Neko Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠶⢦⣤⣤⣤⣤⣤⣄⣤⠶⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠋⠉⠛⠀⣀⣀⡀⠘⠋⠉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢚⣡⠀⠀⣄⠀⣹⣟⠁⣠⠄⠀⢠⡛⢂⡿⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠈⠉⠉⠈⠉⠉⠀⠀⢀⣠⠞⠀⢠⡄⢤⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣶⡟⠁⠀⠈⠛⢳⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠈⠙⠛⣿⠛⠛⢿⡛⠋⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡀⠀⢳⣴⣆⡼⠃⠀⠀⡀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢧⣀⡤⠤⠤⠤⣤⡀⠀⠈⠛⢿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⣷⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠳⢤⣀⣀⣲⡽⠁⠀⠀⠀⢹⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠚⠉⠙⠻⣇⠀⠀⠀⠀⠀⡼⠚⠉⠉⠓⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣯⣤⣧⣤⣼⣤⣤⣤⣤⣼⣧⣤⣿⣀⣷⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡠⢖⡶⠬⣑⠀⠀⠀⠀⢀⣈⣉⣁⣀⣠⣎⣴⣿⣿⠟⡽⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠿⣿⣶⣌⣽⠶⠞⠛⠉⠉⠁⠀⠀⠉⠉⠙⠛⠿⣦⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣆⣹⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⢈⠁⠰⠦⣤⣒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠘⢧⣤⡶⠦⣄⡉⠳⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⠴⠒⠦⣄⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠒⠀⠀⠈⠈⣧⢠⠀⠈⡙⣆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠶⠇⣧⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣟⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠿⠶⠷⠐⡏⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡿⢀⠀⠀⠸⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠙⠟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣥⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⢦⣄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣫⢜⢿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠮⣷⣮⣿⣶⣦⣤⣤⣤⣄⣀⣤⣤⣴⣶⣿⣿⣷⣿⣿⣷⠟⠀⠀⠀⠀⢠⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠙⠢⣍⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣭⣥⣿⡿⠛⢛⠋⠁⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣿⣻⣿⢧⠈⡀⠀⠀⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠂⠀⠀⠀⠀⠀⠀⢆⠀⠚⠿⢿⣿⣿⠞⢀⡅⠀⣠⠆⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⡼⠃⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⡄⠀⣾⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡠⣷⠀⠀⠀⠀⠀⠹⣦⡀⢠⡴⠿⠷⠦⠀⠀⠀⠀⠀⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠘⣷⡟⠀⠀⠀⠀⠀⠐⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠖⠚⢷⡀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠠⠄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠷⠖⠛⠁⠀⠙⠓⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⠪⢭⡒⠤⢀⣤⣤⣤⣤⣄⠠⠴⣊⠭⠢⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢤⡠⠔⠋⠀⠘⠤⠣⠜⠤⠂⠀⠈⠓⠠⢄⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣀⣀⡀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠁⢀⣀⡈⣦⠴⠒⣲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠴⢒⡂⠀⠀⢠⣀⢀⣀⣤⣀⡀⢀⡀⠀⠀⢐⡒⠦⢻⣀⣇⣼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⢁⣏⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠿⣳⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⣻⠟⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠸⡟⠏⠙⢻⣭⣹⠽⠛⢿⡭⣉⣏⡯⠭⠷⠛⠉⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠀⠀⠀⠘⢏⠹⢛⠖⠾⣸⣄⢰⣀⠧⡄⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢘⣆⠀⠀⠀⢌⣻⡈⠌⣨⡬⢯⡍⠉⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠯⡓⢦⢤⠴⡗⣳⡾⣋⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢇⡠⢚⠉⠉⣿⡍⠛⠀⡞⡩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⡔⣡⣿⣧⡀⣀⠧⠷⠁⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⣉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣰⠋⠙⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⡴⠋⠉⢷⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡏⠀⠀⠀⠘⢧⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠏⠀⠀⠀⠈⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠘⢧⠈⢧⡀⣀⣀⣀⣀⣀⣰⠏⢠⠏⠀⠀⠀⠀⠀⢿⠀⣿
⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠲⠶⠶⢦⣼⠃⣿⠀⠀⢀⣠⡤⠶⠞⠀⠈⢋⠋⠉⠉⠙⠉⠁⠀⠙⠷⠦⣤⣀⠀⠀⢸⠄⣿
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣼⠀⣿
⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⢀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⢷⡀⠇⠀⢀⠀⠈⡀⣠⣾⠇⠀⠀⠀⢀⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⣤⣄⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠙⢿⠒⠚⠓⠚⠛⢁⡟⠀⠀⠀⣰⠋⡟⠁⠀⠈⢻⠻⡄⠀⠀⠀⠀⢀⡾⢻⠃⠀⠀⠹⡎⢳⡀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠘⠉⠱⣄⢷⡀⠀⢀⡼⣠⠿⠀⠀⠀⠀⠾⢧⡸⣆⠀⠀⣸⢇⡼⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠠⠄⠀⢸⡇⠀⠀⠀⠈⠉⠐⠂⠀⠀⠀⠀⠀⠘⠆⠔⠃⠀⠀⠀⠀⠀⠒⠂⠉⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⢉⣉⠀⠀⠀⣄⠀⠀⣠⣄⠀⠀⣀⠀⠀⠈⣉⠉⠉⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⢀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠈⠙⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠈⠉⠉⠛⠛⠛⠛⠚⠒⣒⣚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⣰⠏⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡰⠃⠀⢹⢿⠶⠶⠦⠤⠤⠤⣤⠋⠉⠉⠙⢦⠴⠤⣶⠞⠛⠛⠛⠛⠋⠀⠀⢀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠘⠃⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢣⠀⣠⡴⠖⠛⠛⠛⢿⡔⠂⡀⠀⠀⠀⠀⠀⠁⠠⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡞⡄⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠻⣎⣀⠄⠀⠀⠀⢀⠀⠄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡱⡀⠀⢀⡴⠋⠀⠀⠀⢰⣤⣀⠈⠻⢶⣤⣌⣙⡶⢤⣀⣀⣀⣑⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠐⢤⡀⠀⠀⠙⢮⣦⠞⠁⠀⠀⠀⠀⠀⠈⠙⠛⢦⣾⡏⠈⠙⠓⠀⠀⢻⡍⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠙⢦⠀⠀⣴⠋⠀⠀⠀⠘⠿⣦⣄⠀⠀⣤⡾⠉⠙⠳⣦⣤⡀⠀⢸⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠃⣼⠃⠀⠀⢀⣀⣀⠀⠀⣉⣹⣶⣤⣀⠀⠀⠀⠀⠉⠁⠀⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢈⠉⠛⠛⠛⠙⠻⢦⣄⣉⠻⢿⡦⠀⠀⠀⢀⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣄⡘⠿⣦⣀⠀⠀⠀⢦⡈⢙⣿⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡟⣛⠷⣤⣽⠿⢶⣤⣄⠻⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⣠⢤⡤⢤⣼⣆⠐⠟⢰⣿⣠⡾⠋⡛⠶⣬⣙⠁⠀⠀⠀⢀⣴⠯⠤⠤⣄⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠻⡾⠃⣠⡾⠟⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡴⠀⠀⡆⠀⠀⡆⠀⠙⣦⣄⡀⠀⠀⠿⠏⠀⢀⣠⣴⠏⠁⢠⡀⠀⠀⡄⠀⢠⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣧⣤⣤⣧⣤⣤⣧⡴⠞⠁⠈⠉⠛⠒⠲⠒⠛⠋⠁⠈⠳⠦⢼⣧⣤⣤⣧⣤⣴⡧⠶⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠙⣦⠀⠀⠀⠀⢀⡾⠁⠀⠀⢿⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣾⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣄⣾⠜⣤⠀⣶⣠⡵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠛⠀⠉⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⠻⣿⣟⣰⣮⣿⣿⣿⣏⣻⣼⡗⢙⢿⣿⡆⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣯⣵⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⣿⣟⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣗⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣤⣀⡀⠀⣠⡟⣛⣛⢦⡀⠀⢀⣠⣴⣿⠀⠀⠀⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣷⣶⣭⡙⣿⣿⡇⠜⢡⣮⡄⣿⣿⣿⣿⣿⣿⡅⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣬⡹⣷⣄⣋⣛⣵⣿⣿⣿⣿⣿⢹⠇⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡝⣿⣿⣿⣿⣿⣫⠟⠻⣝⢻⣿⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣻⣮⡻⣿⣟⣳⠏⢀⡿⢈⡳⣝⢿⣿⡝⡻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠓⢗⣫⣾⣿⣿⣵⣯⡌⢯⡻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⢈⣭⣾⠿⣻⣿⣿⣿⣶⠶⠙⣝⣯⣿⣿⣿⡏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠉⣵⠿⣾⣿⣿⣅⣀⣲⡦⠘⡎⣿⣿⣿⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣥⣣⠀⠠⠾⠛⣿⣿⣿⢿⣯⠁⠀⡿⣿⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡳⡀⠀⣾⡏⠹⠿⠀⠀⣨⣾⡿⣶⣿⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⡿⠦⢌⣀⣀⡡⠴⢾⣿⣿⣧⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⡴⠖⠢⣄⣀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡴⠒⠤⢄⣀⣀⣀⣀⣀⣀⣀⠤⠔⠲⡄
⠀⠀⠀⠀⢸⠀⠀⠙⠁⠀⠉⠐⠳⣄⡃⠀⠀⠀⠀⠀⡴⠒⠦⣰⠃⠀⠀⠳⠶⠚⢉⡨⣇⠀⠀⠀⠀⠀⢣⠈⣦⠀⠀⠀⠈⠁⠁⠀⠀⢀⡊⡰⠃
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠠⠀⠀⢀⠗⠒⢆⠀⠀⠀⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡨⠃⠀⠀⠀⠀⠀⢹⡇⠀⠠⠀⠀⠀⠀⠠⢀⠀⡟⠁
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⢼⠂⠀⢠⠃⠀⢨⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⠉⠙⡆⠀⢠⠞⠉⣱⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⣆
⠀⠀⠀⣠⠏⠔⠠⠀⠀⡀⢀⠀⢠⠀⠂⢀⠎⠀⠀⢈⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡇⠀⠘⢦⠚⠉⢗⠀⠀⠀⣀⣀⠀⢰⠋⠙⢰⠋
⠀⢠⠚⠁⢂⡀⠀⠈⠀⠶⠀⠀⠀⠀⢀⠎⠀⠀⠀⠘⣄⠀⠀⠀⠀⠉⠐⠒⠒⠂⠉⠜⠀⠀⢠⠇⠀⠀⠘⢷⠀⠈⠢⡀⠀⠤⢀⡤⠋⠀⣰⠃
⠀⢸⠔⣶⣍⠀⠀⠀⠀⠠⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⢳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠴⠋⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠁⠉⠀⠀⢺⡇
⠀⢸⡀⠁⢥⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢣⠀⣧⣦⡔⠀⠀⠀⠀⢀⠄⠀⢀⢸⠀⠀⠀⠀⠀⢿⠀⠀⢄⠀⠀⠀⠀⡠⠀⠀⢸⠇⠀⠀⠀⠀⠀⠸⡄⠸⠷⡀⠀⠀⠀⡸⠃⢀⡞
⠀⠀⠈⠧⣈⣁⡠⠔⠒⠒⠒⢧⣀⣀⣬⠇⠀⠀⠀⠀⠀⠈⢇⣀⣘⣦⢤⣄⣴⣃⣀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡽⠒⠒⢺⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠢⣄⠀⠀⢀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⣀⠀⣀⡀⠀⣀⠔⡆
⠀⠀⠀⠀⠀⣠⠿⠚⠃⠐⠓⠻⢦⡄⠀⠀⠀⠀⠀⠀⡔⡉⠉⠒⠂⠒⠊⠁⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠂⠈⠉⠛⠋⠈⠁⠠⡅
⠀⠀⢜⣉⡹⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠠⠤⣄⠀⠀⣠⠤⠄⠸
⠀⠀⠀⢏⡃⢀⠀⠀⣀⠄⠀⠀⠀⠈⡆⠀⠀⠀⠀⠇⠀⠠⠄⠀⠀⠀⠠⠄⠀⢻⣈⠑⠄⠀⠀⠴⠋⠉⠲⡎⠸⡀⠀⠀⠈⢀⡄⠁⢀⠔⠛⠣
⠀⠀⡘⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⠀⠀⠀⠀⡸⠀⠀⠸⠀⠀⢈⠒⠂⣀⣴⣶⣿⣦⣄⢀⠀⠀⣀⢨⠞⠛⠚
⠀⠀⢇⠀⠀⠀⠁⠀⢄⡲⣤⣄⡁⠀⠣⠀⠀⠀⠀⠜⠱⠄⡀⠀⠀⠀⠀⠀⠴⠁⠀⢠⠃⡞⠉⠙⢀⡴⢿⡛⠻⠛⣛⡿⣧⣀⠂⠠⠊⠠⣶⡰
⠀⠀⠀⡹⠀⠀⠀⡠⠊⣉⠧⣠⡀⠀⣀⠇⠀⠀⡎⠀⠀⠀⠀⠈⠅⠏⠀⠀⠀⠀⡔⠁⠀⠈⡵⠂⣿⡇⢾⡾⠀⠀⢶⡇⢸⣧⠙⡄⡠⠒⠉⢇
⠀⠀⠀⡇⠀⠀⢰⠇⡴⢾⡭⠸⠈⠉⠀⡇⠀⠀⢷⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠢⠤⣿⣍⠲⠦⠀⠠⠴⢞⣱⣿⡆⣾⠁⠰⠟⢸
⠀⠀⠀⢇⠀⠀⢨⠴⣻⣾⠁⡇⠀⠀⢠⠃⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠀⠀⠀⠀⠀⢳⣭⣟⣷⠖⢺⣾⣻⣽⣞⣱⣿⡤⣤⡤⠃
⠀⠀⠀⠀⠙⠳⠤⠓⠠⠄⠚⠤⠖⠂⠁⠀⠀⠀⠀⠑⠺⠿⠒⠒⠒⠒⠒⠛⠷⠒⠁⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠼⠿⠟⠋⠀⠀⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⡿⠿⢷⣦⣄⡀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠿⣶⣿⣿⡟⠉⠉⣿⣿⣿⡏⠉⣿⣿⣿⠛⠛⢿⣶⣦⣴⣾⠟⠋⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠛⠳⡆⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⠁⢀⣿⣿⡏⠀⠀⣾⣿⣿⠇⠀⠀⢰⠟⠛⡄⣿⡇
⠀⠀⠀⠀⠀⣿⡇⢠⣴⠃⠀⠀⠈⢿⣿⣷⡄⠀⣿⣿⡿⠀⢸⣿⡿⠀⣠⣾⣿⣿⠃⠀⠀⠘⢣⡄⠀⡇⣿⡇
⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣀⣀⡀⠀⠉⠛⠃⠀⣿⣿⡇⠀⣾⡿⠃⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠲⠃⢸⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⢿⡿⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣶⣾⣿
⠀⠀⠀⠀⢰⣿⡿⠋⠁⣠⠴⢶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⠦⣍⠉⠛⠿⢿⣿⣿⡇
⠀⠀⠀⠀⣸⡟⠀⠀⡞⠁⣼⡿⠛⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⣿⣿⣿⣧⡈⢳⡄⠀⠀⠉⢿⣿
⠀⠀⠀⠀⣿⠃⠀⠸⡇⢸⣿⣦⣀⣼⣿⣿⣿⣿⠋⡀⠀⠀⠀⠀⠺⢿⣿⡀⢀⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⣿⡇
⢀⣤⣄⣸⣿⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣤⠃⠀⠀⠀⠈⣷⠸⣿⣿⣿⣿⣿⣿⣿⣷⡞⠁⠀⠀⣀⣀⣽⣇⣀⣀
⠈⠉⠉⣿⡿⠻⠿⠷⠄⠈⠻⣿⣿⣿⣿⣿⣿⠿⠋⢀⣀⣀⣀⠀⠈⠻⣽⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠛⠛⠛⣿⡿⠛⠛
⠰⣶⠶⣿⣿⣶⣶⡶⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠘⠿⢿⡿⠿⠃⠀⠀⠀⠈⠉⠉⠉⢀⣀⠀⠀⠀⠸⠿⠷⣶⣿⣧⣄⣀
⠀⠀⠀⢘⣿⣤⣤⣤⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠺⣷⣦⣄⣸⡿⠉⠙⠛
⠀⣶⡾⠿⢿⣿⣋⣡⣶⣿⣿⡿⠁⣠⣶⠀⠀⠲⣦⡴⢿⠷⢄⣴⠞⠀⠀⣴⣶⡀⠀⠈⢿⣿⣿⣦⣀⣈⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠋⢀⣼⣿⣿⠀⠀⠀⠈⣧⠀⠀⡼⠃⠀⠀⠀⣿⣿⣿⣄⠀⠈⢿⣿⣿⣿⣿⠟⢿⣏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣯⣤⣄⣀⣀⣀⣈⣁⣀⣀⣀⣀⡀⣸⣿⣿⣿⣷⣤⣤⣿⠿⠟⠁⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⠙⢻⣿⠟⢛⣛⠛⠛⢿⣿⠛⠛⠛⠛⠛⠛⠉⠉⠀⣀⡀⠤⠤⠛⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠓⠒⣿⣿⠒⠋⠉⠓⠂⢀⣿⣦⣠⣄⡀⠀⠒⠒⠒⠈⠉⠁⠀⠀⢀⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠹⣿⣌⠳⠶⠋⣉⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣤⠈⠛⠿⠿⠿⠟⠋⠀⠀⢀⣴⡷⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⠁⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣶⣶⣾⣿⣶⣤⣤⣀⣀⣀⣀⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⢾⣿⠿⠟⠋⠀⠀⠀⢠⣿⣿⣦
⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠉⠉⠹⣿⣿⣿⠀⠀⠙⢿⣿⣿⡟⠉⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣧
⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠸⣿⣿⣧⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢿⣿⡏⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠛⠉⣿⣧
⠀⢰⣿⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠟⠋⠁⠀⠀⣠⣿⣿⣧
⠀⢸⣿⣀⠞⠁⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆
⠀⠈⢿⣯⡀⢠⡾⠋⠀⠀⣀⣤⣶⡿⠛⠋⢹⣿⠉⠙⠛⠿⠿⠿⣷⣶⣶⣶⡶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠈⠛⠿⢿⣶⡶⠾⠟⠛⢹⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠁⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠻⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣀⣀⣀⣿⣇⠀⢀⣴⣿⣿⣥⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠻⣿⣿⣿⣿⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠉⠙⢿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢿⣿⣷⣶⣾⣿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⣠⣴⡾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⣼⡟⢡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⣿⡟⢁⠴⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷
⠘⢿⣯⣤⣤⣤⣤⣤⣄⣠⣤⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠛⣿⡇
⠀⠀⠉⠉⠉⠙⠉⠉⠉⠉⢉⣿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⠞⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡾⠿⠻⢷⣦⣤⣤⣤⣤⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠋⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣴⣏⣠⣤⣤⣤⣶⣶⣶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠈⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠙⠿⣿⣿⣷⡄⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣶⣿⠟⠛⠁⠀⣤⣀⠀⠀⠈⠉⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⡶⠿⣷⣿⣿⣿⠿⠿⠟⠛⠛⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣷⣄⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⡇⠀⠀⢀⣄⠀⠘⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⡇⠀⠀⢾⣿⣆⠀⠈⠻⠿⠃⠀⠀⠀⣸⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⠿⡧⠀⠀⢹⣿⣿⣷⣄⡀⢀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣀⢘⣿⣿⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⡀⢀⣠⣶⡄
⠀⠀⠀⠀⠀⣽⣿⣿⣸⣇⣿⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⣿⡇⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠛⠁⠈⠛⠛⠧
⠀⠀⠀⠀⢘⠛⠉⠭⠛⠋⠛⠩⠍⠙⣳⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⠀⠀⢰⡦
⠀⠀⠀⠀⢈⡀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠲⢦⡤⣤⣴⠾
⠀⠀⠀⠀⠀⢸⣯⠙⠛⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⣜⣿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢸⣿⡿⠿⠿⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣝⣛⠧
⠀⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⢠⣿⡅⠀⠀⠀⠀⠀⠀⠀⢾⡄⠈⣿⣤⣶⡄⡀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣭⡽⠁
⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠸⠿⠛⠛⠛⠻⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢾⡦⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡤⢠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠙⢿⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⢀⠠⣶⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣀⣠⣀⣀⣀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠱⠦⣤⣤⣤⡴⠞
⠀⠀⠀⠀⠀⠀⢀⠈⠋⠻⠒⢲⡴⠋⠉⠁⠀⠀⠀⠀⢰⣏⠁⠀⠻⣿⣧⣤⣿⠟⠉⠀⣷⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⢀⣥⡈⠀⣤⡅⡙⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⣀⣤⢍⣀⠰⠙⢁⢄⢤⣤⡀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⣤⡄⠀⢰⡄⠀⣿⡟
⠀⠀⠀⠀⠀⠛⠈⠛⠀⠀⠛⠣⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠁⠀⠙⠁

⣿⣿⠿⡿⠿⢛⠿⢿⡟⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣦⣄⠠⣤⣤⣴⣶⣶⣤⣤⣤⣤⣀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⢋⡔⠳⢄⡙⢷⣌⢻⣿⣿⣿⣿⣿⣿⢟⣡⠞⣫⡵⠖⢎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡏⣼⠃⡾⢁⠔⠋⠙⢦⠙⢷⣙⣛⣛⣛⣛⣡⠟⣡⠾⠥⠄⠀⢸⡜⣧⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠇⣿⢸⡇⠐⠀⠀⠀⠀⠳⠈⠏⠉⠉⠉⠙⡏⢸⡁⠠⠨⠀⠀⢠⡇⣿⣨⡥⠶⠶⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⣏⢸⡇⢀⣤⠷⠖⠒⠚⠀⠂⠀⠀⡄⠀⠈⠀⠉⠓⠢⣄⠀⢸⣧⡟⠁⠀⠀⠀⠀⠀⠑⢿⣷
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡟⠀⠀⠀⢠⠀⢀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡿⠀⢀⡴⢂⣀⣀⡀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠈⣧⠀⣀⢀⡼⠀⠈⣧⠀⠀⣼⠃
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡇⠰⠋⠶⣭⣴⣶⣶⣦⡠⠳⡀⠀⢠⠊⠔⣩⣭⣭⣍⡒⣀⠀⢸⡝⠛⠻⣄⣀⣀⣿⣀⣴⢟⣼⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠁⠀⢬⣿⢹⣿⡿⠓⣿⢙⠀⠃⠀⠘⢀⣾⡿⠿⣿⣿⡿⣦⡀⠀⡇⠀⠀⠈⠉⠁⠀⣼⢡⣾⣿⣿⡄
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠉⠀⠘⢿⠷⢶⡖⠁⠀⣀⣀⣀⠀⠻⢷⣴⣿⡿⠤⠛⠂⠀⡇⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⡀
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢻⡇⠀⣀⣀⡀⠀⠀⢀⠀⠀⠘⢿⣿⠿⠁⠀⠀⠀⠀⠧⠥⠤⠤⢴⡇⠀⠀⠀⡀⠀⠀⣿⢸⣿⣿⣿⣿⣧
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡘⣿⣉⣠⠤⠶⠒⠂⠀⠐⠈⠀⣠⣷⠀⠀⠀⠈⠀⢀⡀⠒⠀⢤⡞⠀⠔⠈⠀⠁⢀⠀⣿⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⡿⢄⡀⠀⠀⠀⠀⠀⡐⠊⣙⠛⠛⠒⠁⠀⠀⠀⠀⠉⣢⣾⠁⢀⠀⠀⠀⠀⠀⠁⢹⡏⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⠟⠁⠙⢮⣉⠷⡦⢤⣄⣀⣳⣄⡀⠀⠀⣀⣾⡶⠶⠶⠖⢛⣹⠀⠀⢘⡀⠀⠀⠀⢐⠇⢸⡇⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡟⠁⠀⠀⠀⢠⡟⣿⡦⣤⣄⣀⡀⠀⠉⠉⠉⠁⠀⢀⣸⣬⣼⣺⢿⡀⠀⠀⠀⠳⠒⠒⠃⠀⣸⢇⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣟⠀⠀⠀⠐⠦⢿⣝⢺⡇⣇⠀⠈⣩⠿⠛⠛⢛⣿⡉⠉⢹⡇⢸⣡⠜⣧⠀⠀⠀⠀⠀⠀⠀⣠⡟⣼⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⡗⠀⠀⠀⠉⠈⠂⠉⠳⣷⡸⡄⢰⡇⠀⣶⣿⠄⠁⢹⠀⣸⢀⡿⣀⠠⢚⣷⣄⣀⣀⣀⣤⡾⢋⣾⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠛⢾⣧⡀⢸⡇⠀⣠⠞⢠⢏⡾⢋⡠⢞⡩⡂⣿⡟⢻⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣹⠷⣄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⡟⠻⠿⣯⣁⣠⢿⡾⣑⣥⠾⣋⣴⠞⠛⠻⢶⣝⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⣼⠇⡀⠀⠙⡆⠀⠀⡀⠀⠀⠳⢤⡤⠞⠁⠀⠀⠀⠩⢛⢯⡾⠉⢢⣠⠟⠁⠀⠀⠀⠀⢿⣆⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⢰⡗⠒⢤⡀⠀⢻⡀⠀⠙⢮⣶⣂⡾⠀⠀⢀⢼⣿⠇⠀⠀⠡⡹⣎⢹⡏⠀⠀⠀⠀⡀⣠⠞⢿⡘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⢸⡇⠀⠀⣱⠠⠀⠙⣷⠶⠋⠀⠁⠠⣤⠞⣣⣞⣡⣶⡆⠀⠀⠈⠸⣾⠀⠀⠀⠀⠀⠀⣏⠀⢸⡇⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⡸⣯⠓⢚⠁⠌⠀⠀⢸⣄⠀⣠⣦⠀⠿⠀⡠⢻⣥⢻⠃⢀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠁⠘⠦⣼⠇⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⢻⣆⠈⠀⠀⠀⠀⠘⣿⡀⠀⠀⢀⣀⠙⢧⣠⠾⣟⣴⠿⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡟⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠹⣦⣀⡀⠀⠀⠀⠹⣷⡀⠀⠞⢽⣀⠀⠭⠖⠛⣇⢀⣀⡤⠂⢀⡇⠀⠀⠀⢤⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡄⠀⠀⠀⠹⣗⠀⠀⠀⢸⡶⠖⠒⠛⠛⠉⠉⠀⠀⣼⠁⡀⠀⢠⡀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡀⠀⢸⡀⠀⣸⣷⣄⠰⠏⠀⠀⠀⠀⠀⢀⡠⢚⣼⠁⠀⢹⠀⠀⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠟⠻⠷⠞⠛⠛⠋⠈⠙⠿⣶⣤⣄⣐⣂⣨⣤⡶⠟⠻⢶⣶⠿⣦⣴⡿⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⣤⣀⡙⠲⣄⠀⠀⠀⢀⡴⠚⠁⣠⣤⡄⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢀⣿⣿⣿⣷⣶⣾⣻⣿⣻⣟⣶⣶⣿⣿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⣠⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠉⠓⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡄⠀⢠⠀⠸⠛⠋⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⣀⣠⠬⣉⠉⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢐⡃⠤⠚⠴⠁⠀⠀⠀⠀⠀⠙⠃⢀⣤⣤⡀⠘⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⠀⠈⠛⠋⠀⠀⠀⡄⠰⠀⠀⠀⠀⣸⠶⠛⠉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠿⠃⠈⡄⠀⠀⠀⠀⠀⠳⠤⠤⠖⠓⠤⠤⠞⠁⠀⠀⠀⠀⢠⠋⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⢀⡌⠂⡀⠀⠀⠀⠀⠀⠀⡒⠒⠀⠀⠀⠀⠀⠀⠀⡴⢇⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣾⣾⡿⠁⠀⠘⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠺⣷⣨⡏⠉⠉⠙⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠋⠀⠀⠀⢀⣿⣿⠁⢰⣷⠐⢰⡆⢰⣿⡁⠈⣿⣆⠤⠛⠅⢿⠓⠦⠤⠤⠤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⠠⠊⠀⠉⠁⠛⠛⠰⣜⣷⠘⠛⠓⠈⠉⠀⠀⠀⠈⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠂⠁⠀⠀⠀⠀⡎⢠⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠘⢦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⠜⠁⣠⣰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣾⣿⣿⡟⠀⢸⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⡞⠀⠤⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠻⠇⢒⡀⠀⢈⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⠓⠈⠈⠁⠑⠲⡀⠀⠀⠀⠀⠀⡠⠆⠐⠀⠀⠀⡸⠛⠋⠉⠙⠻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡅⠀⠀⠀⢠⠀⠀⣸⣦⣀⣀⣀⡎⠠⠀⠀⡄⠀⢠⡧⢀⠀⠀⠀⡆⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⡛⠻⠿⢿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣤⣧⣴⣿⣿⣿⣷⣶⣾⣿⣿⠿⠿⢛⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠤⠬⠭⢉⣉⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⡭⠭⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⢦⣄⣤⣴⡶⢲⣶⡶⣶⠶⣤⣄⣤⠖⠛⢳⡄
⢠⠶⠛⠋⠛⡲⣄⠀⢸⡇⠣⠀⠀⠸⣿⠀⢸⣿⢸⡿⢠⣿⡟⠁⢰⠉⡘⡇
⣏⠀⠀⢱⠈⣉⠈⢷⣸⣇⣉⣠⣄⡀⠉⠃⣾⡇⠾⠃⠛⠉⠀⣀⣀⣑⣁⣷
⢻⡈⠁⡤⠒⠄⠀⠸⣿⡿⠛⣉⣩⣥⣤⣀⠈⠀⠀⠀⢀⣀⣤⣭⣉⠛⠻⢿⡄
⠘⣇⠸⠀⠀⠀⡀⠀⣿⠀⢀⠱⣿⣿⣧⣼⣧⠀⠀⢰⣿⣤⣿⣿⡷⠑⠀⠈⣇
⠀⢻⡀⠣⡀⢀⠷⢶⣿⣤⡀⠑⠨⠛⠛⠻⠋⢀⣀⠀⠻⠟⠛⠯⠒⠁⠰⠶⣿⠶⠂
⠀⢸⡇⠀⠀⠀⠰⠶⣷⠶⠂⠀⣠⡄⠀⠀⠀⠙⡋⠁⠀⠀⠀⢠⣄⢠⣍⣛⡿⠶⠄
⠀⠀⣧⣠⣤⡄⠠⠶⠿⣟⣡⣾⣿⢃⣴⠀⠑⠞⢃⠎⠀⢸⣧⠘⣿⣷⣬⡟⠳⠄⠀⠀⠀⠀⣀⣤⣶⣶⡖⠲⠶⣤⡀
⠀⠀⢿⠟⠉⢀⣀⠀⠀⠈⠛⠿⢧⣾⣿⣄⣀⣀⣀⣀⣀⣸⣿⣷⠼⠟⣿⠀⠀⠀⠀⠀⣠⠞⢉⠀⠸⣿⠁⣠⡄⢸⣿⣧⡀
⠀⠀⢸⣷⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢀⣀⣠⣤⣴⣾⣿⡆⠀⠀⢀⡼⢿⣧⠸⣿⡶⠶⠴⢿⣇⣛⣭⡀⢷
⠀⠀⠀⢻⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠟⠛⣇⠀⢀⡾⠡⣶⣦⣴⠋⠀⠀⠀⠀⣸⡟⣉⣀⣸
⠀⠀⠀⠀⠙⢦⣈⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⠀⣼⣿⣦⠈⢻⠃⠀⠀⠀⣰⣾⣿⣶⣌⠛⢻
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⣷⠏⢩⣭⣤⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡞
⠀⠀⠀⠀⠀⠀⠀⣯⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠈⠻⢿⡇⠀⠀⠀⠀⠈⠛⠻⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡇
⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⣤⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⠿⢿⣿⡟⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣤⣤⣼⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠉⠙⢻⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⢶⣾⣿⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠐⠀⠀⠀⠀⢘⡧⠤⣤⣤⢴⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣼⣄⣀⣼⣤⡼⠃⠀⠀⠀⠀⠻⢤⣤⣤⣠⣶⠾⠃
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠉⠉⠉⠛⠿⣶⣶⠶⠶⠿⠷⠶⢶⣶⡾⠛⠋⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠠⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⠛⠛⠛⠗⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣤⣤⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠠⣤⣄⢀⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢠⡴⠖⠀⠀⠸⣦⣄⣠⡿⣧⣄⣤⡾⠃⠀⠀⠲⣦⠀⣼⠋⠁⠀⠈⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣤⠀⣴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡿⠉⠀⠀⠸⠶⢿⣶⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠈⠉⣿⠋⠁⠈⠻⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⡀⠀⠀⢿⣀⢾⡆⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⢉⣛⣛⡉⠀⠀⠀⠀⠘⠳⠦⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠙⠳⣾⣟⠋⠉⠉⠙⠛⢷⡄⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⢀⣹⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢷⣄⡀⠀⠀⠠⣤⣹⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠈⠙⠛⠶⣶⠛⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⡀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠛⠉⠉⠉⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠉⠙⠳⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⣶⠀⢠⡆⠀⢈⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣆⠀⢰⡆⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠻⠿⠿⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣶⣦⡀⠀⠀⡠⢒⣩⣴⣶⣿⣶⣦⣝⢆⠀⠀⠀⠘⠀⠐⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⡛⠁⠈⠉⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣁⠠⢦⡀⣀⣖⡛⠉⠓⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⢛⣟⣉⠠⠭⠟⠠⣵⠖⠿⣧⣦⠀⠀⢀⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⣇⠹⣄⠀⠀⠀⢾⣿⣿⣿⣿⣯⡤⠴⠋⠸⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠙⣆⠈⠑⠒⣶⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠸⡀⠀⢰⣿⣿⠟⠉⠉⠉⠉⠉⠉⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠠⠐⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠒⠂⠐⠒⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡌⠑⠢⠤⣀⣀⡠⠔⡊⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠘⣆⠀⢠⠟⣍⢣⠀⡇⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠋⠀⠀⠰⠀⠀⠈⢆⡏⡜⠀⠁⣷⡇⠀⠀⢸⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡇⠀⠀⠘⡗⠁⣰⣶⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠙⣛⢰⡇⠀⠀⠇⠀⠀⢀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⢿⢿⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸⠀⠀⢸⠀⠀⣶⣼⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⢁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣙⠁⡇⠀⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠺⠯⢁⣀⢩⡟⠀⠀⠀⠀⠀⠀⢸⠀⠘⢿⠇⣇⠀⠀⠀⠀⠀⠸⡭⢍⣉⡑⠶⠀⠀⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢆⠑⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⡴⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⣀⣀⡠⠊⠀⠀⠉⠰⠊⠀⠀⠀⠀⠈⠢⠘⡋⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠛⠳⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠇⠀⠀⢠⠀⠈⠛⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠀⠆⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⣄⠀⢀⡀⠤⠤⣀⠀⠀⠉⣀⡠⠤⣀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢁⡠⠔⠃⠐⠒⠒⠒⠀⠠⠤⠎⠀⠀⠈⠁⠈⠀⠀⠀⠈⠓⠒⠊⠁⠀⠀⠈⠀⠒⢤⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡌⢡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣆⠈⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢃
⠀⠀⠀⠀⠀⢠⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⣧
⠀⠀⠀⠀⠀⠘⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠝
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠤⢀⣀⣀⠀⠀⠀⠤⠀⠤⠄⢀⣀⡀⢠⡤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣴⠄⢀⣈⡁⠀⠀⠒⠒⠛⠓⠀⢀⣉⣁⢠⣤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣴⠀⠀⠀⠉⠀⢹⡇⠀⠀⠀⢺⠁⠉⠀⠁⠀⣘⣦⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⠃⠈⠑⢤⡀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢀⣠⠞⠉⠱⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢠⠁⠀⠀⠀⠀⡹⠂⢸⡇⠀⠀⠀⢸⠀⠰⡈⠀⠀⠀⠀⢣⡀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠠⠆⠀⠀⠀⢠⡖⠁⠀⢸⡇⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⢴⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑⣄⠀⢸⠇⠀⠀⠀⢸⠀⠀⡠⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠣⣸⠀⠀⠀⠀⢨⣀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠈⠶⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⣠⠶⠆⣩⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠙⠦⠔⢛⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢩⡄
⠀⠀⠀⢀⠴⠉⠀⠙⠶⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠴⠊⠁⠑⢄⡀
⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠙⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢤⣶⠋⠀⠀⠀⠀⠀⠀⠈⠳⢦
⠳⢄⡤⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⢀⣠⠶⠛⠛⠻⠶⣦⣤⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠸⣀⣴⠗⠀⣀⠀⠀⠐⠛⡇⠙⡄⠀⣾⠃⠀⠀⠀⠀⠀⠸⣿⡆⠀⢸⡇⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⠀⡷⠀⠀⠀⣀⠀⢰⣄⣈⠇
⠀⠀⠀⠙⢀⠰⢏⣀⠘⠻⣤⢷⠔⡟⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡇⠀⠀⡇⠀⠀⠀⠀⣀⣶⣠⡀⠀⢸⣿⡀⠰⡖⢃⡀⣹⢆⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⠂⠻⣆⣀⡀⠀⠀⢀⣴⠏⡇⠀⢸⡇⠀⠀⣇⠀⠀⠀⣴⠁⠘⢿⣷⠀⡿⠤⣶⢀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⣖⠸⠟⠛⠛⠿⢿⡶⠚⠿⠃⢤⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⠈⠢⣤⣼⣷⣼⠃⠺⠇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡿⠦⠾⠿⠿⠖⠛⣇⠀⠓⠤⠚⡇⠀⢸⡇⠀⠀⣏⠀⠀⠀⠀⢠⠾⠤⠼⠿⠦⠛⠲⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⡏⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⢀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠑⠒⠒⠂⠒⠊⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠈⠒⠂⠒⠂⠒⠊⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠶⠶⠶⠶⠖⠶⠶⠶⠶⠶⠖⠲⠶⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢤⣶⣞⣛⣋⣉⣉⡛⢻⠗⢰⡷⠙⣿⣛⣛⣛⣛⣛⣻⢷⡶⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠰⠎⠉⠛⠉⠉⠛⠛⠻⠂⣸⡇⠰⠎⠛⠉⠉⠙⢉⣉⣠⡦⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠠⢀⣀⠀⠀⠀⠀⣀⡀⠄⡹⢎⠰⢀⣀⠀⠀⠀⠀⠀⣀⠠⣞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡴⠛⠋⠉⠉⣠⠦⣌⢳⡍⠑⢤⡈⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠅⠐⠲⣤⡴⣋⡉⠸⠁⠀⠈⠿⠂⠀⠀⣷⠀⠄⠲⠶⢬⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡀⠀⠈⠉⠉⠙⠳⢶⡋⠛⠲⣦⡀⠀⠉⠁⠀⢰⠀⠀⡘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⢀⠴⢆⡀⠀⠀⠘⣦⢀⣠⠤⠀⠀⠀⡹⠐⢦⡈⢿⠉⠉⠙⢶⠏⡀⠀⢳⣠⠤⠴⡀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠂⠀⢀⡉⠁⠀⠀⣸⠋⢠⠆⠀⠀⠀⠀⣀⡘⠁⠸⡆⠀⠀⣀⣀⣘⣆⠾⠧⣄⠀⠘⢂⡬⠀⢁⠀⠈⠉⠉⠠⠈⡄⢹⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣇⠀⠈⠉⢳⠀⠀⡏⠀⢿⠀⠀⠀⢀⠤⢄⠙⢦⠀⠀⠀⠈⠉⢁⡉⠙⢦⠀⠈⢧⠀⢘⠃⠠⠁⠒⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⡄⠙⢦⡀⠀⢸⠀⠀⠇⠀⠈⠙⠀⠀⢀⡀⠠⠀⠘⡇⢀⠴⠒⠲⠄⠹⡄⠈⡧⠐⠒⣦⠻⠟⠃⠀⠀⠀⠸⡀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠐⢄⢄⡙⠒⠸⣄⠀⠈⠀⠀⢠⡤⠶⠃⢰⡀⠐⠰⢧⣤⠖⠒⢦⣤⠾⠁⠀⠛⢀⡀⣤⠆⠀⠉⠆⠀⠀⠀⠀⠹⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢄⠉⢳⡀⠈⠳⠤⠀⠀⠀⢰⠂⣀⣾⠀⠂⠄⣀⡀⡀⢀⣀⡀⠄⠰⠒⠛⢋⠐⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡄⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢃⠀⠀⠀⠒⠒⠶⢼⣧⡀⠉⠂⢄⠀⠀⠀⣧⠀⠐⢷⠆⡰⠄⠂⠁⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠐⡀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⣀⡐⠋⠁⠈⠐⢀⠑⢄⡀⠬⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⣀⣀⣀⣀⣼⡿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠋⠉⠋⠃⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠊⡠⠊⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢸⠀⠠⡍⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⡀⠀⠀⣽⣆⠀⠀⠀⠀⠀⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⠈⠉⠣⡀⠀⠀⠰⠄⢀⠃⣲⣶⡤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠀⡼⠀⠀⠡⢀⠀⠀⠀⠈⢉⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠈⠉⠀⠀⠐⠂⠁





⣶⣶⣶⣶⣶⡿⣾⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⢷⣷⢶⡶⠶⢶⣶⡶⠿⠷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⢰⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⣧⠀⠀⠀⢹⠀⠀⠀⢸⢀⠀⢸⠀⠀⠀⡏⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠘⡆⠀⢆⠈⠆⠀⡇⠈⢸⠀⡈⢠⠃⠸⠀⢀⠀⣰⠋⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡀⠳⡄⠘⢆⠘⠀⠘⡀⠀⠀⠇⠀⢸⠀⠀⡘⢠⣶⢦⡉⡰⠃⢀⠄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢦⡀⠘⢦⠘⠀⠈⢆⠀⠀⠁⠀⠀⢠⠀⠘⠀⠀⠁⣾⠃⣸⡇⠁⣠⠋⡠⠋⠀⡀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠑⢤⡙⠀⠀⠓⡀⠀⠈⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢸⣿⡴⠛⠀⠐⠁⠀⠀⠰⠊⣠⢖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠄⡀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⢸⣿⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠈⠛⠲⢤⣀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⣰⠏⠠⠒⠉⢸⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣧⠀⠀⢀⣼⠏⠀⠀⠀⡀⢸⣿⠐⢁⣀⠤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⣏⢸⣿⣧⡿⠈⠳⠶⠋⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⢀⣤⣴⠆⠀⠀⣰⣿⡶⡟⢰⣿⡇⠀⣿⠀⠙⠛⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣉⣁⣀⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣼⣧⠀⢀⣿⣿⠀⣿⠈⠻⡷⠼⠃⠀⠀⠀⠀⢹⡇⠀⣤⡀⠀⠀⠀⠀⢀⢸⡿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⢰⣿⡇⢀⣿⣿⣆⡾⢿⣿⢀⡿⠀⠀⠀⠀⠀⠀⣀⡀⠀⣸⣧⣼⣻⠇⠀⠀⠀⠰⡄⢠⡄⠀⢀⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⠃⣿⣿⡃⠈⣙⠛⠁⠀⠀⠀⠀⠀⣸⣿⢻⡄⣿⣿⣿⠟⠀⠀⠀⠀⢸⡇⠘⠷⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠙⠛⠁⠀⣼⣿⣇⠀⣿⣧⠀⢀⣀⠀⣤⠀⣿⡯⠀⢰⡿⣿⡟⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⠀⣸⠏⣿⣿⠀⢸⣿⡀⢸⣿⡀⣾⠀⢹⣿⣠⡞⠁⢿⣧⠘⣿⡄⠀⣸⠇⠀⠉⠒⠦⠄⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⠄⢀⡠⢠⡟⠀⣿⣿⠀⢸⣿⡇⠸⣿⡇⣿⣇⡾⠁⠀⠀⠀⠀⠀⠀⠈⠻⠶⠏⠠⢄⡀⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣡⠖⠉⠀⣾⢀⠄⢸⣿⠀⠸⣿⣇⣼⡿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⢄⠙⠂⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣀⠔⢰⡏⠀⠀⣼⡏⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠑⢆⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⡔⢠⡟⠁⡀⢠⠈⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⢀⠘⣆⠀⠱⣄⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⠳⠴⠋⠀⡜⢀⠏⠀⡄⢸⠀⢀⠀⡄⠀⠀⢃⠀⢀⠈⣆⠀⢳⡈⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⡼⠁⠈⠀⢰⠀⠟⠀⢸⠀⡇⠀⣆⠘⡀⠀⢆⠈⠀⠀⢳⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡏⠀⠀⠀⣾⠀⠀⠀⢹⠀⠀⠀⠸⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠃⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⣀⣀⢀⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡇⣾⣿⣧⠻⣿⣿⣿⣿⣧⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣴⣿⣿⣿⡏⣿⣿⣿⣿⣿⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣭⣿⣿⢟⣛⣸⣿⣿⣿⢛⣭⣙⠿⢿⠷⣈⣙⣩⣿⣿⣧⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢋⣿⣿⣿⣾⣿⣿⣿⣿⣿⣌⡛⣿⣿⣶⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣿⣿⣿⣿⣿⢻⣿⣿⣯⣭⣵⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣾⡿⣿⣿⣱⣶⢉⣾⣿⣿⣿⣿⡟⣩⣥⣾⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠁⣿⣏⣛⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⡀⠀⠀⠤⣤⣄⠀⠢⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠁⠘⢁⠀⠀⠀⡌⠀⣀⣀⣀⢠⠖⠄⠂⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⡶⢶⣿⣭⣀⡎⠀⠀⠀⠀⣠⣽⡿⢶⣤⣀⣬⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠛⣿⠊⠀⠘⢻⣿⣶⣶⣌⡿⣏⢀⡀⠀⠃⠹⣿⣿⣿⡀⠈⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣿⡀⠀⠙⢯⠖⠲⣷⡿⣅⠀⢹⡇⠹⢀⡷⠤⢤⣔⣿⡿⢿⠁⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡁⢀⣘⣧⡼⠀⡤⢤⣹⡾⠃⠈⠟⠉⠻⣄⣟⠀⠀⢉⣼⠟⠄⢈⠐
⠀⠀⠀⠀⠀⡠⠂⠀⠐⠀⠀⢡⣄⡀⢈⠯⠝⠗⣶⠞⠙⢣⢸⠀⠀⡞⠀⣨⠟⠓⡶⠋⠁⢠⠎⡴⠬⠂⠀⠀⠉⣀⠀⢈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣄⣀⡀⠀⠀⠀⠛⠸⣇⠀⠿⣼⠁⠀⠀⣧⠀⡀⠸⠦⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠐⣟⣶⠤⣤⣄⣀⣀⡀⠀⠐⠬⢧⡀⠘⠆⠀⠉⣁⣀⣤⠤⠴⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢤⣠⣬⣩⡉⠉⠋⢐⢒⠾⢶⡖⠛⠻⣇⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠘⠓⠀⠀⠀⠈⠀⠀⠙⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡶⠶⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠌⠠⢖⡤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⣠⠀⡀⢠⣠⡄⡀⢰⡄⣤⡄⠀⠠⠀⣶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡤⠒⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠂⠙⠈⠸⠆⠑⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⣄⠸⣵⠤⠚⡀⠀⠀⣠⠊⠑⣤⠀⡾⠁⠈⢀⠌⠀⠀⣸⠀⢠⢶⣄
⠀⠀⠘⢇⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠣⡼⠾⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢅⡀⣀⣨⠴⠃⠀⡏⣸⠁⠀⠀⠀⣠
⠀⠀⡀⠘⣆⠇⠀⠀⠀⣰⢆⣠⠊⠀⠀⠰⠛⢧⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠒⡉⠋⠉⠛⠻⣏⠉⠁⠀⠀⠀⠀⢇⠸⣀⣀⠴⠻⣮
⠀⠀⠙⢦⢹⡃⠀⠀⣶⠁⠀⠙⠷⠀⠀⠀⠀⠀⠀⣠⠝⠒⠒⠒⠠⠖⠷⠆⠾⠃⠀⠀⢘⠠⠁⠀⢅⠀⠈⡀⠈⡇⠀⠀⠀⠀⠀⠘⢦⣀⣀⠤⠚⠁
⠀⠀⠀⠀⢻⡅⠀⢀⡼⢦⣀⣠⡷⠀⠀⢠⣲⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⢇⠂⠁⠂⠌⡀⢀⡴⠃⡠⣲⠶⠄⣠⡀
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⣀⣾⠞⢀⡴⠖⠒⠒⠚⡓⢖⠚⠒⠚⠉⠀⡎⡞⠀⠀⡴⢱⢃⢞⡆⢀⣀⡀
⠀⠀⠀⢀⠀⢱⡸⠃⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠁⠀⢹⠀⠀⠁⠀⠀⠂⠈⣳⠀⠀⠀⠀⢇⣉⡽⢾⢁⡈⠉⡼⠀⢸⣿⣟
⠀⠀⠀⠉⢣⢸⡇⠀⢠⠒⠒⠒⣦⢞⡉⠂⡇⠀⡀⢀⣠⡴⠖⠲⠒⠶⠖⠤⣬⣯⣀⣀⣀⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⠛⠁⢟⡴⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢿⠀⠀⠘⣦⠀⣴⢃⣄⠉⢲⣇⠀⠁⠉⠑⠀⠀⢀⠀⠁⠀⠐⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠉⡇
⠀⠀⠀⠀⠀⢸⠁⠀⢀⠈⡻⣅⠀⠉⢠⠟⠈⢣⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣠⠴⠋⠀⠀⠀⠀⠐⡿⠷⠀⡿⢻⠀⠐⡾⡖⠀⠀⢸⢸⢀⡇⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠉⢀⡇⠀⣉⡶⠧⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⣿⣾⠀⠀⡇⡇⠀⠀⢸⠰⢨
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⡶⡄⠀⠀⢀⡔⠢⡀⠀⠀⣿⡇⠀⣿⣿⠀⠀⣧⡇⠀⠀⢸⠈⢸
⠀⠀⠀⠀⠀⠀⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠤⠃⡇⠀⠀⢸⡘⢳⢣⠀⠀⢿⠇⠀⠿⠿⠀⠀⢿⠇⠀⠀⢸⡀⢸⠀⡀
⠀⠀⠀⠀⠉⠀⡇⠄⠘⠂⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⢸⠀⡤⠦⡀⡇⢀⡀⢸⡐⠞⡜⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠘⡇⠀⠀⠀⣇⠇⠀⠱⠇⢠⡇⠀⠑⠊⠀⠀⠀⢀⡜⠁⠀⠀⢀⣠⣶⣄⠀⣠⠚⠛⢤⡀
⠀⠀⠀⠀⠀⠀⣹⡃⠀⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠲⢖⡖⠉⠀⠀⠈⠿⠁⢀⡀⠀⠙⠇
⠀⠀⠀⠀⢀⣔⣁⠳⡀⠀⠀⠀⠀⠀⠀⡮⠀⣰⠀⠀⢳⠀⠀⠀⠸⠇⢶⠀⠀⠉⢉⡉⢹⡀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⡞⠑⠸⠟⢧⠀⠀⠀⠀⠀⣸⣷⣶⣾⣷⣶⣾⣇⠀⠀⠀⠀⢽⠀⠀⠀⢸⡇⠀⣉⠉⠉⣭⠉⠉⠙⣄⣀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⣇⠀⠀⠀⡸⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢠⠠⢼⠀⠀⠀⢸⡇⠀⢸⠀⠀⣾⠀⢀⣄⠀⠀⠈⣭⡍⠉⢹⡅⠀⠀⠉
⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⡀⠐⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⢸⠀⠀⠀⢸⠁⠠⢸⠀⠀⣿⠀⢸⣾⠀⠀⠀⡏⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⢈⣿⣆⣀⣠⢿⣦⡤⠤⠄⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⢸⢰⠀⡏⠀⠀⣿⠀⢸⡏⠀⠀⠀⡇⡇⠀⢸⣧⠀⠀⢤⡀
