Reaper Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠋⠒⠀⠚⠰⠲⠭⡒⠠⠂⠐⣂⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢛⠀⠈⠠⠄⣆⠀⠾⠶⠦⠐⠒⠓⠈⠓⠒⠊⠙⠛⠛⣷⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠷⠶⠿⠷⣶⣤⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⣤⠈⠩⠾⣯⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⡀⠘⠃⣰⣾⣿⣿⠃⢿⣶⣤⡀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣀⣤⣾⠃⠀⢠⡉⢛⣿⠛⣃⠉⢏⠛⠿⣓⢶⡃⡒⢀⠑⠢⠤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠩⣭⣙⠲⢦⡈⢿⣿⣿⣿⡿⢀⠈⡀⠀⢨⠉⠘⠈⠀⠢⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠴⣾⠋⠀⣉⡛⠶⣕⡌⢿⣿⣿⣿⡿⡄⠀⠀⠀⠰⢔⣄⡤⠛⠢⣄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣠⠟⠀⠐⢀⣴⡇⠀⠺⣿⣿⣶⣄⠑⠈⢿⣿⠏⣀⠈⠡⢄⡤⣼⣾⠀⠀⠀⠀⠨⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⢀⡾⠁⠀⢀⡄⢈⣿⢡⡀⢄⠈⢿⡿⣿⣿⣶⣤⣹⠶⠷⣄⢠⢂⣶⣿⣿⣿⡀⣥⣤⡉⣀⡠⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠀⣾⠀⠀⣠⠟⠀⣪⣿⠈⣷⣌⣧⡀⠙⠂⠙⠻⣟⢠⠆⢸⠿⠃⣾⣿⣿⣿⣿⣿⣿⠿⠻⣾⡇⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⣿⠀⣰⠏⢀⣾⣿⣿⣀⠀⠀⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⡟⢃⣴⠟⠁⠀⠀⠀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⢀⣤⠤⠤⠤⠦⢤⣿⣄⣏⡀⢸⠛⠉⢳⡶⠀⠀⠀⠛⢛⣻⣷⣦⣤⣴⣦⣦⡀⣌⣿⡟⢡⣼⣿⡏⠑⠋⠀⠀⠐
⠀⠀⠀⢀⡟⢁⣠⣤⡀⣠⣀⣀⣀⠈⠉⠙⠲⢄⡈⠓⠀⠀⢀⠩⢭⣀⠀⠈⢻⣿⡇⣿⠟⢾⣿⣷⠏⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠓⠚⠑⡾⢀⡟⠀⠈⠉⠙⢓⣦⠄⢀⡷⢀⣤⡐⢮⣿⣦⣽⣿⣿⣾⣿⣧⣉⣷⣤⡀⣠⠀⠀⠀⣀⠀⠀⠀⠀⣈⡉⠁⣀⣀⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⣸⠁⣼⣧⣤⠤⠶⠚⠉⣀⣤⣾⠟⠉⠙⣷⣿⣿⡿⠿⠋⢩⣿⠟⠋⠁⠈⢛⠉⢳⣀⡴⠋⠉⢳⣤⠖⠋⠩⠟⠋⢙⡆
⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⣠⡤⠴⠒⠋⢡⡞⡡⠊⢀⣼⣿⠟⣉⡄⠀⣠⣟⣠⡖⢀⡴⢿⣿⠇⣸⠋⡴⠃⣠⠾⣥⡖⠀⣠⠖⣠⠞⠁
⠀⠀⠀⠀⢀⡟⢠⣦⡀⠈⠓⢦⣀⠀⠀⡟⠀⢠⡶⠿⠛⢡⠾⠛⠁⢰⣿⣿⡟⢀⡞⠰⠟⢁⣼⠃⠀⣤⠾⠗⣻⡟⢀⠞⠙⠛⠁
⠀⠀⠀⠀⠘⠧⠞⠀⠉⠳⢤⣀⠈⠙⠲⠿⠦⢠⡴⠲⣦⡤⠴⠚⢧⣀⣉⡿⠁⡾⣆⣠⡴⠋⠀⠳⣤⣤⠴⠚⢹⣁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠔⠚⠀⠀⠀⠀⠀⠀⠀⠈⢸⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡼⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⠀⣰⣾⣿⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⠶⢤⣴⣾⣷⣿⡿⠿⠿⠷⢾⣿⣾⣿⣷⣤⣴⣶⣶⣦⣀⣰⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⣟⣋⣭⣥⣤⣤⣤⣴⣦⣴⣖⣒⣶⢮⣭⣝⣻⠿⣿⣾⣍⡁⠀⣀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⠿⠿⣿⣟⣻⠿⣿⣿⣿⣿⣿⣿⣯⣽⣳⠿⣿⣿⣶⣽⡻⣿⢿⠟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠋⣠⣤⣀⣀⣨⣭⣭⡤⣶⣤⣈⠉⠙⠛⣻⣿⣿⣿⣿⣶⣍⡻⢿⣿⠃⡍⣀⣌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣾⣛⣛⡛⠛⠿⠿⠿⠿⠿⣿⣷⣦⣄⣨⡙⠛⢻⢿⣿⠟⣚⣛⣚⣛⡛⢾⣿⣃⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⡊⣿⣿⠛⢧⡀⠀⠀⠀⠀⠀⠙⠛⠻⣯⣄⣀⣸⣾⡷⢊⣽⡿⢨⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡾⣋⣀⡸⣷⡀⢄⣈⣙⢶⣄⣤⣰⣌⠛⠛⠿⠋⠀⣯⣾⣷⣿⡾⣿⣋⣿⣿⣉⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡟⠿⢟⡌⢛⣶⣝⢿⣿⣾⣿⣻⡿⡶⠁⣀⣤⡘⢿⣿⠉⢿⣷⣾⣿⣿⣿⣿⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣩⣤⣼⣿⣿⣯⢻⡎⣿⣿⣟⣯⣿⡜⢿⣿⣿⣷⣼⣿⡷⢶⣦⣉⠛⠿⢿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⣿⡏⣿⣿⣿⡿⢋⣬⣜⣠⣿⡿⣿⢯⣳⡽⠿⣿⣟⣭⣿⢿⣷⣾⣷⣭⣤⣶⣦⣄⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⣧⣿⣿⡍⣠⣾⣿⣿⡿⢋⢔⣭⣿⣭⣉⣓⠈⠹⣿⠻⣿⣿⣿⣯⣿⣷⣶⣷⣮⣤⣤⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⠹⣿⣿⡿⣽⡿⠉⢊⢔⣽⢿⢋⣿⣿⣭⣿⡆⠀⠈⠀⠙⢍⠛⢿⡙⠿⣷⠮⠻⣿⣹⣿⠿⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡟⢿⣿⠟⢰⡿⢀⡴⣱⡿⠿⡐⠗⣣⣿⣷⣿⡇⢹⣶⣄⢤⡀⠀⠀⠈⠂⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣤⣿⣾⡟⡾⢂⠴⣫⣚⣿⢣⣠⡚⣻⣿⣿⣿⣿⣿⣄⠹⣿⣧⣿⡌⣆⢶⡤⣤⣤⣾⢇⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣾⣿⡿⣍⢃⣔⣽⣎⠁⢘⣿⢢⡍⢱⣿⠟⣸⣿⠉⢻⣿⠀⣌⢿⣿⣇⣿⢸⣿⣾⣿⣿⢸⣟⣷⡀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⣿⣿⣯⡻⣌⣷⣾⣿⣿⣷⣬⠙⠛⠒⠉⣠⠀⣿⣿⣷⠸⣿⡎⣿⣿⣿⣿⣿⢸⣿⣿⣿⠛⠇⠻⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡛⣶⡿⢟⣧⣹⣯⣹⡯⠈⠻⣿⣿⣷⣄⡀⠺⢋⡼⢋⡟⣹⡇⢹⣷⢸⣿⣿⣿⣿⣿⣿⡟⠉⣴⣦⣴⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋⢀⣽⣿⣿⣿⣿⣶⡇⠰⣈⠻⣿⣿⣿⣖⠀⠀⢈⡜⣹⡇⣾⣏⢸⣿⡇⢹⣿⣿⠏⠻⠄⠙⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⠏⢀⣶⣿⣥⣿⣿⣿⣿⣿⣿⡄⢽⣧⣿⣿⣿⣯⠙⠛⣋⣴⡟⣵⡘⣿⡈⣿⣇⣫⠛⠻⡇⢸⣿⣾⣿⣿⣿⠦
⠀⠀⠀⢀⣾⡿⢃⣴⣿⣷⠏⢸⣿⡇⣿⢿⣿⣿⣧⠈⣿⣿⣿⣿⡿⠁⠐⠲⠟⣰⣿⣿⣌⣷⠈⠳⠻⣆⢳⣘⣀⠙⣿⣿⠉⠁⡠
⠀⠀⣠⣾⣟⣿⠟⢿⣿⡶⠀⢸⣿⣿⡿⢸⣿⣿⣿⠀⣾⣿⣿⡿⣷⡤⢝⣻⠿⢋⣵⣿⣿⣿⣧⣄⢴⣮⠀⠛⣿⣿⣿⣧⠀⠀⣹⠆
⠀⣸⣿⡿⣻⠃⠀⠈⠉⠀⠀⢸⣿⡘⡅⢸⡏⠿⣿⡄⣼⣿⡏⣠⣿⣷⡄⣠⣴⣿⡿⠯⣬⣿⣿⣿⣯⣿⣷⣶⣼⣿⣿⣿⣿⡿⠃
⠴⠟⠿⠋⠁⠀⠀⠀⠀⠿⠀⣾⣹⣧⣄⠘⡇⣰⣿⢰⣿⡏⢰⣿⣿⣿⠟⣻⣿⠋⠀⣤⣿⣾⠋⠉⠛⠿⠟⠛⠻⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣧⣹⣧⣸⢻⣿⡄⢿⢰⡿⢻⡏⣿⡷⢛⠁⠀⣸⢟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⡼⣾⣿⡇⢈⣸⣷⣼⠇⠘⠷⠏⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⣿⣇⢸⣿⣿⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⢿⣾⣿⣧⡀⣠⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠙⠦⠊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣤⣶⣶⣶⣄⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⠏⠐⢄⠀⠀⠘⢦⡈⠢⡀⠀⠀⠀⠀⣧⢀⣴⡿⠿⠿⣿⠿⠿⢿⣦⠀⡿
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠳⣄⠀⠀⠙⢶⡵⡀⠀⠀⠀⣾⠈⣏⠤⣄⠀⣄⠀⣠⠅⣿⠀⣧
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠻⣦⡀⢀⣝⡗⢀⣤⡞⠸⠀⢻⣸⠿⢏⣭⠻⠿⣣⠏⠈⠃⢲⣤⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡏⠀⢣⠀⠀⢦⣿⣅⠈⠉⠁⣹⣯⡖⠀⢀⠌⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⡫⠺⢿⣁⡝⡲⣄⠳⡀⠈⢿⡛⠿⡿⠿⢛⡿⠁⢠⠎⠀⢀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡝⢀⡴⠋⣿⢌⠑⠈⠳⡙⢆⠈⠻⣄⣀⣴⠟⠀⡴⠃⠀⢠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⡂⠀⠀⠀⣿⣿⣿⠳⢀⡏⠀⢰⣿⡆⠹⣴⡋⠙⢦⣄⠀⠈⠉⠁⢀⡠⠖⣁⡴⣿⣿⣿⣿⣿⡇⢱⡀
⠀⠀⠀⠀⠀⢃⠀⠀⠀⣿⣿⣿⢢⡾⠇⠀⢸⣿⣿⣦⣼⣿⣄⠀⠻⣝⢦⣄⣸⣷⣶⣭⣥⣴⣿⣿⠿⣿⡉⠁⠸⢷
⠀⠀⠀⠀⠀⠈⡄⠀⠀⢻⣿⣿⡼⡁⠀⣼⢸⣿⣿⣦⣍⢿⣿⡳⣄⠈⠳⣝⠟⠛⣹⣿⣿⡿⠋⠁⠀⠀⠙⡆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⢿⣿⣷⣣⠀⣿⣾⣿⣿⣿⣿⢸⣿⡇⠈⠳⣄⠼⢳⡿⢿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⢻⣿⣷⡱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⡜⠁⠡⠦⡉⠻⣯⠉⢽⡆⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠙⢿⣿⣮⡻⣿⣿⣿⣿⣿⣿⡏⢀⡇⠀⠙⣦⣠⠊⠻⣿⣿⣶⣾⡇⢰⡀⢠⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⠻⣿⣮⡻⣿⣿⣿⡿⠀⣸⣇⡇⠀⣿⠏⣷⡄⠈⢻⣿⣿⡇⣸⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠈⠙⠻⢶⣝⡻⣁⢠⣿⣿⣷⣶⣿⠀⣿⣿⣦⠀⠙⣿⣣⣏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠲⣤⣀⡈⠙⠳⠭⣛⠿⣿⣿⣿⣇⠹⣿⣿⣷⡄⠈⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⣿⠉⢙⡖⠒⠤⠤⢀⣉⣙⣛⣦⣼⣿⣿⣿⢆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠃⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢹⣿⠃⣸⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡴⠋⠀⢳⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠘⣣⣶⣿⣆⠀⠀⠀⠀⠀⠘⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠜⣴⡿⠉⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⢫⣾⢿⣦⠀⣠⣤⣘⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⡹⣇⢰⣷⢤⡄⠘⢈⣉⣩⣿⣱⣜⠀⠀⠀⠀⠀⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⠃⣿⣾⣿⡌⠙⢡⣼⣿⣿⣿⣹⣿⣷⡀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢀⠂⣀⡵⣦⣶⠓⡋⠸⣿⣿⣿⠶⢾⣿⣿⣿⣣⣿⣿⣿⣿⣄⣄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣂⠀⠀⢻⣷⠸⠇⢿⠇⠀⠸⣿⣯⣠⣿⣫⣾⣷⣿⣿⣿⣟⣿⣿⣿⣧⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠸⣄⣠⠟⢿⣦⠶⠿⠟⠋⠀⠀⠉⠿⢿⡿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢸⣫⣤⣀⡿⣾⣿⣿⣿⣦⣠⣦⣄⣀⣠⣄⣀⣤⠂⣹⣿⣿⣿⣿⡇⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢳⠴⣾⣿⣿⣿⠇⣿⣿⣿⣿⣿⠿⣛⣿⣭⣽⣶⣶⣶⣿⣿⣿⣿⡏⢹⡇⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡴⠛⣹⣿⡿⢀⣿⣿⣿⡏⣶⣿⣿⣿⣿⣿⠀⣙⢿⣿⣿⣿⠿⣡⡿⢁⡟⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣶⣿⣟⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢿⣿⣿⣿⣿⣿⣿⣿⡴⢛⡈⢈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣱⣶⣿⣟⢿⣿⣿⣿⣿⣌⣛⢛⣿⣿⣿⣿⣿⣿⡹⣿⣧⠀⠈⠉⠉⠉⠉⠀⠫⣴⣿⣟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣌⣩⣿⣿⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⢾⣧⠀⠀⣠⠤⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠈⢉⣩⣿⣿⣧⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣇⠀⠱⠔⠸⢿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣋⣭⣵⣿⡿⣿⣿⣿⣷⣄⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠌⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣩⣭⠶⠋⠀⢩⣿⣿⣿⣿⣷⣄⠁⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠀⠀⠀⡀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠙⠙⠛⠧⡤⢤⡴⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢀⠔⢀⡀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣿⣿⣿⢻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣶⡤⡀⢀⣠⡞⠀⠎⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⢡⠴⠚⠻⠟⠋⢁⣾⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠰⢰⡶⢧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢀⣿⣿⣿⠟⣱⣿⠟⠁⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠋⠁⠀⣠⠾⠟⠁⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣞⡉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣴⣬⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⣹⠋⡩⠍⠚⣍⠻⣄⣸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣻⡀⢿⡋⢀⣀⡠⠬⣑⣄⣹⣇⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⡰⠉⢦⡀⠙⡇⠀⠀⠀⢠⣭⢑⢠⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⡜⢀⡠⠼⢻⣯⠆⠀⠀⠀⠀⢼⡶⠛⣷⡄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢫⠞⣰⢏⠀⢀⢠⡁⠀⠼⡢⠅⠀⠀⠀⢀⠹⠳⣄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢰⠏⢰⣱⡀⠀⠸⢶⣥⣒⣂⡤⢚⣡⣤⡦⢿⠀⢸⣿⠳⣄⠑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⠇⠀⡟⠈⢱⣴⣿⣌⢿⣿⢿⡀⠻⠿⠿⣣⣀⣾⣧⠛⠀⢫⠑⣬⣙⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡾⠀⠀⡗⠀⢀⣰⣽⣿⣤⣦⢠⣿⡆⢲⣶⠤⢻⡿⢹⢀⠀⠘⡄⠾⣯⡻⣋⣽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠇⠀⢰⠃⠀⠲⣼⣬⣿⡖⠿⣄⠀⢉⣨⣵⣾⣿⢰⣼⡀⡀⠰⡇⠹⣟⣯⣿⡷⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⢰⡿⠃⠀⣷⣸⣄⢋⣼⠿⣿⢿⢿⣟⣭⣶⣿⢸⣿⣽⣡⠋⠝⠛⠻⣻⢿⡗⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢀⠊⣴⠶⣾⢁⣿⣩⣿⣣⣦⢾⣿⠠⡿⢟⣛⣿⣷⣿⠽⢃⡞⠀⠰⣎⣥⢻⡈⢄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⢃⣴⡏⢫⣿⣥⠈⢉⣿⣾⣽⠀⣿⣿⣿⣿⣿⠋⠀⡾⣧⠀⠀⢸⣿⡇⢇⠈⢂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠏⠘⡅⠈⠻⣿⡻⣿⣉⣉⣹⠀⣷⣾⢿⣿⡷⡀⢸⡇⠘⣆⠐⡀⢿⠀⢸⡀⠈⣇⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⠀⣁⢰⡀⠀⠘⢷⡈⣿⣟⣻⣤⣿⣻⣿⣯⡇⡇⠸⣇⠀⠸⡆⠁⢸⠀⢸⡇⠀⢹⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⠀⠀⡀⣷⠀⣆⠈⢷⠸⣿⣿⣿⣿⣿⣿⡿⢰⠃⡟⢸⠀⠀⡇⠀⠀⡄⢸⠇⠀⠀⢧⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡀⠀⠰⢸⡄⢻⡀⠈⡄⣿⣿⣿⣿⣿⣿⢁⡞⣸⡇⠈⡆⠀⡇⠀⢠⡇⣼⠀⠀⠀⠸⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣷⡀⠀⢺⡇⢸⡇⠀⣷⢹⣿⣿⣿⣿⣏⡾⢠⠇⡇⠀⠁⠀⢀⡌⠌⣰⠇⠀⠀⠀⠀⢷⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⢿⣶⣿⡇⣿⠁⠀⡿⢸⣿⣿⣿⣿⡟⣠⢿⠀⠃⠀⠀⢀⡟⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⣿⢁⣿⡀⢰⡇⠸⣿⣿⡿⠟⠉⠃⡟⠸⠀⠀⠀⢀⣠⢴⠏⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡌⣿⠗⣾⢻⣤⣽⡟⢀⡾⠁⢁⣄⠀⢀⠀⢠⡏⣱⠏⠀⠀⠀⠀⠀⠀⠀⢸⣷⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⡀⣽⡹⡀⠿⣌⢿⡿⠰⠁⠀⣠⢾⣿⡿⠁⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⠳⢌⡻⠴⢶⡿⠟⠐⣠⣴⢟⡵⠿⢋⡤⣾⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣌⡤⠟⢀⣰⣚⣋⣡⣤⣀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠎⡀⣈⡑⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣘⡄⡌⠀⠀⠈⡀⡼⠋⠉⠲⢄⠀⠀⠀⠀⢄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠂⠉⣁⠀⠀⠀⠀⠀⠀⠀⠈⣃⣉⣱⠀⡅⣨⠃⣀⠀⣀⣀⠳⡀⠀⠀⠈⣼⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣀⠬⠤⠒⠒⠒⠚⠛⠃⠒⠢⠚⡇⠁⠀⡰⠀⢳⣿⣿⠁⢿⣿⣿⢇⠀⠀⠀⡿⠀⡆⠀⠓
⠀⠀⠀⠀⠀⠀⢰⠵⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠐⡇⠀⢦⣉⡠⠿⢆⣉⣁⡎⠀⠀⢠⠏⠈⠳⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠢⣔⢿⣥⠤⠤⢄⡾⠁⣀⣠⠎⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠣⠙⢶⣦⡾⠁⡔⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠇⠀⠟⠁⠈⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣦⣀⢀⣀⠜⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣿⠏⠉⠌⣼⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠁⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡘⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢸⠁⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡇⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠒⠦⣤⠟⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣇⣠⠞⠙⠒⠦⠴⠚⠛⠲⣄⠀⠀⢀⣠⠤⠤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⣠⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣷⣄⠀⠀⠀⢠⣿⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⢰⣿⣷⡀⢹⣿⡄
⠀⠀⠀⢀⣼⣿⠃⠈⠻⣦⠀⠀⣼⣿⡄⠀⣿⣿⣿⠃⠀⢠⣦⠀⠸⣿⣿⠇⣼⣿⠃
⠀⠀⠀⣼⣿⠇⠀⠀⠀⠘⣷⡄⠈⠻⣿⣦⣌⣉⣁⡀⠀⠈⠉⠀⢠⣬⣤⣾⣿⠟
⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠈⠻⣆⠀⠈⠙⠻⠿⣿⣷⣴⣦⣶⣶⣾⣿⠟⠋
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⢿⡿⢿⠟⢻⣿⣿⣿⣿⣿⣿⠛⢿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢳⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠙⠛⠋⠉⠉⠉⠛⠛⠛⠛⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⡴⢖⣻⣯⣿⣿⣿⣿⣿⣻⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠷⣿⣵⣿⣿⣿⡿⠛⠛⠻⠿⣿⣿⣷⣦⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣿⣿⣿⣿⠏⠉⠀⠀⠀⠈⠁⠈⠉⣻⣿⢿⣿⣿⣷⣄⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⡇⣿⠟⠛⢋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡯⠀⠻⣿⣿⣿⣿⣦⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠛⠀⢹⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠻⢿⣿⣿⣮⣻⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠀⢀⡆⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠙⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣿⣿⣿⣥⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠄⠀⠀⠂⣤⡙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⣿⣿⣿⠟⠀⠀⡄⢀⣠⡀⢠⡤⠀⠀⠀⠀⠀⢤⣷⡄⠀⠀⠀⠀⠘⢦⣄⠉⠻⣶⣝⢿⣿⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣁⣤⢠⣾⡇⢰⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠀⢿⢺⣦⣤⡙⠷⣤⡙⢿⣿⣿⣿⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⣿⣿⣿⠋⣼⣿⣿⣿⡿⠇⠘⠁⠀⣀⣤⣶⣾⣿⣿⣶⣤⣀⠀⠠⡄⢸⣿⣿⣽⣿⣦⣌⠻⣦⡻⣿⣿⣿⣏⢿⡀
⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⡟⣡⣾⣿⣿⡿⠋⠀⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⡇⢈⠉⠙⣿⣧⡽⢿⣗⣬⣿⣼⣿⣿⣿⣿⣣⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⣾⣿⣿⣿⠏⢀⣴⣾⠃⠘⠛⢷⡄⠈⠛⢻⠀⣹⠇⠸⠋⢉⠛⠀⣿⣷⣦⡈⠻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣄⡀
⠀⠀⠀⣷⡿⣿⣿⣿⣿⡏⣶⣿⣿⡿⢃⣴⣿⡿⢿⡀⢁⡀⠀⠀⠀⡀⢸⡁⣴⠀⠀⠀⠀⢀⠀⠨⢙⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣧⡿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⢫⣿⣿⣿⠏⣠⣿⡿⠛⠃⠀⠇⠈⣁⢀⡁⠄⠀⠘⡇⠉⣀⢠⡶⢀⡸⢦⠀⡎⠉⠙⠛⢿⣄⠙⢿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣹⡦
⠀⠀⠐⣿⣿⣿⣿⣯⣿⣿⣿⢏⣴⣿⠋⢀⣠⣤⣴⠇⢈⡉⠈⠛⠷⣄⠶⠱⠀⠋⠈⠐⠋⠀⣼⠀⣷⣶⣶⣦⠈⣿⣷⡈⠻⣿⣿⣿⣷⡜⢿⣿⣿⣧
⠁⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⠀⢼⣿⣿⣿⠀⠸⣿⣦⡀⣀⣸⣶⣦⣷⣀⡀⣴⣶⣿⠏⠀⢿⣿⣿⣿⠄⣿⣿⣿⣆⠈⠿⣿⣿⣿⣾⢿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⡿⢡⣿⣿⣿⠟⠀⢸⣿⣿⠃⠀⠐⣿⣿⣷⡼⠿⠿⠟⠿⣿⣸⣿⣿⣏⠀⠀⠘⣿⣿⣿⠀⢻⣿⣿⣿⣧⡘⢿⣿⣿⣿⣏⢻⣿
⠀⠀⢠⣿⣿⣿⣿⡟⢡⣿⣿⣿⣿⠀⡆⢸⣿⣿⠀⠀⠀⢻⣿⣿⡱⡆⣀⣀⡀⠻⡿⣿⣯⣿⡄⠐⢀⣿⣿⡿⠀⢈⣸⣿⣿⣿⣷⡌⢿⣿⣿⣿⣏⢻⣶
⠀⢠⡟⣽⣿⣿⡿⢁⣿⣿⣿⣿⣷⣤⡀⠀⢉⠛⢿⣶⢶⣿⠟⢃⣾⢰⣿⣿⣿⡄⢻⡜⠻⡿⠷⢶⣋⠁⠀⣠⠖⢉⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣎⣿
⠀⠋⣸⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣾⣥⣀⢈⣀⠀⠀⠈⢡⠐⡻⢇⣼⣿⣿⣿⣷⡈⢬⡢⡌⠛⠀⢹⠀⣠⣶⣯⣿⣿⣿⡿⣿⣿⣿⣿⠸⢻⣿⣿⣿⣿
⠀⢠⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣏⣤⣶⣦⣤⣬⡟⢺⣿⣿⣙⣿⣿⡏⣸⣿⢶⡤⣤⣰⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣦⢣⣿⣿⣿⡟
⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠌⠀⠸⣿⢟⣹⢿⡟⠋⠉⠀⢹⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⠸⣿⣿⣿⠁
⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⢀⣆⡤⠀⠀⠈⢁⠈⠀⠀⠄⢠⣜⣻⣿⠛⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⢸⣿⣿⣿⠃
⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⡿⠻⣿⡟⠻⢆⡺⢷⠗⣀⡶⣰⠖⠟⠻⠋⣿⡟⠘⣹⣿⣿⣿⣿⣿⠇⡤⢿⣿⣿⣿⡿⣿⡿⠟⠁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠌⢻⣿⣿⡟⣿⣿⣆⠣⣿⣧⢂⣸⡇⡈⠀⢙⠀⣿⠀⣄⡶⢶⡿⢀⠀⣿⣿⣿⣿⣿⢏⣴⣾⡿⠿⠿⣿⣴⡟
⠀⠀⠀⠀⠙⣿⣻⣿⣿⣿⣿⣿⣶⣿⣿⣷⢹⣿⣿⣆⣬⣿⣾⣫⡟⣻⡶⣿⠶⡛⢚⣩⡷⡾⠁⣎⣼⣿⡟⢹⣿⣧⣾⠟⢻⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣾⣿⠁⣿⠍⠻⣿⠿⣿⣧⣹⣿⣿⣿⡙⢁⠉⠀⠋⠰⠛⠄⠱⠌⠉⣀⡆⣰⣿⣿⣋⣴⣿⣿⣽⡟⠀⢈⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠁⠀⠀⠈⢻⣿⢉⣿⡿⣿⣿⣿⣾⣳⣄⣠⣴⣶⣦⣤⡀⡠⢏⣴⣿⣿⡿⠛⠿⠛⢻⣿⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣿⠛⠻⣇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⠏⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠇⠀⠛⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⠀⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣿⣿⣿⣿⣿⣿⡟⠈⠟⠀⢸⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⡿⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠋⠉⠉⠀⠀⠀⣰⡵⠶⠦⣄⠀⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢠⡷⣤⡶⣶⣊⡃⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⣾⣶⣄⠱⢫⣭⡇⠂⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⡇⠿⣞⠲⣦⣿⠇⣄⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡏⡠⣼⣧⣶⣿⣿⣿⡿⡰⣠⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣇⠛⠙⢝⣿⣿⢡⡵⣡⠣⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⣤⣤⡙⠻⣿⣿⣿⣰⡋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣽⣧⢻⣿⣿⣦⠈⠻⠛⠷⢽⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⣿⡈⣿⣧⡹⣷⣄⢦⣠⡀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢸⣿⢹⡇⢹⣇⢿⣿⣷⡈⢿⡻⠿⣿⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣹⣸⢻⠸⢇⠀⢿⡜⣿⣿⣿⣆⠙⢔⢦⡄⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⣿⣿⢸⠀⢸⢀⠈⢷⡈⢿⣿⣮⠳⣌⠳⣤⢸⡟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣿⣿⣿⠘⠀⣼⠸⣆⠘⣿⣌⢻⡿⣷⣿⢽⡠⣾⡇⠘⡎⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣾⢿⡏⣿⠀⠀⣿⠀⣿⡄⠘⣿⣷⣅⠘⣿⣧⡹⣿⣇⠀⣿⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡟⢸⠀⡇⠀⢰⣿⡄⠸⣿⡀⠈⢷⡌⠀⢹⡟⠻⠽⠃⣼⢿⠸⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⠀⢸⣿⣇⠀⢻⣷⠘⣦⠙⠀⠠⡏⠀⢠⣾⡟⠀⠇⢻⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⣿⡟⠹⡆⠀⢻⣇⠈⢷⡀⠀⠃⢠⣿⡇⠃⠀⠀⠈⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠹⣄⠈⢿⡀⠈⢻⣦⡀⢾⣿⣿⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠉⠀⠘⠇⠀⠀⠉⠳⠀⠸⡌⢿⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣍⡑⣦⣤⢀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⢏⢀⠇⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⣿⣿⣮⡳⡀⢕⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡟⢊⠏⠀⠀⠀⠀⠀⠀⠀⣰⣟⠟⠁⠀⠀⠙⢿⣿⣌⢳⡘⠳⣄
