Isis Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣀⣤⠤⠴⢤⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆⣠⠞⣩⠖⠛⠛⠛⠲⣝⢦⡀⠎⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡏⡾⠁⠀⠀⠀⠀⠀⠈⣧⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⣿⣸⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡼⢷⡹⣆⠀⠀⠀⠀⢀⡴⢣⡟⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⡇⠀⠙⢦⣝⣓⣒⣚⣩⠶⠋⠀⣿⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠛⠢⠤⣤⣤⣭⣭⣭⣤⣤⠤⠖⠋⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣦⣤⣀⠀⠈⢻⣿⠏⠁⢀⣠⣤⣶⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⠀⠀⣿⠀⢰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠿⢿⣿⣿⣿⣿⣿⣷⣄⣴⣿⣄⣠⣿⣿⣿⣿⣿⣿⠿⠿⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣭⣜⣃⣖⠰⠠⠍⣭⢡⡝⡼⠿⡻⢩⠉⡍⠥⠰⠄⣖⣘⣬⣵⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠋⣩⣿⣿⣿⣒⠚⠛⢒⣚⣿⣿⣯⣉⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⠀⣤⣤⣦⣤⡉⠁⠀⠈⣩⣤⣤⣤⣀⢸⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢻⡆⠈⠓⠛⠓⠙⠆⠀⠘⠃⠛⠛⠋⠁⢸⠓⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢡⡹⣇⠀⠀⠠⡤⠦⡖⠴⡤⠀⠀⠀⣾⢣⡜⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢣⣏⣧⢹⣦⡀⠀⠈⠒⠒⠒⠁⠀⢀⣾⠇⣮⣵⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢶⠰⢶⠰⣿⠛⠦⣄⣀⣀⣀⡤⠞⢹⣿⠰⠶⠶⠖⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣾⣾⣿⠀⠀⠀⠉⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢧⠘⣮⢿⣿⣿⣿⣿⣿⣿⣿⡗⠬⢉⣁⡒⠒⣂⣉⡩⠄⣾⣿⣿⣿⣿⣿⣿⣿⡿⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡘⢯⡿⣿⣿⣿⣿⣿⣿⡏⠑⠃⠶⠰⠴⠄⠆⠒⠈⣻⣿⣿⣿⣿⣿⡿⢯⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢮⡛⠭⢛⠻⢻⢣⡆⡄⣀⢠⢀⡈⡌⣄⢦⢻⡟⠟⡛⡍⣫⠴⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⡳⢯⣜⣰⠨⠌⡅⣃⢘⢈⡃⡃⣈⠨⠱⣐⣦⠵⢚⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣭⣙⣛⠲⠶⠾⠶⠷⠷⢒⣛⣫⡭⠶⠚⠉
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡉⡉⠉⠉⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⠿⠿⠿⠿⠟⠻⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣻⣻⣿⣻⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠠⠁⠀⠀⠠⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⢀⣀⣠⣠⠤⠤⠴⠃⠈⠤⠤⢤⣀⣀⣀⠀⠀⠀⠘⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠖⠚⠉⠁⠀⣀⣀⣀⡄⠀⠀⢁⣀⣀⡀⠀⠈⠙⠲⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⡀⠈⠙⣦⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠘⠐⠐⠁⠁⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠁⠛⠛⠞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠔⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠂⠄⠂
⠀⠀⠐⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢱⣿⡦⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠥⣲⣿⣇⢌⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⣸⡏⠀⢹⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⡐⠀⠱⣶⣿⣇⠀⣾⣿⡶⠃⣀⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠱⠛⠀⠀⠈⡟⠙⠛⠉⡛⠀⠀⠘⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠠⠘⠀⢖⠀⠁⠄⠂⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠘⣤⣮⣠⣥⣴⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠉⠉⠉⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠻⣄⣀⠾⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢮⢠⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⡿⡿⣻⡄⠹⢿⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣾⣿⣦⡀⠀⠀⠀⣿⣿⣿⣧⡀⠙⠀⠀⠀⢀⣴⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⠃⠙⢿⣤⣶⣤⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⢱⣾⣿⣿⣿⣿⣿⣿⣿⡿⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠭⢿⡻⢿⣿⣿⣿⣿⣿⡀⣼⣿⣿⣿⣿⡿⡿⢟⡿⠭⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⠿⠿⣧⣸⢉⠂⠌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢸⡿⡟⠃⠛⣹⣹⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⡹⣷⠈⢰⡷⣯⠀⠀⠀⠀⠱⡀
⠸⣧⠬⢵⣒⣂⠀⠀⣠⡆⢀⡀⠀⠀⡁⣒⡂⠀⠈⠹⣿⣾⠀⢰⣿⣿⠁⠁⠀⣒⡂⡈⠀⠀⣀⠀⣦⣀⠀⢀⢒⣒⡭⢤⡿
⠀⠹⣧⠤⠄⣐⣒⠢⠿⣭⣙⠩⢙⠻⠧⣯⣏⣙⣍⣻⡿⠯⠬⠬⠿⢿⣛⣙⣉⣭⡥⠿⠛⠩⢝⣩⡼⠯⢐⣒⡂⠠⢤⡿⠁
⠀⠀⠘⠻⣥⣄⣒⣊⠭⠔⣛⠿⣶⣄⣂⢀⢔⠍⠍⠍⢟⠀⠀⠀⠀⠏⢉⢍⢁⠀⠨⣐⣤⣶⠿⣛⠬⢍⣐⣒⣤⡽⠟⠁
⠀⠀⠀⠀⠀⠙⠳⣶⣪⠥⢂⡩⢔⡫⣛⢟⡿⡺⠺⠶⠾⠀⠀⠀⠘⡷⡶⡷⡗⢟⠫⡫⢕⠦⣁⠂⢭⣰⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢥⣎⣵⣫⢎⡔⡰⢡⠃⠀⠀⠀⠀⠱⡐⡀⢌⣢⣑⣌⣦⡽⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠧⠀⠀⢸⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠖⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⢤⣤⣶⣾⠀⠀⠠⠇
⠀⠀⠀⠀⠈⢦⡀⠀⠙⣿⣿⣷⣄⡀⠛⠀⠀⠀⠀⠀⡘⢿⣿⣿⡆⣀⠀⢤⠀⠀⠀⠀⠘⠁⣀⣴⣿⣿⡿⠃⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠑⠢⢀⠈⠙⠛⠿⣿⣿⣶⣶⣤⣄⣐⡃⠀⠉⢛⡳⠟⠀⣈⣁⣤⣴⣶⣾⣿⡿⠿⠛⠉⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠂⠠⠄⠉⠙⠛⠻⠿⣿⣷⣄⣄⣤⣤⣿⠿⠿⠛⠛⠉⢁⠤⠄⠂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⡏⠀⡜⢀⡞⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢳⡀⢣⠀⠹⠀⢀
⠀⠀⠀⠀⢰⠀⡇⠀⡄⢰⠃⡾⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⠘⡇⢸⠀⢸⠀⡦
⠀⠀⠀⣀⠘⡆⢸⠀⢃⢸⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⡇⢸⠀⡞⢠⠃⢀
⠀⠀⠀⠸⡄⠡⡈⣇⡼⢾⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡷⢧⣸⠁⡌⢀⠏
⠀⠀⠈⢧⠙⣄⢳⡞⢧⢸⣐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡇⣸⢳⡜⢠⠊⡼⠃
⠀⠀⠠⣌⠳⣌⢺⠳⡌⢦⣻⡇⡀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡷⣤⣴⡀⠀⠀⠀⠀⠀⢀⢼⣟⡴⢡⠞⡗⣡⠞⣡⠄
⠀⠀⠀⣌⠳⣌⣻⡣⣜⢲⢻⣷⡊⠀⠀⠀⠀⠀⠀⠸⠘⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣾⣿⡷⣣⠜⣟⣡⠞⣁
⠀⠀⠀⢈⡓⠦⢽⡝⠪⢽⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢡⠾⣿⡿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⡯⠕⢋⡯⠴⢚⣁
⠀⠀⠀⠀⣉⠓⠚⢯⠝⣺⢿⣿⣿⣧⣀⣀⣐⠟⠀⢸⣿⣿⣿⣿⣿⣿⡾⣆⣀⣀⣴⣿⣿⡿⣗⡫⣽⠓⠚⣉⡀
⠀⠀⠀⠀⢩⣉⣉⣉⣷⣒⣚⡵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠮⣑⣒⡾⣍⣉⣉⡉
⠀⠀⠀⠀⠀⠠⡤⠤⢒⣹⢶⣏⠴⣫⢿⠛⠛⢿⣿⣿⡟⠛⠛⠛⣿⣿⣿⡿⡛⡿⢝⡦⣹⡶⢏⡒⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠐⠛⣉⡤⠊⡩⢛⡶⡿⡄⢦⡀⠉⣿⡏⠀⢀⣴⣿⣿⣣⣱⣽⢮⠶⡛⢇⠑⢦⣉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡴⢋⡼⣱⢻⡌⢳⡄⢘⣿⣿⡿⠿⡙⣿⣿⠘⡘⢆⢣⡙⢦⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠛⣹⠞⠛⠁⠀⣘⡄⢐⣈⣿⠛⠃⠙⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⡐⠚⠛⠋⣻⣟⣟⡙⢷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡴⠟⠛⠛⣛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠞⠛⠛⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣿⣿⡟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠷⠶⠾⠿⠿⠿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢻⣿⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣟⣛⠻⠷⢶⣶⣶⣤⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣤⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢰⠎⠁⠀⠀⣸⣭⣛⠿⢷⣶⣤⣌⣉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⡱⢲⡤⠄⠀⠀⠀⠀⢀⡤⢶⠸⣠⠋⠀⠀⢀⡼⢧⣍⡛⢿⣶⣬⣙⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⣍⣿⣿⠧⠔⢊⡠⣋⠈⣀⣠⠀⡖⢸⠁⣇⡼⠖⠁⠀⠀⡠⢮⡛⢶⣝⠻⣶⣌⡙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⣿⣒⠙⡬⠀⡏⢱⢃⣸⣀⣧⠼⠛⠉⠀⠀⣠⡴⢿⡹⣦⡙⢦⣝⠷⣦⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠤⡀⠛⠋⠱⢼⡅⠛⠓⠛⠉⠉⠀⢀⣀⣠⣴⢶⡻⣏⢻⣌⢻⣎⠻⣦⣙⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣆⢛⣋⣍⠀⠀⠉⠢⢶⣶⡖⣞⢿⢹⡏⣷⢹⡎⣷⠹⣆⢻⣦⣹⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⠙⢷⣤⠰⠜⠆⣀⠀⠈⠑⣇⢻⢸⡏⣷⠸⣇⣿⣸⠧⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠳⣯⡩⡬⠷⠄⡀⠀⠉⠳⢟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⣻⣦⡀⠈⠙⢶⣖⢃⣙⡠⣄⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣡⡾⢛⣦⡀⠀⠙⠻⣧⡴⠞⢊⣢⡤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣾⡏⣰⡿⢡⡿⢷⣄⠀⠈⠙⠻⢯⣤⣰⣚⣃⣤⡤⠠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⣰⡟⣰⡟⣹⡷⣦⣀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡟⢠⡟⢠⡿⢹⡿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⢁⣿⠃⣾⠇⣼⡇⣽⡏⣿⡖⣶⡶⢶⡖⢿⡟⢿⣍⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣿⢀⣿⠁⣿⡇⣿⡇⢸⣧⠘⣿⣄⠻⣦⡙⢷⣤⡙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⠀⣿⡇⢻⣿⠀⣿⡆⠘⣿⣆⠙⢷⣌⠻⣿⣦⡉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡇⢸⣿⡀⢸⣿⡀⠘⣿⣦⠈⢻⣧⡈⠻⣿⣦⣨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠂⠴⠒⠛⠉⠉⠉⠉⠉⠻⠷⣾⣿⡇⠀⣿⣧⠀⠸⣿⣧⡀⠙⣿⣦⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠾⣿⣦⣄⣽⣿⡷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣠⣤⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣈⣁⡼⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣊⠓⠂⠀⢶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡛⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠠⠄⠐⠓⠆
⠀⣿⣍⠉⠛⣛⠛⣿⣯⣈⠁⠁⢔⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣹⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡐⠈⠁⣩⣿⡿⠗⠒⠒⣒⣹⡂
⠀⠸⡆⠀⠠⠤⠤⠼⠽⣯⣭⡀⠀⣰⢦⣶⣮⣅⠀⠀⠀⠀⠀⠀⠀⠀⡰⣶⡆⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣊⣭⣤⢄⠁⠀⣐⣻⡿⣿⣅⣀⡀⠀⠤⡞
⠀⠀⠹⣗⠒⠒⠚⠛⠓⣹⣶⡦⠀⠈⢙⡛⢿⣿⣿⠀⠢⡄⢤⣀⣀⣀⡀⢸⣿⣶⣿⣿⠿⠋⠁⣀⣀⡀⣠⠤⢲⣿⣿⡟⠟⠛⠀⠠⣭⣿⠷⠒⠒⠒⠒⠲⡾⠁
⠀⠀⠀⠈⠻⣍⣉⣁⠠⠕⣾⢿⣯⣥⡤⠐⢊⠽⣿⣷⣀⡉⠒⠨⠭⡝⢟⣀⡀⣭⣉⢄⣒⢚⣿⣫⠠⠘⠀⢠⣾⣿⠛⠤⢍⣀⣰⣿⣿⠯⢍⡀⠐⢒⡾⠋
⠀⠀⠀⠀⠀⠈⠙⢤⣖⣋⡤⠚⢁⠝⡻⠯⣵⣊⠝⡻⢿⡷⣲⣶⣤⣥⣿⣷⣼⢟⣰⣾⣿⣇⣥⣠⣤⢴⣿⣿⠫⡓⢬⣰⣶⠿⡉⠓⢮⡙⠢⣤⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠾⣥⣞⣡⠔⢁⣿⣿⣷⣿⣴⡧⣞⣥⣡⡞⣿⣿⣿⣿⣿⣿⣈⢋⣻⣍⣧⣽⣶⢷⠻⣍⠣⣀⠑⣌⣳⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⠧⣾⣿⣿⣿⣿⣿⣷⣯⣝⢊⣹⢿⣿⣟⠟⢟⡻⡟⢟⢻⠙⢇⠹⣄⣳⣌⡷⠾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣍⢵⠾⠶⠷⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠿⠻⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡤⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⡀⢼⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢉⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠋⠀⠜⠆⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡤⠖⠉⡀⠀⠐⠉⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠉⠀⠀⠄⠆⡄⢀⣀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢉⡡⣠⠷⠊⠁⠒⢀⠀⠀⠒⠈⢁⣵⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡄⠀⠀⠀⠡⠀⠀⠀⠉⠈⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⢶⣾⣇⠀⡀⠁⠨⢽⠂⠐⠀⠀⠀⠀⠀⠀⠉⣉⠔⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠂⠀⢡⠀⠀⠀⢸⠁⠀⠀⠀⣀⣀⣠⡤⠴⠚⠉⠀⠀⢀⣀⠿⠛⠋⠀⠁⢐⡾⠁⠀⠀⠀⠀⢀⡐⢒⣛⡱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⠘⠀⢰⠿⠗⣲⣿⡉⠉⠁⠀⠀⠀⢀⡀⠀⢀⣪⡾⠋⠁⠀⠀⠀⣶⣾⠁⠐⠀⠀⠀⠂⠀⠠⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠴⠂⠁⠀⠀⠆⠃⠀⢀⢻⣿⡧⠀⠄⠀⠂⣈⣤⠖⠚⠉⠀⠄⠀⣀⣀⢤⢿⠍⠀⡂⠐⠂⢀⠐⣬⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠠⠀⠠⢜⠀⠀⠀⢀⣠⣾⣁⣀⢤⠶⠒⠊⠉⠀⠀⠀⠀⣀⣠⡶⠛⣫⢡⢢⠠⠀⠑⠁⢤⣰⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡧⠀⠀⠸⠤⠶⠛⣿⣫⡾⠋⠇⠂⠀⠀⡀⠈⢀⣠⡴⠞⠝⠀⡠⠁⠡⠠⠠⠀⠑⡀⣬⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⢢⢥⠠⠀⠀⠀⠀⠙⠀⠀⠀⠘⢠⡠⡴⡚⣉⡁⠉⠂⠀⠀⠁⡑⡀⣠⡠⠴⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⠉⠿⠂⣔⠀⠀⠀⠀⠈⠢⠄⠀⠀⠓⠤⠈⠐⠢⢀⡀⢄⣧⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢠⠀⠆⠚⠃⡦⣅⡤⠀⠀⠀⠀⠠⠀⠀⠈⠓⢤⡤⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⠀⠀⠱⡄⣀⢀⠐⠲⣄⠀⠀⠀⠀⠀⠀⢀⢠⡟⡖⠒⠊⠉⠐⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠲⠴⢠⠂⢠⠀⠈⠔⠸⠀⠣⢳⢥⣀⠄⠀⠀⠈⠙⠃⠠⠀⠒⣤⣄⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠓⠇⢈⢀⠀⠈⠋⡆⢐⠠⡀⠁⠢⢠⣀⠀⠀⠄⠀⠐⢁⠬⠛⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠰⠇⠀⡈⠁⠨⠄⠁⠌⣸⠄⠀⠉⠕⡒⠒⡏⠁⠐⢄⠲⠄⠐⠠⠉⠑⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠡⡌⠀⠀⠀⠈⠸⠀⢐⢈⢠⠱⠀⠀⡄⢄⠀⢀⠢⡁⠀⠀⠁⠀⠀⠀⠈⠙⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢠⠀⠀⢸⠀⠘⠈⠈⡀⠂⠠⠈⠀⠐⢄⠠⡀⠀⢀⠀⠀⢀⠀⠀⠐⠶⣚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠸⢀⠀⠀⡁⢁⠐⠈⠀⠀⢀⠀⠡⠀⠀⠀⠐⠀⠀⠙⠲⣚⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠐⠄⢄⣀⠀⣀⣀⣀⡀⠄⠠⠔⠀⠀⠠⠀⠈⠑⠹⢈⣆⢃⠈⢦⡀⢢⣀⠀⢢⡀⠀⢄⣈⡳⣓⠋⠁
⠀⠀⠀⠀⠀⣀⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠮⡏⠲⠷⠙⠲⠼⡇⠚⠓⠈⠉
⠀⠀⣖⣋⣉⣁⣀⣀⣀⣀⣠⡤⠤⠤⠤⠬⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠛⠋
⠀⢀⠀⠀⠀⢀⣀⣀⡀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠻⣿⣿⡇⣤⠀⢠⡀⠀⠀⠀⢠⣄⠀⣀⣀⡀⣀⣀⣀⠀⠀⠀⢀
⠀⠀⢣⠀⠀⠈⢿⣿⣿⣿⣿⣷⣶⣿⣦⣤⣤⣤⣤⣧⣄⢀⣉⣝⣫⣴⣾⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢠⠃
⠀⠀⠀⠁⢄⡀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⡿⠿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠈⠑⠀⠠⠠⠄⠀⠀⠤⠤⠄⠀⠀⠄⠀⠀⠀⠀⠠⣷⠄⠀⠀⠀⠠⠠⠤⠤⠀⠀⠠⠀⠐⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆
⠀⠀⣄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡤⠀⠀⠀⠀⠀⠠⡄⠀⠙⠿⣿⣇⣠⡇⠀⢰⡆⠀⠀⠀⠀⠀⠶⠇⠀⠀⠒⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡄
⠀⠀⠘⡤⠄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣲⣽⣩⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠤⡜
⠀⠀⠀⠘⢪⡄⠀⡀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠽⠷⡏⠀⠀⢀⣤⣾⠳⡆⠀⠀⠀⢀⠀⢀⣀⡠⠔⠊⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠉⠫⢖⣀⣀⡀⡀⡀⡁⠉⠉⢉⠉⠉⠉⠉⠉⠁⠉⠈⠈⠈⢱⣤⣴⣾⠋⢉⠹⠻⠿⠋⢀⠎⠉⠉⠉⡉⡁⡀⡀⡀⢀⣀⣐⠮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⠈⠀⠈⠙⠿⣿⣿⠀⠀⠀⡸⠄⢰⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀⠄⢊⣉⣤⠶⠳⠦⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠛⠀⠈⠀⢀⣀⣀⣀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠥⢨⣭⣁

