Renewable energy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⡅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⠑⠅⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃
⠀⠀⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁
⠀⢀⣿⠀⣦⢤⢤⡄⠀⡤⡤⡤⠀⠠⣤⢤⡄⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁
⠀⠀⡆⢀⣿⣹⣿⡇⠀⣏⣿⣿⠀⢸⣹⣟⣿⠀⢀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⢀⣴⣄⣀⡀
⠀⢈⣠⣼⡟⠀⢘⣠⣴⠉⠛⣁⣠⡎⠉⠉⠁⣠⡿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣇⣈⣹⣶⣤⡀⠀⠀⠀⠀⣸⣿⣻⣿⢿⣿⣷⣴⣤⣄⣀⡀
⠀⠀⠿⠟⠁⠀⢿⣿⡿⠀⡏⣿⣿⡇⠀⢰⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣸⣏⢹⣍⣿⡿⠇⠀⠀⠀⢠⣿⣟⣿⣿⣾⣿⣏⣿⣿⠿⣿⣿⣷⣶⣦⣤⡀
⠀⠰⠒⠂⠀⠀⠻⠟⡁⠀⠷⡿⢟⠁⠀⢸⣿⣿⡷⠟⠋⠀⠀⠀⠀⠀⢀⣿⡏⢸⣿⡀⠀⠀⠀⠀⠀⣼⣿⢻⣿⣷⣿⣟⣻⣿⣿⣶⣿⣿⣉⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⢿⣿⣿⣼⣿⠿⣿⣿⣧⣿⣿⡿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣾⣿⡇⠀⠀⠀⢀⣽⡿⣿⣿⣽⣿⣿⣿⣿⣟⣹⣿⣿⣷⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⣿⣿⠀⠀⠀⠘⢿⣷⣿⣟⣿⣿⣷⣿⣿⢿⣿⣿⣯⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠈⠻⢿⣿⣯⣽⣿⣿⣾⣿⣿⠻⣿⣿⣿
⠀⠀⠀⠀⡠⣄⠠⠉⢳⡌⠀⣿⠐⢉⡟⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣿⣿⣿⣾⣿⣿⡛
⠀⠀⣠⣀⣃⠈⢻⣆⢈⣷⣠⣿⢀⣼⠏⢀⣼⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣤⣿⣿⣿⠇
⠀⠀⠃⠈⠙⢷⣄⣿⣼⣷⣿⣷⣿⣵⣴⠟⠉⢁⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⡏
⠀⠟⠛⠛⠶⣶⣿⣿⣿⡿⠿⣿⣿⣿⣾⠶⠟⠛⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃
⢀⣨⡴⠶⠶⣮⣿⣿⣿⠀⠀⢈⣿⣿⣿⣛⠶⠶⠟⡋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠈⠃⢀⣤⡴⠶⢿⣿⣿⣿⣶⣿⣿⣿⣿⠿⠶⣤⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠰⡟⢁⣠⡴⠟⣯⣿⢿⣿⢿⣟⣷⠛⢷⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⢠⠟⢀⣰⡟⠀⣿⠉⣿⡁⠹⣦⡀⢩⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠉⠀⣾⡀⠜⣿⠀⠘⢧⣀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⣈⣀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⡖⠒⠤⠤⣄⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣹⡏⠉⢉⣉⣉⣀⣀⣀⡀⠀⠉⠘⡇⠀⠀⠀⠀⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠞⡇⡇⠀⣞⣀⣇⣀⣿⣀⣇⠀⠀⡠⠈⠲⠤⠔⠊⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡇⡇⠀⣇⢀⡇⠀⣹⡀⣹⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣇⣠⠭⢭⣽⣉⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⣉⣀⣀⠀⠀⠀⠀⠉⠒⢄⠀⠀⣰⠦⣄⢰⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⣡⠊⣡⢤⡏⠈⠓⠒⢤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣀⣀⡠⠇⠀⣠⠖⠂⢰⠁⡜⠁⠀⢀⣾⠀⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢇⠀⠀⡏⢰⡃⠀⢀⣞⣘⢒⡗⠀⠀⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠘⢦⣀⣀⣀⣀⣀⣀⡙⠃⢹⠀⢇⠀⠀⠁⢸⡞⠀⠀⢠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠳⣈⠳⢤⣀⡈⠀⡶⠔⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠁⠀⠀⠈⠓⠦⢤⣽⠘⢦⣴⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠆⠈⠁⠧⠘⠂⠀⠘⠀⠐⠀⠉⠸⠀⠀⠀⠶⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⠆⢸⣿⠀⠀⠀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⡏⣿⣿⣿⣥⣼⣿⣤⣄⠀⣼⣿⣿⡇⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣰⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣯⣾⣿⡟⢠⣿⣿⣿⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠻⠿⠿⠿⠗⠘⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⢛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⣸⠀⡇⠙⠢⣀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠉⣩⢿⡟⠉⢻⡯⡉⠉⢷⠒⠒⠒⠒⠉⠀⠀⢠⣤⣤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡥⠚⢁⣶⣼⠶⡥⣧⡈⠒⢤⣀⡀⠀⠀⠀⢀⣔⣙⣛⣛⣉⣉⣉⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠀⠀⠀⢿⠋⠁⠀⠈⣸⣞⣣⣀⣜⣫⣇⠀⠀⠉⢹⠆⠀⠀⠀⡏⣿⣿⣿⣿⣿⢹⠁⡇
⠀⠀⠀⢀⠖⢉⠁⣠⢉⡑⢆⠙⠒⢤⡀⠈⠀⠀⠀⢰⢉⡳⣞⠉⣳⣞⠙⡄⠀⠀⠀⠀⠀⠀⠀⡇⣯⣿⣿⣿⣿⢸⠀⡇
⠀⠀⠀⢸⠀⣠⠋⢉⢫⠄⢸⠀⠀⠀⡇⠀⠀⠀⢀⠟⠛⠒⠚⠛⠓⠒⠙⠳⡀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⢸⠀⡇
⠀⠀⠀⠀⢣⠀⠗⡶⠳⢀⠎⠀⠀⠀⡇⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⡇⡏⣿⣿⣿⣿⢸⠀⡷⠂⠐⠲⢄
⠀⠀⠀⠀⠀⡇⢸⠀⡰⠃⠀⠀⠀⠀⡇⠀⡤⠤⠀⡠⠠⢄⢠⠀⠀⠀⠀⡰⡄⠀⠴⠒⢦⠀⠀⡇⣇⣿⣿⣿⣿⢸⠀⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⣿⣚⣖⡿⠀⠀⠀⠀⢀⠇⠀⠣⣄⢰⠁⠀⠨⢸⠀⠀⠀⢰⣁⣹⡀⢤⡤⠜⠀⠀⣿⠷⠿⠿⠿⢿⣾⡯⠃⠀⠀⡜⠁
⠀⠀⠀⠀⠀⢻⣒⣂⡇⣀⢀⣀⣀⠜⠀⠀⢄⡠⠗⢆⣀⠜⠸⠄⠀⠀⠏⠀⠀⠣⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⢤⣤⢤⣠⢤⢠⠤⢄⠤⣤⠀⡄⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⢀⠀⠓⠐⠊⠘⠙⠒⠘⠀⠘⢺⠇⣹⠀⠀⠀⡟⣒⣒⣒⠒⡋⡇⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠊⠐⢢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⡇⠉⠉⠉⠀⡇⣇⣀⣀⣀⣀⣀⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣇⠀⠰⢠⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠀⠀⡇⡏⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠙⡺⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡷⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⡏⠉⠙⣏⠉⠙⣍⢉⣫⣉⣣⡤⠤⢤⠀⠦⣤⠤⠤⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⣿⡉⠉⠙⢏⠉⠙⡄⠈⢣⠀⢳⡀⠸⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠛⠀⠙⡵⡒⠒⠛⢟⠂⠹⡉⠉⢯⠉⠱⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⡶⠾⠷⠶⠿⠻⡿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉

⠀⣄⠀⠀⠠⠀⢀⠀⠀⠃
⠀⠀⠚⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠗⢄⢡⣀⢰⣶⡦⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⣴⣿⣿⣿⠆⠀⠷⠈⣀⣀⡀⠀⠀⠀⠀⠀⠉⠠⠼⠷⡗⢸⣿⣗⠋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠉⠽⠿⠿⠿⠧⣤⠼⠿⠓⠒⠛⠂⠀⠀⠀⠀⠠⠤⢥⣿⣟⣿⣄⣭
⠀⠀⠠⠀⠠⠀⠂⣀⢡⣬⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠬⠤⡄⢐⣀⣴⠀⡲⠀⠠⠀⢠⠩⢙⠻⢿
⠀⠀⠀⠐⣶⠛⢷⠛⣏⠻⡎⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⢐⠒⢰⡤⠼⢇⣀⣃⣀⡌⠁⡐⠁⢐⡎
⠀⠀⠀⠀⠈⠁⢸⣆⢸⡆⠸⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢀⣀⠀⢀⠂⢀⡾⠀⢀⠀⠀⡈⠀⢩⠁⠁⡀⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⠘⠷⠶⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠿⠿⢶⠞⠠⠤⠆⣀⡐⠀⠀⠆⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠾⠂⠂⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣦⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⢀⠄⠂⠠⡇⠀⠀⠀⢱⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⡴⠻⠓⠒⡟⢣⠀⠀⠀⠀⠀⠀⣼⣿⣶⡄⠀⠀⠀⠀⠀⢀⣀⣀⣨⢨⣦
⠀⠀⠀⠰⠀⣤⢀⠧⠄⣷⠠⠞⠀⠀⠀⠀⢸⣯⣠⣿⡇⠀⠀⠀⠀⠀⢤⣴⣾⣥⣍⠹⠀⠀⠀⠀⠀⠀⠏⠈⠻⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡰⢫⠏⡍⠝⢢⠀⠀⠀⠀⢀⣀⡀⠀⢐⣤⡀⠀⠀⠀⠀⠄⣠⡶⢢⣤⠠⠀⠀⠀⠀⠀⠀⢘⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢤⣶⣬⡐⠂⡘⢸⠁⠀⠀⠀⠙⣿⣝⢷⡈⠉⢣⠀⠀⠀⠐⠀⣿⡥⢠⣿⠂⠂⠀⠀⠀⠀⠀⣨⣤⣥⠀⠀⠀⠀⠀⠀⠀⢻⢽⣼⢿⠁
⠀⠀⠀⠀⠀⠻⢷⡝⠨⠵⠋⠀⠀⠀⠀⠀⠈⠛⠓⠁⠒⠀⠀⠀⠀⠀⠁⠬⠁⡛⠡⠈⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡤⠤⢍⢹⠴⣤⠀⠀⠀⠀⢀⣴⣯⠏⠑⢄⠀⠀⠀⠀⠀⣀⣤⣀⠀⣴⡖⠀⠀⠀⠀⠀⢦⣤⣀⣤⣴⠀⠀⠀⠀⠀⠀⢀⡴⠳⣾⡀
⠀⠀⠀⠀⠀⣙⢮⡰⣎⠴⡁⠀⠀⠀⠀⢸⠈⢣⡄⣴⣾⡆⠀⠀⠀⠀⠘⢷⣟⡧⠉⠁⠀⠀⠀⠀⢸⢴⣾⠤⣿⣧⡀⠀⠀⠀⠀⢴⠋⢰⣦⡈⢳⡄
⠀⠀⠀⠀⠀⠳⠒⣋⣱⠒⠛⠀⠀⠀⠀⠈⠣⠤⠽⠛⠟⠁⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠁⢈⣾⡈⠀⠁⠀⠀⠀⠀⢸⣄⣈⣟⣁⣴
⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠔⠊⠃⠀⠀⠀⠀⠀⠀⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⡄⠀⠀⠀⠀⠀⠀⣠⡶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣶⡶⠶⠶⠀⠀⠀⠀⠀⢠⡝⠁⠈⢡⡄⠀⠀⠀⠀⠀⣿⣄⣴⣿⣧⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⠿
⠀⠀⠀⠀⠀⠀⢸⠫⢙⡇⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⣶⠀⠀⠀⠀⠀⠛⣥⣤⢠⣤⡍⠀⠀⠀⠀⠀⠈⣿⣿⠿⠇⠀⠀⠀⠀⠈⠻⠙⠛⠋⠟⠉
⠀⠀⠀⠀⠀⠀⠘⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣠⠴⢒⡒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠃⠺⡗⠉⢱⢹⡄⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⣠⣤⡏⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣦⣴⣦⣮⣦⡀
⠀⠀⠀⠀⠙⣅⠦⣘⡗⠔⣰⠀⠀⠀⠀⠠⠞⠃⣦⠛⠶⠄⠀⠀⠀⠀⠙⣿⡇⢰⣿⠋⠀⠀⠀⠀⠀⣙⠁⣿⠀⠉⠀⠀⠀⠀⠰⢿⠿⠿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣿⣿⣶⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠈⠉⠛⠛⠒⠀⠀⠀⠀⠀⠤⠄⣿⣿⣿⣿⣿⣿⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⠀⠀⠀⣤⡄⣤⡄⣤⡄⠀⠀⠀⡙⢿⣿⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⢠⣭⣥⣭⡅⣭⣅⠀⠀⠘⠁⠀⢠⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠈⢩⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⣶⣶⣶⣶⣬⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣭⣭⣭⣝⢿⣿⣿⣿⣿⣷⣄⠀⣠⣾⡆⣶⣘⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⡿⠿⣿⢫⡾⣫⣶⣴⡿⢻⣿⣷⣝⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢟⣭⣭⣵⣾⣿⣏⣾⣿⡏⣾⢳⣿⣿⡿⣱⠸⠟⣿⣿⡏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣧⣙⣿⡇⣿⡸⣿⣿⣥⣭⠻⣱⣿⣿⡇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣝⢿⣿⡸⣷⡹⣿⣿⣿⣰⡿⢿⢟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣮⠻⢶⣭⣝⠻⢳⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣭⣭⣶⣿⣿⠟⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⣄⠀⠿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣿
⠀⠀⣀⣀⣀⣠⣾⣿⣷⡀⠴⠂⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿⣿⡿
⠀⠸⣿⣿⣿⣿⣿⣿⠟⡐⠲⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠹⣿⣿⣿⡿⠿⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⢿⣿

⠀⣀⣴⣶⣦⣤⣶⣶⣶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠠
⠀⠉⣭⡤⣤⡰⣶⠰⣶⠰⣶⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⣿⡇⣿⡇⣿⠀⣿⢈⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠆
⠀⣀⣿⣧⣿⣧⣿⣦⣿⣶⣿⣶⠀⠀⠀⠀⣿⠀⣠⣦⣧⡀⢸⡇
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣶⣶⣿⡶⣿⣿⣿⣷⢾⣷⣶⣶⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠾⠿⠿⠷⠿⠷⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⠀⣤⡄
⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠛⠛
⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠟⠒⠶⠤⢤⡄
⠀⠀⢠⣧⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⢸⣇⣿⣼⣻⡇⣠⣤⢀⡀⠀⠀⠀⢠⣿⣿⣿⣠⣤⣶⣶⡶⠆⠀⠀⠀⠠⡔⠀⠄⠀⠀⠀⠀⢀⣤⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣤⣾⣿⣦⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠄⠂⡀⡐⠠⠀⠀⠀⠀⠀⢰⣖⡒⣒⣒⣖⣦⠀⠀⠀⠀⠀⣠⡤⡄⠠⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⢀⣶⣶⠂
⠀⠀⠀⠐⠏⠁⠈⠆⡀⠀⠀⠀⠀⣿⣭⡅⣭⡍⣭⣽⠀⠀⠀⠀⣼⣳⣓⣇⣀⡀⠀⠀⠀⠀⠀⣰⣄⠀⠉⠁⠀⢠⡄⠀⠀⠀⠀⠀⠾⣿⣷⠖
