Sneeze Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⣹⡿⠂⠀⠰⠿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⣫⣿⡿⠟⠉⢸⣿⣥⣶⣦⡄⠀⠀⠀⣸⣿⣿⣿⣧⠲⡌⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⢸⣿⣿⡟⠃⡀⠀⠀⢀⣀⠘⠋⠀⠀⠀⡽⢿⡉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⡰⢋⣤⠴⠛⠛⠉⠀⠀⠀⠀⠀⣄⣸⡇⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠙⢧⣻⠿⠓⣒⣈⣉⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡔⣡⠞⠏⠀⠀⠀⠈⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣠⡇⠀⣀⡴⠚⢩⣼⣿⣧⡀⠀⠀⣀⠀⠀⠀⠻⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠓⠛⠀⢀⣙⣒⣎⡁⠈⠟⠫⡴⢋⡞⢉⡿⠁⠓⠀⠀⣿⣄⠀⠀⠀⠙⢲⠇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⣈⡷⢀⡴⣾⠟⣽⣿⡿⠿⠀⣤⡀⠀⣟⣠⠞⠁⠀⠀⠀⢀⣿⠇⠀⠀⠀⣠⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢰⡿⢚⡡⢞⣥⢠⣞⣠⣴⢂⡉⢀⡿⠀⠉⠁⠳⣄⠀⠀⣠⠞⢯⣀⡀⠀⣰⠃⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣙⡶⠋⢐⣎⡼⠟⠛⠻⣯⠉⠁⢾⣁⣀⣀⠤⢤⣈⠙⠛⠁⠀⣠⠟⠉⣿⠃⡰⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠾⢫⡄⢱⢋⡞⠑⢒⡤⣄⡈⠑⠒⠲⢦⡀⠀⠀⢀⠜⠀⠀⠀⠀⠷⠒⢶⡇⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠷⣾⢸⡇⡸⢏⣸⣿⣥⡤⠼⠿⠀⠀⠀⠘⣷⢠⠞⠋⢀⡶⣶⠄⠀⠀⢀⣼⣀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡶⠫⣄⣿⣅⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⡓⣚⠽⠛⠁⠀⠀⠀⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡤⠈⢿⣉⣙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠶⠶⠶⣤⣤⣄⣀⣸⡃⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣭⢀⣀⣤⣤⣤⣤⣀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠉⠉⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⠶⠛⠉⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣸⣇⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣏⣹⣍⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠇⠀⢀⠔⠁⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠤⠂⠁⢀⠞⠀⠈⠉⢉⣥⠏⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⢀⠀⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢄⠂⢄⡀⠐⠒⠒⠒⢆⡁⠀⠀⠀⠀⠀⠈⠒⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠀⠀⢲⢀⠀⠀⠀⡀⠈⠐⠠⠀⠀⠀⣥⠑⠒⠒⠦⣤⢤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠈⠐⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢃⠀⠀⠈⣷⠀⠈⠀⣤⣤⣴⡾⠏⠀⠀⠀⠀⠀⠙⠻⠶⠖⠂⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢢⠀⠀⢉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠨⠴⢠⣕⢄⡀⠀⠀⠀⠀⠘⠳⣦⡄⠀⠀⠀⠀⠀⣴⣿⣁⠀⠀⠀⣄⠸⠀⠀⠊⡀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠙⠄⢈⠐⠀⠀⠀⠠⠖⠛⠁⠀⠀⣀⣀⡀⠀⠈⠉⠀⠀⠀⡉⠲⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⢹⠆⠈⠂⣄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠈⠒⢀⡀⠀⠀⠀⠀⠀⢀⠔⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠉⠣⡀⠀⠀⡔⠀⠋⢀⡎⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠒⠦⠀⠀⠀⠀⢀⠤⠖⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠐⠁⡀⠤⠄⢤⣄⠀⠀⠀⠈⢀⠀⣘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⣤⠤⠤⠀⠐⠈⠇⠀⠀⠀⠀⠀⡈⢐⡌⠀⠀⢀⣾⢿⡗⠀⠁⠀⣸⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠤⣀⣀⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠒⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣤⢄⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⣠⡞⠣⡈⢢
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣷⣄⠬⠌⠘⠘⠳⠀⠀⠀⠒⠢⡤⠄⢣⠴⣎⠀⢴⠀⠘⠺⡄
⠀⠀⠀⠀⣿⣿⠟⣋⡀⠆⠈⢉⠳⡀⠰⠀⡀⠠⠐⠒⠒⣒⡂⠀⠘⡴⢷⡇⠀⠢⢀⣶⠃
⠀⠀⠀⠀⢙⠋⣆⠬⠌⠊⢘⣸⡷⠈⡂⠉⢡⠀⠀⠀⠀⠰⠄⠀⡠⠘⠤⣨⡡⠔⡋⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⣿⣷⣵⡶⠥⢺⡟⠁⠀⠇⠀⠀⡀⠀⠠⠄⢀⠄⡪⠊⡠⠊⠀⠑⠒⠁⠀⠀⠢⡀⠈⢂
⠀⠀⠀⠀⢀⠅⠋⠁⠑⢠⠿⣷⠀⠀⠀⠀⢀⠇⠀⠀⠔⠁⠀⠀⢰⠁⠀⡰⠄⠀⠀⠀⠀⢀⡇⠀⡸
⠀⠀⠀⠀⡎⠀⠠⠔⠂⢫⠀⢸⠀⢐⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠒⠁⢺⠀⠀⢠⣎⣁⢈⠕⠁
⠀⠀⠀⠀⢃⠀⠀⠀⣀⠼⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡛⠲⢤⣀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡒⠶⠶⠦⠤⠤⣤⢽⣄⡀⠉⠳⠏⠰⢴⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠛⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⢶⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡤⠚⠉⠓⢦⡴⠚⠉⠁⠀⠀⠀⠀⠘⣷⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠤⢤⡀⠀⠀⠀⢀⠴⠒⠲⠄⠀⠀⣟⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⣏⠀⣞⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⢉⣕⣬⠿⠀⠀⠀⠀⢶⡋⠀⠈⠀⠀⠀⢰⠃⡼⣫⠍⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡰⠚⡇
⠀⠀⠀⢠⣠⠶⠋⠁⠀⠰⡀⠀⠀⠀⡟⠒⣄⢲⣤⣾⠤⠤⣄⠀⠀⠀⠀⠀⠸⠶⢡⣌⡠⠟⠐⠒⠒⠤⠤⠤⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀⡃⠛⠀⣰⠗⢚⡆
⠀⠀⠀⢽⠃⠀⠀⠀⠀⠉⢉⠒⠒⠲⢹⡆⠉⠀⠀⠠⠤⡴⠋⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠐⠢⠀⠀⣀⠀⠀⠉⠉⠉⢉⡍⡇⠀⠈⡇⠀⠞⢁⣠⠞
⠀⠀⠀⠈⠳⣄⡀⠰⠤⠈⠉⣀⠀⠀⡈⣇⠀⠀⠐⢒⡿⠤⠤⣤⣤⡤⣴⡞⠋⠀⠀⠈⠉⠛⠲⠤⢄⣀⡀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠰⢯⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣥⣀⣀⣳⣈⠲⣉⡿⢁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠛⠃⢸⠷⠤⠖⠓⠢⢄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡽⠉⠙⠉⠁⣿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣼⠁⠀⠀⠀⠀⠁⣆⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠒⠠⠊⡟⠀⢠⠃⠀⠀⠀⢰⠒⠀⠈⢱⣀⠇
⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠉⡇⡎⠀⠀⠀⠀⠈⠲⣀⡴⠋⡌
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠚⠉⠀⣧⣤⣤⣀⣀⣀⣀⣠⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠶⡤⠤⣤⡼⣼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠑⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣇⠀⣠⠴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠙⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⣤⢤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣃⢹⡎⢣⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⡤⠽⢿⣌⢇⠈⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⣾⠁⠀⠀⠀⠈⢾⡰⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⢤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣆⠉⠛⠒⠲⠒⠚⢛⣉⣡⠶⠋
⣀⣀⡀⢀⣀⢀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣈⣉⣉⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠎⠉⠉⠱⢤⡄⠶⠀⠀⠰⠦⣄⠀⢠⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠹⠖⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⡀⣀⡀⣤⠒⢤⠀⢀⣀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠐⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠀⠀⠀⠀⠀⠀⢠⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠄⠒⡤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢀⣀⠀⠀⠀⠀⠈⢶⡉⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠓⠀⠀⢸⡆⣰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢹⠀⠀⢀⠊⠀⠀⢓⡄⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠛⠃⠈⠀⠀⠀⠀⢃⠀⢸⠀⠀⠛⡀⠀⠰⠋⠙⠢⠤⠀⠀⠤⠤⠀⠒⠒⠛⠛⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠘⢆⠸⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠰⠿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⢄⠠
⠀⠠⠄⠚⠁⠀⠀⠀⠀⠀⠁⠷⠀⠀⠀⠀⠇⠀⠀⢀⡳⠆⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢡⠔⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢣⡀⠀⠀⠰⠀⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠰⠇⠀⠀⠀⢸⠀⠀⠀⠀⣧⣀⠤⢿
⠀⠀⠀⠀⢀⡴⠁⢀⠎⠁⠀⠀⠀⠙⢄⡴⠃⠀⠆⠀⠀⠀⠀⡴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠦⡀⠀⠀⠀⡞⠢⠤⠔⠃⠀⠀⠀⡸⠁⠀⢀⡌
⠀⠀⠀⠀⠸⣇⡠⠛⠀⠀⠀⠀⠀⠀⠸⠆⢤⣄⣸⣄⠀⠀⠀⢧⡀⠀⢀⣀⠀⠀⠀⠀⠑⢄⣹⡀⠀⠀⠀⠱⡄⢀⡜⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⡿⠶⠄⢀⡀⢹⠀⠀⠀⠀⠀⠉⠑⢦⡀⠀⢱⠄⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣠⠞⣠⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⢣⣀⡀⠀⠀⠀⠀⠀⠘⠶⠶⠀⠀⠀⣼⡀⢰⠀⠀⠀⣀⠰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠋⠘⠋⠙⢧⠀⠐⢲⠀⠀⠀⠀⠀⠀⠹⣶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠉⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠶⢤⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡄⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠙⢤⣤⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠤⠄⠒⠂⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠸⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠉⠀⠑⠒⠒⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠒⠠⠤⠤⠤⠤⡤⠋⠀⠀⠀⠀⠀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠠⠤⠤⠀⠀⠤⠼⡄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⡶⠠⠤⠄⠘⠃⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⣰⣆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠋⠀⠈⠛⠁⠀⠈⣦⠀⠀⠀⣀⠔⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠙⠙⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠦⠀⠀⠋⢹⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠙⠦⠀⠀⠘⠃⠐⠀⠒⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠳⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢟⡿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢔⣆⠀⢀⠐⢦⠄⣀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⠠⠈⢁⠀⢧⠀⠙⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠤⣤⠒⣉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠐⠒⠀⠑⢄⠀⠀⠀⢂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢢⡀⢀⡀⠀⠀⠀⢃⣼⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⠤⠤⠀⡀⣑⠪⢠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠐⡄⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠑⢄⡑⢄⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⣎⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠵⢄⠱⡤⠂⠀⠁⠉⠟⠂⠀⠤⢀⠀⠀⠀⢀⠀⡈⣢⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢀⠀⠁⠈⠀⠀⠁⠀⠜⠁⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢻⡾⣦⣾⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣶⣶⣶⣶⣶⣿⣷⣤⠀⠀⠀⠀⠀⠀⢎⠠⣄⠒⢶⠶⣶⣦⣄⡀⢀⠀⣠⣴⠆⠀⢳⣾⢧⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⢀⠀⠀⠀⠈⠀⠀⠸⠀⠑⠉⠢⣹⠈⠁⢸⣿⡁⣀⡤⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠏⠉⠻⠋⠉⠁⠀⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠒⠉⠀⠀⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠎⡥⠀⠀⠠⠀⣀⠛⣀⠀⠒⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣑⣀⡐⣠⣬⣄⣀⣀⣀⣀⣈⣵⣶⣦⣤⣤⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⢲⡴⠋⠉⠓⠋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢻⡀⠀⢘⡞⠁⠀⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠓⠀⢸⡇⠀⠀⠀⠀⠈⢳⡀⣠⢦⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘⣇⠀⣤⠀⠀⠀⢸⣷⣧⡆⣷⣄⠀⠀⢨⣧⡀