⣿⣿⣿⢿⣿⠿⢿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣤⣶⣶⣴⣧⣾⣯⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⢠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡄
⠀⠀⠀⠀⢸⡂⠀⠈⠉⠳⡶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣴⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡤⢶⡖⠉⠉⠀⠐⡇
⠀⠀⠀⠀⠘⣏⠀⠀⠀⠒⢧⠐⠉⠴⠄⠉⣦⣤⣀⠀⠀⠀⠀⠀⣠⣾⢿⢟⠟⡟⣿⠡⠾⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠉⠀⠦⠉⠐⣼⠒⠀⠀⠀⣻⠃
⠀⠀⠀⠀⠀⢻⡥⠀⠀⠀⢈⣖⡀⠀⠀⢻⣿⢾⢅⠉⠙⠒⠢⢤⣼⣧⣿⠘⣡⣿⣆⠀⢠⢋⣀⣤⡤⠴⠒⠊⠉⡨⡷⣿⡟⠀⠀⠀⣲⠁⠀⠀⠤⢨⡟
⠀⠀⠀⠀⠀⠘⣧⣤⣥⢶⡄⠀⠡⡄⠀⠀⠙⢆⡄⠈⠀⠤⢠⣼⠀⡎⠉⠟⢻⣧⣷⠘⠛⡋⢱⠀⣣⡄⠄⠂⠃⢈⡰⠋⠀⠀⢨⠌⠀⢐⡶⢦⡤⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡿⠛⣋⣄⠀⠈⠣⣀⠀⠀⠙⠦⡀⠀⠆⠉⠛⠙⢀⠘⢸⣯⣿⠤⠊⢀⠋⠓⠉⠡⠀⣀⠴⠊⠀⠀⣀⠔⠁⠀⢠⣙⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢟⣩⡶⢛⣤⢀⠈⠐⠀⡀⠀⠈⠓⠦⣄⡂⠀⠈⡆⠘⠍⠙⠀⠐⠀⡂⢐⣀⡴⠒⠉⠀⢀⠀⠒⠁⠀⣤⡚⢶⣍⡻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⣡⣾⢃⣶⢀⣤⢉⡑⠠⠄⢌⢍⠙⣒⣷⠀⠀⠀⠀⠀⡞⠋⣩⣁⠠⠄⢚⡉⣠⡀⣶⡘⢷⣌⡿⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⠧⣾⣡⣿⣰⣶⢠⣶⢰⡎⣇⠀⠀⠀⠀⡇⣲⡆⣶⣆⣿⣌⣷⠼⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣿⣲⣲⢲⣶⣻⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⡉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠐⠅⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⢀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠘⣆⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠃⠀⠈⠉⣋⣚⡀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⣰⠁
⠀⠀⠈⠢⡀⠀⠈⠲⠄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠿⠟⡃⠀⢀⣤⡾⡆⠀⠀⠀⡀⢀⣀⠠⠖⠁⠀⣠⠔
⠀⠀⠀⠀⠈⠑⠤⠤⠠⠠⠀⠀⠀⠈⠀⠁⠁⠈⡀⢈⠀⠈⣧⣶⣿⡁⠀⠙⠿⠁⠈⠈⠀⠠⠀⠠⠄⠠⠄⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠿⠇⢀⣄⣚⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠩⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⡿⡿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣶⣶⣮⢝⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡿⢹⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢷⣎⢸⣿⣿⣿⠿⠛⣆⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡿⠁⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠀⡛⣛⣩⣭⠴⢾⣿⣿⡆⠀⠀⠀⠀⠀⢀⠏⢀⡾⠁⣰⠏⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣖⠀⠀⢚⣭⣭⣤⡄⠀⣿⢍⣁⣴⣼⣿⣿⣿⣿⡀⠀⠀⠀⢀⠎⢀⡞⢁⡼⠃⣰⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠁⠀⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡜⢠⠟⢠⠞⢁⡾⢃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡐⣠⠏⡰⠋⡴⠋⣠⠞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡜⡴⢣⠞⡡⠊⣠⠞⣡⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣟⡵⡣⠎⡠⠊⣡⠞⢉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡮⢚⠔⣋⠔⢋⡤⠞⣋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣟⠵⣊⠴⢚⡡⠴⢚⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⡿⣓⠩⠀⣊⣡⠴⠖⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣿⣿⣿⣿⡿⢿⣿⡿⣩⢛⡅⣼⣿⣿⠽⡖⠚⠩⠤⠴⠶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣛⣭⡭⠵⠶⠚⡫⣡⢺⢏⡾⣽⣿⣿⡯⠭⠥⠴⠶⠶⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠁⠁⠐⠂⣋⠽⣫⣸⣧⢫⢞⣿⣿⣿⣟⣛⣛⣛⢛⣛⣛⣻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠠⠤⠒⣫⣵⣿⣿⢟⣵⣿⣿⣿⣿⣛⡒⠒⠢⠤⠤⣭⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠶⠋⣵⣿⠟⢁⣵⣿⡿⢯⣛⡓⠦⠬⣍⣙⣛⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣯⣴⣾⣿⣟⠯⣝⣓⠦⠭⣑⣒⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠝⠍⡂⢍⡒⠬⣍⡒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⢀⡨⣐⠬⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⡇⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣀⣀⣀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣏⣉⣉⣉⣉⣙⣻⣿⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣍⣉⡉⠉⠙⠛⠛⠋⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣀⣉⣉⠻⠷⣶⡆⠀⢠⡿⠁⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠉⠉⠁⠀⢠⡿⠃⠀⣼⠇⠀⠀⠈⢿⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣄⠀⠀⠀⢸⡇⠀⠸⠿⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣥⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣾⡿⠟⠻⣧⡀⠀⢸⡇⠀⢼⣿⣿⡶⣶⣤⣀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠈⢿⣄⠓⢈⡛⠷⢿⡇⠀⢸⣿⠇⠀⠀⠈⠻⣧⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠙⠳⣦⣤⣀⣾⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣶⣶⣴⣶⣦⣤⣾⣿⣿⣇⣀⣾⣷⣤⣤⡄⠀⠀⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋⠁⣼⣿⣇⠀⠀⣿⡀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣿⠀⠀⢻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢀⣿⠙⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠁⠀⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣾⠇⠀⢸⡇⢰⣶⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢿⡄⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢰⡟⠀⠀⣼⡇⢸⡟⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⢀⣾⠃⠀⠀⣿⠇⣸⡇⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠈⣿⡄⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⣼⠇⠀⠀⢀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣷⡀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣠⡾⠋⠀⠀⠀⢸⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⢿⣄⠀⠀⠻⣦
⠀⠀⠀⠀⢀⣼⠏⠀⢀⣴⠟⠁⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠈⠻⣦⡀⠀⠹⣧⡀
⠀⠀⠀⢀⣾⣇⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⣸⣷⡀
⠀⠀⠀⣾⠋⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⠘⣿⡀
⠀⢠⡾⢿⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣿⢷⣄
⢰⡟⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢻
⣿⣱⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⠆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⡄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣈⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠉⠉⠙⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⢷⣿⣿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠃⠀⠀⣸⣿⠋⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠋⢁⣴⠶⠾⠛⠋⠀⠀⠀⠀⢸⣷
⠀⡀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠙⠻⠿⠛⠛⢻⣧⣴⣶⣶⣶⣶⣶⣶⣤⣿⠃⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠀⠁⠀⠀⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢿⣳⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣽⣲⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⡾⡽⢿⣷⣾⣿⡏⣹⢿⡿⣦⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⠃⠀⣈⢃⡛⢺⡧⠆⡥⠄⡿⠒⢡⠀⡀⠙⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢘⣄⠅⡟⢳⠈⠃⡞⢻⡂⠁⣄⠀⠀⠀⢱⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⡀⠀⠀⣴⠃⢃⡾⢁⣤⠙⢋⣽⠯⣿⠀⠈⣡⡀⠀⡄⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⢀⣨⣾⣵⢸⣏⠎⣿⠉⠉⡆⢀⣿⠄⠀⣠⣅⡀⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡠⠖⠚⠛⢿⠀⣨⠈⠀⢃⡂⣿⠸⢷⠿⠙⠉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣤⡄⠈⢋⠜⠀⡈⢀⡷⣄⡀⣀⣀⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡿⢛⣺⣿⣸⠀⢿⡜⠀⣿⡄⣙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠊⠇⢀⣺⣿⠙⠷⣊⠑⠀⢻⡏⠏⢻⡏⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⣿⠳⡄⠈⠈⠑⠲⡜⣿⡊⠈⢏⠀⠈⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⣼⠇⠀⠈⢑⣤⡀⡀⠀⣿⣷⢠⠈⠠⠀⡋⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢄⡜⢀⢀⣼⠟⡠⠖⢊⣿⣿⡏⠙⣦⠇⢻⣆⠃⠀⠙⣰⡅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠞⠀⡇⢀⡔⠙⠈⢡⠀⢸⠀⠈⢿⣔⠀⠀⠈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠋⠀⡀⡧⠈⠠⣄⢤⣴⡿⠆⠀⠀⠈⣿⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⡁⢉⠂⠅⡊⠐⠢⠤⢤⠼⣎⣩⡐⠠⣁⢘⣿⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡄⣹⠐⡪⢂⠍⠀⠸⡇⢸⠃⠠⣇⠊⢔⡀⡈⠈⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡜⣀⠄⡰⢓⠀⣘⠑⠈⣀⣐⠑⠡⢂⢄⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣥⢄⣤⣤⠄⡤⣤⠀⢠⣄⠤⡤⡼⣦⢤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠓⠒⠚⠓⠒⠚⠚⠒⠓⠓⠒⠚⠒⠒⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢢⢰⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣜⢊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠙⠗⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢰⣉⠠⠴⣤⡄⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡀⠀⠀⠀⣀⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠖⠉⠁⠀⣼⣿⠀⠀⣸⣿⠟⠈⠠⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠖⠻⠿⠷⠶⠿⠃⠀⠀⠀⣡⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠀⢠⡄⠀⠀⡀⢀⠀⠄⢀⣴⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠂⡾⢻⡀⠀⠀⠀⠀⡈⡟⢸⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠞⠀⣰⠁⠈⣗⠀⠀⠀⠀⢸⡇⠈⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠚⠁⢀⠴⠁⠀⠀⣿⢀⣀⠠⠄⠚⡇⢠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠤⠞⢁⡠⠒⠁⠀⠀⠀⠀⣏⠁⠐⠒⠐⠂⢹⠸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⠠⠘⠁⠀⠀⠀⠀⠀⠀⢠⡏⠄⠑⠀⠀⠀⢹⠀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⢀⠀⠀⠀⠸⡜⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠁⠀⠀⢀⡗⡦⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅⠀⠀⣀⣄⡈⠠⠄⣗⠸⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡸⠁⠀⢣⡀⠀⢸⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⡇⠀⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⠀⠀⠀⢱⠀⠀⢁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡞⠀⠀⠀⠀⠸⡄⠀⢸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠀⡼⠁⠀⠀⠀⠀⠀⣇⠀⣸⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠇⢰⠁⠀⠀⠀⠀⠀⠀⢹⡀⣸⠸⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⠃⠀⠀⣀⠔⠊⠀⠀⢎⠀⠀⠀⠀⣀⠤⠒⠁⠀⢸⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠒⠐⠒⠋⠀⠀⠐⠧⠴⠒⠒⠒⠒⠊⠀⠀⠁
⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣻⣿⣟⣻⣿⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢳⣈⣅⣧⣅⣉⣙⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡠⢞⠙⣿⣿⡿⣿⣿⡟⠉⠶⣈⠳⣄⠀⠀⠀⠀⠀⠀⣸⣅⡀
⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⡀⠀⠀⢀⡲⢁⣎⠑⡼⣥⣴⣤⣤⣤⣤⣤⡼⣤⠉⣦⠘⣆⠀⠀⠀⠀⣰⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⡇⡀⠊⠀⠀⠀⣬⠁⡞⢢⣿⣧⣿⣿⣿⣧⣿⣿⣧⣿⣼⣯⠐⢃⠸⡆⠀⠀⠀⣽⣿⣹⣿⠇
⠀⠀⠀⠀⣰⠿⠻⠇⢰⠀⡀⠀⠀⡇⢨⢁⣸⠿⠟⡟⢻⣿⡍⣹⣿⣛⡟⠿⢿⣓⣹⠀⠓⠀⠀⡿⣿⣻⢿⣹⠃
⠀⠀⠀⠘⠁⠀⠀⣠⣎⣄⠀⢠⠀⠁⠸⢤⣿⣤⡇⢧⣨⡝⣫⣿⣋⣄⡷⣃⣦⢿⢼⠀⠀⡄⠈⠉⣿⠟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢦⡼⢅⠀⠀⠀⣠⠄⣿⠇⣸⡟⠋⠁⠀⢘⠁⠀⠈⠙⢿⡃⢼⡏⢠⡂⠀⠀⠀⠧⠖⠈⠛⠣
⠀⠀⠀⠀⠀⠀⢁⠞⠁⠀⠀⠀⠂⠀⠈⢀⢸⡿⠃⠀⠀⠀⢸⡄⠂⠀⠀⠈⠛⣀⠈⠁
⠀⠀⠀⢀⡀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡄⠈⣿⠀⠰⢆⡉⠀⠈⠀⠈⢤⠖⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙
⠀⠀⠴⠊⠁⠀⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡠⠀⠀⠀⠀⠀⠀⣿⣾⣳⣾⣷⡤⣣
⠄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠛⠀⠈
⠀⠀⠀⣸⣿⣿⣿⠄⠀⠀⠀⠀⠀⠰⠧⣶⣾⣿⡇⢀⠀⠀⠀⠄⠀⠀⠂⢠⣿⣷⣶⠄
⠀⠀⠀⢸⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠹⡧⠻⣷⠀⠐⠂⠀⠅⠀⢀⠀⢾⡟⢸⠏
⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢧⡀⢶⣦⡄⠀⠀⠀⣼⠿⢂⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠐⠀⠀⠹⡄⢹⠃⠘⠀⠀⠉⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⣄⣠⣴⡖⣹⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠐⣦⣞⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣟⣫⡟⠁⠀⠀⠀⠀⠠⣞⣀⡈⣠⣤⠈⢧
⠀⠀⠀⠀⠀⢨⡃⠀⢻⣿⣿⣿⣷⣄⡜⠀⠀⠀⠀⠀⣿⣿⣯⡉⣿⠎⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢀⢸⠇⠀⠀⢶⡽⢿⣿⣿⣿⠃⠀⡆
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠈⠋⠀⠸⣼⡆⠀⠀⠀⠠⣄⠉⠉⠀⢀⠐⠁⠀⢺⣦⡀⠀⠀⠀⢠⣤⡶
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠐⠀⣺⠟⠃
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠘⢣⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠘⠀⠀⠀⠒⠉⣀⣀⣀⣤⠂⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣤⣶⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠠⠾⠿⢿⣿⡁⠀⠀⠀⠈⠓⠄⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡠⠞⠋⠀⣸⣿⡿⠛⠛⠂
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠹⣤⣦⣀⣈⣛⠃⠈⠛⠛⠳⢿⣦⣿⠳⠶⠛⢋⣤⣀⢠⣼⡿⠁
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠓⠖⠚⠀⠈⠃⠀⡀⣀⡀⠈⠀⠈⠳⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⢰⠃
⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠆⢠⣐⡏⠄
⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⠁⠀⠉⣀⠳⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁⠀⣄⣀⡀⠀⠈⠀⠈⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠁⠀⠀⠀⢀⠤⠴⠀⡀⠐⠒⠲⠶⠶⠶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠴⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢠⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣿⣽⠼⣾⣬⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡆⢂⢅⡛⢰⡶⣶⠙⣃⢅⡆⢢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡕⢄⣧⡲⠤⣓⠈⢓⡏⢐⡡⢟⣭⣔⠰⣝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⠳⣣⡴⠞⠚⠛⢛⡳⠶⣜⠳⢋⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⣥⣅⣴⠋⠀⠀⣀⠴⢋⡀⠀⠈⠳⡷⣤⡔⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⠛⣿⠁⠠⢔⣃⠀⠰⠀⢀⣑⠤⢀⢹⡛⠩⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡋⡗⠲⠶⠬⣅⢛⢀⡭⠴⠶⢺⡏⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡀⠰⠶⢘⡇⡏⠰⠆⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠓⠒⣋⠀⠈⠒⠒⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⣄⠀⠉⠒⠃⠀⢠⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠹⣄⠈⠛⠛⠛⠛⠛⠉⣀⠜⢀⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⢈⠒⠢⠭⢛⣛⣛⣛⣛⠫⠥⠒⠂⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⢲⡄⠓⢤⣀⠀⠀⠀⠀⠀⣀⡤⠚⠁⡴⣐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⣇⠰⡶⠤⠍⣁⣐⣈⡩⠤⢴⠖⢰⠇⡏⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠸⡄⠈⠲⠄⣀⣀⣀⡠⠔⠁⠀⡞⠀⡷⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣗⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡟⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡯⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⣀⣀⡴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⣀⣠⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣓⡙⠛⢛⣫⣍⣙⠛⠻⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠥⠄⠡⣉⠑⣾⣿⣿⣿⣷⣴⢃⣴⣧⠱⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣠⠒⡖⢀⠀⠈⠀⣿⣿⣾⣿⣿⠇⠀⠀⠒⠂⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⢀⢛⣊⡹⠾⣿⢁⠀⠀⠀⠈⠛⡛⠟⠋⠀⠀⠀⠀⠉⠲⡤⠤⣖⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟⠃⠀⠀⡰⠚⣻⡭⠍⡀⠋⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⣦⣼⡿⡃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡟⢿⡼⠁⡘⢠⡿⠀⠉⠀⠉⠑⠦⠀⠀⠐⠊⠉⠀⠉⠀⠀⣿⣥⣷⢾⢿⡭⣍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢠⠚⣻⠊⣀⠝⠃⢴⣤⣤⣤⣀⠀⠄⠰⠀⣀⣤⣤⣤⡦⠀⠙⣾⢤⣛⣲⠟⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠸⠟⠻⠈⠑⣀⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠸⡘⣧⢿⡡⢤⣤⣿
⠀⠀⠀⠀⠀⠀⠂⠄⠐⠀⡀⠀⠱⠔⢾⣿⡋⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡹⣜⣼⡓⠛⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠠⢀⢨⠋⠙⡶⠝⠀⠀⠀⠀⠀⠑⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⢳⢻⠟⢻⣇⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⠊⡙⢳⡾⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣿⣿⣾⡄⢠⡏⣁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⢻⣫⣼⣷⠝⣧⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠀⠂⠈⣧⣤⠛⠻⣎⢿⣘⠣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢼⡻⣍⡻⣦⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢿⢿⢠⣆⡹⢫⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡣⣆⠙⡟⣵⠓⢾⡟⠘⠃⠀⠀⣀⣀⡄⠀⠀⠁⢰⣤⣶⡠⣤⡋⠹⠻⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣽⠠⢼⣶⢝⢿⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠙⣧⡿⠋⠰⡎⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣄⠘⣿⣾⣿⡟⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡷⣀⡿⠿⢄⡸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⢽⡧⠌⣩⣽⡏⠀⠀⠀⠀⠀⠀⠀⠈⡁⢠⣩⢅⣳⣶⠺⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠄⢸⡽⢿⢧⣗⠀⣀⠀⠀⠀⣺⣾⢭⠎⣿⣾⣯⡭⠏⠁⠀⠀⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⡝⡫⠙⢀⠘⠛⠒⡶⢄⢀⠦⡅⢀⢩⣯⣠⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠡⢀⠊⢡⠬⣀⠋⢰⡛⢢⠼⠒⠒⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠁⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣷⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⢠⣷⠀⢸⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢳⣷⣿⢼⣿⣿⣿⣶⡏⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠋⠀⠀⠈⠙⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⠻⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡿⣁⠠⠄⠁⠤⢄⣘⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣤⡜⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⡟⣭⠀⠀⠀⠀⠠⡼⢻⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⠄⠀⠀⠀⠘⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⠷⠄⠀⠀⠀⠀⣿⣿⡀⠀⢠⣿⣿⣿⣿⣿⡇⠈⠀⠠⠀⠄⠀⠁⢸⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⠀⠀⠀⠰⠾⠟⠻⠇
⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⣴⣶⣶⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⡟⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣇⢸⡇⠀⣸⠀⠀⠀⠀⠀⢿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣠⣿⡿⠁⠀⠀⠀⣇⠀⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠉⠲⢿⣿⣷⣄⣿⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣠⣾⣿⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⣿⡿⠟⣩⡿⢿⣽⠿⣯⡙⢿⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⣠⡿⠋⠁⠀⠀⢠⣶⣏⢀⣼⠋⣠⡿⠙⢢⡈⢳⣄⠙⣷⡄⠀⠀⠈⠙⢿⣧⡀⠀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣦⣼⣿⠁⠀⠀⠀⣴⣿⠃⣸⠟⢧⡀⠸⣗⣠⠝⢀⣠⠽⢿⣏⢻⣦⠀⠀⠀⠈⣿⣿⣦⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⣼⣿⠃⣼⠏⠀⠀⠙⢦⣸⣯⡴⠋⠀⠀⠀⠹⣦⢻⣷⠀⠀⠀⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⡇⣸⣏⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⣻⣇⣿⡇⠀⠀⢸⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⠟⢦⣀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠖⢿⣼⣿⣧⣴⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⠓⢦⡀⢸⡧⠀⣠⠞⠋⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣿⣿⣿⠙⣿⣿⣶⣤⣴⡇⠀⠙⢾⣷⠞⠀⠀⣷⣤⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠙⣿⣿⣿⣿⣁⣴⠶⠞⠙⠶⣷⣄⣹⣿⣿⣿⡟⠁⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⣿⣦⠤⠠⢦⣼⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⠩⣽⣿⣿⣿⣿⣿⠇⠀⠀⠀⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⠏⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣯⠉⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠘⠁
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠻⣿⣿⡿⠛⠙⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⠉⣯⣭⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⣌⠄⣀⣥⣡⣸⣦⡀⢠⣧⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡞⣿⠀⠸⡏⣻⢻⠹⢇⢈⡿⢏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣜⡻⠇⠈⣼⣭⣿⣧⣼⣿⣼⣯⠀⠻⡞⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⢠⣧⣠⣴⠿⠟⠭⢻⠋⠍⠻⢿⢦⣀⣽⢁⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣨⣿⡟⡁⢉⠀⠀⠀⠸⠄⠀⠀⠀⠁⢈⢻⣿⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣾⡟⢿⣿⠟⣩⠁⠀⠀⠐⣰⣄⠂⠀⠀⢈⡝⠻⣿⡿⢻⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣽⣿⠞⢉⠋⠎⠀⠄⠉⡛⢩⡏⢩⡝⢛⠉⠄⠀⠠⠙⡉⡹⣻⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣯⡿⢏⢻⡀⡠⠀⢀⡠⠄⠋⠀⠀⠑⠀⢄⠀⠀⣀⢀⡟⡽⢿⣽⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣙⡶⢻⠿⢵⡄⠁⣀⣒⠤⠀⠠⢤⣂⣀⠈⢠⡜⠿⡿⢶⣋⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢌⠢⣧⠟⠋⠁⢠⢠⠟⠀⠀⠙⠋⠁⠀⠀⠘⠛⠋⠀⡀⢻⣤⠀⠈⠙⠷⢿⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⡀⠀⠀⢾⡙⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠘⣣⡿⠀⠀⢠⡾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡿⡦⡤⠸⢿⣦⣤⣥⡀⠀⠀⠀⠀⠀⠀⢀⣮⣤⣴⠿⠧⠤⡴⢿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠱⡔⡶⠆⠊⠀⢠⣾⠻⠦⠀⠀⠀⠀⠴⠏⣷⡄⠈⠃⠠⡶⣠⠏⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠂⣄⠀⠙⡡⢃⡘⠀⢉⠙⠀⠂⠑⢌⠛⠈⣀⠰⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠑⢦⣀⣬⡾⢧⡄⢀⢼⡆⠀⢠⡼⢶⣄⣀⠷⠚⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⢿⣛⣛⣛⣛⡿⠿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⡀⠀⠀⠀⠀⠐⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣾⣿⣭⣾⣿⣷⣶⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⣀⡀
⠀⠀⠀⠀⠀⡀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⢀⡀⠄⣀⠀⣀⠠⣀⡀⠬⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣗⣺⣿⣿⣦⡀
⠀⡀⠀⢘⠠⠤⡀⠤⡀⠤⠄⠃⠀⡀⠀⠀⠀⠀⠀⠨⠭⠁⠀⠀⡇⠀⢼⠀⠈⠈⠭⠄⠀⠀⠀⠀⠀⢀⣴⣿⣥⣟⣟⢯⡽⣿⣻⣧⠿⣦⡀
⠀⠂⠐⢏⠈⠢⢆⣀⠴⠌⠈⡹⠀⠆⠀⠀⠀⠀⠀⣉⣀⠁⠀⠀⠷⠀⠿⠀⠀⠈⣀⣉⠀⠀⠀⠀⠀⢻⣿⣇⣎⢾⣏⣧⣼⣽⢿⣺⣸⣾⣿
⠀⠴⣶⠳⢤⣄⠺⣿⠇⣠⡤⠷⣶⡇⠀⠀⠀⠀⣠⠓⢦⣈⠅⢶⠀⣶⠀⡶⠨⣀⡴⠺⣄⠀⠀⠀⠀⠸⠿⡀⢙⠈⠉⠉⠉⠉⠉⣉⠀⡿⠏
⠀⠘⠿⠧⠀⠨⠄⠋⠰⠄⠀⡟⠟⠁⠀⠀⠀⠀⡏⠀⢠⠷⠂⠉⣳⣀⣞⠁⠐⢾⡄⠀⣹⠀⠀⠀⠀⡟⠙⡇⢢⡤⠤⡀⢀⡤⠦⡔⢸⡋⢹
⠀⢈⡨⢴⡄⠀⠄⣀⠀⠀⣸⡄⣀⠀⠀⠀⠀⠀⠙⢪⣏⠀⣀⢀⢠⠀⡄⡀⡀⠀⣿⡔⠋⠀⠀⠀⠀⠹⢌⢹⡄⠹⡄⠀⠀⢀⠎⢠⡟⢡⠞
⠀⠀⢀⠐⠁⣄⠀⠀⠀⡠⠁⠂⠈⠀⠀⠀⠀⠀⠀⠘⡹⣿⠰⢘⡀⠀⢐⡃⠄⣿⢟⠉⠀⠀⠀⠀⠀⠀⠀⠉⣍⠀⠀⡐⢐⠀⠀⢨⠉
⠀⠘⡂⠙⠚⡘⠏⣽⠝⢐⠓⠋⡘⠁⠀⠀⠀⠀⠀⠈⠹⡅⢀⣭⠤⠀⠤⣭⡀⢨⠏⠁⠀⠀⠀⠀⠀⢀⠐⠀⢋⢢⡀⢨⢅⢀⡴⡛⠂⠂⡀
⠀⠀⢡⠖⠀⠈⠇⠀⠞⠀⠐⢦⡅⠀⠀⠀⠀⠀⠀⡠⠀⡇⢸⣬⠷⣒⢾⣤⡇⢸⠀⣄⠀⠀⠀⠀⠀⠀⢀⠰⠦⠉⠀⠶⠲⠄⠉⠱⡆⡀
⠀⠀⠀⠀⠀⠀⢴⣀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡑⠚⠮⠀⢋⠀⠵⠓⢋⠃⠘⡆⠀⠀⠀⠀⠀⡣⠄⡰⠠⣾⡏⢙⣷⠄⢆⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠁⠋⠑⠿⢳⡶⠿⠇⠙⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢠⣦⠀⠀⡀⡀
⢀⢀⡰⠄⠀⢀⠀⢸⠆⠁⠀⠀⠱⢀⢀⡀⠀⠀⠀⠀⠀⢠⠚⠃⣤⣲⣿⣦⡄⠛⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⣥
⣴⢨⠓⣀⠀⡀⠋⠘⠋⠋⡀⠀⣁⠫⡐⡇⠀⠀⠀⠀⠀⢆⣿⢐⡒⣰⣴⡂⢒⢸⣏⡆⠀⠀⠀⠀⠀⠀⠀⠾⡆⢹⠼⠀⠒⠸⢧⡃⠰⠂
⣰⠁⠀⠁⠐⠘⠃⠀⠐⠚⠁⠂⠈⡀⠙⡄⠀⠀⠀⠀⠀⢁⠉⠒⢗⢴⣠⢶⠗⠊⠁⡁⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠀⠈⠀⠀⠀⠑
⡟⠳⣔⣄⣠⠀⠲⡟⢷⠆⢄⣄⡴⣠⠚⠃⠀⠀⠀⠀⠀⣼⢦⣄⠀⢠⢥⠀⠀⡠⣿⡆⠀⠀⠀⠀⠀⣠⣰⠁⠀⠀⠠⡋⠯⣻⡄⠀⡀⠨⡄
⠀⢂⣿⡮⠁⢠⣨⠀⢠⣣⠄⠉⢿⣗⠀⠀⠀⠀⠀⠀⠰⢳⣅⠈⢷⠈⠊⢠⡟⢀⣼⠻⠀⠀⠀⠀⠀⠀⣅⠃⠤⣄⢀⠉⠖⢁⣀⣠⠅⠀⡁
⠀⡭⠝⣄⠀⠉⠉⠀⠀⠉⠉⠀⡨⠪⠥⠀⠀⠀⠀⠀⠀⢶⣌⣡⠌⡲⠐⡂⢥⣉⣴⠦⠀⠀⠀⠀⠀⢀⠘⣀⢘⡭⠵⠆⠀⠶⠯⢝⠂⡄⢃
⠀⠶⡆⠸⠀⡈⣶⣦⣴⣞⡀⠀⠇⣰⠆⠀⠀⠀⠀⠀⠀⣿⠈⢘⣦⣅⣀⣥⣞⠈⢈⡅⠀⠀⠀⠀⠀⠈⠙⠃⡆⢀⡴⢒⠶⠚⢤⠀⣄⠹⠋
⢰⢷⢹⡢⣌⣴⣖⣚⢓⣰⣌⡤⢞⣧⠾⡆⠀⠀⠀⠀⠀⣼⡅⣹⣯⣙⣖⣫⣽⡁⣼⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠⠘⠓⣷⠚⠣⡤⢀
⠈⠛⠚⠻⢗⣛⢒⣺⡗⡖⣛⡻⠟⠻⠛⠁⠀⠀⠀⠀⠀⢀⣧⣏⠙⠉⠟⠙⠋⣧⢇⠀⠀⠀⠀⠀⠀⠀⠀⣌⢳⣀⡀⢀⣶⡠⢀⣰⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢦⣼⣏⢛⣻⢤⡼⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⡿⣷⠾⡁⠉⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣼⣁⡨⣋⣛⣼⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⣾⣍⢷⣿⣿⣿⣷⣝⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⣴⣿⣱⣿⣦⡿⣤⣤⣌⣿⠹⡢⢽⣿⣶⣿⣦⠀⠀⠀⣤
⠀⠀⠀⠀⠘⢠⠘⢳⠄⡂⠀⢠⣯⣙⣿⡉⣿⡗⠛⠻⣿⠛⠣⢽⡞⠛⣏⠙⡻⠇⠰⢾⡆⢳⡄
⠀⠀⠀⠀⠀⠟⢠⠟⠁⠀⠠⣌⠉⣍⡈⠉⢿⡏⠶⠆⡴⠶⢠⣿⢡⠉⢀⡦⠁⣠⠄⠈⠣⡈⠃
⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠻⠀⠘⡂⣸⡞⠿⡄⢲⡗⢀⡼⠿⣼⡀⢘⡃⢈⠁⠀⠀⠀⠘⡣⢤⣄⣠
⠀⠀⢐⠂⠈⠀⠀⠀⠀⠀⢀⣴⣟⣴⣩⣝⠧⠔⣏⣀⣀⣹⣧⣠⡯⣔⢣⠘⠳⣄⠀⠀⠀⠀⠘⢾⣿⠁
⠢⢼⡀⡄⢀⡀⠀⠀⢀⢴⣯⠋⡜⢲⣿⠏⣀⣼⡄⠩⠍⢀⣿⢳⢰⣿⢄⡁⠑⠈⣃⠀⠀⢀⣈⠀⣙⠒
⠀⢠⠋⡁⢈⣧⠀⠀⠸⣬⠿⢤⣍⣿⡿⣠⡝⣿⡟⠒⠖⠋⣿⢈⠸⣿⡦⡤⠀⢊⡀⠳⠀⡸⠽⢾⡏
⠀⠀⠀⠹⠅⠈⢁⣦⠶⠿⠻⢾⡧⢾⣧⡛⠁⢻⡅⣄⠀⢰⡟⢈⢾⣿⡇⣥⣠⠤⣿⣦⣴⣄⡀⣰⠁
⠀⠀⠀⠀⠀⠀⣾⠂⢀⡴⠷⠾⠟⠊⣧⡆⠰⢊⡕⠈⠀⢸⠁⡜⢸⣿⠂⠷⢲⣿⠿⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠉⠲⠃⠀⠀⠀⠀⣀⢥⢵⢂⢹⡇⢀⠐⡈⠙⢧⢹⠉⠀⠀⠈⠀⠀⠉⢱⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢳⣿⣿⡬⣀⣙⢸⢸⣇⠀⣤⢱⣰⡞⠚⣶⡆⢤⣤⣤⣤⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡼⣿⠹⡗⢻⣼⣿⠀⣽⠀⣿⡐⡏⢃⠀⣿⡧⢴⡁⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⢢⡄⣿⠐⣋⣻⣿⡟⠳⣯⡀⡼⠿⢃⠄⠁⣿⣷⠰⡅⢿⣿⡚⡿⡄
⠀⠀⠀⠀⠀⠀⢀⠞⢀⠢⡈⠏⢠⢄⣿⣿⡗⢄⠈⠛⢦⡀⠀⠀⠀⢻⣿⠂⠂⠘⣿⣇⠈⠙⢆
⠀⠀⠀⠀⢀⠤⠋⠠⠠⠁⡐⠀⠴⢸⣿⡟⣮⡀⠀⠀⢀⠉⠘⠦⣴⠸⣿⣗⠚⡀⠹⡟⠂⠀⠀⢹⣦⣄⡀
⠀⢀⡠⢆⠡⠀⠀⠁⠀⢐⠂⢀⢠⣿⡞⠁⠨⠻⢆⡀⠀⠈⠀⠈⢌⠂⣼⣿⣗⣣⠀⠑⡄⠀⠀⢐⠧⣿⠉
⠘⠀⠱⠈⢦⠀⠀⠀⡠⠀⢀⢀⣴⣿⠁⡀⠀⠀⠀⣹⡖⣤⣀⠀⠀⠀⣺⢿⣿⢀⣥⠀⠈⢢⡀⡞⠀⡿
⠀⠁⡀⠙⢄⠳⣤⠍⠀⠀⢁⣼⣿⢿⠐⡀⣤⠶⡁⣼⡟⣿⠉⣹⢦⡄⡿⠘⣿⣦⠤⢆⠀⠀⠙⠋⢠⠏
⠀⠀⠈⠢⡈⠑⠎⠀⠠⢀⡴⣿⠋⢸⡏⠝⠠⠐⣼⣿⡇⡻⢂⠀⢊⢹⡇⠀⢸⣿⡗⠚⡄⠀⠀⠈⠿⠅⡼
⠀⠀⠀⠀⠈⠒⠀⠐⢠⡾⢿⠏⠀⢸⡂⠀⢀⡜⠃⢸⠀⠁⠷⣀⠀⣺⠇⠀⠀⢹⣿⣗⡑⢄⠀⠀⠐⡴⠁
⠀⠀⠀⠀⠀⠀⠈⣰⡺⠟⠃⠀⠀⠸⣋⡠⠋⢀⣀⢀⢀⣀⣠⠵⣂⣿⠀⠀⠀⠀⢷⣿⡦⠘⡀
⠀⠀⠀⠀⠠⢀⣼⢩⡝⢀⡀⠀⣤⠘⣯⡁⠐⠠⠬⠿⣭⡯⢿⠞⣩⣩⡄⠀⠤⢀⠀⢻⣿⣖⣉⢆
⠀⠀⢀⠀⢠⣾⡿⠊⡅⢎⣁⠛⢤⠌⡃⠈⠙⠚⢦⠶⠶⡶⢾⣭⡀⣄⣈⠛⠴⢄⡠⢈⢿⡿⠦⠄⢃
⠀⠀⠂⠄⣸⣿⣁⠂⢽⠀⣥⡚⠠⠌⣙⠉⠉⠀⣾⠀⢸⠿⣀⡀⠹⡇⣡⠛⢰⢌⠁⡼⠀⠛⠟⡓⠋
⠀⠀⠀⠀⠈⣹⠇⡁⣹⠀⡧⠌⡁⠦⣩⠄⠐⠊⣽⡀⢸⡅⠀⠈⢻⠶⢁⡘⠢⢌⠀⢠⠀⠁⢺⡇
⠀⠀⠀⠀⠀⠘⠋⠀⡾⠀⠃⠄⣉⠶⠘⠒⠀⠈⢁⡟⣅⠋⠀⠈⡛⠰⠆⡌⠁⠦⡄⠀⡀⣀⠚
⠀⠀⠀⠀⠀⠀⠀⡹⠃⠚⠃⠀⠈⠀⠈⠁⠀⠘⡆⠘⠁⠀⠈⠈⠉⠀⠐⢦⡙⠂⠈⠘⠄
⠀⠀⠀⠀⠀⠀⠚⢻⠟⡗⠊⡤⠶⣶⠀⢷⣿⢺⠇⢤⠀⢭⡖⣶⢲⣿⢾⣷⠿⠒⠶⡶⢻⠦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢤⣶⠶⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣍⣩⣴⣿⡿⢿⣜⣯⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡶⠛⢉⣁⣄⠉⠑⠁⠈⣉⡆⠖⠋⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⢠⠖⠉⠀⠀⠀⠀⠀⠙⢍⢻⣟⠳⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣟⣀⣀⠀⠀⢀⣀⣀⠀⢸⣈⣧⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣯⠴⠌⠁⠀⠈⡡⠶⠅⣾⠉⠛⡇⢼⠧⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠐⠀⠢⢄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⢻⡟⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡇⠀⠀⢷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣓⡀⣾⣧⣙⠿⣿⡀⠈⠉⠁⠀⠀⣠⡴⢟⡵⠚⠁⠀⣠⣿⣧⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠲⣄⡨⣿⠘⢸⠋⣿⣧⢸⣿⣤⠀⠀⣰⠎⠱⠞⢁⡔⢫⣶⣿⡿⠃⠀⡉⠉⠉
⠀⠀⠀⠀⠀⠠⡀⠙⢈⣍⢠⠎⢠⣽⣷⣾⡿⣿⠀⠀⢻⣀⢾⠀⠼⡦⢼⡯⠿⠿⠟⠍⣶⣦⡀⡄
⠀⠀⠀⠀⣰⣶⡿⢿⠶⢿⢋⣠⣾⡿⡿⠁⢀⡝⠄⠀⠀⠠⠀⠙⠲⠶⠛⠓⠙⠁⠲⣦⡄⢸⡇⡿⢧
⠀⢀⠔⢋⣭⣶⣶⣞⣶⠽⠟⢋⡉⠠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠌⡏⣸⣿⣧⣴⡆
⠀⡜⠀⣿⠛⡹⡏⠈⠁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣾⣷⣿⣏⠛⢻⣗⡦⡀
⠀⠸⣦⡉⢂⡈⠀⠐⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠘⠻⠋⠀⠈⢯⡘⠂
⠀⠀⠻⣿⢿⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣔⡈⢉⠂⠁⣨⡿
⠀⠀⠀⠙⠈⢻⣴⡜⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠏⢠⣶⡟
⡔⠉⠁⠀⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠁⠀⣾⢿⣄
⠹⣄⣀⡠⣀⣾⠃⠀⠀⠀⠀⢀⠜⠉⠉⠁⠀⠀⠈⠑⣈⡀⠀⠀⠀⣴⡇⠀⠀⠀⢀⣟⠘⢦⣌⠛⢾⢿⣶⠐⠂
⠀⠈⠙⠛⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠙⠀⠀⠀⠀⣸⣿⠀⠀⡹⣷⠀⠆⠁
⠀⠀⢀⣴⠋⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⢐⠜⠃
⠀⠀⠹⡇⠀⠀⠀⢀⠀⠉⠃⠄⠀⠀⠀⠐⠆⠀⠀⠀⢤⣀⠀⠀⡀⠀⠀⠀⠀⢸⡟⠀⡕⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠈⠀⠀⠀⠀⢀⡜⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠓⠤⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠏⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣇⠚⣋⢐⠃⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠃⠨⣿⢻⡟⣿⡇⠈⡇⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⡀⠛⣷⠀⣩⠟⠻⣍⢀⣼⠟⡀⣁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣄⣀⣀⠀⠤⢿⡾⠦⢤⣠⠉⢲⢾⣿⣿⣧⣙⡷⣧⢆⡕⣬⢳⣿⣰⣷⣿⣿⠶⠉⠄⣠⢤⠦⣮⠤⠄⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠘⠿⠝⢒⡖⢆⣳⣤⡀⠻⣦⡘⣿⣿⣿⡿⠟⢛⣯⣿⣻⣿⡛⠿⢿⣿⣿⣿⠟⣠⡾⠁⣤⣴⣀⢲⡔⡪⠽⠋
