⡽⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣗⠉⣛⢷⣮⣙⠳⣮⣕⡢⣄
⠀⠀⢰⣿⣾⢣⢣⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠱⢱⡹⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠿⡻⣦⣿⣿⣗⣷
⠀⣔⣽⢯⢣⣳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷⣧⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠋
⡴⣿⢯⡷⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢿⡆⢷⡱⡀
⢻⣳⡿⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣿⡘⣧⣧
⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁
⠀⣀⡀⠀⠀⡀⢀⠀⡀⢀⠄⡀⠀⢀⠠⠀⢄⠀⡢⠐⠖⠠⠀⠔⠠⠂⢄⠠⢀⢄⠠⡀⢄⠄⡠⡀⠔⡠⢀⢄⡠⣀⠄⡀⡀⢀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⡰⣶⣿⣿⠆⠀⢠⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢱⣿⣿⢏⣾⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣿⣿⢡⠂⡿⣿⣿⡌⢻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⢿⠇⡈⢠⣇⣿⣿⣿⣸⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠄⣼⣿⣿⣷⣄⠀⠀⠀⢸⣿⡏⠙⠻⡿⢸⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⠀⠀⣸⣿⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠂⣀⣠⡀⠀⠀⠀⢀⠖⠀⠀⠈⣿⣿⣿⣿⡀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⡛⠛⢃⢠⡿⠁⢻⡇⠀⠀⠰⠀⠀⠀⠀⠀⠟⠿⠿⠿⠇⢸⣿⠇⠀⠀⠀⠀⠀⠀⠉⠀⠶⣶⡖⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⢸⣿⡇⠀⠀⠻⣶⣠⣷⠀⠀⠀⠀⠀⢀⡀⠉⠀⠀⣸⣿⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⣿⠀⠀⣿⣿⠀⠀⠀⣸⣧⡀⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣿⡀⢷⣄⡉⠀⠀⢰⡿⣿⠀⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣿⣷⢸⣿⡇⠀⠠⣿⠀⢸⠂
⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠘⢿⣆⡟⠛⢿⣿⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠙⢿⣄⣼⣿⣿⣷⣶⣤⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠗⠒⠨⣙⢂⢄⡀⠠⡐⣰⣆⡀⠐⠚⠲⠤⢄⡀⠐⠂⠀⠀⢠⠄⠀⠄⠰⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⢐⠳⡴⣿⣤⣽⣛⢿⣶⣽⣿⣿⣿⣤⡔⣒⣻⣄⠵⠒⡀⠐⠲⣉⠀⠈⠂⠈⠀⣀
⠀⠀⠀⠀⠀⠀⢄⢀⡀⠀⣮⣶⣴⡯⣿⣷⣽⡾⣏⣻⣾⣿⣿⣿⣿⣧⠍⢈⣿⢶⣷⣉⣷⢦⡾⣴⣅⠀⢀⡆⠈⠠
⠀⠀⠀⠀⠀⢀⡘⣾⣄⡀⣹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣯⣠⠎⣈⠆⠠⠂
⠀⠀⠀⠀⠀⠀⠸⡟⡁⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢦⣻⣿⣿⣆⢨⢺⡄⠁
⠀⠀⠀⠀⠀⠰⠆⡠⢻⣿⣞⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢁⢾⡀⠒
⠀⠀⠀⠀⠀⠀⠘⡳⣼⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢹⠼⠁
⠀⠀⠀⠀⠀⠀⠈⣧⣾⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠾⣏⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡧⢻⢗⠠⠄
⠀⠀⠀⠀⠀⠀⢀⠄⢧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣯⣿⣨⢨⣄⠃
⠀⠀⠀⠀⠀⠀⠀⢣⢼⣾⣻⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⡟⣅⠌⢈
⠀⠀⠀⠀⠀⠀⠀⢲⠰⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣋⠗⡸⡇⠁
⠀⠀⠀⠀⠀⡄⠐⠘⠀⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⣿⣯⡞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠔⢉⣷⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⠘⣻⠋⢹⣿⡿⠾⣿⣻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⠳⣻⣍⣣⣿⢞⣿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢟⣉⠛⣉⣟⣦⣟⠟⢙⣾⣩⣿⣿⡿⢫⣿⣟⣿⣮⣿⣿⣾⣧⢷⣿⠙⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠙⠖⠻⢩⣡⡬⠟⠛⠿⢟⠹⠏⠙⢺⠃⠘⠘⢹⠏⠣⣭⡟⣻⡻⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠉⠻⠧⠉⠁⠐⢫⠞⠀⡐⠁⠈⠉⠁⠂⠙⠯⠁⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⡠⠞⠓⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⢄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⠢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠶⣷⣦⣤⡤⠀⢀⣀⣈⣩⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢌⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣧⠈⠢⡱⣄⠈⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡀⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣜⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢢⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠂⠀⠔⠀⢉⠀⠈⠀⠀⠀⠡⠀⠁⠈⡁⠲⣠⠀⠄⠤⠀⠀⢀⡌⠈⡥⠀⠠⠀⠈⠀⠆⠁⢠⡀⠁⠀⠸⠀⣀⣀⠠⠳⡁⠀⠀⠀⠀⠐⠈⠁
