Heatwave Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠞⠛⠛⠛⠛⠛⠓⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⣿⡿⠋⢀⣤⣾⣿⣿⣿⠿⠛⢋⣉⣉⣭⠭⠍⠉⠛⠻⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⡿⠋⢀⣴⣿⣿⡿⠟⣉⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⡄⠸⣿⡟⠀⣠⣿⣿⡿⠋⣠⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⣿⡀⠉⠀⣰⣿⣿⠟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⣿⡄⢸⣷⠀⢰⣿⣿⡏⠀⠟⠛⠉⠉⠉⠀⠉⠉⠉⠙⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢠⡹⣷⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣿⠁⣼⡇⠀⣿⣿⡟⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣷⠙⠆⡄⠀⣼⠀⠀⠀⢀⣿⡄
⠀⠀⠀⢸⣿⠀⣿⡇⢰⣿⣿⡇⠀⣾⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢀⣼⣿⣿⠀⢸⣿⣄⢿⡆⠀⡀⢺⣿⠏
⠀⠀⠀⣿⣿⠀⣿⡇⢸⣿⣿⡇⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣼⢁⣴⣿⣿⣿⣿⠀⢸⣿⣿⣾⣷⣠⣷⠘⠃
⠀⠀⠀⢿⣿⠀⣿⡇⠈⣿⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⡇⢀⠀⣆
⠀⠀⠀⢸⣿⡄⢹⣧⠀⢿⣿⣷⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⡴⠈⣁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣧⣾⣾⣿
⠀⠀⠀⠘⣿⣇⠘⣿⡄⡘⣿⣿⡆⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢸⡌⣡⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠉⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⡄⢹⣿⡀⠘⣿⣿⡔⠹⣿⣧⡠⠶⠶⠶⠶⢛⣛⣩⣤⠶⠊⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠻⠿⠿⠛⠁⠀⠀⢀⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠈⣿⣷⡀⢻⣷⡄⠘⣿⣿⣆⠙⢿⣷⣔⠚⠛⠛⠋⠉⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣩⣤⣤⣦
⠀⠀⠀⠀⠀⠘⣿⣷⡀⠹⣿⣆⠈⠻⣿⣧⡀⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢀⡀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠤⠤⠤⠤⠄⠀⢀⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠘⢿⣷⣄⠙⢿⣿⣦⣄⠙⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⡛⠿⣷⣭⣛⡲⢤⡀⠀⠈⠙⢷⣤⣄⣀⠀⠀⢀⣠⣶⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡉⠻⣷⣤⡈⠻⢿⣷⣦⣀⠉⠛⠿⢷⣶⣤⣤⣀⣀⡻⠷⠶⠬⣭⣿⣛⢒⣒⣒⣂⣤⠤⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣄⡉⠛⠷⣤⣈⠛⠻⢿⣶⣦⣤⣀⣈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢉⣀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣶⣤⣬⣉⣁⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⣁⣀⣠⣤⣴⣶⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠁
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⡌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⢛⣉⡻⣿⠟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⢸⣿⣿⣿⡿⢸⣿⣿⣡⡆⣮⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢇⣬⣙⡛⠿⠿⠿⠟⣛⣩⣴⣿⣿⢰⣿⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⣰⡜⣿⣿⣿⣿⣿⣷⣬⣛⠻⠿⠿⢋⣾⣿⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⢸⡕⠿⠟⣸⣿⣿⣿⣿⣿⣿⣿⡇⢾⣿⣦⢹⣿⣿⡿⢋⣼⣿⣧⡻⢧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠿⢏⣼⣿⡟⣰⣿⢟⣛⡻⣿⣎⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⢿⣿⡸⠿⢃⣿⡿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣎⡻⢿⣿⡿⢟⣱⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⣛⣉⠁⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⢟⣡⣾⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣍⣛⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢠⡄⠀⠻⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠋⠉⠁⠀⠰⢶⡄⣠⡶⠟⠛⠛⠷⣦⡀⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣯⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⢰⡟⣴⠟⠀⠀⠀⠘⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠉⠉⠀⠈⠙⣷⠀⠀⠀⠈⠋⢸⣇⠀⠀⠀⠀⠀⢀⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⢠⣴⠖⠹⢷⣤⣀⣠⣴⠟⠁⠷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⣠⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠈⣩⡍⠀⠿
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠙⢷⣄⠀⠀⠀⠹⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢀⣀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠻⢶⡄⠀⠀⠉⠙⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠙⠻⠾⠟⢹⡇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠹⢷⣤⡀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠁⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⡏⠁⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢘⣷⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⢀⣤⣴⠾⠟⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⠇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣀⣼⠏⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣀⣀⣀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠉⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣴⠟⠛⠉⠉⠉⠙⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⠀⠀⠀⠘⣷⠀⣀⣀⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

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



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

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

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢀⡴⠋⠉⠁⠈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠉⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠈⠉⠳⢤⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠉⠉⠙⢦⡀
⠀⢠⡏⠀⠀⢂⡀⢀⡀⠀⠀⢻⡀⠀⠀⠀⠀⡼⠁⠀⠀⢂⣂⠀⠀⠀⠘⡆⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⠀⢀⠀⠈⢧⠀⠀⠀⠀⢀⡞⠀⠀⣀⠀⣘⣿⠃⠀⢹⡄
⠀⢸⡁⠀⡁⠞⢠⣄⠑⡁⠀⢈⡇⠀⠀⠀⠀⡇⠀⠀⣼⠇⠀⠹⠆⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢈⢸⣿⠀⢒⠀⠀⢸⠀⠀⠀⠀⢸⡁⠀⣾⠏⠀⢈⣿⡀⠀⢈⡇
⠀⠘⣧⠀⠀⠄⠀⠉⠀⠀⠀⡼⠁⠀⠀⠀⠀⢳⡀⠀⢉⠐⢿⣆⠄⠀⣠⠇⠀⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢧⠀⢉⠐⠿⣎⡉⠀⠀⣼⠃
