Monastery Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠇⠀⠀⠀⢴⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⢀⢸⢸⣰⠀⠀⠀⣀⠀⢀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣀⠀⣰⣿⣦⡀⢠⠏⠁⠀⠀⠉⠀⠀⢁⠒⠾⠛⠻⣦⣉⠉
⠀⠀⠀⠀⠀⢠⣾⣿⣶⠀⠀⠀⡀⠀⠙⠳⣿⡿⢿⣭⣿⣿⣿⣿⠁⠀⠀⠀⢰⠀⠀⠀⣿⡇⠀⡆⠠⠘⠿⣿⠿⠙⡦
⠀⠀⠀⠀⠀⣬⢶⡶⢶⡆⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⣿⡅⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⣼⣷⣾⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⡅
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢠⡆⠀⠀⣿⡟⢸⣿⣿⣿⣿⣿⠀⠀⢰⣿⣠⠄⠀⠀⠀⠀⠀⡀⠇⢰⣿⡇⣀⡀
⠀⠀⠀⠀⠠⠿⠿⠿⠿⠇⢰⡆⠘⠇⠀⠀⢿⠤⠸⣿⣿⣿⣿⣿⢀⠀⢸⣿⡏⠀⠀⠀⠀⠀⢸⣇⣀⠘⠛⠓⠺⠷
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠖⠠⠀⡄⢀⣀⡀⠀⠀⠀⠀⠀⢸⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠃⠀⠀⠀⠀⠀⡀⠀⠁⠈⠀⠁⠀⡠⡆⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⣷⡄⢰⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⢴⣶⡆⣀⠄⠄⠈
⠀⠀⠐⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣱⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⡴⡟⠃⠶⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⣧⡾⣩⡾⠿⢷⣾⣦⣿⢀⣀⣄⠀⠀⠠⡒⢁⣠⡞⣁⠈⠂⣰
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣷⠋⠁⠐⠢⣤⣤⣿⣶⣿⣿⠀⠀⠀⠀⣸⡟⣹⣿⡀⡀⣏
⠀⠀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣻⡖⠒⢷⢶⣿⣿⡤⣭⣯⠇⠀⢀⠀⠀⣿⠁⣿⣿⠿⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆⠀⠀⠀⠀⠉⠀⠈⠁⠁⡦⠘⠢⡟⢻⣿⡿⠀⢀⠀⢠⡼⠁⠀⣠⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⢼⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡗⡇⠀⠀⢀⣽⡗⠀⠎⠀⣾⣧⣿⣿⣟⢃⡼⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡀⣰⣶⣶⣤⣭⠏⠀⠀⠀⠛⣙⡇⠸⠀⢸⣾⣿⣿⣿⣯⣡⡉⢀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠀⠠⣌⠻⣿⣿⡅⣿⡟⣤⡄⠀⠀⠁⢋⡡⠃⠃⠀⠉⠉⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡅⣝⡛⡻⠓⡿⢷⣦⣤⣄⣹⣿⣿⡅⢸⣶⣶⣿⣿⣿⣿⣟⡀
⠀⠀⠐⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣷⣭⣥⣴⣬⣯⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⣻⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⣄⡈⢻⡏⢿⠻⠯⢭⡽⢿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⡆
⠀⠀⠰⣤⣤⣤⣄⡀⠀⠀⢠⣤⠛⠇⠐⢠⣴⡇⠸⣷⣶⠥⠶⠶⠖⠺⠿⠛⠻⠾⠿⢾⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠿⣆⡀⠁⠸⡦⠀⢸⣶⣿⡇⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣸⣿⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣷⣄⡀⠸⢿⣿⣷⣀⣀⣀⣀⣰⣶⣦⣬⠁⣁⣐⣶⣾⣿⣿⣿⠿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠿⣵⠤⠍⠁⠂⠲⢤⠤⣍⡉⠃⢤⣄⠀⡀
⠀⠀⢀⠀⠀⠀⠈⠳⢶⣤⡈⠻⣷⡦⢽⣿⡶⣴⣶⣦⣄
⡀⠀⠂⡀⠀⠀⠀⠈⢀⣿⣿⠞⠛⠛⠺⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠺⣿⡿⠗⠒⢠⣼⣿⣿⣀⠀⠀⠀⠀⠺⣟⣿⣞⣿⣿⣿⣦⡀
⠀⠀⠀⣈⠂⠀⣿⠿⣿⢩⡍⠁⣲⣶⣤⡤⠈⡿⠿⣿⣿⣦⠙⢏⠉⠲⣤⡀
⠰⣶⣤⣤⣏⠀⣰⣾⣿⢸⡇⠀⢻⣿⣿⡇⠀⢹⣏⠛⣏⡻⢧⣮⣿⠄⠸⡷⣍⣿⡗⠄
⠀⠉⠘⠫⠉⢹⣿⣿⣿⠞⣛⠓⣤⣥⣉⡁⠀⢸⣟⢺⡚⠇⠀⠊⣿⣧⡀⠘⣀⢿⠉⢳⡀
⡠⠵⠂⢀⣴⣿⣿⢸⣿⢈⣿⠀⢻⡇⢸⡏⢹⣞⣯⣟⣷⡀⠐⢄⢸⢿⣆⠀⠘⡄⢁⢠⣱⣄⡀
⣿⣿⣿⠟⢡⠀⠈⢿⡿⢛⡛⢱⠊⡉⠘⠃⠸⠇⠙⠛⠛⢷⣤⣄⡄⢄⣿⣦⡅⢴⢸⡅⠢⡻⠻⡀
⠟⣿⣿⡎⣿⣧⠀⢸⣧⣽⡿⢼⢰⡆⢰⣆⣰⠀⠓⠄⣀⠀⠈⠙⣻⣇⠻⣿⣷⡠⠄⠻⢰⣯⢷⣷⡀
⠈⠉⠛⠂⠸⠿⢄⣽⣿⣿⣿⣸⠀⠃⠘⠛⠨⠀⡀⠀⠀⠀⠐⣤⢿⣿⢲⣽⣿⣧⡖⠎⠳⡇⡨⢓⡀
⡶⠆⠀⠀⢠⣿⡆⢸⣿⣷⡦⣿⡧⡆⠐⣧⢰⣦⢠⡌⣄⣀⠀⡈⢪⣿⣧⡙⣾⣿⣿⠛⡄⢸⠇⢈⡁
⠀⠀⢀⡠⠘⠿⠧⣈⣻⣶⣆⣀⠀⠀⠀⠋⠈⡩⢘⡃⠛⠻⠀⠏⠘⣿⡟⢛⣻⡏⠛⡇⠧⡘⠇⠈⢳
⠀⠀⣉⣶⣶⣤⣄⣀⡈⠉⠙⠻⣧⡀⠀⢀⡀⣿⠁⡇⣿⠀⣠⠀⠨⣿⠋⠈⠹⣿⠀⠀⠦⠓⠘⡀⠘⡄
⠈⠛⢿⣿⠛⢿⣿⣿⡿⠶⢤⣤⠈⢿⣀⣼⣿⢿⣤⡇⠿⠀⠜⠀⢀⣿⣠⡀⣰⣿⠀⠀⠱⢧⡇⠙⣤⣜⢢⡀
⠀⠀⠀⠈⠲⠛⠻⠻⠳⣶⠀⠀⠀⠀⠻⡛⠛⠓⢴⣠⣾⣶⠄⠀⣈⠙⣿⣿⣿⣿⡻⡅⡄⠀⠀⠀⢻⡆⠦⢻⡀
⠀⠀⠀⠠⣤⣄⣠⣄⡀⠘⠯⠁⠀⠠⣀⣑⠐⡄⢾⣹⣿⣻⡄⠀⠁⠀⢟⡟⠻⣿⣿⣶⣷⡀⢀⠰⢿⠿⢄⠂⠨⡳⡄
⠀⠀⢀⣀⡀⠀⠠⠀⠙⠲⢤⣄⠈⠲⠻⣯⠃⢹⠀⠘⠻⣿⣿⠀⣷⠀⠀⣿⣶⢘⠿⣿⣿⠟⠀⠀⠻⠶⠚⢅⠢⡈⢿
⠀⠀⠀⠉⠀⣒⠛⢶⢦⡄⠰⠎⠙⣁⣶⣤⣀⠰⣷⣶⣴⣮⡛⠀⠟⢀⡠⠛⠛⠨⠀⠈⠨⣅⡑⠀⠦⡀⢒⡆⠀⠺⠾⣆
⠀⠀⠠⠾⠿⠇⢠⠁⠀⢰⡖⠃⢺⡫⢿⢿⣿⣄⣻⢿⣿⣿⣷⣤⣷⣮⣀⠀⠀⠀⠀⠠⠀⠐⠹⢒⢦⡐⠦⠐⣀⣰⡆⢊⣢
⠀⠘⣆⣀⠀⠀⠀⠤⠀⠀⣅⡀⣴⢡⣶⣿⣿⣿⣽⢟⠚⢻⣿⣿⣿⣿⣿⣅⣀⠀⣠⣀⡀⠀⠳⣦⣬⣾⠳⣆⠉⢻⣷⣿⠷
⢀⡀⣹⣿⠃⠀⠀⣠⣤⢰⣿⣟⣿⣾⣿⠿⡭⢹⡟⣎⢿⣶⡿⣿⣏⡉⠿⡿⢦⢿⣿⣿⣿⣧⠄⢀⠉⠛⣒⠺⢿⣿⣿⣿⣾
⡼⣻⣿⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⡷⣺⣅⣲⠀⠀⠀⠀⠚⠿⢿⣿⣷⠒⢶⣿⣾⣷⣍⠁⠉⣀⠀⠡⠀⠈⠑⠋⠁⠩⡻⠿

⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠠⣼⢸⣶⡾⢢⣦⣦⣤⣠⢀⡀⢠⣙⣦⣀
⠀⠀⠀⠀⢾⡏⠙⠟⢀⣿⠃⠉⠡⠛⢸⣡⣼⡏⠻⡌⠶⢄
⠀⠀⠀⠀⠀⠀⢰⡞⢹⣿⣶⢂⣿⡇⢸⠋⣿⣇⡀⠀⠀⢨⢄
⠀⠀⠀⠀⢠⣆⣿⡟⣿⠟⣿⣿⢹⡇⠈⣡⠤⣿⣻⢂⣶⡧⡶⣷⡄
⠀⠀⠀⠀⣮⣿⢹⣧⣿⣤⠌⠉⠀⣴⣿⣧⢀⣿⠭⠉⣤⠀⡶⢰⡇
⠀⠀⠀⢸⠇⠉⠠⠬⣿⢩⣴⠶⣷⡻⣿⣿⡟⣿⣦⣠⣟⣰⣴⣾⣷⡛⢦⡀
⠀⠀⠀⠀⠀⢀⠤⢼⡟⢸⡯⠀⠈⠀⣿⠈⣧⣻⢸⣯⠸⣯⢶⣿⠋⣾⡦⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠴⠆⣧⢤⣶⣴⣆⡴⣿⣿⣿⣿⣿⣿⣦⣽⣿⣯⣼⣿⣶⠾⢷⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣴⣻⣿⠿⠿⣿⣿⡋⢻⣿⠋⠉⠉⡇⢸⡏⠀⠀⣤⡄⡇
⠀⠀⠀⠀⠀⣠⣠⣿⠟⠿⣿⣿⠀⠀⣿⠟⠵⠸⣷⡀⠀⠀⡇⠘⠀⠀⠀⠻⠇⠇⣄⠀⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠋⠘⢻⡆⠀⣿⡿⠀⠀⢻⠀⠀⠀⣿⣇⠀⡀⣤⠀⡇⠀⠀⠀⢀⣿⣿⡆⠀⢺⡟⠃
⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⣿⣿⡄⠀⢹⡂⣀⢼⠟⢱⣖⢷⣦⣴⣿⣶⣶⣞⣻⣿⣿⣷⣿⡿⡓
⠀⠀⠀⠀⠀⠠⠿⠛⢃⣘⠋⠉⠋⠉⢫⣯⣵⣾⢿⣯⣾⣙⡚⡿⡟⣹⣳⣻⣿⣼⣽⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠩⢇⠀⠀⠀⠉⠊⠑⠻⣿⣿⣿⣿⣿⣭⣿⣯⣶⣿⣿⣮⣿⣑⣾⡄⢤⡤
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠛⠛⠛⠋⠉⠟⢱⣏⣩⣿⡭⢿⣣⡤⢭⣻⡿⢿⣯⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡓⠀⠘⠺⠀⠀⣉⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠾⠚⠒⠛⠉⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⡆⠈⣿⠀⠀⠀⢿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠠⠀⠀⣸⠇⠀⣿⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣶⣴⣧⣴⣄⣠⣄⣼⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡤⠀⠀⠠⢦⡆⠶⠄⢒⣦⣬⣽⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠰⠚⡛⠛⡍⠛⢁⠈⢐⢾⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡀⠀⢀⡻⠷⠈⠆⠀⠀⠀⠘⠃⢤⣿⢿⣿⣿⣿⣿⣉⣽⡿⢏⣿⣻⣦⡀
⠀⠀⠀⠀⠀⣀⣀⣁⣀⡁⡀⡈⠀⢩⣦⠀⠀⣶⠀⠀⠠⡄⠀⠀⠘⢈⠀⠈⣠⣿⣦⣿⣿⣿⡇⠀⠀⠀⠈⠀⢀⡀⠀⠀⠹⠿⣹⣶⣿⣯⡿⠿⣧⣀⡇⢼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⠠⠈⠿⠿⠦⣿⡀⠠⢀⡄⣤⠘⣿⢾⢇⣼⣿⣿⣿⣿⣿⣿⣿⠀⠶⡎⠀⠀⠐⠂⠁⠀⠉⠈⡋⠉⠋⠀⢡⣰⣿⣿⠇
⠀⠀⠀⠀⢐⡇⠀⠠⣧⠀⠈⡿⠀⠀⠀⠒⠀⠈⠉⣶⠀⠃⠸⠀⠛⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠛⠉⠃
⠀⠀⠀⠀⠰⠷⠾⠷⠾⠿⠛⠓⠒⠒⠒⠂⠒⠃⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⠀⣀⠀⠀⣀⠀⡀⠀⢤⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢁⠀⠀⣀⣂⠘⣉⣁⣉⣁⣲⣀⣻⠟⢈⣯⢖⣠⣀⣀⡀⠀⠀⠀⠈⠋⢻⠟⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠄⠀⠀⠉⠈⠀⠀⠀⠀⠐⠚⠋⣍⢱⠣⣷⣿⣿⢿⡋⠟⠉⠉⠉⠀⠐⠣⠾⠻⢗⣶⣦⡀⠀⠘⠃⠈⠛⠉⠑
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠐⠖⠴⠒⢌⠍⠙⢩⢿⡿⠟⠊⠀⡀⢐⠀⠠⣾⣏⡞⠿⡟⠈⠛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢺⣷⠀⠀⠀⠀⠀⠂⠀⠈⠃⠀⠁⠀⠀⠀⢔⡀⠂⠀⠔⠁⠿⡭⣖⡆⢰⠂⠙⠶⠟⢺⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣿⡄⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡄⠀⠤⠷⣯⣀⡈⡥⠀⠀⠀⢸⡿⡏⣿⣿⡧
⠀⠀⢀⠀⢀⠀⠀⠀⠰⠄⠀⣄⠀⣿⣀⣀⣀⣀⣈⠂⢸⡇⠈⡧⠀⣤⠀⠸⣧⠀⡤⢈⣿⣤⣀⣰⣿⣿⣷⣿⡀⡀⠀⢸⡟⠻⣟⠿⣇
⠀⠀⢀⣤⣸⣥⣤⣴⣦⢼⡈⣯⣹⣿⣿⣿⣿⣿⢸⣀⣈⣿⣆⣃⣀⣀⣀⣦⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⣤⣼⣿⣿⣿⣷
⠀⣠⣛⠛⣛⣛⣛⡻⠷⢾⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣟⣛⡛⣻⣿⡟⠉⠉⠉⠉⣿⣟⠉⠋⠉⠉⠙⣿⣟⣿⣟⣏⣹⣿⢽⣹⠁
⠀⠉⠀⠀⠀⠀⠀⠤⠿⠯⠯⠇⠂⠀⠐⠸⣿⣦⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠲⡶⣶⣿⣿⣦⠁⠀⠀⣀⣿⣯⣿⣿⠟⣿⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠀⢈⠀⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡠⣤⡰⠀⠀⡀⠀⢀⠰⢺⡿⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣿⣏⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣿⡇
⠀⢀⣤⣤⠀⢾⣿⡟⣿⣿⣶⣦⡁⠀⠈⠀⣁⣱⣿⣿⣦⠀⠐⢦⣤⣀⠀⠀⠀⠈⠠⣿⣽⣿⣏⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⢋⣉⠉⢉⡉⠉⠉⢉⣀⠀⣾⣿⣿⣿⣿⣿⣷⠤⠤⢽⣿⣿⣆⠀⢾⣿⣿⣿⣿⣿⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠁⠀⠉⠉⡁⠀⠀⠀⠀⣴⣿⣿⣿⣀⡜⢛⣻⣿⣿⣷⣿⡀⠀⢸⣿⣯⠀⠀⠀⠉⣿⣯⣷⣿⠄⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡇
⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⣠⣴⣿⣿⣿⡟⠈⠻⢿⣿⣿⣿⠇⠀⢹⣿⡇⢰⣿⣿⡆⠨⠇⢙⣿⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⠛⢿⣿⡏⢰⣿⣷⠸⣿⠟⠁⣴⣤⠸⠋⠉⠈⠈⠈⠂⠈⠁⠚⣀⠀⠠⠀⢀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠀⠓⠶⣆⣰⣶⣷⣷⣿⣿⣿⣶⡟⣢⣾⣿⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢀⣠⡀⠡⠜⢿⣿⣿⣿⣿⣿⣷⠂⡐⠂
⠀⠀⢢⣷⣣⣄⡀⠜⠁⢠⣼⣿⣿⣿⣿⣿⠿⣋⣵⣾⣿⣿⣏⡀⠀⠀⠀⠓⠀⠄⡀⠐⠒⠚⠒⠈⠉⠵⢤⣤⣌⠙⠻⢿⣿⣿⣷⠰⣿⠟⡖
⠀⠀⣾⣿⣿⣿⣿⣤⣾⣷⣿⣿⢿⡛⢉⣤⣾⣿⣿⣿⣿⣿⣿⣧⣤⡦⣄⠀⠀⠈⠀⠀⠀⠀⠙⠶⣶⣤⣤⠀⠠⣭⣀⣀⣩⣷⣿⠯⠷⠆
⠀⠀⠀⠈⠹⠾⠽⢿⣿⠿⠊⠀⠤⠳⠯⢿⣟⣯⣹⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⡠⠤⠀⠀⠀⠠⠤⠼⠕⠂⠈⠘⠝⢈⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡴⠟⠋⠉⠉⠉⠙⠛⠷⣦⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡀⠀⣤⠀⠀⠀⠤⡄⢰⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠅⣆⣿⠀⣾⠀⠀⣷⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠲⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⡿⠀⢾⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢸⡇⠀⢨⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢸⡇⢰⣬⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⣥⣧⠾⢓⠚⠉⠻⢦⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠛⠁⢉⠅⠐⠂⠘⠂⠀⠀⠀⠈⠙⢲⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⣶⠀⢠⠞⣀⡒⣦⠀⠀⡁⠀⢀⠀⣀⢠⠋⢩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⠿⢖⣛⣿⣦⣍⡉⣃⣀⣠⠶⠾⠶⣄⠼⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢿⣧⡀⠈⠉⠁⠄⠿⠛⠉⠉⠁⢀⠀⠀⠈⢀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣅⠤⠤⠌⢙⣧⡼⠿⠶⠶⣦⣄⣤⠴⠶⠖⠚⢛⡟⠓⠳⠶⢤⣴⡀⠙⠷⣤⡀⠀⠀⠠⠖⠦⣂⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣀⣀⣠⡤⠤⠐⠾⣿⠄⠄⠚⡲⣤⣤⠁⠖⠂⠒⠂⠀⠚⠒⠒⠂⠀⠁⠈⡻⣤⣄⠀⠀⠀⠀⠀⠀⣃⠀⠠
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠐⠛⠛⣉⣁⣩⣄⡠⣤⠦⠀⠘⡷⠶⠚⠛⠆⠀⠀⠓⠋⠀⠁⠀⠈⠀⠀⠀⠀⠀⣤⣼⡇⠽⠛⠉
⠀⠠⢠⣾⡆⠀⠘⠛⠒⠀⠀⠀⠀⠀⠽⠝⠿⠛⠒⠊⠉⠀⠘⡇⢀⠀⠂⠒⠀⢤⠀⡄⠀⠀⠀⠀⡀⠠⠄⠠⠀⠈⢸⠇⠀⠀⠀⢴⠄⠴⠴⡀⠀⣰⣆
⠀⡄⢈⡽⣻⣦⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣴⠀⡇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠓⠀⠘⠟⠋⢡
⠀⣿⢀⣬⣵⢨⣿⠒⢦⣄⣤⢀⣀⡀⢈⠁⠀⠈⠛⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠀⠃⠀⠀⣠⠀⠀⣴⠀⠀⠀⠀⠀⡤⠀⢀⣤⣄⣩⣿
⡇⠻⠇⣠⣼⠟⠋⠈⠘⢯⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⢀⠀⣴⡇⢸⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠀⠀⠀⠀⣿⠀⠀⢸⣤⣄⠀⠀⠀⠀⠒⠊⡙⠋⢹⣥
⣧⠓⠀⠀⠀⠶⡂⢀⣀⣾⠤⢠⠟⢻⣿⡇⠔⣶⡀⢠⠛⣿⢻⡉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠄⠀⠀⠉⠀⠀⢘⣿⣿⣿⣷⣶⣾⢶⡲⠻⡀⠈⢙
⡏⠀⠀⠀⠀⠀⠀⠎⠉⠉⠀⡞⣠⣿⠹⡇⢀⣿⡀⢠⡀⣿⠀⡇⠀⠀⢠⡀⠀⠀⠀⠀⡀⢤⢰⣧⡀⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣭⠎⠀⠁⠀⠘⠛⣫
⠹⠀⠀⠠⠀⠘⢀⢈⣶⠀⠀⣀⢸⣿⣄⣅⣈⢈⣁⣠⣥⣠⡄⣉⣀⣄⣠⣧⣤⣤⣀⣀⡄⠀⠻⡟⠗⣾⡿⣧⡐⠀⠀⠉⢉⣼⣇⣹⡆⠰⣄⠀⣤⠠⣤
⠀⣀⣴⠄⠀⠂⠉⣿⣿⠂⠈⠙⠛⠛⠛⠛⠛⠹⠛⠋⠻⠋⠀⠈⠿⠏⣙⠋⠀⠉⠉⠁⠀⠀⣰⣶⠻⣝⡆⢿⡟⠷⣤⠦⠄⠁⣼⠟⣿⢴⣗⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠿⠷⠓⠀⠀⡠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠈⠋⠁⠀⠉⢃⣈⡁⢀⣀⡀⢠⡦⠀⠀⠁⠈⢻⡶⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠁⠟⣀⠀⠀⠀⠀⠄⡀⠤⠄⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⢀⣀⣉⣏⡀⠈⠍⠉⠀⡙⠿⣀⣐⠀⢀⣁⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣥⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⣀⠉⠙⢿⠋⣿⣽⢧⠀⡁⠀⠀⠛⠀⣼⡁⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠃⠀⠀⠃⠀⡄⠰⠀⠿⣀⢀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⢐⣈⡀⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⡧⠠⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠐⠂⠀⣠⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠰⡀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠶⠻⠛⠛⠓⢦⣀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⣾⣿⣿⣆⠀⠟⠀⠀⠀⠀⠀⠀⣭⡛⣻⣾⣿⣯⣮⠀⢀⡀⢀⣀⠀⠀⢠⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠶⠿⠿⠿⠿⠤⠤⠀⠀⠀⠀⠀⠰⣯⣭⣭⣿⠛⣫⢵⡏⠠⠀⣿⣿⣿⣣⡸⣿⡿
⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣷⣾⣭⣿⠾⣿⣿⡟⢃⣴⣿⣟⣵⣿⣖⣿⡧
⠀⠀⠀⢂⡻⣞⣽⠗⣶⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣠⣭⠾⣧⣜⣿⣿⣭⣿⣿⣿⣿⡇