⠀⠀⠀⠤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡾⠀⠉⠀⠀⠀⠈⠛⠡⠴⠃⠈⢧⣠⠏⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣨⠴⠒⠛⠛⠛⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣨⠟⡶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣧⣀⠀⠀⢀⡴⣣⠞⠁⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠚⠷⠶⠤⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣏⠉⡿⠉⠉⠉⠀⠀⠀⠙⢦⣀⣀⣤⠰⡄⣶⢼⢧⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⣷⣿⡾⢮⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠋⠀⠈⠛⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠛⠛⠓⠒⠒⠲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠑⠦⣄⡰⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⡯⠉⠉⠉⠙⠒⠚⣒⣀⣈⡁⠉⠹⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⢀⡤⠒⢤⣀⡠⠔⠊⠉⠁⠀⠨⣧⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⣄⡀⠀⠀⢀⡤⠖⠦⠀⠀⣽⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠤⡀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⡯⢀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠁⢸⡴⠛⠁⠀⣀⠘⠟⠀⠀⠀⠀⠀⣇⠞⠥⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡔⢉⡇⣀
⠀⠀⢠⡶⠋⠀⠀⠠⠧⣄⣀⣸⡅⠱⠜⠛⠛⣋⡳⠀⠀⠀⠀⠀⢉⡴⠒⠋⠀⢈⡉⠉⠙⠒⠒⠶⠦⠤⠶⡖⠒⢲⠀⢀⠞⢉⡼
⠀⠀⠈⢧⡀⠀⣀⡀⠖⠀⠀⢀⢡⠀⠀⠠⢥⣞⡀⣀⣀⣀⣠⠴⠋⠑⠒⠢⢤⣀⡀⠀⠐⠃⠀⠀⠀⠀⠋⡇⠀⢸⡄⠀⣴⠋
⠀⠀⠀⠀⠈⠑⠲⠤⢤⣍⣀⠘⣆⠢⣒⡖⠉⠀⠈⠉⠉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⣄⡀⠐⠖⢠⡧⠤⠼⠢⢄⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢩⠏⠙⠋⠁⠇⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⡏⠀⠀⠀⢰⠸⢤⠀⡰⠁
⠀⠀⠀⠀⠀⣜⠉⠉⠾⡄⡸⠀⠀⠀⠸⣉⠀⣠⢇⠃
⠀⠀⠀⠀⠀⠪⠤⡠⠄⠃⣇⣀⠀⠀⠀⠈⢚⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣙⣍⣭⣭⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡆⠈⠑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⠀⣴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢇⠈⠍⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣦⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡎⣧⠱⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⠔⠚⠷⡝⡀⡇⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠁⠀⠀⠀⠈⠗⠁⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣚⠳⠤⣄⣀⣠⠤⠴⢚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠖⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⢀⡄⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣯⠉⠉⢠⡄⢠⡄⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣄⠀⠀⢠⡄⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣼⡷⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⢀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣴⠋⠀⠀⠀⠉⠋⠀⠀⠀⠀⠙⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⣆
⠀⠀⠀⠀⠀⢀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠲⠤⣤⠴⠞⠙⠶⠖⠦⢤⡤⠞⠁
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡼⠓⠀⠀⠀⠀⡾⣁⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⡀⠘⡇
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠸⡅⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⢳⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣳⣬⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⠲⠺⠯⣭⣛⠋⠉⠁⠀⣠⠏
⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠱⣜⠛⠓⠶⣄⡀⠈⠉⠛⠒⠚⡇⠀⠀⠀⢀⣀⣀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⠞⠁⠸⡿⢿⡦⣄⠀⠙⠲⣄⣀⣀⡼⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠹⣄⣀⣀⡀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣳⠀⠀⣇⠸⣷⠈⠙⢦⡀⠀⠀⣿
⠀⠀⠀⠀⠉⠉⠙⠲⠴⠶⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢹⡄⠀⠀⠀⠀⢹⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⣧⠀⠀⠀⢀⡾⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⡀⠀⠀⠀⠀⠀⢰⠛⠉⠀⠀⠀⠀⠀⠀⡴⠛⠒⠦⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⠲⠶⠶⠶⠿⢦⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⡼⠁⠀⠀⠀⠀⣸⠇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢳⡀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⡇⣰⠃⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣿⠃⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⠃⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠐⠫⠘⠓⢣⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠤⠄⠀⠀⠠⢄⠀⠀⠀⠀⢘⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⢀⣀⡀⠤⠀⠀⠤⡀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⠁⠀⠀⠤⠀⠀⠀⡠⠀⠀⠀⠀⡠⠃⠠⡨⠐⠉⠄⠐⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⡆⠀⠈⠀⠀⠀⡠⣂⠙⢦⡐⠊⠐⠀⠁
⠀⠀⠀⠀⠀⠀⢰⡄⠜⠀⠉⠀⠀⢆⠆⠈⠉⠑⠄⠢⠊⠱⡔⠤⠠⠙⠒⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⡂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢈⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠒⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢄⢇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠱⣁⣈⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡶⣶⣶⣶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⣦⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣴⠾⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠴⣾⣿⣿⣿⠏⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡩⢓⡥⠀⠀⠰⣶⣿⡿⢫⣾⣿⣿⣿⣿⣟⣉⣿⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠛⠛⠹⣿⣿⠿⣛⣿⣿⣿⡿⠟⢹⡷⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣹⡿⠂⠉⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⣿⡿⠟⠉⠀⠀⣾⣰⣡⣾⣿⣿⠶⠆⠀⠀⠀⠀⠀⣾⠿⣿⣿⠿⡜⠎⠉⢢⡀⢹⣼⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⢸⡇⣿⠈⠐⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⣠⠔⠂⠀⢠⣴⣶⠾⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣧⠀⢩⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⡼⢁⣴⡾⠟⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣸⠏⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠈⠳⣄⠳⡿⠿⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠀⡨⠛⣠⣴⠒⠋⠉⠉⠉⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡞⠁⠀⡱⠀⠀⠀⠀⠀⢀⣠⠤⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⢀⣴⠃⠀⠀⢀⡴⠞⠋⢀⡦⣾⣿⣿⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣧⡴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢰⠶⠀⠀⠀⠙⠦⠤⠖⠋⡀⠙⢠⠶⣋⢀⣀⠏⣡⠞⠛⣸⠇⠘⢷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⠀⠀⠀⢀⣠⢶⡿⢢⢞⣵⣾⡁⠀⠀⠀⠈⠛⠀⡼⠃⠀⣴⠏⠀⠀⠀⠀⠀⠀⡾⣼⡇⠀⠀⠀⠀⠀⠀⣼⠃⢀
⠀⠀⠀⠀⠀⠛⣠⡄⢀⣤⠛⢃⣤⠖⣋⠴⢋⡴⣣⢎⡾⠹⠃⠀⣐⣦⡀⠀⢸⡇⢠⡞⠁⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⡼⠃⢀⡞
⠀⠀⠀⠀⠀⢀⡈⠀⣾⣯⠖⠋⣠⠞⣡⠀⢀⡴⠃⢀⣠⠤⠒⠉⠀⢨⡷⠀⠈⠉⠁⠙⣦⡀⠀⠀⠀⠀⢀⡞⣿⠀⠀⠀⠀⠀⢀⡾⠁⢠⠞⣸
⠀⠀⠀⠸⢣⣬⣡⢖⡋⢀⡴⠟⢁⢺⠙⣦⡾⣲⣏⣃⠬⠴⠶⠂⢠⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠶⠋⠀⠈⠙⡶⠶⢤⣀⡞⠀⢠⠎⣰⠃
⠀⠀⣀⣴⠞⠋⠠⣀⡵⢋⠀⢀⣸⣆⡽⠋⠉⠉⠉⠹⣇⠀⠀⠀⠺⣧⣀⣠⡤⠤⠶⠖⠲⣤⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠟⠀⢠⠏⣰⠃
⠀⠀⠉⠀⢀⣠⠞⠋⡆⠀⢳⠟⢡⠞⠙⠒⠦⣤⣀⠀⠈⠙⠓⠒⠲⢦⣄⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢿⠤⠀⠤⣾⠀⠀⠈⢰⠇
⠀⠀⢀⣴⡟⢁⣤⠀⡇⠀⢹⣶⣋⣐⣒⡂⠉⠁⠈⣹⣦⡀⠀⠀⠀⠀⢹⣷⠀⠀⣠⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏
⣠⡶⠋⠛⠋⢛⡵⠀⣇⢠⠞⠑⠤⠴⠶⠶⠶⠚⠋⠉⠋⠀⠀⠀⠀⠀⠀⣿⡆⣼⠋⠀⠀⣀⡼⢋⡽⠃⠀⠀⠀⠀⠰⠾⣤⡤⣤⡘⠁
⠉⠀⠀⢀⡴⠋⣴⠄⣽⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡈⠓⠚⣉⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣿
⠀⠀⠀⠈⠀⠀⠀⠚⢁⠈⠙⣿⡶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠾⣧⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⢸⣷⠦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠉⠉⠛⠛⠿⣷⣶⣶⣶⣤⣷⣤⣿⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⠄⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠈⠉⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣶⠒⠾⣁⣤⡾⠟⠋⠁⠀⠀⠀⠈⠉⠙⠳⢦⡿⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣛⣋⣀⡀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣛⣻⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠂⠉⠁⠦⠐⠀⠐⠢⢄⡄⠠⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⡀⠀⢀⠤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠴⠀⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠑
⠀⠐⠀⠀⠀⠘⣀⡌⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢐
⢀⣀⡀⠀⠀⠀⠀⡅⠀⠂⠀⠠⢄⠈⠉⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠐⡀⠇⠀⠑⢴⠁⠀⠉⠀⠈⠁⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⣤⣒⠀⠀⠀⠀⠀⠈⠦⢄
⠀⠒⠈⠀⠀⠀⠈⠓⡀⠀⠘⠀⠀⠔⠂⠀⠤⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⢈⠀⠀⢂⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡠⠒⡰⠂⠀⠐⢄⡰⠀⠈⠀⠀⢠⠆⠀⠀⠀⠀⠀⠠⠂⠐⢄⡀⠀⢈⠄⠀⠜⠀⠀⠘⠁⢈⠃
⠀⠀⠀⠣⠔⠁⠀⠀⠀⠘⠒⠦⠴⡀⠀⠐⣀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠑⡠⠂⠀⠀⠀⠀⡐⣁⡠⠂
⠀⠀⠀⠀⠀⠀⠄⠲⠀⠀⠀⠀⠈⠁⠀⠀⠲⣀⠀⠀⠀⠈⢢⠤⠀⠀⣀⢁⠀⠀⣀⠠⠚⠈
⠀⠀⠀⠀⠀⠘⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠉⠂⠐⣢⠀⠀⠀⠀⠲⢆⠁⠀⠀⠀⢀⠀⠀⠀⢋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠑⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠁⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⠆⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⢢⣀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂⠀⠀⠒⢄⠀⠀⢀⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⡄⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠈⠀⠈⠆⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠄⡎⠀⠀⠀⠀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣧⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠲⢇⣀⣀⣠⡀⠉⠻⣦⣀⣨⠀⠀⠀⢴⣀⣉⠋⠐⠒⠤⣀⠀⠊⡀⢠⡾⠗⠈⠢⡄
⠀⠀⠀⠀⠀⠀⡠⠤⠤⢼⣄⣀⣤⡄⠀⠈⢻⣿⣿⣶⡾⠋⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⠁⠀⣾⡿⠋⢀⣦⣾⡷⠾⠖⠒⠢⡀
⠀⠀⠀⠀⢠⠞⢀⣴⣄⡀⠀⢈⣿⣿⣦⣄⣀⣿⢋⠹⠀⠀⠀⠀⠀⠀⠀⢠⠁⣹⡇⢡⠀⠇⠀⣸⣿⠁⠀⣿⣿⠏⠀⣠⣶⣾⣶⠡
⠀⠀⠀⠀⠁⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⡋⢡⠋⣶⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠛⣻⢲⣀⣼⠋⡇⠀⣸⣿⡏⢀⣴⣿⣿⣿⣿⡆⢃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣘⠸⠛⠉⠉⣶⠀⠀⠀⠀⡇⠀⠄⠀⢼⡂⠉⠀⠀⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠘⡄
⠀⠀⠀⢐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣎⣠⡏⠀⢀⣴⠟⠁⠀⠀⠀⠀⣇⡀⠀⢀⣀⡄⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢳
⠀⠀⠀⢰⠀⠀⢀⣡⣾⣿⣿⣿⣿⢫⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣧⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⡆
⠀⠀⠀⠸⠰⣾⣿⣿⣿⣿⢿⡿⢱⣿⣿⣿⣿⣿⠋⡜⠀⠀⠀⠀⠀⠀⠀⠻⠯⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻
⠀⠀⠀⠀⡄⢹⣿⣽⣿⣟⢸⡇⢻⣿⣿⡿⠿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠋⢙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢸⡄
⠀⠂⠀⠀⢵⣦⣼⣿⣿⣿⣌⣷⣈⠻⠉⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣬⣻⡿⣿⣿⣿⣿⣿⣶⣶⡄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣤⣤⡤⠶⠖⠛⢻⣄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⢠⣤⡤⠶⠆⠀⠀⠰⠶⢤⣤⡄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣠⠀⠀⣀⣀⣠⠄⠀⠀⢠⣄⣀⣀⠀⢠⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠠⢦⠶⠀⠀⠀⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠹⣦⡀⠀⢀⣀⣀⣀⡀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣧⠀⢸⡟⢦⣈⠉⠙⠻⣿⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸⠃⠀⠙⠳⠶⠶⠛⠙⣯⡉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠉⠳⣆⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠉⠉⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠶⠋⠉⠙⠛⠓⢶⣄
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡿⣻⣿⣷⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠟⠛⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣿⣦
⠀⠀⠀⠀⠀⢶⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣤⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⣀⣤⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠿⠟⠛⠉
⠀⠀⠀⢸⢇⡼⣿⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣹⣿⣧
⠀⠀⠀⠈⢸⡎⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣹⣽⠇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠊⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠩⠉⠉⠙⠻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⡀⢸⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣙⠛⠶⠶⠶⠶⠶⠛⣩⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸⡇⢸⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⡾⠉⣠⣼⣿⣧⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠀⣼⣿⣿⣿⣿⣿⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢿⠀⢻⣿⣿⣿⣿⣿⡏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢷⣀⠙⠛⠿⠛⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠉⠓⠲⠶⠖⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣻⣽⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣭⣭⣭⣿⣯⣭⣯⣽⣍⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣁⣀⣀⣀⣀⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣩⡤⢤⠔⠦⣤⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⡇⠀⠀⠀⣱⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠰⢥⣠⣋⣀⣀⣀⣙⡆⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠸⡯⠩⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠉⠓⠋
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠤⣴⢶⣤⣬⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡁⠴⢶⡄⣀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢡⣎⣿⣿⣿⣿⣟⣻⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⡟⠀⠟⠀⣽⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠔⢯⡿⡧⠟⠉⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠁⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢳⠈⠁⠀⠀⠀⠀⠀⠒⠒⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠙⠒⠊
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡆⠀⠀⠒⠀⠈⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠂⠤⠐⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠚⢀⠀⠁⠀⠈⡃⠒⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢀⠓⠄⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⢒⣄⠀⠀⠀⠀⠀⠐⠒⠢⠤⠔⠀⠀⠑⠰⠄⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢨⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡌⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⡟⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠰⠁⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠔⠀⠀⠀⠀⠀⢉⡀⠂⠠⠤⠤⠤⠀⠲⠂⠉⠀⠀⠀⠒⠒⠲⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠠⠄⠤⠄⠀⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⠤⠤⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣻⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣯⣭⣿⣿⣽⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠚⠀⠀⠀⠙⠿⠃⢠⣶⣆⠘⢿⣶⣎⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⣾⠂⠀⠀⢠⣴⡆⠀⠀⡰⠋⠉⢰⡧⠄⠀⠙⠻⢿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⢷⡀⠀⠀⠀⢉⣴⠶⡜⠀⠀⠦⠞⠛⢠⠀⠀⠀⢠⣿⡗⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠁⠀⣿⣄⠀⠀⢄⣀⣾⣧⡀⠀⠉⠀⠀⠐⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⠀⢘⣿⠿⢳⣴⠀⠀⠀⠰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠠⠀⠾⠄⣼⣿⣿⠿⡿⢿⠁⠀⠀⠁⠈⠙⠛⢀⠞⠀⢀⡿⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣶⣼⣆⣀⡀⠐⡋⠀⢀⣿⡉⠛⠂⠀⢸⠁⠀⠀⠀⢀⣤⡴⠃⠀⣠⣿⠁⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠿⣿⠅⠀⠈⠿⢋⣽⡿⠶⠒⠈⠉⠻⠉⢠⣉⡽⠷⡀⢀⣴⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣥⣤⡤⠤⠶⠦⠼⠿⣷⣿⣯⡅⣲⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣶⣼⣿⣷⣾⣷⣶⣆⢀⣀⣀⣼⣿⣏⢀⣹⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢿⣿⣿⣯⠻⠛⠁⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣷⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⢀⡀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠦⠶⠈⢙⣫⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⠄⢀⡤⠖⠃⣀⡤⠤⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠃⠀⠀⣠⢿⡆⠀⠀⠀⣛⣠⠄⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠐⠚⠛⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⣸⠇⠠⠄⠈⠁⠀⣀⠀⠀⠀⣀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢰⣡⡴⢛⣁⠒⠂⣤⠀⠉⠉⢩⢤⣤⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣄⠀⠥⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠈⢷⡄⢤⠀⠀⠈⢙⡓⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣸⡇⠀⢀⠐⠦⠄⠉⠛⠀⠀⠀⠀⠈⠓⠲⠶⠄⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠀⠀⠈⠙⠦⡀⠐⠲⢤⡀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠃⠀⠀⠀⠈⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠈⠑⠀⠀⠉⠛⠂⠀⠀⠀⠘⠲⢤⣀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣤⣄⣀⠀⠀⠄⠠⢚⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠻⠆⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣂⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠒⠀⠀⠔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⠁⠀⠀⠀⠀⠀⠀⠈⠰⠒⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠤⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠆⠀⠀⠀⣠⣾⠇⢰⣴⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⡀⠀⠸⠀⠀⠀⠚⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡌⠆⢀⠁⠀⠀⠀⢀⣠⠀⢰⠴
⠀⠀⠀⣀⡤⠤⠤⠤⢀⡀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠝⠋⠀⠀⠢⡀⠀⠀⠀⠀⠀⡀⠤⠐⣠⡙⠆
⡞⠙⠛⠁⠀⠀⠀⠀⠀⠈⢩⠒⠂⠀⠀⠀⠀⠀⠀⠸⣇⡴⢷⡴⠃⠀⢠⢐⣠⠥⠔⠂⠈⠭⠐⠁⠀⠀⠈
⠘⠦⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢼⣶⠾⠏⠃⠁⠈⠁
⠺⡄⠀⠀⠀⠀⠀⠀⢐⣺⣿⡇⠀⢀⣀⣀⠀⠀⠀⠋⠁⠰⠆⢠⠊
⠼⠁⠀⠀⠀⠀⠀⠀⠶⠟⠏⠀⠀⠀⠀⠈⠑⢦⡀⠀⢀⣀⡤⠾⡀
⢠⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢆⢀⡤⠼⠖⠒⠶⠤⠤⣀⡀
⢸⠆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢢⡀⠀⠀⠀⠀⠳⣴⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⡴⠚⠂⠀⢀⡀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀
⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⢤⣄⣀
⠠⡄⠙⠀⠀⠀⠀⠀⠀⠴⢺⣇⠀⠀⢀⡼⠃⠈⠓⠢⠤⢤⣤⡤⠶⠛⠉⠒⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣄⡀
⠀⢳⠈⠆⠀⠀⠀⠀⢀⣴⣿⣿⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦
⠀⠀⠒⠒⠒⠂⠈⠉⠀⠀⡀⠝⠋⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣶⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣀⣀⣐⣒⣒⣒⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣒⣚⣉⣁⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⣁⣀⣀⣀⣠⣶⠶⠶⣦⣄⠀⠀⠈⠉⠛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣀⣴⠟⠛⠉⠉⠛⠋⠀⠀⠀⠀⠙⠟⠛⠛⠻⢶⡄⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣶⠟⠋⠉⠀⠀⣠⣤⣤⣄⠀⠀⢠⡶⠟⠛⠻⣦⣀⡈⠿⠶⣦⣄⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡁⠀⠀⣠⡶⠾⠟⠁⠀⠁⢀⣠⣤⣤⣄⠀⠀⠈⠉⢿⡄⠀⠀⠙⣷⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⣾⠛⠁⠀⠀⣿⡀⠀⣠⣴⣦⣴⡟⠉⠀⠀⠙⠿⠀⠀⠀⣸⡇⠀⠀⠀⠻⢷⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠸⣧⠀⠀⠀⣾⠏⠁⣾⠋⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⣼⠟⠋⠀⠀⣤⠀⠀⢘⣷⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠙⣿⡆⠀⢿⣄⣠⡿⠂⠀⢠⣄⡀⢀⣴⡟⠛⠛⠛⠋⢀⣀⣀⣴⠟⠀⢀⣼⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣧⣄⣀⣽⡏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⢉⣿⠁⠀⠀⢀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠘⠋⠀⣽⠍⠙⣷⣀⡀⠀⣀⠀⠀⠀⠀⢠⣶⣤⣀⣀⣤⡾⠛⣶⡶⠶⠟⠁⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⢠⡶⠀⠠⡿⠀⠀⠛⠙⢻⡿⠛⢷⣶⣶⣶⣿⠇⠈⠉⢩⣿⠀⢰⡟⠀⢰⡖⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠈⠁⠀⣤⠀⢀⣿⠀⠀⣀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠸⠃⠀⣈⠁⠀⠿⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⠋⠀⣈⠁⠀⢰⡟⠀⢠⣿⣿⣿⣿⣶⣶⠂⢰⡆⠀⢸⡏⠀⢠⡀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠈⠁⠀⠘⠛⠛⣿⣿⡿⠁⠀⠿⠃⢀⣌⠀⠀⠛⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠀⠀⣰⡆⠀⠘⠃⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣀⣀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⠴⠶⠚⠻⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡾⠃⢀⣤⣤⠶⠆⠀⠀⠰⠶⢤⣤⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢡⡀⠀⣀⣀⣠⡄⠀⠀⠠⣄⣀⣀⠀⠀⣌⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⢰⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⢦⡴⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢧⡀⠀⠀⣀⣀⣀⡀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣷⠀⢸⡟⢦⣄⠉⠉⠻⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢸⡇⠀⠉⠳⠦⠴⠞⠙⢷⡉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⠛⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠁⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠶⠟⠉⠙⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⣤⠤⠤⠤⢤⠈⠱⡄⠀⠀⠀⠀⠀⠀⢠⠞⡵⠒⡲⠊⠲⢢⡉⢦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢿⣁⣀⣀⡴⠃⠀⢱⠀⠀⠀⠀⠀⠀⡏⠈⢓⠚⢦⠤⠤⠤⠟⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠈⠁⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡖⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⣠⠤⠊⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢋⣐⣢⣞⣷⣶⣬⠳⡀⠀⠀⠀⠀⠀⠀⢀⠞⣁⣀⠚⣳⢀⣀⠙⢆
⠀⠀⠀⠀⠀⠀⢰⠁⣷⢿⡿⡿⡟⠿⠷⠀⢡⠀⠀⠀⠀⠀⠀⡎⠀⠀⢏⠈⡃⢰⠋⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠈⠓⠛⠛⠑⠁⢀⣀⡀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠶⡏⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡔⠒⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠒⠢⠤⠃