⠀⠀⠈⠳⣤⣀⡀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⠀⣀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠘⠶⣄⣀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⢁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀
⠀⠀⣠⠏⠁⢀⡀⢀⡀⠈⠳⣄⠀⠀⠀⠀⠀⢠⡞⠉⠀⢠⣤⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣶⡄⠀⠉⢳⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄
⠀⢰⠇⠀⣴⡟⠀⠀⠹⣦⠀⢸⡆⠀⠀⠀⠀⡏⠀⠀⠀⢸⣿⠀⠀⠀⠈⣇⠀⠀⠀⠀⣼⠁⠀⠀⣾⣿⣿⠀⠀⠀⢹⠀⠀⠀⠀⢠⡇⠀⢠⣤⠀⠀⣰⡆⠀⠸⡆
⠀⢸⡄⠀⢟⠀⢶⡆⠀⠘⠀⢰⡇⠀⠀⠀⠀⡇⠀⠀⠀⣼⣿⣄⠀⠀⠀⡿⠀⠀⠀⠀⢿⠀⠀⠀⣏⣹⣙⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⢸⣿⣤⣾⣿⡇⠀⢠⡇
⠀⠀⢳⡀⠈⠑⠠⠀⠛⠀⢀⡞⠀⠀⠀⠀⠀⠹⣄⠀⠈⢿⣿⠟⠀⢀⡼⠃⠀⠀⠀⠀⠘⢧⡀⠀⢿⣿⣿⠀⠀⣠⠏⠀⠀⠀⠀⠀⢳⡄⠈⠋⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠙⠲⠤⣤⣤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣤⠤⠞⠋


⠀⠀⠀⣠⠴⠒⠛⠛⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠛⠛⠒⠦⣄
⠀⠀⡼⠁⣀⣀⣀⣀⣀⣀⠘⢧⠀⠀⠀⠀⠀⣰⠋⠀⢠⡖⠀⠲⡀⠈⢳⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⣰⡆⠀⠀⠙⣆⠀⠀⠀⠀⠀⡴⠃⠀⡆⠀⠐⠀⠀⠈⢧
⠀⢸⠃⠀⢿⣿⣿⣿⡿⠿⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠈⣀⣷⡆⠁⠀⠀⣷⠀⠀⠀⠀⣾⠀⠀⠁⠸⢛⠿⠠⡄⠀⢸⡄⠀⠀⠀⢸⡇⠀⣠⡶⠈⠱⣴⡆⠀⢸⡇
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣧⠀⠀⠀⠸⡿⠁⠀⠀⢀⡏⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠸⣇⠀⣿⠧⣶⣤⣼⣷⠀⣸⡇
⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠳⣄⠐⠛⠛⠓⠀⣀⡞⠃⠀⠀⠀⠀⠀⠹⣦⣈⠳⠾⠟⠛⢛⣴⠟
⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⡴⠚⠉⠁⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠀⠉⠙⠶⡄⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠈⠉⠛⢦⡀
⠀⢠⡟⠀⢀⣾⣿⣿⣤⣄⠀⢳⡀⠀⠀⠀⠀⡼⠁⠀⠀⣀⣀⡠⠀⠀⠙⡆⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢰⣿⡆⠈⢧⠀⠀⠀⠀⢀⡞⠀⢀⣠⠄⣤⣾⣀⠀⢻⡄
⠀⢸⡅⠀⣿⣿⣿⣿⣿⣿⠀⢨⡇⠀⠀⠀⠀⡇⠀⢀⠰⣿⣿⣿⢈⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⠋⠉⠀⠀⢸⠀⠀⠀⠀⢸⡁⠀⣾⠃⣀⡟⠉⠿⠀⠈⡇
⠀⠘⣇⠀⠙⠛⣿⣿⡿⠀⠀⣼⠁⠀⠀⠀⠀⢳⡀⠀⠠⠉⠋⠡⠀⠀⣠⠇⠀⠀⠀⠀⠸⡄⠀⠙⠯⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⣧⠀⠡⢈⠛⣧⠄⠀⠀⣸⠃
⠀⠀⠈⠳⣄⣀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⡀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⢀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⢠⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⣀⢀⣴⠂⠈⢲⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠲⣤⣿⠃
⠀⠀⡀⠘⠃⢠⣤⡄⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⠀⠉⠤⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢠⣿⣄
⠀⠀⠀⢀⠀⠀⠙⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⢴⣶⣦⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⢰⣶⣦⡜⢛⠁
⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⡉⠻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣉⣹⠇⠊


⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶
⠀⠀⣰⣿⠇⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⣴⡄
⠀⢸⣿⠋⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣠⣿⡁
⠀⠸⡌⠁⠸⣿⡃⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢾⢀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣼⣿⣿⣿⣇
⠀⠀⠘⠢⢄⣀⡀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⣀⡀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢉⣾⣯⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆⠡⢰
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠨⠁⠰⣶⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡾⠿⣿⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⠑⢦⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠀⣶⣶⣦⣰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠉⠷⠙⠿⠋


⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣾⣿⣿⡟⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢠⣤⣿⣧⡀
⠀⢰⣶⣬⣛⣿⣿⡿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠋⠉⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢸⡟⠞⡛⣿
⠀⠘⣿⣿⣿⡟⣿⣿⣶⡉⠃⠀⠀⠀⠀⠀⠀⠐⠀⠻⣿⣿⡿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠓⠠⣶⣦⡇⠀⠇⠁
⠀⠀⠈⠉⠉⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡈⢹⠧⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡿⢻⡻⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⢸⣿⣿⡄⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠛⠛⠛⠃⠀⠼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣄⣀⠀⠘⠋⣁⣴⣶⣾⣿⣿⣷⣶⣦⣀⠙⠁⢀⣠⣤⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⣤⣴⣶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠟⠛⠉⠀⣄⠉⠻⠿⢿⣿⣿⡿⠿⠛⠉⣠⡄⠀⠉⠛⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡶⠀⢠⣤⣤⣤⠀⠰⣿⣿⣿⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠘⣿⣿⡏⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⢄⣰⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣆⠀⢀⣤⡒⠉⠉⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⡀⠀⠸⣦⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢧⣀⡞⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁
⠀⠀⠀⠀⢣⠀⣠⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⡀⠀⠀⠀⠙⢦⡀⣿⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⡠
⠀⠀⠀⠀⠈⢷⣇⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⣇⠀⠀⠀⠀⢈⣿⠇⠀⠀⠀⠀⢀⣿⠤⠤⠤⠲⠯⠄
⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣈⣿⣆⠀⠀⣤⡾⠋⠀⠀⢀⣠⡴⠟⠁
⠀⡀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠋⠉⠉⠉⠉⠉⠻⣿⣧⣼⣟⣠⣶⣾⠿⠿⠿⠷⢦⣤⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄
⠀⠈⠒⠒⠒⠒⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⡷⠶⠶⠶⠞⠛⠣⢄
⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠖⠀⠀⠀⡄⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣶⠛⠛⠉⠁⠀⣿⠃⠈⠉⠛⠻⠿⢿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠀⠀⠇⠀⠀⠀⠀⠀⠘⢿⡀⠉⠀⠀⠀⠀⠈⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠉⠉⠁⠀⠉⠉⠉⢙⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣷⠀⣿⠛⠛⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⢿⡄⠀⢸⡇⠀⢰⡟⠀⠀⢀⣴⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠿⠟⠛⠛⠻⠿⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠠⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠀⣿⠀⠀⠀⠀⣿⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠈⠉⠋⠀⠀⠀⣰⡟⠀⠀⣀⣠⣿⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢻⣷⢶⣤⣤⡶⣾⡛⠉⠻⣦⡀⠀⠀⠀⣼⠏⠀⢠⡾⠋⠉⠉⠉⠛⢷⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠾⠃⠀⢸⡇⠀⠘⠷⠀⠀⠈⠻⠆⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⠻⣦⣤⣀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠉⠉⠉⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠄⠀⣀⣠⣤⣀⡀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⢸⣿⠃⠀⠀⠀⠀⠈⣿⣧⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⢸⣿⣄⠀⠀⠀⠀⢠⣿⡏⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣷⣦⣴⣶⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠈⢉⡉⠁⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⣠⣶⣷⣦⡀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⣀⣴⣿⠿⠋⠉⠻⢿⣷⣄⠀⠈⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠁⢀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠉⠻⣿⣶⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⣠⣶⣿⠟⠋⠀⠀⣠⣶⣾⣿⣶⣦⡀⠀⠈⠛⢿⣷⣤⡀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⢸⣿⣏⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⣸⣿⠋⠀⠀⠈⢿⣿⠀⠀⠀⠀⠉⠻⣿⣶⣄⡀⢈⣿⣿
⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠟⠁⠀⢠⣤⡀⠀⢀⣤⡄⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣰⣿⠟⠀⣠⣿⡟⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠻⣿⣄⠀⠹⣿⣆⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣽⣿⠂⠀⣽⣿⠃⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢾⣿⡁⠀⢾⣿⡃⠀⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢿⣷⡄⠈⢿⣷⡄⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣰⣿⠟⠀⣠⣿⠟⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠹⣿⣆⠀⠹⣿⣦⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠘⠟⠀⠀⠘⠟⠃⠀⠘⠿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠢⠀⠠⠤⣄⡀⠀⠀⢀⣀⠀⠀⠀⣄⣀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⢅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡂
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠒⠢⢄⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠰
⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣨⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣦⠀⠀⣀⣛⣃⡀⠀⢀⣴⡷
⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⡿⠟⠛⠛⠿⣷⣌⠋
⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷⠀⠀⠀⢀⣀⣀⠀⣾⡟⠀⠀⠀⠀⠀⠘⣿⡆⢀⣀⣀
⠀⠀⠀⣿⠇⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⢸⣿⠀⠀⠀⠘⠛⠛⠃⣿⣇⠀⠀⠀⠀⠀⠀⣿⡇⠛⠛⠛
⠀⠀⠀⢀⠀⢸⣿⡋⠉⠉⠉⣿⡇⠀⠀⣿⡏⠉⠉⠉⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⣀⣀⣠⣾⠟
⠀⠀⠀⣿⡇⢸⣿⡀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⢀⣴⡿⠀⠙⠛⢛⠛⠛⠁⠸⣿⣆
⠀⠀⠀⣿⡇⠘⠿⠿⠿⠿⠿⠿⠇⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣿⡇⢸⣿⠟⠛⠛⠛⣿⡇⠀⠀⣿⡟⠛⠛⠛⢿⣿⠀⣸⣿
⠀⠀⠀⣿⡇⢸⣿⠂⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⢸⣿⣰⡿⢿⣦
⠀⠀⠀⣿⡇⠸⣿⣷⣶⣶⣶⣿⡇⠀⠀⢿⣷⣶⣶⣶⣾⣿⡿⠁⠈⢿⣧
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⣿⡇
⠀⠀⠀⣿⡇⢰⣾⡿⠿⠿⢿⣿⡆⠀⠀⣾⣿⠿⠿⠿⣿⣿⣄⡀⢀⣠⣿⠇⠀⠀⢰⣶⠀⠀⠀⠀⢀⣶⡆⠀⠀⠀⠀⣶⡆
⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⢸⣿⠛⢿⣿⠛⠁⠀⠀⢠⣿⠏⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⣿⡇⢸⣿⣦⣤⣤⣤⣿⡇⠀⠀⣿⣧⣤⣤⣤⣼⣿⠀⢸⣿⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⣿⡇⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠛⠛⢻⣿⠀⢸⣿⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢨⣿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣾⡟
⠀⠀⠀⣿⡇⢸⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁
⢀⣤⣶⣿⣷⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠙⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⣴⣶⣶⣶⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⡀⢀⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠠⡰⠧⠌⠂⣢⡀⢐⣄⠈⢀⠀⠀⠀⠆⠀⠀⢰⠀⠀⠀⠀⠆⠄⠂⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠂⠀⠄⠀⢰⠒⠀⡼⡁⠀⠀⢹⠀⠸⠈⠶⣤⢨⢆⢸⡀⣲⡄⠐⣤⠀⢢⡀⢠⡀
⠀⠀⠀⠀⠀⠂⠀⠀⡂⡀⠀⠈⢀⠀⡇⠀⠀⠀⠀⡇⠀⠈⠠⢡⠀⠀⡞⠹⠈⠀⠰⠼⠷⠚⢷⠶⣷⣷⣶⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⡐⢚⢻⠀⡷⣠⣨⠀⠀⢯⠀⠀⢄⢨⡄⡀⣤⢐⣘⣛⡀⣃⣂⣘⢘⢀⣉⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡅⣱⣸⣈⡄⣧⣰⢨⡀⠐⠾⠀⠀⡘⠠⠇⠣⢥⠈⠸⠓⠀⢻⠃⠘⠨⠈⡂⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠚⣉⡁⠻⠈⠃⢃⠀⠻⠆⠀⠂⠉⠫⠠⢼⡰⠀⣆⡔⢢⣰⠸⡀⠂⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⠖⠒⣄⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⢸⠀⣿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⢸⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠄⢀⣀⣉⣁⣀⠀⠀⠀⠀⣿⡇⢸⣀⣀⣸⠀⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠲⢦⣄⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⢸⣿⣿⣿⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⢸⣿⣿⣿⠀⣿⣅⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⢸⣿⣿⣿⠀⣿⣁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⢸⣿⣿⣿⠀⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⢸⣿⣿⣿⠀⣿⡋⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⢠⣾⠟⢁⣼⣿⣿⣿⣄⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⠉⠉⣉⡉⠉⠀⢠⣿⠁⣴⣿⣿⣿⣿⣿⣿⣷⡄⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢿⠁⠀⠀⢸⣏⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⠃⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⠻⠿⠿⠿⠛⣁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⡶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⢼⡀⠀⢀⡠⣄⠀⢀⡷⠤⣀