⠀⠀⢸⡉⠁⠽⢲⣻⣽⢻⣷⣤⠄⣀⠀⢀⣄⣸⣿⡇⠀⠀⠀⢀⣴⠟⠛⢦⣄⠀⠀⠀⢈⣹⣯⡁⠿⢻⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⣻⣼⠒⢴⣾⣽⣯⣼⣿⣿⣶⣿⣿⣿⣿⠃⢀⣴⡞⠉⠀⢠⣶⠀⢈⣳⣦⠖⢋⣡⣤⣿⣄⣘⡛⢿⣝⣿⣿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⣠⣿⠹⢦⢀⣸⣿⣿⠿⠿⠿⠿⢿⠿⢿⢿⣠⣾⣿⡀⠀⠀⠈⣩⣶⣷⢤⡁⡾⠋⠉⠁⠀⠀⠉⢻⣯⢙⠛⠿⢿⣟⣿⣵⣿⣿⡇
⠀⠀⠀⢠⣤⣴⡶⢶⣶⣶⠒⠒⣶⣶⣏⠋⠉⠉⠉⠉⠉⢹⣇⣤⣶⡿⡟⠛⠉⢸⡇⣛⠂⠀⠀⠀⠀⠀⠘⠹⡿⢿⣿⣶⡶⣭⣭⣙⣛⠻⠆
⠀⠀⠀⠀⠛⠛⠃⠚⠟⠻⠃⠘⠛⠛⠛⠀⠀⠀⠀⢠⣴⣿⡿⡛⡉⣶⡀⠀⠐⠘⡃⠌⢀⣄⠀⢠⡄⢠⡤⠀⣷⡀⡹⣇⣾⣷⣿⣿⡿
⠀⠀⠀⣰⣾⣶⡀⢠⣶⣷⡆⠀⣠⣶⣾⣿⣷⣶⣴⣿⣿⣿⣿⣿⡄⢈⠀⠀⡀⠀⣶⡄⠘⠋⠀⠈⠁⠈⠁⠀⢹⣉⣋⣿⣿⣟⣍⣿⣥⡀
⠀⠀⢠⣬⣭⣉⣡⡬⡭⠥⠵⠶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⡏⠀⡟⡏⠁⠀⠀⠀⠀⠀⢀⠀⠘⠋⠉⠉⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠄⣴⣿⡳⠄⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡰⠲⠶⠦⢤⡆⢠⡇⡠⠄⠀⠀⠀⠀⠀⢟⡀⠐⠒⠒⠒⠒⠒
⠀⠀⠀⠂⣿⣿⣿⡒⠂⠉⠉⠁⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⠊⠀⡮⢧⡀⠚⠃⢸⣏⡁⠚⠛⠀⠀⠀⠀⢯⠁⠀⠀⠀⠀⣤⣤⠀⠀⢠⣤⠄
⠀⠀⠀⠈⣿⣿⣿⠛⠛⠛⢀⢠⣶⣶⣀⣀⣄⣀⣀⣀⣿⣶⣄⣄⣉⣹⣿⣛⣇⣼⣃⣧⣠⣀⣜⣛⣃⣀⣋⣀⣄⣶⣀⡀⣀⡈⠁⠀⠀⠉⠙
⠀⠀⠀⠰⣿⣿⣿⣿⡿⠿⢿⣾⣿⣿⠟⡿⢿⣟⣿⣿⣿⣷⣿⠏⠿⢻⠏⠿⠛⣿⣿⣿⠃⢻⡿⣿⡿⣿⡟⢟⣿⣿⡿⣿⣯⣿⢏⣿⣯⠉⠉
⠀⠀⠀⣤⣿⣿⠿⠿⢷⡶⠶⠿⢿⢿⣶⢳⣾⣿⡙⢻⡛⠋⠉⠀⡀⣸⠉⠉⠛⠛⠛⠛⢰⡘⣛⢋⡙⢛⡙⠉⠉⠛⠛⡛⠛⠛⠛⠛⠛⣲⠒⠆
⠀⠀⠀⠀⣠⣤⡀⠘⠋⠁⠀⠀⠀⠛⠛⢻⣷⣿⣿⣼⣛⣀⣀⠛⠛⠛⠀⠀⠀⠀⠂⠀⠿⠿⠿⠿⠿⢿⠿⠷⠚⠑⣾⣇⣀⣠⣶⣶⣷⣿⡀
⠀⠀⠀⢀⣛⣛⣁⣀⣀⣀⣀⣉⣉⣉⣩⣽⣿⡭⠭⠥⠬⠠⠤⠀⠐⡀⠐⠀⠂⠒⣚⠛⠛⠛⡛⢛⡛⠛⠛⠻⠛⠛⠛⠿⠿⠟⣻⠛⢿⠿⠝⠃
⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠐⠀⠐⠀⠀⠙⠃⠀⠈⠃⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠉



⡿⢿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡫⡎⢱⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣩⣮⣀⣇⣸⣀⣷⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⢀⣸⣧⢸⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡤⠤⠤⠼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡯⠥⠀⠬⠭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡏⣡⣿⣷⣤⢩⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡭⢍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡣⠚⠁⠀⠙⢏⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣎⡉⠀⠀⠈⠉⠉⠑⣌⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡫⠞⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⢌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡫⠚⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡻⣋⠘⠑⠂⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠂⠀⠈⠀⣀⢠⠝⣍⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢋⠄⠥⠄⠀⠉⠀⠀⠀⠀⣠⡼⢛⡿⣧⣀⠀⠀⠀⠀⠤⠈⠈⠀⠑⢌⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢏⢴⠁⠀⠀⠀⠀⠀⢀⡀⣠⣾⢏⠟⠃⠀⢢⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⠟⠁⠀⠠⠀⠀⠀⠤⠤⣨⡾⢋⡄⠁⠀⠀⠀⠀⢈⠢⡙⢷⣄⠄⠀⠀⠀⠀⠀⠀⠀⠙⢍⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠀⠀⠀⠀⠐⠀⠀⢠⣾⢏⡴⠡⠄⠀⣀⣀⠀⠀⠀⠀⠈⣦⡿⣷⡄⠀⠀⠠⡀⠀⠀⠀⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⢰⣿⣿⡆⠁⠉⢙⣿⠂⠒⠒⠀⠀⢀⣀⠀⠀⠐⣿⡋⢉⠀⠀⣨⣾⡍⡀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢉⣷⣿⡏⠀⠀⢨⣿⠉⠀⣴⠤⠤⠴⠶⣤⠀⠀⣿⡅⠀⠀⠈⢽⣶⡇⡅⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠠⠴⠒⣿⣿⡧⠀⠀⢹⣿⠒⠀⣏⠁⢡⡄⠃⢹⠀⠀⣿⡆⠀⠀⠀⢾⣿⡟⠂⠀⠐⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠂⠀⢀⣿⣿⡗⠠⠄⠸⣿⠂⠀⠇⡀⢸⡇⡁⢸⠀⠐⣿⡆⠀⠀⠐⣺⣿⡇⠆⠒⡆⢰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠉⠀⠈⣹⣿⢿⡇⠀⠀⢈⣿⡁⠀⡃⡇⢸⡇⠐⢻⡀⠉⣿⡇⠀⠀⣈⣹⢿⣷⡃⠐⠂⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⣿⣛⣧⠀⠈⢸⣯⡀⠀⣅⠀⣀⢀⣂⢸⠀⠀⣿⡁⠀⠀⢠⡟⠻⣿⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠄⠀⠠⠤⠤⢤⣤⠄⢨⢿⡤⠆⠿⠤⠤⠤⠥⠾⣤⣠⡿⡦⣀⡀⠤⠤⠤⠤⠤⠀⠀⣠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣿⣍⣉⣉⣭⣉⣉⣙⣿⣿⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡁⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠄⢀⡾⠀⠁⢲⠄⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠦⠀⠀⣤⠀⠀⠈⣧⣄⡁⡺⠂⠈⢡⡄⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢂⣀⠀⣸⣿⣄⣀⣠⠙⠃⠀⠊⠀⠈⠘⠃⢦
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢀⠠⢴⡄⠈⠙⢩⣿⣟⣭⠁⠰⠀⠀⠚⠑⠀⣴⣶⠀⠀⠤⣋⣤⡠⣦⢠⡄⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡜⠀⠂⠀⠉⣶⣦⢾⢺⢶⢨⢟⣿⡃⠀⠀⠀⠀⢠⡄⠀⠿⠿⠀⠀⠀⡹⠷⠓⠶⠾⠇⢨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣆⠄⠃⢠⠀⠠⢄⣠⣴⣾⡏⠬⣼⣿⣿⡗⠂⠀⠂⠀⣩⣬⣈⣠⣤⣀⣤⡀⠠⢤⣠⣤⢤⣤⠤⠠⠌⠤⠄⣀
⠀⠀⠀⠀⢠⣄⠂⡏⠀⡆⢰⠀⠀⢠⣙⡿⠻⣵⠾⠿⡹⢿⡇⠀⠂⠀⠀⠉⠁⣦⢰⠎⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⣤⠀⠀⠃
⠀⠀⠀⠀⢻⡿⢱⣧⡆⣸⣬⠀⠀⠀⠿⠇⠀⠀⣀⣀⣀⠀⠁⠈⠀⠀⠈⠁⡄⠁⠈⢰⡄⠀⠀⠀⠀⠀⠀⠀⣆⠘⣿⡀⠀⡆
⠀⠀⠀⠀⣽⣿⠃⠟⠋⠉⠀⠀⠀⢠⡄⠀⠀⠀⠟⠩⢿⡇⠀⡄⠀⠀⠀⠀⠀⣿⣧⠙⠐⠆⠀⠀⠀⠀⠀⠂⡏⢀⣭⠀⠀⡍
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⣀⡀⠀⣀⢀⡷⠀⠀⠀⠘⠁⠀⡇⢀⠀⠀⠀⢠⠟⠿⠴⠤⠤⠤⠤⠤⠤⠤⠴⠅⠘⠓⠖
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠯⣆⠀⠀⠀⢠⣎⡅⠈⠁⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⢸⠀⠀⢀⣿⣀⡂⢠⠀⣰⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⢸⠃⠁⢦⠨⠉⠉⠉⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠠⠀⠀⠀⢀⡠⠋⣟⡐⠀⠐⣰⡍⠁⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⠧⠞⠂⠀⢰⣉⢑⠃⠉⠁⣶⠀⠁⠈⠀⢰⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠀⠀⠀⠆⡀⢠⡇⢀⡀⢨⣇⣀⣟⠒⢸⠀⠀⠸⣿⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢤⠀⠀⢀⠀⢈⡇⠀⠀⣶⠀⢸⠈⠀⠀⢚⣛⠃⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⡀⡀⠒⠚⡛⠐⠂⣀⠀⢠⠐⡶⡶⠉⢭⢰⢰⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠠⠤⠀⠀⠛⠀⠤⠤⠏⠁⠴⠶⠬⠈⠁⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⢛⡻⣛⡛⣛⣛⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠀⠀⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠀⢀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⢸⣿⣿⠿⠛⠻⣿⣿⣿⣿⠟⠛⠿⣿⣿⡇⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⡏⢰⣿⣧⠸⣿⣿⠃⣼⣿⡆⢹⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠀⠀⢸⣿⡇⢸⣿⣿⢀⣿⣿⠀⣿⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣀⣀⣈⣉⣁⣈⣉⣉⣀⣉⣉⣀⣉⣉⣁⣈⣉⣁⣀⣀⣉⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢉⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⡉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⡿⠋⣉⣤⣤⣈⠙⢿⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣉⣉⣀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣀⣉⣉⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⡀⢀⠀⠀⠀⡀⠀⢀⡀⢀⣀⠀⣀⡀⢀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠒⠁⢈⠀⡐⢸⠀⡄⠠⠭⡄⠁⢢⠀⡇⠀⡂⠂⠰⠠⠁⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠁⠀⠈⠉⠉⠁⠀⠈⠙⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣾⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣷⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣤⣿⣿⣦⣤⣤⣿⣿⣤⣤⣴⣿⣿⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣴⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣦⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⣿⣿⡏⠀⠀⠀⣠⣴⣶⣤⣠⣶⣦⣄⣤⣶⣦⣄⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠋⣀⣀⣀⣀⣸⣿⡇⠀⠀⣼⣿⠟⠙⣿⣿⠋⠙⣿⣿⠋⠻⣿⣧⠀⠀⢸⣿⣇⣀⣀⣀⡀⠙⣿⣷⣀⣀⣀⣀⡀
⠀⠀⣴⣾⣿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⢿⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⣿⣷⣦
⠀⣾⣿⠋⣀⣀⣀⣀⣸⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⣄⣀⣿⣿⣀⣀⣿⣿⣀⣠⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⣿⣿⣇⣀⣀⣀⣀⠙⣿⣷
⠀⣿⣿⣿⡿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⣿⣿
⣠⣿⣿⠋⠀⣀⣀⠀⢸⣿⡇⠀⢀⣀⡀⠀⢸⣿⡇⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⢸⣿⡇⠀⢀⣀⡀⠀⢸⣿⡇⠀⣀⣀⠀⠙⣿⣿
⠻⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠉⠁⠀⢸⣿⡇⠀⠀⠉⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⡇⠀⠀⠉⠀⠀⢸⣿⡇⠀⠈⠉⠀⠀⣿⣿
⠀⣿⣿⠀⠀⣤⣤⣤⣼⣿⡇⠀⢠⣤⣤⣤⣼⣿⡇⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⢸⣿⣧⣤⣤⣤⡄⠀⢸⣿⣧⣤⣤⣤⠀⠀⣿⣿
⠀⣿⣿⠀⠈⠛⠛⠛⢻⣿⡇⠀⠘⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠃⠀⢸⣿⡟⠛⠛⠛⠀⠀⣿⣿
⠀⣿⣿⠀⠀⣤⣤⠀⢸⣿⡇⠀⢠⣤⡄⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⢠⣤⡄⠀⢸⣿⡇⠀⣠⣤⠀⠀⣿⣿
⠀⣿⣿⠀⠠⣿⡿⠀⢸⣿⡇⠀⠸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢸⣿⡇⠀⢸⣿⠇⠀⢸⣿⡇⠀⢿⣿⠀⠀⣿⣿
⣀⣿⣿⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣿⣿
⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⣿
⣿⣿⣀⣀⣀⣀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣿

⠀⠀⠀⠀⠀⠄⢠⡄⢤⠀⡶⢄⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠃⠀⢫⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⢉⡛⣿⣿⣿⢦⣤⡀⠀⢴⣻⣼⣹⠆
⠀⠀⠀⠀⠀⠀⠰⣧⣻⡗⠀⠀⠀⢘⣻⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⢴⠻⠛⣿⣯⣿⣿⣿⣿⣷⣿⣧⠿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⣿⢉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡋⠀⢀⣇⣹⣿⣿⣿⣿⣿⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢘⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠮⠿⢻⣿⣿⣿⠿⠿⠍⠤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠰⠄⠴⢿⣿⣿⣿⣿⡇⠾⠧⠶⠦⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⣀⣬⣿⣿⣿⣿⣿⢰⡀⢀⣀⠀⠀⢀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠘⠂⠀⠀⢀⡴⠿⠀⠀⠀⢀⣠⡼⠀⠀⣠⠀⢨⣭⣯⣿⢣⠸⡿⣧⡟⠛⠚⠉⢹⢩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⢯⣿⢳⡿⣿⣻⡯⢯⣭⣽⣯⠈⠀⠀⣾⡆⠀⠀⠀⠘⡶⠠⣦⣾⠀⠀⠚⠛⠛⠛⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠙⠛⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠚⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠛⠛⠓⠒⠒⠒⠛⠓⠒⠚⠚⠒⢺⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡄⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠾⠦⠤⠄⠀⠀⠇⠀⢀⣠⣄⣠⣄⣠⣤⣴⡤⣿⠿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣁⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⠞⠉⠉⠉⠉⠉⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠤⢤⠀⠒⠶⠽⣿⡷⠀⠀⢐⣼⣗⠒⠒⠒⠒⠶⣶⣶⠾⣿⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠠⠋⠉⠁⠈⠀⠀⠁⣠⠉⠀⣿⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⢋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠒⠻⣶⡤⠤⢤⣤⣄⣀⣀⣀⣀⠊⠛⠀⠀⢿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣻⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⠂⠀⠇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠗⣀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠄⣻⡇⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣿⣇⣤⣶⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⢺⠃⠀⣿⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⡛⠉⠔⠊⣭⡈⢹⣿⢤⣦⣼⡆⠀⠀⢸⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠆⠈⠁⠀⣼⣇⣽⣶⣶⣶⡆
⠀⠀⠀⠒⠐⠂⠀⠀⠄⠀⠀⡀⠀⠀⠀⡠⠃⠾⡆⠐⣻⣿⣼⣿⢸⣿⡇⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⡿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠆⡇⠀⣿⣧⢸⣿⠸⠿⡏⢀⡐⠀⠸⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄⣴⣴⣺⣿⡏⣥⣿⣿⣿⣟⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠟⠋⠈⠁⠀⠀⠒⣼⣇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⠟⢻⣵⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠚⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⣴⣿⣿⣿⠿⠯⠉
⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣿⣿⣷⡹⡏⠉⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⢰⢸⣟⣿⣿⡿⠿⠿⠛⠛⣹⣽⠁⠀⠀⠀⣶⠆⢀⠀⢀⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠼⣰⢿⣶⣷⣞⡩⠿⠭⠄⠒⡛⠆⠀⠀⠀⠦⠌⠙⠛⣻⣿⣿⣿⣿⣿⣼⣇⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠚⠿⢽⣿⣯⡳⠶⣶⣦⠖⠒⠋⠀⠂⠠⣤⣾⣶⢾⠻⠿⣿⡟⠪⠿⣙⢛⣿⣺⣧⣄⣹⣿
⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠙⠻⠗⣿⣿⣷⠆⢰⡦⠄⢁⣾⠯⢿⡀⠀⠀⠀⠀⠀⣶⣸⠟⠅⠈⠉⢻⣿⣿
⠀⠀⡀⢠⡀⠀⠀⠀⠀⠈⠈⠀⠠⠎⠉⠁⠀⠘⢁⢀⡼⠃⠀⠈⠍⠓⡖⠀⡀⠘⡏⠙⠀⠂⠀⢠⡼⠟⠁
⢀⢼⣧⣼⡷⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡀⠄⠂⣸⠹⠇⣮⠁⠈⠀⠀⠀⠀⠁⢰⡟⠠⣰⣦⣤⣀⣦
⠈⢲⣿⡿⡏⠀⠀⠀⢠⣄⡀⣧⣤⣼⣭⡗⠚⠈⣾⢸⠧⠅⠀⠀⠀⠀⠀⠀⢠⣄⣴⡶⢻⣿⣿⡙⠛⢄
⠀⢈⣉⣵⣾⡷⠃⣀⠀⡄⡁⣿⣿⣿⠧⣾⠷⠒⠉⢠⣤⠀⠀⠀⠀⠀⢰⣠⠷⠛⠉⠻⠿⣿⣧⡛⣋⡍
⠀⠀⠀⠻⠍⠋⠻⠿⠀⠀⠁⣿⡟⠛⠉⠁⣤⠀⣠⣿⢻⣶⠀⠀⠀⠀⠈⠀⠀⣀⡤⣤⣤⢀⣬⠩⡟⠃
⠀⠀⠀⠀⠀⠀⠛⠿⠂⡰⠀⣡⣄⡀⢈⠂⡼⣿⣭⠜⠿⠿⠀⠀⠞⠀⠀⠈⠀⠈⠂⠉⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡴⢶⠶⠄⢉⡉⡉⠀⠔⠖⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠒⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠞⡳⢾⡿⠟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⡆⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢁⢚⣅⣠⣤⣡⣂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢡⠀⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣟⠿⠛⠛⠲⢶⣯⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠙⠋⢉⠉⢡⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢉⣻⣷⣿⣿⣿⣿⣿⡆⠀⠒⠐⠂⠰⠃⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⢼⣿⣀⠀⣾⣿⣿⡿⠋⣀⣀⢀⡀⣀⣀⣤⣤⣀⠸⣿⣿⣿⣿⣿⣷⠀⠀⣠⣄⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣀⠈⠁⡰⢠⣿⣿⣿⡇⣾⡿⣿⣿⡇⣿⡿⢿⣿⣿⡷⢿⣿⣿⣿⣿⣿⠀⠈⠉⠥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠙⠛⠉⠉⢹⣿⣿⣿⠆⠀⠈⠟⠁⠇⣿⡆⠚⠛⠁⠀⢠⣿⣿⣿⣿⣇⠐⡀⠠⠀⡀⢄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠸⣿⣿⣟⣇⠀⠀⠀⠀⡁⢿⣷⠀⠀⠀⢠⣾⢛⣼⣿⣿⡟⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀⠀⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⣀⣧⣾⣿⡆⡄⠠⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠸⣿⣿⣿⡦⠀⣟⡯⢽⣭⣿⣷⣾⣿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠻⣿⣽⣿⣦⣻⣤⣀⣠⣿⣿⣿⡿⢺⣿⣿⡧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠒⠀⠂⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣹⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⢿⣿⣥⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠄⠀⠤⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢚⣿⠿⡇⠈⠙⢿⣿⣿⣿⠿⠟⠉⢀⣷⡄⣯⢻⣤⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢛⣡⣶⠟⠁⠀⣏⠀⠀⠀⠀⠀⠛⠀⠀⣤⣿⣿⢃⡿⠂⢯⠹⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣫⣴⣿⡿⠋⡀⢠⠀⠘⠄⠀⠀⠀⢀⡤⠐⠋⠉⣷⡏⣼⠃⡈⣰⡦⣖⣮⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⣸⣿⢰⡏⠀⣫⣷⣾⣿⡏⣤⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠝⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣶⣶⣶⣿⣿⡇⣼⠃⡂⡱⡟⣿⣿⡇⢛⡴⢞⡉⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢧⣀⠀⠆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡟⠘⠀⡝⢠⣿⣿⢁⣤⢀⡜⢀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡤⡀⡠⠀⣸⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⢡⡆⠐⡀⢸⣿⡟⡸⢁⢊⡆⣸⡿⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣖⢿⠀⢟⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡆⠓⢠⡇⣿⣿⡇⠁⢂⣤⢰⣿⣧⢻⡿⣿⣿⣿⣷
⠀⠀⠰⠆⠠⠰⠤⠄⣼⣿⣿⣏⣿⣿⣿⣿⡆⠹⠆⣿⣿⣿⣿⡿⠁⠾⣻⣿⣿⣿⣿⣿⣿⢡⣟⢷⣀⣹⢠⣿⣿⣇⠰⣿⢧⣿⡟⢸⣏⣿⣿⣿⣿⣿
⠀⠀⡂⣀⢊⡀⣀⢀⣿⣿⣏⢿⣿⣿⣿⣿⠀⢱⡎⢹⣿⡟⢉⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣾⣿⣿⣼⣿⣿⣿⣀⣻⣼⣿⣅⣸⣽⣿⢿⣿⣿⣿⡇
⠠⠀⠷⠄⣀⣤⡀⢸⣿⣿⣿⣎⣿⣿⣿⣿⠀⢼⡇⢸⣿⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣯⣬⡭⢽⣧⣌⣩⣉⣽⡛⣉⡃⣿⣟⣿⡿⣜⣽⣿⣿⣇
⠀⠛⣋⣸⣿⣏⠉⣿⣿⠹⣿⣿⣼⣿⣿⣿⣘⢸⡇⣾⡏⠀⠋⡈⠉⢍⣩⣿⣿⣿⣿⣿⣎⢹⣿⣿⣷⢾⣿⣶⣿⣿⣿⣥⣬⣵⣿⡿⠉⣴⣿⣿⣿⣿⣿
⠀⠋⠙⠋⠉⠟⢰⣿⣿⣆⡸⣿⣿⣿⣿⣿⣽⡼⢠⣿⠃⣀⡠⣷⣶⣿⢀⣿⣿⣿⠿⢻⣏⣸⣿⣿⣿⠉⡉⠉⣉⣉⣉⠁⣿⣿⣿⠁⠀⠇⠘⠋⣉⣹⣿⠄
⠀⠓⠞⠻⠤⠄⣸⣿⣿⣯⣈⣿⣿⣿⣿⡿⠿⣟⣻⣿⠀⠀⠀⣹⠿⠃⢸⣿⣿⣏⢰⣿⡿⣿⣿⣿⣿⠀⠁⠀⡀⠛⠁⡀⠈⢸⣿⠀⡼⠀⠘⡩⢄⡵⣿⡆
⠀⡗⠂⢀⢀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠗⠀⠀⢠⠁⠀⣠⣿⡿⠿⣭⣭⣧⡈⢙⠿⢿⣿⣀⣦⡀⣸⣿⣿⣀⣼⡿⣟⣴⣁⡴⢪⠔⣩⣾⣿⡇
⠀⣍⣤⣄⡀⢠⣿⣿⣿⠿⠛⣿⣿⢏⣿⣿⣿⣿⠏⠀⠢⠀⣀⣠⣾⣿⡟⣿⣷⣄⠙⢾⡿⣻⣷⣤⣀⠉⠛⢁⡿⠛⠋⠈⠹⣷⣿⠟⡁⢂⢱⣎⠠⢜⠿⣧
⠀⠷⠆⠀⡉⢠⣿⣿⠁⣠⡼⢟⣱⣾⣿⣿⢻⡇⣰⠍⣠⣞⠉⠠⠬⠍⠉⢉⣡⣌⣥⠾⠧⠿⠟⠉⠀⠀⠀⠀⠱⣔⢦⣀⣴⣿⣿⣯⣴⣶⣶⣶⣶⣿⣿⣷
⠀⡒⠋⢀⠀⢸⣿⢣⣶⣏⣿⣿⣯⣿⣿⡟⢿⠛⣿⣿⠻⠿⠿⢿⡷⣶⣶⣶⣦⣤⣿⠦⠠⠤⠄⠀⠀⠀⠀⠀⠀⠈⢹⣿⡈⣍⢹⡟⠛⢻⡟⢿⣿⣿⣿⣿⠀⠄⠠
⠀⠙⠀⠐⠀⢸⡿⠋⣿⣿⣿⡟⣾⣿⣿⠟⠌⢲⣿⡇⢰⡮⢀⠨⠛⠀⣒⠒⢠⣤⣿⠓⠒⠐⠂⠠⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡌⢿⠘⡎⢿⣄⢈⠻⣿⣿⠀⠀⡀
⠀⠘⠀⠀⠀⢼⣥⣿⣿⣿⣿⣾⣿⣿⣿⣾⡄⣼⣿⣻⣿⣦⣤⣭⣥⣤⣬⣙⣉⣛⣈⣿⣿⣟⣷⣶⣼⣄⣰⣤⣤⣀⣾⣿⣿⣿⣷⢸⡇⣿⢸⣿⣾⣷⣼⣿⡄
⠀⠁⠈⠀⠀⠉⠉⠉⠉⠉⠉⠙⠉⠛⠛⠛⠛⠛⠉⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠓⠲⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⣿⣦⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⣶⣶⣾⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣰⣒⡶⠒⠶⠶⠶⡶⢲⠶⠶⢶⣶⠶⡴⠂⢀⣀⡀⠀⠐⢢⠴⣴⣶⣶⡶⠀⣤⣶⣤⣦⣴⡶⣦⣦⣾
⠀⠴⡿⠷⠾⠯⢤⠝⠓⠦⣤⠬⣵⠮⠴⢴⡇⢰⡟⣉⠿⣷⠀⢰⡬⣤⣤⣤⣤⣤⢦⣤⠴⠓⣩⢶⣶⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠠⢀⣀⣀⡀⠐⠐⢻⢧⣔⡿⣿⠶⠂⠀⠀⠀⠀⢀⠄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⡉⠁⠀⢀⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠛⠒⢊⣻⠇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡁⠒⠲⠀⣲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠠⠆⢠⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠐⠞⠒⢲⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠄⣴⣒⠸⢛⡇
⠀⠀⠀⢠⠖⡷⠆⠐⡒⠀⠀⠀⠀⠀⠀⠁⠀⢹⠉⡇⠀⢸⣇⠀⠀⠀⠀⠀⠠⠐⠀⠐
⠀⢀⠔⣛⡿⠀⡰⢛⣭⣳⡄⠀⠀⠀⠀⠘⠀⢼⢀⡀⠀⣾⢻⠀⠀⠀⠀⠀⠀⣾⡿⢿⣆⠀⡀⣴⣮⠂⡄
⠀⠈⡼⠛⢃⠀⠸⢾⣈⣹⠀⠀⢂⠀⣴⠀⠀⣤⠭⢤⣄⠁⢜⠀⢰⡆⢲⡆⠀⢻⣟⠈⣏⣠⣧⣭⡟⣥⣧
⠀⢺⣷⢹⡀⠠⣤⣍⠣⣽⢒⢷⡒⠞⠒⡗⢠⣹⡀⢠⣿⣷⠿⠘⠿⠓⠁⠒⣷⣿⡿⣧⣸⣿⣿⣿⣷⣦⣿
⠀⢰⡏⠛⣿⣈⣿⠉⠑⣞⢿⡇⢻⡄⡸⢃⣼⠇⡇⠸⣿⡇⠀⠀⠐⠀⠀⣴⡿⡏⣼⣵⣿⡗⢻⢹⣯⣿⣿
⠀⢸⣶⡀⠸⠛⢿⡄⠠⠘⣿⡿⢨⣾⡄⣏⢹⠀⢃⠀⣿⡧⠀⣿⢸⠐⠀⣏⣸⠃⣻⠯⣿⠿⢸⢻⣿⠏⠓
⠀⠸⡏⢳⣶⣄⣼⠻⣆⢐⠃⣸⠸⠿⠇⠀⢀⠀⡜⡄⢹⡇⠀⠀⠀⠀⠀⠻⣿⣿⣟⢼⢹⣼⡜⢨⣿⢼⣏
⠀⠀⣧⡆⠀⠙⣻⡇⠙⠻⢶⣾⠠⠀⢠⠀⠀⠀⣼⢁⢸⡇⠀⠀⠀⠀⢈⡀⠙⢻⡿⣛⣾⡟⣠⣿⣿⡀
⠀⠀⣿⡞⠀⠈⢹⡇⠀⠀⡤⠏⢸⣦⠰⠀⠂⣀⣽⡗⡸⣇⣀⣀⡀⠪⢿⠟⢀⡇⢧⢹⡿⠣⣄⣿⣯⣹⠆
⠀⠀⣿⢄⠀⠀⢼⡥⠀⡀⢠⡗⢼⡿⠋⠀⢰⣿⡵⡾⢷⠈⣿⣿⠁⠀⠈⠀⢸⡷⣳⣿⠃⣦⢹⣿⡿⢾⠃
⠀⡀⢸⣯⡀⠀⣼⣷⠿⠀⢺⣇⣿⡕⠀⠀⠛⠛⣿⣷⣿⡿⠟⠃⣤⡀⠀⠀⢸⣿⢹⣿⠆⢹⣇⣿⢧⠈⣀
⠀⠀⠸⣿⠂⠀⢸⣿⡄⠀⡄⡼⢸⡏⠀⢠⣶⠀⣿⡇⢼⡇⠀⢀⣿⣷⣴⡆⣏⢻⣾⣷⠀⢀⣿⢻⣮⠂⠁
⠀⠀⠀⡟⢠⣾⢃⣿⡀⣰⢠⠇⢸⣷⠶⠞⠟⠀⣽⣾⣾⡇⠐⠿⢿⠿⠛⠀⢻⣿⣿⣇⠀⣺⡟⣸⣻⠴⡆
⠀⠂⢰⡇⢡⠏⣼⡇⢸⣣⣿⠀⠸⠁⣀⡖⡁⠀⣻⢿⢺⣇⡄⢐⣾⠁⡀⠀⡀⢿⣿⣯⢀⡟⣠⣿⣿
⠀⠱⣺⡷⢃⣾⣿⣶⢸⡇⡇⠀⠀⠴⠘⣏⣥⣴⠛⣻⡷⢿⣶⣿⢟⣶⣬⣼⡇⣤⠻⣿⣋⡼⣿⣿⢿⣀
⠀⣻⣿⣷⠁⣿⣽⠋⠐⢣⡇⠀⠸⡿⢛⣿⡛⢛⣿⡧⢦⠰⣿⣍⣉⣥⠛⠛⢿⣾⣦⢿⣵⣟⣡⣴⣿⣿⡧
⠀⢱⣿⡇⠀⣿⣸⠀⢸⠁⡇⠠⢸⣾⣿⣤⣼⢉⣽⡿⠛⢿⣷⣷⣿⣟⡛⠻⢿⡛⠓⣹⣡⠉⠀⠑⠦⣸⠇
⠀⢈⣉⡁⠙⠻⠿⢐⠻⠆⠉⠐⢚⢋⢿⡟⣁⣾⣿⣧⣩⣾⣿⣿⣿⣿⡛⣻⡿⠁⡉⠛⠀⡀⢔⣧⠀⣁
⠀⠻⣯⣵⠶⠀⢰⠛⠷⠚⠙⡿⠿⠻⠞⠛⣋⣿⠿⠛⣿⣿⣿⠿⢿⠛⠡⠼⣿⣿⣧⣤⡉⡿⢿⣿⣭⣿⡇

⠀⣠⣾⣿⠷⢶⣶⣿⣿⣿⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣆⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠂
⠀⣿⣿⣶⣿⣷⡌⣿⣿⣿⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣷⣾⣞⣀⣴⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⣬⣯⡤
⠀⢻⠿⠟⠡⠛⢃⣿⣿⣿⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⣀⣀⣠⣤⣤⣾⣷⣦
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣦⣿⣯⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⠿⢿⠚⠒⠒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠔⠰⠖⠀⠖⠂⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣯⣬⣯⣿⣿⣿⠀⠀⠂⠀⠐⠂⠘⠓⠘⠛⠀⠂⠀⢸⣿⣿⣿⣿⡭⢾⣻⣿⣿⡇⠀⠀⠀⠀⠐⠀⣀⣠⣤⠴⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡈⢀⡈⢉⣈⢉⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⣤⣶⣾⣏⣉⣀⡀

⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⠖⠀⠀⢲⣶⣶⣦⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢟⡋⠁⢀⣀⣀⣠⣿⣿⣿⠀⠘⠋⠉⠙⠛⠛⠛⠛⠛⠛⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠚⠒⠒⠒⠛⠛⠛⠛⠛⠂⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀
⠀⣲⣖⠒⠒⠲⠶⠆⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣯⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣏⣿⣿
⠀⣌⣁⡖⠂⠀⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠛⣿⢻⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠂⠀⠀⠶⠐⠀⠀⠀⠀⣿⣿⣿⡿⣿⣻⡇⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣿⣿⣿⢿⣿⣯⡇⠀⢸⢻⣿⣿⣿⣿⠿
⠀⣀⣤⠴⠆⣀⣤⠀⠀⠀⣷⣿⣟⣛⣹⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣿⣩⣿⣿⡇⠀⠈⠈⠙⣻⣿⣿⣷
⠀⠴⠦⠤⠤⠤⠥⠄⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠿⠛⠿⠏⠀⠀⠀⠀⠰⠿⠿⠿⠿

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⠀⠀⠀⢾⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣹⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠛⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⢀⣰⣟⡷⣠⠅⣤⡠⠀⣤⣀⠀⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠙⠛⠙⠊⠙⢈⠉⠁⠁⠋⠀⠀⠀⠈⣡⡾⣿⣟⡴⢠⢄⡄⢀
⠀⠀⠀⠀⠀⡠⢜⣶⡤⠊⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠐⠀⠉⠝⡉⠉⠈⠀⠁⠎⠰⠅⡴⠊⠁⣠⣦
⠀⠀⠀⠀⠤⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢜⠟⠏⠤⣩⣷
⠀⠀⠀⠀⣠⠞⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡳⣣⣯⢪⣾⣿⠟
⠀⢀⣠⡿⠁⠀⠀⠀⠉⠈⠋⠛⠛⠛⠳⠖⠒⠦⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠑⠁⣵⣿⡷⠃
⣴⣿⣿⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢠⣶⣦⢤⠀⢈⠷⠒⠒⠲⢶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣼⣍⡼⣠⣾⣿⠟⣵⡄⡀
⠀⠉⠉⠉⠋⠛⠛⠛⠻⠿⠿⢿⢏⣾⡟⣡⣿⣶⣿⣦⢀⣀⠀⠀⠀⠉⠻⠗⢈⢵⣒⣦⠤⣲⡟⣩⠇⣴⣿⣽⣻⡞⢃⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⣿⡻⣿⣥⣿⣻⣿⣿⣿⣶⣦⣄⣀⠉⠈⠀⠈⠹⡏⣼⢯⣾⣯⠞⠉⠉⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠬⢡⡟⠉⠛⠛⠛⠋⠁⠀⠈⠉⠙⠓⠾⢯⣟⣳⡶⣤⣤⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠯⣯⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣀⣽⣷⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣙⣛⣛⣛⡛⠛⢿⣷
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣟⣛⡛⠛⣿⡇⣸⣿
⠀⠀⢰⣾⡿⠟⠛⠋⠉⣿⡇⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣤⣿⣯⣉⡁⠀⡿⡇⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⢶⣿⣯⣭⡥⠀⣿⡇⣿⡇
⠀⠀⠀⢻⣧⠀⠀⠘⠉⣿⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠒⠒⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠿⢿⣿⡿⡿⠁⣿⡇
⠀⠀⠀⠸⣿⡀⠀⠀⠂⣿⡇⠀⠀⠀⠛⠉⠛⠛⠓⠒⠚⠃⠀⠀⠀⣿⠀⠀⠀⠀⡤⠤⢤⣤⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠉⣿⡇⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠁⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠖⠶⠤⠤⠤⠤⠤⠤⠤⢤⣀⣠⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠸⣿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠤⠤⡄⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠈⠁⠁⠀⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠄⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠻⠿⠿⠿⠿⠟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⣀⣀⣠⣤⣴⣶⣶⠿⠿⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⣤⠤⠤⡄⠀⢤⠀⢠⡄⠀⡤⠀⢤⠀⠀⡄⠀⢠⠀⠀⡄⠀⡤⠀⢠⡄⠀⡄⠀⢠⡄⠀⡄⠀⢠⠄⠀⠄⠀⠤⠀⠠⠀⠀⢤
⣁⡏⠉⠉⠉⠁⠁⠈⠉⠉⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡖⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢸⡟⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣏⣇⣴⣦⣶⣆⣄⢠⣦⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡀⣄⠦⡏⡎⠆⠀⠀⠀⠀⢀⠤⢀⡀⠀⣰⣤⢤⢄⡰⢢⣀⣤
⣍⡿⣸⣷⣻⡟⣰⠟⠿⢦⠈⡺⠦⠐⣐⣄⢀⡤⢀⡀⠀⠙⣶⣿⡀⢿⣷⣈⡗⠀⢀⡀⠸⠛⢣⣵⠖⠍⠃⢻⠉⡇⠘⢸⣀
⡏⡿⠁⢫⣿⠟⠛⠳⠟⣫⠯⣻⣅⣽⢛⠊⢡⣦⠻⠘⠂⣄⠠⡻⡇⢸⣯⡽⠷⣽⣿⡑⠶⠌⠻⡌⠀⠀⢀⠑⣈⡻⠀⠰⣄
⡏⣿⣾⣀⡽⣦⢀⡀⠴⣧⣴⢈⣿⣿⡾⣦⠞⢸⣖⣖⣪⣿⢣⡐⡇⢨⡏⠀⠀⠙⣿⣋⠀⠀⠹⣇⠀⠃⠘⠆⠋⠧⣨⢦⣇
⡏⣿⡉⠁⠂⢾⠤⠠⠆⡏⡇⢿⣿⡟⡟⠚⣛⣽⡿⠚⠓⠃⠀⠨⢷⡿⠂⠀⠀⠀⠠⠖⠖⠀⠂⠀⠀⠀⠀⢲⠙⡶⢴⢻⣏
⠈⢷⣰⣀⠀⢇⢆⣰⡄⠁⡄⠀⣿⡷⡄⢸⣙⡙⢁⣛⠁⠀⣀⣀⣀⣤⡄⠀⣰⣦⢀⢠⢀⡀⣴⠈⢠⣀⠖⠚⠀⢂⢀⡀⣇
⠘⠛⠉⠛⢳⣚⠋⠉⡉⢀⢠⡟⠉⠁⢃⡾⠋⠒⠛⠤⠀⠀⠉⠉⠀⣸⡅⢀⠀⠀⠀⠅⠉⠉⠙⠓⣼⡃⠀⠴⠆⢾⠻⣿⡏
⠐⢻⢿⣶⣾⢙⠀⣞⡁⠀⣈⣥⠀⢲⡍⠀⠀⢀⣠⡘⠀⢒⣶⣯⡴⣭⢐⠉⠀⣠⡀⠀⠀⠔⠦⢀⡏⠏⡀⠁⡀⣼⣴⡧⡏
⠐⢋⠁⠀⢧⣿⣠⡾⠄⣀⣠⠇⣀⣼⡿⢷⠚⠻⠛⠛⢫⠉⠈⠋⠙⠓⠮⠽⠛⠘⡯⢕⡤⢁⠠⢸⠄⠀⣠⡾⠿⣿⣿⣿⡏
⠀⢘⣠⠢⣾⣿⣯⢶⡘⣈⣦⣀⣹⣏⣰⡆⡤⠚⠜⠩⡃⠉⣈⠉⠨⢩⢻⢦⡳⢈⢘⡀⠁⠀⠀⠨⠥⣐⣦⣷⣾⣿⣿⣷⡏
⠐⢨⢥⡼⠇⣬⣧⣤⡿⢀⣴⣤⣼⠇⢠⡐⡤⢠⠛⠀⣴⠄⡜⠃⢠⡷⢤⡟⠄⢤⢠⠤⡘⠣⠤⠧⠤⠟⠀⢸⠤⠼⠣⠄⡄
⠀⢺⡖⡓⠶⠖⣷⡀⠲⠄⢸⣲⣲⠑⠲⠂⣏⣖⣷⡾⢞⣷⠘⠈⠉⣿⡟⠅⢀⠁⠘⠠⠀⠁⠘⣿⣮⠉⠔⢺⣎⢡⠄⡙⡃
