Madam Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⡿⠋⠁⡈⠁⡀⡇⢹⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣹⣿⣿⣿⡟⡀⠀⢰⠃⣰⠁⡹⡌⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡴⣩⣿⣿⣿⣿⣵⠇⣠⠏⣰⠏⣰⣻⣷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⡟⣰⣿⣿⣿⣿⡿⣫⡾⣣⣾⣡⣾⢳⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠃⡾⢱⣿⡿⣹⣿⣿⣿⠟⠛⠉⠹⣯⡌⢠⣦
⠀⠀⠀⠀⠀⠀⠀⣿⢰⠇⣾⢹⡇⣿⣿⣿⣿⠀⠀⠀⠘⠉⠁⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢠⡏⣾⠹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⠀⣿⢻⣇⣀⠀⠀⠀⠀⠀⠀⠘⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⢠⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠸⡇⢙⡀⢙⡜⢿⣏⠀⠀⢠⣤⣤⣤⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠙⠳⠤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠻⡄⠷⠈⣧⡀⠱⠈⠙⢷⣦⠸⣿⣿⡿⠉⢀⡎⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⢹⠀⣆⠘⣷⡀⣥⣀⣾⠟⠀⠙⠛⠃⣠⠟⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣬⣶⣼⣿⠿⠟⠁⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⡼⡷⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠓⠰⢾⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢾⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢠⢶⣄⣀⠉⠀⠰⡇⡴⣆⣀⡀⠀⠀⢠⠟⣠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣄⠀⠑⠢⢄⡀⠀⠀⣸⠻⣦⣄⡀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠉⠙⠼⢷⢲⣄⣀⠀⠙⠱⡟⠁⣠⠏⣼⣿⠃
⠀⠀⠀⠀⣠⣴⡾⢟⣥⣢⣿⣷⣤⣀⠀⠈⠉⠉⠁⢀⣿⠩⡿⣿⣦⡀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⢰⠏⣼⣿⠃
⠀⠀⣠⡾⠋⠙⡛⢿⣾⣿⣿⣿⡟⢿⡉⠀⠒⠒⠒⠋⢹⣌⢿⣷⣿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣼⣿⠏
⠀⢠⡟⠀⡾⢂⡐⢠⢿⣿⣿⣿⠟⡌⢿⡄⠀⠀⠀⠀⠈⣿⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⣽⠏
⠀⣼⠇⣿⢀⣿⣹⣾⡬⣿⣿⣿⣷⣿⠈⣷⠀⠀⠀⠀⠀⢹⡇⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡹⣽⠃
⠀⣿⢰⡏⢺⣿⣻⣟⢠⢿⣿⣿⣿⣿⠇⢹⡆⠀⠀⠀⠀⢸⣗⠰⣿⠛⢲⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⣀⡶⡄⢠⡿⠑⢱⠏
⠀⢿⣿⣯⠾⠟⠿⠿⠾⢸⣿⣿⣿⣿⣤⡌⣷⠀⠀⠀⠀⠀⣯⠾⠚⠋⣁⣿⠌⣛⣶⣤⣀⠀⠀⠶⡏⠹⡆⠸⡇⠸⣿⠡⢠⠏
⠀⢸⣿⣿⣤⣚⣷⠶⣼⢸⣿⡇⣿⣿⣿⣿⣿⡄⠀⠀⠀⡰⠋⠀⠀⠸⠟⠛⠛⢉⡡⢿⠋⢿⣾⣆⢱⡀⢳⠀⠁⠀⢹⡁⠎
⠀⢸⣿⣹⣿⣿⣋⠀⣁⣿⣿⣿⣿⣿⣻⡿⠘⣇⣀⠤⣴⠃⠀⠀⠀⠀⠤⠒⠚⣉⠴⣻⣳⣦⣝⣿⠈⠃⠀⠀⠀⠀⢸⡟
⠀⣸⡇⣿⣿⡟⣿⠈⣛⣯⣿⣿⣿⣵⣦⡾⠛⠛⢷⡀⠸⡄⠀⠀⠀⠀⠐⠒⣉⡡⣾⣿⣿⠿⠻⡏⠀⠀⠀⠀⠀⠀⣼⠁
⠀⢸⣇⢼⠛⡛⠟⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⢹⡀⠱⠀⠀⢀⣀⣘⣉⣥⣾⠛⠾⡟⡉⢹⡛⢦⣀⣀⣀⣠⠞⠁
⠀⠀⢻⣾⣷⡀⣾⢿⣿⣿⠋⠄⠀⠀⠀⠀⢀⣀⣀⠀⣧⠀⡿⠛⠛⣉⣹⣿⣾⣿⡇⠞⡑⣬⢦⠳⣤⣤⡭⠉⠁
⠀⠀⠸⣿⣿⣿⣤⣿⣿⣅⣠⣀⣤⣴⡾⣿⣿⣿⣿⡇⣿⣿⣧⠀⠀⠳⠸⣿⠿⣾⣥⣤⣤⣴⡶⠟⠉
⠀⠀⠀⢻⡈⢟⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣭⣥⣴⡿⢹⡟⠠⠐⠆⠀⣿⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠉⠀⠈⠙⠻⠿⠿⢧⠾⠻⠿⠛⠛⠋⠉⠹⠇⠸⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣶⣶⣿⣷⣶⣤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣗⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡅⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣟⣁⠀⠘⠿⠿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣥⣤⡀⠀⠠⠾⠿⠛⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠈⠻⠇⠁⠀⠀⠀⠁⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣦⠀⠠⢤⡶⠀⢀⣴⠟⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠉⠀⠀⠀⠀⠰⠾⠉⢀⣠⢯⣶⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠂⠀⣼⣿⣿⡔⣾⣿⣡⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣠⠄⠈⠁⣀⢰⣻⣿⣿⣷⣟⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡏⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⣿⣿⣿⣿⣿⡿⣦⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⣿⠀⠀⠀⠀⢧⡀⠀⣤⢀⣇⠸⠃⣼⣿⣿⣿⣿⣿⡿⢡⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠣⠀⢸⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⡇⢰⣿⣿⣿⣿⣿⣿⠏⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⣿⢃⡟⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⣿⣿⣿⣿⣿⣦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢙⣷⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢳⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⠆⢰⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⠉⣠⣿⣿⡛⠷⠀⠀⠀⠀⠀⠹⠟⠿⠀⠀⠜⢡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢙⠟⢀⣴⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢀⠄⢀⠞⣷⡼⣦⡀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢈⠐⣏⠐⡿⡁⢆⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⢿⣲⠉⠯⣿⣍⠦⣀⠀⠀⠄⠀⠀⠐
⣶⣴⣶⣤⣶⣤⣤⣴⣤⣦⣤⣶⣼⣿⣷⣶⣿⣿⣿⣷⣆⣿⣿⣾⠁⠓⠀⠂
⣟⣿⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠱⠏⢹⣿⣿⣿⣿⣿⠇⠀⣀⠘⡀⢡⡀
⣿⠛⠓⠐⠚⠛⠓⠛⠓⠋⠐⠓⠋⠉⠋⣸⣶⣿⣿⣿⣿⣿⣿⣿⡂⠘⣻⡶⣹⠀⣇
⠋⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠙⠛⣿⠛⢻⣿⣿⣿⣿⣇⣼⡟⢣⠃⢦⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣾⣿⣿⣿⣿⣿⣿⣏⡞⣸⣮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣇⣿⣿⣿⣿⡏⣾⣿⠀⡽⡆⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣿⢿⣿⣿⢻⣿⣿⠀⡁⢱⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣾⣿⡟⢸⠃⢃⡶⠀⠀⠀⠘⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢹⡀⣿⠀⡇⢸⡇⢸⠀⡿⢀⡇⢸⠇⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠸⡇⢹⠀⣇⢸⡇⣾⠀⡇⢸⡇⣾⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⡇⢸⠀⣿⢸⡇⣿⠀⡇⢸⠀⣿⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⢀⡇⢸⠀⡟⢸⡇⣿⠀⡇⢸⡄⣿⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⢸⡇⣸⠀⡏⢸⡇⢿⠀⣇⢸⡇⢿⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⢸⠀⡿⠀⡇⢸⡇⢸⠀⣿⠈⣇⠸⡇⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⢀⣤⢆⡴⢂⡴⢠⡖⢠⡆⢰⡄⢶⡄⢦⡀⢤⡀⢤⡀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀⡿⠁⣾⠁⣾⠃⣼⠇⢸⡇⢸⡇⠘⣧⠈⣷⠀⢿⠀⢻⡄⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⣿⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⣿⢿⣷
⠀⠀⠀⠉⠐⠁⠀⠀⠀⠀⠀⠀⢸⡇⠈⣷⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⣿⠀⢸⠁⢸⡇⠀⠀⠀⠀⠀⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⠀⣿⠀⢻⡄⢸⡇⢸⡇⢀⡟⠀⣿⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⡄⢹⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢠⡏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⠇⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⡀⣿⠸⣧⢸⡇⢸⡇⣾⠁⣿⢀⡟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠸⣇⢻⡄⣿⢸⡇⢸⠇⡿⢰⡇⣸⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠻⠘⠇⠻⠘⠇⠻⠐⠇⠸⠁⠟⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠄



⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠒⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⡶⢶⡥⠀⢠⠠⠶⠉⠄⡴⠀⠀⠁⠔⠐⠀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠈⠄⠐⠈⠀⠀⠸⠈⠡⢀⠃⢠⠈⠀⠱⡇⢶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢑⠵⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠁⠙⠌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢚⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢠⣰⠞⢘⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⢤⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡆⠀⢀⣺⣟⡄⡛⢿⣿⠟⣁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠄⡀⠀⢐⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠻⣄⠀⠿⣿⣟⣟⠟⠁⣸⡟⠃⠻⠿⠸⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣸⠟⢉⣠⠖⠸⣿⣷⣾⣿⣿⣴⠅⠹⣋⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢁⠴⠛⠁⠀⠀⠈⠻⣿⣿⡇⢠⣀⡇⣿⣿⠏⠁⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡉⠀⠀⠀⢀⣠⣄⡀⠈⠉⠀⠸⡏⣳⠌⠁⣠⡾⢡⠂⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⢾⠟⣇⠀⠀⢀⣼⣷⣶⠦⠀⣀⣠⡼⠛⢁⣠⣾⡿⠁⠎⢀⠀⢦⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠊⠉⢹⣿⣿⣷⣞⡋⠁⠀⠒⠻⣿⠟⠁⠁⠎⠀⠀⢸⠃⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠙⠀⠀⠀⠀⠀⠁⠀⠀⠛⢂⠀⣤⡤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⠀⢀⠛⠋⢡⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣄⣴⠆⠀⠀⣢⣄⡤⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡴⠀⠀⠀⡞⠉⠉⠻⡿⠀⠀⠀⣿⡿⠁⠀⠀⡀⠠⢲⣦⠄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡴⢄⡀⠀⠀⠁⠀⠀⢀⡇⡀⠀⠰⠟⠀⣀⠀⠀⠀⢺⡟⢫⣤⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣎⢀⠀⠀⠰⠂⠀⠈⠓⠋⡄⠀⡀⠀⠉⡔⢶⠆⢸⠀⣦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⠞⢁⡟⠀⠂⠘⠃⣗⠀⠘⠉⠀⠂⣷⠀⣶⠀⢸⠀⠙⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢀⡞⢁⡟⢁⡀⢀⡷⢠⡄⢸⡗⠐⡇⠀⡿⠀⡟⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜⠀⣾⠁⢺⠁⢸⠃⢨⠃⠀⡇⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠃⠀⠚⠀⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠁⠁⡇⠀⠙⢻⢠⠋⠀⢺⠀⠀⠀⣠⡴⢿⡿⣟⠀⠀⠀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⢳⢀⣠⣾⡎⢡⣄⠸⡄⠀⠀⠀⠀⠀⢻⡾⠀⠐⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⠀⠀⠀⣸⣿⣿⣿⣿⠆⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣴⣷⣄⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⡀⠀⢸⡆⠀⠀⣸⣿⣇⠀⠀⠉⣽⣿⣿⠟⠀⠀⠀⠀⠀⠶⣤⣄⢰⣿⣦⣽⡿⣿⣿⣿⡿⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⠰⠀⢉⢿⣿⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠸⢹⣿⠿⠷⣞⣿⣷⠏⠍⠛⠀⢁⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣆⠈⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠁⠐⢰⣷⠘⢾⣟⢢⡄⢸⣹⡏⢠⠄⠁⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⠀⠀⠄⠐⠓⠴⠂⠀⠀⠀⠰⠂⠀⠘⠙⢿⣿⣿⣿⣷⣿⣿⣧⡘⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣀⠲⠃⠀⠀⠀⠒⠀⠀⠠⠐⠀⢀⠀⡀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣷⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢦⣼⣯⣵⣿⢄⣀⣤⢤⣤⣤⣴⡆⢠⣴⣿⣎⠔⠀⠀⣿⣿⣿⣿⣿⢁⣿⣿⣿⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣴⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣸⠿⢿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣻⣿⣿⣿⣿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⡿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⡿⢿⣿⣩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣇⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣶⣶⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠹⡻⠿⠋⡉⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠉⠉⡇⠀⠀⠀⣀⣄⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢹⠀⠀⡼⠁⡼⢰⣇⡄⢠⢴⣦⣞⡗⣸⠋⡘⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡄⢸⡇⠀⠀⡼⢱⡇⢿⡭⠘⠗⠂⠉⠀⠉⠁⠈⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠁⠈⠓⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢙⠷⣶⡾⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡆⢀⣀⢀⡄⣤⢶⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⡤⠞⠋⣠⣿⠶⡇⠀⢠⢾⠇⣾⡟⢠⢴⡆⡴⢹⢡⣇⣏⢸⠡⠇⠓⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⣠⢎⣿⣤⡾⠁⢀⡏⣼⡼⢸⡃⠻⠘⠂⠓⠉⠁⠁
⠀⠀⠀⠀⠀⢀⣠⠴⣶⣿⣯⠄⠀⠀⠀⢀⠼⠁⢿⣾⣿⠁⠠⠞⠀⠛⠁⠈⠉
⠀⠀⠀⣤⣶⢫⠰⢹⠙⣿⣿⣯⡁⢔⣊⣁⡤⢴⡛⠋⠀⣙⣒⡄
⠀⠀⠀⠺⡁⡷⠂⢼⠀⡟⣿⣿⣿⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠙⢧⣴⡿⠀⡧⣯⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠶⢿⣡⠾⠛⠁

⠀⠀⠀⠀⠀⠀⣤⡄⠠⡦
⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄
⠀⠀⠀⠀⠀⠀⢈⡁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢈⣁⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢂⠀⠀⠁⠀⡇⠀⠀⠀⠀⠁⠀⢰⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠉⠈⡉⠀⠀⠀⠀⡀⠰⡀⠀⠈⢄⠀⠆⢀⠸⡄⠁⠀⠀⡇⠐⠀⠀⡄⢀⠃⠀⠀⠎⢀⠄
⠀⠀⠀⠀⠀⠀⠈⠋⠈⠃⠀⠀⠀⠀⠑⣄⠘⠄⠠⠘⣧⠀⠀⠀⢇⠀⠀⢠⡇⠀⠘⢠⠃⠀⡀⢠⡖⢀⠊⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠃⢀⠈⠐⢄⠀⡈⠳⡌⠀⠀⠘⣇⠀⠀⢸⠀⠀⢸⡇⠀⠀⡜⠀⠀⢠⡟⠀⢀⠀⣠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠰⠦⠀⠆⠀⠑⠢⣄⠑⢌⠢⡙⢦⡀⠀⠈⢆⠀⠀⠂⠀⠘⠃⠀⠠⠁⠀⢠⠋⠀⠀⣠⠞⠁⠐⡡⠀⣀⠔⠂
⠀⠀⠀⠀⠀⠀⠰⠆⡸⠷⠠⢀⠀⠌⠓⢦⡀⠀⠀⠑⠄⠀⣀⡤⠶⠒⠛⠛⠛⡓⠶⢤⣄⡀⠀⢀⠔⠁⢈⠄⢈⡤⠞⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠠⠤⠰⡶⠰⢤⣀⠀⠀⠀⠉⠲⠀⢀⡴⠟⠁⢀⡀⠷⣴⣤⣀⠀⠀⠀⠈⠛⢦⡀⠀⢀⡤⠚⠉⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣤⡀⠀⠈⠉⠒⠤⣀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠁⠉⠀⠀⠐⡀⠙⣦⠈⠀⠀⢀⣠⠴⠚⠋⣁
⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⠤⣌⣀⣀⠀⠀⠀⢠⡟⠀⠀⣹⠀⠄⠒⢦⣄⠀⠀⢀⡴⠋⠉⠂⠟⠂⠈⣧⠀⠉⠁⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠂⠀⠀⠀⠉⠉⠀⣾⠀⠠⠒⢁⠈⠻⠶⣦⡁⠀⠀⠀⢠⡴⠾⠋⠐⠆⠀⢸⡀⠀⠄⠒⠒⠒⠚⢋⣁
⠀⠀⠀⠀⠀⠀⠈⠉⣉⣭⣤⡤⠤⠤⠀⠀⠀⣿⠈⠀⡰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠉⠓⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣸⠁⠀⠉⠉⠉⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠘⠛⠘⢋⡤⠤⠖⠒⠈⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠐⠀⣰⠏⠀⠢⠤⣀⡀⠀⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠐⠋⠙⠇⠐⠂⠈⠀⣀⡤⠒⠀⠈⢷⡀⠀⠰⠀⠀⠀⠻⢶⣶⣶⠋⠀⠀⠀⠀⣴⠋⠠⣄⠀⠀⠀⠈⡙⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠰⠤⠤⠶⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠙⡶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠈⠑⠦⣄⠀⠄⠈⠐⠤
⠀⠀⠀⠀⠀⠀⢠⡤⢤⡔⠊⠀⠐⡁⠀⠀⠀⠀⢀⠄⠐⠁⠈⠁⠒⠢⠤⠴⠶⠶⠒⠊⠁⢠⣷⠄⠀⠂⡐⢄⠀⢌⠙⠂⢄
⠀⠀⠀⠀⠀⠀⢠⣤⢠⡄⢀⠄⠊⠀⠊⠀⠀⠀⢸⠆⠀⠀⠀⡐⠛⠳⠶⠤⠶⠶⢆⠀⠀⠀⠀⠆⠀⠀⠈⠢⡀⠀⠉⠢⣀⠈
⠀⠀⠀⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠓⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠛⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠚⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠚⠗⠐⠆