⠀⠀⠐⡀⠢⠄⠔⢁⠂⠀⠀⠀⠀⠓⠒⠒⣶⠖⠒⠚⠁⠀⠀⠀⢻⣽⡭⡯⣽⣭⠃⠀⠀⠀⠀⢿⣿⡷⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠉⠓⠓⠛⠁⠀⠀⠀⠀⠀⠈⠉⠳⠀⠸⠛⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢺⢛⡻⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⡿⣻⣤⣼⡟⣿⣇⠀⠀⠀⠀⣴⣿⣇⠀⡀⡀
⠀⠀⠀⣸⣄⣈⣽⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⡴⠚⠶⠋⠀⠀⠀⠀⠀⠉⢰⣂⣇⣒⣸⠉⠁⠀⠀⠀⠀⢻⠹⠃⠸⣿⡏
⠀⠀⠀⣿⣿⣾⣻⡇⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠞
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⣴⣿⣦⣠⡄⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣶⣾⡄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⡴⠶⠶⡄⣠⣤⡀
⠀⢀⣤⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⡏⠉⠉⠉⠉⣉⢹⡇⠀⠀⠀⠀⠀⡽⠿⣿⡂⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⢸⠀⠀⠀⠀⠀⠀⣇⣉⣁⡗⠘⡟
⠀⠈⣿⡟⠛⣿⡟⢻⣿⠁⠀⠀⠀⣇⣀⣀⣀⣀⣁⣸⠇⠀⠀⠀⠀⠀⢰⣶⣿⠄⠀⠀⠀⠀⠀⠀⢰⣯⣭⣭⣽⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠄⠇
⠀⠀⠿⠷⠶⠿⠇⠘⠿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

⠀⠀⠀⢀⣠⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢡⣶⣶⣶⡎⠂⠀⠀⠀⠀⣀⡔⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⡷
⠀⠀⠀⠘⡛⠛⢛⠃⡀⠀⠀⠀⠀⢸⣉⣿⠛⢻⣏⣷⠀⠀⠀⡔⢀⡐⢹⣆⠄⠘⣃⠐⡀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⢨⣿⣷⠀⢀⣄⢀⡀
⠀⠀⠀⢬⡯⠤⢽⡡⠀⠀⠀⠀⠀⠈⠿⠉⠉⠉⠿⠏⠀⠀⠀⠣⢈⠠⠋⠋⠀⠢⢈⠤⠃⠀⠀⢴⡿⠦⠤⠤⠞⣷⠄⠀⠀⠀⠈⠿⠀⠀⢿⡿⠟⠁
⠀⠀⠀⠈⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⢀⢀⣺⡄⠄⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠄⠄⠀⡀⡴⠂⠀⢔⡢⠀⠀⠀⢀⠂⢰⡄⠐⡴
⠀⠀⠀⠀⠠⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣁⡀⡝⡶⠖⠁
⠀⠀⠀⠀⢀⠠⣴⡀⠀⠀⠀⠀⠀⠀⠤⣖⣄⠘⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣀⣠⡀⠀⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠥⠼⡬⠟⠺⠫⢥⡈⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠁⠉⠀⠠⣄⠀⠘⠶⡆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢉⡖⠁⠀⠀⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⡗⢨⠘⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⡠⠈⢀⡀⠄⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠉⠉⢐⡌⢖⣅⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠄⢫⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢀⡀⠀⠀⠀⡀⠐⢀⡀⠀⠀⠀⠀⡀⠆⠀⢸⠀⠀⠂⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠿⠆⢰⠇⢸⢳⢰⢿⡇⠀⠀⠀⠀⣿⡶⡇⢸⡀⣗⠰⠾⡄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣄⣥⠊⠀⣾⡄⠸⠁⠘⡾⠀⠀⠀⠀⠀⢻⠏⠁⢸⣟⣮⠀⠑⢮⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⠂⠄⠀⠛⠁⠀⠀⢀⣠⢲⣦⠴⣴⡴⣀⣀⡄⢸⠈⠙⠃⠰⠐⠟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣴⡀⠄⠠⠇⠩⠄⠸⠇⠽⠏⣀⣨⠿⠈⠿⠁⠨⠆⠀⠠⠄⠦⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣤⣤⣀⣀⡀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠟⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠈⠛⠂⠂⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⢤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡀
⠀⠀⠰⣦⣼⣧⣴⠆⠀⠀⠀⢰⠃⢐⠂⠐⡂⠘⡆⠀⠀⠀⠘⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣯⡄
⠀⠰⠶⣿⣿⣿⣿⠲⠆⠀⠀⣌⣀⣈⣁⣈⣁⣀⣡⠀⠀⠴⠛⡏⠉⢂⣼⠦⠄⠀⠀⢸⣦⣿⣦⣷⣼⣿⡇
⠀⠀⠰⠟⢻⡛⠻⠆⠀⠀⠀⠉⠉⠉⢹⡏⠉⠉⠉⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⢸⠦⡷⢼⡦⣿⣿⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠂⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠁⠀⠀⠀⠚⠒⠛⠚⠓⠛⠛⠓
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠀⠀⠀⠀⣠⣤⣄
⠀⢀⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⣠⣿⣿⣿⣦
⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⢲⣶⣦⣿⢿⠿⠋⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⡇⠀⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⣤⣶⣿⣿⣶⣤⠀⠀⠀⢸⣿⠟⠛⠛⠻⣿⡇⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣷⡖⢲⣿
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⢠⠄⠠⡄⢸⡇⠀⠀⠀⠘⡄⠀⠀⢠⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠛⣿⠛⠛⣿⠛⠀⠀⠀⢸⣷⣄⠀⠀⣠⣾⡇⠀⠀⠀⠀⠈⣶⣶⠁⠀⠀⠀⠀⠀⠀⣿⣟⣻⣿
⠀⠀⠀⠛⠀⠀⠛⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛
⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣿⡿⠇⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘⠿⠏⠹⠿⠃
⠀⠀⠀⣠⠂⡖⠐⡀⠀⠀⠀⣠⣴⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⠆⠀⠀⠰⣿⣷
⠀⢀⣿⠿⣿⣿⣿⠿⣷⠀⠀⢹⣿⠛⣿⣤⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢷⣶⡆⢰⣶⣾⠇
⠀⠀⠑⠒⠉⠉⠉⠒⠁⠀⠀⠈⠛⠀⠛⠛⠛⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠈⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢸⡇⢸⡇⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣦⠀⠸⣇⠹⡆⠸⡇⢸⠇⢸⠇⣸⠇⠀⣤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡈⢷⣄⠹⣆⢿⣀⣷⣾⣀⡿⣰⠏⣠⡾⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡈⠙⢶⣽⣶⠟⠋⠉⠀⠀⠉⠙⠻⣶⣯⡴⠋⣁⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣌⣙⠳⢶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⢞⣋⣡⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣈⣉⣙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣋⣉⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣤⣤⣬⣭⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢭⣥⣤⣤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⡴⠟⣉⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⠛⠶⣤⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⠾⢋⡾⠋⣿⢲⣤⣤⣤⣤⡖⣿⡙⢧⡙⠷⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢁⡴⠿⠷⡾⢧⠿⠶⡿⢿⠶⠿⡼⢷⠾⠿⢦⡈⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣱⣁⣀⣠⢛⣇⣀⣀⡄⢠⣀⣀⣸⡚⣄⣀⣈⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣜⣀⣀⣀⡌⣎⣀⣀⣀⡅⢨⣀⣀⣀⣱⢡⣀⣀⣈⣣⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⠊⠉⠉⠉⡹⢰⠉⠉⠉⠉⡧⢾⠉⠉⠉⠉⡆⢏⠉⠉⠉⠑⡜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⡡⠋⠉⠉⠉⢉⠃⡎⠉⠉⠉⠉⡃⢘⠉⠉⠉⠉⢱⠘⡋⠉⠉⠉⠙⢌⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠛⠓⠛⠛⠛⠛⠒⠓⠚⠛⠛⠛⠓⠚⠛⠛⠛⠓⠚⠓⠛⠛⠛⠛⠛⠛⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠉⠉⢹⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠶⠶⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠲⠤⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣠⡾⠟⠛⠛⠛⠳⢶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢘⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠙⣿⣦
⠀⠀⠀⠀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⣴⣶⡄⣶⣶⣶⣶⠶⠖
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣼⠋⠉⠉⢹⠋⠉⠉⢹⡏⠉⠉⠙⡏⠉⠉⠉⣧⠀⡇⠀⠀⢀⣶⣮⣭⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠰⠿⣧⡀⠀⠀⠀⢰⠿⠒⠒⠲⣿⠖⠒⠒⢾⡷⠒⠒⠲⣿⠖⠒⠒⠿⣆⠘⠀⢀⣾⡟⢹⣿⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⢈⣷⣦⣀⢠⡟⠀⠀⠀⣰⣇⠀⠀⠀⣸⣇⠀⠀⠀⣸⣇⠀⠀⠀⢹⡄⠀⡾⠋⠀⢸⣿⠀⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⡾⠛⠛⠛⠛⣿⠟⠛⠛⠛⢻⡟⠛⠛⠛⠻⣿⠛⠛⠛⠛⢷⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠈⣧⠀⠀⠀⢸⣿⠒⠒⠒⠒⠒⣿⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠿⠆⠀⠀⢸⣿⠀⠀⠀⢐⣿⣿⣿⡿⣿⠉⢹⡻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⣿⠀⠀⠀⢰⣿⣿⡟⣴⣏⡀⠛⣿⡎⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠸⣿⣿⣧⠻⣿⡇⠘⣿⢇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠰⣿⣿⣿⣷⣬⣽⣼⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⡤⠠⡄
⠀⠦⠤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣇⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡄⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⠀⣛⣛⣛⣛⡋⢘⣛⣛⣛⣛⣛⠀⣛⣛⣿⣿⡿⢛⣩⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⡋⢘⣛⣛⣛⣛⣻⡇⣸⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⠀⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠃⠿⠿⠿⠿⠿⠿⠀⠿⠿⠷⠸⣿⣿⣌⣋⣴⣿⣬⣉⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣷⠀⠀⠙⢿⣿⣿⢹⣿⢹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⠀⠀⠀⠈⣿⣿⢸⣿⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⢀⡀⡀⣀⠀⢀⣀⣀⢀⣀⣀⠀⣀⣀⡀⢀⠀⠀⠈⠈⠀⠌⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠸⠀⠀⠃⠀⠀⠀⠛⠀⣿⠉⠀⠈⠅⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡬⡁⡀⠀⠁⠘⠰⡆⡉⡉⢉⠍⠉⢔⣏⠶⠔⠖⠓⠔⢀⠀⡀⠀⢐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠓⠉⠀⠀⡀⢀⠄⠀⠁⠀⢀⠤⢈⠔⠈⠀⠀⠀⠀⠀⠀⠃⢓⠁⠀⠠⠄⠀⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⢈⠂⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣤⠀⡄⠱⡤⠄⠤⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠨⠄⠀⠂⠁⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠅⠠⠀⠠⠠⠀⠀⠔⠀⠈⢢
⠀⠀⠀⠀⣀⢠⡠⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠂⠀⠀⠀⠁⠀⢀⠀⡀⠀⠀⠁⠑⠀⠘⢠⠃⠈⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠘⠙⠁⠀⠀⠀⠆⠀⠀⡀⠀⠐⠁⠀⣀⠀⠀⠀⠀⠀⠁⡐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠄⠀⠀⢨⣄⡀⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠔⠀⣀⡀⣀⡠⠠⠢⠠⠠⠒⠒⠒⠐⠐⠑⠐⠐⠀⠊⠉⠉⠩⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣶⣶⣶⣦⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠿⠟⠛⡛⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠁⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⠛⠛⠉⠉⠁⠁⠁⠀⠀⠁

