Indra Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡟⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢦⡶⠐⠈⣡⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠼⠌⢓⣴⠛⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⡄⠀⠀⠀⠀⠀⡴⠦⡆⠀⠀⢛⡐⠋⠵⣨⡍⠀⠀⠸⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⣟⠀⢠⣞⡖⡟⠁⠀⣴⣿⡷⠴⢶⡟⠀⠷⢤⡀⢮⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⣺⠓⠀⠹⣄⠻⡴⡄⠁⣸⠶⠆⡾⠹⢠⠀⢨⢷⣠⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢀⠃⣄⠲⡤⠁⣾⡀⢀⡧⢶⡞⠃⣴⣼⠂⢼⠑⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠁⠉⠓⡙⠻⡧⠄⣴⠰⣼⡢⣼⢇⠺⠹⠋⠐⣩⣠⢎⡀⣀⣠⣤⢖⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢠⠴⣤⠶⢟⡛⢦⣀⣴⠃⠠⠋⢠⡀⡈⣃⣀⣠⢴⢻⣄⠡⠷⠬⢭⡍⢴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠉⠋⠁⠀⠀⢺⡏⡇⢨⡾⡯⠟⠁⠀⠀⠉⣀⡈⠛⣡⣅⣠⡘⠁⠀⠀⢐⠀⢙⡯
⠀⠀⠀⠀⠀⠀⣰⢿⡇⠀⢴⡖⠤⣄⠀⢸⠃⡠⠀⣵⠀⠀⠀⣴⣛⣩⣭⣴⣾⣟⣍⡟⢼⢢⣐⣬⣶⣋⢾⠃
⠀⠀⠀⠀⠀⢠⣼⣋⡁⢾⡋⠀⠀⢨⠋⢉⡻⠁⢈⣿⠀⢰⢊⣿⣿⣿⣭⡿⡿⢿⣿⡧⠹⣘⠈⡐⢌⢺⡟
⠀⠀⠀⠀⠀⠈⣿⡛⠀⢩⠁⠀⢀⣧⢀⣸⠁⣀⢸⢏⡠⢃⡾⣿⣿⣿⡿⣿⣾⡫⢿⠃⡀⢨⠇⢌⣂⡉
⠀⠀⠀⠀⠀⠀⢉⠃⢀⡌⠀⠀⠀⠋⠬⠤⠇⠙⣾⢇⢹⣿⣟⣲⣷⣿⣯⣾⣟⣳⢷⠀⡁⢀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢀⠎⢀⡜⠀⠀⠀⠀⠈⡇⢰⠰⣀⣈⣈⣭⣽⠙⢩⢏⡉⠙⠝⡷⠛⢾⠎⠄⢸⣀⢀⣀⢸
⠀⠀⠀⠀⠀⠸⠠⠞⠀⠀⢠⣄⠀⠀⠹⡌⡀⠀⠈⠉⠋⢀⠄⣛⣲⣶⠃⠄⠩⡥⢘⣦⡾⢌⡟⠾⡿⢹
⠀⠀⠀⠀⠀⠢⢨⡀⠀⠀⡆⡾⠀⣰⠀⠘⢻⡄⠀⠀⠀⡼⠒⠸⢿⡋⠴⢷⠴⣿⡀⠉⠉⠉⠀⢠⡇⢨⡀
⠀⠀⠀⠀⠀⠘⠦⣉⣀⣠⡴⠃⠀⡇⠀⠘⡿⢱⡀⠀⢈⡇⠀⠳⠾⠗⠀⠀⢣⠀⠙⣦⡀⠀⠀⢸⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣾⠁⠀⠃⠀⠸⡧⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⣷⠀⠀⠸⣇⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢤⢠⡟⠀⠀⣾⣶⣞⡽⠀⠀⠀⠀⠀⠀⠀⢸⠆⡄⢿⣄⠢⠐⣸⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢊⢀⣐⠚⠁⠀⡠⡋⣀⢿⠁⠀⠀⠀⠀⠀⠀⢀⣞⣀⠈⣿⠋⠀⣀⢷
⠀⠀⠀⠀⠀⠀⠀⣨⡧⡅⣬⡮⣭⢩⡴⠁⠮⢃⠍⡭⢍⢉⡆⡽⢨⡧⡍⣫⢭⡩⢌⡧⠭⡅⢨⡀⢨⡄⠇⠋⢇⡀
⠀⠀⠀⠀⠀⠠⣾⣛⣐⣸⡒⠒⠒⣟⡆⣚⣚⣓⣟⣖⣓⣚⣙⣐⣒⣓⣛⣐⣣⣺⣀⣳⣓⣐⣂⡑⣚⡀⣒⢰⡚⣿⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⢀⡄⠀⠀⡎⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠤⢀⣀⠀⠀⣀⠀⠤⡤⠤⡤⠀⣠⣾⣿⣿⠈⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⠞⠀⠹⣧⠀⠈⠀⠀⠀⠈⠀⣰⣿⡿⢿⢹⣧⢤⣀⡀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⠀⢀⠀⣿⠀⠀⠀⠀⠀⠠⡞⠁⠀⠀⠀⠈⢿⣮⣘⡻⣶⣄⡀
⠀⠀⠀⠀⢸⢿⣷⠶⢶⣿⡜⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⣿⣿⡿⡏⠉⠻⣦⡀
⠀⡀⠀⠀⢈⣀⣢⣴⣯⣼⣷⣆⣠⣴⡟⡀⠀⡀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠉⠀⢩⠀⢀⠀⠈⠙⣿⣄
⠀⠲⠥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡇⠀⠁⠀⢀⠸⠿⠄⠀⠀⠀⠀⠀⠐⠚⠀⠘⡁⠀⠐⢿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⠟⠁⠈⠙⠉⠉⠀⣊⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⢸⣿⣷
⠀⠀⠀⢰⣿⠋⠁⠀⠀⠀⢰⣶⣾⠁⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⣇⡠⠀⣼⣿⣿
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⣩⣯⣿⣏⣉⣿⣿⣿⣏⣙⣛⣳⡖⠀⠈⠉⠆⠀⣼⣿⡏
⠀⠀⠀⣻⠃⠀⠀⠀⠀⠀⠘⠀⢿⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⢀⣿⣿⠃
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠨⣾⣿⡟⠙⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠠⣶⠀⣠⣿⣿⡿
⠀⠀⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢟⣴⣿⣿⣿⣷⡀
⠀⠀⠈⠛⣣⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣻⣷⠤⣤⣄⡀⣤⣤⣤⣤⣤⣤⣤⡔⠀⠀⢸⡏⡘⣾⡇⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⠻⡗⠰⢾⡿⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⡐⢠⢶⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⣿⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣡⣦⣿⣿⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠉⠈⠙⣿⣆⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢀⠀⠀⠀⠀⠀⢿⡏⢰⡆⠘⠲⣶⠘⣿⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣖⠟⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⣤⡀⠀⢰⣿⡄⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡈⠷⠙⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠈⠀⠰⣿⣷⣿⡆⠀⠀⠀⠀⠀⠹⠳⢶⠄⠤
⠀⠀⠀⠀⠀⠀⠀⡙⠋⠀⠀⠀⠀⠀⠀⣴⠿⢿⡾⡿⠷⡀⠀⢀⣰⣿⡾⣿⡿⣧⡀⠀⠀⠀⠀⡂⠛⡤⠆
⠀⠀⠀⠀⠀⠀⠀⣉⣦⡄⠀⠀⠀⡠⠊⢠⠶⢻⣀⣿⣦⣄⠿⢏⣠⣿⣇⣼⠓⠤⢿⡆⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠘⣷⠞⠃⠀⠀⣴⠛⠀⠀⠠⠋⠉⢹⡉⣷⠒⠻⣏⡟⠉⠉⠀⠀⠀⠈⢃⠀⠀⠰⣷⣦⡃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⢿⣤⠀⣴⣦⠂⠀⠈⠑⢻⡏⢡⢟⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⡄⠈⣷⢀⠀⠀⠂⠀⠱⣄⠾⠁⠀⠘⢠⣔⢶⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠺⣿⠟⠀⠀⢠⡀⢰⡏⠀⡀⠀⠀⠘⠋⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠈⠉⠛⠙⢿⡅⠚⠋⠈⢠⣦⣝⠀⠰⣷⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣦⣤⣦⣼⠊⠑⣀⣤⣤⡾⣿⣯⣟⣀⠀⢀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠙⣿⡛⠉⢁⣀⣩⣿⣧⡀⡀⠀⠹⡏⡽⠃⠘⢁⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠆⠀⠀⠀⠀⢀⣀⣠⡷⢿⠿⣙⣻⡿⣿⠟⣩⠖⠰⢇⣠⡆⠀⠂
⠀⠀⠀⠀⠀⠀⢃⠀⣀⡀⠀⠀⠀⠀⠀⣾⡏⠀⠚⠋⠉⠈⠙⠃⠀⠀⠀⠉⠀⢻⣿⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠁⠄⠁⠀⠀⠀⠀⠀⣠⢿⡁⠀⣤⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⢸⣿⣤⠀⠀⠀⠀⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⣼⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⡏⠀⠈⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢤⣿⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⣷⠲⣄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣄⠀⠀⠀⠊⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡠⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡞⠁⠀⠀⠀⠀⠀⠉⠈⠳⡄⠀⠀⠀⠀⠀⠀⢠⠖⠙⠉⠀⢀⠀⠀⠀⢸⣦⡀⠀⡐
⠀⠀⠀⠀⠀⠀⠁⢠⡬⠁⠀⠀⠐⠀⡍⠠⠤⠤⣄⠀⠀⠀⠀⠃⠀⠀⢀⣀⣀⣀⡀⣀⢉⠀⠂⠀⠛⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠠⡞⢳⠀⠀⢀⡰⠂⠁⠰⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⢨⡥⠍⠙⠿⠞⢻⣆⠀⠀⠀⡆⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠋⠁⠀⠀⠀⠀⠀⠰⡄⠰⡄⠀⠀⠀⠀⣼⠆⠀⠀⠠⡶⠄⠙⠷⢦⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⢀⠂⣀⣦⡈⠶⠉⠳⡶⠆⠋⠁⠀⠀⠀⠀⠚⣻⣦⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠉⠉⢸⣷⣤⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣈⠀⠀⠴⠄⠀⠙⢳⢾⣜⠻⠉⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⣦⠀⠀⠠⠀⠀⠀⠀⣦⠀⢀⡖⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠘⣷⠂⠀⡀⠀⠀⠀⠀⠁⠑⠈⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢠⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡛⣶⠤⠤⡤⠤⠴⣾⡄⢰⣷⡒⡂⠤⡤⠤⠶⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠷⠞⠿⠀⠈⠿⠩⠵⠶⡗⡚⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠰⠀⠀⠀⢠⡆⠀⠠⠀⠀⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⢻⣿⣿⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠀⠀⠘⣿⣿⠀⠀⠙⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⢲⣶⣤⣹⣿⡆⢀⣴⣶⠟⠻⠌⢿⣿⣿⣿⣿⣿⣿⡆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣤⣄⣂⡩⢍⠀⢻⡇⠀⢉⠤⣢⣤⠤⣤⡿⣿⣿⣿⣿⠈⢿
⠈⠢⢤⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⣿⣿⠁⢿⣄⣾⣿⣹⠄⠁⠀⢳⠀⠡⣊⣿⣿⣴⠟⢀⢽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⡿⠛⣿⣿⣿⣿⣿⢸⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢸⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⣿⣿⣿⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣸⣿⣿⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠠⣤⣤⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡀⠁⣀⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠒
⠀⠀⠀⠀⠀⠐⠁⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⠳⣄⠀⠀⠀⠀⠀⠀⠀⡠⠎⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⢻⣿⣿⣿⣿⣿⠃⠉⠠⢿⡓⠤⠤⠤⠄⠠⠚⠁⢰⢿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠈⢿⣿⣿⣿⡇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠛⣿⢿⠃⠀⠀⠄
⠀⠀⠀⠀⣰⣿⡿⠛⠉⠀⠀⠈⡏⠀⣿⡇⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠂⠀⠔
⠀⠀⠀⠐⠋⠉⠀⠀⠀⠐⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⣬⠴⠖⠂⠀⠀⣀⠋⠀⠀⠀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣿⣶⠄⠀⠀⠀⢀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣭⣿⣿⣿⣿⣿⢏⣾⠆⣤⡠⢌⣹⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠿⠟⠙⣟⡿⣆⣿⠿⣿⠟⣣⡜⠋⢈⣸⣿⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢸⡕⢹⣿⠃⢠⠷⣯⣿⡰⢛⡻⠿⣯⢲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣘⡇⢠⣿⣿⡟⣷⣏⢹⠃⢮⣧⣦⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠿⠿⡿⠛⠀⠀⠹⣿⣿⣿⣃⣭⣴⣶⣿⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣍⡉⠛⠂⠈⡠⣴⣾⠗⠃⣠⣤⣥⣭⣽⡿⠆⠈⢲⣤⡶⠤⠘⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣺⣿⣷⣆⣴⣶⣿⣷⣿⡆⢀⡿⠉⠀⠀⠀⠀⠀⢹⣿⣏⣙⣷⣀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⣿⣿⢻⣿⢿⣿⣿⡟⣰⡿⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⣿⡟⣾⣿⣿⡿⢷⣾⡿⠹⣷⣲⣶⠤⢤⣤⣴⣾⣿⠀⢻⣿⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⣿⡿⣧⣿⣿⣿⠁⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣏⠁⠀⢼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢪⣻⣿⠇⢠⣿⣿⣿⠂⢸⣿⣿⣿⣿⠙⣿⣿⣧⡀⠘⣧⣤⣵⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡷⠤⢬⡹⣠⣟⣙⣿⣿⣷⡿⢻⣿⣿⡏⣸⣿⣿⣿⡿⠀⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠛⠛⠉⠉⠁⠚⠛⠛⢛⡉⠾⠟⠋⠉⠉⠁⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣬⣿⣶⣤⡴⠆⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⡰⠇⠚⠛⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠄⡀⠀⠀⠀⢰⠃⠀⡀⡠⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢄⠀⠀⢀⡈⠋⠀⢀⡇⠈⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⣎⠇⠀⡇⠧⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠛⠟⠀⣼⣇⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⡀⠀⠉⠛⣧⠩⠈⠁⠀⠀⠀⠠⣶⡀⣐⣤⣾⣠⡏⢸⡄⠈⠀⠀⠀⢄⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⡀⢱⡄⠸⡦⠀⠠⠀⠀⠀⠈⠀⠀⣹⡿⠟⠩⠇⢸⣷⣦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡄⠀⢿⠀⠈⣷⢸⠛⢦⡅⠀⠀⠀⠀⢠⠀⢠⡼⠃⠀⠀⠀⢀⡈⠳⠙⣄⠘⡿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡄⠸⣿⠀⡄⢸⠇⠀⠀⡀⠀⠈⠀⢻⠀⠀⠀⠀⠀⠈⡤⠐⠀⢉⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠉⠃⠠⡏⣠⠁⣼⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡰⠁⠀⢀⣠⠀⣶⢀⡜⡁⢀⡇⢠⡇⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠶⡾⢦⣾⣠⠾⢿⣿⣿⣴⡞⠁⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠈⡃⠀⠀⠀⣰⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠀⠀⠀⠋⠋⣉⠠⠤⠾⡇⠀⢒⠀⠀⠁⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠼⠇⠀⣧⣤⠤⡤⢽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣠⡤⠀⠀⠄⠀⠖⠀⢀⣠⡠⠏⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣼⣄⠀⠀⠀⣠⠈⠁⠀⠀⠀⠀⣴⡿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣿⣿⣿⣿⣿⡷⠶⠛⠛⢂⠀⠀⠀⠀⢸⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢸⠀⠀⢠⡅⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢼⠀⠀⠘⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⢸⠋⡄⠀⠀⣸⡇⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠐⢿⣷⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠆⢠⡏⠀⠘⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠔⠀⠀⢀⣾⣤⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠈⠙⠛⠒⠛⠛⠛⢷⣦⣤⣽⡿⢿⠿⠿⠶⠖⠂⠀⠀⠀⠾⠿⠶⡿⢦⡶⣤⠼⠷⠿⠿⠿⠿⠛


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⣉⡁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠳⡐⢈⡇⢠⠇⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠇⠀⢡⠇⠀⢘⠂⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠣⡀⠀⠉⠒⢢⠎⠁⡀⠈⢢⣩⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⡂⠀⠀⡴⠀⢀⠆⢠⢤⡄⠀⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⢘⠠⠻⠀⢠⠀⣠⠎⢄⢀⡄⣨⡀⠀⠀⠘⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠛⠋⠠⡄⢈⡷⠀⠀⠀⠀⡰⠼⠆⢀⠠⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢷⠀⠀⣉⣀⡆⠜⠀⠀⠀⠀⠀⠱⡦⠴⠀⠀⢱⠀⢠⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⠉⠈⠃⡐⠒⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠈⠁⢄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠶⠂⠀⠀⠀⠀⠰⠊⢶⠡⠉⠈⠈⠀⠀⠀⠀⠀⠀⠑⠒⠠⡄⢀⠁⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁⠀⠀⠀⠀⢈⡰⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⢶⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠠⡞⠿⠀⠀⠀⠀⠀⠠⢒⠃⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⢘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠠⠖⠤⠤⠠⢲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣏⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠙⠔⢦⣶⣴⡦⣶⠴⠴⢦⣦⣶⡇⠀⠀⠀⠀⢀⡴⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠄⠀⠀⠀⠀⡠⠃⠀⠀⠀⠙⠖⢢⠀⠀⠀⠀⠀⢈⡀⠀⠀⣈⠁⣀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠂⠠⠤⠬⠂⠀⠀⠀⠀⠈⠀⠀⡐⠁⠀⠈⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠐⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓



⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⢄⣀⡀⠀⠀⣠⠀⡀⢀⣀⣀⡀⢀⢀⠀⢀⠀⡤⠄⠀⡀⡀⡄⡀⠀⣄⡄⣀⢠⠠⡀⡀⡄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠈⠁⠀⠀⠀⠀⠉⠈⠁⠉⠉⠈⠉⠁⠈⠀⠁⠁⠀⠉⠀⠁⠁⠀⠉⠁⠂⠈⠀⠁⠀⠁⠈⠉⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣏⣉⣉⠟⣩⡛⠛⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣯⣛⣢⡀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠰⠿⢿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠋⠀⠘⠻⡏⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⡿⠿⠿⠾⠿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡶⠤⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡈⣿⣶⣦⣄⣀⠀⠀⠀⠀⢰⡿⢶⣿⣃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣧⣿⣿⣭⣟⣻⢻⡷⢦⡀⣼⣿⣽⣿⣿⣄⠀⠀⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡺⢃⣿⣿⢦⡿⢻⣿⠀⡀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠉⠁⠀⠐⠺⣟⡻⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣾⣿⣶⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢫⡽⣿⣿⣿⣿⣏⣽⢛⣿⣯⣽⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⣿⣿⣿⣇⠉⠙⠳⢮⣽⣻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣾⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⣿⣿⣄⣿⠀⠀⠀⠀⠀⠀⠻⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⠀⣿⣿⣿⣿⢃⡿⡿⠃⠀⠀⠀⠀⠀⣠⣤⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡙⣼⣿⣿⣻⣦⣟⣿⣷⣶⢶⣿⣿⡟⢹⣿⣿⢇⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢰⣿⣿⣷⡿⣷⣽⣿⠻⣧⣻⣿⣿⣿⢫⡿⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣆⠀⠀⠀⠀⠀⠈⢻⣿⣼⡿⣼⢿⣷⣏⣻⣿⣷⣹⣯⢻⣿⣏⣼⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣆⡀⠀⠀⠀⠘⣿⣟⢷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣾⣿⣦⣀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣶⣿⣿⣿⣷⣘⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣟⣧⣿⣗⣨⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⣿⣿⣿⣿⣯⢻⣷⢸⣿⢻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣸⣿⣰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⠿⠿⣯⣥⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⢾⣿⣶⣶⣶⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣌⠿⣿⣻⣯⠶⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⣷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⣀⣠⣰⣖⣾⣿⣿⣿⣿⡓⡂⠀⠀⠀⠀⠀⢀⡀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣱⣾⣿⣿⣿⣿⣿⣿⠁⣧⡄⢀⡀⠀⠀⠛⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣽⣿⣿⣿⣿⣿⢟⠟⣿⣿⣿⣏⣉⣴⡿⠁⣼⠟⠠⠁⣦⣾⡇⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠻⠛⠛⠁⠀⢉⣙⡻⢿⡀⣿⠓⠹⡿⠇⢍⣤⣄⡉⠀⢀⣠⣼⣿⠇⣰⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢀⡿⠱⠚⣿⢹⡞⠀⡁⠷⡮⢋⣿⠈⢠⠿⠏⢻⣿⣿⣡⠦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠻⣿⣀⣀⣽⢢⠸⣧⣸⣷⠁⠀⢒⠁⠂⢹⡟⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣶⣄⣀⣩⡆⠠⠿⣿⣿⣿⡃⣾⣿⣄⠈⣇⠀⢊⣾⣴⡷⣾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⠷⠊⠀⠀⠀⢱⣿⢛⣿⣿⠟⢂⣈⣤⣤⣴⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠓⠒⠦⣤⡀⠍⢡⠄⠀⢀⡰⣦⣈⡟⠋⠉⠙⠛⠙⠟⠛⣿⣷⣎⠉⠳⠄⠀⠀⠀⢀⡀⠀⢴⣶
⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⣥⠀⠀⠈⠁⠀⠀⢴⢚⡟⣿⠟⠉⠀⢠⣶⣶⣾⡶⠿⠿⠿⡟⠻⢀⣀⣐⣿⣿⣿⡙⠃⠂⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠿⣶⣾⣳⡄⣀⣽⣶⣿⣿⣧⣿⣷⠄⠀⢸⡟⠉⠀⠀⠀⠀⠀⢱⠀⠀⠉⣿⣿⣅⢉⡙⢷⡀⢠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣮⣭⣷⣿⣿⣿⣿⢿⣿⣿⣵⡟⢀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⣿⣿⣿⣾⣻⣷⣿⣟⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣀⡉⣶⢶⣿⣧⢊⣿⣏⣹⣿⣿⠿⠟⢁⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠀⢬⡄⢻⣿⡿⢋⣾⠻⣿⣿⣿⠋⢿⣿⣿⡿⠠⣭⣟⣛⡶⡶⠶⠶⣶⣦⣤⣼⣼⡿⡇⠀⠸⣿⣿⣿⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠘⣿⣿⣿⣇⢸⣿⣿⣿⣿⠏⠀⢸⣿⣿⣷⠀⠀⢹⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⠓⠃⠀⢠⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠈⠈⠁⣩⣿⣆⡹⣷⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⡇⠀⣸⣿⣧⣿⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⣿⡿⠿⠿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣬⣹⣿⡿⠃⠀⣴⣿⣿⣿⣿⡇⠀⣻⣿⣿⣿⣿⣿⡟⠉⢹⣿⣿⣿⣆⡀⠀⠸⣷⣤⣤⣵⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣷⣀⣀⡉⠉⡇⢀⣼⡟⠛⢿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡉⠉⠰⢄⣿⣿⣴⣿⣿⠿⠿⠛⣛⣃⣨⣿⣿⣿⡟⠤⣾⣿⣿⣿⠿⠿⠛⢀⣜⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠚⠛⠛⠛⠋⣉⡀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠂⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣈⣷⣶⣤⣀⡀⠒⠤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠋⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣴⣺⣿⣿⣿⡲⠂⠀⠀⠀⠀⢀⣀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣿⣿⣿⣿⣿⣿⡇⣧⡄⢀⠀⢀⠛⢻⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⡿⠿⣿⣿⣿⣏⣵⡿⠁⣾⠇⡍⢰⣾⡁⠞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠒⡺⠟⠻⠑⢈⣛⡿⣇⣾⡇⢿⡿⢨⣥⣄⡁⠀⣠⣾⣿⢃⠖⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢃⣿⠐⢻⡏⡗⠀⡱⢶⠫⣿⠁⠼⠯⣻⢿⣯⡶⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⣧⣴⡷⣜⠷⢼⣯⢠⠘⣁⡀⣿⣃⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣺⣷⣶⣤⣼⣄⠚⢻⠿⢿⣸⣿⣦⣨⡇⠈⣾⠿⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣤⣾⣿⣿⢿⠿⡿⣿⡿⠋⠀⠀⠀⢻⣏⣿⣿⣋⣠⣴⣶⣾⣿⣦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠉⣉⠙⠛⠲⡦⠂⠲⠀⣠⣜⣷⡼⠉⠀⠈⢉⢙⣉⣻⣿⣆⠈⠓⣀⣄⣰⡦⠀⠺⢿⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠚⢿⣿⣂⡀⠀⢠⢐⣯⡝⣿⡁⠀⠀⣿⣿⠿⠛⠛⠛⡏⠋⢤⣤⣿⡿⣿⣅⠉⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣉⣙⢿⣿⣷⣤⣿⣿⣿⣿⣿⣿⠀⢸⠏⠀⠀⠀⠀⠀⠰⠀⠀⣿⣿⣦⣤⣝⣧⣬⢿⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢤⣘⣳⣾⠻⣿⡿⢿⣿⣷⣿⠇⣴⡟⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣧⣿⣧⣿⣿⣯⣥⣶⣟⠧⣤⣀⣀⣀⠀⠀⠀⠀⢠⣿⡟⠛⢿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢷⣼⣿⡇⣾⣯⣿⣿⠇⢹⣿⣿⡉⠙⢷⣶⣯⣭⣉⣻⣿⣻⣿⣿⡇⠀⢸⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠳⠘⣿⡟⣧⣹⣿⣿⡿⠀⢸⣿⣿⣧⠀⢘⣿⣻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠼⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠳⢆⠻⣿⣿⠇⢀⣾⣿⣿⣿⠀⢼⣿⣿⣿⣿⡿⠻⣿⣿⣿⡀⠀⠘⣿⠉⠉⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣆⠈⠛⠿⠏⠀⣾⢿⣿⣿⣿⣀⣾⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣦⠀⢹⣿⣿⣿⡿⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠻⠦⢁⣾⣏⣴⣾⣿⣿⡿⠟⢹⣿⣿⣿⠁⣼⣿⣿⣿⣿⡯⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠺⠛⠛⠉⠉⠀⠠⠾⠿⠿⠟⠛⠹⠸⠿⠛⠉⠉⠉⠀⠜⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢻⣤⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠿⠿⠟⠿⠖⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⣀⣤⡿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⡟⠃⠀⠀⠀⣾⣿⣝⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠘⡇⠀⠀⠀⠀⡍⠀⣿⡿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⣰⡷⠄⠀⠀⣀⢴⠧⠴⣿⠛⡁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⢻⣄⠙⠟⠀⡰⢢⢝⢳⢾⣃⣡⡇⠀⢰⠲⡆
⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠀⠀⠀⠿⢷⣸⡲⢊⣟⣺⡴⠸⠄⢈⡃⡔⢉⠛
⠀⠀⠀⠀⠀⣠⠁⢠⡷⠀⠀⠀⢄⠈⠈⡉⡉⢯⢠⣿⡑⢲⡶⣾⡱⣗⠁
⠀⠀⠀⠀⣴⡁⠀⢀⣄⡀⠀⠀⠀⡇⠐⢁⡈⢸⠊⠀⠉⠉⠀⡸⢷⡈⠓⢆⢄⣀
⠀⠀⠀⢸⠑⠁⠀⠾⢤⣴⡆⠁⠀⣱⡀⠘⣡⢂⡤⠤⠀⠊⠉⠀⠹⣧⡀⠀⡽⠀⠑⣦
⠀⠀⠀⢸⣇⠒⠓⣞⠉⠯⠔⣧⣞⡕⣷⣊⣠⣞⣀⣠⣤⣤⡤⡤⠼⠇⠑⢨⠃⠀⣴⠋⠑⣄⠀⢠⢤
⠀⠀⠀⠘⣏⢋⠢⡈⠣⡀⢠⣏⠁⠚⣖⣸⠀⢸⣦⣤⣤⡤⣴⣡⣤⣬⣰⠃⠀⣸⢷⣶⣶⠞⠷⠾⣾
⠀⠀⠀⠀⠘⡆⡇⢰⠀⡇⠀⢯⣬⠽⠛⠉⢤⠤⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣐⡇⠀⠉⢹⠀⠀⢠⠏
⠀⠀⠀⠀⢠⠃⣇⢈⣄⡛⡀⠜⠁⠀⠀⣠⣿⢦⣄⣀⣀⣀⣀⣀⣀⣤⣴⡀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠈⠪⣮⠗⠐⠁⢐⠀⠀⠀⣰⠛⢿⡀⠀⠀⠀⠉⠀⢹⠀⡿⣿⣷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡖⠱⣄⡀⢁⡖⠀⠀⠀⠀⢸⠆⠠⡞⠀⠙⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢸⠁⠀⠈⠳⢾⣨⠇⠀⠀⢀⠾⠄⢠⠇⠀⠀⣟⠀⢸
⠀⠐⠲⠒⠒⠒⠒⠒⠺⠧⣬⣠⣯⣤⣤⣤⡤⠤⠤⠤⠤⠤⠾⠶⠶⠾⠶⠶⠶⠧⢦⣼⠶⠶⠶⠶⠶⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠠⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⣠⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢄⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⠀⠀⠀⠀⠢⠆⠉⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡦⠀⠀⢠⠒⠀⠰⡏⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⡆⠠⣶⡗⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⣉⡕⢿⣴⣿⣿⡝⢿⠷⡀⡀
⠀⠀⠀⠀⠀⡴⠄⠂⠉⢻⣿⠟⣓⠲⣦⡖⠚⢻⡟⠝⠙⠉⠙⠁⢸⣿⣿⣿⢿⣧⣌⣁⠉⠳⠾⢦⣤⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣾⠁⠀⢾⣿⣄⠈⡄⠀⠀⠀⠀⣴⠿⢿⠋⠙⡈⢿⣟⠿⣿⣦⣄⡈⠉⠙
⠀⠀⠀⠀⠀⢀⡄⠀⠀⢾⣿⣻⠀⠀⠇⣿⣿⠀⠃⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⡿⠻⣷⣄⡀
⠀⠀⠀⠀⢰⣿⣿⣶⢿⣿⣿⣿⠀⠘⠀⣾⡿⠀⢸⠀⠀⢨⠀⡎⠀⠠⠀⠀⠀⠀⣭⠿⢿⡛⢳⠀⠘⣹⣿⣆
⡀⠀⠀⠀⠘⢟⠼⢣⣾⣿⣿⣿⣦⣈⣴⣿⢳⡆⢸⡀⠀⢸⢰⣿⡀⠀⠀⠀⠈⠀⠁⣔⢾⡇⠘⡆⠀⠊⣽⣿⣧
⠈⢕⣶⡀⢐⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⠏⢸⡇⠨⠁⢀⣀⣸⣿⡧⠀⠀⠄⠐⠀⠀⠊⡙⡇⠀⣇⠀⠐⢻⣿⣿⣇
⠀⠀⠈⠙⣿⣿⣿⣿⡿⠛⢻⠿⢿⠿⠋⠠⠺⡇⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠐⠀⠠⠉⣵⠀⣿⠈⠈⠉⣻⣿⣿
⠀⠀⠀⠀⣿⣿⠟⠋⠀⠀⠀⢦⣤⣤⡶⠚⠛⡧⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀⡀⠀⢚⣦⣸⣤⠴⠤⢿⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠙⠖⢲⣞⣛⣚⡚⢿⣿⣿⣾⣷⣿⣾⣷⣶⠚⠃⠙⢱⠀⠀⣾⣿⡟
⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⢀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣸⡄⣰⣿⣿⡇
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣏⢹⣿⡆⠀⠩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⣀⣤⠣⣴⣿⣿⣿⠁
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢹⠘⣿⣿⠀⠀⢹⣿⣿⠁⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢉⣸⣴⣿⣿⣿⣿⡄
⠀⠀⠀⢽⡇⢀⡀⠀⠀⠀⠀⠀⠈⠀⢼⣿⡇⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⡷⢿⡟⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⢧⣿⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⣿⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠘⣿⣧⣶⣶⣦⣤⣤⣦⣤⣴⣤⣤⣶⣶⣴⠃⠀⢸⣿⣶⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⠶⠿⢟⣤⣴⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⣻⣲⣞⣿⡿⠿⠿⠿⠟⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡇⢳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⣠⠞⠀⠈⢣⡀⠀⠀⠀⠀⢰⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⣇⠀⠀⢀⡾⠁⠀⠀⠀⠐⠘⢦⠀⠀⢀⡾⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠐⠼⠧⣄⣾⠀⠀⠀⢀⡀⠀⠀⠈⣧⣠⠾⠔⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠂⠢⡀⠀⠀⠀⠀⠀⠀⢠⠂⠲⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣠⠤⠶⣄⠈⠺⠅⠀⠀⠀⠀⠀⠀⠸⠂⠀⡤⠦⢤⣀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠘⡇⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡟⠀⠀⠀⢸⡆⠀⠀⣦⠀⠀⠀⠀⣄⠀⠀⢸⠀⠀⠀⠀⣧⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠄⠸⣅⠀⠀⠀⠈⡇⠀⠠⠹⡆⠀⠀⣸⠳⠀⠄⠘⠀⠀⠀⢀⣼⣆⣧⣀⡾⠁
⠀⠀⠀⠀⠀⠀⣸⣣⣬⣀⠀⠙⢦⠀⠀⡇⠀⠐⠀⢳⠀⢀⡇⠀⠂⡇⣭⠀⢀⡄⠉⢀⣨⣯⣻⡅
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠑⢦⡀⢳⣿⡑⣄⠀⠀⢸⡄⢸⠁⠀⠀⡴⢋⣷⠏⢀⠔⠋⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠻⠇⢸⠀⠀⠀⡇⣾⠀⠀⢰⡇⠻⠏⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⡏⠀⠀⠀⡇⣿⠀⠀⠀⣿⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠧⠟⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣤⣄⣀⣀⣀⣠⣀⣀⣠⣤⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣻⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⣿⣷⡟⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⡀⠀⠀⠀⠀⠀⠛⠚⠶⣶⢾⡟⢀⣀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠹⠋⡆⠀⠀⠀⠀⡀⠟⠀⠀⠻⣿⡿⠄⢉⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢇⠀⠀⠀⠙⢀⣀⣴⣾⣿⠛⢁⡌⢁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀⣟⠈⠀⠀⠀⠀⡜⢻⣿⣻⣿⣛⡍⢙⡶⢵⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡃⢁⣸⡄⠀⠀⡆⠀⠸⡬⠿⢋⠊⠹⣿⡧⢠⢾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠛⠀⣼⡁⢚⠖⠄⡆⣆⢀⢘⣿⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⡇⠁⠈⠀⠰⡆⠿⠯⣽⣿⣂⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣻⡅⠀⠀⠀⢈⣹⡙⠀⡀⠀⣀⣀⣠⣴⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⣀⠀⠰⣽⠄⣴⠋⠫⣿⣁⡠⠶⢊⠉⡙⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢂⠠⠠⠀⣭⣙⣷⣶⠬⠍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢸⣧⣤⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⢁⡀⠀⠊⣿⡛⠋⠀⠀⠀⠠⣤⣤⣴⣴⣶⣶⡶⡴⠾⢿⡍⢛⠫⣀⠀⡀⠀⢻⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠨⣶⣾⣤⣎⣤⡀⣯⠭⣿⣦⡄⠀⠀⢰⣏⡴⠋⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢹⣿⣣⡀⠀⠉⠙⢶⣄
⠀⠀⠐⠒⠚⣟⢛⡄⠀⢺⡟⠚⠁⠀⣝⡟⣲⣿⡿⠁⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⡿⣿⣆⠱⣦⡀⠙⣦
⠀⠀⠀⠀⠁⠉⠈⠀⢠⣿⠁⠀⠀⠘⣁⣤⣿⣻⠃⠀⣠⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣻⣧⣿⣿⣷⣾⣿⣷⣸⣇
⠀⠀⠀⠀⠀⣠⠆⣸⣿⠇⠀⠀⠀⠀⣩⡷⠿⠏⣠⣾⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠐⠛⣴⡿⠋⠀⠀⠀⠀⢠⡟⠒⠶⢶⣾⣯⣄⣹⣇⢀⣀⣠⣀⣐⣠⣤⣤⣤⣀⣀⣀⣀⣠⣸⣿⣿⡟⠀⠈⠙⠻⢿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⠃⠀⢈⣶⠃⠀⡼⠀⠀⢠⣺⣿⣷⡄⠿⡏⠈⣉⠙⠉⢉⣹⠉⣁⢉⣉⠉⠉⢉⡉⠉⡉⠉⠑⠀⠀⠐⣆⠹⣿⡇⢹⡇
⠀⠀⠀⠀⠀⠘⣿⡀⢀⢼⠏⠀⡰⠀⠀⠀⣾⡿⠋⠉⢃⠒⠀⠻⠻⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⠃⠀⠀⠀⠘⢻⡿⠃⠸⡇
⠀⠀⠀⠀⠀⠐⠮⠽⠶⠋⠀⢠⠃⠀⠀⢠⣿⠁⠀⠀⠈⣿⠀⠀⢸⡿⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣴⣿⠁⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⡿⠃⠀⠀⠀⢀⡟⠀⢤⣾⠁⠀⠀⠀⠀⠀⠀⣾⡿⠛⣿⣿⠏⠀⠙⣦⡀⠀⠈⢙⣿⡄⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢸⣧⠀⠀⠙⣆⢀⠀⠀⠀⠀⠀⡿⠀⠀⣸⡟⠀⠀⠀⠈⠻⡄⠀⠀⣽⣧⠀⠀⠀⢸⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢐⡟⠀⠀⠀⠀⠀⠀⠈⠳⢃⣤⣼⠃⢸⠀⠀⠀⠀⢠⠧⠀⢨⡟⠀⠀⠀⠀⠀⠀⣹⡄⠠⢸⣿⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠃⠀⠀⠀⢀⣞⠀⠀⣨⠁⠀⠀⠀⠀⠀⢀⣿⣷⡶⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠾⠷⠼⠿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡘⠹⣏⣭⡿⠀⠀⠀⠀⠀⠀⢜⡛⠛⡓⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣶⣎⠉⣠⠤⢄⡀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢶⣶⠿⢷⡾⣽⣉⣻⡿⠭⣉⣉⠩⣉⠙⢦⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⠊⣰⠋⠀⠈⢡⣾⠿⢛⣻⢦⣠⣤⣤⣀⠀⠙⢄⢃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠃⡆⢸⢠⢏⢃⣾⠟⠁⠙⠒⠲⢦⡉⢳⡄⠈⠢⠀⢳⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢴⣏⣼⠴⢊⠜⡠⣣⠏⠀⢱⢿⠀⠀⠀⠀⠀⠀⢱⡀⢣⣐⠠⢶⣈⡥⠶⣒⢲⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⢴⠶⠛⢉⡴⢞⡣⠵⠚⠁⡠⣢⡿⢸⠀⢀⣤⠀⠀⠀⠀⢳⠤⠈⠩⠉⠒⠈⢁⣀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣠⢁⡏⢀⡜⠁⣰⠋⠀⠀⢀⢾⣫⠞⢁⣾⣄⠀⠁⣠⣴⠶⠟⠻⠿⢶⡶⠖⠲⢟⠀⠈⡁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⢈⠀⣼⠀⠀⠇⢠⠪⣮⢖⡿⣷⡚⠿⣦⡉⠀⠀⠉⣰⠿⢿⣿⡖⢸⣿⡸⢀⢌⡴⢊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⢠⣿⠀⠈⡄⢸⠘⢡⣿⠀⠈⠛⠛⠛⠁⠀⠀⠀⠛⠛⠛⠋⠀⣼⢋⡵⢡⠊⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⠁⡞⣆⠀⢳⡈⢧⣆⢿⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢻⡾⠀⠸⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡘⢬⣢⡀⢣⠈⡟⡎⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⢷⡀⠀⠈⠙⠲⢤⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣎⠹⣿⡆⡆⡇⢸⠈⣧⠀⠀⠐⠋⣙⣛⡙⠒⠀⠀⣠⣿⣷⣝⠛⢟⠲⢦⠐⠀⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣜⣽⢁⠁⡸⠀⠟⣷⣄⠀⠀⠀⠉⠀⠀⣀⣼⣿⣿⣿⣿⠇⣸⡼⢸⠀⣠⠏⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⠝⠡⠂⣠⠃⡜⠀⣿⣿⣿⣦⣤⣤⣴⣾⡿⠿⣛⡻⣭⡷⠞⣉⣴⡥⢚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣏⡬⠂⣠⠔⡡⠊⢀⣴⣿⣿⣿⣿⣿⠟⢉⠴⠊⠁⡠⠚⣡⢶⣛⠭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾⢁⢾⡟⠗⠈⢹⣿⣿⣿⣿⣿⣿⣇⢠⡏⠀⢠⠞⣠⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠈⡜⠷⠶⠼⠾⠿⠿⢿⣿⣿⣿⣿⣆⢳⠀⡞⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠑⠲⠤⢄⣀⠀⠀⠑⠘⣿⣿⡿⢸⠃⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠚⠛⢦⠀⠀⢫⠒⠴⠋⢠⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡀⠑⠢⠐⢁⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣈⠐⢤⠶⡫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠤⠤⡀⠀⠀⢀⡀⠀⠀⠀⠀⡰⠁⢷⡀⠌⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣠⣦⠵⠒⠀⢢⡀⢠⣄⣩⡽⠁⢳⡿⢿⣿⣾⠇⡄⣁
⠀⠀⠀⠀⢀⠠⠀⠨⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢣⢬⣬⣽⠃⣾⡟⠀⠀⣼⣿⣾⣴⣈⣩⣶⣤⣀⠀⡀
⠀⠀⡤⣊⠐⣖⡂⠙⠂⠀⠀⡖⠨⠆⠀⠀⠀⠀⠀⠈⢯⣽⢇⣶⢻⣿⠶⠙⠫⠟⢿⡟⠙⠻⣿⠟⠻⠇
⠀⠀⢧⠉⡘⡎⢉⠀⠀⠀⠀⡀⠀⠂⠔⠈⠀⠐⠀⠀⠘⡧⠿⢿⡞⠁⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⡶
⠀⠀⠈⢆⠁⣽⡌⠀⠀⢀⠠⡀⠀⠄⠓⠐⠂⠔⠁⠈⠀⠹⠀⢰⡆⠀⠀⠁⠀⢤⢄⠀⣿⣶⣄⡀⡼⢁⡀
⠀⠀⠀⠈⠆⢟⣷⠠⠀⠀⠀⣈⠁⠀⠀⠀⠀⠄⢉⡀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠺⣿⣿⣿⣿⠿⣿⣆⡀
⠀⠀⠀⠀⠘⣤⢿⣧⠀⠢⠄⠤⠀⠀⠀⢀⠀⠀⠠⠄⠁⠂⠈⢦⣀⠀⠂⠄⢠⣤⣾⡅⣀⡞⣽⠿⢿⣷⣿⣿⠷
⠀⠀⠀⠀⠀⠘⡜⣿⣆⠀⠀⢐⢃⡀⠀⠀⠀⡀⠓⢄⠠⠀⠀⠘⣟⡳⣦⣤⣸⣟⣹⣿⠏⣼⠳⣶⣾⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⡌⣿⡄⠀⠀⠂⠀⠉⠈⡁⠀⠀⠄⠀⠀⠀⢀⡹⣠⠏⠛⠻⠻⢻⡿⣼⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡘⣷⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣠⠀⠀⢈⣀⠡⠈⠉⠉⢈⡁⣹⣏⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧⠀⠀⠀⠀⠤⠰⠀⠀⠈⠀⠀⠰⢀⣩⣄⣠⠄⣤⡈⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠛⡯⠙⡛⠹⢗⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠢⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠈⠨⠋⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣯⣽⠄⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡏⢻⠿⢷⣰⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦⣿⣿⣿⣼⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⢿⣿⣷⣿⣑⡙⢛⣿⡿⢿⣿⣿⣿⣦⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢶⡄⠠⠀⠀⣻⡛⣿⠛⣿⡧⠿⢿⡿⠿⠦⣸⡿⢿⡟⠻⢿⠀⣀⣾⠱⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸⠇⠀⠀⣘⠛⢛⠛⠛⣻⣿⣴⡄⣼⣶⢸⣾⡗⠂⠁⡘⠋⣀⠀⠛⣄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠁⠀⠀⠀⢹⡄⠘⠇⢠⡾⢷⡀⢰⡷⠀⣼⣿⣆⠀⢸⠁⠸⠃⠀⠀⠈⠻⢄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠍⠀⠀⠀⠀⠀⣠⣿⣲⣛⣹⢧⡴⣟⡐⢃⣺⣅⣠⣯⡔⣊⠻⢧⡀⠀⠀⠀⠀⠻⣿⡝⠁
⠀⠀⠀⠀⠄⡒⢀⠀⡇⠀⠀⠀⣠⣾⡕⢩⠃⣿⡟⢀⣶⡌⢉⡉⢩⣿⠇⣴⣿⠘⠁⢂⠹⡄⠀⠀⢀⡀⠉⠷⠤
⠀⠀⠀⠀⠀⡸⢋⠀⣻⠄⠀⠀⢧⡿⢦⣜⣻⣿⢢⣿⣿⡟⠦⠶⢛⡟⠀⣸⣿⠤⡄⠀⣏⠘⡆⠀⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠈⢻⠁⠉⣰⣴⠾⡿⢶⣧⢼⣿⡟⠈⢹⡇⠀⠀⢸⡇⢦⣿⣿⣙⣁⣤⢼⣿⣤⣤⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⠾⠷⠛⠮⢧⡴⠐⢊⡝⠉⠁⢸⠇⡇⢰⣿⠠⠷⣲⡿⠋⡉⠁⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠀⠀⠀⣀⢸⢽⠆⠹⡗⢐⡔⠉⠙⣿⠛⠉⠉⠁⠉⠀⠉⠁⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡦⢣⢨⣷⡆⣿⠀⣶⢠⣴⡟⢻⣶⡖⢤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡟⣿⢹⣾⣿⠁⣯⠀⣿⣴⡏⠃⣸⣿⡇⢦⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢙⡏⣿⠀⡇⣹⣿⣿⠙⢿⣦⡼⠛⢣⡖⠁⣿⣗⢺⠝⣿⣿⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢌⠑⡿⢀⠧⣾⣿⡏⠢⡀⢛⢶⣖⠊⠀⢀⣿⣿⡚⠆⠹⣿⣇⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠉⢀⢀⠀⡱⠆⡘⢲⣿⡏⢳⣅⠀⠀⢀⡈⠿⠿⣴⠛⣿⣏⢙⠀⢻⡜⠆⠀⠈⢿⢦⣄⣀
⠀⠀⠀⠀⡄⢰⡚⡍⠂⠀⠀⠀⡠⠍⠀⠠⣿⣿⠁⢠⠙⠻⢦⡀⠑⠀⠈⠔⠁⣿⣿⣅⠧⠀⠹⣄⠀⠀⣿⣸⡇⢸⠁
⠀⠀⠀⠀⠑⡀⢳⡘⢆⠀⢀⡤⠀⠠⠦⣼⣿⣁⡦⠀⣁⣀⢾⣿⣷⢤⣤⢶⡂⡟⢿⣿⠴⢃⠀⠈⢦⣬⠃⣸⠇⢸
⠀⠀⠀⠀⠀⠐⣄⠙⢦⣱⡞⠀⠀⠣⣼⣿⠉⣯⡶⡿⠏⣼⣿⡿⣿⢂⣫⠿⣴⡇⠘⣿⣷⠊⡄⠀⠀⠙⣔⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠃⢄⡙⠀⡀⠁⣜⣿⠏⠀⡇⠘⠀⢐⡾⢻⡇⠿⣃⠀⠟⢽⠀⠀⢹⣿⣏⢰⡄⠀⠀⢸⡡⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣉⣾⣿⠃⠀⠀⣧⠄⡰⠏⠀⠘⠁⠀⠻⢂⡀⡾⠀⠀⠀⢻⣿⣦⠘⡄⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠯⡽⠁⠀⠀⢀⡷⠮⢁⡐⢦⣤⠶⣶⢾⣯⡧⡗⠀⠀⠀⠀⢻⣿⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠠⣈⣞⣿⠿⢄⠖⣠⠛⠡⡝⠲⢤⣈⣉⣉⣉⣛⣹⡝⠒⠞⢣⡜⣁⠖⠀⢿⣿⣗⡊⣢
⠀⠀⠀⠀⠀⢀⠐⠀⣽⡟⢃⣳⡌⠶⢠⡜⠁⢶⡤⠤⠆⣽⠉⠹⣯⠙⠛⢶⡳⠆⣤⣉⠶⠂⣼⠻⣯⣥⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣾⣀⢉⡇⢸⠃⠎⣁⠶⣀⣀⠤⢿⠀⢰⠏⠑⠢⣼⣱⠛⣤⢉⡴⠐⠇⢀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢤⠇⡸⣉⠖⢈⡰⡥⠄⠰⠟⡦⠛⠶⠤⠤⡿⢡⡛⠰⢏⣐⠀⡗⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠠⠥⠉⠰⠦⠀⠆⠠⢰⠌⠱⠏⠠⠲⠴⠗⠠⢌⡁⠮⠉⠦⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⢦⣤⡠⣤⣤⡤⣤⣤⣴⣦⣀⡤⣦⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣷⣀⣡⣼⣿⣤⣼⣿⣤⣤⣿⣯⣤⣼⣥⣬⣿⣤⣿⣧⣼⣤⣿⣤⣧⣤⣥⣽⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⢀⡐⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠐⢋⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡠⠐⡫⢥⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢔⠀⣄⠀⠐⠂⡄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⡈⠁⠚⠫⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠘⢫⣤⠄⠰⣀⠀⠘⠠⢄⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠤⢌⠡⢂⣰⡦⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⡆⠀⠀⢊⡀⠉⠐⣀⠀⡊⠀⠁⠀⠀⣀⠀⠀⠀⠐⠂⠈⠃⠔⠦⠘⡌⠀⠀⠀⠼⠿⠁⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⠀⠀⠈⠀⢀⡴⠊⠀⠀⠀⢏⠀⠁⢂⠴⠁⠀⠁⠐⢢⡀⠀⠀⠀⠀⠉⠀⠀⠠⢤⣄⡀⠀⠀⠀⠰⣧⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⢀⠁⠠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠴⢦⡀⠀⠀⢀⠀⢀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣄⠠⠆⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠘⠆⣄⠀⠀⠀⠀⠠⡆⠳
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⢁⠀⠀⡜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⢠⠧⡈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠤⠎⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⡠⠂⣰⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠴⢶⡖⢠⠿⠯⠐⣿⠀⡁⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠂⣠⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⢲⣤⠼⡃⠤⠀⣴⠒⣀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⠁⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⠃⠀⠀⠐⠉⠱⠀⠀⠀⠐⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡔⠀⠀⠀⠀⠁⠑⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠓⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠐⠀⠛⠀⢠⢀⠀⠁⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠀⡀⢀⣈⡀⢀⠖⠠⠌⠂⠀⣀⣀⣀⡀⠞⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠉⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⣄⠁⢀⡾⢋⣬⡙⢧⡀⣶⣀⡴⠋⢀⣨⡉⠠⣄⢠⠀⠀⠀⣿⠀⠀⠀⡰⠁⠠⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠐⠓⠀⠀⠀⠈⢿⡷⣤⣤⣤⣤⣀⡋⡴⠊⠉⢶⠌⣷⢸⡟⢀⡰⠋⠉⠉⠳⠊⡠⠷⢴⣿⡟⠀⠀⠀⠀⠈⠀⠀⠈⣠⠆
⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠙⠛⠣⠳⠞⠞⠁⠁⠀⠀⠀⣾⢙⣷⠁⣸⠁⠀⠀⠀⠀⠀⠛⠓⠓⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⠄⠰⢷⠿⠛⢟⠱⠙⠃⡰⠖⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⢠⡤⠀⠘⠠⠀⣀⠀⠀⠀⠀⣤⣤⣄⠈⡁⢆⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠏⡤⠤⠴⠾⠀⠀⣁⠀⠀⠈⠛⠁⠀⡀⠀⠁⠀⠈⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤⢤⠶⢦⣴⠄⠀⠀⠀⠀⠀⠀⠁⠈⠴⠿⠀⠀⢀⠶⠤⠀⠀⠀⠀⠀⠀⠀⣀⡠⡴⠛⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠁⠀⠖⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠄⠈⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠘⠣⣄⡀⢀⣠⡀⠀⠈⠅⠀⠠⠀⣞⠁⠻⠀⠀⠀⡸⠀⢐⠀⠀⠀⠀⠾⠀⠀⠂⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⡀⠀⠀⠂
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠛⠁⠀⠙⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠎⠉⠀⠀⠠
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠊⠀⠀⠄⠀⠀⠀⠰⠄⠀⠀⠀⠠⠀⠀⠠⢀⡀⠀⠀⢀⡨⠁⠀⢤⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⣈⢩⣦⣔⠀⡄⠀⠂⠰⢀⠀⠀⠀⢀⣄⠀⠀⠛⠈⠁⠈⠃⠈
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠈⠀⠀⠀⠀⠐⠀⢀⡀⣾⣥⡟⢲⡾⣉⣽⣾⢿⣾⡖⣿⡫⠙⠖⠀⠈⡂⠒⢀⠈
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣀⠾⠛⠁⠈⠙⠛⠁⠟⢀⠛⠀⠙⠉⠀⠛⡆
⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠳⠀⢀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢀⡀
⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠰⠊⡽⣩⡌⢥⣿⡉⠀⣀⡈⢉⡁⠀⢀⢀⡄⠀⢡⠀⠁
⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠘
⠀⠘⠁

⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠐⢠⠀⠀⠀⠊⠉⠉⠀⡀
⠀⠀⠹⣿⣿⣿⡷⠀⠀⠄⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡾⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⣿⠂⠀⢀⡔⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠦⠀⠀⠀⠼⣿⡆⣠⣿⢉⢓⣀⠀⠀⠀⠀⠀⢠⣴⡦⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠀⠀⠀⠀⣸⣧⢋⢇⣼⣿⠇⠀⡀⠠⠤⠶⠛⠉⠀⠀⠀⠀⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⣿⣿⣿⣾⡏⠁⠀⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⡿⣿⣿⠃⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣇⣹⣾⣿⣿⡀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⢀⡴⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢘⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣄⣀⣠⣀⡀⠀⠀⠀⠑⠾⠋
⠀⠀⠀⡇⠂⡒⢤⣄⣀⣠⡀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⢿⣤⣮⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠂
⠀⠀⠀⢸⡉⠉⢉⠻⠋⠁⠀⢀⣤⢤⣖⣋⠉⠀⠀⠀⠀⣀⢐⠀⢠⣾⡿⠄⢠⣾⣏⣀⣀⢀⢲⢩⣵⡗
⠀⠀⠘⢾⣷⣄⠘⢿⣷⣤⣾⣿⠿⠿⣿⠋⠀⠀⠀⣀⡐⠉⠸⢀⣄⣿⣿⣗⡟⠘⣿⠁⣹⡿⠛⢷⣆
⠀⠀⠀⣺⣷⣿⣿⡺⢿⣿⣿⠋⢾⣿⣿⣦⣴⡂⠀⣿⣛⠛⠂⠀⣿⣟⣏⡙⣿⣽⣿⣿⣴⣧⣤⢶⡌
⠀⠀⢀⣡⡼⣿⣿⣿⣾⣿⡿⢿⣾⣿⣟⣿⠋⠀⠀⠻⢿⣷⣌⣾⣿⣿⣿⣿⣋⣙⣿⡮⣤⣯⡶⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷⠟⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠁⠀⢀⣠⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢶⣶⣶⣦⣤⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠛⠉⠉⠉⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀⣀⣀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⣠⣶⣿⣿⣿⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⠿⠛⠉⠀⠈⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢻⣿⣿⣶⣤⣤⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣤⡂⠀⠀⣰⣷⡀⠀⠀⣢⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣆
⠀⠀⠀⠀⢠⣶⠾⢶⠂⠀⠀⠉⠛⠋⠈⠐⠗⠈⠈⠛⠋⠁⠀⠀⢠⣶⠛⣿⣿⣿⣶⡄⠀⠀⢰⣿⣿⡿⣿⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠾⠛⠛⠻⠞⠁⠀⠀⠀⠰⢛⣹⣿⢿⡿⡟⢁⣗⠀⠀⢸⣋⣿⣿⡿⡇
⠀⠀⠀⢀⠔⣛⣣⣤⣤⣴⣶⣶⣤⣄⡀⠀⠀⡀⢀⡀⢠⠀⠀⢰⣾⠏⠁⠀⠀⢂⣾⣯⡇⠀⢸⣯⣿⣿⣿⡇
⠀⠀⢰⣿⣴⣿⣿⣾⢿⣿⣿⣿⠿⢿⣟⣦⡀⠻⣬⣥⡾⠀⠀⢸⡇⠀⠀⣀⠀⡼⠃⡉⠇⠀⠈⣿⣿⣿⣿⠃
⠀⠀⣾⠻⠍⠈⠉⢻⠛⡁⠙⠁⠀⠈⠉⠈⠀⠀⠀⠀⣀⡀⠀⢸⣿⡃⠤⠩⡊⢱⣟⡾⠇⠀⠀⣿⣿⣿⠃
⠀⣰⣿⠿⠚⣿⣦⠈⢸⣿⠏⠻⢿⡆⠀⠀⢀⢄⣀⣤⠞⠁⠀⡘⠉⠛⠚⠃⠦⠞⠉⠀⢀⡀⠀⠈⢻⠋
⠀⠘⠿⠆⠀⠈⠻⠶⠈⠻⠦⠀⠘⠁⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⡆⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⠋⠁⣠⣄⣧⠤⠀⢄⠁⠈⣉⢐⣂⡞⠰⣴⣾⣷⣶⣶⡀⡉⠂
⠀⠀⠀⠀⠀⠀⡀⠀⡂⠁⣀⡴⠶⠛⠉⠁⠀⠀⠀⠀⢇⢘⣉⣻⡟⢠⣶⠏⠀⠉⣷⣧⣞⣨⢛⢂⣤⣀⡀
⠀⠀⢀⠤⠂⣥⠤⠐⡎⠉⠀⢀⠤⢰⡀⠀⠀⠀⠀⠀⠈⣞⢻⡟⢀⡼⣿⣦⡄⠾⢿⡿⣽⡿⠻⣿⣿⣿⣿⡦
⠀⠀⢇⠛⠂⣇⠺⠂⠀⠀⠀⠈⠁⠀⠀⡠⠐⠀⢀⠀⠀⠘⡟⣥⣿⣥⠟⠁⠀⠀⠁⠀⠑⠁⠀⠈⠛⠀⠀⢡
⠀⠀⠘⡆⠳⢸⡄⠆⠀⠀⠀⠐⠀⢀⠀⣅⢀⡀⣁⠆⠠⠄⢻⡀⠈⡍⠀⠙⠳⠀⠀⠀⠀⠐⣤⣀⠀⠀⢠⠯⠀⢀
⠀⠀⠀⠘⡄⣻⣷⡀⠀⠀⠂⠉⢠⠊⠁⠀⠀⠀⠑⠤⠀⠀⠄⢆⠀⠱⠀⢀⠀⠀⠀⠙⠂⠀⣿⣿⣷⣦⣯⣴⣷⣄
⠀⠀⠀⠀⠸⡜⢺⣧⠁⢀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠈⣄⠀⠀⢈⠀⠀⠀⠀⣠⠂⠈⣻⢿⣿⣿⣏⣻⣿⣶⡠
⠀⠀⠀⠀⠀⠹⡝⣿⣗⢀⠉⠈⢣⠀⠀⠀⠠⠀⠀⣈⠁⠀⠀⠀⡘⣷⣄⡀⠈⠀⢶⣿⣿⣴⡴⢣⣟⣉⣹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠱⡘⣿⡄⠀⡀⢂⡰⠂⡄⢤⠄⠂⠀⠱⠈⠀⠀⠀⢹⡦⠹⣿⣶⣿⣦⣾⡿⢡⣿⣨⣹⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠱⡸⣷⡀⠀⠁⠄⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⡆⢣⠘⡄⢄⠉⠩⠿⢧⣿⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⢴⠌⠀⠀⠀⠀⢴⢀⣞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣇⠀⠀⠀⠀⠐⠘⠀⠈⠀⠀⠀⢐⣃⣤⣶⣤⡖⠀⣶⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢝⠀⣩⠁⠛⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠢⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄⠀⠰⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠁⠀⠀⠀⠈⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣓⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠀⠠⢠⣬⣤⡤⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠀⠇⠀⠀⠀⢷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠰⠹⠃⠿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠀⣬⣷⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⢀⣠⣤⡘⠰⠇⠀⠀⠀⠀⠎⡷⣄⡀⠀⠀⡼⠀⢷⡀⢄⣰⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⠦⠀⡈⠉⣷⠀⠀⠈⠀⡇⠀⠂⠤⠜⡀⢮⠓⢨⠁⠐⣆⣸⠃⠀⠉⠗⢠⣎⢀⡠
⠀⠀⠀⠀⠀⠀⠁⢀⣬⠄⡻⠤⠁⢰⢠⠔⠉⢀⣄⠀⠀⢻⠀⠀⠂⠳⣾⠄⠈⠁⠀⠀⣽⣇⣀⢀⡀⠠⢃⠿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣴⢀⣠⡴⠆⠢⡘⠛⠡⠀⢸⠏⠀⡐⠉⠀⠀⡄⡠⠀⢨⣩⠃⠻⠆⣹⢦⣍⠀⠈⡃
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠋⠙⢉⡀⠤⣀⠈⠁⠀⠒⢉⣲⣿⡣⡀⡠⢀⡇⠀⠐⢥⣊⠇⠀⡔⠁⠘⠟⠟⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⡄⣰⡊⠁⠲⠬⠭⠅⠤⡄⠉⢀⠀⠙⠺⣵⡿⠁⠀⢠⠏⠉⠀⠜⠀⠀⡈⠲⣷⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡯⡤⠀⠀⠀⠀⠀⢰⢡⠠⠘⠃⡠⠀⢸⡇⠀⣰⠆⠀⢠⠂⠚⠀⠠⠂⠀⠀⠹⣤⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡜⡶⣡⠃⠀⠀⠀⣠⠚⠈⠀⠻⠃⠀⣀⣾⣪⠷⠯⠄⠀⠻⠀⣿⠃⢤⣧⠀⢀⠀⢈⣹⣿⠧⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠼⣱⡣⠆⠀⡐⢉⡁⠀⠀⠀⠀⣁⠀⡟⡣⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠤⡌⢼⣿⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢀⡾⠻⢼⣢⣋⡁⠀⢀⣞⡢⠏⠁⠀⠀⠀⠀⠀⠀⠑⠒⠶⠢⣴⣶⣷⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡤⡤⡄⢀⠀⠠⢠⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠿⠿⠛⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣴⠇⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣏⣾⣧⣀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠛⡟⢯⡛⡛⢏⠂⠀⠀⠀⠀⠀⠀⠀⠘⠙⣾⣷⡋⣿⠀⠀⠁⠀⠠⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣶⣨⣆⡓⠄⠸⠀⠀⢠⠀⠀⠀⠀⢀⢰⣶⣏⣿⡷⣶⡤⣄⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣼⣿⣿⡿⢩⡇⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⣆⢻⣿⡿⠁⣾⣧⡹⢧⡀⠀⠠⠀⠠⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡇⢙⡿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⣶⣿⠙⣿⣿⡷⠀⠀⠀⠀⠈⢹⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢨⣿⣿⣿⣷⣤⣴⡼⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢲⣶⣤⣿⣿⣏⠀⠈⢻⣷⡀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠈⠀⣿⣿⣿⣿⡟⣅⣾⣿⡇⠀⠀⢀⣀⣄⠀⠈⡘⡆⠀⡸⠞⢿⣿⣽⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⢠⣀⡄⠀⠠⠀⠀⠉⠀⠀⠠⠟⠉⠀⠡⢂⣽⣿⣿⡇⠀⠀⢸⣉⠉⠄⠀⠸⡀⢀⠃⡘⠚⠉⢩⢿⡟⣷⠀⠀⠘⢿⠄⠀⠀⠀⣠⡍
⠀⠀⠀⠀⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⠉⣿⣷⠀⠀⠀⠇⠀⣶⡆⠀⣅⢨⠀⣵⠈⢰⢘⡭⡇⡟⠀⠀⡉⠈⢻⠂⠀⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠞⢩⡿⠿⡿⡟⠀⠹⣷⣄⠀⠀⡀⠀⠀⢠⡀⠹⡇⠀⠹⠼⣒⡉⣈⠇⡇⠀⠀⣉⣹⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠱⣄⠱⣾⣦⠀⠘⠻⣴⣄⣁⠈⠀⣾⠳⠀⣿⡄⡀⢸⠖⢹⠉⢸⠁⠀⠀⢹⣿⢿⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⠈⠳⠿⣿⡇⠐⠂⠘⢿⣿⣿⠒⠻⡿⣶⢿⣿⣴⣾⠀⠘⠴⠇⠀⢸⡀⠉⠇⢈⠀⠉⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣷⣿⣿⠷⣄⡀⠈⣿⡀⠀⠀⠀⣷⡄⡀⠀⢱⡜⣾⠛⠛⠃⠀⠀⠀⢸⣦⡼⠏⢠⠀⢈⡀⢠⣷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢘⠁⢹⡟⠀⠈⠙⣦⢨⣷⠀⠀⠀⢹⡻⣷⡄⠘⣿⣏⢰⡀⠀⠀⠀⠀⠈⢿⣿⣦⣶⡤⠾⠀⠈⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠊⠀⡞⠀⠀⠀⠀⣘⡃⠏⠀⠀⠀⠀⣿⣻⣷⠀⠉⢉⡜⣿⡌⠁⠀⠈⠀⠈⣿⣿⠁⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⡤⠖⠚⠉⠀⠀⠀⠀⠀⢸⠃⣷⠈⠐⠛⠚⠋⠻⢁⣀⡴⢿⡿⠃⣯⣦⣬⠉⣷⣄⠸⣽⣿⣇⣀⡀⣿⣧⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠿⠿⠛⠻⠿⠏⠀⣿⣿⣿⣷⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀

⣠⣤⣤⣤⣤⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣠⣤⣤⣀⣀⣄⣀⣀⣀⣀⣤⣤⣤⣀⣠⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⠀⠀⢠⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡇⠀⠀⠀⣸⡁⣀⣲⠶⣆⡠⢸⣀⠀⢨⢿⠯⡀⠀⠐⠀⢰⡵⢮⣄⡀⠀⠀⠀⠀⠀⣶⡆⣿⣧⠀⣤
⠀⠙⠒⠂⠀⠑⠬⣀⣌⣊⣠⡀⠵⠃⠀⠀⠈⠀⠀⠂⠀⣐⠥⠄⠀⡤⢳⠈⢁⡀⠀⠀⣿⡇⣿⣿⢸⣿⠄⠐
⠀⠀⣤⣴⣄⡀⠘⢾⠻⠿⢫⠜⠀⢀⣀⣀⠀⠀⠀⢈⠠⢒⡀⠀⠀⠀⡘⠃⠠⠀⠀⠀⠛⠛⠂⠀⠘⠛⠃⣖
⠀⠐⣿⣞⡽⠂⠀⡸⠀⠀⠸⡄⠀⢇⣸⣷⡛⠓⠀⠀⡐⠉⠘⡳⠒⡋⠡⡄⠀⠀⠀⢀⠀⢸⡆⠀⣦⠀⠀⠿⠆
⠀⠀⠀⠈⢀⠤⠊⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⠀⡀⠠⡶⢄⠀⠀⠀⡀⠃⠀⠛⠀⢰
⠀⠀⢀⠞⠁⠀⠀⢐⡆⡀⡦⠄⠀⠀⠈⢢⡀⠀⡐⠀⠀⠀⠀⠀⠀⢰⠁⠀⢻⣼⣧⠸⠀⠋⠀⢀⡀⠀⠈⡄
⠀⢠⡏⠀⠀⠀⠠⠲⡼⣚⣕⡃⡇⠀⠀⠀⢳⡀⣆⣀⠀⣀⣠⠀⠀⠈⡀⠀⠀⠋⠉⠃⠀⠀⠀⢶⠦⠀⠀⠀⠇
⠀⣼⠀⠀⠀⠀⠠⠱⠒⠚⠶⠖⠁⠀⠀⠀⠀⡇⢉⡉⠶⢙⡁⠀⠀⠐⠗⠀⠀⠸⠀⢴⠄⠀⠀⠒⢀⡀⠀⠂⠆
⠀⢿⠀⢀⢀⡀⡀⠀⠀⠁⠁⠀⠀⠀⣠⠀⠀⡇⠘⠓⠀⠚⠀⠀⠀⠈⠉⠀⠀⠀⠡⡀⠀⡀⠀⠀⠀⢠⣤
⠀⠸⡆⠀⠀⠄⢉⠠⠀⣄⡀⠄⡨⠠⠑⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠢⠀⠑⣄⠀⠀⠀⠀⠙⢋⠌
⠀⠀⠱⣕⠀⠀⢀⠈⠂⠉⠐⡀⡀⠀⠀⡠⠃⢀⢄⠀⠀⠀⠀⢸⣶⣦⠉⠁⣴⣶⠀⠀⢸⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠈⠓⠤⣀⡀⠁⠀⠀⢀⣀⠴⢊⡔⣸⠏⠈⢿⡑⣀⠀⠀⣻⠇⠀⠀⢸⡏⠀⠀⡀⠀⠀⠖⠄⠀⠇
⠀⡃⠀⢘⠀⠀⠀⠉⠉⠉⠉⠁⠀⢠⣊⣿⠃⣠⣤⡀⠻⣿⣠⠀⠈⢶⡆⣶⡖⠁⠀⠀⡇⠀⠠⣤⠄⠀⢀
⠀⠼⠛⠥⠀⠐⠊⠉⠐⡀⠀⠀⠀⡜⣰⠃⠈⠬⠥⠀⠀⠹⣘⡆⠀⢸⡋⢹⠁⠀⠀⠈⠑⠒⠒⠒⠒⠒⠋
⠀⠀⠒⠀⣰⡀⠀⠀⠀⣤⡀⠀⠀⠸⠥⠤⠤⠤⠤⠤⠤⠴⠽⠀⠀⠀⣼⣸⠀⠀⠀⠀⢀⡄⢒⣋⣉⣉⡝⠙
⠀⠀⠀⢸⡉⠃⢀⡄⠀⠍⣱⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⣒⠤⡀⠀⠀⢸⡇⠀⠀⠀⣠⢇⣴⣾⣉⣇⣟⡅⡀
⠀⠀⠀⠀⢧⣤⠀⠃⢠⣤⠆⠀⡀⢗⡀⠀⠀⠀⠁⠀⠀⢸⣕⡈⢆⠀⢸⡇⠀⠀⠀⡟⢰⡏⣺⣿⣿⣿⣷⡟⡆
⠀⠀⠀⠀⠈⠹⠀⠀⠨⠁⠀⡀⠀⠀⣤⣤⡀⠀⠀⠀⠀⢦⠃⠆⡸⠀⠀⢀⠀⠀⠀⢇⢘⣿⡿⣿⣿⠿⣿⢋⠇
⠀⠀⠀⠀⠀⠀⣜⣻⡄⠀⠈⠊⠈⠀⠿⡾⠇⠈⠦⡀⠀⠎⣊⠔⠁⠀⢘⣱⡾⠀⠀⠈⠣⢼⣙⡓⠓⣛⣩⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠋⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⡀⠀⠀⢀⠜⠅⠁⠀⢠⣀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠳⣄⠀⠀⢀⢀⠀⠀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡅⠀⡀⣼⡯⠄⠛⠀⢀⠀⠀⠀⠈⠈⡒⢲⡉⠀⠀⠀⠀⣀⠄⠚⠂⢽⣯⠀⠀⢀⠇⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡁⠀⠀⠿⣷⡤⠀⠀⣀⣈⣦⣤⣤⣾⣯⣾⣷⣦⣤⣤⣈⣀⠀⠀⢤⣾⠯⠀⠀⢈⠀⡔
⠀⠀⠀⠰⠄⠀⠀⠀⠀⠇⡐⢀⠀⠀⢁⣤⠄⠀⠀⠀⠘⡩⢿⢁⣀⣀⣈⣿⢯⡃⠀⠀⠀⠠⣴⣉⠀⠀⡀⠐⡰⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣻⠛⠛⢡⣄⣠⣤⠄⣬⢴⣿⠆⢬⣧⢰⣿⣦⣤⡢⣤⣄⣠⣄⠛⠛⢿⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡨⠀⠀⠈⠀⠁⠀⠀⠘⠽⠓⢔⣓⠖⠿⠃⠀⠀⠈⠀⠀⠀⠀⢍⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⢀⡀⣠⠀⠜⢀⠃⠐⡃⠘⡆⠱⠀⢄⢀⣄⡀⣀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢭⡀⠀⢸⠤⢅⡀⠤⡕⠀⢀⡩⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢗⠂⠀⣘⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁


⣶⣖⣲⡶⢶⣶⠶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠈⢿⣿⡿⠋⠀⠀⠀⢀⣠⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⣄⣀⣉⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠃⠀⢀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣤⣀⣀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠁⠀⠠⠄⢀⣨⣴⣶⠆
⠀⠀⠀⠀⢀⡈⢠⡁⡢⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⢀⣤⠾⠋
⠀⠀⠀⠀⠀⢼⣦⣼⣿⡶⢉⡖⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⠀⠀⠂⠀⠁⠀⠀⠀⠀⢀⡴⢋⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠜⣿⣿⡽⠷⢿⠃⡈⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠈⢀⣤⡐⠀⠀⠀⠀⠀⠀⢀⡤⢻⢶⢨⠻⠒⠊
⠀⠀⠀⠀⠀⠀⠓⣿⣉⣱⣖⡾⠇⠤⠀⠀⢢⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡀⠈⠁⠁⠀⠀⠀⠀⠀⠐⠛⠀⠼⢤⢀⡀⢠
⠀⠀⠀⠀⠀⠄⠀⢈⣹⣮⡭⣷⡠⠀⠠⢰⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠐⠀⠀⠀⠀⠀⠀⣠⣴⡀⠀⠾⠶⠐⠂⠐
⠀⠀⠀⠀⠀⠂⠀⠀⠹⢟⣷⣚⠳⠁⠘⢦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⡀⣀⢀⠈
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠙⣭⣭⡄⠐⢻⣏⠐⠈⠀⠀⠀⠀⠀⠠⣀⠂⡡⠀⠀⠀⠀⠀⠬⠉⠁⠘⠀⢠⠯⠠⠴⠀⠀⠀⠄⠀⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠷⡇⠀⠘⣿⠆⠡⠀⠠⣠⠀⠀⠀⠛⠈⠓⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠋⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠿⡀⠀⠁⠀⠀⠀⣀⠐⠀⠀⠀⢰⠀⠄⢐⡄⠀⠶⣤⠤⠄⡐⠂⠀⢆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⢛⣟⡉⠉⠉⠀⠀⢀⠌⢁⡦⠒⠦⠽⢁⡀⠀⠀⠳⠀⠀⠈⠛⠉⠉⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡀⠀⠀⠀⠀⠀⠁⣀⠤⠒⠁⠀⠀⡛⠲⣼⣦⠄⠀⠙⢆⠳⠆⡀⠐⠰⠀⠊⢁⠀⠀⠀⠈⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠄⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢀⡗⢦⡅⢸⣧⠁⠀⢄⠀⠈⠓⠀⠖⠈⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⣍⣉⣉⢩⡿⡀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠘⣾⠯⠤⣈⠛⠍⠩⠄⠀⡴⠁⠀⠀⣤⡀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢻⠶⠖⠦⠛⠆⠐⠂⠀⠋⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⣤⣀⣀⠴⠞⠆⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠁⠀⢀⠀⠀⣎⠾⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠀⢷⣄⡀⠶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⡜⠀⠀⠀⠀⠀⠀⢸⠏⠀⢻⣶⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⠐⠃⠀⠀⢰⠆⠀⠈⠋⠁⠦⠀⠀⠀⠄⠀⠀⠠⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢰⠆⠀⠀⢘⣿⠀⠀⢡⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠀⠀⠀⢸⠆⠀⠀⢘⣥⠀⠀⢸⡇⠸⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⠄⠠⠴⠠⠖⠀⠾⠀⡍⠀⠀⠀⢼⡀⠀⠀⠀⣳⠆⠀⠘⡇⠀⠠⠆⠻⠧⠀⠄⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⣐⢀⣀⣴⠄⠀⠀⡀⡿⠣⠀⣀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠁⡀⠉⡉⠂⠀⠀⢡⠀⢤⠀⠀⠀⠀⠀⢀⡠⠈⠁⠀⠰
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠁⠘⠁⠠⠀⠃⡈⢀⠘⠑⢀⠀⠠⡜⠫⠃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢠⠉⠀⠊⠐⠀⠡⡄⠀⠶⠀⢠⠤
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠡⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠘⠀⠀⠀⠀⢶⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠶⡄⠀⠂⣠⣄⠀⠀⢨⡠⠄⠁⠀⠀⠀⣤⠀⠀⠀⠈⠈⠀⠀⠀⠀⢴⡄⠀⠀⢲⠂
⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠿⠀⢀⣶⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠠⠄⠐⠀⡘⠒⣈⠉⠠⠙⠃⠀⠀⠀⠀⣾⡄⠀⠿⣁⠀⠀⠀⠁
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣤⠀⠛⢷⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠁⠀⠀⢸⠟⠃⣤⣀⠈⠏
⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠽
⠀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠠⢀⣄⠸⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⢰⡿⠁⢰⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⠚⠣⠾⢇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠇⠀⠠⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣦⣴⠶⣤⡄
⠀⠊⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣮⡇⠀⠈⠁⠀⠲⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢧⡀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠠⣿⢀⠁⣀⠙⣿⣿⣿⡞⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠛⢿⣿⣅⠉⠀⠀⢀⣰⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠀⢙⣿⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡇⣀⣀⣨⣇⡁⠀⠀⠀⠀⠀⠀⠰⣶⡄⢀⠀⠀⠀⠀⠈⣾⣿⣦⣤⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠙⣿⣿⣃⠈⠀⠀⠀⠀⠀⢠⣀⠹⠟⣾⠗⠀⠀⠀⠀⡟⠻⠿⠿⣧
⠀⠀⠀⢙⠿⣧⡀⠀⠀⠀⠀⠀⠙⠇⢀⡑⢆⡈⠀⠀⠀⢠⠀⠠⣿⡇⢀⣠⣦⣄⢠⡤⠚⠁⠀⠀⢀⣈⠇
⠀⠀⠠⡎⠀⠈⠁⠀⠀⠀⠀⠀⠋⢠⠞⢿⣄⠁⠀⠀⢠⠄⢀⡄⠉⠀⠻⠙⢻⠁⠀⠀⠀⠀⠀⠀⠀⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢷⢄⠈⠀⠀⠀⡆⠠⢅⣹⣤⡄⠘⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡄⠈⠀⠀⠀⠺⠀⠀⠀⠸⠿⠃⠀⡾⠁⠀⠰⠦⠀⡀⠀⠀⢢⡀⠄⠄
⠀⠀⠀⠀⡜⠀⠉⠀⠀⠀⠖⣠⠗⠈⠀⠀⠀⣀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠸⠿⠦⠀⡀⠋
⠀⠀⠀⠀⠱⠀⠀⠈⠋⣀⡰⠀⠀⠉⠀⢠⠞⠉⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣄⢰⣶⣦⢔⠰⡒⣤⠄
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⡈⢉⣻⢿⠟⠛⢻⢦⠋⡉⠈⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢠⣿⡶⣷⣶⣾⠢⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣀⣀⣀⠄⠀⠀⠀⠀⠿⠿⠋⠉⢿⣰⡿⠿⢿⣄⠁⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡈⢡⡆⠀⠀⠹⠀⠐⠠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣀⢦⣴⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠈⠃⠀⠻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠠⠶⠀⠀⠀⠠⣎⠉
⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣧⡀⠈⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠐⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠐⠶⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠶⢀⠀⢀⠀⠈⠋⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⣤⠂⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠊⠻⠇⠄⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⡀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡁⠐⠀⠉⠀⠐⠀⠂⠐⠷⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠂⣠⡆⠀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠀⠀⠈⠁⠀⡄⠂⣈⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠁⠒⠀⡉⠀⠀⢀⡄⠀⠀⠦⠀⠀⠰⠟⠀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣀⣐⠨⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⡀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠹⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣴⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠙⠻⠟⠙⠟⠻⣟⠟⢻⡿⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠐⠫⣤⣤⢤⣀⡀⠀⠀⣒⣀⠀⢀⠘⠲⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠉⠀⠀⠰⠆⠀⡉⢀⡠⠀⠀⢠⠉⠱⢧⡀⠀⠴⠓⠘⢇⡀⢈⡒⠄⣉⢳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢠⡸⣷⠀⢀⡠⠀⢠⡲⣷⠶⠋⠁⢀⣤⠿⠀⠤⠬⣧⡄⠀⠀⠠⠄⣈⢂⠱⡄⠀⠉⠙⠳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣮⡁⣻⡞⣤⣶⣿⠾⠉⣴⠤⣄⠠⢂⠀⠀⠀⠀⠀⠨⡦⠀⠙⠐⢠⡀⠈⠑⢷⡀⢀⠀⠀⢿⣾
⠀⠀⠀⠀⠀⡄⢀⠷⠈⢻⣿⢟⡿⠋⠄⢀⡊⠔⠒⠐⠦⠉⠁⠀⠀⠀⠀⠀⠚⢃⣶⢀⢸⡄⠐⠄⢸⡏⢻⣄⠀⣜⢿⡀⣰⣆
⠀⠀⠀⣶⠀⠇⠈⠀⣴⡟⣿⣼⡵⡛⡉⡁⡀⠀⣠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠾⢢⣧⣦⣀⡀⡏⠈⣟⢣⢸⠘⣿⡁⣿
⠀⠀⢀⢸⡄⣸⡄⢰⣯⣼⢏⣿⡦⡷⢔⢖⠳⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⢳⡈⠉⡿⢇⣼⣦⢻⡜⢻⣿⠏
⠀⣀⠘⡀⢳⡱⢈⣿⣣⠏⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠙⢦⣧⠀⠐⠻⣿⣦⣼⣿⠰⠆
⠀⠸⡄⢹⠈⣿⡾⣣⡿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢦⡀⣿⣧⢹⣿⠓⣎
⠀⠀⢻⣆⠐⠸⣿⣟⣰⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⢿⣿⠈⣿⡄⡧
⣤⠀⢠⠿⣧⣤⣿⣏⣼⡟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠺⣿⢦⣿⣯⣗
⠘⣷⡌⡟⡼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢳⣿⣆⣿⡻⠇
⠀⠘⣧⢹⡁⣿⣿⡏⢀⣀⠀⢠⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣹⣿⣿⣿⠃
⠀⢀⠸⣯⣧⣿⣿⡇⣻⠛⢈⡮⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣦⢄⠈⠳⢼⣿⣿⡿⣆
⠠⣼⣇⣿⣇⣽⣿⣯⠁⠀⢺⣿⢗⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⢶⣿⢼⣿⣿⡇⢺
⠀⠨⡯⡞⢿⣿⣿⣿⡄⠀⣝⣻⣟⡇⢀⣦⣷⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⠧⠐⢾⣿⣿⠃⣾⠀⢀
⠀⠀⢹⠔⣨⢻⣿⣿⣷⡀⢘⣜⣷⠥⣸⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⠥⢠⣿⣿⣿⢀⡏⠀⡌
⠀⠀⠀⢻⡕⡀⢿⣿⣿⣷⡄⠱⡿⢹⣿⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⡀⠀⠀⠀⠀⠀⠀⢰⢶⣿⢉⣿⣿⣿⣿⡾⠀⠈
⠀⠀⠀⠀⠈⢷⣼⣿⣿⣿⣿⣎⣻⣿⣿⣭⣶⣶⣾⣄⠀⠀⠀⠀⠀⠀⠀⢩⡿⣿⢦⠀⠀⠀⣼⣿⠎⣿⢓⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣿⣿⣿⣿⣬⣄⢺⡗⣏⣿⣿⣧⡄⠀⠰⠀⠀⠀⠺⢺⣻⣖⣾⣶⣧⣟⣜⣽⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡉⠉⠙⢿⣿⡿⠛⠛⠛⠛⠛⠓⡦⠶⣖⡗⡰⠚⠋⠄⣤⣶⠄⢨⣿⣿⣿⣿⡿⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠓⠒⣼⣿⣷⡖⠐⣠⣼⣔⠈⠀⠀⡠⠛⢢⣀⣸⣧⡤⠿⠶⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠀⠙⣿⣿⣍⣤⣭⣿⣶⣍⣠⠖⠛⣏⡞⠉⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⢿⠁⢠⠊⠀⠑⠠⡜⠑⠀⠀⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣸⠀⢀⣠⣤⣄⣠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣄⣠⣤⣄⡀⠀⡆⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣫⡆⢰⢁⣸⣿⣟⣻⣿⡿⣿⡟⠛⠋⠁⠈⠙⠛⢻⣿⢿⣿⣟⣻⣿⣇⡀⡆⢰⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡇⢸⡸⣿⣿⣾⣿⣻⣿⣾⠇⣤⣶⠄⠠⣶⣤⠸⣷⣿⣟⣿⣷⣿⣿⢇⡇⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢃⠸⠷⣿⣿⣿⣷⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣾⣿⣿⣿⠾⠇⡘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡜⡄⢿⣿⣿⣿⡿⢿⣿⣿⣿⣄⠀⠒⠂⠀⣠⣿⣿⣿⣿⢿⣿⣿⣿⡿⢠⢣⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠱⡌⠻⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠈⢰⣿⣿⣿⣿⣿⣿⣿⠟⢡⠆⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡜⢦⡈⠻⡿⠟⠛⠛⠋⠀⡀⢠⡀⣀⠀⠙⠛⠛⠻⢿⠛⢁⡴⢣⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣍⡞⠀⠀⠀⠀⢔⡴⣾⢟⡿⡷⣤⠀⡀⠀⠀⠀⢳⣩⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠊⠓⣬⣝⢫⣤⠚⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠜⢿⣿⣿⡏⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠀⠀⢹⣿⣿⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠢⡙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⠔⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠲⢄⠐⠶⣤⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⣤⣤⠴⠂⡠⠖⠁⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠲⢤⣉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣉⡤⠖⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠉⠑⠒⠲⠶⠶⠶⠶⠖⠒⠛⠉⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡇⠰⠀⢸⡀⢀⡔⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠤⣀⠀⠃⠉⡤⡈⡶⠛⠛⢲⢃⢠⡉⠁⠏⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠐⢄⡤⢇⡀⣼⠧⠘⣣⡐⣣⣼⠋⠨⣧⣀⡠⢧⡠⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣘⣤⠋⠀⡒⢼⣿⢿⠳⣤⡍⠉⣤⠞⠻⢿⣧⢐⠀⠙⢦⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⡽⠁⢀⡴⠏⠇⡽⠐⢄⣢⡉⠍⣴⡠⠊⢮⢈⠛⠦⠀⠈⢿⠍⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠃⠀⡼⡤⣤⣿⡥⠄⠀⠀⠀⠀⠀⠀⠀⢤⣽⣦⣦⢷⠀⠈⣷⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢉⡉⣼⠀⠀⣧⣣⣼⡃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⣿⣼⣩⠃⠀⢻⢉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣁⣻⠀⠀⣿⣌⣽⡇⠐⠏⠗⠀⠀⠈⠸⠹⠃⠀⠸⡯⣁⣽⠂⠀⣸⣈⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠽⡄⠀⢻⡞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢱⡿⠀⢀⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢂⣳⡀⠀⢿⣼⣿⡀⠀⠀⠐⠀⠒⠀⠀⠀⠀⣾⣵⡟⠁⠀⣼⠁⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠳⣄⠀⠙⢿⣧⠀⠀⠩⡍⡍⠁⠀⢀⣼⡿⠋⠀⢠⠾⡍⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⠚⢳⣄⡀⠈⠳⣄⡀⠀⠀⣀⡤⠊⠉⠀⣠⡔⠛⠄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⢰⠋⡳⢦⣤⣍⣉⣉⣡⡤⡤⢖⢻⡇⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠜⠉⠈⡇⠰⠀⢸⠁⠹⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠠⠂⡄⡰⠋⣁⡠⢒⡄⢠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡆⠐⡷⠀⠘⡃⠀⢸⡅⠂⠙⠅⢨⡗⢀⣸⡤⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠢⣤⠀⠘⡧⠀⢐⣁⣈⠁⠀⠏⠀⠀⢀⠬⠿⠂⠀⣙⠃⠤⣾⠐⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠢⣇⠀⠘⣂⣤⣶⣾⠟⠁⠀⠀⠓⠶⠴⠶⠦⠴⠖⠛⡡⠄⠈⠱⣦⣍⡐⠏⠁⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢦⠀⣊⣴⣾⣿⠿⠿⠛⠀⢃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠳⢀⠀⣿⣿⣿⣷⣬⡛⢀⢠⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢦⠄⣡⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡀⠀⣀⠠⠎⠀⠁⠈⠙⠻⣿⣿⣦⡑⣀⣠⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦⡠⢀⣾⣿⡿⣝⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢽⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⢹⣿⣷⣌⠣⡀⢠⡇
⠀⠀⠀⠀⠀⠀⢀⠀⢡⡀⡀⢡⣿⣿⣟⠓⠄⠧⢌⡄⠀⠀⠀⠀⠀⠀⠄⢸⣏⠁⢨⢽⡇⣰⠀⠀⠀⠀⠀⠐⢀⠰⡶⠃⠀⣿⣿⣿⣧⠙⠁⠀⣀⠁⠠
⠀⠀⠀⠀⠂⠀⠀⠆⠀⠸⢠⣿⣿⣿⡏⠁⠀⠀⠘⠂⠃⡆⣀⢠⡀⡄⠐⠈⠳⣧⣼⠟⣁⣄⡁⠀⠀⠀⡀⠰⡍⠃⠁⠀⠀⣿⣿⣿⣿⣧⠘⠟⠛
⠀⠀⠀⠀⠀⢄⠀⠀⣻⢁⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⡀⠀⠀⠀⠀⠙⠂⠙⠚⠓⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⢰⠤⠶⠅
⠀⠀⠀⠀⠀⠀⠷⣰⠇⠼⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⡆⠀⢦⡀⢀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⣄⣠⡔⠂
⠀⠀⠀⢀⡠⠃⢁⣠⡴⠦⢶⡤⣌⡙⢿⣷⣤⡀⣀⠀⠀⠀⠀⠉⠈⠁⠀⠈⠀⠀⡀⠉⠀⠀⠀⠀⠀⠁⠲⢶⣤⣤⣴⠿⠋⢉⣉⣉⣉⠙⠻⣷⠈⠉
⠀⠀⣰⠎⣠⠖⠋⠠⠋⣠⠞⠀⠀⠉⢢⡘⣯⠁⠀⢀⡀⠄⠒⠀⠀⠀⠀⠀⡈⠈⡣⠔⠃⠸⠒⠐⠤⣤⣄⣀⣨⢟⣡⠶⠋⠉⠁⠀⠀⠉⠲⣈⠻⡛⢃
⠀⡴⣧⠞⠁⠀⠀⠀⠨⡗⡀⠀⠀⠀⠀⠑⣌⣷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣷⠞⢳⡄⠀⠀⠀⠀⠀⠀⠂⠈⣆⢱⡖⠃
⣾⣾⣇⢠⣆⡀⠀⠀⠀⠈⠀⠂⠂⠤⣰⡆⠈⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣿⡶⢺⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠹⣆
⠈⠙⠳⢷⠚⠁⠗⠂⠆⠶⠐⠂⠂⠓⠉⠙⠉⢡⠀⠁⠒⠠⢄⡀⠀⠆⠀⠀⠈⠁⠀⠨⠘⠀⣀⠔⣒⣉⣤⣴⠁⣯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣈⠆
⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⡍⢩⣍⠐⡟⠄⠀⠀⠀⢤⣀⠀⠁⠀⡼⢁⠜⢉⣴⣀⢸⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠢⠤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠐⠊⠘⡇⠘⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡄⠀⢀⣴⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠀⠀⠀⠀⢀⠐⢛⣿⣿⣿⣦⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⠌⠛⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡴⢖⣉⣽⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣟⠛⣤⣀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣋⣥⠐⢻⣿⣿⠖⠠⣀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣸⣉⡻⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠶⢈⠜⠁⠀⣠⣿⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣧⢿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣤⠀⡸⠟⣿⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⠝⠊⠀⢐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣉⡗⢸⡇⣷⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠘⡃⠀⣰⡿⠿⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠉⣇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠑⠚⠓⠴⡴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠜⠃⠀⣀⠀⠀⠀⣠⠤⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢈⢦⡆⡠⠾⠐⠖⠾⢠⣠⡆⡠⡹⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⡀⠀⡀⠀⡳⢚⡙⠲⠄⠀⠄⠀⠈⢀⠐⠏⣑⠗⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢑⣺⠊⠸⡉⡄⠋⠑⠀⠀⠀⠀⠀⠀⠱⢿⢃⠑⣎⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠁⢀⠦⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢾⡇⠈⠾⠥⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡃⠀⠨⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⢰⣀⡀
⠀⠀⠀⠀⠐⠒⠒⠚⠉⠉⠹⠀⠀⢰⠀⠀⠀⠛⠀⠒⠀⢀⠘⠀⠓⠀⠀⠀⢀⠀⠀⡆⣀⣠⣀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠇⠀⠠⡁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠼⠀⢘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⡀⠀⢡⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⢈⢁⠉⠀⢀⠶⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠊⣱⢄⠀⠉⠳⢄⣀⠀⠀⠀⠀⣀⡠⠖⠐⠀⠠⡯⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢈⢵⢄⡀⠀⠈⠁⠀⠀⠈⠁⠀⢀⡠⢎⠀⠁⠀⠈⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠈⠸⠃⠂⣴⠤⠄⡤⠒⢸⠣⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡳⠀⠀⠇⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠀⠀⠁⠐⢀⠉⠃⠀⠉⣉⡀⠀⠀⠀⠐⠈⠠⢶⡷⠆⠀⠀⠉⠁⠉⠉⠏⠉⠛⠳⣶⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡎⡀
⠀⠀⠀⠁⠀⢀⡤⠒⠩⠀⠀⠀⠒⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⡅
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⠀⠈⢮⣐⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠘⠁⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡤⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠖⢻⠇⠀⠀⠀⠐⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠋⡁
⠀⠀⠀⠀⠤⣾⠁⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣓
⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⠋⠀⠀⠁⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⣉
⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣹⡄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢉⠁
⠀⠀⠀⢠⠼⠀⠀⠀⠀⢀⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠐⣀⣀⡀⠀⠀⢀⡞⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠄⠰⡇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⣤⣶⠟⠋⠉⠉⡙⠛⣦⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢈⣭⣶⣶⣦⣤⣄⠁⠀⠀⠀⠀⢀⡇⠀⠀⣀⢀⣰⠏⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢧⠀⠠⠴⠀⠐⠏⠋⣈⡠⠟⠋⠀⠀⠀⠀⣼⠀⠀⢐⣭⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠷⣼⣶⡄⠀⣀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⣠⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣾⡇⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⡿⣽⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡟⢇⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣅⣅⠙⢃⠰⡄⡄⠀⠀⡀⢠⣾⡏⠀⣁⣼⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣒⣀⠡⢘⣄⣠⠁⢃⣙⡀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠁⠈⠛⠲⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⢀⡌⡀⠈⢧⠀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡤⠰⠛⠉⡀⠈⢁⠙⠷⣄⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡇⠋⠀⠈⠀⠀⠁⠁⠈⠀⠁⠈⠙⢷⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣀⠀⠄⠓
⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢀⣘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡔⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⠦⢄⣀⣀⣀⣤⣤⠖⠊⢹⠈⠉⠉⠀⠒⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⡔⠙⡆⠀⠘⠀⠙⠢⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠙⣤⠃⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⡤⠄⠀⠀⠀⠀⠀⠀⣀⢠⡊⠄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣗⡂⠉⢛⠛⠛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠁⣡⠤⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠎⡁⠊⠄⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠅⠴⠓⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢉⣙⣤⣀⢠⠀⠈⠠⠈⠛⠿⠛⠛⠻⠟⠉⣀⠠⢩⢆⣴⣌⣁⢉⡿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⢉⣻⣿⣿⠷⠒⠐⡤⠀⠊⠀⠀⠲⢶⣤⡈⠀⠀⠝⠃⢺⣿⣿⣿⣶⣤⡄⡸⢿⣿⣿⣿
⣿⣿⣿⣿⠿⠛⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⣿⡀⡀⠀⡀⠀⠈⠉⠙⣿⣿⣿⣦⡀⢿⣿⣿
⣿⣿⣿⣷⣴⣾⠟⠿⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣼⣿⠃⠀⠐⠀⠀⠀⠀⠐⠛⠿⢿⣿⣧⣶⣿⣿
⣿⣿⣿⣿⡿⢋⡀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠑⠘⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⣿⣿⣿⣿⣿
⣿⣿⣿⡿⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢤⡄⠴⠶⠰⡄⠀⣀⡀⠀⠀⠀⠀⠀⠔⠛⠛⣿⣿⣿⣿
⣿⣿⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢲⡂⠁⠂⢰⣥⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⡰⠁⠀⠠⢻⣿⣿⣷⠠⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⢻⣿
⣿⣿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣀⠀⣀⠈⠉⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣬⠀⠀⠈⠇⠀⠀⢀⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣶⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⡁⠀⡀⣠⠆⠈⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠌⢁⡼⢋⡍⢳⡴⣇⡴⠋⣡⣍⠰⡰⡀⠀⣰⠇⠀⠀⢀⠄⢀⣀⣾⣿
⣿⣿⡀⠉⠀⠈⠁⠀⠀⠘⢿⣶⣶⣲⡤⠰⠉⠘⣳⢿⡾⢀⠞⠀⠀⠉⢠⣿⡿⠏⠀⠀⠀⠀⠀⠀⠼⣿⣿
⣿⣿⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠐⢀⠀⢠⡮⠞⡗⠸⠂⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣸⣽⣿
⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⢀⡀⠐⠊⡠⠄⠀⠁⢠⡥⠀⠈⢠⠤⢄⣈⢒⡤⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿
⣿⣿⣿⣟⠻⠃⠀⢀⣀⣀⣀⡈⠐⠂⠋⠀⠀⢀⢠⡀⠀⢀⡀⠀⠀⠀⠀⠈⠁⣀⡤⠄⠀⠀⠀⢟⣻⣿⣿
⣿⣿⡟⠿⠷⠀⠀⠈⠑⢟⢛⣻⣷⣄⠀⠄⠀⠀⠁⠀⠀⠀⠁⠀⠠⠀⣤⡶⠿⡷⠂⠀⠀⠀⠚⣿⢟⣻⣿
⣿⣿⣵⣤⣤⣠⣤⡄⠀⠀⠋⠤⠄⠹⠆⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠼⠛⠛⠛⠃⠀⠀⠀⠀⢀⣀⣌⢹⣿
⣿⣿⣿⣿⣍⠉⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣋⣽⣿⣿
⣿⣿⢟⡋⠙⠁⠀⠀⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠌⣁⣠⣤⣄⢰⣶⣿⣿
⣿⣿⣿⣳⣤⠐⠚⠀⠀⡀⠀⠀⠁⠀⠠⡄⠁⠐⠀⠀⠀⠃⠀⠀⠀⠀⢠⡠⠄⠀⠠⠈⠈⠋⠙⠹⣛⢿⣿
⣿⣿⣿⡿⠂⠀⠀⢀⠀⠀⠀⠀⠐⠘⢢⣞⡷⣰⣔⣤⣤⣤⣴⡶⠳⠀⠁⠠⠄⠠⠀⢀⣠⣤⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣣⣶⣿⣿⣿⣄⢀⣀⢀⠾⠋⠉⠘⠋⠘⠉⠃⠙⠉⠙⣦⣤⣀⣤⠰⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣂⢠⡤⠤⠀⠄⣀⠀⢀⣤⡀⠀⠉⢻⣛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣶⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡜⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠂⠀⠀⠴⠀⠀⠀⠘⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠁⠀⢀⡸⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡘⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠈⠀⠀⡜⠁⠀⠀⠀⠀⡄⠀⠀⠀⠇⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢰⠇⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣧⠀⠀⠈⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡟⠀⢸⡄⠀⠀⠇⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⢸⡃⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⢠⠀⠁⡀⠘⠁⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢇⠀⢺⠀⠀⠸⠇⢰⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠆⠈⢣⠀⠷⡀⠀⢀⡈⠀⠀⣠⠇⠀⠀⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠢⣀⠐⠀⠉⠠⠞⠁⢀⡴⠃⠀⣤⠦⣀⠀⠀⠈⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢶⠀⠀⠒⠉⠀⠀⠀⠀⠰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⡀⠀⠁⠀⢰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡀⠖⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠉⠁⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠛⠓⢠⡀⠶⠀⠀⠰⢀⡆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠶⠁⠀⠀⠀⠀⠁⢀⡇⠸⠀⠘⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣾⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠰⡀⠰⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠖⣀⣶⡤⠞⠁⠀⠀⠀⠀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠲⠻⢶⣤⣭⣉⣁⠀⠈⠉⠉⠉⠉⠉⠀⠈⢉⠙⣆⡰⠃⣠⠶⠛⠓⠐⠒⠈⣉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠂⠀⠱⠖⠲⠾⠋⢉⣍⣀⣤⣤⣄⣀⣤⣤⡤⣤⡄⠈⣤⢈⣷⠌⢡⣤⠄⠚⠛⠓⢠⡀⠃⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠁⣀⣀⣠⡤⠀⠀⠀⠀⠰⠶⡄⠐⠂⠈⢉⠤⠟⠛⠛⠛⠁⠀⠀⠁⠀⠸⣞⠡⠞⠃⠄⡀⠀⠀⠀⠀⠈⠶⠀⢰⡄