⠀⠀⠀⠀⠀⠀⠰⠦⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠟⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣀⣀⣀⡀⠀⣀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠐⠿⣿⣿⣿⣿⠇⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠙⠉⠁⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠹⠇⠀⠀⠀⠀⠀⠀⣠⣾⣠⣄⣠⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢈⣀⣉⣽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣶⣤⡴⠟⠉⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣦⢤⣿⠟⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠧⠒⠀⠀⠀⠒⠦⣜⣸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣧⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣽⡀⢼⡀⠀⣥⡀⠀⣦⢠⡄⢀⠠⡄⠀⡀⠈⢛⣻⣶⣟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⡁⣸⡇⠀⠹⣺⡀⣿⡀⣿⠈⣇⠁⠀⠈⡀⢀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠓⠚⠋⠀⠛⠓⠛⠓⠛⠛⠛⠓⠓⠈⠂⠒⠀⠀⠀⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⠀⠀⡄⢠⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⣾⡿⢿⡄⠀⣓⢧⣬⣿⣾⣧
⠀⠀⠀⠀⠀⢀⢀⣸⣇⣾⣿⠾⣮⡧⣀⣯⡤⡿⠻⣿⣿⡆
⠀⠀⠀⠀⠀⢋⣼⣷⠞⢻⡍⠀⢻⣧⡌⠱⠐⣆⠀⠈⠟⠳⣶⠄⠀⠂⠀⣀⠄⠤⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠊⠙⠛⠋⠭⢙⣷⣾⣿⣷⡴⢾⣷⠶⠄⠀⠞⠁⡀⢠⠀⠀⠈⠐⠉⡈⢻⣿⡟
⠀⠸⡏⢻⡏⠙⠇⠀⢀⡄⠚⡻⠿⣅⣠⣀⣼⠸⣄⡄⠀⠀⠀⠳⣬⣦⣿⡀⣰⢰⡁⠙⠏
⠀⢀⡇⠀⠟⠚⠛⠛⣻⣷⣆⡀⣀⠀⠙⢋⡿⠀⣇⠀⠀⠀⠀⠀⠀⢉⣻⠉⠛⢿⠉⠀⣸⠂
⠀⠀⠐⠲⠷⠾⡷⣶⣾⣿⣿⣿⣿⡇⠀⠈⠃⠀⠇⠀⠀⠀⠀⠀⠀⠚⣿⡿⠃⠈⢛⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠙⠻⣷⣟⣠⣼⣮⣧⡀⠀⠀⠈⡄⠀⠚⠛⠁⢀⡴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣖⡶⣾⣿⣿⣿⣿⣿⡿⠛⠎⠰⠻⣾⣷⢠⣴⢾⣿⡑⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⢿⣿⣿⣿⣿⠁⠈⠀⠀⠀⠀⠸⠿⣿⠈⡀⢻⡝⠎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡇⠀⠙⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⠁⠀⠀⢷⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣈⠁⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢛⣻⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣆⣴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⡇⢀⠀⠀⠀⢠⡅⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⡃⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⢹⣿⠋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣦⣀⠈⠀⠁⠀⠀⠘⣧⣄⣸⣿⡞⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡴⢷⣽⣯⣽⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⣛⢻⣿⡎⣸⣿⣿⣿⣿⣿⣝⢻⢯⣬⣿⡿⣟⣿⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠟⠛⠻⣿⣿⣿⡟⠃⢸⣿⣿⣟⠿⠽⠑⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠺⢿⣿⣄⣀⡈⣿⣿⣿⠟⠛⠲⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠈⠀⠉⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢾⡁⢶⠰⣨⢰⡆⡖⡆⠤⠆⠀⢸⢨⠲⠆⢿⢰⡆⡖⣴⡂⡶
⠀⠀⠀⠀⠀⠀⠀⡄⢰⠖⠀⡀⡄⣀⠀⢀⢀⡴⠀⠰⠆⡀⢀⠀⠀⠀⡠⠰⢀⣠⢀⢀⡀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠉⠉⠈⠁⠀⠈⠚⠃⠀⠘⠀⠙⠁⠁⠁⠀⠁⠀⠉⠃⠈⠉⠀⠁⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢉⣠⡀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡖⣿⣁⣻⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣷⣾⣿⣿⡷⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡾⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡉⠉⠉⠉⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠐⠘⠄⠀⠀⠐⠐⠀⠁⠄

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⡟⢛⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⣿⣿⣏⢀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢻⣿⡅⣄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡹⢀⡇⢠⡹⣿⣿⣿⣿⣿⣿⠛⠛⠋⠉⠉⠉⠛
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⢸⠃⣾⡇⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠋⠀⠸⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⣦⣤⡼⣿⣿⠟⠉
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⠏
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⡇⠏
⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⠀⠀⢠⣶⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀
⣿⡿⠿⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣟⡿⡀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⠀⠈⠙⠻⠿⠿⠟⠻⣿⣿⣿⣿⣿
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙
⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣯⣉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾
⣦⣄⣀⣀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢶⣶⡶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⢹⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣤⣧⣥⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣭⣥⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⡀⢀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣥⣥⣭⣥⣿⣤⣧⣿⣬⣤⣬⣬⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣇⣼⣿⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⠛⢛⣿⣿⣿⣿⡿⠈⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠴⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⠁⠀⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡃⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠓⠒⠂⠐⠒⠉⠀⠀⠈⠛⠛⠃⠀⠀⠀⠐⠒⠛⠛
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣶⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣰⠄⠀⠀⣀
⠷⠒⠊⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⡃⢸⣶⣾⠟⠃⠀⠀⣼⣿⡿⣿⣿⡄
⠀⠀⠰⣦⠒⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⢃
⠀⠀⠀⣿⣆⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⡤
⣀⡀⠀⠙⠧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠁⠁⢹⣿⣿⡏⠀⠀⢲⠶⠂
⠀⠀⠀⠀⠀⠀⠐⠈⠁
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⠀⠀⠀⠀⣿⠃⢰
⠀⠀⠀⠀⠀⡄⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠈
⠀⠀⠀⠀⠾⣁⡷⠒⡌⢑⠄⢀⠀⠀⠀⠀⣸⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠐⠒⣥⡄⣼⡟⢺⠟⣡⠂⠀⢠⠛⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠇⢘⡛⠉⢡⣀⡨⠀⠀⠀⠸⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣧⠀⡐⢠⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡇⢀⠔⠁⠀⠀⠃⠀⣸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠀⠀⢹⡄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠸⡷⡔⠟⠳⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⡇⠀⡈
⠀⢀⢀⢖⡀⠠⡁⠰⣶⣦⣤⣤⣤⡴⣾⠀⠀⠠⠠⠁⡀⢀⣴⣧⡌⠀⠀⠊⠑⠀⠋⠠⠀⠀⠀⠀⡰⡄⠀⠀⠀⠆⠀⠀⠀⣸⡇⠀⠃
⠀⣿⣷⣦⠹⡄⢙⠶⣍⡳⣀⠀⠀⠀⠘⣄⣤⠖⡠⢊⡙⢛⠀⢀⣙⣀⡈⠀⠀⣎⠀⠘⠆⢀⠖⠀⠐⣄⡠⠀⠀⠣⡀⢠⣴⡿⣧⢦⡇
⠀⣿⣿⡟⠀⠃⢸⠀⠈⠑⢤⣳⠀⠀⠀⢿⣧⠂⠀⠀⢁⢡⣸⣿⣿⣿⣿⡆⠀⢻⠀⠀⠀⢆⣠⡀⠀⢸⣧⡄⠀⠀⡆⣸⣿⣇⠀⠀⠰⡀
⠀⢹⣿⡟⠛⢶⡟⠀⠀⠀⠘⠉⠀⠀⠀⢸⡜⣧⠀⠀⡸⠀⠸⣿⣿⣿⣿⠃⠀⣿⠀⠀⡀⠘⠆⠓⠀⠘⠁⣄⡀⠐⠁⠓⢿⣿⣯⡀⠰
⠀⠘⣏⠁⠧⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⣷⣶⡾⣦⡄⣠⠄⠉⠛⠶⠤⡏⠘⠦⠀⠱⣔⠙⠂⠠⢼⡇⠀⠁⠀⡇⢐⠀⠀⠈⠙⠻
⠀⢀⠛⠲⢾⡜⢣⠀⠀⠀⠀⠀⠀⠀⠀⣽⠯⣉⣽⡷⣿⣇⣧⠀⠀⠀⢠⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠐⠀⠀⠀⡹
⠀⠈⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠛⢿⣫⠭⣿⣿⠀⠀⠀⠀⠖⢚⠀⠀⠀⠀⠿⠄⠀⠀⠀⠐⠐⠀⠀⠈⡗⠂⠂
⠀⢀⠆⠀⠀⠀⢸⠛⢳⣶⡤⢴⣶⠒⠋⢱⡗⠀⣿⣿⣿⣿⣿⣤⣀⣀⣀⣀⢨⠀⠀⠀⣰⠈⠁⢰⢂⠂⣠⡦⣤⠴⠏⢰⡄⠀⠐⠒⠞
⠀⠘⠓⣶⠶⢶⠁⠀⢰⣴⠀⠀⢐⠀⠀⢾⣿⣶⣿⣿⣿⠇⠈⡀⢀⡀⠀⠹⠞⠓⠶⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣼⣄⢸⡀⠶⢛⣿⡀⣷⣿⣀⡀⠀⢹⣩⣋⡏⠀⠀⠀⠁⠈⠃⠀⠸⠢⠦⠤⠖⠃⣀⣄⣀⢀⣄⠀⠠⣤⢄⢠⣄⡀⡀
⠀⠀⠀⠛⠛⠉⠛⠁⠘⠻⢿⣿⡟⠻⠿⠂⢸⡿⣇⡷⢤⡀⠀⡆⠀⡆⠀⢲⣤⡖⠐⣲⡄⠡⠽⠿⠶⠦⠥⠶⠙⠋⠉⠋⠁⠁⠈⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠭⣉⡻⠿⠀⠙⠒⠻⠔⠋⠁
⠀⢀⠔⠁
⠔⠁
⣾⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣤⣤⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠩⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣫⢟⢴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡍⠽⢛⡛⠻⠷⣶⣦⣤⡀⠖⣰⣪⣴⣾⣟⡫⡕⢕⣋⠾⣛⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡈⠁⠀⠀⠀⠀⠁⠀⠀⠈⠉⢹⠟⡻⣫⢟⢵⣊⢞⡫⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠹⠿⣖⣤⢄⣐⡒⠲⠀⣀⠀⠀⡯⣐⢝⣫⣶⣿⡟⠫⡓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠻⣦⡀⠀⠈⠉⠋⠚⠻⠷⠶⣤⣤⣿⣾⡿⠟⠉⠮⠀⠀⠁⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡄⠙⢷⣔⣂⣀⣀⣀⣠⣤⣤⣤⣤⣬⣥⣬⣥⡤⡤⣤⢤⢴⣶⣶⡶⡶⣭⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠈⢮⣦⠀⠈⢻⡏⡛⡫⡸⡽⢝⢕⢕⣩⡪⡪⡊⡾⡁⢕⢕⢕⢕⠽⡪⡪⣺⠇⣦⣀⡀
⠀⠀⠀⠀⢀⣤⣲⠱⠤⠀⠀⠀⠀⠀⠀⠑⣿⣗⣸⢣⡎⡪⡪⣊⢔⢕⡕⡡⣪⣪⣊⣪⢐⣕⣵⡕⡵⠡⠪⢾⣞⣼⣽⡿⠃
⠀⠀⠀⠀⠀⢻⠶⠬⢥⣄⣐⣠⠄⣀⡀⠀⣈⡛⠯⠿⣾⣯⠭⠭⡭⠼⠷⠶⠶⠶⠶⠰⢒⡾⠛⠓⠪⠽⠛⠋⣩⣿⣻⡇
⠀⠀⠀⠀⢠⣮⣄⡀⠀⠀⠌⠹⠓⠾⠬⠅⠲⡝⡕⠈⢞⣭⣶⣿⣯⢿⣥⣤⣔⣒⣠⡀⠤⠤⠤⣀⡀⠀⣠⣪⢎⣺⣷⡇
⠀⠀⠀⠀⠈⠒⠂⠭⠽⣓⣶⡤⢤⣤⣀⣂⣀⠇⣏⣞⢾⡇⢨⢩⠍⠁⠈⠉⠙⠛⠓⠛⠉⠉⠉⠑⣺⣛⣋⣿⣯⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠉⠑⠒⠶⠶⠴⠾⡿⠟⣾⣻⣤⢄⣀⡀⠀⠤⠤⠶⠤⣄⠀⠀⠀⠠⡟⠿⣿⢾⣿⡿⠛⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡉⠛⠹⠖⢭⣭⣽⣒⣒⡤⠬⠭⢀⣀⣷⢞⣵⣿⣾⣶⣖⣾⡮
⠀⠀⠀⠀⠀⠀⡸⢺⠿⠿⠿⠭⠽⠿⠿⠍⠉⠛⠒⠲⠶⠒⠛⠓⠒⠐⠚⠛⠛⠛⠉⠩⠍⠉⣙⠓⡾⣫⣁⠉⠀⠉⢡⡫⢂⠅
⠀⠀⠀⠀⠋⢕⡇⣿⢰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠲⣤⠉⠀⠀⠀⠀⠘⣎⠕⡢
⠀⠀⠀⠀⠀⠀⠣⡻⣧⣐⣒⣒⣒⢂⣀⣀⣀⣀⣀⣀⣀⠤⠤⢤⣤⣤⣤⣀⣀⣒⣒⢒⠒⢒⣐⣀⠀⠠⠴⠶⢶⣶⣶⣿⣿⡪⣔⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⡻⢛⠾⡫⢏⠽⡻⠿⡿⢗⠶⡲⢖⠖⠢⠖⠰⠂⠖⠩⠟⠛⠋⠟⠝⠩⠋⠝⠱⠋⠋⠈⠈⠁⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠉⠙⠦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⠀⠀⠈⠢⡑⢤⡀
⠀⠀⠀⠀⠀⢀⠆⠀⢴⠆⠀⡶⠀⢰⡆⠀⢶⢄⡴⣋⠔⠁⠀⣠⣴⢶⣶⢦⣤⡀⠈⠢⡙⢦⡠⡶⠀⢰⡆⠀⢶⠀⠰⡦⠀⠰
⠀⠀⠀⠀⠀⢸⣀⣀⣟⣀⣰⣇⣀⣸⣁⣀⠟⣩⠞⠁⠀⢀⣾⠋⠀⢸⡇⠀⠈⢿⣆⠀⠈⠢⣝⠻⣀⣈⣇⣀⣸⣆⣀⣻⣀⣀⡇
⠀⠀⠀⠀⠀⡏⠉⢹⠏⠉⣽⠉⠉⣿⠉⠹⡞⢹⠀⠀⠀⢸⣧⣀⣀⣼⣧⣀⣀⣨⣞⠀⠀⠀⢘⢳⠍⠉⢿⠉⠉⣯⠉⠹⡏⠉⢱
⠀⠀⠀⠀⢰⠓⠒⠛⠒⠒⠛⠒⠒⠓⠒⠚⠓⣺⠀⠀⠀⢸⡏⠉⠉⢹⡏⠉⠉⠉⣟⠀⠀⠀⢸⠚⠓⠒⠚⠒⠒⠛⠒⠒⠛⠒⠚⡆
⠀⠀⠀⠀⠀⠫⡭⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⢽⠀⠀⠀⢸⣷⣤⣤⣼⣷⣤⣤⣴⣯⠀⠀⠀⢸⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠝
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⢸⡟⡟⢻⠀⠀⡟⣿⢻⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣹⢹⡇⠀⣿⢹⡋⡇
⠀⠀⠀⠀⠀⠀⡇⢸⡗⡷⢾⠀⠀⡗⣿⢺⡇⢸⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⢸⠀⡏⢻⢙⡇⠀⣿⢹⡋⡇
⠀⠀⠀⠀⠀⠀⡇⢸⣧⣧⣼⡄⢠⣧⣿⣼⡇⢸⠀⠀⠀⢐⣿⠟⠟⢿⡽⠛⠿⣿⡇⠀⠀⠀⢸⠰⠷⠿⠾⠗⠀⠿⠾⠷⠷
⠀⠀⠀⠀⠀⠀⡇⢠⣤⣤⣤⠀⠀⣤⣤⣤⡄⢸⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⠀⡶⢶⢶⡆⠀⣴⢶⠶⡆
⠀⠀⠀⠀⠀⠀⡇⢸⣇⣇⣸⠀⠀⣇⣾⣸⡇⢸⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⠀⡧⣾⢼⡇⠀⣿⢼⡦⡇
⠀⠀⠀⠀⠀⠀⡇⢸⡇⣏⢹⠀⠀⡇⣿⢸⡇⢸⠀⠀⠀⠸⡟⠒⠒⠚⠓⠒⠒⢻⡇⠀⠀⠀⢸⢀⣇⣼⣸⣇⠀⣿⣸⣅⣇
⠀⠀⠀⠀⠀⠀⡇⠙⠛⠛⠛⠁⠘⠛⠛⠛⠃⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠈⠉⠉⠉⠁⠀⠁⠉⠁⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⢸⡧⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⢀⣀⣀⢰⣇⢀⣀⣀⣀⣀⣀⣸⣇⢀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠸⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡌⠉⢀⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠛⠛⠛⠛⣛⣛⣛⢁⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⢤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⣿⣿⣿⠿⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⢤⡖⠀⠀⠀⠀⠀⠀⠀⠠⠀⡴⠫⠟⣭⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠨⠀⣠⡿⢄⡀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠸⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠋⢉⠀⠉⠈⠀⢨⠊⠁⠀⠀⠙⢆⢠⣀⣄⠀⠀⠀⠁⠒⠊⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⢷⣤⡞⠋⠙⢧⡀⢀⣼⣷⣶⣷⣶⣄⡼⠘⠿⠏⠀⠴⠦⠤⠀⠀⠀⠙⢿⣦⠀⣶⡆⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⣠⡾⠉⢧⣄⣠⣞⡁⠈⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠐⠢⣤⣀⠀⠀⢠⡄⣀⠉⡀⢉⠐⠂⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡌⢁⣀⠀⣀⠈⠁⠈⠀⠀⠙⠻⣿⣿⣿⣧⢀⡄⠒⣄⡄⠒⣤⠙⢷⣄⠘⠁⠉⠀⠃⠈⠀⠀⣤⠔⠋⠉⠉⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣉⣱⠏⠀⠀⠀⠀⠀⠀⠀⠛⡛⠛⠛⠓⠗⠒⠈⠑⠒⠟⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠩⠙⢦⠀⠀⢰⣶⠀⠀⠀⣯⡽⡀⠀⠀⠀⠀⠀⠀⣀⣀⡞⠃⠀⠘⣧⠀⢀⣀⠀⠀⣠⣴⠟⢋⣡⠄⠀⠀⢀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠚⠂⠘⠃⣴⣤⣤⣦⣤⣤⢸⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⠆⠀⠀⢻⡆⠹⠟⣠⣾⠋⠁⠀⣾⣿⠇⣀⣤⡴⢄⡀
⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⢨⡗⠒⢦⡀⠀⠙⣿⣿⣿⣿⣿⣄⣿⣿⡆⠀⣤⣄⠀⠀⠙⠛⠋⠀⠀⠀⢸⣇⠀⣰⡿⠁⠀⠀⠀⠀⠁⠈⠉⣴⡤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⠤⠄⠻⠀⠘⠿⠿⠿⠿⠿⠏⠸⠿⠗⠀⠙⠁⠀⠀⡀⠛⠀⠘⢀⡀⢸⣯⢀⣿⠃⠀⣿⣿⡇⠀⠀⣀⠀⠁⠀⠀⠀⠈⠂⣀
⠀⠀⢀⠂⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠦⣤⡀⠀⠁⠀⠀⠀⠈⠁⣿⡇⣼⡿⠀⠀⣿⣿⡇⠀⡘⠟⠀⠀⠀⠀⠀⠀⠀⠟
⠀⣰⣯⣀⣀⠀⠛⠃⢠⡀⠴⢀⡴⠀⠀⠀⠈⢷⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠀⠘⠀⢰⣿⢁⣿⡇⠀⠀⢉⣛⣁⣸⣿⠀⠐⣄⡀⠀⢀⣠⠁
⠀⠀⠉⢩⡟⠁⢀⠓⠄⠀⠀⠀⠀⠚⣁⠀⠀⢸⡇⢀⣬⣤⣤⠀⠀⠀⠀⠂⠀⠀⠀⢻⣧⠀⠀⢀⣿⡏⣼⣿⠀⠀⠀⢘⣿⣿⣷⢿⡇⠀⠛⠁⠀⠈⠋
⠀⠀⢀⠏⠀⠀⢉⡁⠀⠀⠀⠀⠀⢋⣁⠀⠀⢸⡇⣾⣿⣿⣿⡧⠀⠀⡀⠃⠄⠀⠀⠈⣿⣇⣠⣾⡿⢱⣿⡇⣰⡗⢀⣀⠉⠉⠉⢈⣁⣤⣶⣷⣶⣶⣶⡦
⠀⠀⠈⣀⡀⠀⠈⠁⠀⢀⠀⠀⠀⢠⠵⣄⠀⣿⠇⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⡿⠀⠿⠛⢻⡼⡀⠀⠐⠙⠛⠛⠛⠛⠛⠋⢁⠒
⠀⢠⣾⠟⠁⢰⣤⣄⠀⠸⣿⡗⠀⠀⢰⠋⠀⣿⠀⠚⠿⢿⣿⣿⣿⣿⢿⠶⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠀⢠⣤⣠⠞⣰⣅⣀⠀⠉⠀⠀⠒⠒⠀⠉
⠀⠌⠁⣀⠀⠘⢿⡟⠀⠀⠀⠀⢀⣤⠀⠀⠀⣿⡆⠀⠀⠘⠿⣿⠿⢟⡀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠀⠀⠀⠼⠿⠦⠀⢉⠟⡀⠀⠀⣀⠀⠀⠀⠀⢠⠄
⠀⠀⠻⠿⣶⠄⠘⢁⡀⠀⡀⠀⣾⠷⢶⣙⠄⠸⣿⣦⡀⠀⠀⠀⣴⣿⡆⠀⠀⣠⣿⣿⡿⠋⢠⠴⠎⠀⠀⠀⠀⣀⠄⠀⠈⠀⠀⠰⠀⣀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⢀⣀⠀⣼⠋⠀⠀⠈⠻⣿⣷⣦⣄⡙⠛⠁⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠐⡀⠉⠐⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢹⣿⣅⠀⠀⠀⠠⠀⠈⠙⢿⣿⣿⣷⣤⣾⣿⣿⠁⣄⣉⢈⡌⣁⣠⠏⠀⣄⠈⠉⠀⣤⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⠀⠉⢠⡤⠀⠀⠀⡄⠀⠀⠈⢿⣿⣿⣿⣿⣧⠘⠄⠐⣦⠄⠐⠃⠀⠀⣿⠀⠀⠀⣴⣄⠀⠀⢰⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⣘⣿⣿⣦⠀⠀⠀⢸⡀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠠⠿⠄⠀⠀⠀⣼⡟⠰⠖⠀⠉⠀⠲⠦⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⣰⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⣀⣤⠤⠴⠶⠶⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣰⣏⣤⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣄⡀⢸⡇⠀⠀⢀⣠⣤⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⢦⣤⣄⣻⡄
