Cuff Emojis & Symbols

⠀⠀⠀⠀⠀⢰⣩⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣰⠷⢦⡄⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣴⣺⣟⣺⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠋⠉⠉⠉⠉⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠸⡀⠈⠻⡣⡖⢫⡇⣑⣾⣭⣇⡁
⠀⠀⡴⣫⠝⠓⠛⠑⠒⠯⣕⢦⠀⠀⠀⠀⠀⠀⠀⢀⡷⡇⢀⠤⠒⠒⠚⠉⠉⠓⠲⡄⣷⣵⠀⠀⠀⠀⠀⠱⣄⠀⣡⡷⢀⡾⠓⠈⠉⠐⠻⣧
⠀⠐⣵⠁⠀⠀⠀⠀⠀⠀⠈⣇⡇⠀⠀⠀⠀⠀⠀⠈⣧⠟⣧⡀⠀⠀⠀⠀⠀⠀⠀⡸⣇⡸⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⢧⣀⠀⠀⢀⣠⠟
⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⣀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⣹⠓⣤⠖⠰⣤⣴⠛⣇⡼⠁⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠈⠉⠉⠉⠁
⢠⠟⣡⡌⠀⠀⠀⠀⠀⠈⠊⠠⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢤⡿⣦⣀⣸⣿⠴⠋⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⣱⠂
⠸⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠦⢽⠋⠀⠀⠀⠀⠀⠀⠠⡲⢶⡄⠀⢡⡴⠋
⠀⠙⠲⢤⣀⣀⣀⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⡆⠀⠀⠀⠀⠀⠀⢰⡄⠀⣥⠀⡟⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⣇⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⡜⠙⠋⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣾⠀⠀⠀⠀⠀⠀⡤⢲⣞⣇⠱⢄⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣴⣡⣢⡀⣗⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠹⡀⠀⠀⠀⠀⢘⠀⠀⠉⣼⡆⠀⠀⠈⠀⠀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠜⣹⠤⡴⢋⣽⠿⣯⡈⢦⠤⢾⢣⠀⠀⠀⠀⢰⣶⣶⣶⣶⣇⠀⠀⠀⠀⠈⢦⠤⠬⠟⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠨⠇
⠀⠀⠀⠀⢀⡴⠁⡴⠁⠒⠷⠿⠷⠶⠾⠿⠴⠒⠈⢆⠳⡀⠀⠀⡘⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠒⠦⠔⠺⠆
⠀⠀⠀⠀⠘⡍⠉⠁⠈⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠇⠀⠀⠿⠛⠛⠛⠛⠛⢧⠀⠀⣰⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⣀⣰
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⣄⣀⣀⣀⣀⣀⡞⠀⠀⢸⣾⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠁⢨
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢫⡉⠉⢩⠏⠀⠀⣠⢊⠩⠤⢬⣉⠲⡄⠀⠀⠀⠀⠀⢘⠉⠧⠖⠘
⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠼⠀⠀⠀⠀⠀⠀⠀⠱⡿⠃⠀⠀⢰⣇⡏⠀⠀⠀⠈⢧⢹⡀⠀⠀⠀⣰⣫⣤⣤⣤⣬⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣀⠀⠀⠀⢀⡞⣼⠁⠀⠀⢀⣿⡿⠛⠉⠉⡛⣿⣇
⢀⡞⣭⠛⠛⠲⡦⡤⢦⡴⢚⡉⠋⠰⡦⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢬⣁⣀⣰⡭⠈⠁⠀⠀⠀⣿⡟⠰⢶⣤⣾⠟⠘⣿⣿
⠸⡇⢁⣄⠘⡁⠖⣿⣰⣎⢈⣅⠘⠃⣶⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡡⠀⠀⠈⠀⢀⡜⡽⠉
⠀⣿⠆⢁⡘⢋⡼⡿⠉⢺⡛⢥⡛⢀⣤⠎⠀⠀⠀⣀⡴⢚⣋⡦⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠱⢭⣂⣤⣐⣯⡞⡇
⠀⠈⠳⣞⡛⢛⠟⠀⠀⠀⠙⠣⢽⡯⠋⠀⠀⣠⠞⡡⣾⢩⠏⠀⠀⠀⠀⢀⢴⡫⠂⢹⢋⠀⢨⣹⡍⠪⣢⡀⠀⠀⠀⠀⠘⢾⠀⠀⠀⢸⠟
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠊⢸⠃⡏⠀⠀⠀⣠⢐⡕⡉⠀⡀⠈⠉⠀⠀⠙⠁⢀⣈⢪⡢⡀⠀⠀⠀⢸⠒⠚⠂⢸
⠀⠀⠀⠀⠀⢀⣀⠤⢲⣇⠀⠀⠀⠀⡰⢋⠾⠥⢄⡈⢸⠁⠀⢠⡾⣳⣿⣿⣿⣿⣯⣷⡄⢀⢔⣭⣿⣿⣿⣯⣭⣻⣦⡀⠀⢸⣀⣀⡀⢸⠁
⠀⠀⠀⢸⠉⠀⠀⠀⠀⠻⣦⣀⡤⢊⡴⠋⠀⠀⠀⢳⠘⡆⠀⠻⣄⣿⣿⣿⣿⣿⣿⣿⡟⠊⣿⣿⣿⣿⣿⣿⣿⣟⡼⠁⠀⢸⠁⠀⠀⢸⠄
⣶⣶⢶⣿⣶⡶⠶⠶⠶⠾⢛⣩⠔⠋⠀⠀⠀⠀⠀⢸⣦⡁⠀⠀⠈⠳⣝⡻⠿⠿⠿⣟⠗⠙⣟⠿⠿⠿⠿⣛⡭⠚⠀⠀⠀⢸⠄⠒⠦⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡎⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡞⠀⠀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⡞⣤⣰⣔⣾⡄⠀⢠⠀⠉⠒⠦⣀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡾⢹⣟⣷⡾⣿⣷⣷⣿⣿⣴⣦⣤⣀⡁⠢⢀
⠀⠀⠀⠀⡼⠁⠀⠀⢠⠏⢈⠱⣱⡀⠑⢌⠋⣿⡿⢻⣿⣿⣿⢿⣿⣷⣦⣑⠢⣄
⠀⠀⢀⠜⠀⠀⠀⢠⠏⠀⢆⠀⢈⠉⠀⠀⠀⠉⠈⠃⠀⠈⠃⠈⠈⠹⣼⠟⠑⠈⠓⠒⠀⠤⢀⡀
⠀⣰⠃⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⢄⣀⣀
⣨⣁⣽⣄⡰⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠒⢢⣽⣟⠭⢛⡡⢲⠦⠀⠀⠀⠀⠀⠈⠉⢹⣶⣶⣴⣤⣤⣤⣄⣀⠀⠀⠀⣀⣀
⠙⢿⣷⡖⣲⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣿⣿⣿⢿⢹⡟⣿⡿⣟⢿⣿⠿⢿
⠀⠀⠀⠻⣷⣷⣿⣬⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣿⣿⢶⣿⢸⣷⣽⣿⣼⢺⣿⣷⢽
⠀⠀⠀⠀⠀⠙⠻⣿⣯⣿⣴⣤⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣟⣛⣎⢸⣿⠿⠿⢿⣹⣿⣟⣾
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣿⣽⣿⣷⡶⠇⢌⣑⣂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣋⢿⣿⣸⣏⠿⣿⣿⣻⣿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣭⣷⣶⣖⣼⡐⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣭⣧⡟⣿⣜⠿⣏⣿⣿⣯⡺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣶⣿⣬⣷⢾⣉⠤⣒⢦⣒⠔⡂⢀⠔⠀⡠⣀⢔⠄⡠⣢⣘⢿⡟⣭⣝⢻⣿⣿⣿⣛⢿⣿⣿⣟⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣶⣿⣿⠯⣒⣽⣺⣵⣴⣯⣾⣿⣷⣯⣾⣿⣯⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⣠⣼⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠰⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⢹⣯⣿⡿⠛⠛⠻⢷⣄
⠠⢦⠖⠠⢿⣭⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣠⣾⠟⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⣿⡆
⢿⣅⠀⠀⠈⠉⠉⢀⣀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠛⠻⣶⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢠
⢾⣿⡄⠀⠀⠀⠀⠀⠙⠛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⢀⣼⠃⠀⠀⠀⣠⡶⠻
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⢀⣼⡟⠀⠀⠀⢀⣴⡾⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠞⠋⠁⠀⠀⠀⢰⡏
⠀⠀⣿⠇⠀⠀⠀⢀⡴⠛⠋⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠛⠋⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⣴⡾⠋⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠙⠷⣤
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⣼⣿⣾⠋⠓⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣶⣾⣿⡏⠀⠀⠀⠀⢀⣴⠾⠟⠋⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⡶⠿⠋⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡟⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢻⡄
⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡔⠆⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⣤
⠀⠙⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠋⠁⠀⠈⠙⣿⡄⠀⠀⠀⠀⣠⣴⡶⣶⣿⣟⣿⡁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⡄⠐⠛⠁⠀⠀⢠⡾⠉
⠀⠀⣿⣟⣿⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠏⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⠿⠃⠀⠀⠀⠀⠀⢸⡇
⣠⣼⡿⠛⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠕⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀
⠋⠁⠀⠀⠀⢀⣼⡷⣿⣤⣄⡀⠀⠀⠀⠀⠙⠷⢤⣤⠶⠟⠉⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣴⣶⣶⣶⠶⠶⣦⣄⠀⠉⠙
⠀⠀⠀⠀⠀⢀⣿⡿⠛⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢀⣴⡶⠶⢶⣦⡀⠀⠀⠀⠸⣿⣾⠏⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠈⠿⣤⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⣤⣀⣠⣴⠟⠁
⣤⣄⡀⠀⠀⠀⠈⠛⠿⠶⠿⠋⠁⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠦⠐⠠⠛⣷⡿⠇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣀⣤
⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣀⣤⡾⠋⠀⠀⠀⠀⣠⣼⣯⣄⠀⠀⠀⠀⠀⠺⠶⢶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠖⠸⠛⠿⠋⠉⠉⠉⠀⠀⠀⠀⢀⣴⠿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠈⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣠⣤⣴⠿⠁⠀⠀⣾⠋⠀⠀⠀⠈⢿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⠀⣀⣀⣤⣶⠶⢶⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⣸⣷⣿⠋⠚⡥⠀⠀⠀⠀⠀⠀⠀⠀⢒
⢿⣟⣿⠏⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣦⣀⠀⢀⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⣀⣠⣴⡿⠃⠀⠀⠀⣲⣶⣶⣄⣀⡀⠀⠀⠀⠈
⠙⢻⣟⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠈⠉⠛⠙⠉⠀⠀⠀⢀⣴⠾⠛⠳⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠈⢿⣷⠟⠋⠉⠛⢷⡄
⠀⠈⠻⢦⣄⣀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣶⣤⣴⡾⠛⠀⠀⠀⠀⢀⣤⠶⠾⠷⣦⣤⣤⣀⡀⡄⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠿⠋
⠀⠀⠀⠀⠀⠀⣴⡟⠛⠉⠛⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣙⡻⠿⠋⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠘⣿⣽⣿⠉⢑⣣⣀⣤⣀⣀⣀⡀
⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣶⣶⣦⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣙⠛⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢻⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⣰⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢀⣼⣿⣿⣿⣿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⢀⣾⣿⣿⣿⠟⠁
⠀⣾⣿⣿⣿⡏
⢸⣿⣿⣿⡿
⣾⣿⣿⣿⡇
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠈⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇
⠀⠈⢿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣦⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠄⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡴⠚⠋⣀⡃⣤⡺⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⣠⠜⠋⠁⣠⣰⠞⠋⠉⡻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⠀⠈⠀⠀⣶⡵⠋⠀⠀⡀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣤⡴⠿⠶⢦⡄⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠘⠁⠀⡀⠠⣀⢹⣷⣆⣤⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠞⢁⢀⡄⠀⠀⡝⠳⣦⣤⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢳⢭⡄⠀⠀⢠⢀⡷⠋⠁⠀⠙⢽⣆⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⢴⠿⡚⠣⢴⣶⢤⡀⠊⠙⢷⣦⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠘⠓⡀⠀⠈⡡⠖⠋⣀⣤⡀⠘⣯⡇⠀⠀⠀⠀⠀⠀⠀⠁⢔⡿⠁⠘⢿⠀⠀⠙⠲⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠀⠀⢉⡫⠟⠁⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⡸⢡⣴⠷⠆⠀⠄⠺⣶⣤⢉⠋⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡰⠇⣿⠿⢧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠋⢀⣀⣀⠀⡀⠀⠨⡙⠿⣧⠀⠀⢟⡄⠘⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠁⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠛⣋⡀⠾⣿⣦⠀⠀⠀⠈⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠓⠄⠋⢿⣦⠈⠛⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡸⢀⣿⠇⠁⠂⠀⠀⢠⠝⡢⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⢀⡴⠐⠠⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⡄⠋⡤⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣺⠀⢈⡰⢅⠀⠀⠀⠀⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡱⠀⠀⠀⠀⠀⠀⠀⢴⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣇⠘⣏⠊⠀⠀⠀⠀⠀⠐⢄⠌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢈⠀⠀⠀⠀⠀⠀⠐⣙⡄⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⣞⢆⠘⢦⡀⠀⠰⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄⠀⠀⠂⠀⠀⠀⠘⢡⣴⢿⡆⡌
⠀⠀⠀⠀⠀⢀⡠⡔⠁⠀⠙⣿⠫⣵⣄⡑⠤⣀⠀⢀⠀⡠⡀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⠀⡂⢰⡀
⠀⠀⠀⡠⠀⢀⢿⡱⠡⠀⠀⠈⠁⠈⠛⢽⣦⣌⠁⠰⠂⠞⠴⢃⠗⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⡠⢀⢾⠁
⠀⡠⠈⡀⠀⠙⠀⠉⠓⢤⡘⠐⠀⠀⠀⢀⣍⠛⡯⠶⣶⣶⣶⡦⣄⡀⢀⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⠀⠔⠀⢀⠔⡰⠊⢐⣴⣿⢷⣀
⠊⡀⢘⠄⠀⠀⠀⠀⠀⠀⠀⣀⠢⠄⢠⣉⢟⣾⠀⠀⠀⠀⠀⠉⠚⠟⠉⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⡐⣥⢶⣶⣾⣦⠀⠀⠔⠠⠂⠘⣀⢴⣿⠏⢫⠖⠛⠡⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠒⠐⢄⡀⠀⠂⠀⢈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣁⣀⣠⣤⡶⢿⠟⡽⠁⠄⠁⢀⢀⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠀⠈⣀⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠦⣄⠀⣠⡶⠛⣿⡷⠶⠛⠋⠉⠉⠙⠛⠛⠉⡠⢈⠈⠀⡠⠂⠄⢑⣠⠊⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⡋⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠠⡴⠛⠀⠀⠀⠠⠞⢇⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⡂⠀⠰⠂⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠠⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠉⢩⣷⣄⣴⡖⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣶⠛⠷⠶⠛⠋⠈⠿⣥⣀⣾⣧⡀⠀⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⣽⡿⢿⣿⣿⣿⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠛⠛⠛⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁
⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠘⢿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣦⣤⣤⣦⣾⣿⡿⠟⠁
⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠉⠉
⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠘⢿⣿⣷⣤⡀⠀⠀⠀⢀⣀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠻⢿⢽⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⢞⣩⣤⣾⠾⠿⠿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⠿⠷⣿⣤⣭⡳⣦⡀
⠀⠀⠀⠀⣰⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⣆
⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣀⣀⣀⣀⣀⣤⣾⣿⡿⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣕⢿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⣏⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣩⣿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡀⠉⠋⠀⠀⠀⠁⠘⠋⢁⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡄⠀⠀⠀⠀⠀⠀⢠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣦⣄⡀
⠀⣠⣶⣿⣿⠟⠋⠉⠀⠀⠈⠉⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠉⠉⠉⠉⠙⠻⢿⣿⣶⣀
⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠙⢿⣿⣦⣤⣄⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣿⣀⣽⣿⣿⣿⣿⡏⠙⣷⣤⣤⣤⣤⣤⣾⠛⢙⣿⣿⣿⣿⣉⣹⣿⣿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠿⠋⠁⠈⠛⠷⣿⣅⣀⣿⣅⣠⣿⠷⠟⠁⠀⠙⠿⠿⠟⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠭⠭⠉⢉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⡴⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⡼⢀⠤⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⢠⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⢸⡰⠀⢀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⡇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢧⠐⠀⠀⠀⡴⠒⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠈⣆⡆⠀⠀⡇⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠈⠑⢶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠈⠹⡄⣠⠇⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠈⠓⢤⡀⠀⠣⡀⠀⢀⡤⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠑⠢⢄⡑⠋⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⢀⣀⣀⣀⣀⣀⠀⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡾⠿⣿⡏⠀⣠⣶⡾⠛⠛⠉⠙⠛⠛⢿⣶⣄⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠉⠁⠀⠀⢀⣿⣷⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣦⢸⣿⠀⠀⠀⣠⣤⣀⡀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣿⣯⣿⣿⡿⠿⣿⡇⠀⠀⠈⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⣻⣿⠋⠉⠁⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣷⣀⡀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠁⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣿⣇⣈⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⢿⣦⣀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣴⣶⠿⠛⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠉⠉⠉⠉⠁⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⣶⣶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣠⣾⠿⣷⣾⣿⣿⡿⠿⠿⠿⠿⠿⣶⣦⣤⣀⠀⠀⠀⠻⣷⡀
⠀⢀⣾⠟⠁⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠹⣿⡄
⢀⣿⠏⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠹⣷
⣸⡏⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⢿⡇
⣿⡇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⣿⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⣸⡿
⢸⣿⡀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢠⣿⠇
⠀⠻⣷⡀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⢠⣾⠏
⠀⠀⠙⢿⣦⡀⠙⠻⢷⣶⣦⣤⣀⣀⣀⣀⣀⣤⣴⣾⠟⠁⢀⣴⡿⠃
⠀⠀⠀⠀⠙⠻⣷⣤⣾⡟⠉⠉⠛⠛⠛⠛⠛⠉⠉⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⢌⡖⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠣⣤⠒⠲⢤⣐⡄
⠀⠀⠀⠀⠸⠀⠈⣠⠴⠲⠾⣁⣀⠄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣶⣶⠛⡴⠒⠦⢤⣰
⠀⠀⠀⠀⠸⡔⠊⢁⡤⠔⠢⣽⣇⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢹⣷⠞⢀⡴⠋⠐⠀⠇
⠀⠀⠀⠀⠀⣄⠀⠁⢀⣠⠧⢈⣿⡦⠀⡈⠀⠀⡴⠀⠀⠀⠀⣿⠀⠀⣸⣿⣶⣏⠀⢾⠇⢠⠊
⠀⠀⠀⠀⠀⠘⢆⠀⡹⠷⠞⠻⠻⣷⣰⠁⠀⢀⡄⠀⠶⠀⠀⠹⣆⠰⠋⠉⠀⠈⠑⠈⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⣳⢿⢀⡴⢯⡴⠢⢤⣀⣀⣤⣼⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠁⠸⡞⢳⠋⠀⠀⠶⠫⠵⠾⢻⣷⣄⠀⠠⣼⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⣀⣣⠏⠀⠀⠒⠀⠀⠀⠀⢸⠈⠙⠻⢤⣘⣊⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠏⢠⣤⡾⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣿⣷⣦⣀
⣿⣿⣿⣿⣷⣦⣄
⢿⣿⣿⣿⣿⡿⠻⣷⡢⣀
⢈⡋⠿⠝⠯⠀⠃⠀⠀⠙⠷⣦⣄
⠀⠀⠀⠐⠃⠀⠀⣀⣣⣶⣤⣽⣿⣿⣶⣄⠀⠀⠀⠀⣠⠤⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠚⣹⣿⣿⣿⣿⡾⣿⡿⠁⠀⠀⠀⠇⣤⣴⣀⣤⣀⠈⠑⢦⣀⡀⠀⣀⣴⠾⠿⠦⡀
⠀⢀⠃⣀⠠⠤⠐⠒⡿⠟⠛⣋⣉⣽⠟⠀⠀⠀⠀⠀⠾⠿⠋⠈⠉⠉⠓⢶⡏⠛⠻⣿⣿⡏⠀⠀⠀⣹⣦⡀
⣼⣿⣿⣾⣿⣷⣶⣦⣶⣾⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⣿⣿⣿⣦⣤⣾⡿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠈⣿⣿⣿⡿⡟⠉⠀⠘⠃
⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡟⠁⣠⠶⢄⣠⠒⢁⡼⠒⣆⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠋⢀⡾⠛⠁⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠙⠛⠿⣿⡿⠀⠀⠻⠞⠋⢿⣷⡏⢁⣴⢋⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⢁⣼⠃⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⢤⠀⠀⠈⡳⣞⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣈⣏⣀⡀⠀⢀⣄
⠀⠀⠀⠓⢾⠲⠚⠉⠁⠈⠉⠓⠖⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⢊⣭⠀⠶⠀⣭⠲⡞⠉
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣴⡄⣡⡰⠷⠀⣭⠰⡿⡄
⠀⢞⣱⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣏⡇⠀⠀⢿⡦⠖⢹⣏⠀⠉⠉⠲⢴⡿⠀⠀⠐⠿⠒⣇⣤⣀⣍⠰⠷⢠⡍⢴⡆⡗⠺⠗
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠀⠀⠀⡜⢁⣀⣈⠀⠀⠀⠔⡟⠈⢣⠀⠀⠀⠀⠀⠹⡉⣬⡏⢀⠿⢊⠱⠶⣰⠃
⠀⠀⠀⡤⢼⠤⢄⡀⠀⠀⢀⡠⢬⡧⢤⠀⠀⣤⣴⠃⢈⡛⠆⠀⡤⣄⠀⠀⠀⠈⣦⣤⠀⠀⠀⣤⠟⠪⢄⣛⣃⣸⠷⠚⢣⣤
⠀⠀⠀⠙⠚⠀⠀⠈⡵⢏⠁⠀⠀⠓⠁⠀⠀⠀⠈⣷⡯⠄⠀⠀⠑⠋⢠⣙⣶⣴⠁⠁⠀⠀⠀⠉⠀⠀⠀⢠⣧⠀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⣹⡴⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢠⠖⡆⠀⠉⠈⡧⡞
⠀⠀⠀⠀⠀⣤⠀⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⠀⠛⠋⠙⠒⠿⣧⠤⠒⠋⠙⠛⠀⠀⠀⠀⠀⠀⠈⢙⡝⠚⡽⢻⡄⠙⠢⣄⣰⣙⡆
⠀⠀⠀⠀⠀⠀⣳⢔⡒⢩⣭⠒⡢⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⠏⠀⠀⠉⠏⠆⠀⠀⠀⢺⠉⠁
⠀⠀⠰⢤⣀⡜⣁⠘⠋⣨⣍⠘⠚⣨⣧⣀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠘⠧⠟⡅⠀⠀⠀⠀⠀⠀⠀⠠⣀⣘⡇⣀⡀
⠀⠀⠀⠀⢸⠉⡛⠘⠟⣨⣅⢘⠝⣩⡉⡇⠀⠀⠀⠀⠀⣀⡀⠀⣀⡠⠴⠃⠧⢤⣀⠀⢀⣀⠀⠀⢸⡀⢀⣠⣣⠄⠀⠈⠙⣧⢾⠚⢧⠍
⠀⠀⢀⣀⠼⣟⠛⣺⠗⢨⣦⠛⣣⡍⢱⠧⣄⡀⠀⠀⠀⠙⡎⠋⠁⠀⠀⠀⠀⠀⠀⠙⢫⡞⠀⡖⢾⠥⣌⠲⠼⠀⠀⠀⠀⣴⠃
⠀⠀⠈⠀⠀⠘⢦⡛⠠⣶⠀⣶⣌⡵⠃⠀⠀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠙⠊⠀⠀⠉⣦⡔⠒⠊⠉⡞⣲
⠀⠀⠀⠀⠀⣤⠋⠈⠉⠒⡖⠉⠁⠙⣤⠀⠀⠀⢀⡤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡀⠀⠀⠘⠥⠕⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⡤⢅⠀⠀⠀⠀⠉⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡒⣆⠀⢀⣀⡧⣞⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣠⠄⢰⠄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠉⣝⣛⠆⠈⠑⠦⢴⣉⡧⠀⣴⡣⢤⣀⠀⠀⠀⠀⠀⢀⡤⢌⣦⠀⠀⡀⠚⡧⢜⠓⢉⣙⠚⠦⣜⠂⡀
⠀⠀⠀⠀⠀⡤⣄⣠⠋⠀⠀⠈⠏⠓⠀⠀⠀⢸⡀⠀⠀⠈⠁⠀⠈⠉⠓⣤⡒⠉⠁⠀⠀⠁⠀⡌⢹⠋⠀⠻⠃⣨⡋⠸⠽⢈⣳⡉⡀
⠀⠀⠀⠀⠀⠳⠚⢦⠀⠀⠀⠀⠀⠀⢀⡰⣤⣄⢇⣠⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢈⣉⡇⠻⠇⢖⡦⣙⡁⠰⡲⠙⠋⣏⢉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡤⣧⠂⠀⠀⠈⠧⣺⠚⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢉⡧⣶⠆⣴⡄⢛⡃⣖⠆⠫⠇⡏⢙
⠀⠀⠀⠀⠀⠀⣖⢺⠖⢤⡑⠚⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣹⣄⢰⡮⢀⣘⠊⣀⡺⠟⣼⠉⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⡷⡔⠛⠉⠉⣇⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣨⠗⢤⣘⣚⣀⣳⠷⢎⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠼⠄⠘⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⢲⡄⠀⠀⠀⠀⠀⣤⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡟⠛⠛⠛⠋⢰⡏⢰⠇⣰⢂⠀⣠⡄
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢹⠀⠀⠀⢰⠇⡼⢣⡾⢱⡆⢀⣿⢀⡿⢸⢳⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠈⠀⠸⡷⠊⠀⢸⡇⣼⢁⡟⠘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠿⠀⠈⣷⣀⡀⠀⠀⠀⠁⠘⠇⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠠⠟⣸⡤⠖⠚⠛⠛⠛⠶⢼⢃⡶⢰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣯⠁⡀⡆⢸⡆⢰⡆⢀⡿⠃⠛⠛⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⣴⠃⢸⡆⣿⠸⣇⠇⣿⠀⣾⠁⣾⠃⣼⢠⡆⢀⣤⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡆⠁⠲⣄⠀⠘⣇⣿⠀⣼⠀⢿⠘⣇⠰⠏⢸⡇⣾⡃⣼⠃⣼⠋⣹⢻⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢷⡄⡏⢿⡄⠙⠂⠘⠇⠉⠀⠀⠀⠀⠈⠁⠙⠺⠃⣼⠃⣼⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠓
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⢿⣦⡀⠀⠀⠉⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠈⠉⠻⣶⣄⠀⠀⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣦⣶⣤⠞⠉⠙⠃⠀⠀⠐⠀⠺⢿⣷⣄⠤⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣤⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢐⣦⣄⠀⠀⠀⠀⠀⠀⢶⣶⣤⣿⣮⣿⣿⣿⣦⡀⠀⠀⡆⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠀⠀⠀⢀⡀⣀⠀⢉⠤⠈⠁⠈⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⠀⡀⠀⠀⠄⠨⣿⣨⠶⠆⠂⠀⠀⠀⠀⢸⣿⣦⣅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠘⢿⠀⡄⠀⢀⡄⠁⠰⣞⣏⡻⠀⠀⠀⡾⠻⠝⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠘⣷⠀⣤⠁⣀⠀⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⠆⠀⠀⠰⠏⠀⠀⠄⣸⣆⠀⠘⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠰⣿⣏⠈⠙⠲⣦⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣀⡀⡿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⢠⣤⡴⠂⠀⠀⣿⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠀⠋⠀⠀⠰⡾⠏
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡗⢚⠉⢋⡡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠰⠞⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣠⣴⣶⣾⣿⣶⣶⡶⠄⠀⠀⠀⣠⠀⠀⠀⡠⠉
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠋⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣽⣇⡀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⢶⣤⣤⣄⣀⣤⣿⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣼⠏⠉⠛⠻⠿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠘⠳⢦⣄⡀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠸⣧⣤⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠻⣦⣠⡶⢶⣄⣀⣀⡀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢷⣄⠀⠀⢀⣴⠟⠻⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡋⠀⣠⡿⠋⠉⣻⡄⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠉⠻⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡟⠁⢀⣴⠟⠛⣷⠀⠀⠀⠻⣶⣄⠀⠀⠙⢷⣦⡀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣴⡿⠁⠀⣰⣿⣀⢶⣄⠀⠀⠙⢷⣄⡀⠀⢹⡿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣾⠋⠈⣹⡆⠙⢷⣤⡀⠀⢹⡿⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠿⣦⣀⣀⣿⠛⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣄⣀⠀⢀⣤⣄⠀⠀⠉⠛⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠋⠉⠀⢠⣿⣿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⣩⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠴⠾⠟⠋⠀⠀⠀⠘⣧⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢻⣿⣿⣿⣿⣸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⠏⠹⡇⢸⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡉⢄⠀⢷⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⣿
⠀⣶⠀⠀⠀⠀⠀⠻⣼⢦⠘⡇⠀⠒⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣠⣿⣿⣿⣿⣿⣿⣿⡁⠁⠀⠀⢸⣿⣇⠈⠉⢛⣫⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿
⠀⠘⣧⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⠀⠘⣿⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠹⣷⣦⣀⠀⠀⠀⠀⠈⢻⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠀⠉⢻⣿⣿⣿⣿⡆⠀⢿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿
⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠱⢦⣀⣠⣤⠀⠀⠈⠛⠿⣿⣿⣇⡜⡘⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⣄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⠀⠀⠩⠿⢿⣶⣶⡇⠀⠀⠀⠙⢿⡼⣷⢿⣿⣿⡷⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣧⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠈⣿⣹⣤⣿⠿⡥⠃⠀⠀⠸⣿⣿⣿⡇⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣈⠻⣷⣦⣤⣴⣿⣿⠟⠁⡆⠀⠀⠀⠀⣰⣿⣿⣿⡇⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣧⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⣷⣼⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⡇⠀⢰⣾⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣦⡀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡴⠛⠿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣌⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⢻⣿⣿
⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⢶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣵⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⣻⣿⣿⣿⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⣀⡼⣿⣹⣿⣿⡆⠀⠈⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡝⣯⡇⠈⢻⣿⣿⣿⣿⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣼⣷⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⢈⠆⠀⠀⠀⢣⡼⠃⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢪⢧⠀⠀⢀⠽⡇⠰⡁⢀⡸⠀⠀⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⠀⢀⠤⣄⡠⠇⢷⠒⠊⠚⠓⠒⢾⣾⢄⣀⠀⣺⣠⡸⢈⣁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⠸⡀⠀⢀⠼⠊⠁⠀⠀⣘⠣⡀⠀⠀⠀⡈⣹⠀⠀⠉⠓⢧⡔⡽⠃⡎⢀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠑⡌⠓⢡⠒⠉⣹⠀⠀⠀⠁⠐⢀⣀⠈⠠⠾⡥⡂⠀⠀⠀⠈⡳⢶⡕⠓⠒⠁
⠀⠀⠀⠀⠀⠀⢀⡠⠤⡄⠘⠧⢼⣦⠖⠀⠀⠀⠉⠙⢿⣇⠀⠀⠀⡜⠁⢀⠈⠦⠀⠀⠂⠀⠠⠔⣫⡞⠉⠙⣷⠀⠀⣠⠒⢆
⠀⠀⠀⠀⠀⠀⠸⡀⢢⣌⣲⢄⣰⠁⠀⠀⠀⠀⠀⠀⠈⣏⠓⡀⠀⠈⢢⣩⡽⠃⠀⠀⠀⢠⠒⠢⢍⠐⠃⢈⠃⠉⡭⠓⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⢓⠮⣕⠒⠀⠀⠀⠀⠀⠀⠒⢬⠖⠒⠒⢤⡻⢌⣊⠀⠀⢀⡆⠰⡤⡸⠀⡠⢣⢎⠆⣡⠀⠔⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣤⡨⠯⣝⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠿⡀⠶⠆⣢⠇⠀⠈⠀⠀⠙⣚⠳⠞⠁⠀⠔⡱⠏⠙⢾⠋⠛⠛⢁⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡇⠀⠀⠀⡇⠠⣬⠼⢻⠦⠀⠀⠈⠙⠳⢗⠀⡔⠒⠒⣄⠀⠀⠀⠐⣂⣶⠟⡗⡦⠥⣾⠿⠭⠽⠟⢶⣦⠆
⠀⠀⠀⠀⠀⢀⡤⣀⠀⢀⣈⣇⡠⠄⠀⠈⠒⠁⠀⢐⠀⠀⠀⠀⠀⠀⠂⠘⢄⣒⣑⡼⠀⢀⡠⣊⠔⡑⣊⡬⠴⠒⠉⠑
⠀⠀⠀⠀⠀⢸⠀⢨⠥⠐⠶⡴⠒⠒⠀⠀⠀⠀⢀⡔⣠⣤⠀⠀⠀⠀⡆⢠⠈⢕⡩⠊⢀⣥⠾⠓⢦⣔⠍⡹⢉⣙⠿⠎