⠀⠀⠀⠀⢀⣿⣿⡟⢀⠎⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⠆⠀⠀⠀⠀⢀⣹⣿⣦⡙⣆⢮⡳⡄
⠀⠀⠀⠀⣠⣿⣿⡳⡌⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣤⣿⣷⢒⡜⠻⠿⠋⢻⡿⡜⢯⣿⣿⠦⠀⣀⡀
⠀⠀⠀⠀⣹⣿⣿⣷⠁⠀⠀⠀⠀⠀⢀⣏⣴⡯⣿⣿⣀⣀⣄⠈⠛⠀⣴⢲⣿⣧⠀⣺⡟⣌⠀⡠⠀⠺⡀
⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣄⣐⣾⡟⢈⣿⣿⣼⣿⡅⢿⡋⠀⠐⠀⡗⠠⠂⠖
⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⡠⢋⣿⣿⣿⣿⣧⢿⣿⣿⣷⣿⣿⠏⠀⣠⣿⣿⣿⣿⠻⣤⡌⢿⢠⡀⢠⣱⠇⡄⢀⡀⠘⡄
⠀⠀⠀⣿⣿⣿⠀⠀⠀⣠⢞⣴⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⡿⠃⢄⣉⣉⣿⣿⣿⢿⣾⣄⢉⣀⠙⢇
⠀⠀⠀⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣽⣿⣝⢿⣿⡀⢘⣿⣿⣿⠀⠀⠘⣿⣿⠈⣿⣰⣮⣿⠇⠠⠀⠄⡞⠒⢤⡀
⠀⠀⠀⣿⣿⡏⢀⣴⣿⣿⣣⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⡛⠁⠀⣶⣶⣿⣧⣤⣿⣘⣿⣿⣄⣄⠀⣼⠃⠁⣰⠄
⠀⠀⠀⣿⣿⣧⣾⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⠎⠛⠋⠱⠍⠉⠿⠿⠟⠃⠀⠈⠀⡀⣈⣉⣙⣛⡛⢹⣿⠟⠉⢻⣿⣴⠛⡁⢆⢸⡆
⠀⠀⠀⢿⣿⣿⣿⣿⠈⣼⣿⣿⣿⣿⣿⣿⡙⠸⣦⣄⣀⣰⣄⣀⣀⣠⣦⣴⣄⢰⣿⣿⣿⣿⣿⣿⢴⣟⢀⣠⣶⣹⡟⠀⠁⢠⡟⠁
⠀⠀⠀⣾⣿⣿⣿⡇⠀⣿⡿⢻⣿⣿⣿⣿⣿⣷⣮⣭⣭⣽⣝⣛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⡇⣿⣿⣿⣿⣿⣿⣦⣀⣼⡠⣝
⡀⣠⣴⣿⣿⣯⢹⣧⠀⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣷⣮⣙⡻⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡟⠉⢥⣀⡾
⠹⣸⣿⣿⣿⣿⠂⠻⣦⣘⣿⣦⡙⠿⢿⣿⣿⣿⣿⣋⣴⠂⣼⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣷⣾⡇⢀⣿⣿⣿⣾⡇⠚⢶⠔⠋
⠀⣼⣟⣄⠛⣹⠷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣧⣻⣿⣿⣤⢾⣋⣀⣀⡀
⠀⠻⣿⡃⣾⣿⣶⣌⢹⡟⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⣴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣾⣻⣿⣯⣿⣷⣿⢿⣿⣉⣡⡴⠆⣀
⠀⠀⠈⠳⣾⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣭⣾⣿⣿⣿⣿⣿⣿⡟⣡⣿⣿⣿⣿⣶⠟⠁
⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⢀⡠⣶⣦⠀⠀⠀⣼⣿⢸⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣦⣤⣠⡤⠀⢰⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣏⣀⣀⡿⠀⠀⣰⣿⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣴⣿⣿⣏⠙⠛⠛⠉⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣟⡿⠛⠉⠁⠉⠀⠀⢀⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠀⢀⣾⣿⣿⣿⣿⣷⣦⣤⣤⣴⣿⣟⣠
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢁⠔⠁⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⢆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢀⠔⢁⣴⣾⣿⣿⣿⣿⣿⣿⣷⣬⣉⣛⠛⠛⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡘⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⢀⡠⠚⢁⣴⣿⣿⣿⣿⣿⣿⢧⡈⠻⣿⣝⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠹⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⢀⣠⠖⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠻⠿⣷⣶⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠻⣄⣠⣤⣤⠾⠛⠛⠉⠙⠉⠀⢀⡠⠖⠈⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠅⠀⠈⠀⠀⢹⡀⠀⠀⢀⡠⠔⠊⠁⢀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠘⣠⣄⠘⣦⠀⠙⠂⠉⠀⠀⡠⣔⣮⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠠⣿⠄⠘⣷⣄⢤⣀⡤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⡿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣤⣿⣮⣌⠩⡟⠃⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⡀⠙⠿⠿⣿⠿⠷⣦⣄⠉⠛⠷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠉⢿⣤⠇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡙⢿⣿⣿⣿⣿⣇⠈⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⠻⣿⣿⣿⡻⣆⠀⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠻⢿⣷⣌⡁⠀⠀⠙⠻⢿⣷⣄⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠈⠉⠉⠀⠒⠢⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣿⠂⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡘⠃⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣷⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⣿⣿⣿⣿⣿⣷⣶⣦⣴⢀⣶⣶⣿⣿⣿⣿⣿⣧⢿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⣧⣿⢿⣿⣿⣿⣿⡿⣿⠆⠀⠉⣿⣿⣿⣿⣿⣿⣿⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡘⠂⠐⠛⠿⢛⡛⠃⣀⡀⢀⡉⠟⠉⠻⠿⠟⠁⠈⣥⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠉⠁⠀⢀⣠⣽⣷⣾⣿⣿⣆⠀⣠⣄⡀⠀⠀⣠⣮⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣶⢟⣿⣿⣿⣿⣿⣿⣽⣿⠀⢛⣯⣭⠶⠶⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⡁⠙⠉⡀⠚⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢱⡀⠀⠀⠀⠀⠘⡻⠿⣿⢿⣿⠳⡀⢄⢁⠀⡠⠈⠀⠀⠀⠀⠀⢀⣴⠿⠀⣈⠇
⠀⠀⠀⠀⠀⠀⢀⣱⡠⢤⡄⠘⠛⠷⣤⣄⡀⠀⠙⠶⣇⡁⠀⢙⠈⠈⡀⣂⠄⠀⠀⠀⢀⣤⣤⣿⠃⢀⣤⠝⠒⢄
⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⣀⣀⣧⣄⡀⠀⠛⠛⠒⠤⢈⡉⠁⠉⠉⠁⢉⣀⡤⠤⡲⠿⠛⠉⠋⠁⢀⣄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠈⠓⠙⠟⠛⠙⠛⠛⠙⠿⢶⡀⠀⠀⠀⠀⠁⠲⠶⣉⡁⠀⠀⢀⠀⣤⡤⠴⠎⠓⠛⠉⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣀⡠⠜⠋⠓⠦⢄⣀⠀⠀⠀⠉⠻⠶⣯⣍⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠻⣿⣿⠟⠛⠋⣀⣤⣤⡴⠖⠚⠙⠉⠐⠲⠤⣀⠀⠈⠉⠛⠻⠿⠶⠒⢒⡖⠛⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠘⠦⣄⣠⣤⡾⠛⠁⣺⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠥⣷⣦⣠⣴⣶⢴⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠃⢈⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣇⠀⠁⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡴⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠉⠁⠀⠈⠉⠉⠛⠒⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠻⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣿⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⢠⣾⣿⣿⣿⣷⣤⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀⢻⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⡟⠀⣀⡀⠀⢻⣿⣿⣿⣿⣿⣷⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⢿⣻⣿⣿⣿⡿⠁⢸⣿⣷⡆⠘⣿⣿⣿⣿⣿⡿⠀⠙⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠉⠉⣉⣭⡤⠀⣾⣿⣿⣇⠀⢤⣉⠉⠉⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠈⣿⣶⣔⠃⡼⠟⢀⠈⠿⠋⠻⠏⠀⠈⢿⡟⠒⠒⣋⣵⠃⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⢈⣷⠀⠀⠀⠀⢸⣿⣿⠑⣤⡀⢀⠀⠀⠀⡀⢀⠀⠀⢀⣷⠛⣿⣿⠇⠀⠀⠀⠀⣾⢁⠀⠈⣷⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⠶⣾⣿⠀⠀⠀⠀⠀⣿⠘⣦⣿⣛⠀⠀⡃⢀⠀⡎⢠⠀⣸⣇⣸⡿⡿⠀⠀⠀⠀⠀⣹⣿⠶⣾⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⣰⠚⠿⠅⠛⠿⠗⢂⡀⠀⠀⣧⠀⣿⣏⢻⠿⠟⢿⠻⠛⠿⠛⠻⣿⣿⠏⠀⡗⠀⠀⠀⡀⠲⠿⠛⠓⠿⠓⢦⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢙⣈⠳⠶⠴⠶⠾⣿⣆⠀⠀⠻⣄⠸⣿⠏⠰⡇⠸⢴⠄⠆⠴⠐⠻⡟⢀⡰⠃⠀⠀⣠⣿⠷⠶⠦⠴⠖⢀⡮⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠳⠮⠶⢶⣤⣈⡛⢷⣄⠀⠈⠳⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣠⣾⠟⣁⣤⡶⠶⠴⠞⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣝⡛⠰⠤⡈⠢⢀⣳⡀⠀⠠⠀⠚⠩⠝⢀⡠⠖⢛⣋⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡉⠢⣄⡀⠀⠀⢀⣠⠄⠈⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣈⠑⠞⢥⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢋⣽⣿⡿⢶⣤⣌⡙⠒⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠚⠭⣀⣠⣾⠿⠛⠁⠀⠀⠈⠙⠿⣿⣤⣀⠩⠓⣢⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢠⡴⠞⠷⢒⣀⡬⢭⠉⠁⠀⣀⡰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠈⠉⡩⢧⣀⠒⠾⠛⠦⣄⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢻⡄⠀⣠⣌⣉⠁⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆⠀⠀⠈⣉⣡⣄⡀⢠⡿⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠛⢿⡏⠉⢀⣰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠈⠹⣿⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⣀⣤⣾⡿⢁⠹⣿⣦⣄⣀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠻⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⠛⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠀⠀⠀⠀⢻⡿⣿⡿⣿⡿⣿⡿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠘⠇⣼⡇⣿⡇⣿⡇⠟⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠛⠻⠿⣿⣷⣦⣤⣀⠈⠁⠉⠁⠉⢀⣠⣤⣶⣿⡿⠿⠛⠋⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣤⣴⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⠿⠛⠋⠙⠻⢷⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠛⠛⠋⠉⠓⠲⠤⣀
⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⡴⢡⡾⠋⠉⢉⡛⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣽⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡎⠙⠗⠒⣤⡀
⠀⣷⡀⢠⡤⠀⠀⠈⠍⠦⡄⠀⠀⠀⢀⣼⠋⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠜⢷⣄⠀⠀⢀⣠⡾⠽⡏⠀⡧⠀⠀⠡⢻
⢠⡌⢱⣿⠃⠀⠀⠀⠀⠀⢀⣀⠀⢀⣾⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⡶⠾⠋⠉⠀⠀⠙⡀⠂
⠘⣯⡄⢹⣇⡀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣦⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣿⡇⠀⠀⠀⣻⣆⠀⠛⠂⠀⠀⠀⣀
⠀⠈⠙⠛⠛⠻⠦⠶⠒⠶⠄⢀⣀⣸⣿⡇⣼⡇⢀⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣏⣿⣿⣷⣾⠿⠒⠛⠻⠦⣶⣀⣠⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⠗⣿⠇⢈⣣⣤⣤⣤⣴⣭⡉⠀⠀⠀⡄⢠⣴⣶⣶⣶⣦⣬⣟⠘⣷⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⢺⡄⢻⢿⣿⣿⣿⣥⣤⣿⠀⠇⢸⠀⣿⣤⣬⣿⣿⣿⣿⢿⠁⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡇⠈⣷⣌⡛⣿⣿⣿⠟⢻⢃⠸⣦⣻⣿⣿⣿⠟⠋⣴⡟⠀⣹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠛⠃⠀⠈⠉⠉⠉⠁⢀⠀⣸⣏⢹⣆⠀⡀⠈⠉⠉⠉⠉⠀⠀⠙⡄
⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⠤⠒⠋⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⢋⣱⡏⣿⣏⢻⡜⠃⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢰⡿⠃⠈⠂⠶⠒⣋⡉⠀⠀⠀⠀⠈⢟⡷⠶⠤⢤⣀⣀⣀⣀⠀⣛⢤⠇⠳⠬⡋⠃⠀⠀⣠⣤⣤⣿⣿⣿⡀⠀⠈⠑⠢⠤⣀⣀⣀⣤⣎⡀⠛⠏⢢
⠀⢸⡇⠀⠀⣶⠖⠈⠿⠃⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⢻⣿⠈⡎⠠⠈⠀⠀⠀⠀⠀⠀⣄⢠⡿⡇⠀⠙⠳⢤⡄⠀⠀⠀⠀⠀⠀⠁⠘⠻⣿⣿⣄⣤⣾
⢀⡴⠇⠀⠰⠗⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢦⣧⣰⠀⠀⠀⠀⠀⠀⣄⣼⣾⢿⠇⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⢀⣠⡟⣿⡿⠃⢀
⡏⣠⡶⠾⢂⠃⠀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⡋⢹⠛⢿⠃⠻⠉⠋⣹⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣦⡈⢻⠃⢿⠝⢀⠎
⠀⢉⣷⠀⢫⡁⢄⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠐⠚⠀⠒⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⠦⠀⡌
⣤⣬⣽⣷⣦⣼⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣥⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠉⠁⠀⠈⠀⠘⢿⣿⣿⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⠀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡄⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣶⠁⠛⠻⢤⣁⡒⢞⡛⠋⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣤⣶⣶⣶⡍⢻⠉⣶⣾⣿⣿⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⢳⠜⣾⣿⣿⣿⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠘⢿⢟⣋⡴⠂⣤⣼⣟⣿⣿⣯⢴⣿⣿⡿⠀⠀⠀⠀⠴⠄⠱⡄
⠀⠀⠀⠀⢀⣟⣠⡤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣴⡴⢰⣿⣿⣿⣷⣶⣶⢿⣿⡟⠀⠀⠊⢁⣠⣶⣦⣤⣅⠀⢰⠐⡀
⠀⠀⠀⡀⢺⣿⠁⠀⠀⠈⠙⠛⠶⢤⣄⡀⠘⠐⢧⡀⢧⢾⣿⣿⣿⣿⣿⣥⡾⠛⣁⣠⡴⠚⠉⠉⠁⠀⠀⢹⣷⣊⣴⠃
⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠴⣁⠀⠃⠸⠙⡉⣿⣿⣿⡷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣬⣝⣸⣩⣩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀
⠀⠀⠀⠀⢠⠀⠐⣄⠁⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⡉⣠⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣶⠦⣀⡟
⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⠋
⣀⡀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡀⡀⢀⠀⡀⡀⣀
⣿⣿⠛⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠒⠛⠛⠛⠓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡟⣠⣶⣿⣿⣶⡄⠀⠀⣠⣾⣿⣿⣷⡄⢻⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣹⠃⢹⣿⣿⣿⣿⠇⠀⠀⠹⣿⣿⣿⣿⠇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠻⣦⣀⠀⣿⠀⠀⠻⠿⠟⠁⢀⣾⣷⡀⠈⠛⠟⠋⠀⠀⣼⠀⠀⣀⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣯⣤⣤⣌⡙⠳⢾⣦⣀⡀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⣀⣴⣯⠴⠚⠋⢀⣈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⠳⢦⣌⣿⢟⣷⠀⠀⠛⠋⠘⠛⠀⠀⡞⠿⠿⠋⣠⣴⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠘⣿⢶⢦⢤⣤⣤⡤⡶⣾⣿⠂⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡀⠻⣿⣺⠾⡴⡧⡷⣟⣿⠏⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⠶⠟⠋⢉⣳⣄⠈⠙⠛⠳⠓⠛⠉⠀⣠⣞⡉⠛⠶⣤⣄⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⣀⣤⠶⠞⠋⠁⠈⠳⣦⣀⡀⢀⣀⣠⠾⠋⠉⠛⠷⣤⣀⠉⠉⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⢀⣴⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⣿⠀⠈⠉⢻⣷⠟⠻⣷⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣷⣤⡾⠟⠛⠻⢶⣤⣼⣧⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⣿⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣿⡄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠙⠛⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⢀⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⣝⣴⣩⠝⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣈⠀⠈⠘⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⡞⣽⡾⠋⠁⠀⠈⠙⢯⣬⠷⢤⣀⡀⠀⠀⠀⣀⣠⡴⢺⣯⠆⠀⠀⣐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠂⠠⠤⡠⠊⢡⡀⢨⡵⣏⣶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⣯⣤⢟⣩⠝⣩⢏⣴⠵⠋⠁⠎⠀⡇⠈⠃
⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠴⠊⣁⠂⠂⠁⠆⡄⡌⡄⣼⠁⢀⢄⠝⢮⡓⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠁⠀⠀⠀⠀⠀⠔⡜⡼⡝⣄⠔⡡⢊⢔⡍⡓⡒⢪⠃
⠀⠀⠀⠀⠀⠀⡠⠃⠤⠉⠀⠀⠀⠀⠀⠀⠀⢞⣜⠞⡴⡸⣻⣟⣵⣫⣊⢞⣴⡃
⠀⠀⠀⠀⠀⢰⠁⠌⠀⠀⠀⠀⠀⠀⠀⠨⣡⣪⢊⠞⡴⡡⢃⠏⡟⡏⡹⡏⣿⠻⡄
⠈⠁⠀⠀⢀⠃⠬⠀⡀⠀⠀⠀⠀⢀⠎⢔⡕⡱⢣⢎⡔⢡⠎⡜⡸⢠⢣⢃⢿⠠⡇⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢸⠀⠬⠀⢥⡄⣀⣤⠲⢁⡴⣫⢎⡔⢡⠎⣠⢃⡞⡰⢣⢃⠎⡸⠈⠰⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⠀⣒⠄⠀⢊⠉⢀⡴⣫⠞⡡⢋⠔⢁⡔⢡⠊⡰⡱⢃⠞⢰⠃⠀⠱⡇⠀⠀⠀⠀⠉⠁
⠒⠀⠀⠀⠀⢧⠐⡭⢖⡩⢖⡥⢊⡤⢊⠴⢋⡴⢋⡔⢡⢞⡕⣡⠎⡰⠁⠀⢄⢻
⠀⠀⠀⠀⠀⠈⢧⡘⡩⠔⡩⠖⣡⠔⣁⠔⢋⠴⣋⠔⣡⢊⠔⣡⠞⠀⢀⢸⢩⠇⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠋⢔⣪⠔⢊⡥⢚⡡⠊⡡⢚⠕⢁⠜⠁⡀⡰⠜⡠⠋⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠢⢔⣊⠡⠖⣁⠔⣫⢔⡡⣾⢕⠕⠕⣁⠵⠊⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠶⠮⠶⠖⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠠⡠⢀⠄⣠⢀⡀⢀⠀⡀⢀⡄⣀⠀⡀⢀⠄⣠⢄⡄⡠⢀⡄⡠⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠚⠵⢪⠔⣡⢚⡥⢊⠔⡡⠚⡵⢪⠔⠡⠊⠔⠩⠊⠰⠋⠚⠑⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⢿⣿⣿⡿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⢀⣿⣿⡀⠀⠀⢈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣿⠏⠹⣿⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠹⣿⣿⣶⣶⣿⣿⠏⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠙⠛⠿⠿⠛⠋⠀⠀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣿⣿⣿⣿⣿⣿⣋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣿⡿⠟⠛⠉⠁⠈⠉⠛⠻⢿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠴⢶⢤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⢻⢛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠻⣿⠇⣀⣀⣤⣄⠀⢠⡤⢀⣴⣶⣤⣴⣲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⣿⣾⣿⣿⣿⡿⣇⣼⢷⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢸⣇⡇⢹⣿⣿⣿⠇⢼⣸⢸⣿⣿⣿⣿⣿⡿⢻⣿⣷⡀⠀⠀⠀⠀⡄⠐⣆
⠀⠀⠀⠀⠀⠀⡠⠠⣄⠀⠚⠣⣬⠉⠹⣄⡀⠸⡌⠀⠈⠛⠋⠥⠚⠁⢹⣿⣿⣿⣟⣉⣩⣤⣾⣿⡟⠀⠀⡠⠔⠊⠙⢁⡨⠦⢤⠄⡄
⠀⠀⠀⠀⠀⠐⠀⣰⣿⠶⠛⠻⠿⣷⣦⣄⡉⠐⠰⢦⡐⠌⠁⠘⠂⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠂⠀⣀⣤⣾⡿⠿⠷⣦⣼⠃⣠
⠀⠀⠀⠀⠀⢣⣰⣿⠇⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⣄⡈⠙⢳⡄⠀⠐⠙⠿⠿⢿⣿⣿⣿⣿⣿⣯⡶⠶⠛⠉⠀⠀⠀⠀⠀⢿⣷⡶⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⡧⣀⡀⢀⣀⣀⣸⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠷⣾⣬⡭⠽⠾⣿⣽⣿⣿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠋⠁⠀⠀⠀⠀⠉⠛⠻⢦⣤⣉⠑⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠔⠀⠀⣀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣄⣀⠀⠀⠀⣰⠮⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠞⣹⣦⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡞⠓⢆⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⡇⢠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⢿⡀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠠⠤⣧⡌⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⢰⠀⡛⣒⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢴⠖⠈⠈⠁⣿⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⣃⣀⣀⣀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠠⠄⣿⡀⠀⠀⠀⠀⠀⠀⢸⣀⠀⡀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⢸⡇⠀⠀⠀⠀⠀⠀⢠⠠⠄⠄⠌⠀⢹⡆
⠀⠀⠀⠀⠀⠈⠁⠁⠁⠈⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⠀⠀⠀⠀⠄⠰⢀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⡆⠂⠢⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⣦⠀⠸⠁⠀⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⣉⣩⡤⠿⠀⠀⠀⠀⠀⠀⢠⡶⡤⣤⠤⣤⠤⣴⣧⠀⠀⠀⠀⠀⠀⠈⠠⡀⣨⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣿⣿⣿⣿⣿⠦⠀⠀⣀⣠⡤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⠟⠳⠶⠶⠛⠉⠁⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⡇⣿⣿⠉⣹⡏⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡷⣮⣤⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⢿⣿⣶⡷⣣⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣀⣀
⠀⠄⠀⠀⠀⢀⣀⣰⣖⣀⣁⡀⠉⠉⠉⠉⠑⠒⠀⠀⠀⠀⠤⠤⣄⡀⠀⣀⣀⢀⡀
⠸⣄⠀⠀⠀⠈⠽⠟⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠀⠉⠃⠀⠀⠀⠀⣶
⠀⡉⠓⠢⣄⣀⣀⣤⣄⣠⣄⣤⣤⣄⣀⣀⠀⢀⡀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⢀⠁⠀⠀⠘⠀⠉⠄⣍⡹⢿⠻⠿⠟⠋⠍⠍⠉⠙⢩⣿⠋⠉⠙⠋⠛⠛⠟⠟⠛⠛⠒⠚⠋⠀⢹⣿⣿⡇
⠀⣀⠀⠀⢸⣦⣀⣼⡛⣧⢨⡀⠠⢀⠂⠀⠀⢰⣰⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢦⣀⠄⢸⣿⡿⠁
⠀⠈⠓⢄⣸⣿⡿⡏⣯⣿⣬⣼⢠⡤⠦⣶⣿⠻⢿⡾⢻⢼⠿⠷⢶⡤⠄⣄⣤⡤⢹⢿⠋⡿⠁⢸⡟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣉⢳⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣄⡿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣝⢣⠪⣳⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠎⡝⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠈⠻⣿⣷⡙⢧⠙⢦⡀
⠀⠀⠀⠀⠀⣼⣿⡿⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⣾⣹⡿⠀⠀⠀⠀⠀⢈⣿⣷⣄⢳⡔⣝⢦⡀
⠀⠀⠀⠀⢰⣽⣿⡷⣜⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣧⣸⣷⡶⢆⠸⢯⡿⠹⣟⣷⡻⣜⣿⣿⠆⠀⢀
⠀⠀⠀⠀⣼⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⡼⢁⣶⣿⣿⣍⠈⣉⡁⠚⠇⢀⣤⣤⣿⡎⠅⣾⣟⡏⠀⣀⠉⣷⡀
⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⣤⣾⡟⢸⣿⣗⣾⣟⡏⣿⡌⠁⠀⠀⣷⢀⠄⠀⠀⡀
⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⢀⡴⣻⣿⣿⣿⣿⡿⣿⣿⣿⣽⣿⣿⠃⠉⣠⣿⣽⣿⣿⢿⣁⠙⣷⠀⠀⢠⣷⡆⠀⠀⠀⠚⡀
⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⣠⢊⣼⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣷⡶⠾⣿⣿⣿⣿⣷⠆⡉⠓⠛⣾⣷⣾⣟⣰⡁⠚⠁⠒⣿
⠀⠀⠀⣿⣿⣿⠀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣟⢿⣿⡧⠀⠽⣿⣿⡿⠁⠀⢹⣿⣿⠹⣿⢩⣜⣿⡟⠈⢙⠂⣼⠤⣀
⠀⠀⠀⣿⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⠿⠋⢀⣤⣤⣿⣋⣀⡏⢸⣿⣿⡇⡈⠀⢰⡞⠀⢈⡇
⠀⠀⠀⣿⣿⣏⣰⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣞⢿⡿⠛⣻⠻⢻⣿⣟⣿⠃⠀⠂⠀⠀⠛⠛⠿⠿⢿⢿⣾⡿⠟⢷⣷⢀⡾⠇⡀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⡿⠝⣆⡁⠀⠠⣄⠀⠁⠀⢀⣐⣄⠀⢰⣶⣶⣶⣷⣶⣶⣼⡿⠀⢀⣘⢻⣿⠐⠬⠉⣼⢿
⠀⠀⠀⣹⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣷⣤⣜⡻⠷⠶⠿⢿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡝⣷⣴⣿⣿⣷⣧⣰⠀⢸⢇⠲⡄
⠀⠀⢀⣿⣿⣿⢿⣯⠀⣿⣯⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣶⣮⣭⣟⠻⢿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⡷⠟⠋⢹⠆
⣴⢲⡟⣾⣿⣿⡈⣿⡄⠹⣿⣆⠻⣿⣿⣿⣿⣿⣿⣿⠟⣉⠀⣿⣿⣿⣿⣿⣿⣿⣶⡌⣻⣿⣿⣟⣿⡇⠸⣿⣿⣿⣟⡿⢧⣀⠙⣳⠋
⠙⢺⣿⠛⣽⣿⣄⡈⢻⣦⣿⣿⣷⣬⣽⣿⣿⣿⣿⣧⣾⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣧⣠⡟⣿⣿⣿⡇⣀⣽⠋
⠀⣟⣿⡏⣀⣼⣉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⣿⣿⣿⣿⣷⣦⣤⠆
⠀⠈⠻⣇⢻⣿⣿⡷⣸⠷⠈⠉⠉⠛⠛⠋⠉⠉⠀⠀⣰⡿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠿⣫⣾⣿⣿⣿⣷⣿⣿⡿⣻⣿⣶⣶⣟⣁⡴
⠀⠀⠀⠉⠻⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣽⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋⣼⣿⣿⡿⠿⠟⠛⠁⠀⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⣀⣠⠔⠻⢿⡄⠀⠀⣸⣿⡇⠛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⣿⣿⣿⣶⣶⠞⠁⢠⣇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⡶⠦⡼⠃⠀⢰⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣾⣿⣿⣷⣄⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠀⣠⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡤⠂
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⡠⠊⢀⣼⣿⣿⣿⣿⣿⣻⠿⣿⣿⣟⣛⡉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢻⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⡤⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣍⣉⣙⣋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⢷⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⣀⠔⢉⣠⣾⣿⣿⣿⣿⣿⣿⠻⠌⠙⢿⣮⣙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡘⣧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⣀⠔⠊⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠙⠛⠿⢷⣶⣶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠼⢦⣤⡴⣤⡴⠛⠋⠉⠈⠁⠀⢀⡠⠔⠉⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠉⢃⠀⠀⠀⠀⠘⣇⠀⠀⢀⡠⠔⠊⠁⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠓⣤⠀⢳⡄⠈⠛⠁⠁⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣜⢧⠀⢻⣦⣤⡤⣤⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⣼⢗⣽⡉⣽⠚⠉⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⡀⠙⠿⢿⣿⡿⠿⣶⣄⡈⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠋⠉⠻⣧⣽⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣇⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣧⡈⢿⣿⣿⣿⢿⣄⠀⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠙⢿⣿⣷⣌⠁⠀⠀⠙⠿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⠿⠟⠛⠻⠿⠿⠻⠿⠿⠻⠿⠛⠻⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⢸⣿⣿⡿⢿⣿⣿⠿⠿⡿⠛⠻⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⣿⡇⠸⠟⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⣰⣿⠋⢰⡆⢸⣿⠃⠀⠀⠀⠀⠀⠹⣿⠇⠀⠉⠀⠙⠻⣿⣷⣹⣿⡆⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢠⣾⣿⠁⠀⠘⠇⠘⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡆⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿⣿⣆⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣼⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⢻⣿⣇⠀⠃⣸⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⣀⣿⡷⠘⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿⣿⣧⠀⠁⢹⣿⡄
⠀⠀⠀⠀⠀⢠⣿⠇⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠹⣿⣧⠀⠀⠹⣿⣄
⠀⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠸⣿⡀
⠀⠀⠀⢀⣿⡇⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⢸⣿⡀
⠀⠀⢀⣾⡏⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄⠙⢿⣿⣧⠀⠀⢹⣿
⠀⠀⣾⡟⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠹⣷⠀⠈⢹⣷⡀
⠀⣾⡏⠁⣤⡟⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⡀⠀⠹⣷
⣿⣿⣤⣴⣿⠃⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠈⣿⣷⣤⣤⢿
⣿⡇⠀⣹⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣀⣀⣿⣶⠟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣾⣛⠏⠁⢈
⢻⣿⠀⠻⣿⠀⠀⢺⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⢶⣶⣿⣿⣿⡟⠉⠐⢿⡗⠀⣶⣶⣤⠜⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢹⣿⠀⠀⣼
⠈⣿⡄⠀⣿⡇⠀⢸⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠛⠿⣶⣴⣿⡿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿
⠀⣿⡇⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⣾⣿⠀⢸⣿
⠀⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢻⡟⠀⢸⣿
⠀⣿⡇⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢸⣿
⠀⣿⡇⠀⢸⣇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣿⠀⠀⢸⣿
⠀⢸⣇⠀⠈⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⣾⡇
⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⠀⢰⣿⡇
⠀⠀⣿⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠇
⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⣿⡇⠀⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸⣿⠀⢸⣿
⠀⠀⣿⡇⠀⢻⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⣸⣿
⠀⠀⢸⣇⠀⠸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⣦⠀⠀⣿⡏
⠀⠀⠈⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠃⠀⢸⣿
⠀⠀⠀⢻⣇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⠀⣼⡿
⠀⠀⠀⠈⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⠀⢀⣿⠃
⠀⠀⠀⠀⣿⡄⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⡟⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⢸⣿
⠀⠀⠀⠀⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠁⠀⣾⠏
⠀⠀⠀⠀⠀⣿⡀⠸⣿⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿
⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡟⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⣸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⢰⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠀⢸⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢸⣇⠀⠀⠀⠀⠈⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢀⡿⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠈⠁⣸⣷⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣷⣻⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⢹⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠰⣶⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡇⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠘⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣿⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢘⣿⡀⢀⣾⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣠⣴⣦⣤⣾⡇⢀⣿⡟⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠼⣿⣿⠿⠟⠛⠛⠛⠓⠾⠿⠶⠶⠶⠿⠿⠷⠞⠛⠛⠻⠿⠿⠷⣾⣿⣷⠞⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⡿⠿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢀⣀⡀⠀⣀⡀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣦⣤⠏⢧⣤⡾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡘⠛⢣⣎⠛⠋⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣦⣀⠙⠿⠿⣋⣤⣾⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠘⠿⢷⣄⣀⣴⠿⠋⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠀⢉⣻⡷⣿⣏⠁⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣷⠶⢈⡁⠰⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣶⣾⠟⠁⠀⠀⠙⢻⣶⠦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠉⢠⣶⣿⣿⣷⣤⠉⢉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡛⠉⠉⢓⠶⢤⣀⠀⠀⠀⡀⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠁⠀⠘⠀⣀⣐⠁⠀⢀⠈⠝⠦⣠⠉⠢⠞⠁⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⡴⠚⠉⠉⠉⠙⠲⢦⣀⠀⠔⠈⠙⠢⡄⣲⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠟⡆⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠳⢤⣄⣸⠁⣨⡦⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠉⠀⠀⣾⠀⠈⢠⡟⠲⢤⡀⠀⠀⠀⠀⠀⠀⠘⠒⠲⡄⠰⣀⠤⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢹⠋⠳⢤⡀⠀⠈⠛⠒⠋⠀⠀⢀⣽⡖⠢⣄⡀⠀⠀⠀⠀⢰⡇⢀⣁⡀⠀⢠⢤⣸⠁
⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠈⠣⣄⣀⠉⠙⠓⠒⠒⠒⠒⠚⠉⣸⠃⠀⠀⠉⠳⣄⠀⠀⠀⠛⠉⠀⠹⢤⠏⠀⠉
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠚⠛⠛⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢤⡀⠀⢠⠎⠀⠈⠓⠦⠎⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⠀⠀⢀⠤⣄⠀⠀⡰⠃⡀⡀⠀⠉⣲⠃⠀⣀⡔⠰⢄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣴⠒⢄⡀⢠⠏⠀⠈⠑⠖⠁⡈⠀⠘⡄⠈⠁⠀⢘⣁⡀⠠⠞⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠜⠀⡄⠀⠈⡁⡔⠢⡀⠀⠠⢀⠃⠀⠀⠱⡜⠀⠀⡤⠀⠦⠔⠁⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠈⢛⠁⠀⢃⡇⠀⢡⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠤⠞⠁⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣇⠀⠀⠀⠀⣸⠃⠀⢀⣠⠤⢠⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡤⠥⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡀⠀⠀⠀⠈⠓⠒⠒⢩⠏⠀⢠⠛⣸⡈⠓⠉⣉⠇⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣡⠵⠶⣄⠀⠀⠀⠀⠐⠃⠀⠀⠸⣉⠏⠚⠋⠁⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⢠⢦⡀⠀⠀⠀⠀⣀⡤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⠀⠀⠉⠀⢀⡠⠤⠾⠿⠶⢏⡉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣉⣐⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⡶⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⠀⠀⠐⠒⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⣤⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢇⣠⠔⠀⠀⠀⢀⡸⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣀⡤⠖⠊⠁⢀⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠎⠁⠀⢀⡤⠞⠉⠉⠙⠦⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⢸⡀⠀⠀⠀⠀⢸⠟⢴⣾⣿⣿⡟⠂⣀⣶⣿⣿⡇
⠀⠀⠀⠀⠈⠠⠐⠠⡀⢳⣄⣤⣰⣿⡿⠀⠻⣿⣿⠿⠟⢸⣿⣿⠿⣿⠇⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠰⢆⠀⠘⠄⠉⠋⠉⣜⠁⠀⠀⠀⠀⠀⠀⢿⣻⡿⠐⠀⠀⠀⣀⠠⠆⠀⠀⠁⢦
⠀⠀⠀⠀⠀⠀⠈⠀⠂⠒⠓⣤⡀⠸⣷⣦⣤⣶⣄⠀⠀⠀⢀⣁⢸⣿⣾⠋⢀⣀⣀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠾⠻⠛⠛⠻⠷⠶⠿⣋⠉⠛⠙⠛⠦
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉
⠀⠀⠀⠀⠀⠠⠄⠤⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⠿⢟⣋⣥⡀⠀⠐⠢⠄⡉⠛⠿⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⢛⡩⡴⠞⠛⠉⠀⠀⠀⣀⣀⠀⠀⠉⠂⠠⡈⠉⠛⠿⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⣋⡥⠖⠚⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣶⣤⠀⠀⠀⠀⠀⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⠛⣉⣥⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢀⠜⠁⠀⠀⠀⠀⡜⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⡀⠄⠚⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⡸⠀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⡰⠋⠀⠀⠀⢀⣴⡿⢿⣶⣤⣀⠀⠀⣠⣶⡿⠛⠀⠀⢀⠔⠁⠀⠀⣀⠀⠀⠀⡰⠁⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠁⣼⡅⠀⠀⠀⠰⢿⣋⠀⠀⠈⠙⠻⢿⣾⡿⠋⠀⠀⢠⠜⠁⠀⠀⠀⠀⠛⠁⠀⢠⠃⣸⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡟⢁⣼⠟⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⢴⣧⣀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢢⣿⡿⢻⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⣿⠏⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀⠀⠀⠀⢀⠂⠎⣼⣿⠁⣸⣿⢡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢠⠞⠓⠦⣀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⡸⢠⣿⡏⠀⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢀⡔⠁⠀⠀⠀⠈⠉⠲⢤⡀⣰⠃⠀⠀⠀⠀⡇⢸⣿⠃⠀⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⣴⠏⠀⠀⠀⠀⠄⠀⠀⣠⣾⡿⠋⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⣠⠃⣸⢃⣾⡟⠀⢠⣼⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢀⡾⠃⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⡶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠊⡠⢊⣴⣿⣿⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢠⡞⠁⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠔⣋⠔⣡⣾⠟⣻⢹⣿⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⠋⣰⠟⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⡵⢊⣵⡿⠟⠁⠀⣾⣿⣿⣆⣾⠇
⠀⠀⠀⠀⢠⣼⡿⠁⣴⠋⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⡥⢚⣡⣾⠿⠋⠆⠀⢀⣼⣿⡿⣿⣿⠃
⠀⠀⠀⢀⣾⡟⢀⡾⠁⠀⠂⠀⠀⣠⣿⠏⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡥⠞⣡⣴⡿⠛⠁⠀⠀⠀⢠⠸⣟⣿⣾⣿⠇
⠀⠀⢠⣿⠏⢠⡞⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢚⡥⠞⣡⣴⡿⠟⠁⠀⡄⠀⠀⠀⠀⣰⣶⣿⣿⠟⠁
⠀⣰⣿⠋⣰⠏⠀⠀⠀⢠⣾⠟⠁⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⡥⠞⣉⣴⡾⠟⠉⠀⠀⠀⠀⢠⣶⠀⣠⣿⣿⠿⠋
⣰⡿⠁⠰⠁⠀⠀⠀⢰⡟⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⣁⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣈⣵⣿⠿⠋
⣿⣅⠘⢄⡁⠀⠀⠀⠀⠈⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⢀⡤⠾⣍⣴⡾⠟⢉⠄⡄⢠⠆⠀⠀⠀⢀⣴⣾⠟⠋
⠸⣿⣿⣦⣝⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢁⣤⡾⠟⠁⠀⢠⢎⠞⠀⠁⠀⢀⣤⣾⠟⠋
⠀⢿⣿⡙⠿⣿⣦⣈⠒⢤⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⣰⠣⠏⠀⢀⣴⣾⠟⠋⠃
⠀⠘⣿⣿⡀⠀⠉⠛⢿⣦⣌⡐⠳⢠⠴⣻⣿⠿⠋⠀⠀⠀⠀⠀⠰⠃⢀⣤⣾⠿⠛⠋
⠀⠀⠈⠻⣷⣤⡀⠀⠀⠈⠙⠻⠷⣶⣾⢛⠁⠄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠈⠹⢿⣷⣦⣀⠀⠰⠂⠀⢸⡇⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⠀⠀⣇⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣼⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⢀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⢸⣿⣦⡀⠀⡟⢷⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠹⣆⠹⢦⣽⠀⣿⡏⢸⣿⣆⠀⠀⢀⣀⣀⣀⣀⡀
⠀⢠⣶⣶⣿⣖⠻⠷⠈⠙⠁⠀⣀⡤⠄⠛⠛⠉⠉⠉⠉⠉⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⣉⠉⠉⠛⠛⠓⠶⣦⣀⠀⠀⠀⠀⣰⣿⠶⢻⡟
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣈⡙⠻⣆⣀⣀⠀⠈⢙⡿⠛⢳⣴⠋⠀⠀⢾⡦⢤⣶
⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⡶⠟⠛⠉⠳⠞⠋⠀⠀⠀⠙⣆⢀⡀⠀⣴⣞⠁
⠀⠀⠀⢀⡼⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠘⢏⠁⠀⠀⠉⠙⣦⣄
⠀⠀⢰⡏⠀⠀⠀⢠⠁⠀⠀⠀⡤⢤⣤⡀⠀⠀⠆⠀⠀⠀⠀⠰⠋⠀⣴⠟⠁⠀⠀⠀⠙⢶⡀⠀⢸⣁⣤⠖⠒⠈⠉⠉
⠀⠀⢸⡇⠠⠀⠀⠀⢀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡖⢲⠀⠈⣷⠀⠸⣿⣤⣤
⠀⠀⢸⠇⠀⠀⠀⣰⠟⠉⠁⠀⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠉⠁⠀⣠⡏⠀⢰⠋⠰⣏
⠀⠀⣸⠀⠀⠀⢸⡇⣠⠦⣀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣀⠀⠀⠸⡿⣦⣤⣀⢀⣤⠾⠋⠀⠀⣼⠒⠀⣽⠀⢰⡾⣶⠿⣷⣶⣤
⠀⠀⣹⡀⠀⠀⢸⣇⠉⠲⠋⠀⠀⠀⢀⣼⡇⠀⠀⣼⢆⠀⣿⣥⠀⠀⠳⣤⡈⠉⠁⠀⠀⠀⠀⠀⣸⠶⠾⠋⠀⣼⡇⢸⡄⣿⠀⣿⣀
⣴⠶⠛⡇⠀⠀⠈⠻⣦⣀⡀⢀⣀⣴⠟⣹⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣿⢻⢸⡇⢸⡀⣿⠋
⢿⣄⠀⢸⡄⠀⠀⠀⠀⠉⠛⠋⠉⣀⠴⠋⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⡶⠶⣶⠚⣿⠉⠀⠀⢰⡟⠀⠀⠀⠀⠀⣿⡸⣴⣧⣸⣷⡏⣸
⠀⣿⡃⠀⣽⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢶⠀⢻⠋⠀⠀⠀⠀⡇⠀⢀⡄⠉⡄⠀⢀⡿⠁⠀⠀⠀⠀⠀⠹⣧⣀⣹⠛⢶⣴⡟
⠀⠈⠛⠛⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠘⠉⢣⣠⠞⠳⠎⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⢡⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠷⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠦⢤⣤⣄⣠⡤⣦⡄⠀⢠⡶⠚⠋⠉⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠾⡇⠀⢘⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⣷⠤⠤⠶⣶⠶⠶⠿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠛⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⣾⣧⣤⠤⡴⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣽⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⢤⣄⡀⠀⣤⠶⠾⠛⠛⠃
⠀⠀⠀⠀⢀⡴⠚⠁⠀⠛⣶⣤⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠏⠉⣋⠀⠀⢀⡋⠀⣿
⠀⠀⠀⢠⠟⠀⠀⢀⣴⠞⠉⣟⠀⢀⣠⣴⠎⠁⠀⠀⠀⠀⠀⢠⠀⠈⠉⠉⠉⠉⢃⡀⣿
⠀⠀⢰⡏⠀⠀⠀⣾⠃⠀⠈⠿⠿⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠀⠐⠖⠒⠚⢻⡏
⠀⠀⣾⠁⠀⠆⣾⠏⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⢻⡇
⠀⠀⢿⠂⢘⡆⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠿⢸⡇