⠀⠀⠀⠀⠀⠀⡰⠊⠀⣠⠖⠊⣉⠛⠉⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⠴⣆⠀⠀⢀⣄⡀⠁⠀⠀⣀⠀⠀⠀⣇⣸⠁
⠀⠀⠀⠀⢀⠞⠀⣠⠞⠁⣠⠛⠀⠀⠀⠀⠀⠀⢀⠈⢠⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢀⡿⣦⡀⠘⠏⠰⠾⢦⣄⠈⠀⠀⣶⠉⣿
⠀⠀⠀⢀⠉⠀⣘⠃⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⡇⠀⠀⠀⠀⠈⠙⠃⣠⡞⣀⠇⡸⠃
⠀⠀⠀⠘⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢴⠀⠈⠓⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢀⣤⠾⢯⠚⠁⡸⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⣄⣀⡀⠛⢣⣶⣶⣶⡄⠀⣶⣴⠾⠟⠁⠀⠘⠋⡰⠛⠂⠀⠀⠀⠠⠾⠁
⠀⠀⠀⢀⠀⠀⠈⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠛⠳⠶⠶⠀⠀⠉⠁⠀⠀⠀⠀⠰⠎⠀⠀⠰⠶⠖⠈⠀⠀⡀
⠀⠀⡄⠈⢀⠀⠀⡆⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⠀⣀⠃
⠀⠀⠁⠀⠀⠀⠀⠙⠦⣀⣀⡤⠃⢀⡠⠦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢤⣤⠿⠏⠁⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣴⠎⠀⠀⠈⠉⠉⠁⠀⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠀⢀⣉⣉⠲⠂⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠓⣤⠀⠀⠀⠀⠀⠀⠛⠛⠓⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣤⣤⣤⠤⠒⠒⠛⠋⠀⠀⢀⣀⣠⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠶⠆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠶⠶⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠙⢿⡿⠁⠀⣀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀⠹⣿⣷⣶⣶⣶⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠛⠿⠿⠿⠿⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠉⠉⠀⠉⠻⣿⣿⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⣰⣿⣿⣿⡿⠋⠉⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⡿⠋⠀⠀⣠⣿⣿⣿⡟⠁⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢰⣧⡀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⡿⠀⠀⠘⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣀⣽⣭⣭⣭⣭⣯⣯⣭⣽⣍⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠈⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢤⠄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠹⠉⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⣶⣦⢰⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⣝⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠙⢻⣿⣶⣶⣿⠏⠋⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠈⠉⠉⠀⠀⠀⠴⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋⠉⠀⠄⠀⡄⠀⠀⠀⠈⡀⠉⡀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠚⢀⢠⣶⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢈⠀⠂⠀⠀⠀⠀⠄⡘⠃⠀⠀⠣⠥⠔⠒⠢⡤⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡭⠁⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠱⡶⠀⠁⠀⡀⠁⠀⠀⢐⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢡⢷⡀⠀⠂⠀⠀⠀⠠⢦⡄⠀⠀⠀⠑⢈⡑⢄⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠤⠄⢀⠟⠆⠀⠐⠀⠀⠋⢈⠠⡁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢸⠁⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠐⠒⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢄⠁⠀⠀⠐⠄⡀⠀⡔⠃⠀⠀⡆⠀⠙⡦⠰⡋⠀⢘⠀⢀⠀⠀⠀⠀⠀⠲⠂⠈
⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠆⠲⠀⠀⠀⠀⠀⠄⠀⡴⠀⠀⠀⠀⠸⠗⠛⠀⠀⠀⠀⠈⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡄⠀⠀⠂⠐⠀⠀⠀⢐⡇⠠⠾⠗⠒⠖⠒⠀⢀⡀⠤
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢀⡴⠁⡸⠁⠀⠀⠀⠀⠀⠀⣚⠁⠩⠔⠂⠠⠒⠂⠀⠀⠊⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠗⣀⠀⠐⡡⢤⠾⠃⠀⠀⠀⢠⢀⠀⠀⢀⣼⢀⡖⠲⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠶⠇⠀⠀⠀⠐⠄⠀⠀⠀⢄⢸⠀⠖⠒⠒⠲⢾⣥⡼⢰⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠚⠂⠘⠠⠘⠂⠀⠀⠀⢂⠂⠀⣰⡇⠀⠻⢦⣸⣋⣠⠄⢆⡀⠀⠂⢀⠀⠀⠀⠀⠁⢌
⠀⠀⠀⠠⠠⠄⠠⠀⠰⠤⣤⡀⠄⠛⠄⣀⠁⠀⠂⠈⠀⠉⠋⢧⠀⠨⠀⠀⠐⠄⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠢⠍⠀⠀⠀⠀⠀⠀⠀⠀⠉⠕⠀⣀⢴⡿⢂⡀⠰⢤⠀⠁⢘⡄⠀⠀⡄⠀⠀⢀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠂⡴⠚⠉⠀⠉⠋⠉⠹⣷⣀⠉⠀⠈⠻⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⣖⠃⠀⠂⠀⠀⠀⠀⠀⣍⡹⣿⣲⡃⠀⠈⠃⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠤⠤⠀⠠⠤⠔⢓⡿⢹⠁⠀⠀⠀⠀⢀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠄⠒⢠⢀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢰⣄⣚⢀⡜⠀⠀⢀⢠⠐⡀⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠙⠌⠀⠌⠃⠁⡀⠀⠀⣠⢛⡽⠧⠋⠄⠌⠼⠋⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⢄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡰⠳⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⠧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢴⡆⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡒⢲⣞⣛⣿⠟⠿⠿⣿⠿⠿⠿⣿⣛⣳⡖⢒⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⡇⣠⠖⠉⠀⠀⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠙⢿⡄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡄⣠⠀⣠⠾⠛⢩⠭⣉⠉⣱⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣙⣿⡞⣿⡞⠁⢀⡤⣀⢳⡈⠑⢎⠀⠙⢄⠀⠀⢀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣠⠞⠋⠙⠉⢻⣷⣿⣿⡿⣶⣽⡄⠙⣇⠀⠈⢆⠀⢸⡠⠖⠉⠀⢠⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢼⡇⠁⠀⠀⠀⠈⢳⠀⠹⡀⠀⠸⡀⢸⠀⠀⠀⠀⢸⠤⠒⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢷⣄⠙⢦⡀⠙⣾⠀⣀⡠⠖⠁⢀⣼⠖⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠜⠙⡄⠀⠀⠀⠀⠀⠀⡸⠀⠉⠓⢚⣿⣽⠉⠉⠉⠑⠒⠊⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠦⣤⠤⠴⠁⠀⣤⠶⠋⠁⠘⢟⠢⣄⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡄⡄⢰⢰⡐⠄⡆⢰⢠⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⢇⢐⢇⢱⢣⢱⠸⡄⡇⠘⡇⢸⢈⠀⡆⡆⡼⠐⣇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢦⡳⣌⢫⠻⡄⣧⢻⠝⣞⡇⣿⠀⡇⢣⡘⢸⢸⢹⣹⢱⡇⣏⠞⡴⢁⠄⡀
⠀⠀⠀⠀⠀⢴⣵⣮⡢⡙⢌⠳⡹⡇⢻⡌⢾⢑⢳⣿⠀⣧⠈⡇⣉⣾⠞⣃⡞⣠⢫⠞⡴⢡⢞⣡⡄
⠀⠀⠀⡀⢬⡑⢷⡄⢍⠛⠦⡓⡌⠣⡈⢿⣦⣌⠐⡏⢠⣿⠀⢧⠏⣠⡾⢫⠞⡱⢃⡼⠖⣋⣡⢊⠴⢂⡀
⠀⠀⠐⠌⡓⠬⣑⠿⣦⠙⢦⣌⠙⠲⢽⡀⢹⣿⢲⡟⡾⠹⡆⢼⣴⣿⢡⣏⡾⠓⢁⣴⢞⡽⢋⡥⠞⣩⠔⣂
⠀⠠⣍⡒⠮⣕⠢⢍⡪⢵⣄⠙⢷⣦⣤⡟⢇⡩⠔⠒⠐⠒⠒⠬⢄⡛⢻⣯⣠⣾⠟⢡⡾⣚⡥⢖⡩⠖⣋⣥⠆
⠐⢯⡉⡛⢕⡢⠝⢊⣉⣑⠛⠦⢀⡟⢢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠿⣅⡰⠛⢚⣛⣛⣛⣒⣫⢅⣞⣋
⠨⠥⢽⢦⣄⣒⣉⣉⣍⡉⠛⢷⣮⠁⡣⠔⣒⠾⢲⡀⠀⠀⣰⠯⠟⢒⠦⠑⡌⣹⣶⠟⠛⢛⣛⣋⣭⠴⠭⠤⢔
⠨⠭⠉⢀⣐⡂⣒⣂⣀⣿⠷⢸⠆⡇⠰⠏⠸⠿⠤⠀⠀⠀⠠⠼⠿⠉⠻⠀⢱⢈⣵⠶⠯⠭⠅⢖⣒⣒⣉⣉⣭
⠐⣠⡶⠶⢒⣒⣉⣭⣥⣶⣶⣟⡁⡇⠀⠀⠀⠀⢀⣀⠀⢀⣠⠀⠀⠀⠀⠀⠈⠠⣼⠿⠷⠶⠶⠶⠶⠤⠍⣉⠒
⢙⡳⠤⠤⢤⢤⣤⣀⣉⣉⣉⢹⡆⠁⠀⠀⠀⠀⠀⠀⣓⢋⡀⠀⠀⠀⠀⠀⣠⠒⡟⠠⢶⣖⣒⣚⣛⡛⠃⠬⠽
⠐⢚⣉⠩⠤⠖⢒⣋⡭⠛⣩⣼⢷⡂⠂⡀⠀⠀⠀⠉⢩⣩⠍⠀⠀⠀⠀⡠⣉⣻⠿⠷⣦⡈⠻⣖⣆⣬⢭⣉⣓
⠈⢀⠤⢔⣚⣭⡥⠊⣠⠞⣩⠴⡾⠙⣧⡌⠀⣀⡀⠀⠉⠉⠙⠀⢀⣤⠸⣦⣽⡉⢦⡐⢌⡛⠷⠦⠭⠥⢖⢄⠙
⠀⠐⠊⡵⢊⠵⠖⣛⣡⠞⣡⡞⢠⣾⠟⠛⠻⣧⣾⣁⣇⢀⣌⣱⣤⣼⡋⢡⡙⠻⣆⠹⡦⢉⠳⣘⠋⣍⠣⠍⠃
⠀⠀⠈⡧⢰⠞⣩⠟⡡⢺⢏⣴⠟⣁⣴⠃⣾⠟⢡⡆⢸⣿⠃⢙⢦⠹⣷⠀⢿⡲⣌⠳⡜⢦⡑⠠⣑⠢⠉
⠀⠀⠀⠈⠁⢊⡵⢋⢴⠋⢜⡡⡞⢡⢻⡀⣟⢀⢇⣧⠀⣿⠀⡟⡎⠄⣿⠀⡷⠘⢦⡱⣄⢀⢳⢕⡬⠁
⠀⠀⠀⠀⠀⠀⠐⢱⣧⡔⡟⢰⠇⡎⡌⡇⢹⠘⡎⠉⣇⢹⢀⢇⢹⠐⠙⢦⡑⣌⠢⡑⢌⠣⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢁⠏⡜⠰⢰⣱⢋⢴⢘⢐⢸⡀⢸⡜⡄⢣⠸⡑⢦⡈⡗⢌⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⢨⢡⠋⢸⢸⢸⢀⢇⢧⢃⢹⡈⡄⠃⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣷⣄⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⡀⠀⣀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⣀⢀⣀⡀⠀⣀⣀⢀⣀⣀⣀⡀⣀⠀⢀⡀⣀⣀⢀⢀⣀⣀⣀⢀⣀⢀⣀⢀⠀⢀⣀⣀⠀⣀⣀⠀⢀⣀⡀⠀⢀⡀⠀⡀⢀
⠀⢰⠉⠀⠀⠀⠀⣀⠁⠉⢁⡉⠉⢀⡀⠀⠀⣄⢈⠀⣀⠈⣩⡀⠉⠀⣀⠈⠁⣀⡈⠉⡉⢁⢈⣁⣀⣈⣁⡀⢉⣉⠁⠉⢉⣀⡠⢘
⠀⠀⠦⠀⢻⠗⠛⠉⠁⠛⠋⠁⠛⠋⠉⠋⠋⠉⠛⠋⠉⠉⠉⠉⠉⠙⠉⠉⠋⠉⠁⠉⠉⠈⠈⠉⠉⠁⠉⠉⠉⠁⠈⠂⠙⣿⠃⣼
⠀⠀⡆⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘
⠀⢠⠑⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢄⠀⠀⠀⡀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣮⠀⠘
⠀⠀⡆⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠈⠠⠀⡇⠀⠀⠘⠛⠀⠀⡹⠀⠀⢀⡴⣉⠀⠀⠀⠁⢀⠀⠀⠀⠈⡀⢸
⠀⠀⡀⠀⢺⠁⠀⠀⠄⠈⠃⠀⠀⠐⠀⢸⡆⡀⢲⢵⡆⠼⠤⢁⡞⠀⠀⠀⠀⠀⠠⠁⠀⠀⢈⡄⠀⠰⡄⡂⠀⠀⡀⠀⠀⣿⠀⢸
⠀⠀⣷⠀⣰⠆⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⣤⡁⢊⠤⠂⠀⠀⠘⠀⢒⠀⡐⠖⢀⠁⠀⠄⠠⣀⠀⡀⠀⢁⠀⡠⠀⠀⠀⣿⠀⢈
⠀⠀⡧⠀⢹⠀⠀⠀⠂⠀⠐⠀⠆⢀⠀⠀⠀⢈⠘⠳⠂⠀⠐⢀⠃⡄⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⠐⡂⠂⠀⠀⠀⠀⣿⠆⠈
⠀⠀⡇⠀⢻⡀⠀⠀⠀⠁⠀⡂⠀⡇⠀⠀⠂⠘⠁⠐⠀⡁⠀⠐⠂⠀⡝⢀⠄⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⣀⠀⠀⠀⠀⠀⣭⡀⠘
⠀⠀⡟⠀⢼⠂⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⠅⠀⠂⠀⠒⡇⢀⠀⡅⢸⠋⠡⠠⠀⠀⠀⡀⢂⠏⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⣏⠀⠸⠄
⠀⢀⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⣄⠀⠀⠀⢀⡇⠈⠄⣃⠈⠄⡠⠴⠀⠁⠀⡠⠋⠀⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⢿⠃⢀⠁
⠀⠀⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠆⠀⠠⠀⠱⢄⠋⢹⠀⢨⠃⢀⢀⡀⠀⠀⣠⣤⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⣟⠀⠈
⠀⠈⡅⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⠆⠀⠳⠃⠀⠸⠄⠊⡤⠀⠐⣼⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠸
⠀⠀⡧⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡟⢀⡾⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠟⠀⢀
⠀⠀⣧⡀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠠⣈⡉⠳⡀⠀⡀⢡⣶⠶⢶⢤⣵⣀⣃⠀⢐⣠⠞⠁⠈⢳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⢚
⠀⠀⢋⠀⢹⡆⠀⠀⠀⠀⠆⠀⠀⠢⡀⠓⠠⠈⠋⠀⡀⢠⠀⠀⠀⠩⡿⠉⠉⠉⠀⠀⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢠
⠀⠀⡏⠐⢾⠃⠀⠀⠀⠀⡙⠀⠀⢦⠠⠀⠀⠀⠀⠸⠁⠰⠇⠒⢀⣈⣁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⡟⠀⢠
⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⣠⡶⠿⠛⠉⣯⠀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⡿⠀⢸
⠀⠀⣿⠀⢸⡃⠀⠀⠀⢚⠒⠂⠀⢀⠈⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠘⢀⠀⠀⠀⠀⠘⢂⠀⠈⠀⠀⠀⠀⠀⠀⠰⣇⠀⢘
⠀⠀⠇⠈⢻⡇⠀⠀⠀⠈⡖⠁⠀⠀⢀⡼⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⣐⠁⠀⠈⠁⠀⠀⠀⠀⢀⡓⡀⠂⠀⠀⠀⠀⠀⢀⣯⠀⢠
⠀⠀⡎⠀⢼⠇⠀⠀⠀⠀⠛⠀⢀⠶⠡⠀⠉⠀⠀⠀⠀⠄⡀⢀⣀⣠⣤⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠀⠀⠀⠀⠀⠷⠀⢸⠂
⠀⠀⣷⡀⠐⡆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⠀⠀⠀⠀⠀⠛⠛⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠈
⠀⠀⣃⠀⢻⡅⠀⠀⠀⠀⠀⠀⠘⣦⢣⡀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢸
⠀⠀⡏⠀⢾⡃⠀⠀⠀⠀⠀⠒⠢⠙⢦⡑⢴⡄⠀⡠⠀⡁⠀⠀⠀⠀⠀⠙⡂⠁⠀⠐⠀⠐⠀⠁⠀⠠⠆⠤⠀⠀⠀⠀⢀⡄⠀⢸⠂
⠀⠀⣿⠀⣸⡇⠀⠀⢀⠀⠠⠀⠀⠀⠀⠁⠒⠙⠒⢤⡤⠦⠀⠀⠠⡀⠈⠀⠉⠀⠀⠰⢀⠀⠀⠠⢀⡄⠂⠀⠠⠀⡀⠀⠘⣧⠀⢩
⠀⠀⡿⠀⣸⠀⠀⠀⠀⠠⠀⠀⠂⠠⠃⠴⣄⡠⠤⠠⢄⡀⠀⡨⠀⠀⡀⠄⠉⠀⠀⠠⠀⢐⠘⠂⠈⠠⠀⢄⠀⠀⠀⠀⠸⣇⠀⢺
⠀⠀⣇⠀⢹⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⡀⠈⠒⠚⢀⠕⠂⠀⠚⠂⠠⢀⠈⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠰⡗⠀⠼
⠀⠘⡏⠁⢼⡃⠀⠀⠀⠀⢀⠀⠀⠐⠑⠂⠀⡠⠀⡀⠀⠀⠀⠈⠐⠀⠀⠒⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠀⢠
⠀⢰⠓⢀⣸⣧⣬⣤⣠⣤⣀⣠⣤⣄⢤⠤⣄⣀⣀⣀⣠⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⡀⠀⣀⣀⠀⡀⡠⠄⡠⠄⠀⡀⠀⡺⣇⡀⠦⡀
⠀⠈⠀⣸⠟⢯⡁⠀⠀⣀⠀⠁⣉⠀⠀⢀⠀⠈⢉⠀⠈⢀⠀⠉⢀⣀⠈⢡⡀⠈⢁⡀⠀⠀⣀⠀⠀⣀⠀⢑⡀⠀⠑⣀⠀⠹⣇⣀⠂
⠀⠀⠀⠈⠀⠀⠉⠁⠀⠉⠉⠉⠀⠀⠉⠉⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠁⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠤⠈⠱⠒⠒⠴⣀⠀⠐⣠⡀⡄⠙⠕⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣰⣄⠀⢀⠀⢉⣥⣤⠶⠀⢀⣼⢀⣉⣶⡀⠁⠀⠉⠂⡐⡄⠐⠺⠷⣄
⠀⠀⠀⠀⠀⠈⢮⡃⣷⢂⣵⡶⠟⢠⠤⣀⠀⡉⠀⠀⠀⠈⡅⠀⠒⢠⡁⠙⠺⡄⠀⠀⢸⣾
⠀⠀⠀⠀⠀⡸⠀⣻⡿⠻⢋⣀⠤⠀⠊⠐⠆⠉⠀⠀⠀⠀⠘⢣⡆⡘⡄⠐⠀⣇⢱⣄⠠⢻⣀⢶⡄
⠀⠀⢸⡀⢀⠀⡸⢻⡷⡟⣉⢀⢠⣠⡸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡗⠴⣥⡄⣏⢢⡨⣿⣼⠃
⠀⠀⡄⢧⡊⢲⡟⡿⣼⠛⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠗⠙⢦⣉⠘⠻⣷⣇⣿⡇⡤
⠀⢣⠘⠎⣧⣟⡿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣄⢸⣯⢻⡗⣖
⠀⠈⢷⡀⢸⣿⢵⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣹⣿⠸⣷⡺
⢳⡀⣏⢿⣾⣿⣾⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⣿⡻⣿⡿
⠀⢻⣸⡎⣿⣿⠋⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⣿⣿⡟
⠀⠈⢿⡧⣿⣿⣸⠟⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢄⠙⢧⣿⣿⣷
⠐⢾⣼⣧⣿⣿⡅⠀⣿⣟⡄⠀⢠⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣗⡾⠷⣿⣿⢀⡇
⠀⠹⣟⠹⣿⣿⣧⠀⠂⣿⣇⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⠁⢸⣿⣿⢨⠁⡐
⠀⠀⠱⡪⡸⣿⣿⣧⡘⣻⠇⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⣿⣿⢲⣿⣿⣿⡎⠀⠁
⠀⠀⠀⠘⣦⣿⣿⣿⣷⣜⣾⣿⣭⣶⣾⣄⠀⠀⠀⠀⠀⠈⡿⡷⡄⠀⠀⣶⡦⣿⣳⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⣿⣶⣄⣟⣹⣿⣿⣖⠀⠐⣀⠀⢈⡬⠷⠿⠾⢾⣷⣵⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠹⠳⢤⣀⣹⣿⣉⣀⢀⣀⠈⠉⠙⢻⢄⠁⢠⡐⣻⣡⣼⣿⣿⣿⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠙⢿⣶⣛⣙⣷⣤⣂⡠⠞⣟⡟⠋⠁⠀⣀⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠻⡏⠠⠋⠀⠂⡜⠀⠀⠐⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠒⠒⠀⠢⠤⣀⠀⠀⢀⠀⠂⠀⠉⠉⠀⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⢁⡠⠄⠉⠀⠀⠒⠐⠠⢀⢍⡔⠁⠁⣢⠀⡀⠀⠀⠀⠀⣀⠈⡠⡀
⠀⠀⠀⠀⠀⡐⠁⡈⠊⠀⢀⠠⠠⠀⠠⠄⢀⢀⡙⠐⠠⠘⠠⠂⣵⡄⠀⢀⠀⠀⠀⠀⠚⢀
⠀⠀⠀⠀⠀⠄⠀⠃⠀⡜⡡⠠⠒⠈⠐⠶⠒⣯⠟⠐⣂⠀⠀⠀⠈⡔⣷⢾⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠈⠃⢉⡄⢘⠀⢰⠀⠀⠀⠀⠁⢠⣶⠀⠶⣀⠄⡶⠀⠠⡀⢖⠹⣤⠂⠀⠀⢰⠀⡃
⠀⠀⠀⠀⠀⢃⠀⠇⠀⠢⣀⡤⡀⠀⠀⠀⠀⠉⠒⡃⠍⠀⠄⢠⠘⠻⠁⣆⠀⠀⢰⢶⣯⣅⠅
⠀⠀⠀⠀⠀⠀⢦⠚⣠⠄⠼⣗⣤⡤⠲⠖⣦⠀⠀⠀⠀⠀⠀⠀⣀⣂⠙⠁⠐⠀⠀⢸⠇⡏⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢈⣠⠻⠳⠝⠊⠀⠀⠀⠠⠄⡰⠂⣾⣟⢶⢐⢶⣻⡀⢨⣠⡭⡧⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⣨⡗⠃⠀⠀⡀⠈⠁⠄⠀⠀⠀⠀⠈⢰⡆⢿⣾⣿⡿⢺⣲⢷⡉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣿⡟⠉⠁⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠹⣸⠟⡩⢒⡍⠕⢀⠌⠐⢯⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠁⣿⡇⠀⠈⠂⡀⢾⣶⠌⠁⠉⠐⣋⡉⢰⣗⣤⠚⠋⠀⠀⠀⠀⣠⠔⢱⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡧⣠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣟⡿⢋⣠⠤⣤⡄⠃⢀⣷⡄⠘⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡟⡉⠓⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⣗⡆⢈⣤⡆⠀⠀⠀⠀⠉⠀⠀⣿⠇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⠀⠐⡀⠀⠀⠀⠀⠢⠀⠁⡀⠀⢀⢾⣿⡇⣷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⡀
⠀⠀⠀⠀⢰⣿⡿⠛⠛⢣⡛⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⠐⢣⣼⣿⡇⢸⣆⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠋
⠀⠀⠀⢰⣿⠋⠀⣘⠂⠶⢧⡀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠈⣶⣿⣿⠺⣏⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠄⡄⠰⠎⣛⠐⣶⡈⠀⠀⠈⠡⣛⣠⠝⠁⠎⠉⠐⣨⣵⣶⣦⣶⣾⣿⡗⠷⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢾⠴⣶⡶⠟⠋⣡⠛⠿⠟⠟⠙⠛⢾⣆⠀⠀⠀⠀⠀⠀⠉⣯⠙⢍⠗⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠆⠦⠰⣄⢀⣀⣀⣀⢴⡾⣪⣄⣀⠀⠂⠀⠀⠀⠁⢟⠶⢄⣀⠀⠀⠀⣝⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⠯⢉⠋⡙⠰⠁⠻⠿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⣻⣵⢣⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⡀⠁⠈⠁⠀⠀⠀⠀⠀⣤⣀⠘⠟⡀⠀⡀⠀⠀⠀⠀⠀⠘⡿⣿⠖⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⢀⣿⣶⣦⡜⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡐⠰⠊