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⠛⠉⠉⠉⠉⠙⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⣀⣤⡴⠶⠛⠛⠛⠛⠷⠶⢦⣤⣀⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⣠⣾⣋⠀⠀⣠⠟⠋⠛⣷⣦⡶⠶⣦⡉⠛⢶⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠾⢻⡟⠉⠉⠛⢻⡿⢦⣴⡟⠉⠈⢻⣦⣸⣇⣀⠀⠙⣧⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣼⠃⠀⢸⡇⠀⠀⠀⣾⠁⣸⠏⠙⣷⣾⠋⢩⣿⡍⠉⠛⢦⣸⡇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠀⠀⣨⣷⡟⠛⠛⣿⢶⣿⣤⣼⣏⡙⣷⡟⠀⢻⡀⠀⢀⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠸⣧⢠⡟⠈⠻⢦⣴⣇⣼⠃⢨⡏⣉⡿⢿⡷⠶⢾⡷⠶⠛⠁⢻⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠈⢻⣧⣀⡀⠀⣸⣏⣿⣻⣿⡿⠿⠶⢾⣧⣤⠀⢻⣄⠀⢀⣾⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⣷⡉⠙⢻⡟⠋⣿⠁⢹⣇⠀⢀⣼⠃⢸⡇⠀⠉⠛⠋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠈⠛⠾⠋⠀⠀⠹⣦⡀⠙⠛⠛⢁⣠⡾⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣶⣶⠶⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠙⢦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠉⠉⠙⠢⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠠⠀⠀⠀⠀⠀⠈⡆⢀⠀⠤⠄⡀⣠⢂⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⢻⡀⠀⢸⣦⣶⠄⠀⢀⠀⠀⠀⣤⣰⣶⣀⣴⠴⠂⢐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢁⠀⠀⠳⢀⠤⢒⠏⠀⢀⡔⠁⣐⠉⠛⠓⠒⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉⠂⠀⠀⠀⠁⠀⠀⠀⡈⠓⠒⡊⢡⡿⢄
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣜⠒⠎⠱⠶⣲⠇⠀⠀⠀⠀⠠⢊⠔⡝⢀⠈⢧⠀⠀⠀⠆⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⡠⢃⠎⠠⠀⠀⠈⢷⠂
⠀⠀⠀⠀⠀⢰⠉⠛⠙⡆⠁⡠⠀⠀⢀⡤⣬⣈⣳⣁⣌⣀⠾⢶⣄⠀⠒⠄⠸⡄
⠀⠀⠀⠀⠀⠘⠦⠤⠒⠁⠊⢳⣤⠀⡜⡇⡇⡍⠛⠉⠁⠀⠂⡏⡟⡄⠀⠀⠀⡇⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡠⠃⢠⡿⡇⢣⠀⠀⠀⠀⠀⠀⡇⡿⡇⠀⠀⢠⢳⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠎⠀⠀⠈⠁⣇⢸⠀⠀⠀⠀⠀⢀⢇⡇⠀⠀⠀⡇⢸⡀⠀⠀⠀⠂
⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⢒⠀⠀⢹⢸⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠈⡆⠀⢣⡀⠐⠠
⠁⠈⠀⠀⠉⠁⠉⠓⠂⠀⠈⠁⠀⠀⠄⠘⣾⡀⠀⠀⠀⢀⢼⢸⠀⠀⠀⢀⣱⠐⠀⠱⡄⠀⠀⠀⡀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠣⣷⣞⣀⣠⣄⡬⠞⠀⠀⠙⠛⠉⠀⠀⣴⠟⠀⠀⠙⠒⠋
⠀⠀⢀⠤⠀⠀⠀⠈⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠠⡄⠤⠄⠀⠀⠀⠤
⣶⣶⣦⣶⣦⣶⣶⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⣶⣤⣄⠀⠀⠀⢹⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣿⣾⣿⣿⣿⣿⣿⣆⠀⠀⢳⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡇⠀⠀⠀⠀⢠⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⣵⣭⣛⡇⣩⣭⣭⣭⡹⡇⠀⡞⠁⠀⠀⠀⠀⢸⡇⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡅⠘⠇⠀⠤⠤⠤⠤⣼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣭⣝⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢺⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⠀⠀⢀⣴⣿⠇⣟⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⢸⣿⡇⣿⣿⡧⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠘⢿⣷⣬⣭⣵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠸⠇⠹⠇⠿⠸⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆⠄⠒⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣜⣏⡀⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣰⠴⡟⢛⡳⠈⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⣈⡡⠀⠀⠀⠀⠀⠀⠁⠰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠀⠀⠀⠀⠀⠔⠊⠴⠠⡀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠲⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⢠⠈⠃⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡔⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠺⡌⣿⣦⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⠉⠛⢍⣹⠆⠀⠀⠀⠀⠂⠀⠠⡼⠄⠀⠀⠀⠀⣸⡀⢲⠁⣿⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⢈⠉⠈⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡘⠐⠠⣠⡄⠀⡇⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⠀⣴⠠⠤⠏⡶⠀⡗⠀⠲⠆⢡⠡⠀⠀⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠓⢆⠜⠀⡆⠡⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⡀⠠⣄⠀⠀⢀⣀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⣼⠃⠐⠀⡉⠀⠉⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢣⣿⠀⠀⠀⠀⣀⣠⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⢠⠁⣾⣟⠀⠀⠀⣨⠉⡁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣦⢤⠉⣫⠇⠀⠀⢺⣝⣁⡝⠂⠀⢲⣿⣾⣷⡿⠿⢿⠿⣟⠁⢑⣆
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠼⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢙⣿⠿⠯⠶⠶⠬⠿⠹⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⠀⢀⣀⢀⣀
⠀⠀⠀⡖⡀⠀⠀⢒⣲⢀⢲⠀⡆⢸⠠⣒⡆⡃⠀⢰⢒⠀⣒⡆⠒⠒⠂⠀⠀⢮⡄⢴⡇⢈⢸⠀⡁⡇⢀⠎⢈⠜

⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢰⣲⣀⠠⠜⠣⢄⣀⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣀⢤⠐⣓⠠⢄⡰⠆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢰⣡⢘⡡⡯⢘⣣⣵⡀
⠀⣏⠶⡁⠀⠀⠀⠀⠀⠀⢈⠶⡱⠀⠐⢷⠔⢺⣏⠉⠁⡢⣾⠂⠀⠰⠒⢇⣬⣨⡑⣖⣬⠱⠆⡗⠺
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⠃⣴⡤⠀⠀⠈⠋⠈⡄⠀⠀⠀⠘⣤⠾⢠⡝⣤⣛⡜
⠀⠀⠸⠽⠉⠒⢢⡔⠒⠉⠯⠇⠀⠘⢻⣤⡂⠁⠐⠽⠀⢢⣄⡟⠛⠀⠀⠛⠀⠉⠚⣖⠊⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠓⢋⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⣦⠆⠀⢈⣝⠀⠀⠀⠀⠀⣠⢤⠀⠉⢸⡽⠂
⠀⠀⠀⠀⢦⠀⢀⣸⣀⠀⢀⠆⠀⠀⠈⠋⠈⠑⢻⡖⠊⠁⠙⠁⠀⠀⠀⠀⠈⡽⠉⢝⣱⠈⠑⠤⡮⠧
⠀⠀⡀⠀⡠⠋⡧⢰⠆⣭⢣⡀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣔⣢⠜⠀⠀⠈⠈⠀⢀⠀⠇
⠀⠀⠉⢹⠙⠃⠶⣙⡁⠶⣛⢻⠉⠁⠀⠀⢀⡀⠀⣀⡠⠻⢄⣀⠀⢀⠈⠀⢇⠀⢀⣄⠀⠀⠺⣋⡯⢞⡇
⠀⠀⡤⠼⡟⢑⡗⠨⠜⢣⡍⡸⠤⡄⠀⠀⠘⡏⠉⠀⠀⠀⠀⠀⠙⢹⠁⢔⡾⠤⣑⠛⠀⠀⣀⡜⠁
⠀⠀⠀⠀⢈⠮⢅⣗⣘⠧⢎⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠁⠀⢀⠵⡏⠉⠀⠯⠃
⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢀⡃⠀⠀⠘⠟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠻⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡔⡆⢀⣀⠧⡏⠀⠀⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠄⢤⠀⡄
⠀⠀⠀⠀⠀⠀⡨⠋⠱⢭⡆⠈⠑⢺⠬⠂⠰⠕⠢⢄⣀⠀⣀⡠⠔⢾⠆⠀⣄⡹⢚⡉⢭⠍⣓⢎⣠⡀
⠀⠀⠀⠠⣋⢖⠁⠀⠀⠀⠈⠀⡀⠈⡄⢀⠀⠀⠀⠀⠈⠾⠀⠀⠀⠀⠀⠖⡞⣤⢈⡁⠶⢀⣉⠴⢧⠴
⠀⠀⠀⠀⠀⠈⡄⢀⣴⡄⠀⠈⠺⢽⠘⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⡤⣨⡁⢟⢠⡍⠶⢸⠒
⠀⠀⠀⠀⠰⣊⠗⠪⣒⠁⢀⣀⣠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⣤⡾⢠⡝⢡⡘⣣⢎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢱⠚⡥⢺⡋⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠿⣿⣦⣀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⡿⠿⠛⠉⠁⠀⠀⠈⠻⢿⣷⣦⣄⣀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠸⢿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠁⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡧⠀⢀⣀⡀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⣶⣾⣿⣿⠿⠟⠋⠀⠀⠈⢿⣷⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⢠⣤⡄⠀⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠈⣿⣷⣀⣤⡀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⠿⠃⠈⣿⣿⢠⣴⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠉⠙⠿⢿⣿⣶⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣧⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣶⣯⡄⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⢛⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣯⣤⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⡿⠿⠛⠋⠉⠛⠛⠛⠋⠉⠉⠉⢿⣿⡉⠛⠛⠛⠿⠿⣿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢀⣴⣿⡿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⣿⣿⡿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠀⢹⣿⣆⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣶⣶⣶⣶⣿⣿⠿⠟⠋⠀⠀⠀⠀⠙⢿⣿⣶⣶⣶⣿⣿⣿⡿⠁
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢡⣿⡿⠁
⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁
⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋
⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟
⠀⢹⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⡏⠛⠿⣿⣷⣦⣤⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠸⠿⠇⠀⠀⠀⠉⠙⠻⠿⣿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣔⣒⣀⠒⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣽⠟⢉⣩⣤⣤⣤⣤⣤⣬⣿⣒⠂⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠯⠛⠁⠢⢴⡿⠟⠙⣓⡚⢓⠛⡛⠳⠮⢤⣀⣄⡐⠻⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡫⠂⣰⡆⢴⣆⣤⣤⣈⠛⠿⢛⠉⠁⢀⣅⢀⡀⠀⠈⠙⠻⣦⣌⠐⠕⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠏⠠⣺⠟⣡⠀⠸⠟⠉⣤⢶⣳⣬⡛⢿⠿⣶⣦⣬⣉⠒⢦⡀⠈⠻⣷⣄⠲⡡⢀
⠀⠀⠀⠀⠀⣰⣳⠋⠁⢀⡼⢁⠞⡱⠃⡨⢤⣼⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⢾⡀⡈⢿⣷⣔⡹⣷⡀
⠀⠀⠀⠀⢰⢳⡏⢀⢠⡿⢠⢃⠠⠀⠐⣁⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⢩⣍⡈⠛⠻⠎⢻⢳⡀
⠀⠀⠀⠀⣿⢿⠇⠆⡟⠁⠋⢁⡄⡸⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆⢻⡟⣦⣶⠀⠈⣏⣧
⠀⠀⠀⠀⣿⡞⢠⠁⠀⠁⠠⣿⢣⠃⣿⡇⣀⡤⡄⠀⠀⠀⠀⡠⣄⣀⣀⣃⡀⠄⠀⠀⢸⡘⡈⠗⢇⡀⣴⢻⢸⣿⡀
⠀⠀⠀⣶⣿⡇⡀⢸⣠⡆⣄⣈⠘⢢⠟⠋⠁⢀⠇⠀⠀⠀⡜⠀⢿⣿⣶⠏⡘⠀⠀⠀⠈⠓⠐⣿⢹⠆⣿⠸⢸⣿⡇
⠀⠀⠀⣿⣿⡇⣷⠈⢿⡇⡇⢻⠞⠁⠀⠀⢀⠎⠀⠀⠀⠀⣇⠀⠀⠀⣠⠜⡤⠊⠙⠀⡘⢠⠰⣿⠈⡄⡿⢀⢹⣿⡇
⠀⠀⠀⢻⢿⠀⢸⣇⠸⣟⢰⣸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⣜⣠⣄⡸⡰⢃⠃⣦⠁⠘⠄⠁⡈⣸⣻⠃
⠀⠀⠀⠸⣾⡇⠈⢏⡆⢿⣖⠗⣷⡀⠀⠀⠈⠷⢄⠀⠀⠶⠮⣥⣦⠀⠀⠈⠉⣩⡞⠁⢃⣼⣿⠃⠈⠀⢠⢡⠇⡿
⠀⠀⠀⠀⢿⣿⣆⠘⣺⡜⢿⢆⠻⡷⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⡤⠤⠤⢶⠻⠗⢀⡀⠾⠙⠁⠞⠀⠐⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢻⣎⢆⠱⡹⣎⠻⣧⣿⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⡲⢤⣴⣋⣤⠴⣪⡵⡣⢦⠄⠀⠀⠠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠛⢯⣧⡙⢿⢧⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣤⣤⠴⠾⣋⠤⠖⢁⠔⡹⢋⡴⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠪⡻⣟⠎⠳⣝⡻⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠷⠾⠋⣀⠤⣚⡵⢊⠕⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⣤⡀⠙⢳⠀⠀⠀⠀⢸⠍⢉⣳⣄⣀⣴⣛⡤⢄⣠⣶⠟⢉⣪⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡻⣶⣧⠀⠀⠀⠈⣗⣒⡲⠞⠒⠒⢒⣉⣉⣥⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡀⠀⠀⢸⣽⡿⠿⠟⠛⠛⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣄⡀⠀⢀⣥⣤⡄

⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡄⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠒⠓⠲⢸⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⣀⡠⠠⠔⠂⠄⠒⠶⠦⣀⣀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⢸⠈⠀⠀⠀⠈⢶⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢦⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢱⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⢠⠎⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⠧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⢸⠀⠀⠀⠀⠀⠀⢰⠀⡜⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⢇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⢲⠀⡀⠀⠀⠀⠀⢰⠀⡇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢺
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠡⠀⠀⠀⡰⠁⠀⢳⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠏
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠂⠁⠀⡠⠖⠛⣄⣠⠞⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠏
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⢠⡇⠀⠀⢘⠁⠀⠀⠀⠀⠀⠃⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡹⠦⡤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⢄⣀⣀⣀⡤⣄⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢠⠀⡇⠃⠘⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠇⠀⡇⠀⡀⠀⡇⠀⠀⠀⢸⢄⠀⡤⠀⠀⣀⡀⠁⣀⣀⠀⠘⣆⣀⣼⡃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠀⡇⠘⠁⠀⣷⠀⠀⠀⡇⠀⢙⠁⠀⠀⠡⠀⠀⢠⠁⠀⠀⠛⠁⠀⡁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠐⠳⠦⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⢱⠀⠈⠀⠀⠀⠰⠀⠁⠀⠄⠀⠀⡄⠀⢰⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠄⢀⠌⠀⠀⠀⠀⢇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠌
⢸⠀⢀⣀⠀⣠⣤⡄⢀⠠⠂⡇⠀⠀⠀⢰⣄⠀⢀⡀⠀⠤⢤⡀⠒⠛⡀⠀⠀⠀⠀⠀⢡⣀⡡⠄⠀⣀⣀⠀⠀⠴⣁⣰⠃
⢸⠀⢜⡄⠀⠀⠑⠏⠁⠀⠀⡇⠀⠀⠀⠀⠀⠉⠈⠒⠒⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹
⢸⠀⠀⠙⢦⡀⠀⠈⢂⡤⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠙⠲⠶⠊⢴⡶⠚⠂⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠈⠡⠤⠀⠈⠉⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⢉⣀⡉⠀⠛⠈⢀⣈⡉⠙⡃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠊⠀⠀⠉⠢⣠⠆⠁⠀⠉⠺⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠤⠤⠤⠤⠠⡯⠠⠤⠄⠠⠤⢠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠁⠀⠀⠉⠉⠉⠉
⢸
⢸
⢸⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢸⠀⠀⠀⠉⣿⡏⠁⣠⣶⢶⣶⡆⣿⣇⣰⡶⠀⢠⣴⣶⣦⡀⠀⠀⣼⣿⣤⣶⡆⠀⣶⣦⣦⣴⣦⢤⣦⣴⣦⡀⣴⣶⣦
⢸⠀⠀⠀⠀⣿⡇⠀⣿⣇⠀⣹⡇⣿⣿⣿⡀⠀⣿⡿⠶⠿⠇⠀⠀⢸⣿⠀⢿⣇⢀⣿⢹⣿⠁⠀⢸⡟⠀⣿⡇⠙⠿⣦⡀
⢸⠀⠀⠀⠀⠻⠇⠀⠙⠛⠟⠻⠇⠻⠇⠘⠿⠂⠘⠻⠿⠟⠁⠀⠀⠘⠿⠀⠈⠻⠿⠋⠘⠿⠀⠀⠸⠇⠀⠻⠇⠛⠿⠟⠑⠿
⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⠷⠶⠀⠀⠀⠀⠙⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠛⠻⠟⠋⡡⠤⠀⠉⠀⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢄⠀⠈⣧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠⠀⣼⣿⣶⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠸⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣄⠀⡠⠊⠀⠀⠀⠐⠀⠀⠀⢷
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣇⠀⠀⠀⣀⣀⡀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣷⣄⠈⠉⠉⠁⣀⣴⣿⡿⢿⣿⣿⠓⣇⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠉⠋⢉⡀⠄⠁⠀⠀⠹⣆⢀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠆⠀⠀⠀⠀⠁⠉⡽⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⢉⢡⠐⠀⠀⠀⠀⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠈⠙⠛⠁⠀⠀⠀⠁⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢠⠤⠤⢠⡀⢀⡄⣤⠤⠄⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢸⠓⠒⠈⢧⡼⠀⣿⠒⠂⣾⠲⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠈⠁⠀⠉⠉⠁⠈⠀⠈

⠀⠀⢠⣄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀
⠀⠀⠀⠈⠓⢤⣀⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⡤⠞⠉⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠐⠒⠤⣀⠀⠀⠀⠀⣀⠤⠔⠂⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢈⠶⠀⠀⠀⠀⠀⠀⠶⡁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄⠀⢸⠃⢀⡤⢐⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⣛⣿⣿⣿⣷⣄⣰⣯⣾⣿⣛⣉⣙⡆⠀⠒⢤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡩⠒⠂⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⣦⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣿⣿⣿⡿⠟⣏⢈⠀⠙⢿⣿⣿⣿⣿⠻⠿⣿⣿⣶⣄⠙⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠊⢩⣿⢛⣽⣿⡿⢻⣠⠞⡏⢸⠀⠀⠘⣿⣿⠹⣿⡄⠘⠾⡙⢿⣿⣷⣤⣄⣀
⠀⢇⠈⠉⠐⠒⠈⠁⠀⠀⠀⠀⠀⡟⢁⣾⣿⡿⢁⢸⠫⠔⠁⣀⣤⠀⠁⢸⠏⠄⡿⢘⣀⠘⢹⠀⢻⣿⣿⣆⠈⠙⠒⠤⣀⡀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⡇⠀⠃⠀⣴⡿⢟⣋⣁⡀⠀⠀⠀⠀⠹⣿⣷⣤⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠄
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢠⡞⠍⠎⠂⠍⠀⠀⠀⠀⠀⠸⡹⠏⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⡤⠀⣿⣿⣿⠇⠀⠀⠀⣾⠁⠀⠈⣈⣠⠄⠀⠠⣦⠀⠀⡇⠀⠀⠀⣿⣿⣿⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠠⣃⠎⡠⢶⡿⠿⣟⠀⠀⠀⠀⠸⣆⠀⠁⠀⠁⠀⣀⠀⠀⢀⡴⠁⠀⠀⣸⣿⣿⡟⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⡾⠀⢀⣃⠀⠀⠀⠀⠀⠀⠐⠬⣭⠶⠋⠀⠀⠈⣳⠾⠯⠔⠁⠀⠀⢻⡙⡛⡙⢳⠀⠷⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣇
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⡗⢹⢸⡇⠒⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⢀⣼⠃⠀⢉⡼⠁⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣴⡿⣄⠀⠀⠀⠀⣀⣴⡶⠖⠲⠴⠚⠁⠀⠀⣀⣤⠶⣿⣥⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⠀⢀⣀⠤⢜⣫⠭⢑⣰⠋⢀⣀⡀⠀⢀⣤⣶⣾⣿⣿⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠃⠀⠀⠀⣼⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⢀⡠⢴⣮⡥⠖⢊⡡⡴⠾⠛⠛⠛⠻⡟⠀⠀⡾⠁⠀⠉⠻⣿⠀⠀⠀⠈⠑⠲⢄⡀⠀⠀⠀⠀⢠⢣⠇⠀⠀⠀⢀⡿
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠆⠱⠶⠞⣡⠴⠊⠉⠀⣳⠟⠈⣹⣷⣄⠃⢀⣎⣴⣿⠉⠓⠚⢃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣞⡞⠀⠀⠀⠀⣼⠇
⠀⠸⣷⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⢠⠾⣫⠊⣴⣿⡟⢙⣧⣾⠁⢿⣿⡆⠀⣄⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣸⡏
⠀⠀⣿⡄⠀⠀⠀⢀⣀⣀⡠⠒⠒⢄⠀⡔⠁⡼⢁⣼⣿⡿⠁⢸⣿⣿⠀⠸⣿⣿⡀⠘⣆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣰⡟
⠀⠀⠸⣧⠀⠀⢰⠁⠀⣼⠁⠀⢀⠀⠀⢌⣸⠁⣼⣿⣿⠇⠀⣾⣿⣿⠀⠀⢻⣿⣷⡀⢘⣇⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⡟
⠀⠀⢀⣿⡦⠶⢾⠀⠀⢸⡄⠀⢸⠀⠀⢸⠉⠉⠉⠉⠙⠒⠦⢿⣿⡿⠶⠚⠋⠉⠉⠉⠉⠉⠙⠓⠚⠷⢤⣀⠀⠀⠀⠀⠃⠀⢀⣾⠏
⠠⠖⠉⠀⠀⠀⠀⠳⣄⡠⠷⢄⡼⠓⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⠁
⣿⠿⠿⠿⣿⢿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣻⣟⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠩⠥⠬⡭⢭⡅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣥⣬⡭⣥⠄
⠀⠀⠀⠀⠀⠀⠐⢒⠂⡐⠒⠐⠒⠂⡁⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠒⠂⣒⠂⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠉⠉⠁⠇⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠄⡀⠀⠠⠉⠀⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡆⠀⠀⠀⣘⣿⣿⣷⡆⠐⠙⠃⠉⢸⠀⢀⡤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠹⢹⠆⠀⠀⡇⠀⠀⠘⣿⠀⠋⢹⡿⠈⠀⠀⠀⢸⠀⣈⠃⠃⣻
⠀⠀⠀⢀⣀⠀⠀⠐⠾⣉⠴⠃⠀⠀⠃⠀⣀⣰⣿⣦⣤⣿⣆⠀⠀⠀⠀⠘⠈⠟⢢⡵⢿
⠀⠀⠀⠈⠈⠗⠒⠛⠦⠡⠀⠈⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⣠⣴⡆⠀⡀⠀⠀⣄⣀
⠀⠀⠀⢸⠆⠀⢀⣀⠀⡀⠀⠈⠿⣿⣿⡿⠋⠙⠛⢟⠿⠟⠋⢻⣿⡿⢺⣲⡇⠀⠀⠀⠀⠿⡿⠇
⠀⠀⠀⣤⠶⠂⠈⠻⠆⠀⠀⣘⣯⣭⡿⠇⠀⠀⠀⠊⠀⠀⠀⠀⢻⣿⣶⣼⡧⣄⡀⢂⣆⣙⠭
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣿⣿⣿⣯⣽⣭⣷⣓⡿⡿⢶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡿⢿⡟⣿⣻⣏⣿⢻⡏⣿⢻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⢀⣀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣇⡀
⠀⠀⠀⠀⠀⡀⡀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢻⠗
⠀⠀⠀⠀⢈⠏⠡⣿⡛⠿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠉⠘
⠀⠀⠀⠀⠈⠀⠚⠂⡀⠔⠊⠁⠀⠀⠀⠀⢀⢀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣉⠉⢚⠯⡀⠁⠁
⠀⠀⠀⠀⠀⠰⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠦⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⡤⠶⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠏⢀⣀⣀⣀⡼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠘⠋⠩⠔⣚⣿⣿⣿⣿⣿⠟⠋⢻⣿⣿⣿⣿⣿⣧⠀⠐⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠡⢖⣻⠿⠛⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⣰⠁⠀⠀⠴⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠤⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢨⣿⡆⠀⠀⢠⠤⠀⠀⡆⠀⠉⠁⠀⡀⡠⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣾⣿⣧⠀⢀⢀⠀⠀⠈⠃⠀⠀⠀⠈⠐⠁⣹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⢿⣿⣿⡗⠁⠂⠀⠀⢔⣉⠭⠵⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⠻⣿⢿⣿⢷⡀⠀⠀⢸⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⢶⠲⠔⠃⠀⠀⠀⠙⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠙⢶⣶⣶⡶⠖⠋⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸

⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⣄⠀⠀⠀⠀⠀⢀⣤⢶⣶⣿⢶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠓⢶⣤⡀
⠀⠀⠀⣰⣿⡏⣻⣿⣿⡮⣿⣿⣧⠀⠀⠀⢰⣿⣿⣟⡿⣿⣿⣯⣿⣷⠀⠀⠀⢠⣾⡟⠋⢌⢤⣷⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠛⡃⠀⠀⠀⣿⣿⣿⣿⣧⣿⠿⠿⠟⠻⠀⠀⠀⣼⣿⡶⣀⣷⣿⠿⠿⠟⢻
⠀⠀⠀⢿⣿⡟⠛⠿⠁⠀⢠⠄⢡⠀⠀⠀⢻⣿⣿⠛⠻⠃⠀⢠⠄⠰⡀⠀⠀⢸⣿⣿⠛⠻⠏⠀⠀⡄⠘⡄
⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⡏⠀⠀⡤⠤⠠⠜⠀⠀⠀⠀⠀⠀⡉⠁⠀⢠⠤⠤⠴⠁⠀⠀⠀⠀⠀⢹⠁⠀⢠⠤⠄⠤⠃
⠀⠀⠀⣀⣴⣇⣀⣀⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣇⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⢀⣠⣮⣀⣀⣸⣄⡀
⠀⠀⠚⠿⠿⠿⣿⠿⠿⠿⠛⠀⠀⠀⠀⠘⠿⠿⠿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠐⠻⠿⠿⢿⡿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⠉⠉⠉⠉⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⡄⠀⠀⣀⣀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⡀⡄⠀⢀⠉⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⣠⠃⠀⠙⠃⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡜⠧⠀⠀⠠⡂⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⠣⡀⠀⢀⡴⠛⠒⠀⡴⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠉⠉⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣄⢀⡠⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⢄⠀⠀⠀⠀⣀⡼⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⡹⠁⡀⠀⣽⣏⠉⠀⡰⠀⢸⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡇⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⠠⡇⠀⡅⡈⣽⠩⡀⠐⠀⠰⠚⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢇⠀⠀⠀⠀⢠⠞⡇⠀⠀⠀⠀⠀⡠⠇⢀⡟⢰⣿⠄⡷⢀⠀⠰⠀⠀⠀⠀⠀⢠⠖⠒⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⢠⠏⠀⡇⠀⠀⠀⠀⠀⡅⠀⢸⠀⣾⣿⡖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡀⠀⢀⡟⠘⡀⠃⠀⠀⠀⠀⠀⢡⠀⢸⢀⣿⡟⠀⠀⠤⣤⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⡞⠀⠀⢃⢠⠀⠀⠀⠀⠀⠸⠀⢸⢸⡟⢀⡴⢠⠞⠁⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡦⢇⡬⣄⠀⠹⠢⣶⡄⠀⠀⠀⠀⠀⡄⢸⣘⡴⠋⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿
⠀⠀⢸⠯⠤⠤⠭⡇⠀⠀⠤⠒⠒⠃⠚⠀⢬⠗⠀⠁⢨⣇⠀⠀⠀⠀⠀⢁⢸⠏⢀⡼⠐⣷⢠⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡹⠈⠛⠒⠒⠶⢊⠀⠀⢠⠏⢸⡄⠀⠀⠀⠀⠘⠃⠀⡼⠄⠀⠀⠀⣠⢧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⢱⠀⠈⠉⢉⣀⡏⠀⣠⠏⠀⠀⣇⠀⠀⠀⠀⠀⡄⠀⡷⠂⠀⠀⠠⠃⠈⠓⢖⢲⠒⠒⠬⠼⣯⠁⠲⡀⠀⣿
⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠘⢦⡐⠒⠒⣒⠃⣴⢃⣀⡀⠀⢻⠀⠀⠀⠀⠀⠃⠀⠸⡒⠀⠀⠒⠀⠀⠀⢸⠈⡇⠀⠀⠀⠈⢷⠀⢈⣦⣸
⠀⠀⠈⠀⠀⠀⠈⡆⠀⢴⡉⠉⠉⠉⢭⠉⠁⠀⠈⠉⠉⠉⠭⠯⠉⣙⠒⠒⠢⠤⣄⠉⠛⠒⠒⠤⠤⠤⠋⡰⠁⠀⠀⠀⠀⠈⠃⠀⠈⢹
⠀⠀⠀⣥⣦⡖⣸⡇⠀⠀⠉⠓⠲⠦⣤⠀⡈⢀⣀⣀⡀⠀⢀⣁⣁⣀⣀⡠⠤⠴⠒⠉⠉⠉⠛⠒⠒⠲⠦⠤⠤⠤⣄⣀⣀⣀⣀⡀⠀⣸⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠉⠉⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠰⢿⣇⢸⡇⣸⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢐⣷⣤⣽⣿⣿⣧⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⡀⠻⣿⣿⣿⣿⡿⠃⣀⠀⠀⢡⣤⡞⢻⡟⢷⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣰⣿⣿⣦⣌⣉⣉⣉⣤⣾⣿⣿⣦⠀⠙⠁⠾⠷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⣿⣿⣿⠻⣿⣿⣿⣿⣿⡋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⠀⣿⡿⠿⣿⣿⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⣶⣿⣿⢠⣄⡉⠛⠿⠿⠟⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠿⠿⠿⠿⠿⢸⣿⣿⣷⣦⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⣶⠆⣠⡀⠀⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠻⣦⣿⣤⡿⠛⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⢸⠿⠟⣿⣿⣿⡿⠻⡯⠀⠀⣼⣿⣿⣿⠿⣿⣿⣿⡆⠀⠐⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢶⣾⠋⣿⠉⣿⡶⠀⠀⢀⣿⣿⣿⣿⠀⢿⣿⣿⣷⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠃⠈⠀⠀⠀⢸⣿⣿⣿⡇⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣷

⣿⠟⠋
⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢢⣴⣾⣿⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠁⠀⢀⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡜⡡⠂⢀⠀⠀⣿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠘⠃⢠⠆⢀⠰⠈⠇⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢧⡄⠀⢀⠘⠀⠀⠀⢄⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣧⠀⡄⠀⠐⠀⠣⡀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠀⠀⣠⣾⣤⢴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣾⠶⣿⠿⠉⢿⡄⠙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠘⢦⡀⠀⢁⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠤⣦⣤⡴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣶⡟
⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣁⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⢀⠀⠀⠀⠀⣀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡇⠀⠉⣷
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⣀⡇⡇⠀⠀⠀⠀⠀⢠⠀⠀⢻⡃⠀⢸⡄⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣿⠀⠀⢻
⠀⠀⠀⢹⣿⣶⠶⣀⢰⡀⢀⣤⣾⡇⡇⣴⣶⣄⣤⡴⣤⠀⠀⠀⠀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⣻⡀⠻⠛⢀⠀⢸
⠀⠀⠀⠀⢀⣭⣆⢀⣀⡃⢿⣟⣸⡇⣿⣘⠋⣴⣍⠶⠋⢠⡥⠴⠀⠀⢀⠀⠐⠀⢀⣤⣦⡄⠀⠀⢤⣤⢄⡀⠠⣯⠠⡆⡔⠹⣇⠨
⠀⠀⠀⠀⢿⡇⢐⠸⣸⡙⣾⠀⠹⠇⡿⢩⣶⣬⡤⠀⠀⡻⠃⢈⣁⡴⠃⠀⠀⠀⠘⠁⠈⠟⠘⠂⠀⠛⣿⠇⠀⠘⣼⣿⡖⠀⠀⡀
⠀⠀⠀⠀⢠⣄⠸⣇⡛⠈⠈⠀⠀⠀⢀⣛⠿⠏⠐⠀⠀⠁⠀⡼⠋⢐⠀⠸⠀⠀⢀⡀⠀⡄⢠⡠⢊⠀⠀⠐⣶⣤⣎⣿⣁⡴⢀⣿
⠀⠀⠀⠰⣿⣿⣷⣄⡹⠄⠀⠀⣠⣶⣿⣿⣿⣦⣀⠀⠐⠀⠀⠀⠐⠉⢓⠀⠠⢠⡙⠛⠀⠀⣠⣿⣇⠀⢀⡀⠘⢿⣿⡿⠟⣠⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣧⡠⢣⡤⣿⣿⣿⣿⣿⣿⣿⣦⣶⡄⡀⠀⠈⠢⠈⠀⠀⠃⠥⠀⠀⠀⣩⣤⡿⠀⣸⣿⠀⣸⣯⣾⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡖⠿⠟⠂⠀⠒⢠⣶⣇⢹⣿⣿⡆⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢻⣧⣿⣿⣿⣠⣧⣴⣟⣿⣶⣿⣿⣿⣽⣯⣠⣤⣟⣿⣏⢻⢸⡏
⠀⠀⠀⠀⠀⡿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠠⣿⢸⣿⣿⠻⢿⣾⣿⣿⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣕⣬⢀⡏
⠀⠀⠀⠀⠐⠦⢧⡿⣽⣿⣿⣿⣿⣿⣟⣻⢸⣿⣿⣿⣿⡛⠁⠀⠘⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⢁⡏⡿⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣦⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣷⣦⢙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠑⢳⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢿⣽⣿⣿⣿⣯⣿⣿⣿⣷⠻⣿⣾⣿⣤⢭⣤⣀
⠀⠀⠀⠀⠀⠒⠿⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠛⠘⠛⠿⠿⠟⠛⠛⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⢷⠞⠚⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠇⠀⠂⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠⣦⠀⢠⣆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣘⠋⠀⣘⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⣀⣀⠀⢀⡀⠀⢀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠁⠀⠈⠉⠁⠉⠉⠀⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠀⠠⠀⠤⠤⠤⠤⠶⣅⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠢⣀⠀⠀⠤⠀⠀⢰⠊⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣊⢢⠠⡈⠠⠈⢂⣀⡠⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢀⠅⢠⠈⠋⠐⠠⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⣨⠉⠉⠀⠀⠈⠉⢳⢠⠄⢤⡀⠈⢦⡀⠀⠀⠀⠀⢀⠗⠊
⠀⡀⢀⡀⠀⣦⠀⠂⣠⣚⡥⠤⠐⠒⢻⣉⢳⡀⠀⠀⠀⠀⠀⢸⡀⣀⠗⠢⢤⣙⣆⠀⡀⠀⣸⠶⠒⡆⣀
⠀⡯⢩⠁⣄⣉⣁⡈⡄⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠸⠑⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⢈⡀⣀⠃
⠀⣏⣹⠀⣟⣁⣀⠊⠇⠀⠀⠀⠀⠀⢀⠸⣀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠉⠒⠒⢀⣠⣤⠴⠖⠀⡉⢁⠀⠀⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣦⣬⣥⣥⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠾⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣘⡇⠀⠀⠀⠀⠀⣀⠀⣀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⡤⠄⠳⣾⣿⣆⠀⠀⠀⠀⢰⣷⠄⠀⣾⣿⣿⠀⠀⠀⠀⢀⣐⣲⠠⡦⠘⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⢀⣿⣿⣿⡀⠀⠀⣰⡿⠧⠖⠲⢿⣿⣿⡇⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠘⢻⣿⠿⠃⠀⠀⢻⡇⠀⠀⠀⢸⡏⣿⠁⠀⠚⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠘⠟⠀⠀⠀⠀⠸⠇⠀⠀⠀⠟⠀⠿⠀⠀⠴⠿⠃⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠈⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠘⠁⠹⠃⠉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⠺⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠉⢸⡏⠀⠀⠀⠀⠀⠀⣹⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣿⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠠⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡟⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠔⣂⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡤⣄⠰⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠋⠺⡿⢿⠟⡆⠀⠀⠀⠀⢠⣤⣼⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠋⠉⠀⠃⠀⠀⠀⠀⠁⣿⣇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⢀⣠⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⢹⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢿⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⢶⣚⣿⣿⣿⣿⣿⣿⣶⣤⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣭⣿⣿⣿⣿⠿⣿⣿⣿⣯⣭⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣾⣿⣿⣷⣄⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣽⣿⣿⣯⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⢀⡠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠌⠂⠀⡠⠖⠈⠉⠁⠒⢤⠀⠀⠀⠙⣿⠃⠎⡁⠀⢠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠁⠤⠀⠀⠈⢱⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠃⢰⠁⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠈⢨⠎⡀⠀⢠⠱⣀⡀
⠀⢠⠒⢢⠀⠠⠃⠀⠀⠀⡠⡀⠀⡀⠀⠃⠀⠃⢀⠀⣤⢲⠀⠸⠎⠀⣠⢲⠀⠀⠀⠠⠀⠀⡞⡇⠀⠀⠀⡀⢈⠈⣐⣧⠀⠀⠀⢰⠉⢇⣁⡀⠈⢀⡬⠓⡆
⠀⠀⢀⢈⠀⠘⠀⠃⠀⠀⠀⠀⠀⠙⢆⠰⢀⠞⣡⠀⠉⠙⠂⠐⠀⢠⠁⠀⠀⠀⠀⠀⠃⠐⢱⠀⠀⠀⢀⠁⢉⠀⡰⠀⢄⠀⠀⢸⠀⢎⣹⠋⠉⠉⢑⡄⡇
⠀⢀⠀⠘⣆⠸⣣⠀⠘⠇⠀⠀⠟⠀⢸⣻⡁⠰⠁⠀⡩⠀⠁⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⣀⠴⠓⢆⡶⠜⠓⠚⢩⡗⢲⡄⠁⠔⠘⢋⡖⢿⠿⢿⡶⢛⣛⢟⠀⠂
⠀⠘⢄⡸⠁⢀⠜⠣⣀⠀⠠⠀⢀⡴⠛⢼⡸⠂⠤⢤⠁⠆⠀⡔⠌⠰⠀⡈⠂⠀⠀⢠⠊⠁⠀⠀⢸⠀⠠⣄⣠⡞⠖⣻⠟⠀⠀⠀⢞⣀⣈⣒⣊⣐⣛⣛⣋⣣
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢗⢄⡀⠘⣄⣒⠢⠒⡄⠐⣶⠃⠀⠀⠀⠘⡌⡄⠀⠀⠸⢆⠀⠀⢀⣸⡾⣷⣿⡲⣷⡖⠒⡆
⠀⠀⠀⠀⡆⠀⠀⣀⣨⣗⠀⠀⣽⣀⡀⠀⠈⠦⠉⢲⠀⠙⠛⠛⠉⠻⡁⠀⠀⠀⠀⠀⢱⠸⡀⢠⠃⠀⠉⢉⣽⢟⣿⣟⠭⣉⠩⣣⠊
⠀⠀⢀⡀⢡⡀⠀⢎⠀⢡⠀⢸⠀⢹⠀⠀⣰⠀⠚⢉⡄⠀⣀⣬⠆⠀⢱⠀⠀⠀⠀⠀⠀⢣⢡⢸⠀⢀⠐⡟⠀⠻⠿⠟⠂⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠒⠛⠛⠛⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⠑⠚⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⡦⠤⢽⡃⠒⠒⠚⠑⠛⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⡁⠉⢩⠿⡀⠀⢀⡠⠏⠣⠇⠀⠀⠀⠀⠀⠹⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠃⠀⠈⠀⠀⠘⠀⠈⠋⠁⠃⠘
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⡎⠈⠃⠀⠀⠔⠉⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠤⠀⢀⣀⡀⠠⠃
⠀⠀⠀⠇⠀⢀⠀⣘⡰⠀⢖⠀⢠⡄⡀⢰⢸⡅⠀⡦⠂⢱⡠⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡐⠒⠚⠒⢶⡒⣆
⠀⠀⠀⡇⠀⠘⢉⣤⠁⠠⠆⠀⣠⠀⣄⣜⡠⠔⠒⠑⠢⡀⠀⠀⠰⠉⠉⠑⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠴⢀⠄⠂⡀⠀⠈⠠
⠀⠀⠀⠑⠐⢱⢁⠤⠓⡴⡔⠢⣀⣠⠞⢺⠶⠛⡀⢠⠀⠈⡆⠘⠧⡀⠀⣠⠁⠀⠀⠀⠀⠀⡉⡖⠒⠊⠉⠁⠒⠒⠠⠤⣀⡀⠀⠀⢀⢪⣞⡎⠘⠀⠰⠀⠀⢈⠢
⠀⢠⠄⠀⠀⣶⠁⢀⣤⠓⣆⠼⡁⠙⡶⡇⢀⡐⠁⠸⡀⠀⡆⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠐⠓⠓⡤⠤⠒⠁⡒⠲⡄⠀⢋⡀⠉⠐⡏⢸⠉⠛⢤⣀⣀⠀⠒⢺
⠀⢸⠀⠀⠘⢴⠀⠁⢈⠖⢹⠆⠓⡔⢣⣘⠾⢟⣀⠤⣽⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢠⢎⠤⡂⠀⢁⠈⣂⡀⠀⢇⠀⠉⠀⠀⢀⣿⠀⠀⢸
⠀⠸⠀⠀⠀⠸⠄⠤⠊⠀⢠⠀⠀⣰⡎⠀⢠⣀⠂⠀⠀⠙⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⡑⢄⣔⣿⠈⣉⠤⠬⠅⢹⡼⣒⠊⠉⠀⠈⠉⠁⠐⠀⠀⢸
⠀⠜⠁⠀⣄⠆⣀⠤⠠⣤⠤⢍⠋⠀⡌⢁⠭⡍⠀⡠⠆⢀⣸⠠⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠡⡌⠙⣄⠢⠒⠒⣂⠌⠀⢋⠇⠀⠀⠀⠀⠀⠰⠀⠀⢸
⠀⠀⠀⠀⡟⣾⠁⠀⢠⠁⠀⠈⡖⡤⠇⣧⠺⣄⣌⠀⠀⠀⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠃⢛⡓⢰⠇⠈⠉⠁⠀⠓⠀⢰⢆⠒⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⢀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠀⠪⣇⣼⠀⢨⠀⠰⠀⣆⣀⡈⠃⠠⣀⣀⡀⢦⠤⣀⣀
⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠠⣺⣁⡈⢣⠼⣷⡧⢴⡀⣈⡹⠇⠀⠀⠉⠈⠙⠃
⠀⠢⠀⠀⠧⡼⠤⢀⣇⢀⡰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢀⠉⠀⠉⠉⠈⡁⠀⠀⠒⣄⡀⠦⣀
⠀⠀⠀⠀⠪⠄⠤⣎⣀⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡎⠀⠀⠀⠸⡆⠀⠀⠀⠀⠘⣴⢦⠀⠀⠈⠳⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⣉⡙⠢⡄
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣦⡀⠀⠖⢋⢳⠳⣀⡀⠀⠀⠀⢀⠀⠶⠀⠀⡞⣤⣴⠒⠿⠧⠱⣄⠸⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠿⠙⠻⣿⣖⠛⡌⠈⠀⠁⡃⠀⠀⠀⠙⠁⠶⠒⠀⢻⠉⢢⢠⠰⠎⠀⠈⠚⢧
⠀⠀⠀⠀⢸⣿⣿⠉⢡⡀⣐⣧⣠⡃⡜⢣⠀⠑⠀⢠⠠⠶⠒⠒⠒⢒⣚⡑⠀⠸⡴⢿⠯⠀⠀⢢⢀⡖⠋
⠀⠀⠀⠀⠀⡿⠛⠤⡼⡫⢑⡠⢄⢙⠃⠀⠹⠂⠉⠉⡄⠀⠀⠀⠉⠡⡄⠀⠀⣀⠑⢄⣉⣓⣀⠤⡞⠋⠋⠒⠢⢄
⠀⠀⠀⠀⠀⠘⠒⠢⣇⠨⠄⢒⣡⡎⣀⡀⠀⡇⠀⠀⢇⠀⠀⢀⡠⠀⢀⡠⢚⠕⠀⡠⠊⠙⢤⡠⠃⠀⠀⢀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠻⣍⠀⣴⣷⠀⠀⠉⠁⠀⠀⢸⠀⠀⠈⢉⠴⠋⠐⠁⢠⠎⠀⠀⣄⠀⠀⠀⠀⠀⠀⠙⡄⠀⠘⡆
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⡸⠉⠛⡇⠀⣀⣀⣀⡠⠊⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⣰⣙⡆⠀⠀⠀⢀⡠⠴⢥⣀⡼
⠀⠀⠀⠀⢰⠁⠀⠰⠒⠉⠀⢱⠀⢠⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣏⠀⡀⣷⠀⠀⠀⣯⣤⣠⡠⠊
⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠛⠁
⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⣾⡄⠀⠤⢄⣀⣠⠀⢀⠀⣀⡤⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⡀⠰⠆⠬⠀⠀⠈⠩⡁⡞⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠧⠀⠀⠀⠐⠁
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠊⠀⠀⠀⠀⠃⠀⠀⠂⠈⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⡄⠀⡀⠀⠄⠀⠀⠇⠀⠀⠀⠀⠀⢠⡲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠂⠀⠀⠀⠀⣀⡤⠞⡃⢸⡶⠺⠦⠴⠷⠶⠛⠺⢷⡆⠀⢳⣄⡀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⣿⠘⠗⡦⢲⠔⢦⢤⡤⠰⠋⠀⢠⣟⠈⠻⣆⣠⢖⡉⢤⣿⣮⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠸⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣼⠀⣰⣏⣼⡞⣿⢾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢀⡀⠀⠀⡇⢹⣧⡀⠀⠀⠀⠀⠀⢀⢴⡟⢰⡇⣸⡻⡿⠎⢇⣸⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⣧⠀⠀⡇⠀⣷⠈⣴⣦⠒⢒⡿⣧⡾⠁⣾⢠⡿⠗⣽⣢⠞⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢻⠀⠀⣷⠀⢸⡟⠀⠙⣾⣿⡄⡼⠃⢀⡟⣼⣹⡏⣿⠏⡆⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⠀⣷⠀⠀⣿⣯⣿⣧⣠⡾⢿⣿⡏⠟⠙⢺⠇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣧⣹⡀⠀⡟⢿⣫⣶⡏⡔⣴⣿⡷⢀⢠⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠒⠲⢦⣄⢠⠀⡌⠀⠀⠀⠀⡟⢹⡇⢠⡇⢰⣿⠛⣿⠛⠋⠁⠀⣾⢸⣥⣗⡉⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡞⠹⣄⠛⣆⠀⠀⢸⡇⢸⡇⢸⡇⢸⡇⢀⡟⠀⠀⢠⠶⡇⣸⠉⢷⡙⣦⠀⠀⢸⠇
⠀⠀⠀⢀⢀⣀⡀⣸⡄⠀⠀⢸⠁⠠⢸⡇⣿⠀⠀⠀⣷⢸⡇⢸⡇⠈⣧⣼⣁⣀⡀⣿⣴⠋⠁⠀⠈⣧⢹⠀⠀⣼
⠀⠀⠀⢠⣘⡛⣛⣛⣻⣶⣤⣼⣧⡐⠂⠙⣿⢶⣶⠞⣉⣍⡙⠛⠛⠛⠋⠭⠡⣄⠙⢿⣿⠀⠀⠀⢀⣿⣋⣀⣼⡟⠋⠛⠛⠃⠛⠃
⠀⠀⠀⠈⠋⠁⣀⠀⠠⠍⣳⠞⠙⢷⣦⣶⠟⣲⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⢿⠳⠶⠶⠿⠿⠿⠿⣿⡐⣀