⠀⠀⠀⢀⣤⣤⠴⠛⢻⡿⠁⠀⠾⠎⠀⠐⠗⠁⠀⠀⠀⠀⢠⣀⠠⠤⣼⠤⠤⠤⠾⠼⠼⠷⠦⠤⠿⠤⠴⠧⠧⠤⢤⡷⠷⠶⠾⠷⢼⡆⠚⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠈⠀⠃⠁⠀⠉⠉⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣽⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⢰⣶⡶⠿⢿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣆⠘⠁⠀⠀⠀⠈⠳⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣽⣏⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⡄⢀⡤⣄⠀⣄⣤⠠⣄⣠⠄⠀⣠⢤⡄⢀⣤⡀⠠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⢿⣤⠇⠸⣧⡤⠀⡟⢧⠀⢹⡏⠀⠀⢿⣤⡄⣼⠿⣧⠀⣿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠖⠐⠶⠖⠂⠀⡂⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠡⠣⣚⠁⢀⣀⡁⠀⠈⠢⠄⠄⠨⠀⢀⣈⠟⠀⠐⣶⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⢀⢾⡷⣏⠀⠀⣀⠝⠉⠉⠒⠚⠀⠑⢤⣁⠄⡀⠺⠀⠀⢁⣼⡗⠢⡀
⠀⠀⠀⠀⠀⠀⡀⢊⠝⠀⢰⡀⠀⡜⣄⡠⠊⠁⠀⠐⢁⠀⠀⢀⢠⠜⠁⡀⠨⠖⢤⣠⠞⠁⡀⠀⢈⡦⡀
⠀⠀⠀⠀⠀⣘⠁⡠⠤⠄⠀⡁⠀⠝⠏⠃⠀⡀⢀⠀⠀⠄⠀⠃⠀⠀⠈⢀⡀⣀⠴⠋⢳⠾⡀⠘⡅⢐⠈⣄
⠀⠀⠀⠀⢸⠉⠉⠀⠠⡀⠀⢀⠀⡸⠀⡠⣮⠞⢙⠉⡂⠀⠀⠀⢈⢀⡶⢚⣉⡀⠀⠀⢀⠄⠹⠆⠀⡀⣁⠈⠆
⠀⠀⠀⠀⢸⠁⠀⠑⢄⠈⠐⠂⡄⠠⠔⠀⠈⡙⠻⣝⢿⣭⢝⡾⡟⠊⢀⣀⠠⠀⠀⠀⠀⠑⠀⠄⠁⠢⣀⠤⠘
⠀⠀⠀⠀⢘⠖⠂⠀⠆⠀⠁⠰⢅⠀⠀⠊⠉⢀⣠⣀⠷⠮⠭⠵⣘⡄⠣⡀⠀⠀⠀⠤⠀⠑⠀⠠⠜⡄⠘⠂⢠⡇
⠀⠀⠀⠀⠘⡄⠀⠄⠀⠠⠀⠀⢈⣠⠘⣸⡿⠁⢙⠉⠀⢀⡀⠀⠛⣗⢄⠈⠀⠀⠢⠤⢶⠀⠂⠀⢀⡥⠤⠂⡰⠁
⠀⠀⠀⠀⠀⠙⢫⣄⣀⡔⣀⣠⠿⠳⠊⠅⠀⢀⡈⠀⠀⡼⠇⠀⠀⠀⠀⢞⡯⠒⠉⠀⡆⠀⢀⡁⠀⢠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠑⠚⠅⣀⣲⠈⠁⠀⢸⠙⠒⠒⠒⠂⠀⡚⠀⠠⢀⠉⠂⢤⠆⡀⠀⠀⠉⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣈⠉⠛⠳⠤⢉⣤⡂⠀⢀⣱⠀⠈⢁⣄⣀⣥⣘⣌⣡⣼⡦⠄⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠠⠄⠁⠙⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢚⣋⣭⣥⠤⠤⠤⢤⣭⣍⡛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣥⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣍⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠟⠀⠀⠀⠀⢀⡴⠶⠾⠋⠉⡻⠛⠉⠻⣦⣄⡀⠀⠀⠘⢧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢰⡏⠀⠀⠀⢠⡶⠞⠁⠐⢦⠀⣠⣇⣀⣀⣰⠀⠀⠹⣆⠀⠀⠈⣷⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⢿⡠⣄⣠⠶⠚⠛⢯⣀⡉⠹⣅⣰⡃⠀⠙⣷⠀⠀⠸⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⡇⠀⠀⠀⢰⡏⠀⢸⡧⢤⡀⠀⡸⠁⠉⠀⠀⠉⠉⠛⢂⡿⠀⠀⠀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇⠀⠀⠀⠀⢻⡆⣠⠷⣄⠀⠀⢀⡴⠒⠶⠶⠞⠛⠛⠋⠁⠀⠀⠀⡗⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠈⠳⣤⡀⠈⣧⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣧⠀⠀⠀⠀⠀⠈⢷⡀⢸⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢧⡀⠀⠀⠀⠀⠘⠳⠾⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣣⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣙⠳⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢬⣭⣙⣛⣛⣛⣛⣩⣥⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠓⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠑⠦⣀⡀⣸⡇⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣰⠋⠀⠈⢻⡆⠀⠀⠀⠉⣿⠓⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⡠⠃⢀⠔⠒⢦⡀⠀⣿⠀⠀⠀⠀⣷⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠉⠀⠈⠁⠀⠀⢺⣴⠀⠘⡇⡠⢻⣿⡀⠀⠀⡟⠀⣼⠃⢀⣿⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣴⡆⠀⠀⠀⠀⣸⣿⣒⡺⠋⠀⢸⣏⠙⠓⣾⠓⠋⢿⣧⠞⠙⣿⣄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⠔⠁⢉⣉⡀⠀⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⢀⠀⠈⠛⠷⠦⠤⠔⠁
⠀⠀⠀⢀⠴⠒⠛⠛⠛⠛⠛⠛⠓⢻⡟⠋⠉⠀⢠⡞⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡏⢀⣤⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⢀⠏⠀⠀⠀⠀⣰⠏⠉⢹⠀⣾⡗⠁⣾⠁⠀⠀⠀⠀⣀⣀⡀
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠑⣾⡧⠤⠄⠒⣡⠀⠀⣾⠀⠀⣿⠀⠀⠀⢠⣿⣷⣞⣁⡀⠀⢀⠔⠋⠁⠈⢻⡆
⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⢰⣿⠀⢰⢿⣦⣤⠴⠃⠸⠇⠀⠈⢙⠿⣧⣤⣀⣀⣠⠞⠁
⠀⠀⠀⠙⠷⣶⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⡸⣿⣤⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠔⠒⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢘⠈⡰⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠂⠊⢐⠟⠉⠁⠢⡀⣀⡀⠀⠈⠂⠄⠺⣄
⠀⠀⠀⠀⠀⠀⢠⡎⠉⢳⣴⠋⠉⣦⠀⣼⣷⣿⣶⣤⠇⠛⠃⠐⠒⠂⠀⠀⠙⢷⠄⣶⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠰⠗⠶⠋⠈⠓⠖⠧⠀⢻⣿⣿⣿⡇⠀⠀⠀⠈⠒⢦⡀⠰⠆⠆⠶⠠⠈⢀⣀⠴⠷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣖⠒⡸⠀⠀⠀⠀⠀⠈⠿⠿⠷⠼⠤⠛⠠⠼⠀⠙⣦⠀⠀⠀⠀⠀⠈⠁⢀⣠⡬⠭⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡄⠀⠰⠆⠀⠘⣿⡆⠀⠀⠀⠀⢀⣤⣞⠀⠈⣧⢀⣤⠀⢀⡴⠞⣩⡅
⠀⠀⠀⠀⠀⣀⡀⠘⢎⣉⡀⠁⢾⣿⣿⣿⡗⢿⣿⠀⠀⠀⠀⠺⣿⡿⠀⠀⢹⡄⠉⣴⠋⠀⠀⠻⠃⠴⣒⠁⠀⡀
⠀⠀⠀⠀⠀⠉⠀⢰⠃⠤⠹⡄⠰⣿⣿⣿⣿⠻⣿⠇⠘⠟⠀⠀⠰⠀⠂⠀⢸⡇⣼⡇⢰⣶⡆⠀⠀⠀⠋⠁⠈⠛
⠀⢀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠒⠲⢤⣀⠈⢀⠀⡀⠁⣾⢣⣿⠀⢸⣿⡇⢀⡛⠀⠀⠀⠀⠀⠘
⠀⠾⠶⣤⠈⢁⠐⠀⠀⠊⡠⠀⠀⣷⠀⠰⠀⠀⠀⠀⠀⠀⠈⢷⡌⠀⠀⢰⡿⣸⡇⠀⠀⣬⣶⣼⣇⠀⡦⠀⠠⡆
⠀⠀⡼⠁⠐⠒⠀⠀⠀⠀⠒⠀⠀⣿⢰⣿⣿⣇⠀⠀⠐⠀⠀⠀⣿⡆⣠⣿⢣⣿⢁⡄⠀⠛⠛⠋⢛⣀⣤⣤⣤⣤⡀
⠀⠀⣀⡀⠀⠁⠀⣀⣀⠀⠨⢦⠀⣿⠀⣉⣩⣥⣤⣀⠀⠀⠀⠀⣼⣿⣿⣷⣿⠏⠺⠛⣫⢁⠀⠘⠛⠛⠛⠛⢋⠁⠂
⠀⠾⠋⠀⢶⣶⠀⠙⠟⠀⠀⠋⠀⣿⠈⠙⣿⣿⣿⡟⠛⠀⠀⢠⣿⣿⣿⡿⠃⠀⢲⣶⡁⠷⣤⠀⠀⠀⠈
⠀⠠⣶⣤⠈⠏⠀⠀⠀⣸⣧⡴⡀⢻⣆⠀⠀⠉⣩⣴⠀⠀⢠⣿⣿⠟⣁⡤⠀⠀⠈⢉⠁⠐⠁⠀⠄⢐⠀⠀⠀⡌
⠀⠀⠀⠀⠀⢀⣄⣀⠀⣀⣠⠏⠀⠀⠛⢿⣶⣤⣙⠋⠀⣰⣿⠟⢁⠀⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⢂⠉⠐
⠀⠀⠀⠀⠀⢰⣿⣿⠀⠸⠷⠄⠀⠀⠀⠀⠉⠻⣿⣿⣾⣿⡏⡰⠀⡐⠁⡤⠁⢰⡄⠀⠀⠆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⢠⣼⣍⠀⠀⢡⠀⠀⣸⣿⣿⣿⣷⡀⠀⣿⠀⠀⠀⣼⠇⡤⠘⠛⢠⡄⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠈⣠⣾⣿⣿⣿⣿⣿⠿⠷⠦⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣼⣿⣿⣿⣿⣿⡟⢁⣠⠤⠖⠒⠚⠛⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢀⡼⠁⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⢀⣾⠴⠒⠋⠉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠓⠲⢤⣸⡇⢀⣠⠤⠖⢒⣛⣋⣉⣉⣉⣉⣉⣛⣒⣲⣦⣼⣆
⠀⠀⢀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠲⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡄
⠀⣠⣺⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣶⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣷⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣿⣷⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣠⣤⣤⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⣡⣤⣤⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠑⢄⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⠏⠐⠈⠉⠁⠀⠀⠉⠉⠐⠈⢿⣿⣿⣿⠇⡘⠀⠀⠀⠠⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡫⠋⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠌⠫⡛⢀⠃⠀⠀⡆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠑⡋⣿⢿⡅⠄⠀⠀⠀⠀⡊⡿⣿⡍⡗⠁⠐⡆⠀⠀⡠⠃⢄⠆⠄⠄⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠈⠉⠈⢀⣤⣤⣤⣄⠈⠈⠉⠈⠀⠀⠀⢠⠀⡐⠀⠀⠀⠉⠘⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢁⣀⠈⠥⢰⠁⠙⢿⠟⠁⢳⠨⠍⢁⡈⠉⠀⠸⠠⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢁⠠⠒⠉⠹⠀⢀⠜⢄⠀⡸⠉⠐⠢⡀⠉⢀⠃⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⠀⠀⠙⠢⠤⠤⠚⠀⠀⠀⠀⠀⣰⡇⠘⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡟⡷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠃⠃⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠑⠻⠾⠭⠿⢿⣿⣿⠿⠿⠿⠟⠛⠉⠢⡀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢄⠀⠀⠀⠀⠀⠈⢦⣀⣠⣮⣐⢄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠂⠀⠑⠄⣠⡀⣀⣠⣿⣿⣿⣿⣿⣷⡑⡀⠀⠀⠠⠂⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⢂⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⠊⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⡠⠄⠂⠀⠐⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠠⡀⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣨⠻⣿⣿⣿⣿⣿⣿⣿⣿⢫⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡂⣂⣂⡣⠗⠓⠱⢍⠻⠿⣿⡿⠿⢛⠵⠁⠒⠦⣆⡆⡄⣄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠈⠃⠀⠈⠁⠂⢼⡀⠀⠀⠀⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⠰⣀⢃⢩⡘⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⢂⡞⠁⠔⠘⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢲⡀⠀⠀⠰⠌⠀⢩⢨⡅⠅⠀⠀⢀⡅⠀⠀⠀⢀⣄⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠓⣠⠤⠶⠮⣍⠒⠁⢹⡦⠂⠐⠎⠷⠀⢒⣰⣏⣰⡀⠀⠀⠸⣸⣀⣰⡷⠃⠀⠀⡔⢲⠀⠠⣄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⣄⠐⠀⡀⠄⢛⠨⢍⡔⣀⣤⣀⠉⠉⡉⠙⠛⠀⠀⠀⢈⠉⢹⠀⠀⠀⠈⠁⠈⠉⠐⠇⢀⡶
⠀⠀⠀⠀⠀⠀⡇⣾⠁⣠⠖⠄⠶⠀⣰⠋⠀⣸⣴⠁⠀⠈⠆⠉⣉⣑⣴⠇⠀⠀⠸⠀⠀⠀⠀⠀⠏⠁⠘⠣⣤⡀⠙⢂⡲
⠀⠀⠀⠀⠀⠀⠻⠘⠉⣠⠴⠒⠂⢀⣯⠀⠀⠀⣩⡤⠖⢂⡴⠊⠉⠀⠀⠐⣀⠀⠀⠀⠀⠀⢀⡔⠀⣤⠂⠀⠘⠷⠀⠻⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣄⠀⠀⣀⣠⣀⣾⠃⣠⠰⡏⠀⠀⠀⠠⢄⣀⡄⠠⠄⠀⠀⠤⠀⠏⠀⠉⠁⠀⣰⡟⠀⢾⡀⠁
⠀⠀⠀⠀⠀⠈⠁⠀⢰⠁⠀⠙⢶⠊⠁⠀⢠⡇⠸⠁⠀⠑⡀⠀⠀⠐⢄⠈⢻⡌⣭⣭⣭⡍⠘⠀⢸⡆⠰⠎⠁⠀⢴⡂⠚
⠀⠀⠀⠀⠀⠊⠉⠙⣸⡄⣰⠀⣀⡀⠀⠀⠘⡀⠀⡠⠚⢁⡀⠀⠀⠀⢘⡇⠀⢧⠈⠛⡏⠀⠀⢀⡟⠂⣠⠤⠘⣏⠀⠛
⠀⠀⠀⠀⠀⠀⢠⠏⠁⢠⠟⠉⠉⠉⠁⠀⠀⠁⣸⠁⢰⡁⠀⠀⠀⠠⠋⠀⠀⠘⠀⠀⢻⠀⡠⠋⠀⡀⠛⠀⠀⠁
⠀⠀⠀⠀⢀⠀⢋⡀⢠⡏⠀⡴⠁⠴⠒⠒⠢⠀⢻⠀⠈⢧⠀⠲⢤⣀⡀⠀⠀⠀⠀⢀⠏⠈⣠⠞⠁⠃
⠀⠀⠀⠀⠀⢺⠉⠀⠘⡇⠀⠳⣄⢀⠴⠒⠒⠶⢼⣆⠀⠈⢧⠀⠀⢠⠀⠀⠀⠠⠖⠉⠀⢰⠁
⠀⠀⠀⠀⠀⠘⡄⠠⣄⣑⣄⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣸⣀⡴⠖⠒⠂⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⡀⠀⣤⡆⠀⠀⠁⢸⡏⠈⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠈⠦⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠦⢀⢸⣇⢂⠀⠀⠀⠀⠀⠒⠂⠀⠀⠒⠒⠂⠀⠀⠀⢀⡦⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣸⣿⣿⣿⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣦⣤⣴⣶⣼⣿⣷⣄⣠⣾⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣯⣽⣿⣿⣭⣿⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣿⣿⣿⣯⣿⣿⣯⣿⣿⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣾⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣻⣿⣟⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⣿⣿⡏⠉⠻⣯⣅⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠛⠉⢹⣿⣿⣿⣿⠛⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⠿⠿⠿⠷⠶⠶⠖





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣼⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣽⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡿⣿⣿⡿⣿⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣿⣿⣿⣿⡏⠉⠉⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⣽⣿⣿⣟⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢤⡀⣀⣀⠀⠀⣠⣔⠒⠉⠁⠀⠨⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⢠⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠋⡀⢢⣀⣀⣀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⢸⠗⢤⣀⡜⢸⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢇⠀⠉⠐⠊⠓⠒⠠⠤⠤⠴⢒⠒⠲⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠤⠀⠀⠀⠀⣀⠬⠤⠶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣠⡄⠀⢰⡎⠘⡇⢀⣀⠀⠀⠀⣇⣤⡶⠛⠉⠉⠉⡿⢻⡅⠀⠤⠀⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠃⠘⠁⠑⢼⢰⡉⠳⠼⠁⠀⠻⡀⠀⠃⣠⠇⠀⢙⠦⠤⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠥⡀⠀⠀⠀⠀⠉⠙⠉⠁⠉⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠤⣤⣄⢀⣀⣀⢤⠤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠐⠢⢧⡐⣰⣇⣀⠇⡅⠀⠠⡒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠙⠳⡃⠀⣿⣉⣆⡴⢓⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡄⠀⠀⠀⠀⠘⠋⠁⠀⢸⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠓⣤⣀⠀⠀⠀⠀⠀⠸⣤⡴⠚⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢠⠈⠉⠒⠒⠒⠒⣿⠁⠀⠀⠀⢱⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡟⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠘⠁⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⢳⠤⠤⠔⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⢮⡀⠀⠀⠀⠀⢀⡇⢀⢀⣀⣠⠤⠒⠋⠀⠀⠨⡍⠁⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⢟⠁⠉⢉⢉⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢡⠀⠀⠀⢀⠔⢡⠀⠀⠓⠤⢤⡞⠉⠒⠒⣹⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⢰⠁⠀⠀⠈⡖⠶⠶⢶⡉⠐⢆⡉⢀⡟⠢⡀⠀⡠⠊⠀⠀⠈⠒⡄⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠱⢤⡠⠜⢡⠀⠀⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠣⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⢀⣠⠜⠁⠀⢆⠀⠀⠀⠀⠀⡜⠀⡄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠓⠒⠒⠒⠂⠁⠀⠀⠓⢤⣀⣀⣀⣀⡤⠊⠀⠉⠉⠀⠀⠀⠀⠀⠀⠑⠠⠦⠐⠋⠀⠀⠁

⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣷⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠄⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⣾⣿⣿⣿⡟⢹⣿⡿⠃⢉⣻⢿⣅⠈⢻⣿⣿⣿⣿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠋⣀⡚⠉⠀⠀⠘⢿⡆⢸⠇⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣼⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡓⠀⠀⠀⠀⢀⣰⣤⡀⠀⢀⣠⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢿⣷⣄⡀⢰⣾⣿⣿⣿⡇⠀⠈⠉⢉⣽⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣼⣿⣯⡀⠙⠛⠛⠋⠀⠀⠀⢠⣿⣿⣿⡐⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣷⣶⣆⠀⠀⠀⠈⢧⣬⣙⡡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⢶⣤⣄⡾⠛⠿⡛⠛⠻⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⢻⣿⣷⣦⣄⣠⣶⣶⣾⣿⣴⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠛⡟⠻⠻⠿⠟⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⣼⣿⡿⠛⠀⠀⠀⢀⠂⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣦⣄⣠⣼⠿⠋⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠉⠉⠉⠉⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⣀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣽⡱⣯⡕⢢⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠙⠛⢿⣿⣿⣿⡠⠀⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠹⣿⣿⣿⣄⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠓⠀⠀⠀⠙⣿⣿⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠘⣧⠀⠀⢸⡇⠀⠀⣼⠃⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⢻⡇⠀⢻⡇⠀⣸⡟⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠻⣧⠀⠀⠛⠀⠘⠁⠀⠋⠀⠀⣼⠏⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢷⣦⠈⠃⣠⡾⠟⠛⠛⠻⢦⡀⠈⢁⣴⠾⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠈⠙⠳⣶⣤⡀⠁⣼⠋⠀⠀⠀⠀⠀⠀⠹⡄⠈⢀⣤⡶⠞⠋⠁⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠙⠻⢷⣤⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣴⠿⠛⠉⢿