⠰⠮⠭⠭⠭⠩⠏⠁⠀⠀⠸⠄⠠⠄⠀⠰⠀⠀⠁⠀⠘⠀⠀⠀⠀⠃⠀⠑⠀⠀⠒⠀⠈⠂⠀⠃⠀⠘⠀⠘⠃⠀⠃⠀⠐
⣶⣶⠿⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⠀⠀⢀⣤⣶⣿⣿⣿⣷⠀⠀⠀⠀⢀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣴⣿⣿⣿⣽⣿⣿⡿⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣿⣮⣿⣿⣿⡅⢀⣾⣿⣿⣟⣿⡽⢝⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣹⣷⡿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣯⡧⣷⣳⢾⣇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣿⣿⣿⣬⣿⣿⣿⣿⣼⣿⣿⣺⣿⣿⣿⣿⣷⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⣿⣿⣟⠛⠛⠉⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⠀⠀⣼⣿⣿⠿⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⡟⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⢿⡇⢀⣾⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣯⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡇⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⣿⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣧⢻⣿⣿⣿⡏⠀⠀⢹⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣯⢿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠸⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⣿⣿⣿⡏⠉⠉⠛⠛⠻⠿⠿⢿⣿⣦⣄⠀⢀⣀⣤⠤⠶⠶⣆⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣷⠀⡷⠀⠙⠛⢏⣉⣘⣿⣇
⣿⣿⣿⠁⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⠟⠉⠉⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⠃⣀⣨⠀⠀⠫⠽⠟⢿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣟⢋⣽⣷⣋⣔⣉⢄⡀⠀⠀⠀⠸⣇⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⣯⣅⡼⠻⣆⣼⣮
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⣿⣿⡁⣀⠾⣿⣶⡐⡀⠆⢻⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣶⢴⣹⣯⠙
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣻⣿⠁⡬⠟⠰⠌⠻⣿⣧⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠋⠁⣀⣨⣿⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠒⠀⠀⠁⢀⣿⣿⣯⣽⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣤⣾⣿⣿⣿⠟⠉
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠠⠄⠀⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠊⣹⢱⣿⣿⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣤⠶⣶⣿⣻⣿⣿⡛⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠘⠢⠀⠀⠀⠀⠙⠻⣏⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢜⢥⣾⠿⠙⣿⣿⣮⣝⡷⢦⣄⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢂⠀⠀⠀⣀⣨⣤⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠙⠛⠀⡻⠟⣿⣿⣽⡆⣸⠟
⣿⡟⠓⠒⠀⠀⠀⠀⢀⣀⠀⠀⠀⢿⣀⣀⣤⣬⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡦⣀⠀⠀⣴⣿⣿⣿⣿⣶⣄⠀⠐⢿⣿⣓⡶⠈⢻⣷⠏
⡇⣠⡶⠶⠿⡟⢀⣀⣠⡉⠻⣆⠀⣾⣏⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣷⣾⣿⡿⠿⣿⣿⣿⣿⣿⣧⣤⣍⣙⣣⣔⣿⠏
⣾⣿⢃⣈⣶⣵⣖⠈⢿⣿⣼⡿⢿⡧⠾⣤⣶⣤⣤⣅⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠃⠘⢻⣿⣣⣆⠀⢞⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⡟⢫⠀⢽⣯⣈⣿⠇⢸⣿⣤⣄⠸⣡⠤⣶⣾⣿⣿⣿⡿⠟⣿⣿⣿⣖⣆⡘⣮⣝⣥⣴⡿⢃⡹⠿⣷⠂⢀⠼⢧⣉⡻⢿⣿⣿⣿⣿⣦
⠙⣿⣧⠈⣭⣿⣇⣠⣾⡟⠛⠿⠅⠀⢰⣬⣿⣿⣿⣿⠀⠀⣿⣿⣟⣏⣯⣬⣤⡼⠉⠙⡷⣴⡃⠀⡈⠙⠚⢿⣌⠙⢋⣸⣿⣿⣿⣿⣿⣷⣄
⣬⣀⠑⠲⡿⠿⢿⠙⠊⣄⣶⣤⣄⠐⠊⠈⢉⣻⢿⡏⠀⠀⣿⣿⣭⣬⣬⣯⣥⡇⢰⡄⣷⣤⣽⢿⣯⣄⡀⠔⠉⠙⢻⣏⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣷⠆⠀⠀⠙⢶⣿⠱⢿⡽⣽⡇⢠⡴⢋⣿⣿⠃⠀⠀⣿⡌⢤⠉⠻⣿⣿⣿⣾⣿⡋⢡⣒⢺⣿⣿⣿⣷⣶⣆⣡⡿⠉⠛⠻⠿⢿⣿⣿⣿⣿
⡿⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣷⣷⣦⣬⣡⣸⣿⡟⠀⠀⠀⠿⠷⠿⠶⠾⠿⠿⠿⠿⠷⠶⢶⣷⣾⣿⣿⣿⣿⣿⡟⠙⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡟⣹⣻⣿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢁⣿⣿⣹⣿⣿⣿⣿⣿⡇⣾⣿⣿⣶⡆⠀⠀⠀⠀⣀⣨⣿⣿⡻⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣡⣤⡶⣾⣟⣿⣿⣿⣿⡿⢫⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣜⣿⣿⣽⣿⣿⣿⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢾⠋⠉⠉⢸⣿⣿⣿⡿⢻⣿⣧⠼⣿⡟⣯⣿⣿⣯⣽⡟
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣘⣓⢢⠎⢙⠻⢿⠇⣾⣿⣤⣴⣾⡿⣿⣶⣆⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠏⡞⣀⣀⡄⡀⣸⣿⣿⣿⣿⣿⣷⣿⣿⣟⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⢋⣼⠀⠉⠙⠛⠲⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⡰⠿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣛⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠍⠀⠂⠀⠈⠀⡀⠀⠄⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠥⢈⡑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢠⢦⣀⠀⣀⠤⠔⠒⠉⠀⢀⣠⣴⡶⡆⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠻⠶⣶⣤⣀⠠⠜⠋⠉⠀⣀⣤⠀⢰⠟⠛⠉⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⢀⣤⡴⠞⠛⠉⢩⠀⢸⡇⠀⠀⠀⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠅⡂⠀⠠⢀⣾⠀⢸⣅⠀⢀⡐⠁⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢢⡇⠀⡤⠂⠸⠀⢼⡗⠀⠝⠔⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢸⠌⠇⠀⠁⠀⠀⠀⣸⠁⠁⠎⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⣧⠀⠎⠀⠀⠀⣾⡆⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡇⢸⣧⡷⠤⠦⠠⠆⠀⠛⠂⠒⠂⢈⠥⢤⡄⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡂⢘⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⢠⣤⣤⣤⣤⣄⡄⠀⡴⠴⠶⠶⠾⣐⡾⠇⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⠁⣦⡀⠀⠀⢀⠀⣷⠀⠠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⢸⡆⠀⠀⠆⢸⠀⣿⠀⠀⠀⡐⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠈⡂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠇⠰⠃⠄⠘⠀⣿⣈⠀⢬⠂⡀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠨⡀⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢸⣤⠈⠀⢄⠀⠀⠀⣿⠴⠺⠣⠈⠁⠀⠀⢼⡗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⣿⡲⢀⣠⣴⡇⠀⣿⠁⡐⠀⠂⡆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠓⣂⡇⠀⣿⠀⠀⢀⡀⡇⠀⠀⣋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢰⢨⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⠼⠾⠦⣬⣀⠀⠀⡀⠀⠁⠀⠛⠳⠖⣀⠤⡇⠀⠀⢵⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠕⠄⠰⢸⠀⠀⡀⢸⠀⠀⠀⠀⡀⠤⡐⠪⢉⢁⣈⠀⠂⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⡀⠁⠰⢸⠀⠀⠂⢸⣀⣠⠴⠊⠀⠁⠀⠀⠀⠀⠈⣁⢀⡀⠀⠒⠐⠀⠀⠈⠉⠉⠒⣢⣄⣀⠀⠀⠠⢸⣽⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⢀⠈⡀⢸⣸⠀⢐⣩⡼⠿⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠄⠈⠉⠛⠲⢍⣸⣿⠠⣀⣀⠠⢤⣆
⠀⠀⠀⠀⠀⢈⣡⡤⠚⠛⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠈⠄⢐⡀
⠀⠀⠀⠴⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢀⡄⢸⠀⠀⣠⠤⢤⣄⠀⢸⡇⣤⠀⠐⣦⠀⠀⠀⠀⠀⢠⡖⠀⢠⡄⣿⠀⢀⣤⠤⢤⡀⠀⢸⠀⣄⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠁⢸⠀⣾⢡⡞⢶⠘⡇⢸⡇⢸⡇⠀⢹⡀⠀⠀⠀⠀⣼⠀⠀⣿⠀⣿⠀⡟⢰⣾⣦⢹⡆⢸⠀⢹⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡄⢸⠀⠹⢺⡇⢸⠸⠃⢸⡇⣸⠃⠀⣼⠀⠀⠀⠀⠀⢸⡄⠀⢻⡀⣿⠀⠳⢼⣿⣿⠺⠁⢸⠀⡾⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⠀⠀⢸⡇⠸⠛⢷⠾⣧⣤⡀⠈⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣶⣿⣤⣄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡾⠁⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠛⠛⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⢸⠀⠀⣀⢂⢂⡀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣿⣿⣿⢿⡽⡽⣿⣿⣿⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣸⠃⢸⠀⠀⠀⠆⠆⠀⠀⢸⡇⢹⡄⠀⢻⠀⠀⠀⠀⠀⢸⠃⠀⣼⠁⣿⣿⣿⣿⣽⣽⣿⣿⣿⠀⢻⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⡀⢸⠀⠀⣠⠶⢦⣄⠀⢸⡇⢸⠇⠀⣸⠀⠀⠀⠀⠀⢹⡀⠀⢿⠀⣿⣿⣿⡿⠿⢿⣿⣿⣿⠀⣸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠈⠁⢸⠀⢸⡇⠀⠀⣹⠀⢸⡇⠉⠀⠰⠏⠀⠀⠀⠀⠀⠈⠧⠀⠈⠁⣿⣿⣿⡁⠀⠀⣿⣿⣿⠀⠋⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣙⣒⣚⣁⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⡠⠊⠑⢄⡀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡐⠁⠀⠀⠀⠀⠈⢂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠑⢄⠀⢱⡄⠄⠤⠄⠠⢠⡎⢀⡠⠋⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠰⠈⠁⠆⢸⡀⢰⠈⠀⡇⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠂⠀⠂⣧⠀⠀⠁⠈⠁⠈⠰⠀⣴⠐⠆⠐⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡆⠀⡆⠈⢠⠃⠁⣀⡀⠈⠙⡄⠁⢐⠀⢰⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠃⡄⢫⠀⠀⠿⠿⠀⠀⡝⢀⠘⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣁⣀⡰⡀⢸⠾⣆⢀⠤⣀⢀⠰⣷⢏⡀⣅⢁⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡩⠄⠀⠄⠠⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠦⠬⢯⠗
⠀⠀⠀⠀⠀⠀⢀⡤⠤⢴⡎⠀⠀⣤⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠤⠀⠀⢸⡤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⡸⢠⣤⢾⡇⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⡆⠀⠘⡇⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠈⠀⠀⢸⠃⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡀⠀⠀⠀⠀⠀⠰⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠒⠓⠒⠒⠓⠒⠒⠒⠒⠒⠒⠛⠚⠛⠚⠓⠚⠒⠒⠒⠒⠒⠒⠚⠀⠀⠈⠃

⠀⢰⡶⠶⠶⠶⠶⢦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⠃⠀⠀⢰⠶⠶⠶⠶⠖⠒⠒⠒⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⠀⠀⣿
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡆
⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⣟⡽⠯⠭⠿⠿⠛⠳⠙⡇⠀⢸⣏⡭⠭⠭⠭⠽⠭⢍⣶⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⢰⣧⡄⠀⠀⠀⠀⠀⢸⣽⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡅⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠸⣿⡇⠀⠀⠀⠀⠀⢸⣼⠀⠀⢸⡀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣻⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⡇⡧⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⣿⠂⠀⠀⠀⠀⠀⢸⣾⠀⠀⠘⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣿⡆⠀⠐⣿⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢼⠀⠀⠀⠀⠀⠀⣿⡁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⡇⠀⠀⢀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢷⢸⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⣧⠆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠘⡇
⠀⢸⡇⠀⠀⠘⡇⢀⢤⡀⢸⣜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠘⡇⠀⠀⠀⡇⢼⠀⠇⠸⠯⠭⠿⠿⠷⠿⠟⠿⠆⠀⠀⠫⠿⠶⠶⠾⠶⠿⠿⠿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⡜⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠳⠤⠞⢀⣄⣀⢀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢤⣄⣤⣄⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⢀⠁⡆⢸⢹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠋⠛⠉⢻⢸⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠈⠒⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠈⡇⠀⠀⠸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠎⡄⠀⠀⡇⠀⠀⠀⡿