⠀⠀⠀⠀⠀⣠⠊⠀⠀⢗⠊⠉⠀⠀⠉⡹⠃⠀⠈⠑⢄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⢆⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⡎⠀⠀⢦⠀⠀⠈⠳⠤⠴⠃⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢸⠁⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡀
⠀⠀⠀⠘⣄⣀⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⣣
⠀⠀⠀⠀⣷⠈⡏⠉⢹⡑⠦⡀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠒⠋⠹⡀
⠀⠀⠀⠀⢹⠉⢣⠀⠀⢳⠤⠤⠒⠒⠒⠒⠒⠒⠉⠀⠈⢧⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⢸⠀⠈⢧⠀⠸⡀⢀⠾⠽⢦⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢈⡇⢧
⠀⠀⠀⠀⠀⡇⠀⠈⢧⠀⣇⢸⣇⡠⠖⠛⠉⠓⠒⠊⠉⢹⠀⠀⡜⠺⠛⢿⡿⠀⠉⠓⠢⠤⣀⠀⠀⠉⠓⠒⠒⠋⠉⠙⣾⣆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢘⡼⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠉⠉⠉⠉⠉⠉⠉⠁⠉⠙
⠀⠀⠀⠀⣀⢽⠟⠛⠉⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⢠⠎⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠔⠋⠀⠈⣇⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡠⠔⠃⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣠⡤⠤⠤⠴⠒⠋
⢀⠏⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⡴⠁⠀⠀⢎⡴⡆⠀⠀⠀⡗⠒⠒⠲⠤⢤⣤⡤⠤⠤⠤⠤⠤⠒⠉
⣸⠀⠀⠀⠀⠀⠀⠀⠈⢉⣟⣟⣒⡲⠤⠤⣀⣰⢁⢆⣤⡴⠁⠀⠀⠀⠀⠀⠀⠉⠑⢦
⢇⣀⠀⣀⣀⣀⣀⣀⠤⠋⠁⠀⠀⠈⠙⠦⠖⠋⠛⠚⠋⠑⠒⠒⠒⠦⠤⣄⣀⠴⠒⣼
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠟⠟⠃⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⠀⠀⠀⠀⠀⡈⣋⠈⠙⠃⠀⠀⢀⠀⣀⡀⠀⠀⢀⣀⡀
⠀⠀⣠⠾⢶⠀⠀⠀⠀⠀⠸⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⣤⡠⣤⠙⣴⠀⠀⠺⡩⠘⠆⠀⠀⠄
⠀⠀⣦⠀⠹⣤⠀⠀⡀⣧⣄⣼⡆⠀⠀⠀⠀⠀⠀⠀⡀⣣⡼⠋⢉⣷⣿⣼⣿⣿⠆⠀⠺⢲⣿⣤
⠀⠚⠇⣀⡘⡷⣴⢞⣵⣿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠶⣿⣿⣿⣿⣿⣿⣦⣠⡆⣿⣿⣿⣧⠈
⠀⠘⢀⣼⠳⢳⢾⠞⣿⣿⣷⡎⠀⠠⠄⠀⠀⣀⣠⠾⠀⢀⠀⣼⣿⣿⣟⣹⣿⣧⠀⣹⣷⣿⣿⣿⣿⡀
⠀⠘⠻⠏⠀⢨⠘⢰⣿⣿⣿⣇⠀⠀⠀⢀⠠⠛⠁⠀⠀⢤⣮⣼⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⡟⠃
⠀⠀⢀⣠⢂⢺⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢸⡇⢸⣾⡿⣿⣿⣿⣿⡆⣄⠀⠀⠀⠀⠀⠈⠀⠀⢻⡛⠉⢹⣿⣏⢸⣿⠟⢹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢾⢷⠨⠁⠀⠉⠻⠿⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠸⣿⣿⣶⡿⠀⡘⣿⠃⢹⣯⢹⡿⣧
⠀⠀⣼⠏⢀⢀⠀⠀⣄⠘⣻⡿⠃⠀⠀⠀⠀⠀⠀⢀⢠⢘⣣⠄⠀⣼⡾⢻⣿⣶⡿⢿⠀⠀⣏⢨⡆⢘⣆⡆
⠀⠀⠀⠀⠈⠙⢷⢌⡻⢺⠛⠑⠛⠉⠁⠘⢋⠿⠽⠬⠣⠞⢁⡠⠶⠿⠀⠙⠻⠋⢠⣿⢋⣼⡿⠛⠁⠈⠙⠃
⠀⠀⠀⠀⠀⠀⢉⣀⣐⣮⡀⣴⣶⣶⣤⡀⢀⠀⠘⠿⣿⣿⡿⠿⣦⡀⢀⡤⠤⠶⠚⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⢷⣎⣬⣛⣳⣿⣧⣄⣈⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⣦⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⣶⡀⢸⡇⠀⠈⠁⠀⠰⠿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣾⣿⣶⣤⡀⠀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡇⠀⠾⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠻⠿⠛⠋⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣠⡄⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⡇⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢀⣿⣿⡇⠸⣧⡀⠀⢀⡀⢀⣀⣤⣤⣬⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣾⣿⣿⣷⣆⠙⣿⡀⠀⠙⠛⠛⠛⠁⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠲⢶⣶⣶⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⠿⣿⣿⠿⠋⣰⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣥⡀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠘⣧⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠙⢦⡀⠈⠓⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣠⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣤⡀⠀⠈⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣦⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠻⣿⣿⣿⡆⢀⣴⣿⣿⣿⣿⣿⡿⢟⣥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣬⣍⣘⣻⣿⣿⣿⣿⠿⠋⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠀⠀⠀⠀⠉⠙⠛⠻⠟⠋⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢐⣏⣴⠛⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣴⣦⣸⣇⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣧⣤⣀⣉⣘⠃⠈⠉⠙⠛⠂⠒⠀⠂⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢛⡛⠛⠛⡏⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⡷⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⡂⠀⠀⠀⠀⠀⠀⢠⠀⠰⡇⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠢⠄⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢤⣤⣿⣿⣷⠀⢀⠀⠀⡐⢀⣨⡀⢲⣿⣠⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⠀⢰⢠⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠘⡇⠀⠄⠀⠀⠀⠀⠀⠀⠰⣣⣾⣶⣿⣿⣿⣿⣿⣿⡇⢈⠀⣰⡇⠀⣼⣿⣿⣿⣿⣹⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠀⠀⢀⣾⣠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣾⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⢸⡀⠸⠼⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠚⠳⠾⠟⠉⠁⠀⠉⠛⠿⠋⠀⠀⠀⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⡧⠴⣾⣶⣴⡀⠀⢸⠁⣹⣓⠂⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⡀⠴⠂⢘⣿⣿⣿⠀⡀⠛⣿⣿⣿⠁⠀⢀⣽⣿⣿⡿⠂⣽⠀⣿⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣤⣤⡄⠄⠈⣇⣐⣶⣶⣴⠋⣄⠀⠀⠀⠀⣀⢀⣉⣷⣶⣌⣿⣿⣧⡶⢁⡼⠙⣿⣏⡀⣀⣾⣿⣿⣿⡀⡀⣾⣶⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢻⣷⣬⣵⣦⡀⠀⣉⣩⣿⣿⣿⣿⣿⠏⣿⢃⣠⠶⡟⡟⠊⢁⠀⠈⠙⠉⠐⠛⠘⣧⣿⣿⣿⡇⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡍⠋⣹⡶⢰⣦⠐⠀⣅⢀⠀⠀⠆⢸⠀⣇⡼⡹⠻⣧⠀⠛⠋⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣇⠀⢸⣿⣇⠀⣙⡇⢀⣈⣶⣿⣶⣽⠛⠃⢞⣥⡘⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⣿⠿⠿⠛⠛⠷⠿⠿⠿⠿⠿⠟⢵⢾⣾⣯⣽⣿⣿⣿⣿
⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⢝⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣸⣿⠷⠶⠷⠶⢾⠿⣷⣐⣠⣿⣷⣾⣾⣿⣿⣬⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⢠⣾⠇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⠠⠀⠶⣴⠾⠶⠾⠿⠿⠿⣿⣿⣿⣱⣿⢿⠿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⣿⣧⣴⣿⣿⠀⣽⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⢢⠀⠈⢳⣔⣦⠱⡄⠀⠇⠀⠠⠀⠈⡙⠀⢿⣿⠇
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠀⠈⢳⣄⠀⠘⠻⣦⣹⣦⠀⠀⠀⠀⠒⠀⠀⢸⡟
⠀⠀⠀⠀⠋⠻⠟⢋⠛⢻⣿⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄⠀⠀⠹⣷⡀⢀⡉⠻⣽⣧⣄⣀⣀⣀⣀⣀⣈⣿⡆
⠀⠀⠀⠈⠀⠀⠈⡁⠤⠄⣠⠦⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣷⣿⣿⣦⣈⡛⠿⣿⣇⠉⢉⠻⣿⣟⣻⣿⣻⣿⣿⣿⡄
⠀⠀⠀⠀⠄⠀⠒⠀⠐⢀⠠⣄⣤⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣋⣩⡾⢿⡿⠛⠷⣭⣭⣌⣁⠾⢿⣿⣿⣿⣿⣾⣿⠉
⠀⠀⠀⠀⠠⢤⠀⠀⠤⠌⠀⣤⠈⢉⣁⡐⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢭⡿⠛⢓⠛⠂⣀⡾⢛⡒⠒⡂⣀⠀⢀⡀⠒⠂⠀⠈⠉⠀⢠
⠀⠀⠀⢸⣷⣦⠐⠀⠀⠐⠶⠄⠀⡬⢉⣠⣾⡿⠉⢉⣙⠛⠛⢉⡝⠛⢋⣤⡚⠀⠒⡒⠒⠀⣀⡼⠋⠄⠈⠄⠀⠈⠁⠤⠬⠁⠀⠌⠁⠀⠉⡰⠃
⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⡆⢀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⡀
⠀⠀⠀⠈⡇⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⡏⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣇⢠⡇
⠀⠀⠀⠀⢻⡄⢹⡄⢻⡄⠀⠀⠀⠀⠀⠹⣆⠹⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⢸⠃⣼⠃⡷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⣷
⠀⠀⠀⠀⠈⣇⠀⣧⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⢠⡏
⠀⠀⠀⠀⠀⡏⠀⠟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢳⠀⠇



⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⡞⠀⡏⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢳⡠⡏⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼
⠀⠀⠀⠀⢷⡀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠈⢷⠈⢷⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⢯⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡄⠻
⠀⠀⠀⠀⠀⢳⠈⢷⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧⠘⠇⠀⠀⠀⠀⠀⠀⠹⢀⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⣆
⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡼⠁⣼⠀⢻⣄⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⢰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⡀
⠀⠀⠀⢧⡀⠘⠷⣄⠈⠙⠆⠀⠀⠀⠀⢻⡄⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠰⡜⣆⠳⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡷
⠀⠀⠀⠈⠻⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢳⣌⢷⡙⣆⠇⠀⠀⠀⠀⠀⠀⢸⡀⠙⣆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢠⠄
⠀⠀⠀⠀⠀⢦⠀⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠀⢻⡀
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠻⣦⠙⢷
⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡟⣸⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡞⠁⡼⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠜⠁⠜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠡⡎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡇⠀⢆
⠀⠀⠀⠀⠀⢸⡄⢠⡏⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⢋⡵⠲⣄⠀⠀⠀⠀⠀⠀⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣼
⠀⠀⠀⠀⠀⠘⣇⠸⡇⢰⡏⠀⠀⠀⠀⠀⠀⣠⠶⣄⠈⢳⠈⢯⠒⠃⠀⠀⠀⠀⠀⠀⢰⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠧⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠳⠘⠁⠀⠀⠀⠀⠀⠀⠙⠾⠈⠇⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄
⠀⠀⠀⠀⠀⡏⠀⣸⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡇⢀⡄
⠀⠀⠀⠀⠀⢻⡄⠁⠀⣸⠀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠏⣠⠞
⠀⠀⠀⠀⠀⠀⢹⠄⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⡇⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠉⠉⠢⣄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⢦⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣇⠀⠀⢰⠋⠁⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠓⠒⠒⣖⠒⠒⠒⣞
⠀⠀⠀⢸⠀⠀⠸⣄⡀⠀⠀⠉⢳⠤⠤⠶⠖⠒⠒⠒⢳⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⡖⠒⢄
⠀⠀⠀⠈⡇⠀⠀⢀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣠⠶⠷⠶⠶⠶⠷⠀⠈⢆
⠀⠀⠀⠀⢷⠀⠀⠀⠙⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠙⡤⠤⠤⠤⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠳⡏⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠣⣤⠤⠤⠤⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣀⣀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠳⢤⠤⠤⠄
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⢤⡀
⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣼⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⣿⢀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿⣿⣿⣿⡏⣿⣿⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢸⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⢹⣿⣿⡇⠀⠁⠀⠀⠀⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⣠⠟⣸⣿⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⣸⢣⣾⣿⣿⣿⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⢻⠸⣿⣿⣿⣿⡿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⠈⠳⣍⣛⣛⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⣱⢸⠀⠀⠀⠀⣱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣋⣸

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙⠳⠆⢠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢀⣤⣶⣶⣿⣷⣌⡻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⢛⣛⣛⣩⣭⣭⣭⣥⣬⡉⢭⣭⣭⣤⡰⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣦⡙⢿⣿⣿⣶⣤⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⣠⣬⡙⣛⣛⣋⣵⣶⣝⠿⣿⣿⣿⣿⣦⡻⣿⠟⢁⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠾⠿⠂⠀⠀⠀⠐⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣶⡍⣭⣭⣭⣿⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠛⠿⠿⠋⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⡆⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢶⣶⣶⣶⣶⣶⣶⣶⠶⣶⠀⠀⠀⠸⣿⣶⣌⢛⣛⢿⣿⣿⣿⡟⠈⠛⠋⠀⠛⠒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⣤⡄⢀⣠⣾⣶⣭⣍⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⡻⠿⠿⣛⣩⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠉⠙⠲⠄⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⣴⣶⣾⣷⣝⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⣭⣷⣶⣖⣾⣿⣷⣮⡻⣷⣶⣍⡛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⠀⣠⡟⢛⣛⣫⣮⡻⢿⣿⣿⣎⢿⡟⢣⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠻⢳⣿⣿⣿⣿⣿⣷⣄⢾⣾⣷⢁⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢣⣮⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⢿⣷⣝⣛⠻⣿⣿⠏⠙⠋⠐⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⣝⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣻⣭⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⣠⣶⡦⠀⠀⠠⠄⠠⠤⠠⠀⠤⠠⠄⠀⠄⠆⠀⠀⠄⠀⢠⣦⡠⢀⣤⣀⣀⡀⡀
⠀⠻⣿⡆⠀⠀⠠⠠⠀⠀⠀⠠⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠻⠐⠟⠱⠿⠯⠿⠿⠶⢶⣤⣆⣀
⠀⠀⠈⠀⠀⠈⢰⢲⣠⢦⣀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠆⠈⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⣆⣰⣀⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢛⠈⠁⠁⠁⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠟⠻⠘⠃⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡤⠀⠀⠄⠈⠀⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡶⠩⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠒⠞⠀⠀⠀⠠⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠋⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠗⠒⣀⠀⠀⠀⠀⢀⣾⣄⠄⠀⣀⠀⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢤⡤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠈⠀⠈⠉⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⢀⣼⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣄⡀
⢀⣴⣾⠟⢿⣶⣄⠀⢀⣴⡾⠟⠛⠛⠛⠻⢷⣾⠟⠋⠁⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠉⠻⣷⡿⠋⠁⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢻⣦⣀⣀⣀⣀⢀⣶⡀
⠀⠀⠀⠙⠁⠀⣀⣿⠃⠀⠀⠀⢀⣠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠛⠿⢶⣦⣄⡀⠛⠛⠉⣙⣿⣿⠻⣷⣤⣀
⠀⠀⠀⢠⣶⠿⠛⠛⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠘⠛⠿⠿⠷⣶⣤⡈⠙⢿⣦⡀⠀⠉⠛⢿⣶⣿⣟⠉
⠀⠀⢠⣿⠁⠀⠀⠀⢀⣴⡟⠁⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢿⣆⠀⠀⠈⠿⠉⣿⡄
⠀⠀⣼⣿⠀⠀⠀⢀⣾⠏⠀⠀⠈⢻⠓⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⣿⡇
⢀⣾⠏⠀⠀⠀⠀⣼⡏⠀⠀⠀⠐⠿⠋⠻⠗⠀⠀⠀⠀⠀⣀⣤⡾⢿⣦⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠹⣷
⢸⣿⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠿⠛⠉⠀⠀⠙⣷⣄⡀⠀⠀⠸⣿⠀⠀⠀⠀⢻
⠀⢿⣆⡀⠀⠀⢸⣿⣶⣶⣶⡶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣿⡆⠀⠀⠀⣸
⠀⠀⠙⠿⠂⠀⢀⣿⣃⣿⠃⠀⣀⣀⣀⣠⣀⣀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⡀⠀⢿⣏⣿⡏⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⣸⡿⢻⣿⠀⠀⠛⠋⠉⠉⠉⠛⠀⠀⠀⠀⠙⠋⠉⠉⠉⠛⠃⠀⢸⣿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠁⢸⣿⠀⠀⢠⣤⣄⣀⣤⡄⠀⣠⡄⠀⢀⣤⣄⣀⣤⣦⠀⠀⢸⣿⠀⢻⡇
⠀⠀⠀⠀⠀⢸⣿⣆⠸⣿⠀⠀⠀⠈⠉⠉⠉⠀⠀⣿⠇⠀⠀⠈⠉⠉⠉⠀⠀⠀⣸⡟⢀⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠻⠷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠿⢻⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢸⡿⠀⣀
⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⣴⡿⠁⠀⢀⣿⠇⠀⢻⠗⢺
⠀⡀⠀⢀⠀⠀⠈⢿⣦⡀⠀⠘⢿⣆⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⣠⣾⠟⠀⠀⣠⣾⠏⠀⠀⠾⠃⠻
⠘⢿⠒⠿⠃⠀⠀⠀⠙⠻⠷⣶⣤⣽⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣥⣴⡶⠿⠋⠁
⠰⡿⠓⢿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⠿⣶⣶⡿⠿⠛⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠑⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣭⣷⣶⣶⣶⣿⣭⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⠔⠲⡋⣀⣀⡤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⡿⣫⣭⣽⠿⠿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠎⠀⠀⡇⢧⣀⣠⡼⠤⢄⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⢸⣜⣿⣟⣻⣛⣻⢿⣿⣿⣿
⠀⠀⠀⢣⠀⠀⡎⠙⠢⣅⣀⠤⠔⢣⠀⢰⡭⢿⡹⣆⠘⠦⠀⠀⠀⣿⣿⣿⡿⣿⣿⢱⣾⣝⡾⠿⣟⣻⡼⣿⡏⣒⣒⣆⠹⣷⣹⣿⣿
⠀⠀⠀⠈⢢⡀⢈⠩⣀⠎⠀⠀⠀⢸⠀⠈⡯⢽⠀⡇⠀⡀⠀⠀⠀⣿⣿⣿⣿⣝⢿⣷⣞⠿⣳⣿⣿⣿⣇⣿⣿⢘⡆⣿⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢉⠊⠀⠈⢢⠤⠤⠔⠃⠀⠀⠈⠙⠤⠘⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⣵⣿⣷⣝⣻⣟⣫⣾⣿⣿⣷⣶⣛⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠓⠶⣤⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠉⠓⢦⣐⠋⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⣠⠶⠚⠛⠋⠀⠉⠳⣤⡀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣀⣀⣀⣤⣤⣤⣤⡤⠿⢦⣤⠤⢤⣤⣤⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠀⠀⣦⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠬⣍⣉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⣨⣧⡀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⣩⠗⠀⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢀⡾⠋⠉⢻⠶⠶⠖⠛⠁⠀⠙⠦⣄⡀⠀⠀⠀⠀⠙⢦⡀⢤⡞⠡⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠸⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡓⠒⠒⠛⠈⠻⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⣠⠙⣦
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠙⣿⠁⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡟⣠⣀⣀⡿
⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠈⣧⠀⠀⠀⠛⠚⠛⣦⣀⡀⠀⠀⣰⠏⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠈⠉⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠓⠒⠶⠶⢤⣤⣄⠴⠋⠈⠙⠲⠦⢤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢈⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠶⠛⠉⠁⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⡤⢤⣀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⠀⢀⢀⣀⣀⣀⣀⣀⣀⢀⡠⠤
⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡄⠠⡀⡄⠠⡄⢀⣄⠀⡤⢠⢠⠀⠀⡄⠀⠀⡀⠂⠀⡄⠀⠔⣄⢠⠀⢄⢠⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⣇⠀⠇⡇⠀⡇⠼⠘⠆⣏⠱⢸⡀⠀⡇⠀⠀⢈⡲⠀⠇⠀⠓⠧⠣⡀⡸⢸⠛⠄⣏
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠴⡋⢈⡶⠀⢀⡴⠊⠉⠀⠉⠉⠒⠉⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⡎⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⣀⡀⠀⠹⠴⠲⠤⣤⠶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠁⠀⢀⡴⠊⠁⠀⠀⠀⠐⠒⠤⠤⣄⠈⠓⢄⡀⠀⠘⠣⣀⢾⠃
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡰⠋⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠉⠈⡆
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡸⠁⠀⠀⢠⠀⡅⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠱⣄⠀⠀⠀⢱⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠸⡤⢤⠤⠴⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠑⢦⢄⣘⠆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣃⡜⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠄⠀⠘⣄⣣⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣇⠀⠀⠈⠉⠉⠁⠀⡎⠀⠀⠉⠉⠉⠀⠀⢀⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠑⢺⡀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⠀⣸⠒⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡰⠃⠀⢀⡜⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠈⠑⠢⠤⠬⢦⣄⡀⠀⠀⠀⠀⣀⣤⠮⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢻⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⢿⣧⠀⠀⢠⣾⡟⠛⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠻⠿⠿⠿⣿⣿⠿⠿⠿⢿⣿⡋
⠀⠀⠀⠀⢸⣿⡀⠀⠸⣿⣄⠀⠀⠈⠙⠻⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣧⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠈⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠿⠿⠆⠀⠀⠀⢀⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠙⣿⡆⠀⠀⠀⠀⠀⢈⣿⣇⣀⣀⣀⣹⣿⣁⠙⣿⣆
⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢴⣦⡀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠘⣿⣦
⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠻⣿⣆⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⢻⣷⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⢻⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠘⣿⣧⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⠿⠿⠂
⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠻⣿⣦⡀
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⡜⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣫⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢇⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠰⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢹⣿⣿⡆⣛⣛⣛⣛⣛⣫⣭⣥⣄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⢛⣽⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠸⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣬⣛⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡌⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃


⣿⣛⣿⣻⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣻⣿⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠱⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣖⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣼⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣻⣿⣯⣈⢗⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣦⣾⣽⣶⣿⣿⢧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣽⣭⣹⣿⣿⣿⣿⣿⣿⣿⣶⡿⢁⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢿⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢷⣾⡆
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⠀⠀⢀⡀⢜⣉⡹⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠉
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⡶⢛⠻⢿⣿⣿⣿⣏⡉⠹⣿⣦⣴⣿⣧⡄⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⠁
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣶⣿⡗⠾⢿⣿⣿⠟⢻⣿⣿⣿⣿⠟⠁⠀⠃⠀⠀⠀⠀⢹⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⢏⠤⠔⢲⣶⣤⡀⣈⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣿⣛⣿⣿⣿⣤⡖⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠈⢻⣿⣿⣿⣿⣿⠀⠉⠉⢇⢰⣽⣶⣽⠿⢿⣿⣿⣧⣀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⢹⣿⣿⣿⣿⣾⣿⠹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣠⣾⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⡏
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢶⣼⣿⣿⣿⣿⣿⣿⡷⢸⣿⣿⣿⣿⠿⠿⠗⠰⠾⣿⣿⣿⣿⠿⣿⣆⠘⢿⣿⣿⣿⣷