⠀⠀⠀⠀⠀⠀⠉⠁⠀⡠⣄⡸⠀⠀⢀⣠⣀⠔⡥⠊⡭⡪⠄⠀⡠⠒⠁⢿⡀⢔⡨⠀⢛⠝⣆⠴⢀⡡⢊⣴⡕⡉⣲⣤⡖⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡴⠊⢢⠠⣏⢀⠀⡞⠀⠀⢒⢂⡤⠀⢣⡈⣲⠜⠂⠀⢔⡳⢤⠊⢝⡋⢁⠔⠉⢫⢎⢰⢁⣀⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣼⠷⡈⠛⠊⢀⡀⠤⣳⡫⠤⢒⡩⠜⡡⢔⡪⢐⢕⠝⣣⣑⠒⡔⠁⣂⣘⠦⣡⠛⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⠇⠀⡘⣡⠆⣸⣄⢐⡸⠐⡪⠕⣂⠭⢂⠁⡰⣁⠅⡊⢼⡉⠓⠓⠂⠺⣟⣿⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡪⠞⠁⠑⢆⡁⠔⣊⠁⠐⡩⢒⠩⢈⣧⡇⠐⣊⠔⣫⣔⣉⢄⣿⡽⠋⠁⠀⠙⡏⡉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⡏⠀⠀⠀⣠⡿⠋⠑⡞⣰⣋⠭⠐⡨⢒⣞⡝⢦⠔⣪⣖⢩⣵⠏⢣⢧⠀⠀⠀⠀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣁⡼⠃⠀⡰⣱⠃⠈⠁⢶⡶⠘⠦⠥⠊⠙⣯⡀⠀⠀⠀⠀⢫⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢱⠃⠀⠀⢠⠾⡳⡄⠀⠀⠀⢠⣭⣎⡃⠀⠀⠀⢸⣐⢄⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡸⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡠⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣂⠤⢒⡠⠔⢊⡠⠔⢊⡠⠔⣊⠄⢊⡡⠔⣉⠤⢊⡡⠔⣉⠡⢒⡡⠔⡩⠔⣉⠔⣊⠔⡪⢊⠵⠫⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠊⠁⠔⠊⠡⠒⠉⠤⠊⠁⠒⠉⠠⠒⠁⠔⠊⠀⠒⠁⠐⠉⠐⠊⠁⠊⠁⠈

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⡜⢄⠉⣇⣀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⡠⠚⠢⢠⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠳⢘⡌⠢⡨⠊⠀⠀⠰⠝⠊⢀⠀⠀⠀⠊⠈⢃⡀⠤⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡏⡭⡢⡀⠙⠢⠅⢀⡴⢊⠁⠀⠰⠇⠀⠈⠀⠀⠀⠀⣠⡄⠀⠀⢀⠀⠀⠀⠰⡹⠁⠹⠂
⠀⠀⠀⡖⠢⡀⠘⢄⡚⡀⠀⢀⠔⠉⠀⠣⠐⠀⠀⠀⠀⠀⠐⠂⠀⠄⠻⢀⡠⠐⠁⠀⠀⠀⠛⠃⢀
⠀⠀⠀⠘⢄⠑⠄⡄⠀⠀⠘⠀⠀⠀⠀⠀⣠⡾⢑⢄⠀⠈⠐⠁⠀⠐⠂⠁⠀⠎⠐⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⣰⢯⣲⠃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠞⢛⣼⠃⡐⠀⢀⠀⡰⡡⢀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠂⠀⠀⠀⠀⠠⡒⠁⠀⠀⠀⠉⢤⡾⠋⢉⠔⠀⠀⠈⢄⣔⢲⠇⠀⠂⠂⠢⡀⢀⠾⠆⠀⠞⢂⠄
⠀⠀⠀⠀⡀⢄⡀⠀⠈⠈⠁⠀⠀⢀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠀⠀⢀
⠀⠀⡄⠊⡤⠊⠑⡀⠀⠀⠀⠀⠒⠀⠆⠀⠀⠀⠰⡟⠋⡰⠀⠀⠈⢇⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠈⢎⠀⠀⠅⠅⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⣵⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡁
⠀⠀⠀⣊⡇⠀⠀⠀⠀⢀⠊⠀⠉⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠊⠉⠉⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⠀⠀⣈⠀⠀⠀⠀⠂⠀⠀⠠⠒⢀⠀⠀⠒⠀⠀⠠⡀⠀⠀⠀⡞⠂⠂
⠀⠀⠀⠀⠀⢶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠐⠀⠀⠐⢄⣀⣶⣠⡔⠁⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠄⠄⠌⠍⡒⠀⠀⠀⠀⢀⠴⣗⢄⠀⠀⠀⠀⠀⠀⠀⠌⠍⢿⡇⠀⡃⠀⠀⠀⣀⣀⡀
⠀⠀⢠⢆⠀⢊⠈⠌⠀⠀⠀⠀⣰⡁⠀⠈⡪⠇⠀⠀⠄⠀⠀⠀⡂⠀⢺⣇⣀⡁⠀⠸⠛⠃⠉⢁
⠠⠀⠀⠀⠁⠆⠆⠀⠀⠀⠀⠀⠀⠀⢠⢊⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⠘⠃⠀⠀⠂
⠀⠀⠄⠆⠶⠲⠀⠀⠀⢀⠀⠀⠀⠀⠈⠈⠀⠀⠘⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠘⠤⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢠⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⡀⢠⡜⠻⡅⠀⠀⠀⠠⣄⡀⠦⠀⣴⠀⡀⠀⠀⢴⡤⠾⠚⠠⢼⡆⠀⠀⢈⢞⡋⡁⠙⢲⡆⠀⢠⣆⠢⣼⢗⠚⠲⣏⣀
⠀⠀⢠⠏⣆⠀⠐⠀⡀⠀⢰⢳⠀⠀⢰⠉⠉⠇⠀⣂⡜⢰⠐⠶⠓⠎⣷⠂⠙⡯⡁⠆⠈⡆⠠⡋⠁⢨⡅⣞⣉⣭⣭⠀⠈⠦⠄
⠀⠀⣽⠀⠂⠐⠒⢉⣁⠀⢈⠿⢴⠖⠖⠙⢋⠖⠀⠀⣨⣀⡁⠈⢁⣰⡏⠀⠘⢹⣵⣚⡂⣀⣰⣉⠀⠐⣿⡝⠛⠋⠁⣀⡼⠳
⠀⠀⠘⢢⣀⣀⠠⡌⠀⠀⠀⠟⠐⢶⠒⠐⠛⠀⠀⠀⠉⠁⠸⠽⠀⠉⠀⠀⠀⠈⠀⠘⠁⣙⠀⠀⠀⠀⠀⠉⠹⠏⠹⠃⠉
⠀⠀⠀⣀⡀⢀⢀⠀⠀⠀⠀⣄⠀⢰⣀⢀⡄⠀⠀⠀⠀⡰⠛⣑⡒⢤⡀⠀⠀⠀⣀⡷⠫⣭⣿⢶⠀⠀⠀⣄⡤⠖⢺⣄⡀
⠀⣠⢂⠿⢁⡔⡑⡜⠃⠀⠀⢉⠏⡔⠮⠉⠑⡶⠀⠀⡸⢁⣨⣵⣱⣇⣹⢆⠀⠀⢩⣦⢀⡬⠶⠋⡀⠀⠀⣷⡴⢩⡉⣾⠁
⠘⠿⢠⣘⡼⠝⣿⠷⢷⡆⠈⠉⢢⠈⠄⠅⢀⠛⠂⢸⣃⠐⠸⣿⣿⡟⠛⡿⠀⠀⠈⡿⠀⠉⠙⢻⡁⠀⠀⠘⡜⣦⣾⣿⠃
⠐⣾⡛⠒⠒⠉⠀⠀⣱⡄⠀⠸⠚⡐⠂⡒⠟⠁⠀⠀⠉⠒⢤⣀⣥⠔⠊⠀⠀⠀⠘⠻⣤⣄⣀⣤⠋⠀⠀⠀⠙⡿⠿⠃
⠀⠀⢸⠢⠤⠒⠐⠒⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢦⡆⠀⣎⠄⠀⠀⠀⣀⣄⣠⣁⣤⡅⡄⠀⠀⠀⠀⠀⢸⡄⣤⠆
⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⣠⣂⡹⣅⣭⣿⣓⡄⠀⠠⣀⣴⣓⣞⣣⡄⠀⠀⣘⡻⢺⡴⢆⠟⠣⣞⡀⢀⠀⢘⢦⣿⣿⢿⣗⣆⢀⡀
⠀⠀⢠⠊⠁⠀⢀⠁⢆⠀⢬⠜⢦⣌⣉⣉⣭⡄⠀⣠⢇⣻⠿⠿⢿⢫⡁⠠⢼⠀⢀⡈⠀⢀⡄⣸⠆⣈⡛⠁⠈⠁⠉⠀⠉⠀⢹⠤
⠀⠀⢸⠀⢂⣚⣗⠃⠾⠀⠻⣀⠂⠈⡉⢆⢴⠃⠀⠐⣇⡀⠀⠀⠐⣾⠂⠀⢛⡀⠉⠀⠀⠀⢠⡏⠀⠀⣩⠆⠀⠀⠀⠀⣠⠤⡞⠆
⠀⠀⠈⠬⣈⣁⣀⡰⠚⠀⠀⠁⠛⠒⠲⠈⠀⠀⠀⠈⠀⢉⠋⡈⠃⠀⠀⠀⠈⠙⠋⠓⠲⠲⠉⠀⠀⠀⠀⠰⠋⢲⠒⠺⠈⠀⠁
⠀⠀⠀⠀⢀⡀⣄⢀⢀⡀⠀⠀⢠⠞⣔⠒⠆⠀⠀⠀⣀⣜⣾⡥⡆⢤⠀⠀⠀⠀⢀⡶⠢⡀⢠⢀⠀⠀⠀⢠⡀⠈⠀⣠
⠀⠀⠉⡞⢐⠛⠓⣾⠄⣣⠂⠀⡜⠤⢛⠄⢳⡀⠀⡌⠀⠈⠈⢂⣀⠀⠀⡄⠀⢰⡒⠂⣢⡎⢡⢘⠇⠀⠰⣾⠽⡇⢛⠶⣿⠃
⠀⠀⠘⣇⣘⣍⣾⢩⡶⠏⠀⢎⡁⡄⠀⠼⢧⡆⠀⡅⠀⠀⠂⠀⠈⠀⢀⡇⠀⠈⠳⠂⡥⠝⠻⠈⠶⠀⠀⢩⡵⡬⢧⡰⣬
⠀⠀⠈⠀⠙⠉⠋⠈⠃⠀⠀⠻⠴⢃⣬⠤⠚⠀⠀⠘⠒⠤⠄⠠⠄⠤⠈⠀⠀⠀⠛⡄⡤⠎⠲⠥⠚⠀⠐⡎⠈⠚⠟⠃⢈⡇
⠀⠀⠀⢠⣀⡐⣢⢤⣀⡀⠀⠀⢀⡀⠀⡃⡀⡤⢀⠀⠀⣀⢀⠷⠴⢧⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠩⠦⣤⢀⡤⠺
⠀⠀⠀⣈⢸⠜⠢⠣⢬⣡⡄⢠⢄⡱⣬⣧⣀⠷⡋⠀⢤⢜⣍⡂⣐⣠⡹⡆⠀⢾⢁⡾⢦⢷⡓⡂⠀⠀⠀⡠⢤⣼⠢⠶⣤
⠀⠀⢀⢀⠁⠀⢀⠀⠀⣉⠄⢀⣠⡍⢶⣂⡡⢃⠇⠐⠚⠀⣡⡀⣀⣤⠌⡅⠠⠊⠇⠈⣉⡁⢸⠙⠄⢠⡤⠹⠅⢧⡺⠗⠊⣢⡄
⠀⠀⠈⠩⣠⢜⠀⢩⣁⢯⡁⠀⠁⡼⠒⠤⡄⠈⠒⠂⠀⠷⡀⠀⠈⠀⢀⡁⠈⠒⡎⠲⠠⠔⠚⡞⠁⢀⣧⠂⢘⣿⣗⣠⢀⢸
⠀⠀⠀⠈⠀⠋⠉⠓⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡴⠐⢲⠍⠀⠀⠀⠉⠈⠉⠑⠊⠀⠀⠈⠀⠉⠟⠈⠘⠞⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣳⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣄⣀⣤⣶⠟⠃⠀⢿⣿⣿⣿⢻⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣇⣷⢹⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⢹⣿⣦⢠⣤⢀⡀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⠦⣠⣶⣦⣤⣤⣤⣵⣦⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣏⣾⣿⣿⣶⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⢠⣀⣴⣿⣿⣿⡿⢉⠀⠀⠀⠰⣦⣤⣄⣀⠀⠀⠉⢩⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⣶⣴⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠇⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⡄⠀⠀⠀⠀⠀⠀⠀⠉⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣼⣿⣿⣿⣿⣿⣿⡇⣹⡁⢀⣴⡀
⣿⣶⣄⣠⣾⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⠉⠿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣯⣷⣿⣷⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠴⠶⠦⠤⠤⣀⡀
⠀⠀⠀⠀⠀⢀⡄⢀⡤⠖⠋⠉⠀⢀⡠⠴⠒⠚⠛⠛⠛⠒⠀⠉⠛⢦⣄
⠀⠀⣀⢶⡒⠈⡷⠉⠀⢀⠀⠀⠈⠁⠀⠀⣀⣠⠤⠴⠒⠒⠀⢀⣀⣠⠽⢚⡥⣰⠃
⠀⠀⠉⢢⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⡤⠶⠊⠉⠠⠒⣩⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣤⡤⢤⣤⣀
⠀⠀⣰⠃⠄⠀⠀⠀⠀⠐⠂⠤⠤⠤⠤⠤⠤⠐⠒⠉⠀⠀⠀⠀⣠⡴⠞⠁⡌⣇⠀⠀⠀⠀⢀⣀⡠⠔⣉⠤⠒⠉⠀⠀⠀⠀⠀⠀⢉⣲⣄⣀⡴⠋⡙⠛⢶⡄
⠀⢰⠃⡜⠀⢠⠂⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠶⠚⠋⠁⠀⠀⠀⢱⣹⠀⠀⣠⠞⠩⠎⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⢿⣀⡠⠀⠹⡄
⠀⡟⢠⡆⠀⡇⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣆⢷⡠⠔⠀⡇
⠀⡇⢸⡇⠀⢀⠄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠆⠀⡇⢰⢻⠸⡀⠀⠀⠀⠀⠀⢈⣉⣉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⣠⠟⡾⠇⠀⣸⠃
⠀⣧⠀⠇⣠⠋⣠⣤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⢀⠓⢦⣄⣦⠤⣄⣀⠀⠈⠙⠲⢤⡀⠀⠉⠓⠶⠶⠾⣫⠀⡷⠴⠚⠁
⠀⢻⡀⠀⡏⣾⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠀⠀⣸⠁⠘⣇⣸⠰⠀⠀⣤⠀⠀⠉⠙⢦⡀⠀⢀⠙⣆⠐⠀⠀⠀⠊⠁⡔⡇
⠀⠘⣷⠀⠀⠙⢮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⣰⠃⠀⠀⠈⢿⠀⡄⠀⠈⠀⠀⠀⠀⠀⠃⢸⡍⢱⡈⠀⠀⠀⢀⠄⡼⢡⡇
⠀⠀⠸⣧⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⢞⡵⠃⠀⠀⠀⠀⢸⡘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⠼⠁⠀⠀⠂⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⣄⡙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⢉⡴⠋⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⡤⠎⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠒⠉⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠺⠷⠦⢄⣀⡀⠀⠀⠀⠀⢀⣀⣀⡤⠖⠋⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠂⠤⠤⠖⣋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄⠈⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠙⠒⠤⣄⡀⠀⠀⣀⣠⣷⣤⠤⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⠤⠍⠓⢤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⣀⣀⠀⠀⠀⠀⠀⢀⡨⢆⣀⢎⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠰⡄⠀⠀⠀⠀⢡⠈⠧⠀⠀⠀⠀⠹⣿⣿⣿⡿⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⡆⠣⢄⡀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠉⠙⠒⠶⡶⠜⣦⠤⠴⠚⠉⢈⠀⠀⠀⠀⠀⠀⠸⣄⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠲⡄⠀⠀⢀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠤⠿⠗⢒⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢑⣠⠀⡀⠀⣀⣠⣼⢰⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠞⠉⢂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠾⢦⢰⣴⡉⡿⠾⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⣤⣄⣀⣤⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⡤⡀⠀⣏⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢮⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠜⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠰⠩⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣤⣤⣤⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠺⠿⡿⠛⢻⣿⡟⢉⣡⠔⢀⣀⣤⣀⠀⢀⣤⠄⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠤⣤⣬⣿⡛⠛⠛⢶⣶⣿⣅⠀⠀⣴⣿⠁⣠⣴⣿⣿⢀⣾⠏⢠⣿⢏⣴⣿⠟⠛⣏⢀⣾⣿⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⣨⣿⠛⣿⣦⢀⣾⡟⠩⣿⡷⢸⣿⠃⣴⡿⡫⠿⠃⣾⠏⠀⣾⠏⣾⣿⠋⠀⣰⠟⣼⢿⣿⣰⡟
⠀⠀⠀⠀⠀⢀⡾⣿⡏⠀⣰⡿⠁⣠⡿⢫⣿⠏⣠⣾⠟⢡⡿⢁⣾⠟⠀⣀⠀⣼⠟⠀⣼⠏⢠⣿⣇⣀⣼⡟⣸⡏⠀⣿⣿⠃
⠀⠀⢀⣀⣠⣿⣷⣿⠗⣴⣟⣥⠾⠋⣰⣿⣧⡾⠋⠁⢠⡿⢁⣾⢏⣠⠞⠀⣼⠏⠀⣸⠏⠀⠀⠻⠿⠟⠉⢠⠏⠀⠀⠹⠁
⠀⠀⠉⣸⠏⠀⢸⢇⣼⡿⠛⢁⣠⣴⠿⠛⠁⠀⠀⢠⠞⠁⠸⠿⠟⠁⠀⡰⠏⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡜⠁⠀⠠⠟⡽⠁⠀⠀⠋⡹⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣷⣶⣶⣶⠶⠶⠖⠚⠀⠀⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⠴⠿⣿⡿⠟⠛⣿⣿⠋⠁⢀⣴⣿⣿⡆⠀⢠⣶⡗⢠⣿⣿⠁
⠀⠀⠀⠀⠀⣠⢠⡶⠒⠀⠀⠀⣴⠃⠉⠛⠉⣛⣿⡿⠉⢉⣿⣿⢇⣾⣿⠃⠀⣼⣿⠃⢀⣴⣿⠟⣹⣿⠃⢠⣿⡟⠀⣼⡿⠃
⠀⠀⠀⠀⣰⠃⠘⣧⠀⢀⣀⣾⡿⠀⠀⠀⢠⣿⣿⣃⣴⣿⠟⠋⣼⣿⠃⠀⣸⡿⠁⣠⣿⡟⠁⠐⢋⢤⣴⣿⣿⣷⣿⣿⠓
⠀⠀⠀⢠⠃⠰⠒⠋⠀⠀⠋⠐⠃⠀⠀⣠⣾⣿⣿⣿⣏⡁⠀⢠⣿⠃⠀⣰⣿⠇⢠⣿⡟⠀⠀⣀⡴⢢⣿⠏⠉⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠏⠈⣹⣿⣿⢠⣿⠃⠀⢰⡿⠁⢠⣿⣿⣀⣤⣾⠏⠀⣾⡏⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣴⣿⡿⠛⢁⣿⠋⠀⢠⡿⠁⠀⠘⢿⣿⡿⠛⠁⠀⣸⠏⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠟⠛⠋⠁⠀⢀⡞⠁⠀⢠⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣄⣼⡿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⡏⢟⡇⠻⢿⠿⣇⠊⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠋⡿⡿⠿⢶⢷⡌⢀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠶⠿⡏⠂⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡿⠂⠁⠀⠀⢠⡀⣀⢂⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣬⡙⢋⠇
⠀⠀⠀⠀⠀⣠⣶⡿⠿⢫⠙⠿⠶⠒⢂⠐⠛⣄⠻⠿⠻⠁⠀⠀⠠⣆⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⢦⢢⣦⣼⡇
⠀⠀⠀⢀⣾⣷⣮⡀⣠⣀⡄⣀⠀⣰⣺⣠⣀⢿⠋⠀⠀⠡⠀⠀⠀⠈⠝⠀⠀⣿⡛⠶⣤⠀⠀⠀⣼⣿⣶⣿⣿⡿
⠀⠀⢀⣾⣿⣿⡟⠁⠈⣛⣾⡿⠒⠺⣿⣿⣉⠀⠀⠀⢀⡴⡆⠇⢠⠀⠀⠀⠀⠈⠛⠂⠈⠁⠀⢸⡿⢿⠿⢿⡿⠁
⠀⠀⢸⣿⣿⣟⠂⠀⣼⢿⡿⠁⠀⢠⣿⣿⡇⠀⠀⢀⠊⢠⠇⠀⠠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢹
⠀⠀⢸⣾⣿⡏⠀⠀⢻⣸⣄⣼⠯⠛⢱⣶⣬⠀⠀⠈⠀⠁⠀⠀⠄⠈⠒⠦⠄⠀⡀⠀⠀⠀⠀⠀⠒⠃⡠⠂
⠀⠀⠘⣿⡜⡿⠁⠀⠸⣷⡜⣿⡷⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁
⠀⠀⠀⢿⡷⣘⣴⣦⢀⠀⣤⣿⠃⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠘⠬⢀⡀⣀⢀⡀⠤⠄⠐⠊
⠀⠀⠀⠘⣿⡹⣿⣷⢀⢸⣿⣇⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⢀⠈⢀⠘⣿⣷⡍⢻⣶⣿⡉⠑⠚⠁⠀⠀⢀⣠⠤⠖⠊⠁
⠀⠀⠈⣾⣶⣿⡇⣾⣿⣽⣿⣿⡿⠇
⠀⠀⠀⠙⠝⠟⠿⣿⣦⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⠞⠙⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣩⣿⠿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⡧
⠀⠀⣠⠴⠚⢉⣤⡴⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠁⠀⠀⠜⠁
⠀⠀⢿⠀⠿⠟⠉⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⡻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⢑⢀⣴⡁
⠀⠀⠈⠳⠦⠖⠋⣁⣀⡤⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⠹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⢘⡵⣿⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠀⠈⠛⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢲
⠀⠀⠀⢻⣷⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀⠤⠒⡠⢄⡀
⠀⠀⠀⢠⣿⠇⠀⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⣤⣤⣤⣤⣥⣬⠽
⠀⠀⠀⢸⠸⠠⢚⠕⠋⣠⡴⠚⠉⠢⠀⠀⠀⠀⠀⠀⢸⣷⠂⠐⣸⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣭⣭⠽⠛⠛⠛⠛⢓⡄
⠀⠀⠀⠈⠲⠀⣡⠴⠋⠁⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⡇⣿⠦⠄⣿⠀⠀⠀⠀⠀⠀⠀⢸⡍⠉⠉⠉⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⢾⡙⣧⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠓⠚⠁⠘⢦⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⠀⠀⠠⠤⠧


⠀⠀⠀⠐⣏⣽⡷⠄⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⢻⣼⣿⡾⡿⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⣃⣀⣐⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣴⠾⢛⣁
⠀⠀⠀⠀⣸⣙⣟⠛⣽⣷⣶⡟⠂⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄⠊⠹⠍⠉⠯⢹⠋⠁⠀⠀⠀⠀⢀⡴⠋⡀⠈⠛⠛⠊⢡⣄⠑
⠀⠀⠀⠀⠉⢿⠙⢿⣟⡀⢀⢹⠀⠀⠀⠀⠀⠀⢀⡾⣟⠻⣿⢢⣟⣠⣤⣷⣸⠀⢀⠀⠀⠀⠀⡾⠀⠾⣤⡖⢠⣤⠘⢇⡙⠗
⠀⠀⠀⠀⠀⠸⡄⠿⠥⣔⣊⣀⣀⠀⠀⠀⠀⠀⢸⡗⣿⠗⣇⢸⠿⣭⣨⠾⣽⠀⢸⠀⠀⠀⠀⣧⠀⠀⢉⡴⠒⠒⠲⣌⠉
⠀⠀⠀⠀⠀⠀⢹⣿⣯⠭⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠈⠲⠖⠋⠀⠀⠀⠀⠈⠑⠢⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣴⠶⠶⠶⢦⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⢀⡿⠀⠀⠀⢸⣇⠘⠛⠛⠛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣉⣛⣛⣛⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⣄⠀⠉⠉⠉⣉⣭⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠳⣄⠀⢸⡏⠀⠙⠳⣦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠈⠳⣌⠻⢦⣄⠀⠀⠙⠳⢦⣤⣄⣿⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⠚⠛⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠄⠀⠀⣠⣤⣤⡤⣿⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠠⠠⠀⡀⠆⠠⠠⠠⠀⡀⠀⡄⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣷⣶⣄⡀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠛⠉⠀⢀⣤⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣏⠻⠟⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⠛⠛⠉⠁⠀⣴⣶⣦⣤⣤⣄⣀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢛⣥⣶⣦⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢿⣿⣿⣿⣶⣝⠻⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣝⠿⣿⣿⣿⣷⣄⡀⠀⢀⣴⣾⣿⣿⣿⠿⣋⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⢿⣿⣿⣿⠟⠋⣴⣿⣿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⢀⣀⣀
⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠘⠟⠋⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠀⠈⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢉⠄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠙⠂⠀⠀⠈⠉⠀⠠⠤⠼⠀⠀⠀⠀⠀⠀⡠⠂⠈⠤⠀⠂⠀⠀⠀⠀⢀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠐⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠽⠀⠀⠤⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠐
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⣠⣤⡤⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠂⠀⠀⡐⢢⠀⠑⠁
⠀⠀⠀⠀⣚⠟⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠄⠁⡀⠀⣀⡏⠀⠀⠀⠀⠀⣠⠀⠀⡠⠀⡄⣰⣠⢂⠔⡀⡄⣠⢀⠆⣰⠀⠀⠀⢤⠀⠀⠀⠀⠀⠻⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠎⠂⠀⠀⠀⠀⠀⠓⠑⠑⠊⠑⠚⠂⠘⠐⠃⠈⠁⠁⠀⠉⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁
⠀⠀⠀⠀⠀⣹⠟⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡠⣴⣿⡶
⠀⠀⠀⠀⠀⠌⠸⢷⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⣤⣦⠀⠀⠀⡀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠁⠀⠀⠀⢒⠀⠀⠉⠟⠃⠉⠈⠀⠛⠃⠌⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢰⣿⠇⠈⠻⠛⠀⠀⠀⢀⠀⠀⠤⠀⠠⠀⠉⠉⠉⠉⠁⠈⠈⠥⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢣⢩⣬⡄⢘⠈⠑⠘⡁⢘⢓⠓⠃⠀⠀⠘⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠏⣥⣀⠠⣀⠀⣀⡂⣖⢒⡒⠁⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠀⠀⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢿⠟⢿⠟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣽⣤⣽⣳⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⢿⣿⣤⣿⣦⣾⣦⣾⣿⠇⠈⠉⢀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⡿⢿⠿⠿⠛⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⢀⣀⣀⢀⣀⣀⢀⠀⣀⢀⣀⡀⢀⣀⡀⠀⢀⡀⢀⣀⡀⢀⣠⣀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠟⣰⠿⣇⢸⡶⣟⠀⣿⠀⣿⠳⣿⢸⣗⡂⢸⠶⡟⠀⢸⡇⢯⣀⡀⢿⣀⣽⢸⡟⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠈⠀⠈⠀⠈⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡾⠁⠀⢠⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⢈⠏⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠺⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⣴⠞⠋⠉⠁⠀⠀⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⡴⠇⠀⣀⣀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠳⢼⡇⠀⠀⠺⠤⠜⠘⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣼⠁⢀⣤⣤⣄⣀⣀⣹⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⢄⢻⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⡏⠓⢾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⠷⢤⡾⠀⠀⢀⡔⣶⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡾⠀⠀⢀⠀⠀⣸⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠐⠭⠇⠀⣿⠃⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣯⠤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢁⡈⠉⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣴⠛⠛⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠛⢻⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣄⠀⠚⢻⠟⠁⠀⣠⡟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠙⢷⡀⠀⠀⠙⠀⢤⣤⣤⡴⠞⠋⠀⠀⢀⡾⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣸⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣇⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⢴⣰⢰⣇⢴⢰⢲⢴⣶⠔⣦⢰⣆⠀⣶⣲⢠⣶⢰⡆⣟⢰⢲⣤⣶⠆⡖⢲⡆⡟⣦⣾⢰⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠁⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠉⠁⠀⠈⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢀⣀⡀⠀⠀⢹⣶⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣈⣏⣀⠀⠀⢀⣀
⠀⠀⠀⠺⢴⡣⠔⠊⠉⠀⠉⠓⠦⢼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡦⢚⣭⠡⣶⠌⣩⡢⡜⠋
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣤⡈⣉⠰⣶⠀⣉⢡⣺⡆
⠀⣰⢳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡚⡆⠀⠀⢶⢦⡤⢒⣮⠉⠉⠓⠢⣤⢶⠆⠀⠀⢴⡶⢴⢁⣉⠈⡛⢱⡦⢀⡛⢡⣄⢸⠴⣶⠄
⠀⠈⠋⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠙⠁⠀⠀⡸⠃⠀⠀⠋⠃⠀⢠⢶⠎⢳⡀⠀⠀⠀⠀⠀⢏⠛⣘⡛⠀⣶⠟⢫⣮⢁⠎
⠀⠀⠀⢀⣨⣆⡀⠀⠀⠀⠀⠀⣀⣸⣁⡀⠀⠀⣀⣰⠃⠐⠿⡟⠁⢀⡀⠀⠉⠀⠀⢧⣀⡀⠀⠀⢀⣈⠷⣝⡁⢾⠆⢶⣦⡥⢏⣀
⠀⠀⠀⠹⠬⠇⠈⠙⠲⣰⠒⠋⠁⠹⠥⠃⠀⠀⠛⠺⣤⣔⡃⠀⠀⠫⠼⠀⠐⢤⡄⡿⠛⠃⠀⠀⠘⠋⠀⠀⠉⠙⣟⠉⠁⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⢲⢀⡄⠈⠉⠛⡻⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠘⠛⠰⣏⣱⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣶⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⠝⠢⣄⣺⡿⣀⣀⡤⠺⣵⠄⠀⠀⠀⠀⠀⠀⠺⢴⢧⡤⢤⣴⡋⠧⣄⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠉⢧⣀⣀⠤⠿⠤⣀⣀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠧⡴⡃⠀⠀⠙⢚⠗⠚
⠀⠀⢀⡀⠀⢀⡴⠫⣶⠄⠯⠇⣴⡍⢢⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⣦⠞⠀⠀⠀⠀⠁⠉⠀⠀⠀⠈⡆
⠀⠀⠘⠙⠒⡞⢶⡆⣤⡔⠿⠂⡤⡔⠿⢷⠒⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⠀⠀⠀⠀⠀⠈⠉⠈⡇⠀⠀⠀⡀⠀⠀⠠⢌⡶⢲⣓⣤⢲
⠀⠀⠀⠀⠀⣇⣤⣀⣍⠑⠿⢢⣬⠱⡶⢸⠀⠀⠀⠀⠀⠀⢴⠦⣠⠴⠚⠉⠁⠈⠉⠒⠦⣀⡴⡦⠀⠀⣹⠀⢰⡚⢶⠁⠀⠀⠀⠓⡻⠁⠉⠊
⠀⠀⢰⠖⠊⠳⡈⣨⡍⠀⠻⠃⡁⢶⢆⠞⠑⠲⠆⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠘⣍⡷⠙⠢⣍⡉⠀⠀⣀⣤⣸⣁
⠀⠀⠀⠀⠀⠀⢙⠮⣅⣻⢇⣘⣿⠴⡋⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠵⣏⠉⠀⠀⢯⠴⠃
⠀⠀⠀⠀⠀⠲⠏⠀⠀⠀⣽⠀⠀⠀⠹⠖⠀⠀⠀⠠⣖⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣲⠄⠀⠀⠈⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⠒⢦⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡹⣀⣀⣀⡼⠳⣅⠀⠀⢀⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⠀⢴⠂⢰⡂⢰⡦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠈⠰⣏⣽⠂⠀⠙⠢⢞⠤⠗⠀⢴⡡⠲⢄⣀⠀⠀⠀⠀⣀⡤⠶⣝⡦⠀⢀⡄⢉⠶⣊⡉⢩⢭⠉⣒⢾⡉⢠
⠀⠀⠀⠀⠀⡴⠦⣠⠎⠀⠀⠀⠘⠈⠃⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠉⡇⣞⠉⠁⠀⠀⠀⠀⢠⣌⡸⢃⠀⠓⠃⡬⡅⠘⠿⢀⣽⡅⣡
⠀⠀⠀⠀⠀⠑⠚⢳⡀⠀⠀⢀⠀⠀⠀⢀⣲⡤⣄⢧⡠⢤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣌⣨⡇⠳⠃⢾⠆⣉⡁⠰⣷⠈⡊⢹⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⢼⡴⠂⠀⠀⠘⠴⡿⠋⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢸⡷⣿⠢⣶⡄⢛⡃⢞⠆⠸⠟⣸⠉⡃
⠀⠀⠀⠀⠀⠀⢰⠛⡵⠢⣌⠒⠋⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣱⡄⣴⡮⢀⣘⠚⣀⡘⠝⣠⡋⠙
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢙⢦⡖⠛⠉⠉⣎⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣨⠦⢄⣘⣚⣀⣳⠧⢾⣁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠾⠀⠽⠅⠸⠆⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡝⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⢠⢡⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣹⣀⠤⠔⠲⣄⣀⣀⣀⣀⣀⠀⠀⠀⠠⠊⡆⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠁⠀⢀⡠⠔⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡎⠁⠈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣩⠇⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡌⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⣼⢀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠐⣻⠀⠀⠨⠭⠭⠭⠭⠥⠤⠤⠤⠤⠀⢻⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⣿⠀⠀⢀⣒⣂⣀⣐⣀⣀⣀⣀⡐⠐⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⢀⡔⠊⠙⠒⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠀⠀⢈⣉⣉⣉⡉⠉⣉⣉⣹⣉⡁⠀⠀⢸⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⣀⠤⠋⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠶⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠂⡔⢲⡀⣆⢠⡆⣶⠀⣶⢰⠐⡖⢠⡆⣰⡆⣖⠀⣦⣠⠐⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠑⠚⠀⠉⠃⠃⠃⠋⠘⠘⠀⠃⠘⠘⠁⠃⠓⠀⠁⠙⠀⠉




⣿⣟⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⠡⠂⣉⠁⠐⢢⣬⣑⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⠔⡁⠚⠁⠈⠙⠲⢯⣄⣨⣷⡦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢁⢔⡵⠊⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢥⣌⢷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⢔⡴⠋⢀⡴⠊⡉⢉⠉⠁⠲⢤⡀⠀⠀⠀⠈⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⡔⡡⠊⢀⠔⠁⠀⠘⠉⠇⣴⠄⠀⠀⠃⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⢊⠔⠀⢀⠎⠀⠀⠀⣰⣄⠂⡀⠀⠀⢀⠇⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⡰⠡⠊⠀⠀⢸⠀⠀⣶⣄⢀⠈⠘⠃⢀⠔⢁⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡔⡐⠁⢀⣀⠠⠔⠀⢀⡀⣈⣙⣁⠤⠐⢁⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⢍⡌⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡱⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⠁⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠓⠒⠀⠠⠤⣄⠀⢀⣀⣀⡀⠀⣴⣿⠃⠀⠀⠀⢀⡤⠞⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠀⠀⠀⠀⡞⠁⢙⠃⠀⡍⢭⢻⡟⠁⠀⠀⠀⡠⠁⠀⠀⠀⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠤⠤⠥⠤⠥⠤⠴⠌⠦⠋⠀⠀⠀⠀⠀⢡⡀⠀⣀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠛⠃⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢈⣀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠂⠀⠀⠀⠀⠻⡈⢂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠉⠀⠀⡇⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⠀⣠⠃⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⡐⠀⠰⠃⠂⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠤⠠⠒⠈⠀⠀⠀⠀⠀⡰⠛⠒⠂⠀⢰⡅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠠⠐⣋⡀⣤⡚⠁⠠⠭⠁⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⡈⠭⠉⠀⠠⠄⡒⠂⠈⠀⢠⣙⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⢂⡡⡐⢾⠀⠐⠢⠄⠉⠀⢠⠋⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⠰⣌⠢⡑⠙⠆⢻⡀⠐⣈⠡⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠺⡆⠁⢀⡤⠖⠒⠛⠉⠇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠠⡑⠈⢠⠎⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠂⠀⠠⠑⠀⠀⠎⢀⠀⡀⠀⠀⠀⠀⢃⠀⠈⠤⢐⠀⢣⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠠⠊⠠⢊⠔⠁⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢣⡀⠠⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠊⠤⢂⡀⢤
⠀⠀⠀⠀⣠⣤⠴⠒⣖⣲⣲⣲⣒⣦⣠⣄⣄⣀⣀⣐⣂⣒⣒⣒⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠉⠐⠀⣠⣄
⠀⠀⠀⠸⣵⡿⠻⠋⠛⠉⠍⢍⠍⠌⠡⢉⠊⠌⠈⠁⢠⠀⠉⠉⠉⠙⠓⢤⠟⠀⠛⠰⠶⠶⠶⠦⠒⠁⠌⠑⠀⠁⠢⡴⠂⠐
⠀⠀⠀⠀⠈⠙⠓⠳⠶⠤⠬⠈⣈⣠⣀⣆⣈⣰⣠⣠⣄⠼⠦⠼⠖⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠉⢙⣣⠤⠖⠒⠚⠛⠒⠲⢤⣜⡋⠉⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠶⠋⠁⠀⢀⣀⣀⣀⣀⡀⠀⠈⠙⠶⣄⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⡾⠁⢀⣠⠶⠋⠉⠀⠀⠀⠀⠉⠙⠶⣄⡀⠈⢷⡄⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠧⠀⣴⠋⠀⣤⢿⡁⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⣈⡿⣆⠀⠹⣆⠀⠼⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣸⠇⢀⡾⠁⠀⠙⠛⠉⠹⣦⣀⣀⣼⠛⠛⠛⠉⠀⠈⢷⡀⠸⣇⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⣰⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠘⣆⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⡟⠀⣾⠁⠀⣤⠀⠀⠀⠀⢰⡆⠀⠀⢰⡆⠀⠀⠀⠀⣤⠀⠈⣷⠀⢻⡄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣸⠃⢸⡇⠀⠀⢹⣇⠀⠀⠀⣼⠃⠀⠀⠘⣧⠀⠀⠀⣰⡟⠀⠀⢸⡆⠘⣇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣿⠀⣾⠀⠀⠀⠀⠙⣷⢶⡾⠻⠂⠀⠀⠐⠟⢷⡶⣾⠋⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡏⠀⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⠀⢻⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣧⠀⣷⢀⡔⠋⠀⠈⢧⠀⠀⢰⡆⠀⠀⢰⡆⠀⢠⡞⠁⠀⠙⢦⠀⣾⠀⣼⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠖⠀⣿⠀⢿⠠⣆⠀⠀⢀⡼⠀⠀⠸⣧⡀⢀⣿⠁⠀⠰⣆⠀⠀⢀⡼⢀⡟⠀⣿⠀⠶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢹⡄⠸⡇⠈⠙⠛⠋⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠈⠙⠛⠉⠀⢸⠇⢠⡏⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣼⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠸⣆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠃⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⠀⠀⠹⣆⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣰⠏⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠀⠹⣆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠏⠀⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠳⣄⠀⠙⠳⢤⣀⣀⡀⢀⣀⣀⣤⠞⠋⠀⣠⠞⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠉⠉⠉⠉⠁⠀⢀⣤⠞⠁⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣤⣴⡏⠛⠶⠶⠤⠴⠶⠶⠚⢹⣦⣤⣀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⣀⣀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠄⠠⠄⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠁⠀⠀⠀⠂⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡐⠁⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢠⢜⡂⠈⠁⠆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⡖⠠⡆⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠐⠢⣠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣰⠋⠁⠀⠀⢴⡤⠖⠓⡦⠄⠠⠀⠘⠀⠚⢃⡤⣀⣀⠀⠀⠀⣟⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⠒⡐⢆⠀⠀⠀⠈⡆⠀⠀⠀⠘⡶⠁⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠂⠀⠸⣇⠀⠀⠀⢸⡇⠀⡄⠀⠐⠀⢁⠈⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠠⠀⠀⠐⠂⣨⡀⠀⠀⡄⣿⠀⠀⠉⠘⣷⡐⠁⠆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡏⠰⠀⢀⠀⢸⡁⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡂⠐⢼⡇⠀⠀⠐⠚⣧⠀⠀⠀⢀⠀⠀⠀⠠⠀⠂⠘⡾⠀⠀⠀⠀⢬⢸⡗⠁⠰⠉⢐⠁