⣿⠿⠿⠿⠿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠈⠛⠻⠛⢻⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⢰⣶⣶⣾⣶⣶⣶⡆⠀⠀⠀⢰⣷⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠾⠿⠿⠿⠿⠿⠇⣴⡆⢰⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆⣿⡇⠸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠭⠭⠭⠭⠭⠤⠭⠭⠭⠭⠥⣾⡇⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣭⣭⣭⣭⣭⣬⣭⣭⣭⣭⣥⣼⡇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣛⣛⡛⠟⢛⣛⡛⠟⣛⣛⡛⢿⡇⣿⡇⠈⣿⢹⡏⣿⢹⡏⣿⢹⡏⣿⢹⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⢸⡇⣿⡇⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠛⠃⠀⠛⠛⠃⠀⠛⠛⠃⠘⠃⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣴⣾⣿⣿⣶⣦⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⡦⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⢴⣶⣶⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⡄⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣷⡆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡆⢰⣶⣶⣶⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡄⢀⣿⣿⣿⣿⣿⣿⡇⢠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⠂⠉⠱⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠁⠀⠀⠀⢀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠠⠤⠤⣀⣀⡠⣀⣀⡠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⢀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣯⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠢⡄⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠐⠚⠀⠀⠀⠘⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⡖⠉⠉⠲⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠒⠁⡇⠀⠀⠀⡇⠀⠀⡎⠀⠀⣈⣄⣀⣀⣀⣀⣀⣀⣀⣀⣸⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠤⠀⠐⠂⡇⠀⠀⠀⣇⠠⠊⠀⠀⠀⠇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⣻⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡖⢒⣒⠒⡂⠀⠀⠀⠀⣏⣀⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⢘⠒⠂⠀⠒⢺⠀⠀⠀⢸⠒⠒⠀⠀⠐⠒⠚⠁⠈⠒⠂⠀⠀⠐⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⣟⣐⠒⠚⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠒⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠈⠀⠀⡚⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⠀⢈⡱⠀⣎⡃⠀⡏⡇⠀⡏⠆⠰⣉⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢸⠰⡆⢰⠆⢰⠆⢰⠆⢶⠆⢶⠀⢶⢸⡆⠀⡖⡄⡖⠓⠂⠂⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠐⠒⠚⠓
⠀⠀⠀⠀⢸⢰⡆⢰⡆⢰⡆⢰⡆⢴⠄⣶⠀⣶⢸⡁⣀⡉⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⣀⣄⣠⣄⠀⠀⠀⠀⠀⣤⣀⣀
⠀⠀⠀⣤⣤⣤⣴⣶⣶⣶⣤⣤⣤⠀⠀⠀⢠⡏⠀⠀⠘⡆⠀⠀⠀⠀⢸⣿⢿⣿⢸⣿⡀⠀⠀⢀⣼⣳⢣⢯⡟⠳
⠀⠀⢸⡿⡿⠇⣿⣿⣿⣿⡸⡿⢿⠧⠀⠀⠈⠳⣄⣠⡴⠃⠀⠀⠀⠀⠼⠿⢽⣿⠽⠿⡧⠀⠀⠀⡗⢿⣯⡟⣰
⠀⠀⠀⠀⠁⠀⠉⠹⠏⠉⠀⠈⠁⠀⠀⠀⠰⠀⠀⡀⠀⠲⠀⠀⠀⠀⡔⠒⠒⠁⣀⣀⣀⠀⠀⠀⠙⠺⠿⠇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠃⠊
⠀⠀⠀⢀⣀⡀⣀⡀⣀⣀
⠀⠀⠀⢘⣿⣟⣿⡟⣿⣟⠀⠀⠀⠀⢐⢐⡂⣒⢐⡂⡂⠀⠀⠀⠀⠀⡴⠛⠙⠳⣆⠀⠀⠀⢀⣿⣻⣿⢻⣿⢿
⠀⠀⠀⢘⣿⣟⣿⣟⡿⢛⡀⠀⠀⠀⠨⢨⡅⣭⢨⡅⣥⠡⠀⠀⠀⠸⡇⠀⠀⠀⣼⠁⠀⠀⣼⣿⣿⣟⣟⣻⣃
⠀⠀⠀⠘⠿⠟⠿⠏⣿⢿⢉⡆⠀⠀⠧⠼⠤⠶⠤⠦⠴⠐⠀⠀⠀⠀⠙⠲⠶⠞⠉⠀⠀⠀⠙⠛⠻⠯⢺⣇⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⢉⡇⠀⠀⡀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣷⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠐⠀⠀⠀⠀⡞⠀⠀⢠⠃⠀⠀⠀⠀⠙⢿⣿⣿⠯⠁⠀⠀⠀⢸⣴⣷⡧⣾⣾⡦⠀⠀⠀⠀⢸⣿⣶⠆
⠀⠀⠀⠩⠉⠭⠉⠯⠩⠍⠹⠅⠤⠈⠀⠀⠀⠀⠁⠹⣈⣀⡀⠀⠀⠀⠬⠻⠿⢏⠻⠿⠫⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⡴⠚⠓⣄
⠀⠀⠀⠹⠤⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⡐⠄⠀⠀⠀⡀⢄⣆⡄⣀⠀⠀⠀⠀⣤⣤⣤⣄⣤⣄
⠀⠀⠀⠀⠀⠀⣸⡿⢳⡿⢣⣿⡿⠁⠀⠀⠻⣿⡿⢿⣿⡿⡀⠀⠀⣐⣽⡿⠻⣿⣗⡀⠀⠀⠰⣿⡷⣿⡷⣿⣷
⠀⠀⠀⠀⠀⣬⡿⣡⡿⣡⣿⡽⠁⠀⠀⠀⣾⣿⣷⣾⣿⣷⠄⠀⠀⠀⣻⣿⣶⡿⡓⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠉⠉⠑⠉⠙⠉⠉⠁⠀⠀⠀⠀⠊⠉⠉⠈⠉⠑⠃⠀⠀⠀⠁⠁⠃⠁⠈⠀⠀⠀⠀⠘⠓⠚⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣭⣭⣭⣭⡅⢀⣭⣭⣭⣭⣭⡅⢨⣭⣭⣭⣭⣭⡀⢨⣭⣭⣭⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⣤⣤⣤⣤⣤⣤⡀⢀⣤⣤⣤⣤⣤⣤⠀⢨⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣠⡖⡆⡇⡰⡆⡶⠀⢰⢲⣄⣶⣴⣶⣤⣶⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠁⠉⠁⠁⠀⠘⠉⠀⠉⠀⠀⠁⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣤⣾⣿⡆⠀⣀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⣠⢠⠔⢀⣤⣽⣿⣿⣿⣿⣿⢸⣿⢿⣿⣿⠿⠇⠀⡀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣀⣀⣠⣿⣾⣶⣮⣿⣿⣿⣿⣿⣿⢿⣾⣿⣾⣿⣿⣄⣤⣤⢡⠀⠒⠒⠐⡦⢠⡤⣴⣦⣴⡶⠾⣱⣿⣿⣦⡁
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁⠀⠀⠁⠁⠀⠈⠛⠳⢿⣿⣿⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⠀⠀⠘⡧⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⡤⠐⠂⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⢤⣿⢻⢿⣳⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⣦⣴⣿⣿⣿⣯⣯⢿⣶⢢⣎⠄⠀⠀⠁⠀⠒⠆⠐⡂⠠⢀⡄⠀⠀⣠⠟⡿⠄
⠀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣟⢵⣷⢛⠽⣩⢟⢒⠮⡴⢀⣀⠀⠀⠀⠈⠀⠁⣾⡗⠒⠐⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠿⣗⠿⠻⡳⢢⡜⡽⡣⢂⠀⢡⠆⡐⠁⢀⡼⠀⢀⡞⠹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠅⣊⠞⠖⠔⡜⢁⠞⡠⣦⣿⣯⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠪⢔⡉⣨⡟⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠂⠁⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⣀⣀⢀⣀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢁⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⢄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⠀⠀⠁⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠰⠉⠉⡂⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⡁⡄⠀⠀⠀⠀⡰⠀⠀⠴⠦⠤⠤⠤⠤⠤⠤⠼⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⢥⠀⠀⡄⠀⠀⠘⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠻⠀⠀⠇⠀⠀⢸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⢸⠀⠀⠀⡼⠈⠀⠀⣀⣐⣂⣒⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠃⠀⠀⢸⠀⠀⢠⠧⢄⣬⣭⠉⠀⠀⠀⠈⢭⣥⠬⡆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⠀⠀⠀⠆⠰⠂⠀⠀⠰⠀⠀⡜⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲
⠀⠀⠀⠀⠀⠀⠀⢐⡢⠀⣲⠀⣒⠀⢐⡂⠀⣲⠀⣒⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⢙⠀⡋⢘⡁⣋⢘⡃⣛⠀⡃⠇⠀⠝⠈⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠁
⠀⠀⠀⠀⠉⠀⠉⠈⠁⠉⠈⠁⠉⠀⠡⠉⠉⠉⠡⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⡀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠼⠶⠆⠐⠠⠐⣀⣠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢨⠂⠀⠀⠀⡆⢠⢻⣗⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣰⡡⠀⠀⢀⡏⠀⠀⠀⡘⢀⠆⠀⠹⣔⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠤⠦⣄⠀⠀⠀⢠⡁⠉⢢⢐⢊⠠⠠⠤⠴⢁⡎⠀⠀⠀⠹⡧⠜
⠀⠀⢀⠀⢤⢼⡤⠣⢴⡤⠺⠇⠀⠐⠀⡜⠐⠀⠈⢰⠁⠀⠀⢨⠂⡜⠀⠀⠀⠀⠀⢹⡌⠡
⠀⠀⠀⠁⠠⣭⠇⠀⠈⠀⠠⣐⡲⣤⡴⣅⣀⡀⣠⡁⠁⠀⢀⠆⡼⠀⠀⠀⠀⠀⠀⢸⠿⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⣧⡈⠉⠉⠐⠊⠡⠭⠱⠒⠎⡰⡇⠀⠀⠀⠀⠀⠀⢸⡠⢀⣨⠯⠦⠄
⠀⠀⠀⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣄⣤⠶⠚⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⢴⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡀⣤⣢⣸⢃⡔⢴⠰⣄⠀⠀⠁⣼⢀⠟⡴⢢⠠⡆⣤⢠⢰⢰⠇⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠘⠉⠣⠊⠸⠘⠇⠻⠀⠓⠀⠀⠀⠟⠀⠀⠳⠚⠚⠃⠻⠘⠆⠚⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣦⢰⢶⣶⣶⣶⡆⠀⢶⣦⣶⢠⣴⣶⣴⣶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣀⣀⣀⣀⣀⣾⣃⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣹⣆⣀⣀⣀⣀⣀⣣⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠈⠉⢹⡏⠈⠉⠉⠉⠈⣽⠁⠉⠈⠈⠁⠁⠉⠈⣿⠈⠉⠁⠉⠉⠈⣿⡈⠉⠉⠉⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⣿⡛⠛⠛⠛⠛⠛⠻⣟⠛⠛⠛⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⣼⣧⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣴⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣷⡀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⢸⣧⣦⣶⣤⣴⣦⣤⣤⡾⣧⣤⣤⣦⣤⣴⣤⡼⣧⣦⣤⣤⣤⣤⣤⣤⣤⣴⢿⣴⣦⣤⣤⣤⣤⣤⡿⣦⣦⣴⣦⣤⣴⣤⣴⣷⡄
⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠃
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⡟⢩⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⢀⣌⣽⣿⣽⣿⣯⣿⣿⣿⣿⣷⡀⢀⣶⣶⣶⣶⣆⠀⠀⠀⠩⣿⡇⠈⠁⠀⠀⢠⣶⣶⣶⣶⡄⠀⣴⣄⣿⣿⣿⣽⢿⣿⣿⣯⣿⡄
⠀⠀⠾⠶⠾⠷⠾⢷⣶⠾⠷⠾⠷⠶⠷⠈⠉⠉⠏⠉⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠹⠉⠉⠁⠸⠷⠾⠷⠶⠷⣶⡾⠷⠶⠿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⣒⠛⠿⠧⠘⠓⣒⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡴⠶⣤⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣰⠟⠁⠀⠀⠀⠈⢷⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠻⣆⠀⠀⠀⠀⢀⣼⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠶⠶⠾⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡍⠉⠉⠉⠛⠛⡟⠛⠻⠿⠿⠿⠷⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⠉⠀⠀⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⠉⠉⠹⡍⠙⠛⠛⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠉⠉⠉⠛⠛⢻⡖⠒⠒⠒⠢⠤⠾⣦⠤⠤⢤⣀⣀⣵⣄⣀⣀⣀⠀⢹⣧
⣴⣶⢦⣤⣤⣤⣤⣀⣀⣀⣀⡀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⢻⠀⠈⠉⠉⠉⢻⣧
⠹⣿⣆⠀⠀⠈⡄⠈⠉⠉⠹⡉⠛⠛⣿⣿⣟⠙⠓⠒⠒⠚⢿⠖⠒⠶⠤⠤⠤⢷⣤⠤⠤⢄⣀⣠⣧⣀⣀⣀⣀⡀⣿⣆
⠀⢻⣿⡄⠉⠉⠙⡃⠀⠒⠒⠳⠒⠂⠺⣿⣿⣆⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠉⠉⢿⡆
⠀⠀⢻⣿⡄⠀⠀⠶⡄⠀⠤⠤⢧⡄⠀⢹⣿⣿⡖⠒⠒⠒⠢⠾⣶⠤⠤⠤⠤⠤⠬⣷⣤⢄⣀⣀⣀⣸⣦⣀⣀⣀⣀⣘⣿⡄
⠀⠀⠈⢿⣷⡀⡀⢀⣰⣀⡀⠀⠀⣂⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⡅⠀⠀⠉⠉⠙⣿⡀
⠀⠀⠀⠈⣿⣧⠀⠀⠀⢣⠀⠀⠀⠉⠏⠉⠉⢿⣿⣷⡒⠒⠒⠶⠶⢿⣦⠤⠤⠤⠤⠤⢼⣦⡤⠤⠤⢤⣄⣽⣄⣀⣀⣀⣀⣹⣷⡀
⠀⠀⠀⠀⠘⣿⣏⠉⠉⠈⢛⠀⠀⠀⠚⡖⠐⠘⣿⣿⣧⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠹⣿⣦⣤⣤⣬⣦⣤⣤⣤⣴⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢤⣤⣿⢤⣬⣭⣭⠉⠉⠉⠉⠁⠉⠁⠉⠉⠉⢿⡉⠈⠉⠈⡏⠈⠁⠉⠁⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠗⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠿⣿⣛⣿⣛⣳⣶⣶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠶⠶⠖⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡁⣠⣄⡾⡠⢤⢠⡀⠀⠊⣿⢸⢦⢤⢠⣤⡄⡠⢸⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠋⠃⠋⠓⠓⠛⠈⠃⠀⠀⠋⠀⠘⠚⠚⠁⠋⠓⠘⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠆⠳⠶⠶⠶⠀⠶⠶⠆⠶⠶⠷⠾⠂

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣯⣩⣩⣍⣍⣿⣁⣁⣉⣉⣽⣅⣈⣀⣀⣀⣀⣿⣁⣁⣈⣁⣹⣍⣉⣉⣌⣹⣧
⠀⠀⠀⠀⠀⢠⣟⣀⣀⣀⣀⣰⣏⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣈⣷⣀⣀⣀⣀⣘⣧
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢠⠏⠉⠉⠉⠉⠉⣹⠏⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⢿⡉⠉⠉⠉⠉⠙⣧
⠀⠀⢠⣿⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⡼⣧⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣇
⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣀⣀⣀⡀⠀⠀⠾⣿⡇⠰⠀⠀⢀⣀⣀⣀⠀⠀⣤⣤⣤⣤⣤⣶⣶⣶⡆
⠀⢀⣮⣽⣯⣽⣤⣼⣯⣽⣭⣧⠐⠛⢻⠛⠓⠀⠀⠀⣿⡇⠀⠀⠀⠚⠛⠟⠛⠃⣴⣯⣿⣽⣯⣤⣿⣬⣯⣽⡄
⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣆⣉⣻⣇⣈⣀⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠶⠶⠆
⠀⠀⠀⠀⠀⣾⣾⡇
⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣄⣠⣄⣠⣤⣄⣠⣤⣄⣤⣄⣌⠛⠛⠛