⠀⠀⠀⠹⠃⣿⣿⣿⠀⠀⠚⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡦⣶⠶⣷⣾⣿⣿⡟⢼⣷⣿⣿⣦⣠⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣶⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠻⣿⣿⣿⣿⠿⢿⣿⣿⣷⣄⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣇⣀⠀⠀⢻⣿⣿⣿⣄⠀⠈⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠙⢿⠟⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⣿⣿⣿⣤⣀⠠⣤⣌⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣤⣌⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣷⣦⣡⣶⡀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣷⣝⣻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠓⠶⠶⠖⠒⠛⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⢀⡤⠤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠒⠒⠒⣶⠐⠖⠀⠁⣰⠋⢰⠖⡆⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡾⠀⠀⠀⢀⡏⠒⠆⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⡴⠃⠀⠀⣠⡼⠑⠲⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⢀⡴⢋⣉⢳⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠻⢥⣤⡤⠤⠾⢧⠀⠀⠀⠀⠀⠀⣿⣄⣀⠉⠀⠀⠀⠸⣇⡼⠁⠿⠀⣸⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⠀⠀⠀⠀⠀⠤⠶⡞⠀⠀⠀⠀⠀⢸⡃⣀⠉⠉⠉⠓⢦⡞⠁⠀⠀⢠⡞⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡿⠁⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⣋⠀⣀⣠⡷⠖⠀⠀⢸⠇⠙⠲⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡤⠶⠒⠒⠒⠋⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⢀⠄⠀⡀⠀⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⠒⠁⢄⠀⠐⠀⠠⠆⠦⠈⢐⠀⡇⠀⠀⠠⣀⠀⠇⠐⠀⡃⠀⠠⢀⠀⠟⠣⠀⠀⠹⠀⠢⡀⠄⠠⡀⠄⠐⠒⠅

⠀⠀⠀⠀⠀⠀⠠⠤⡤⠄⢠⠤⠤⢀⣄⠀⠀⣤⠀⡤⠤⣄⢠⠤⠤⠀⡤⠤⣄⠀⢀⣄⠀⠤⢤⠤⠀⡄⠀⢠⠀⡤⠤⣄⢀⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠒⠒⢰⡟⣆⡜⢹⠀⡧⠤⠞⢸⠖⠒⠀⡷⠴⣎⢀⣾⣸⡆⠀⢸⠀⠀⡇⠀⣸⠀⡷⠴⣎⢸⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠋⠈⠁⠘⠁⠈⠀⠁⠀⠀⠈⠛⠛⠀⠁⠀⠈⠈⠀⠀⠉⠀⠈⠀⠀⠉⠛⠁⠀⠁⠀⠈⠈⠛⠛

⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢸⡇
⠀⠀⠀⠄⠠⣄⠀⠐⠉⠉⠢⠀⠀⢃⠀⢘⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⡄⠀⢸⡧
⠀⠀⠀⠀⢘⡟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⣧⠀⠀⠀⠀⣰⠶⣄⠀⠀⡌⠘⠀⠀⢸⡇
⠠⠂⠐⠖⠠⠤⠶⠶⠶⠴⠧⠼⠼⠬⠤⠴⡄⠹⡀⠀⠋⠁⢳⣶⣿⣷⡋⠀⠎⠈⣉⠉⠉⠙⠛⠋⠙⣛⣛⣛⠛
⡆⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠷⢤⢀⡀⡎⢹⣿⣷⠗⡔⠀⡔⣸⡆⠀⠰⠀⠀⠀⠉⠉⣿⠇
⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠻⡂⣄⣏⢽⣿⣿⠟⢱⡜⠀⠐⠿⢭⢧⠀⠀⠀⠀⠀⣴⣶⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢈⡶⣼⣯⣟⠻⣤⣱⣷⣤⠶⢾⣿⢾⠛⣿⡟⡶⣶⡶⢶⣶⣶⡶⢶⠆
⡇⠀⠀⠤⠤⠤⣤⡴⠋⠁⠀⠀⠀⠀⠀⠸⣫⣙⣹⣿⣌⣏⠻⠫⢌⡃⢸⡗⣈⣍⣉⣿⣁⣼⡁⢋⣄⡍⣁⢸
⡇⣀⣀⠁⠀⠿⡿⠇⣀⣀⣀⣀⣠⣴⣶⣶⣿⣿⣿⣯⣸⠉⢀⢶⣾⡅⢸⡏⠳⠛⠋⠫⠟⣹⡄⢸⢒⢵⡂⢘⡼
⡃⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣿⣿⢿⣿⣿⣿⣿⡿⣿⢿⢀⣀⡈⠉⠀⢸⣃⠀⠔⣳⠀⠀⣬⠃⣈⠛⠋⠁⣠⣿
⠇⠀⠀⠀⠀⠀⠒⠀⢀⣴⣿⣽⣷⣃⣴⡿⣻⠻⣿⡿⠃⠚⢏⣽⢛⣛⠟⢻⠛⡞⣻⡶⠷⠛⠟⢙⣻⣛⣻⠛⠺
⡆⠂⠀⠄⡁⠁⠈⣠⣿⣩⣿⣿⣿⣿⣟⣽⣿⣾⣿⣦⣬⣨⣥⣃⣀⣛⣐⣈⣁⣒⣛⣒⣋⣂⣘⣓⣒⣒⣚⣛⣀
⠁⠀⠀⠀⠀⠀⢴⢟⡽⢻⣯⣳⣾⣯⣿⣿⠾⢻⣿⣿⡏⣿⡝⢙⡋⢩⢹⢈⣈⡉⠀⠀⠈⠡⠈⢉⠉⡉⠉⢹⣿
⡅⠀⠀⠀⠁⢰⡇⢼⣽⣨⡿⣏⣿⣻⣿⡅⠀⢸⣿⣿⣟⣿⣈⣶⣄⣂⣾⠸⢅⡹⠔⢃⣂⠀⠴⢿⣁⣮⠐⢸⣿
⡇⠀⠀⠀⢠⡟⢡⣾⢟⣿⡀⣧⢿⣿⢿⡇⠁⢸⣿⣿⣿⢷⡞⣺⠛⠲⣶⣎⢀⠘⠄⡿⠻⢾⡶⠀⡉⠁⠀⢸⣿
⡇⠀⠀⢀⠟⢨⢣⡿⢸⠋⠁⢹⡎⣿⣿⡿⡄⣸⣿⣿⣋⣷⣙⡛⣁⣺⣞⠁⢈⢀⠈⢿⣶⣞⡥⢀⡅⢐⠐⢸⣿
⡇⠀⢠⠆⠀⣤⡟⢠⢿⠆⣤⢸⣧⡘⣿⠡⢰⣿⣿⣿⡉⡧⠎⡔⢿⣯⣿⡣⢊⢩⣶⣠⣶⢞⣲⣎⣓⡼⡖⣼⣿
⡇⣠⣄⢀⡼⣃⢀⣼⡐⠀⠋⠀⣿⣶⣾⣶⣿⣿⣿⣿⣥⣻⣟⣉⣩⡹⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⢻⡟⡕⣽⣿
⢏⣉⣉⣉⡛⠛⠟⠻⢿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⣭⣁⡏⠌⢲⣧⣏⣀⣀⣀⣀⣐⣀⠀⠀⠈⢿⣷⣿⣿
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡄⠀⢸⠀⠀⣠⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢤⠀⠀⢳⠀⠀⢷⠀⢸⠀⠀⡏⠀⢰⠇⠀⣠⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢧⡀⠈⢧⠀⢸⠀⢸⠀⢰⠇⢀⡏⠀⡴⠁⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠦⣀⠀⠈⢦⡀⠳⡀⡘⡆⡞⣧⣽⢸⣸⢰⣼⣄⡼⡁⣠⠞⠁⢀⡴⠂⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠈⠳⢤⡀⡙⢿⣽⣴⣹⣄⣿⣾⣠⣿⣹⣿⣞⢞⢜⠅⣠⠞⠁⢀⣠⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠦⣀⠽⣺⣷⣽⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⢟⡵⡊⣕⡤⠖⠉⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣐⡪⢽⣺⣽⣿⣿⣿⣿⠿⠛⠿⠿⣿⣿⣿⣿⡿⠏⣓⣪⠤⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⠤⠤⢤⣄⣀⣐⣾⣯⣝⣺⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣯⣕⣒⣶⡤⠤⠤⠤⠴⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠻⢭⣭⣻⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⣶⠋⠉⡓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠵⠚⠭⠀⠀⠀⠀⠀⠈⠙⠻⠿⢤⣀⣀⣀⠀⣠⠴⢚⡿⠟⠋⠁⠀⠄⠀⠀⡠⠉⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠁⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠂⠈⠉⠀⠈⠙⠦⢤⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠨⠥⠤⠤⠤⠤⠤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢛⠁⢀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⣿⡿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠸⡎⣿⣿⡟⡭⠥⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⡀⠀⢠⢖⣘⡿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⠛⠿⣟⠋⠀⠀⣀⣨⡄⠀⢠⣒⣛⡗⡒⡄⠙⢄⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣤⣄⠤⣶⣿⣿⠿⠇⢿⣿⡟⢭⡤⢩⢭⢻⣮⡻⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣤⠂⠈⣱⣾⣿⣿⣿⣷⠀⠈⡯⠭⣿⣿⡙⠄⠈⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠈⢛⣽⣿⠎⠉⠀⠀⠀⢸⣿⣿⢘⣛⣻⡏⣮⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⡇⠑⢾⣿⣿⣿⣿⣿⠇⠀⠀⠹⣭⡿⣿⣇⠠⠤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⠀⣸⣮⡉⠀⠀⠀⠀⠀⣸⣿⣿⣇⣶⣨⠆⠸⣟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢹⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⡆⣤⣤⣤⣶⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿





⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣼⣧⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠘⠳⠂
⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠨⢿⣿⣦⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠛⠙⠀⠀⠈⢾⣷⡄⢀⡀⠀⠀⠀⠀⠀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠎⢀⠀⠀⠀⢰⠀⢈⠐⢳⠂
⠀⠀⠀⠀⠐⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡠⠎⠀⠀⠀⠀⠀⠀⠓⠊⠀⢢
⠀⠀⠀⠀⠀⠀⢄⢨⣇⡓⠀⣀⠀⢀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⡄⡀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠐⠀⠀⢏⠈⡄⡂⡹⢦⢩⣷⢀⠀⠐⡆⠘⡀⢢⣄⢀⣆⣰⡀⣤⣀⠠⣃⠀⢄⡀⡀⠀⠈⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠉⠀⢸⠉⠀⠀⠰⢷⠀⠁⠣⠁⡁⠇⣼⠟⢉⠁⠀⡜⠇⠙⣷⢾⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠠⠼⣧⢸⡯⣄⡆⠈⣿⡂⠠⣀⡄⡇⠂⣡⠀⢸⣂⠀⡆⠀⡀⠠⢀⠉⠃⣽
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣃⡏⣹⣨⣍⢬⣣⠀⢠⡇⠀⠐⠃⡇⢪⢽⡰⠨⠦⠀⢧⠃⠹⠤⢘⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⢛⣙⡋⡧⠽⠚⡀⠘⡧⠀⠸⠙⢳⣠⢸⣠⠀⡆⠤⢸⣤⢰⡀⠌⣎⡁