⠀⠀⠀⠀⣰⡿⠧⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣈⣙⣾⣧⣶⣾⣭⡥⠤⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠾⠿⣿
⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢀⣜⣼⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣡⡀
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠛⠛⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⠿⠿⣿⣿⣿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣴⣿⣿⡋⠀⠀⢀⣠⣴⣶⣤⣄⠀⢀⢀⣀⣀⣀⡀⠀⠀⠐⠷⣤⡶⠀⠀⠀⢀⣤⣄⡀⠀⣠⣶⣿⣿⣷⣦⡀⠀⢀⠔⡀
⠀⠀⠀⠀⠈⠙⢿⡿⣋⣀⡀⢠⣿⣿⣿⣿⣯⣿⣆⠸⡸⠟⣋⣀⣀⠀⠸⣦⡞⢿⣇⠀⠀⠀⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣷⠀⠊⠺⠿⢦⡀
⠀⢠⣾⢻⣟⡛⣮⠑⣿⣿⣿⠸⣿⠻⠿⣿⠯⢻⡿⠀⣰⣾⡟⢉⡻⣿⣦⠙⠓⠟⠛⠀⣀⣤⣤⣤⣄⡀⠘⣿⣿⣙⣉⣿⣿⡏⠀⣎⣤⢒⠀⠉
⠀⣿⣿⣼⡟⣽⣿⡇⢉⣉⣁⡀⠙⠿⣿⣿⡿⠟⠁⠀⣿⣿⣾⣿⣷⣿⣿⡆⢸⠷⢠⣾⣿⠿⣿⡿⢿⣿⣆⠈⠛⠿⠿⢛⡋⠀⠢⠯⣢⣏⠳⢀⣠
⠀⠘⢿⣿⣿⡿⢟⣴⡿⠿⠿⢿⣦⠀⠀⠐⢶⣠⡦⠀⠻⣋⣀⣉⣀⣉⡿⠀⣠⠄⣾⣿⠀⢀⠀⣀⠀⢸⣿⡆⠀⣰⡿⠿⠿⣷⡄⠚⠁⠀⠀⠸⣿
⠀⣠⣴⣦⣄⡀⢸⣷⡄⠀⠀⣰⣾⡇⠸⣦⡞⣿⡁⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⡇⢹⣿⠀⠈⠀⠈⠀⢸⣿⠁⠀⢿⣆⠀⠀⢸⡟⠀⣠⣶⠛⠓⣦
⣾⡟⠷⠶⢻⣷⠈⢿⣧⣿⣿⣿⡟⠁⠀⠙⠓⠛⠃⠀⠀⣠⣾⡿⣿⡿⣷⣮⡉⠀⠀⠙⠿⣿⣿⣿⡿⠟⣡⣴⠶⢾⣟⠿⠿⢛⣠⠀⣿⣸⠀⠀⣿
⢻⣇⣠⣄⣸⡿⠀⣀⡈⠉⠉⠁⣠⣾⣿⣿⡿⢦⡀⠀⣼⣿⡿⣷⠈⢰⣿⣿⣿⡄⠀⠀⣀⣀⣀⡀⠀⢰⣿⣤⣶⣦⣽⡇⢰⣿⣿⠇⠈⠛⠻⠛⠁
⠀⠉⠛⠛⠉⠀⣶⣿⣿⣶⡀⢰⣿⣿⡛⠉⣰⣿⣷⠀⣿⣿⣷⣌⠃⢈⣻⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⠈⢿⣁⣉⣈⡹⠃⠋⠛⠯
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠀⠈⢿⣧⣝⣾⣿⣿⠏⡀⠹⣿⣿⠛⠈⠛⣿⣿⡟⠀⢾⡿⣻⣿⣿⣻⣿⠀⢀⣬⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡛⠁⢼⡿⠂⠈⠙⠻⠿⠿⢛⠉⢀⡀⠈⠷⣶⣶⣶⡶⠏⠀⣿⡋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⠀⣠⣶⣾⣷⣶⣄⠀⢄⢸⢞⢣⢤⡀⢠⣤⣍⠁⢀⣰⡆⠙⠳⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⡟⠉⠉⠉⠉⢿⡆⠀⢍⡷⢛⠆⠁⠻⣿⡿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣤⣄⣠⣤⣸⠃⣠⠀⡠⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⡄⠀⢸⣿⣎⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠈⠙⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⠿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿⣿⣶⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣼⣄⣈⣉⣋⣉⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠶⠟⠙⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⢉⡉⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣛⣶
⠀⠀⢀⡄⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⣿⣿⢻
⠀⠀⠀⠳⡀⠀⠘⠛⠋⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠟⠋⠀⢘⡆⠀⠀⠀⢀⡀
⠀⠠⡈⢆⠱⠴⣒⣒⡚⢊⠴⢒⡈⠥⠴⠀⠀⠀⠀⠀⣀⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠠⣄⠀⠀⠒⠉⠀⠀⠀⠀⣸⣿⣷⣶⣶⣤⢗⣤
⠀⠀⠱⣈⣻⠭⠥⠤⠭⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⡿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⢯⠅⣠⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⠛⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣘⠷⢟⡻⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⣿⢛⢛⣯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡧⠀⠀⠀⠀⠀⠀⠸⡿⠿⠧⠉⠿⠃⣿⣋⡬⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⢀⣥⡀⠀⢸⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡀⢺⢻⣿⡿⡗⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠋⢻⢸⣿⣇⡇⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣩⣔⣚⣿⣭⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠉⠉⠉⠉⠉⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠁⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⡷⠶
⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⢟⣛⣛⠿⢛⣛⡻⠿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⡇⠸⠄
⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡟⠛⡟⠛⢻⠛⢻⠛⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣦⣖⣤⣭⣉⣥⣴⣶⣷
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⢳⠒⡗⠒⢺⠒⢺⠓⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣯⣽⣿⣋⣭⣴⣿
⠐⠐⠒⠒⢲⣿⣿⠒⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡚⢗⢻⠒⡟⡲⢛⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠋⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟
⠀⠀⢣⠀⠀⠾⠟⠁⢀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣧⣀⠀⠀⠀⠀⡆⠀⠀⠀⠸⠿⠋⢰
⠀⢀⠠⠱⣖⣒⣒⡩⢒⠀⠅⠂⠀⠀⠀⣨⣭⣭⣭⣭⡀⠀⠀⠀⠁⠀⠀⢠⡄⠀⠀⠈⠀⠀⠀⣿⣷⣶⣦⣶⣤
⠀⠀⠢⠑⠒⠀⠒⠚⠁⠀⠀⠀⠀⠀⠰⠟⠋⣻⡛⠻⠧⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣗⣤⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⠼⢶⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠄⠀⠀⠀⠀⠸⠿⠯⠙⠟⢸⣏⠝
⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣠⣴⣤⡀⣿⠀⡀⠀⠀⠀⠀⠀⠀⢰⣦⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣿⣿⣿⡧⠟⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⠀⣛⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⣠⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⣥
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢲⠛⠟⠒⡟⢻⠖⢺⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣍⣙⣉⣥⣴
⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣘⣇⣛⣈⡏⣸⢁⠟⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣯⣿⣯⣵⣾
⠈⠀⠀⢹⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣼⣼⡽⠥⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠿⠿⠿⠟⠛
⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢠⢰⣶⠄⠀⠀⣰⣿⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣤⣘⠁⠀⡐⠋⠰⣿⣿⣷⣍⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣿⣷⣾⣷⣯⣭⣴⡌⢙⣦⠀⢸⣿⣿⡿⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣿⣿⣿⣿⣿⡏⠁⢡⣌⠋⠀⠸⠿⠉⠁⠀⠀⠀⠀⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⣿⡿⠿⡄⠀⠀⠀⠀⠀⠀⣄⠘⣷⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣈⡾⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⣶⣶⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠟⣒⡀⠤⠤⠐⠒⠒⠀⠘⠂⠀⠒⠒⠒⠤⠤⢄⣀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠤⠔⠒⢒⣉⣉⣉⠀⠀⠐⠀⡉⣉⣉⡐⠒⠆⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢚⣉⡉⠤⠄⠀⠄⠀⢈⠠⠀⠠⠤⢌⣉⢉⠒⠆⠰
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⡥⠤⠔⠒⠂⠀⠂⠀⠠⠀⠀⠐⠒⠂⠴⢨⡅⣈⡁
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠠⢈⠀⠀⠀⠀⠅⠀⠀⠀⠤⠤⠔⠒⢊⣉⠀⠀⠀⠀⠐⠀⠀⠀⢉⣡⣼⣤⣦⣤⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⡆⠀⠀⠀⠀⠒⢋⣉⠤⠤⠐⠂⠀⠀⢈⠀⠐⠒⠠⢼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣉⠤⠤⠒⠂⣁⡀⠀⠀⠠⠀⠀⣈⡐⠒⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠤⠔⠒⠉⠁⠀⠀⠀⠐⢐⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠾⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠶⠶⠦



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⢀⠀⠀⡄
⠀⠀⠀⠠⠀⠀⠦⠀⠀⠠⠀⠢⠌⠀⠈⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠠⠁⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⠀⣿⠀⠀⠠⣀
⠀⠀⠀⠀⠰⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁⠀⠀⠰⠀⠀⠀⠀⠀⣿⡇⠀⣿⠄⣿⠀⣶⠄⣿
⠀⠁⠀⠀⠀⠀⠀⣠⡄⠀⠑⢀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠉⡉⠁⡉⠉⡍⠉⡋⠐⠂⠂
⠀⠰⠤⠀⠀⠀⠈⠁⠀⢀⠐⢁⠔⠠⠀⡄⠀⠀⠀⠀⠀⠀⠐⣉⠀⠢⠘⠃⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠉⠄⠀⠀⠂⠁⠀⠀⠀⠐⠒⠀⠉⠁⠀⠠⣄⠀⢠⠄⠉⠂⠈
⠀⠀⣀⠀⠀⠐⠐⠀⠀⠀⠀⠀⡀⠀⡀⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠐⡀⠀⠈⠁⠀⠀⠁⢦⣤⣄
⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠄⠂⠀⠀⠀⠀⠀⠀⢠⠂⠐⠀⠢⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠷⢴⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠘⠂⢀⣨⣥
⠀⠀⠀⠀⠀⠀⠄⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠀⠘⡃⠵⠀⠀⠀⠀⠀⠣⠀⠔⠉⠙⠉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⢨⠌⠀⠀⠀⠠⢄⠀⡆⢠⣄⠀⡄⠀⡀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠀⠀⠁⠀⢀⡁⡙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠣⠀⠀⠀⠀⠀⡄⢀⡄⠀⡆⠓⡀⢀⠔⠐⣠⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠓⠺⠁⠀⠃⠀⠀⠘⠛⠛⠋⡀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢀⢡⠀⠀⠀⠀⢰⣤⣼⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡀⠀⣀⠠⠀⣤⠠⠀⢠⠀⠄⠄⠰⠀⠆⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠉⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢤⡀⠀⠀⠀⠀⠀⢝⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠲⠄⠀⠀⠀⠀⠀⣀⣀⡇⠀⢸⣀⡀⠀⠀⠀⢀⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠬⠍⠀⠀⠀⠀⠀⣿⠈⠁⠀⠈⠀⡿⠀⠀⠀⢀⠥⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠰⡦⠀⠀⠀⠀⠀⠙⠒⡆⢠⡖⠒⠀⠀⠀⠀⠠⢅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠷⠼⠇⠀⠀⠀⠀⠀⢲⣍⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢁⣀⣀⣀⡤⣤⠤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠰⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠟⢉⡤⢊⡡⠔⣡⠞⡠⠊⠀⠈⠢⣕⡠⣉⠛⢍⡓⠲⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣫⣖⡯⠒⣩⠔⣊⠔⠈⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣍⠲⣜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣡⠔⣋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢊⡤⠊⠀⠠⠤⠤⠄⣀⡀⣀⠀⢀⣀⢀⣀⣀⣀⣀⠀⠘⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡏⠉⠡⠈⠀⠀⠀⠀⠢⢀⠶⠓⢶⡄⠄⠀⠀⠄⠐⣌⣭⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⡀⡘⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣧⡂⠀⠀⠀⢀⢿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣻⡅⠈⠛⠤⠤⠭⠭⠖⠃⠀⠀⠀⠈⠓⠤⠤⠬⠄⠊⣿⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡄⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠉⠓⠦⠤⠤⠤⠖⠋⢩⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⡶⠲⡖⡺⡻⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⡝⠲⠶⡶⠶⠶⡶⡤⣤⣄⣀
⠀⠀⠀⠀⢀⣤⡖⢿⠋⠁⠂⠈⠀⠀⠀⢀⡞⠁⠀⢸⢣⣄⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠘⢦⠀⠀⠀⠀⠀⠀⠳⠹⠹⣻⠳⣤⡀
⠀⠀⠀⡴⢏⣼⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⠀⠋⠹⠓⠒⠶⠒⠊⡉⠉⢰⠃⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⡻⣦⡀
⠀⠀⡞⢉⠢⠉⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡇⠀⡑⢎⡡⠀⣀⡶⠡⠀⡼⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣩⣷⡀
⠀⡸⢍⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣤⣄⡀⠀⣇⠀⢌⠣⠁⣐⣾⠃⠀⢠⠇⠀⠀⢀⣤⡖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡫⢝⣿
⢠⡓⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡾⠃⠀⢸⠀⠀⠀⠰⣼⢃⠂⡰⡞⠀⣠⡶⠋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⣻⣇
⣸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⡆⠀⠀⣹⢃⠢⡰⣹⠃⠀⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡥⣿
⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⢱⠀⠠⠃⠂⣰⢣⡏⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣒⣿
⡇⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣦⡀⠀⠀⠈⡆⠀⠀⢰⣿⣾⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡊⢽
⣇⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⢱⡎⢀⣷⣹⠇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡊⣹
⡧⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⢷⣄⠀⣇⢸⢣⡟⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻
⣏⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠸⣧⡿⠀⠀⢀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⣃⣀⣀⣀⣀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣟⣓⣒⣒⣚⣒⣒⣖⣒⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤
⣿⣿⡿⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠛⠳⡟⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡄⠀⠤⠬⠠⡀⠀⠀⢠⠿⠚⠛⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠁
⠀⠀⠀⢀⣴⣄⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁
⠀⠀⣼⢾⡁⡉⠉⠁⠀⠀⠔⠉⡁⠀⠀⠀⢀⠉⠀⠀⠀⠀⠉⠒⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣿⠀⠀⠀⠈⠙⠁⠀⠀⠀⠁⢠⠀⠀⠈⠁⠀⠀⠸⡄⠀⠀⢀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠸⣆⠀⠀⠀⠈⢃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⢡⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⠕
⠀⠀⠀⠈⠳⠶⢶⣀⠈⠑⢄⡀⠀⠒⠒⠒⠊⠁⣠⢴⣥⠴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠁⠄⡀⢀⠠⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠐⠀⣿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⠇⠀⠀⠹⠶⣤⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠈⠀⢀⠀⠀⠀⠀⢱⠀⠙⠦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⢛⡔
⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡆⠀⠀⠼⠧⣤⣀⣀⣤⡤⠶⠖⠛⠉⡀⠈⠀⠁⠱⡄
⠀⠀⣠⡾⠋⠀⠢⠀⡀⠀⠀⠄⠀⠀⠈⠠⠁⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⣀⡤⠖⠃
⢀⡴⠋⠀⠀⢀⡴⠋⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⣄⣀⣀⣠⣤⠔⠀⠀⠀⠉⠋⠉⠁
⣿⠁⠀⠀⡴⠋⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠈⠉⠉⠩
⠘⣧⠀⠀⠹⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠀⠂⠀⠀⢠⡔⠋⠉⠉⠉⠉⠳⡄