⠀⠀⠀⠀⠀⠀⠀⡈⠶⠱⣾⢿⣟⠥⠤⣾⡳⢿⣷⠋⢀⡼⣻⠟⠋⠙⣿⣎⠑⢶⡙⢻⣿⠶⣿⠦⠬⣟⣽⢯⡆⠸⢀⡀
⠀⠀⠀⠀⠀⠀⠐⣿⢇⠙⠿⠿⢳⠐⡂⢯⣽⣿⡇⢠⣿⡞⣁⠤⠒⠚⠉⢣⣢⠈⢻⡈⣿⣿⣭⢁⡂⡼⠻⠿⠛⣑⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⢤⣬⣷⢻⣾⣿⣿⣇⣿⡸⢼⠃⠀⠀⠀⠀⠀⠙⢷⡈⢷⣼⣿⣿⣼⣳⣧⣤⢶⣖⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⣠⣰⣶⣴⡿⣿⠃⣿⢿⡇⢳⣿⣀⡀⠀⠀⠀⣀⣤⣄⢑⣘⣿⣿⣿⢿⣷⣧⣦⣄⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣴⠟⠉⣠⣷⣤⣄⠉⠀⠀⠈⢁⣤⣤⣄⠙⢿⠻⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠙⠿⠿⢿⣾⣷⣇⠠⣾⠋⠙⠛⠛⠃⠀⠀⠀⠓⠚⠛⠁⢀⢰⠃⠂⠹⣿⠿⠿⠋⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣡⠴⠷⠶⢻⡿⣽⣿⡃⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⠀⠀⠀⣿⡟⠷⠾⠦⣄⡁
⠀⠀⠀⠀⠀⠀⢀⣾⣯⣠⣶⣤⠴⠁⡙⣿⡁⠁⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠁⠀⡘⣰⠇⢳⣴⣶⣄⢩⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⢤⠐⢿⡾⣳⡒⠦⢿⣇⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡇⢀⠌⣴⠷⠒⣞⢳⡿⢇⣠⠌
⠀⠀⠀⠀⠀⢀⣴⣪⡝⢦⠉⡹⠛⠀⣴⡟⣻⠃⡦⣾⢹⡆⠀⠀⠀⠀⠀⠀⠀⢸⣴⡇⡈⣼⠛⣦⡀⠛⠿⠉⣤⠅⢽⣲⣄
⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⢾⠷⠶⠺⠙⣀⣿⣰⢧⠏⣼⠇⠀⠀⠀⠀⠀⠀⠂⢻⢻⣷⡗⣿⡉⠈⠱⠶⡾⠷⠆⠀⠈⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠛⣩⣱⡏⡞⢠⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⢺⡋⣽⣏⢷⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠁⣰⠟⡿⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⢻⣌⢎⠐⣦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠃⠀⠅⡎⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠃⡎⢰⡟⠾⣟⡜⠂⢹⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠰⢠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⣣⠏⠀⢀⡇⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠁⠀⠆⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢀⡚⠀⡴⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⢰⡖⡠⢊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠗⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣤⣠⣤⣠⣤⣤⣤⣄⣀⣄⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⡶⠴⠀⠀⠤⢦⡜⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠊⣁⣘⡠⣀⡤⠄⣒⢄⣃⣈⠓⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡼⣛⣭⠥⢴⢰⠁⠀⠀⢘⡎⡤⠬⣽⣻⢧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡻⡯⢁⢸⣸⠾⡸⣆⠀⢀⣰⢧⠳⣇⡇⡈⢽⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⣿⡤⠿⠒⠻⠮⢟⡯⠵⠛⠒⠯⢤⣿⠞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣧⠃⢉⣠⠴⠚⠈⠐⠀⠀⠀⠀⠂⠁⠑⠶⣄⡉⢘⣼⣀⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠉⢀⣀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠩⠈⠪⢳⣸⢁⡤⠶⠶⣶⣦⠀⠀⠀⠀⣴⣶⠶⠶⢤⡈⣇⡞⠕⠑⠉⠢⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⢀⠔⠢⠅⣲⡗⣁⣴⣶⣴⣢⢤⠀⠀⠀⠀⡤⣔⣶⣶⣦⣌⢺⣖⠌⠔⠢⡀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⣀⠤⢸⠗⠟⡛⢿⣭⣍⢿⡇⠀⠀⢸⡿⣹⣯⡽⢛⠻⠺⡇⠤⣀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⠙⠶⠶⢟⠞⠁⠀⠀⠀⠳⡻⠶⠶⠋⠀⢰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⢷⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⢄⠀⢀⣷⡄⠀⠀⢇⡰⠀⠀⠀⠀⢆⡰⠀⠀⢠⣾⡀⠀⡠⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⡲⡄⠈⠁⠀⠀⠀⠀⠈⠀⠢⢖⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠲⣉⠜⡻⢢⡄⠀⠀⠀⠀⢠⡜⣟⠣⣀⠖⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣍⠛⠥⣧⣴⡀⠀⠀⠀⠀⠙⢬⠛⢩⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣠⣹⣄⠀⠀⠀⠀⠀⠉⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⠤⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⡀⣀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣶⣨⢫⡛⣉⡤⠦⣍⡙⡿⣁⣴⣦⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⣿⣿⠛⢋⣴⣠⣟⠛⢿⣿⠗⠛⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠖⠻⠃⠀⢘⣿⣫⡟⠀⠘⠛⠷⢿⢠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣯⠻⡄⠀⠀⣶⣟⢻⣿⡄⠐⠀⣚⣼⣿⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠛⣽⠋⢠⣾⣴⣿⡝⢏⣿⣧⣾⣄⠉⡾⡻⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣠⣴⣉⠾⠟⣻⢿⣻⣿⡻⣿⠋⠼⣃⣶⣁⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣈⡦⠀⠄
⢀⠆⠁⢦⡀⠀⢠⣶⠀⠀⠀⠀⢤⣒⣂⣀⣀⠻⣃⣸⣏⡴⢾⣿⣸⠁⠀⣹⢾⣿⠶⣬⣏⣀⠿⠃⣀⣐⣒⣢⡄⠀⠀⠀⠀⠘⣧⣀⣶⠉⠀⡠⠆
⡇⠀⠁⠀⠙⠂⡻⢿⡀⢸⣳⣠⠟⠉⠁⠹⣟⢳⣿⣿⣿⠻⡄⠻⣮⣳⣶⣩⡾⢁⡾⢛⣿⣿⣶⠻⡿⠀⠉⠙⢦⣀⡀⠀⠀⣀⠀⢻⣿⣦⠂⡀⠂
⠡⠀⡄⠀⠀⢨⣹⡞⠁⠠⣿⡏⠀⠐⠃⢠⣅⢘⢙⣷⣬⢳⡈⢀⠀⢀⡾⠀⡄⠈⣴⣋⣵⡟⢃⢐⣅⠀⠡⠂⠘⣿⣧⠒⠰⣥⢾⠀⠈⠛⠉⠉⠃
⠀⠛⠀⠀⠀⢸⠏⠉⠀⠀⠙⢧⠀⠀⠀⣿⣿⣏⡷⣽⣹⡷⠄⢰⡆⠀⠀⠠⣟⠀⠺⣯⣫⡟⣹⣿⣿⠇⠠⡀⢠⠟⠀⠀⢠⠿⠊⡠⢀⣀
⠀⠀⠀⠀⢠⡀⠌⠀⠀⠀⠀⠘⢦⡀⠀⠈⠛⠋⣰⠈⠁⠀⠀⠉⠁⠀⠀⡀⠉⠀⠀⠀⠁⢴⠉⠛⠋⠀⣠⣡⠇⠀⠀⠀⠀⠀⠘⠑⢚⡇⠂
⠀⠀⠀⡣⢀⣿⡄⠀⠀⠀⠀⠀⠀⢠⡀⠠⠄⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⣴⡆⠀⡰⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠹
⠀⠀⠀⢀⠀⡙⠀⠀⢀⠀⠀⠀⠀⢠⡷⠂⠀⠀⠓⣇⠐⠀⢠⠁⠀⠀⠀⠀⠀⠀⣤⠆⢰⡓⠁⢰⣄⠘⠃⠀⠀⠀⠀⡸⠋⠀⠀⠀⣀⠆
⠀⠀⠀⠘⣀⠁⠀⠀⠹⢦⠀⢀⠀⠐⠁⢀⠀⡀⣰⡿⢓⢀⡶⠀⠀⡀⢀⠀⠀⣰⠆⢴⠻⣇⠀⣴⡇⠈⢲⣤⣀⡀⢰⣿⣦⡀⢠⡾⠁⠀⠁
⠀⠀⠀⠀⢃⣴⡀⠀⢠⡾⢦⡘⣶⠀⡀⣣⠀⠄⠋⠠⠄⢼⡇⠀⠀⠀⠀⠀⣴⠋⠐⠋⣀⠉⠛⠁⣟⠀⠘⢿⣿⡆⠀⠙⢋⣽⣟
⠀⠀⠀⢰⣿⡏⢻⣞⠛⢉⠂⠉⠻⡗⠉⠀⠀⠀⢇⠀⠀⢼⡇⠀⠀⠀⠀⣼⠃⡔⠉⢉⠉⠙⢦⠀⢨⡁⡀⠘⠿⣿⡀⠠⠦⠀⠈⢀⡂⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⣦⡀⢀⠀⠀⠀⢀⡠⠎⠰⣇⢾⡇⠀⠀⠀⠀⢿⠀⠀⠀⠈⢦⠀⢸⡇⢸⡇⠀⣠⣼⡏⠁⢶⡀⠞⢠⣄⠐⣈⢱⣶
⠀⠀⠀⠀⠀⠀⠀⠨⡆⠀⠆⠹⣿⡀⠀⠘⠋⠀⠀⣰⣷⡾⢿⣆⠀⠀⠀⠀⠑⠂⠀⠀⠼⠀⣼⡁⣸⡇⠀⣿⠀⠉⠈⠉⠓⠂⠀⠉⠉⠉⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢳⣿⡀⠀⡀⢀⣤⣾⣿⣌⠳⡈⠛⢦⢀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣻⣿⣴⠂⠀⠁⠀⠀⠁⠀⠀⢠⣄⠀⠁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠸⠋⠾⠳⡶⢿⠇⠈⢿⣽⣧⡘⣤⡀⠁⠘⠃⠐⠒⠚⢉⡴⢉⣶⣯⡿⠉⠠⣾⣠⡼⢶⠂⠐⠂⠀⠰⠂⠠
⠀⠀⠀⠀⠀⠀⣠⠤⠚⢡⠀⠀⠀⠈⠓⢀⡟⠀⠀⠈⢻⣿⣿⡟⢿⣷⣄⣤⣄⣴⣿⠟⣱⢿⣿⠟⠀⠀⠀⠹⡇⢳⣄⠀⢀⣠⠈
⠀⠀⠀⠀⣰⠋⠈⠀⣘⠂⠠⣤⣀⣀⡤⠊⠀⠀⠀⠀⠀⠹⣿⣷⡤⠻⣿⡏⣩⣿⠻⢴⣽⣿⠁⠀⠀⠀⠀⠀⣇⠀⠙⠛⠟⠃
⠀⠀⠀⡘⣅⠐⠤⠜⠁⠀⠀⠈⠹⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⡟⣡⠋⢱⡌⣳⠂⣹⠁⠀⠀⠀⠀⠀⠀⣿⣷⣶⡿
⠀⠀⢠⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⡷⡀⠀⠀⠠⢿⡛⠉⠀⠀⠀⠀⠀⠀⠀⣼⡇⠑⠒⠒⠒⠂⣀⣀
⠀⠀⠀⢘⣧⠀⠀⠀⠀⠀⠀⠀⠠⢜⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠻⢦⡴⠗⡎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠿⢀⣤⡶⠮⠙⠃⠉⠙⠷⣦⣀
⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣄⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢋⠔⠂⠀⠀⠀⠀⠀⠘⡎⢶
⠀⠀⢡⣼⠀⣆⠀⠀⠀⠀⠀⢰⡀⠀⠈⢺⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠁⢀⣠⠔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠀⢻⡄⡄⠀⠀⢂⠀⠂⠀⠀⠈⢿⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀⣤⣾⣭⣶⡿⠛⠀⠶⠒⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⣿⡄⠀⠀⠰⡀⠀⠈⢣⡀⠑⠀⠀⠈⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣢⣤⣲⣬⡴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠉⣿⠀⠀⢢⠹⣦⣄⡀⠙⠲⢄⣀⣀⣤⣤⣶⣿⣿⣿⣶⡶⣶⡞⢃⣩⠤⠈⠉⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡴⢎⠳
⠀⠀⠀⡇⢸⣇⠀⠀⠣⡈⠉⠭⣻⣿⢟⡽⢋⠝⣡⡴⣶⣟⠕⣋⣴⢿⡟⢫⠁⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⠐⠢⠤⠤⠤⠤⠄⠒⠎⠋⢀⠌⠁
⠀⠀⠀⠹⣦⣿⣆⠀⢀⡈⠛⠚⠉⠁⢀⡔⢡⣞⠏⡰⠿⠿⢞⢩⡏⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠙⣠⠴⠉
⠀⠀⠀⠀⠈⠙⢿⠷⣀⠙⠛⢲⣶⠾⠁⢠⡿⠋⣰⡃⣤⠀⠂⠀⢷⠀⣦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂⠀⠒⠴⣶⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠢⣄⡘⣋⢉⡁⠤⣸⡿⢿⣾⣇⡟⠉⣡⠀⠀⠈⠶⣌⣧⠀⠑⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣦⣤⣴⣻⡟⢻⡾⢟⠟⣥⠊⠈⠃⠀⠀⡠⠉⠙⠷⠶⠾⠿⠿⠶⢶⣄⠐⠤⢤⡀⣰⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣧⠾⢋⣵⠇⠀⣿⠃⢄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⡟⣁⣠⠼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣠⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⠘⢀⠃

⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠂⠈⠀⠀⠈⠀⡁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠖⠀⢠⢠⠄⠀⠀⠀⠀⠠⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⢤⣸⠅⣤⠃⠀⠀⢄⠀⠀⡐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠡⠀⠘⠰⠆⢤⢰⡆⣁⠀⡄⢀⠀⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠘⡤⡓⠀⢰⣎⠇⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠦⠀⠣⠞⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⢀⠦⠠⠄⠚⢂⠾⢶⡉⠁⠀⠂⠢⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠓⠒⢦⠸⡎⢠⠈⣧⢠⠖⠉⠁⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠠⠀⡂⠀⠀⠀⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡘⠀⠀⠀⠀⠀⠀⠀⠈⢃⡀⢀⡀⢠⡴⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡀⠀⠀⢀⠀⣤⠘⠛⠀⠁⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⡀⢞⠁⠀⠀⠰⠁⠈⠈⠀⠅⢲⡀⠀⠀⠈⠳⠀⠀⢝⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⣀⡀⣤⣄⢸⡄⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇⢘⠁⠈⠁⡏⢸⠁⡄⠐⠀⠀⠀⠠⣄⡀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⠇⠻⢄⠀⠀⢿⣠⡄⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢉⡃⠈⢣⠀⠀⣿⡁⡅⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠠⡗⠀⣸⠁⣉⡇⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠠⠂⠀⣠⢄⠴⣋⠁⡜⠀⠐⠓⣠⡿⠃⢀⠏⡉⢀⡄⠠⠄
⠀⡀⠀⠀⠀⠀⠀⡀⠿⠀⠀⠀⠔⢢⡄⠠⠀⠘⢁⣴⡅⠐⣇⠀⢿⢸⡳⠀⠀⠀⠈⢦⠐⠀⠀⠤
⠀⡄⠀⠀⠀⠀⠈⢩⠄⢀⣢⠜⣂⠀⠉⢺⠀⠛⠋⠀⢁⢰⣟⠀⠀⠑⠃⠘⣀⡰⠈⠈⠃⢀⡄⠀⠀⠀⡤
⠀⡂⠀⠀⠀⠒⠀⠀⠀⠀⠀⢪⠉⡅⢀⠀⢃⡈⠢⠀⠘⢈⡑⣳⠈⠓⠏⠉⠛⢤⠀⠀⠑⠈⠡⢄⠅⠉⢀
⠀⠁⠀⠀⠀⠐⡂⠀⠀⡀⠀⠠⠚⠀⠊⣠⠄⠛⠳⠒⠞⠐⠀⠩⡠⡞⠀⠀⠀⠀⡀⡐⠄⠁⡐⠈⠒⠂⡤⠄
⠀⡅⠀⠀⠀⢄⣰⣀⠄⠀⢁⠀⢒⡴⠺⢋⠄⠀⠀⠢⠘⠂⠰⢠⠈⢠⡀⠧⡀⣂⠁⠀⢤⡍⠋⠤⠀⠀⡀⣠
⠀⡆⠀⠀⠀⠀⠀⠀⠈⠃⠀⠐⠊⠅⠀⠀⠤⠀⠀⠄⠀⠁⠀⠃⢀⢖⣁⠾⠀⠁⠀⠑
⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀⢀⣀⢠⣽⡄⠀⣀⢀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡿⣦⣻⢟⣠⠘⣦⢸⣿⣐⡿⣥⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡇⠀⡀⣿⡰⣎⡁⠀⠀⠀⠀⠀⢈⣡⡇⣻⠏⡀⢰⡇⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠤⡀⢈⠃⣁⠀⠷⠹⣿⣿⣋⣀⣀⣀⣀⣀⡉⣿⣿⠏⠐⠐⡉⣀⣄⠀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⡀⢇⣀⠁⡀⣈⡉⠙⣦⠘⣪⣻⠷⢛⣹⣿⣿⣟⡿⠿⣇⠘⣞⣣⠈⠁⣀⡠⣀⡸
⠀⠀⠀⠀⠀⠀⠈⢀⠙⠿⢤⣤⣤⣤⣤⣷⡝⢁⣼⣻⠛⠉⣻⣿⠊⢣⡌⢶⣯⣤⣶⣮⣤⠾⠋⡁⠊
⠀⠀⠀⠀⠀⠀⠠⠀⢄⣁⢐⣞⠑⠀⢠⣿⢁⡾⣿⡵⠈⠁⠁⠈⢶⡄⢹⡄⡍⠡⠐⠲⠇⡈⠠⠄
⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀⢀⣄⣥⠈⡟⣿⣾⢳⣹⠁⠀⠀⠀⠀⠀⠙⢆⢻⣶⠂⢨⣁⣀⣀⠀⡀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⣿⡭⡿⣰⡿⠟⢰⣉⠓⠦⠀⠀⠖⠚⣁⠐⢷⡽⣿⡫⣿⣶⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣴⣿⢶⣷⡏⢠⡶⠛⠿⠿⠆⠀⠀⠰⠿⠿⠋⠂⡕⡌⠑⠾⣧⠉⡈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⢟⣎⢾⣿⡃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡟⠀⠀⣱⡞⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⡷⠾⢿⠉⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢀⠀⢾⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⡀⢸⣦⢠⢨⣷⡀⠀⠀⠀⠀⠀⠀⢰⣿⡁⡠⣡⠆⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⣿⢸⢾⠋⡇⠀⠀⠀⠀⠀⠀⢸⣼⣧⢡⣿⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠼⢫⡟⡎⢸⠁⠀⠀⠀⠀⠀⠀⠘⢸⡟⣼⢧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡾⡿⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⠳⠏⣐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡎⠀⡅⢠⠀⢸⠀⠀⠀⠀⠀⠀⠄⡄⢡⡷⣿⣇⢃⢸⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢁⠙⡄⣼⠀⠀⠀⠀⠀⠀⢠⡇⠋⠀⢸⢃⠸⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠄⠃⡇⠀⠀⠀⠀⠀⠀⢸⠟⠀⢀⠊⣠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠅⢠⠀⠀⠀⠀⠀⠀⠀⢻⣜⣸⢣⠞⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢠⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⢰⢰⡆⡆⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⣝⢎⣼⣼⣧⣱⡱⣫⢞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠢⢟⣾⣿⠫⡉⣩⢙⠿⣽⡫⠖⠃⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣣⠇⣼⠀⢳⡱⠈⣙⠠⣾⡟⢻⣯⠔⡫⢄⠞⣼⠇⠹⡜⢆⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣝⠦⢱⡈⢮⡰⣝⠧⣀⠹⠷⠾⡃⣠⠝⣫⡮⢉⣇⣙⠸⣮⡛⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠐⢻⣿⢻⣶⢁⡌⢓⣂⣠⣤⣬⣄⣒⠺⢁⡵⣿⡟⣿⡯⣁⠙⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢁⡴⠀⠠⡙⠛⢉⡴⠞⠛⠉⠉⠉⢉⡉⡹⠛⡿⣷⡛⠛⠉⣤⢄⠰⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢧⢉⣐⠖⢜⣼⠛⠉⠀⠀⢀⠠⢒⡱⠊⡀⠀⠀⠉⠹⣶⡰⠉⣊⠱⠞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⡁⣿⣿⡿⣜⠃⠀⣀⠤⢚⣤⣾⡿⣿⣷⣬⡓⠤⣀⠀⡸⢧⣿⣿⣷⡈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡼⣉⠆⠡⣼⠁⣂⣍⣒⡻⠷⣬⡙⢧⡾⣋⣤⠾⢿⣾⣥⣀⢈⡞⠉⠰⡔⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢾⡏⠻⣿⣿⣿⡿⢶⣬⣻⣄⣸⣏⣥⣶⣾⣿⣿⣿⠋⢿⡿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠂⠀⠈⠉⢀⣴⣶⣤⡙⣯⡿⢁⣴⣶⣤⡈⠉⠀⠀⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣷⡀⠀⢿⣏⠀⣹⡇⢸⡄⣿⡁⠈⣿⡇⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣼⣷⣤⣈⡛⠛⡋⣴⣿⣷⣌⡛⠛⢋⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⣽⡟⣼⣿⣿⣿⡏⠠⣿⣿⢿⣿⣿⣿⣿⡮⣿⡇⣁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⣿⣟⢹⣿⣰⣷⣤⣴⣾⣿⣿⢻⣿⣷⣿⡇⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠘⣿⣿⣿⣦⣈⠉⠉⠁⠉⠉⢉⣠⣾⣿⣿⡿⠁⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣤⣤⡀⢡⠀⠈⠻⣿⣭⣛⣿⣿⣿⣿⣿⣿⣛⣽⡿⠟⠀⢠⠊⣠⣤⣤⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣿⣿⠇⢠⠚⠓⠶⠤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠶⠂⢺⠀⢿⣿⡿⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⡄⢤⣐⠊⠲⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡴⠋⠑⣤⠄⢤⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢰⡆⢿⡆⠀⠈⠙⠲⢤⣀⣀⡀⠀⠀⣀⣀⡤⠶⠛⠁⠀⠀⣸⡇⣺⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇⢸⢧⠀⢻⡓⠶⢤⣤⣀⣉⣉⣉⣁⣀⣤⡤⠖⣾⠇⠀⡟⡇⢺⠀⡝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⢸⡇⢸⢸⡄⠀⠙⢦⣀⠀⠀⠈⠉⠉⠁⠀⢀⣠⠞⠁⠀⣸⢃⠀⢹⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡇⠀⡀⣷⠀⠀⠀⠈⠙⠲⠖⠶⠖⠶⠊⠉⠀⠀⠀⠀⡿⢸⠀⣹⡀⢸⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠀⢸⡇⠀⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠀⢿⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠀⢸⡇⠀⢀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆⠀⣽⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⠀⢸⡇⠀⠘⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠃⠀⣻⠀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀⠈⠁⠀⢀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⠈⠀⠀⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣤⣤⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⣾⡇⠀⠀⠙⡀