⠀⠲⠁⣀⣤⡤⡢⠞⠂⠀⠄⠀⠀⠀⠀⠀⢀⡤⠖⠢⣀⣠⡄⣤⠊⠠⠀⠀⠀⠀⠀⡀⠈⠀⢳⠄⠫⢤⣁⡤⣤⣀⣔⡋⠑⢪⣠⠐⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠸⠃⠀⠑⣀⠀⣀⣆⡈⢳⠀⢂⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣃⢰⡇⠀⢦⠃⠎⠀⡀⠀⢀⣠⠋⠄⠐
⠀⠱⡀⠀⠀⠠⠀⠈⠀⠀⠡⠀⠘⡦⡀⠀⢹⠢⣀⠞⠀⠀⠈⠀⠀⢸⠀⠀⠀⠐⠀⠈⠀⠉⢹⡏⣠⡏⠀⠄⠀⡁⠀⠸⠁⡠⠀⠙⠎⠉⠁⠀⠰⠄
⠀⠑⠠⢄⡀⣀⣀⠀⠒⠁⠀⠐⠀⡄⠁⡴⢀⡒⠀⡁⠀⢤⣀⣐⣀⠀⠒⣏⠀⠀⠪⠀⠀⠁⠐⠣⢾⡳⢤⣀⣀⡰⣄⢀⠴⠌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⠈⠁⠀⠀⠠⢨⠅⠉⠄⠀⠁⠉⠁⠆⠀⠠⠁⣤⠀⡩⠂⠀⠀⠀⠀⠡⡐⠉⠉⠁⠙⠂⠐⠀⠐⠄⢀⡤
⠀⠀⠀⠀⠀⣆⠂⠀⠀⠀⠰⠦⢄⡈⡀⠀⠀⠈⠒⠀⠀⢀⠀⠀⠀⣀⢆⠀⠩⠖⢫⣋⡈⠀⠀⠤⡆⣄⣁⣠⠀⠀⠀⠀⠀⠀⢠⡖⠈⠀⣀⢀⢀⠆
⠀⠀⠀⠀⠀⡆⠘⠷⣀⢄⡀⣷⠔⢿⣹⠉⠁⠀⠀⠀⠀⣀⢀⣀⠜⠉⢚⠊⡠⣃⢀⡜⣿⣤⡀⠀⢰⡡⣴⠁⠂⠀⠀⠂⠤⣵⠏⡐⢀⠀⠐⡦⣀
⠀⠀⠀⢀⠬⠀⠀⣀⠂⠀⡘⠀⠊⠙⣟⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⣰⡟⢈⠁⡁⠈⠙⡏⠀⠉⠁⠐⠀⠉⠠⠚⠃⠀⠂⣠⠟⢂⠁⠀⠀⠘⠁
⠈⠀⡀⠀⠀⠠⠈⠛⢠⠌⠀⠰⢢⡟⡍⠐⠁⡒⠠⢤⠀⠀⢀⠶⠊⡅⠱⠄⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠣⡀⠀⠐
⣀⡞⠀⣀⣤⢖⡲⠚⠅⢰⣦⠀⠐⠂⠿⠇⠘⣄⠀⠀⠀⠤⠤⠐⠋⠀⠢⣀⠒⡀⠀⠄⠀⠀⠐⣀⠀⠀⠀⡀⠀⠂⠘⠁⠀⠀⢐⡰⠁⢠⠄⢀
⢌⡁⠀⠀⠀⠀⠀⢀⡴⠋⠙⢊⣀⠀⠀⡐⠄⠱⣄⢲⡌⠀⠀⠀⠀⠀⠀⠙⠈⠲⠀⠀⠀⠀⠀⠛⣄⢿⠋⠀⠀⠀⠀⠄⢀⠔⠈⠗⢀⢈⠀⠀⠉⢀
⠂⠠⡰⠀⠀⠀⠀⠋⠀⢀⠡⠐⢉⠓⠀⠗⠰⢿⡄⠚⠀⠤⢄⠀⠀⠃⠀⠀⠀⠖⠀⠉⠉⠓⣸⢯⡡⠞⠐⠷⢴⡧⠄⠈⡄⠀⣠⠿⠀⠈⠉⠒⠎⠉
⠀⠲⢄⡀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠈⠂⠧⠀⠀⡷⠀⠠⠄⠂⢈⡁⠁⠤⠀⠀⠀⠀⠈⠰⠁⠀⠙⠀⠀⠀⠀⠡⡠⠂⣠⠞⠁⠀⣀⣄⡀
⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠐⡀⠰⠀⠀⡀⠚⠁⢴⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣵⠨⠰⣇⡒⠱⢀⠉⢄⠰⠀⠀⡀⠈⠀⠀⠙
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠂⡀⠁⠀⠸⣄⠀⢀⡀⠓⡇⠀⠐⠀⠈⠑⠀⠠⢠⡤⠙⠲⠄⣀⠈⢱⣇⠈⠀⠀⢂⠠⠔
⠀⠀⡀⠀⠠⣅⠀⡄⢠⠀⠀⢧⠀⠴⠀⠀⠄⠀⠀⠀⣄⠠⠧⡤⠀⠀⢼⠀⠒⠄⢀⡁⠢⣀⡀⠀⠀⠀⢀⢸⢷⠀⢘⡙⢠⣆⡎⡀⠀⠀⠀⠄
⠀⢀⠀⢀⠊⠀⠀⣀⡏⠀⠀⠀⠀⠀⢀⡐⠀⢀⠀⠀⢨⠐⣇⢁⠀⣀⡬⠂⢀⠴⠀⠁⢀⠀⠀⠀⠀⠀⡬⠰⠾⣗⠙⢀⣠⠃⠹⠧
⠀⠃⠀⠀⠀⡠⠖⠏⠁⠀⠀⠀⡀⠉⠌⠀⠀⠀⠉⠁⠀⠀⠠⡬⠟⠿⣅⠠⠙⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢡⠀⠛⡻⢭⠀⠀⠑⣆⠚⠂⠀⠀⠐
⠀⠘⢀⠀⠘⠂⠀⠒⠀⠀⢠⣄⡀⣤⣬⣖⠀⣀⠀⠀⠀⣶⠖⠓⠉⠁⠺⠙⣲⣄⣤⣀⣤⡀⠣⡀⠹⡀⠈⡠⠘⣿⠟⠉⠒⠶⣢⣴⣍⣥⣤⣈
⣀⡄⠋⠀⠀⠀⠀⠀⠠⡔⠁⠢⠀⠀⠀⢸⡛⣿⣉⣶⡃⢨⣠⠴⠀⠀⠀⠓⠋⠑⠀⠀⠀⠁⠀⢏⡐⢗⠺⠀⢠⡏⠐⠈⢀⠶⠊⣷⠀⢀⠀⠉⠀⢀
⠉⡟⠐⠀⠤⠤⠲⣤⠀⠀⡶⠂⢀⠁⠁⠞⢻⡿⠩⡷⠠⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⠃⠀⠘⢤⣀⣸⡹⠄⠘⢈⠀⡠⢀⠙⡍⠉⠈⠛
⠀⠠⠠⠀⠀⢀⠀⢀⢣⠌⠉⣅⢉⣀⣴⣃⣎⢃⠀⢟⠠⠀⠀⢀⠀⢐⠀⠀⠀⠀⢈⠀⢠⣄⠈⠂⠀⣠⠂⠈⡀⢇⡀⠃⢤⡴⠐⠓⢻⣂⣀⢈
⣦⡈⠂⠐⢀⠤⠆⠊⡨⠂⡠⡗⣠⠨⠁⠁⠜⣹⣆⠙⠄⠀⠀⡀⠰⠃⡀⠀⢀⠀⠀⠈⠁⠁⠉⠲⢫⠁⠀⠀⠀⠁⠠⡢⡀⢠⣲⣤⠚⡩⠿⡀⠙⠆
⠀⢬⣓⠀⣀⣀⠒⣤⡜⠒⠀⠀⠀⠐⠀⢀⡞⠉⠠⢍⡛⢦⡀⣀⠂⣠⡅⠀⠁⡀⠀⠈⠀⠀⠛⠘⠠⢭⡐⢆⣈⣠⠂⢐⣵⠀⢹⠀⠳⠀⠀⠂⠀⠘
⠀⠀⠻⠀⠀⡇⠀⡀⢣⠄⠀⢱⠀⠐⠀⠀⡀⠀⠀⠀⠙⠀⠙⢀⠰⠁⢠⠀⠊⠄⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠛⠁⠀⠁⢸⠓⠂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠶⠚⢛⣡⠶⠞⣛⣉⣉⣛⡛⠶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠁⠀⣴⠟⣡⠾⠛⠉⢉⣩⣭⣝⣷⣯⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠃⠀⠀⣼⢃⡾⠁⠀⠀⠀⣾⣁⣴⠟⣷⠈⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠻⣥⣤⣴⠏⠀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣌⢷⣄⠘⣧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣦⣙⠳⠘⣧⡙⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⢁⣽⠗⠂⣼⠻⣦⣍⡛⠶⠶⠶⠶⠞⣋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⠁⠀⡟⠁⠀⣼⠃⢠⡟⣽⠛⢻⡖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⢟⣴⠟⠁⢰⡟⠛⠃⠀⣰⠏⢠⡟⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢋⣵⠟⡁⠀⣾⠛⠃⠀⠀⣰⠏⢠⡟⣸⠃⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⣿⠙⠉⠀⠀⠀⣰⠏⢠⡟⣸⠇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⢰⡟⢀⡟⣰⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⣱⠏⠐⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣏⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⢿⠿⡿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣻⣿⣛⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠛⠛⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢀⣤⣄⡀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⢿⣯⣿⣿⣷⡶⠾⠛⠿⣷⣶⡶⠾⠿⠷⢶⣦⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣽⡿⠋⠀⢀⣀⣠⣾⠟⠁⠀⠀⣀⠀⢀⣨⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⠀⠀⢀⣾⡟⠀⠀⢀⣿⢿⣿⠏⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⢸⣿⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣻⠏⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣵⠋⠀⣼⣿⣆⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⢀⣼⠏⠈⠻⣷⣦⡤⠤⠤⠬⠿⣷⣦⣄⣀⡤⠖⢊⣡⣴⡾⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠃⢀⣾⠃⠀⠀⠀⠀⢹⣧⣤⣤⣤⣤⣾⣿⠟⢻⣷⣿⡟⠉⠹⣧⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣿⢣⡿⠁⠀⠀⠀⠀⠀⠈⣿⡇⠀⣹⠀⣿⠆⠀⠀⠉⠛⣿⡷⠀⠹⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢻⠘⣿⠄⠀⠀⠀⠈⢿⣧⣄⠀⢻⡆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠈⢻⡆⠀⠀⠀⠀⠈⠙⠿⣦⡀⢻⡄⠘⣷⡀
⠀⠀⠀⠀⠀⢠⣿⡟⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣻⣷⠀⢿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⢸⣆⣿⠇⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠈⣧⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⢰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⢸
⢸
⢸
⢸
⢸
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢸⠀⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡴⠂
⢸⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⢀⠀⠀⠀⣀⡤⠄⠀⠠⠴⠋⠙⢶⡖⠚⠉⢨⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⠀⢄
⢸⠀⠀⠹⠛⠀⣤⣤⣄⠀⠀⠀⠀⠒⢾⣟⠛⠙⠉⣠⢪⡍⠁⠀⣌⠁⠀⠀⠀⠀⠀⡀⠀⠈⢙⣳⣖⢸⣀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠿⠠⣿
⢸⠀⢀⣀⠟⠻⣿⣿⣿⣧⠀⠀⠀⠀⠈⠛⠳⡶⠞⠋⠀⠀⠀⠀⠀⢁⣰⣤⣤⣄⣀⠷⠀⣲⡟⠉⠩⣸⣿⢳⣄⠀⣀⢀⣀⣠⣿⣳⠀⣾⣿
⢸⣾⡿⡅⠀⠀⣹⣿⢷⡄⢠⠀⠀⠀⠀⠦⠤⣤⣤⣾⣖⠀⠀⠀⣰⣿⣿⣿⣿⡏⠉⣹⣿⣟⡿⣤⣤⢿⣿⡟⠛⠃⠈⠿⣯⣉⣵⣿⣿⣏⢹
⢸⡏⠀⢹⣶⣿⣿⣿⣦⠁⢸⠂⠀⠀⠀⠀⠀⣠⣶⣶⣄⢀⢀⣾⣿⣿⠋⣿⣿⣿⣆⠉⠻⢿⣯⣿⣼⣾⡅⡿⠀⠀⠀⠀⣽⣿⣿⣿⣿⣗⣼
⢸⠁⣰⣿⣿⣿⣿⣿⢹⡇⢸⡇⠀⠀⠀⠀⠀⣿⣿⡿⠿⢾⡎⣽⣿⣯⣀⣿⣿⣿⢟⠀⠀⣾⡇⢸⣿⣿⣿⡇⠀⠀⠀⢸⡏⠘⢻⣿⣿⣿⣾
⢸⣰⡿⣻⣿⣿⣿⡿⠶⣿⢼⣇⠀⠀⢀⣛⣂⣨⣿⢃⣀⣈⣽⣿⣿⣿⣿⣷⣿⣿⣿⣄⠀⢻⣿⣼⣷⣿⣿⡧⠀⠠⣿⣿⣿⣿⣿⣿⢻⡏⢹
⢸⠉⠈⣸⣿⣿⣿⠃⠀⠀⠈⣿⢀⣴⢸⣿⣿⣿⣿⠿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣯⣿⡿⡇⠀⠀⠙⠿⠿⢿⣿⣿⣼⡟⢸
⢸⣇⠀⣿⣿⣿⣧⠀⠀⠀⣤⢵⣿⣏⣸⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⠿⠛⣿⠋⠸⣿⣿⣿⣿⣇⢿⣿⣥⣤⣶⣶⣾⠂⢀⣾⡟⢩⣿⠃⠈
⢸⠋⠀⣿⣿⢿⣿⣦⣠⡾⠃⠀⠉⣻⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⢻⡇⠀⠀⠁⠀⠀⣿⣇⡉⢻⣋⣸⣿⠿⣿⣿⣷⣿⠆⣿⡏⡴⢿⣿
⢸⠀⢰⣿⡏⠀⠙⢿⣿⣧⡀⠀⠀⣿⡿⣿⣿⠃⠀⢸⣿⡏⠻⢿⣿⣧⡀⠀⠀⠀⠀⠋⠉⠛⣿⣿⣿⡇⠀⠘⣿⣿⣿⣎⣿⣇⠀⠘⣿⡀
⢸⠀⢸⣿⡇⠀⠀⠀⣹⣿⡇⠀⠀⣿⣇⣿⡏⠀⠀⣾⣿⠁⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⡐⣿⣿⣾⠃⡀⠀⠸⣿⣿⣿⣷⣿⡀⠀⠈⢿⣦
⢸⠀⣾⡿⠀⠀⢠⠾⣿⣿⡅⠀⢀⣼⣿⣿⣧⣀⢀⣿⣿⠀⠀⠈⠉⣿⣿⠃⠀⠀⠀⠀⠀⢛⠫⣶⣿⣿⣿⡄⠈⣿⠑⢿⣿⣿⣟⢷⣶⠀⣽
⢸⣼⡿⠁⠀⠀⠀⢠⣿⡿⣿⣤⣾⣿⠿⠿⠻⢿⣿⠟⠇⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⣿⣿⣿⣄⣿⣃⣼⣿⣿⣿⡨⢟⠘⠋
⢸⡿⠁⠀⠀⠀⠀⣾⣿⡇⠈⠻⠛⣿⡀⠀⢠⣿⠏⠀⠀⠀⢀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣏⣽⣟⠋⠁⣸⣿⣿⣿⣿⡇
⢸⠁⠀⠀⠀⠀⠀⠙⠿⣷⡀⠀⠀⠉⠁⣠⣿⠏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢻⣿⡧⢵⡘⢀⣼⣿⠟⠉⠉⣿⣧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⡽⣦⣞⣿⣷⣿⣿⣷⣿⣿
⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠿⠿⠶⠤⠤⠤⠤⠤⠶⠿⠿⠿⠿⠿⠿⠷⠦⠤⠤⠤⠾⠿⠿⠿⠿⠿⠶⠾⠿⠿⠷⠿⠤⠤⠿⠿⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠉⠙⠁⠀⠙⠛⠛⠛⠉⠉⠉⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣀⠈⠆⠀⠀⠀⠀⠀⠀⢫⡁⠁⠃⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⠅⣤⢠⣿⣿⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⣤⣤⣤⣭⠙⡛⠿⣿⣷⠄⠀⠀⡀⢡⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣷⡞⢉⣿⣿⣿⡯⡷⠊⠟⠁⣰⡇⠸⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀⠀⠚⢩⠟⢡⢿⡛⠛⠀⠁⠀⣤⡴⠊⣀⠂⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣿⣿⣶⡄⠀⠀⠁⠀⡇⠀⠀⠀⠀⠉⠠⣾⠹⡀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠛⠛⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡜⠁⠀⢇⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠆⣼⡆⠀⠀⠘⡀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⠌⠀⠀⠀⠀⠀⢃⠀⠼⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠉⠉⠉⠛⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⣀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠲⢤⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣴⣯⠭⠽⠷⣝⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠹⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡞⠀⢠⢖⣲⣶⣄⠀⠀⠀⠀⠘⠢⠤⠤⠴⠋⠀⠀⢧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣇⡇⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢧⠀⠹⡧⡀⠀⠀⠈⢫⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠈⢆⠀⠈⢶⢄⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⢠⠞⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣆⠀⠀⠀⣀⡈⠢⡀⠀⠙⠷⣶⣤⣾⠞⠀⠀⠀⠀⣀⠔⠁⣠⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⠞⠁⢀⠴⢋⡜⠁⠀⠈⠓⠤⣀⣀⣀⣀⣀⣀⠤⠔⠊⠁⣠⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠚⠀⢀⡠⠚⣁⠔⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⣥⠞⠁
⠀⠀⠀⠀⠀⢠⡤⠤⠞⠀⣠⠔⢋⡠⠚⠁⠀⠀⠀⠀⣠⠞⣹⢬⣉⣒⣒⢒⠒⣚⣉⣤⠶⠋⠁
⠀⠀⠀⠀⠀⡼⠀⠀⡠⠞⣁⠔⠋⠀⠀⠀⠀⢀⡠⢊⡴⠋⠁⠀⠀⠉⠉⠉⠉⠉⠉
⢀⡴⠓⠒⠊⢁⡴⠋⡠⠞⠁⠀⠀⠀⠀⣠⢔⣩⠞⠉
⡏⠀⠀⣠⠔⢁⡤⠊⠀⠀⠀⠀⠀⡠⢊⡵⠋⠁
⣇⣠⠞⢁⠔⠋⠀⠀⠀⠀⢀⠤⣪⠞⠁
⣿⣁⡔⠁⠀⠀⠀⠀⡠⢖⣥⠞⠁
⠈⠓⠮⣭⣖⣒⢶⣫⠶⠋

⠤⠀⠂⠐⠻⡂⠀⠀⠔⢒⠶⣶⠄⠴⠦⠤⠞⠉⠀⠀⠀⠈⢹⡄⠀⡀⠀⣴⠉⢲⡀⢀⠄⠈⠀⠀⡀⠐⠒⢁⢀⠐⡆⢀⠀⠀⡂⠀⠀⠄⠰⠄
⡀⣂⡗⢀⡆⢱⢀⠒⠲⢝⡖⠈⠉⠐⠂⠉⢒⠐⠚⢧⠁⢧⠈⠁⠀⠁⠰⠬⠤⠌⠃⠈⢂⠞⢈⣤⠈⠹⡄⠀⠈⠈⣀⡈⠀⠀⠀⠀⠀⠀⠂⠀⠑
⡀⠀⠀⠸⣇⢘⠀⠀⠀⠀⠈⡱⡄⠀⢲⡀⢈⡇⢀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⢸⡇⠘⠘⠄⢈⠀⠀⢀⡇⠀⠀⠀⣼⠃⠀⠀⢀⣠⡴⠀⠀⢐
⣿⣧⡀⠀⠉⠘⣠⣶⣿⣦⠀⠀⠈⠃⠀⡈⢸⡇⠐⠻⢤⠀⠀⠀⠀⠁⢸⠁⠠⠘⢧⠀⠀⡖⠉⢠⡴⠞⠁⠀⢀⡾⠃⢸⠀⠀⢺⠁⠀⠢⠀⡈⢠
⣿⣿⣷⣀⣀⣴⣿⣿⣿⠟⠀⣴⠀⠀⠀⡀⠈⠌⠂⠀⠀⠀⢀⡀⠀⠁⡟⠀⠀⠀⠀⠔⠀⠀⣠⠏⠀⠀⠀⣶⠛⠁⠸⠋⠃⠀⠀⠃⠀⠀⠀⠁⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠉⠁⠀⠀⠀⠂⡈⠀⠀⠀⠀⣿⠀⠀⠈⠐⠀⠈⠁⠀⠀⠀⠀⠋⠐⢤⡀⠀⠀⠾⠿⣦⡀⣀⠀⠀⠀⠀⠀⠀⢦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠌⠀⠠⠀⣤⠀⠉⠃⠀⢀⣠⠴⠛⠀⠀⢠⠆⠀⡀⠀⡑⡙⡄⢐⣆⠀⠘⣷⠙⠉⠀⡆⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⣴⠒⠀⠠⠞⠃⠀⠑⠶⠾⠛⠁⠀⠀⠀⢀⡇⠀⠀⠃⠀⠋⡇⠈⢁⡀⠀⣰⠃⠀⢠⠀⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠰⣦⠐⠊⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠆⢀⡇⠀⡆⠈⣿⡍⣷⠀⠀⢰⠁⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⡅⠀⢣⡰⠇⠇⡋⢀⠀⢸⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢴⠌⡄⠆⡄⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣌⠀⠀⠀⣤⡀⠀⣀⢹⡄⠀⠁⠀⠀⠀⠀⡀⣾⢸⠇
⣿⣿⣿⣿⣿⠏⠉⠉⠀⣠⠄⠀⣤⠄⠁⢀⡄⠀⣠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣶⠐⠻⡇⠀⢠⠈⢸⠀⠀⠀⠀⢀⡤⢠⠏⠘⠀⠤⢄
⠛⠋⠉⠛⠋⠀⡰⠀⠋⡞⢭⢀⡀⠀⡀⠀⠀⠀⠈⠁⠀⠎⠀⠀⣠⠀⣰⡆⠀⠀⠀⡏⠀⠀⡇⠀⠘⠀⠀⠓⠀⠀⣾⣋⠀⠀⠀⠀⠀⣄⠀⢳
⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠠⠀⣁⠀⠉⠀⠁⠀⡄⠀⡤⠀⠀⠀⠁⢠⡟⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⢰⣿⡄⣿⣿⡇⠀⠚⠀⠀⠁⠀⠀⣧
⠀⠀⠈⠠⡔⠠⡀⠀⠀⠒⡏⠠⣌⠉⠓⠦⣤⡼⠃⠀⠘⢠⡀⢀⠀⢀⠄⠑⠶⣄⠀⡇⠀⠈⠁⠀⠒⠀⠈⠺⣿⡓⠛⠉⠀⠀⠀⠀⢄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⣁⠀⠈⠆⣀⡀⠀⠀⠀⠀⠀⢀⡷⠀⢀⠂⠀⠂⠀⠟⣆⠁⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⠳⣄
⠀⠀⠀⠆⡀⠀⠒⠄⠸⠄⠋⠀⠀⠐⠂⠬⠑⠒⠂⠤⢄⡈⠀⠀⠠⠄⠀⠘⠁⠘⠛⠂⠀⡠⠋⠀⠀⠠⢂⡷⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⢳⡀
⠀⠘⠀⠀⠀⠁⠒⠤⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠂⠉⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠩⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠂⠂⠀⠈⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠀⠁⠀⢀⠀⠀⠠⠤⢤⡼⠃⠀⠁⠀⠀⢠⡟⠀⠀⠘⠆⠸⡅⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⢠