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⡖⣶⣶⣶⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⡆
⠀⠀⠀⢴⡓⠒⢒⣲⣖⣶⣓⣟⣟⡟⣓⡆⠀⠀⠀⢀⣗⡀⠀⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⢿⣣⣾⣷
⠀⠀⡿⠭⠉⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠇⢰⡀⠈⡋⠀⠀⠀⠀⠀⢿⠀⠼⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣧⣿⣿⣿⣧
⠀⣀⣓⣀⣀⣐⣂⣀⣀⣒⣂⣰⠐⠀⢲⣰⡆⢨⠁⠐⢋⣤⣀⠀⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⣿⣏⣙⢻⡏
⣀⣧⣤⣤⣤⣥⣤⣭⣬⣟⣦⣼⣀⣀⣰⣾⣃⣀⣀⣀⣻⣟⣻⣹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣚⣻⣧⣾⣿⣶⣾⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⣠⢦⣄⣀⡀
⠀⠀⢠⣶⣶⢴⣿⣿⣿⢶⣶⣦⠀⠀⠀⣾⠀⠀⠈⡇⠀⠀⠀⠀⣿⡟⣿⣿⣿⡄⠀⠀⠰⣯⣯⣞⡾⠉
⠀⠀⠚⢻⠛⠸⠿⡿⠿⠞⢻⠛⠃⠀⠀⡈⠓⠶⠚⢁⠀⠀⠀⠐⠛⢓⡻⠓⠛⠓⠀⠀⢸⣏⣿⣮⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠘⠀⠀⠀⠀⠤⠤⠤⣶⣶⢢⠀⠀⠀⠀⠉⠛

⠀⠀⠀⢸⣿⢾⣷⢾⡷⠀⠀⠀⠀⡠⠤⠠⠄⡄⠄⠀⠀⠀⠀⢀⣤⠶⠤⣄⠀⠀⠀⢠⣶⣴⣶⣶⣶
⠀⠀⠀⠸⣿⢾⡿⢿⡿⠀⠀⠀⠀⣃⣙⣘⣀⣇⣀⢀⠀⠀⠀⣼⠁⠀⠀⢹⠆⠀⠀⣸⣟⣿⢷⢿⡿
⠀⠀⠀⠸⠿⠻⠿⢻⢿⠛⡆⠀⠀⠄⠧⠼⠤⠧⠬⠠⠀⠀⠀⠈⠷⠤⡴⠛⠀⠀⠀⠛⠿⠿⡹⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⡅⠀⠀⡀⠀⠀⠀⢐⣾⣿⣿⣖⠀⠀⠀⠀⠸⣿⣿⢿⣿⡿⠀⠀⠀⠀⣸⡟
⠀⠀⢀⣀⣀⣀⣀⡜⣀⣀⡰⠀⡀⠀⠀⠀⢻⢿⠿⠃⠀⠀⠀⠀⡷⣿⣿⣾⣿⣷⡄⠀⠀⠀⠿⣿⡟
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⢀⣈⣀⣀⡄⠀⠀⠀⠓⠒⢒⣖⠛⠛⠃⠀⠀⠀⠠⠏
⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠁⣤⡆⣤⢄⣤⣴⠀⠀⢰⣿⣿⢾⣿⣷⠀⠀⠀⢘⣾⣾⣾⣞⠀⠀⠀⢸⣿⣿⣿⣻⣿⡀
⠀⠀⠀⠀⢀⣾⢟⣾⢛⣾⡳⠁⠀⠀⢠⣿⣿⣼⣿⣧⠀⠀⠀⠻⣿⣦⣿⡿⠃⠀⠀⣸⣿⣾⣿⣾⣿⡆
⠀⠀⠀⠀⠚⠛⠚⠛⠚⠛⠁⠀⠀⠀⠨⠛⠛⠘⠛⠫⠀⠀⠀⠈⠙⠹⠙⠉⠀⠀⠀⠉⠹⠥⠿⠼⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⡜⠘⡌⠣⣄⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠯⢤⣴⡷⠶⣷⣶⠶⠿⣆⣀⣤⣤⡤⠤⠞⠀⠀⢀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⡴⠋⡟⢧⣠⠞⡏⠳⣄⠘⠃⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⠶⠧⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠶⠶⣤⡴⠶⠮⠥⠤⣼⣫⡟⠙⢯⡿⡤⠤⠽⠶⢶⣄⠀⠀⠀⠸⣷⣶⣶⣶⣶⣶⣶⡞⢻
⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⣀⠘⣄⠀⠀⠀⠹⠃⠀⠀⠀⢰⣿⠤⣷⢶⡧⢽⢷⠀⠀⠀⠀⠟⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⡇⢹
⠀⠀⠀⠀⡎⠀⢄⡠⢇⡴⠈⣧⠀⠉⠓⡄⠀⠀⠀⠀⢀⣏⡤⠛⢇⣀⡝⠣⣌⣇⠀⠀⠀⠀⠀⠀⠀⠀⡗⣿⣿⣿⣿⣿⢿⡇⣼
⠀⠀⠀⠀⡇⠤⢏⠀⣬⡗⠆⣼⠀⠀⠀⡇⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⢸⡇⣿
⠀⠀⠀⠀⠙⣄⠸⣹⠏⠓⡰⠃⠀⠀⠀⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⢸⡇⣼⠒⠒⠒⠒⣄
⠀⠀⠀⠀⠀⢸⠀⣇⢀⡞⠀⠀⠀⠀⠀⡇⠀⡤⠒⠄⢠⠖⠲⡄⢰⠀⠀⠀⠀⡰⢆⠀⢰⠒⠒⡄⠀⠀⡟⣿⣿⣿⣿⣿⢸⡇⣻⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⢸⣯⣿⣾⡇⠀⠀⠀⠀⢀⡇⠀⠳⢤⡀⡏⠀⠀⢸⢸⠀⠀⠀⢰⣇⣸⡆⢸⣤⠴⠃⠀⠠⣟⣻⠿⠿⠿⣿⣾⡯⠟⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠸⣗⣒⣺⠇⠀⠀⠀⣀⠜⠁⠀⢠⣀⡽⠳⣀⣠⠏⢸⣀⣀⡀⡎⠀⠀⠱⠸⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⣀⡤⣄⣠⣤⢠⡤⢄⣤⣤⡀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠸⠭⠥⠇⠘⠫⠭⠸⠀⠘⠴⡇⢱⠏⠀⠀⢰⠿⠤⠤⠬⢽⠟⡇⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠴⠢⢤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠊⠀⠀⠀⢸⠐⠛⠛⠓⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡇⠀⢀⠄⡇⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣴⠀⢸⡇⡗⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠦⣬⠞⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⣏⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⠚⠀⠀⠁⣤⠤⠤⣤⠶⠶⢤⠶⠶⡶⠶⣄⠀⠀⠀⠀⢸⠀⠈⠁⠀⢸⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡤⠤⠬⣦⠤⠬⢧⠴⠾⡶⠚⢯⠉⠉⡆⠈⠉⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣃⣀⣀⣀⣀⣸⣷⡀⣀⣈⣆⣀⣈⣧⣠⡽⣤⠬⢧⡀⠳⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠈⣧⡀⣘⣆⣀⣙⣄⣀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣟⡟⠛⠛⠛⠛⣏⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⢺⠃⠈⠋⠈⠙⡆⠀⠐⠀⠀⠀⠀⠀⠂⠀⢰⠁⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⢂⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠴⡆⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⠀⠀⡇
⠀⠀⠀⠀⢸⣶⣶⣶⣶⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡀⠀⠀⠀⢈⠀⠀⡇
⠀⠀⠀⠀⠘⠛⠙⠋⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣷⣿⣔⠀⠀⠃
⠀⠀⠀⢀⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢹⣓⣚⣃⠀⡀⡶⠂⠐⡂⠀⠀⠀⠀⡄
⠀⠀⢀⣴⣶⣀⣶⣿⣦⣤⣴⣦⣤⣤⣶⣤⣤⣴⣤⣤⣤⣼⡄⠀⠈⢯⣽⣭⣭⣭⣭⣭⣭⣭⣤⠀⠀⠰⣷⠀⠀⠀⠀⢠⡄
⠀⣶⣿⣿⣿⣿⡿⣿⡿⠋⠛⠋⠉⠋⠛⠙⠉⠉⠛⠉⠙⠛⠙⣷⠾⠿⠯⠿⠿⠯⣿⣽⣿⣿⠉⠀⡗⠠⡧⢀⠀⠀⠀⢘⡇
⣰⣿⣯⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣅⣀⣁⣀⣐⣿⣿⣺⣷⣦⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠘⡆⢀⠀⢤⠀⠠⠀⢠⠃⢀⠌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠀⠠⡌⡇⢸⠀⢰⠀⠘⢀⡄⣰⠃⠠⠊⠀⢀
⠀⠀⠀⡀⠀⠢⣀⠀⠳⢀⠙⡄⠳⠹⠘⠀⠘⠀⠃⠜⡰⢀⠔⠀⠀⠄⠉
⠀⠀⠀⢈⠓⠆⢀⠐⢤⡈⠃⢀⡤⠖⠚⠉⠉⠙⠒⠦⣄⠉⠠⠞⠁⠠⠖⠋
⠀⢀⡀⠈⠉⠀⠦⣅⡠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠚⣁⡤⠴⠂⢀⣀
⠀⠀⠀⠈⠉⠃⠐⠤⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠱⠂⢈⣁⣁⡀
⠀⠀⠀⠉⠉⠙⠀⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⣉⡉
⠀⠀⠐⠒⠒⣀⡉⠩⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⣀⠉⡉⠁⠈⠁
⠀⠐⠒⠀⣉⠁⠖⠊⣠⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⠲⠄⢀⠉⠉
⠀⠀⠀⠁⠀⠠⠖⠈⣠⠆⢀⠙⠦⣄⡀⠀⠀⢀⣀⡤⠞⢁⠐⢦⡀⠲⢬⡉⠓⠆
⠀⠀⠀⠀⠀⠀⠤⠈⢀⡄⢁⡖⠤⢀⠈⡉⢉⠉⡀⡄⠰⡈⠢⡀⠈⠃⡄⠈
⠀⠀⠀⠀⠐⠃⠀⠠⠎⢠⠌⡴⠀⡞⠀⠇⠸⠇⠧⠠⡀⠑⡀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⢀⠁⠠⠄⠀⡆⠐⠀⢳⠀⢈⢤⡤⣥⢤⠤⡤⣤⢤⠤⡤⣤⢤⡤⡤⢤⢤⡤⡤⢤⢤⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠷⠀⢀⡾⡴⠤⠦⠦⣼⢥⠧⠬⠴⢤⠯⠴⠴⠤⠆⡼⠴⠥⠦⡴⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠖⢲⠒⡖⢒⡖⣲⠒⡞⠛⠗⣻⠚⡖⢻⠛⠺⠓⡿⢻⠚⠳⠓⡞⢺⢷⠆⡖⠰⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⡞⢒⠓⠲⠓⡾⢲⠓⡺⠒⡞⢲⠟⡾⠒⠞⢲⠓⣾⢓⠖⢲⠃⡾⢒⡟⠞⠺⠲⠓⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡼⠠⡞⢠⠇⣼⠣⠞⢰⠇⡼⢠⡿⠰⠃⡾⢠⠆⢴⠇⡾⢠⠇⢴⠃⡞⠙⢰⠃⠎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢴⠧⡼⢤⠮⣴⠧⡼⢤⠧⣴⠥⡾⢤⠧⣴⠥⡾⢤⡯⡴⠤⡾⢤⠧⣼⠻⢁⠋⡘⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢥⠧⣴⠥⡼⢠⡯⣴⢥⡮⢤⠧⣼⢥⡮⢤⠧⣼⢤⡾⢤⠥⡼⢤⡮⣴⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣃⣎⣠⡇⣰⢁⣾⣠⣅⣼⢀⡎⣰⣇⣼⢀⡆⣠⢃⣼⢡⡇⣰⢁⣼⢠⡏
⠀⠀⠀⠀⠀⠀⢠⣇⣸⣀⣎⣠⣃⣼⣁⣆⣰⣁⣜⣀⣏⣰⣁⣜⣀⣇⣸⣃⣌⣠⣃⣰⣁⡞

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⢶⡉⣭⢩⣤⣩⣭⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣼⠀⠀⠀⢸⠛⠃⠛⠀⠉⢘⣻⡻⠤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⣤⡄⠀⢀⠀⠀⢀⣋⣹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡖⣾⠻⠞⢷⣿⣽⠁
⠀⠀⠀⠀⠀⢰⡟⠋⠉⠀⠀⡇⠀⢸⣻⠀⠸⠟⠻⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡇⠀⠠⠯⠭⠭⠵⣾⣭⠀⢸⣶⢞⣓⣒⣒⣚⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⡗⢠⣂⠀⠀⠀⠀⣿⣶⡀⣸⣤⣸⡇⠀⢀⡎⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠃⠛⠘⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡯⠌⠆⠰⠠⠆⠤⠰⠀⠆⠴⠰⠀⠆⠴⠠⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠐⠻⠤⠤⠤⠄
⠀⠀⠰⡒⠒⠃⠀⠀⠀⠀⣶⣆⢰⣶⡆⣶⡦⢰⣶⡆⢰⠆⡦⢲⡀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡆⡇⡇⡇⡇⡇⡇
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣭⡇⢸⣽⡄⣿⡇⢸⣿⡄⠀⠈⠀⠀⡅⠀⠀⠀⡆⢀⣀⣀⣀⡈⠁⠁⠁⠁⠁⠁⣵⡆
⠀⠀⠀⠓⠒⠒⠒⠐⠒⠂⠛⠓⠘⠛⠃⠉⠁⠈⠉⠁⠒⠒⠒⠒⠛⠀⠀⠒⠃⠚⠛⠛⠛⠃⠒⠂⠀⠀⠀⠀⠛⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠋⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠙⡿⠁⠀⠀⠀⠀⠀⠈⢿⠋⠀⠀⠀⠀⠀⠈⢻
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣦⣀⣀⣀⣀⣀⣤⣿⣷⣄⣀⣀⣀⣀⣠⣾⣷⣄⣀⣀⣀⣀⣀⣴⣿⣦⣀⣀⣀⣀⣀⣴⣿⣦⣀⣀⣀⣀⣀⣠⣾
⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠈⠙⡿⠋⠀⠀⠀⠀⠀⠙⢿⠋⠁⠀⠀⠀⠀⠈⢻
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣄⣀⢀⣀⣀⡀⣠⣾⣦⣀⢀⣀⣀⣀⣀⣼⣷⣄⢀⣀⣀⡀⢀⣴⣿⣄⡀⣀⣀⣀⢀⣠⣾⣦⡀⡀⣀⣀⢀⣀⣼
⣿⣿⠟⠉⠉⠉⠉⠉⠙⢿⠟⠉⠉⠉⠉⠉⠙⢿⡿⠋⠉⠉⠉⠉⠉⠻⣿⠟⠉⠉⠉⠉⠉⠙⣿⠟⠉⠉⠉⠉⠉⠙⢿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣄⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⣠⣷⣄⠀⠀⠀⠀⠀⢀⣾⣄⠀⠀⠀⠀⠀⢀⣼
⣿⣿⠟⠛⠛⠛⠛⠛⠻⣿⡿⠛⠛⠛⠛⠛⠛⢿⡿⠛⠛⠛⠛⠛⠛⢿⣿⠟⠛⠛⠛⠛⠛⠻⣿⡿⠛⠛⠛⠛⠛⠛⢿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⢀⣼⣀⠀⠀⠀⠀⠀⠀⣸
⣿⣿⡿⠒⠒⠒⠒⠒⠿⣿⣿⠓⠒⠒⠒⠒⠺⣿⣿⠗⠒⠒⠒⠒⠚⢿⣿⡿⠒⠒⠒⠒⠒⢿⣿⡿⠓⠒⠒⠒⠒⠺⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⣸
⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⠎⠁⠈⠀⠀⠀⠐⠉⠑⢄⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠢⠀⢀⡚⣿⡢⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⡂⢸⣿⣿⠇⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⠁⣠⢁⣸⡌⠁⢠⠁⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⠔⠤⠀⠀⠈⠁⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠈⢻⠐⠛⡛⠉⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠸⢤⠔⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣛⣛⠓⠒⠶⠶⠦⠤⠤⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣭⣭⣭⣉⣙⣛⡒⠒⠲⠶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⠒⡤⠒⡄⠀⠙⣶⠶⠶⠤⠤⢤⣭⣍⣉⣉⣁⠉⠠⠀⠔⠁⢶⠐⢦⠤⣄⡀
⠀⠀⠀⡖⠉⠉⠹⣡⣁⣊⣠⠊⢀⠔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠣⡀⢑⣄⣡⣨⡼⠋⠉⠲⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡇⠀⠀⢦⡴⠀⠀⠀⠀⠀⠀⠀⠇⠰⠞⠀⠀⠀⠶⠃⠀⠇⠀⠀⠴⠆⠀⠆⢠⠀⠰⠛⠀⠀⠀⡜⠀⠀⠀⠁
⠀⠀⠀⡇⠀⢠⡀⠀⠀⠈⠀⠀⠀⠀⢸⠀⢠⠇⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠀⠀⠀⠎⠀⠖⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⡇⠀⠀⠀⡌⡀⡞⠀⡠⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⢠⠀⠀⡄⠀⡄⢀⠀⠀⢀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⡇⠀⠈⠋⠁⠈⠁⠈⠁⠀⠙⠁⠈⠉⠁⠀⠀⠉⠉⠁⠋⠉⠉⠉⠉⠁⠉⠉⠉⠀⠈⠋⠀⠈⠛⠁⠈⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠀⢸⣿⠀⠀⠀⠀⢠⣤⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣄⠀⣍⣽⣿⣿⣿⢀⣸⣿⣀⣀⠀⠀⢸⣿⣿⣿⡇⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⣿⣿⣿⣿⣠⣽⣿⣿⣿⣿⣿⡻⣻⣿⣿⡇⠀⣼⣿⣿⣿⡇⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠃⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠈⢷⡆⠀⢙⠁⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠠⠂⠀⠀⠀⢠⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⢠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⣶⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⠇⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠘⣿⣿⣿⡿⣿⣿⣿⣿⡋⠁⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣿⣇⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⣿⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡏⠀⠉
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡀⢰⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠘⠛⠛⠃