⠀⠀⠀⠀⢀⣤⣤⡤⠤⠶⠶⠦⣤⣀⠀⢀⠀⠢⣝⣿⣦⣥⣀⣙⢦⡀⠀⠀⠀⠀⢀⣤⠤⠶⠶⠶⠤⣤⣤⣄
⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠀⠈⠈⢲⡈⢻⣽⠻⡿⡻⣿⣿⣦⡀⠀⠀⠈⠀⢀⣀⣀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠈⠉⠉⠻⠤⢽⣿⣯⠽⠦⠀⠀⠀⠈⡔⢻⣵⠀⢸⣌⢪⢓⠝⣿⠄⠀⠰⠾⢭⣿⣿⠤⠼⠋⠉⠉
⠀⠀⠀⠀⠀⠰⣓⣀⡠⠖⠋⢿⠀⠀⠀⠀⠀⠀⢰⢃⣿⢦⡌⠀⠘⢧⠎⠀⠀⢀⣀⠀⢸⠉⠳⠤⣀⣐⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠮⢼⣎⠱⡀⠌⠬⠀⠀⣴⠋⠉⢳⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠉⠖⠁⠀⠀⠀⠈⠁⣠⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠃⠘⢦⣄⣤⣄⠀⠀⢠⣬⣧⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠏⠀⠀⠈⢻⡽⣬⣳⣖⣫⢟⡼⢯⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⢌⠀⠀⠀⠀⠀⠹⣶⣭⢭⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⠀⠀⠀⣀⠀⠈⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠈⡤⠀⢙⣾⠁⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠉⠀⠀⢠⠃⠀⠘⠉⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⡐⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠄⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠽⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣯⣯⣵⣭⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣾⣿⣶⣶⣾⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢸⡟⠋⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡥⣾⠯⣿⠟⢉⣤⡿⠛⠉⣉⣻⣶⣶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠞⣷⣀⣼⣇⢠⣾⣤⣿⠋⢀⣴⡿⠟⠉⣁⣤⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣶⢿⣿⣿⣿⣿⣿⡿⠁⣰⡿⠋⣀⣴⡿⠟⣋⣩⣤⣴⣾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢿⢾⠇⢺⣿⠋⣿⠃⣼⡟⢁⣼⠟⣫⣴⢿⣿⣿⣿⣿⣥⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⢸⣷⣸⣿⠀⠀⠀⠈⠁⠙⠃⣾⢟⣵⣿⣿⣿⣏⣉⣉⣠⣬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⡤⢿⣿⡷⠒⠒⠒⠤⣄⡀⢠⣴⣿⣿⣿⣿⡿⠛⠛⠛⢉⣉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣼⣥⣤⣄⣠⣤⣶⡶⠿⠷⣾⣿⣿⣿⣿⣿⣿⣿⠷⠾⠿⠿⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣉⣽⣥⣮⣿⠃⡟⠻⣾⣿⣶⠟⠛⠿⣿⣿⣿⣫⣿⣶⠶⠶⠶⠿⠿⠿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣛⣉⣹⠋⠉⠋⢸⠇⠀⠀⠀⠀⢀⣤⣴⣾⣿⢛⣷⣼⣷⣶⣶⣶⣶⣶⠶⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣉⣉⣹⣦⡸⢿⣿⠶⠟⠀⣀⣀⣿⣿⣿⣿⣿⡘⠿⣿⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⣛⣉⣉⣹⣧⣾⣽⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣅⣠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⠛⣛⣋⣉⣹⣦⣤⣴⣶⣤⣿⣿⣿⣿⣿⣿⡿⣿⣿⣉⣉⣉⣉⣩⣭⣭⣭⣤⣤⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠟⠛⠛⠛⠛⣛⣏⣸⣧⣌⣩⣿⣿⣿⣿⣿⠟⠀⣿⣿⣉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠟⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣏⠀⠀⣿⣿⣉⣩⣭⣭⣭⣭⣭⣭⣤⣬⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⢏⣿⣰⣿⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⢻⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⢫⢋⣾⣍⡈⣾⣿⣿⣿⠏⣠⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢴⣿⡇⢸⣿⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠔⣱⢿⢷⣈⣿⣟⣿⡿⣁⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠈⣿⣇⢿⣶⢿⣿⡷⣽⣳⣦⣹⣧⣖⣪⣵⠿⣿⢟⡼⣻⣵⡿⣫⣶⣿⣿⣿⣿⡿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠻⣿⣿⣿⣿⣍⣴⠋⣼⠏⣯⠙⢹⡇⢹⡇⣛⣽⠾⣋⣵⣿⣿⢟⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⠙⢿⡻⣿⣿⣿⣟⣻⣾⣿⣷⣿⣿⠾⠟⣩⣴⠿⣻⡿⠟⣡⠞⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠙⠉⠛⠛⠛⠉⠉⠀⠐⠛⠉⠀⠚⠉⠀⠊⠀⠀⠀⠂

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠩⠀⠴⠔⢢⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢀⡀⢎⣈⡑⢚⠉⠇⠀⠀⠁⡂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⣈⠧⠉⢸⠗⠆⠘⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠙⣡⠀⣌⠙⢀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⢰⠄⠠⠰⠊⣴⢴⠀⠁⠠⠐⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠙⠒⠤⡀⡎⢠⠈⡇⣠⠒⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠐⠀⠀⠐⠄⣦⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠐⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠁⠄⢀⡀⢰⠆⠀⠀⠀⠀⠀⠠⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⠀⠀⡀⢶⠜⠋⠀⡀⠀⠀⠀⣄⠀⠀⢠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢎⠃⠀⠀⢀⠃⠀⠀⠀⠀⢻⡀⠀⠀⠈⠓⠀⠻⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠠⡟⠲⠶⡟⠶⢹⠀⠀⠀⠀⠈⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⢴⡆⠀⠀⣧⢈⢸⠀⠠⠀⠀⠀⠙⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠘⡺⠈⢆⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⣹⠆⡮⠀⣰⢙⣻⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢀⡠⡠⢞⠃⡐⠁⡐⠃⣸⡿⢀⠸⡉⢁⣄⠠⡀
⠀⠀⠀⠀⠀⠀⣀⠸⠁⠠⠀⠒⡂⡀⡀⠀⢁⣤⡇⠐⣆⠀⣇⢿⠇⠀⠀⠀⢧⡐⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠉⠄⢈⠲⢊⣠⠈⠉⠇⠘⠉⠀⢐⢸⣟⡀⠈⠉⠀⣁⡰⠋⠀⡑⣀⠆⠀⡀⣠⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠳⢠⠀⡤⠈⢤⣑⠂⢈⠠⠈⡿⠈⡛⠉⠉⠒⠀⠀⠈⡀⠐⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠀⠁⠂⡄⠐⣁⠀⢠⡶⡈⠉⠋⢋⠊⠀⠑⢾⠅⢀⠀⢀⢠⠘⠒⢐⡤⠀⠉⠱⠂
⠀⠀⠀⠀⠈⠚⠳⠄⣄⠐⠀⡴⠉⠁⠚⠀⠀⠈⠠⡉⠉⡄⢀⡳⢈⡢⠴⠀⢀⠙⠆⠐⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠈⠓⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⡟⠻⢿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⣴⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢀⠀⠀⠈⢛⠃⠀⠔⣿⣿⡗⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⡀⠸⢿⠿⡿⠃⢀⠀⠀⠀⣠⠂⣸⣿⣇⠀⡀⠀⠀⠀⢘⢹⣿⣧⠀⢺⠀⠀⠀⠹⣦⣤⣶⣶⣿⣶⣦⣤⠞⠀⠀⠀⢻⣀⣿⣧⣠⠇
⠀⠀⠀⠙⠶⣿⣿⣿⡷⠋⠀⠀⠀⢈⣿⠂⠀⠰⣿⣁⠀⠀⠐⢦⢿⣿⣿⣇⣀⠇⠀⠀⢤⣤⡼⣿⣿⣿⣿⣿⣥⣾⠗⠀⠀⢀⡼⠉⢿⣿⠂
⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⡀⣾⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣀⢀⡃⢰⠠⣾⡄⢠⠀⠀⠀⠀⠀⠀⢠⣶⠄
⠀⠀⠀⡄⣄⣨⣿⣷⡤⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠠⣄⣠⠻⠟⣢⡤⠖⠀⠀⠉⢁⣑⣾⡞⠿⢳⣾⣚⣁⠀⠀⠀⠀⡚⠿⢃
⠀⠀⠀⠀⡨⠛⠻⠿⡅⠀⠀⠀⠀⠃⢸⠿⣿⡝⠳⠆⠀⠀⠀⠀⣨⡿⠖⠉⠛⣧⡤⠀⠀⠀⠀⢀⡼⠣⣤⡜⢦⡀⠀⠀⠀⢀⡼⠁⠀⠙⢦
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⡤⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄


⠀⠀⠀⢀⡴⠀⡀⠆⠀⣤⠀⠀⠀⠀⢰⣶⠀⣄⠠⣷⡄
⠀⠀⠀⢰⠀⠀⣰⡄⠀⢹⡄⠀⠀⢀⡔⠀⣴⣤⡆⠀⣤⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢰⡆⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠈⠧⣤⣿⣿⣤⠞⠀⠀⠀⠀⠹⢤⣟⣛⣫⡴⠃⠀⠀⠀⠀⢲⢾⣿⡇⠀⠀⠀⠀⠀⢳⣾⣿⣷⣤⢾⣷⠀⠀⠀⠀⠀⣙⣜⣁
⠀⠀⠀⠀⠀⠸⣿⣿⢧⣄⡀⠀⠀⠀⠀⠐⢿⣿⡿⣤⣀⠀⠀⠀⠀⠾⢿⣿⣿⡇⠀⠀⠀⣀⡼⠿⣿⡿⢧⣠⣥⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢘⠁⠉⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⡘⠋⠉⠙
⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⣿⠀⠀⠀⠀⠀⢠⣾⠀⣿⢸⡇⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢺⣶⣿⠿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣀⣀⣨⢿⣿⣿⣶⣿⡇⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣯⡸⠟⣍⣹⡿⠻
⠀⠀⣜⠉⣘⠀⣠⡶⠛⢿⡟⠻⢶⡀⠀⣀⡏⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣈⣾⣿⠀⡟⠀⢠⣄⣠⡄⠀⢳⠀⣿⣷⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣥⣿⣿⣿
⠀⠸⡥⢿⡿⣄⠹⡄⣼⣿⣿⣇⣠⠃⣰⢿⡿⢬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠳⣩⢿⣿⢦⣸⣿⣿⣿⣿⣧⡶⣿⠿⣉⠊⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠐⢊⡝⢻⣿⣿⣿⣿⣿⣿⡟⢯⠕⠁⠀⠀⠀⠀⠀⠀⠘⠈⠿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⡿⠃
⠀⠀⠀⠀⠀⠈⠃⢾⣿⣿⣿⣿⡷⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡾⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠙⠽⠿⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⢠⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢧⣾⣿⣿⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣮⣽⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣿⠁⣿⣿⣿⣿⡋⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢸⣿⣿⣿⡀⢹⡇
⠀⠀⠀⠀⣀⡀⠀⠀⠾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣾⣿⣿⣿⡇⢼⠇
⠀⠀⠀⠀⠻⣷⣦⣀⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣶⡄⢠⡄
⠀⠀⠀⠀⠀⠈⠻⣥⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢘⡇⢸⠃⣧⠘⡕⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣮⣽⣿⣷⡿⣻⣭⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣛⣋⢕⣾⣖⢍⣛⣫⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⠇⠀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⡇⣿⣿⡿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣯⣿⣿⡇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣧⣿⡿⣷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣿⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣤⣀⣤⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠞⢥⣀⡌⠀⠀⣁⢾⡄⠂⠁⢴⣆⣙⠿⣟⣩⣴⣾⣿⡿⠟⣛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⣖⣂⡐⢤⣠⣏⠸⣣⠆⡀⢸⡅⣂⣁⠀⣅⢸⡇⠸⣷⣟⢝⣵⡾⠩⠵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⣦⡄⠴⣤⣜⣢⡾⣿⣻⡾⠻⡌⢁⣾⢿⡽⣫⣠⣿⢸⢼⠀⣹⢣⣾⢛⣍⣞⣁⠘⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡌⡁⢌⢩⣟⣿⣿⣿⡈⢻⣇⢣⣧⡘⢞⣻⣟⣿⣋⣵⡫⠈⢸⡇⣾⣧⡘⠏⢼⣿⣇⢡⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣾⢯⣿⣿⣿⣿⣿⣧⣽⣷⡉⠳⠶⠴⠟⠋⣁⣿⠄⢀⡼⠻⠟⢧⣋⣒⠷⠶⠇⢠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣬⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⡤⠖⠛⠯⣤⢴⡖⢉⢔⣤⣒⡖⢋⢉⠹⣦⣬⣥⡄⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⡿⢩⣿⢽⡿⠾⢧⣭⣭⣿⣭⣿⠿⢿⡟⣖⣾⢡⠊⡀⢰⣾⣿⣿⣧⣍⠈⡗⢬⡿⠿⠟⣫⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣹⡿⢿⡟⠉⠀⠀⠀⠀⠀⠀⣤⠤⠶⠂⠉⠻⡄⡄⣄⠓⢻⠇⣈⠻⣟⣉⡶⢀⡞⢸⣿⣿⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⣻⡏⠉⠙⠒⠀⠀⠀⠀⢀⡠⣴⣦⣤⠀⣷⣌⣏⢀⣳⠧⣥⠽⣏⣉⣚⡅⠐⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢛⣿⣹⣿⠛⢿⡿⠆⠀⠀⠀⠘⠀⠛⠋⠁⠀⠸⣿⡟⢷⣙⣷⡌⠀⠵⠯⠀⣠⢧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣫⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣖⣶⢿⣯⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⢠⣤⣐⣠⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡈⠻⣭⠭⠉⠁⠀⠀⠀⠀⢀⠔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣏⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠁⠀⠈⠉⠉⠛⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠳⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⡻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣸⣷⣿⣿⡁⣤⣦⡾⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⡀⠀⠀⠀⢠⡍⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣴⣾⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⢟⡳⢀⣠⣠⠰⣟⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⡏⠘⠉⠀⠀⡹⡅⠀⠈⠙⡌⣿⢠⠀⣶
⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠁⣁⠀⠆⠻⣾⠸⡶⢀⡶⣖⠰⡾⢠⣷⠃⠆⢀⡁⠁⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠘⣄⣀⡀⠈⠁⠈⢢⠁⡜⠉⠁⠆⠀⠀⠂⠉⠈⡅⢁⠎⠀⠉⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠬⠙⠲⠶⠶⠶⠶⡖⠉⡠⢴⠀⠀⠀⢠⠦⠈⠁⣶⢶⠶⠶⠶⠚⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠞⠉⠘⠳⢲⡲⡛⠀⠙⡒⠀⠀⠀⠀⠋⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⢈⣀⡤⢚⣷⡶⠖⡖⠁⠁⣶⠒⣖⢏⠢⣄⣈⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠩⠛⣵⣊⠀⡤⠈⡾⢡⠞⣧⢔⡤⡝⡮⡙⡏⠀⣄⠁⣳⣍⠫⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼⡱⠁⡘⠀⢡⢻⡄⢨⡄⢬⠔⣰⠁⣰⢃⡈⠘⡀⠙⡼⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⣿⢴⠋⢠⢿⡂⠙⢦⡉⠲⠊⢡⠜⠁⢸⡧⠊⢞⣄⡿⠲
⠀⠀⠀⠀⢃⠀⠴⠒⠒⠒⣧⡀⡨⢲⡾⠁⠀⣀⡹⣄⡴⣋⡀⠀⠘⣷⠪⡀⣠⡷⠒⠒⠲⠆⠀⠎
⠀⠀⠀⠀⠚⠀⠀⣀⠀⡀⠡⡀⢅⢹⠁⠀⢸⣾⣃⣿⣇⣻⣾⠆⠀⢸⣇⡅⢠⠇⡀⢀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠈⡞⣳⢠⠸⠩⡿⠇⠿⡿⠝⢠⡐⡏⡎⠀⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢼⣿⡈⡼⡀⡄⡧⢸⡀⢸⣟⢧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⡀⠀⠂⡎⠌⠒⡀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⡇⣷⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠰⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⢠⠀⠈⠻⠿⠇⡤⠀⠀⠆⠀⠀⠀⠀⠸⠃⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⢀⡔
⠀⠀⠑⣂⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⢶⠆⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⡒⠁
⠀⠀⠀⠈⠢⢄⡀⠀⠉⠐⠀⠀⠠⠠⠄⠄⠤⠠⠄⠄⠀⠤⠄⢭⠉⣉⣭⠄⠄⣤⣾⡿⢹⠤⠤⠄⠠⠄⠐⠒⠈⠁⠀⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣷⡄⠈⠈⢹⠀⡘⠀⠐⠂⠒⠒⠒⠂⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠠⠙⣃⠠⠶⠥⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠴⠭⠡⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣷⣺⣯⣙⣿⣯⣫⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠆⠒⠒⠀⠤⠤⣀⡀⠀⠀⢀⠀⠀⠀⠉⠉⠀⠀⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠎⠁⣀⠄⠂⠀⠀⠤⠐⠆⣀⢉⢁⠔⠀⠂⣥⢀⠀⠀⠀⠀⠀⠀⠂⠁⣄
⠀⠀⠀⠀⠀⠀⠜⠀⢺⠈⠀⠀⣀⢠⠀⠄⣀⢀⠀⠁⠏⠀⡀⠖⢩⠄⣣⣄⠀⠀⠀⠈⠀⠈⠠⠇⡀
⠀⠀⠀⠀⠀⣈⠀⢰⠁⢀⡴⢉⢀⠠⠐⠂⢄⣤⢵⢛⡆⠠⠀⠀⠀⠀⠙⡣⢔⣤⡇⠀⠀⠀⠒⠠⢁
⠀⠀⠀⠀⠀⡦⠰⠄⠀⠞⠁⢩⠀⠀⠀⠀⠄⠀⢸⡄⢀⣜⠃⢀⣠⠀⡀⠁⣨⠹⡃⢀⠀⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⢠⠀⢺⡀⢲⡀⠊⠀⠀⠀⠀⠀⠀⠻⢇⣤⠀⠆⠀⠩⠀⢨⣷⡴⢢⠙⠃⠀⣀⣠⠭⠄⠇