⠀⠀⢸⡆⠈⠁⢻⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠛⢷⠆⠂⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢺⡏
⠀⣠⣴⡿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⣠⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠸⡇
⠀⠙⠛⣿⠀⣿⡂⣧⣨⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢻⣦⠿⣷⠿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⢤⣤⠖⢢⣴⠛⠉⠛⢿⡷⠒⠂⢤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣣⠀⠀⣦⠀⠀⠀⠈⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠀⣿⠀⠀⠀⠀⣯⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣤⣤⢿⣧⠀⢀⣤⣿⣤⠤⠤⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡄⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣼⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⣀⠴⠿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⢀⠴⠶⢀⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⡇⠀⣇⠀⠀⠀⠀⣿⣀⣠⡈⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡤⠄⢋⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⢸⡠⠀⡢⠀⠀⠀⠑⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠚⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⡄⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠂⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠀⠀⣠⢂⠊⠀⠀⠀⠈⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠉⠢⡥⠴⠒⠋⠁⢀⠀⠀⠀⡰⡁⠁⠀⠀⠀⠀⠀⠀⠁⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⣈⠀⢀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⢀⡀⠼⡀⠀⡘⡜⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠀⠀⠈⠁⠠⣁⠀⠀⠀⠀⠑⠢⣀⡖⣂⣩⠥⠐⠈⠀⠀⢧⠃⠀⠀⠀⠤⢾⠶⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣄⠀⠀⠀⠀⠀⠀⠑⠦⡁⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢈⠒⢈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈
⠀⢀⣠⠤⠖⠊⠉⠙⠒⠆⠠⢤⣄⡙⠢⢄⡀⠀⠀⠀⠀⠀⠠⠑⠆⡙⠢⣄
⠀⠀⠞⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⡉⠐⠠⠀⠀⠀⠀⠀⠀⠈⠡⠈⠑⡄⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠌⢐⡀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠑⠒⢤⣁⠂⠤⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠢⡀⣀⠀⠀⣉⡐⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠁⠀⣀⣉⠀⠘⠀⡘⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠈⠫⣀⠂⠈⠉⠑⠲⠆⠀⠀⠀⠅⢐⠠⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣇⠀⠀⠀⠀⠀⠈⡍⡁
⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⠀⢤⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⡀⠀⠀⡘⠓⠤⢄⣄⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢌⣐⠀⠄⣈⣀⡀⠤⣒⠽⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠤⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠤⠞⠈⠓⠒⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣠⡀⠀⠀⠙⠛⣻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡉⠀⠀⢰⠊⠁⠉⠻⡤⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⢣⠀⠀⠀⢐⡧⠀⠀⢀⣩⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢶⣄⠀⠀⡿⡍⢱⠉⠃⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⢻⣦⣈⣧⡀⠀⠀⣶⠶⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠙⢻⡾⠛⢷⣦⡙⣿⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢼⡧⣈⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⢾⡏⢠⣾⣅⣌⣿⣧⠹⡿⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠙⠛⠛⠳⡄
⠀⠀⠀⣴⢋⡁⠀⣧⠈⣿⠓⢲⡄⠀⠀⠀⠀⣠⡾⠟⠉⠀⡀⢸⠳⢤⣙⡻⠿⢛⣩⣼⣷⣄⠀⠈⠛⢿⣶⣄⠀⠀⠀⣸⡇⢀⠁⠀⡄⠀⢀⠀⢻⡄
⠀⠀⢸⡇⢼⡃⠀⣼⠀⣽⠀⢸⡇⠀⠀⣠⡾⠋⠀⠀⢠⣾⡇⣿⠀⠀⠈⢭⣽⣿⣿⣸⣿⣿⡇⠀⠀⠀⠙⢿⣷⣄⠀⢿⣤⣘⣷⣂⣻⣤⣨⠇⠀⠹⡆
⠀⢠⡿⠁⠀⠙⠻⢿⣍⡉⠙⣿⠃⣠⣾⠿⢦⣄⠀⠀⠸⣿⣷⣝⢷⣴⣦⣼⣿⣿⣿⣿⣿⡿⠁⠀⢀⠀⠀⠀⣻⣿⣦⠀⠙⣧⣤⣭⣷⣴⣤⠶⠃⠀⢻⡀
⠀⣾⡧⠀⠀⠈⢻⣿⡿⠟⠛⠁⢠⣿⠃⠀⠀⠘⠃⠀⠀⠙⢿⣿⣿⣿⣿⣿⣟⣿⠟⠝⠉⠀⠀⠀⣾⡀⠀⢐⢽⣿⣿⣇⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠈⢷⡀
⢰⡟⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠸⣿⠀⣠⣤⣄⠀⠀⠀⠀⠀⠬⣽⣿⣿⣶⠞⠋⠀⠀⠀⠀⢀⣠⣼⣿⣦⣺⣿⣿⣿⣿⣿⣶⣶⣶⣿⡄⠀⠀⠀⠀⠀⠈⣷
⣿⡇⠀⠀⠀⠀⢀⣿⣿⠿⠿⠿⣿⣿⣇⢩⣿⠛⢷⡀⠀⠀⢀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⢣⡟⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢿
⢿⡇⠀⠀⠀⠰⠿⠋⠀⠀⢀⣴⣿⡇⠘⠘⣿⢠⣿⣷⡀⠀⢸⡇⠀⢠⡆⠀⠀⠀⢀⣠⣴⣶⣶⣄⡀⠀⠀⢸⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⢸
⠈⢿⣆⠀⠀⠀⠀⠀⠲⣶⣾⣿⣿⡇⠀⠀⣿⣧⣝⣚⣻⣾⣾⡇⣆⢸⣤⣤⣠⡴⠟⢿⣿⣇⡴⣛⣉⣁⣤⣫⢞⣿⣿⣿⣶⣄⣄⣤⣴⣯⣠⣴⣴⣆⣠⣦⠞
⠀⠈⠛⠶⢾⣿⣶⣿⣿⣿⣿⣿⣿⣷⣦⡞⠁⠀⠈⠙⠃⢈⣿⣿⠛⠛⠛⠿⣭⣙⠛⠒⢋⠽⠋⠉⠁⠘⣻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⡇⠀⠀⡦⠀⠀⠉⠳⣬⡀⢀⣴⣤⡿⠋⠉⠀⠁⠀⣀⣈⣽⠿⠌⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣔⣄⠻⣶⡦⣤⣀⠐⠿⠿⠟⠁⠀⣀⡤⠖⠛⠉⢩⢇⡞⢀⣤⣿⣿⡟⢛⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⣿⣿⣿⣿⣄⠈⠛⠒⠒⠒⠋⠉⠁⣀⠤⠤⠞⣡⠞⣠⣿⢿⠟⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣿⠉⠉⣿⣿⠉⠙⠛⠛⠛⠋⠉⢀⣀⡴⠚⠁⣈⠜⡱⠋⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠈⠉⠓⠒⠶⠶⠖⠒⠛⠉⢁⣠⣴⣾⣷⣫⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣤⣤⣤⣤⣤⣶⣾⡿⠿⠟⠛⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⣀⠀⠀⠀⠠⠭⢿⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⠏⠿⠟⢿⣿⣿⣦⡀⢐⣛⣻⣿⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢀⡴⢙⡇⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣿⣧⠐⠒⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢸⣀⠾⠃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠙⣿⡄⠨⠭⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣾⠟⠛⠛⠻⠇⠀⠀⠀⠸⣇⠀⢚⣛⣿⡇⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠇⠀⣠⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⢻⡀⢒⣲⣾⣷⡾⠛⠁⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⡟⠀⣴⠟⠉⠉⠉⠛⢿⣿⣿⣆⠀⠀⠀⠀⠘⣇⠨⢭⣿⠟⢀⣴⣶⠓⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⡉⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⢁⡾⠃⣰⣦⣀⠀⠀⢈⣿⣿⣿⡄⠀⠀⠀⢀⣿⢠⡾⠁⢠⡿⣋⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠈⠉⠙⠒⠦⣬⣶⣤⡀⠀⠀⢀⣾⣇⠀⠙⠛⠿⠃⠀⣸⣿⣿⣿⠇⠀⣠⣴⡿⢃⡞⠁⢀⠏⠸⢫⠟⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣶⣶⣀⣀⣤⣤⣭⣻⣿⣶⣶⣿⠿⣿⣦⣄⣀⣀⣤⣾⣿⣿⣟⡟⣠⣾⠟⠋⠀⠀⣀⡀⠈⠀⣠⡶⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠹⣧⣀⣀⣀⣴⣿⠃⢹⡏⠁⠀⠀⠈⠙⠻⢿⣿⣛⣫⠽⣫⠞⠀⡏⠁⠀⠀⠀⢀⣨⠿⠿⡟⢉⠃⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⠯⠉⠁⠈⡝⠀⡿⢠⣶⣤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠘⠁⠀⠀⢀⣴⠏⠀⠀⠀⢱⡀⠹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⣤⣴⠈⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣦⣼⡆⠀⢸⣿⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⡄⠸⣿⣿⠿⠟⠛⠿⠟⠀⠀⠀⠀⠀⠀⢺⡿⠉⣷⡀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠛⠋⢁⡀⢰⣿⣶⣌⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣠⣤⠀⣿⣷⣾⡟⠀⠙⠉⣷⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠿⠟⠉⠁⠀⠀⠀⠀⢻⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⡿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠈⠙⣿⣿⠋⠁⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⡀⠀⠀⣠⣿⣿⣄⠀⠀⢀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹⣿⣿⣿⣾⣷⣿⣿⣿⡏⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠻⠿⠿⡿⢿⠿⠿⠟⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢻⣿⣿⣶⣶⣿⣿⡟⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣾⣿⡿⠿⠛⠉⠉⠛⠿⢿⣿⣷⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣴⡆⠈⢰⠆⠀⠀⠀⣤⠒⠠⣠⡀⠀⠀⠀⠀⠀⣰⢗⣢⣄⡀⠀⠀⣤⣄⠈⡄⠀⠀⠑⢀⡀⠀⠀⠀⠂⠁
⠀⠀⢀⣀⢠⣦⣄⡙⠃⠀⠀⠀⢀⣬⣡⣝⣧⡀⠀⠀⢀⠾⣧⠭⢤⡀⠨⡂⠌⠿⠟⠀⠐⠀⠀⠀⠀⠁⡄⠀⠀⠀⣄
⠀⠁⠘⠛⠀⠉⢁⡄⣴⣶⣦⡀⣨⠛⠁⠈⠈⠉⠈⠆⠀⠀⠈⠀⠀⠉⠲⣤⡘⠃⠀⠀⠀⣀⣠⣄⣤⣄⢠⡄
⠀⠀⠒⣀⠀⣐⣷⣭⣌⣛⣻⠏⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠈⠀⠀⠀⠀⣽⣿⣿⡿⠘⢳⡧
⠀⠈⠀⠉⢙⠿⢿⣿⠿⠁⡆⣠⠄⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⡀⢈⢻⣿⡿⡿⠂
⠀⠀⠀⠀⠈⠦⣄⡈⠻⡇⢷⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⡰⢿⣦⠋⠀⡶⠀⠀⡀
⠀⠀⠀⠀⠀⠛⠈⠛⠿⠄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⢱⠔⢁⠡⢠⣄
⠀⠀⠀⠀⠀⠐⢡⣴⡖⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⢶⣆⠘⠟⠉⠀⡞⠸⢂⠀⠀⠀⠀⠀⠀⠃
⠠⠀⠀⠀⠄⠀⢼⣿⡏⢢⠁⢀⣰⡒⠐⢤⣠⣄⣠⠆⠈⢠⣴⡌⠉⣀⠠⠎⠁⠀⢶⠀⠀⣷⠐⠀⠑
⠀⣾⣿⡆⠀⠀⣰⣽⡇⠹⠄⠈⠹⢦⣀⡤⠼⡟⠙⠀⡀⠀⠈⠓⠺⣿⠃⠀⠀⠀⠈⠁⢠⡟⣴⡄
⠀⢼⡟⠁⠀⠀⢀⠸⣇⢶⣀⠀⠀⠀⠄⠀⠀⠀⠁⣀⣿⢆⠀⠀⠑⠄⠀⠀⠀⠀⠀⣆⠉⣼⣿⣿⠆⠀⡀
⠀⠀⠀⠐⠀⠀⠀⠑⣼⡌⠺⠂⠀⠀⠀⠀⠀⠀⠀⣿⣟⠈⠷⡆⠀⠀⠀⠀⠀⠀⠆⢨⣴⣟⣿⡖⣒⠀⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⡄⢹⠲⡄⠀⡆⠀⠀⠉⠈⠀⠀⠐⠀⠀⢀⡆⠀⠀⡐⢸⣿⠈⠙⣿⡿⠣⠙⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠟⡛⣿⣿⣾⡄⠐⠀⢡⣀⠀⠰⡀⠀⠀⠀⠀⣀⡴⢷⠀⠀⠀⡾⠿⠂⠰⠾⣶⡃
⠀⢀⠀⠀⠀⠀⠶⢀⢀⣥⣽⣿⣿⣿⣦⡀⠀⣿⡴⣤⠀⡄⢤⠒⣲⣼⡇⢸⠀⠀⠈⡀⠀⠀⢄⣈⡙⠃
⠀⠈⠁⢀⣤⠀⠀⠛⠀⣠⡼⠿⡏⠹⢿⣿⣄⠙⠀⠘⠿⣷⠾⠟⢛⠉⠀⠈⠀⡄⠀⡇⡀⠈⠘⢻⣿⡀
⠀⠀⠀⠀⠉⠀⠐⠀⠈⠑⠀⠈⠇⠛⡌⠉⠀⠀⠀⢀⠀⠙⠀⠀⠈⠀⢀⠀⡀⢀⡠⠛⣷⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⠠⠆⡀⠀⠀⠀⡀⠢⠀⠀⠀⠀⢀⣀⣉⠀⠘⠃⢸⠅⠿⠀⠀⣄⠈⢂⠄
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⣿⣦⡄⠀⠀⠀⠁⠀⣡⡀⢄⠈⠀⢈⡁⠀⣰⡦⠜⠀⣠⣧⡀⠀⠠⠤⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠂⠐⠀⢛⣿⣿⣿⣾⠆⣹⡿⠁⡄⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⢉⠇⢀⡏⠱⠀⠀⠈⠄⠀⠊⠉⢅⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠑⠄⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠄⠀⠠⠀⠐⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠖⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠤⠤⢠⠤⠤⡤⣤⣤⣤⣤⡄⢤⣤⣤⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣦⢠⣤⠀⠘⠺⣿⣿⣿⣿⣿⡿⠛⠟⡛⣛⠛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢸⣿⠀⠀⠀⠞⠻⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣀⡅⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⡿⠞⠋⠀⠀⠀⠀⠀⠰⢯⣍⠹⠻⠾⠿⣿⣿⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡿⠛⠁⠀⡏⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⡈⠃⠿⢿⠆⠀⠀⠀⠀⠀⠀⠠⠴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣠⢟⢀⣶⠄⠀⢀⠀⣼⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠂⠈⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢸⠇⠹⣷⡐⠄⢀⣰⣼⣿⣮⡉⠙⠤⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠒⠚⠋⢀⡿⠛⠋⢀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣰⡆⠀⡀⢸⣷⣤⣧⠙⠿⣿⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣄⠤⣀⠀⠤⠏⠉⣇⣿
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⣴⣿⡇⢀⣿⣿⣷⣌⠁⠀⠐⠁⠈⢣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⣀⡀⠀⠙⡿⠀⢈⣠⣿
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣠⡟⠁⣼⠀⣾⣿⣮⠉⠀⠸⠶⠴⠆⠀⣤⣼⠃⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣷⠀⠀⠀⠀⠠⣴⣾⣶⣷⠟⢻⡃⠺⡏⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠀⠀⢸⢸⣿⣬⡉⠀⠰⠶⠶⠶⠆⠀⢈⡁⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠧⠠⢎⡷⠀⠀⠀⠀⢀⣾⡟⠡⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⡏⣸⣷⣍⣁⠀⠲⢶⣆⠀⠦⠀⠠⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢌⢁⡴⠀⠀⡿
⠀⠀⠀⠀⢴⠋⠁⠀⠐⠀⠀⠀⢠⡇⣿⢦⡁⠌⠶⠘⠿⠖⠀⠋⢀⠛⠀⢈⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠃⠀⢀⡇
⠀⠀⠀⠀⣈⢣⡓⢠⠁⠀⠀⠀⣼⠀⢟⡳⠣⢶⡟⢃⠠⣌⠐⠀⠱⢄⠈⠆⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⢴⣾⠁
⠀⠀⠀⠀⠻⡺⠁⠴⠀⠀⠀⠀⣿⠀⢿⣝⢲⣤⢵⣅⠲⣌⡛⢤⣀⢰⡁⣤⠁⠀⠀⠀⢐⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠜⠀⠸⡿
⠀⠀⠀⠀⠀⣧⡆⢰⠀⠀⠀⢰⡇⠀⢿⣯⠙⠿⠀⠉⠁⠈⠑⠚⠉⠈⠉⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢂⡀⠂⡄
⠀⠀⠀⠀⠀⢹⠇⣸⠀⠀⠀⢸⠀⡀⢺⣿⠁⣸⠂⡀⠀⠀⠠⣬⡐⠢⣽⣿⠿⠏⠀⠰⠬⠤⢤⠀⠀⢠⠋⠀⠀⠀⠀⣀⠐⠚⠀⠀⣾⣧⠘⠃⢀
⠀⠀⠖⠒⠄⢸⠆⣯⠀⠀⠀⠘⢀⣧⡈⠻⣄⠈⠀⠒⢡⣶⣶⣤⠈⠒⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠐⠁⠒⠉⣡⣤⠀⢀⡀⠉⢹⡆⠰⠸
⠀⠀⠀⠀⠀⢸⣷⠉⠀⠀⠀⡇⢸⣿⣿⣷⣄⠈⠁⠄⠘⣿⣿⣿⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⢀⣀⣠⣤⣶⡿⠛⠧⠴⠊⠙⠂⠐⣀
⠀⠀⠀⠀⠀⠀⣿⠘⡄⠀⢘⠃⠀⢹⣿⣿⡿⠆⢀⡀⠘⣿⡿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠶⠶⠈⠉⠀⠀⠈⠉⠀⠀⢀⣿⡄⠀⣠⠟⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⣸⠂⢠⠀⣸⠀⠀⠀⣿⠛⠙⢦⣤⠀⠀⠸⠿⠋⠁⣠⣶⣶⠶⠲⠋⠉⠀⠀⠀⠀⠀⠀⣠⡤⠶⣾⣿⣿⠷⢼⣿⠁⠐⣽⣿⣶⠃⠀⠈
⠀⠀⢀⣠⣤⣶⣿⡆⠈⢻⣿⢠⣿⣿⣿⡇⠀⠸⣷⣄⣰⣄⠐⠷⠀⠋⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠉
⠀⢀⠀⠀⠀⠀⠈⣧⠀⠼⠃⠘⠿⢿⣾⣆⠀⢀⡈⠙⠛⠛⠓⠴⠶⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⣽⣷⡀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠑⠚⠻⢷⣶⣦⣄⣀⠀⠀⠀⠉⠙⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⣿⣿⣷⡀⣿⣷⡀⣤⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⡉⠛⠶⢦⣄⣀⡈⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡇⠠
⠀⢿⣯⣿⣷⣉⡟⠁⣙⠛⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠉⠙⠛⠷⣦⡄⠀⠀⣀⠀⡄⠀⠀⠀⠀⢀⡏⢸⠇
⠀⣿⣿⣙⢧⠀⠀⠈⣿⠀⣯⣦⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣠⡀⠸⠁⠀⠀⠀⠀⢸⠁⠈⠀⠀⠀⠀⠀⠀⢀⣾
⠀⢸⣷⣾⣧⡀⢀⠰⡇⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡈⠉⠳⠚⠆⠀⠀⠀⠀⣾⠀⡆⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠙⠻⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠇⠀⠀⠀⠀⠀⣀⣤⢖⣩⠴⠋⣀⣤⣶⠆⠁⠀⠀⢀⣸⠃⠀⠁⠀⠀⠀⠀⠀⠀⠴⠟⡏⠁
⠀⠀⠀⠀⠙⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⡠⠤⢚⣩⠶⢛⠡⠴⠮⢿⣇⠀⠀⠀⠀⠀⢪⡅⠀⣠⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁
⠀⢠⣤⣤⣤⡄⠀⣿⠀⢠⡀⠀⠀⠀⠀⣠⡇⠀⠀⠈⠳⠋⠀⠰⣉⠔⠚⠉⠠⠚⠁⠀⠀⡀⠀⠛⣦⠉⠈⠂⣀⣿⣽⣄⠉⠢⠀⢠⣾⣷⣤⣄⣀⡀
⠀⠈⠛⠿⢿⣥⢠⡏⢀⡼⠷⠀⠀⠀⠀⠛⠋⢀⡠⠔⢀⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⣧⣠⡎⠉⠛⢿⣿⣦⠀⢢⠀⠈⠉⠉⠛⠛⠛⠒⠊
⠀⠀⠀⠀⠀⠀⢸⠇⢺⣧⣤⣀⣀⠀⠆⠀⢠⡉⠡⣴⣄⢀⠄⢀⡀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠛⠦⢤⣭⣗⣤⡀⠤⢀
⠀⠀⢢⣄⠀⠀⡟⠀⠀⢻⣿⣿⡏⠀⠛⠿⣿⣏⡀⠀⢹⠏⢀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠣⠔⢊⠰⠄
⠀⠀⠀⠉⠳⣦⠁⠀⠀⠀⠈⠛⠿⣦⣤⣀⡀⠉⣄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣧⣤⣋⣡⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⢈⠀⠈⠀⠁⢀⠁⠀⠀⠀⢀

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠀⠉⠉⠁⠀⠙⠻⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣰⣿⣶⣶⣆⣴⣧⣶⣴⣶⣤⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠛⠛⠉⠉⠉⠍⠀⠀⠀⠐⣿⣤⡀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠘⡇⠀⠀⢠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⠤⠀⠀⠀⠤⢄⡠⡺⠀⢀⡃⠀⡀⢀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠂⠐⠀⠐⠀⠠⢀⣤⣄⡀⠱⠀⢿⣋⠀⠡⢊⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⣄⣤⣿⣤⣤⣶⠟⠛⣿⣷⡄⡀⠘⡇⠀⣰⢏⣴⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⣸⡏⢹⢻⠻⣦⣤⣿⣿⢃⠄⠀⠈⠙⠛⣸⣿⢏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣷⣌⠈⡀⠀⣍⡉⢡⣊⠀⢀⣶⣿⣦⣙⣉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠛⣠⠈⢦⢠⣾⣽⣿⣶⡿⠏⣿⣿⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣁⣀⣀⢋⣜⣿⠿⣿⣿⡟⠁⠘⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣯⡯⠧⣾⣿⣿⡂⠘⡈⢠⠀⣼⢣⣸⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠭⢭⠋⡙⢩⠘⠁⢀⣄⣴⣿⠃⠈⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠈⠈⠀⣴⣿⣿⣿⡟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣀⠀⠀⢐⢾⣿⣿⣿⡟⠡⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠈⣿⣿⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠒⢒⣚⣛⣉⣉⣭⣭⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠁⡼⣏⠙⠉⠋⠉⠀⠘⠃⠹⠳⣄⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡻⠁⡾⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⣝⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡴⣁⡼⢁⠾⠀⠀⠀⣤⠀⣀⣠⣤⣄⡟⠀⠀⠀⠀⠀⢙⢷⡎⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡼⢡⡿⢁⡎⠀⠀⠀⠀⠘⣧⠋⠉⢾⣏⣧⠀⠀⠠⠀⠀⠈⠃⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⣾⣷⡿⢁⠞⠀⠀⢸⡗⠤⠤⠛⠀⠀⠀⠈⠀⠶⢶⣆⠀⠀⠀⠀⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣧⠸⡚⠂⠀⠀⣧⠐⠀⠁⠀⠀⠀⠀⠀⠀⡀⠿⠤⠀⠀⢀⢸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡿⣿⣧⠱⠀⠀⠀⢹⡆⠀⠀⠀⡀⠀⠀⠀⡥⡶⠒⠺⠀⠀⠈⠘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣧⠙⢻⣻⣇⠁⠀⢦⠀⢻⡴⠚⠟⣿⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⠈⢇⣿⡄⠀⠀⠀⠀⠁⠀⠀⠸⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⡄⠈⢎⢿⡄⠀⠀⠀⡂⠀⠀⠀⢻⡀⠀⣿⠰⡀⠀⠀⠀⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡌⠈⠈⣷⡀⠀⠀⠀⠀⠀⠀⢸⠇⠴⣛⡡⠕⠂⠀⠀⠘⡀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⢿⡄⠀⣸⣧⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠃⠘⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢻⡄⠘⡿⣇⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢿⡄⠰⡹⡆⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢳⣄⠑⢻⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢿⡄⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢿⡀⠘⣷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢷⠀⢸⣧⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠩⢧⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⣿⡈⢿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⣷⠀⢻⡘⡄⠀⠀⠀⠀⡀⠀⠀⠄⢤⠭⠞⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣮⣷⡀⣸⣃⠰⠐⠃⣀⣀⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣧⣠⡴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄

⠀⠀⠀⢀⡀⠀⠀⣼⡄⠀⢀⡀
⠀⠀⠀⠈⣿⢢⡀⠏⠷⣠⠛⣧
⠀⢀⣠⣄⣸⣷⡙⠿⠀⠋⢀⣹⠷⠶⠶⠶⠶⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠉⠉⠁⠒⢤⣀⡠⠒⠁⠀⠀⠀⠀⠀⢀⣞⡛⠛⠒⠒⠤⣄⡀⠀⠀⠀⣰⡿⠴⡷
⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣍⠛⣦⣀⣀⠀⣹⠟⠛⣶⠋⠀⠸⠷⢴⠆
⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠉⠋⠀⠀⠀⠈⢣⡤⠀⠸⢧⣄
⠀⠀⣴⠋⠀⠀⡄⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⡔⠂⣠⠖⠛⠙⠓⢄⠀⠀⠧⢀⣀⠤⠬⠗
⠀⠘⣇⠠⠀⠀⠁⣀⣀⣄⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠰⡄⠈⡷⠀⢸⣫⣄
⠀⠀⡇⠀⠀⢀⠞⠉⠁⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠀⢀⠟⠀⢸⠁⢾⠀⠀⡀⢀⡀
⠀⠸⡇⠀⠀⣿⠠⡒⡄⠀⠀⠀⢸⡀⠀⢀⡀⢰⣄⠀⠸⡛⠶⠦⠴⠞⠉⠀⠀⢏⣁⡾⠀⢸⢻⡏⣿⢻⡆
⢀⡴⢇⠀⠀⠹⣆⠀⠀⠀⢀⡴⢟⠃⠀⠺⠐⠀⠋⠀⠀⠉⠒⠀⠀⠀⠀⠀⢰⡟⠉⠀⢰⣿⡄⡇⢹⢸⡷
⢯⡀⢈⡄⠀⠀⠉⠛⠛⠛⢋⡡⠎⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠴⣶⠀⠀⢰⡞⠁⠀⠀⠘⣇⣿⣷⣸⣼⢁
⠀⣿⣀⣼⠶⣤⠀⠀⠀⠀⠀⢀⣠⡤⢲⡆⢹⠋⠀⠀⠈⠀⠀⡆⠉⡀⢀⡾⠁⠀⠀⠀⠀⢻⣄⣩⠛⣶⡿