⠀⠀⠀⣀⣤⣤⣀⣀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢶⣾⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⢋⣉⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣴⣖⠲⣶⡆⢺⣿⡅⢻⣿⡌⣿⣧⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⣿⣿⠀⣿⡇⢸⣿⡇⢸⣿⡇⣿⣯⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠙⠒⢀⡆
⠀⠀⠀⣿⣿⠀⣿⡇⢸⣿⡇⢸⣿⡇⣿⣟⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠀⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⠉⠁
⠀⠐⣤⣿⣿⣶⣿⣷⣾⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣧⠀⢴⣾⣶⣷⣦⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⣤⣤⣤⣿⣿⣤⣼⣿⣿⣿⣿⣦⣼⣿⣧⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣴⣷⣤⣤⣴⣦⣤⣤⣤⡄
⠀⠀⠈⠛⠿⠿⠟⠛⠛⠛⠿⠿⠟⠋⠉⠁⠀⠀⠀⠀⢲⣖⣿⣽⣿⣶⣾⣓⣺⣯⣝⣳⣞⣛⣷⣓⣈⡃⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠰⠀⠈⠉⠉⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢶⡦
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⡀
⠀⠀⠀⠀⣿⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠟⠛⢷⡶⠦⣤⣤⡀
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⢰⣿⠾⣷⢶⣶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⣿⣐⣿⣸⣿⣸⡇⢀⣴⣶⡄⢀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⢰⣶⣶⣿⣿⣿⠰⠆⠀⠀⠀⠀⠀⢠⡌⠃⠡⠤⠀⠀⠒⠀⠀⠀⠀⢠⣤⣦
⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⠀⠀⢀⣀⣀⣠⣿⣷⣀⣀⣈⣀⣀⣀⣈⣀⣉⣉⣉⣀
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠁⠉⠉⠉⠈⠁⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠐⡶⠣⠌⣁⠒⠠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠄⠀⣠⢃⢀⠀⡜⡀⠀⡠⠁⠀⠉⡱⠒⠠⠤⡈⠁⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠉⠐⠧⠹⢒⣢⢮⢄⡑⠰⢠⢀⠀⠜⠀⠀⢀⠎⠀⠈⠁⠖⠠⠨⣀⣂⣥⡒⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠂⠀⡠⠊⠀⢀⠌⠀⠈⡰⠊⠕⠂⠆⢮⣈⠲⠠⠂⠆⠀⠀⠎⠀⠀⠀⠠⠃⠉⠑⠲⠄⠤⣀⠈⠁⠒⠀⠄⡀
⠀⠀⠀⢀⠄⠒⠉⠧⣀⢀⠔⠡⡀⡠⢣⠀⢀⠞⠀⠀⢠⠂⠈⠁⠢⠍⣓⠢⢤⣁⠎⠸⠄⡀⠠⢁⠀⠀⠠⠁⠀⠀⣰⡅⠀⠀⠀⠀⠀⣩⡦
⠀⠀⠀⠈⠂⠢⢄⡀⠀⠉⠛⠒⠸⢅⡁⡠⢂⢁⠀⡔⡁⠀⠀⡔⠁⠀⠀⢉⠐⠠⢍⡐⠢⢤⣁⠈⠆⡠⠁⢄⠀⣠⢉⡿⠒⠀⢀⡤⠞⢉⠃
⠀⠀⠀⠀⢀⠀⢀⠈⠀⠂⠄⠐⠄⠀⠈⠉⠒⠤⣌⠔⠄⢀⢎⢀⠀⠀⠠⠃⠀⠀⠀⠁⠋⠐⠢⠍⠑⠣⢤⣈⣰⠏⠀⠀⣠⠞⠋⠁⠀⠈
⠀⠀⠀⠀⠀⢂⠀⢦⠄⢀⠀⠀⠈⠐⠨⢐⠀⠀⠀⠉⠑⠢⢌⡀⠃⠔⠁⠂⠀⠀⡐⠁⠀⠀⠀⣼⡡⣀⠀⠀⠀⣠⡴⠛⠍⠀⣀⠂⢠
⠀⠀⠀⠀⠀⠈⡀⠈⠄⠈⠗⠂⠴⣀⠀⠀⠈⠑⠢⠅⡂⠄⡀⠈⠙⠲⠤⣄⣀⠌⢢⢃⠄⢀⠎⠙⠃⠀⢀⡴⠛⠇⠐⢀⢠⠃⣍⠀⠆
⠀⠀⠀⠀⠀⠀⠐⡀⠈⠄⢈⡀⠀⠁⠈⠳⠤⣰⠀⠀⠀⠑⠠⢅⡂⠤⢀⠀⠈⠓⠦⢤⡠⠋⠁⢀⣤⣮⠋⠁⠈⢀⠖⠁⡀⢠⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠀⢤⡀⠰⠀⠈⠁⠂⢦⡀⠀⡀⠀⠑⠂⠬⡒⠠⢄⡀⠀⣠⡴⠽⠋⠀⠄⣠⠀⠁⠆⣠⠊⢰⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠰⠄⡄⠀⠄⠈⡖⠤⡅⠀⠀⠀⠑⡧⢀⡀⠄⠀⠀⠉⠑⠪⠵⠛⢁⠀⠌⡬⠂⠁⠀⡪⠀⠁⢀⣄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠁⠘⠙⢴⡀⠁⠀⠈⠉⠰⢄⠄⠀⠀⠀⠓⠠⢀⡀⠀⠀⠰⢀⢈⠄⠈⠐⢠⡧⢊⡁⢠⠆⠘⢨⠡⠄⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠁⠀⠀⠙⠂⢤⡀⠈⠀⠈⠓⠤⣂⠀⠀⠀⠁⠀⠀⢠⠀⠌⠊⠰⡀⢛⠀⢸⢄⡧⠀⢁⠂⠀⠀⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠉⠂⢥⡀⠀⠀⢀⠀⢔⡠⠚⢀⢎⠔⡋⠈⠀⢃⡜⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢁⠀⠀⠀⠀⠀⠀⢈⠀⡠⠠⠁⣦⠃⠈⠀⢀⠴⠋⠀⡐⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡜⣀⠀⠁⠢⣀⠉⠂⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠘⠐⢀⡁⠊⠀⠂⢀⡜⠉⠈⠀⠀⣀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⡂⠀⣆⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠠⢁⠁⠀⢀⡔⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠓⠃⠀⠈⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⠀⡴⠊⢀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⡇⠀⢀⠎⠁⠀⠀⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⢢⣤⠀⠀⠀⠁⠢⡀⠀⠁⡴⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢀⣠⡤⠤⣤⡀⠴⠋⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠙⣆⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢺⠀⠀⠀⠀⠀⠀⣿⠐⠒⠒⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣃⣀⡀⠀⣠⡼⠃⠀⠲⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠩⠉⠉⡭⠉⠩⡍⠉⢩⠍⠉⢭⠍⠉⣭⠉⠉⡭⠉⠩⡝⡆⠁⡀⠙⠷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡇⡇⠀⠏⠀⠠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡇⣧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠒⠒⡗⠒⠒⡗⠒⢺⠒⠒⢺⠒⠒⣿⠒⠒⡗⠒⠒⡇⡇⢸⡗⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡇⡟⠋⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡧⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣶⣶⣷⣶⣶⣷⣶⣾⣷⣶⣾⣶⣶⣿⣶⣶⣷⣶⡠⠴⠆⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡏⠉⠉⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠞⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠷⠶⠶⠟⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠤⠤⠤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁⠀⠀⠈⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠠⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⡠⢠⣾⡿⢡⣿⣿⢠⣿⣿⠘⣿⣷⡙⣿⣦⡑⠀⠀⠀⠀⠀⠐⣡⣿⡟⣱⣿⡟⢸⣿⣧⢸⣿⣧⠹⣿⣦⠠⡀
⠤⠴⠿⠿⠥⣾⠿⠯⠼⠿⠿⠤⠿⠿⡷⠼⠿⠿⠦⠄⠀⠀⠤⠾⠿⠿⠴⡿⠿⠧⠼⠿⠿⠬⠿⢿⡦⠽⠿⠷⠤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⢀⣶⣷⡖⢰⣶⣶⠆⣶⣶⣶⠰⣶⣶⡄⢷⣶⣦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⠋⣰⣿⣿⡟⢠⣿⣿⡿⠀⣿⣿⣿⠀⣿⣿⣿⡄⢻⣿⣷⣆⠻⡛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣾⣿⣿⣯⣤⣿⣿⣿⣧⣴⣿⣿⣿⣦⣼⣿⣿⣿⣤⣿⣿⣿⣷⣬⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⡤
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣠⠴⠋⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⢉⡴⠖⠒⢦⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⢠⡟⠉⠙⠀⠀⠀⠀⢹⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠉⠉⠉⠓⢾⣉⣥⣀⠀⠀⠀⡴⠛⠛⢶⡀⠉⠹⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣀⣀⠀⠀⠀⠁⠀⠈⣧⡀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡴⠋⠁⠈⠙⡶⠒⢦⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠖⠛⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠴⢮⡴⠋⠉⠙⢦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡀⣠⠴⠶⠶⡤⠴⠾⠦⣄⡀⠀⠀⠀⢀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⡤⠦⣤⠖⠦⡄⠀⠙⢦⠀⠀⠚⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣤⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⢦⠀⠘⠳⢦⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⡤⢏⡥⠭⣷⠤⢤⡤⠤⠤⠤⢤⡤⠤⠾⠤⣤⠤⢼⡧⠤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠳⡖⢻⡖⠚⡗⠒⠒⠒⢺⡗⠒⠒⠒⣿⠒⣾⠓⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠚⡷⠒⡗⠒⠒⠒⠺⡗⠒⠒⠒⡟⠒⡟⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⣷⠤⣷⠤⠤⠤⢤⡧⠤⠤⢴⡧⢴⡧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣷⠀⠀⢠⠞⢳⡀⠀⢰⡇⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠀⣿⠀⣠⢏⡞⢧⡳⡀⢸⡇⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⣰⢣⠏⠀⠈⢳⡹⣼⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠛⡿⣳⣯⣷⡄⣠⣿⣷⡽⣟⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡼⣱⠃⠉⠉⣹⡏⠉⠁⠹⡜⢦⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡟⣼⣥⣤⣤⣴⣿⣿⣤⣤⣤⣽⡎⢧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⠗⢲⡖⠒⠒⠒⢲⡖⠒⠒⠒⣶⠒⢾⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣼⠀⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⢸⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⡇⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⣧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡽⠉⢉⡏⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⣏⠉⠹⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣴⣇⣠⣼⣅⣤⣤⣤⣀⣨⣧⣠⣤⣤⣄⣼⣤⣄⣿⣄⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⣏⣠⣠⣯⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣠⣧⣀⣈⣧⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⡤⢿⡤⢤⠾⡤⠤⠤⠤⠤⠤⠼⠧⠤⠤⠤⠤⡤⣼⢦⡤⠬⠧⢤⠷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⣶⣶⣶⣶⡆⢠⣴⣶⣶⣶⡦⢀⣴⣶⣶⣶⣶⠀⣤⣶⣶⣶⣶⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠸⣿⣿⡿⡿⠏⠰⣿⣿⣿⣿⠟⠀⢿⣿⣿⣿⠿⠁⢾⣿⣿⣿⡿⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⠀⣠⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⡦⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠿⠿⠿⠿⠟⠀⠿⠿⠿⠿⠿⠃⠸⠿⠿⠿⠿⠃⠸⠿⠿⠿⠿⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣴⣶⣶⣶⣶⠆⢠⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⡶⠀⣴⣶⣶⣶⣶⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠺⠿⠿⠿⠿⠃⠸⠿⠿⠿⠿⠋⠐⠿⠿⠿⠿⠟⠀⠿⠿⠿⠿⠿⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⣾⣿⣿⣿⡷⢀⣶⣿⣿⣿⣷⠀⣴⣿⣿⣿⣿⠆⣰⣿⣿⣿⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠿⠿⠿⠿⠋⠀⠿⠿⠿⠿⠟⠀⠻⠿⠿⠿⠟⠃⠸⠿⠿⠿⠿⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣶⣿⣿⣿⣿⠀⣴⣾⣿⣿⣿⠆⢰⣾⣿⣿⣿⡗⢠⣾⣿⣿⣿⣷⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇⡌
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠿⠿⠿⠿⠟⠁⠸⠿⠿⠿⠿⠃⠘⠿⠿⠿⠿⠋⠀⠿⠿⠿⠿⠟⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡌⣰⣿⣿⣿⣿⡇⢰⣾⣿⣿⣿⡷⠀⣾⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⠇⡎
⠀⠀⠀⠀⠀⠀⢠⢁⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⡿⢠⠁
⠀⠀⠀⠀⠀⠀⢎⠘⠟⠿⠟⠟⠋⠐⠛⠿⠟⠟⠋⠀⠛⠿⠛⠟⠛⠁⠚⠟⠻⠛⠟⢁⠆

⠀⠀⠀⡠⠀⠤⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣴⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠠⣇⠀⡹⠀⠀⠀⠀⣏⣴⡿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠴⠀⣿⡇⠔⡆⠀⠀⠀⠀⢤⣤⣤⡸⡃
⠀⠀⠠⣧⢀⠏⠀⠀⠀⠀⠙⠏⡼⠥⡯⢸⡇⠿⠄⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠸⠿⠼⢯⡼⠯⠽⠧⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢤⡀⢠⣶⣴⣷⣤⡄⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀
⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣇⣿⣄⣀⡀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠹⠀⠀⠀⠀⣿⢛⡇⢸⣿⣿
⠀⠸⣿⣻⣿⡿⣛⢿⠁⠀⠀⠀⢸⣼⣿⣧⣯⣿⠀⠀⠀⢠⡄⠀⡿⡇⠀⠀⠀⠀⠀⢠⣧⣧⣶⢾⣿⣿⡄
⠀⠀⠈⠋⠀⠀⠙⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠘⠓⠚⠓⠛⠒⠒⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢤⡄⠀⠀⠀⠀⠀⢰⣤⢤⠤⡤⢤⢰⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠐⣇⠺⠂⡗⠂⠀⠀⠀⠀⣿⠁⠈⠐⠁⡟⣿⠀⠀⠀⠀⣀⣰⣿⣆⣀
⠀⠀⠀⣘⣿⡇⠀⠀⠀⠀⠀⠀⠄⠉⠹⠉⠁⠄⠀⠀⠀⢰⠇⢀⠀⠀⢸⢷⣿⠀⠀⠀⠈⠿⠉⠏⠉⠯