⠀⢠⠀⠀⢂⠀⠠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⠃⠀⡈⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠫⣄⠀⢠⠀⠀⠂⠙⢄⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠁⠈⠀⠩⢲⠀⠦⡀⠀⠀⠀⠀⠠⠿⡎⠂⠀⡿⠁⠠⡄⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡈⣦⠀⠀⠀⠀⡀⠈⠳⣄⠀⠉⠉⠉⠉⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠓⠲⠀⠀⠀⠀⠀⠰⣶⡆⠒⠊⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠈⠁⠀⠀⠈⠳⣄⠁
⡀⠐⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣀⠀⠀⠙⣇
⡀⠀⣠⡈⠁⠁⠀⠘⠛⢦⠀⠀⠀⠀⠀⠀⠀⡤⠀⢐⡖⠀⠀⠀⠀⣦⣴⠆⠀⢀⠀⠈⠀⠀⢀⡀⠀⠀⠘⣧⠀⠀⠉⠀⠀⠀⢻⠀⠀⠂
⠀⠀⠉⢀⡆⠀⠀⠀⡴⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣧⠰⢟⣆⠀⠀⠀⠈⠀⠀⠀⠀⣹⠀⢠⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠈⠐⣠⠆⠀⣆⠸⠋⠀⠀⠀⠀⠀⣤⠀⠀⠻⣿⡀⢀⣿⣷⣶⡆⠀⡄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⣀⠠⡀
⠀⠀⠀⢰⠀⠀⣠⡤⠴⠚⠂⠉⠌⠂⢀⣀⠉⣧⠀⠀⠀⠀⣀⠸⣆⠀⠀⠈⠁⠺⣿⣿⣿⡇⠘⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠈⠁⣘⠃⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠏⢁⠀⣀⠘⡁⠀⡀
⠀⠀⠀⢀⢀⣾⣶⣄⡻⡄⢀⠀⡸⢥⡀⠀⢰⡄
⠀⢀⡄⠀⢈⣹⣿⢯⠀⠠⣤⣴⠁⢀⠖⣦⡄⠀⣄⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣼⡋⡀
⠀⠈⢇⡈⠝⢙⡏⠀⠁⠀⣿⡇⠀⠈⢠⠿⣿⣦⡌⠀⠀⠀⣼⣿⠋⠁⠀⠀⠀⣿⡆⢠⠂⠀⠇⠐⠲⣤⡄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⢿⡀⠀⠀⠸⠀⠈⠉⠀⠀⠀⢰⡟⡀⠀⠉⠁⠀⢠⡞⠃⠈⠀⠀⠀⠂⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⠸⠁⠀⢢⠀⡄⠐⠐⠀⠀⠀⠀⠀⠐⠂⠀⢸⣆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠉⠁⡇⠀⡀⠀⢀⡠⠤⠀⠀⠐⠰⣟
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⠀⠀⢀⣿⡘⠁⠀⠚⠗⠂⣤⡈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⣿⣷⠀⠀⠀⠀⢸⣯⣭⣤⡄
⠀⠀⠀⠀⣿⠀⠀⠀⢀⡼⣯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⠰⠀⠀⠘⣿⣿⣷⣾⡟⠀⢸⣿⣿⣿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⠄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⢢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠷⣵⣦⣤⡤⠀⢀⣀⣈⣡⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢄⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⠈⠣⡱⣌⠨⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡄⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣌⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢪⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣶⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠶⠛⠛⠲⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠏⠀⠀⠀⠀⠀⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⡄⠀⠀⠀⠀⢀⣿⡀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡄⠙⢲⡶⠀⠀⠀⠀⠙⣦⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⢈⣿⡀⢸⡇⠀⠀⠀⠀⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠀⢀⣠⡴⠞⠋⠉⠳⣄⠳⣄⣀⠀⣀⣴⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠙⠳⣤⣉⣉⣉⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢳⡶⠚⠉⠁⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠻⡄⢳⣤⠴⠚⠉
⠀⠀⠀⠀⣀⣤⠶⠚⠋⠁⢀⣠⣴⠞⠛⠾⠃
⢀⡴⠞⠛⠉⠀⣀⣤⠶⠛⠉⣴⢋⣀
⢸⣄⣠⡴⠖⠻⠧⢶⡆⠀⠀⠈⠉⠹⣆
⠀⠈⠀⠀⠀⠀⠰⣏⠀⠠⡖⠛⣧⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣻⡄⠘⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠘⠀⠀⡀⠤⣀⠠⠤⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣀⡈⠳⣤⣄⡀⠀⠦⡄⠈⠀⠹⠒⠢⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠁⣠⣨⠆⠀⢠⠋⠀⠀⠰⣤⡇⡀⠐⢄⠀⠤⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠑⠂⡤⢀⠊⠀⠀⢛⠀⠀⠀⣸⢣⠀⠉⠀⠈⡡⢔⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣉⡀⠀⠪⠀⠒⢀⠁⠈⢉⡀⠶⠊⡀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠂⠀⠀⠀⠀⠀⠁⠀⠈⢉⠒⠀⢤⡀⠀⠈⣱⡔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡇⠀⠀⠀⢰⡆⠀⠀⠀⠘⡧⠀⠀⠰⠁⠀⣤⡟⠐⡄⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠃⠀⢀⠀⠀⠈⠲⢄⣄⠟⠙⢂⠙⢙⡣⣤⠯⠇⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠗⡙⠀⠀⠀⡠⠀⢈⢨⠱⠚⢉⣝⣯⣄⣉⡹⠽⠁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠈⠀⠀⠈⠀⠀⠀⡀⠀⢀⡎⠚⠉⠀⢹⡷⣅⣱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠸⢄⡀⠀⠠⢟⠶⠛⡁⢀⣠⡴⠉⣠⡌⣫⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠉⠢⢭⣠⡀⠀⠓⠤⠄⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠷⣤⣀⣀⠤⠤⠒⠉⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣋⠬⠭⠬⣍⡛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠕⠉⠀⠀⠀⠀⠀⠙⢷⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢧⣤⣤⣤⣄⣀⡠⠤⢒⣚⣻⡶⢯⣅⡐⠊⠩⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠍⠚⡇⢸⣥⠈⣿⠉⠛⢝⡻⣯⣐⠲⢦⣌⡐⠬⡓⢤⡀⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⡁⢄⠳⡹⣮⣿⣤⣿⡤⢄⡀⠉⣪⢲⣙⢳⣬⡙⢦⡈⢪⢳⣄⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⣈⠳⠁⠀⠘⢿⣽⣿⣿⣦⠱⠀⠤⠧⢹⡧⡹⣍⢌⢳⡄⠑⢻⡄⠀⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢐⠒⠠⠀⠀⠀⠠⡸⣇⣿⣿⣿⢀⢄⡒⠂⡀⡇⠠⢹⡄⢆⢿⠀⡇⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⠑⠬⠽⠟⠕⠁⡲⢌⠀⡇⡿⠘⢸⡇⠈⢸⡇⠀⣿⢀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢬⡑⠀⢡⡇⠀⢸⡇⢠⢹⣥⢃⣟⡋⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠀⠀⢈⣢⡀⠀⠀⠀⠀⢄⠣⡱⠌⠂⢠⢃⡞⢀⡌⡾⠀⣌⡾⢣⡞⠣⡙⠸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡞⠃⣠⢤⡖⢁⡜⠈⠢⡀⢠⢌⠢⡣⠑⢀⠔⣡⠊⡠⢊⡾⢁⢜⡞⣱⣿⡄⠀⣽⡄⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡴⠡⠊⣰⣿⠂⠀⠀⠧⢤⠤⣄⣐⣪⠔⣋⠔⣊⣴⡋⢔⣥⡞⣹⣧⡹⣧⡀⠘⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢊⠜⣡⠟⡉⣻⣷⣾⢛⠟⠫⣿⠋⢡⣈⣥⠲⣿⠸⠠⣾⣿⣿⠇⠸⡈⢿⣌⢳⣄⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡵⢃⡾⠃⢐⡼⢃⣿⠇⠋⠀⠸⠇⠂⣸⣿⣟⢸⡿⠸⠀⣿⣿⠿⠀⠀⢇⠀⠹⣿⡮⣧⡀⠘⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⢊⡴⠋⢀⡼⠋⠀⣌⡗⡘⠁⢰⣟⢘⣶⣾⢿⠀⢼⡏⢰⠚⢿⡱⠆⡆⠀⠘⡄⠀⠈⢿⣕⢅⠀⠀⠻⠶⣄
⠀⠀⠀⠀⠀⠀⣀⠼⣡⠞⣠⠏⠀⣰⠏⠀⠀⣸⡿⢡⠁⠀⡾⠋⠁⣼⡏⣽⠀⣸⡏⡃⠀⠀⣩⢀⠰⠀⢀⢱⠀⠀⠀⠹⣾⣺⣦⠀⠑⣜⣇
⠀⠀⠀⠀⠀⠸⢃⡜⣡⡞⠁⢀⠞⠁⠀⠀⢠⣟⠇⠄⠀⣸⣷⠀⠀⣿⡇⡟⠀⢿⡷⠁⠀⠀⠻⣟⠀⠆⠀⣺⡆⠀⠀⠀⠘⢿⣷⡱⡄⠈⢆⢻
⠀⠀⢀⣴⠖⡡⢀⢴⠉⢀⡴⠁⠀⠀⠀⠀⣾⠏⡐⠀⢰⡟⠁⠀⠀⣿⡇⡇⠀⣿⠁⠀⠀⠀⠀⣿⡸⠸⠀⠉⢥⠀⠀⠀⠀⠈⠻⣶⠿⣆⠀⢻⡶⢤⡀
⠀⠀⣷⢣⠊⡠⡿⠀⡠⠋⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⡇⠀⣻⡇⠀⠀⠀⠀⠲⡆⠇⠃⠀⠪⡆⠀⠀⠀⠀⠀⠙⢷⡝⣧⡈⠿⣈⢻⡆
⠀⠀⢸⠁⣴⣗⣱⠎⠀⠀⠀⠀⠀⠀⢀⣿⢃⠁⠀⣼⣟⠀⠀⠀⠀⣿⠇⡇⠀⡿⠀⠀⠀⠀⠀⠰⣿⡀⠸⠀⠁⢷⠀⠀⠀⠀⠀⠀⠀⠻⣎⠟⢦⡙⣶⠁
⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⣰⠟⠃⠀⠀⠀⢀⣿⠢⠇⡞⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠃⢹⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠠⠀⢠⣿⡀⠀⠀⠀⠀⠘⣿⣴⠀⢱⡄⠀⠀⠀⠀⠀⠀⢸⡁⠀⠘⠘⡏⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣾⣣⠃⠀⠀⠀⠀⠀⠈⢮⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢷⣧⣀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠁⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣦⣤⡀⠀⢀⣠⣴⣶⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⢿⣿⠀⣉⣿⣿⣷⣿⣿⣍⠀⢸⣿⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠘⣿⣶⣿⣯⣤⣄⣤⣽⣿⣷⣿⠏⠀⠀⢙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠀⠀⢠⣿⠿⣿⣍⠻⣿⠟⣋⣾⡿⣿⣆⠀⠀⢨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣾⣿⠀⣉⣿⣿⣿⣿⣿⣋⡀⢸⣿⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⠿⠛⠉⠀⠈⠙⠻⠿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⣠⡀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠟⠋⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣦⡀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⣤⣀⢀⣤⠶⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠉⢻⣿⣧⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⢹⣿⣧⠀⠀⠀⠀⢠⣾⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡼⣏⢿⣿⡆⠀⠀⠀⢛⡾⠁⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣅⣀⣤⡴⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣴⣿⣿⣿⣿⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣾⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣅⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⢋⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⣦⣴⠞⠉⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⡾⣡⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣷⣄⣠⡶⠋⠻⣄⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣏⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠉

⠑⠀⠈⢠⡀⣧⠛⠾⡓⠀⠀⠀⠀⠀⠀⢀⠸⡃⠀⢀⠀⡀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠐⣠⠄⠀⣠⠊
⢀⣶⣕⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠁⠀⠐⠁⠘⢆
⡉⠀⠁⢠⠶⠈⠀⠀⠀⡀⠆⠘⠖⠒⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠓⠀⠀⠀⢀⠐⠑⠆⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠐⠂⠁⢀⠀⣀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠁⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠋
⠀⢀⡄⠀⠂⢘⠀⠀⠀⠀⠏⠈⠀⠀⠢⠀⠀⠀⠁⠐⠀⠀⠀⠸⠂⢈⠄⠋⠈⠁⡆⠓⡀⠀⢄⠀⠀⠀⠐
⡈⠀⠀⠀⠀⠠⠛⠈⠁⢐⠀⠊⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠰
⠉⠁⠀⠀⢀⠤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠠⡄⠀⠀⢨⠥
⠓⠠⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰


⣉⣉⣀⠤
⣉⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣉⣤⡴⠖⡊
⠉⣥⠶⢃⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢿⣶
⣈⣴⣮⣅⣤⣴⣎⣠⣠⣄⣤⣄⣠⣇⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣀⣤⣤⣈⣤⣌⣵⣦⣈

⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⢠⣾⣿⣏⣀⣾⣷⣄
⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⡿⠋⣠⣦⡀
⢀⣴⣤⡀⠙⢿⣿⣿⣦⡀⠙⠀⣾⣿⣿⡟⠀⣀
⠺⣿⣿⣿⣦⣾⣿⣿⣿⠃⣤⣄⠈⣻⣿⣦⣾⡿⠃
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠋⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣤⣤⣀⠀⣠⣤⣶⣶⣶⣶⣦⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⠀⣀⣀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣴⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⢶⢾⡁⣤⣶⣶⣦⣄⣠⠄⡀
⠀⠀⠀⠀⠀⡀⠀⣤⢼⣿⣿⣿⣿⣿⣿⣯⣾⡄⠀⠀⠀⠀⠀⠀⠀⠐⠙⣻⢾⣼⣿⣿⣿⣿⣟⣿⣥⣾⠟⡆
⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⠊⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣳⠟⠁
⠀⠀⠀⠀⠀⢼⣼⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢢⡀⠀⠀⠀⠀⠀⠋⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡠
⠀⠀⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⢾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈
⠀⠀⠀⠀⠀⠀⢸⣎⣿⡷⣿⣿⣿⣿⣿⠿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡧
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠇⢏⠏⠓⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⢟⣿⣿⣿⣿⣿⡟⠋⠀⣠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⠘⠷⠃⡹⠓⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣿⣿⣿⣷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡴⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣯⣿⣿⣿⣿⣿⣿⣿⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣿⣶⣶⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣬⡭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠛⠉⠁⠁

⠬⠒⠃⠀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⢀⠀⠘⠦⠀⠀⡀⠀⠅⠀⠁⢧⡀⢹⠋⢻⣬⢺⣣⣼⣼⡿⠉⡽⠁⡨⢋⠁⠃⠼⢁⠞⠹⠃⢈⡼⠋
⣅⡀⠀⠀⠀⠀⠀⠈⠀⠈⢢⣠⣤⣄⣤⣀⡀⠀⠠⠀⠀⠀⠀⣠⠖⠾⣇⡀⢱⢋⡷⠻⠋⠀⠈⣷⡼⢁⣀⠀⡈⠀⠀⢀⡀⠀⣀⠐⠁⠀⠈⠀⠀⢀⡴
⢤⢉⠁⠤⣀⠀⠀⠀⠀⠀⣾⠉⠀⡀⢀⣉⣻⡖⡦⠥⡀⣀⣴⠧⠁⠀⠙⠉⢨⠮⣤⡀⠀⠀⠰⠟⠗⠛⠉⠙⠉⣹⠋⠁⠹⠏⢉⠷⡄⠀⡀⠔⠊⠉⠉
⣓⠀⠀⡀⠀⠐⠀⠀⠀⠀⢻⡍⠀⠈⠯⡉⠁⠀⠁⠀⠀⠉⠉⠗⠲⠄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠉⠀⠀⠸⠀⠀⣈⡗⠂⠁⠀⠀⠠⠐
⠦⣠⠀⠁⠉⠒⠀⠀⠀⠀⠻⣆⡄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⣲⢤⠤⢤⠀⠀⠀⠀⢲⡋⠀⠀⠀⠀⠠⣄⠼