⠀⠈⠉⠁⠀⠙⢷⣀⣀⠀⠀⠈⠁⠣⠎⠑⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠈⣿⢻⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣄⡀⠀⣀⣀⣀⡀⠀⠀⣠⠷⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡉⠭⠭⣿⠀⢸⠷⠶⠖⠒⠒⠒⠒⠒⢲⣤⠤⢤⣤⣤⢾⡚⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠘⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣀⣾⣤⠤⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⣀⣤⡤⠷
⠀⠀⠀⢀⣤⠞⠁⠘⣠⡄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠐⠀⠂⠀⢉⠃⢸⠇
⠀⠀⢀⠎⠁⠀⣠⠞⢩⡇⢀⣠⣴⠋⠀⠀⠀⠀⢀⠂⠀⠁⠁⠈⢆⣸
⠀⢀⡟⠀⢀⣸⠇⠀⠈⠛⠋⢉⡟⠀⠀⠀⠀⠀⠈⠈⢣⠀⠘⠛⢈⣿
⠀⢸⡇⠰⢀⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣤⣠⣤⡦⣿
⠀⠈⡇⠐⠴⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⠀⠀⣿⠀⠀⠀⠀⠀⠈⠻⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠃
⠀⡴⠛⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠘⢧⣴⡆⣤⢠⠙⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡟⠋⠀⠀⠀⢸⡇
⠀⠀⢸⣇⣿⣼⠿⠋⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢀⣤⣼⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢻⠋⠙⠋⠀⠈⢹⡏⠁⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⡇⠀⠀⠸⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠤⠤⢿⡄⠀⣤⠦⠤⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢸⡇⠀⡅⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⠈⡇⠀⡇⠀⠀⢀⡸⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠁⠀⠀⠀⠀⣧⠀⡇⠀⠀⠁⢀⠤⢤⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣤⣤⣤⣤⣤⣤⠴⠿⠀⣇⣀⠀⢀⣿⣤⣆⣹⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠦⣄⣀⣠⠤⠶⡒⠲⡖⢶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⣍⠋⠁⣴⣉⣉⡲⡤⠒⠒⠒⠚⠚⠒⠧⣇⣹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠒⠋⣤⣙⡄⠀⠀⠀⠀⠀⣀⣈⣿⣼⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠵⠚⢻⡆⣠⠦⣀⣿⠈⠁⠀⠀⠰⡟⠛⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠉⠀⠀⠀⢀⣠⣧⡤⠶⠀⠀⠀⠰⠾⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡇⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢋⡠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠈⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⢿⣏⡉⢩⡟⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣴⠋⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠁⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣼⠃⣠⢸⡇⠀⠀⡄⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⣾⣿⠘⣧⣤⠜⠧⡔⠋⣡⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢰⣿⡟⢠⡿⠁⠀⠀⣇⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢿⣆⣉⣴⠟⠁⠀⠀⠀⢻⡜⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣗⡒⠛⠛⠛⠋⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠂⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠉⠉⠛⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠈⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠀⠀⠀⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⣠⣴⠞⠁⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠈⠙⠛⠻⠻⣏⡉⠉⠉⢷⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠷⠷⠛⣿⡈⠙⠒⠬⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⠤⠤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣤⡤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣋⣀⣤⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⠉⠉⠉⠙⢿⡿⠋⠉⠉⠉⠛⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⢀⣼⢧⡀⠀⠀⠀⠀⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣴⣿⠃⠈⢿⣦⣤⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⢈⣿⣿⣿⣿⣿⠿⠁⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠿⢿⣿⣦⣌⡛⠿⠿⣿⣿⣿⣿⠿⠿⢟⣣⣴⣾⣿⠿⠟⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣤⣀⠀⠀⣀⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠟⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣤⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠶⡶⠒⡒⡖⠒⠂⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⠋⠁⠹⠀⠀⠀⠐⡃⢀⣄⣠⠋⠁⢠⠌⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠁⠸⢦⠀⢠⠆⠀⢱⠖⠚⠊⡁⠠⠀⣤⠀⢀⠀⠐⠃⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢱⠒⠒⠂⠀⠴⠧⠤⡔⠛⠒⠒⠨⢥⠴⠋⠒⠛⠳⣤⡥⠒⠊⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠁⠀⠈⠀⠀⠀⡆⢳
⠀⠀⠀⠀⠀⠀⠀⡔⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⢠⠄⡤⠠⠤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⣐⠦⣤⣤⣀⡄⠀⠤⣀⣀⡀⠠⣤⣤⣄⣀⡀⢀⠠⠀⠀⡀⠀⠀⢀⠀⠠⠀⠐⠀⡀⣸⠀⠰⢄⠓⠁⠐⠒⡸⠁
⠀⠀⠀⠀⠀⣠⣾⣥⡶⣀⠀⠈⠣⢠⠖⢿⣳⣕⢦⠀⠀⠉⢟⢿⠁⢀⠀⠀⠀⠀⢀⢠⠃⠀⠁⠀⠀⢁⠇⡠⠙⠀⠠⣉⠛⠒⠁
⠀⠀⠀⠀⠠⣦⣼⣿⣿⣼⡂⠀⠀⣿⣷⣿⣿⣿⡟⡇⠀⠀⢸⠀⠀⠻⢆⠰⣷⣠⡆⠃⠀⠀⠀⠀⢠⠎⠀⠁⠀⠈⠐⠤⡠⠎
⠀⠀⠀⠀⠀⢻⣿⣿⡿⡹⠁⢀⠄⠛⣿⣿⣽⡫⠞⠁⠀⡠⢋⠀⠀⠀⠀⡆⢨⣿⣿⠁⠠⠀⠀⡰⢋⡔⢁⠱⠖⡰⠢
⠀⠀⠀⠀⠀⠀⠙⠷⠊⣀⡴⠁⠀⠀⠈⡙⠳⠤⠐⠒⠉⠐⠁⠀⠀⠀⡴⢅⣈⠔⠁⠀⢀⡠⠊⠀⠊⢚⠂⢴⣀⠞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⣀⣀⠀⡸⠀⣠⠶⠿⠛⠿⣶⣤⣤⠞⠀⠀⠀⠀⠀⠘⡏⠀⣜⠍⠒⠤⢈⠒⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠓⠛⠛⠛⠻⠐⠊⡡⠄⠒⢠⠀⣼⠛⠀⠀⠀⠀⠀⠀⣠⠤⢂⠤⠃⡠⠉⠐⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⡍⢩⡙⣊⣍⡠⡤⠶⢛⡄⠙⡟⠀⠀⠀⣀⣬⠴⠅⠒⢌⠂⢄⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢩⣩⣶⠃⠀⠀⣠⢚⠉⠊⠀⡀⠀⠁⣨⠞⡔⠁⢆⠀⠢⡉⢢⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠂⠘⠠⢱⠀⠀⢰⠀⠃⠀⠀⠀⢐⡩⠿⢩⠀⠈⠲⣄⠑⠠⢃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⡀⠀⠀⢈⣣⡀⠀⢧⠀⠀⢠⡊⢩⡄⢀⠎⠀⢀⠄⠈⠃⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠠⡀⠀⢱⠀⠈⡍⠀⠀⢧⢿⠇⣎⡠⠔⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⡀⢀⠂⠁⠀⢠⢁⣈⠥⠊⠘⠆⠈⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠄⠀⠐⠋⠀⠣⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠉⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⢀⣤⠞⠋⠉⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣄⠀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⣀⡴⠟⠉⠀⠀⢀⠀⠀⠀⢀⡾⠛⠛⢶⣦⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠋⠀⠀⠀⢀⣴⠋⠛⠳⠶⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠃⠀⠀⠀⠀⠺⢧⣤⣀⠀⠀⠀⠀⠀⠛⠳⢦⣤⠀⠀⠀⢀⡿⠀⠀⠀⣼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣠⣤⣤⣀⣀⡾⠃⠀⠀⠀⣸⠃⠀⠀⢰⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣴⠋⠀⠀⠉⠉⠁⠀⠀⠀⣠⡟⠀⠀⠀⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⣾⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣠⡾⠋⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣠⠞⠉⠀⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠠⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⡀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⣠⡾⠋⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣽⠀⠉⠙⠛⠶⢦⣄⣀⡀⠀⠀⠀⠈⠉⠛⠾⠋⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣄⣀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣄⣀⣿⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣀⡀⠀⢀⣀⡀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣀⡽⣴⢯⣀⡹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⣀⡽⢯⣀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠘⠒⡇⠘⠾⠃⢼⠒⠃⡠⣄⠀⠀⠀⠀⠀⠀⣀⡀⠙⠉⣿⣧⣼⣿⡋⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⡈⠣⣄⠙⠲⠶⠒⠋⣠⠜⠁⣈⡷⠀⠀⠀⠀⢾⣿⣷⣄⠀⠙⠛⠛⠋⠀⣀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣝⣲⡤⢖⣫⠴⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⣠⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⡥⠒⢯⡓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⢉⡴⠊⠁⠀⠀⠀⠈⠓⢮⠉⢉⡧⠀⠀⠀⠀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠄⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⢾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⠀⠀⠀⡱⢮
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⣴⣿⣷⡆⠀⠀⠀⠀⠀⠊⠀⠀⠱
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠢⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃
⠀⠀⠀⢰⠤⠠⠤⠤⠄⠠⠀⠀⠤⠄⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⣠⣾⣧⡀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠿⠿⠿⠶⠖⠶⠖⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⣀⣀⡀⣄⠀⠀⣀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠹⣷⣦⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⣿⣿⣿⠇⢾⣿⣿⣿⠿⠿⢿⣿⣿⣷⣀⣀⢿⣿⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁⠀⣠⡿⠋⠀⢈⠛⠟⠑⠀⠀⠀⣹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠮⠀⢠⡟⢠⣤⡀⠾⣥⡀⠁⠀⠀⠐⠛⣹⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⡀⠀⢀⣿⣿⣿⣷⣌⠿⠀⠀⠀⠀⠘⢹⡿⣿⣿⣿⣿⣿⣟⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠈⢻⣟⣩⠟⠋⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣶⣾⣿⣿⣿⣷⣾⣷⠀⠀⠀⠘⡿⠏⢛⢽⣯⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣿⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡅⠀⠀⠟⣮⠉⠛⣉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠉⠉⠉⠛⠛⢻⣅⠀⠀⠀⠁⢠⠀⢸⠟⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣶⠦⣤⠶⣤⣌⠛⡤⠀⡄⠀⣨⡶⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠀⢀⠀⠀⠙⣉⣁⣘⣯⣴⣿⠂⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠏⠐⠂⠀⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠟⠋⠛⠛⠁⠀⠀⠀⡀⢀⣀⣈⣏⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠩⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⢾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠉⢁⡼⣿⣷⣂⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠀⠸⠟⠁⠀⢁⣺⣧⡄⣀⠀⠀⠘⠁⠉⠀⠀⠀⠀⢀⠠⠤⠄⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⣤⣼⣿⡗⠀⠀⢸⡄⣠⠄⠀⠀⠀⠐⠂⠉⠀⣀⣠⡤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣶⠆⠀⠛⠛⠛⠛⠛⠁⣀⣠⡾⠋⠁⠀⠀⣀⠀⣤⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢂⠀⠀⠀⠀⠀⢠⣄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠌⠀⠀⠀⠀⣴⠟⠛⠳⣦⡰⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠈⠀⠀⠀⢀⣸⡟⣀⣀⠀⣼⣷⣔⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⢀⣢⣿⣿⣉⡅⠃⣠⢴⣯⣸⢏⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⢉⣼⣿⡿⣘⣇⡀⣰⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣶⣿⣿⣿⣿⣾⣿⣿⣟⠙⣿⡿⠃⢤⣼⣿⣛⣷⠀⠌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡾⠛⠁⠰⢾⣧⣌⣼⣷⡀⢰⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿⣟⢡⣄⣀⣀⣀⣡⣤⣠⣶⣶⣶⣼⡇⣠⣻⠇⠂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⢸⡏⣿⣿⣿⣿⣶⣾⢿⣿⣿⣿⠿⣿⣿⣿⡏⣿⣿⣿⡷⢶⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⣧⡹⣬⣷⣝⣿⣿⣿⣯⠆⣼⣿⣿⣿⣿⣺⣿⣿⣧⣹⣿⣿⢑⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢢⣾⡝⡿⠻⠿⠿⠿⠟⠉⣼⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣆⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⢀⣀⠀⠀⣼⡿⣱⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠟⣱⣿⣟⣛⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠿⠤⠝⠀⣼⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⣏⡉⠉⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠁⣠⣾⣿⣿⡿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢁⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⣠⣾⣿⣿⠟⠻⣿⠶⠯⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢂⠀⡀⣿⡿⠿⠟⠉⠁⠀⢀⣤⣾⣿⣿⣿⣿⣧⠀⠈⠛⠒⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠀⢉⠙⡄⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⠄⢳⣄⢀⠠⣴⣿⣿⣿⣿⣿⣿⢿⣿⣟⢿⣿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠳⣇⠊⠀⠀⠈⢿⣿⡿⣿⣿⣿⡏⢿⣿⡄⠙⠛⠛⠓⢌⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠿⠌⢻⣿⡷⠄⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢢⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡿⠀⣠⣤⣤⣤⠀⠀⠉⠀⠀⣀⣠⣤⣤⡈⢿⣆⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⣼⣿⣿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣿⣿⣿⣿⣿⠇⢰⣿⣦⠀⣿⣿⣿⣿⣿⡇⠸⠿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣣⡀⠈⠉⠉⢉⣁⠀⣼⣿⣿⡀⢨⣉⠉⠉⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⠀⠈⢷⣦⣔⠡⠿⠁⠀⠻⠋⠛⠃⠀⢻⡟⠂⢐⣨⠎⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢈⡷⠀⠀⠀⠸⡿⢏⢰⣄⠀⠀⡀⠀⠀⠀⠀⢠⡶⢹⣿⡟⠀⠀⠀⠀⣯⡀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣙⣿⣇⠀⠀⠀⠀⡷⠸⣾⣽⣤⣀⣄⣠⣀⣠⣤⣼⣧⡾⠻⠃⠀⠀⠀⢀⣿⣟⢻⣏⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⢉⣀⠉⣉⣠⡆⠀⠀⣷⠀⣿⣤⠉⡏⠉⢉⠉⠅⡉⣽⣿⠁⢀⠀⠀⠀⣦⣉⣉⠁⣈⠉⢱⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣝⣉⣍⡙⠻⣄⡀⠈⠣⣈⠇⠀⠉⠈⠉⠀⠀⠁⠈⢃⡴⠁⠀⢀⣴⠟⢉⣩⣙⣡⡴⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣌⡻⠖⢄⡈⠀⡀⢠⠀⡀⣀⠠⣒⡢⠊⣀⠤⠾⣋⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠑⠢⣅⡀⠀⠀⢀⡤⠐⢉⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣦⣈⠐⠺⢤⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⣩⣽⣿⠿⠶⣦⣌⡑⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣄⡠⠴⠚⠃⢀⣰⠾⠛⠉⠀⠀⠀⠀⠙⠻⢴⣂⡈⠛⠲⠦⣀⣤⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠡⡶⠆⠉⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⡀⠀⠁⠲⠦⠋⠉⠙⣦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣼⡷⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠢⣿⣶⠾⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⡽⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠀⢀⡉⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢤⡀⢿⡇⠀⠀⠀⣰⣿⢲⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣰⣀⠇⢨⡇⠀⠀⠀⢃⡼⠸⣎⣡⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇⠀⠋⠀⠀⢉⡁⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⣶⠀⣶⠀⣶⠀⠀⣦⠀⠀⠀⠀⡀
⠀⠀⠀⠡⡜⢨⡷⢶⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⢰⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⠀⡟⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠹⡄⢣⠹⡆⠀⢻⠀⠀⠀⠀⢈⠁⠀⠐⡇⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢯⠀⠿⠀⠿⠀⠠⠷
⠀⠀⠀⠀⠀⢳⡈⣆⢳⠀⠸⡄⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢷⠘⡜⡇⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢧⠱⣹⡀⢸⠀⠀⠀⢸⠀⠀⢰⡇⠀⠀⠘⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⣶⠀⣶⠀⢲⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠈⢧⢣⢧⠈⡄⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡊⡆⠇⠀⠀⢸⠀⠀⢸⡃⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠰⠀⠀⠀⠸⠀⠀⢸⡇⠀⠀⠀⢹⡤⠤⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣧⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠂⣴⣶⣶⣶⡶⠓⠀⠀⠐⠀⡀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⣿⣄⠈⠻⢿⠟⠀⠀⠀⠀⠀⢄⠈⢠⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣿⣷⠀⠈⠈⠁⠀⠀⠀⢀⣸⣧⢀⣿⡃⠀⠀⠂⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣤⣶⣿⣇⣮⡻⣦⣤⣰⣶⣿⣿⣿⣿⡿⠃⢀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠹⣿⣿⣿⠀⠀⣞⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⠀⠀⢿⣿⣿⠀⠀⣹⠄⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⠁⠀⢰⡏⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠈⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣰⠋⠀⠀⡿⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠟⠛⠛⠀⠀⢠⡟⠀⢀⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡵⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠜⣆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣠⡾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠘⢦⡀⠀⢐⣿⣿⠏⠉⠙⠛⠛⠛⠛⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠻⣤⡛⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⢿⣿⣷⣄⡀⠀⠀⠉⠓⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠛⠋⣿⣿⣿⣿⣦⣄⡀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠷
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢐⣒⣈⡁⢈⡉⠉⠉⠉⣈⠉⠐⠒⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣼⣷⣾⡟⠟⠛⠿⠿⢥⣟⣁⣀⣀⡸⠿⢿⣿⢇⡴⠀⢀⣀⡉⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠑⠋⠉⠀⠀⠀⣀⡤⠒⠊⠉⠃⠠⣰⡏⠀⠀⠀⠉⠑⠊⠤⣄⢾⣿⣿⡷⢀⣀⠉⠢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⣚⡛⠿⣶⣔⢾⣾⣦⣹⡏⠓⡲⣄⠀⡀⠀⠀⠀⠈⠒⢍⠾⣿⣿⠷⢐⣩⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣵⠿⠛⠛⠓⣦⠙⣧⢿⠸⣿⡏⢀⣼⠟⠉⠉⠓⠦⣀⠀⠀⣀⢵⢈⣴⣾⣿⣿⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⣤⡤⣾⠿⣈⣷⡽⠌⠀⠟⣵⡿⢃⣤⣶⢶⣭⣻⡾⢗⣻⡵⠟⢻⣿⠛⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⣬⣛⡴⠟⣿⣿⡀⢀⣸⡟⠀⣿⡿⠗⡋⡝⣿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠉⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠛⠳⢿⢧⣤⣼⣿⠋⠇⢸⣿⣠⣴⣿⣿⣜⣈⣴⣾⡌⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢶⣶⣄⣀⠀⠙⢿⡟⣂⡴⢞⣡⣾⠛⣿⣿⣿⣿⣿⣿⢃⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠏⢀⣴⣶⣶⣤⡤⣟⣫⣵⠞⠉⢹⣿⠀⣿⠉⣿⣿⡿⣿⢸⡟⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⢋⢡⠢⢻⣿⣿⣷⣌⠳⡄⢸⣿⠘⢁⡀⢻⡿⠁⢃⡿⢱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⢖⣯⣿⣧⣤⣖⣽⣿⣿⣿⣿⣆⠙⡜⠿⠟⣿⡇⠿⠁⠀⣊⣴⡿⠋
⠀⠀⠀⠀⢀⡠⣔⣯⡵⠶⢏⡁⠀⠩⣭⣭⣭⡭⠿⠟⠛⠿⡆⠸⠀⢀⡟⠀⣀⣴⣾⣿⠏
⠀⢠⣴⣺⣭⠾⠛⠉⠀⠀⠀⠙⠲⠤⠤⠤⠤⠤⠴⣶⢶⣶⣤⡀⠀⣼⠿⠿⠛⢻⣿⡟
⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣻⣿⣿⢀⣤⣀⣤⢶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⢻⣿⡿⠱⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⡼⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠛⠛⠷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣤⣄⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠿⠿⡿⣿⡿⠿⠯⠿⠛⠒⠀⠁⠀⠀⠐⠀⠀⣬⣻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⣀⣀⡀⡀⠀⢠⣀⠀⠀⠀⢀⣀⣀⠀⠀⠤⠄⢌⣻⣿⣿⣭⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⣿⣿⣿⣿⣿⣿⡷⢠⣾⣿⣦⣼⣿⣿⣿⣿⣿⣿⣶⡄⢪⡦⣿⣿⣿⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠛⠛⢿⣿⣿⠃⠸⣿⣿⣿⣿⠋⠉⠙⠻⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⣀⣠⣰⣾⡿⠁⠀⠀⣭⠻⠟⡉⠃⠀⠀⣀⣬⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣾⣶⣫⣽⠟⣠⣀⡐⢾⣿⣅⡈⠃⠙⠲⣶⢾⡿⢟⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠁⠈⠃⢰⣿⣿⣿⣦⣍⢻⣷⠂⠠⠀⠀⠊⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣶⡀⠀⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⢀⡆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣍⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇⠀⠈⢿⣯⡦⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢀⠀⡀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣽⣿⣷⣿⣿⣷⣦⣾⣶⣆⠀⠀⠀⢺⣿⣿⢻⣿⣿⡟⣿⣿⣿⣿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⠁⠀⣏⣿⡿⠿⢿⣿⡿⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⡏⠀⠈⠸⢻⣿⣦⢈⣡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⠙⢋⡉⠉⠙⠋⠛⢻⣿⡆⠀⠀⠰⠃⢀⣶⠀⢿⡟⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣵⣄⣾⣦⣀⣽⢿⣇⣠⠀⡄⠞⢨⣏⣴⠘⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠉⠙⠉⠉⠙⣿⣷⠯⣅⣼⣷⣶⣼⣿⣥⠒⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣦⣄⣠⣄⣆⢴⣨⣷⣶⣶⣾⣿⣿⣿⣧⠈⠀⢀⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓⠛⠋⠰⠖⢂⣿⡶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⣿⡿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣯⡝⡱⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⢠⣤⠤⠂⣀⣀⣀⠀⠀⣀⠤⠶⠒⠛⠋⠉⠙⠛⢿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣩⡘⣿⠿⠿⠐⠒⢉⡀⠀⠀⠀⠒⠒⠒⠂⠙⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⢡⡄⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠢⢤⡀⢀⣠⢤⣄⡠⢄⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠚⠉⠀⠁⠀⠘⠋⠀⠀⠈⠀⠈⠁⠀⠘⢇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠚⢂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠃⠈⠓⢢
⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⡄⠀⠀⠀⠀⠀⠘⠳⢄
⠀⠀⠀⠀⠀⠀⢀⣞⡀⠀⢠⠀⠀⠀⠀⠀⠀⡤⠴⠂⠀⠀⠠⠒⡂⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⡀⠀⠀⠀⠀⠾⠿⢿⡄⠀⠀⠀⠀⠀⠈⣿⣷⣶⡴⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⡀⠀⠒⡀⠀⠀⢻⢀⠀⠀⠀⠀⠀⠁⠈⠋⣀⠤⠄⡲⠤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠚⠧⠤⠤⢾⡄⠀⠀⠀⠀⡸⠄⠴⠒⠉⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣀⡴⢲⡇⠀⠀⠀⢠⡧⠄⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠂⠀⠀⠐⠈⠧⠦⡄⠤⣸⡇⠠⠀⠀⠒⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣀⣄⣤⠶⣀⠈⡀⣠⠤⠤⠟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⢄⡀⢀⣠⠂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠁⢁⠺⠍⠉⣹⢀⡠⡠⢏⡀⢉⠑⠀⠀⠈⠓⠶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠀⠊⠁⠀⠄⠈⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡣
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠒⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠘⠳⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠀⠀⠀⠐⠤⠐⠢⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠉⠀⠛⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠁⠀⠀⠀⢸⡶⠀⢿⣦⠀⠀⠀⠀⠀⠀⢴⠗⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠘⠀⠀⠀⠈⠀⠆⠐⣹⣿⢇⣠⣬⠁⠈⢴⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡈⠀⠀⠀⠀⠀⠈⠲⠇⢀⡾⠛⠁⢀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡄⠀⠀⠀⠂⣶⠋⠳⠆⢀⠄⠀⠀⠀⠀⠾⠃⠀⢺⣧⠀⠀⠀⠐⠄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢱⣤⣒⣲⡶⠏⠉⠀⠤⠍⣼⠂⠀⠀⠀⠀⠘⣆⣾⠛⠈⠐⠤⣀⡀⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⠤⣤⡦⠄⣼⠁⠀⠀⠀⠀⠀⠸⣅⠀⠄⠀⠀⠀⠉⠻⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⢋⣰⣾⡏⠀⠀⠀⠀⠀⠀⠀⢿⣿⡶⠀⠀⠀⠀⠀⡏⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣿⣿⣿⠁⠀⠀⠀⡀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠛⠆⠀⠀⠙⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠌⠂⠖⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠂⢤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣠⢀⠈⠀⠀⠀⠀⠀⠀⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣠⡴⢧⡄⠀⠀⠀⢀⣴⣷⣼⠟⡙⠣⠈⠈⣡⠀⠀⠀⠀⠀⠀⣿⠁⠙⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠘⠹⢯⠄⣿⠃⠀⠀⣠⠜⠛⡡⠤⠄⠀⣆⠈⠋⡀⠀⢀⠀⠀⢄⣂⣸⡿⠷⡌⢁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢈⡀⢀⠆⠀⠀⠀⠀⣄⣠⣾⡋⠀⡄⠑⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠐⠋⠉⠁⠀⠀⠀⢀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠲⡄⣨⡩⢆⠀⠀⠀⠀⣸⡿⠋⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⠈⠀⠐⠀⠃⠚⠀⠀⠀⠀⢽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠕⠀⠀⠀⠀⠀⠀⡜⠀⠤⠀⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣠⠒⠉⠀⠀⠀⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠇⠀⣠⠆⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⡓⢣⡀⢀⡏⠀⠀⠀⠀⠀⣸⣶⣶⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣿⠸⡇⢛⡿⠟⠀⠀⠀⠀⠀⠺⣿⠋⠀⡄⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠺⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡞⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣇⡀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⣿⣷⣦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠻⣷⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣠⡤⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢐⠀⠈⠠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠉⢻⡲⢦⣀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⠶⢶⠛⠋⠁⠀⠀⠀⣹⣾⣿⡿⢿⣿⠇⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⣁⠈⡁⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠻⠶⠾⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠻⠍⠂⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⣀⣤⣶⣦⣄⡀⠀⠑⠒⠬⣑⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⢀⣄⣠⡞⠻⣉⠁⠺⢭⣛⠷⣶⡄⠀⡸⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⢰⡺⢟⡉⠓⠁⣐⣦⣉⡒⢤⡀⠀⡼⠀⣰⢻⣿⣿⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣀⡴⠖⣛⡉⠙⠲⠉⡰⠋⠀⢈⣷⠩⡗⠀⡼⠁⢰⢣⣿⣿⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡁⠠⢾⡁⠉⠒⠤⠎⠀⢀⣴⣿⢯⣑⠀⠀⠀⢠⢇⣾⣿⢸⣿⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⣶⠢⠬⠁⠀⠉⡳⠀⠀⠀⠀⠻⣟⠯⡓⠀⢠⠀⢀⠎⣼⣿⢻⣿⣿⢿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡰⠋⠋⠓⠀⠀⣠⠞⠀⠀⣠⣶⣦⣤⣀⢹⡆⢠⠃⢀⡞⣼⣿⡟⣾⣹⣿⢸⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⡗⠀⠉⠃⢀⡴⠁⠀⣠⣾⡟⠣⠍⠙⠻⢟⡁⠀⠀⡜⣸⣿⣿⣟⣿⢹⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⣾⠏⡀⠙⠀⣠⠎⠀⢀⣾⡟⠧⡍⠂⠀⢀⡴⠋⢀⡤⢊⣷⣿⣿⡇⡇⡗⢀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣾⠋⢀⠈⢀⡞⠁⢀⣴⣿⡓⠌⠃⠀⢀⡴⠋⢀⡴⢋⣴⣿⣿⣿⣿⣇⣯⡟⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⡿⢃⡀⠲⣜⠮⣑⠦⣾⣿⡦⠉⠀⠀⡠⠋⢀⡴⢋⣴⣿⢟⡏⡼⣹⣿⣿⣿⣇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣴⡿⠁⠢⣌⠀⠀⣙⠮⣑⢤⡉⠀⠀⠠⠊⢀⡴⢋⣴⣿⢿⠇⠼⢰⢡⠇⡞⢿⣿⣯⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣼⠟⠀⠲⢤⠀⠀⢦⣈⠃⠈⠓⠀⠀⠀⢀⡴⢋⣴⣿⢻⢇⡎⡴⢠⠃⡞⣸⢃⠟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣾⠋⡈⠑⠦⠀⠀⠠⣄⠈⠓⣠⠖⠀⢀⠔⢉⣴⡿⢻⢃⡟⡜⢰⢃⡏⡸⢡⠇⣼⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⣾⠃⢀⠙⠲⠀⠀⠙⠂⠄⣠⠞⠁⣠⠞⢁⣴⣿⢻⢁⠏⡜⠀⢠⠇⡼⣰⠃⣞⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡾⠁⠀⢀⣀⠳⠄⠀⢉⡠⠊⠁⡠⠞⢁⣴⣿⢫⢃⠏⠎⡼⢡⢄⡎⣸⢡⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡾⠁⠀⢀⠲⢌⣓⣦⡽⠋⢀⡴⠋⢀⣴⣿⢫⠇⡎⠞⠀⡰⢡⠏⡼⠀⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠓⢦⡀⠙⠲⢍⣲⣽⣶⠟⠉⢀⡴⠋⢀⣴⣿⢃⠎⡞⠘⠀⡤⣰⢃⠏⢘⡥⠊
⠀⠀⠀⠀⠀⠀⠈⢦⣄⠈⠳⢄⡀⠉⠛⠁⢀⡴⠋⢀⣴⢿⢇⡏⠜⠀⢠⠄⡼⢱⠃⣊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⠢⣄⠙⠲⣤⠞⠁⢀⣴⠟⠋⠘⠀⡀⢰⢃⠏⡼⢡⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⠙⠢⣜⣠⣶⣿⣧⡖⡼⡜⡸⢡⢃⠏⣸⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡈⢻⣿⣟⡜⡼⡼⣱⢡⢇⡾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣹⣿⣾⡱⣱⢃⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡷⠟⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⢄⡠⠀⠠⠉⠙⠲⢤⣀
⠀⠀⠀⠀⢀⠄⠠⡄⠂⠀⠀⠈⢄⠙⠄⢀⡤⠔⢊⠁⠈⣑⢄⡀
⠀⠀⠀⠀⠈⠦⢀⡈⣦⠀⠀⠀⠀⣆⠸⣥⠔⠊⢀⠴⠊⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠈⡄⠛⢤⠎⣥⣀⣤⣀⡤⠀⠬⢆
⠀⠌⠐⠠⢀⠀⠄⡀⠀⠀⢧⠀⠀⠀⠁⢸⠈⢦⠤⠤⠤⠤⠤⠴⠒⠺⡄
⠀⠀⠠⣀⠢⢌⡑⠂⠭⣑⣚⣆⠀⠀⢀⠸⠀⠘⣏⣉⣁⣉⣀⡀⠠⠀⠃
⠸⠉⢀⠏⠏⠒⠤⣍⣒⠦⠤⣙⡄⠀⠸⠀⡇⠀⠸⠤⠤⠶⠀⠐⠀⠀⡀
⠸⡆⡌⠈⠀⠀⡜⢠⠋⢉⡟⠒⢳⠀⢀⠀⡇⠀⠀⣧⠀⠀⠀⡀⢀⠀⡇
⠀⢻⠁⠂⠄⢰⢁⠎⢠⠏⠀⠀⠀⡇⠈⡆⢣⠀⠀⢸⠐⠄⠀⠀⠀⡽
⠀⠈⢦⠀⠀⠇⡜⢀⠎⡠⠀⠀⠀⡇⠐⠘⣄⠳⠄⠸⠀⠠⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀
⠀⠀⠀⠳⡀⠀⠇⡸⡰⢠⠊⠀⠀⢰⠀⠀⠈⠁⠀⡎⠀⠈⡨⠓⢤⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⡱
⠀⠀⠀⠀⠈⠲⢄⣳⢡⠃⠀⠄⠀⠸⠀⠀⠀⠀⠀⣁⠼⠋⠀⠀⠀⠙⢆⠐⢤⡀⠀⠀⠀⠀⣀⠔⢊⡉⠥⠀⠀⠙⠂⠀⣀⠤⠒⢊⣉⠤⠒⠊⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⠄⢤⣧⡤⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠑⠒⢒⡪⠕⠈⠀⠀⠀⢀⡠⠴⢊⣉⠤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⠀⠀⣀⣤⣐⣊⠥⠶⠮⠥⠤⠤⠄⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠊⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠸⠛⠀⠀⠘⠿⠃⠀⠀⠀⠙⠁⠀⠀⠀⣦⡄⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⢀⣔⣿⠣⠂⠀⠀⠈⠋⠭⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⠁⠠⣴⠁⢂⠉⠀⠀⠐⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣹⠟⠃⠀⠀⣠⠄⠄⠀⢀⡉⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢨⠒⢔⣅⠄⠀⠀⢀⣀⡤⠊⠽⠿⠛⡃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠁⣠⡤⠖⠛⢿⢄⠈⠀⠀⠐⢒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⣺⡟⠧⡀⠀⠀⠀⡀⠂⣀⣐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⢤⢜⠋⣲⣇⣦⣄⡀⠀⠀⠀⠀⠙⠿⢅⠈⠀⠆⠀⠀⠚⢁⡲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢿⣃⠄⡁⠄⠀⠉⢉⣦⠀⠀⠀⠀⠀⠀⠙⠛⠛⠷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⡟⠠⢾⠷⠉⠁⢀⡀⠀⠀⡠⢴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸⠣⢴⢉⠀⠀⠀⢉⣀⠀⠶⠿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠘⣿⠟⠁⣀⡤⠖⠿⡅⠈⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢷⣵⣿⠻⣄⠀⠀⠀⠠⠂⣀⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙⠻⢄⠁⡐⢶⢶⢞⣑⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠉

⠀⠀⠀⠀⡀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⠿⠿⡿⠻⠿⠿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⠾⠛⠛⠛⠛⠛⠛⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡶⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠀⠚⠛⣿⠀⠀⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⡀⠠⣤⠤⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⣦⡀⠀⣾⠟⠀⢀⣰⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣾⣿⠈⢿⡶⠙⣄⣰⠞⢁⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣁⣀⠀⠁⠸⠟⠁⣴⠿⢷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢀⣿⣧⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠶⠶⠖⠈⠉⠉⠉⠛⠀⠉⣿⠶⢆⡀⠀⢀⣠⠶⠆⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⣷⡄⠉⠙⣷⣀⣿⠀⠀⣀⣀⣀⠰⠶⠶⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠿⠃⠀⠀⠁⠶⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⠛⠁⠀⣴⠿⠿⠿⠗⠊⠈⠉⠁⠀⠀⠀⣀⣠⣤⣤⣴⡾⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠂⠀⢠⣿⣄⣀⣠⣤⣤⣤⣄⠾⠿⠟⠛⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠀⢀⠙⠉⠛⠉⣤⡍⠉⠛⢠⣤⣄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣦⣾⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠹⣧⡀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⣤⠄⠠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⣿⠀⠀⣿⠀⠀⣴⢙⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠸⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⢻⡄⠀⠿⣀⠀⠈⠉⠁⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁⠀⠀⠀⠙⠻⣆⡀⠀⠀⠀⢹⠁
⠀⠀⢠⣤⣴⣾⢷⣶⣶⡄⠀⣸⡁⠰⢿⡆⠀⠀⠙⠷⣄⣀⡀⣀⣀⣴⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠀⠘⠇⠀⠀⠀⠈
⠀⠀⠛⠉⢹⡇⢸⡇⢸⣇⠀⠙⢷⣶⣶⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣡⡆⠀⠀⢠⣾⠃⢀⡶⠆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛⠻⠃⢠⡆⠀⠀⠀⢸
⣾⠿⣷⠀⣸⠃⠀⢷⡿⣿⡇⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠈⠉⠀⠸⠷⠂⠀⠀⠈⡹⢷⣄⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⢻⣷⣤
⢿⡀⠛⣴⡏⢐⣇⣸⣇⣸⡇⠀⠀⠀⠀⠻⣶⡀⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⠛⠒⠲⠶⠿⠟⠁⠀⠀⠀⢀⠀⠀⠈
⠘⣿⣀⡿⢿⣿⠿⠃⢹⡿⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⡘⠉⠈⠉⠀⠙⠛⠛⠛⠛⢢⣤⣤⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⢰⡿
⠀⠈⣿⣿⠈⣡⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⢀⡟⠛⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⣧⣄⣴⠇
⠀⠀⢹⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣶⠀⠀⠁⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠤⠛⠀⠀⠀⠀⠀⠀⣀⠀⣸⡟⠃
⠀⠀⢸⣿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠛⠛⠋
⠀⠀⠸⣿⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡉⠉⠉⢱⣶⡄⠀⢀⣶⡶⠶⠤⠤⠶⠶⠶⠶⠞⠛⠋
⠀⠀⢠⣿⠋⠹⠿⠶⢶⣶⡶⠶⠤⣶⡟⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠸⠃⠀⠸⡷⠦⠀⠀⢀⣀⣀
⠀⠀⠀⠹⠾⠧⣀⣀⣀⣬⣦⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠚⠀⠀⠀⠀⠀⠉⠙⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⠿⢶⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣄⡀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⡆⠀⡼⠛⠁⠠⣧⣤⣤⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠙⠓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⣷⠤⠤⣤⣤⠀⠸⠆⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢸⣿⠿⣄⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⣄⠁⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣄⣀⣈⣧⠀⠘⢳⡄⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠙⠛⠛⠋⠀⠀⠈⢿⣆⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⠿⠿⠶⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢰⡿⠗⠚⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⢀⢠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠙⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⡞⢡⡀⣤⡄⢺⣷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣷⣼⡃⣿⡇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠳⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⠐⠒⠐⢾⡿⠟⠋⠙⢷⣤⠶⠂⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡋⠀⠀⣿⠁⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡌⠀⠀⠀⠀⢻⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⢸⣇⡀⠀⠀⢀⣿⡇⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠙⠁⠙⠛⢻⡆⠀⣾⡟⠙⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠇⠀⣿⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⠀⣿⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠳⠂⡀⠀⠀⠀⢸⡇⠀⠘⠀⠀⠀⠀⠀⠀⣿⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣀⠐⠢⣀⠀⠀⠀⠀⠀⢸⡇⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠁⣀⣀⣹⣤⠀⠀⠀⠀⢸⠃⢸⣧⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡿⢿⣧⣴⣶⣶⣶⣞⠁⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠋⠙⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣄⣀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⢿⠿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠟⠛⢢⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣇⢀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠛⢧
⠀⠀⠀⢠⣳⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸⡏⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⣣
⠀⠀⢠⠋⡉⢻⣶⠦⠌⠻⣦⡀⠀⠀⠀⢹⣷⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⣿⢁⣿⠇⠀⠀⠀⣠⡾⠋⠤⢶⣾⠋⡉⢣
⠀⠀⠀⢧⣼⣄⣍⠑⢦⣄⠈⠙⠤⡀⠀⠈⣿⡄⠘⢃⣤⠀⠀⠀⠀⠰⠶⠀⠠⠤⠄⠀⠐⠒⢿⢷⣟⡇⢸⡇⠀⠀⡠⠜⠉⢀⣤⠖⢋⣅⣼⣤⠇
⠀⠀⠀⠀⠈⠉⠉⠑⢦⣉⠢⣄⠀⠈⠲⢄⣹⣷⠢⠊⠁⣀⣀⡀⠀⠰⡄⠀⡀⠀⠀⢀⣠⣀⣀⡸⠞⢳⣿⢣⠴⠊⠀⢀⡤⢊⣡⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡳⢤⡀⠀⢹⣇⣠⣶⣾⣿⣿⣿⣷⣞⠱⠋⠹⣶⣼⣿⣿⣿⣿⣿⣦⡜⢻⠀⠀⣠⠴⣫⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢽⣲⣴⡇⠘⢿⣿⣿⣿⣿⣿⢏⡆⠀⠀⣼⣝⣿⣿⣿⣿⣿⠟⠓⣬⢶⣚⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁⠙⠛⠛⠿⢏⡨⣧⣾⣷⣽⡇⡙⠛⠛⠛⠛⠁⠤⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠄⣀⣀⣂⡰⣿⠀⣿⣿⣿⣇⢵⣶⣦⠀⠀⢀⣀⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⠖⠹⠗⠛⣻⠋⠙⠚⠗⠸⠿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⣿⣇⡀⡀⡀⢀⠐⢀⠀⡄⠀⠀⣹⣿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⣝⣩⢩⠎⠋⢹⠉⡏⢿⠉⢻⣿⡟⡂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢀⣹⡆⢞⢿⣽⠴⢤⠄⠼⠠⠧⡔⠄⡫⠿⢁⣱⣇⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⢁⣠⡶⠋⠈⠣⠀⠀⠈⡉⠘⢐⢔⠐⠲⢱⡺⢅⠀⠁⠃⠈⠳⢦⣀⠙⠲⠤⣀
⠀⠀⠀⢀⠀⠀⣀⣀⡤⠖⠋⣁⣠⣴⡶⠛⠁⠀⠀⠀⠀⠀⠑⡄⠀⠁⠀⠀⠀⠀⠉⣣⡼⠃⠀⠀⠀⠀⠀⠙⠳⣶⣤⣀⡉⠓⠦⣄⣀⡀⠀⢂
⠀⠀⠘⢿⡀⠀⠟⠁⠴⠚⢻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠚⠁⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠛⠲⠄⠙⠇⠀⣸⠟