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠁⠁⠈⠈
⠀⠀⠀⠠⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⢰⡆⠀⢁⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⠂⠀⠀⠀⠀⠀⢸⣘⣛⡃⡇⣰⡆⠀⠀⠀⠀⣠⣷⣌⠁⠀⢀
⠀⠀⢸⡇⠀⣼⡇⠀⠀⠀⢰⣿⣿⣍⡀⢘⣃⣀⠀⠀⠀⠀⢸⠰⠶⠆⡇⠿⡿⠀⠀⠀⠀⣿⣿⣿⣷⣄⠋
⠀⠀⠘⢇⢸⡿⠁⠀⠀⠀⠈⠉⠉⠉⠋⣿⣿⣿⡇⠀⠀⠀⢸⢭⣭⡅⡇⠀⠁⠀⠀⠀⣸⣿⡿⠿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣙⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠉⠛⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠤⠀⣿⣿⣿⣿⣿⣿⣿⡆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣅⠀⠀⠀⣀⣀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⣿⠀⠀⠀⠛⠛⠘⠛⠃⠛⠓⠀⠀⠀⠰⠂⠈⠙⢛⠛⠉⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣿⡏⢼⣿⡇⢿⣿⠄⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣶⣾⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣭⣭⣭⣭⣙⠻⣿⣿⣿⣿⣿⣷⣦⡀⠀⣠⣼⡷⢰⡆⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⣿⣿⣿⣿⣿⡿⣡⣾⢛⣥⣁⣿⡟⣿⣿⣦⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⠿⢟⣡⣿⣿⢏⣵⣶⣿⢡⣿⢡⣿⣿⣿⡟⡄⣿⣿⣿⣿⣆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⣾⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⢸⡇⣾⣿⣿⡟⣼⢧⣭⢠⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣿⢸⣷⠸⣿⣿⣶⣶⡆⢇⣾⣿⣿⡟⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣌⢻⣿⣧⠻⣧⡙⢿⣿⣿⣅⣾⡟⢿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣷⡝⠻⣷⣬⣝⡛⡿⢰⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣩⣵⣿⣿⣿⠟⠁⠀⠀⠉⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣴⣿⣿⠀⣾⣿⣷⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⣿⣿⡏⢸⣿⣿⣿⡆⢹⣿⣿⡄⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠐⠿⠿⠿⠀⠿⠿⠿⠿⠷⠈⠿⠿⠷⠀⠀⠀⠀⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣴⠟⣿⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠶⠟⠁⠀⣿⡇⠀⠙⢷⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⡀⠰⢶⣾⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠈⠇⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣾⣿⣿⣿⣷⣦⣌⡙⠻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣰⣿⡿⢋⣡⣌⠹⣿⣿⣿⣿⣦⠘⣿⣿⣿⣿⣿⣿⣿⣷⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⣾⣿⣿⡄⠻⠿⢿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⠋⣰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠸⣿⣿⣿⣿⣷⡄⢹⣿⣿⣤⡙⠛⠿⠿⠟⢃⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣌⡙⠛⠛⠛⣠⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⡏⢙⣶⢶⣋⢹⡄⠀⠻⠇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣉⠀⠀⢈⡿⣇
⠀⠀⠀⢰⠒⠓⠒⠒⠒⠒⠛⢲⡆⠀⠀⣾⠉⣷⠀⡟⠉⡇⠀⠀⠀⠀⡼⠀⢈⡷⠾⣍⡀⢻
⠀⠀⠀⢸⠀⣟⣹⡆⢸⣛⡇⢸⡇⠀⠀⡿⠉⢿⠀⡟⠉⣇⣾⠛⠛⠛⠛⠛⠛⣧⠤⠬⠽⢾⣦⣀⡀
⠀⠀⠀⢸⠀⣉⣉⡀⢈⣉⡁⢸⡇⠀⠀⡇⠀⢸⠀⡇⠀⣿⠉⣿⠉⠉⠉⠉⡟⠡⠤⠤⠤⣤⣷⣬⡽⢿⣶⠆
⠀⠀⠀⢸⠀⢯⣼⠇⢸⣭⡇⢸⣿⣦⡀⡇⠀⢸⢀⡇⠀⣿⠀⡏⠀⣸⣇⠀⣿⢀⣤⠖⠋⠁⢹⠙⠃⠘⠋
⠀⠀⠀⢸⠀⣤⢤⡄⢠⠤⡄⢸⡇⠙⣿⠉⣉⣉⠉⣉⣉⡉⣿⠇⠰⣿⡟⠀⢹⠀⠉⠛⠲⠤⣼⡇
⠀⠀⠀⢸⠀⠳⠾⠃⠸⠶⠇⢸⡇⠀⣿⠸⣧⡽⠀⢯⣼⠇⣿⠀⠀⡟⠀⠀⠈⣇⠀⣠⠴⠞⠋⣧
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠚⢳⠠⢤⣄⣀⢸
⠀⠀⢰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠛⠛⣷
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⡤⠤⠀⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡆⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢙⣛⣛⣛⣛⢹⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣇⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣿⣿⣿⣿⢘⣛⣛⣛⢘⣛⣛⣛⡃⣛⣻⣿⢟⣫⣭⣾⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢼⣿⣿⣿⡇⣿⣿⣿⡟⣾⣿⣿⣿⡇⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣴⣶⣶⡆⣶⣶⣶⣾⡇⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⢾⣿⣿⣿⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣘⣛⣛⣛⣃⣛⣛⡛⣛⣃⣛⡛⠸⣿⣮⡥⣿⢮⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠛⠛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠃⣙⣛⣛⡁⠀⠘⣿⡇⣿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⢸⡇⠿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣭⠍⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀
⠀⠀⠀⠀⠀⠈⡀⠠⠄⠤⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢠⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⣠⣮⣀⠀⣠⡶⠋⠀⠀⣠⠆⠀⠀⢠⠄⠈⢄⢀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣽⠏⠘⠶⢦⡀⢾⣋⡀⠀⡴⠁⠀⣨⡦⠀⣴⢄⠡⠀⠀⠠⠀⠀⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⢶⠔⢿⣁⠀⢀⣤⠞⠈⠈⢉⡏⠀⠵⠖⠺⢋⡰⠊⠀⠈⢧⠐⡀⠄⠀⠀⠈⠀⠄⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⠄⠀⣠⠦⠈⠁⠉⣭⡉⢷⣤⡀⠴⠋⠀⠀⣤⠀⡠⠊⠀⣀⢦⠄⠀⠣⡈⢣⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣈⠙⠊⠡⣄⣀⡰⠞⠁⠀⢀⡼⠖⠈⢙⠒⠛⡡⠞⠀⠀⠀⠹⣸⠀⠀⠀⠙⢄⠱⡀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⢆⡀⢀⠀⠓⠲⠆⡋⠀⠀⣠⡾⢃⠞⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠣⡘⢆⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠂⢥⢀⣄⡈⠙⢊⡕⡰⠋⠀⠀⠠⢀⠀⠀⠂⠐⢲⠄⠀⢸⣹⠀⢙⢜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠉⣬⠨⠀⠀⠐⡀⣀⣸⡆⠀⢴⣨⣼⠅⠀⠐⣼⠀⠈⣠⠂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠑⠲⣄⠀⢀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣽⠀⢐⡛⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⢂⢀⡀⠀⣯⠇⠀⡀⢈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠘⠳⢶⡸⠀⠂⠀⠀⠀⢠⠒⠒⡏⣱⠀⠀⠄⣠⣄⣫⡵⠆⠀⠇⠁⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄⡀⠀⠀⠀⡄⢀⡂⠰⠀⠀⢸⠠⠮⠥⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡀⡸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⢁⠀⠀⠀⠀⠀⢸⠀⠀⣀⣇⣀⣀⣀⣤⠤⠤⠖⠒⠚⠓⠋⠁⠠⠤⠶⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣀⠂⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⣈⣁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣴⠞⠛⠉⠉⠛⠳⣦⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠦⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠴⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠀⠀⠀⠈⣿⠉⠁⠀⠀⠀⠀⠀⠀⣿⠁⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣦⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠂⠀⠂⠀⠀⠐⠰⠐⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⣶⣚⣫⣍⣭⣍⣭⣿⣖⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣾⠁⠀⠀⢹⡿⠿⠛⠛⠛⢻⢿⣿⣧⠤⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⠛⢻⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣿⣾
⠀⠀⠀⠀⠀⠀⣶⠟⠻⡆⠉⣿⠀⠀⢻⣾⠀⢸⠿⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠐⣯⠭⠁⠀⠀⢹⡟⡇
⠀⠀⠀⠀⠀⠀⡏⠀⢰⣷⣲⣟⣳⣦⣿⣿⠀⢸⣤⣼⣦⣤⣤⣼⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣻⢦⣼⠁⠁⠀⠈⣸⣿⣿⠀⣞⣹⣿⡏⠉⠉⣽⡀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠸⠀⣇
⠀⠀⠀⠀⠀⠀⡇⡘⠛⠋⠉⠉⠉⠙⠋⠙⠓⠛⠛⣿⢳⠆⣾⢻⣷⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠐⠂⢸
⠀⠀⠀⠀⠀⣼⡃⣒⢒⡐⣒⢒⡒⡐⢂⠒⡐⣂⢒⡒⣂⢒⡐⠒⠚⢳⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⣀⣼⣇⣀⣀⣀
⠀⠀⢀⣤⣤⣯⣡⣉⡈⣁⢉⣈⡁⣁⣈⢀⣁⡈⣈⣁⡉⢨⡅⣄⣠⣸⠀⠀⠀⠀⡷⡈⠀⠀⠘⣦⢰⢰⢠⣤⡄⡄⡄⡇
⠀⠀⠈⡇⠀⠀⠀⣤⡄⠀⢘⣻⠃⣛⣛⢘⣻⡃⣻⣻⠃⡜⣦⢻⡜⣷⠀⠀⠀⢰⠃⣧⣄⠀⠀⠛⠸⠸⠸⠻⠇⠇⣇⡇
⠀⠀⠀⣇⣀⣀⣀⣿⣇⣀⣸⣿⣃⣿⣿⣸⣿⣇⣻⣿⣃⣦⣿⣼⣿⣿⣠⣖⣶⣿⣷⣿⣿⣷⣾⣀⣀⣀⣀⡀⢀⢀⣿⡇



⣿⣟⣻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡔⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⠀⠀⠑⠀⠉⠉⠉⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠤⠄⠒⠒⠠⢄⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⠠⠔⠊⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠘⢄⠀⠀⠀⢀⠔⠁⠈⠉⠁⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡰⢁⣾⣿⡿⢁⣿⣿⡟⢘⣿⣿⣇⢹⣿⣿⣇⠻⣿⣿⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣼⣿⡿⢃⣿⣿⣿⢈⣿⣿⣟⠘⣿⣿⣏⠹⣿⣿⣆⠳⡄
⠀⢠⠊⣰⣿⣿⡟⢁⣾⣿⣿⠇⢸⣿⣿⣷⠀⣿⣿⣿⡆⠹⣿⣿⣦⡘⢆⠀⠀⠀⠀⠀⢀⠞⣠⣾⣿⡿⠁⣾⣿⣿⡇⢰⣿⣿⣿⠀⢿⣿⣿⣆⠙⣿⣿⣷⡈⢢
⣴⣡⣾⣿⣿⣿⣤⣾⣿⣿⣿⣤⣼⣿⣿⣿⣤⣼⣿⣿⣿⣤⣽⣿⣿⣿⣦⣱⠄⠀⠀⣴⣥⣾⣿⣿⣿⣥⣼⣿⣿⣿⣤⣼⣿⣿⣿⣤⣼⣿⣿⣿⣦⣼⣿⣿⣿⣦⣱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣧⣠⡤⠤⣠⣤⣤⡠⠤⣤⣤⣤⡤⠤⣤⣤⣤⡤⠤⣧⣤⣤⡤⠤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⣬⣤⣤⣤⣤⣤⣤⣤⣤⠎⢠⣼⣿⣿⡿⠁⣼⣿⣿⣿⡇⢠⣿⣿⣿⣯⠀⣿⣿⣿⣿⡄⠹⣿⣿⣿⣤⠈⢢⣤⣤⣤⣤⣤⣬⣤
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠟⢁⣰⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣷⡀⠘⣿⣿⣿⣷⣀⠙⢏⠙⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⣾⣿⣿⣿⡿⠁⢰⣿⣿⣿⣿⡏⠀⢼⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣦⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣄⣤⣤⣤⣄⣄⣀⣤⣤⣤⣤⣀⣤⣤⣤⣤⣅⣠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⢤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠄⠠⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⢸⡆⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡠⡀⢠⢠⣠⠄⠀⠀⠀⡤⢠⡄⡠⢠⠀⢤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⠛⠻⠛⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠑⠋⠃⠑⠘⠂⠉⠂⠀⠀⠀⠁⠁⠈⠁⠘⢉⡾⠘⡺⠀⠒⠒⠒⠒⠒⠒⠚⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁
⠀⠀⠀⠀⢸⠃⠀⢼⣅⠀⠀⠀⠀⠀⢠⠞⢉⣵⣯⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢐⣏⣗⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⢯⡄⠀⣰⣃⠀⠀⠀⠀⠀⣺⢠⣿⡿⡿⠓⣃⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠏⠛⠰⡧⡷⠐⠋⠇⠀⠀⠀⠀⠀⠀⠲⠶⠶⢶⠦⢚⣀
⠀⠀⠀⠀⣼⡀⢀⡽⠋⠀⠀⠀⠀⠀⠈⠳⡝⣨⣭⢡⣭⡅⣽⣭⢩⣭⡄⠀⠀⠀⠀⠀⠀⠀⢠⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣯⢻⡆⠀⠄
⠀⠀⠀⠀⢸⣣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢰⣶⠆⣶⣶⠰⣶⣦⠀⠀⠀⠀⠀⠀⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⠌
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠒⣶⡞⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⢀⣃⠞⢄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡠⠊⠀⠀⠀⠈⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢀⠐⠉⠒⠒⠤
⢤⣄⠀⢀⣴⣷⣾⣿⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⣡⡀⢈⣄⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡆⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠁⠀⠀⠀⠄⠒
⠀⢹⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣿⠋⠁⠀⢻⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡿⢷⡾⢿⠶⣶⢶⡆⠀⠀⠀⠀⠀⠛⠉⠀⢰⣀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⡻⢦⠴⢿⢀⣿⣿⣿⣇