⣾⣷⠾⢾⠾⠶⢿⠶⢶⣾⢿⣿⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣷⣷⣷⣷⣾⣾⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠖⠒⠒⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⢀⠀⣤⠀⠀⣿⡀⠀⢀⡄⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⣷⣄⠀⡟⢧⣸⣿⠳⣆⣼⣇⣠⢴⠇⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠰⣦⣀⠀⢹⣯⡻⡇⠈⠻⣿⡆⠙⢿⣿⠃⣿⣀⣴⡾⣁⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠘⣯⡙⠻⢿⡇⠘⣦⡀⠀⠙⣦⣀⣉⠀⠹⠿⢋⡞⡁⠀⠀⢀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡈⠳⣶⣤⣭⣿⣶⡤⠤⠞⠁⢈⡟⠉⠁⠀⠉⠉⠲⣄⣸⣧⡴⣶⠟⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠢⠤⣥⣛⣷⣬⣅⣀⣀⡤⠴⡶⠋⢀⣀⣀⣀⠀⠀⢀⡈⢧⣤⣾⣋⣀⣀⣀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠄⣀⣀⣈⣻⡿⠞⠁⢀⣼⡃⠀⢩⣭⣭⡍⢳⡌⣭⣭⢹⡅⣭⠴⠋⠉⠠⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠛⠿⢿⣭⣉⣀⣀⣀⡤⣾⣿⣷⣄⠀⠀⠀⠀⠈⣷⠀⠀⠈⡇⢈⣹⡶⠶⠖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣒⣠⣤⠭⠽⠿⣿⡟⠀⢸⣿⣿⣿⡀⠀⠀⠀⠯⠿⠶⠂⢰⠟⢯⣉⡀⠐⠒⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⣴⡾⠋⢀⡠⠋⢻⣿⣿⣧⡀⠀⠒⢒⣶⡒⠀⡞⣠⠤⠬⠿⠯⠄⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠴⠚⢋⠡⣴⠞⠉⢀⣀⠀⢹⣿⣿⣷⣤⣀⣀⡉⣠⢾⣀⠙⠳⡤⢀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢁⣾⡥⠖⡛⣿⣿⠀⣸⢿⣿⠛⠿⢿⣍⠉⣷⣄⡉⠛⡓⠿⢆⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠛⠉⠠⠊⣰⣟⣡⢾⣿⢸⣿⡇⣼⣆⢹⣆⢻⠈⠛⢦⡀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣰⠟⠉⠂⢸⣯⡜⣿⡷⠇⢹⡞⠿⢿⡄⠁⠀⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠞⠁⠆⡿⠁⠀⠀⠃⠀⠀⣁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⠤⠤⠤⠶⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣄⣀⣤⠯⠼⣤⣀⣠⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢤⣀⣠⠟⣋⠁⠀⠀⠀⠀⢰⡆⠈⡙⠻⣄⣀⣠⢴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠟⠱⣤⡉⠻⣦⣀⣀⣀⢸⣧⣰⠇⠀⠈⠹⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡤⠤⡼⠁⠀⠀⠈⢿⡶⠛⠉⠉⠉⠉⠛⢧⡴⠞⣫⡄⠈⢧⠤⢤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠛⠓⢶⣶⠏⠀⣀⠀⠀⠀⠀⣀⠀⠙⣶⣿⣤⡤⠌⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣿⠀⠒⠚⠻⡟⠘⢫⣭⣽⢧⢰⢯⣭⡝⠃⢹⡀⠀⠀⠀⣿⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢿⠠⠤⣤⣀⣷⣄⡀⠀⠀⣸⢸⠀⠀⠀⣠⣼⣿⠛⠛⠒⣿⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡄⠀⣠⡿⠻⡌⠙⣦⢀⣭⣨⡁⢠⡟⢁⣾⡾⠷⠶⢠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠳⡀⠥⠴⠞⠛⢦⡉⠀⠒⠒⠂⢈⣠⣞⠹⣇⠀⢀⠞⠒⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣦⡀⠀⢸⠆⢻⡗⠒⠒⠒⠋⠉⠙⠳⢌⣰⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠗⠋⠉⠙⣦⣞⡀⠘⠇⠀⠀⠀⠀⢀⣠⣴⠋⠉⠙⠺⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠉⠛⡖⢲⠛⠉⠉⢣⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣿⣿⡉⢻⣿⠙⠦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣋⠁⠀⠘⣿⡿⣿⣿⢿⡟⠀⠀⠴⠓⠽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠂⠠⡄⠀⠀⠀⠋⠤⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠡⡑⠀⢸⡀⠐⡄⢀⠀⠄⡆⠰⡰⢀⠁⡠⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⢈⠂⡐⢌⢜⢄⢳⣀⣿⣼⣴⣥⣇⣅⣁⠆⡔⢡⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢜⢄⣩⡾⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⣞⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡈⢱⡟⠁⢀⣖⡽⠿⣿⣿⣿⣿⡿⣿⣿⡉⠙⢻⣅⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠐⣰⠏⢀⣞⣿⣷⣷⢞⣙⣭⣭⣟⠻⣿⣿⣿⣦⠀⠹⣆⠡⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⢠⡟⢀⣿⣿⣯⡋⢽⣧⣿⣿⣿⣿⣿⡎⣿⣾⣿⣧⠀⣿⣆⠉⠡⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⢼⣧⣼⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡆⣿⣿⠤⠄⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢼⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⡿⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠄⠒⠊⠉⡀⠀⡺⣿⣿⣿⣿⣿⣿⣿⣬⣻⣛⣛⣿⣿⣿⣿⠿⠛⢿⣿⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠐⠙⣿⣿⣿⣿⣿⣋⡛⠻⠟⠿⠟⠛⠋⠀⢀⣠⣾⣿⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣽⡟⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣒⠠⠭⢒⣀⠀⣀⣀⣀⣐⡀⠠⠄⠒⡺⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠐⠒⠒⠒⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢾⠁⠀⠀⢈⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢄⡀⠀⠙⢦⠔⠋⠀⠀⣠⠞⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⢄⡀⠙⢄⠉⠓⠒⠤⠴⠒⠒⠉⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠉⢢⠈⠓⢤⣀⠀⠀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⠤⠒⠒⠒⠢⡀⠀⡇⠀⠀⠈⠉⠉⠉⢉⡭⠕⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠋⠀⠀⠀⠀⠀⠀⢹⠀⡞⠀⠀⠀⠀⠀⡰⠋⠀⢀⠴⠒⠒⢤⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⠃⣴⠁⠀⠀⠀⢀⡜⠁⠀⡰⠋⠀⠀⠀⠀⠱⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠉⠀⢀⡤⠊⢿⢄⡀⠀⣠⠎⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢣⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠒⢌⡲⢍⣉⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢦⠀⠱⡄⠀⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⢀⠇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⢇⠈⠦⣀⠀⢀⡠⠋⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⠤⣀⡀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣀⠀⠀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠃⢄⢉⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡉⣠
⠃⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇
⠀⠀⠀⠈⢻⣦⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣴⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⢀⠀⠀⠀⣴⡃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠙⠁⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⢙⣶⡆⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢀⣀⣤⡀⠁⠐⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢁⡀⠀⠙⠒⠄⢈⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣬⠿⠃⠒⠀⠃⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠓⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⢀⣀⢀⡀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⡟⢡⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠀⣟⠉⠀⡸⠃⠀⣠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠁⠀⣴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣺⠔⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀
⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠳⠄⢀⢀⠸⢘⠌⠀⠁⠇⠀⠀⠄⠸⠈⠁⠇⠀⠀⠀⠀⠹⣿⣿⣦
⠄⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠻⣧⡀
⡀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⡄⠊⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡔⠀⠀⠀⡄⠀⠀⢡⠀⠀⠀⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠈⠃⠀⠀⢠⣾⣿⣿⣿⣿⣶⡄⠀⠀⠘⠁⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⣿⣿⣃⠰⠄⣘⣿⣿⡄⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⣠⣿⠟⢩⠙⠁⠈⠛⡉⠻⣿⣄⠀⠀⠜
⠀⠀⠀⢠⣀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠠⣶⠓⣠⣾⣿⣯⠀⠂⠄⠈⠀⠠⠐⠀⣽⣿⣷⣄⠂⣖⠄⡀⠀⠀⠀⠀⡀⠀⠄⠠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⠀⠠⠊⠙⣾⢸⣿⣿⡟⠟⢧⣀⣤⡄⣠⣄⣀⡼⠻⢿⣿⣿⡃⣷⠋⠑⠄⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠀⢢⠤⢇⡏⠈⢿⣫⠃⣠⣄⠨⢹⡎⢱⠏⠁⣠⡄⢘⣝⡿⠁⢹⡰⡦⡔⠀⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⣄⠈⢿⣄⠀⠹⠻⣄⡞⠁⠰⣏⣩⡮⠀⢻⣡⠿⠏⠀⣠⡟⠃⣠⠃⠀⠘⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⢀⣈⣙⣛⢻⠇⠀⢠⡸⡍⠃⠈⢫⡍⠁⠐⢩⣆⡄⠀⠰⡟⣛⣋⣁⡀⠀⠀⠀⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⢤⣷⡎⠀⢀⠈⠻⠺⠇⠐⠀⠀⠃⠸⠗⠟⠁⡀⠀⢱⣾⡤⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠏⠤⠾⠛⠹⠶⣦⣤⣤⠄⠀⠀⠀⠀⠠⣤⣤⣴⠶⡏⠛⠷⠦⠹⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠄⣀⣀⠀⢀⠞⢻⣷⣧⠀⠀⠀⠀⠀⠀⣼⣾⡟⠳⡀⠒⡐⣀⠠⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠡⠀⡀⣿⣿⣷⡲⠶⠶⢖⣾⣿⣿⢀⠀⠍⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⣿⣧⡞⢿⣼⣿⣿⣿⡀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠃⢀⡀⠛⠿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⣄⠉⠲⣌⢦⠀⢀⣿⡇⠈⡆⡆⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣈⠹⣦⠸⢼⣒⣻⢿⠀⣼⣣⠇⣴⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠙⠲⣬⠗⠁⠻⢮⣿⣯⡏⠁⠀⢑⢿⣃⡴⣡⠃⠀⠀⠀⢔⠒⡄
⠀⠀⠀⠀⠀⡖⠛⠳⠶⠾⠿⠿⣷⣶⠰⣦⡤⠤⢤⢘⡟⣡⢄⣤⣥⡷⢳⣞⠁⠀⣀⣤⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⡠⠀⠈⠏⠶⣆⠙⠛⢀⡄⡄⣄⠈⠋⢠⠟⠫⠿⢿⠿⠿⠋
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠘⡠⠀⠀⠀⡀⠈⠒⠒⢁⣀⣇⣠⠉⠉⠁⠄⠀⠀⠀⠠⢀⠀⠓⠐⢀
⠀⠁⠀⠂⠀⠢⠀⠀⠆⢠⠄⠀⠐⠀⠀⠲⠖⠺⠋⠈⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠈⠉

⠛⢈⡉⠉⡈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⢀⡉⢁⡀
⡆⢈⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣟⠁
⠀⠈⠉⠙⠋⣠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣄⠛⠋⠉⠁
⠆⠀⠀⠀⠀⠙⢿⣷⣤⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠠⠀⠁⣠⣠⣾⡿⠋
⡄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠁
⡀⠀⠀⠀⠀⠀⠂⠈⠻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⠘⢿⣿⣿⡿⠃⠀⣾⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⣿⣿⣦⡀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣴⣶⡶⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⣦⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠈⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠴⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⠀⢸⣷⠀⣠⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡿⢿⣿⣿⣿⣿⣿⣿⡿⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⢹⡿⠁⠈⠛⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⣤⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡄⠀⠀⠀⡆
⠄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠄⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠄⠀⠀⠀⠸⣷⣦⣀⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣠⣴⣾⠃
⠆⠀⠀⢀⣀⣬⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣥⣀⡀
⠄⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣀⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛
⠂⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣉⣽⣿⣿⣿⡆⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣯⣙⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠉⠛⠿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⠟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠿⠋⠉⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⢼⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢻⣿⣄⠀⠀⠀⠀⣰⣿⠏⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣾⣿⣿⣦⣤⣤⣾⣿⣿⣷⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠃⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠐⢆⠀⡄⢤⡀⠀⠰⠠⡆⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠻⠰⠆⠀⠙⠀⠛⠀⠻⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠃⠀⠀⠀⠀⣴⡿⠛⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠻⣿⣦
⡀⠠⣤⣴⣦⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠁⢶⣦⣤⠄
⠃⢠⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡠⡀⠀⠀⠀⠀⠄⠠⠀⠀⠀⡈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠐⡐⢄⠑⠄⠀⢰⠀⠀⡄⠀⠠⢀⠁⡇⠀⠀⠀⠀⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⠄⡀⠀⠰⠀⢂⠈⢂⡀⠀⠀⢀⠐⠁⠘⠀⠃⠀⢀⠄⡪⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠑⢌⠑⠠⡇⠀⢁⠈⠃⠀⠀⠀⠀⢀⠃⣸⠠⠊⡡⠊⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⢀⣀⡀⠀⠑⡀⠑⢀⠈⠐⢄⣶⢀⠠⠀⡀⣠⠤⠂⠁⡠⠊⢀⠊⠀⢀⠤⢂⠅⠖⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠊⠁⣒⠀⠀⠀⢀⡀⠙⢄⠀⠑⡄⠂⠁⠀⠀⠀⠀⠈⠐⢠⠊⠀⡀⠉⠀⢀⠤⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠁⡀⢦⠈⠠⠐⣒⠀⠀⠀⠄⣒⠢⠄⠡⣬⡄⠒⠈⡀⠄⠈
⠀⠀⠀⠀⠀⣀⣀⡀⠄⠀⠀⠒⠂⠈⠉⠁⠀⠘⠀⠀⠉⠛⠁⠀⠀⠈⠛⠋⠀⠀⢃⠀⠀⠉⠁⠐⠂⠀⠀⠠⢀⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠤⠀⠀⢀⡀⠁⠁⢡⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠀⡈⠈⠀⢀⡀⠀⠠⠤⠀⠐⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⣁⠬⠙⡛⢀⠀⠀⠠⢶⣒⣒⡶⠄⠀⠀⡀⠷⠍⡀⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠐⠉⠀⣀⠌⠀⡠⠂⢄⠀⠀⠀⠀⠀⠀⡠⠐⢄⠀⠡⣄⠀⠀⠒⠀⠀⠈⢐⡠⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡔⢑⠠⠐⠈⠁⡠⠁⡠⠊⢀⡠⠒⠋⠀⠐⠂⠀⠿⠐⢄⠀⠑⢄⠀⢄⠉⠁⠀⠀⠉⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡠⠊⡠⠐⠏⢠⠁⠀⡀⠀⠀⢠⠀⢀⠀⢘⠂⢄⠑⢄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡪⠐⠁⠀⢰⠀⡄⢀⠌⠃⠀⠀⠈⠡⡀⢢⠀⠄⠀⠉⠢⢁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠉⠀⠀⠀⠀⢠⢀⠁⠂⠀⠸⠀⠀⠃⠀⠑⢄⠑⡀⠄⠀⠀⠀⠉⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠖⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠶⡧⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⡀⠀⢠⡀⢡⠀⣤⠀⠀⠐⠀⠀⠀⠀⠀⡌⠀⠀⠀⠼⠁⠂⢄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⢂⠀⢀⠒⢛⠀⠂⠓⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠴⠄⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⣄⠈⡅⠦⠀⠀⢀⠀⠐⠀⠈⡃⠀⡐⠗⠀⠀⠀⠀⠠⠀⣦⡀⠀⡀⠀⠀⣀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠑⢠⠄⠀⠀⠀⠀⠆⡠⠀⢠⠈⠀⠀⠀⠠⠆⠀⠁⢀⠀⠉⡤⢤⡀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠠⡔⢁⠀⠀⡀⠰⡆⠀⢀⠠⠴⠀⠀⠠⡄⠂⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠇⠀⠘⢀⠇⠀⢀⠀⠰⡀⠒⠀⠀⠀⠀⠀⣜⢠⠔⠀⠀⠀⠀⠀⠁⠀⡠⠓⠤⠤⠤⣀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⣸⠀⠀⠈⠀⠀⢉⢁⠀⢀⠀⣇⠀⡋⠀⠄⠆⠀⠀⠀⠀⡀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠋⠡⡀⠂⠀⠀⠀⠀⢹⠀⢸⠀⣇⠀⠣⠀⠠⠤⢠⠀⠀⠊⡀⠊⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠘⣄⠀⠀⠀⢈⠐⡀⢈⠔⠹⠀⠀⣸⠀⠀⠀⠀⠄⠠⠁⠀⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠐⠈⠂⠠⢇⣀⠀⠈⠰⣀⠎⠀⠀⡇⢀⡀⠐⠃⣀⠀⢠⡟⠙⡧
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⡈⠉⠀⠀⠋⠀⠀⠀⠋⠀⠀⠞⠁⠀⠀⣼⠀⢀⡟⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠩⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡏⠀⡼⠁⠀⠀⠄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⢉⠙⢦⠆⠀⠀⠀⢠⣴⡶⣶⣦⣄⣤⡀⠰⠁⠀⠀⠀⣠⠞⠀⠀⠓⣄⠀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⠈⡄⠀⠑⠲⠊⠉⠁⢆⡀⠀⠀⠈⠙⢻⡷⠲⠶⠒⠊⠁⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⢀⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠁⠰⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⢀⠀⠀⡀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠈⠀⠀⢠⡶⠟⠛⠉⠉⢸⡄⠀⢠⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣒⠲⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠁⠀⠀⠀⠀⠀⠳⠄⠂
⠀⠀⠀⠀⠀⠀⠠⢦⠄⠀⠀⠀⠀⠀⣺⠁⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⢀⠄⠚⠁⡤⠀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠆⠀⠑
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⢤⣤⠶⠶⢦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡙⠋⠁⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠠⡈⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢚⣄⠹⣆⢂⠀⠂⠀⢊⡀⠂⠀⠀⠀⢀⡀⠂⠐⠦⠤⠖⠐⠀⢀⠀⢀⣀⠰⠀⠀⠀⢀⢀
⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠈⠳⠈⠙⢣⠄⠀⠊⠀⠃⠀⠀⠀⠀⠀⠁⢀⢀⣠⠀⠀⠀⡁⠀⠀⠀⠐⠂⡀⠸⠃⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠁⠐⠛⠀⠶⠄⢰⡀⠄⢀⠰⢀⠀⠀⠀⠀⣀⣀⠀⠃⠀⠄⠀⡄⠐⢠⡎⠰⠀⠈⠠⠄⡀⡀
⠀⢠⠀⠀⠀⠀⠀⡀⠀⠠⠂⠠⠊⡀⢤⡀⠀⠤⠄⠰⠄⠀⠀⢈⡄⠀⠀⠂⠠⠀⡁⠀⠀⠀⠄⠀⠈⡄⠁⠀⠀⠀⠀⠀⠀⠥⠰⠀⠁
⠀⠼⠀⠀⠀⠀⠀⠄⠀⠁⠀⡔⠀⠀⠀⠀⠀⠀⠈⠡⡄⠈⠐⠢⠤⠴⠀⢓⠠⠄⠀⠬⠤⠊⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠕⠃⠀⡀⠀⠀⠀⠄⠀⢀⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⢀⡤⢷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⠖⠛⠉⠀⢸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠀⣴⣤⠀⠀⡼⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡉⣰⣿⠃⢠⣾⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠉⠉⠉⠉⠉⠁⣿⡇⠰⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠻⣦⣌⠳⢦⣄⡀⠀⠸⣿⠁⠀⠀⠀⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢙⡻⠶⣤⣭⣉⣛⣓⡒⠒⠒⠒⠒⠒⣂⣈⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠛⠳⣦⡉⠉⠉⠛⠛⣛⣛⡛⠛⠛⠛⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣇⡀⣀⣴⠟⠋⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠈⠛⠷⣶⣤⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠾⠟⠛⠁⠀⠀⠀⠀⠈⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠃
⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣿⣟⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⣶⣶⣤⣤⣤⣶⡀⠈⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡌⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⡏⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠈⠀⠄
⠀⠀⠀⠔⡈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⠄⡀
⠀⠀⠀⢆⣿⣿⣟⣤⡀⠀⠀⠀⢠⣀⣠⠀⠀⠀⠀⠀⠀⡀⠀⣠⣦⢂⣄⢹⣿⣿⠉⢠⠄⠀⠀⠀⠀⠀⣳⣀⣄⡀⢀⣤⣤⣵
⠀⠀⠀⠈⠻⠿⠿⠟⠉⡲⢬⣿⣿⣿⣷⢦⢀⣀⣀⣦⣤⣶⣿⣿⣿⣿⣿⢸⡿⣯⣧⣴⣦⣀⠀⠀⢀⣼⣿⣽⣿⡽⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⣉⣉⣠⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠛⣿⣶⣽⣻⣿⣿⣿⣿⠉⠀⠀⢯⡉⠛⠉⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠓⢄⠀⠈⢳⡀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠈⠀⠀⠢⣿⣦⣦⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⣿⡛⠛⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠠⣅⠀⠀⠰⣅⠀⠀⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠛⢻⣤⠚⠛⣻⣶⠈⠉⢹⣧⣘⣋⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢁⢀⢸⣿⢉⣡⣿⣏⣁⣉⣁⣼⣿⣿⣯⣉⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡏⠀⠀⢸⣏⠀⠀⠀⠀⠸⣿⣇⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⣖⣈⣄⢀⡐⢫⣠⡴⣶⣶⣶⢶⣿⢛⡁⣀⣉⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡛⢻⣿⣉⡙⠛⠁⠀⠁⣰⣤⡀⣤⣤⣽⡀⣹⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠈⠛⠧⠏⡷⣶⠇⠐⠶⠛⠂⡿⠷⢿⣿⣿⣿⢹⣻⣿⡀
⠀⠀⠀⠀⡠⠀⠀⠈⠈⠙⡄⠀⠀⠀⠀⣶⣀⣀⣸⣿⣦⣴⠙⢛⡄⠀⣠⣴⣠⣷⣤⣼⠛⡛⢿⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⢰⠁⢀⣤⣼⣷⡏⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡿⢋⠀⠅⠐⢰⣿⣿⣿⣿⣿⡇⠘⡰⠃⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⣤⣴⡀
⠀⠀⠀⠈⠓⢾⣿⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⡛⣶⠶⣿⣇⡈⢉⣥⣶⣾⢦⣶⣾⡿⣿⡟⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠉⣶⡁⠀⠀⠀⠀⢀⣠⣾⠁⣿⣿⣿⣄⠉⢁⠀⢿⣎⠋⡭⢶⡟⢙⢠⣿⣿⡆⣹⣇⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⢻⣶⠀⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠈⠿⣾⢰⣿⣿⣿⠈⡄⢠⣴⣟⣱⣄⣀⣡⣶⠃⢸⢏⣿⢲⣿⡇⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣞⢿⣷⣄⡀⢂⠀⡀⠉⢀⣿⣿⣿⣇⠘⢶⣿⣿⡿⣿⠁⠙⢧⠀⣿⠋⠙⣿⣿⡃⢿⣿⣿⣿⣿⡆⠖⠀⠀⢾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠭⠟⠃⠀⠉⢀⣤⣀⣾⣷⣿⣇⠐⠸⣏⠛⢳⠿⠀⠆⠘⠀⣿⢻⣞⣿⣿⠁⣻⣿⣿⡿⠿⣷⠀⠀⠈⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢁⠀⣀⣤⣾⠟⠉⠉⠙⠫⠉⣻⣇⠀⢧⣴⣉⠀⣀⡀⢀⠀⠿⠛⣿⣫⣾⣷⣿⣉⣃⠐⢶⣄⠀⠀⣠⣿⠃⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣾⣿⣿⣿⣿⠟⢹⣷⣉⡴⢋⣠⣶⡈⣹⣟⠉⠀⠉⡹⣿⣿⣦⣻⣷⣶⠟⢀⣀⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡎⣿⣿⠀⣿⡿⢹⣿⣿⣿⣿⣿⣿⡏⣿⡇⢸⡧⠛⣤⡿⢋⣿⣷⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠞⢠⠋⣸⣻⣿⣿⡿⢡⣿⣿⣿⣿⣿⣾⢿⣿⣏⡁⠈⢀⣴⣿⣿⣿⣿⣿⣿⡟⠛⡆⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡀⠞⠁⣴⠃⣴⢏⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⠒⠝⢛⣣⡞⠛⠛⢋⣵⣿⣿⡿⠟⠁⠘⢾⡄⣿⣿⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠃⢠⠞⢁⣼⠋⡼⠋⠉⣠⣿⣿⢛⣿⣿⣿⠟⠛⣤⠎⠉⠑⠀⠈⠻⠟⠛⠛⠉⠀⠀⠀⠀⠉⣿⡉⣿⣿⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠃⠀⢠⡾⢃⡎⠀⢀⣼⣿⡿⣯⣿⣿⡿⠉⢷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⢿⡟⠷⠄⠀⠻⣿⣿⣿⣿⣿⠉⠉⢻
⠀⠀⠀⠀⠀⠢⠌⢡⣿⡆⢀⣿⣿⡟⣰⣿⣿⠛⠀⠀⠀⠙⣿⣿⢶⣤⣤⣠⠊⢀⣀⠄⠄⠂⠀⠀⠀⠀⠀⠸⣿⠀⢱⡀⠀⠀⠀⣿⡟⡿⢿⡇⠀⠺⢸
⠀⠀⠀⠀⠀⢢⣴⡿⠋⣠⣿⣿⣿⣴⠿⠛⠱⣄⠀⠀⠀⠀⠈⠻⢷⣶⣿⣿⠀⣼⣯⣶⣿⣀⣀⣀⡀⠀⠀⠀⠙⣧⠼⢿⣆⣴⣶⣿⣿⣀⠞⠀⢀⡇⣾
⣀⣀⣀⣤⣤⣾⣿⠁⣰⣿⣿⣿⡿⠁⠀⠀⠀⠘⢦⡀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⢳⡬⣿⣿⣿⣿⣷⡿⠀⢀⡬⣨⣿
⣿⡿⠿⣿⣿⡟⢁⣶⣿⡿⢹⣿⣄⠀⠀⠠⠂⢠⣾⣿⠷⣤⡀⢠⣿⠿⠟⠋⠁⠐⠐⠒⠈⠀⣠⣴⣿⠟⠁⠀⠈⠉⠻⣷⣻⣿⡀⢩⣿⣷⣴⣿⣶⣿⣿
⣛⣿⣡⣿⠛⣠⣿⣿⣿⠁⠀⠻⣿⡀⠀⢀⡼⠟⣿⣿⢰⠏⠛⠉⠓⠒⠶⠢⠤⠶⢒⣶⡾⠟⠛⠃⠀⠀⠀⠀⠀⠰⣾⣿⣧⠸⢿⣴⣿⣿⣷⣿⣧⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⡷⠶⢶⠝⢷⣀⣾⣄⣈⠛⠋⠀⠀⠀⣀⣀⣤⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢿⡽⣿⣧⣩⣿⣿⣿⣿⣿⣿⣿⣿
⠂⢀⡀⣤⣼⣾⣤⣴⣭⣷⣄⣾⠁⠀⠙⠛⠛⠛⠁⣠⣦⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⡀⠀⢀⣿⢙⣿⣿⣷⣿⣿⣿⣾⣛⣻⣿⣿
⡆⢸⣇⣭⣽⣿⠉⠉⠉⠉⠉⢉⣀⣐⠂⢀⣶⢶⣾⣿⠏⠀⠀⠀⣰⠀⠀⠀⠀⠀⢠⣷⣄⡀⠀⠀⣼⡿⢿⡇⠀⢙⣋⣸⡿⠏⠉⠉⠉⠉⣿⡿⣿⢻⣿
⠃⠘⠇⠙⢿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠟⢻⠉⠼⣿⠳⣆⣀⣤⡼⣿⢣⣀⠀⢀⣠⡾⢹⣿⠛⠿⢾⠿⠃⡼⠃⠀⢸⣿⠻⡇⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿
⠀⠂⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣷⣤⣤⣥⡞⠁⣈⣙⠛⠛⢉⣤⣟⠁⣰⣷⣶⣶⣾⣷⣤⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿
⠀⠐⠂⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢿⣿⠉⠉⠀⠳⠿⠛⠛⠻⣶⡟⠋⠙⠻⠟⠁⠉⠉⠙⠁⠈⠉⠉⠋⠀⠀⠀⢀⡀⠀⠀⡿⠲⠾⠿⠽
⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣴⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣦⣦⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⣀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⡀⠈⡀⠀⠁⢠⢤⣶⣶⣦⢰⢠⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠈⠁⠃⠀⠀⠀⠋⠉⠉⠉⠈⠉⠛⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠘⠛⠀⠟⠀⡴⠀⢠⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠈⠁⣄⠀⣀⡀⣠⡄⠘⠦⠀⠿⠀⠸⠀⠀⣶⠀⢠⣤⠀⠀⡉⠁⠰⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢶⠆⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠃⠀⡶⠀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠀⠦⠀⠆⢠⠄⠠⠀⢤⠀⠆⠀⡄⠠⡄⠠⠄⠀⠀⠀⠀⠘⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⡇
⠀⠀⠀⠀⠀⠈⢴⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣆⡇
⠀⠀⠀⠀⠀⠀⠤⠤⠬⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠬⠤

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⡇⢂⣀⣀⣘⣃⣃⠸⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢻⣿⢉⣿⣏⣽⣿⣉⣿⡏⢛⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢿⣧⣨⣿⣶⣿⣿⡿⢿⣿⢿⣿⠟⣿⡿⣿⣿⢿⣿⣾⣿⣵⣿⣃⣹⡿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡿⢿⣏⢹⣿⣀⣿⣧⣼⣿⣶⣿⣿⣿⣷⣿⣿⣤⣿⣧⣈⣿⡿⢻⣿⢿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢩⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣾⣷⣿⣷⣿⣦⣿⣾⣿⣼⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⢸⡷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣦⣼⣿⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⢼
⠀⠀⠀⠀⠀⢸⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⢠⢀⡀⡆⢠⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⠀⢀⡧⢠⡃⢘⡄⢼⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢇⢀⣠⣦⡏⡤⠃⠙⠋⠘⢄⢹⣤⣄⠀⡘⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠠⠀⣨⣿⢟⡋⢀⠀⠀⠀⠀⠀⠈⢸⢻⣛⢝⠁⠀⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠙⣄⠀⣱⣿⣟⣽⣿⡈⠀⠄⠀⠀⣀⠔⢁⣯⡻⡛⠦⢫⠄⡠⠋⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠂⡐⢌⣷⣿⣽⢛⡿⢻⣻⣦⣀⣀⡀⣀⡴⣿⣮⡳⡜⠀⠘⡞⠁⡀⠈⠀⢀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⣄⠈⢲⣿⣿⡇⣫⣾⣿⣿⠟⠃⡀⡀⠀⠙⠦⡹⣇⣻⡄⠈⣿⣦⡐⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⡀⠀⠁⠢⢁⠪⠑⢾⣟⣿⣿⣿⣿⣯⣤⣄⠀⠙⠁⠀⣠⣤⣴⣿⣿⡇⣆⠙⢿⠿⡅⠀⢔⡀⠀⢀⣀⠤⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡈⠐⣿⣿⣿⣿⡟⣿⠉⠓⠛⣁⠠⠤⢈⡙⠛⣿⢻⣿⣿⠘⢮⣦⠀⠀⠀⡈⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠌⣀⠦⢙⡽⠻⣻⡟⣹⣷⣿⡒⠊⠉⠀⠀⠀⠀⠈⠋⣼⣿⠹⣯⡀⠀⠉⠻⢄⠀⠀⠸⡂⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠂⠠⠞⣠⠞⢁⠴⢞⣩⣾⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⢰⣿⡿⡀⢿⡏⠲⣤⣀⠀⠀⡀⣸⡧⠐⠒⠒⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢇⣾⢃⣶⣶⡿⠋⣙⣿⣿⣿⢻⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄⢸⡇⠀⠈⣿⣳⡹⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⢳⢋⡿⠁⠐⢸⡿⠋⠈⢃⡿⣽⣿⣷⣤⣀⣀⣤⣾⣿⣿⣿⣮⠈⡳⠀⠀⠈⠃⣥⢸⡯⣭⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⡟⣧⡾⡼⠀⢰⠛⣀⠀⢀⡾⠻⢛⢩⣿⣿⣿⠟⠿⠻⣿⣿⣿⡞⣷⠁⠀⢀⣴⡆⠟⠀⡗⢆⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢳⣽⡗⠃⠀⢢⠏⠁⢠⠟⠀⠀⠎⠸⣿⡏⠁⠀⠀⠀⣿⠁⣿⣷⠏⠀⠜⠋⡉⠀⠀⢠⠙⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡤⡠⠁⢻⣿⣏⠀⡟⣠⣾⡿⡀⢀⡴⠀⢀⣿⡃⡆⠠⠀⣠⣧⡴⢋⣡⣠⠴⠞⢋⣁⠀⠀⢸⡄⠈⢄⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⡐⠁⠀⢠⠓⢉⡻⢋⡵⠋⢀⣤⣞⡡⣠⠋⢻⣵⣇⠀⠀⡸⡏⣮⣿⣿⣵⣾⡻⠭⠖⢀⠀⡋⠁⠀⠈⢂⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⠀⠀⢨⢀⠜⢠⣿⣷⣿⡿⠿⡿⣾⡥⠀⡹⠋⡟⢢⣠⠏⢹⢹⣿⣿⣿⣍⠉⠳⠔⠤⠊⠀⠀⠀⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⢘⡀⠂⣀⠒⢊⣟⡴⠿⢿⠏⠕⣴⣿⡿⠛⠉⢁⣴⡖⠳⢸⠿⠳⣄⣷⣿⡟⢯⠛⠳⣆⠀⣒⠀⢐⡒⠀⣀⠀⢂⡃⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠁⠈⣿⠇⢠⠇⡆⠀⣿⣿⣷⠀⣰⡈⠁⠁⣤⢆⠀⢀⣿⡿⠹⠀⠘⡇⠀⠈⠙⢍⠁⠈⠁⠀⠉⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⢷⣇⠉⠁⠀⡴⠛⠛⠻⠸⠿⠟⠊⠉⠁⠀⢀⠄⣸⡃⠀⡀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠘⠿⠶⠟⢰⢠⠇⠀⡂⣼⠿⠥⠴⠶⣶⣯⡴⠾⠿⠞⠋⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢈⠟⠛⡓⠛⠁⠐⠤⠀⣤⠌⠋⠁⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠈⠳⠏⢀⠀⡶⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢠⠏⣠⠘⣆⠀⠀⠀⢀⣤⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣉⣥⣤⠘⣷⡀⠀⠀⢯⣴⠃⢦⣼⠆⠀⢀⣾⠃⠠⢤⣍⡙⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⣉⠤⠶⣩⠹⠇⣬⣷⡘⣷⠀⠀⢾⡏⢠⠘⡿⠀⠀⣼⠃⣾⡗⠘⠻⠿⠷⣦⣬⣙⠻⢶⣄
⠀⠀⠀⠀⠀⣰⡟⣡⠴⠿⢃⡀⠂⠁⢀⡎⠀⠻⢧⠸⡷⠃⡞⢀⣿⡆⠸⡆⢻⡏⣸⣿⣁⡀⠀⠀⠀⠐⣉⡉⠵⢦⠙⣷
⠀⠀⠀⠀⠀⢹⣇⢹⡄⠈⠁⠀⠀⡀⠁⠀⢀⣢⠾⣇⠱⣼⠁⠸⠿⠿⠀⢳⡜⠰⠛⠳⠀⠘⠒⠠⠀⠀⠀⠰⣇⡄⣰⡟
⠀⠀⠀⢀⡴⡒⡛⣆⠹⡄⣀⢀⠔⠡⠶⠲⠁⣀⣀⠼⠂⣉⣨⣭⣭⣭⣉⣛⠁⠿⣴⣮⣿⡦⣀⠀⠀⠀⢶⣿⡟⣰⢟⢒⢦⡀
⠀⠀⠀⠸⣜⠄⢜⣼⣦⠙⣟⡖⣶⣆⣄⣤⡾⢋⣥⣾⠁⢀⢹⡏⡀⠈⣿⠁⠀⢳⣦⣉⠣⣀⡼⠃⠀⡘⢻⠏⣴⣧⡣⠠⣣⣇
⠀⢀⣴⠞⣛⣉⣙⡛⠻⢷⡘⣤⠒⣊⣤⡞⡀⠈⠉⠹⣓⡷⠿⠟⠾⠾⠿⠾⣗⠻⠋⠋⠳⡘⣷⡤⢴⣿⠇⡼⠟⣋⠭⠤⣭⣙⠻⣦⡀
⢠⡿⠁⡜⠉⠀⠀⠀⠠⠠⣁⣰⣮⣁⡟⠀⡔⠐⡄⡼⡁⠐⣒⢦⠀⢰⢖⡒⢀⢷⡀⡐⡰⡹⡌⣧⣼⣯⣀⡔⢠⠀⠀⠀⠀⠉⣄⠘⣷
⣾⠃⢸⠃⠀⠀⠀⠀⠈⠀⠉⡙⣿⣿⢁⣼⠵⠼⡇⢧⣝⣀⣯⠞⠀⠐⢬⣃⣘⣵⠇⡷⠰⢥⡇⢸⣿⡏⣀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹
⣿⠀⢺⢶⣀⠀⠀⣦⣦⠀⠈⠁⢸⣾⠘⠁⠈⠑⡄⣦⠼⠿⠇⠀⣀⠀⠠⠿⠿⢤⡆⠞⠉⢀⠹⢸⣿⠀⢀⠀⠠⢶⣆⡀⢠⣴⢻⠇⢸
⢻⡆⢺⣆⠉⠑⢻⣡⠔⠀⠈⢁⣼⣿⠀⡜⠢⠔⣣⣿⡶⠂⠀⠀⠉⠁⠀⠀⠲⣾⣧⠳⠤⢚⡄⢸⣿⣆⠉⠀⠈⠢⡼⠉⠀⣠⣾⢃⣿
⠀⢻⣄⠛⠷⣶⣤⣤⡤⠔⣂⠀⣿⠉⣇⢻⡕⢨⣿⣿⡇⢶⡖⠀⠀⠀⠀⣶⠆⣸⣿⣧⠀⣼⠃⡏⢹⣧⠠⣑⠒⠦⠤⠾⠻⠛⣡⡾⠃
⠀⠀⠙⠻⢶⠶⠶⢶⣶⡿⢃⡜⢳⡾⣽⣆⠻⣧⣸⠛⢿⣶⣤⠀⠀⠀⢠⣴⣾⠿⠻⣀⣿⢋⡜⢩⡴⠚⣧⡙⣿⣷⠶⠶⣿⠛⠉
⠀⠀⠀⠀⣏⠎⢉⢇⡿⢁⠀⢉⠟⠑⠳⢿⣷⣌⠻⣦⣤⡟⢻⣆⠀⠀⣾⠛⣣⣤⡾⠟⣡⠎⠤⠁⠈⠾⣏⣷⡈⠃⠏⠹⢸⠆
⠀⠀⠀⠀⠈⠓⣲⡟⢠⣿⡟⢈⢱⣻⡅⢀⡈⣿⠃⢠⡍⠛⠛⠛⠒⠚⠛⠛⡉⢡⡄⢊⡌⣀⣠⢆⢇⣤⠹⢯⣿⡄⢳⣞⠁
⠀⠀⠀⠀⠀⢠⣿⢀⣀⣸⡇⢃⠨⣓⠮⠌⢻⢟⣳⡿⢠⣧⡽⣟⡇⢸⡻⣯⣵⡈⢧⣪⣄⡴⣦⡼⠛⠉⠀⣈⣉⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠻⣦⡙⠻⣯⡈⠛⠊⢙⡂⣉⡰⣯⢁⡟⠷⠿⠯⠇⠸⠽⠿⠞⢳⡘⣧⡼⠷⣶⣤⣀⣊⣴⠿⢛⣡⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣍⡒⠶⣿⣾⣉⣹⠃⣾⡇⢀⠀⡀⡀⡀⢀⣠⣠⣈⣷⡘⢧⣤⣿⡿⠟⣋⣥⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣌⣉⣁⣾⠋⠙⠙⠮⡿⡋⣙⢷⠛⠉⠁⠘⣧⣀⣉⣥⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⢧⡃⠸⣹⠃⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⢠⣿⡿⢛⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡛⢿⣿
⠀⠀⢸⣿⣷⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣾⣿
⠀⠀⢸⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⢠⠀⠰⠜⠘⠃⠈⠁⠈⠈⠈⠈⠈⠁⠚⠀⠷⠀⡤⠀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠦⠈⠋⠀⠀⠀⠀⣤⡀⢤⡄⢰⠀⡆⠀⢰⠀⢠⡄⠀⡀⠀⠀⠙⠃⢰⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⢶⠂⠃⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⡆⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠁⠘⠀⠃⠐⠀⠃⠐⠀⠙⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⡇
⠀⠀⠀⣿⡿⢛⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣿⣿⡛⢿⣿⡇
⠀⠀⠀⣿⣷⣬⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣥⣾⣿⠇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠉⠉⠓⠦⡀⠀⠀⢀⡤⠖⠲⠦⣄⣠⠴⠚⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⣀⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠛⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠲⣄
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣠⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⡤⠶⠛⠁
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠲⠦⡴⣶⠋⠈⠛⠶⢤⣤⣤⣤⣤⣤⣴⠾⠦⣤⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠉⠓⠶⣤⣤⣬⣥⠶⠟⠉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⡀⠀⠀⠀⠀⠀⠠⠀⠛⢛⣿⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡠⠂⠀⠾⠓⢫⠟⠀⠀⠀⠀⠀⠀⠀⠠⣰⠟⠁⠀⢀⠄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⠞⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⡰⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢀⣼⣯⡴⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣀⣠⡄⠀⠀⠀⠀⠀⢈⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢁⡞⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⢠⠞⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⡴⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠂⢀⣴⣿⣅⠈⡏⠁⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⣦⠶⣻⡉⠠⠸⣟⠁⠀⠀⠀⠀⠀⢡⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⣿⠳⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣷⣆⡀⠉⠲⣴⡟⠀⠀⠀⠀⠀⠀⣴⠀⢀⠆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠁⠁⠈⠙⢿⣷⣄⢈⢷⡄⠀⠀⠀⠠⠴⣟⡁⣇⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠘⠇⠀⠀⠀⠀⣐⡛⠛⠑⠰⡛⠀⠀⠀⠀⠀⢻⠀⠀⠡⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⢀⢀⡀⡀⠀⠀⠀⢀⠿⠯⠑⢆⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡴⠇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁⠖⠈⡽⠁⡀⠀⠁⠈⠐⠄⠀⠀⠀⢀⡴⣀⠀⠀⠀⠀⢸⡂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣴⡂⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⣀⣩⣵⣏⡀⠀⡀⣀⡹⣄⣠⠐⣋⣴⠏⠀⠀⠀⠀⠁⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄⢈⣦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠹⣟⠿⣾⢹⣟⠿⢹⡙⣧⢯⠆⠉⠛⢥⡀⠀⠀⠀⣸⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡜⠁⠀⠡⢄⠀⠀⠀⠆⠑⢄⠀⠀⠙⠘⠿⠀⡈⠿⢨⢣⣿⢃⠀⠀⠀⠈⡷⠀⠀⠠⠛⠳⣇
⠀⠀⠀⠀⠀⢠⣦⣦⠄⠀⠀⠀⠙⠀⠈⠄⠀⠀⠑⢄⣠⢁⣼⠀⣳⡀⠀⣆⣰⡟⠁⢠⢆⣟⠃⢸⠀⣤⣤⣀⡅⢀⠔⠁⠀⡈⢸
⠀⠀⠀⠀⠀⠀⣡⢟⠶⠀⠐⢤⣀⠀⠀⠈⢀⠀⠀⠈⠁⠀⡉⢠⢿⡇⠸⣢⡈⠀⢀⠏⣼⠇⠒⢹⡟⠛⠞⠛⢳⠎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠤⣴⡏⠀⠀⠀⠈⠉⠘⠁⠀⡠⢁⣾⡟⠑⢤⣤⡐⠤⣞⣸⣃⣀⣦⠮⠦⠄⠀⠀⢸⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠠⣀⡀⠀⠀⣰⠁⡾⣹⠃⠀⠨⡍⠁⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠤⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⠉⢉⣅⢨⠏⡏⢀⡀⠀⡼⠂⣶⡟⠙⠁⠒⢲⡆⢤⡤⠄⠒⠁⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢀⠤⠤⠀⠒⠀⠀⠀⠀⠀⣀⠠⠤⠒⠀⢬⡿⠄⣧⣀⣀⡞⢁⣾⢏⣠⠔⠒⠛⠛⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠈⠐⠤⣀
⠀⠀⠀⠀⠠⠃⠘⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⢠⡿⠑⠀⠈⠙⠇⢁⣿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡸⢰
⠀⠀⠀⠀⠎⠠⠁⠀⠀⠀⡠⠊⠀⠀⢀⡀⠄⢀⣬⣏⣀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢠⠀⠀⣀⠤⠤⠤⠠⢀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠳⠠⡀
⠀⠀⢀⠈⣀⣉⠠⠀⠀⢘⠁⠀⠐⠈⠀⠀⠀⠐⣿⠀⠉⠉⠉⠉⣿⡿⢷⡎⠉⢲⡛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⠁⠠⠀⠀⠀⠀⠀⠱
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⢨⣽⡀⠀⠀⣰⣾⣋⣀⣠⡇⠀⠈⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠀⠀⢄⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠤⠤⠤⠤⠿⠷⢶⠿⠟⠉⠉⠉⠉⠉⠒⢤⠄⠑⡀⠉⠀⠀⠀⠒⢂⠴⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠀⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠀⠀⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⣰⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣶⣶⣶⠆⠀⠀⠀⢀⣶⣶⣶⡆
⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⡀⠀⠀⢠⣿⡟⣀⣀⣠⣤⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⣸⣿⣿⣷⣾⣿⠃⠀⢠⣿⡿⠿⠟⣿⡟⠁⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⣼⣿⣿⣯⣶⠆⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠠⠿⠟⢻⣿⡿⠁⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⣴⣿⣿⣧⣤⡤⠀⠀⠐⠛⣻⣿⣿⠏⠀⠀⠀⢰⣿⣿⣧⣶⣾⠟
⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⢀⡾⠃⣀⣀⣀⠀⠀⠾⠿⢻⣿⣿⠟⠁⠀⠀⠀⢠⣿⣿⣏⡀⠀⠀⠀⣼⣿⠿⢻⣿⠋
⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠟⠛⠉⠉⡱⠃⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠠⠿⢻⣿⠟⠀⠀⠀⠀⠋⠁⢠⡿⠁
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⢲⠶⠦⡄⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢖⡮⠅⠠⢄⠀⢀⣖⣷⣿⣍⣾⣦⠀⢀⡤⠀⠭⢶⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡰⠁⡴⠒⠲⡄⠱⣼⠧⠿⠧⠿⠧⠽⣴⠃⡠⠒⠲⣄⠙⡄⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢃⠀⢆⠀⢀⡜⢀⠟⡒⠒⠒⠒⠒⢐⢿⠀⢇⠀⢀⡼⠀⡇⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⢄⣉⣁⠤⠊⠀⠱⡀⠀⠀⢠⠎⠀⠳⢄⣉⣁⡤⠊⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢓⠖⢒⠖⢒⡶⠒⢒⣲⠶⠶⠖⠒⢒⣒⣒⣒⢚⡗⠚⣗⢒⣒⣒⠒⠒⠒⠶⠶⠶⣒⠒⢖⠒⠲⡒⠲⡚⢦
⠀⠀⠀⠀⠀⠀⠀⡜⢠⠏⢰⠃⣠⢋⡴⠊⣡⠴⠒⠊⠉⣉⣀⣀⣀⣀⣀⡇⠀⣇⣀⣀⣀⣀⣈⡉⠉⠒⠲⢤⡉⠢⣕⡄⠘⡆⠙⡄⢱⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡎⢀⠃⢠⢡⠋⢠⠊⢀⡤⠒⣉⣩⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣌⣉⠑⠢⣄⠉⢦⠈⢻⡄⠸⡀⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠀⡇⢸⠀⡶⡆⠀⡇⠀⡏⢠⢪⠋⠁⣿⣿⡎⠙⢫⢢⠀⠀⡴⢻⠋⢡⣿⣿⡌⠙⡝⢦⠈⡆⠈⡆⠀⡇⠀⡇⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢱⡀⠱⡀⠑⢌⠛⠷⢦⣿⣿⡧⠶⠗⠋⠀⠀⠈⠺⠶⢬⣿⣿⠴⠞⠓⣁⠔⠁⡰⠁⡰⢳⠀⡇⠀⠇⢸
⠀⠀⠀⠀⠀⠀⢠⠀⡇⢸⠀⡇⠀⠓⢄⠙⠢⣀⠉⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠊⣁⡤⠚⣁⢴⠁⠈⠀⡇⢰⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⠀⡇⠀⠉⠒⠤⠍⣁⡺⠁⠀⠀⠀⣠⠔⠀⠀⠀⠢⡄⠀⠀⠀⠹⣂⡩⠥⠔⠊⠁⢸⠀⡆⠀⠇⢸⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⢃⢀⣆⣰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣇⠀⠀⠀⠀⠀⣸⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣇⣰⡀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⢘⡶⠋⠁⢀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠧⠤⠽⠦⣀⠴⠯⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢀⡀⠈⠙⢤⡇
⠀⠀⠀⠀⠀⠀⡜⠀⡰⠋⠀⠉⢢⠀⢹⡀⠀⠀⠀⠀⠀⢀⣠⢤⡤⣤⢤⡤⡤⣤⢤⡤⣤⢤⠤⡤⣀⠀⠀⠀⢠⠇⢀⡞⣁⣈⠑⡄⠀⡇
⠀⠀⠀⠀⠀⠀⢇⠀⢰⡀⡔⠂⠚⢄⢸⠁⠀⠀⠀⠀⠸⠃⠚⠒⠃⠓⠚⠒⠓⠚⠒⠒⠒⠚⠒⠓⠚⢻⠀⠀⠨⡄⣸⠋⠀⠈⢣⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⢸⣀⡠⣤⡺⣷⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣠⠀⠀⠀⣹⠛⢫⣶⣲⣾⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠉⢿⣆⣸⡇⠀⠀⠀⠀⠀⠓⠾⠤⠧⢿⠀⠀⠀⠀⠀⣧⠥⠼⠤⠧⠞⠁⠀⠀⢀⡿⢤⣿⠃⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢺⠛⠷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠱⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⣀⢀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢡⠣⠤⣪⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢭⡼⠧⣄⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⡴⠯⡑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣈⢏⠷⢦⣄⣀⠀⠀⠀⠙⠦⠤⠴⠊⠀⠀⣀⣠⡴⠶⢍⢏⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⡼⢶⠛⠛⢳⡒⡲⠶⡖⡲⠛⠛⢳⠮⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠑⠶⠃⠈⠒⠊⠁

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣋
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠃
⠀⠀⠀⠀⢸⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣿⣶⠟⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣁⠀⠀⠀⠀⠀⠀⣾⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶
⠀⠀⠀⠀⠀⠺⣿⡷⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠘⢥⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠚⢿⡟⠃
⠀⠀⠀⢠⠀⠀⢠⢖⡄⠀⡇⢸⠀⠀⠀⡅⠀⢀⢰⠀⠀⠀⣄⠀⠀⣴⣷⡄⠀⣿⡇⠀⠀⢸⠀⠀⢀⣼⡀
⠀⠀⡐⠉⢧⠆⠠⠀⠆⠀⠐⠌⠀⠀⠂⠀⠂⠈⠀⠈⠀⡸⠿⡤⠀⠿⡿⠇⠀⠻⠇⠀⠐⠀⠐⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⢐⣉⡆⠀⠀⣌⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⣄⠀⠀⠈⠉⡇⠀⠰⢤⠀⠀⡉⠁⠀⡀⠀⠀⠀⠰⡀⠀⠀⢸⣿⡇⠀⢿⣿⠀⢸⣿⠀⢠
⠀⠐⠚⠃⠸⠀⢴⠀⢃⡄⢰⠀⡀⠀⡇⠀⢐⠙⡂⠀⢰⣾⣶⣶⠀⡼⡿⢣⠄⢸⣿⠀⢸⣿⡄⣿⣿
⠀⠘⠂⠀⠚⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠀⠀⠈⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⠎⠁⠀⣰⣣⠀⡷⣄⡇⡌⢉⡇⡆⠀⢸⡯⠁⠀⠀⠀⡏⢉⡆⡞⠳⡄⡔⢦⡄⡖⢢⠀⠀⠀⢲⣒
⠀⠈⠳⠶⠲⠃⠈⠗⠃⠘⠇⠷⠚⠑⠓⠶⠘⠶⠂⠀⠀⠰⠟⠉⠀⠓⠚⠁⠳⠚⢡⠷⠋⠸⠦⠦⠸⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠤⠀⠀⠀⠐⠓⠠⠄⠀⠀⠀⠀⠶⠈⠐⠴⠂⠀⠅⠒⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠺⣿⣭⡿⡟⠛⠒⠒⠒⠲⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠉⠁⠀⠀⢀⣶⢴⢤⢤⢤⢴⣦⡄⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠰⡇⡦⢰⠞⣰⡞⠶⠾⠧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠄⠐⠀⠀⠀⠀⠆⢠⡞⠒⠒⠀⠀⣿⡇⠀⠀⠐⢺⡀⠀⠀⠀⠔⠀⠀⢀⠀⠂⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⣠⣄⠘⠀⠀⠀⠀⣯⣕⠀⠀⠐⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡞⠀⠉⠁⠀⠀⠀⠐⣇⣨⠄⠀⠀⠀⠈⠻⣿⣷⣦⣄⡐⠀⠀⠀⠀⢀⠄⠀⠐⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⣶⣿⡟⠁⠈⡀⠀⠠⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⠄⠀⠀⡀⠈⢳⣄
⠀⠀⠀⢠⠞⠀⠀⠠⠀⡈⠀⠀⠀⠀⣴⣾⣿⣿⣿⣇⠄⠐⠀⠀⠀⠀⣦⡞⠉⠀⢽⣶⡄⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⡀⠆⠀⠠⠀⠹⣦
⠀⠀⢠⡏⠀⠀⠐⠀⠠⠀⢀⣀⢴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠀⣤⡿⠀⠁⣤⠈⠹⣧⡄⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣷⠄⠤⠄⠀⢈⡀⠀⠀⠀⠸⣇
⢀⣀⣏⠀⠀⢒⡛⠈⠈⠉⠉⣉⠉⠛⠿⢿⣿⡇⠀⠀⠀⣀⢀⣀⣀⣼⣇⠐⢠⣤⠖⠀⣹⣁⠀⡀⠀⠀⠁⠀⣿⡿⠟⠋⠡⠐⠒⠂⠀⠀⠺⠶⢦⣀⠁⠀⠸⣆