⠀⠈⢷⡀⠀⠘⢦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠹⣦⠀⠈⢧⡤⠶⠧⢀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⣀⣀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠹⣤⠤⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡤⢠⣀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⣷⡀⠀⠀⠀⠀⠀⡨⠟⠓⢤⣰⣶⠀⢰⠁⠀⠀⠈⢃⠀⠀⠀⢀⠄⠠⡀⢈⠀⠁⠀⠀⠠⠀⠀⠢⡑⡀⠀⠈⢂⢀⠐⡄
⠀⠀⢿⠹⣆⠀⠀⠀⡜⢀⡀⡈⠀⢹⠉⠀⠘⣄⠀⠀⢀⡆⠀⠀⠀⣵⣶⣶⣶⠃⠀⠐⠀⡀⠀⠐⠀⠀⠑⡘⢄⠀⠨⠈⠀⠐⠄
⠀⠀⢸⡄⠘⢿⠀⠀⣹⣄⠀⠀⢀⡜⠀⠀⠀⠈⠉⠉⠁⠹⣷⣄⠀⠙⡿⢿⣯⣿⣿⣿⣾⠀⠘⠀⠀⠀⠀⠈⢄⠒⠁⡄⠀⠀⢈
⠀⠀⠸⠇⠀⠀⠰⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⡄⢠⡶⠀⠀⠈⠛⠃⠀⠀⠀⠂⠉⠉⠉⢣⣄⠀⠀⠀⠀⠀⠀⠀⠢⡀⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣼⣥⣤⠀⠀⢀⣴⣶⣶⣦⣤⠄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⢀⣀⠀⠀⠀⠀⠀⣼⠋⠙⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠰⢿⣿⣷⣿⣿⡁⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⢾⣿⣿⣶⣲⠀⠘⠳⢾⡷⠾⢶⣄⠀⠈⠻⠿⠿⠋⠀⠀⢰⣿⣿⣿⠿⠗⠀⠧⠰⡷⠀⠀⠰⠆⠀⠀⠠⠊⠀⠀⠀⠀⠑
⠀⠀⠀⡽⠉⠉⠀⠀⠀⠀⠸⣇⡀⣠⡟⠀⠀⢀⠀⠖⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣴⣧⠀⠀⣤⠴⢶⡀⠀⠉⠛⠉⠀⢠⣾⣦⣤⡀⠀⠐⢳⡀⢀⠠⢀⣸⠀⣆⡀⠤⡀⠀⠀⠀⠠⢢⠀⢃⠀⠀⠀⠂⠢⢀⠆
⠀⠀⠀⠉⠀⠀⠿⣤⡼⢋⡀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣴⡀⡇⠀⠑⢤⣼⣿⣧⣤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠁⠂⠄⠠⠐⠁
⠀⢤⣦⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠈⢄⠙⢿⡄⠀⡿⡱⠁⠀⣊⠀⠼⠓⡧⣤⣑⠄⠀⠁⠀⠸⠁⠀⠀⠀⢀⣴⣤⡀
⠀⠈⠻⡟⠀⠀⠀⢲⣿⣿⡇⠀⠀⠀⢀⠀⠀⠉⠦⢦⡔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⡠⠤⠀⠀⠀⠀⠉⠛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠶⠷⠒⠀⠀⠶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠂⠂⠁⠀⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⡖⣶⠆⣦⡀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠉⠛
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠯⠽⠀⠒⠀⠀⠀⠀⠀⠰⢿⡟⢩⠧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⡞⠛⠛⠛⠛⠛⠛⠉⠛⣷
⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠠⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⣻⣟⣛⠛⠛⠋
⠀⠀⠀⠉⠉⠙⠓⠒⠦⠤⢄⣼⠁⠀⠀⡒⠉⠙⣶⣄⠀⣤⠘⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠉⠉⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⣿⡖⠁⠙⠟⢣⠀⠀⠀⣴⠟⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⡴⠋⣹⠆
⠀⠀⠀⠀⡠⠄⠀⢀⡀⢰⠡⠀⠀⢇⡘⠉⠑⣤⣤⡄⡘⠀⢰⠟⠀⠀⠀⠀⠀⠶⠆⠀⠀⣾⣿⡧⠀⠀⠄⠐⣏⣠⠞⡡⣪⣶⣄
⠀⠀⠀⠀⠠⣰⡀⠘⠀⠀⠀⡀⠂⠈⠻⣦⠀⠛⡿⠋⠀⠀⠀⠀⠀⠠⡶⢠⡜⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⢻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⣟⠲⠦⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠐⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠁⡔⠂⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⣀⠴⠚⠁⣀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢊⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠋⠀⣠⡴⡚⣕⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⠠⡐⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠑⠁⢀⢤⢶⢯⠾⠛⠈⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⢈⠂⠀⠀⠘⠂⠂⠀⠀⠀⠠⠀⠐⠒⠆⠈⠀⡠⣠⣩⠵⠻⠃⠁⣀⡤⢞⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣼⠥⣤⣤⠒⠆⠈⠉⠐⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⢄⣢⡦⠞⠉⠀⢀⣠⢶⣫⡻⠊⠋⢩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠠⠯⠘⠋⠐⠀⠀⠀⠀⣀⢠⡄⠖⠚⠩⠉⠛⠛⠛⠓⠲⠶⠶⠛⠉⠀⣀⢄⣖⡽⠷⠑⠁⢀⣤⠶⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠔⠀⠀⠀⠀⡀⢀⣄⣴⠼⠓⠋⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣂⣮⠾⠏⠋⠀⣠⡤⠺⠋⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⢠⣄⠰⠄⠗⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⢒⡒⠶⠶⠶⠤⠤⢤⡴⠞⠋⠁⢀⢀⠔⠞⠉
⠀⠀⠀⡠⢀⠀⡄⡰⠄⠃⠈⠀⠀⠀⢤⡞⠠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡠⡢⠚⠁
⠀⠀⠄⠀⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠒⠒⠀⠀⠄⠀⠀⠀⠔⠋





⡿⠿⠿⠿⡿⠻⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣟⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣒⣶⣤⣄⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣋⣉⠉⢙⣿⣷⣤⠀⢠⣋⠋⣉⠉⣑⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⣿⣁⣴⢟⢛⠛⠿⠃⡠⠁⢰⣏⡱⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣿⡏⠛⠁⠈⠛⠉⢀⠞⠀⠀⡾⡏⠀⡀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣳⡾⠏⠀⠈⣦⠐⠋⣠⣎⠀⢀⡜⠀⠙⢾⠶⢶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡹⠟⠁⡍⠀⠀⠚⡇⠀⠊⠱⢆⡵⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣼⠁⠀⠀⠁⡀⠀⡀⠉⠐⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠁⠀⢹⡄⠀⢰⣾⠐⠂⢸⡇⣀⠟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡠⢁⠉⢉⡋⠉⢿⡄⠀⠙⢟⠛⢹⠇⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⡀⠀⠰⢠⠙⢦⡀⠀⠣⠤⠀⡀⠁⠀⠆⢠⣤⣄⣀⣒⣒⡒⢒⣖⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠆⢀⢠⠎⠀⠀⡇⠘⡄⠈⣙⠢⣤⣀⡀⠀⠈⠁⠐⠈⠛⣽⣿⣏⣩⣍⣩⣭⣵⠏
⠀⠀⠀⠀⠀⠀⢲⠉⠀⢠⠀⣀⡆⠠⢮⣁⡠⡎⠀⠙⢯⠉⠋⠉⠙⣿⠉⠉⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⢂⢰⡟⢻⡿⡏⢹⡆⠀⠀⢄⠁⠀⠀⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⢀⢲⡇⣵⡼⡇⠀⠀⠀⣀⠐⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢸⠁⢻⣿⠀⠀⠀⠀⠹⡂⢸⡇⣈⣇⠀⠀⠀⠀⠀⠀⣠⣶⢤⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠔⠆⠀⣿⠀⢸⣿⠀⡀⠄⢂⣈⣧⣽⢶⣯⣿⣠⣴⠖⣒⠒⢄⠀⠹⣦⠙⠢⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⡖⢰⡏⠀⢘⣿⠎⣠⠞⠉⠉⢏⡁⡞⢨⣿⢹⡉⠀⢝⣡⠔⠀⢀⣿⡧⢠⡿⠀⣴⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡼⠀⢸⡗⣩⣨⣿⡶⠅⠀⠀⠀⢨⡥⠁⣸⣿⢘⣒⣿⣟⢁⣀⣴⡿⡁⠘⣿⡇⠀⣿⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⠏⢹⣿⣿⣗⠀⠀⠀⠀⣮⠡⢀⡏⣿⠈⢡⣶⣭⣻⢿⣿⡟⢄⢠⣿⠀⣠⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⣿⣿⢾⣒⡀⠀⢀⢂⠀⢸⠇⣯⣤⣦⣺⠟⠛⣯⡍⠉⢻⡿⢿⣯⠵⢩⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠢⣼⣿⣯⣇⠙⠁⠀⠁⠊⢀⣿⠀⣧⣾⣿⣿⣶⣾⠵⠃⡖⢢⣶⣿⡧⡦⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠄⣿⡿⣿⣯⡄⢧⡐⠈⠡⣼⡇⠀⢱⣿⠟⡁⠀⣯⡲⣸⣿⠟⢉⢃⢎⢞⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⠲⣼⡇⢿⠙⠠⢀⢉⠁⠀⢹⠃⠀⢾⣭⢎⠵⣶⣿⣾⡟⣸⣴⠋⢘⢠⣏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡴⠊⣕⣧⠛⢂⠠⡉⠠⠀⠀⣿⣠⠴⠟⢴⣯⣾⣿⣿⡷⣾⠟⠁⠀⢘⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⣄⠠⠛⡜⡄⠈⢄⠘⠀⠀⠀⣟⠣⣤⠆⢈⣿⣠⣉⣁⠀⣽⣤⣠⣀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠆⢀⠴⠏⠄⢛⡼⢄⠀⠂⠀⠀⠀⣻⡞⠁⠚⠋⠋⠑⠾⠛⠂⠛⠉⠙⠏⠉⠙⡆
⠀⠀⠀⢀⣠⣔⡊⣁⣤⣎⡀⣀⣀⣤⣶⠟⠢⣄⠀⢀⣀⣥⠚⠉⠃⠸⡒⠒⠲⠶⠖⠒⠒⠂⠀⠩⠔⠃⠀⢷⣄⡀
⠀⠀⠐⠥⠤⠤⠀⠀⠀⢀⠭⠉⠛⠻⢯⣄⣞⠟⠛⠽⠋⡙⠛⠒⠛⠓⠛⠚⡃⡒⡚⣂⡀⢀⣀⡀⢦⣤⣼⣶⠿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠛⠛⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠷⠀⠀⠺⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⣠⣶⣿⣿⣿⣶⣄⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣿⣿⣿⣿⣿⠏⢻⣿⣿⠏⡸⣿⣿⣿⢱⣆⢻⡇⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⢸⣿⠀⠹⣿⣶⣶⣶⣾⡘⣱⣶⣾⡇⢰⣶⣾⣧⡹⣦⣄⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣿⣿⢸⣿⠀⠀⠙⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈⠉⣛⡛⣩⣵⡆⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡘⣿⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣿⡇⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⣿⣿⣇⢻⣧⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣸⡿⢱⣿⣿⡇⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣦⠻⣷⡀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⢀⣴⡿⣡⣿⣿⣿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡘⣿⣿⣿⣷⣝⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢋⣴⣿⣿⣿⡟⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣭⣭⣝⡻⣿⣷⣬⣛⠿⢿⣶⣶⣶⣶⣶⣾⠿⢛⣩⣶⣿⠿⣛⣭⣭⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠃





⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⢀⡀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡿⠛⠉⠉⠻⣿⠄⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⣶⣤⠀⠙⠀⠀⠀⠀⠀⣿⣈⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡇⠀⠰⠀⠀⣹⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠐⠃⠈⠛⠋⠀⣤⡆⠀⠀⣴⠶⣿⣿⣿⢶⣦
⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣱⢤⣀⣀⢴⡇⠀⠀⠀⢻⣻⣟⣻⣟⡟⠀⠀⠀⠀⢴⣦⡄⠀⠛⠁⠀⠀⣿⣤⣿⣿⣯⣿⡿

⠀⣠⠴⣶⢲⡶⣄⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⢲⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⢀⣶⣶⣆⠰⡆⠶
⢰⣛⣶⣿⣿⣷⢛⡇⡇⠀⠀⠀⠀⡟⢏⠀⠀⠀⠀⠀⡧⢤⡤⢤⡬⣼⠀⠀⠀⠀⡎⠙⠧⠀⠀⠀⠀⠀⢰⣿⣟⣿⡀⢠⠂
⠈⠻⣿⠛⣿⣏⡿⢰⠃⠀⠀⠀⣸⠁⠸⡄⠀⠀⠀⠀⡷⣿⡷⢾⡷⢾⠀⠀⠀⠀⠘⠢⡤⠀⠀⠀⠀⠀⣺⣟⣖⣿⡆⢸
⠀⠀⠙⠶⠛⠋⠀⠘⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠳⠿⠷⠾⠷⠞⠀⠀⠀⠀⠲⠾⠿⠷⠀⠀⠀⠀⠛⠓⠒⠚⠓⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⡄⠠⠀⠀⠀⠀⠀⠀⢷⡀⣠⠏⠀⠀⠀⠀⡥⠀⣤⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢀⠃⠙⠆⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠀⠀⠀⡜⡥⡔⠓⢃⡀⣌⡅⠀⠀⠀⠀⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡇⣏⢱⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⡀⠴⢦⡄⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡃⣴⡄⢹⠀⠀⠀⠀⢠⣾⣿⢿⣦⡁⠀⠀⠀⢸⢻⡿⡄⠀⠀⠀⠀⠀⢀⢈⣱⣿⡿⠸⡄
⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣳⣦⣴⡋⠀⠀⠀⠀⠘⢿⣯⣿⣿⠃⠀⠀⠀⢸⢸⡇⠹⣆⠀⠀⠀⠀⠘⣿⡙⢻⣷⣦⠁
⠀⠀⠼⠿⠿⠿⠷⠀⠀⠀⠀⠀⠛⠏⠻⠓⠀⠀⠀⠀⠘⠶⠬⠉⠁⠀⠀⠀⠀⠸⠽⠷⠤⠼⠧⠀⠀⠀⠀⠀⠱⠟⠋⠁

⠀⢀⣤⣄⢰⣿⣿⣆⠀⠀⠖⠂⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⢠⣤⣤⣦⣤⣦⣤⣤⠀⢀⣠⣴⣾⣿⣶⣦⣄
⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⣋⣉⠉⢁⣀⣉⣉⡁⠀⠀⠀⢀⡇⢸⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⠟⠛⠻⡟⠁
⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠅⠀⠤⠤⠤⠤⠤⡅⠀⠀⠀⠘⠣⠾⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠈⠙⠛⠋⠁⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠒⠂⠤⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠈⠉⠁⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠼⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠞⠀⠀⠉⠀⠀⠹⠀⠀⠀⡅⠀⠈⠁⠀⠈⡇⠀⠀⠜⠀⠀⠉⠀⠀⠁⠀⠀⠀⡁⠀⠈⠇⠀⠈⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢘⠒⠒⠚⠒⠒⡇⠀⠀⠀⡗⠒⠒⠓⠒⠚⠂⠀⠀⠘⠒⠒⠓⠒⠒⡇⠀⠀⠀⡗⠒⠒⡗⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠤⠀⠀⡇⠀⠀⠀⡧⠀⠠⠇⠀⢤⡀⠀⠀⢨⠄⠀⠤⠀⠀⡇⠀⠀⠀⡧⠀⠠⠇⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠃⠀⠀⠀⠑⠒⠒⠒⠒⠊⠀⠀⠀⠈⠒⠒⠒⠒⠒⠃⠀⠀⠀⠑⠐⠒⠒⠒⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠜⠉⠉⠛⠋⠉⠳⠀⠀⠀⡛⠉⠙⠋⠉⠙⡆⠀⠀⡔⠉⠋⠛⠉⠙⠃⠀⠀⠀⡛⠉⠙⡋⠛⠙⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠴⠀⠀⡆⠀⠀⠀⡇⠀⠠⡆⠀⠰⠀⠀⠀⢰⠀⠀⠦⠀⠀⡆⠀⠀⠀⡇⠀⠠⡆⠀⠰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣀⣀⣀⣀⣀⡇⠀⠀⠀⣇⣀⣀⣄⣀⣠⠀⠀⠀⢠⣀⣀⣀⣀⣀⡇⠀⠀⠀⣇⣀⣀⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⠀⠓⠒⠒⠒⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⢢⠀⠀⠀⠖⠒⠒⠖⠒⠒⡄⠀⠀⡴⠒⠒⠖⠒⠒⠢⠀⠀⠀⠖⠒⠒⠖⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⠀⠀⡄⠀⠀⠀⡇⢀⢀⡄⠀⢠⡀⠀⠀⢡⡀⠀⣤⠀⣀⡄⠀⠀⠀⡇⠀⢀⡅⠀⢠⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡃⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠺⠁⠀⠈⠀⠀⠇⠀⠀⠀⠇⠀⠈⠁⠀⠸⠂⠀⠀⠺⠀⠀⠉⠀⠀⠇⠀⠀⠀⠇⠀⠀⠁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠀⠉⠀⠒⠤⢄⡀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⠑⢬⡀⠱⠾⢅⡀⢇⡀⢇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⡞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠀⢠⠖⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠈⠉⠙⠁⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀⢸⡄⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡂⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣇⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠃⠜⢙⣻⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣟⡉⠃⠘⠋⠛⠙⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣁⡖⢸⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢣⡸⠂⠀⠀⠘⠁
⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⠿⠿⠟⢁⣾⣿⣿⣷⡈⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⣶⣌⠳⣄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡴⢋⣴⣶
⠀⠀⠀⠀⠀⠘⣿⣿⣆⠹⣆⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣿⡿⢀⣙⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢘⣁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⢡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⠻⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠿⠋⠁⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡿⢋⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢦⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠳⢶⠋⠀⠀⠀⠀⠹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣀⣉⣉⣑⣤⠀⠀⠀⢠⡷⠾⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠑
⠀⠀⠀⠀⣠⡆⠀⢠⡟⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠢⢤⡊⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄
⠀⠀⢰⣴⡏⡛⠛⠋⠀⠀⢀⠜⠓⠀⠀⠀⠀⠀⠓⠀⠀⠁⠉⠒⠦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈
⠀⠀⣿⠈⠁⠀⠐⡖⡆⠀⠀⠀⠘⠀⡀⠀⠀⠚⠀⠀⠀⢳⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⡆⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠁⢂⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠖⠀⠀⠀⡀⠅⠀⠐
⠀⠀⠀⠹⢦⣤⣠⡤⠐⠳⢄⠀⠀⠠⢄⣠⡤⠔⠀⢀⣠⣂⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠰⠱⢀⡀⠀⡠⢈⠔
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠾⠗⠂⣤⠤⠤⠤⣤⠒⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠰⠴⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣧⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⡟⠋⠀⠀⠀⢨⠉⡏⠻⢦⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢹⡀⠀⠉⣻⣆⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠀⠐⠳⠺⣄