⠀⠀⢸⡟⣫⡻⣿⣿⢛⣝⢻⡇⠀⠀⠀⠀⠀⢀⣿⣾⣷⣿⣶⣿⣾⡇⠀⠀⠀⠀⠀⣶⡆⠀⣮⠿⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣧⣾⡜⣿⣿⣿⣿⡀
⠀⠀⠀⠈⠳⠃⠈⠁⠘⠞⠁⠀⠀⠀⠀⠀⠀⠘⠟⠟⠛⠛⠛⠛⠛⠳⠀⠀⠀⠀⠀⠟⠳⠐⠟⠖⠺⠂⠒⠴⠆⠀⠀⠀⠰⠟⠻⠛⠻⠟⠗⠟⠛⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠛⢶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣆⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤⣤⣤⣤⣤⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡏⢁⣄⠉⣧⣀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢁⡜⠁⡜⠁⣼⢰⣷⠀⠀⠀⠀⠀⠀⠀⢀⣻⣟⡀
⠀⠀⠀⠀⠀⡿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⡈⠋⣠⡟⠉⠀⠀⠀⠀⠀⠀⢸⡿⠀⡸⠀⠈⠀⠈⡋⣸⡏⠀⠀⠀⠀⠀⣴⣦⣾⣽⣿⣷⣶⣦
⠀⠀⠀⠀⢰⢏⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⣁⢉⣋⢁⣉⣀⡀⠀⠀⠀⠀⠀⣾⡇⣀⠁⣀⣁⠀⣹⢷⣿⣿⠀⠀⠀⠀⢀⣙⣯⣉⣹⣏⣉⣿⣋
⠀⠀⠀⠠⠼⠧⠧⠽⠦⠄⠀⠀⠀⠀⠀⠀⠀⠴⠤⠦⠤⠦⠤⠦⠴⠄⠀⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⠛⠸⠿⠟⠀⠀⠀⠀⠠⠦⠥⠦⠵⠤⠴⠬⠴⠄

⠀⠀⠀⠀⢀⡔⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⢠⣸⠁⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠶⠶⠆⣿⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠴⠋⠀⠀⠀⢀
⠀⠀⠀⢸⣿⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⢀⣤⣤⡀⣴⠟⠛⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠶⠶⠆⣿⠀⣿⣿⣀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⣄⠀⢀⣴⠋
⠀⠀⠀⢸⣿⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⡀⠀⢘⣣⣤⣀⠀⠀⠀⠀⠀⠀⢸⠸⠿⠿⠇⣿⠐⠿⢿⡟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠘⣿⡀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠛⠿⢡⣯⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⢘⣛⣛⡃⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠈⢃⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⣘⣛⣛⣃⡿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠋⠀⠙⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢂⡠
⠀⠀⠀⠈⠀⠀⠈⠉⠩⠈⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⣾⠂⠀⣠⡴⠃⠀⢀⡤⠁⢀⡐⢀⡀⠠⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠂⠍⣹⠎⠳⢤⡄⢶⣃⠀⡠⠋⠀⣠⠕⣠⣌⢂⠀⠀⠀⠀⠈⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⠿⡗⣺⣅⣀⣠⠞⠉⠈⣹⠅⠚⢖⠚⡡⠎⠀⠈⢆⠢⡐⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⠈⠬⣝⡡⡾⡋⠀⣠⠮⠚⠳⡆⠺⣁⣀⡴⣣⠞⠀⢰⣲⡆⠈⠣⡘⢄⠀⠀⢄⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⡘⠓⠚⢧⣀⢠⠞⠁⢀⣴⢉⠞⠁⠀⠀⠈⠽⠇⠀⠀⠘⢌⢦⡠⠎⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⠴⣠⣄⡘⠲⠴⡻⡱⠃⠀⣀⣀⠀⠀⠠⢤⠀⠀⣶⠄⢣⣿⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠙⠳⣮⣽⠀⠀⡆⢠⣾⡄⠾⠾⣾⠄⠠⣿⠆⣿⡷⢥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⡙⢢⠀⢀⣀⠀⠀⠄⢸⠀⠀⠀⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⣁⡏⣼⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠚⠀⢸⣿⡈⠆⣀⢸⠀⠀⣀⣀⡀⡀⠀⠐⠉⢉⡥⣔⠀⡿⠀⠐⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⡀⠀⠀⠈⢙⠀⠃⢰⠀⠀⡆⣰⣧⡇⠀⠐⠚⠓⠋⠁⢀⠇⠈⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢘⠀⠃⠀⠀⠀⡇⠁⢸⣃⣀⣀⣀⣤⠤⠴⠾⠖⠛⢘⡠⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⠒⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣖⣻⠶⠀⠀⠀⠀⠀⢴⡧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⠀⠀⢀⣤⡀⣺⡗⠀⠀⠀⠀⢸⣯⠀⠀⣤⣤⣤⣭⡍⠛⠋⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⢰⡟⠛⠛⢻⣄⣀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠐⣶⣶⣶⡖⠀⢼⡧⠀⢠⠶⠶⢾⣿⠶⠶⣿⣿⣿⣿⠁⠀⣀⣀⣀⢰⡇
⠀⠀⠀⠀⠀⢸⡇⣴⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡀⣿⣿⣿⡇⠀⣻⣏⣠⣾⣶⣶⣾⣿⡄⠀⣿⣿⣿⣿⡄⠀⣿⣿⣿⢸⡗⠐⡆
⠀⠀⠀⠀⣀⣸⡇⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⡗⣺⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣧⢠⣿⣿⣿⣼⡇⢀⣧⣀⡀
⠀⠀⣤⣼⣿⣿⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢷⣭⡭⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢸⡿⡿⡇⢤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠀⢰⡟⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣀⣿⡀⠀⠈⣇⠀⠀⠈⠉⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠞⠋⠉⠉⠙⢳⡄⢻⡀⠠⣄⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣿⠈⢷⡀⠈⠳⢦⣄⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠷⢤⡤⠤⠤⢤⡤⠟⣤⣌⠛⠶⣤⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣧⠈⠛⠓⠶⣦⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⡐⢦⣀⠀⢸⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢠⣶⡀⠀⣾⠙⢦⣌⠳⢦⣭⡿⠋⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢠⣿⡌⢳⡄⣏⢷⡀⢸⡇⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠿⣷⡼⠃⠙⠳⠿⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣤⣤⣤⣤⣤⡴⠖⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠐



⣏⠉⣻⣛⣛⣛⣛⣻⣛⣛⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣿⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣄⠀⠀⠻⠀⠀⠘⠃⠀⠀⠟⠀⠀⣠⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠙⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣤⡶⠛⠋⠉⠉⠉⠉⠙⠛⠶⣤⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠰⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡦⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢴⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣀⣀⣾⣁⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣹⣆⣀⣀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠹⡏⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⣀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠿⠶⠶⠿⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡏⢸⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣊⣁⣰⣅⣨⣇⣀⣙⣦⣄⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⣠⢞⡿⡁⢈⡾⡗⣄⠀⢲⡎⠉⠉⠉⠉⠀⠀⠀⢠⣶⣶⣆⣸⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣔⣊⣁⣸⣦⡿⢾⡔⣧⣀⣑⣢⣄⣀⣀⠀⠀⠀⢀⣶⣉⣋⣻⣋⣡⣀⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⢙⡟⠉⠀⠉⠉⢩⣷⣛⣄⣠⣟⣬⣏⠉⠉⠈⠙⣷⠂⠀⠀⠀⢹⢹⢹⢹⣹⣿⣿⢹⠁⡇
⠀⠀⠀⠀⠀⡴⠊⠉⠉⡍⢉⠲⣄⠙⠢⢄⡀⠀⠁⠀⠀⠀⢠⡟⢻⣭⠋⠙⣯⡝⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⢸⢸⣻⣿⣿⢸⠀⡇
⠀⠀⠀⠀⢸⠀⠈⡷⠚⠳⡎⠀⢸⠀⠀⠀⢹⠀⠀⠀⠀⠀⡼⠾⠭⠬⠷⠴⠥⠭⠷⢷⡀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⢸⢿⣿⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢇⠁⢣⣄⡣⣋⠉⡸⠀⠀⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⢸⢸⣿⣿⢸⠀⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⢆⠈⡜⠁⢈⠞⠁⠀⠀⠀⢸⠀⠀⢀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⢸⢹⣿⣿⢸⠀⡏⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣸⣄⣿⣠⡏⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⣶⠀⢠⠇⢸⠀⡿⣄⠀⢸⢠⠏⠉⠙⢆⠀⠀⠀⣟⣿⣿⢼⢸⢿⣿⢸⣠⣇⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⢸⡷⠾⢿⡇⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⡜⠈⣇⡜⠀⢸⠀⡇⠘⣆⡂⢸⡄⠀⠀⡸⠀⠀⠀⠿⠉⠉⠙⠙⠙⠉⠿⠖⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠸⣭⣭⣽⣃⣀⣀⣀⡠⠞⠁⠀⠀⠀⠘⠁⠀⠙⠁⠀⠸⠀⠃⠀⠈⠃⠀⠧⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⣴⠶⣄⡴⣦⢠⠶⢠⠴⡤⡄⢠⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢖⠚⠀⠛⠳⠖⠘⠀⠘⠶⡇⢘⡏⠀⠀⢰⠟⢒⣒⡒⠒⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣗⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⢸⠀⠙⠛⠛⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡀⠀⠀⢤⣀⡤⠒⠒⢦⡀⢰⠒⡄⠀⢀⡄⢸⢀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⡀⠀⢸⢸⠃⠂⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠲⢤⣀⣀⡤⠤⠄⠀⠙⠒⠊⠀⠀⢋⡀⣸⣞⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣽⠁⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠉⣩⠝⢻⡿⣗⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠋⠀⠀⣸⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠢⡼⡇⠬⠓⢤⣈⠟⠈⠉⠉⠉⠉⠉⢱⠀⠀⠀⠉⢻⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢀⣤⡦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠧⠼⠤⢊⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠒⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⢸⣿⣿⠀⠀⠀⠀⠀⢀⣴⣶⣿⠿⠷⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⣰⣿⣿⣿⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⢸⠏⠀⠀⠀⠀⠐⠟⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡜⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠉⠀⠀⠀⢰⠋⠉⠙⣦⠀⠀⠈⠃⠀⠀⠀⠀⠀⣠⣶⡿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠤⠀⠀⣽⠀⠀⢀⣀⣀⡀⠀⠒⠀⣶⣶⣿⣿⣿⣿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⣷⣤⡀⠀⠀⢰⠋⠙⣷⠀⢠⡿⢠⡾⠋⠉⠈⠙⣆⠀⠀⠀⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠠⠀⢷⣿⣿⣿⠟⠋⠉⠁⠀⠀⠈⣁⣴⢟⣴⠟⣱⠟⠁⠀⢀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⣭⡷⢛⣵⠞⠋⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠘⠻⢿⣽⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⠾⣛⣭⠶⠟⠋⠁⠀⠀⠀⠀⠀⣿⡟⠉⡉⠉⠉⡍⠉⠉⡉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡀⠀⠀⡄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠶⡶⢶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡀⠀⠀⠄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡄⠀⠀⠄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠄⠀⠀⠄⠀⠀⠠⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠂⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢠⠀⠀⠀⠄⠀⠀⠠⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡁⠀⠈⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⠀⠀⠀⠄⠀⠀⠠⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠄⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⣿⠃⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⣸⣿⠀⣀⠀⠀⢀⡀⠀⠀⢀⡀⢹⣷⡀⠀⢀⣠⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣀
⠀⠀⠀⠀⠀⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠛⠛⢻⣿⣿⠟⠛⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛⠛⠻⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⣆⣿⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡇⠀⠀⠀⠀⠀⠀⣼⠡⠂⠙⣦⣀⣀⠀⠀⢰⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⣀⣠⠴⠖⢋⡽⣶⢠⡞⠲⣄⡉⠉⠓⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠟⠋⠉⢀⣠⠴⠋⠀⣿⠘⡇⠀⠀⠉⠳⢤⡀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⡄⣷⠛⠛⠉⠁⠀⠀⠀⣿⠐⡇⠀⠀⠀⠀⠀⢻⡀⣦⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡛⢦⡀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⢠⡗⠛⢮⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢚⣧⣤⡼⣍⡛⠒⠦⣄⠀⡇⠀⡇⠀⢀⣀⡤⠶⢚⣧⢠⡾⣍⡛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⢋⣁⣠⠶⠋⢸⠀⡇⠀⠉⠳⢤⣈⠳⣇⣀⠿⠚⢉⣁⣤⠞⠉⣿⢸⡇⠈⠙⠲⢤⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠉⠳⡾⠄⣿⠉⠉⠁⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠉⠳⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣀⣸⡇⠄⢼⡀⠀⠀⠀⠀⠀⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢠⡴⠾⠃⠀⠀⠀⠀⠉⠙⣷⡶⠦⣄⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⢦⠟⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠁⠉⠃⠐⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠀⠀⠀⠁⠈⢆⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢿⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣴⣶⣶⣤⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⡇⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢿⣿⣿⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⠉⢹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢠⡄⠀⠘⠃⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⠟⢻⣿⡟⠉⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠋⠀⠀⢸⣿⣷⠀⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⢸⡟⠛⠛⢿⡆⢾⣿⣿⣿⣧⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠿⢿⣿⠀⣿⣿⣿⣷⣿⡇⢸⡟⠛⠛⢿⡆⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⣿⠛⠛⠛⢻⡇⠸⣿⣿⣿⣿⣿⡀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠛⠛⢻⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⡿⠟⠛⢻⣇⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠟⠛⠃⠘⠛⠛⣿⡟⠛⠛⠀⠙⠛⢻⡟⠛⠋⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠷⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠏⠥⠯⠴⠶⠦⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⢙⡿⠿⢿⠿⢿⣿⢿⠻⢿⠿⣿⣿⣿⢿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣾⣿⣷⣿⣾⣶⣶⣶⣾⣾⣾⣿⣿⣿⣿⣾⣿⣷⣶⣶⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⠄⡠⠒⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⡠⠖⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣘⣖⣒⠊⢀⣳⣒⡒⠁⠐⠂⠒⠁⠀⠀⠀⠀⠀⣘⣛⣛⠋⢉⣻⣛⡛⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⢸⠛⠛⠻⠂⡟⠛⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣟⠂⣺⣿⣿⡓
⠀⠀⠀⠀⠀⠸⠀⢠⠄⡇⠃⠀⡤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡆⢐⠍⠂⢁⠀⡢⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⣏⣿⣏⣿⣿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣲⣶⣿⠿⠀⠤⠤⣈⠘⠀⠀⠒⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⣻⣿⠋⠁⠀⠀⠚⠀⠀⠈⠑⢄⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⢻⠇⢀⣼⣿⣧⠤⠦⣤⣤⡀⠂⠀⠀⠀⠀⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⡀⠀⢰⠉⣾⣿⠏⠀⢀⣤⠀⠙⢳⣄⠀⠀⠀⠀⠀⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣀⡴⠃⢠⣿⡟⠀⡀⠈⠈⡥⠀⠀⡉⢻⣄⣀⠀⠀⠀⠀⢹⡀⠀⠀⠱⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠂⠀⢲⡞⠀⠀⣾⣿⣷⡄⠁⢀⡘⢁⡀⣈⠁⠒⣟⢿⣇⡀⠀⠁⠈⢷⡀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢸⡁⢻⣿⡀⠀⠚⢳⡇⠰⣟⣀⣀⠈⠀⢙⣧⣆⣀⣀⠄⢳⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⢼⠃⢨⣿⣧⡟⠀⣿⠺⠿⠟⣻⣿⣯⡐⣷⠛⣹⣿⣿⠀⢨⠛⠃⢠⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⡆⣼⡀⠀⢹⢿⣿⣄⢿⡄⣰⣎⠙⣿⣿⣷⣏⠀⠈⣹⡏⢀⡀⢼⠀⡼⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢡⣼⣿⣦⡀⠠⠙⡿⢿⢿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣶⠟⢷⠃⢰⠄⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡶⠟⠿⢷⣾⣟⣿⡿⣿⣿⣶⣮⣽⣿⣿⣿⣿⣿⣿⠒⢪⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠈⠹⢿⣶⣾⣯⣭⣄⣬⣿⣿⣿⡿⣿⣿⣿⣿⡾⠂⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠈⠁⠀⠈⠉⢹⡿⢿⣿⣻⣿⣿⣿⣿⡍⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠄⢤⢚⣴⣠⣿⣿⣿⠋⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢦⣴⠒⢋⠁⢉⣻⠆⠀⠘⣓⣇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠞⠋⠉⠉⠄⢀⡸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠒⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⢀⡴⠊⠁⠀⠀⠈⠐⢀⠀⠰⡀
⠀⠀⠀⠤⠐⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠
⠀⠀⠀⠈⠂⢄⠀⠢⠀⠀⠀⠀⠀⠀⡠⠎⠀⢀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡎
⠀⠀⠀⠀⠀⠘⠋⠀⡀⠀⠓⣤⠊⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠂⠀⠀⠐⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣁⠀⠆⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢁⡀⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠚⠒⠂⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢀⣀⣀⣀⡀⠀⠀⠀⠁
⢰⠂⠐⠈⠁⠈⠀⠈⣀⢀⡀⣰⣤⣴⣾⠀⢠⠀⠄⠀⡄⠀⠄⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢢⡄⠀⠀⠀⡀
⠉⡆⠈⡀⢁⠀⣁⣀⣀⢀⣁⣈⡋⠉⠙⣦⡤⠤⠤⠤⢰⣤⡤⠤⠦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠊⠀⠀⠀⠀⠇
⠀⢉⠉⡉⠉⠥⠿⠛⠣⠀⠀⠀⠄⠀⠀⣴⠀⠂⠀⠂⠈⡉⠈⡀⢉⠀⣱⠄⠤⠤⠤⠤⠤⠤⠄⢷⠤⠤⠼⠀⠀⠀⠀⠨⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⠀⠠⠀⠄⠠⠄⠐⠒⠒⠀⢸⣭⣭⣿⠀⢈⠀⢁⠀⠁⠀⠁⢀⣀⣐⠀⠀⠀⠀⠀⠀⠀⠀⢻⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠘⣆⣠⢤⠤⠤⠦⠤⠦⠰⠲⠒⠀⠀⠀⠃⠀⠉⠉⣉⠉⠉⠩⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠈⠀⠀⡁⠀⢀⡁⠀⠀⢰⣾⠀⠀⠀⠁⠀⠀⡀
⠀⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠧⠀⠀⠈⠁⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣨⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠙⠹⢋⠋⠙⠩⠋⠙⠉⠙⠉⠝⠁⠍⠩⠉⠀⢁⠌⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡠⠀⠈⡀⠁⠈⠀⠀⠠⠂⡠⠀⠌⢀⠈⡀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣝⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣌⡛⢦⣝⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢶⣞⠛⠉⠉⠙⠶⣬⡛⢦⣝⠳⣦⡀
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡖⠛⠉⠉⠀⠀⠀⠉⠳⢦⣀⠀⢀⣀⣽⡷⣬⡙⠦⣍⡳⢦⣀
⠀⢀⣴⣖⠛⠋⠉⠀⠀⠈⠙⠶⣄⡀⠀⠀⢀⣀⣤⡴⠿⠻⢯⣉⠀⠀⠀⠙⠳⢮⣙⠶⣍⡳⢦⣀
⠀⠘⣷⣍⠳⢦⣀⠀⠀⠀⢀⣀⣬⠿⠶⣟⡉⠉⠀⠀⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠉⠳⢮⣙⠶⣍⡳⢦⣀
⠀⠀⣿⢸⡟⢦⣍⠳⢶⣛⠉⠁⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⢛⣦⣤⡴⠞⠛⠉⠛⢮⣝⠶⣍⡓⢦⣄
⠀⠀⣿⢸⠃⠀⠈⠛⢦⣍⠳⢦⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣀⣠⡤⠶⠚⠋⠁⠈⠛⢦⣄⠀⠀⠀⠈⠛⠦⣝⡳⣬⡛⢦⣄
⠀⠀⣿⢸⠀⠀⠀⠀⠀⠈⠛⢦⣍⠳⣦⣀⢀⣀⣤⠴⠖⠛⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⣈⣹⡶⣝⡳⣬⡛⢦⣄
⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠻⢯⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⣀⣬⡿⠶⣟⡉⠀⠀⠈⠙⠶⣝⡳⣬⣛⣦
⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠳⢦⡀⠀⠀⠀⠀⠀⠈⢙⣶⣤⡶⠞⠋⠉⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠈⠙⣳⣯⡁
⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣝⠳⣤⣠⡤⠶⠚⠋⠁⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⢙⣶⣤⡶⠛⠉⠀⠙⠳⣤⡀
⠀⠀⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠓⢦⣉⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⣀⣤⠶⠛⠋⠁⠈⠙⠳⣤⣀⣠⡴⠞⣿
⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠉⠙⢻⡿⢦⣙⠳⣤⡀⠀⠀⣀⣤⠶⠞⠛⠷⣤⡀⠀⠀⠀⣀⣤⠶⠛⣋⣡⠴⠞⠋
⠀⠀⡇⣿⢀⣀⣠⣤⠴⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⠳⢦⣙⠻⣯⡉⠀⠀⠀⠀⠀⣀⣹⡷⠞⢋⣩⡤⠶⠛⠉
⠀⠀⠛⠛⠻⠿⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⢸⡏⠉⠉⠛⠛⠛⠳⢦⣙⠳⣤⡤⠶⠛⣉⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠦⢽⣷⣾⡧⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠳⠳⡤⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⡴⢦⣀⡀⢰⣄⣠⡀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠂⠐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡒⠠⠅⣑⡚⠂⠠⠀⠀⠀⠀⠡⠠⠪⠊⠀⠀⠀⠉⠁⠀⠀⠉⠑⠀⠰⡆⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠓⡦⡥⣇⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡀⠺⠌⣉⠑⠒⠤⣪⣂⡀⠐⠀⠀⢀⠄⠇⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠙⠓⠶⣤⣍⣀⡘⠤⠄⡉⢙⠗⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣄⣠⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠐⡐⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠇⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣀⣦⡀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⣿⣧⠇⠀⠀⠸⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡗⢦⣀⠀⠀⡠⠎⠁⣀⡀⠠⣤⣦⡂⢷⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⢀⠀⠀⠀⠈⠦⠟⣞⣻⠄⢸⡮⡏⠀⠴⣒⠐⠋⢍⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡀⠀⠀⠀⠀⠀⡀⠀⠁⠆⢈⣨⣄⠀⣦⢴⠀⣷⣻⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣠⠀⠀⠀⠀⠄⠀⠠⠅⣚⠀⠸⠘⠁⠘⠚⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠁⠀⠀⡀⡀⣀⣠⠤⠚⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠴⠱⠿⠈⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣂⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠁⠀⠈⢆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣴⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢼⢹⠀⢸⣉⣏⣉⣏⣹⠀⠀⠀⠀⠁⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣀⣇⣀⣇⣸⠄⠀⠀⠀⡠⠔⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠼⠖⠒⠒⠚⠼⣄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠒⠒⠢⡄⠀⠀⡄⠉⢀⠀⠀⢇⠀⢰⡉⡱⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡜⢠⠴⠳⠦⠾⠦⡀⠈⢦⠀⢸⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠊⠉⠁⠀⢸⠀⠀⠀⠀⠀⠁⠀⢸⡦⠼⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⠤⠤⠤⠘⢄⠀⠀⢀⡰⠀⠀⠀⠓⢤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣉⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