⠈⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⡀⠉⠡⣀⣶⣞⡯⠧⠟⠓⠙⠁⠀⠈⢸⠀⠀⡻⢿⡽⢟⣳⣦⣀⠔⠉⢀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣽⣄
⢰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣄⠀⠻⠞⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡀⢀⠀⠀⠀⠈⠁⠓⢻⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾
⣸⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠀⠀⠀⠄⠀⠀⠀⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢸
⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠸
⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠃
⣇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢸⠀⠀⣤⣤⣤⣄⠀⠀⢨⠀⠀⠠⠀⠀⢀⣠⣶⡦⣤⠀⡌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡯
⣿⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠠⡜⢿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠟⢛⣛⡰⠃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⢸
⢸⠀⠀⠉⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠛⠂⠚⠀⠀⠀⠂⠀⠀⠀⠃⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⣸
⠸⡆⠀⠀⠀⢉⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⠀⠀⠀⠀⡟
⠀⢷⠀⠀⠂⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣣⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⣄⣼⢁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠑⠂⠐⠀⠀⣸⠃
⠀⠘⣇⠀⠀⡀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠏⠘⣿⠃⠀⡄⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠘⣿⡇⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⡀⢀⠀⠀⣰⠏
⠀⠀⠹⣆⠀⠈⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠜⠁⢀⣼⡿⠀⠘⡇⠀⠀⠀⠤⠀⠀⠀⠀⢀⡗⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⣠⠏
⠀⠀⠀⠘⣧⡀⠀⠀⣐⠃⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠃⠀⣰⡇⠀⠀⠠⡄⠀⠀⠀⠀⣼⣧⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⢳⣄⠜⠉⠀⠀⠀⠀⡀⠀⠒⠘⢿⡿⠟⠁⣠⣾⡿⠁⠀⠀⠤⠀⠀⠀⠀⣰⣿⣿⣷⣄⣸⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠯⣀⡀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠒⠻⣿⡿⠑⠀⠠⠆⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠀⢀⡀⠀⠀⠀⠙⡄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠑⠀⠀⠀⠀⡀⠀⠀⠐⠼⠁⠂⣤⠀⠄⠀⢀⣴⠿⣿⡛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠁⠂⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠈⠀⠀⠀⠂⠠⠀⡀⠄⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⢠⠀⠂⠀⠐⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠁⠀⢸⢁⠀⠘⠀⠀⡃⠀⠀⠀⣦⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⣘⡄⠀⠙⡀⠠⠓⠠⠄⠀⡈⠒⡘⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠉⠠⠆⢠⠆⢇⣉⣾⣤⣤⡶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⡀⠀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⣀⠀⠠⠖⣦⢹⡿⣷⣼⢿⡏⣴⠲⠆⠀⣀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠸⣯⣁⠀⣰⣧⣿⠐⠃⠘⠆⣿⣼⣇⠀⣈⣽⠇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣤⣤⣻⣝⡓⢶⣾⢿⣿⣿⣿⣿⡄⢠⣿⣿⣽⣿⣿⣷⡶⠚⣫⣟⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡻⣿⠻⠀⠁⢰⣿⣿⣿⣿⢿⣿⣁⣈⣿⡿⣿⣿⣿⣿⡗⠈⠀⠟⣽⢛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢳⣸⣷⣴⣶⣌⠙⠻⣿⡓⠀⠉⠻⠟⠉⠀⢚⣿⠟⠋⣡⣶⣦⣶⣇⡾⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⣈⣿⣿⣿⣿⣿⣦⣾⣿⡗⠂⠠⢤⡤⠄⠐⢺⣿⣧⣴⣿⣿⣿⣿⣿⣁⡀⠰⡄
⠀⠀⠀⠀⠀⣠⣤⣤⣈⣛⣩⣿⣿⣿⣿⣿⡳⠍⠙⠟⠓⢦⢉⡐⢂⡉⡵⠺⠻⠋⠩⢞⣿⣿⣿⣿⣿⣍⣛⣃⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠻⢯⣯⡥⠀⠻⠿⠿⠿⣿⣷⡶⢶⣶⣶⣤⡷⢾⣤⣶⣶⡶⢶⣾⣿⠿⠿⠿⠟⠀⢬⣽⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⠲⠄⣤⣶⣦⣽⣿⣿⢷⠺⣺⠿⢛⠃⠈⡛⠿⢗⠗⡾⣿⣿⣯⣴⣶⣤⠠⠖⣻⣳⣤
⠀⠀⠀⠀⠀⠓⠿⠛⢋⣭⣛⣿⣟⣿⣿⣿⣿⣷⣄⠁⢀⣄⠜⠇⠸⠣⣠⡀⠈⣠⣾⣿⣿⣿⣿⣻⣿⣛⣭⡙⠛⠿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠙⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⠩⢶⡶⠏⠙⣷⣾⣿⣿⣿⣿⣿⣿⣿⠋⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣼⢹⡿⠻⠿⠟⣩⣵⠛⠉⢀⡾⣤⣤⢷⡀⠉⠛⣯⣍⠻⠿⠟⢿⡏⣧⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣥⡾⣠⠆⠀⢸⡿⣡⣶⣶⣿⢼⡉⢉⡧⣿⣶⣶⣬⣿⠇⢀⠐⣆⢷⣬⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠛⣽⢿⣥⠼⢿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⡿⠧⣬⡽⣯⠛⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠱⣞⡋⠁⠸⡏⢿⡀⣄⣠⢀⡿⢻⠇⠈⢙⣳⠆⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠉⠁⠐⠦⠟⢸⣗⡿⢿⣺⡇⠻⠴⠂⠈⠉⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⢾⠁⠀⢘⡦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠲⠤⣘⢮⡑⢦⣑⣔⣊⣠⣔⠚⠡⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢔⠉⡄⠀⠀⠀⠀⠀⠀⠀⠈⠵⡕⠤⣀⣀⣀⢀⣈⠠⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣂⡤⠴⠒⠒⠢⡄⠀⠀⠘⡄⠀⢀⡤⠚⠋⠁⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡧⢄⣀⣀⣠⠔⠁⠀⠀⢀⢇⠔⠁⠘⡖⠀⡠⠒⢺⠀⠓⡀
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⢠⠓⡶⠁⠀⠀⡆⢂⢥
⠀⠀⠀⠀⠀⡆⣾⠀⠀⠀⠀⠈⢆⠀⠙⠋⠀⠀⠀⠀⠈⠛⠋⠀⢀⠖⠁⠀⠀⠀⣷⡈⢸
⠀⠀⠀⠀⠀⣏⡩⣇⠀⠀⠀⠀⠈⠣⠔⠐⢄⢀⠀⠀⢲⠀⣖⠊⠁⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠰⠐⢭⣦⡀⠀⠀⠀⠀⠀⠀⢨⣄⣁⣄⠈⢸⠈⠳⣄⣀⣀⢀⠤⠊⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠣⡀⠄⠉⠒⠤⡄⣀⡤⠐⠁⠀⠂⠀⢀⠀⢤⡆⠈⠀⢉⠀⠠⡃⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣉⠤⡢⠀⠀⢀⠀⠀⠀⠀⣰⠏⠀⠈⠣⡀⠰⠨⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠼⠤⠈⠪⠴⠄⠊⠁⠀⠀⠀⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣠⣴⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢸⣇⣀⣀⢀⡾⠋⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣠⡾⠛⠉⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠛⢷⣄⣠⣤⠶⠶⠶⣤⣄⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀⣀⠀⠈⣷⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⢷⣼⡇
⠀⣤⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⣤
⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹
⠀⠈⠙⠻⠶⠶⠶⠶⢶⣶⠶⠶⠶⢶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⣠⣿⣁⣘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⣰⣟⣁⣀⣀⣈⡉⠉⢉⣿⠋⠙⠿⠶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡿⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠠⠾⠿⠶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠶⠶⠶⣶⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢉⡟⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠑⢆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⡀⡜⠀⠀⠀⠀⠀⠀⠈⠦⣰⢂⠜⠀⠀⠀⠀⠀⠀⠀⢣⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠢⠤⢩⣯⠤⠤⠤⡄⠀⣠⠖⠒⠒⠿⣠⠤⣄⠀⠀⡰⠒⠒⠒⠴⠧⠤⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⠋⠀⡿⠤⠒⡆⠀⠠⠦⣾⠀⠀⢸⡄⢠⠖⣢⠀⣀⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⠀⣿⠀⠀⢸⠈⠁⠀⡇⠸⠀⡇⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⠀⠁⠀⣿⠀⠀⢸⠀⠀⠀⠉⠊⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠣⠤⠄⣀⣀⡀⠤⠋⢲⠀⠣⣄⣀⠀⠀⢀⡠⠇⠀⢠⡾⢄⡀⠀⠀⠀⢀⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠐⣒⠂⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠙⠦⠤⣄⣉⣉⡭⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠋⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠢⡀⠀⠀⠀⠀⠀⠀⢠⠋⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠂⡅⠀⠀⠀⠀⠀⠀⠘⡄⢠⡠⠃⠀⠀⠀⠀⠀⠀⠀⢸⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠤⢼⣭⣤⠤⠤⢤⠀⠀⡠⠤⠬⣿⠥⠤⢄⡀⠀⠀⢀⠤⠄⠀⠛⠥⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡀⢠⠖⢆⣠⠲⢤⡞⠁⢸⠓⠒⠐⡄⠀⡔⠒⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢇⣀⠇⠀⡇⠀⠀⢸⢱⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠂⠠⡀⠀⠀⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⡇⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢣⠀⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣉⡐⠒⠒⠒⠒⠒⠊⠈⣃⠈⠒⠠⠄⠀⠠⠤⠚⠁⠀⣴⠋⠢⠤⠄⠀⠀⠤⠐⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡑⠒⠒⠀⠀⠈⢀⣀⣀⣈⠉⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠈⠉⠉⠁

⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣤⡴⠀⠀⠀⠀⠐⢶⣆⠀⠀⠀⠀⠀⢠⡆⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⣾⣿⠖⠀⠀⠀⠀⠀⢀⡿⠂⠀⠀⠀⠀⢀⠛⢻⠀⠀⠀⠀⠀⣰⣧⡖
⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠊⠽⡿⠃⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠘⡗⠞⠀⠀⠀⠀⠈⢡⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠸⠷⡾⠀⠀⠀⠀⠀⠸⣿⠂⠀⠀⠀⠀⠀⠾⣶⠀⠀⠀⠀⠀⠸⢿⠖⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠝
⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢰⣷⣦⠄⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⠀⢼⣗⠀⠀⠀⠀⠀⠀⢧⠆⠀⠀⠀⠀⠀⠸⡷
⠀⠀⠀⠚⡿⠃⠀⠀⠀⠀⠀⠈⡹⠃⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠼⡆⠀⠀⠀⠀⠀⠘⡙
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠸⠂

⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⠀⠀⢀⢄⣠⠀⠀⠀⠀⠀⢠⣞⡁
⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢺⣿⡥⠀⠀⠀⠀⠀⣨⣯⠀⠀⠀⠀⠀⠘⢻⣇⠀⠀⠀⠀⠀⠹⣿⠆
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢋⠟⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠟⠏⠀⠀⠀⠀⠀⡰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠇⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠈⠛⣃⣴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣦⣤⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣷⡀⠀⠈⠙⠛⠛⠋⠉⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⠿⠛⠉⠀⠀⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠟⠛⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⡿⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⢰⣂⠀⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣶⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢧⠛⠋⠈⠎⠁⠟⡡⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⡛⠀⠀⠀⠀⠀⠀⠀⠠⠛⣍⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠀⠁⠀⠩⠿⢿⣿⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⢆⢀⠀⠀⠀⠁⠀⠌⠀⠓⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣣⣦⠀⡆⡀⢀⢢⡝⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣼⣇⣿⣼⣿⣿⣾⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡏⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣟⣦⠀⠈⡇⢿⠀⢀⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⢦⣄⣀⣹⡌⢧⣰⣇⣘⣧⡞⣸⠁⠀⣀⡤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠉⣹⠷⣚⣩⡭⢭⣭⣓⠿⣞⠋⣡⠞⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⡚⠋⠙⠓⢿⡞⣡⠞⠁⠀⠀⠀⠀⠈⠓⣎⢻⣧⣤⠴⢖⣋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⣄⡞⡼⠁⣠⠴⢤⡀⢀⡤⠶⣤⠈⢧⢳⣴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠶⠚⢹⢧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⡏⠛⠲⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⣺⡏⡇⠀⠀⠀⠠⠤⠶⠀⠀⠀⠀⢸⢣⣟⡛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣁⣷⡹⣄⠀⠀⠰⢤⡤⠴⠀⠀⢠⢏⣾⣈⣙⡓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⣡⠏⠱⣌⠳⣤⣀⡀⢀⣀⣠⠶⣫⠞⢻⡀⠈⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡤⠖⠋⡟⢲⣬⡭⠭⣭⣶⢻⡟⠲⣄⡙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⣸⢣⠞⠘⡇⢠⡇⠘⣆⢧⠀⠀⠉⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⣧⣼⠀⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠋⢁⡃
⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠄⠀⠀⠀⠀⠀⠀⡾⢃⠀⢰⣿⣄
⠀⠀⠀⠀⠀⠀⠜⣿⣯⡀⠀⠀⠀⠀⠀⠀⠸⣧⣺⣇⠘⢹⣿
⠀⠀⠀⠀⠀⠀⡞⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠦⠝⢿⣶⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣴⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠇⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡓⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢶⣤⣾⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠠⢀⠃⠀⢀⠔⠀⠀⠀⠀⠰⡎⠀⠀⢀⡷⠀⠀⠀⣠⣟⣰⠿⣦⡀⠀⠀⠀⠀⢀
⠀⣠⢠⠌⣦⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⡓⠀⠐⣆⣉⣦⣸⡄⠒⡣⠆⠀⠀⠀⠀⠰⣌⠉⠋⢁⠜⠀⠀⢾⣷⡦⢭⠿⢿⣛⠷⠀⢠⣀⣿⣤⡤
⠀⠉⢘⠀⢹⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⣈⣱⠖⠉⠁⠀⠈⠋⢳⣖⠰⡦⠎⠀⠀⠀⣠⠫⡝⣣⡀⠀⠀⠀⣿⠈⠉⠉⢰⡏⠀⠀⢠⡽⣿⢧⣄
⠀⡿⠀⠄⢽⠀⢰⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⢉⣂⡓⢤⡏⠰⠶⠆⠀⠐⠿⠆⡹⠂⠢⠤⠤⡄⠀⠉⠀⡇⠈⠁⠀⠀⢀⠇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠸⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⠈⠀⣪⣥⡄⠀⠀⠀⠀⠀⠀⠀⠸⠯⣑⣀⠀⠀⠀⠀⠘⠃⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠓⠠⢀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠘⠀⠠⡇⠀⠀⠰⠎⠐⡦⠼⠦⣀⠈⠀⠁⣀⣴⡃⠲⠤⠊⠓⠀⠀⠀⠀⣀⡴⠟⠈⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢢⡀
⠀⠀⠀⡄⠀⠀⡂⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⡜⠕⠂⠒⢯⠹⢏⠉⢻⠤⠤⣵⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠐⠀⠈⠃⠀⠺⣭⣼⣯⠽⠆⠀⠀⠀⠀⠈⠉⠀⠀⢀⡾⠔⢈⡔⠋⠀⠀⠈⠀⠀⠀⠀⢀⢾⠃⠀⠀⠀⠐⠈⢧⣀⠀⠀⠀⠀⣠⡟⠐⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⢀⢀⡀⠀⠀⣾⠇⠀⠀⠀⠀⠘⡇⢀⡙⣿⣷⣾⡟⣉⠀⡼⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⣾⣷⠀⠀⠀⠀⠠⡁⢈⣿⠂⢠⣿⠀⠀⠐⠂⠀⠀⠸⣄⣁⣼⣿⣿⣧⢈⡴⠁⠀⠀⣤⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣿⢹⡀⣿⣹⡇⢠⣖⡆⠀⠈⠈⠁⠀⢸⣿⠀⠀⠈⠁⠀⠀⠀⣸⢿⢟⣻⣟⠻⡿⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⡴⠒⠒⠀⠀⡷⠄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣿⣾⡇⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⣏⡆⠀⠀⠀⠀⠀⢠⣃⡇⠓⢿⡷⠋⣆⡱⡀⠀⠀⠀⠀⠀⠀⣾
⠀⣹⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⣾⣇⠋⠉⠁⢸⣿⣧⠀⣤⡀⠀⠀⠀⠹⡜⡄⠀⠀⠀⠀⡎⢻⠇⠀⢸⡇⠀⢿⡏⢣⠀⠀⠀⠀⠀⢰⠏
⠀⠘⢦⣀⣤⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠹⣌⢦⡀⠀⠀⠃⠘⡧⣄⣸⣇⡠⣞⠀⠸⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢸⠀⠀⢸⡆⠀⠀⡇⠀⢀⠸⠿⠇⠀⢠⠀⠀⠀⠈⠳⣕⠢⠀⠂⠀⢁⠀⠀⠀⢀⠇⠀⠔⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⡍⠉⠉⠉⠉⠉⠉⣡⣤⣄⠀⠀⢈⠀⠀⠸⠇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⡀⠤⠤⠤⢌⡛⠦⣄⣀⡀⠈⠁⠈⠃⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠐⡒⢶⠒⠀⣦⡠⠄⠀⠀⠀⠸⣧⠘⣷⠀⢸⠀⣲⡴⠀⠄⠂⠀⡀⠘⠃⠀⠅⠀⠀⠨⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁⠀⠀⣀⣀⡀
⠀⠀⣈⠋⣨⡋⣚⣁⠶⢠⡀⡆⠀⠀⠀⠹⣿⣿⣧⡈⠠⠀⠀⠀⠀⠶⢀⠀⠀⢸⠆⡁⠀⠀⠦⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢼⣭⣨⡽⢀⠤
⠀⠀⣀⠛⣬⠾⠱⠵⠎⣴⠨⡕⠀⠀⠀⠀⢩⠁⠀⠃⡀⠀⠀⠄⢲⣖⠦⠀⠀⠀⠄⠉⠀⠀⡛⠿⠟⣤⠻⠿⠗⠀⠀⢀⡴⢄⠀⠀⠀⠀⠀⣸⢲⢿⡧⣷⣾
⠀⠀⠀⢠⠄⠎⠃⠙⠳⢶⠑⠉⠀⠀⠀⠀⠸⠀⢀⢠⡈⠀⠀⠐⠲⠂⢀⠀⠀⢠⡄⠀⠀⠀⠐⢶⡀⠉⠁⣶⠊⠀⢀⠞⠀⠈⢆⠀⠀⠀⠀⣙⠟⠀⠃⠐⣏
⠀⠀⠀⠀⠀⠈⠉⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠉⠁⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⡁⠀⠀⠀⠈⡉⠉⠉⠩⠂⢠⠎⣠⣤⣤⣌⢣⠀⠀⠘⡯⠁⠀⠀⠀⢽⡆
⠀⠀⢀⡀⢰⠻⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢣⡀⢀⠠⠄⠀⠀⠀⠀⢰⣲⡦⢀⠇⠀⠀⠀⠀⠈⠀⠉⠀⢠⠃⢎⠀⢟⡾⠈⠀⢳⡀⠀⢫⠀⠀⠀⠀⣸
⠀⠀⠀⠁⢸⢸⠀⠀⠀⢀⡴⠚⠉⠁⢪⣍⠲⣄⠀⠹⣄⠀⠀⠠⠄⠄⠀⠘⠛⢉⠜⠀⠀⢀⠄⠀⢄⠀⠀⣰⠃⠀⠀⠙⠒⠒⠁⠀⠀⠱⡀⠸⡧⠀⠠⢠⡏
⠀⠀⠐⠃⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣥⠈⣆⠀⢸⠓⠤⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠆⠀⡿⢀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣷⠀⠀⢸⠁
⠀⠀⠀⠀⢰⣾⠀⠀⠀⠀⠀⠀⠀⠀⣆⠼⢠⠀⢿⠀⢸⠀⡀⠀⠂⠀⠂⠀⢸⠀⠀⠀⠀⠘⠒⠒⠈⠀⠀⠀⠀⠀⠐⠀⣀⣀⡀⠀⢀⠀⠀⠀⠘⣦⢰⠏
⠀⠀⠀⠰⠾⡿⠆⢀⠀⠱⣀⠀⠀⠀⡘⡀⠂⣠⠃⠀⢸⠀⠀⠀⠎⠑⠀⠀⢸⠀⠰⢡⠬⠙⣀⢀⣴⡿⢿⣦⠀⠀⠀⠰⡇⣶⣾⠄⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⢦⣤⣥⡤⠜⠀⠀⠈⠳⢂⣤⣭⠤⠚⠁⠀⠀⣼⠀⠀⠀⣀⡁⡀⠀⢸⡄⠐⠌⠋⠠⠋⠀⠻⣷⣾⠟⠀⠐⠀⠈⠙⠉⠊⠀⡀⠀⠀⡤⠶⣿⣒⣒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⣀⣁⣀⣈⣁⣀⣤⡸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢤⣿⣿⣩⠭⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠃⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢰⡄⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢛⣿⣿⣿⣛⣋⠀⠀⠘⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⢍⣍⢳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢵⣿⣿⣿⡏⣻⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⣿⣿⣿⣧⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠠⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣭⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢿⣯⣿⣵⣿⣿⣿⣿⣿⣿⠏⢺⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⣭⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣾⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣾⣷⣾⣷⣎⡹⠛⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣶⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣷⣽⣿⣟⣿⣿⢗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⣤⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⣿⡿⣽⣿⣿⡿⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢩⣿⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠐⢿⣿⣮⣿⠯⣵⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢼⡳⣯⡛⣉⣾⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡶⠝⣣⣶⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣿⠉⠉⠈⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣄⡀⣀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣄⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣭⣭⣭⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