⠀⠀⠀⠀⠀⠀⢂⠈⡥⠀⠙⢶⣏⠂⠀⣀⡀⢀⠀⠀⠘⠀⠀⠀⠁⠘⠂⣈⢧⡿⠀⡀⠠⠋⣿⢻⠧
⠀⠀⠀⠀⠀⠀⠈⢇⡠⠎⠁⠙⠮⠚⢾⣈⢉⢗⠂⠀⠀⠀⠀⠀⢀⠀⢴⣤⠀⠀⣀⣁⣀⠀⢿⠸⢆⡔⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢨⡵⡍⠈⢂⡉⠀⢀⠀⠐⠂⠁⠌⠁⠹⠻⡙⣄⡝⣭⣧⡀⡴⢯⡓⣖⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⡴⣏⣁⣀⣀⠠⢄⣀⠀⠂⠀⠀⠀⠀⠀⠘⣇⢸⣿⣿⡿⢋⣯⠽⠻⣄⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠠⠙⣿⡟⠁⠀⠈⠁⠈⠉⠻⣦⠀⣀⠀⢀⣀⠘⢪⠟⡩⢚⣥⠐⠎⡀⠆⠉⠪⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠁⢀⠀⣿⡶⠈⠀⠀⠈⣉⣀⢠⣗⣭⡆⠛⠁⠀⠀⠀⠀⣠⠖⢰⣾
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡧⡠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⡿⣿⠋⣀⡤⢤⣤⡀⠃⢈⣷⡄⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣷⡟⡛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣇⠈⣁⣤⠀⠀⠀⠀⠀⠉⠀⠀⢻⡗
⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⠋⠀⠡⡀⠀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⣴⣿⣯⢿⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠐⣾⡁
⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⣏⠾⠄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠁⠠⢎⣸⣿⡯⠸⣄⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠁⠹
⠀⠀⠀⠀⣼⣿⠋⠀⢦⠀⣜⣅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠘⣭⣿⣷⣤⣟⠃⠀⠀⠀⠀⠈⠀⠀⠀⣀
⠀⣀⢀⣈⠿⠇⣀⡘⠃⠀⠈⠙⠂⠤⠄⡠⠆⢀⠶⠂⣁⠒⣪⣀⣀⣀⣉⣿⡿⣧⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄
⠃⠂⠀⠁⠐⠂⠘⠓⠀⠀⠀⠀⠈⢶⣿⣡⠄⣌⡀⣠⣰⡿⢻⣿⣿⣿⣿⢿⣧⣩⣅⠀⠀⠀⠀⠀⠀⠀⠶⡄⢤⡤⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠉⠋⠋⠛⠉⠈⡤⠞⠈⠉⠉⡁⠀⠀⠙⠻⢦⣀⠀⠀⠀⠀⠀⠀⢷⠀⠑⠎⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⢈⠷⣂⣤⣤⡶⠖⠻⠩⣿⣦⣤⣤⠀⠀⠀⠀⠀⠘⠍⠑⠢⡤⣀⣀⣤⡷⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⡓⡐⢌⣨⠁⠃⠠⠛⠛⠿⠿⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣹⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣶⡤⠈⠉⠀⣀⠠⠀⠀⠀⠀⠀⠀⠈⢟⣿⠿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣸⣿⣿⣶⠞⠁⠀⠀⠀⠀⠈⠣⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡰⢒⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶
⣿⣿⣶⣶⣷⣾⣶⣷⣶⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⣫⣽⡓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣁⢲⣾⣟⢙⣿⣖⣊⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⣿⣿⠞⠻⣿⣿⣿⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⣶⣿⣿⣾⣿⣿⣿⣷⣶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢖⠉⠀⠀⠁⠀⠉⠒⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠆⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢀⣤⡄⢀⣿⠀⠀⠀⠀⡀⣆⠀⠀⠀⠀⠀⠀⣠⢂⠄
⠀⠀⠀⠀⠀⠀⢺⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠸⡿⠛⠃⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣐⣦⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠷⣝⡿⠒⠀⠀⠀⠀⠂⢾⣻⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⠀⠀⠀⠀⠈⠙⠛⠂⠀⠀⠀⡄⠈⠁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠑⠦⠶⠚⠀⠀⠀⠀⠀⠘⠻⠿⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣶⣤⣄⣀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠛⠛⠛⠛⠟⢻⣿⣿⣿⠿⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⠉⠈⠀⠀⢀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣛⠛⠲⢤⢤⣦⣤⣤⣤⣤⣤⣶⣶⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣉⡛⠓⠶⠶⠍⠀⢹⣿⣟⣿⣿⣾⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣛⡻⠿⠿⠶⠦⠀⢸⣿⡇⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠀⢸⣿⣷⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠈⣿⣿⣎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⠀⡀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢒⠢⠄⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠒⠪⣿⠁⢂⠀⣠⣿⣧⡀⠸⢿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⣿⣿⣿⣿⠄⠀⠒⠁⠀⠈⢰⣿⣿⣿⣿⠀⠸⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠇⡠⡀⣀⠘⠿⠿⠇⠰⡆⠀⠀⠛⠀⠀⠙⠿⠟⠁⣤⠀⠋⠃⠈⠅⢀⡀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⠜⠸⠌⠠⢀⣄⠀⢀⠀⠻⢿⣶⣄⠘⣦⣀⢰⣶⡆⣉⣴⠃⠀⠤⠤⠼⣷⡄⣤⡤
⠀⠀⠀⠀⠀⢀⣀⣫⣠⣃⣀⠀⢠⣶⣾⣿⣦⠚⠊⠁⠀⡁⠀⠀⠙⠿⠮⠉⠵⠿⠋⣀⣀⣀⠀⡀⠈⡇⠉⠀⠘
⠀⠀⠀⠀⢀⠀⠉⠋⡟⠋⠉⠐⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠉⡏⣛⠀⠁⠀⣾⣷⡶
⠀⠀⠀⠀⠛⠀⠐⠀⡇⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠬⠧⠿⠀⠐⣤⣍⣭⣡⡴
⠀⠀⠀⠀⣤⣷⡀⠀⠁⢀⡴⠲⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣖⠀⠀⠉⠉⠉
⠀⠀⠀⣼⣿⣿⣿⡀⠦⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣀⠀⡀
⠀⠀⠀⠸⣿⣿⡿⠀⢁⠈⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣧⠀⠐⣶⣶⣶⣶⡶
⠀⠀⠀⠐⠀⠁⠀⢰⣿⣄⣠⣄⣠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⡁⠀⠈⢉⣟⠉
⠀⠀⠀⢀⣤⣤⣀⠘⢿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠉⠁⢠⣦⠈⠉⠀⢶⠆
⠀⠀⠀⠸⣾⣾⡎⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠆⠀⢠⣿⣤⣤⣤⣄
⠀⠀⠀⠀⢙⡟⠀⢀⣀⡘⠛⡁⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠐⠀⠚⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠸⡇⠀⢹⣿⣇⠀⠇⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⣦
⠀⠀⠀⠀⠀⠀⠆⠈⣿⣿⣇⠀⣤⣾⣿⣿⣦⡄⠰⣤⣼⣤⡴⠀⠀⠰⠿⠶⠿⠿⠷⠶⠇⢠⣜⢿⢟⡤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠉⠛⠋⠁⠈⠹⠿⣿⠿⠿⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠉⢺⠊⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⢀⣿⣿⣿⣿⣿⣿⣦⠀⢸⠀⢀⣠⡀⢀⠀⣠⡀⠀⢸⣿⣿⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠻⠿⠿⠿⠀⢸⠀⠀⢼⣧⣿⣿⣿⣿⠄⢸⣯⣿⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⠈⠙⢿⠋⠹⣿⠀⠘⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⣩⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢋⣥⣖⣽⠇⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢘⣷⣿⡟⢰⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⢠⣿⡏⣿⣿⢀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⣰⣿⡿⢡⡿⣱⣿⣿⣿⡿⢋⣭⣤⣤⣬⣭⣭⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⣼⣿⣿⢧⡟⠼⣿⣿⣿⣿⠉⣾⠟⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣻⣿⣿⣾⣽⣿⠶⢬⣭⣛⡛⠏⠸⠿⠿⠿⠿⠿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣻⣿⣿⣿⣧⣤⣍⣽⡧⠀⣤⣤⣭⡓⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢧⢧⣿⣿⣿⣿⣿⣿⣷⣿⣴⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⣰⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⡏⣿⣿⣿⡇⣿⣿⣿⣏⢝⣙⢛⡻⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⢿⣿⣿⡇⣿⣿⣿⣿⣻⣛⣛⣻⣿⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣎⠻⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡁⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⠰⣝⠻⠿⢭⣛⣛⡿⠿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣶⣶⡦⠌⠉⠙⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠙⠛⠦⣄⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢱⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿
⠀⠀⠀⠻⠿⠛⠛⠀⠀⠀⠀⣇⠀⠀⠀⢺⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⢦⣉⠓⠒⠶⠶⢒⡿⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠈⠙⠒⠒⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⡿⣿⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠻⣍⣷⣼⣷⣿⣾⣧⣾⣩⡟⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢯⣹⣿⡿⣟⣿⠋⠀⠙⣿⣻⢿⣿⣏⡽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⣾⣿⣻⠽⠾⠃⢰⣭⠆⠘⢷⣯⣞⣿⣷⠟⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣹⡟⠳⠏⠐⠗⣀⠛⠛⠛⢀⠾⠙⠟⠚⠻⣏⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡽⠀⠀⠀⣀⠀⠿⢠⣿⡐⠛⠂⠀⣀⠀⠀⢫⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣷⠦⡄⠠⠿⢃⡠⣔⣽⠿⣯⣢⢄⡙⠿⠄⢠⢴⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣷⣸⣷⠿⠶⠟⠊⠁⠀⠈⠙⠫⠶⢿⣷⣅⣿⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠛⣿⠀⣷⣦⣶⡶⣆⠠⡀⢄⡴⢶⣶⣶⡆⠀⣿⠋⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡄⢠⣍⢻⠑⠀⢫⡟⣭⡄⣰⠇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⣿⠘⠀⠙⠶⠷⢟⠀⠀⠈⠻⠾⠞⠁⠀⠂⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢹⡄⠀⠀⣄⠀⠈⠑⣒⠁⠀⠀⣀⠀⠀⢠⡏⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⢲⣄⠀⠻⡄⠀⠛⠷⣶⣿⣿⣷⡶⠾⠋⠀⢠⠟⠀⣠⣴⠒⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠠⣩⡡⠈⡗⠀⠙⠷⣄⣀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⣌⠣⠀⣉⠀⠊⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠌⢄⢀⡟⠒⠶⢄⣈⡉⠛⠓⠒⠚⠛⠉⢁⣀⡠⠤⠩⡆⡀⠍⢀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢺⠲⣿⠰⣄⠀⠀⠀⠉⠉⠉⢓⠒⠉⠉⠉⠀⠀⣀⡶⣿⠖⣳⢲⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⢸⠀⢹⡆⣬⡙⠶⣤⣀⡀⠀⠰⠀⠀⣀⣠⡤⠞⣉⣠⡟⡀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠇⢸⠁⠌⣷⠈⢯⡓⠦⢬⣉⣉⣛⣛⣉⣉⡤⠔⣺⠟⣸⠃⠀⣿⠀⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠀⢸⠀⠀⢸⡆⠀⠙⢦⣀⡀⠀⠀⠀⠀⣀⣤⠞⠁⠀⡟⢰⠀⣿⠀⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠸⠀⢸⠃⠘⠀⣧⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠀⢸⠇⠀⠀⣿⠀⢰⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡀⠀⢸⠄⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⡀⠀⣿⠀⠈⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠃⠀⢸⠁⠀⠚⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⠃⠀⣿⠀⠀⠂⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⠀⢀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⢤⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠷⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⡀⠀⠀⣿⣿⠀⠀⢀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⣰⠃⠀⢰⣿⣿⡆⠀⠈⣦⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢀⣼⡇⠀⣠⣿⣿⢿⣿⣆⠀⢸⣷⡀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⢀⣠⣶⠟⠁⢳⡾⠋⠀⢠⣄⠀⠙⢷⣞⠈⠻⣷⣄⡀⠈⠻⣿⣦⣄
⠀⠀⠰⠶⢿⣿⣿⣅⠀⠒⢿⣿⣿⣤⣴⠏⠀⠀⠀⣿⣿⠀⠀⠀⠙⣶⣤⣿⣿⡿⠖⠀⣨⣿⣿⡿⠶⠆
⠀⠀⠀⠀⠀⠈⠉⠛⠷⢦⣴⡟⠻⢿⣷⣀⠀⠀⠀⣿⣿⠀⠀⠀⣀⣼⡿⠟⢻⣦⡴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠉⠻⢷⣦⣀⠘⠇⣀⣴⡾⠟⠉⠠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠙⣿⣿⣿⣿⠋⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠉⢻⡆⠸⣿⣿⠇⢠⡟⠉⠈⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⡇⠀⣿⣿⠀⢸⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣿⠀⢸⡇⠀⢸⡟⠀⢸⡇⠀⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡇⠀⢸⡇⠀⢸⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡵⡋⣰⣆⡤⡄⠀⠀⠀⢀⡄⢶⢤⣶⠄⣶⣳⣀⢀⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⣈⡾⣵⣫⣾⣗⣃⣀⢀⣴⣟⡱⠶⢶⣶⠠⣶⣿⣷⣿⡿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠛⠿⠿⠿⠿⠻⠯⠰⠿⠏⠀⢶⣿⣿⢾⣿⣯⠙⠛⠁⠉⠁
⠀⠀⠁⠀⠀⠀⢀⡥⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠈⠡⠀⠈⠁⠁⠀⠀⠈⠀⠀⠴⠖⠒⢦⢄⣄⢀
⠀⠀⠀⢠⣴⠛⠁⡔⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠁⢩⠻⢦⡀
⠀⠐⣰⠣⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⢰⣘⡀⠀⣇⣠⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⢸⠙⣤⠂
⠀⣼⡏⠀⠀⠀⠠⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡛⠉⠉⠉⠁⠑⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠰⢸⡧
⢢⣟⠂⠀⠀⠀⠀⢱⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠀⢀⠀⢀⡤⠚⠛⠉⠀⠀⠀⠀⠀⡸⣷
⠸⡇⠀⠀⠀⠀⠀⣈⠀⠀⠨⣛⠂⠠⢀⡀⠀⢀⠀⣠⣑⣤⠀⠀⢀⠀⣀⠈⡄⣀⣀⣀⠤⠒⠁⣈⠂⠀⠀⠜⠀⠀⠀⠀⠀⢊⠼
⠀⣯⠀⠀⠀⠀⠀⢀⡀⠀⠀⠱⡀⠀⠀⠀⢀⠁⠉⠉⣌⠑⠰⠀⠘⠉⠉⠀⡁⠉⠀⠀⠀⠀⡰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢺
⠀⠸⣆⠀⠀⠀⠀⠀⢢⠀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠀⠈⠂⠠⠀⣠⢪⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠀⣪⠃
⠀⠀⠱⣒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠁⠒⠒⢦⣶⣦⣤⣶⡦⠀⠀⠀⡆⠀⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⠀⠤⣴⠉
⠀⠀⠐⠹⡩⢄⠀⠀⠀⠀⠀⠀⠀⠦⣌⠀⠀⢠⠢⡀⠈⠀⠀⠀⠀⠁⠀⠀⢀⣈⠀⡀⠠⠈⠉⠀⠐⠄⠀⠀⡀⠠⢬⠟⡀⠀⠁
⠀⠀⠀⠀⠘⠱⣦⣠⢀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠐⢤⠦⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣖⠟⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠱⠧⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⢈⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⡢⠓⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢛⡻⠷⠒⠛⠉⠀⠀⠄⠰⠀⠐⠀⢧⣠⡆⣴⡀⣂⣆⠶⠾⠟⠕⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠧⠬⣄⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣀⣀⣤⣤⣤⣤⣀⣒⣒⣒⣒⣒⣒⣐⣚⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡃⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣻⡏⢻⡾⠿⣒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⣿⣹⣿⣰⡿⢆⢰⡆⢀⣀⠀⢀⣄⣷⢟⣼⣡⣿⠀⠀⣤⣀⢻⣇⣙⣗⣶⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡟⢏⣿⡇⣿⣿⣿⣵⣇⢹⠃⠸⡿⢿⡿⢿⠃⣡⣞⣿⢿⣰⣷⡝⣿⣾⣿⣿⢦⣸⠎⣿⢿⣶⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢿⡇⡟⣿⣇⣼⣿⡿⠙⠇⣿⣿⢻⡞⣧⣄⠀⠸⡿⢸⡧⠱⢟⡉⣸⣿⡿⣿⣿⣿⣟⠻⣦⡙⠿⣽⣼⣿⢻⠿⠟⠓