⣾⠛⠒⠶⠶⠶⠤⣤⣏⣀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡴⠟⠉⠀⠀⠀⢀⣈⣧⡤⠴⠶⠶⠒⠒⢻
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀⠀⠈⠻⣆⠀⠙⠳⠤⣤⡤⠴⠊⠁⢀⡾⠋⠀⠀⣀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠈⠳⡄⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⠷⠶⠾⠿⠿⠿⠿⠿⠶⠶⠶⣦⣤⣄⣀⠀⠈⠒⢄⠈⢆⠀⠀⠀⠀⠎⢀⠔⠋⠀⢀⣀⣤⣤⡴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤⡀⠁⠀⠀⠀⠀⠀⠀⠁⡠⠴⠚⠉⠁
⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠀⠀⠀⠀⢀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠂⠙⠽⠹⠍⠀⠐⠁⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠠⠶⠄⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⡀⢀⣀⡀⠀⡀⠀⢠⠄⣏⣀⣋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣡⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢹⠿⠲⢶⣲⠶⢶⣖⣒⣶⠾⠟⠋⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⡄⠀⠘⢯⡫⠛⢶⡄⠦⣳⠅⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢽⠤⣀⣽⢃⡔⠉⠙⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⠊⠁⣿⠉⠁⢊⡇⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠓⠢⠭⠤⠤⠄⠐⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⢄⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠒⠀⠘⠇⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢰⣿⣷⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⣿⣿⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⠀⢸⣿⣿⣿⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⣤⠀⠀⠀⠈⠛⠛⠃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢁⣤⣾⣿⣿⣶⠂⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣤⣫⣜⠿⣿⣿⣿⣿⣿⣦⣠⡔⠁⣀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⡇⡌⠉⢙⣿⣷⣟⡛⢿⣿⣿⣿⠏⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡇⠀⠀⠀⠀⣠⡿⠉⢤⡝⡛⢛⡏⢿⡻⣿⡿⣿⡿⠋⠀⠰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⠀⠀⢰⡟⢤⣰⣮⣿⣦⣀⠷⠿⢟⠉⣸⡟⠁⠀⠀⠀⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⢹⣾⣿⣏⣿⢿⣿⣿⡶⢛⣶⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠿⣮⣿⠷⣦⣿⠇⣠⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠛⠿⣿⣿⣯⡽⠟⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠶⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡔⠈⢸⡴⠋⠑⡄⠒⠲⣄⡴⠒⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⢸⠀⠀⠈⣡⠤⠄⡠⠚⠉⠉⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⢄⠀⢰⢳
⠀⠀⠀⠀⠀⠀⠜⠁⠀⣄⠀⣀⣤⣤⡞⢁⠤⣎⠀⠀⠀⠤⢄⣼⠀⠀⠐⠚⢀⠀⠀⠘⠼⡤⠼⠃
⠀⠀⠀⠀⢈⢤⡀⡇⢀⠘⠞⠁⠀⢰⠁⠛⢀⡈⠀⠀⠀⠰⡄⠙⡄⠀⣴⠊⠁⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⠴⠓⣼⠤⠤⠄⠀⠈⠀⡰⠃⡄⠀⠀⠀⠠⠇⠀⠇⠀⢰⠀⢀⡜⠃
⠀⠀⠀⠀⠀⠏⠀⣼⠁⡰⢀⡠⠤⢄⠀⡇⠀⢧⠀⢄⡀⠀⠀⠀⠀⠀⣸⠐⢉⠔⢰⠆⠀⠀⠀⢐⣀⣡
⠀⠀⠀⠀⢶⠉⠀⣇⠀⠧⣀⣠⠔⠲⠤⢵⡀⠈⢧⠀⠈⡉⠀⠀⠠⠚⠁⠠⠇⠠⠀⢻⠿⠿⠟⢛⡛⠉
⠀⠀⠀⠀⠘⠀⠦⣌⠤⡀⠀⠃⠀⠀⠀⠀⠀⠀⢈⠀⢀⣇⠴⠒⠠⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣭⠅⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠞⠀⠠⠃⠀⠀⠀⠁⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣀⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠠⠀⠀⠐⠒⠀⠠⠀⠤⠀⣀⠠⠀⢈⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠗⠒⠒⠲⠦⠶⠶⠶⠶⠚⠉




⡟⠉⣻⣛⣛⣛⣛⣟⣛⣛⣻⢛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣛⣿⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⠄⡀⣄⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠒⠉⣂⡬⣁⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡴⢊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠀⠀⡴⡫⠊⢁⠄⠠⢄⠀⠀⢀⠔⠉⠲⠀⠀⠀⠀⠀⠀⣎⡉⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡜⡁⢠⡞⡔⠀⢀⡤⠀⠀⠀⢣⠀⡎⠀⠀⠐⠥⢀⠀⠀⠀⠀⠰⠀⢭⠤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡾⡸⢠⡓⡜⠀⠐⠀⠀⠈⠢⣀⠀⣣⠅⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⡐⠀⡼⠳⠁⡞⠜⠀⠀⠐⢄⡀⠀⡀⠘⢣⣽⣷⢃⣀⣢⠔⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠁⢰⡷⠃⢰⢍⠂⠀⠀⠀⠀⠀⡉⢭⠓⢿⢿⡿⢟⡉⡉⠉⠑⢆⠀⠀⠀⣘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⣼⣠⠀⢸⠪⠀⠀⠀⠀⢠⠊⠀⠈⠶⠉⢸⡗⠆⠙⠑⠀⢀⡼⠀⠀⣰⠃⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠆⠀⢯⣼⠀⢰⢽⠀⠀⠀⠀⠈⠢⠤⠄⠀⠀⡘⢃⠀⠀⠀⠀⡅⠀⠀⣰⠓⢀⠿⡔⠀⡜⠁
⠀⠀⠀⠀⠀⠰⡆⠀⢸⠙⡆⠈⣟⡆⠀⠀⠀⠀⠀⠀⠘⠂⠒⠁⠈⢳⣤⠔⠊⠀⢀⡜⠃⣠⠟⡩⠀⣼⡙
⠀⠀⠀⠀⠀⠀⣧⠀⠘⣷⠚⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡑⢄⡠⠋⢀⣴⢋⠟⢀⡜⡬⠁
⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠈⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⠀⣴⣿⡷⠃⢠⣾⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠗⠾⡀⠐⢤⡀⠀⠀⠀⠈⠲⠄⡀⠀⣀⣀⡠⠔⠀⢁⣠⣶⠚⡲⠋⢀⡴⣿⣾⣷⡶⠶⣲
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠾⠛⣉⠕⠋⠀⡠⡛⡥⠋⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠒⢄⡈⢙⠲⠦⣀⡀⠀⠀⠀⠀⠀⠓⠉⠀⣀⠰⢫⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣠⠈⠑⠢⠤⢀⣨⣇⡄⠀⢀⣀⡤⠔⣋⣥⣖⠉⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠓⠮⢦⣠⣠⡻⠲⠗⣚⣙⡡⠜⠚⠁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠂⠡⠀⠨⠐⡠⢀⠄⡉⢍⠡⡀⠄⡀⢔⠠⣐⠄⡂⠄⡂⠅⡂⢅⡪⠑⠌⠂⠅⠪⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⡟
⠀⠀⠛⣲⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠓⠞⢻⡏⠉⠉⠉⠉⠉⠉⠉⠉⡻⠗⠛⠚⠳⠒⠛⠛⠛⠚⠒⠚⠒⠒⠒⡋⢹⠛
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠒⠂⠀⠐⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘
⢠⠶⣸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡆⠀⢈⡄⠀⠀⠀⢠⠚⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠱⣖⢩⡇⠀⠀⠀⢸⠒⢤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡁⢸⡆⠀⠠⡏⠑⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢼⢃⡀⣴⡀⣤⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢛⢨⠀⠀⠀⠉⠀⢹⠀⣞⣷⣔⢷⣶⠄⢀⣤⣤⢀⣤⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠜⠀⠀⠀⠀⠀⠀⢸⡀⠋⠁⠈⠀⢉⡆⢨⡿⠛⠚⠓⢿⠀⠈⣿⠿⣼⣽⣿⠀⣴⣶⣖⣶⣦⡄⠀⢀⣤⣤⣤⣄⡀⠀⢸
⠀⡿⠛⠒⠀⠠⠤⠤⠼⠦⡄⣀⣀⣀⣨⡇⢸⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠈⠀⡼⠋⠉⠉⠉⡇⠀⢸⠿⠛⠚⠻⠏⠀⢸
⠀⢥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠛⠛⠒⠚⠂⠾⠦⠤⠀⢀⣠⣀⣥⣀⠀⠀⠀⣟⠀⢈⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠒⠦⠶⠦⠀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠀⠒⠶⠶⠾⠄⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠉⡘⠀⠀⠀⠀⠉⠉⣛⠉⠑⣶⠒⠲⠖⠒⠠⣤⣤⣤⡤⡀⠀⢀⣀⣀⠀⣸⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⠸⢠⡄⠀⠀⠀⠈⠉⠉⠁⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⢡⠌⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢿⡀⠀⠀⡇⠀⠀⠏⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢠⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠸⠂⠀⠸⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢲⠀⠀⠸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡜⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠀⠀⣠⢁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⠀⠀⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⠆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠇⠀⠀⠟⢰
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⡘⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢰⠄
⠀⠀⠀⠀⠀⢠⠁⠀⠀⣠⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇⢰
⠀⠀⠀⠀⠀⠆⠀⠀⢰⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠀⡆
⠀⠀⠀⠀⠚⠒⠒⠒⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢶⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⡆⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⢸⣰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⠲⠤⠏⠈⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⡶⠶⠶⠶⠶⢶⡌⢳⠆⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⡇⠀⠀⢀⡤⠼⠃⠿⣄⡴⢉⡇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠈⠓⢲⡄⠀⡴⢦⠘⠧⣴⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡷⠶⡄⢠⠶⢾⡷⠶⠃⠘⠶⢶⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠳⠞⠀⠘⠧⢤⡀⠀⠀⢸⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡖⠖⠃⠀⠀⢸⡇⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⡄⠀⠀⠀⠸⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠼⠇⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⣼⠋⠀⠀⠐⠁⠜⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣙⠒⠀⢉⡉⢷⣶⣿⡔⣠⢂⡄⠀⠀⠈⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⠀⠀⣁⡒⢬⣓⢬⣻⣳⢡⢎⡴⠁⠀⠈⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⡁⠐⠢⢭⣓⠮⣕⢾⡇⠃⠊⠰⢁⠔⢈⡼⢁⡤⡶⠛⠟⢻⠓⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆⠤⢉⡓⠪⡝⠢⠅⠸⠀⠀⠀⠀⢰⡿⣡⡟⢠⠁⠀⠀⠀⠀⠁⠜⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠳⡲⢦⣙⠷⣶⡬⢍⠂⠀⠀⢰⣇⡞⡀⣰⠏⡴⠛⠀⠂⡰⢡⠎⡠⢀⠄⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢇⠀⠀⠀⢀⠑⢄⠙⠷⢦⡉⠳⢦⡀⠠⣸⣇⣜⡼⢡⢞⠀⡀⠀⠀⠠⢋⢞⡴⣣⢆⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠑⠀⠐⢄⠑⢤⡑⠄⠀⠀⠀⠉⡳⢤⣉⠳⣌⣧⣞⡵⠻⣋⣸⣥⠧⠔⠶⣕⠯⣘⢷⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠢⣍⡲⢍⡂⢀⡐⢌⣱⣬⣱⣭⣷⣾⣿⣿⣞⠻⢝⠋⠁⠀⠐⢬⡲⣝⠮⡑⢌⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣦⣽⣶⢯⢖⡝⡁⡀⠈⠈⠉⠝⣡⠞⢩⠿⡜⢮⡙⠲⣄⠀⡐⢌⠢⡙⢮⡳⣌⠢⠁⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣱⢫⢎⠞⡴⣡⢠⢄⢀⡼⠃⣰⠏⡼⣿⠒⢝⠢⣌⠛⢮⣦⠓⠈⠂⠑⠈⠂⠐⣌⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⢡⢋⡞⡼⢱⢃⠏⣾⡾⢁⡼⡁⠀⠀⠞⡏⠲⠀⠈⠛⣦⣙⠳⣬⣦⣱⣌⣦⣱⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠂⠘⠰⠁⠁⡈⣰⢟⣴⠿⢊⠔⡀⢀⠀⢸⠁⠀⠀⠠⢄⣉⠳⣄⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣰⢠⠁⡆⣸⣰⠶⢫⡞⡁⠐⡡⢚⡵⣣⢢⢞⢤⡓⢬⡑⠀⠀⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⡟⠉⠀⠀⢸⣉⠄⢈⠴⣫⠞⡵⣣⢟⡳⢍⠢⢌⠒⠀⢀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⢸⡥⠂⠠⠚⡡⢊⠜⣱⣿⣯⠢⠑⠀⠀⠀⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⠁⠀⠀⠀⠀⠈⢿⣡⠔⣀⠔⣁⣠⠟⠉⠻⣦⣄⠀⣄⣸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡏⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠚⠋⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠠⠤⠀⢀⣤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠸⠶⢤⡄
⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢛⣧⣖⡯⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⡅⠀⠀⠀⣻⡇
⠀⠀⠘⠇⠠⠀⡀⠀⠀⠀⢠⣠⣴⡶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠈⠉⠲⡆⠀⠀⠀⠀⠀⠀⠀⠸⣶⣤⣴⣿⠇
⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⣷⠀⠒⠲⠎⢹⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉
⠀⠀⠀⢈⠒⠶⣤⣤⣤⣀⣀⣀⣠⣴⣾⠿⣛⣫⡽⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠈⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣈⠓⠦⠀⠉⣑⢲⣚⠩⠕⢚⣩⠵⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠉⠢⢄⣐⣄⡭⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠘⠗⣱⡄⠀⠀⠀⠀⠨⣥⣟⠀⠠
⠀⠀⠀⠀⢔⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢨⣔⣀⣀⣠⣤⡴⠞⠛⠉⠀⠀⣀⠄⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡆⠸⣷⢰⡆⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠃⡳⢚⣷⣌⠐⣄⣀⠄⠊⠀⠀⣀
⠀⠀⠀⣿⣠⣿⣾⠀⢄⡄⠀⠀⠀⠄⠀⠀⢀⣾⡄⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⣠⢴⣾⠋⠀⠀⣠⠀⠀⠀⠂⠀⣤⣠⣤⣠⠤⢀⠀⠢
⠀⠀⠀⠸⡏⢻⣿⡗⣼⠀⠀⠀⠀⠻⢇⣀⢸⡆⣉⠛⠶⣶⣤⡄⣀⡀⣀⡠⠔⣚⡭⠊⠀⠐⣪⢽⠁⠀⠀⠒⠡⠤⠊⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⣿⠀⠀⠈⠓⠀⠀⠀⠀⠀⠢⣽⣼⣇⣄⠀⠰⣇⡈⠻⠷⣤⡐⠈⡁⠦⣬⡥⠔⠋⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠳⠶⠶⠦⠿⠷⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣦⡀
⠀⠀⠀⠀⢸⡄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠈⠛⢤
⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⡀⠀⠀⠀⠀⠉⣉⣉⣀⣀⢀⣀⣀⢀⢈⡉⡉⠉⠉⠁⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⣣⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢳
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠀⡠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⡆⠀⠀⠀⠀⠀⠀⢀⠁⠈⣿⠋⠸
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠⣔⡉⢸⣵⢄⠀⠀⢠⣈⣠⠠⠄⡀⠑⠄⠀⢰⣖⡀⠀⠀⢀⣧⠀⠠⣳⣄⠀⠀⠀⠀⠀⠇⢀⣼⠇⢠⠓
⠀⠀⠀⠈⠂⠀⠀⠀⠰⠊⢤⣀⣷⢟⠁⢀⠰⠀⠞⠁⠀⠀⠀⠀⠐⠋⠀⠈⢁⠠⠀⡠⠀⢀⠀⠀⠈⣻⡳⣦⣠⣤⢶⠶⡞⢁⢀⠐⠁
⠀⠀⠀⠀⠀⠁⢼⡁⠐⣤⡾⠋⠄⠠⠂⠀⠀⠀⠀⠀⠠⣄⠀⠀⣀⠀⠀⠀⢭⠩⡉⡏⡍⢽⠀⠀⠀⠉⠛⠿⢿⣽⡷⠦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢾⡋⢰⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⡿⠃⠀⠀⡺⣐⢓⠅⠂⣢⠀⠀⠀⠀⠀⠀⠐⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠊⠉⠀⠀⠀⠀⠨⠠⠠⠥⠀⠜⠀⠀⠀⠈⠙⠀⠄⠤⠠⠛⠋