⠀⠀⠀⢀⡴⠟⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠈⣇⠀⡀⠉⠉⠛⠲⠖⠚⠛⠉⠉⠀⡠⠐⠀⠀⠀⠀⣸
⠀⢀⡴⠋⠀⠀⠀⣠⡤⣄⠀⡈⠀⠀⠀⠀⠃⠀⠀⠀⢸⡶⣤⡀⠀⠀⠀⠀⠀⠀⡀⠐⠑⠳⣄⣤⠴⠒⠋⠁
⣠⠞⠁⠀⢀⣠⠞⠁⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⠶⡦⠴⢶⠞⠉⠂⠀⡀
⢻⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⠀⠀⠀⠈⠁⠀⣀⣀⣤⣄⣀⣀
⠀⢻⡄⠀⠈⢷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⣄⣉⣉⣉⣡⠏⠁⠀⠀⠀⠀⠀⠹⣌⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀
⠀⠀⠹⣦⠀⠀⠹⣄⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠳⣄⠀⠙⠶⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣀⣙⣦⣀⣠⣤⣀⣤⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠧⢤⡄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⣀⣀⣀⣀⣀⣀⣸⣣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⢛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢀⣀⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣄⠈⠛⠁⢴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⢀⣴⠀⢰⡆⠀⣠⣆⡐⢢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡏⣼⠁⠀⠀⠀⠀⠙⣿⡿⣷⣿⣤⣶⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⢰⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⢀⣿⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⠟⠁⠛⠀⡀⢹⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠛⠁⠀⣼⡇⠀⣄⣲⠿⠿⠿⠿⠟⠛⠁⠀⡀⠀⣀⣽⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⢸⣿⣿⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢬⣭⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⠇⠀⠀⠀⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠁⠀⠀⠐
⠀⠀⠀⢠⢀⡄⢸⣿⣿⣷⡆⡄⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⡆⠛⠤⠀⠀⠀⢁⠈⠀⠀⠀⠀⠦
⠀⠀⠀⠀⣨⣸⣴⣽⠻⣿⣇⠀⣷⣦⣈⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡷⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠿⠻⣿⣿⠟⠀⠛⣼⣿⡟⠁⣻⣿⡿⣹⣿⣿⡗⠀⠀⠀⠀⠀⠀⠠⣿⡇⠐⠘⠀⠀⢀⡄⠟
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠻⠀⠀⠈⠉⠀⣿⣿⡿⠁⠀⠀⠀⠀⠈⠀⠀⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⢀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠶⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠀⠀⠊⢀⡆⠀⠀⠀⠀⠀⠊⠀⢀⠜⡄⢐⠐
⠀⠀⠀⠁⠀⠀⠠⣠⣆⠀⠀⠀⠀⠀⠀⠠⡀⠌⠀⠖⠀⢀⣸⡇⠀⠀⠀⠀⠀⢄⠐⠁⡌⡁⠀⠚⠰⠒
⠀⠀⠀⠀⠁⠀⠀⠘⢄⠓⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠘⠀⠊⠂⠀⠀⠀⠀⠀⠠⠀⠸⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⡀
⠀⠀⠀⠀⡤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⠈⠀⢂⠀⠀⠀⠀⠀⠀⡸⠜⡀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠇⣐⠦⠚⢃
⠀⠀⠀⡄⠀⠀⢀⡄⠠⠂⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠴⠤⣀⡴⠮⠄⠠⠧⢅
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠀⠀⠀⠀⠀⠀⡀⠠⡔⠈⠁⢢⠂⢀⠀⠀⠀⠀⠐⠣⠜⠃⠓⠀⠃⠘⠀⠚
⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⡐⠲⠞⡆⠠⡆⠉⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠐⣿⣿⣿⣿⠇⠀⠀⠀⡐
⠀⠀⠃⠀⠀⠀⡕⢁⡍⡹⢂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⠀⠀⠀⠈⠛⠛⠫⠐⠁⠀⠈
⠀⠀⡄⠀⠀⠀⢃⠤⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠠⠇⠟⠄⢰⠂⠁⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡈⢀⣾⣿⣶⠾⡿⢿⣯⣅⣲⣄⣠⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⡀⣠⣄⣷⣿⣿⠁⠀⠙⢀⡹⠆⣶⡾⢿⡏⡠⠼⢫⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠉⠉⠉⠙⣲⠚⣡⠔⠋⠉⠓⠺⢿⠿⣾⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠤⣄⣠⣶⣾⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣹⣋⣜⣡⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣗⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢋⠙⠿⠋⣹⡿⠃⠀⣀⡀⠤⢤⣀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣶⠉⡄⠀⠈⠁⢐⣨⣍⠛⢷⠀⣰⣿⣧⡀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡟⠋⢼⠀⠀⠀⠀⠉⠧⠿⠁⠀⠀⠙⠉⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⢃⡀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⣀⣰⡆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢣⠀⠀⠀⠀⠀⠀⠀⠀⣥⢮⣵⠄⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⡵⠚⠛⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡄⠈⠙⠦⣄⡀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⡴⠃⠀⠀⠀⢿⡀⠀⠀⠀⠀⠉⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⢦⠈⢷⡔⠋⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⢸⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣬⣎⢢⣷⡌⢻⣆⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡹⣿⣦⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡜⢿⣷⣄⠙⢷⣤⡀
⠀⠀⢠⢠⠀⣠⠀⠀⠀⠀⠀⠙⣿⣦⣙⣿⣷⣤⡈⠙⠳⠦⠄⣀
⠀⠀⠘⠘⠃⠛⠛⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠚⠉⠛⠛⠛⠓⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡀⣸⠇⠀⢸⣆⣀⡀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠁⢸⡇⠀⢸⠋⠉⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣀⣀⣤⣬⠿⠷⣯⣄⡀⢀⣀⣀⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠋⣉⡥⠀⠀⠀⠀⠀⠀⢹⡏⠁⠈⠉⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⢀⡤⠚⢁⣀⠀⣠⠄⠀⠀⡼⢸⡇⠰⡄⢠⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⢣⠾⠚⢋⣡⡽⠞⢁⣤⢠⠞⣱⠏⢷⡀⠻⣦⠘⣆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠞⢋⣡⡴⠞⣹⣃⡤⠞⠁⠀⠈⢷⡄⠙⠀⠘⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⠋⣡⠤⠴⠿⠉⠀⠀⠀⢀⣠⢤⣝⢦⡘⣧⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣴⡏⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⢰⣶⠈⠀⠙⣿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢹⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣿⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠀⢸⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣆⠀⠀⠀⢀⡀⠀⠀⠉⠀⢀⠀⠀⠀⠀⡿⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⣦⣤⣤⠶⠋⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⣤⣤⣤⣤⣴⣾⠋⡾⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡇⠀⠀⠀⠀⣸⡟⣦⡇⠀⣶⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⢸⡇⠀⠀⠀⠀⢻⠇⠈⠻⣦⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣿⠁⠀⠀⢘⣇⠀⠀⠀⠀⣿⡀⠀⠀⠘⣟⠻⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⣙⠛⢶⡶⠋⢻⡶⣄⣀⢴⡏⠙⢷⣤⠶⠛⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡀⠀⠀⠀⣾⠁⠀⠈⣷⢾⡵⣾⠇⠀⢀⡽⠃⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠈⠻⢦⡀⠘⣷⢠⡏⢀⡴⠋⣀⣀⣀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⠹⣦⡸⣏⡴⢯⡤⠾⠧⢶⣿⢦⡼⣧⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠀⢸⣷⣿⠂⠿⣟⢸⡇⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⠀⢸⣧⣭⣤⣬⣭⠼⠇⣸⢤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣧⡀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⢠⡄⢿⠀⠀⠀⠀⠀⠀⠠⣾⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠃⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠼⠧⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠶⠶⠶⠶⠿⠶⠶⠶⠾

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠈⠁⠈⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡠⠀⠀⡀⠀⠀⢱⢤⠒⠀⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠉⠀⠈⠑⠒⠠⠀⣸⠃⠄⠰⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⢷⠈⠂⠃⠊⢈⠡⠃⢀⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡤⢀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠆⢀⠴⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⣤⠕⠐⠠⡀⣀⡀⣀⠀⠠⠛⣦⣨⢆⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠈⠀⠀⠀⠀⠣⠘⢸⠔⠃⠀⠀⠀⠑⠄⠄⠀⢠⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠰⡀⠀⢁⡀⠀⠀⠀⡖⠠⠀⠀⠀⠀⣧⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠃⠀⠀⠀⣣
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠉⠂⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⡤⠄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢥⡴⠡⠐⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⣠⠾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣠⣾⣿⣷⣮⣿⣿⣿⣿⣿⣿⠿⠿⠟⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣷⣶⣶⣶⣾⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡧⢻⡏⠀⠀⠐⠛⠲⠄⠀⠀⠀⠀⠀⠀⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠘⡄⠀⠀⠙⠳⣦⣦⠀⠀⣀⠛⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⠻⣇⠀⠀⠀⠀⠀⠐⠀⠴⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣴⠏⠀⣿⣆⠀⠀⠈⠛⠓⠶⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⢻⡄⠀⠸⣿⣷⣄⡀⠉⢉⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠒⠚⠛⠋⠁⣿⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⠇⢠⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⢈⡏⠀⠘⣿⠶⣤⣄⡀
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣷⡀⢀⡞⠀⠀⠀⡟⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢂⡟⢰⡏⠉⠙⠛⠙⠛⠛⠛⠛⠛⠓⠓⠓⠓⠚⠛⠛⢻⣿
⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⣰⠃⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⢠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡖⠒⠂⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢘⣷
⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⣡⣿⡃
⠀⠀⣼⣋⣠⠤⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⣼⠃⠀⠀⠀⠀⣠⠞⣿⠀⠀⠀⢀⡞⡿⢀⣴⡆⣼⠟⠁⢀⣿⣄
⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⠁⠀⣼⠃⠀⠀⠀⠀⡴⠃⢰⠇⠀⠀⣠⡟⢰⡧⠊⡼⢿⣏⡴⠚⣁⣀⡙⢷⡄
⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣷⣴⠏⠀⠀⠀⠀⣼⠁⠀⠘⠷⠒⠛⠉⠠⢏⡤⠊⣠⠟⢿⣿⣛⣉⣉⠀⣀⠙⣦⡀
⣠⠇⠀⠀⠀⠀⣀⠴⣚⡩⠉⠻⢿⣿⣿⣿⠟⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠴⠿⢋⣠⣾⡏⠀⠀⣿⠉⠉⠙⢧⣌⡀⠘⢷⡀
⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⢛⠙⠛⠓⠒⣤⣤⠾⠃⠀⠀⠀⠀⠀⢀⣀⣤⠶⣶⠋⠁⠀⢰⠇⠀⠀⠀⠀⠈⠛⠀⠈⢷
⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣰⠃⠀⠀⠀⢀⣠⡤⠖⠊⠉⠀⠀⣸⣏⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢰⣯⡴⠶⠒⠛⠛⠳⠶⣶⠶⠶⠶⠶⠛⠉⣿⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠉⠛⠛⠛⠿⠶⣦⣄⣀⣾⠀⠀⠀⠀⣾⢣⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠹⣆⡀⠀⣀⣀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠷⠶⣤⣄⣤⠟⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣠⠞⣿⠀⠀⠈⠛⠋⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⣬⣁⡀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣠⠞⠁⣀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠠⣶⣬⣍⡛⠓⠀⠀⠀⠀⠀⠀⣱⠀⠀⠁⠀⠈⠁⠀⠘⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡉⢆⢺⡒⠶⢌⣪⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⠢⣝⣋⣾⣥⡿⠟⠋⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣽⡿⢋⣩⡁⠀⠀⣤⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣾⣿⡆⠈⠉⠀⠀⠀⠉⠁⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠉⣧⠞⢳⡶⠋⡿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣄⠀⠠⣤⡤⠀⣰⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⣰⠏⠀⡾⠁⣸⠃⣤⣶⡄⣠⠶⣦⠴⡆
⠀⠀⠀⠀⠐⢿⣯⣱⣿⣿⣿⣿⣿⣿⣷⡄⠄⣤⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⣼⠁⢰⡇⠸⣁⡾⢳⠏⠰⠃⢸⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⢦⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠹⣿⣿⣿⣿⣿⢿⣿⣿⠏⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢈⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠻⢿⣿⣿⣿⡟⠁⠀⣤⠀⠀⠸⡀⠀⠀⠀⠀⣸⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⢷⡀⠈⠙⠿⠋⠀⢀⡼⠙⣆⠀⠀⢻⡀⠀⣠⡾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⢻⠀⠀⠴⠀⠀⢨⡇⠀⠸⡄⠀⢀⣷⡞⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⠤⣾⡀⠀⠰⠂⠀⠈⢧⠀⠀⣹⣶⠟⠉⢳⡄⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠈⠙⠓⢦⡀⣀⢠⣞⣦⡾⠋⢱⡀⠀⠀⠹⣄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢄⣀⡀⠀⡸⠁⠀⢸⡿⠛⣆⠀⠀⢳⡄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠛⣻⣦⣤⠿⠏⠀⠈⢦⡀⠀⠱⡀⠀⠀⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⡾⢿⣿⣿⠀⠀⠀⠀⠀⢹⣤⠀⠙⠶⠞⠁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣀⣸⣿⣿⣦⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⢀⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠛⠛⡿⠛⡿⠛⠻⡟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⢰⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣷⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣾⠇⢸⣿⠟⣿⣌⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠈⠛⠀⠙⠻⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣄⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⢞⣷⣹⣮⣽⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡀⠀⠀⠀⠀⠀⠠⠤⠚⠛⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠀⠀⠀⠘⣻⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠃⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⡴⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠠⠀⠂⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣰⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡐⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠰⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣄⠀⠀⠐⠠⠀⠀⠀⠂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠑⠞⠃⠀⡀⠁⡀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⣀⡀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⡄⠀⠄⠀⢠⠾⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠊⠠⠋⢠⠖⠏⠐⡄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠆⢀⣈⠓⠀⠀⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠒⠀⠀⠀⠀⠤⠀⠘⠂⠀⠲⠀⠀⠖⠀⡞⠋⠁⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⠀⠀⠀⠐⠀⢰⠀⠄⠀⠁⠀⠎⠀⠀⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⠀⠣⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡅⠐⠀⠴⠀⠹⠀⠀⠁⠀⠀⠀⠄⠀⡄⠀⠀⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠰⠂⠀⠀⠀⠈⠀⠞⠁⠈⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠒⠄⠀⠀⠀⠀⠄⠀⠀⠀⡐⢰⠖⠂⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠠⠀⠀⢀⡠⠦⠀⠀⠀⠃⣀⠀⠀⠀⢀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠉⣁⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠓⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⠀⠈⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣭⠽⡉⠉⠉⠉⠉⢩⡽⣏⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣤⣤⣤⣿⣿⣷⣤⣀⡀⣀⠀⠀⠀⣏⠀⣀⠀⠀⡈⠀⠀⠞⠣⠼⢷⣿⣿⣶⣿⣻⣿⣾⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣽⠛⠛⠛⠛⢿⣿⣯⣴⣿⡆⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡴⣶⣯⣉⠭⣿⣿⣿⣿⣿