⠀⠀⠀⠀⠀⠂⡀⠈⠹⣤⠀⠀⠉⢀⢸⡏⠀⠀⠀⠀⠘⠳⣅⠀⣸⠀⠀⠀⠀⠆⣠⠞⠁⠀⠀⠀⠀⠀⣻⣠⠀⠀⡷⠈
⠀⠀⠀⠀⠐⠀⠀⠂⠠⠷⢄⣀⡀⠈⢹⡷⠀⠀⠀⢈⠀⠄⣌⠿⢧⣄⡀⢤⣰⠟⠁⠀⠀⠀⠂⠠⠂⠈⣿⠀⣠⡾⢁⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠉⢳⣼⡇⠀⠀⠀⠀⠁⠀⠁⢀⢀⣩⡿⠿⣧⣞⣀⠀⠀⠼⠀⠁⠀⣸⣧⡾⠋⠀⠈⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⡇⠀⡀⠃⠀⠀⢦⣠⡾⠛⠁⠄⠀⠀⠉⠙⠿⢿⣦⣀⡄⠀⣿⣿⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢙⢀⣴⠟⠛⠲⠀⠂⠰⠁⠀⠀⠀⢤⡀⡬⠻⢿⣦⣿⣿⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠘⠀⠀⢻⣆⠐⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠈⠀⠀⠹⣿⣿⡇⠀⢀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠁⢀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠰⠀⢠⣿⣿⡇⠀⠀⠀⡵⠚
⠀⠀⠀⠀⠀⠀⠉⠑⣮⡀⠀⠀⣜⠁⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⣿⠋⠠⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠹⣆⡀⣾⠁⡄⠀⠀⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠾⣯⠈⠀⠈⡄⠂
⠀⠀⠀⠀⠀⠀⢀⡰⠆⠈⠉⠛⠛⠻⢷⣤⣿⣿⡇⠀⠀⠀⠀⠀⣠⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⣼⣿⣧⠈⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⡟⠁⢹⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⣿⣿⡇⠀⠈⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡇⠀⠀⢀⣿⢸⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⡆
⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⠾⠿⠿⠷⠶⠶⠶⠾⠿⠿⠟⠛⠛⠿⠿⠛⠛⠳⠶⠶⠤⣤⣄⣀
⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡷⠀⢠⣤⣤⣤⠀⠰⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡉⠀⣀⣀⣼⣿⣅⣀⣀⠀⢹⣿⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢀⣿⣿⠿⠛⠋⠉⣍⠉⠙⠛⠿⣿⣯⡀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⢛⣥⣶⡶⠂⣠⣄⠈⢳⣤⠞⠉⠀⠀⠀⠀⣀⣀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣡⣿⡿⢃⣴⡿⠋⣡⣴⡄⠉⠀⠀⠀⠀⠀⠀⠈⠉⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠛⢋⣼⣿⣿⣤⣾⠏⣠⣾⣿⣿⡟⣠⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⠟⣰⣿⣧⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣧⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢿⣿⣆⠻⣿⣿⣿⣿⣿⣿⣧⠹⣿⡇⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣿⣿⣿⣿⣿⢋⣴⣷⠀⠀⠙⠛⢰⣿⣿⣿⣿⣿⣿⣿⣇⠙⣰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣤⣤⣴⣾⣿⡇⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣛⣛⠻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣆⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣦⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣇⠹⣿⡄⢻⣿⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣌⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⡄⢻⣧⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⣿⣿⣧⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣦⣄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⡒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠
⣆⠀⢀⠀⠀⢀⣀⠌⠀⠀⠀⠀⠀⠀⠈⠳⠄⠄⠔⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠄
⠈⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣀⡀⡀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⢀⣀⠀⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡆⠀⠀⠀⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠐⠀⠉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⣀⠤⠂⠀⠁⠀⠀⠀⠀⠀⢀⠸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⡀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠝⠀⠀⠀⠀⠀⣀⠔⠈⠁⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⣀⠤⠤⠒⠂⠋⠉⠁⠀⠀⠀⠀⠄⢀⠖⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠊⠀⠀⠀⠈⡄
⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠂⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⢀⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠤⠒⠒⢲⣄⡀⠀⠀⠀⠀⢸⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠈⠂⠤⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠴⠊⡅⠀⠈⢁⠀⠐⢀
⢄⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⡀⠀⠀⠀⠀⢀⡀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠤⠣⡀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠡⡀⠀⠀⠑⢄⠠⠖⠉⠀⠀⠀⠀⠀⢂⠀⠀⢀⡏⠄⠄⠈⠀⡀⠢⠀⠀⠀⠀⡠⠂⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢷⠂⠉
⠀⠀⠀⢡⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠢⣀⠀⠀⢀⠠⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢆⠀⢀⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠒⠤⢤⡀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡠⠤⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂
⠀⠁⠀⠀⠁⠀⠀⡠⡞⠙⠢⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠦⠤⢤⡈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡠⠃⠠⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⡗⠀⠀⠐⠒⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠃⠀⠀⠀⠀⠀⠐⠀⠤⠁⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡠⠃
⠤⠀⠀⠀⠀⠈⠢⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡰⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡃⠀⠀⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡌⢠
⠀⢀⠔⠋⠀⠀⠁⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐
⣆⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⡀⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣄⣒⣈⣁⣀⣀⣀⣀⣀⣐⣈⣀⣀⣈⣁⣘⣃⣀⣀⣀⣀⣀⣐⣁⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠒⠀⠠⠤⠀⣀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⡀⠀⣀⠔⠃⠀⠤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⢀⡠⠤⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠎⠈⠠⣀⠀⠀⠀⠂⠁⠢⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠐⢄⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡜⠀⠀⢠⠦⡄⠀⠀⠀⠠⡀⠀⠀⠑⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⠀⠀⡆⠀⢨⠀⢄⠀⠀⠘⢄⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⡀⠀⠠⡀⠀⠑⢄⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡟⢻⡶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠙⣿⢸⡇⢸⡇⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠾⠷⠶⠶⠶⠶⠶⠾⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠋⠈⠁⠈⠁⢸⡇⢀⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡿⠋⢉⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠗⠲⠶⠶⠶⠶⠶⠿⠶⢶⣄⠀⠀⠲⣶⠶⢶⡶⠲⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢦⣄⠈⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡀⠙⠳⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠶⠿⠦⠀⠀⠙⠷⢶⣶⠶⠶⠶⠶⠶⠶⢶⣴⠟⠁⠀⠀⠀⠀⠀⠀⣿⣶⡄
⠀⠀⠀⠀⠀⢀⡿⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣁⣠⣾⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠉⢸⡇⢀⡀⢀⡀⣠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠶⠶⠶⠶⢶⡶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⡇⢸⡇⣿⣄⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⣧⣼⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⣤⠶⣄⢠⡴⢦⡀⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠷⣤⠟⠸⣦⡼⠃⢯⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠀⣀⣀⠀⣀⣀⠀⣀⡀⣀⣀⣀⣀⡀⠀⣀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⡏⣹⠆⣯⢉⡇⢸⡇⢸⡉⠉⠈⣿⠁⢸⡇⣾⡉⣹⢸⡟⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠉⠉⠀⠉⠉⠀⠉⠁⠈⠉⠉⠀⠉⠀⠉⠁⠈⠉⠉⠈⠁⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⢹⢶⡀⠀⣠⢾⢹⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢸⢸⢸⢹⢸⢹⢸⢸⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⣀⠀⠈⢸⢸⠈⠀⠀⣸⡟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⠴⠃⠳⣄⠈⠧⠀⣸⠸⡄⠀⠞⢁⡤⠃⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠙⢮⠁⠀⠀⡤⣾⣓⣒⣒⣷⣤⣟⣒⣒⣚⡆⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣼⣄⣰⣾⠛⠙⠋⠉⠙⠃⠀⠋⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠚⠉⠉⠉⢁⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡤⣄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡴⡫⢒⡭⣝⠲⣝⣆⠀⠀⠀⠀⠑⢡⠖⠚⠓⠚⠒⢮⠀⠴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⢹⢷⣃⠔⡡⢺⢸⢸⠀⠀⠀⢀⣼⣉⣉⣉⣉⡉⠉⠉⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣎⢧⡱⠮⠤⣊⢞⡼⠀⠀⠀⢸⢹⣁⣠⢤⣀⣠⠤⣈⣻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠩⠡⠠⠈⠓⠯⣭⡭⣵⠋⠀⠀⠀⠀⢸⢸⡇⠀⡀⠠⠀⢂⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⣀⣁⣁⣁⣁⣁⣀⣁⣀⣀⡼⠃⠀⠀⠀⠀⢸⣾⡇⡀⠀⠀⠐⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⣒⣒⣲⣴⢰⣖⣆⣶⣒⣚⣷⡀⠀⠀⠀⣠⢺⡯⡗⠒⠒⠒⠒⠒⠒⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠷⡜⢦⣀⡞⠁⢸⡇⣗⣒⣒⣒⣒⣒⣒⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠃⠀⠉⠀⠀⠀⠳⠼⠾⠿⠿⠿⠿⠿⠵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⢻⠖⢂⡇⢀⠴⠒⣲⢖⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⢠⢋⣴⣃⣮⠶⢭⠴⡷⠶⣭⢤⣎⡥⠤⠼⢷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠿⡏⣰⣣⠞⢿⡉⠑⡌⠢⠄⠷⠞⢄⡸⠈⣇⠙⡗⡶⣧⠤⠬⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡇⣠⠿⢏⡉⣱⠴⠋⠉⠀⠀⢀⡖⢲⡄⠀⠀⠈⠉⠁⢧⠈⡟⡶⣏⣉⡑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⣠⡷⠃⢲⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣯⣀⠀⠀⠀⠀⠀⠀⠹⠗⡁⢹⢦⣉⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢠⡏⣩⡯⠉⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠉⠁⡜⢹⡶⢄⣑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠛⢼⣏⠠⠜⠀⣞⢙⣆⡴⠒⠒⠒⢸⣀⠀⠀⠀⠀⣠⡵⠒⠒⠒⢦⣰⡏⢳⠀⠣⠔⢱⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢇⡾⠒⢦⠀⠀⠈⠉⡞⠀⠀⠀⠀⢸⠈⢙⡲⢔⡋⠁⡇⠀⠀⠀⠀⢳⠉⠁⠀⠀⡰⠁⣿⢦⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣑⢼⠧⠒⠷⠄⠀⠀⠀⢧⡀⠀⠀⣀⡼⠚⠉⠀⠀⠉⠓⡇⣀⠀⠀⢀⡼⠀⠀⠀⠀⣱⣚⣹⡤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣾⡞⢉⡆⠀⠀⠀⠀⠀⣙⠦⣍⡁⢸⠀⠀⠀⠀⠀⠀⡇⢈⣹⠲⣍⠀⠀⠀⠀⠀⢹⠋⣸⠷⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡗⠺⠖⣉⡀⠀⠀⠀⠀⡞⠁⠀⠀⠉⢸⢤⣀⠀⠀⣀⡤⡟⠉⠀⠀⠈⢳⠀⠀⠀⠀⢀⣉⣸⣇⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠒⢌⡟⢀⡇⠀⠀⣀⣀⣧⠀⠀⠀⠀⢸⢀⣨⠕⠮⣅⡀⡇⠀⠀⠀⠀⡼⣀⡀⠀⠰⣋⣠⢾⠏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⠿⣎⡤⢄⣀⢯⣸⠏⠳⠤⠤⠤⢞⠋⠀⠀⠀⠀⠉⡇⠤⣤⡤⠞⠹⣅⡽⢀⣤⡤⣤⣇⣧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⠢⣽⡄⡜⢛⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠹⠦⣾⣿⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢒⡛⢴⡁⢹⠀⡀⠀⠀⠀⠀⠀⠀⠉⣻⣟⠉⠀⠀⠀⠀⠀⠀⢀⣰⣋⣉⣹⢃⣋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣈⣙⡳⣆⣺⠉⡇⢀⡀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⡔⠚⠫⣭⠋⡏⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠱⣃⠒⠻⣷⣇⣏⠈⢶⣦⠒⢤⠀⡴⠢⣤⣶⡋⠑⢢⣽⠖⡹⢉⠆⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⠶⠊⢛⡿⠞⣶⣤⣧⠵⢦⡼⡶⠚⡩⠋⣥⡀⢧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠰⣉⠠⢴⣡⡊⣡⠜⠀⠸⠥⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠉⢉⠱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣆⢴⣏⣀⣀⣠⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⢿⠻⠿⠃⠁⠝⠉⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠻⠟⠁⠀⠀⠀⠀⠑⢐⠀⠀⠀⢀⡈⠳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣆⠦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡗⣤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⢿⣿⡁⠬⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠲⣶⡀⠀⠀⠀⠀⠀⠀⢺⣤⠋⠀⠀⢀⢻⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣦⡝⠁⠀⠀⠀⠸⠘⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠀⠢⢹⡄⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⢀⣾⡄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠻⢷⡀⠀⠉⠀⠀⢀⠀⠀⠀⠀⣠⠟⠛⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠖⠛⣳⠀⡤⠀⠀⠀⡀⡤⠔⣻⠟⠰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⣤⠶⠉⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⣀⠀⠀⠀⠀⢀⣀⣤⣾⠃⠀⠠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠛⠛⠛⠋⢻⣿⣏⠀⠀⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⡄⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠿⠉⠛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠏⠁⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⣼⣧⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⡀⢸⣿⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢻⣿⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⠂⠀⠀⠀⠀⠘⢻⡇⣿⠀⠀⠀⠀⠀⠀⢸⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠿⠀⠀⠀⠀⠃⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⢻⠿⡿⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣟⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠟⠁⠀⠀⣠⣿⢃⣾⠃⠀⣾⠇⠀⣠⡿⣧⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⡜⠁⠀⠀⢠⣿⣧⣃⣤⢾⡟⠀⢀⡆