⠀⠀⠀⠀⠀⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣆
⠀⠀⠀⠀⣼⣿⣿⡿⢿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢤⣶⣾⣿⠿⣿⣿⣿⡄
⠀⠀⠀⠰⣿⣿⡗⡀⢨⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢾⡁⠛⢙⠆⠀⠀⠀⠘⣿⣿⣯⠀⡀⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣁⣀⣷⡖⠋⣉⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢠⡋⡀⠘⣿⢤⡬⠂⢀⣠⡄⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⣠⣶⡶⢶⣼⡟⠀⣀⣀⡀⠀⠀⠀⠉⠑⠚⢹⡄⠀⠀⣼⣿⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠀⣼⣿⠏⠀⢀⣿⢃⣾⣿⣏⣻⣄⣠⣾⡿⢷⡀⣸⣇⣤⣤⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢠⣿⣿⡏⠙⣿⣿⡇⠀⣿⣿⣤⣴⣿⠇⣾⣿⠃⠀⢹⣿⣿⡏⠙⢺⡟⣿⣿⠏⢹⣿⠀⠀⠀⢠⣿⣿⣿⠉⢿⣿⣷
⠀⠀⣸⣿⣿⣀⣸⣿⣿⠁⠀⢈⣉⣭⣴⡿⠚⣿⣿⠀⣠⡟⣿⣿⠀⠀⣾⢹⣿⡏⠀⣾⡏⠀⠀⣠⠟⢹⣿⣿⣄⣸⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣾⡿⠋⢁⡿⠁⣀⣙⡛⠛⠋⠀⠻⣿⣦⠞⠁⠘⢿⣷⡴⠛⢷⣤⠾⠋⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⡾⢁⣾⠛⣿⣿⢀⣤⡆⢀⣤⠀⠀⣠⣤⣄⠀⢠⣷⠀⠀⣰⡿⠃⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⠉⢻⣿⣿⠛⠿⠟⠋⠀⣾⠃⢠⣿⠏⢸⣿⠃⢸⣿⠀⣼⡟⢰⣿⠀⣼⡏⢀⡴⠏⠀⠀⢀⣿⣿⡏⠹⣿⣿⣿
⠀⠀⣿⣿⣿⣀⣼⣿⣿⠀⠀⠀⠀⢰⣏⣴⡿⠋⠀⣿⡿⠀⣸⡟⢸⣿⠁⠀⠁⣠⣿⠿⣯⠀⠀⠀⠀⢸⣿⣿⣇⣰⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣾⡿⠋⠀⠀⢠⣿⣧⣤⣿⣇⣼⣿⠀⢀⣴⣿⡏⠀⣿⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣄⣠⣴⠟⠛⠛⠋⠈⠛⠻⣿⣿⠟⠁⠿⠃⠀⢹⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣿⣿⣿⡏⠹⣿⣿⣷⡀⠀⠀⠙⠛⠉⠑⠶⢤⠀⠀⢀⣶⡄⠀⠠⠶⠖⠂⠀⠀⠀⠉⠁⣠⣿⣿⣿⠋⢹⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣧⣀⣿⣿⣿⣿⣄⠀⠀⠠⠤⢤⠤⠄⠀⠻⣏⠁⢉⡿⠃⠠⠶⠶⠶⠆⠀⢀⣴⣿⣿⣿⣇⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣠⠴⠂⠀⡷⠖⠾⡇⠐⠲⠦⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢈⡉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠘⠀⢢⣀⣀⡯⠌⡡⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢇⢤⡷⠞⠋⠉⠉⠙⢾⠖⠶⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣻⠏⣠⠤⡄⠀⠀⠀⠀⠰⣤⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣇⠀⠁⠀⠉⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⢀⠀⣥⡼⠂⢠⢎⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠖⠃⠈⠀⠀⠀⢸⢻⡄⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢁⣀⠀⠀⠀⢀⡟⢸⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠁⣸⠂⠀⠀⡼⠁⣼⣿⠀⠀⠀⠀⣿
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⠞⠁⠀⢀⡼⣇⣼⠟⡟⠀⠀⠀⠀⣿⣄
⠀⠀⠐⣯⠀⠉⢢⠀⠀⠀⠀⠀⠷⠶⠋⢻⠀⠀⠙⠋⠀⠘⠓⠚⣡⠇⠀⠀⢀⡇⠈⢳⡀
⠀⠀⠸⣝⢦⣀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣼⡇⢀⠑⢷⡀
⠀⠄⡐⡙⡓⡟⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⢤⣄⣴⠃⠀⢀⣤⠴⢺⡇⡇⠈⠳⣄⠙⣦
⠄⠱⣌⢇⡴⠠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡟⠀⠐⡄⢈⠻⠶⠟⢁⡀⡼⠰⠀⠀⠀⠈⢳⡈⢷⡀
⢠⠵⠼⠸⠄⠀⢸⠤⣤⣀⣠⣤⡤⠖⠋⢁⡾⠀⠀⠀⠹⡇⠀⠀⣍⠀⣹⢁⠂⠀⠀⠀⠀⠀⢳⡄⢻⡄
⠀⠉⠉⠲⠤⠤⠼⠤⣄⣀⣀⣠⣤⠤⠖⣿⠁⠀⠀⠀⠀⡘⢦⠀⡷⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢌⠳⠏⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠤⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠨⠭⠥⢒⣒⡀⠄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠖⢚⣓⠶⠎⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⡇⢰⢰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⢸⠁⠱⣸⠼⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⡏⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁⡧⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⡀⠀⠠⠄⠀⠴⠞⠢⠤⠿⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠷⠤⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠶⠖⠒⠻⠤⠤⠷⠀⠀⠀⠉⠛⠯⠭⣭⣄⣀⣀⡠⢦⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣤⣀⠀⠀⢀⣀⣀⠀⠀⢀⣠⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣄⡈⠙⠛⠛⠉⠉⠉⠙⠛⠋⠁⣠⣤⣄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠉⠙⠁⢀⣀⣀⡀⠘⠉⠉⠙⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠐⠒⣃⠀⠀⣀⠀⠘⣻⡟⠁⢀⡄⠀⢀⣘⠲⢰⡿⠞⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠁⠀⠀⠈⠛⠛⠋⠙⠛⠋⠀⠀⠀⠉⣡⠟⠀⢀⡀⢀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠁⠀⠲⣼⣧⣼⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⠃⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆⠀⠘⠷⣽⡵⠞⠀⠀⠀⢠⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⣤⡴⠶⠒⠲⠶⢦⡀⠀⠀⠉⠙⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣙⣦⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⢦⣄⡀⠀⠰⣬⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠛⡟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⡤⠶⠶⣤⡇⠀⠀⠀⠀⠀⠀⢸⣤⠴⠶⠦⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡄⠀⡄⠀⢹⡄⠀⠀⠀⠀⢠⡟⠁⢠⡀⢠⡌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠷⠶⠷⠶⠟⠛⠳⠶⠶⠚⠛⠻⠶⠾⠷⠾⠷⠟⠁


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

⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠖⠚⠛⠛⠛⠒⢢⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⢀⡴⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿
⠀⠀⠀⠇⠀⠀⢀⣎⠄⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⡿⠿⡇⠀⠀⠀⠙⢔⠃⠈⠢⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢐⠾⠁⠀⢰⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⡧⠆⡇⠀⠀⠀⠀⠀⠑⣄⣠⣾⣦⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠲⠄⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠁⡇⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⣦⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⣥⣤⣤⣤⣤⣤⡼⠀⠀⠀⠀⠀⠳⣄⡉⢀⣠⠞⠀⠀⠀⠀⠘⣇⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠋⠀⠀⠀⠀⢸⡋⡉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠚⠁⣂⣄⣃⠙⠆⠀⠀⠀⣠⡾⠶⢿⣤⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣞⡄⠀⠀⠀⢰⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠐⣿⣿⠇⠀⠄
⠀⠀⢴⠈⢼⢡⣶⢹⠌⢑⠄⠀⠀⢼⣇⣛⣸⣷⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⠇⠀⠀⠀⠸⣖⠤⠀⣠⡿⠀⠀⠀⠀⠀⠈⣠⣤⣬⢏⣤⣀⡈
⠀⠀⠀⣈⠁⡓⠖⣋⢈⡅⠀⠀⠀⠀⠙⠉⠋⢰⣯⣷⠀⠀⠀⠀⠀⢹⣿⣿⠇⠀⠀⠀⠀⠀⠈⠙⠛⠉⢿⣦⡀⠀⠀⠀⠈⢙⢿⠏⠈⣿⢟⠈
⠀⠀⠀⠈⠀⠑⠖⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠝⡻⣦⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠴⢶⣾⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⠛⢠⣾⣿⣿⣷⡖⢒⡶⠀⠀⠀⠀⢠⡶⣿⣿⠾⠟⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⡆⠀⠀⠸⠟⠛⠛⠿⠟⠛⠉⠉⠀⠀⢸⠀⢸⠀⠀⢀⣤⠚⢹⣿⣿⣟⣫⠟⡉⠀⠀⠀⠀⣴⢶⣿⣶⠿⠁⠀⠀⠀⠀⣾⡏⠀⠀⣿⣿⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁⠘⠛⠀⠀⠀⣀⣿⣕⣪⣷⠀⠀⠀⠀⠀⠀⢿⣧⣠⣤⣥⡅
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⠉⠁⠀⠀⠀⠀⠀⠀⠼⠿⠷⠬⠍⡥⠄

⣿⢿⢿⠿⡿⡿⡿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠸⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣓⣠⣿⣿⣿⠀⠈⠉⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠘⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⡇⠀⢀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⢸⣉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⣿⣿⡇⠀⠉⠀⠀⠌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⢈⡁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⢹⡍⣷⣦⣴⡖⠀⠀⠠⣄⠙⠻⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⣾⠀⢠⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⣓⣿⢿⡿⣷⢸⣿⡀⠈⠳⣤⡈⠙⢿⣿⣿⣿⠀⢸⣿⣿⣿
⠸⡄⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⢦⣽⠇⠀⠀⠈⠻⣦⣄⠙⠻⣿⠀⢹⣿⣿⣿
⠒⠻⠒⠒⠂⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠈⠻⣿⣦⣼⣄⠘⠿⠿⠿
⠀⠀⠀⠀⣶⡀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢈⠻⣿⠙⢷⣄
⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⣿⠟⠁⠀⢠⠏⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠳⣽⡀⠀⠈⢣⡀
⠀⠀⠀⠀⢸⡿⢸⣿⣿⣿⡏⠁⠀⠀⢠⣿⢠⡿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡀⠀⠀⢸⠙⢄⠀⠀⠙
⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣦⡀⠀⣾⣧⣾⠇⠀⢸⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⡿⣷⣄⠀⢸⣧⠈⢦⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣟⣾⡟⠀⠀⠸⣿⣿⣿⣿⣿⢈⣿⣿⣿⣿⣷⣼⣿⡕⢾⣿⡆⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿⣿⡇⠘⠻⣿⡿⢟⣧⢸⣿⣿⣾⣿⣿⣀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠿⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢻⡿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠑⢿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢀⡄⠀⠀⠀⠾⠿⠿⠟⠛⠁⠀⢰⠞⣿⣿⣿⣿⣿⣿⣿⠁⣿⡇⠀⠘⡷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢰⣶⣦⣾⣆⠀⠀⢀⣸⣯⣥⣿⣿⣿⣿⣿⣿⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⠛⠛⠉⠈⣿⣿⡿⠛⠿⣿⣿⣿⠀⡏⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⡟⠀⠀⠀⠀⠉⢻⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⣦⣀⠀⠀⠀⠀⢸⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⣿⣷⣜⢶⣤⣼⣿⣿⣿⡟⢶⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠁⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠸⣿⣿⣷⣽⣿⠙⢿⣿⣿⡄⠙
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣴⡆⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⠀⠀⠹⣿⠁
⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿

⠀⠀⠀⢀⠤⠤⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢄⠀⠀⠀⡀⠀⠀⠀⠀⡤⠤⢄⣀⣀⣀⣀⣀⡠⠤⢤
⠀⠀⠀⠨⠀⠈⠀⠀⠈⠑⢼⠀⠀⠀⢀⡒⠒⡜⠀⠀⠑⠒⠉⠘⡄⠀⠀⠀⠡⣰⠀⠀⠀⠉⠀⠀⠀⣀⠆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢎⣈⡣⠀⢸⠀⢸⠀⠀⠀⡀⠀⠀⠰⡥⠤⡀⠀⡠⠼⡀⠀⠀⠀⠀⠀⢀⠧⢄
⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠌⠀⠘⠄⠀⠀⠀⠀⠀⣠⠀⠀⡇⠘⢤⠒⢣⠀⠀⠀⠀⠀⡞⠂⡜
⠀⡔⢁⡠⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠘⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠁⠀⠈⢇⠀⠀⠀⠀⡠⠞⢀⡜⠁
⠀⡏⠝⣃⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠱⡀⣮⣤⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠸⡀⠀⡀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⢣⠀⠆⠀⠀⠀⠴⠀⡌
⠀⠀⠑⠤⠤⠊⠉⠁⠳⠤⠴⠃⠀⠀⠀⠀⠱⠤⠵⠢⠤⠮⠤⠜⠁⠀⠀⠀⠀⠈⠤⠤⠏⠀⠻⠤⠔
⠀⠀⠀⠀⡀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⢀⡵⠞⠂⠒⠲⣴⠀⠀⠀⠀⢀⠄⠑⠂⠀⠒⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠉⠘⠋⠈⠀⡇
⠀⠐⢥⡌⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢈⠄⡀⠀⠀⢀⠀⡀⡠⡆⠐⠒⠄⠀⠔⢂⣠
⠀⢀⠂⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢈⠓⠈⡆⠀⢡⠀⠈⣁⣴⣀⠀⠀⠁⠀⡬⠤⠅
⠀⠠⡀⠁⠀⠀⢄⣠⠄⠀⠑⡀⠀⠀⠜⠢⠀⠀⠀⠀⠠⠎⠀⡰⢰⠀⢂⣠⢿⡛⠛⣛⠷⣀⣂⠀⠁⣶⠆
⠀⠀⠈⠀⢀⡔⢡⣽⢲⠤⠤⡃⠀⣌⠀⡀⠀⠘⠈⠀⠀⠀⠌⠀⠀⡌⢨⡧⣘⠃⠀⢛⡃⣷⡈⡠⠊⣬⠄
⠀⠀⠀⠀⢀⣥⣮⡝⡈⠀⠀⠇⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠙⣟⣮⣤⢤⣬⣶⣿⢣⣇⣀⡡⠂
⠀⠀⠈⠐⠀⠓⠀⠐⠁⠒⠈⠀⠀⠀⠑⠚⠒⠀⠐⠐⠚⠒⠀⠀⠀⠀⠀⠈⠛⠷⠼⠿⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⠀⠀⢀⡀⠀⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡤⠒⢩⣥⣄⢰⣶⡎⢉⠂⣠⠞⠁⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠳⡄⢰⡍⠱⣶⡖⣠⣤⡍⠐⠦⡀
⠀⠀⠀⠀⢀⠊⢀⣼⣷⣾⠿⠊⠉⠀⣠⠟⠡⡾⠃⢴⣿⣇⢈⠀⠋⠙⠇⠀⠀⠀⠀⠸⠋⠋⠹⡉⣰⣿⡷⠘⠷⢄⠻⣦⡀⠉⠓⢻⣧⣶⣧⡀⠙⡄
⠀⠀⠀⠀⠈⠀⢸⣿⣿⠃⠀⣠⣶⡿⢋⠆⠁⠑⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠠⠄⠀⠐⣉⢿⣷⣄⡀⠈⢿⣿⣷⠀⢠
⠀⠀⠀⠀⡇⠀⣨⡍⠁⢀⣾⡿⠟⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡹⣿⣷⡄⠈⢩⣅⠀⢸
⠀⠀⠀⠆⢱⡆⠹⣷⡀⠺⠟⣥⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⣾⣿⡧⠀⢀⣤⡸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⠗⠀⣼⠏⢠⣶⠘
⠀⠀⠘⡀⠀⠛⢶⣈⠛⠷⠿⠿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠏⠁⠀⠙⠛⠃⠀⠴⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠿⠿⠾⠛⢁⣴⠟⠁⠀⠆
⠀⠀⠀⠘⢢⣤⣀⠉⠷⠆⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠾⠉⣀⣤⡔⠂
⠀⠀⠀⠀⠘⠶⢹⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⢀⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡟⠶⠂
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⠓⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠇
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡀
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣤⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⢿⣶⡄⢁⣈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠛⠢⠍⢠⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢠⣤⣤⣤⡄⠀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢹⡇⢠⣷⣤⠀⣿⠇⣤⡀⣤⣤⣄⣘⠋⣩⣽⠃⣠⣴⡄⢠⣤⣤⣄⠰⠀⣾⣄⣽⡧⠀⣿⣧⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⣿⠀⣾⡇⢸⡧⠿⠁⣿⣶⡟⠁⣿⠈⢿⢻⣆⠀⢀⡰⣧⠈⠻⢨⡏⢸⡇⠀⠀⠀⢁⡿⠃⢸⣇⣿⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠿⠄⠻⠇⠘⠿⠶⠂⠿⠛⠿⠄⠹⠿⠃⠀⠻⠿⠿⠃⠙⠿⠃⠸⠟⠸⠇⠀⠀⠰⠿⠷⠆⠀⠩⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⣉⣉⡠⢤⡤⢠⡄⢤⡤⠀⠀⣿⣉⠁⣠⢤⡀⣴⢦⣀⠶⢄⠠⡄⢠⡶⣄⢠⠦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣼⠇⣼⣇⣸⡁⠀⠀⣈⣹⠇⣿⣈⢿⣄⣴⠈⣳⡾⣦⡇⢻⡄⡏⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠉⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠙⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡄
⠀⠀⠀⠀⠀⠘⠀⣄⠀⠀⠀⠀⢀⡀⠀⡀⢀⡀⢀⡀⠀⢤⠀⣀⢀⠀⠄⠀⠀⠀⣀⡀⡀⢀⡀⠀⣀⣀⠀⡀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠉⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⡀⠀⠀⠄⠀⠀⢀⡰⠇
⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢀⠀⠀⣀⣀⡀⢀⠀⢀⠀⣀⡀⢀⣀⣀⡀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⢠⠶⣸⡿⠇⠀⠀⠀⠀⠀⠾⠋⠘⠛⠃⠸⠛⠀⠀⠀⠛⠃⠈⠛⠛⠀⠛⠀⠀⠼⠇⠀⠀⠘⠄⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⡘⠿⣧⠾⡄
⠀⠀⠀⠀⠚⠃⢁⣠⡾⠃⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠿⣄⡈⠘⠃
⠀⠀⠰⠀⢀⣴⠟⢁⡴⠾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⣿⠷⢦⣌⠻⣦⡀⠀⡆
⠀⠀⠀⠆⡹⠇⣴⡟⠀⢰⣦⡛⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠇⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠛⣴⡶⠀⢹⣦⡘⢿⡀
⠀⠀⠀⠀⡄⠀⢼⣧⡀⠈⢿⣿⣷⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⡀⣀⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣽⣿⡿⠁⢀⣬⣯⠀⢸⠃
⠀⠀⠀⠀⢁⠀⢺⣿⣿⣄⠀⠉⠻⣿⣬⠢⡀⡰⠒⢤⣀⣠⡀⠀⠀⠀⠉⠋⠙⠋⠉⠉⠀⠀⠀⠀⣄⣀⣤⠒⠒⢀⡔⣩⣾⠿⠋⠀⣠⣾⣿⡷⠀⡘
⠀⠀⠀⠀⠈⢢⠀⢻⡿⣿⣶⢶⣄⠀⠙⣷⡈⢿⡄⠺⣿⣫⡌⠛⠛⠻⠇⠀⠀⠀⠀⠸⠛⠛⠛⠃⣨⣿⠟⢠⠾⢋⣼⠏⠀⣠⡶⣾⡟⢿⡟⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠙⠦⠜⣛⣉⣘⣛⣋⠰⠆⠙⠦⣀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄⢀⡤⠃⠸⢇⣙⣻⣇⣈⣛⡃⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠉⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠛⠛⠻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠉⠀⠀⢀⣀⣀⣀⡉⠛⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠈⠙⠻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠉⠀⠀⠀⠀⢰⣾⣿⣿⣶⡄⠀⠉⠁⠀⠀⢀⣤⣤⣤⣄⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣾⣿⡋⠁⠀⠀⠀⣤⣤⣦⣤⡀⠙⠛⠛⠋⠀⠀⢀⣀⣀⠀⠙⠿⣿⣿⠿⠃⠀⠀⠈⢙⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣤⣀⠈⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⣴⣶⣶⣶⣄⠈⠉⠉⠁⠀⠀⢀⣠⣴⣿⠟⠋⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣟⠁⠉⠻⣿⣿⣿⣿⣷⣤⣀⠙⠛⠛⠛⠋⠀⠀⠀⣀⣤⣶⡿⠛⠉⢀⣴⣾⣷⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣠⣴⣾⠿⠋⠁⠀⠀⠀⣿⣿⣿⠏⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣷⣦⣾⣿⣿⡏⠀⠉⠻⣿⣿⣿⣿⣿⠟⠉⢀⣴⣾⣷⠀⠀⠀⠙⠋⣁⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⢹⣿⣿⣿⣿⠀⠀⣾⣿⣿⠏⠀⠀⠀⣠⣾⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⠿⣷⣶⣾⣿⣿⣿⣿⠀⠀⠙⠋⣁⡀⠀⠀⠀⣿⣿⡿⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠿⣷⣿⣿⣿⣇⠀⠀⠙⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⠀⠀⠀⠈⢁⣤⣄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠈⢻⣿⣿⣿⣆⡀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⡿⠃⠀⠀⠀⣴⣿⣿⡿⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⡀⣸⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢁⣤⣄⠀⠀⠀⢿⡿⠟⠁⣠⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⣰⣿⣿⡿⠀⠀⠀⢀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣧⣄⣀⣸⣿⣿⣿⣿⠀⠀⢿⡿⠟⠁⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠀⠙⠛⠓⠦⣄⣠⠴⠚⠛⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣤⣀⡾⠛⣧⣀⣤⡀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣤⢤⡀⠀⠀⠀⢠⠞⠳⠶⠖⠒⠦⣤⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢿⡀⢉⣤⠤⣤⡉⢀⡟⠀⠀⠀⠀⠩⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣸⠆⠀⡴⢛⠛⣦⠀⠙⣧⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡶⠋⢰⡏⠀⠀⠀⠹⠄⠛⢶⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⢠⡇⢀⣴⠇⠈⠀⠛⠶⣄⢸⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠛⢶⡘⣧⡀⠀⠀⡼⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣇⢸⡅⢶⠀⣀⣀⣁⡿⢸⡇⠀⠀⠸⣇
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢿⣤⣤⣍⠋⣡⣤⣤⡿⠀⠀⣠⡤⠶⠶⠒⠛⠒⠶⠶⢤⣄⡀⠀⠙⣦⡙⠓⠚⠉⠉⠉⠀⢻⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⠙⠶⠋⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀⠀⠙⠳⠶⠶⠖⠓⠶⠛⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⣹⠇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠰⠏⠀⠀⠀⠒⠛⣆⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠛⢻⡆⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⢈⡗⠀⠀⠈⠀⠀⠀⠀⢀⡾⠁⠀⠀⡇⠀⢠⡾⠋⠙⠓⠦⣀⡀⠀⠀⠀⢀⣠⠞⢳⡄⢸⡄⠀⠀⠀⠀⢰⠏⠀⠀⠀⣠⡄⠉
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣧⠀⡞⠀⠀⠀⠀⠀⠈⠙⠛⠒⠛⠋⠁⠀⠀⢻⣾⠀⠀⠀⠀⠀⠠⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣤⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠉⠁⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠉⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣤⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠀⢀⣀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣀⠙⠶⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⡆⠀⠀⠀⠀⠀⠙⡷⣤⣀⣀⠀⢀⣀⣤⠶⣿⠀⠀⠀⠀⠰⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠶⠖⠚⢻⡍⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣽⠛⠓⠲⠶⠶⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠻⢦⣀⠀⠀⠀⢀⣠⠶⢋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣍⣉⣛⣋⣉⣥⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣏⣈⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣏⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢰⠋⠱⡄⠆⠈⠒⠁⢰⢦⠀⢜⣑⠆⢠⠀⢠⡎⠁⠀⢰⡊⠀⠈⡟⠁⡞⢱⠄⡔⠙⢦⠀⣆⠔⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⢸⣀⣠⠇⡄⠀⠀⠀⠗⠘⠆⢩⣈⠆⣧⣀⠰⣇⡀⠀⢀⣈⠇⠀⡇⠀⡏⠹⡀⢇⣀⡼⠈⡏⢦⠸⣇⡀
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠢⡄⠐⠒⠒⠊⠥⣀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠰⠊⢀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠑⢄⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠰⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠂⠁⢨⠉⠢⠔⠸⡑⠤⡀⢀⡀⠀⠀⠀⢀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⣼⠀⠀⠀⠀⢣⠀⠀⠁⣀⠉⠉⠈⢁⣂⣸⣄
⠀⠀⠀⠀⠀⠀⣇⡄⠀⣄⠈⡄⣀⠀⠀⢸⣀⣤⠖⠒⠒⢶⢺⠁⢠⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠁⠐⢆⡎⠑⠊⠀⠘⣄⣈⣠⢏⢀⠑⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡀⠀⠀⠀⠀⠀⢉⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠍⣟⠈⣜⢙⣝⠀⠀⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠛⢏⠈⣯⢡⡤⢳⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠢⣀⠀⠀⠀⠁⠀⠀⡄⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡗⠢⠤⢀⡀⣤⠿⠋⠀⠣⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠐⢃⠀⠀⠀⠀⠘⡂⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⠀⢀⠃⠀⠀⠈⢸⠀⠀⠀⠀⠀⢹⡖⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⠢⢤⡀⠤⠴⢦⠄⠐⡖⠉⠁⠀⠀⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡠⢔⠁⠣⢄⣀⠞⠠⠤⣃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⡌⠀⠀⢰⠒⠀⢾⠁⠲⡈⡉⠓⠄⠀⠔⠁⠀⠈⠢
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡀⢃⠀⠀⠀⠓⠀⠊⢆⠀⠀⠀⠀⡸⠰⠀⠀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠀⠈⠢⡀⠀⠀⠀⢀⠜⠂⠀⠐⠈⠀⠀⠂⢄⣀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⢀⣶⠞⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣿⠃⣠⣤⡀⠙⣷⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠙⣦
⠀⠀⠀⠀⠀⣿⡀⣿⠀⠹⡄⢸⡇⠀⠀⠀⢰⡏⠀⢀⣄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢹⣇⠸⡆⠀⢻⠀⣧⠀⠀⠀⢸⡇⢰⠏⠉⣻⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄⢸⡆⣿⠀⠀⠀⢸⡇⣸⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠙⢷⡀⠀⠀⠀⠀⠀⠀⣰⢟⣉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠻⡈⣷⠀⡇⢿⠀⠀⠀⢸⡇⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠞⢦⠻⣄⠀⠀⠀⠀⡼⢣⠞⠉⣧⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠃⠀⠘⠒⠋⠙⠛⠁⠀⠸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠀⠐⠃⠉⣾⡋⠛⠐⣶⠈⠈⠓⠋⠈⠃⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠀⠄⠀⠀⠀⠀⠀⠀⢀⢠⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⡸⣱⢛⡶⢠
⠀⢀⠰⣖⢲⡻⡄⠀⠀⠐⠋⠉⠓⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠀⠀⢳⣿⣼⡅⢰
⠀⢱⠀⣽⣬⣷⠇⠠⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠐⠀⠀⠀⢸⠃⠀⠀⢸
⠀⢸⠀⠀⠀⢿⡀⠀⠒⠀⠀⠀⠀⠦⠚⠳⠴⠂⠀⠀⠀⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⠉⠈⠋⠀⠀⠀⠀⠀⠀⢠⠏⡠⠖⠒⣼
⠀⢸⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣻⠀⣧⣀⣠⡟
⠀⠘⣇⠀⠀⠀⠐⣏⠛⠳⢶⡶⢦⡤⢤⠤⣴⠶⠖⠚⢛⣡⠞⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢦⣍⣉⠛⠛⣳⠖⣒⣾⡛⠛⡉⢉⣠⠶⠃⠀⠈⢀⡟
⠀⠀⠘⢦⡀⠀⠀⠈⠛⠲⠶⠤⡜⠐⠒⠒⣻⠧⠴⠖⠋⠁⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠟⠉⠙⠛⢻⣈⣥⡠⠿⠏⠉⠻⣷⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠉⠀⠀⠀⣤⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠐⠒⢄⠀⣧⠊⠁⠀⢀⡌⠉⠀⢈⣉⠀⣠⠇⠀⠠⠏
⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⣠⠔⠀⠀⠒⠒⣊⣷⡶⣮⡴⠂⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣠⠘⣮⠱⠶⢤⣤⣮⣤⣤⡬⠷⠚⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⡰⠁⠀⠀⠀⢠⣾⠟⢃⡀⠛⣾⠩⡤⢶⡟⢣⠀⠀⠀⠀⠀⠀⠀⠻⣧⠚⢁⢠⡼⠂⢀⣤⡈⢷⣧⡀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣆
⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄⠀⠀⠀⢠⢿⡏⢡⡗⡿⢠⣶⣛⡆⠠⣷⠘⡄⠀⠀⠀⠀⠀⠀⡴⠈⠛⡞⠋⣤⠄⢹⡌⡃⠀⢻⣧⠀⠀⠀⠀⡆⢆⠀⠀⣷⢸⡄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⣎⣿⣀⡾⠀⣾⣾⠉⢹⡇⠰⡏⠀⡇⠀⠀⠀⠀⠀⢠⡇⠐⢢⣧⠈⣹⡞⠀⢯⡿⠂⠉⢿⡇⠀⠀⠀⡙⢦⣀⡤⠟⠀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⢰⣿⠇⠤⠄⠀⠀⠁⠀⠟⣀⣹⠃⢠⡇⠀⠀⠀⠀⠀⢸⡇⠈⡆⣿⡔⠀⠉⠀⠀⢒⢦⠄⢘⣿⠀⠀⢰⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠹⢤⣸⣿⣉⠀⢴⠄⠐⣽⡟⠆⢠⡟⠀⣼⠁⠀⠀⠀⠀⠀⠈⣿⣮⡴⠋⢷⡂⠀⢷⣆⢙⣿⣷⠈⣿⠀⠀⠀⣧⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠈⣿⠀⠤⣼⡂⠼⣿⠋⣈⡿⠁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠘⢷⣀⠛⠏⢀⡉⠅⣺⠏⠀⠀⠀⡿⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⢠⡎⣠⢀⡄⠀⡆⠀⠛⢦⣄⠀⠍⢙⡋⣰⠞⢡⠰⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡶⢠⠄⠛⣶⣌⣀⣤⣾⣥⣤⣤⣄⣀⡇⠀⡆⢰⣸
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠚⠁⠀⠀⠀⠈⠉⠛⠉⠉⠙⠓⠛⠚⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⣶⣶⣶⣶⣶⣖⣲⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣒⣶⣶⣶⣶⣶⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣟⡿⣿⣿⠿⠿⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠻⢿⢟⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⡄⠉⢳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⢐⡀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⢀⣠⣀⡈⢷⡄⠀⠐⠀⠀⣰⠃⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡃⢉⣿⠀⠙⠷⠶⠶⠚⣡⣏⡹⠈⠛⣦⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⣙⣓⣿⠀⠀⠀⠀⠀⠐⣯⠢⡅⣢⠜⣸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢺⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡤⡤⠴⠋⠀⠠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣇⡀⢸⣒⣔⣶⣘⣆⡠⡤⠀⠀⠀⠀⣰⣶⣶⣼⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣒⣻⣿⣿⣻⣶⣒⣦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠉⠉⠉⠹⠿⣛⢛⡛⠛⢛⣛⡛⣛⣛⣛⣛⣛⠛⠛⢻⠋⠋⠋⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄⡇⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⡉⠉⠉⢁⣀⣀⣀⣈⣉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠇
⠀⠀⠀⠀⠐⣶⡀⠀⣹⣦⠀⣴⠁⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅
⠀⠀⠀⠀⠀⢻⣿⡷⣾⠗⣿⡿⡄⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣷⢸⢹⣿⠃⣇⣀⣀⣀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣮⣼⣿⠀⠠⠀⠀⡆⢰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣠⣤⣦⣤⣥⡼⠿⠻⠛⠻⢳⣶⣶⣶⣶⣶⣶⠶⠿⠾⠿⠿⢦⣷⣶⣦⣬⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣯⣤⣤⣤⣤⣤⠜⠛⠛⠛⠛⠻⠛⢤⣤⣤⣤⣤⣤⡿⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠹⢷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡟⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢻⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣾⣏⠁⠀⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠀⠀⣠⡴⠶⠶⣤⡀⠀⠀⠀⠈⣹⣷⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⢠⡟⠁⠀⠀⠀⠈⣷⠀⠀⣾⠁⠀⠀⠀⠈⢻⡄⠀⠀⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠼⢧⣤⣤⠾⠁⠀⢸⡇⢸⡇⠀⠈⠳⣤⣤⡼⠧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣀⡀⠀⠙⢷⡄⠀⢸⡇⢸⡇⠀⢠⡾⠋⠀⢀⣀⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣹⠿⢉⣿⠃⠀⠀⠀⢸⡇⠀⣼⠋⠉⠁⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠈⠉⠙⣧⠀⠸⡇⠀⠀⠀⠘⣿⡉⠿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠿⠷⣤⠀⠀⢸⡇⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢰⡇⠀⠀⣤⠾⠿⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⡀⠀⠐⣦⣀⣀⣀⣀⡀⢸⡇⢸⡇⢀⣀⣀⣀⣀⣴⠂⠀⢀⣴⠟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⠄⠀⠀⠀⠀⠉⠉⠉⠉⠙⢿⡅⢸⡿⠋⠉⠉⠉⠉⠀⠀⠀⠀⠠⣄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⠳⠆⠀⠀⣠⡴⠶⠆⠀⠀⢸⡇⢸⡇⠀⠀⠰⠶⢦⣄⠀⠀⠰⠞⠋⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢤⣤⣤⣤⠾⠛⢶⣄⠀⠀⢦⣼⡇⢸⣧⡴⠀⠀⣠⡶⠛⠷⣤⣤⣤⡤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⡀⠀⠉⠀⠀⠀⢻⡇⢰⡏⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣼⡇⠀⠀⠀⠀⢀⡆⠀⢸⡇⢸⡇⠀⢰⡀⠀⠀⠀⠀⢸⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠰⠦⡤⠾⠁⠀⣾⠁⠈⣷⠀⠈⠷⢤⠴⠆⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⣀⣠⡾⠃⠀⠀⠘⠷⣄⣀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⡿⠿⢛⣫⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣿⣿⠿⢛⣋⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣶⣤⣍⣙⠛⠿⢿⡿⠿⠛⣋⣩⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡃⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣿⡇⢉⣛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣉⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣉⠸⣿⡇⢸⣿⣿⣿⣶⣬⣉⠻⢿⡿⠟⣉⣥⣶⣿⣿⣿⣿⠿⠛⠉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢶⣶⣶⣦⣤⣉⡙⠛⠿⢿⣿⣦⣴⣿⣿⣿⡿⠟⠋⠁⢀⣠⣴⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣦⣤⣉⡙⠻⠿⢿⣿⣶⣶⣬⣍⡙⠛⠉⠁⣀⣤⡶⠾⠛⠉⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣉⠛⠻⠿⣿⣿⣶⣦⣬⣉⡛⠻⠿⣿⣤⡶⠟⠋⣥⣶⣿⣦⠶⠟⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣷⣶⣤⣌⣉⠛⠻⠿⣿⣿⣶⣦⠀⣀⣠⣶⠶⠛⣿⣿⣧⣠⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣶⣦⣬⣉⡛⠛⠋⠁⢀⣠⣴⠽⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣤⡶⠟⠋⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠂
⣀⡀⠀⠈⠀⣀⡰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀
⠀⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠖⠀⢤⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠄⠄⠂⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡐⠀⠀⠀⠀⠀⢄⢰⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⡋⠠
⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠰⠂⢀⣀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⣷⠈⠻⠀⠀⠀⠤⠀⣁⠤⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠀⠀⠱⠀⠀⠹⠆⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠊
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠁⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⡻⣮⣕⢤⡀
⠀⠀⠀⠀⠀⠠⢊⣴⢄⠀⠀⠀⠈⢿⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣝⠷⣮⡳
⠀⠀⠀⠀⠀⠀⠈⠱⠋⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⢠⠁⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⣀⠄⠀⠀⠊⠉⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠀⢀⠤⠀⠀⠀⡀⠀⢀⠀⠚⠲⠂⠑⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠔⠒⠈⠁⠀⠀⠀⣠⠃⡀⠈⡆⠀⠀⠈⡆⢰⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠉⠋⠀⠘⠉⠂⢀⡤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠄⠒⠈⠉⠀⢀⠀⠀⠈⢤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠾⠏⠀⠘⠳⣦⠴⡖⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⢤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⢟⠻⢿⣯⣼⣲⣄⣀⡀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⢠⣼⠷⠞⢷⠖⢿⠟⠛⠿⢿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⠀⠀⠀⠀⣾⣶⣿⠖⢂⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣽⠦⣤⣴⣾⣿⣿⠋⢠⣿⣿⣿⣿⣿⡿⠟⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⣠⣿⡿⠻⠿⠛⠉⠀⢀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠛⠁⠀⣀⣠⣤⣶⣾⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⠇⣼⣿⣿⣿⣿⣿⠇⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⡟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⠿⢛⣿⣿⣿⣖⠀⠀⠈⢿⣿⣿⣿⣄⣴⣶⣶⣶⣦⡀
⠀⣴⠞⠀⠀⠀⣀⣼⡿⠋⠀⠀⢸⣿⣿⣿⣿⠟⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⠿⠁⠀⠒⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢠⣿⣿⣿⣿⣿⡿⠟⠻⠿⢿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⢨⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠃⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⢀⡀⠀⢀⣠⠴⠖⢚⠛⠋⠩⠍⢁⠐⠂⠤⢀⠀⠀⡠⠤⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣀⢼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠑⠂⢄⡀⠉⠢⢄⣀⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⠈⢀⠊⠀⢰⠙⣆⠉⠒⠠⢀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠈⠐⠤⣀⠉⠢⢑⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⡰⠁⠀⢀⠄⠀⠈⡳⣄⢀⠀⠈⠐⠀⠀⠄⢀⡀⠀⠈⠁⠂⠀⠀⣈⣤⠶⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⢸⡔⠀⠀⡠⠊⠀⡠⠊⠀⠀⠉⠪⠁⠒⠠⠤⠀⢀⡀⠉⠉⠄⠹⠟⠋⣠⡴⣂⣽⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⢴⣉⣀⠔⠈⣀⠴⠎⠠⠀⠀⢀⣠⠖⠒⠠⠛⠒⢦⠤⠶⠶⠤⠒⠲⠂⣉⠈⢀⣈⣹⠂⠀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⡌⢡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣭⠜⡋⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⠁⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠀⠆
⠀⠀⠀⣠⠂⠀⠀⠌⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠐⡀⠀⠘⡄
⡔⠂⠈⠀⠀⡠⠊⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠑⠄⠀⠈⠑⠂
⠈⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢄⡀⠀⠌⠄⠲⡁⠀⠈⠙⠢⢌⣁⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣍⡉⠌⠀⠀⠀⠉⠒⠄⡀⠀⠈⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡾⡄⠀⢀⡀⠀⠀⠠⠄⡀⢀⠁⠂⠞⢢⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡐⢠⣡⠔⠉⢸⠀⠀⡄⠂⣹⡇⠑⠄⣠⡳⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠉⠁⠏⠀⠀⠀⠀⣀⣤⣈⣏⡐⠁⠀⠀⠀⢑⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠤⣣⣤⡄⠀⠀⠀⠀⠸⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⣀⣀⠰⠄⣀⠀⠀⠰⠆⠛⠠⠆⠀⠀⣀⣠⣁⡀⣀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⠦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⡚⠛⠳⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡀⠉⠒⠤⣀⠀⠀⠀⠀⠀⡘⠌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⠀⡟⠀⢔⠒⢽⠲⣤⢄⣘⡬⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⢻⡇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣸⠃⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢰⠊⡦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠳⣦⣴⣾⣻⢀⡀⠄⣀⢾⠀⢸⠀⢸⠂⠀⠀⢀⣀⣀⣎⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠁⠀⠀⠈⠀⢸⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠢⣀⣀⣀⡠⠂⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠐⢄⣀⣀⣀⠌⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡤⢔⣒⣒⣶⣶⠶⠖⣾⠲⠞⣓⣛⣛⣒⣶⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡋⠉⠓⠒⠂⠉⡡⣏⢌⠋⠉⠀⠀⠉⠙⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠀⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠲⠖⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⡋⢉⣺⠃⠀⠀⢶⣴⣷⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠙⠋⠁⠀⠀⠀⠈⠉⠁⠀⢰⣢⣿⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠐⠚⢆⠀⠀⠀⠀⠀⠠⠘⠁⠀⠉⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢸⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠣⢕⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠁⠒⠅⠒⠶⣂⣐⣒⣂⡟⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⡠⠞⠀⠀⢀⠍⠑⢄⠀⢄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢄⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⢲⣒⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡾⠉⢍⡢⡱⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⣋⠍⠹⡇⠒⠠⢀
⠀⠀⠀⠀⠀⠠⠋⠀⠀⠸⡗⡄⠤⠓⣽⢐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⡕⠡⠤⡄⡿⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠡⣁⠀⢀⡿⠚⠺⠦⠄⠀⠀⠀⠤⠤⠒⠉⠚⣆⠀⢀⡰⠁⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠘⡄⠑⠂⠤⢀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⡀⠤⠐⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠉⠉⠀⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠁⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠤⠄⢀⣀⣀⡀⠀⠀⠀⠀⢀⡀⣀⡀⠠⠤⠐⠂⠉

⠀⠀⠀⠀⠀⣠⠴⠤⣄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀⠀⢀⣀⣀⡀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⢀⠇⠀⠘⠊⠉⠉⠉⠶⣅⠸⠄⠀⠀⠀⠀⠀⡤⠤⣄⣰⠇⠀⠈⢣⢤⠴⠚⠉⣇⠀⠀⠀⠀⠀⠸⡈⢢⡀⠉⠉⠉⠙⠉⠉⠉⠁⢠⠊⢰⠃
⠀⠀⠀⠀⢸⠃⠀⠀⡀⠀⠀⠀⠀⠀⢹⠤⢤⡀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣱⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⢻⠖⠲⢦⠀⠀⣠⠖⠳⡄⠁⠀⠀⠀⠀⠀⠀⠀⡰⠓⠦⡀
⠀⠀⠀⠀⡼⠢⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⡏⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢻⠀⠐⢇⡠⠖⢧⡀⡀⠀⠀⠀⠀⠀⢠⠟⠤⣀⠟
⠀⠀⡠⠞⠄⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢣⠀⠀⠀⠈⠐⠠⠠⠤⠤⠤⠔⠺⠃⠀⠀⡸⠀⠀⠀⢧⡀⠈⢣⠀⠀⠈⠉⠀⢀⠏⠀⢠⡟
⠀⢸⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢀⡴⠃⠀⠀⠀⠈⢳⠄⠀⠉⠂⠈⠤⠚⠁⠀⢤⠋
⠀⢸⠁⠍⠳⢉⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠈⣆⠀⡹⣹⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠘⡇⠀⣤⡀⠀⠀⠀⠀⣠⡄⠀⡼
⠀⠀⠸⡄⢹⠟⠛⢀⣀⣀⣀⣠⠂⠀⠐⢦⠃⠀⠀⠀⠀⠀⢳⡀⠀⢣⡀⠀⠀⢀⡞⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⣄⣀⣀⡴⠉⠀⢰⠁
⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠉⠒⠶⠔⠋⠀⠀⠀⠀⠀⠀⠀⠓⠶⠒⠛⠒⠒⠚⠦⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠞⠃⠀⠀⠙⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⢩⡓⣄⠀⣠⣞⡍⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⠀⠀⣀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠤⢤⣤⡤⠤⠔⠉⡇
⠀⠀⢀⡠⢄⡴⠛⠉⠠⠠⠄⠉⠛⢇⠀⠀⠀⠀⠀⠀⢠⠡⠀⠀⠈⠉⠉⠀⠀⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠉⠀⠀⠀⠐⢇
⠀⠀⠸⣤⣴⠁⠂⠐⠀⠀⠀⠀⠐⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣀⠀⠀⠀⠀⢀⢀⡀⠀⡴⡏⠀⠐⠒⠄⠀⠀⠔⠒⢂⢨⡄
⠀⠀⢀⠇⠇⠀⠂⠐⠀⠀⠄⠂⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠐⠆⠀⠀⠀⠰⠂⠀⢹⡤⡈⢣⠀⠀⠰⡃⠀⠈⠻⠀⢠⡀⠀⠀⠀⠸⠀⠀⢐⢋⢉⢱
⠀⠀⡜⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠀⠀⠀⠀⡸⠀⠀⢸⠀⣀⣀⡏⠈⢠⣶⣾⣿⣼⣶⡄⠄⠀⠀⠄⢼⠉⠉⢹
⠀⠀⢣⡀⠀⠀⠈⠀⠠⣈⣣⠆⠥⠄⠈⢆⠀⠀⠀⣠⠋⠓⠤⠀⠀⡀⠀⠀⠠⠞⠁⠀⣠⠃⢸⡀⠈⣁⣴⠻⡯⡙⠛⢋⢭⠻⣦⣜⣀⠔⠃⠰⣿⡼
⠀⠀⠀⡜⠀⠀⠀⣠⠊⣩⣹⡦⡤⣀⣀⡼⠀⠀⢰⠁⠀⠀⠀⠀⠘⠄⠅⠀⠀⠀⠀⡔⠁⠀⠀⡹⠃⢸⣏⠘⠷⠇⠀⠸⠾⢂⢻⡦⠹⡄⡴⠋⡈⡇
⠀⠀⠀⡇⠀⠀⠘⡇⢸⠶⢯⠀⠇⠀⠀⢸⠀⠀⠘⡶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠛⠄⣾⢧⡙⠒⠀⠀⠄⠒⣃⡼⣿⡄⡿⠀⠸⠟⡰
⠀⠀⠀⢣⠀⠀⠨⡷⢿⣗⡇⣸⠀⠀⠀⡜⠀⠀⠀⢃⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠀⠀⠀⠀⠀⠸⣮⣿⣻⡟⠛⣿⣿⣧⣷⣋⣼⡷⠴⢦⠔⠁
⠀⠀⠀⠀⠙⠳⠄⠞⠂⠖⠚⠣⠰⠛⠋⠀⠀⠀⠀⠀⠓⠾⠿⠒⠒⠒⠒⠒⠻⠿⠒⠁⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠤⠿⠿⠛⠁⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⢄⠀⠀⡠⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⣡⠃⠀⡠⠤⡥⢬⠤⢄⠀⠘⣌⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠵⠀⠶⣁⠀⠛⠁⠀⢰⡇⠀⠈⠛⠀⣈⠖⠀⠪⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⡠⢂⡀⢬⠀⠐⢤⠋⢫⡝⠑⡤⠂⠀⡧⢀⠐⢄⣘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠉⣃⠈⠀⢈⠀⢀⠌⠀⣸⣇⠀⠡⡀⠀⡅⠀⠁⣘⠉⣺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣰⣡⠂⠀⡈⠀⠠⢀⡀⢣⡞⢀⡀⠄⠀⢑⠀⠐⣌⣆⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠁⠀⠈⠲⠚⠣⠤⠤⠄⠈⢣⡝⠁⠠⠤⠤⠜⠓⠖⠁⠀⠈⠑⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠒⠒⠙⢤⣀⡆⢀⣀⣀⡐⠀⠸⠃⠀⢂⣀⢀⡀⢰⣄⡰⠋⠒⠒⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡷⠠⢤⡀⠈⠀⠘⠁⠀⠀⣱⡀⢰⡆⢀⣌⠀⠀⠈⠁⠀⠁⢀⡴⠄⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠧⣠⠊⠉⢧⠒⠂⠤⠤⠞⠀⠈⢃⡘⠁⠀⠓⠤⠤⠐⠒⡼⠉⠑⣄⠼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠆⢀⠀⢰⡀⠠⢤⠛⢢⠔⠀⡀⢀⠀⠤⡔⠓⡤⠄⢀⡆⠀⡀⠱⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⡏⠺⣄⠈⠑⢄⡈⠀⢸⠀⢠⠣⠜⡇⠀⡇⠀⢁⡠⠊⠁⣠⠗⢹⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⣧⠀⢀⡅⠀⢺⡀⢂⡀⢀⡐⢀⡗⠀⢨⡀⠀⣬⡀⠀⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⡑⢄⣁⣈⣂⣨⠄⠈⡅⢨⠁⠠⣅⣐⣁⣘⡠⠊⡨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠧⣿⢦⡠⠍⢩⣇⣸⡍⠩⢅⡴⣿⠼⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠁⠈⠛⠛⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⣿⡇⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠈⠛⠋⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡷⢀⣤⣄⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢿⣿⣿⢀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⣭⣤⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡛⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⠏⠀⠈⢻⣿⣿⣿⣿⣷⠋⠉⠛⣿⣿⡇⣈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣿⣦⣀⣀⣼⣿⣿⣻⣿⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠎⠉⠉⠏⠉⠉⠥⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡀⠀⠀⠰⠇⠄⠶⠀⠀⠀⢰⣿⣿⠋⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⢻⣿⣿⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⣸⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⣸⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋⣀⡄⢀⣠⣾⣿⣿⣿⣧⣤⣶⡶⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣌⠙⣿⣿⣿⣿⣿⠟⣥⡴⢾⣟⣵⡌⢻⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢹⣿⣿⣿⣿⡇⠰⠶⠚⠛⢁⣤⡄⠻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢀⣾⣿⣿⣿⣿⣇⠀⢠⣶⠞⣿⢿⡇⠀⢹⣿⣿⣿⣿⣷⣄⠻⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢾⠇⣈⣵⠿⠆⢻⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠐⢚⣿⣷⣿⢷⠄⣿⣿⣿⣿⣿⠟⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣦⠀⢸⡇⠛⠋⠀⣠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