⠀⠀⠀⠀⠻⢿⠟⠁⠐⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠒⠀⠙⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣀⣀⠀⠠⡀⠀⠀⠀⠀⢁⠎⢹⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⠀⠀⠉⠃⠀⠈⠥⠒⠀⠀⠠⠆⡠⢨⢾⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠤⠖⠂⡀⣄⠰⣱⡝⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⢁⡠⢐⣲⢭⡰⢹⣦⣸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⢠⠠⢐⡭⢔⣪⠽⣺⠕⠚⢡⡤⣛⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢎⡁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⢁⠠⢔⣪⠡⢐⡬⢴⡚⠍⠂⠁⠀⣴⣡⢖⣷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⡤⢂⠁⡀⠀⠈⠑⠢⠀⡤⠖⠈⠁⢀⠄⢐⡩⠔⡊⠄⢒⠣⠕⠂⠈⠀⠀⠀⣠⣼⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠃⡨⢑⠄⠑⠢⢄⠀⠀⣀⠠⢐⡭⠔⡊⠄⣂⠡⠐⠈⠀⣀⡄⡐⣟⣴⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠐⠤⡪⢐⠢⢈⡨⠵⢪⠁⣀⠤⠒⠉⠀⠀⠀⣀⣤⡽⣿⡯⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣈⠑⠩⢃⡜⠭⢂⢉⠀⢀⡺⢃⣤⡶⣟⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣽⣦⣀⢠⠀⡙⢎⣶⢗⣫⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣧⣻⠽⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠉⠁⠀⠈⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⢀⡀⠀⠀⠤⣤⣀⡀⠀⠀⢀⣀⡠⠤⠀⠀⠀⢀⡾⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢼⠁⠀⣀⣀⣠⣠⡄⠈⢁⡀⠉⣠⣤⣤⣀⣀⡘⣾⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣰⣿⣿⣿⣿⣿⣿⡄⢘⡇⢰⣿⣿⣿⣿⣿⣿⣾⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⢿⣿⣿⣿⣿⣿⡿⡏⣉⠉⡿⣿⣿⣿⣿⣿⣿⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⡈⠻⢿⠿⢿⠿⣣⢱⣿⡇⢣⣈⠻⠿⣿⣿⠟⢁⢨⠗
⠀⠀⠀⢀⡤⣤⠤⣄⠀⢳⣀⣸⠛⠛⠛⠓⢺⢧⣿⣿⣿⡄⡏⠛⠛⢢⢖⣾⣅⡼⠀⠀⢀⣤⠶⠦⣄
⠀⠀⠀⣏⠀⠀⠀⢸⡆⠈⠛⠉⠉⠉⢳⡀⢸⡼⠿⠛⠟⢃⣧⠀⠐⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸
⠀⠀⢠⠜⠒⠦⢤⡈⠳⣄⠀⠀⠀⠀⢸⣤⡈⠀⠀⠀⠀⠀⠘⠀⡓⠀⠀⠀⠀⠀⣀⡼⢁⠎⠀⠚⠓⢦
⠀⠀⣧⠀⠀⠀⠀⠑⠄⠈⠙⠢⣄⡀⠀⠣⣜⣙⡛⢺⢲⣲⣙⡭⠗⢀⣠⠴⠚⠉⠁⢀⡀⠀⠀⠀⣀⡜
⠀⠀⠈⠳⢤⣤⠤⠖⠒⠲⠦⣄⠀⠉⠒⠦⠤⣀⡈⠁⢁⣀⡤⠖⠚⠉⡀⣠⠴⠖⠊⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣀⣠⠤⠛⠋⠩⠒⢀⣠⣴⣛⢉⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠶⠒⠲⠤⠤⠤⠖⠋⠁⠀⠀⣀⡤⠶⣞⡉⠁⠀⠉⠉⠀⠀⠀⠀⠈⢹⠄
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⢀⠖⢁⡴⠖⠋⠁⠀⠀⠀⠈⠑⠢⣀⠀⠱⠄⠒⠶⣶⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠓⡖⠀⠉⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠⣤⣤⣄⣀⠀⡀⢠⢆⣞⣩⣙⣣⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⡙⢽⢿⡿⣹⡏⣡⢾⣍⠀⠀⠀⠈⠐⢶⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠀⠙⠦⠈⠳⠅⣷⣿⣶⣿⣷⣦⡐⠄⠀⠀⠙⡛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠨⠭⠭⠛⠉⡏⠉⠀⠀⠀⠀⠀⠁⠢⣄⡈⠛⠚⠉⠛⠙⢀⠂⠀⠀⢹⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣦⣾⡞⠋⠀⠀⠀⠀⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⣴⡖⠀⠀⠀⠀⠀⠘⠶⡆
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠯⣿⣫⡳⣄⠀⠀⠀⠀⠀⢳⠲⠄
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡂⠦⢤⣀⣀⢀⣺⡿⠿⠉⠁⠁⠨⢷⡀⠀⠀⠀⠀⠱⡚⠢⡀
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⡄⠀⠀⠀⠀⠈⠒⢉⡳⣄
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠁⠋
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⡏
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⢸⠃
⠀⠸⣄⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢰⠇
⠀⠀⠹⣎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠎
⠀⠀⠀⠘⢧⡀⠓⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡲⠃
⠀⠀⠀⠀⠀⠙⢦⡈⠦⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠂⠀⠈⠁⣀⠤⡀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⢀⢀⣀⠤⡔⡐⢎⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣮⣊⠓⣄⡀⠙⠂⣀⡀⠐⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⢊⢊⡥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠭⣟⣤⣤⣈⣉⠒⠒⡠⠤⠨⠤⠀⣀⣀⡠⠔⠛⠉
⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣬⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠂⠀⠀⠉⠀⠒⠒⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠙⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡤⡞⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡆⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣷⣿⠃⠀⢀⣤⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⠈⢻⣿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣟⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣀⠠⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⡿⢿⣿⣿⣿⣿⣿⣿⠀⢸⣾⣿⣧⢧⠀⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠻⡿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠄⠙⠻⠿⠿⠛⠛⠛⠁⠀⢠⣿⣿⣿⡞⠀⠈⡛⠛⠛⠿⠿⠓⠋⠐⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣆⡀⠀⠀⠀⡀⢆⣼⡶⠂⠀⣿⣿⣿⣿⣷⡀⠀⢿⣦⣄⠀⠀⠀⠀⠀⡴⢛⠝⠁
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢹⣾⣷⣖⠛⠡⡾⠛⠀⡀⠀⠛⠛⠈⠛⠋⠼⠀⠈⢻⣏⠙⠂⠐⢋⣩⡾⠁⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⡾⠁⠀⠀⠉⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢦⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡴⢲⣾⣿⣿⠃⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠹⡆
⠀⠘⣿⡄⠀⢀⣻⡿⠁⠀⠀⠀⠀⠀⢸⣯⠹⡇⢸⣏⣿⠰⠋⢠⠀⠀⠀⠀⠸⠃⠀⠀⡙⣿⠁⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠻⣿⣂⠀⠀⣴⡟
⠀⠀⢈⣿⡉⣭⣿⣷⡀⠀⠀⠀⠀⠀⢸⡿⠀⢻⣾⣧⣿⣐⣄⣘⣀⣀⣄⣀⣘⣀⣦⣀⣧⣿⣴⣿⠃⢻⠁⠀⠀⠀⠀⠀⠀⣸⣿⣯⡉⣽⣏
⢰⠋⠛⠯⠔⠒⡻⠯⠏⠒⢦⠀⠀⠀⢸⣧⠀⢨⣿⡏⢹⠟⢟⠛⢻⠛⣻⠉⢻⠋⣿⠛⡟⣿⢻⠁⠀⢸⠀⠀⠀⠀⢠⠒⠊⠭⠿⠓⠲⠤⠟⠋
⢸⠀⠁⢦⣤⣤⣤⣄⣴⣷⣿⡀⠀⠀⠀⢿⡄⠀⢿⣷⠸⢁⢸⡆⢸⣀⣿⡀⢸⠀⣼⠘⡷⣿⠇⠀⠀⡞⠀⠀⠀⠀⣼⣷⣧⣤⣤⣤⣤⡤⠆⠁
⠈⠲⣷⣄⣈⣓⣈⠀⣈⠉⠻⣷⣄⡀⠀⠈⠻⣦⡈⠻⡄⠀⠀⠈⠃⠀⠉⠁⠀⠈⠁⠀⠀⠋⢠⡾⠋⠀⠀⠀⣀⣼⠟⠋⢁⠀⢈⣒⡉⣀⣴⡶
⠀⠀⠀⠉⠑⠒⠒⠶⠶⣤⣆⡈⠹⢿⡢⣄⡀⠀⠙⢦⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⠏⠀⠀⣀⡴⣺⠿⠉⣀⣦⡺⠶⠗⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣉⠃⠉⠶⢤⡀⠓⢤⣨⣭⣤⠀⠠⠤⠄⠈⡁⠤⠞⠁⣠⠴⠊⠁⢛⣡⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣔⡨⢙⠓⠦⣄⣤⠒⢋⠡⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣷⣯⣀⡀⠉⠙⠷⢿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠊⢋⣠⣿⣿⣿⠿⢶⣶⣦⣄⡈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠒⢋⣁⠤⢔⣪⣿⡿⠟⠋⠁⠀⠀⠉⠛⠿⣿⣿⣒⠤⢄⣙⠓⠢⣤⣀⡀
⠀⠀⢀⣤⣤⡤⠤⠤⠶⠖⣚⡻⠭⠑⠒⠉⢡⡴⣚⠭⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⢝⡶⡤⠉⠑⠒⠭⠽⣛⣒⠶⠦⠤⠤⣤⣤⣄
⢠⣾⠉⠉⠉⠉⣣⣤⡭⠥⠔⠒⠀⢀⢠⡠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠐⠒⠤⢭⣥⣄⠋⠉⠉⠉⢹
⠸⣿⠂⡀⠠⣠⡄⢀⠀⠀⠀⠀⢠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠀⠀⠀⢀⡁⣤⡀⢀⡀⢼
⠀⠈⠛⠲⣏⣿⠋⠄⠀⠀⣴⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡄⠀⠀⠉⢻⣿⡿⠚⠋
⠀⠀⠀⠀⠸⡿⡀⠀⢸⣿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠙⠷⢶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⣉⣬⡉⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠅⠀⠐⡫⠔⠒⠦⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⠲⣄⢰⠀⠀⠐⠀⢹⢆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣂⣀⡼⠀⠳⢤⡤⠴⠣⠞⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣁⣀⣤⠒⡞⡖⠦⡀⠀⣰⢿⡗⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢃⡩⢁⢍⣡⠞⣰⠁⠀⢣⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⠉⠉⣁⣀⣰⣻⠀⢃⠀⠡⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡝⠀⢀⡏⡆⠘⠀⠀⡇⠠⣄⠀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⠀⢀⡞⢀⡷⠲⠦⢤⡁⠀⠀⢣⠀⠀⠀⣽⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣤⣎⠴⠋⣰⢛⡱⡄⠉⠒⢖⠊⠉⣶⠒⠛⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠸⠷⠴⣶⡵⠟⣪⢄⡀⠀⢈⡆⠀⢸⠖⠲⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢋⡤⠚⠁⠀⢠⣏⣀⠜⠰⡞⣡⠚⠁⢿⡤⠚
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⠉⠀⡰⠋⠀⠀⠀⠀⠀⡏⠀⠀⠸⠧⢟⡀⡠⢾
⠀⠀⠀⠀⠀⢀⠠⠼⡆⢣⠁⠀⢣⠀⠀⠀⠀⠀⢰⣃⠀⠀⡗⠀⠈⠻⠀⢸
⠀⠀⠀⠀⠀⢸⠰⡺⣧⠞⡀⢀⡸⠤⡀⠀⠀⠀⠘⠁⢳⣱⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢀⣾⡤⠃⣇⠸⣱⠘⢯⠓⠜⠀⠀⠀⠀⠀⠀⢹⠷⡤⣤⣴⣏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡡⠱⣠⠇⠀⠀⠀⠀⠀⠀⠀⠸⡖⠀⠸⡄⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⡗⣄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⠈⢧⠈⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡤⠾⠉⠸⠑⠂⣌⣇⡹⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠛⢧⡒⠀⠀⠀⠀⠍⣏⠄⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠠⣄⠀⠀⠀⠀⠀⠑⣤⣭⡶⠂⢠⡬⣆⠠⠒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠑⠦⣀⠀⠀⠀⠐⠀⠀⠀⣈⣧⢐⣂⡭⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣀⣉⣑⡒⠒⠒⢒⣫⠵⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⡤⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢻⡇⠀⠀⠀⠀⠀⠀⠀⠂⢠⢾⣷⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⢇⣀⣠⣄⡀⣄⠀⢀⣤⣀⡀⢯⡐⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣿⣿⣿⣿⣧⠼⣤⣿⣽⣿⣿⡼⡇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⣼⣿⡇⢻⣿⣿⣿⣿⣀⡆⢇⡻⠿⡿⠁⠳⢃⠀⢀⡴⠶⣊⠔⠀⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠉⠉⠂⠀⢀⠈⢿⣿⣶⣤⣬⣿⣿⣿⡄⠀⣴⠀⢀⡀⣡⠡⠔⢋⣤⣼⡧⠦⢤⣻⡀⠠
⠀⠀⠀⠀⠀⠳⡖⠋⠙⠻⠳⠦⣤⣉⡻⣯⣭⣻⣿⣿⣿⣿⣷⡔⠉⣴⠞⠋⣁⡤⠾⠛⠉⠀⠀⠀⠀⠹⣤⡴
⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⠉⠛⠁⠀⣠⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣿⣯⣿⠴⠼⣽⡖⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⣠⡴⠞⠛⠉⠀⠀⠀⠀⠉⠓⠲⢤⣀⡀⠀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⡀⢂⠀⠀⠀⢀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⠈⠉⣟⠓⠆
⠀⠀⠀⠀⠀⠠⠀⡁⠖⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡴⣶⠒⠒⢒⣲⣶⣤⣄⣀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⣹⣿⠛⣎⢷⠋⣉⣤⠤⠤⣴⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣡⠖⠳⡄⠈⣧⠘⡎⢏⢁⡠⠜⠟⠻⠷⠒⠚⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⠴⠚⢳⡀⠸⡆⢱⠘⣇⣀⡤⠤⠒⠒⠒⠒⠲⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠱⣞⢁⡠⠖⠚⣇⠀⢻⠈⡆⢻⢀⣠⡤⠴⣶⠒⠲⢶⡒⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢦⡈⠻⣄⡰⠚⢻⠀⢸⠀⣇⢸⠉⠙⠦⣄⠀⠙⢦⡀⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠳⡄⠙⢦⡀⠙⢶⣹⠆⢸⠀⡟⢸⡗⢤⡀⠈⠳⣄⠀⠙⣆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢦⠘⢦⡀⠙⠢⣄⠈⠓⢿⣀⡇⢸⣧⠞⣹⣦⡀⠈⢳⡀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠳⣄⠙⠢⣄⠈⠙⠦⣄⣈⡽⠋⣠⠞⠁⢀⡷⣄⠀⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡏⠳⣄⠈⠑⣦⡴⠚⣁⡴⠊⠁⣀⡴⠋⢀⡼⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡜⣡⠟⢋⣡⠴⠚⠁⣀⡤⣾⡷⢦⣴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡼⠛⠻⠯⢤⣤⣖⣋⣁⣤⡶⢿⡜⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⠛⠛⠋⠉⠉⠀⠀⠈⣿⣿⣿⠿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⠉⠁⠀⣿⠧⠄⠀⠀⠀⠀⠀⠀⠘⠻⠃⣿⡞⠀⣍⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠛⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣟⠋⠛⣦⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⣿⣿⡟⠛⠀⠈⠷⣦⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣀⠀⠒⢀⣠⢀⣤⣾⣿⣿⣏⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠄⠀⠀⠀⠀⣷⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣿⣶⣶⣾⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣷⣿⡛⠓⠈⢡⠀⢀⣠⣿⣿⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠸⠿⣿⠃⠀⠀⢀⣾⣿⣦⣿⣿⣴⣿⣿⣿⣯⠉⣠⣾⡿⣽⠟⣏⡠⣤⡇⢸⣿⢶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡾⠁⠀⠠⣿⡠⠴⢤⣼⣟⣻⣿⡿⣿⣿⣿⣿⠿⣿⣷⠉⢉⣰⣿⣿⣦⣹⣿⣿⠘⣿⣟⣛⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢼⣠⣾⡇⠀⠀⠀⣿⡇⠀⣠⣿⣿⡟⠿⣿⢾⣿⣿⣿⣷⠛⠀⠠⢿⣿⣿⡿⢿⣿⣿⠟⠠⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠻⢦⡎⠀⣿⡇⠀⣿⣿⣿⣿⣶⣿⣦⡽⣿⣿⣿⣆⠀⢀⣼⡿⣟⣾⣿⣿⣿⣶⣤⡈⠹⣿⣽⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⢺⣷⡀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⠆⢈⠙⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⢋⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⠄⠹⣿⣷
⠀⠀⠀⠀⠀⠀⢰⡿⠿⢿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣀⣀⣟⣹⡿⠿⠿⢿⣿⣿⡟⢀⣠⡌⠀⢿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⣀⠙⡿⠷⣾⣿⣷⡀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠻⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠙⢿⣧⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠠⠀⠟⠁⠘⠛⠉⠙⣿⣿⣿⡿⠿⠿⣿⣿⣶⣶⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⠙⣿⡿⠉⣿⠃⣹⣿⣻⣿⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⢀⣾⣿⣿⡀⢸⣿⣿⣿⣿⡇⢰⣿⡃⠀⣿⠀⣿⡞⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠘⢃⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣿⣿⡟⣡⠀⣿⣿⣿⣿⣁⣿⣿⠃⠀⣿⣰⣏⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠈⢷⡄⠀⠀⠀⠀⠀⠈⣠⣾⣿⣿⣿⡇⢻⣦⡸⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⡇⠀⠀⠻⣄⢀⣀⠆⣀⠆⠙⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣨⣷⣤⢶⣆⠠⣞⠛⠋⠁⠀⣰⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣧⡀⢿⡀⢹⣇⡀⠀⣠⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⠘
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠙⢿⣿⣧⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⠃⢾⣿⣿⣿⣿⣿⣿⣿⢣⡌
⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠛⠿⣶⣄⠈⣿⡁⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢸⣿⠀⠀⠙⠛⠛⢧⠐⠳⠄⠜⠃
⠀⠀⠀⠰⠟⠙⠞⠀⠀⠀⠀⠀⢸⡿⠀⢻⣷⣄⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣷⡄
⠀⢀⣴⡿⠿⢿⣷⡆⠀⠀⠀⠈⠿⣿⣶⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣿⠋⢻⣿⣷
⠀⢸⡏⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⣿⣿⣿⣿⣿⣄⢸⣿⠀⢈⣽⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣠⣾⠟⠁
⠀⠀⠹⣿⣦⣀⡐⡀⢀⡤⠀⠀⠀⠃⠀⠀⠻⢋⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⡄⠀⠀⠛⠻⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠈⠙⠿⠃⢻⣾⠁⠀⢠⣀⣀⡴⠀⢠⣿⢿⣿⣿⣿⣿⣿⡿⠀⠇⠀⡇⠀⠀⠀⠀⠈⢻⣿⢻⣧
⠀⠀⠀⠀⠀⢠⣀⣾⡋⣤⠀⠾⢿⠟⠁⣰⠏⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠙⠟⠻⣿⡇⢀⠀⠀⣰⠋⠀⠀⠀⣸⣿⣿⣿⣿⣿⣀⣶⠀⠀⠀⠀⠀⣀⠀⠀⠐⠸⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣀⣼⠃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣀⣼⣿⣿⠟⠙⠿⣿⣿⣿⡟⠀⠀⣾⠀⠀⣿⣿⣿⣀⣄⡀⠀⠀⠙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣿⡿⠟⠋⠀⠀⠀⠀⠈⠛⣿⡇⣾⡆⣿⠀⣿⠋⠙⠻⢿⣿⣿⣦⣄⣀⠀⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢸⠿⢡⣴⡏⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⢲⡀⢉⠛⠀⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠁⠀⠐⠊⠈⣇⡈⠀⠤⠀⠀⢈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠈⠀⡠⠀⠀⠀⠀⠈⢧⡀⠀⡀⠀⠀⠀⢹⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⣤⣀⡐⠀⠉⢀⠀⠀⡀⢀⣉⣄⣀⣹⣠⡄⣵⣺⠇⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⣹⡛⠓⠛⠛⠛⡟⠉⠉⠉⠀⠀⠁⠀⠀⠀⢸⢄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠐⠀⠒⢒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣰⢸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠐⠘⠔⠋⢀⣀⣀⣀⣀⡀⠀⢀⢀⣀⡘⢀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣧⡟⠘⠀⠘⠉⣽⣿⣯⠀⠄⠀⣼⠙⣿⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢻⡇⠀⠀⠀⠓⠿⠿⠋⠀⠀⠀⠛⠖⠛⠋⠨⡗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠁⠤⠐⠀⡀⠀⢀⠔⢀⡀⠀⠀⠀⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢧⠅⠀⠈⠆⠀⢠⠀⠈⠀⠉⠉⠀⠀⠁⠁⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡀⠀⠉⠀⣶⣄⠀⠀⢂⠤⠄⠂⠀⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠈⠹⠛⠛⠿⡞⠖⠚⢻⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠰⠂⠈⠋⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣬⣱⡀⠉⠙⠻⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣝⣵⡿⠇⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣈⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⢞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶⣶⡾⣣⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡴⣒⣒⣒⣒⣒⣒⣒⣒⣂⣀⣲⣽⣏⣙⣛⣟⠉⣞⣛⣛⣛⣛⣛⣛⣛⣛⣛⡿⠿⠿⠿⠿⠿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⡾⠾⠻⠿⠿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠮⢢⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠶⠶⠦⢶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠢⠤⠧⠤⠤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠚⠛⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣤⡴⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣷⣮⣾⣶⣬⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⢠⠄⠘⠀⠀⠃⠀⠂⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠒⠀⠀⠀⠀⠀⠢⠏⠀⠀⠄⠀⠉⠣⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡧⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⠌⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⡇⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⠬⠐⠒⢒⠤⡀⠀⠀⠀⢀⠤⡒⡒⠒⠼⣆⡇⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⡿⣇⠤⢓⠚⠚⠓⢷⡀⠀⡀⡾⠚⠃⠘⠒⢤⣸⠯⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⢸⡀⠀⠀⠀⠀⢘⡗⠠⢻⡇⠀⠀⠀⠈⣶⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⡈⢿⣀⡀⣀⣠⡾⢱⠐⠈⢿⡢⠀⠀⣀⡾⢁⡆⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡠⠸⡄⠈⠝⢋⠏⠐⣸⣀⠀⢀⠻⡙⠫⠁⢠⠇⢿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡇⠀⠈⠒⠒⠁⠊⡐⣥⢠⡕⡄⠡⣌⠒⠒⠁⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢦⣀⣀⡴⠁⠀⠈⠁⠈⠁⠉⣆⠈⠱⣄⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⣼⣿⠀⠀⠀⠀⠀⠀⢀⠂⢸⡂⠀⣹⢫⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⡇⠀⢸⣀⣀⣀⣀⣀⡑⠚⡄⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠀⠂⣿⣏⣩⣍⣿⡇⠀⡇⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠇⠉⠉⠉⠉⠉⠄⠀⢧⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠠⠄⣀⣀⠀⠞⠈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡀⠀⠀⠘⠋⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⢸⣟⢀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠤⠀⠠⠠⠀⠀⠤⠄⠀⠤⠄⠤⠤⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠋⠉⠉⠙⠛⠛⠲⠶⢦⣤⣤⡄⠀⠤⠤⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⢛⣉⣥⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣫⠵⠶⠾⠿⠶⠦⣬⣍⡛⠻⣿⡿⣷⠶⠶⠿⠿⣤⣤⣄⣀⠀⠀⣾⠃⠀⡀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⡞⢡⣶⣶⡶⠤⣀⠀⠀⠀⠉⠀⠀⠙⠊⠳⡄⠀⠀⠀⠀⠉⠛⠛⢾⣇⣤⣼⣶⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⠶⢿⠀⣾⢟⠁⠀⠀⠈⠿⣦⣤⡀⠈⠹⣧⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢠⠉⠙⣿⡿⠃