⠂⠀⠅⠀⠂⠀⠀⡀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⢀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⡀⠀⠠⠆⠀⠈⠹⠀⢀⣸⠃⠀⠀⣀⣠⠇⠀⠀⠀⠀⠀⠀⢀
⠙⠃⠀⠀⠀⠀⢀⢀⣠⣔⡓⢺⡋⣷⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠠⠄⠀⠀⠱⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⠞⠁⣴⣃⣠⡀⠀⠀⠀⠀⠄
⠚⠛⠓⠲⠆⠀⢰⣏⡀⠉⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠹⠻⡟⢲⡄⣀
⣛⠖⠃⠀⠀⢀⣿⠉⠀⣀⠴⠚⠁⠀⠙⠂⠀⠀⠀⠒⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠀⠁⠀⠀⢀⡼⢋⣑
⠻⠍⣠⣶⠋⠙⠇⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶
⠉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠀⠀⠀⢠⣤⣄
⠤⠼⢿⣓⣤⣰⠯⠀⠸⣧⣀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠂⢠⠀⢀⡀⠀⠀⠀⠀⠁⠀⠉⠉
⠃⠉⠀⠀⠀⠉⢳⣀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠘⠆⠀⠀⠁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣸⠇⠀⣠⡼⠟⠂⠒⠠⠤
⣴⡖⠊⠀⠀⠀⠀⠈⠛⠷⠖⢡⡀⠀⠀⠀⠀⠺⢤⠴⠦⠄⠀⠀⠀⠀⠘⡄⠀⡀⠀⠀⠀⠠⠀⠀⠠⠨⠆⠀⢀⠀⠀⠀⣹⡝⠀⢙⠀⠀⢄⣀⠀⠘⠑
⡩⠀⠀⠀⠀⠀⣀⠄⠈⠁⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⢆⣀⡔⠀⠀⠰⣤⠤⡀⠙⠆⠀⠀⢀⣄⠀⠀⡀⠘⠒⠋⢹⣷⣛⠁⢠⣴⡏⠀⠀⠀⠀⠉⠀⠈
⠂⠀⢀⡀⠄⠊⠀⠀⣀⡤⡾⠊⠳⠴⠤⠴⠻⣄⣀⣰⣯⠉⣅⡀⡐⢀⣽⡆⠁⠘⠲⢤⢶⡞⠉⠓⠚⠻⣤⣤⡽⠛⠀⠉⠙⠋⠁⠀⠀⠀⠀⠑⠢⢈⡐
⣪⠕⠉⠀⠀⣀⠔⠛⡉⠈⠁⠀⠀⠀⠀⢀⠀⠀⠁⡰⠃⢠⠻⣤⠶⣞⠈⡻⣆⠀⡀⣀⡾⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⢂⡠⠋⣡⡴⡋⢀⣄⣠⠊⡰⢂⠔⣉⡴⠀⣰⠃⣰⣦⣢⠘⣶⡔⣾⣧⣨⣃⢹⡉⡀⠀⡀⠀⠪⠀⠰⣄⠀⠐⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠐⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢉⡙⣓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⢶⣾⣷⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠠⡔⡀⡀⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⡠⠁⣿⣿⣿⣿⠉⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⢿⡿⠿⢧⢀⠘⠛⠿⠟⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠑⠄⢣⠀⠀⠐⠙⠀⠤⠀⠊⢣⠀⠀⠀⠀⠀⠀⠠⡤⢆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢈⠀⠀⠀⠊⣤⠤⠀⠐⠂⠐⠀⠒⠸⡀⠀⠀⠀⠀⠒⣦⢥⣎⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠦⠊⠀⢸⢻⠀⠀⢨⠄⠭⠂⠀⠀⠈⢣⠀⢠⠄⢀⠐⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠄⢸⠀⣠⣿⣿⣿⣿⣿⡷⣤⠄⠋⠀⠈⠛⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠰⠈⠙⠾⠛⣅⡿⠿⠿⠟⡸⠄⠀⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⣀⣀⣠⠀⠠⡉⠀⣄⣀⡽⠁⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢤⣔⡉⠂⠀⠌⠉⠉⠉⠉⣁⠅⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠳⠦⣍⠲⡁⠉⠃⠈⠋⠉⠡⢍⠎⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠀⠀⠂⠙⢜⡀⠀⠀⠄⣰⠽⠁⠂⠀⠀⢑⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣴⣞⠁⡀⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠊⠈⢲⡀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠤⠤⠖⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢽⣿⠿⢖⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡒⠀⣅⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⢘⡩⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⠛⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⢤⣀⣀⡀⠀⠅

⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⡀
⠀⠀⢀⡴⢋⡴⠖⠛⠛⠶⣄⠹⣦
⠀⢀⡾⠁⣾⠀⠀⠀⠀⠀⠹⡆⢹⡄
⠀⡾⠁⢀⣿⡀⠀⠀⠀⠀⢠⡇⣸⠁
⢸⠇⣴⠋⠀⠀⠀⠀⢶⡖⠋⢠⣿⣄⡀
⢸⡀⣇⠀⠀⠀⠀⠀⢸⡇⢀⣿⡁⠈⠙⠓⠶⣤⣀
⠘⣇⠹⣤⣀⠀⣀⣠⠞⣠⠞⠉⠙⠳⢦⣄⣀⠀⠉⠛⠳⢦⣤⣀
⠀⠈⠳⣤⣉⣉⣉⣤⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⡀⠈⠉⠓⢶⡞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡞⢠⠟⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠳⣦⣄⡀⠈⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣦⠉⠛⠶⣤⣀⠀⠉⠛⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠀⢰⡶⠼⠟⠲⢦⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣼⠛⢲⠄⠀⣹⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⢠⣏⠀⢰⡏