⠀⠀⠀⠰⠶⠶⠛⠡⠾⠦⠽⠿⣿⣿⠿⠷⠦⠀⠋⠿⠾⠿⠿⠿⠾⠷⠿⠦⠶⠿⠍⠁⠀⠉⠛⠉⠉⠁⠘⠓⠶⠿⠧⠬⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠉⠀⠂⠀⠈⠄⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⣀⣤⣶⣲⣲⣆⣤⣀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⣴⣟⡿⣫⣥⣭⣻⢿⢻⣤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡸⣚⡩⣭⢟⣹⣏⣣⣭⣝⣻⣇⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣽⡍⣑⣽⡋⢛⢏⡎⢞⣯⡿⣤⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⡥⣧⣧⢋⣀⣉⠁⠈⣁⣈⡳⣵⡼⢜⡦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⠀⠐⣿⢳⡟⣿⠰⠶⠎⠀⠀⠴⠶⠆⣟⣿⢾⡿⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⣟⣷⣽⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⣀⡅⣬⡀⠀⣸⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⢠⡾⣡⠟⣳⣄⠈⠑⠈⢀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⣧⣦⡙⢧⠉⠒⠚⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣼⡯⣿⠞⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢧⣿⢤⡙⢲⡄⠀⠀⠀⠀⠈⠓⠦⡤⣤⣄⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⢹⡘⢼⣻⡞⢱⡞⠁⠀⠀⠀⠀⠀⣠⠞⡰⢱⡏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡤⢺⣷⡈⣻⣟⣿⣿⢤⣀⣠⡤⠴⠊⢁⠌⢀⣿⡳⢄⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⢖⡩⢧⠻⣿⣇⡺⣤⣄⣀⣀⡠⠴⢊⠁⣠⢋⡾⣝⠲⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠈⢧⡈⡿⣯⣞⡒⠒⠒⣒⣩⠴⠊⢁⡞⠁⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣀⠈⠉⠉⢁⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠸⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣶⠄⣉⣤⣭⣥⣌⡁⢴⡞⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡌⢛⡷⠞⠛⠛⠻⢾⣟⢉⣾⣿⣿⣿
⠀⠀⠀⠀⠦⢀⣾⣿⣿⣿⣿⣿⣿⣆⠠⠆⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⠏⠀⠀⠀⠀⠀⠀⠙⣿⣻⣿⣿⣿
⠀⠀⠠⣴⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⡤⠀⠀⠀⣿⣿⣿⣛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⢛⣻⣿
⠀⠀⠀⠀⣤⠘⣿⣿⣿⣿⣿⣿⣿⡟⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⡄⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣿
⠀⠀⠀⠀⢀⣴⡌⠛⠿⠿⠿⠟⠋⣴⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠇⢹⡦⠄⠀⣀⣠⣴⣟⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠚⠀⢶⠆⠘⠃⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣯⣾⡏⢩⣿⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢀⣠⣤⠤⠤⣄⣀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢳⠀⣠⢞⡽⠚⠋⠉⠙⠓⢮⡳⣄⠀⡜⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷⣳⠏⠀⠀⠀⠀⠀⠀⠀⠙⣞⣷⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠠⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣽⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⠉⢧⡳⣄⡀⠀⠀⠀⢀⣠⢞⡽⠋⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠀⠙⠺⢽⣿⣷⣿⣿⠷⠋⠀⠀⣿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠈⠛⠲⠶⣴⣶⣶⣶⣶⣶⣶⠶⠖⠛⠁⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣦⣤⣀⠀⠈⠻⣿⡟⠁⠀⣀⣠⣴⣾⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⣞⣯⣾⣿⣿⣷⠀⠀⣿⠀⠀⣾⣿⣿⡷⣿⡳⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⢿⣾⣳⣿⣽⢾⣿⠀⠀⣿⠀⠀⢿⡿⣯⣻⣮⢿⣽⣯⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣿⣿⣟⣾⣷⣿⣇⠀⢀⣿⡀⠀⣸⣿⣞⣷⣹⣯⣿⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠶⢨⡍⣙⢛⡛⢛⠛⠷⠛⢶⢳⣶⡏⡾⠛⠻⠛⣛⢛⡛⣋⠩⡍⠶⣘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⡶⢶⡬⢥⣭⣌⣧⣘⣁⣳⣭⣝⣁⣛⣼⣧⣩⡬⢥⡼⢶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⡇⠴⠚⠛⠻⠶⣤⡍⠉⢉⣤⠶⠿⠛⠓⠢⠀⡇⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢿⡇⢴⣶⢿⣿⡷⣄⠀⠀⠀⣠⢶⣿⡿⣶⡤⢀⣷⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⢹⡇⠀⠈⠉⠛⠓⠙⠃⠀⠐⠋⠘⠛⠋⠁⠀⢸⡏⣸⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⢸⣧⣾⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢋⢻⣷⠀⠀⠀⣀⣠⣤⠤⣤⣄⣀⠀⠀⠀⣼⣟⡙⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠀⠈⢻⣦⠀⠀⠀⠣⢤⣭⠤⠜⠀⠀⠀⣰⣿⠄⠀⠘⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣧⣦⣤⣴⣠⣿⣷⢦⡀⠀⠀⠀⠀⠀⢀⣤⢾⣿⣅⣦⣦⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠙⠳⠶⠤⠶⠞⠋⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⡷⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢾⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣽⣿⣿⣿⣿⣾⣿⣿⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣮⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢤⣍⣛⠓⠒⠒⠒⢒⢛⣩⡥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠏⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢷⡻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠰⠌⣭⢩⣉⡍⢩⠅⠦⠘⣿⣿⣿⡯⣿⣿⣿⣿⢟⡷⢃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠻⣮⡛⡿⠻⠿⢿⣿⣿⢀⠐⡠⠠⠠⠀⠀⠄⢀⠀⡀⢿⣿⣿⠿⠿⢿⣛⣥⠟⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠶⣍⡰⢠⡄⡃⠞⠸⠃⡟⢸⠀⣧⢸⠈⠇⠻⢘⠁⡄⢶⣩⡴⢋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⣙⡓⠶⢥⣝⣐⣂⡖⠰⠀⠆⠰⣐⣃⣹⣬⠷⠚⣋⣥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢬⣭⣉⣉⣉⣉⣉⣩⣭⠴⠶⠚⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡛⡿⠛⢛⣛⣋⡛⢻⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣿⣟⣿⣿⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢤⣄⡤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡤⢂⣌⡫⠥⠄⠠⠶⠖⠲⠶⠄⠀⠒⠰⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⡀⠀⢁⢐⣒⣠⣶⣠⣴⡾⠶⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⠀⡄
⠀⠀⠀⠀⠀⣰⣧⣾⣄⣴⣵⣿⣤⣼⣿⡿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠦⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⢀⡂⠀⠀⠀⢀⠄⠀⠀⢀⣀⣼⣷⣶⣶⣶⣿⣶⣶⣶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠀⠈⠐⠀⠀⣠⣾⣋⣽⠟⠛⢛⣩⠋⣩⣉⡙⢻⣿⣻⣿⡾⣿⣶⣆⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⡠⣀⣄⣴⣿⢿⣛⣏⣵⣿⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣷⣞⢪⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⢠⣀⣜⣻⣯⣟⣋⣵⣾⣿⣿⣿⣿⡿⠟⢟⣿⢟⣹⣿⣻⣛⠛⢿⣿⣿⣯⣾⣿⣯⣛⣿⣿⣯⣝⡠⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⣨⢫⡜⠛⣼⣿⢿⢿⣾⣿⢿⠇⡐⢷⣼⣷⣿⣿⣿⡛⠉⠉⠒⡿⠉⠻⣿⣿⣿⣿⣻⡯⣩⣿⣿⣍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣹⡼⢈⣼⠟⠋⠁⢸⣿⣿⣾⣤⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⢠⣷⡶⠀⠹⡟⠿⣿⣿⣿⣿⣧⣭⣹⣷⠉⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠸⠃⢒⣴⡟⣠⣾⡃⠀⠀⠀⣼⣿⣿⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠶⣿⣿⣗⠂⢤⡇⠀⠙⠻⣿⣿⣿⣿⣿⠞⠁⢀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠔⢛⣯⣼⣿⣿⡇⠀⢀⠀⢸⣿⣿⣿⣠⣽⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠗⠀⣸⠇⠀⢉⣫⣿⣿⣿⣿⣟⠛⢋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣤⣼⣿⣿⣿⣼⣯⣆⡨⠆⠈⣿⣿⣟⡋⢽⣿⣷⣭⣙⣻⣿⣿⣿⣿⣿⡿⠂⣬⡟⠀⠀⣠⣿⡿⣫⠿⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⣿⣿⣿⣿⣓⠅⢀⠈⠿⣿⣿⣾⣿⣹⣿⣿⣿⣿⣿⣿⢿⢫⣴⣾⢟⣁⣴⣾⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣿⣾⣧⣄⣀⡈⠻⢿⣿⣿⣷⣄⣯⣿⣾⣾⣶⣿⣿⣷⣞⡯⢿⠛⠛⠀⠉⠀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⢶⠶⠶⢷⣿⣿⣿⣿⠿⣿⡿⣿⣿⡛⢯⠄⠒⠉⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠠



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢔⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠵⣌⠳⣕⣄⠀⠀⠀⠀⠀⠀⣶⢚⢳⠒⠖⣶⠄⠀⠀⠀⠀⢀⣴⡵⢋⡴⢗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠳⣬⣳⢮⡳⣕⠀⠀⠀⠀⠀⠛⠁⢎⡑⡈⠛⠀⠀⠀⠀⠐⡱⣫⢿⣫⠴⢋⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢭⣙⠢⢽⡳⢍⠪⣢⠦⡀⠀⠀⠆⠈⢀⠇⠀⠠⠀⠀⣠⠺⡮⣊⠕⣫⡷⣚⡭⢞⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠦⣍⣻⡭⠓⠬⣾⠀⠙⡄⠠⣀⣶⡛⢛⣶⣄⠃⡰⠁⠀⡿⠐⠉⣐⣯⠵⣚⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣝⣒⣾⡍⠒⠤⣶⠀⠀⢘⠘⣟⠛⠋⠘⠛⣻⠇⠁⠀⠀⡿⠒⠉⣵⣗⡩⠭⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⣒⠽⢯⣔⡒⠸⡆⠀⠠⣼⣿⣷⣾⣷⣶⣿⣷⠀⠀⣸⢓⠈⢥⣞⢓⣚⣭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠭⠿⡇⠀⣀⣹⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠧⠄⠀⣸⢛⣒⣲⠎
⠀⠀⡀⠈⠵⢌⠑⠢⣄⢀⣤⡀⠐⡲⠖⠒⢿⠿⣽⠛⠋⠙⡯⠍⠟⠋⢻⣿⣿⣿⣿⡟⠙⠉⠈⠉⡏⠙⠋⣽⠷⠦⠒⠶⢖⠂⢠⣤⢀⣠⠔⠊⡤⠮⠁⣀
⠀⠸⣇⠩⢷⠀⠛⣀⠀⢉⣈⣋⣀⣀⣂⣀⣈⣀⣙⠿⡄⡀⢹⡄⢹⡀⢘⣿⣿⣿⣿⡇⠀⠀⠀⢸⣀⢠⢿⣉⣀⣁⣀⣂⣀⣀⣛⣀⠁⠄⣀⠛⠀⡾⠭⣼⠇
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⢉⣡⡿⠋⣹⣧⡄⢧⠸⠿⠿⠟⠻⠇⠀⠀⣠⢾⡙⢿⣯⣉⠈⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⣀⣤⢶⢣⡟⢣⠎⠀⡿⣒⠌⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢶⠳⣹⢳⢦⣀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠴⠞⢛⣚⣼⡞⢠⠇⠀⠀⠁⠘⣄⠚⣧⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⡆⢹⣎⣜⣈⠛⠶⢤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡎⠈⠜⣈⡼⠛⠸⡗⡞⠀⠀⠀⢸⠀⠘⢄⠈⢢⣀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠚⡎⠘⠿⣌⡌⠆⠘⢦⡈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡼⡙⡠⣰⡿⠫⠀⠁⣸⡆⠇⠀⠀⠀⠀⠷⠀⠘⠖⠖⠷⠒⠀⠀⠆⠀⠀⠀⠀⢸⠀⣿⡀⠀⡈⠻⣖⡀⡎⡷⡌⢆
⠀⠀⠀⠀⠀⠀⠀⣎⡾⢁⢡⣳⠋⢡⠀⠠⠡⡍⡷⢲⠀⠀⠀⣠⠋⡖⣻⠞⠒⠒⠀⠀⠀⠙⡄⠀⠀⠀⡸⢲⠉⡥⡀⠀⠰⠙⢳⣳⢠⠱⣞⡄
⠀⠀⠀⠀⠀⠀⠘⢻⣇⠸⡾⡇⠂⠀⢣⢠⡝⠀⠸⡀⣧⠀⢀⡇⠀⡇⢸⠘⠀⠀⠀⢠⣿⣷⢻⠀⠀⣠⢃⡞⠀⠘⡕⢰⠁⢀⠃⠿⡎⡠⡏⣟
⠀⠀⠀⠀⠀⠀⢠⢸⣏⢷⢣⢠⠀⡘⣄⡛⠀⠀⠀⠙⢍⠳⣼⠁⠀⡦⣿⠀⠀⠀⠀⠀⠈⠸⠘⣇⠔⢡⠏⠀⠀⠀⠙⣣⡆⡌⠰⢰⢳⢇⣿⢘
⠀⠀⠀⠀⠀⢀⢸⣸⢹⠈⡘⡄⢇⢱⠏⠀⠀⠀⠀⠀⠈⠳⢎⠓⠼⣅⣻⢐⠀⠀⠀⠀⣀⡠⠖⣹⠖⠁⠀⠀⠀⠀⠀⠈⠿⢠⠃⡏⡄⣈⢹⣸
⠀⠀⠀⠀⠀⠰⡟⢹⢸⡄⣇⢳⣘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⢍⣉⣸⣇⣉⡩⠽⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣸⢁⡇⣿⢸⠹⠾
⠀⠀⠀⠀⠀⠇⡇⣾⠘⡇⠻⠼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢼⠀⡏⢸⡄⡇⡇
⠀⠀⠀⠀⢸⢰⠇⣿⠠⠱⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠁⢸⡇⣇⢰
⠀⠀⠀⠀⡇⣾⢠⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢡⢸⡈⡄
⠀⠀⠀⠀⠷⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠱⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣶⡮⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⡿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣆⣿⣿⠿⢛⣻⡄⠀⠀⠀⠀⠀⢀⡏⢀⡾⢁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠛⢛⡋⠽⠛⠛⢨⣭⡴⠖⢻⣭⣿⣿⡄⠀⠀⠀⠀⡜⢀⡞⢁⡾⢡⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠀⠀⠚⠟⠚⠃⡘⢯⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⡰⢁⠞⢠⠞⣠⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠠⢡⠏⡴⢃⡔⢁⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⡱⢣⠞⡰⢋⡴⢋⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣟⡵⡡⢊⠔⣡⠔⣋⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⡿⡫⢚⠕⣊⠴⢚⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣯⠊⠕⡈⠤⢚⣩⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢸⣿⣿⣿⣿⣿⣿⣿⡿⡿⠏⣼⣿⣿⣓⡠⠥⠒⣊⣩⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢿⣛⣛⡭⠽⠛⣩⡾⣴⣱⣿⣿⣏⣒⣂⣈⣩⣥⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠉⠉⠍⢒⡶⢊⣾⣜⣽⣿⣿⣿⣛⣒⠒⠒⠒⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠤⠔⣊⣵⣾⡿⡫⣺⣿⣿⣟⣓⡒⠀⠠⠭⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠋⣵⠿⢋⣬⣾⣿⡯⢽⣓⠢⠭⢭⣭⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣿⣿⢿⡺⠭⣓⡂⢭⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠈⠀⠁⡪⢥⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⣀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⡀⠈⣷⣄⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⠀⣼⣿⣷⠀⣿⣿⣿⠃⠀⠀⣶⠀⠀⠀⠀⣀⣴⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣶⣦⣌⢿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡏⢀⣠⣾⣿⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⢠⣄⣀⠀⢀⣀⣀⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣇⣤⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣤⣀⡈⢻⣿⣿⢟⣯⣶⣿⣿⣿⣿⣿⣷⣾⣝⠿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⡟⠉⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣽⣿⡿⣼⣿⣿⣶⣶⣦⣝⣿⣿⠿⢛⣛⠻⢿⣿⣎⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠻⠿⣿⣿⣿⣧⣿⣿⣿⣯⣿⣧⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢸⣿⣿⣋⣉⣀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣟⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠛⠉⣩⣵⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠉⠁⠀⠀⣼⣿⣿⠿⠿⣻⣿⣮⣝⡻⢿⣿⣿⣿⣿⡿⢟⣫⣾⣿⣿⣿⡿⠿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⡏⠙⢿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⣿⣿⣿⡏⣿⣿⣿⡟⣿⣿⣇⠀⠈⠻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⣿⣿⠏⠀⣿⣿⣿⡇⠈⠻⣿⣄⡀⠀⠈⠛⠛⠛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠘⣿⠀⠀⢿⣿⣿⣧⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠉⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⢀⣄⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠶⢤⣄⡀⢸⣿⣿⠟⠀⠘⢿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣡⣀⣀⣄⡄⢜⢩⡿⠲⠶⢦⣦⡀⠰⡨⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠁⣀⣤⣶⣶⣿⡿⠋⠉⢉⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠆⠙⢿⣿⣟⣡⣶⣿⡟⠿⣿⡿⠋⣤⣦⣀⣤⣿⣿⣷⣾⣿⡿⣿⣿⣿⣿⣿⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡀⠀⢀⣵⣿⣭⣽⣿⣧⣾⠏⠀⠀⠻⣿⣻⣿⡿⠿⢿⡷⠄⡀⢠⣿⣿⠿⠶⠞⠋⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⣠⣿⣿⢿⣷⡿⣿⣿⠋⠀⠀⠀⣴⣿⢟⣛⣛⣿⣷⣶⣶⣶⣶⠟⠁⠀⠀⠀⠚⠈⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢧⣰⣿⣯⣵⣾⣿⡇⣿⡏⠰⠯⣤⣈⠿⣿⣿⢿⣿⡿⢡⣿⣯⣿⣿⡀⠁⠀⠀⠀⢀⡀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡿⣩⣿⣿⣷⣿⠘⣿⣶⣫⠅⣿⣿⡗⣾⣟⣧⣿⡿⠫⣿⡇⡇⠀⢰⠄⠀⠘⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⣤⠾⢁⣿⠿⣿⣾⡇⣿⣿⣿⡿⣪⢿⡿⠛⡛⣿⣿⡇⢿⣷⣽⣿⣇⢻⡏⢏⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣎⠈⠀⢸⣿⣴⣿⣿⣿⣾⠿⣿⣿⣷⣟⠀⡄⣿⣿⡟⣧⠈⢧⠀⠙⣿⣿⣷⣿⣿⡆⠐⣄⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⣶⣶⣆⣾⣿⣿⣿⣿⠟⢁⣾⣿⣿⣿⣮⣾⡷⠃⢻⡇⡿⣦⠈⠳⡄⠘⣿⠙⠛⠋⠀⠀⢹⣻⣦