⠀⠀⠀⠀⠀⣀⣬⠵⢂⡾⣥⠄⠀⠀⠀⠀⢰⣧⣐⣇⠀⠠⢤⣄⣀⡀⠀⠠⢀⣚⣡⣴⡞⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⠈⣵⣋⣀⣁⣀⣀⣀⠀⠉⠉⠙⠛⠛⠛⠓⠒⠶⠶⣤⣀⣈⣉⣉⣀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠐⠊⡷⠆⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣋⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣶⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⢠⡶⠾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⣾⢁⣼⢿⣿⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠃⢻⡇⢿⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣉⡴⠟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣺⠡⠒⠒⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⡀⠈⣌⠔⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⡀⡤⢤⣾⣽⡠⠄⠐⠂⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⣠⡌⠛⢀⠎⠀⠈⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡄⠀⠀⠀⠀⢸⡀⠸⡀⠀⠀⢩⡍⠀⠀⡼⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣄⠠⡀⢀⢴⣮⣳⡄⠣⣄⡠⠎⠈⠑⠋⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡟⠉⠋⠠⣬⠉⠀⠀⠀⠀⠙⠢⡀⢠⡤⠔⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⢴⠀⠀⠀⠀⠀⠀⣱⣀⡨⠖⠂⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠘⠁⠈⢀⡤⠒⢡⡄⠠⡄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣀⠀⠈⠉⠒⡂⠀⠀⢀⠔⠉⠀⠀⠤⠀⠰⡕⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⠁⠀⠀⠠⢠⢵⠀⠀⠎⠀⠀⠀⠀⠀⠚⠀⠀⠮⠆⠀⢠⢄⣀⣀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣤⣄⣬⣎⠀⠀⠀⠀⣔⠙⢤⡇⠰⣓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢲⢰⣿⢳⠊⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣍⠉⠙⠯⡉⢑⡿⠀⠀⠀⠈⣦⡸⣧⣠⡽⢦⡽⣦⠀⠀⠀⠀⠀⢀⢇⠀⠴⡋⢸⣿⢠⠓⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣼⠑⠢⣀⡜⠉⢢⠀⠀⠀⠘⠁⠀⢻⠀⠀⠀⠳⡀⡓⢄⣀⣰⠚⡤⠿⠓⠐⡆⢸⣿⣀⣟⠉⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣠⣇⡀⠸⣀⡇⢰⡔⣤⠀⡄⠀⠀⠀⠇⣀⠀⠀⢱⠰⣌⡇⢡⠀⠓⡚⠉⣦⠜⢩⣋⠁⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡞⠀⠈⠒⢬⣠⣦⣭⠀⠛⣤⠷⠀⠀⠇⡹⢅⠇⡀⠆⠡⠈⠈⠓⠒⠁⠀⠀⡤⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠶⠄⠚⠀⣥⠂⠐⠠⠗⢰⠇⠀⠚⠀⠈⠪⠕⠘⡄⢣⠀⠀⠀⠀⠀⠀⢁⢾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⡂⠸⢰⡂⡰⠀⢱⠀⢃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡇⠀⣀⠉⠀⠀⠀⡆⠀⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡖⠢⡀⠀⢣⠖⢻⢦⡄⠤⠚⢣⡾⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⣿⠁⠀⠈⠢⡴⠀⢸⠀⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠾⣼⢿⠀⠀⠀⠀⠀⠀⡖⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡌⣀⣼⡮⠀⠀⠀⠀⠉⡿⠛⠏⠉⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⢢⡴⡆⠀⠀⠀⢰⡄⣼⠻⣷⡀⠀⠀⣀⣼⡁⣠⢀⣄⡀⣠⣧⡤⣄⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠷⠒⠆⢀⣲⡋⠀⠀⠀⠳⢷⣥⠴⠊⠉⠉⠉⢹⡿⣷⣿⢿⣻⠛⠋⠛⠚⡻⢀⣁⣠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡻⠿⠯⠩⠭⠭⠭⠿⠁⠿⠚⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⣰⣿⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⠃⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣿
⣤⣄⠀⠀⠠⣤⡄⠀⠀⢠⣤⡀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣤⣤⣾⡿
⢘⣿⡆⠀⠀⢹⣿⠀⠀⠈⣿⣷⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⠁
⣾⡟⠁⠀⢠⣿⡏⠀⠀⢰⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣀⣀⡀
⢻⣿⡄⠀⠀⢿⣷⠀⠀⠘⣿⣆⠀⣿⡇⠀⠀⠀⠀⠀⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠷⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄
⣼⣿⠃⠀⢀⣾⡿⠀⠀⢠⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣽⣿⡿⠗⠀⠀⠀⠐⠿⣿⣿⣭⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⠀⠀⠐⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⢿⣷⣶⣶⣶⣶⣾⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⢀⠀⠈⠢⠄⠀⣀⣀⡀⠜
⠀⠀⠀⠀⠀⡀⠂⠩⠓⠈⠥⢀⠇⠀⠀⢻⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠂⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠃
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣴⣦⣠⣦⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠙⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠋⠛⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠈⠁⠂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠘⠻⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠈⠀⢀⠁⠀⢠⣆⣀⣀⡀⢀⣀⡀⠀⡁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢂⠀⠀⢀⠀⠆⠀⠀⠀⠙⠋⠉⠛⠛⢩⠃⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠴⣄⠴⠶⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠜⠉⠉⠙⠒⠒⠒⠒⠚⠉⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣠⡤⢶⡋⠀⠙⠲⠤⠤⠷⢦⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢉⡟⠀⠀⠀⢻⡀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋⠀⠀⠀⠀⠈⢷⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⠀⢀⣠⡼⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠎⠁⠀⠈⠉⠉⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠈⠉⠉⠁⠈⠱⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⠴⠶⠶⠶⠶⠞⠛⠛⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠶⢤⣄⡀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠈⣏⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠂⠉⠙⡛⠛⡉⣡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠒⠒⠚⠓⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠃⠀⠀⠀⠀⢀⣊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⡿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣿⣿⣿⣦
⠀⠀⠀⠀⢀⡀⠀⣄⢸⣿⣿⡇⠀⠀⠀⠀⠠⠀⢸⣇⢀⠀⠀⣀⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢠⡀⢸⡇
⠀⠀⠀⠀⠀⢆⠀⠶⠀⠛⣹⠁⠀⠀⠀⠀⠀⡀⠈⠓⠴⠄⠀⠶⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠂⠘⢋⡟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠶⠀⠰⠄⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠁
⠀⠀⣤⡀⠀⢀⠚⡣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢛⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⢱⠄
⠀⠀⠠⠛⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠗⠤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡄
⠀⠀⠀⢠⣿⢷⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠈⡏⢀⡄⠀⢤⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⢘⡅⠀⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣉⠀⢘⡃⣸⠁⠀⠀⠀⠀⠀⠀⠀⢿⠐⣏⠀⢐⠂⢸⠁
⠀⠀⠀⠀⠓⠘⠂⠄⠘⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠂⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⣀⠀⠀⠖
⠀⠀⠀⠀⡀⠁⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠤⡤⡀
⠀⠀⠀⠀⠀⠇⢀⣀⣀⡸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠦⢐⠀⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠂⠀⠂
⠀⠀⠀⠂⠀⠲⠤⠽⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠞⠛⠛⠛⢻⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠠⠆⠀⠶⠜⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣉⠉⢉⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⣿⠐⡂⠀⢐⠂⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠐⡒⠀⢘⠂⢸⠃
⠀⠀⠀⠀⠀⠣⠀⠛⠀⢋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠫⠈⠧⠀⠰⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠋⠄⠘⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⠛⠂⠘⠅⠯
⠀⠀⠀⠂⠀⢐⠒⣀⠠⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⣀⠘
⠀⠀⠀⠀⠁⠀⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠔⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠊⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⠱⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠆⡀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠾⠖⠚⠓⠒⠯⣀⠀⠀⡠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡏⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠄⠒⠒⠒⣠⠄⡴⢋⠜⢓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠊⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⣥⣄⣶⡀⠀⠀⢀⠀⠀⠀⠀⢸⣅⣠⣶⡄⢀⣴⢥⡮⢒⠥⡪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢯⡛⠈⠁⢀⡤⡟⠀⠀⠀⢀⡞⠁⡉⠙⠿⣿⣭⣁⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⠀⠀⠀⠷⠘⣆⢀⠀⠀⠙⠒⢊⣡⠞⠀⢀⣀⠴⣏⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⣀⠀⠀⠀⠈⠀⢣⠀⠘⠀⡄⠉⠉⠉⢉⠀⣠⡟⠳⡄
⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⢸⠒⢲⡖⠓⣦⣤⠒⣾⠀⠀⠀⠀⠀⠁⠀⣠⠞⢡⢞⡜⠀⠀⠹⣆⠀⠤⠤⠀⢴⣉⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠩⠉⠁⠈⠛⢏⠠⡀⠀⠀⠀⠀⠈⢁⡔⢁⠞⠀⠒⠉⠀⢘⣂⣀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠐⢂⠈⠉⠉⠀⠀⠒⠢⠀⡀⠄⢄⢀⠀⠀⡠⠋⣠⠏⠤⠚⠉⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠚⠈⢳⡄⠀⠀⢤⡐⠈⠉⢀⣤⢦⣄⡉⠙⣇⢡⣈⣀⣀⡷⢶⢦⡄⠉⠙⠲⠄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⠜⠁⡠⢾⣅⡀⠀⠀⡜⡟⢸⠀⠻⠭⠭⠙⠏⠉⡠⠆⢸⢸⣧⡀⠀⠀⠀⠀⢹⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⠊⠀⠀⠑⣏⠀⣸⢱⡇⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⡾⠘⡇⣇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠜⠁⠀⣿⡎⣧⠘⡇⠃⠀⠀⠀⠀⠀⠀⠀⡇⢰⠻⣿⠀⠀⠀⢀⠏⣇⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡔⠁⠀⠀⠀⠛⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⡜⠀⢹⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⣠⣀⡀⠀⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠰⡇⠀⠘⡆⠀⠀⢀
⣀⠤⡄⠀⠀⢀⣀⠀⠀⣀⡨⠥⠒⠂⢉⣀⠠⠤⠐⠂⠀⠸⡇⢷⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠀⠀⢷⠀⠀⠙⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣨⠉⠉⠉⠉⠁⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⣧⢸⠀⠀⠀⠀⠀⠀⡘⢸⢠⡇⠈⠀⠀⢀⣀⢘⡆⠀⠀⠈⢷⡐⠂⠀⠀⠀⣀⡀
⠀⠄⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣘⡀⣶⣄⠠⣀⣀⡃⣿⣸⠁⠀⠀⠛⠷⠶⠟⠁⠀⠀⢀⡤⡗⠀⠀⠀⠺⠷⠮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠒⠚⠢⠄⠀⠀⠒⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠚⠓⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠿⠊⢀⡀⠄
⠀⠀⠀⠠⠚⠒⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣆⠀⠀⠀⠀⠀⠀⣤⡠⠤⠤⠤⠤⠀⠀⠀⢤⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣉⣂⣀⣄⣄⣤⣤⣀⣀⣈