⠛⠁⠀⢀⡴⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⣠⡾⠛⠻⠟⠁⠀⠀⢿⣤⡾⠋⠀⠀⢻⣀⣀⣴⠖⠀⠀⠀⠀⢀⡿⢁⡴⠋⠀⠀⢀⡴⣿⣏⡟⠉⠀⣾⠁⢠⠚⠷⣴
⣀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡷⠋⠀⠀⢀⡴⠋⠀⠻⠛⠀⠀⠀⠿⠖⠁⠀⠀⠸
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄
⠀⠀⠀⢠⡾⠛⣦⠀⢀⣠⣾⠆⠀⠀⠀⠀⠀⣴⡶⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⢀⣿⣿⣰⣿⡶⠿⣿⡏⠀⢀⡄⠀⢠⣾⠋⢠⣾⠇⠀⠀⠀⠀⠀⠀⣠⡟⠁⢀⡼⠁⢀⠄⠀⠀⠀⢴⡆⠀⠀⢀⡀⠀⢀⣴⢿⡆⢠⡀⠀⠀⠀⣠⣿⣿⠁
⠀⣴⣿⣿⢋⣿⠋⠀⣸⣿⠀⢀⣾⡇⢠⣿⣃⣴⠿⠃⠀⠀⠀⠀⠀⢀⣾⡿⠁⣰⠟⠀⣠⢷⣠⡤⠊⣠⠏⠀⠀⠀⣼⠃⣠⣞⡡⠋⠀⢸⠃⠀⠀⣰⣿⡿⠁
⡾⠋⢿⣇⡼⠃⠀⠀⣿⡇⢠⣾⠉⠓⢾⡿⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣡⡞⠁⢠⢾⣏⡠⠀⠀⢠⠏⠀⠀⣠⠎⠻⣴⣿⠋⠀⠀⣠⡟⠀⠀⢠⣿⡿⠁
⠁⠀⠀⠉⠀⠀⠀⠀⠻⢷⠿⠁⠀⠀⠸⣇⣀⣀⣤⡴⠂⠀⠀⢀⣿⣿⠟⠃⢀⡴⠋⠈⠉⠀⠀⠀⣏⠀⢀⡼⠁⠀⠀⢾⣿⣀⣠⣾⠏⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⣟⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣼⡿⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⣠⠟⠀⠀⡠⣄⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⡆⠀⢸⡆⠀⠀⠀⠀⠀⢀⣾⣿⢃⡾⠋⠀⣠⣶⡇⠏⣠⣾⠏⠀⣴⠂⢠⣾⣿⣿⠋⠀⠀⡀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⢠⣿⢻⡆⠀⣠⣿⠃⢀⡀⠀⣠⡿⠋
⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⢀⣴⣿⣏⡟⠋⣹⡟⠀⡴⣿⣤⣿⡿⠋⠁⠀⣠⠎⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⣿⣧⣾⣿⠃⠀⣸⡇⢰⣿⣡⣾
⠀⣴⠏⠀⠀⠀⠀⠶⣾⣿⡏⠀⢀⣴⠏⠸⠿⠟⠀⠀⢿⣧⠞⠁⠀⠹⣿⣄⣀⣤⡾⠃⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⢹⡏⠀⣰⡟⠙⣿⠟⠋⠁
⡾⠋⠀⠀⠀⠀⠀⠀⣿⣿⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣠⣿⠛⢿⠟⠀⠀⣿⡇⣴⡿⠁⠀⢿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠙⠻⠛
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣠⡖⢶⡄⠀⢀⣴⡷⠀⠀⠀⠀⠀⣠⣾⠟⣶⡀⢠⡏⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢠⣿⡇⣼⣧⣶⢿⣿⠃⠀⣀⠀⠀⣴⡟⠁⣴⡿⠀⢸⡇⠀⠀⣠⣾⠟⠀⠀⠀⣠⠖⠀⣸⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠁⣶⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣰⣿⡿⢻⣿⠉⠀⣾⡏⠀⢠⣿⠀⢰⣿⣠⡾⠛⠀⠀⠀⠳⠶⠿⠋⠁⠀⠀⢠⣾⠏⢀⡼⠁⣠⠈⠀⡠⢀⡟⠁⠀⢀⡖⠀⣰⢏
⢠⠇⠀⢠⣿⡿⠃⠀⠀⠀⢠⣾⠟⣿⢃⣾⠃⠀⢰⣿⠃⢠⡿⠉⠳⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣴⠋⢀⣴⠋⡙⠋⢀⡞⠀⠀⣠⢾⣇⣼⡿⠋
⠏⠀⢀⣾⣿⠃⠀⠀⢀⣴⡟⠁⠀⠉⠛⠁⠀⠀⠈⢿⣴⠟⠁⠀⠀⠹⣇⣀⣠⣴⠖⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⣴⠋⠘⠛⠀⠀⡼⠀⢀⡴⠁⠀⢸⣿⠀⣀
⠀⠀⣸⣿⠃⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣠⠟⠁⠀⠀⠀⠀⠀⠿⡶⠋⠀⠀⠀⠈⠛⠛⠋
⠀⠀⣿⡏⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠁
⠀⠀⠻⢧⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣠⣾⠟⠁⢠⣤⣄⠀⠀⣠⣶⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢷⠀⠀⠀⢀⡀
⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣠⣾⡟⠁⠀⢠⣿⣧⣿⠶⢻⣿⠃⠀⢠⡄⢀⣔⢋⣼⠃⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣠⠏⠀⠀⢠⡏⡃⠀⠀⣠⡟

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

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⢲⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢿⣿⡗⠰⣍⣯⡹⠸⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠟⢁⡘⢿⠟⢁⣼⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣴⠻⠿⢃⣴⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⡿⠅⠀⣿⣿⣿⡷⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡆⠀⡿⠿⠋⠁
⠀⠀⠀⠀⠀⣿⣿⢿⣿⠀⠀⠀⠀⢸⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠃
⠀⠀⠀⠀⠀⠻⠟⠈⠟⠀⠀⠀⠀⠘⠃⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⢀⣤⡀⢀⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣀⠀⢀⣤⡀
⠀⠀⠀⣤⣀⡿⣣⡾⠛⢿⣦⣭⣵⡾⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⡄⠀⠀⢈⣻⣧⣤⣿⣥⡄⠐⠄
⠀⠀⠀⠉⢻⣿⣿⡁⠀⣸⣿⣿⠃⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⠠⠀⠀⣴⢿⣿⡿⢻⣿⣿⠀⠀⠐⢀
⠀⠀⠀⠀⢸⡿⢻⡇⠰⡿⠁⢿⠆⣠⡴⠋⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠁⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠈⠁⣦⣀⣀⣤⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠸⣿⡇⠸⣿⡿⠀⠐
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠀⡈⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⠄⠀⠀⠀⠀⠀⣀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⠀⠀⢤⠉⠀⠀⠀⠀⢠⡾⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠃⠈⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⣠⡟⠀⠀⢀⡾⠁
⠀⣠⣤⠴⢤⣄⠀⠀⠀⠠⠦⠶⠶⠶⠛⠉⠀⠀⠀⠀⢀⣤⠾⣿⠁⣤⠶⠋
⣸⠋⣠⣤⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠋⠁⢰⡇
⣿⠀⠿⠀⢸⡇⢸⡟⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠘⣇
⢹⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⢸⡇⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠙⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠁⠀⠀⠀⣤⣀⠀⣀⣀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠰⣄⠀⠙⣿⠋⠉⠛⠋⠁⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢷⡄⠈⠓⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣘⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣤⣄⣠⣴⠟⠀⠀⠀⠀⠀⠀⠻⣆⣠⠶⠛⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⡄⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⡼⠁⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠁⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⣤⣤⣤⠞⠋⠈⠛⠐⠂⠘⠁
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠈⠻⠿⠿⠃⠀⠰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢠⣶⣦⢀⣶⣦
⠀⠀⠀⠘⣿⡿⠀⣿⣿
⠀⠀⠀⠀⢻⠇⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒
⠀⠀⠀⠀⢠⣄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⣿⢷⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠉⠑⢄
⠀⠀⠀⠀⣿⣼⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢿⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡰⠃
⠀⠀⠀⢼⢻⠟⡆⡞⢻⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁
⠀⠀⠀⠸⣼⣼⠁⢹⣸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠙⠃⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠠⠦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤⣤⣤⣤⣠⣤⣤⡤⠴⠶⠂
⠀⠀⠀⠚⡏⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣧⡟⡿⣿⣿⢿⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠲⣿⡏⠀⠈⠁⠀⡂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠃⠁⠃⠋⠘⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⡝⡏⡏⠿⡽⠹⠹⠹⢉⢣⢻⢻⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣⡀⠀⠊⠉⠂
⠀⠀⢀⢀⣞⢁⣜⢙⣇⢉⣦⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠀⠀⠣⣀⣳⣤⣃⣠⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣔⠡⣿⠄⢨⣧⠈⡦⠠⡤⠀⠀⠀⠘⠙⠉⠙⠓⠒⠒⠒⠒⠒⠻⠭⠙⠃⠀⠀⠀⠀⢨⡅
⠀⠀⠀⠀⠀⠐⠉⠢⠊⠁⠊
⡶⢶⠶⢶⡶⠶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⣶⣶⣶⢶⠶⣶⠶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⢀⣴⣶⣄⠀⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⠙⠛⠋⠁⣠⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣠⣤⣴⣦⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠋⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⢻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠐
⠀⠀⠀⠀⢀⣀⣀⠌⠀⠀⣀⣀⡀⠉⠙⠛⠿⣿⣿⣿⣟⡁⢄⠀⣾⣷⠀⠀⠀⠀⠀⠀⣾⣷⠀⡠⢊⣻⣿⣿⣿⠿⠟⠋⠉⢀⣀⣀⠀⠀⠡⣀⣀⡀
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠂⢸⣿⡷⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢸⣿⡇⠔⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣿⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠖⢀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣾⣿⣦⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣷⣿⠀⠀⠀⠀⠈⠁⣤⡀⠰⢷⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣥⠀⠀⠀⠀⠀⠀⠀⠉⣲⣤⡤⠶⠾⠇⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠆⠘⠟⠋⠁⠀⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠷⡄⠀⠠⠄⢀⣤⡀⠀⠀⠀⠘⠃⠈⠀⠀⡄⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣾⠶⣃⠀⠀⠀⣈⠋⠀⠰⠆⠀⠀⠀⢀⣄⠀⢀⡀⠀⠀⠀⢀⠴⣶⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠋⠘⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⢿⡟⠀⠀⠀⠻⣶⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠶⢨⡕⠶⢌⡁⢀⠀⣴⡄⢠⣾⡟⠓⡀⠀⠀⠀⠀⠐⠰⠆⠘⠦⠌⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠀⢶⠀⠀⢀⠀⡅⠙⠁⠦⢀⠀⠡⠼⢶⣥⠀⠀⠀⠀⠐⠆⠘⠲⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⠿⠛⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣤⡈⠅⣤⡐⠊⠀⣤⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠣⡀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⠀⠀⠉⠀⠤⠀⢠⡄⠀⠀⠠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠦⣤⣤⡶⠚⠛⢦⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣷⣬⣿⡶⠆⠁⢀⡀⢀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠘⢿⣷⡄⠀⠀⠙⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⡏⣀⠿⡆⠛⠁⠀⠆⠀⠴
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠈⠳⣄⠙⢄⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣟⣀⡁⠀⠘
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠈⠳⣄⠑⢄⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣷⣍⠋⠄⠀⠁
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠈⢷⣄⠳⣄⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⡛⠃⡀
⠀⠀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠙⢷⣌⠳⣄⣤⠶⠛⢉⣿⣿⣿⡇⣶⠄⢀⠈
⠀⠀⣧⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠹⣷⡇⠀⠀⠀⢸⣿⣿⣿⡆⠻⠀⠉⠁
⠀⠀⠹⣄⠀⠈⠓⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⡆⠀⢀⣠⠬⠟⠀⠀⢀⣼⣿⡿⢡⠀⠀⠀⠁
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠁⠀⠸⠟⠋⠉⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⣠⠞⢉⠁⠀⠈⠉⠳⣄⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⣠⠞⠉⠁⣀⣈⣉⣳⣄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠤⠀⠈⢷⠀⠀⢠⠏⠀⠀⠀⢴⣿⣿⣿⡇⠹⡄⠀⠀⡾⠁⠀⠃⠀⠀⠀⠀⣿⠈⢧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⢸⡇⠀⣿⠀⠀⣀⣀⣺⣿⣿⣿⡇⠀⣿⠀⢸⡇⠀⠀⠑⠒⠶⠒⠒⠋⠀⢸
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠹⡄⢸⣿⣿⠛⠿⠿⠿⠃⢠⠏⠀⠈⣧⠀⠀⣠⠊⠀⠀⠀⠀⠀⡼
⠀⠀⠈⠳⣄⠐⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠙⢮⡉⠉⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠧⣜⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠙⠓⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠋⠁
⠀⠀⠀⠀⣀⡤⠶⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⢤⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡴⢋⡄⠀⢀⣠⣴⣶⡝⢦⠀⠀⠀⢠⠞⠁⡐⠀⠐⠄⠀⠈⠳⡄
⠀⠀⡏⠀⢸⣿⣿⠀⠀⠀⡀⠀⢹⡄⠀⣸⠁⠸⡇⠀⠀⠿⣿⣿⡇⠈⣇⠀⢠⡏⠀⢸⠀⠀⣼⠃⠀⠀⠀⢹
⠀⠀⡇⠀⠈⠻⠋⠀⠀⠘⡿⠀⢸⠇⠀⢿⠀⠀⠀⡀⠀⠀⠈⠏⠁⠀⡿⠀⠸⡇⠀⢠⣄⣀⣸⡀⠀⠀⠀⢸
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠘⢧⠀⠠⡀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢳⡀⠀⣉⣭⣥⣧⣤⡄⢠⠏
⠀⠀⠀⠙⠲⢤⣀⣀⣀⣤⠖⠋⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠙⠲⣤⣀⣀⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣠⠶⠛⠉⠉⠉⡙⠲⣄⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⣠⠔⣋⣉⣉⣉⣛⡶⣄
⠀⠀⣴⠃⠀⠟⠛⠀⠈⠛⠛⠈⢧⠀⠀⢠⠏⠀⠀⣾⣿⣿⣷⠀⠀⠹⡄⠀⠀⡼⠁⠀⣿⣿⣿⣿⣿⠁⠘⣦
⠀⠀⡇⠀⠀⣁⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣷⠀⢰⡇⠀⠀⣿⣿⣿⣿⣿⢠⠀⢸
⠀⠀⣧⠀⠀⠸⠆⠀⠀⠀⠀⠀⣸⠁⠀⢹⡀⠀⠀⣿⣿⣿⣿⠀⠀⢀⡏⠀⠈⣇⠀⠀⣿⣿⣿⣿⣿⠀⠀⣼
⠀⠀⠘⢧⡀⠀⠂⠒⠒⠀⢀⡴⠃⠀⠀⠀⠳⣄⠀⠻⠿⠿⠟⠀⣠⠞⠀⠀⠀⠘⢦⡀⠻⠿⠿⠿⠿⢀⡼⠃
⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠓⠛⠂⠒⠂⠒⠓⠒⠒⠒⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠙⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣞⡽⠓⠲⠤⣤⠞⠉⠉⠉⠉⠙⠛⠒⠒⠒⠚⢯⢳⡀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢠⢎⡞⠀⠀⠀⡰⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣠⢫⠏⠀⠀⠀⡞⠁⢀⣠⠞⠉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⢳⡹⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣰⣣⠋⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⡹⣽⣄⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡜⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠁⠀⣳⠊⠉⢢⠀⠀⠀⠀⠀⠓⢤⡀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡜⠁⠀⡰⠋⠙⡆⠀⠓⢦⡀⠀⠉⢳⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣠⡞⠁⠀⡴⠛⠲⣄⠀⠈⢳⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣞⠁⠀⡴⠋⣳⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⠴⣄⠀⠀⠀⠀⢀⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⠏⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⡾⠀⠀⢠⠶⠒⢦⠖⠲⢦⡀⠈⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠖⠋⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⠷⣄⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠶⣄⠀⠈⠳⣄⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣤⠀⢹⡆⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣳⣼⡟⠓⠒⠶⠶⢦⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢦⣤⡟⠁⠈⠳⢦⣄⣀⣀⡼⠃⠀⠀⠀⠀⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠳⢤⣀⠀⠀⠈⠉⠙⣦⠀⠀⠀⠀⠀⡿⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡟⠓⠶⠦⠶⠃⠀⠀⠀⠀⠀⡇⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⣄⣠⣤⣤⣠⣤⣀⣠⡇⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⡗⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢠⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢸⡇⠀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠙⣟⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠛⢾⡷⠛⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠿⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠄⠀⢠⠀⢀⠀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡀⠂⠒⠀⡆⠒⠀⠂⠀⠸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⡤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠧⠤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⢀⣀⣀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⢀⣀⡀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠐⠁⠈⠃⢸⠀⢸⠀⢰⢻⡀⢸⠤⡇⢸⠀⠀⡇⠀⠀⠀⠧⡀⠀⢸⠀⢸⣀⠇⡜⠀⠈⡆⣧⡜⠀⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣈⣤⣤⣃⣸⣀⣸⣀⣏⣈⣁⣸⣤⣏⣸⣤⣄⣧⣤⣀⣀⣤⣜⣀⣸⣀⣸⣀⣇⣘⣤⣔⣁⣇⣈⣆⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠤⢦⡄