⠀⢠⣼⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠀⣠⠀⠀⠰⠂⠀⠀⠀⢠⢤⡴⣏⢰⣟⣿⣿⣼⣽⣿⣿⣿⣯
⢀⣾⣿⣿⣿⣿⣿⠡⣴⣶⣾⣻⡷⢦⣤⣀⡀⠀⢉⢻⣿⣧⠀⡉⡦⠀⠓⢶⡀⣀⢀⠀⠡⢇⢹⣉⣼⣿⣿⣿⣟⣿⣿⣵⡇
⣿⣿⣿⣿⣿⣿⠇⠀⠙⠿⢿⣛⢻⣷⣾⣿⣷⣷⠃⠀⢽⣊⠀⣻⡹⠊⠀⠀⠉⠉⡝⣀⣠⣬⣍⢹⡙⢿⠿⢯⣼⣯⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠑⠀⠀⠛⠉⠉⣃⢻⣷⣦⢺⣩⡄⠁⠀⠀⠀⠀⠀⢰⣧⣤⣾⣿⣿⣷⠈⠢⣠⠼⣿⣿⣧⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣰⣿⣷⣾⣿⣽⢏⠹⣿⣿⠀⠀⠙⠟⠻⢆⠀⢧⠙⢿⣿⣿⣿⡿⠂⠀⢠⣾⢿⣿⣿⠃⠛
⣿⣿⣿⣿⣿⣿⣏⣿⣧⣿⣀⣀⠉⣙⣛⣿⣿⣟⠀⠀⣿⡇⠀⠀⠀⠰⠆⠀⠀⠀⠑⠒⠲⠀⠀⠀⣤⣴⠛⢿⠀⣿⣿
⠉⢹⣿⣿⣿⡿⠃⣿⠈⣿⣟⣿⣿⣿⣿⣿⣿⣽⣬⣽⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⠀⠀⢹⡟
⠀⢸⣿⣿⣿⣧⣀⡄⠀⠈⣤⣽⡿⢿⣷⡾⠋⠉⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣷⡾⠃⠐⠀⠀⠏⢹⠀⠀⢈⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣤⣽⣇⠀⠉⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⠿⣛⠙⠃⠀⠀⠀⠀⢀⣾⠟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡴⠋⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠁⠀⠐⠀⢀⣴⡿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⣽⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⣀⣠⣷⣿⠟⠀⠀⠀⢠⣾
⠀⣸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣠⠎⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⠇⠀⠀⠀⠀⣿⡿
⣶⣿⣿⣿⣿⣿⣿⣿⣶⡾⠃⠀⡔⠁⠀⠀⣿⣿⣶⣤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⠘⠄⠀⠀⠀⡿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣁⣾⠇⠀⠀⠀⠀⡜
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣠⠔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⠞⠁
⣿⣿⣿⡝⠻⣿⣿⣿⣿⣿⣿⡟⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡉⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⣏⣥⢈⣍⠢⡄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠉⡸⢧⡜⠛⡁⢈⣴⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡙⢂⣀⠔⠁⢶⠷⣄⠓⢄⠻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠐⠂⠘⣿⣰⢶⠃⣠⠄⠒⠒⠢⢴⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠈⠻⣁⠜⠁⠀⠀⠀⢠⡀⢙⣌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢠⠏⠀⠀⡄⠀⣶⠜⠉⠋⠹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠁⠀⢠⣀⡿⠚⠁⠢⢤⠞⠢⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠗⢳⠀⣀⡠⡏⠑⠀⠸⣀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⢰⠀⠐⣤⠚⢲⠉⢰⡁⢸⠛⡇⢉⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⣾⠀⠀⢸⣤⡞⠢⡌⠘⣧⠴⠛⠁⠷⢄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢠⡤⢤⣦⠀⠀⠀⡌⢧⣼⠀⠀⡉⠀⣰⠎⠀⠣⠔⠳⠆⠹⠀⢰⠀⢀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠈⢳⡴⠇⣀⣀⡀⠖⠋⠀⠀⢠⣏⡾⢋⠀⠀⠀⠀⠀⡀⠀⡄⠸⠀⠸⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠸⠧⠀⠠⠤⢤⠤⠤⠰⠲⠗⡤⠸⡏⡇⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⡆⠘⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⠃⠾⢧⠇⠀⠀⠀⠀⠀⠀⠘⠀⢁⠀⠃⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⣷⠃⡆⠗⠀⠀⠀⠀⠀⠰⠀⠇⠘⡀⢸⠀⢡⡀⠒⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢳⡀⢉⡖⢰⢸⢃⠀⠀⠀⠀⠀⢡⡸⠀⢃⠀⡄⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢇⡈⣷⠃⠆⣾⡀⠀⠀⠀⠀⠀⠛⢣⠘⡀⢡⠀⡌⠠⠈⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⡏⡴⢹⢷⡀⠀⠀⠀⠀⠀⠀⠳⢧⠈⡄⢀⠀⢃⠰⡀⠀⠀⠀⢸⡟⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠁⠀⠀⢇⠐⡘⢎⠙⠢⡀⠀⠀⠀⠀⠀⠀⢣⣡⡌⣁⠀⠢⡱⡀⠀⠀⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⡀⠀⢰⠀⣾⠀⢡⠀⠙⣦⢈⠂⠀⠀⠀⠀⠀⠀⠉⢃⠘⡀⠀⠈⠃⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⢡⣀⡜⠀⠝⢠⠀⠄⣠⠀⠺⡥⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠞⣤⡼⠃⠀⢀⡀⠀⣰⢷⠀⠑⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠈⢹⠇⣀⠀⠸⡆⠀⡙⠘⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢠⠀⣶⠠⠈⢀⠡⠀⢀⡝⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⣿⡀⠀⢧⠑⢶⣬⣃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠤⡾⠁⠉⣞⡓⠖⢺⣩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠏⠀⠀⠀⠀⠀⠀⠑⢇⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣀⡐⢁⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠈⠳⠋⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠄⡐⠠⢆⢀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠊⠔⠁⠀⠐⠉⢉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠐⠀⠁⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠁⠀⠀⠀⠁⡠⣀⡀⠀⠀⠀⣠⣶⣙⠀⡾⢋⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠐⠠⠀⠀⢹⣿⣿⡆⠀⠀⠛⠿⠋⠀⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⣀⣀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣨⣄⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄⠀⠀⠀⠀⠀⠀⢸⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠑⠶⠀⢟⡀⠀⠙⢓⡀⠀⠀⠀⡘⠁⠙⠌⠆⠀⣠⡀⠀⠀⢸⠴⠾⢯⡀⠀⠀⠀⠀⠀⣿⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠲⡤⣥⠆⠈⡁⠁⠀⠐⠈⠀⠰⠀⠀⠀⣰⡟⣧⠀⠀⠈⠠⠤⠞⠁⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⠠⡀⡷⡾⠶⠛⢔⠠⠄⠊⠀⠀⠀⢰⠋⠉⠉⣧⠀⠀⠀⠀⠀⣀⠠⠖⠶⠀⠀⣿⠀⡇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠶⠓⠻⣿⠀⠉⠀⡠⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡌⠒⢤⡤⠀⠀⠉⢀⣤⠀⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠲⠃⠀⣿⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠚⠁⢠⠄⠐⠋⠀⢀⡀⠰⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⣄⡒⠀⣀⠀⠐⠂⢀⡀⠀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠃⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡉⠉⠛⠛⠛⠛⠛⠙⠋⠁⠀⠀⠈⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⢸⡀
⠀⠀⠀⠀⠘⠃⠀⠇⠀⡀⠀⠀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠘⠈⠁⠀⠀⠈⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠐⠀⠀⠄⠄⠀⠃⠀⠈⠀⠀⠸⡂⠠⢄⠀⠀⠀⠠⡄⠀⠂⣄⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠁⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢀⠻⢗⡀⠀⠀⠀⠀⠸⢷⠀⠀⠀⠀⠀⠐⣿⠇
⠀⠹⣦⡾⣿⣾⣷⣄⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⡄⢀⠀⣺⣿⣿⡗⠀⠀⠀⣴⠏⣻⡉⢷⡄
⠀⠀⠀⢀⣿⣿⣾⣿⠶⣦⠀⠀⠀⣿⣾⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⡇⠀⠉⠉⢹⣿⣧⠀⠀⠀⠻⢦⣾⣤⠿⠃
⠀⠀⠀⠈⣿⣿⠇⢉⣤⢤⠀⠀⠀⣇⣿⣿⡏⠁⠀⠀⣾⣼⣿⣿⡇⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣼⡿⣿
⠀⠀⠀⠀⢹⣿⠀⢨⣤⣤⠀⠀⠀⠉⢸⣿⠁⠀⠀⠀⠛⣿⡟⣿⡇⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠘⠛⠀⠈⠉⠉⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠻⠃⠹⠃⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠛⠁⠛⠃
⠀⠀⣠⡀⠀⠀⠀⠀⠀⢠⣤⠀⢠⠀⢤⠀⠤⠤⠤⢄
⠀⢀⣶⡋⠀⠀⠀⠀⣀⣉⣙⣀⠀⠐⠛⢻⡾⢂⡀⢸⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⣿⣿⣿⣤⡆⢺⡏⣿⣿⣿⠇⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⢬⢤⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⢸⣿⣯⠉⢩⣶⣦⡏⠉⠉⠉⠀⣤⣤⣤⣤⡄⠠⣤⢠⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣦⡄⣿⣆⣀⣀⡀
⠀⢸⣿⡿⠀⠰⠿⢷⣧⣤⣤⣤⣤⣿⣿⣿⣿⢷⣆⣸⠀⠀⠀⡧⠀⠂⠐⠚⢲⣬⣷⣿⣿⣿⠟⠓⣿⠉⠉⠉⠙
⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⠀⠀⠀⣇⣀⣀⣀⣰⣀⣀⣀⡘⠛⠛⠛⠛⣿
⠀⠀⠛⠃⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠈⠃⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣬⣫⣤⡀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣁⠀⠀⠀⠀⠀⠀⣘⣿⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣻⣋
⠀⢠⣿⣿⣿⣿⣧⠀⢴⣿⡦⠀⠀⣿⡻⠙⢿⡇⠀⠀⠀⠀⢠⣿⣿⡀⠀⢀⣼⡟⠀⠀⠀⠀⠀⢰⡿⢻⡿⢷
⠀⠘⡟⣿⣿⡇⠿⠤⢾⣿⣷⠀⠀⠈⣧⣤⡾⠀⠀⠀⠀⠀⢸⣿⣟⠿⣦⣾⣿⣧⣀⡀⠀⠀⠀⢸⢷⣿⣿
⠀⠀⢳⣿⢹⣿⠀⠀⢸⣿⠟⠁⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⢸⣟⣿⡆⠠⢻⣿⣿⣿⣧⠀⠀⠀⠀⣼⡟⣿⡆
⠀⠀⢸⡿⠸⣿⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⠇⣿⡇⢠⠀⠀⢀⢿⣿⠀⠀⠀⠀⣿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⡀⠀⠀⠐
⠊⠀⠐⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡊⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠠⠀⠉⠀⠁⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠡⠀⡀
⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⣀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠐⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡤⠞⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠈
⠀⠐⠀⢦⣀⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠁⠐⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠙⠳⠶⡵⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠢⠤⠤⠢⠆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠀⠀⢰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⣀⣀⣭⣭⣭⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⣤⣶⠾⠟⠛⠉⠉⠁⠀⠀⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣑⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⡿⠁⠀⠀⠀⠀⠀⢀⡠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⣼⡟⠀⠀⠀⠀⠀⢀⣴⣿⣧⣀⡇⠉⢿⢿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡄⢰⡟⠀⠀⠀⠀⠀⡠⠃⠈⠙⣿⣿⣿⣶⣜⣯⣿⣿⣿⣿⣿⣾⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠰⢀⣿⠃⠀⠀⠀⠀⢰⣷⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠿⠿⠛⠟⠁
⠀⠀⠀⠀⠀⠀⡀⢸⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠃⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡇⠀⠀⣀⡤⢤⣤⣀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠄⠄⢀⡀⠀⢸⣿⣿⣿⣯⡞⠀⠀⠀⠁⢀⣤⣀⠈⠁⠀⠀⣰⡫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠁⠊⠉⠀⠀⠀⠀⠀⠋⠛⠃⠀⠀⠀⠀⡛⠃⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⢿⠿⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣇⣤⣴⣾⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⡤⠴⠶⠶⢶⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠙⢶⣤⠤⣾⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠢⠄⠀⠀⠀⠀⠉⠉⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⡊⣁⢀⣠⣤⣤⣿⡇⠀⡀⠀⠀⠀⠀⠐⡆⠄⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠤⠙⠛⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠐⠢⡄⡀
⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⡀⠀⠉⠐⠠⡀
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠢
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠐⠈⠀⠀⠀⠑⡀⠀⠀⠀⡀⠌⠀⠉⠐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡰⠁

⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢻⡿⢠⠄⠉⠀⠀⠈⢰⣿⡇⠀⠀⠀⠀⠒⠀⠀⠙⠄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣔⠇⠸⡀⠐⣠⣤⣷⣿⣿⡁⠀⠀⠀⠀⠀⠐⡄⠀⠲⣬⠷⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠻⣿⣿⡇⠀⠤⠀⠀⠀⢀⣙⡀⠁⢼⣇⣹⠃⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠳⣦⣄⡀⠀⠀⠀⢻⡅⠀⠘⠟⠿⣦⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⣅⠉⢶⡄⢰⠟⡥⠐⠊⢶⣄⣷⣤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿⠛⢀⡀⠀⠷⣄⣿⣿⣿⡏⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⣸⠀⢼⠟⢡⣀⣿⣟⢻⡿⣧⠨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣾⡅⠀⠀⣿⠀⣦⣀⣸⣿⣿⣿⡘⣷⢻⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣷⣶⣿⢿⣿⠁⠀⠀⡟⠀⡏⠉⢹⣿⣿⣇⠇⢸⡌⢇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠋⠉⢀⡀⠀⠀⠀⠐⠀⠺⠿⠿⠟⠂⠀⢠⡗⠐⢳⣤⣿⣿⠇⢸⠀⠈⢷⠘⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⣀⣀⢀⡆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠻⡽⡿⠀⠀⣇⠀⠘⡆⢿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⡆⢱⠘⣧⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⢸⡇⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⡄⣶⢻⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡜⢋⠀⠀⢚⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠆⢳⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡿⠛⠀⠀⡡⣶⢻⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⡈⢏⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡇⠘⡄⠀⣴⣿⡟⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡇⠈⠃⢀⣾⢿⣿⣎⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⣿⡀⠀⠛⠁⠈⢿⣿⣧⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠃⠀⠀⠀⠀⠈⢻⣿⣷⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠋⠉⠉⠁⠀⠀⠀⠀⢰⣶⣦⡄⠀⠀⠀⠀⠀⠑⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⠄⠀⠀⠐⠀⢀⢀⡤⣤⠤⣤⣤⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧⣀⣠⣤⡀⠀⣤⡀⡀⢀⡀⡤⢤⣬⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠈⣧⡈⣥⣿⣶⡌⠀⣅⡄⠀⠀⠸⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣧⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠉⢿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢸⡆⢿⣿⡇⠀⠀⠀⠂⠀⠀⠀⣠⣤⣿⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⢸⣿⡇⠈⠀⠈⠀⠀⠀⠀⡏⢀⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠀⠀⠀⠀⢹⣏⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⢀⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠋⠛⠋⠀⢠⣄⣀⣿⣿⣿⣷⠠⠶⠴⠶⠶⠶⢾⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠙⠋⢉⠉⠀⠀⠐⠲⠠⠾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠒⠀⠰⣾⠟⠋⠉⠋⠙⠋⠉⠉⠋⠉⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠐⣒⡐⣂⠐⢒⣒⠆⠀⠀⣾⡗⡖⠀⠃⡒⠚⠛⠒⠛⠂⠒⠓⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠙⠉⠁⠀⠈⠉⠉⠉⠀⠉⠉⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣇⢰⡇⠀⢸⢢⢘⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠋⠉⣉⠭⡭⢍⡉⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢀⠎⡡⠼⠹⢤⡙⡆⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡀⠓⢲⢰⠒⠃⡸⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠳⣄⡸⢼⣀⠴⠁⠀⠀⢠⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠈⠛⠶⠶⣦⣤⣤⣬⣭⣤⣤⠴⠶⠶⠛⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠞⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢠⡟⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⣿⣦⣄⡀⢀⣠⡴⣿⠁⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⣴⡇⠀⠈⠙⠋⠁⠀⣿⣤⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠞⢛⣿⡉⠻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠉⢹⡟⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⣠⡾⠋⠉⠙⢷⡜⢷⡄⠀⠀⠀⠀⣠⡞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠻⣆⠙⣦⡀⢀⡼⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠻⠟⠁⠀⠀⠀⠀⠀⠘⣇⣠⡴⢦⣄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸⠇⣽⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣈⣛⢲⣆⣀⣰⡖⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣋⣁⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠁⠀⠀⠉⢻⣿⣿⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇





⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⠆⡯⠁⣯⡷⠺⣍⢰⠉⡆⡷⣸⢀⣾⡄⡇⠀⠀⡏⢣⣾⠍⠈⡏⢠⣳⠐⡇⡇⠐⢯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠉⠉⠊⠈⠉⠁⠁⠈⠈⠀⠁⠙⠂⠀⠁⠁⠉⠀⠀⠁⠉⠈⠁⠁⠁⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠸⣿⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡼⠟⣶⡆⠠⡤⠶⣰⣆⡀⢀⣀⣶⣶⡆⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠟⠻⡇⠀⠒⢦⡀⠈⢁⣌⣡⠛⢿⣿⣦⣼⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁⠈⢃⡘⢻⣤⡼⢻⡏⢀⣸⡧⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢟⡠⠄⠋⠀⠉⠁⠿⣥⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠀⢶⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⣉⠠⠄⠀⠀⠀⢀⠀⠴⠓⢀⣘⠷⠄⠘⡋⠠
⠀⠀⠀⠀⣤⠄⠒⠀⣶⡀⠀⠀⠀⠀⢷⡟⣠⡤⠟⠂⠀⠀⠀⠀⠏⠱⣦⣀⠿⠛⠺⠛⠞⠻⠃
⠀⠀⠀⠈⠻⣦⣶⡾⠏⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠹⠂
⠀⠀⠀⠀⠀⠀⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢛⠁⠀⢀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠀⠐⠒⠒⠀⠚⠀⠈⠁⠀⠀⠘⣀⠀⠀⠐⠢⣤⠿⠀⣀⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢀⠀⠀⠘⡀⠀⠙⣆⠀⠀⠈⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡄⠀⡀⠡⠀⢠⠌⢆⠀⠙⠶⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢡⠀⠀⠀⢲⣇⠀⠀⠀⠀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠠⠀⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠖⠀⠚⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⢃⠀⠁⠄⠀⠀⣀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠆⠀⡀⠠⠀⠈⠳⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠊⠀⠀⠀⠾⠀⠐⠀⠱⣄⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⢠⠀⣶⠀⠘⣄⠙⣧⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⢢⠀⡄⠈⠀⠀⡆⠀⠀⠈⠀⢻⡆⠀⢹⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣠⠆⠒⠀⠀⠀⠀⠈⠀⡞⢀⠀⠀⢻⡄⠹⣤⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠢⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⢸⡇⠀⢹⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠙⠀⠛⠂⢠⡀⠀⠀⢰⠀⠈⢷⠀⢸⣿⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⡀⠀⠸⡄⠀⠀⠙⡆⠀⠞⠀⡘⢁⣀⡈⢿⣀⠀⢰⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠘⠃⠀⠀⢣⡜⠀⠈⠛⣤⠉⠀⢿⡏⠂⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⣀⠈⠀⢰⡇⠀⣾⣿⣶⠂⡄⢀⡀⢠⠞⠂⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠠⠀⠀⠃⣼⣿⣿⣿⣦⣤⣌⣀⣠⡄⠀⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⣤⠔⢲⣿⣿⣟⠉⠻⠟⠻⣿⣿⣿⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢋⡀⠀⠀⠀⢾⣿⡿⠀⠀⠈⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡁⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠐⠀⠀⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢸⡇⠀⠀⢀⣠⡤⠞⠋⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠁⡆⠀⠀⠀⠀⢀⠀⠀⠸⡄⢡⡀⠀⠈⠁⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠒⠃⠀⠀⠀⢀⡿⣀⠀⠀⠃⡀⢹⡀⠀⢀⣠⠄⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠛⠀⢀⣄⣠⠶⠀⠀⠀⠀⠀⠘⠀⠈⠦⠀⠠⡀⠀⠉⠀⠀⠀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠒⠒⠛⠛⠛⠓⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣤⣙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠂⠈⠀⠀⠈⠁⠀⠉⠸⠧⣈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠀⠀⢀⠀⢀⡀⠀⢀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠀⠉⠀⠈⠉⠁⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠉⠐⠀⠉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠐⠀⠤⠀⠀⠤⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⠀⡀⠀⢀⡀⢀⠀⠀⠀⢀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠀⣀⠀⢀⡂⠀⠀⡇⠀⢀⡀⢀⡄⠀⠀⣃⠀⢀⡀⠀⡀⠀⠀⢃⠀⠀⡂⠀⣘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠃⠀⠀⠇⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⠃⠀⠀⠛⠀⠊⠐⠐⠃⠀⠀⠚⠀⠂⠑⠀⠃⠀⠀⠚⠑⠀⠁⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠀⠠⡄⠀⠀⡇⠀⠠⠀⠀⡆⠀⠀⠤⠀⠠⠄⠀⡄⠀⠀⠠⠀⠀⡆⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣀⣀⣀⡀⠀⠀⣃⢀⣀⣀⣀⡇⠀⠀⣀⣀⣀⣀⣀⡆⠀⠀⢠⣀⣀⡆⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠘⠀⠐⠆⠀⠀⠇⠀⠘⠀⠀⡇⠀⠀⠑⠀⠘⠂⠀⡃⠀⠀⠐⠀⠀⡃⠀⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣗⣀⣒⣂⣰⠆⠀⠀⣖⣂⣐⣂⣒⡇⠀⠀⢔⣂⣐⣂⣒⡃⠀⠀⢰⣂⣐⣂⣀⡲⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠘⠀⠈⠐⠄⠘⠒⠀⠐⠀⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠰⠀⠠⡄⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢠⢠⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠈⠁⡏⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⢹⠘⠁⠁⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠧⠏⠡⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠌⠹⠜⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡆⠘⠀⠄⠀⠰⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣤⣶⢿⠦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡤⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡠⠈⣿⠇⠀⠀⠀⠀⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠐⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣠⠌⠀⠀⠀⠆⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠋⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠰⣧⠀⠁⡃⠀⠀⠀⡆⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠐⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠀⢀⠃⠀⠀⠀⠁⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⡠⣿⠐⠀⠂⢰⠀⠀⠀⠀⡅⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⢀⠀⡀⠁⢨⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠆⠀⠃⡔⢁⣼⡿⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⡀⢸⠀⠈⡠⠀⡰⢀⣿⡿⠁⠐⠁⠘⠠⠀⠀⠀⠠⠀⠰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠃⠀⠀⠀⡺⢡⢿⠀⠘⠁⡰⢁⣿⠟⠱⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢶⠄⡀⠀⠈⠀⢻⠘⠀⠀⠀⠃⠉⠌⣠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠉⠀⠀⠀⢨⡈⠀⣠⠀⠷⠀⠜⠁⠀⢀⠄⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⡇⢀⡆⠀⠀⠊⠇⡼⠂⠈⠀⠈⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣐⣶⢰⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⠴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠖⠀⡀⠼⠗⠒⠚⠁⢰⠀⠀⢀⣵⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢭⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠶⠦⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⢹⠛⠿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⡀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣧⣄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠻⢿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣤⣤⣤⡄⢀⣤⣤⣤⡶⢂⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⡀⠢⣌⣛⣛⣁⡘⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢛⣿⠉⠀⠀⠀⣀⣠⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡆⣸⣿⣿⢹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣟⣴⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠍⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡄⠀⠘⠯⣽⣿⣿⣿⣏⣙⡿⣟⣛⡻⢿⠿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡿⠒⠉⠉⠲⣄⢠⠖⠉⠉⠒⢤⣤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡴⠀⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠼⡧⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⣠⠊⠀⠑⠦⢰⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⡤⠤⠊⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠇⠀⢠⠀⠀⠈⠂⠀⠀⠀⠋⢰⡇⠈⠀⠀⠀⠐⠁⠀⠀⡀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⡆⠀⠀⡁⢰⠀⠀⡤⠀⠀⠀⢨⡇⠀⠀⠀⠦⡀⠀⡁⢈⠁⠀⢠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠙⢤⠊⠀⠈⠀⠀⠀⠀⠖⢀⢼⡧⣀⠲⡀⠀⠀⠀⠃⠀⠑⣤⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠘⢆⠀⡀⠀⠀⠀⠀⠀⢀⡜⢣⡀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠬⣉⣀⣈⣂⠴⠊⠀⠀⠙⠦⢐⡀⠀⣈⡡⠞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⢋⣉⣉⣉⣉⡃⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣤⣖⠭⠔⠒⠉⠉⠉⠀⠀⠀⠀⠉⠛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣤⣤⣤⣤⣤⣤⣤⣤⣧⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢧⠀⠀⠀⢀⡜⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠝⢻⡍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠄⠀⠠⠀⣠⡿⠤⠀⡀⠀⠀⠀⠀⠀⠈⢆⠀⠋⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⣁⣀⡀⠀⠀⢠⡋⠀⠀⠀⠉⡀⢤⣤⡄⠀⠀⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠉⣦⣾⠁⠀⢹⡆⠀⣼⣿⣿⣿⣷⣶⡗⠘⠛⠁⠀⠒⠒⠂⠀⠀⠀⠙⢷⠄⢷⠇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠤⠶⠁⠘⠦⠴⠾⠄⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠓⢦⣄⠀⠀⣆⢠⠄⢦⢠⠄⠂⠀⠀⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣲⡒⣰⠆⠀⠀⠀⠀⠈⠉⣿⣿⣿⣧⣰⡋⢉⠶⡉⢉⡆⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠙⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣏⠀⠀⠀⠀⠀⠀⢀⣨⣭⠉⠉⠉⠁⠀⠀⠀⠈⢁⣀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⠹⡆⠀⡈⠛⠁⠀⠀⢿⣻⡆⠀⠀⠀⠀⠀⢀⣶⣾⣅⠀⠀⢹⡆⢠⣴⠀⢀⣴⠟⠉⣠⣾⡁⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⢦⣀⣀⠀⠀⢺⣿⣿⣿⣿⣿⠹⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠘⣿⠀⠉⣴⡟⠁⠀⠀⠻⠟⠡⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⡞⣀⣉⢳⡀⠠⣾⣿⣿⣿⣿⡇⢻⣿⡇⠀⢿⠇⠀⠀⠈⡄⠀⡤⠀⠠⣿⠀⣸⡟⠀⣤⣤⡄⠀⠀⠀⡰⠟⠁⠀⠙⠷
⠀⠀⠀⡀⠀⢠⣤⡄⠀⠈⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⢉⠁⠈⢉⣁⣀⡀⠀⠀⠰⠆⠀⠀⠀⠶⢸⣿⢀⣿⠃⠀⣿⣿⡇⠀⢰⡖⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡞⠀⠀⠈⣿⠅⡀⢠⡄⢀⠀⠀⠀⠙⢶⡀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⡆⠀⠆⠀⣼⡏⣸⡿⠀⠀⠻⠿⠇⢰⣦⠈⢆⠀⠀⠀⠀⠠⠊
⠀⠙⠛⢻⡶⠀⠠⢄⠙⠀⠁⠈⠠⠆⠀⠀⢸⡇⠀⣘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢰⣿⢃⣿⡇⠀⠀⠺⣷⣿⣞⣿⡆⠀⠿⠂⠀⠺⠇
⠀⠀⢀⡞⠁⠀⠒⠂⠀⠀⠀⠀⠀⠚⠁⠀⢰⡇⣸⣿⣿⣿⣆⠀⠈⠐⠆⡀⠀⠀⢻⣷⠀⣠⣿⡏⣼⣿⢀⣦⠀⠈⠛⠛⠛⠙⠃⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠈⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣙⣀⠀⢸⡇⠈⠉⠉⠉⣁⡀⠀⠂⠀⠀⠀⠀⢸⣿⣾⣿⣿⣼⣿⠇⣼⠿⢶⣁⠄⠀⠀⠚⠻⠿⠿⠿⠿⠛⢉⡅
⠀⢠⣶⠿⠀⣬⣐⡀⠀⣿⣾⠇⠀⢡⡞⠁⣾⡇⠰⣶⢿⣿⣿⣿⣿⣶⣤⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⣀⣀⡼⢣⣆⠀⠀⠐⠒⠂⠀⠤⠐⠊⠁