⠀⠀⠀⠀⠾⣿⣿⡿⢹⣷⣿⣿⢣⣶⣿⣿⡿⣿⢿⡿⢇⢰⠀⠀⣷⢻⣜⢷⣄⠐⢾⣿⣷⣤⣤⣤⣶⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢹⢄⠘⣿⣿⣿⣿⡿⢃⠈⠃⢿⠘⡇⣾⣌⠀⣶⡸⣧⢻⣷⠿⡆⠀⠙⣯⢻⣏⣩⠉⠀⠐⡇⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⢲⠂⠸⢂⡰⠀⢷⡘⡟⢂⣹⣿⣿⠋⢰⣮⣵⠀⠀⢹⢸⣿⣷⠀⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣾⢸⠃⢸⠃⢀⣿⠄⣷⣬⡻⣿⣯⠀⢸⣯⣄⣀⣠⣾⣿⣿⣿⣇⣴⣷⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⡟⠀⣿⣿⣴⢹⡘⣇⠘⢿⣆⠀⠙⠻⠿⠿⢻⣿⢿⣿⣿⢿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣼⡏⣹⢸⡈⣷⢻⣷⣤⣙⠛⠶⠖⠂⠀⢀⡄⢨⣿⣿⠀⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡷⣿⢾⣧⡘⢮⣿⣿⣿⣿⣶⣤⣤⣆⠝⣡⣿⣿⣏⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡏⣧⣿⣷⣄⣟⢿⣿⣿⣿⣿⣿⡏⣴⡿⣯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⣷⡀⠉⡙⠓⠓⠿⠷⡄⢸⣷⣿⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⣿⣆⠀⠀⠀⠈⢿⣿⣇⢿⣷⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠈⠻⢿⣶⣭⠟

⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣠⡀⢸⣎⣿⡿⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⡤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠂⢀
⠀⠀⣠⣈⡛⠄⠛⠟⠃⢞⣉⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠗⠁⠀⠀⠀⠀⠀⠀⠘⣋⣨⣌⣛⠂⠀⣰⣾⣿⣷⣦⣄⡀⠀⠀⠀⠀⠰
⠀⢀⣤⣬⠄⠐⠀⠀⠄⠀⣤⣤⡄⠀⠀⠀⠒⢂⠐⠀⠀⠀⠀⠀⠀⠀⣤⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠡⠌⠉⠀⢠⣦⣄⠉⠉⠻⠿⣿⣷⣄⡀⠄
⠀⠠⠤⡤⠤⠀⠀⠂⠀⢤⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣒⣒⡙⠆⠀⠀⠀⠀⠀⠀⡀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠈⠙⡿⠋⠀⠀⠠
⠀⠴⠶⠶⠶⠶⠜⡏⠴⠶⠶⠶⠶⠄⠀⠀⠀⠀⠄⠀⠀⣾⣿⣿⠟⠁⠐⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⣦⡀⠄⠉⠀⠀⠀⠁⠅
⠀⠀⠀⢛⣛⣛⡋⠇⣛⣛⣛⠃⠀⠀⠀⠀⠀⢈⣉⠀⢸⣿⣿⣿⣦⠀⠀⠂⠀⠀⣼⠋⢻⠀⠀⠀⠀⠐⠀⠀⢠⡄⠀⠀⠨⠄⠀⠀⠀⠀⠈⠀⠀⠀⢾⡄⠀⠅⠇
⠀⠀⠀⠈⠭⠭⠇⠀⠭⠭⠍⠀⠀⠀⣀⠔⠄⠀⠂⢀⣿⠿⠛⠛⠿⣷⡀⠃⠀⢠⣈⣀⣈⣀⠀⠀⠀⡄⣰⡀⠈⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠃⠚⠀⠀⠅⠅
⠀⠀⠀⠀⠘⠛⠃⠀⠛⠛⠀⢀⢀⡃⠋⠀⡀⠙⠂⠈⢠⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⢠⠀⠀⠀⠀⠼⠛⠉⠉⣶⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢢⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡎⠈⠁⠀⠀⢀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⠀⠈⠳⣄⢀⡄⠀⠀⠀⡔⠀⣰⡞⠋⠙⠢⠀⠀⠨⠭
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⢉⠀⠀⠀⠀⠠⠤⠀⠀⠀⣸⣿⣀⠀⠀⠆⠀⠀⠀⠀⣄⠀⠀⠀⠈⢬⠃⠀⠀⣀⣠⡾⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣠⠀⣄⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠠⠉⠉⠀⣀⠀⠀⠀⠀⢷⡷⣿
⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠈⠠⣄⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣁⡀⠑⠀⠀⠀⠀⠀⠿⠿⠟
⠀⡐⢤⠀⠀⠐⢀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⢠⠖⡩⠭⢉⠢⡀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡐⠀⠀⠀⠀⠀⣠⠄
⠀⠈⠈⠁⠀⠀⠀⢀⡀⠚⢋⣉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡇⠀⠀⠀⠀⡃⢱⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣘⡛⠓⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠖⢂⡀⢀⠀⠀⠀⠀⠁⠃⠀⠘⡣⡐⠄⠀⠄⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠦⠈⠁⠄⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁⣰⠁⠔⢠⠖⣥⠊⡔⣠⢶⠄⣌⣏⣒⣝⣄⢠⢦⡀⢄⠠⡔⠢⠐⢤⠙⠂⠱⠀⠁
⠀⠀⠀⢠⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠉⠈⠈⠀⠉⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠀⠒⠀⠓⠰⢦
⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⣠⠴⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠆⡀⠀⠀⠀⠐⠒⠄⠀⠀⡄⠾⠀⠀⠀⠀⣿⠃⠈⠠⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠿⢛⣿⡇⠀⠀⠀⠐⠁⠀⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⢄⠲⠈⠆⠀⠀⠀⠀⠀⡈⠀⠀⠇⠐⠂⡤⠀⠀⠨⠀⠈⠁⠀⣄⣀⣀⣠
⠀⠐⠠⠀⠛⠃⠀⠄⠀⠀⠀⠀⠀⢤⡀⣀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⣀⠀⠶⠐⢠⠆⠄⢈⡄⢯⡀⠄⢧⠀⠰⠆⢀⡀⠀⠀⠀⠀⠄⠀⠛⠛⠛⡟⠀⠀⠠⠆
⠀⠀⠀⠰⠀⠐⡆⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠒⠂⠀⠸⡔⢧⡀⠀⠈⡰⠟⡼⠠⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠁
⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢸⡇⠻⠆⠀⠀⠀⢉⢀⠖⣹⠄⠃⠈⠁⠀⢶⡑⢇⢈⠁⠀⠀⠀⡼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠸⠋⣄⠀⠀⠐⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠐⠉⠈⠀⠈⠠⢈⠀⠙⠊⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡐⢂⡀⠀⠀⠀⣘⠘
⠀⠐⠐⠀⠀⢘⡇⠀⠂⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⡄⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠊⣵⡂⠀⠀⠀⠀⠀⠀⠸⠃⠸⠏⠶
⠀⠀⠀⠀⣷⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢴⡷⠀⠀⠀⠀⠀⠔⠦⡌⠀⠠⠨⠆⠀⠀⣤⣄⣤⣤⠀⠀⠀⠲⠀⠀⠁⠂⠂
⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠰⡄⠀⢀⠀⠈⠙⣦⠀⠀⢠⡘⠂⢄⠀⠠⠠⠀⠀⠲⡀⠁⠿⠷⠁⠀⠀⠀⠀⠋⠢
⠀⠀⠀⠀⣿⢸⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠁⠆⠠⠀⠀⠀⠁⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠿⠸⠄⠀⠀⠀⢠⠀⠤⠀⠀⠂⠀⢸⠇⠀⠀⠀⢀⡀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠉⡀⡀⠓⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠋⠀⠙⠋⠀⠛⠋⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣃⣈⣷⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⣫⢼⣾⢼⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⡠⢷⡈⠾⠿⠞⡿⢇⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢒⠟⠀⠘⢸⠘⠂⠀⠚⢩⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠮⡇⠀⢀⡘⢠⣶⠀⢀⣴⣧⢠⡀⠀⢲⠩⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣠⢖⠉⠐⢘⡿⡀⠘⣻⣏⡎⠑⠀⢈⠆⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⢵⠀⠄⢠⠃⡀⠉⠀⡀⢀⣧⢈⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢆⠀⣼⠃⠐⠮⡶⢴⠓⠏⠙⠫⠎⣍⣩⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⡸⣌⠸⣛⠀⠀⠀⠉⠛⠊⠀⢀⣸⡋⢀⡔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠣⠅⠈⢤⡄⠀⠉⠑⢶⣤⣥⣤⣤⡴⠏⠡⠈⠉⢨⠊⢱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠱⠊⠐⠀⣀⠀⠀⠀⠸⡂⡝⡀⠐⣄⠀⠠⡄⣰⡿⠒⢈⠚⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡒⠤⠜⠀⠀⠀⠐⢺⡇⠘⢆⣬⠆⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣕⠖⠂⠀⠰⠀⠀⠀⢿⠀⢨⠁⠀⠀⢈⠟⢃⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⣀⣀⡐⠁⡰⠂⠸⠇⠲⢆⣠⣂⣠⠄⠥⠔⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠝⡒⠣⠔⠢⣭⣓⣈⡠⠟⠐⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣛⣻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣟⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⠤⠼⣧⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣼⠧⠤⢼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡾⠛⠛⠯⣵⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣿⠵⠛⠛⢷⣶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠦⠤⠤⠬⢦⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⡴⠥⠤⠤⠶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣙⣦⣀⠀⠙⢷⣄⠀⠄⢠⠀⣠⡾⠋⠀⣀⡴⣋⣀⣀⣀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠙⢧⣾⣿⣼⠋⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⣼⡟⢹⣷⣥⣄⣤⣤⣶⣶⣶⣶⣶⡶⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠃⠉⢋⠭⠉⠉⠉⡈⢿⣿⣼⣿⢁⠈⠉⠉⠤⣚⠉⠉⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠀⠀⡠⠎⠀⢀⡤⠊⡀⠋⣿⣿⠉⢀⠁⠀⡀⠀⠘⢤⡀⠀⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠔⠂⢀⣄⣠⠤⠲⠋⠀⠐⢀⠂⣿⣿⡀⠀⠣⠀⠈⢓⠠⢄⡠⠀⠀⠂⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⠊⠀⡌⠀⣴⠃⢀⠎⢀⢂⠎⢸⣿⠐⠔⠀⠱⡀⠈⢦⠀⠱⡀⠀⢦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣞⠀⣠⠁⢀⠎⠀⡠⠂⠀⠘⠏⠀⠈⢦⠀⠐⡄⠀⣃⠀⣱⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣤⡾⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⢻⠟⠀⠉⠋⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⢻⣿⣿⣿
⣿⣿⣿⢿⡿⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠹⠿⣿
⣿⣿⢥⠟⠓⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣏⠛⠓⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢳⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠹⠟⠛⠻⠛⠉
⠘⠂
⡌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠒⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣤⠀⠀⠀⠀⠑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣄⠀⡀⠀⠀⠀⠈⠉⠐⠀⠀⠠⠀⠀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠀⠀⠠⠀⠔⠃⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣠⡼⠚⠋⠀⠀⠀⠀⠀⠀⠀⠐⠉⢿⣷⣀⢠⠀⡛⢦⣄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⠞⠉⣰⣿⣾⡒⠂⠀⠀⢀⣀⣠⣤⣶⣾⡟⠿⠛⣤⢻⣦⢈⠳⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣾⣃⣿⣿⣿⣧⡴⣴⣟⡻⠛⠉⢹⣿⡇⠀⢰⡄⠻⣿⣿⡏⣷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣰⣿⣿⢻⣿⣿⣿⠃⣠⣿⣏⡡⠀⠐⠊⠉⠠⣄⠈⢹⡄⢹⣿⡧⢿⣿⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣰⣿⣿⡋⢠⣭⢿⣿⣰⡟⠧⣽⠁⠀⠀⠀⠀⠀⠈⠳⣄⢻⣆⣩⡇⢈⣻⣿⣧⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣿⣿⣥⠱⣺⡟⢠⣿⣿⢇⡼⣯⣀⠀⠀⠀⠀⣀⣠⣀⡉⢀⡻⣿⣷⠈⣭⣿⣿⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⡿⢦⣹⢧⣿⠛⠁⢠⣤⣄⡉⠛⠀⠀⠘⠉⣠⣤⣄⠙⢿⡟⢿⣶⣿⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡇⢸⣿⣷⣶⣶⣿⡺⠁⢠⡶⠛⠻⠿⠿⠀⠀⠀⠀⠚⠿⠿⠋⠀⢀⡃⠀⠀⢿⣾⣿⣿⢸⡄⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⣿⣿⡟⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡀⠀⠀⢸⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⢿⣿⣿⣿⡟⠊⠻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠄⠀⢀⠀⣸⣿⢿⠆⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠘⢋⣡⣶⣾⣿⠀⠀⠆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠀⠀⢂⣴⣎⡙⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⣿⣿⢿⡇⢠⣾⡻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⣾⣿⣿⠋⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⠻⡿⢿⡇⢼⡈⠃⣸⠂⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⡄⣿⣿⠟⢡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠳⠤⠿⣹⣏⠇⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⢃⣿⣧⡖⢡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠾⠃⣴⣴⡟⠸⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣯⠔⣧⠘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠰⠁⢸⠀⠃⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⡄⡄⠸⡷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠏⠈⢉⠃⠀⠀⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠼⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠄⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢠⡾⣠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠀⠊⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣃⣼⣯⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠠⠀⠁⠢⠀⡒⢠⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢔⠀⠉⠉⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢰⠀⡦⠈⠅⢱⠘⡇⠀⠀⠀⠀⠞⠛⡫⠅⠁⠥⣉⠚⢇⠀⠀⠀⠀⠀⠐⢁⣵⡖⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠔⢒⡓⣻⠀⠀⠘⠀⢀⡀⢀⠀⡄⣿⠀⠀⠀⣸⣁⣫⠀⠁⠀⠉⠨⣇⣘⡄⠀⠀⠀⠀⠀⢸⡿⠀⠙⠀⠀⡖⢸⡀
⠀⠀⢠⠐⠃⠀⠀⠠⠀⠀⠀⠇⠘⡂⠘⠀⠁⢸⡄⠀⢠⠉⣉⣉⢣⡀⠀⣰⣋⣉⢉⢑⡀⠀⠀⠀⢅⠀⢻⡄⠀⠈⠑⢣⢼⡇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢇⠀⡀⢰⠘⡇⠀⠈⠓⠦⠤⢼⠃⠀⢻⠤⠠⡴⠋⠀⠀⠀⡰⣿⠀⠈⣷⣄⣠⣷⣇⠀⢻⡆
⠀⠀⢀⠀⠀⢹⡖⠀⠀⠀⠀⠸⠀⠈⠀⠀⠘⡀⣷⠀⠈⠙⢱⠴⢸⣄⠀⣴⠶⢲⠁⠉⠀⠀⠀⠀⢻⠃⠀⢰⠈⠩⠁⢸⠈⠘⣿
⠀⠀⠀⠠⠀⢸⣿⡆⠀⠀⠀⠀⠆⠈⠅⠸⠀⠃⢹⠀⠀⡀⡌⡗⣸⣽⣾⢼⣋⡘⣠⢠⠀⠀⠀⠀⠸⠀⠀⢸⠀⠈⠀⠈⡀⢤⢿
⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠠⠖⠒⠒⠒⢾⠀⠀⠳⢭⣟⣳⣶⣷⣶⣟⣻⠵⠃⠀⠀⠀⠢⠤⣇⣶⣭⣤⣤⣤⡤⠤⠤⠚
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡀⠀⢤⠀⠀⠀⠀⠀⠈⠉⢉⣩⣤⣄⣀
⠀⢸⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⢚⡯⡀⠀⢸⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤
⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠤⢠⣆⡀⠀⠀⠀⢀⢣⠘⡦⠞⠛⠩⠈⢹⣧⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠛⠣⣄
⠀⡟⠉⠁⠈⢁⣀⣠⠄⠀⠀⠀⠁⠀⠀⠀⢀⠸⣿⣿⠃⠀⠀⠈⠋⠀⠻⡄⠀⠀⠀⢐⠟⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡟
⠐⣇⣀⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠠⠌⠛⠉⠈⠀⡀⠀⢆⣀⠀⠐⣓⠀⢲⠀⣮⣄⡟⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⡇
⢠⠋⠀⠀⠈⠇⠀⠀⠀⠀⠀⠐⠆⢄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢈⢠⠿⠦⠍⢇⠀⠂⡷⢼⣇⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⠉⢧
⢸⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡞⠀⠀⠀⠘⣄⢀⠰⡢⢼⡆⠂⣿⢼⢿⡄⠀⠀⡐⠀⠀⡤⡈⣆⠠⠄⣼
⠸⡀⢸⠀⠀⡄⠀⠀⢐⠂⠀⠀⠀⠀⠀⠱⠒⠒⢾⡇⠀⠀⠀⠀⠹⣦⣓⣷⣮⣧⣤⣽⣫⡿⠀⠀⠠⡙⠀⠀⢇⠉⠀⠉⠀⢰⡇
⠀⠀⠈⠑⠤⡏⠉⠁⠙⠁⠀⠀⠀⠀⠀⠠⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢋⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣸⡗
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⣤⣭⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠠⣧⡀⠐⠤⡄⠀⠙⡀⢹⡆⠠⣿⣧
⠀⢠⠔⢲⣟⡇⠀⠀⠀⠀⠀⠀⠀⣺⣭⣭⣿⡿⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣀⣀⣿⡧⠀⠀⠰⡀⠀⢣⠈⢧⠰⣿⡿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠀⠈⠙⠿⡇⠀⠀⠀⢳⠀⠘⡆⠸⣆⣿⣿
⢰⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⢀⣀⣾⡇⠀⠀⠀⠠⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢆⡀⢹⡄⠀⣠⣿
⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠈⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⢰⠀⠈⢿⣿⡧⠀⠹⣿⣿⡄⠀⠪⠀⠀⢀⠀⠘⠉⠀⠀⠀⠀⠀⠂⠀⠀⣀⡀⠀⠒⠂⡀
⣓⣀⣀⣚⣒⣂⣈⣂⣀⣀⣀⣈⣀⣀⣀⣩⣭⣤⣤⣬⣼⣃⣀⣀⣀⣀⣀⣀⣉⣉⣓⣒⣢⣤⣴⣢⣦⣶⣛⣛⣘⣚⣚⣃⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⢯⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠚⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⡙⠻⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣆⠙⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⢰⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣧⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡿⠃⢀⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠠⣘⣿⡿⠟⠋⠀⣠⣾⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