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠑⠲⢤⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢼⠄⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠠⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣀⡠⡔⠉⠙⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⢀⠀⠀⠀⠉⠉⠁⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠋⠁⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠼⠃⠀⠀⠀⠑⢦⡀⠀⠀⢀⣀⠽⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠀⠞⠉⢧⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⣀⠀⡇⠀⠀⠀⠀⠀⠒⠛⠉⠁⠀⠐⣄⠀⡀⢀⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⡞⢉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠉⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠋⠈⠁⡠⢆⠀⠀⠀⠀⠀⣴⣊⠽⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠋⠈⠑⠚⠉⠀⢸⠀⠀⠀⠀⡜⢹⠆⣠⠀⠀⠀⠀⠀⡠⢞⣡⠔⠊⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⢀⣀⡤⠖⠢⣀⢀⣸⠀⠀⠀⠀⣇⣠⠟⠁⠁⢀⡠⠔⠋⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⠨⠍⢋⠀⠀⠀⣀⣈⠤⠤⠒⠉⠓⠒⠢⠤⠤⠚⠣⢄⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⡄⠀⠈⠳⠚⢋⠀⠀⠀⢠⠒⠺⢍⠳⡄⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡴⢡⠖⠁⡼⠀⠀⠀⠀⠰⠌⢢⢀⡠⠚⣀⣱⠚⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡸⠚⠷⢜⣲⡾⠀⠀⠀⠀⠀⢸⠈⢢⣉⠔⠁⠀⢸⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⣸⠀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠉⠉⢫⣏⠉⠄⠀⠀⠀⠀⠀⠧⠊⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢻⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣤⣤⣤⣦⢤⣷⡒⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⡀⠀⠖⢋⢳⢳⣀⡀⠀⠀⠀⢀⡀⠲⠂⠀⣿⣿⣿⡿⠟⠃⠈⢻⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠛⠉⠻⡿⣖⠛⠌⠈⠀⢁⠁⠀⠀⠀⠈⠁⠲⠓⠀⢻⠛⢳⢰⠔⠫⠀⠀⠉⢹
⠀⠀⠀⠀⠸⣿⣿⠉⢀⡠⡐⠧⡀⠀⠜⢣⡀⠑⠀⢰⠀⠒⠒⠂⠉⣉⣉⠭⠁⠘⡔⢿⣟⡀⠀⠀⢀⣾⣿⠷
⠀⠀⠀⠀⠀⡟⠋⣲⣨⠥⠭⠊⠈⣆⠀⠀⠹⠊⠉⠉⡄⠀⠀⠀⠈⠀⠤⠀⠀⣀⣸⣦⣀⣉⡠⠔⢺⠙⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠈⠒⠢⣔⣤⡤⣊⢭⡃⣀⡀⠀⡆⠀⠀⠇⠀⠀⢀⡠⠀⢀⡠⢊⠕⢈⠝⠉⠈⠢⠤⠃⠀⠀⠠⣀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⢅⢀⣴⣷⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⢁⠔⠉⠐⠁⢠⠋⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⣀⡸⠉⠛⡆⠀⣀⣀⣀⡤⠊⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⢀⣰⣉⡆⠀⠀⠀⢀⡤⠴⠣⣄⡰⠃
⠀⠀⠀⠀⢰⠁⠀⠐⠊⠁⠀⢱⠀⣠⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣏⠀⡀⣹⠀⠀⠀⢾⣤⣠⡠⠊
⠀⠀⠀⠀⠀⢇⡀⠀⠀⢀⡠⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⢀⠤⡔⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠉⡇⢈⡀⠀⠀⡀⢠⢠⠔⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⣀⠤⠄⢀⠀⣀⣀⢤⡄⠀⠀⠀⠰⢠⢠⡆⠀⢸⠰⢁⣀⠀⠀⢀⡰⠊⠸⠀⠱⣼⠀⠑⠜⠛⠈⠸⢰⠃⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠇⠀⠘⠤⠄⢀⠘⠸⡅⠠⠔⠇⠰⠃⠒⠚⠘⠐⠈⠒⠊⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⣀⠀⡤⠀⡀⠀⠀⠀⠀⢄⣀⡮
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠘⠀⠙⠀⠃⠚⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠤⠐⢆⠀⠀⠀⠈⠃
⠀⠀⡀⠂⠀⠀⠀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠒⠂⢀⠀⠀⠀⠂⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠂⠀⠁⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡗⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⢠⠀⠀⠠⣀⣀⠀⠀⠠⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠦⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⡸⠀⠀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢡⠀⢀⡀⠀⠀⠀⢰⠀⢠⡀⢄⠀⠀⠐⢦⠒⠈⠁⠐⠒⠀⠀⠀⢠⠁⠀⠀⠀⢠⡀⠀⠋⠀⠋⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡀⢸⠟⢷⠀⠀⠄⠀⠀⠱⢀⠣⠀⢈⡠⠀⠀⣦⣠⣶⠀⠀⠀⠁⠀⡄⡈⠀⠠⠅⣀⠐⠀⠐⠁⡀⠀⢂⠀⠀⠀⠱⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠁⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠉⠀⠀⠠⠐⠀⢰⠁⢀⠀⠀⠀⣴⠀⠀⠀⡀⠀⢸⠀⢀⠀⠀⠸⠲⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠈⠤⡠⠔⠁⠑⡒⠒⠁⢀⠉⠀⢸⠀⠇
⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠄⠀⠉⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⣀⣀⠀⠀⢀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡤⠀⠄⡐⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⡄⡀⠀⣀⢀⣀⣀⠀⣀⠀⡔⢀⡀⣠⢰⣄⢀⣰⢄⡀⡀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠈⠀⠉⠀⠉⠀⠁⠈⠀⠈⠈⠉⠉⠉⠉⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⢀⡾⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⢠⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢾⡉⠻⣦⡀⠀⠀⠀⠀⠀⢀⣠⣼⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⠟⢉⡷
⠀⠀⠀⠀⠀⠙⢦⡈⠻⣦⡀⢀⣴⣾⣿⣿⣿⣀⣈⣉⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⣉⣁⣀⣿⣿⣿⣷⣦⡀⢀⣴⠟⣁⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⠟⢹⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⡏⠻⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠛⢁⡄⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡇⢠⡈⠛⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣴⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣦⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⡄
⢀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⠉⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⠉⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤
⢿⣅⣤⣠⣤⣄⣤⣄⠙⣿⠇⣰⡄⢻⣿⣿⣿⣿⣿⣿⣿⠋⢸⣿⣿⣿⣿⡟⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣆⠸⣿⠋⣠⣤⣤⣤⣤⣤⣨
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⡿⠟⠁⢀⣤⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⣤⣀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢿⠛⠉⠉⠀⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠂⠀⠀⢨⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⣺⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⢸⡧⠀⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⣼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⠗⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣑⣌⠑⢦⡜⣅⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠂⢘⣀⠀⠀⣀⠤⠔⠒⢲⡄⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣁⠀⠙⠉⠀⣠⢄⠀⠀⡯⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢄⠀⠁⠀⠀⢈⣀⡀⠀⠀⡇⢸⣀
⠀⠀⠀⢀⡤⠔⢊⠉⠉⢰⣖⡋⣀⣀⢘⡃⠀⠀⠀⠀⣳⠻⢂⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣽⣀
⠀⠀⢼⠁⠀⠀⠘⡒⠠⠜⡆⠑⠉⠉⢤⠟⠀⠀⠀⢀⣡⡋⠁⠀⠰⠄⠀⡉⠉⠛⠒⢒⣳⠉⢹⠀⠜⣡⠜
⠀⠀⠈⠢⢄⣒⠀⠡⠀⢀⠰⣀⣀⠼⠓⠀⠀⠐⡶⠋⠀⠀⠈⠉⠒⠠⠤⣀⠀⠀⢀⡀⣼⠀⢸⡀⠘⢧
⠀⠀⠀⠀⠀⠀⠈⠁⠐⢒⠗⠺⠛⡶⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠏⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡞⠀⠀⠀⡟⢄⠀⡜
⠀⠀⠀⠀⢈⠉⠁⢧⡸⠀⠀⠀⢧⠀⡤⠲
⠀⠀⠀⠀⠘⠒⠖⠀⢷⣤⣤⣤⣤⣬⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢷⠒⠺⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡆⢀⠬⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⡈⠳⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠱⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡷⡶⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⣲⣦⣻⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢰⠁⠀⠀⠙⠴⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⡞⠦⢄⣀⣀⠤⠔⣊⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣒⣒⣒⣚⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⡀⠄⣀⣃⣀⣀⡀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠤⠒⠡⠀⠀⠀⠀⠀⠀⠰⡿⠉⠙⠁
⠀⠀⠀⠀⡈⠢⣜⡛⣛⣹⣿⣧⡆⠐⠃⠀⠀⠀⠀⠀⠀⠘⠐⠀⠈⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢠⠋⢁⠈⠉⢉⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠠⠉⠀⠀⠩⠄⡌⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠄⠀⡢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣬⣂⣂⣨⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣿⣶⣦⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠘⠇⠿⠿⠿⠷⠟

⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠛⡉⠛⡛⠛⠀⠀⠀⠀⠀⠀⠈⠈⠨⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⡆⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⢀⣤⡤⢤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢠⣾⣿⣽⣿⣶⠀⠀⠀⠀⠰⣶⣷⣦⣴⣷⡶⠀⠀⠀⠀⢀⣴⣵⣴⣶⡄⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠃⠀⠀⠀⠀⠤⠤⠤

⠀⠀⠀⠀⢀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦
⠀⠀⠀⠀⠘⠋⠀⠠⠀⠀⠀⠀⠀⣴⡆⠠⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠉⠀⠀⠀⠀⠀⠀⠉⠡⠉⠩⢉
⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠰⣯⣽⠀⢠⣄⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠁⠀⠀⠀⠈⠉⠙⠉⢀⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣶⠖⠢⣀⣀⣶⠲⠼⢷⡾⠥⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣤⡾⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⣒⡄⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢀⡴⢩⠏⠀⠀⢰⢀⡼⢇⡄⠀⠀⠀⠀⠀⠈⢾⠁⠘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠼⢀⡴⠴⠻⠊⠀⡟⢡⡷⣠⡄⠀⠀⠀⢾⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢸⢳⡀⠀⢀⣤⡖⠀⠀⢀⠀⠈⠑⢤⡄⠀⡾⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣲⣾⣾⡇⠀⠀⠀⣔⡀⠀⠺⠣⠄⠀⠀⠘⠦⠧⡄⠀⠀⠉⠳⡆⠀⠀⢀⣤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠷⣄⣈⠉⠙⡿⢷⡶⠶⠾⣥⠀⠀⠀⠘⠷⠀⣠⠀⣰⠃⢀⡴⠋⢱⠇⠀⠀⣶⠘⠛⠛⢹⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢉⡟⣋⠉⠛⠶⣄⠀⠉⠙⠒⠷⡀⠀⠀⠀⠀⠓⠀⠀⢀⣠⠞⠛⠋⠁⡴⠋⠀⣰⠋⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⡟⠛⢻
⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠘⢧⣏⣧⣄⡶⠼⣶⣶⣒⠦⣤⠞⠲⢤⣤⣀⣤⠴⢾⡋⠀⠀⠀⠀⡼⠁⢀⠞⠁⠀⠀⠀⠀⢸⡀⠀⠀⡟⠀⣈⡗⠒⢾
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠠⢯⡀⠀⠀⠉⠓⠦⣤⣘⡡⠖⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⢰⠃
⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠴⣿⠋⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⢦⡀⠸⣄⠀⠀⣀
⠞⠁⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⡴⢀⡟⠒⠒⠒⠲⠦⣄⠀⠀⠀⢀⡘⢟⣦⣖⠽⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⠀⠀⣠⠀⠘⣦⠀⠀⠾⠱⠆⠹⠷
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⣇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠘⣷⡀⢀⣠⣴⣾⡿⠿⠿⣶⣦⣤⡀⠀⠀⢀⡼⠋⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡟⠀⠀⠀⢠⠂⠀⠀⠀⠀⢈⣿⣿⣿⡋⠉⠀⠀⠀⠈⣿⣿⣿⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠀⠀⠀⠈⠳⠤⣤⣄⣠⣾⣿⣭⣿⣿⣷⣦⣴⡾⠿⠛⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃⠀⠀⠀⠀⡀⢀⠀⢠⣿⡟⣩⣤⣤⠀⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠠⠀⠀⢠⣿⠯⠁⣩⣥⣄⠀⠀⣺⣶⣿⡿⣿⣽⣿⣿⣿⣿⣟⠉⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⡟⠀⢸⠋⠻⢟⡧⠐⡿⢻⣿⠤⠼⣿⣟⣿⣿⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠀⠀⣛⠛⠛⠀⠀⠻⡌⠁⠀⣀⡼⣩⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⢾⣿⣶⣶⣶⡎⢹⣿⠋⠈⠁⠘⣿⣿⣿⣿⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⣀⣤⣽⣿⣯⣭⣀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣾⣿⠟⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡦⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠀⠀⠀⢀⣾⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⡙⣆⠀⠈⣿⣿⣿⣿⣿⣿⠉⠉⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⡆⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢹⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣸⡏⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠙⢿⣄⡀⢀⣼⣿⣿⣿⣿⣿⣿⡟⣴⡟⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠉⠛⠿⠿⡿⠿⠿⠿⠛⡫⠶⠛⢁⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠹⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⣼⣿⡿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣀⢀⣀⣤⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