⠀⢸⡇⠀⠀⢐⡇⠀⠀⠀⠸⢾⠭⠭⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⠭⠭⠭⠭⢶⠇⠀⠀⣗⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢰⣖⣒⣒⣶⣶⣶⢶⣶⣶⣶⣾⣷⡾⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⡗⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠘⡇⠀⠀⢀⡇⠀⠀⠀⠀⡿⣿⠾⣿⣿⣿⣿⡿⠿⠾⠶⠿⠷⢾⣓⠺⠶⣖⠚⠛⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⣠⣧⣤⣤⡤⣷⠶⠦⡶⢦⣴⠴⠦⣤⣶⠶⣦⣶⣶⣶⣶⡶⢶⣶⡦⠶⠶⢶⠿⢿⠿⠗⠒⠧⠀⢤⡤⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠄⠒⠒⠒⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢰⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⡀⠡⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡆⠘⠀⢄⡀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡀⠀⢰⠂⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠄⠀⠥⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠀⠐⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣁⣒⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢣⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣥⣴⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠠⢴⣶⡖⠶⢿⡟⣦⠀⠀⢠⡞⣩⣇⠀⠀⠀⠀⢘⡿⡏⣙⣷⣤⣀
⠀⠀⠈⠉⠉⠙⢦⠀⠀⢹⠀⠀⠸⣏⣋⣽⡄⣀⡀⠀⣾⣄⡿⣟⣷⣦⣽⣦⡀
⠀⢻⠀⠀⠀⢠⠏⠀⠀⣼⢶⠦⣤⣈⣡⠟⠛⠛⠋⠉⣿⢿⣟⣿⣧⡙⠃⠀⡹⠓⣤⣄⣀⣀⣀⡠⠔⠟⠁
⠀⢸⣷⣦⣄⢻⠀⠪⢥⣿⣤⣄⣀⡈⠙⠲⣤⣀⠀⠀⢸⣿⠻⣿⡩⢛⣶⠶⢶⠾⠒⠒⠋⠉
⠀⠘⣟⣿⣟⠀⠑⢦⣤⣬⣥⣌⣉⣙⡟⣖⠷⠬⣷⣦⡈⠻⠾⠛⠛⠋⠀⢀⣿
⠀⠀⠀⠀⠈⢧⢀⡞⠀⠀⠀⠀⠀⡟⢹⠋⠙⠓⠶⢮⣭⣓⣦⣤⣤⣤⡴⢋⣟⡀
⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⠉⠙⠛⠓⢶⣞⣛⣿⣥⣤⠴
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠔⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛

⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⡤⢻⡛⢤⠁⣄⢠⠀⡄⣠⠀⠀⠀⢠⡀⠀⢀⡄⠀⠀⠀⣄⣠⠀⡄⢠⡀⠀⠠⠀⠀⢠⡄⠄⠠⢀⠤⣡⠌⣓⠳⡄⡇
⠀⠀⡝⢲⡎⠑⠊⠉⠀⠉⠉⠀⠁⠉⠈⠁⠁⠉⠁⠉⠈⠀⠁⠉⠈⠀⠉⠉⠈⠁⠉⠁⠈⠈⠁⠁⠈⠉⠁⠈⠉⠂⠉⣶⠘⡇
⠀⠀⠾⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠆
⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠁
⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡁
⠀⠀⠀⠊⡁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠂
⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⡀⠀⠉⠀⢄⠀⠀⠀⠀⢖⠂
⠀⠀⠀⠼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠐⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠳⠆⠀⠀⠀⠀⠀⠀⣂⠇
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⢂⠄⠂⠈⠠⠀⠄⠀⡄⢸⠘⠀⠀⠀⠈⡫⠀⠀⠀⠐⠄⠀⠀⠀⠠⠈⡄⠀⠀⠀⠀⠀⢂⡄
⠀⠀⠀⠄⡃⠀⠀⠀⠠⠠⠤⠨⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⢀⠀⢣⠀⠇⠀⠀⠀⠀⠀⡃
⠀⠀⠀⡺⠅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠂⠀⠀⠀⠀⠀⠍⠆
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠶⠁⠀⠀⠀⠀⠀⡀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠁⠂⠄⠀⠀⠠⠄⠢⠀⠀⠢⠀⠀⠀⠀⣁⡄
⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠈⠶⠀⠈⠳⢄⢀⢀⣀⣀⡀⡧⠀⠀⠀⠰⠁⠂⠀⠀⠀⢄⠀⠀⠀⠀⡂⠀⠀⡆⠀⠀⠀⠃
⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⠉⠋⡍⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡖⠁
⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠒⠅⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡅⠃
⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢴⡀⡁⠀⠀⠀⣲⠔⢀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠏⡁
⠀⠀⠀⠶⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠀⠒⠀⠀⠐⠀⠀⠀⠀⠈⠙⠄⠀⠀⠐⠁⡒⠀⠀⠐⠊⠂⠄⠀⠀⠀⠀⣃⠆
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂
⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡄⠀⠀⡀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢂⡄
⠀⠀⠀⠂⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠂⠀⠀⠠⠀⠀⠘⠀⠀⠀⠄⠈⡆⠀⠀⠀⠀⠀⡧⠄
⠀⠀⠀⡛⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠕⡂
⠀⠀⠀⠄⡃⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄
⠀⠀⠀⠚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡂
⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⢐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠄
⠀⠀⡻⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⡃
⠀⠀⠷⡊⢣⡄⢐⢤⠤⢠⠄⠤⠤⡄⠄⠤⢤⡠⠤⠄⡄⠠⢤⠤⠠⠠⠄⠤⠀⠄⠤⠤⠄⠄⠤⡤⢤⢠⠠⠤⣤⠀⣤⠛⡲⡇
⠀⠀⠣⠭⠾⠶⠩⠌⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠈⠀⠉⠈⠀⠀⠀⠈⠀⠁⠀⠁⠀⠈⠬⠱⠾⠾⠅⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠗⣡⠄⠰⠏⠐⠎⠀⢦⠔⠢⠀⠉⣠⡾⠐⠆⠀⠠⠉⠂⠀⠀⠀⠐⠒⠂⠐⠂⠀⠂⠐⠪⠄⠂⢠⠆⢢⡀⠀⠀⠠⠠⣠⠁⠠⠆⠐⠭⢦⣀⠾⠇
⠀⢀⠀⠀⠀⡓⠸⠊⠈⠀⠀⡀⢋⢉⠈⠁⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠉⠘⠹⠆⠀⠀⠆⠐⢶⢒⣂⠈⠆
⠀⠀⠐⣊⠤⠄⣃⠶⣤⢄⡺⠠⠈⡀⠖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠃⠄⢖⡀⣄⡴⢜⡈⠡⣰⠒⠆
⠀⠀⠁⢀⣼⣀⠄⢠⠐⠋⠤⢠⠉⠀⢠⠄⡄⠀⡔⠂⠀⠠⢤⠀⠀⠀⠀⠁⠐⢳⣸⡃⠀⠀⠈⠀⡤⢠⠀⡀⠀⣀⠀⠫⡉⠁⠙⠎⡀⠩⠀⣶⡉⠴⠄
⠀⢘⣍⠀⢄⢒⡀⢠⠄⡀⢀⠊⠀⠀⠸⢤⠥⡤⠿⠤⠀⠴⣬⠄⠀⠀⠈⠀⠀⠉⠡⠉⢢⠀⠀⠠⠖⠮⣸⣹⣂⣤⣅⠀⠹⡀⢀⡠⣄⠐⠛⢦⢄⣡⠆
⠀⠈⠉⠨⡈⠀⠁⠈⠁⠐⡟⠀⠀⣮⠴⠼⠨⢧⡀⡢⠀⢀⠐⢠⠰⠀⠀⠠⠖⣄⠀⣀⠶⡀⢐⡂⡄⢀⡥⡄⢀⡄⠀⢀⠀⠳⠊⠈⠁⠉⢀⢁⡒⠛⠃
⠀⢰⢰⠐⠃⢈⠒⠋⠁⢐⡆⠀⢠⡇⡀⠃⢠⠼⢳⠀⢦⠈⠀⠚⠀⡀⠀⠀⠀⠀⠱⠈⠄⠂⠠⡧⠷⢳⣦⢷⠣⢡⢅⡆⠀⢨⣀⠐⠓⠦⠀⡘⠪⡀⡄
⠀⠈⠀⠀⢆⡄⠙⠳⠀⠀⠇⠀⠋⠙⠙⡈⠩⢆⢠⡰⡆⠶⠿⠞⠂⠃⠀⢐⠃⠀⠀⢠⡀⠄⠨⠩⢀⠀⠄⠀⠀⡆⠀⡔⠀⢘⠁⠰⠔⠔⠤⠐⠄⠣⠃
⠀⠠⠔⠐⠰⠠⢦⢰⠄⠐⠲⠀⢰⠀⣠⠁⠁⠀⠈⠃⠡⠀⠀⠀⢀⠤⠐⠺⠀⠀⢀⠐⢃⡀⢂⠀⠶⠂⣠⠈⢀⠃⠀⠀⠀⠜⢀⢠⣄⣀⠈⣬⠔⣆⡀
⠀⠘⡀⠀⣉⠤⡍⣦⠂⠢⢄⠧⠀⠉⠤⠀⠢⢀⡅⠙⢻⠀⠀⠀⠖⠲⠠⠀⠀⠀⡄⠀⠘⠁⠀⠀⠐⠀⠙⠐⢣⠀⠀⠀⣴⣐⠄⠉⡽⠒⣐⢋⠈⡜⠆
⠀⠀⠀⢄⠋⢀⣤⠈⣜⠌⠅⠀⠥⢄⠀⠀⠀⠄⠄⠶⢏⡄⠤⠂⠀⡠⠀⠀⡀⠌⠀⠀⠘⠀⠀⡈⠀⠀⠀⠀⠄⠀⢀⢤⠤⣜⠠⢃⠁⣈⠀⠛⣂⠨⠁
⠀⠄⠉⠈⡠⠘⡘⠃⠈⠀⠋⡌⠲⠠⠐⠄⠀⠀⠀⠀⠠⠧⠄⡀⠖⠀⠀⠘⡃⢠⢄⡀⡸⠥⡄⠀⠀⠀⠉⠀⠠⠆⡄⡤⠂⠒⠀⠉⠓⢫⢉⠈⠱⡢
⠀⠀⠀⡀⠀⢊⠙⠀⡐⠀⣀⠈⠥⠂⠀⣤⠀⠒⠀⡀⠀⠁⠀⠀⠐⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⢀⡀⠀⡰⠉⠀⠀⡄⡐⠇⠲⠀⣀⠅
⠀⠈⠙⠂⠈⠈⠀⠐⠁⠨⠁⠉⠁⠐⠂⠉⠂⠀⠀⠀⠀⠕⠂⠄⠐⠀⠀⠀⠀⠀⠀⠀⠦⠐⠤⠁⠂⠘⠣⠵⠉⠤⠀⠐⠀⢙⠀⡘⠀⠘⠀⠄⠩⠾⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠿⡶⠂⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀
⠀⠀⣤⡀⠀⠀⠀⠀⠀⢰⣆⣀⣀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠐⠈⠁⡀⠀⢸⠟⢀⣀⡀
⠀⠸⢿⠇⠉⠉⠈⠀⢠⠀⠉⠉⠁⠈⣟⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣷⢬⠃⠘⠿⠿⠆
⠀⠀⠈⡀⢀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠘⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣤⣀⣀⣚⣿⣫⣿⠤⣂⣠⣤⣤⣿⣴
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠙⠦⠀⠀⠈⠑⠆⠀⠀⠠⠴⠿⠟⠛⠋⠉⠉⠱⠿⠿⠟⠉⠠⠾⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠄
⠀⠀⠀⢸⠇⠀⠀⠄⠂⣿⠆⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⡤⠿⢿⡿⠆
⠀⢠⣿⡄⠈⠉⠙⠛⠂⢏⠙⠿⡿⠗⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡒⠒⡠⠄⣾⣴⣿⣦⣀⣠⣤⣤⠤⢤⢤⠶⠶⡶⠶⣤
⠀⢠⣼⣴⣤⣤⣤⣤⣶⣶⣶⣤⣧⣘⣛⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣿⣿⣿⣿⢿⢿⣷⡤⡤⣶⣾⣛⣟⣛⠿⠛
⠀⢸⣿⡿⠿⠿⠻⠫⠻⠻⠝⠝⠝⠝⠭⠯⠫⠻⠻⠳⠶⣤⡀⠀⠀⠸⠿⠿⠟⠟⠝⠝⠵⠱⠣⠫⠎⠎⠜⠔⠵⠱⠫⠫⠟⠗⠟

⠀⣠⡄⠀⠀⠀⠀⢀⡀⠒⠂⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠀⠁⠤⡀
⢠⣿⣿⠌⠁⠁⠀⠄⠀⠀⢰⣿⣿⠀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢢⣶⣿⣷⣄⠀⠀⠘⣧⣤⣄
⠀⣹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣏⢉⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠉⠻⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠘⠻⠻⠻⠻⠻⣿⣿⠉⠉⠉⠉⠸⠿⠷⠀⠤⠤⣤⡶⠶⢖⢶⠆⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠓⠁⠁⠀⠀⠀⠈⠉⠈⠈⠈⠈⠀⠀⠀⠁⠁⠁⠁⠈⠈⠈⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⣠⣾⣶⣄⣀⣀⣤⣾⠛⣧⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣠⣄⣀⠀⠀⠇⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⡇⠀⠀⡀⠀⢀⣿⣿⣆⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠿⠇⣻⣿⣿⣿⣿⡤⢤⡿
⠀⠈⠉⢻⡏⢳⣾⣾⣿⣶⣿⣶⡇⣴⣾⣿⣆⠘⠻⡟⠟⠀⠀⠀⠀⡉⠉⠁⠩⠌⠉⠉⣉⣉⠉⣙⣿⣶⠾⣿⢿⣿⣿⣿⣶⣶⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣩⣿⣿⣿⣿⣦⣶⣷⣤⣦⡀⠀⠀⣟⣢⢤⢾⡻⡻⡝⢝⢯⢫⡻⡹⡝⢝⢿⢻⡳⡿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠈⠀⠁⠁⠉⠈⠈⠀⠁⠁⠉⠈⠈⠀⠁⠁⠉⠈⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡞⠉⠉⠉⡟⠉⠉⠙⡆⠀⠀⡇⠀⢰⠖⠒⠒⢲⠒⠒⠒⢲⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡟⠉⠉⠉⡏⠉⠉⠉⡇⠀⠀⡇⠀⢸⠒⠒⠒⢺⠒⠒⠒⢺⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⣀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠛⢻⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣧⣤⣤⣤⣧⣤⣤⣤⡇⠀⠀⡇⣶⣼⣀⣀⣀⣸⣀⣀⣀⣸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠁⠀⠀⢸⠁⠀⠀⢹⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡗⠒⠒⠒⡗⠒⠒⠒⡇⠀⠀⡇⠀⢸⠦⠤⠤⢼⠤⠤⠤⢼⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠧⠤⠤⠤⠧⠤⠤⠤⠇⠀⠀⡇⠀⢸⣄⣀⣀⣸⣀⣀⣀⣸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣤⣤⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢠⣾⡿⠁⠀⣠⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣄⠀⠈⢿⣷⡄⠀⠘⣿⣧
⠀⠀⠀⠀⠀⣼⣿⠃⠀⢠⣿⡟⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⢻⣿⡄⠀⠘⣿⣇
⠀⠀⠀⠀⢠⣿⡏⠀⢀⣿⡿⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⢿⣿⡀⠀⢹⣿⡄
⠀⠀⠀⠀⣸⣿⠁⠀⢸⣿⠃⠀⢸⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⣿⡇⠀⠘⣿⡇⠀⠈⣿⣇
⠀⠀⠀⠀⣿⣿⠀⠀⣾⣿⠀⠀⣿⣿⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿⣿⠀⠀⣿⣷⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣿⡿⠀⠀⣿⣿
⠀⠀⠀⠀⢻⣿⡀⠀⢸⣿⡄⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢠⣿⡇⠀⢀⣿⡏
⠀⠀⠀⠀⠘⣿⣇⠀⠈⣿⣧⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⣼⣿⠁⠀⣸⣿⠃
⠀⠀⠀⠀⠀⢻⣿⡄⠀⠘⣿⣧⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⣼⣿⠃⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠘⣿⣧⡀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⣼⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠈⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣧⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉⠉⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢿⣶⡄
⠀⠀⠀⠀⠀⠠⠄⣼⣿⠁⠀⠹⣿⣧⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⠀⢰⣿⡇⠀⠀⢹⣿⣴⠀⠐⢄
⠀⠀⠀⠌⠀⠀⠀⠹⡟⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠸⠧⠀⠀⡀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠘⢿⡁⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⡠⠁⠀⠀⠘⡋⠁⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠁⠀⠐⠐⠆⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠁⠄⠀⠀⠀⠂
⠀⠀⠁⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠎⠇⡴⠰⠀⠠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠔⠁⠀⠈⡀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢠⢀⡀⢆⠀⠀⠀⠀⠀⠀⠀⢂⡀⠂⠀⠠⠀⠈⢂⠀⠀⠀⢈⠔⠀⠀⡘⠄
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⢀⢦⠀⡘⠁⠀⠀⠀⠑⠀⠀⠐⠀⠐⠀⠂⠀⠂⠀⠂⠀⢔⠀⠀⠁⡘⠀⡀⠀⠀⢀
⠀⣤⡇⠀⣤⠀⠀⠀⠑⠀⠐⠁⠀⠱⠈⠢⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠊⠀⠁⠠⢠⠇⠀⠀⠀⠀⢸⣤⡀⠀⣤⡄
⠀⣿⡞⠀⣏⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠁⠀⠈⢙⡇⠀⣿⡇
⠀⣿⡁⢸⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠧⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠄⠀⠀⡇⡇⣽⡇
⠀⣿⡇⣸⣿⠀⣶⣄⢀⡀⠀⠀⠀⠀⠐⠨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣇⢷⣾⡇
⠀⣿⣿⣿⣿⠀⣿⣿⣿⢃⠀⡐⠁⢡⣤⠀⠹⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠐⢀⡸⣤⠉⠈⠜⠀⢰⣾⣿⠀⣿⣿⣿⡇
⠀⣿⣿⣿⣿⠠⣿⣿⣿⡄⢴⣦⣄⠀⠀⠁⠂⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⢰⠀⠀⠀⣠⣾⡆⢸⣿⣿⢀⣿⣿⣿⡇
⠀⣿⣿⣿⣷⣶⣿⣿⣿⣶⣾⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣷⣾⣿⣿⣶⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢊⡈⠁⢨⡉⠉⠉⢠⡈⠁⢁⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⡇⠀⣸⡀⠀⠀⣸⡀⠀⣸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠏⠉⢻⠋⠉⠉⢻⠋⠉⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⡆⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⡗⠒⢻⠗⠒⠒⢻⠗⠒⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢘⡇⠦⢾⡆⠶⠤⢾⠦⠤⢾⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⣇⣀⣸⣀⣀⣀⣸⣀⣀⣸⣷⣶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⣦⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡄⢴⣲⡉⠁⡶⢹⣇⢿⣿⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⢷⣦⣀⡀⠀⠀⠄⠀⠨⠏⠛⡟⠹⡈⠹⣶⠁⠀⣻⣼⣾⣷⣞⣻⠃⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⡿⢅⠍⠿⢻⣿⣝⠳⢶⣄⠐⢄⡈⠟⠻⠵⠟⠶⣭⣷⣤⣘⣿⣿⢛⣅⣴⠀⠄⣷⣍⢹⣿⠂⠀⢀⣠⣠⡄