⠀⠀⠀⠀⠈⢹⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢥⠤⠼⠴⡅⠀⠀⠀⢰⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡆⡈⢘⢀⡄⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠐⠐⠀⠀⠂⠃⠀⠀⠀⢨⠁⠂⠀⠀⠀⠅⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⢇⣈⣈⢈⢈⡨⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠠⣀⠀⠀⠀⢠⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠖⠂⠒⢄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⡇⠀⠀⠀⡈⡷⠂⠀⠂⠀⠀⠐⠀⠀⢉
⠀⠀⠀⠀⢠⡀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈
⠀⠀⠀⠀⡀⢈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠁⠀⠀⠀⠀⠀⠈⢉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⣰⢼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠉⠘⠖⠀⣤⠸⣿⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠄
⠀⠀⠀⡄⠌⠇⣈⡜⠄⡘⠁⠀⠀⣿⡏⠉⡅⠀⣟⡽⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢳⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣏⢨⣿⠾⣢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣻⣸⢻⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣯⣿⣿⡻⡆⠀⠀⢰⠤⣀
⠀⠀⠘⠛⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⢀⣤⣤⣦⠀⠀⠀⣸⡟⠀⠀⠀⢉⣿⠀⠀⠈⠃⠻⣽⡇⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⠠⠀⠀⠀⡤⠄⣀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⢈⠥⠤⠤⠀⠀⠀⢹⠁⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⢡⢀⣽⣾⣿⡿⠗
⠀⠀⠀⠀⠀⠈⠀⢠⡛⠱⡎⠀⠀⠀⠘⠓⠓⠋⠀⠀⠀⠈⠓⠓⠻⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠦⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣀⡀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⣶⣿⣯⣵⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠒⠒⣄⣀⣠⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⡀⡞⠁⠄⡈⢣⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⡀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠛⠻⢿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣿⣿⢁⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣤⡄⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠼⠿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⣀⣉⢩⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠰⠿⠶⡾⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡼⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⢹⡟⠀⠀⠀⢀⣀⣤⣤⢤⡄⠀⠈⠛⢿⣶⠖⠋⠁⠀⠀⠀⢠⡖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣰⣟⠀⠀⠀⢸⣿⠿⠋⣀⣤⢏⣀⣀⣀⡞⡎⠀⠀⠀⠀⠀⠀⠈⣿⣷⣞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠚⠧⣌⢦⡀⢀⣀⣽⠿⠟⠛⣻⠟⠉⠉⠙⢣⡇⠻⠶⢤⣴⠶⢤⣴⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡍⠋⠙⠷⣦⡤⠞⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⡀⠀⠹⣧⠀⠀⢀⣾⠳⣄
⠀⠀⠀⠀⠀⣠⡾⡆⠀⠀⠀⣠⣿⣥⡄⠀⠀⠀⠀⢳⡇⠀⢀⣤⠶⠶⠶⣶⡄⠀⠀⡼⣷⠦⠶⠟⠳⣤⣸⣿⡄⠸⣇
⠀⠀⠀⠀⢰⡟⢰⣧⣀⣀⣼⠋⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⣻⣅⣀⣀⣴⠟⠁⠀⠘⠱⠁⠀⠀⠀⡠⢊⠄⠻⢷⣤⠏
⠀⠀⠀⠀⢻⣧⠟⠓⠀⣿⡇⠀⣠⠿⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡽⠇⠀⠀⣠⠶⢤⡀⠀⠀⠀⠰⠃⡀⠻⣆⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠋⠉⠳⠬⠁⠀⢀⣴⠾⢿⠄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻⣦⠀⠀⠀⣼⠛⢷⠈⠤⠴⠚⢻⣷⣼⡿
⠀⠀⠀⠀⠀⢀⡶⢆⠠⡇⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⢺⠿⠶⠾⠛⠀⠀⣠⣿⡀⢸⡇⠸⣿⠁⠈⠛⠛⠁
⠀⠀⠀⠀⠀⣾⠀⢸⣗⣒⠒⠒⠂⠀⠀⠀⠸⢿⡶⠛⠛⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠿⠁⠀⣿⠀⢀⣀
⠀⠀⠀⠀⠀⣿⣀⡞⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠈⢻⡦⠀⠀⠀⢀⡴⠒⣦⠀⠀⠀⢀⠀⢀⣿⣠⡿⠏⣷
⠀⠀⠀⠀⠀⠈⠉⠀⢾⡇⢰⡟⢿⣦⠤⠖⠋⠀⠀⠀⠀⠀⠀⢸⡻⣦⣤⡾⠁⠀⠀⣸⣿⠁⠀⣸⠀⠀⠀⠨⠭⠭⣭⣿⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⡿⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠉⠀⠀⠀⠀⠈⢿⣀⣰⠃⠀⣀⣄⡀⠀⢀⡏⠘⢷⠟⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠘⣇⠸⣦⣼⠃⠀⠀⣠⢲⡶⢶⣿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠉⢿⡿⠁⢨⠇⠀⣼⠇
⠀⠀⠀⠀⠀⣿⡉⢳⣤⡿⢂⠀⠀⠀⠀⠀⢻⣌⢳⡀⢹⣿⠀⠀⠀⠀⣠⡿⠛⠛⢻⣷⠀⠀⠀⢸⣃⣠⠏⠀⣴⡟
⠀⠀⠀⠀⠀⣿⡀⠀⣿⠟⢯⡀⠀⠀⠀⠀⠀⠙⠓⠛⠛⠋⠀⠀⠀⢰⣇⣀⣠⡴⠟⠁⣬⢷⡀⠀⠉⠠⣄⡘⠋⣠⠴⣦
⠀⠀⠀⠀⠀⠘⢷⣴⠏⠀⠀⢙⣦⡀⣠⢏⡀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠈⠁⠀⠀⠀⢀⣽⠞⢛⡿⢈⡷⢦⣙⣿⠃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠛⢷⣎⠀⠀⠀⠀⣴⠏⠉⠉⠙⢲⣤⠀⡀⡀⠀⢾⣧⣤⣮⠴⠋⠀⠀⢹⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⢻⡛⠳⢦⣄⡙⠶⢤⣤⣶⠾⠛⠀⣿⣷⠶⠶⠚⠉⠀⠀⠀⠀⠀⠸⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠃⠀⢀⣨⣿⣿⠛⠛⠛⠉⠉⢉⡿⠿⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣈⣱⡄⠀⠀⢰⣇⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⣀⠀⠀⢠⡅⠐⠀⠉⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠈⠑⢤⠀⠀⠠⢊⠀⠀⠀⠀⠀⠀⠑⢄⢀⡠⣛⡆⢀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡢⡀⠱⡀⠁⠀⠀⠀⠀⠀⠀⢀⢔⣫⣵⠊⢁⢁⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠀⠀⠀⠀⡀⢔⡬⠒⠉⢁⠸⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡥⠚⠁⠀⠀⠀⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠈⢆⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢆⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⢁⠊
⠀⠀⢀⠄⣒⠃⣿⣶⣤⠄⡪⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠶⠁⠀⠀⠀⠐
⢔⣪⣿⣛⣽⡿⢞⣩⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡀⢈⠕⠉
⠀⠈⠉⠻⣵⠚⠅⠀⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⠠⠂⠁
⠀⠀⠀⠀⢱⠆⣆⠠⠊⠁
⠀⠀⠀⠠⠌⠊
⠀⠀⠀⠀⠀⢠⠀⠀⡠⠠⡀⡠⡢⠀⠀⡰⠂⠀⢀⢄⠀⠀⡀⠀⢀⠐⢀⠀⠀⡀⡀⡀⢀⢀⢀⢀⠀⢀⠀⢄

⠀⠀⠀⠀⢀⠤⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠤⠔⠂⠀⠁⠀⠀⢾⡄⠤⠒⠁⠀⠔⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠂⠀⠀⠀⡜⠀⡀⠤⠠⡀
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢂⠀⡊⠀⠀⡌⠀⢀⡀⢈⠣⠀⠀⡠⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄⠀⠀⠀⠰⢊⡄⠊⠀⠀⠀⡄
⢀⠚⠓⠂⠀⠀⠂⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡃⢰⠁⠀⢀⠠⢈⠤⠋⠀⠀⢀⡤⠊
⠀⠈⡁⠈⠁⠀⠀⠀⠀⡈⢄⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⠀⢐⠄⢀⣠⡦⠦⠤⠀⠀⠄⠀⠀⣀⢀⢃⠂⠀⠐⠉⠀⠁⠀⡰⠆⠀⠁
⠀⠸⠉⠙⠀⡦⠐⠀⠈⠁⠀⠉⠐⠀⠀⠀⠀⠀⠀⠈⠑⢀⣡⠞⠑⠁⠄⠀⠀⠔⠀⠀⠀⢀⠶⠻⠁⠀⠀⠀⡰⠀⠀⣰⣶⣿⡏⢿⣿⣿⣧
⠀⠄⠀⠀⠀⠇⠀⠀⠀⠠⠀⠀⠀⢠⣤⢠⠀⢀⠀⠀⠀⠋⠁⠐⠒⠀⠤⠄⠀⢀⣠⠔⠊⠁⡠⠊⢸⡇⠀⠀⢐⣷⠀⠙⠿⠛⠁⠀⠉⠛⠁
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡄⠀⢄⡈⠁⠈⠃⠈⠀⠀⠀⠒⠠⠄⠀⣀⠀⠀⠀⠀⠀⣀⠔⠋⠀⢀⣾⣿⣤⣤⣾⣷⡀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠈⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠰⠑⡀⠀⠀⠠⠀⠄
⠀⠐⠂⠀⠚⠀⠂⠀⠀⠂⠀⠁⠈⠀⠀⠈⠸⡀⠁⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⢉⠑⢢⠀⠀⠀⠀⢹⣻⣿⣿⣿⡏⠀⠇⡀⠐⡄⣠⠂⠀⠄
⠐⢠⣤⣴⡄⠀⢠⣶⣤⠐⠂⠄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠂⠄⠀⠀⢀⠀⢀⠀⠨⠀⠀⠃⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠋⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡄⠀⠂⡀⠂⠀⠈⠀⠀⡀⠀⡀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠀⡖⠂⠀⠠⠄⠀⠀⢠
⠀⠰⠁⠉⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠄⠀⠀⠀⢀⠀⠀⠄⢠⠃⠀⠀⠀⠈⢹⣿⡿⠉⠉⠀⠈⠀⠀⠀⠀⢀⠀⠉
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠑⠂⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡄⠀⠀⠀⡀⠁⡈⠁
⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢇⠄⠠⡀
⠈⠀⠀⠀⠀⠧⣀⡀⠀⠀⠃⠀⠀⠀⢀⢀⠀⠀⠈⠠⠀⠆⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡆⠀⠀⢱
⠀⢀⠄⠀⠒⠂⠄⢀⢠⣤⣶⡐⠐⣿⣿⠇⢀⠐⠁⠀⠀⠀⠀⠂⠀⠀⠈⠢⡀⠀⠀⠄⠀⠢⠄⠀⢀⠀⠀⠀⠀⢠⠄⠅⠀⠀⡄⠉⠀⠀⡆
⠀⠄⠀⠀⠀⢠⡄⠁⠀⠛⠛⠁⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⢰
⠀⠀⠐⠦⡀⠨⠁⠐⠢⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠂⠐⠀⡉⠀⠈⠀⠀⠀⠄⠠⠖⠂⠀⠀⠀⢀⠀⠂
⠀⢆⠀⠀⠀⠁⠂⠠⠄⠀⠀⠀⠠⠄⠀⠢⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠀⠀⠀⠁⠀⠀⡄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⢰
⠠⠀⠀⠄⠠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠠⠄⠀⠀⠆⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢄⠀⠈⠓⠀⠀⠀⠈⠀⠀⣠⡄⠀⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠒⠆⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠒⠒⠒⢲⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠱⠀⡀⠀⠀⠁⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⡀⠣⢀⠀⠀⢀⡀⠀⠀⠀⠠⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠓⠀⠁⠀⠀⠀⠀⠠⠄⠤⠂⠀⠀⠀⠑⠂⠀⠐⠑⠀⠐⠂⠁⠀⠀⠀⠈⠁⠁⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠴⠄⠲⠂⠐⠒⠀⠶⠆⢤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠓⠈⠁⠀⠀⠀⡀⢀⣀⠀⣀⠀⠀⠀⠀⠈⠙⠰⢦⡀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠉⠀⠀⢀⣤⠄⠖⠃⠉⠉⠈⠉⠀⠉⠁⠘⠓⠠⢤⡀⠀⠀⠀⢀⡴⠋⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⢀⠠⠞⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⢀⡐⠋⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢀⣀⣄⣀⠀⣠⡴⠛⠻⣯⠀⠰⠄⠘⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⣠⠛⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠉⠻⣅⠀⢀⡼⠃⠀⠀⠈⠷⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⣀⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢀⡴⣏⠙⣦⠀⠙⢦⡀⠈⠳⣞⠁⠀⠀⠀⠀⠀⢳⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢐⠃⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣴⠛⣇⠘⢧⡈⠻⢦⠀⠙⢦⣄⢸⡆⠀⠈⢷⡀⠀⠀⢶⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠟⠀⠀⡾⠀⠀⠀⠀⠀⢀⡴⣏⠙⣦⠈⠳⣄⠙⠓⠀⠀⠀⠀⣨⡟⠁⠀⠀⠈⣇⠀⠀⠠⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢨⠁⠀⠸⠇⠀⠐⠃⠀⠀⠀⢀⡴⠻⣆⠙⢦⡈⠳⢦⠈⠁⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢻⠀⠀⠀⣥⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢘⠀⠀⢰⡆⠀⠀⠀⣤⢀⡴⢿⠀⠀⠙⢧⣄⠙⠆⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣙⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡄⠀⠀⠀⠈⢿⡀⠈⢷⡀⠰⣦⡉⠀⠀⠀⠀⣠⠞⢁⣠⠶⠚⠋⠉⠛⠓⢦⣄⠈⠀⠀⠀⠟⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⢈⣷⡄⠀⠙⢦⣌⠳⣄⠀⣠⠞⠁⣠⠟⠁⣠⠖⠛⠛⠛⠶⣄⠈⢳⡀⠀⠀⠃⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡿⢦⣄⠀⠉⠳⣾⠞⠁⠀⢠⡏⠀⡾⠁⠀⠀⠀⠀⠀⠘⣇⠀⢷⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢷⡴⠋⢠⡾⠋⣠⠞⠉⠛⠶⣾⠁⠀⠀⠀⢸⡇⠀⣧⣠⠤⢤⣠⡤⢤⣀⡟⠀⢸⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⣈⣠⠞⠁⢀⡀⠀⠀⠈⠃⠀⠀⠀⠘⣧⠀⢸⣛⡒⢋⣬⠓⢞⣹⠇⢀⡿⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢳⡄⠀⠈⠙⠲⢤⣄⣀⣀⠀⠀⠀⠘⢧⡀⠉⠻⠦⣤⠼⠋⠉⣠⠞⠁⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠙⠳⠶⢤⣤⠤⠶⠛⠁⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠓⠆⢤⣤⢀⣀⡀⣀⣀⢀⡤⠀⠀⠀⠀⠀⠀⣀⡄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢀⣄⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⣀⡀⠶⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠃⠰⠶⠠⠶⠄⠶⠖⠐⠛⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣄⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠙⢷⣤⣶⠟⠁⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣤⣴⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⠙⣷⣆⣀⣀⣀⣀⣠⣤⣤⣄⣤⣦⣤⣄⣀⠀⠀⣀⣾⡟⠁⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣰⡟⠋⠉⠉⠉⢉⣭⡾⠟⠋⠁⠀⠀⠉⠙⠛⠛⠋⠙⢿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠿⣧⣄⣀⣠⣤⡶⣦⣄⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢠⡿⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠉⠻⣦⣄⠀⠀⠀⢹⣷⡀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⡿⣧⣴⠟⠋⢻⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠈⠙⢷⣄⣴⡟⠙⢷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣠⡾⠛⠁⢈⣿⣤⡀⠀⠀⠀⠺⢶⣄⡀⠀⠙⠷⣦⣀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠀⢀⣴⠿⠋⠉⣿⣆⣰⣦⣀⠀⠙⠻⣦⣄⠀⠈⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⠁⠀⣠⡾⠋⠉⢻⡎⠛⢷⣦⡀⠀⢹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⢿⣄⣀⣴⠿⣧⣄⠀⢸⣿⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠙⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠈⠈⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⡀
⠀⠀⠀⣼⡿⣿⡿⢿⣿⣷⡀⠀⠀⠀⠀⣾⣿⡿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⡿⣿⡄
⠀⠀⠀⠙⣀⠀⡀⠸⠿⣿⠁⠀⠀⠀⠀⢻⢏⣭⠀⢤⠸⡏⠀⠀⠀⠀⠀⠀⠫⡌⠉⡀⢸⢿⡿⠀⠀⠀⠀⠀⠘⡇⢠⠈⢨⠀⡿
⠀⠀⠀⠀⠈⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠘⠀⠴⠀⠀⠁
⠀⠠⠄⢀⣀⠚⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠓
⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⡀
⠀⠀⠀⣿⠛⠋⠿⠛⢻⠇⠀⠀⠀⠀⠀⠸⡇⢠⠈⢩⠀⡿⠀⠀⠀⠀⠀⠀⢿⠁⡀⠉⣁⢸⠏⠀⠀⠀⠀⠀⠸⡏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠘⡀⠁⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠗
⠀⠀⠀⠀⣃⠤⡤⢜⠀⠀⠀⠀⠀⠀⠀⠀⡈⡂⣄⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣈⣭⣍⣉⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣬⣤⣀⡁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠘⡏⠈⠉⠉⣿⣿⡟⠀⠀⠀⠀⠀⠸⡟⢉⠙⠛⠉⡿⠁⠀⠀⠀⠀⠀⢿⠁⡄⠉⡅⢸⠇⠀⠀⠀⠀⠀⠸⣏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠉⢁⠈⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠑⡈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡌⠀⠈⡈⠁
⠀⠀⠀⠀⠐⠒⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐
⠀⠀⠀⠀⡠⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⢠⠔⠰⢠⡆⢊⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣤⣮⣭⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠑⢈⡊⠒⠘⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠀⠛⠛⠃
⠀⠀⠀⢲⢤⡠⠆⠀⢦⢠⡄⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡰⢸⠰⠲⠆⠞⠈⠀⠆⠳
⠀⠀⠀⠛⠈⠃⠘⠁⠈⠉⣂⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⢻⡏⣿⢩⡏⡻⡟⣻⡏⣿⢽⢿⣿⡯⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣷⣾⣵⣽⣼⣧⣧⣷⣿⣷⣿⣾⣶⣿⣧⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠛⠛⠋⠛⠿⠿⠛⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣽⣿⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡼⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠗⢦⡣⠤⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠒⠚
⠀⠀⠀⠀⠀⠀⠈⠘⠐⠑⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢸⡆⡷⠴⡄⣾⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠋⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠘⠀⠁⠈⠀⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢤⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠂⢙⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⢰⡘⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣰⠀⡄⢀⢀⡔
⠀⠀⠠⠐⠈⠅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠃⠈⢠⠆⡄⡤⡀
⠀⠀⠐⠒⠒⠒⠂⠉⠉⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⠚⠙⠓⠂
⠀⠀⠀⠀⢠⣟⠁⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠄⠀⠘⠣
⠀⠀⠀⠀⣸⠋⠀⠀⠉⣷⠀⠀⠀⠀⠄⠀⡜⠀⢠⢤⠀⡀⠤⢤⠀⠢⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠘⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⢀⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠢⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣽⠁⠀⢺⡋
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠆
⠀⠀⠀⠀⠀⠀⡀⠸⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠉⠉⣹⠐⠀⠀⠀⠀⠀⠀⡀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠏⠀⠀⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠚⠛⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⠊⠀⠀⠀⠀⠀⠤⠤⠄

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡄⠀⠀⠀⠄⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠑⠉⠀⠀⠊⠀⠀⠀⠀⠆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⣵
⠀⠀⢀⣠⠤⠤⠤⠀⠠⡴⠷⠼⠦⢠⡄⠨⣤⢀⣀⣀⡂⠤⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⢢⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠘⡂
⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠳⠤⠔⠋⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣠⡀⠠⠔⠊⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠣⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡰⠁⠀⠠⠴⠚⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠑⡓⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⡤⠄⠤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠀⠁⠂⢀⢃⠀⠀⣿
⠀⠀⠀⢀⠔⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⢰⠀⠀⡂⠥⢒⡶⠆⠀⠀⠛
⠀⠀⠀⠁⠤⠄⠉⠰⠀⠀⠀⠄⠀⠀⠀⢤⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠘⠤⣾⠶⠚⠉⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠰⠦⠀⠙⢀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡇
⠀⠀⠀⠀⠀⠀⠐⠲⢀⡈⠃⠀⠈⣀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠄⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⢠⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⢠⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⢠⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⡀⠀⠀⠀⠀⠀⣸
⠀⠈⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠁⡆⠀⠀⡀
⠀⢀⠀⠀⠀⣀⠀⢀⠀⣀⠀⣀⠀⡀⠀⠃⠀⠀⠀⢀⡀⡀⠀⠀⠀⠘⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢀⡇⢠⠀⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠑⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⡸⢰
⠀⠘⠀⠀⠀⠀⡴⠋⠉⠒⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢠⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⡌⡗⡱⠃
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⠐⠉⠒⠢⢄⣃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠱⠞
⠀⢸⠀⠀⠀⠀⢂⡀⠀⢀⠎⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣭⣤⢤⣤⣤⣤⣤⣤⣤⣀⣁⡀⢀⣀⠤⠔⠒⡦
⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⡟⣷⣷⣿⣧⣿⣼⣧⣾⣸⣯⣿⣿⣼⣿⣷⣿⣿⠉⢀⣠⠴⠚⠁
⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⢩⠥⠒⢺⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⣠⠤⠔⠒⠻⠿⠿⠿⣿⣿⣿⣿⣽⣿⣯⣿⣽⣿⣿⣿⣿⠞⠉⠉
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡘
⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠄⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠈⢆⠀⡔⠁⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠻⡀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢤⡀⠀⠀⠀⠑⢄⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢳⣄⠀⠀⠀⠈⣇⠐⠄⣠⠤⣄⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⢆⠀⠀⠀⢫⠢⡄⠀⠁⠘⡄⠀⠁⠀⡄⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠘⡄⠣⡀⠀⠀⠈⢆⠀⠀⠀⢣⠻⣄⠀⠀⠰⠀⡇⢠⠁⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠘⡀⠐⡄⠀⠀⣸⡄⠀⠐⠉⡄⠈⢂⢀⣀⡇⠈⠪⢄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⡇⠀⡇⠀⠀⠸⠱⠀⠀⠀⣱⢖⢜⠀⢀⡦⣀⠀⠀⠑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⠁⠰⠁⠀⢀⠃⠀⠃⠀⠀⢸⡌⠛⣼⣡⠃⢸⠑⠤⣀⡀⣈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡌⠀⠀⡆⠀⠀⢀⠘⢄⠀⠀⠑⢄⢆⡰⠁⠉⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠘⠀⠀⠀⠇⠤⠤⡎⢢⣈⠑⢤⡤⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠀⢣⣣⡸⠑⠮⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⢸⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠸⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣸⣇⡀⣀⣤⠖⠛⢃⣤⠖⠛⠓⠶⠦⣤⣄⣠⣤⠼⣧⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣰⠏⠈⠙⠉⠀⠀⣠⠟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣾⡁⠀⢀⣤⠞⠋⠈⠳⣤⡀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣄⣠⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⢿⡀⢠⣀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡈⠉⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⢈⣧⠈⠉⣁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⠓⣴⠖⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠘⣧⠉⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⠟⠉⣹⠶⢦⡀⠀⠀⢀⠀⠘⠳⣤⡀⠀⠙⣷⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣾⠃⠀⣼⠛⢶⡄⠈⠻⢦⡀⠀⣹⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⣇⣠⡞⠡⣤⣀⣸⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣩⣏⢹⣻⣹⡉⣋⢙⣛⡉⣻⣁⣏⣟⣝⢙⣉⣟⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣄
⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣇⣶⣶⡀
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⣷⣿⣯⣥⠄
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡾⠽⢿⣿⣿⣥⠄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⣀⣹⣶⣦⣿⣿⣿⣟⡥
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣐⣾⡞⠁⢹⣿⣿⣿⣯⢾⠏
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣩⠴⢿⣾⠞⣯⢻⣸⣧⣯⣤⠄
⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢤⣰⣿⢻⣀⣰⣿⣋⠙⣿⣿⢿⣿⡿⠆
⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⢁⣏⣓⣿⣿⣿⣿⠛⢾⡿⠴⣿⠋⣩⠽⣇
⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠳⢾⣍⡿⠛⢿⣽⣿⣌⣿⡛⠙⣧⣾⣷⣤⠈
⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⣴⡷⢋⣹⣯⡤⠿⢶⠤⣿⡙⠛⣿⢧⢞⣙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⡏⣸⣿⡁⡞⢽⠳⡍⢳⣸⡄⢸⡝⢦⢸⣟⣉⠉⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠻⣿⣴⠻⣄⡙⣷⣮⠷⢛⡫⠋⡇⢸⠀⢨⢿⡁⣠⣾⡅⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡏⠙⠒⡾⠻⣄⠈⠫⣭⣭⣟⣋⡤⣾⣴⠋⡰⠋⢀⣿⢁⣀⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⡞⢆⢀⡄⣈⠉⢳⣙⣤⣹⣇⠀⢻⡓⠲⠾⠶⠤⠴⠚⢁⡼⠋⢀⣤⠿⣿⠈⣉⣠⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠶⣷⣿⣸⣇⣿⢶⣋⣉⠤⢬⡟⠓⣤⣉⣓⣲⣶⡦⠴⠒⠛⢒⣫⢿⣯⣅⠘⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠒⠶⢿⣿⣿⡛⠛⢉⢩⣿⡿⠂⢲⠞⢳⠤⠽⠓⢤⣤⣤⣔⣋⣁⣀⣁⣈⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣉⠛⣹⠿⣿⠗⣺⣿⢻⡇⠀⠉⠀⣀⣠⠴⣾⡉⠉⠗⣨⠏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣰⠃⢈⡵⢿⣨⡷⠛⢛⣯⠤⠶⠛⠁⠉⠳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡾⠋⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠀⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡄⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣬⣹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠄⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⣿⠋⠉⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⣹⠋⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠐⢛⠇⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠈⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣆⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢿⠀⠀⠀⣀⣠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠑⠠⠁⠰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣑⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣏⠀⠀⣼⣿⠟⠉⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣦⣴⣿⣿⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⢻⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠇⡀⠀⠀⠀⠻⣿⣿⣿⣿⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠈⠿⠿⠛⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠸⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠐⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⡜⠑⠒⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠃⢀⣤⣄⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠻⠛⠛⠛⠛⠆⠀⠀⠀⠀⠀⠀⠀⣠⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠁⣠⢉
⠀⠀⠀⢄⠀⠀⡄⠀⢠⡀⠀⠀⠀⢰⣧⣞⣾⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣰⠈⠂⠀⡁⠀⠜⡧⠀⠀⠀⠸⣝⠟⣳⠏⠀⠀⠀⠀⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂
⠀⠀⢻⠣⠀⠀⠈⠀⣼⠸⠃⠀⠀⠀⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⢇⢑⠀⠤⢊⡄⠀⠀⠀⣠⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠐⠂⠆⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⢠⣠⣤⣤⣄⣄⠀⠈⠉⠲⢀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣴⠛⣤⣿⡯⠥⣉⠙⢻⣆⠀⠀⠀⠠⠻⣿⠲⠄⠀⠀⠀⠀⠀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣄⢸⡃⠀⠀⠈⢆⠀⠻⠃⠀⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣄⡒⣿⣶⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣄⣀
⠀⠀⠀⠠⠂⠐⠀⠀⠘⠐⠰⠀⠀⠀⠀⠀⠀⠩⠿⣿⣿⣤⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠁⠀⠀⠈⠃
⠀⠀⠀⠄⠀⠥⠖⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣬⠛⠿⠑⠲⠂⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠄⠀⣔⠄⡀⠀⠈⠀⢸⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⡤⠔⠃⠀⠀⠀⠀⠘⠉⠀⠀⠐⠀⠀⠈⠀⠀⢜⠀⠁⣀⠠⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠠⠂⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠉⠀⠈
⠀⠀⡤⢈⠒⠀⠃⠀⠇⢠
⠀⢠⠁⠈⠀⠠⢦⠄⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠸⠀⢀⡀⣃⡴⣂⣈⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢛⡨⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡒
⠀⢀⠀⢨⢀⣅⣬⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡈⠁⠛⠁⡀⠀⠀⠀⠀⠀⣤⡸⠀⠀⠀⠀⢈⣴⣿⣦⣤⠑
⠀⠐⢄⠀⠈⢫⣠⡞⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠁⠀⠘⠁⠀⠀⣐⠀⠀⠀⠀⠀⢸⣿⣁⣀⡿⡇⢠
⠀⠀⠐⠂⠀⠀⡑⠒⠐⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠀⠀⠀⠀⠂⠓⠿⡛⠋⢐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠠⣞⠛⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⠔
⠀⠀⠀⠀⣤⡀⠀⠙⠛⠋⠀⢀⣤⠀⠀⠀⠀⠀⡠⣀⠀⠑⠒⠊⠀⣀⢤⠀⠀⠀⠀⠀⡠⡀⠀⠑⠒⠊⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣼⣿⢯⣦⣰⢶⣶⣶⡽⣿⣧⠀⠀⠀⡜⠁⡜⢢⠔⣒⣒⠒⢣⠀⠁⠀⠀⠀⠀⠀⡜⠢⠔⣒⣒⠒⢣⠀⢢⠀⠀⠀⠀⠀⠀⠂⠄⠐⠀⠀⠃
⠀⠀⠀⠙⢋⣿⣻⣷⣿⣿⣿⣿⠙⠋⠀⠀⠈⠓⢾⣶⣌⡊⠀⠀⠙⢦⡓⠚⠀⠀⠀⠀⠺⣄⡌⡊⠀⠀⠑⠠⠓⠚⠁⠀⠀⠀⠐⠀⠌⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠘⠽⣮⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠙⢎⡽⡄⣀⣮⠗⠁⠀⠀⠀⠀⠀⠀⠙⢎⡵⡄⢀⡦⠖⠁⠀⠀⠀⠀⠀⠀⠉⠄⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