⠀⠀⠀⠀⠀⢀⠎⡠⠞⠁⠀⢸⢸⣷⠇⠀⠀⠀⠀⢠⣫⡼⠿⠦⣄⠈⠳⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⢠⣶⠞⠀⠀⠀⠀⢈⢻⢙⣀⠀⠀⠀⢀⡾⠋⣠⣶⣦⡈⢷⡀⠀⠀⠀⠀⢿⣦⠀⠀⠌⠀⣴⣿⠃
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠈⠸⡾⠋⠙⠛⠖⣿⣁⣼⣿⣽⣿⣧⢿⡿⣄⠀⣄⠀⠀⠹⣷⡘⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣧⣴⣟⣿⡟⢡⣦⡀⠀⣈⣡⢤⣞⢡⣿⡄⣿⡀⢠⢸⡿⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⡙⠛⡡⠺⡟⠻⠂⠈⣡⣾⡇⣾⣿⣇⣸⡇⣸⢸⠃⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣿⣿⣿⣤⡄⠤⢤⣀⣪⢾⡟⢁⣼⣿⣷⡿⢁⣿⠄⢠⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡌⢿⣿⣿⢷⣽⣯⣿⣿⡟⢠⣿⡟⢱⠋⠀⢸⠇⠀⣾⣿⡟⠻⡏⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣝⢶⣽⡿⣦⡟⢡⣦⣘⣡⣿⢏⡄⢸⠀⠀⠘⠀⣼⣿⣿⠇⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣾⣶⣮⡳⣿⣿⣿⣿⢃⣾⠀⢸⡄⡰⠁⣼⣿⠃⠸⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢹⣿⣿⣿⣿⣿⣝⢿⡇⣾⡇⠀⣸⣿⠃⢸⣿⠃⠀⠀⡀⠀⢰⠀⢸⡄⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⡀⠀⢸⠇⡏⣿⣿⣼⣿⣾⣰⣿⡇⠀⢻⠇⢠⣿⠃⠀⡇⠀⢻⡀⠈⡇⠘⣿⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⣧⠀⠘⣀⣀⢿⠟⠁⣠⣬⡙⢿⣷⢠⠏⣀⣿⣿⣇⢠⣧⢀⣾⣧⠀⣿⣄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⣿⣠⠊⠁⠁⠆⠀⣼⠋⠃⠈⢀⣘⡯⣾⣿⣿⣿⣿⣿⣿⣿⢻⣿⣾⣿⢹⣿⠏⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡔⢻⠇⠀⠀⠀⠀⢀⣿⠀⠀⣷⡼⠻⣧⡙⣿⣿⣿⣟⠙⣿⣟⠘⣿⣿⣧⠈⣿⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡾⠉⠀⠈⠀⣸⠀⢸⢠⢿⣿⡿⠾⣏⠹⣦⠙⣷⢿⣿⣿⡟⡆⢸⡿⠂⣿⣿⠁⣠⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢡⡇⢸⡆⢠⣿⣶⣿⣾⣌⣿⣿⣄⠙⢷⡼⣿⠿⠛⢿⡿⠈⠃⠘⢿⠀⠸⠏⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⡷⣾⣿⣿⣅⣹⡙⣿⣠⣹⣟⢿⣦⣩⢃⠁⠀⠀⠈⠁⡀⢀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣧⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣿⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢹⢲⣿⣿⣿⣷⡟⢑⣾⣿⣿⣷⡖⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⣼⠿⠿⠛⢉⣷⣦⢌⡛⠿⠿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣀⡀⠀⠙⣿⣿⠀⠀⢀⣀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢻⣹⣀⠀⠀⠀⠀⣀⡏⡟⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢤⠀⠀⠀⠀⠻⠒⠬⠉⠈⠥⠳⠜⠁⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢁⠀⠙⠆⡀⠀⠀⠠⠄⠀⠀⢀⠰⠃⠀⠐⠀⠲⡌⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠶⣄⠀⠁⠀⡀⠈⠑⠤⠤⠤⠤⠂⠁⡠⠀⠀⠀⠀⢀⣈⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠒⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣒⣂⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠸⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣽⣆⡠⠤⠴⠲⣖⣒⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣫⠽⠿⠟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢞⡯⢟⣗⠋⠉⠉⠉⠉⠉⠑⢮⡱⣄⠀⠀⠀⠀⠀⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠕⢻⡱⣼⣼⡀⠀⠀⠀⠀⠀⠀⠀⠹⡜⡄⠀⠀⠀⠀⠀⠘⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡳⠁⠀⠀⠙⢮⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⠀⠀⠀⠀⠀⠀⢳⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡇⠀⠀⠀⠀⠀⢣⠘⡆⠀⠀⠀⠀⠀⠀⠀⠸⡸⡄⠀⠀⠀⠀⠀⣸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠈⢧⠹⡄⠀⠀⠀⠀⠀⠀⠀⠳⡙⢄⡀⠀⢀⡠⢃⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢳⠀⠀⠀⠀⠀⠀⠈⢆⠹⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠭⣭⡭⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢣⠀⠀⠀⠀⠀⠀⠘⣆⢱⡀
⠀⠀⠀⠀⠀⣀⠤⣶⣶⣒⠼⣆⢧⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀
⠀⠀⠀⢀⢞⡕⠋⠁⠀⠀⠉⢻⢸⣆⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⣸⢸⢏⣆⠀⠀⠀⠀⠀⠀⠘⡌⣇
⠀⠀⠀⠘⡼⣄⠀⠀⠀⢀⡴⣣⠏⢸⣼⠀⠀⠀⠀⠀⠀⠀⠹⡜⡆
⠀⠀⠀⠀⠘⠪⢽⣿⣯⠥⠚⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠶⢶⣿⣛⣻⣿⣣⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣷⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢳⣿⣿⣿⣿⣿⠀⠸⣿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⠋⠉⠁⠀⠀⣀⣀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣶⣾⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣸⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⣿⣿⣿⢠⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⣿⣿⣿⣿⣿⣿⣿⣧⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣿⣿⣿⣿⣿⣿⣿⢿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢻⣿⣿⣿⣿⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣿⣿⣿⡿⠿⠇⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣾⣿⣶⣿⡿⠿⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠠⢤⣄⠶⣄⠤⢄⣤⡄⠀⠘⣿⣷⣿⣶⣶⡏⠶⠾⣷⣄⠀⠀⠀⠀⠀⠀⠀⡠⣶⠖⠲⢦⡀
⣺⡎⠀⠀⠀⠈⢀⢸⣿⡄⠐⡟⣤⠚⠉⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⢈⠘⠁⢀⣁⠀⣹
⣟⡇⠀⡀⠀⠀⣿⣿⣿⡇⢨⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡇⠀⠀⠀⣿⣧⣤⣤⣨⣄⣽
⣿⣧⣾⡆⣶⣶⠻⣿⣿⣇⣾⣿⡀⠈⠄⠀⠀⠉⠀⠀⣠⢼⣾⡇⠀⠀⠀⣿⣿⡿⢋⠿⠟⣿
⣿⣿⠿⣣⡦⢡⣤⣹⠷⣿⣿⣿⠁⢠⣦⣘⣁⣀⣀⠀⠻⣼⣿⡁⢀⠄⠂⢙⣿⣷⣶⣶⣿⠗⣦⣀⣀⡴⠖⠛⠉⠉⠴⠄
⠛⣛⣷⣿⣯⣦⡹⣾⡆⣰⣿⣿⣾⣿⣿⡿⠙⢿⣿⣿⣆⡯⣿⣿⣿⠀⠀⢸⣿⣯⣏⣿⠿⣹⡇⣸⠯⠀⠀⠀⠀⠀⢠⣤⣧⠐
⠀⣷⣏⣿⣿⣿⣷⡄⢾⣿⣿⣟⣌⢹⣏⣾⢷⡄⠙⣿⢿⢃⡘⠙⠳⢿⣧⣿⣿⡦⣽⣥⣶⣿⢠⣿⣐⠉⠀⠻⣯⣁⠀⢹⣿
⣦⣿⣿⣿⣿⣿⣿⠷⣾⣿⣿⣿⢿⣾⣿⣿⢧⣉⣼⣏⣳⣾⡏⣆⠀⠀⠈⠻⢿⣷⣿⠇⢙⣡⣿⣿⢱⣿⣿⣿⠿⣿⣿⣾⣿
⠻⢿⣿⣿⣿⣿⣿⣦⣿⡿⠛⠁⠘⡹⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⡀⠀⠀⠀⠀⠉⢻⣦⣿⣿⡿⢿⡞⣟⣿⡷⣤⣝⣿⣿⠍⠒
⠀⠀⠈⠻⣿⣿⣿⣿⠏⠀⠀⠠⡀⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⢀⠀⡀⠀⠀⠀⠈⠁⠀⠀⣾⣿⣾⣿⣶⣯⣽⣿⣷⡆
⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⣷⢿⡘⡝⣿⣿⣿⡿⢹⣿⣿⡀⢰⣼⡤⣇⠀⠀⠀⠀⠀⠀⣼⣿⢸⣿⣿⣿⣿⣿⣿⡿⠃
⣴⢀⡴⠀⠀⠀⢿⡇⠀⢦⣄⢀⣿⣼⣿⣷⣬⣛⣻⣁⣾⣿⣿⡅⠀⡄⠛⣿⠀⠀⠀⠀⢀⡠⠛⣿⣾⣿⣝⣿⡿⢿⣿⡇⠀⢠
⣷⣿⠁⡄⣀⠀⣸⡇⠀⡌⣿⠚⡃⠈⢿⣿⣿⣷⣶⣿⣿⣿⣿⣟⣿⡇⣀⣿⡾⠀⢀⠀⠈⢠⢰⣿⣿⡹⣿⣿⣶⣿⣿⠃⠐⣸
⣿⣿⣰⠀⠉⡑⢁⣼⣆⣸⣿⢀⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃⣿⡏⣿⣷⣿⣠⢀⣿⠀⡀⢸⠀⢟⣻⣿⣟⣻⡿⢃⣾⣅⣰⣿
⣿⣿⣯⠀⠰⠁⣀⣉⣀⣉⠛⢿⣉⡉⣿⢹⣿⣿⣻⡟⢛⣿⡀⣿⠁⢹⣏⣽⢹⣾⣟⠀⠇⣾⠀⢸⣿⢸⣿⢿⠏⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠻⠿⠭⠍⢪⣿⣿⢁⡁⣏⠘⠃⠙⢿⡟⣿⣿⣿⠻⣿⣶⣽⣿⣿⣿⣿⣴⣿⡇⠀⠠⢉⣾⡟⢾⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠁⡘⠳⠒⢢⣤⡙⢿⠸⣿⣿⠟⠛⠃⡸⣿⠋⣿⡁⠀⣶⡟⡴⠃⣿⣿⣿⣿⣿⠇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣻⣿⣷⣿⠇⠀⠁⠉⣽⣿⣿⠗⣤⡏⠘⠛⠀⡄⡇⡠⣿⣿⣷⣿⠿⢠⠼⢋⠉⠻⣟⣫⣾⣴⣶⣶⠷⢾⣛⣡⢿⣽⢿⠁⢸
⣿⣿⣿⣾⣥⡎⠀⣠⣾⣯⣿⠏⠀⢹⣆⣴⡾⢰⡇⣿⡇⠘⣀⣩⡥⢖⣡⡾⣟⣶⣤⡈⠿⣿⣿⣍⣉⠙⠂⠈⣞⣿⠟⠀⠀⢸
⣿⣿⣿⣿⣿⠃⠀⠹⣿⣿⠏⠀⠀⠎⠨⣷⡄⠀⠆⣿⣦⠠⡿⣿⣿⣿⣿⣿⠿⣿⡃⠙⠆⠘⣿⣿⣝⡳⣮⡢⣼⣇⣠⠂⠀⠘
⣿⣿⣿⡟⣋⡀⠀⠀⠹⣿⠀⠀⠀⣼⡆⢹⠇⠀⡀⣁⣽⣦⡛⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⢀⣿⣿⣿⣿⢿⣷⣿⣯⠅⣀⣴⠄
⣿⣿⣿⣫⣥⡶⣦⣤⡀⢿⡄⠀⠀⠻⠀⠾⠶⣿⣿⣯⢉⣽⣷⠘⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣌⠻⢿⣷⠾⠟⣋⣥⣊⣽⣿
⢹⣿⣿⣿⡟⠀⠾⣿⣿⣾⡇⠀⠀⠰⣷⣜⣋⡽⣫⣿⣞⠙⠿⣇⣄⠀⠙⠛⠿⠿⠟⠻⣿⣿⣿⣷⡈⢻⣿⣹⣿⣿⣿⣿⣿⡂
⢸⣿⣿⣿⡧⠎⣠⣿⠿⠟⠁⠀⠀⢸⣿⣿⣿⣿⣤⣿⣿⣦⣝⣿⣧⣤⣤⣥⣤⡶⣾⣷⣜⢻⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⡟⠁⠀⣿⣿⣶⡖⠀⠀⠀⢘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⡛⢿⢿⣿⣿⣿⣿⣿⣦⡀⠙⣿⣿⣿⡇⡇
⣿⣿⠟⠀⣠⠘⣿⡟⠋⠀⠀⠀⠀⠬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣷⣾⣿⡿⣥⣼⣼⠿⣿⣿⣿⣿⣿⣷⣦⠘⣿⣿⡿⠁
⣿⣡⠀⣰⡿⢠⣿⣷⣖⠀⠀⠀⠠⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠉⠿⠀⠀⠹⣿⣿⣿⣿⢿⡆⠘⠻
⢻⡟⣸⣿⡇⢸⣿⣿⣿⠂⠀⠀⡀⠀⢲⣯⣿⣿⣿⣿⣿⣿⣿⣛⠓⠶⠶⣿⣿⣿⣯⠀⠀⠀⠀⠀⠙⣿⣿⣿⡸⣷⠀⠆
⣿⣿⣿⣿⡇⢸⣿⣿⡿⠃⠀⣠⠃⠀⢹⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⡷⣿⣇⣿⣄⢺⣄⠂
⡿⣿⣿⣿⠃⣿⣿⠋⠀⠀⠰⡏⠀⢀⣿⣿⣿⣿⣿⣿⣷⠀⢹⣿⣿⣿⣿⠿⣿⡥⠆⠀⠀⠀⠀⠀⠀⢿⡇⣿⣿⣿⣿⡇⢯⡆
⣷⠻⣿⣿⣇⣿⣷⣶⠄⠀⠸⢀⣪⣩⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣷⣀⠀⡇⠀⠀⠀⢠⣺⣇⢿⣿⣿⠊⣷⡞⢰
⣿⠀⣻⣿⣿⣿⣿⣿⣴⠄⣬⣿⣷⣿⣿⣿⣿⣿⠙⠓⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠁⣸⣧⣱⠀⠀⣸⣿⣿⢾⣿⣿⢿⣿⡁⠈
⠏⠀⠹⠛⠿⠿⠛⠃⠋⠸⠏⠻⠿⠿⠟⠻⠿⠛⠀⠀⠀⠀⠀⠈⠿⠿⠿⠟⠋⠀⠰⠿⠿⠁⠀⠀⠻⠻⠿⠿⠿⠿⠀⠿⠗⠶
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠁⠈⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣦⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠛⠁⠉⢩⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠶⣄⠀⠀⠀⠀⠀⢀⣼⣃⣀⢀⣬⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠚⠛⣹⠻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢀⣴⠋⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢞⣛⣛⣉⣙⣛⣛⠻⢿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣤⠞⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⢶⣄
⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢷⡀
⠀⠀⠀⠀⣰⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⡄
⠀⠀⠀⢠⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣷
⠀⠀⠀⢸⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡆
⠀⠀⠀⢹⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⢸⡆⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⠁
⠀⠀⠀⠀⢿⡘⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⠏
⠀⠀⠀⠀⠈⢳⡜⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠙⢦⡙⢿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣝⡛⠿⠿⠿⠿⠿⠟⣛⣩⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠶⢖⣤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢁⣠⣴⣦⣄⣈⠉⠚⠯⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⣠⣴⣴⠏⡁⠤⣉⠚⠝⢻⠆⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⣀⡤⠜⠯⡓⢀⡴⠚⢻⠢⡍⠀⡜⢀⢾⣿⣿⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠉⠡⣊⠑⠢⠴⠋⢀⣴⡿⣓⠀⠈⠀⣎⣾⣿⣿⢸⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣾⡒⠪⠁⢀⡵⠀⠀⠀⠛⢿⡻⠂⡰⠀⡜⣼⣿⢿⣿⢸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⡎⠠⠈⠀⣠⠊⠀⣠⣾⠻⢶⣤⡿⠐⠁⡸⣼⣿⣿⣿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⡟⠑⠄⢀⠔⠁⣠⡾⣟⠢⠉⠀⡠⠊⢀⠴⣵⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣰⠏⢀⠁⡰⠋⢀⣴⣟⠍⠂⠀⡠⠊⢀⠔⣡⣾⣿⣿⣇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣼⠋⠤⠐⠿⡪⣑⠿⡿⠊⠀⣠⠊⣀⠔⣡⣾⢋⢇⢏⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⡾⠃⣈⠑⠀⡐⠮⠒⢕⠀⠀⠈⣁⢜⣵⣿⢏⠇⡈⡎⡎⡜⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠁⠢⡌⠁⠠⡈⠲⢀⡄⠀⡠⢚⣵⡿⡟⡜⡜⡜⡜⡰⣱⢡⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡟⠀⠉⠂⠀⢘⠂⢀⠔⠉⡠⢊⣴⡿⡻⡸⡸⠈⡰⢱⢱⢣⡧⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⠟⠀⢀⠤⣑⠠⣠⠔⢁⡴⠋⣠⢾⢯⢳⠑⠱⢡⢢⢃⢃⡷⠋
⠀⠀⠀⠀⠀⠀⢰⢇⡀⠻⢄⠢⣲⣽⡾⠛⢁⠔⠁⣠⡞⡟⡎⠆⢀⢠⢣⠣⢃⠔⠉
⠀⠀⠀⠀⠀⠀⠘⢦⡉⠲⢄⠉⠛⠋⣀⠔⠁⣠⢾⡹⠱⠁⡀⢠⢣⠃⡣⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠫⡓⢤⡙⢲⠞⢁⣴⠞⢋⢀⣠⢠⡳⢡⢣⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠛⢶⣿⣿⣎⢮⢣⢣⢣⡣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⣿⣯⢎⢣⢣⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢁⠠⠴⠠⠃⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⠀⣤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠃
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⣴⡀⣀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠓
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠋⠀⠈⠀⡏⠡⢶⠉⢘⡄⠀⠀⠀⠀⢀⠀⠀⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠦⠀⠀⠀⠈⢠⠘⢷⣦⣴⡌⡟⡝⢲⠀⠀⡀⠀⠀⡀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢤⡄⠀⢇⠠⠉⠉⡁⠀⠀⡟⠁⠂⠈⠐⣀⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠠⠤⠄⠀⠁⠀⠴⢠⠞⢄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢰⠇⠀⠄⠉⠧⠀⠊⠀⠀⠀⠀⠐⣈⣂⣼⣛⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⠀⠀⠀⠀⠙⢾⡆⠀⠀⠀⠀⠀⡰⠀⠀⢀⠶⡛⠳⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡟⠀⠀⠀⠀⠀⠐⠳⡀⠀⠀⠀⠈⠀⠀⠰⠃⢀⠾⣿⠄⠀⡄⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠉⠀⠃⠚⠀⠀⠀⠀⠀⠀⡀⠘⡄⠉⠀⡠⠗⠊⡁⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⠂⠀⣀⠀⣲⡶⠶⣦⠆⠀⠀⢀⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⠡⠀⠘⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣾⣉⣾⠀⡔⠒⢤⣤⠆⠀⠈⠃⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠰⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠝⠡⠈⠀⠈⠉⠋⠑⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠐⠄⠩⠁⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡄⠸⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡼⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠑⡀⠀⠆⠀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠐⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣂⣁⣀⣀
⣿⣿⠋⣋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠤⡀⠀⠀⠀⠀⣄⡀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⢳⠟⠵⠃⠀⡀⠀⠉⣄⠀⠀⠀⠉⣁⠀⠀⠀⠀⠂⠀⠀⠀⠉⣷⠂⠀⠙⠁⠀⢸⢩⣄⡶⠌⠀⠀⠀⣤⠁⠀⠙⠿⣷⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠟⢁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⡿⠋⠀⠈⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢠⣬⣿⠛⠃
⠀⠀⠀⠀⢤⡀⠀⠀⠻⠀⠀⠀⢀⡾⣁⠀⠀⢠⣿⣁⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠐⢺⣿⣷⣶⣄⠀⠈⠀⠐⠀⠀⠀⠀⠙⠀⠀⠀⣾⠟
⠀⠀⠀⠀⠘⣿⣦⠀⢀⣠⠀⠀⠀⠀⠹⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⣀⣘⢻⣷⣿⠇⠀⠀⣰⣀⡄⠀⠀⣰⠄⠀⢀⠀⠀⠀⠂⠀⠀⠀⣾⡟⠀⠀⠀⠾⠀⠀⡀⠤⠒⠄⠀⠀⠀⠀⠻⠆⠀⠀⠸⣂
⠀⠀⠀⠀⠙⠿⠛⠛⠋⠀⠀⠀⠨⣿⡆⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠠⠄⠀⠁⠀⠀⠑⠅⠀⠀⠀⠶⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⣠⠖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠳⠤⠋⢀⡀⠘⣆⣠⣤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠀⡴⣄⠀⡴⠋⡇⢀⣀⣤⡄⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢉⡀⡇⠈⠛⠁⠀⠙⠉⠰⠯⢤⣄⡉⠙⠒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢰⠈⠁⠀⣀⢠⠀⡀⠀⢠⠖⢋⣩⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⣣⠀⠀⢻⣛⠛⡇⠀⠈⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠮⢤⣘⣛⡳⠤⠤⡼⠨⢠⠇⠺⠿⣝⢛⣁⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢑⡏⠀⣾⣛⣛⣛⣋⣁⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠈⡸⠃⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣤⠚⠁⠈⠙⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣏⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠁⣠⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⡛⣷⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣼⠏⠀⡴⠋⠀⢀⣿⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠈⢻⡄⠀⠀⠈⣷
⠀⠀⠀⠀⢸⡟⠀⣸⠃⠀⢰⣿⣿⣀⣠⡄⣠⣴⣿⣇⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠈⠉⣿⠁⢹⠻⢿⡿⠟⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⣷⠀⢻⣄⠀⠀⠀⣰⡏⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠹⣦⠀⠉⠛⠓⠛⣋⣀⣀⠘⣷⡀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠉⠉⠹⡷⠈⠛⠷⠶⡶⠟⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⣛⣉⣉⡉⢉⡒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠶⠋⠉⢀⣀⣀⣉⣉⠐⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⡾⠁⠀⠔⠛⢋⡉⢉⣉⣙⣿⣿⣶⣤⣈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣾⣧⣤⣀⣠⡞⠁⠈⠁⣀⣤⣤⣬⣉⡙⠿⣷⣦⣙⠲⢤⣄⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣀⣴⠋⠉⢙⠋⠁⠉⠉⠛⠳⢶⣴⡟⠉⠁⠀⠈⠉⠛⠳⣤⡙⠻⣿⠷⢶⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣷⣶⠾⠿⠛⠛⠛⠛⠛⠿⢾⣷⣧⣄⡀⠀⠄⠀⢽⢿⣷⣄⣠⣤⣤⣀⡀⠈⠻⣄⠘⢷⡄⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⣉⣡⠤⠶⢶⣶⣾⣾⣿⣿⣶⣿⣶⣯⣟⣿⣿⣤⡈⢈⣼⣿⣿⣭⣭⣭⣭⣟⢷⣄⠙⣧⡀⢻⣆⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⡡⠔⠋⠁⠄⠒⠂⠀⠀⢈⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣷⠈⣷⠀⢻⡆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣾⠋⡴⢋⡀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⣠⣾⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣇⠘⣇⠀⢿⣎⣿
⠀⠀⠀⠀⠀⢀⣾⣿⠃⠎⡐⠃⡀⠤⠤⠶⠞⠛⠋⠉⠀⠀⢠⣾⠟⠉⠀⢀⣴⡿⢿⣿⣿⢿⡿⣿⡙⠻⣯⡻⣽⣿⣿⣿⣿⡆⣿⠀⣿⠀⢸⣿⣿
⠀⠀⠀⠀⢀⣾⢣⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⣰⡿⠋⢠⣾⡏⠃⠈⠻⣎⠻⣄⠈⢳⡌⡾⣿⣿⣿⠃⣿⠀⣿⠀⢸⡇⠉
⠀⠀⠀⠀⢸⡏⠈⢀⡟⠀⠀⠀⠀⠐⠒⠲⢤⣤⣄⣤⠶⠋⠀⠀⠀⣰⠏⢀⣴⡟⢻⡄⠀⠀⠀⠹⡆⠙⡆⠀⢻⠀⢻⣿⣿⢠⡏⢰⠇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣼⠇⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠞⣁⣴⣿⢿⣧⡀⠳⣄⡀⠀⠀⠓⠀⠰⠀⠈⠄⢸⣿⡟⣼⢁⡎⠀⢰⡏
⠀⠀⠀⠀⠈⣿⣀⣿⠀⠄⢁⣤⣶⣶⣾⣷⣶⣶⣶⣦⣤⣤⣴⣶⠿⠿⠋⠁⠀⠙⠻⣦⣄⠙⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⢷⡏⡼⠀⢠⡟
⠀⠀⠀⠀⠀⠸⣿⣿⠰⢠⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢀⠀⠇⠀⢨⢠⣿⣿⡾⡐⠃⢠⠏
⠀⠀⠀⠀⠀⠀⢻⣿⡄⡼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣆⢠⢀⣼⣾⣿⣿⠃⠁⢸⠏
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢹⡇⠈⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣿⣿⣿⣿⠇⠀⠀⡿
⠀⠀⠀⠀⠀⣀⣠⣽⡟⠘⣘⣧⣮⣤⣤⣤⣍⡒⠄⣀⣀⣱⡖⠀⠠⠤⠀⣀⣠⣤⡿⠶⠶⡄⠀⠘⣿⣿⣿⣏⡿⢻⡏⠀⠀⢸⡇
⠀⠀⠀⢠⡟⠉⣉⣹⣿⡿⠛⠉⣁⣤⣶⣿⣿⣟⠷⠾⠶⢿⣧⠀⠲⠾⠿⣿⣿⣿⣿⣷⣶⣄⡀⢀⣿⢋⣀⠙⢧⣾⡇⠀⢠⣼⣧
⠀⠀⠀⢸⠃⠀⣼⣿⣿⣧⣶⣾⢿⣏⠀⠤⠀⠈⡷⡀⠀⠀⠈⠀⡄⣠⣾⠁⠀⠀⠀⢹⣿⣿⠟⠉⣿⣿⣿⠂⢸⢸⡇⠀⡞⢻⣿⣦
⠀⠀⠀⠘⣇⠀⢹⣿⣿⡇⠈⠙⠷⣿⢶⣤⣴⣾⣶⣿⠇⠀⣀⠘⡇⠻⣿⣻⠦⠤⠶⠛⠉⢁⡀⢸⣿⣿⡿⠀⠘⠈⣿⣴⡅⠀⠈⠙⠛
⠀⠀⠀⠀⠘⣧⠈⢻⣿⡇⠀⠙⠦⣤⣤⣤⣴⡾⠟⡡⢀⡴⠶⠂⠰⡀⢨⡛⠶⣤⣤⠤⠶⠀⣠⣿⣿⣿⠃⠀⠀⠆⠘⢿⣇
⣀⡀⠀⠤⢞⡽⠷⠊⠛⣃⠀⠀⠀⠀⠀⠀⠀⠠⠚⢁⡞⢀⣠⣄⠀⠈⠢⠉⠓⢒⣒⣂⣤⣾⠟⢿⡿⠃⡠⠀⢈⠀⠀⠀⠙⢦
⠈⠉⠉⠀⠀⠀⠀⠀⢰⠋⠄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⣧⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡶⠶⢶⣤⣌⢠⡀⠀⠀⠀⠸⣿⠿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠸⣿⣿⣿⣝⠆⠀⠀⢀⡴⠄⡤⢤⣀⡀⣠⣤⠄⢴⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⣻⣦⡀⠙⣏⣙⡟⠀⣀⠖⣋⡔⢶⡶⣦⣌⣿⣿⣿⠷⠀⠀⠀⢀⣾⠏⠆
⠀⠀⠀⠀⠀⠀⠀⠊⢠⣱⠏⠈⠻⣷⡿⠍⠃⠙⠻⣿⠟⠛⠻⠟⢋⡉⣿⡯⠝⠀⠀⠀⢀⡾⣿⡐
⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠘⣿⣆⠀⠀⠀⠙⢦⣉⣁⣉⣩⣭⠟⠀⠀⠀⠀⣠⠏⠀⠙⢷⡆⡀
⠀⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠉⠉⡉⠉⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠙⢄⢀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣾⢿⣿⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⠿⣦⣤⣀⣤⣄⣀⣤⣴⠾⠋⠁⢸⣿⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢈⠈⠋⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡊⠀⠀⢸⣿⣿⠀⠀⠐⠂⠀⢸⠞⠃⠀⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣠⣿⡿⠃⠀⠈⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠁⠀⠀⠀⠐⠂⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠉⠀⠀⠙⠓⠤⢄⣀
⠀⠀⠀⠀⣠⠴⠛⠁⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⠀⠀⠠⠀⠀⠀⠀⢀⣴⣆⠀⠙⠀⢀⡔⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠑⢆⠀⢀⠛⠳⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⡴⠋⢹⣿⣆⠀⢹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⡄⠀⠚⠁⠿⠒⠤⢄⡠⢀⣀⣀⣀⣀⣀⡤⠴⠚⠋⠉⠀⠀⡀⠹⣿⣦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