⣾⠶⣶⣦⣄⡀⠀⠀⠀⠀⠻⣷⠀⢸⣿⣿⡇⢠⠾⣿⣿⣶⣬⣵⣤⡠⡤⠴⠔⣯⣈⠿⣿⣦⣽⣿⠋⠙⠛⣧⣤⣶⣻⠛⣻⣧⠢⠴⠚⠉⠀⡀
⣿⣾⣿⣿⣿⣶⠀⠀⢀⠀⡞⡿⠀⠸⠿⠿⠇⠸⡿⣿⣿⢻⡟⠛⢿⣇⠀⠀⠀⣀⡀⡄⠀⣈⣿⣿⣻⣶⣶⣼⣿⢿⣟⢉⣟⣿⣤⣤⣴⠟⠑⠢
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⡷⡟⠀⠐⣁⣉⠀⠙⠀⣿⣿⣿⣿⣿⣶⣿⣿⣷⠦⣼⣷⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣷⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣰⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠀⣲⣿⣽⣿⣿⣿⣿⣿⠩⣿⣿⣿⣿⣿⣿⣿⠽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠛⠛⠛⠛⠛⠛⠙⠙⠛⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠉⠉⠉⠻⠟⢿⡿⠟⣻⣿⣿⣿⣿⣛⣛⠛⢛⡛⠛⠿⠿⠟⠛⠛
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣾⣿⣿⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷
⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⡿⠟⠿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣸⢿⣿⣿⡅⠀⠀⠀⣴⣿⣯⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣟⣿⣿⣿⣿⣑⠆⠀⠀⣨⣿⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠉⠛⠠⠀⠠⠄⠴⠿⠿⠿⠿⣿⡿⢿⣷⡄⠠⠶⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⢾⣿⡿⢉⠋⠩⠭⠩⠿⠉⠉⠛⠛⢿⣛⣿⣿⠛⠛⢻⣿⣿⡿⢿⡟⢛⣿⠛⡉⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⡄⢀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠉⠙⢇⠒⢠⡤⠀⠛⠶⡾⢿⡿⠻⠟⡃⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠈⠁⠀⠀⠁⠀⠉⠐⠀⠀⠀⠈⠿⠿⡿⢿⡿⠿⢿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠴⣤⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠒⠂⠐⠛⠃⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⢸⣇
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣦⡀⠀⣀⣀⣀⡘⠀⠛⠟⢿⣄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⣷⣯⣿⣿⠁⣀⣁⡴⣾⣿⣧⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⠛⣹⣿⣼⣿⣿⣿⣶⣿⣯⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠐⠒⠒⠒⠒⠚⠒⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠹⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⢀⢀
⠄⠀⠀⠤⠀⢄⣀⠈⢡⠀⠀⠀⢠⠀⠀⢸⣯⣿⣿⣇⢸⠀⢹⣿⠀⠇⢻⡿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⠿
⢀⡀⣠⣘⣿⡻⣿⠻⣽⡏⠁⣼⠟⣡⣴⣿⣿⣿⣿⡁⠀⡇⢸⡏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⠐⠈⠁⢀⡀⠶⠚⠛⠉⠀⠀⠀⣴
⣏⡉⣿⣿⣿⠅⢈⠷⠚⠲⠀⣤⣾⣿⣿⣿⣿⣿⣿⣇⣀⣿⡾⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠂⢁⣴⣶⣿⠿⣿⣶⠂⠨⠿
⣟⡊⠿⢿⣧⣤⠀⣄⠗⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡧⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⠴⠟⠛⠉⠁⠐⣀⣀⣠⡄⢒
⣿⣧⠄⠈⠛⣿⠆⡍⠹⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣧⢨⣷⢈⣿⣿⣿⣷⣿⣿⣿⠯⠀⠀⠠⠤⠐⠀⠀⠀⠀⠀⣀⠴⡛
⡟⠁⠀⠀⠀⠹⠗⢹⠇⣴⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⠙⣿⡀⢻⣽⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠀⠀⠠⣽⣤⣤⣛⣻⣾
⡴⡗⣱⠄⠀⠀⠂⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠂⠀⠹⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡒⢤⣤⠤⠀⣤⣤⠄⣈⣿⣿
⣿⣦⠈⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠘⠹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣼⣾⣭⣀⣨⡭⢽
⣿⡁⠀⠀⠀⠀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⠠⠀⠀⠀⢀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡆⢠⠀⠄⢠⣈⢅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠐⠤⠄⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⠓⣋⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⣀⣉⡛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⢀⠒⠂⠀⠀⢈⣉⣀⣤⣶⣶⣌⣛⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⣠⠤⠤⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⢸⢡⠊⠱⢈⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠠⠤⠤⠤⠾⠤⠤⠤⠤⠤⠤⢤⡀⠀⡞⢹
⠀⠀⠀⠀⠀⡇⠈⢻⠀⠀⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠇⠀⡇⢸
⠀⠀⠀⠀⠀⠑⠒⢸⠀⠀⠅⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⠂⠀⠀⠀⠀⠀⠀⠙⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠙⣦⠤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢠⠒⢺⠀⠀⡇⠀⡇⠀⢸⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠉⠉⡇
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠁⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠤⢤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢡⣶⣶⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢺⣿⣿⠎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠆⢸⣿⣿⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⡀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣯⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣻⡿⣻⢿⣿⣝⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣿⣻⣿⣿⣿⣿⣿⣿⠿⠻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠋⠁⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣠⣤⣴⣾⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠩⠇⢼⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣉⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⡀⢀⢀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣀⣁⣀⣉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠈⠲⠀⢠⣟⡀⠭⠭⠤⠤⠤⠌⣉⡉⠻⠄⢀⡠⠊⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃⣀⠈⡟⢋⠘⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠠⠤⠤⣴⠀⠀⣭⣦⠀⢠⠀⡆⠀⢰⣆⢄⣙⠂⢨⡀⢤⠤⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢰⡆⡇⡆⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠀⢘⠀⠀⣿⣿⡇⣸⠀⣁⠀⣸⡇⣸⣿⠀⠈⠀⢠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⡇⣇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠌⡇⢸⠀⢠⡨⡅⠤⡄⠦⠀⠤⣤⣧⢠⣅⡀⠀⠀⠨⡄⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣓⣈⠉⠀⡤⠉⣉⣻
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠦⠴⠷⠾⠦⡾⣶⠦⢴⢦⣖⡑⣤⣤⠴⠤⣬⣥⠴⠦⠴⠤⠶⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢉⢷⣆⣠⢰⡟⠛⣇
⠀⠀⠀⠠⠤⠤⠤⣦⠀⠀⠚⠃⢠⠆⠕⠂⠒⠝⠠⠊⠀⠀⠀⠑⡒⠨⠒⠚⡐⣄⠀⠒⠀⠀⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⠥⢾⠽⢻⠞⡶⢧⣼⣄
⠀⠀⢐⡃⠀⠀⢐⡦⠤⠤⠤⠤⢼⡟⠒⠒⠒⠻⡶⠒⠒⠒⠒⠒⡗⠗⠒⠒⠻⣯⣈⣉⣄⣉⢹⡆⠀⢺⡃⠀⡀⠀⠀⠀⠀⠀⠀⠠⡇⢠⡔⠘⠀⢸⠀⡟⡄⢉⡇
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠄⠀⠀⠀⠀⠘⠉⠉⠉⠁⡠⠳⡀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢸⠀⡇⡇⡎⡇
⠀⠀⠀⠀⠀⠶⠀⠀⠀⠐⢸⠆⢸⡇⠀⢻⠀⠀⡇⡭⠀⠀⢸⠀⠃⠀⢸⠀⢘⡇⢰⠂⡆⠀⠀⠀⠀⠀⡀⡏⣂⡁⠀⠀⠀⠀⠀⠀⡇⢸⡇⣠⠀⢸⠀⣇⡇⣗⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⢸⠅⢈⡃⠀⠾⠀⢀⡇⠳⠀⠀⠸⠀⢐⠀⠸⠀⠀⡇⠸⠄⠇⠀⠀⠀⠀⠀⡥⠗⠚⠲⢄⣀⠀⠀⠀⣠⣇⢸⠀⢀⣀⣸⣀⡏⢁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢀⣀⠀⠠⡆⡀⠄⠠⢄⠀⠠⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠤⡤⠤⠽⠶⡄⠃⠀⢒⠊⠉⠛⠛⠛⢻⣛⡟⠿
⠀⠀⠀⢀⢢⢠⠂⠀⢰⠁⠎⡄⢠⡆⣧⠠⠃⢰⣷⠀⠋⠙⠈⠀⠀⣴⠐⠁⡀⡇⠸⠀⠇⠀⠀⡀⠀⠀⡇⠀⢄⠀⡇⢠⡄⠀⡇⠛⡔⢣⡖⡇⢸⢒⠃⢺⠋⡧⡆
⠀⠀⠀⠸⣠⣸⠀⠀⠸⣤⣤⠇⢸⡇⠯⣴⠄⠰⣿⠀⠀⠀⠀⠣⠸⠹⠚⠣⠥⡇⢠⣓⡠⠀⠀⡇⠀⠀⡇⠀⠃⠀⠏⠈⠃⠀⣷⠀⡇⠸⠇⠇⠸⢸⠀⢸⠀⡷
⠀⠀⠀⠁⠉⠉⠑⠓⠀⠀⠒⠒⠚⠛⠂⠀⠀⠚⠛⠚⠂⠐⠓⠛⠛⠓⠒⠒⠚⠛⠒⠒⠒⠒⠒⠛⠓⠒⠛⠒⠒⠒⠛⠓⠒⠐⠉⠛⠛⠛⠛⠓⠚⠚⠓⠛⠃⠋⠁



⠀⠀⢀
⣿⡟⣉⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢶⡰⠀⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⡄⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠂⢳⣀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢠⠀⢶⣤⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡹⠿⠣⠀⠈⠀⠀⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠘⢉⠀⠀⠈⠠⠄⠘⠉⠀⠀⠀⠀⠀⠀⠀⢠⠤⢼⡀⣿⠀⠀⠂⠀⠀⠀⠐⢀⡀⠀⠀⠐⣂⣼⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠐⠀⠀⠀⠰⠆⠀⠀⠀⠈⢦⡼⢿⠛⠙⢆⠀⠸⠃⠀⢠⡟⣩⣆⡼⠛⠉⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢀⡞⠉⠉⠀⠀⠀⠀⠀⣿⣾⣾⠝⠀⠀⠀⠀⠀⠉⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⠿⢿⢃⠡⢄⠀⠀⠄⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠘⠄⠄⠆⠆⠇⠀⠐⠀⠃⠁⠀⠁⠉⠂⠒⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠘⣤⡀⠀⢀⡀⢀⣠⡤⢠⠀⠠⠤⠦⠤⠄⠤⠀⠀⠀⠢⠂⠆⠤⠖⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⣄⡀⠶⢀⠄⡀⠀⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢴⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠋⠉⠀⠙⢾⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⣟⠓⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠂⣔⠈⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠈⣁⡰⠆⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢈⠉⠑⠈⠙⠈⠀⣫⣿⣿⡿⠒⠒⠀⠀⢠⣤⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣷⣏⠓⠀⠣⠄⢀⡄⠀⠀⠀⠀⣠⡄⠠⠀⡀⢠⠎⠀⠀⠤⡆⢀⣀⠀⠀⠀⡁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠆⠉⠀⣦⣠⡤⠀⢈⡀⠁⠀⠀⠀⠀⠀⠱⢀⠁⢰⠀⠀⠀⡀⠀⠀⠉⠛⠛⠚⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠒⠛⠏⠐⡰⠊⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠⠀⠀⠁⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⣀⠄⠀⡀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠤⠀⠀⠘⠀⠈⠁⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⡀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⠠⣀⢀⡠⡄⠀⠀⠀⠀⠀⢀⠀⠀⡀⠐⡀⠀⢸⠀⠀⡐⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡤⣀⢀⡀⢀⡀
⠀⠀⢸⠈⢌⡏⢲⢤⡨⠚⢥⠓⠉⠀⠄⠈⢸⣇⢠⠈⢠⣀⢿⠈⠀⠄⠈⠑⢣⠝⠪⣀⢴⠊⣏⠌⢸⠀⠁
⠀⠈⠠⠲⢭⠣⡼⣃⡹⢧⡬⣌⣀⡠⠐⡀⣬⠚⠁⠀⠀⢙⢪⡄⡐⠠⣀⣈⡬⢤⠿⣉⡻⡤⢫⠵⠂
⣀⣀⣁⣀⣁⣈⣉⣛⣘⣂⣒⣒⣈⣛⣁⣀⣀⣀⣰⣦⣶⣀⣀⣀⣀⣛⣋⣐⣒⣂⣚⣘⣋⣉⣀⣁⣀⣁⣀
⣿⣇⣿⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠁⠀⠀⠀⠀⠀⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⡤⠶⠶⢦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠏⠀⠀⠀⠀⠙⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⢀⡤⢤⣄⢀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠃⣿⠀⠀⣿⠘⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⣿⠀⠀⣿⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⣿⠀⠀⣿⠛⠳⣮⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⠀⠀⣼⠀⠀⢸⠁⠉⢳⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠉⠻⣦⡀⣿⠀⠀⠿⠀⠀⠘⠀⠀⢸⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠖⢒⣒⡚⠉⣩⠭⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⡉⠀⠀⢐⠆⠈⠉⠉⠀⠀⠀⠀⠀⠙⠲⢄⠊⢧
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠐⡀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠰⣾⡀⢀⡠⢴⣄⣌⠀⠀⠀⠀⠀⠀⠀⡴⠒⠢⠀⠀⠀⠀⠀⠢⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣷⢪⠊⠁⣨⣆⠀⠀⠀⠀⠀⢀⡠⠜⠀⠀⠉⠉⡀⠀⠀⠀⠡⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢀⠪⠔⠉⢂⠀⠀⠀⠀⠀⠂⢀⠠⠀⠀⢖⠁⠀⠀⠀⠀⠐⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢆⠀⠀⠀⠀⠀⠐⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⠀⠀⣡⡀⠀⠀⠀⠀⠀⠀⠑⠒⠈⠀⠀⠀⠀⠀⠀⠀⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⢊⢅⡽⡔⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡱⡫⠊⠁⢈⢄⣈⡁⣐⠆⠠⠤⠤⠤⠤⠤⠀⠰⣍⠀⠘⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡁⣠⠀⢴⡷⠒⠐⠒⠒⢀⣽⣿⣿⣶⣒⣂⣠⣤⡥⠍⠩⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠜⣼⣭⣩⣭⣱⡶⠒⢠⡤⠤⠄⢀⣀⣤⣀⣶⣀⣛⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡭⠬⠭⠷⠶⠖⠚⠛⠛⠊⠉⠉⠉⠉⠉




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⡄⠰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⢿⢀⠰⣟⣾⣴⣤⡄⢀
⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣰⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣆⠀⠀⠀⣨⣷⣿⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢦⣆⣀⣀⣀⡀⠀⠀⣴⠀⠀⢿⣿⣿⣿⣿⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣥⣽⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣛⣿⣠⣤⣠⣼⣷⣤⣄⣠⣬⣧⣼⣿⣿⣽⣯⣄⣤⣤⣿⢶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣽⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠋⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣼⣫⣿⣻⣭⣸⢻⣿⣿⣿⣸⣬⠉⠁⣿⣿⣿⠃⠀⠀⠀⣶⡄⠀⠀⠀⠴⡄⠀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠟⣹⣟⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣷⡀⠈⠉⠉⠉⠁⢀⢶⡀⠀⠸⣿⣿⣿⣿⣿⣿⢿⣿⣿⡏⣼⣿⣿⣿⣿⡿⡿⣿⠿⠿⠿⠟⠛⠛⣙⣛
⣧⣤⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⣿⣿⣿⣿⡧⠀⠀⣀⡀⢀⣙⠈⢀⠀⠀⠉⣿⢫⣿⣟⣿⡑⣿⣯⣷⣿⣿⣿⣿⣿⡷⠘⣿⡏⠁⠀⠀⢠⣼⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⢤⣤⣘⠛⢀⡀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠿⣇⠀⢀⣼⣿⣿⣿⣿
⢀⣬⣿⣿⣦⣺⣒⢲⣧⣼⣿⣿⣿⣿⣿⣿⣿⣟⣹⡿⢘⣠⣤⣼⣿⣛⠛⠓⠂⠘⠻⠟⢇⣤⣿⣿⣿⣿⣯⣿⣿⣽⣿⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣾⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⠟⢡⣭⣥⣤⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠛⢩⣤⠟⣶⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣥⣴⣧⣟⠛⢛⣏⡿⣿⠋⣲⠙⠃⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣠⡌⠛⠗⣺⣿⣿⣿⣄⣼⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿
⢀⣡⣶⠇⠁⣀⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿
⣾⣿⣷⣖⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣌⣤⣤⣤⣤⣴⣿⣿
⣿⣿⡟⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⣰⣾⣟⣂⠀⠀⠀⢀⠀⣀⢀
⠈⠉⠀⡀⣁⣈⣤⣤⣤⠀⠄⠉⠉⠁⠀⠘⠓⡀⠐⣄⠉⠈⠉⠉⠀⢸⣿⠟⠝⠀⢀⠠⠴⡀⠖⡀⠤⣈⠒⡀⠀⠀⠒
⠀⠐⠊⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠄⠈⠀⠀⠘⠀⣾⣿⣳⣿⡶⠀⠀⠁⣬⡦⡋⢀⢜⠚⠐⠐⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠁⠀⢄⠀⠀⠠⠀⠀⢀⠀⢸⣿⠛⠙⣿⣿⣯⠄⠙⢧⠖⠉⠙⠁⠀⢠⣄⡀⠤⢹⣿⠳
⡀⣤⢤⡀⠀⠀⡐⠣⣀⠀⠀⡇⡀⠀⠈⠁⠀⠠⢀⣠⣿⣦⣸⡗⠀⠀⠀⠀⠠⠀⠄⡟⢂⠴⣤⠦⠀⢤⠛⡕⠀⣸⠉⠳⠏
⠀⢄⣄⡘⢷⡮⠇⠐⠒⠈⠑⢿⠡⠀⠀⣠⣼⣿⠮⣿⣽⣠⢟⠰⢀⠀⠀⢀⣀⠥⠈⡲⠛⠀⠧⠠⠀⡀⠀⠀⠈⠀⠀⠀⠠
⡀⣿⣄⣀⠸⣫⢀⣀⣹⣶⣀⣸⣇⣸⣇⣈⡰⢧⣀⠹⠀⠐⠁⠑⠾⠖⢷⣁⣁⣐⣠⣋⣀⣀⣀⣀⣨⣀⣬⣴⣤⣤⣤⣄⣤
⠈⠉⠛⠛⠛⠛⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⢋⣀⡁⠠⠒⠉⢉⡀⠅⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⢚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⢶⣶⣶⢶⡶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣤⣴⣿⣯⠀⠤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣰⠶⣦⡄⠀⣴⣦⠀⣠⣶⣦⠀⣠⣶⠶⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⣶⣠⠴⠀⠀⠀⢸⣿⡇⢀⣴⣟⠀⣿⡇⢰⣿⡟⣴⠛⣿⠏⣼⣿⠃⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠛⠘⠃⠀⠀⠀⣾⣿⠃⢸⡟⠙⢶⣿⠧⣾⣿⡟⠁⠀⠈⢰⣿⠇⠀⣰⣿⡏⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠸⣷⣤⠿⠃⠰⣿⡟⠀⠀⠀⠀⢸⣿⣤⡾⢻⣿⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⢓⣾⡿⢀⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣉⣉⣀⣀⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠁⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠀⠀⣤⣤⣤⡤⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⠘⠃⠸⠃⠀⠛⠘⠘⠃⠹⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣆⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠾⠟⢛⣿⡟⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠷⠟⢻⣿⡟⣹⣿⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣿⣆⠀⠀⣸⡿⠀⣸⣿⢠⢾⡿⢢⣾⢋⡷⢠⣿⢢⢾⣿⠂⢀⣴⡟⠉⡇⠀⣾⡿⢀⣿⠏⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⢻⣿⡀⢠⣿⠃⢠⣿⣿⠃⠈⠁⣿⡟⠛⢁⣿⡿⠁⣿⠏⢀⣾⡟⠀⣼⡿⢰⣿⠇⢸⣿⠚⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣾⡿⠁⢸⣏⡴⢻⡿⠁⠀⠀⠀⢿⣤⡴⢺⣿⠁⠸⣿⡤⠛⣿⣁⢼⣿⠇⣾⡟⠀⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠁⣼⡿⠀⣿⡇⢀⡍⠀⠀⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠈⠓⠛⠁⠀⠀⠈⠳⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠤⣄⡤⢿⠀⢠⡤⡤⣤⢤⡀⡄⠀⠀⠀⢠⡄⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠘⠰⠋⠓⠚⠀⠘⠃⠃⠛⠈⠷⠀⠀⠀⠀⠸⣆⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠙⣦⠀⢠⣶⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⠀⣾⣿⠁⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠉⠁⢸⣿⠃⣠⣤⡀⢀⣬⡁⠀⣠⣤⢄⠀⣾⡿⠀⠀⣤⣤⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠈⣿⣷⡀⠀⣿⣏⡜⣿⡿⠁⣸⣿⠁⢰⣿⣡⠞⢰⣿⠇⢀⣾⡏⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⣿⡇⢸⣿⡟⢠⣿⠃⣠⣿⠇⣠⣿⡏⠀⢀⣾⡟⢀⣼⡟⠀⣠⣿⠃⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣤⣿⠟⠀⢸⡟⠀⠈⠿⠞⠉⢿⠾⠃⠹⠷⠞⠋⠻⡷⠏⠙⠷⠞⠙⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢤⣠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠎⠙⣧⣤⣿⢸⡇⣟⢻⢹⡇⠀⣸⡃⣜⢻⠌⡁⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⣤⣤⣄⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⣀⡀⢾⣿⡿⠿⠶⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⠆
⠀⠠⠿⡇⠸⣿⠷⠇⠷⠶⣦⢳⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠃
⠀⠀⠀⠻⠀⠇⠠⠄⠠⠄⣿⢳⡇
⠀⠀⠲⢷⣶⣿⣶⣤⡖⠀⠿⠛⠃⠀⢠⣆⣀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣷⣾⣯⣿⠋⠉⣁⣀⠀⢰⣶⣾⡟⠀⡝⢿⠀⢀⡀⠀⠀⠀⠀⠀⠠⠦⣄
⠀⠀⣿⢿⣿⣿⣿⣦⠀⡆⠀⢸⣿⣿⣧⡤⠻⠛⢀⣤⣴⣶⣦⣀⠀⠀⣤⣶⢾⣷
⠀⠀⠛⣿⣿⣗⠘⠿⢻⠤⡄⠀⣥⠒⠉⡀⠘⠈⠉⠉⠋⣿⣿⣿⡦⠚⠀⣿⠟⣿⠒⠶⢠⡀
⠀⠀⠀⠻⠛⠃⢀⣠⣤⣦⠅⠘⠈⣄⢀⣁⣀⣀⣀⠀⠀⣿⣿⣻⣟⡃⢠⠈⠉⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⡅⠁⠀⠀⠀⠀⠀⠹⣿⣿⣛⣿⠇⠀⠁⢀⠀⠀⠀⢺⣿⢿⣿⣿⣿⣿⣿⣷⡆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⢀⢀⠀⡈⠸⣿⣷⣾⣤⣤⠄⠛⠏⠹⠃⣸⣷⡀⠀⠀⠻⠏⠙⣇⣠⣘⣛⣦⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢱⣈⠙⣿⣿⡄⠙⣿⣿⣿⠁⠀⠀⠀⠀⢠⣉⣿⡇⠀⢠⠀⠀⡀⠉⢉⡘⢿⣄⡈⠻⣿⡆
⠀⠀⠀⠀⠀⠈⠀⡉⠀⠀⣿⣿⣦⠸⣿⡇⡀⠈⢿⡟⠂⠀⠀⠀⠀⠈⣿⣿⣿⣈⠂⢴⣀⣩⣿⣿⣥⡄⠉⠉⠓⢠⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⣿⣿⡇⢀⡾⠀⠃⠀⠈⡇⢰⠈⠀⡄⠀⠀⠛⠛⠉⠉⠀⠛⠛⠁⠸⠻⠛⠓⠀⠀⡀⢸⢠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠾⠿⠿⠿⠋⠀⠀⢸⡀⠀⠅⠘⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠀⡂⣴⣷⠀⠀⠀⠸⢻⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣀⣀⠘⠁⠀⠠⠀⢠⣿⡷⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠸⣶⣾⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠲⠷⣤⣀⣉⣀⣀⠀⠀⠀⠀⠛⡀⠀⠁⢠⠀⠀⢸⣿⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣟⣶⣧⣄⣀⣉⣀⠀⣀⣀⣀⣸⡟⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣾⣿⣿⡿⣿⣿⣿⣯⣤⣀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣯⣭⣿⠿⠛⠛⢁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠛⠋⠁⠀⠙⠛⠂⠤⠘⠚⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣾⡿⠋⢿⣄⠀⢀
⠀⠀⣦⣄⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣶⣾⣿⣿⣿⡿⠿⠃⠀⠀⠻⣆⣻⣆
⠀⠀⣿⣿⣿⣿⣾⣿⣿⣤⣄⣀⣀⣸⣦⣤⣶⣾⣿⡟⢙⣩⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⣿⣿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠿⠛⣿⡏⠉⢹⣿⠉⠉⢹⣿⡶⠶⠶⠿⢿⣿⣿
⠀⠀⣿⣿⠃⡿⠿⢿⣿⣿⣿⣿⣿⠟⣿⠋⣿⣧⡄⢠⡆⣿⡇⠀⢺⣇⠀⠀⠘⠿⠇⠀⠀⢀⣼⣿⣿
⠀⠀⣿⣿⣷⣷⣾⣦⣿⣛⠘⣿⠈⣤⣭⣷⣟⣿⣿⣿⣷⣤⣤⡀⢸⣿⣿⣧⣶⣶⠀⠐⣶⣼⣿⣿⣿
⠀⠀⣿⣿⣿⣷⣉⢹⣿⣿⣡⣿⣸⣿⣿⣿⣿⣻⣿⠏⣿⣿⣿⠀⢸⣿⣿⣿⣿⢿⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠉⠻⠿⠿⠿⠛⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⣽⣧⣿⠿⣿⣷⣸⣿⣿⣿⣿⣿⣝⣟⣿⣿⣿⣿⣿
⠀⠀⢀⣀⣀⣤⣀⣀⣀⣄⣀⣀⣀⣀⣹⣶⣶⣄⡒⠂⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠋⠙⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⢹⡟⢿⠧⡽⠋⠛⠁⠀⡀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢳⡶⣤⡄⠀⠀⠀⠀⠚⠶⠚⠉⠁⠀⠀⠀⠀⢧⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣧⠛⠁⠀⠀⠀⠀⠀⠀⢦⡀⠈⠀⠀⠀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣻⠆⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠹⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⣀⣠⡀⠀⠈⢿⠀⠁⠒⠦⠄⢀⣠⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⣧⢴⣶⠈⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡶⠁⠙⣿⣃⡀⠀⠘⡗⠐⠀⠀⠀⢀⠀⠀⠀⠀⢠⣿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠓⠯⠘⣿⣦⣤⣾⣿⣦⣀⣾⣶⣶⣦⣴⣾⣿⣿⣷⢶⠏⣯⡵⠄⠀⠀⢹⡀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠆⠀⠀⣾⣿⣻⣿⢿⠻⠟⡏⣉⣿⣿⣿⣿⣿⣿⡟⣳⡄⠀⠄⠁⢠⠄⠀⠀⠻⠈⠰⠄⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠠⢀⡄⠀⢨⣿⡀⠽⠖⣻⡴⢘⣻⣿⣏⡡⢞⠿⢿⢉⠈⠀⠀⠐⠂⠀⣀⣥⡴⠚⠀⠀⠀⠀⠀⠀⠀⡾⠙⠉⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠈⠻⠷⣿⣿⠿⠛⠙⠛⠻⡄⠰⠓⠀⠈⠀⠀⣀⠀⠀⡀⠈⠉⠉⠛⠿⠷⣶⣦⣄⠀⠀⠘⠁⠀⠀⠀⠀⣠⡦⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠤⢤⣤⣤⠀⠀⠀⠩⢁⣠⡮⠘⠂⠈⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⣡⠋
⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡌⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠀⠉⠙⠛⠲⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣰⠛⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠃⠛⠓⠶⠛⠚⠒⠃⠀⠀⠈⠉⠛⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠙⠛⠻⣷⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⠠⣤⣄⠀⠀⠀⡀⢀⣀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠛⠂⠘⠛⠃⠀⠌⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣈⢉⡁⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⠃⠀⠐⠲⠒⠒⠒⠒⠀⠐⢶⠓⠶⣶⠶⠲⠶⠶⠶⠶⠒⠂⠀⠀⠐⠂⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⣿⠿⠋⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠀⠂⠀⠀⠆⠀⠄⠆⠰⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠁⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡁⠠⡒⠴⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⣤⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠌⠀⢰⠀⡇⠀⠘⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠛⠛⢀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⡃⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡂⠉⢳⠀⡄⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⡡⠈⠀⠀⠀⠠⠀⠹⠆⠀⠐⣀⣠⣄⠐⠄⠀⠀⠀⠀⠀⠀⢜⠄⢁⡀⢺⣴⠂⠀⣟⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⢘⣁⡀⠤⠀⡠⠀⠁⠀⠀⡆⢰⡏⠙⠈⣷⠰⠂⠀⠀⠀⠀⠀⠘⢦⣌⡁⠀⠀⢳⠆⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡀⢳⠀⠘⢇⡈⢭⠀⠀⠀⠀⠠⠀⡀⠤⢄⠐⡀⠀⣠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⢠⡖⠜⠋⠀⠀⠀⠁⢺⠑⠆⠀⠀⠳⠀⠆⠀⢸⠀⠰⠎⠉⠳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⡆⠀⠀⠀⠀⠀⠀⠀⠄⠐⢄⠀⠀⡦⠀⡄⠈⠚⢉⡀⢀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠘⠀⠀⠀⠈⠂⠰⠀⢸⠀⠂⠐⠀⠀⢃⠈⠄⠀⠈⠀⠀⣠⠁⠀⠀⠀⠙⡄⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠔⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⢢⠈⠀⢃⠄⠀⡆⠀⡀⠀⠀⠀⠐⠀⢸⡄
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠁⢀⠆⠀⠀⠀⠀⠀⠀⠇⢠⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⢱⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠡⠉⠀⠀⢢⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠘⠁⠀⠀⠀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠄⠀⠀⡀⠀⠀⠀⣴⡀⠁⠀⠀⠀⠘⠀⠀⠀⢀⢴⢀⠀⠀⠘⡄⢆⣀⡉⠍⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁⠀⢸⠀⠙⢀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡄⠀⠀⠑⠀⠀⠀⠀⠐⡀⠀⠐⠀⠀⡐⠂⠀⠀⠀⠀⠀⠢⡀⠐⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠈⠀⠀⠀⠀⠇⣠⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠘⡀⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠁⠀⠀⠀⠐⠀⠤⠖⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡄⠀⠀⠀⠀⠀⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠙⠀⠐⠁
⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠐⢇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡯⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠋⢏⠏⠻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⣿⠀⢸⡇⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣧⣿⣤⣬⣇⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⣄⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⣯⡇⠸⣿⣿⠇⢨⣷⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠒⠉⠀⡈⣇⣄⠀⢀⣄⠈⡇⠈⠑⠪⢗⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣨⠗⠊⠀⣀⡀⠀⠀⡇⣿⣿⠀⢸⣿⠇⡇⠀⠀⢀⡀⠈⠓⢯⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⡴⠻⣿⡏⠢⡀⡇⣿⣿⡀⣸⣿⡇⡇⣠⠊⣷⣯⠓⡄⠀⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣇⣤⠀⣤⡀⡇⡇⡏⢉⣥⣍⠉⠁⡇⡇⣠⣄⢠⣄⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⣿⣿⣿⣻⡇⡇⣇⡇⡞⢸⠈⡆⢠⡇⡇⣿⣿⣟⣟⠇⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⣿⣿⣿⣿⡟⣇⣽⣇⡇⢸⠀⣇⣼⣇⡅⣿⣿⣿⣿⢲⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⣿⡿⠛⣿⣇⠇⡏⡏⡇⢸⠀⡇⠉⡏⣇⣿⣟⠻⣿⡰⠈⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡧⣼⣤⣤⣤⣤⣤⣤⣤⣧⣷⣷⣾⣶⣷⣦⣧⣤⣤⣤⣤⣤⣤⣤⡤⢧⣄

⠀⢠⣿⡿⠶⣦⣄
⢀⣿⣿⠁⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣶⡶⣶⠶⠶⠦⠤⣤⣀⣤⣤⣤⣶⣦⣤⣀
⣸⣿⡇⠀⠀⠀⢠⣿⣿⢻⣷⠀⠀⣠⠖⢛⡿⠻⠫⠀⠀⠀⠈⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⣀
⣿⣿⠁⠀⠀⠀⣿⢸⣿⠀⣿⡆⢸⡗⠊⠉⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣄⣀
⣿⣿⠀⠀⠀⠀⣿⡘⣿⣴⣿⡇⠀⠉⠛⠛⠛⠉⠉⠉⠛⠛⠛⠉⠉⠙⠻⠶⢶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀
⢿⣿⡄⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⣻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠸⣿⣧⠀⢀⣠⣾⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢀⡼⠟⠉⢻⣄
⠀⠹⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣶⠋⠀⣀⣴⠟⢿⡆
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡏⠀⠸⣿⣁⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠻⠿⢿⣿⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡉⠙⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣤⣤⣾⢦⣄⣠⣤⣴⡶⠶⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠭⠭⠿⠿⠿⠿⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⠗⠛⠛⠛⠛⠒⠒⠒⠒⠒⣒⣶⣶⠶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠄⠒⠒⠈⠉⠉⣀⠀⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠒⠒⠂⠉⠉⠉⢸⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠉⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢠⠖⠒⡆⠠⠖⠒⢲⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠈⠀⠀⢸⠀⢸⡇⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⣸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡛⢻⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⠈⠒⠒⠃⠠⢄⣀⣸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠉⠁⠒⠒⠤⠄⣀⣀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⡴⠶⠶⣶⡾⠿⡿⢷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⡵⠊⠉⣁⣩⡳⣦⣦⣤⣹⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢻⣿⣿⡿⠘⢰⣰⣋⡴⠞⠿⠾⠿⠻⠛⠷⠹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠜⠛⢻⡿⣶⣌⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡀⢀⣈⣺⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣟⠋⠈⠀⢀⣀⣠⣤⡴⠀⠀⠀⣠⡤⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡿⣿⣿⣿⣝⣿⡿⠋⠀⠀⠘⣁⣤⣤⣀⠀⠀⠀⠰⣠⡤⢤⣨⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣕⠈⣿⡏⠻⣿⣲⠄⠀⠀⠀⠿⠵⠄⠀⠀⠃⠀⢡⠛⠓⠐⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣥⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣣⠀⠀⠀⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡘⠿⠿⣻⣥⣶⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⣻⣿⣿⣮⣽⡃⡀⠀⠀⠀⠀⠀⠀⢀⢔⡁⠠⣑⡀⠀⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡿⡇⠰⣄⠀⠀⠀⠀⠀⠰⢬⣉⣁⡴⢓⠁⠀⠀⠒⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣇⠀⠘⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠓⠦⣀⠈⠻⣿⣶⣤⣄⣀⣠⣴⣆⡃⠀⠀⠀⢠⠻⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠀⠀⣴⣦⣷⡄⠀⠙⠻⠿⣿⣿⣿⣿⠁⠀⠀⠀⣀⣀⣀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⡧⠀⣿⠛⢿⣷⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠙⢿⠿⢷⣬⣶⣤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠙⢆⠀⠀⠀⠀⠹⡆⠀⠹⣷⣦⣤⣀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡃⠀⠀⠀⢉⣙⣿⣷⡀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠃⠀⠀⠀⠀⠉⠙⠻⣿⡄⠀⢸⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢻⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣈⣤⣤⣤⣬⠤⠤⠤⠄⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠁⣤⠞⠋⢹⡏⠉⠀⢒⢲⡞⠉⠉⠙⡿⠋⠉⢙⡏⠉⠉⠉⡏⢿⢆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣷⠈⢏⣁⣼⢆
⠘⢄⠀⠀⠒⠠⠴⠶⣿⣤⣤⣾⣧⣤⣶⣶⣾⣷⣦⡰⣾⣷⣤⣤⣼⣧⣴⣤⣴⣷⣼⡇⣑⡠⠀⣐⠂⠀⠀⠒⢀⡀⠘⠉⠀⠈⢩⣹⣤⢸⡟⢾
⠠⠉⠀⠒⠀⠀⠀⠀⠀⠈⠉⣭⡉⠉⠉⣀⣁⡛⠛⠓⠚⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠁⠀⡂⠠⠐⠔⠉⠣⠤⠤⢄⠀⠾⠿⠿⣯⡄
⠸⣅⠍⠉⣠⣀⡀⠒⠂⠀⠼⠀⠀⠀⢀⡀⡄⠀⠀⠉⠉⠐⠋⠒⠂⠤⠤⠤⣐⣂⣠⣄⡀⠀⠀⠀⠀⠐⠒⠚⠠⠰⠒⠀⠀⠈⢡⣶⡾⢯⢻
⠀⠈⠐⠚⣻⣯⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠐⠒⠒⠠⠀⠀⠀⠀⢹⡿⢯⣿⡿⣧⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⡄⣿⣭⠽⣾⢸
⠀⠀⠀⠀⠈⢯⢙⣷⡶⠂⠤⠤⣀⣀⡀⠀⢀⠀⠀⠀⢠⣶⣿⡏⣷⣦⡄⠀⠸⡯⠼⣷⣶⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢏⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠉⠡⠂⠔⡀⢀⠈⠈⠉⠛⠠⠤⠤⠾⠿⠿⣷⣺⣿⣟⣀⣀⠛⠛⣛⣚⣘⣋⡉⠉⢛⣛⣋⠉⢉⣁⠰⠶⣶⣾⠟⠛⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠁⠀⠀⠃⠄⠀⠀⠄⠀⠀⠀⡿⣿⣿⣿⡟⠻⠷⠾⠿⠾⠿⠛⠛⠛⠛⠓⠛⢿⣿⠈⠀⢨⣟⣉⠤⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⣿⡿⠃⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠒⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡄⣴⠀⢀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠄⠀⠄⠀⠐⠀⠀⠀⠠⠄⡀⠀⠀⠀⠠⠁⡁⠀⠈⣠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣠⢎⣁⡀⠀⠀⠀⠀⠀⠀⠒⠂⠄⠙⠓⠐⢠⠤⠄⠀⠀⡀⢈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠭⠁⠀⢀⡐⠢⠄⠀⡂⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠤⠠⢤⢜⠕
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⠐⠰⠀⠀⠆⠀⠀⠀⠀⠀⠈⡀⠐⡀⠆⠋⠉⠁⠠⠄⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⢀⣀⠈⢀⠀⠆⠀⠀⠁⠈⠐⡞⠀⠀⠂⠀⠁⠀⠁⠀⢀⠀⢐⣈⡖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠂⠀⠘⠉⠀⠀⡀⠀⠀⠀⠐⠁⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢤
⠀⠀⠀⠀⠃⠀⠀⣐⣂⠀⠐⠒⠂⠀⠀⠀⠀⠀⠠⠅⠀⠈⢠⠐⠀⠀⠈⡖⠀⠋⠀⣀⣘⠀⣀⡀⠀⠀⠀⣇⡘⠀⠀⠀⠀⠀⠀⠖⠒⠒⢂⡀
⠀⠀⠀⠀⠀⠀⠉⡉⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠐⠈⠁⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⡈⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣟⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁
⠀⡀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠼⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠉⠉
⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⢂⠤⠒⡒⠂
⠀⠁⠀⠈⠂⠄⠤⠀⠀⠀⠠⠄⠀⡈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠁
⠈⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⢋⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣷⣚⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⢿⣿⠛⠛⢻⣟⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠋⠐⡓⢲⣶⣿⣿⣟⣿⣄⠀⠀⠀⠀⢠⣮⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧⡀⠀⠀⣺⠿⠛⢩⠘⠻⢧⠿⠿⢿⣿⣿⠆⠀⠀⠀⣛⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠗⠀⠀⡽⢀⠀⠨⢤⡀⠇⠷⠒⡃⢴⣭⡄⠀⠀⢸⣉⣉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡈⠲⠀⠀⠱⠓⠸⠛⢾⡇⠀⠀⣿⣿⡄⢹⠁⠀⠀⠀⠉⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠙⠃⠀⠀⠀⠀⡈⢀⣸⠓⠀⠀⠿⢿⣦⣾⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠦⡷⠿⢦⣤⣤⢀⠀⠀⠉⠉⡑⠀⢰⡀⠀⠩⣼⠇⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⣞⡷⠦⠶⢤⣴⣤⡾⠷⠲⣷⢾⣿⠶⢲⣤⡀⣠⣄⣐⣃⣈⠅⡼⠉⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⡇⠀⠀⠈⣿⣿⡇⢀⠀⢸⠀⣿⠀⠀⠈⠀⣿⣿⡃⠙⠋⠹⣿⡷⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⢰⡯⡁⢸⠀⢸⠀⣿⠀⠀⠀⠀⡙⣏⡁⠀⠀⠀⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣯⣄⣀⡀⠺⣿⣷⣇⡸⢠⣼⠀⣿⠀⠀⠀⠀⣷⣾⠃⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣞⢒⣿⣿⣿⣭⣭⣉⣿⣿⠠⡖⢠⣄⡀⢩⡅⣀⣠⡦⢿⣿⠀⡒⠂⠐⠷⠶⠷⠄
⠀⠀⠀⠀⣠⣤⣴⡿⠿⠻⡍⠉⠉⠉⠉⠛⠛⠛⠛⠛⡷⠙⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣾⡷⢷⣶⣶⣦⣙⣦⣤⣤⣤⠂
⠀⠀⠠⡾⠿⣟⣻⣿⢰⣼⡇⠀⠀⠐⠒⠀⠀⠒⠂⢸⠀⠀⡟⠺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⢮⣈⠉⠉⠉⢉⠉⠉⠉⠉⠁
⠀⠀⠀⣶⣾⣿⣿⡿⠹⠻⡇⠀⠰⣷⠤⠀⠀⣀⣄⣸⡀⠰⠃⠀⣻⣿⢉⣿⣿⡟⣿⣿⣿⣿⣷⡝⣶⣿⣷⡏⠑
⠀⠀⠀⢁⢠⠁⣄⣀⠀⢀⡇⠠⣿⣽⡆⡬⢭⠍⠉⠁⠀⣄⡀⠠⠞⠉⠸⠿⡿⠀⠀⣿⡿⢻⣿⣿⣤⣿⣥⠛⠦⢢⢂⣇⣀
⠀⠀⠀⡿⢿⠏⣿⣿⣿⣿⡇⠀⣵⣆⣷⡶⢀⠀⠀⣦⠄⣯⣹⡇⣐⡁⠀⠃⠀⠀⢀⡇⠀⠀⣿⣯⠉⢹⣸⣷⣾⣿⢀⢀⣀
⠀⠀⢈⡛⣺⣶⣏⢃⣤⣬⣅⠀⢿⣿⣿⠀⣿⠃⢀⣿⢢⣟⡉⣿⣿⣿⣻⣿⣿⣿⡿⠿⠞⢻⢿⣯⣭⣭⣩⣉⣡⣄⣈⣁⡉
⠀⠀⠈⢹⡏⠉⣉⣀⡸⣟⠉⠙⣾⡏⠉⠀⠉⠀⠈⠋⠿⠛⠀⠀⠀⢠⠿⠟⠉⠁⠀⠀⡳⢿⣭⣽⣿⣿⡿⠿⠿⠿⠍
⠀⠀⠀⠘⠁⠛⠛⠉⠉⠉⠁⠀⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⣀⠀⠉⠉⠛⠲⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣴⠋⠀⠀⢀⡀⠀⠀⠀⣰⠏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⡾⠁⠀⠀⣴⠋⠉⠛⠶⠶⠏⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⠏⠀⠀⠀⠀⠛⠶⢤⣤⡀⠀⠀⠀⠀⠈⠉⠙⠛⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⢏⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⠶⠤⣄⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢃⠎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣰⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢁⠎⠀⠀⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⠀⢀⡼⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⡀⢀⡾⢡⠎
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠠⠃⡀⢀⡞⣡⠏
⠀⠀⠀⠀⠀⠀⡼⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠁⡜⢠⡟⣰⠏
⠀⠀⠀⠀⠀⣸⢡⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠜⢠⠟⣰⠃
⠀⠀⠀⠀⠀⡇⡀⢰⢿⡉⠛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⠞⣠⠏⣼⠃
⠀⠀⠀⠀⠀⡇⠃⢺⢸⡀⠁⠒⠤⣄⣉⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⠎⣰⠏⡼⠃
⠀⠀⠀⠀⠀⡇⠀⢸⣿⢀⡈⠁⠒⠀⠀⠉⠐⠀⠈⠉⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⠎⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠘⡄⠀⠿⣤⣀⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⢀⣾⢃⠔⢠⠏⣼⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠉⠛⠶⢤⣀⠀⠀⠀⣶⠲⠤⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⣠⡿⢃⠞⢠⠃⡼⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠀⠈⠉⠒⠄⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡁⠒⢿⢡⠊⢠⢃⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠢⢄⣈⠁⢻⠃⠠⢃⡾⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡇⠈⠑⠢⢤⣀⡀⠀⠀⠉⢸⡤⢁⡞⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣤⡄⠀⢀⡞⠁⠈⠐⠢⢄⣀⠉⠓⠦⣤⣘⣧⠟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠉⢸⡇⢀⡞⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡉⠉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