⠀⡘⠁⠀⠀⠻⣿⡇⠀⠈⠉⠀⣠⡈⠀⠀⢻⣇⠀⠀⠸⢿⣿⡿⠏⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⢨⣿⣦⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠾⣿⣦⠀⠛⠁⠀⠀⠀⢠⣿⣶⡖⡆⠘⣿⣦⡀⠀⠀⠀⣴⣾⡀⠀⠀⣰⣿⣿⠟⠉⣠⠔⠀⠄⠀⠀⠀⡤⠀⠐⠀⠀⠤⠀⣉⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠀⢠⡟⠀⠀⠈⠻⣿⣷⣤⣄⡙⠛⠀⢀⣼⣿⡿⠁⡀⡀⢀⠀⣀⡀⠀⢰⣿⡇⠀⠀⠀⡀⠐⡀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⣿⣯⡀⠀⠀⠀⠂⠀⡙⢿⣿⣿⣷⣤⣾⣿⡟⢀⡄⠀⡰⠀⢀⡔⠁⢠⠀⠉⠀⢤⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠉⠉⠁⡴⠄⠀⠀⢠⠁⠀⠈⢿⣿⣿⣿⣿⡇⠢⠤⢴⡤⠤⠋⠀⠀⣼⡆⠀⠀⣰⣄⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢠⣿⣿⣷⠀⠀⠀⢸⡄⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠚⠓⠀⠀⠀⣠⡿⠀⠶⠁⠀⠀⠑⠖⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠶⠶⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⡤⠶⠶⠒⠒⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢠⡞⢁⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠹⣆
⠀⠀⠀⠀⣠⣿⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠛⠳⢦⣼⢀⣠⣴⢖⣛⣋⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣉⣛⣶⣿⣆
⠀⠀⠀⢠⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡄
⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠀⠀⠄
⠀⢠⡾⠛⠛⢿⣆⠀⡶⠶⠶⡖⢶⡄⠐⠀⠄⠀⠰⠴⢿⠿⠦⠆⠀⠀⠀⠄⣰⣶⣶⡄⠠⠀⠁⢀⠄⠀⠀⢢
⠀⠘⣷⠿⡿⢿⠟⠀⡇⠀⠀⠃⡠⡇⠀⠨⠤⠆⠀⠢⠀⠀⠀⠁⠀⠁⠀⡇⢻⣞⣳⠇⠀⠀⠀⠐⠀⠀⠄⠘
⠀⠠⠻⣟⣻⡿⠃⠀⡧⠶⠶⠤⠴⡇⠀⢠⠀⠈⠀⠀⣐⠂⠠⠆⡤⠠⠀⠇⣶⣿⣷⣦⠀⠀⠀⠀⠈⡐⠁
⠀⠀⢀⠈⠉⢄⠀⠰⣿⣸⣀⣄⣈⡇⢀⠋⠄⠌⢃⠀⠉⠀⠠⠀⠐⠆⠀⢢⠈⠈⠁⠀⠸⠀⠀⠀⠀⠈⠠
⠀⠀⠠⠀⠀⠰⠀⠀⠈⠉⠁⠀⠀⠤⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠰⠂⠀⠀⠄⡀⠀⢖⡂⠂⠀⠴⠀⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⢈⠀⢀⡀
⠀⠀⠀⠐⠀⠀⡀⠉⠀⠀⠂⠀⣠⡐⠂⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢴⣶⠀⠀⠀⡀⠀⠀⠀⠸⢠⠀⠀⡏
⠀⠈⠂⠀⠀⢅⠏⠀⠀⠘⡁⢀⡿⠿⢶⣄⠢⠄⠔⣀⢄⠈⡀⠀⠀⠀⠔⡰⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠃⠤
⠀⢀⠂⠀⠐⠀⠔⠈⢱⠀⠀⢸⠃⠀⠈⢿⠀⡴⣄⠙⠓⠀⢱⠀⠀⠈⠀⠀⠀⠀⠁⡰⢀⡄⣤⢠⡄⣤⢠⠒⠄
⠀⠈⠃⠄⠘⠀⠦⠄⠊⠀⠀⠑⠂⠀⡀⠊⠀⠫⠀⠂⢀⡀⠀⢀⡄⡄⣠⠀⠀⢒⠀⠀⠘⠇⠿⠸⠃⠟⠋⢀⡄
⠀⠆⣲⣶⣰⣶⢸⠓⡆⠆⠀⣤⣄⠁⢑⣤⡄⠀⣴⣾⣿⣧⠀⠀⠷⠇⠟⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠒⠪⠟
⠀⠀⣈⣤⣄⣀⠀⠀⠀⢀⠈⠙⣿⡿⣿⡟⢉⢀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠉⠉⢘⠀⠀⠔⠁⠀⠀⠽⠷⠿⠇⠀⠀⢁⠀⠀⠀⡠⠐⠀⢘⠀⠀⠰⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠈⠁⠠⠀⠊⡀⠐⠈⠁⡀⠀⠀⠀⣠⠀⣀⠄⠈⠐⠀⠊⠀⡀⠀⠀⠀⠅⠀⣀⠈⠀⠒⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠐⣀⣉⠀⠀⠀⠀⠀⠂⠘⠀⠂⠂⠀⠀⠀⠀⠀⠀⠈⢀⠈⠁⠁⠉⠈⠉⢀⠌⠀⢀⡀⠀⠀⢀⠄⠠
⠀⣄⣚⣿⣿⣓⣠⡄⠀⠠⠄⢀⣀⠀⢀⢂⢄⠀⠀⠀⢀⡀⢤⠀⠀⠀⠀⠀⡀⠀⠒⠉⠀⢀
⠀⢻⠉⠀⠀⢙⠆⠁⠀⠄⠀⠀⠀⠀⠀⠓⠁⠴⣱⠂⢈⠂⠈⠁⠀⠀⠀⠾⠄⠀⠀⠈⢀⠀⠀⠀⠀⠀⣀
⢀⠊⠣⡀⡀⠊⠢⠀⠀⢠⣴⣶⠄⠀⠀⡠⠀⠁⠐⠀⠀⠀⣿⣳⠀⠀⠀⠀⠀⠀⠀⠘⠗⠘⠋⠀⠈⠸⣯⠇
⠀⢀⣄⣀⣀⣦⡀⠀⠀⡈⠉⠀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠧⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰
⠀⠛⠿⠿⠿⠟⠓⠀⠀⠈⠁⠈⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠀⠀⣀⡀⠀⠸⣿⢧⡄⠀⠀⠀⠀⢀⣿⡗⠀⠀⠀⠀⠰⠀⠀⠈⠢⠀⣠⠄
⠀⠀⠀⠀⠀⠈⠛⠁⢀⣠⡟⠀⣄⠀⠠⣤⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢂⡀⠘⠃⡈⢦⡘⣆
⠀⠀⠀⠀⠀⣀⠀⠐⠉⠀⠀⠀⠀⠴⠄⠈⠙⣿⣿⣟⠀⠒⠀⠀⠀⠀⣈⡣⠀⠰⣇⣨⡇⢸⣆
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠙⢷⣤⣄⠀⠀⠀⠈⣧⠀⠀⠙⠋⢷⠀⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⣩⡄⠹⣦⣰⡿⠃⠄⢣⠑⢦⣼⣦⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢸⡏⠁⣄⡀⠙⢦⣼⣿⣿⣯⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡄⠀⠀⢸⡇⠸⢿⠉⣦⣸⣿⢙⣯⢿⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⣀⣼⡇⠀⠀⢸⠇⢶⣄⣠⣿⣿⣿⡇⢻⡈⡇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⣶⢿⣿⣿⣿⣿⡇⠀⠀⢸⠀⢈⠀⢈⣿⣿⣿⢸⠀⣇⠹⠀⠸
⠀⠀⠀⠀⠐⠿⠿⠟⠉⣉⢁⣦⠀⠀⠀⠑⠂⠸⠿⠿⠛⠓⠀⢀⡿⠀⠙⣦⣼⣿⡟⠌⡖⡇⠸⡀⢧⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⣀⣀⣀⣀⣦⡀⠀⠀⠀⢀⣀⣼⠃⠀⠀⠈⢫⢿⠳⠀⢱⠎⡄⢣⠘⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⡿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠂⠄⢃⠀⠇⢡⠀⠆⢹⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣼⡇⡿⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠈⠆⠘⠀⠃⠸⠈⢇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⡇⠀⢨⡽⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⡄⠐⠀⠀⠡⠈⡇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠏⣇⠀⠠⢾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠛⠀⠐⠀⠀⠃⠀⣄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠈⣹⠀⠐⣿⣟⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⢿⡄⠉⢀⣴⠏⣿⣷⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡇⢸⣧⠀⠈⠁⠀⠹⣿⣿⣿⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡈⢆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠋⠀⠇⠀⣞⡄⠀⠀⠀⠀⠈⢿⣿⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠸⠁⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠣⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠱⢀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢿⣿⣶⡀⠀⠀⠀⠀⠈⠓⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠌⠀⢧⠀⠀⠀⣀⣀⡄⢤⠠⡄⢴⠐⡦⢴⣼⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣄⠈⢳⣄⠈⠁⠄⠀⠀⠒⠂⠀⠨⡄⠻⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣆⠸⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠁⠀⠹⣆⢽⣿⣿⡄⠀⠀⠀⢀⢀⠀⢀⠇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠀⠀⠀⢻⡌⣿⣿⠃⡀⠀⣘⠈⠀⠀⢨⠆⣼⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠄⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⢸⠀⣿⢋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⡇⠰⠀⢀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⢨⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⠆⠉⠙⠀⠀⠶⢦⣼⣿⣿⣿⣟⡓⠒⠞⠦⠛⠿⢿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠙⠁⢈⣁⣀⣁⣈⣛⣛⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⣻⠿⢋⠃⢩⠉⣉⠩⣀⣀⡐⠀⠁⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢼⣿⣯⣭⣭⣭⣩⣭⣍⡝⠀⠀⢸⣿⣟⣛⣉⢹⡉⢉⡁⠈⠉⣉⣉⠙⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠁⠀⠀⠁⠉⠉⠀⠈⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣶⠀⠀⠀⠀⣀⣈⣀⠙⠛⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡷⠿⣟⠀⠀⡆⠐⠛⠓⠂⡴⠂⠀⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠇⢸⣿⣿⠟⠄⣠⠂⠀⠀⠀⡔⠡⠀⢀⣽⡟⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⣽⠖⠉⠀⠀⠀⠚⠁⠀⡀⠀⢠⡇⠁⡰⣿⣿⠃⠀⡀⠓⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⠃⠀⠀⢀⠖⠦⣄⣀⠀⠀⢀⠞⡁⠀⠐⣿⡏⠀⢸⠃⠁⠁⢹
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⡏⠀⠀⠀⠀⣀⡤⠊⠉⠛⡙⠋⠀⠀⠀⢉⣿⡇⣤⠪⠀⠂⠂⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠃⠐⠀⠈⠀⠂⠀⠀⠀⠀⣸⣽⠃⠀⠀⢀⢀⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠆⠂⠀⠀⠀⠀⢠⠆⠀⢠⣉⣿⠀⠀⠀⠘⠌⠈⠘⡇
⠀⠀⠀⠀⠀⠀⠛⢀⠀⢸⡏⡇⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⡴⢀⠌⣤⣿⡏⠀⠀⠀⠀⠠⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢨⡀⢸⣷⡇⠀⠀⠀⠀⢨⣶⠀⠀⠀⡠⠀⡏⣸⣿⡟⠀⠂⠀⠀⠀⠀⡈⢻
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⢷⣤⠀⠀⠀⣼⢿⣧⠀⢀⡄⣈⣴⣿⠃⢠⠀⠀⠀⠀⠖⠰⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⣷⣄⡀⠁⠀⠤⣂⣾⢻⣎⣽⣿⠟⢋⢻⣇⢠⠀⢰⠇⠀⠆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢿⣻⡻⠳⠀⠀⠐⣻⠈⣿⣩⠸⡀⠉⣘⡿⢸⠀⡸⢀⡆⢠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⠸⣿⠱⣄⠀⠀⠀⣿⣶⠙⡼⣀⠇⢻⣿⢃⠟⢠⠃⡼⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⣿⣇⢹⠀⠀⠀⣿⣿⡄⠀⠟⣶⡿⠃⠈⠀⢀⣜⠡⠞⡓⠶⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⢿⣿⡌⠀⢀⡔⢻⡟⠀⠒⠹⠟⠀⣀⡠⠖⠛⢉⣶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠘⣧⠁⠀⡈⠀⡟⠀⠀⠐⠐⠚⠉⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠀⠀⠁⡸⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⢘⣡⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡉⠁⢀⠈⣆⣲⣄⣀⣀⣠⣀⣀⣀⣠⣄⣀⣀⣤⣤⣤⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣀⣤⣤⣤⣤⣤⡤⠽⣿⣫⣿⡹⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠂⠁⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⣶⣶⣿⣧⣴⣿⣶⣶⠀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠀⠙⠿⠿⠿⠿⠿⠿⠛⠀⠈⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢀⡀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣾⠿⢿⣶⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠉⠁⠀⠀⠈⠙⠛⠛⠛⠛⣿⡇⠙⠛⠛⠛⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⢠⣿⠁⣤⣤⣤⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⣸⣿⣿⣇⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠛⠛⠛⠛⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠉⠉⢹⣿⣿⡏⠉⠁⠀⠀⣿⡇⢠⣤⣤⣤⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠘⠋⠛⠛⠛⠛⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣤⣤⣶⡿⠟⠡⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣒⣒⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠋⠉⠉⠉⠉⣭⡄⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣴⣶⣿⣷⣦⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠉⣿⡏⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣸⠀⠳⣄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣰⠃⠀⠀⠹⣦⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡼⠃⣀⠤⠤⠀⢹⣧⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡠⠚⠁⠀⢀⣴⠟⠀⠀⡤⠄⢀⣀⡆⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⢀⣘⡆⠀⠀⠀⢠⠀⢸⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡎⠀⢸⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢆⠀⠓⢿⠚⠟⠚⠟⠀⡼⠀⢀⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⣠⠜⠀⠀⢀⠎⠀⠀⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⢠⠏⢀⡤⠄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢀⡶⠒⠒⠘⣆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡆⠀⠘⠀⠈⠀⣀⡴⠀⢀⠞
⠀⠀⠀⠀⠀⠀⢺⠀⢰⠓⠀⣹⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢻⡀⠀⠀⠒⠚⢉⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣈⠀⠒⠋⠀⢀⣠⠞⠁⠀⢀⠀⠀⠀⠀⠙⠒⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠉⠉⠀⠀⠀⠀⠈⠳⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡘⡆⠈⠻⡇⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢊⣰⣿⣿⠏⠀⠀⠀⠀⢀⣠⣄⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⡿⠁⠀⠀⠀⢀⣴⡿⠟⠛⠳⣄⠀⢳⡀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⣾⡿⠐⣷⢦⡀⡇
⠀⠀⠀⠀⡠⠘⢷⣿⣿⠏⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠈⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠇⠀⢹⠃⠀⢿⠃⠀⣧
⠀⠀⠀⠀⠃⣀⣿⣿⠏⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⠔⠀⠀⠀⠀⠃
⠀⠀⠀⠟⣴⣽⣿⡟⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠛
⢀⠰⠀⢰⣾⠏⣿⠃⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⡐⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡀⠀⠀⠀⢠⠀⢀⠃⠀⠀⢀⠏⠁
⠘⡀⣱⣿⠏⠀⢻⠀⠀⠀⣼⣿⠁⠀⠀⠀⢀⡀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠠⢃⣤⠀⣠⡸⡆⠈⠀⠀⠀⢨
⢠⢹⣿⡏⠀⠀⠀⠀⠀⢰⣿⡏⢠⠀⠑⢷⣦⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⠠⠬⢀⣉⡁⠀⠀⠀⠀⣠⠃
⢘⢿⣿⠀⠀⠀⠀⠀⠀⠸⠟⠓⠛⠛⣦⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇⠀⠁⠀⠀⠀⠀⠀⠈⠀⣀⢀⠔⠁
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⡀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠳⡄⠄⠀⠀⡁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠒⠂⠁
⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠑⢄⠀⠀⠀⢀⠔⠑⢤⡀⠀⠀⠀⠀⢀⣦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⠀⠀⠀⠀⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣌⣉⣉⣉⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢸⡁⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠛⠛⠻⠛⠇⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⢱⡇
⠀⠀⠀⠀⠠⠂⢠⡖⠒⠲⡄⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠃⠐⡠⢃⡄⠀⠀⠀⠀⠰⠪⢿⣄⣀⣴⠟
⠀⠀⠀⠀⠠⡀⢀⠻⠻⢟⠃⣀⠄⠀⠀⠀⣧⣀⣀⣀⣠⣄⣀⣀⣀⣠⡇⠀⠀⠀⠛⠛⢏⠔⠩⠬⠤⡄⠀⠀⢀⣤⣤⣤⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣶⣶⣶⣀⣀⣨⣶⣶⡆⠀⠀⠀⠀⠁⠀⣤⣶⣶⣦⡀⠂⠀⠀⠀⢀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠉⢻⣄
⠀⠀⠀⠀⠩⡭⡭⡭⡭⠭⠭⠭⠁⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠘⠟⢂⣠⣤⣶⣦⣐⠚⠟
⠀⠀⠀⠀⢀⣃⣓⡓⣓⣒⣒⣒⠀⠀⠀⠀⠀⢀⠈⢻⢸⡟⡼⠃⠀⠀⠀⠀⠀⢣⡀⠀⡐⠀⢀⡠⠀⠀⠀⠀⡏⢌⡟⠛⠛⠛⣿⠩⡃
⠀⠀⠀⠀⠠⠆⠦⠆⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠷⠷⠇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠔⠃⠀⠀⠀⠀⣛⣚⠛⣛⣛⣛⢛⣓⡳
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠛⠗⠀⠠⠀⠀⠀⠀⠀⠀⡠⠀⡴⢁⠀⠲⠄⢤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⣾⠖⠚⠛⠓⢾⣶⡆
⠀⠀⠀⠀⠀⢁⠀⠠⠠⠄⢀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⢢⡀⠀⢀⠏⠀⠀⠀⠀⠀⠲⣶⣶⠆⠀⠀⠀⠀⠀⢨⠃⠀⠀⣀⡤⠂⢹⡀
⠀⠀⠀⠀⠀⣀⣤⣀⣀⣠⣆⠀⠀⠀⠀⠀⢠⡜⠓⣧⡙⠛⡡⠔⠋⣆⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠒⠛⠛⠛⠛⠓⢐⡄⠀⠀⠀⠀⠙⠒⢻⡉⠉⠓⠃⠜⠁⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢘⡦⢤⣤⠤⢾⠁
⠀⠀⠀⠀⠀⠃⠙⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣶⠿⠛⠛⠿⣷⣄⠀⠀⠀⢰⠿⠟⠃⠀⠙⠿⢷⡆
⠀⠀⠀⠠⠤⣶⣶⣿⣿⣿⣿⣶⣶⠤⠄⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡙⢿⣵⢆⡀⠀⠀⠘⣿⡆⠀⠀⢸⠀⣤⣤⣤⡄⠀⢸⡇
⠀⠀⠀⢀⠀⣿⣶⣮⣭⣭⣵⣶⣿⠀⠀⠀⠀⣿⡇⠐⠂⣃⡀⣠⡄⠀⠀⠀⠡⠐⢝⢿⣵⣆⡀⢀⣿⠇⠀⠀⢸⠀⠶⠶⠶⠂⠶⢸⡇
⠀⠀⠀⣸⠀⠻⠿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠿⠧⠿⢧⠀⠀⠀⠀⠈⠢⡁⢝⢿⣯⡾⠋⠀⠀⠀⢸⠀⣏⣉⣉⡁⣋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⠙⠟⠁⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠼⠃