Kuromi Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣴⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⡀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣠⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⣠⣾⣿⣿⣿⡿⢡⡀⠰⠎⣿⣿⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣮⣄⣤⣼⠟⠁⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢹⣿⣿⣿⠏⠀⠀⠀⠈⠉⠛⠁⠀⣀⣴⡟⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠰⢶⣶⣶⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠘⣿⣿⠀⠀⢀⣀⠀⠈⠛⠁⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⡀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀⠀⠀⠀⠀⠈⠉⣀⣠⣶⣿⣿⣶⣤⣔⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡟⠻⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡿⠛⠋⠉⢿⣿⡿⠋⠀⠉⠛⡿⢀⠀⠱⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡞⢫⠀⠀⠀⠀⠀⠃⠊⠀⠀⢱⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠲⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠈⠢⣀⣀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⣏⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠻⢶⣤⣴⡾⠓⠲⡏⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣼⠃⠀⠈⠓⠦⠤⣄⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⣾⣿⡿⣿⣽⣛⣏⣿⣛⣿⣿⣷⡀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢿⣾⣿⣿⣾⣿⣿⣿⣿⣾⣷⣿⣾⣿⠀⠀⠁⠀⠁⢾⣶⠀⠀⠀⠀⠀⣴⡇⠀⠄⠀⠄
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠐⢄⠈⠁⠀⠀⣀⡀⠀⠈⠀⢠⠅⠊
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣠⡄⣒⠀⠂⠀⠠⠀⠀⢈⡁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠺⠳⠀⠰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⢀⠀⡩⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠁

⠀⠀⠀⢀⣤⣤⣤⣄⣠⣴⣦⣤⡀⣄⢀⡀⠀⠀⣤⣄⠄⣠⣶⣾⣶⣶⣦⡀⠀⢠⡀⣀⡀⢀⣀⣠⣤⣄⣄⢦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⢿⡇⠀⠄⠈⠛⣾⣿⣏⣴⣷⡾⠻⣿⡎⣿⣿⣏⣵⣿⣿⣿⣿⣿⣿⣦⡀⢀⡀⠘⢃⣦⡁⠀⣀
⠀⠀⠀⣼⣿⣿⠒⠉⣙⠶⠺⢟⣡⣤⣴⢇⡈⠈⠀⠀⣿⠿⠿⠟⠀⠁⣿⢸⡇⣾⣿⢸⣻⣿⣿⣿⡿⢿⣿⣿⣿⢸⣿⣿⣿⢻⣿⣿⣿
⠀⠀⠀⢈⠩⡀⠀⠀⠉⠀⢀⠙⠟⢻⡿⠊⠁⠻⠣⡗⠀⣝⡋⠃⠀⠀⠀⠎⠙⣛⡃⠿⠏⢰⡉⠈⢉⣴⣶⢙⣛⠃⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠉⣰⡶⣦⣥⣬⣶⢀⡐⠂⠀⠛⠀⠀⠀⠁⠀⠈⠋⠐⡁⠀⠀⠀⠀⠘⠿⣿⣿⣾⣿⣿⢶⣿⣿⡁⣿⢿⡷⠾⣫⣭⣭⣽⡻⠇
⠀⠀⠀⢴⣿⠋⣹⣟⠻⣿⣏⠁⠄⠠⠀⡀⠠⠄⠀⠀⣈⣠⡈⠻⣿⣶⠰⠿⠃⢠⣜⢿⠿⣿⣿⣮⠉⠙⠻⣰⡿⢶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠛⠉⠁⠀⣻⡃⠀⢸⣿⣷⣾⣿⠟⠃⠀⠀⠀⠘⠂⠀⠘⣿⣦⣴⡄⠿⠟⣤⠆⠈⠛⣿⣿⢿⡯⠉⠙⢿⣿⣿
⠀⠀⠀⠀⠂⠐⠂⢾⣿⣯⣥⡄⠒⢿⠗⠒⠢⡸⠿⠿⢿⡆⢀⠀⠀⠀⢄⠀⠀⠀⠘⣿⣿⣿⡐⠀⠀⢠⣴⡀⢶⡀⢲⠲⠛⠇⣿⠙⠃
⠀⠀⠀⠸⠀⠀⠀⠈⠉⠉⠛⡇⠚⣺⡒⠂⠀⡅⣀⢄⡈⠁⠛⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⢻⠇⠀⠀⣬⣭⣵⡍⠋⠶⡐⠄⠠⢂⣤⣀
⠀⠀⠀⠹⠂⠨⠄⠰⢀⣀⡀⠇⠨⢙⣛⠶⢿⡇⠰⢒⣢⡔⣰⣶⣶⣶⣀⡀⢀⡀⡀⠁⠉⠁⢤⣤⣀⠹⣿⣿⣿⣾⣣⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠐⢠⣤⣴⣾⣿⣿⣿⣷⡤⠆⢻⣿⣶⡾⢷⣿⣿⠇⣿⣿⣿⣿⡿⠋⠈⠷⡀⠀⠀⠀⠘⠙⠻⠇⠻⡛⠿⡻⣮⠻⠿⢿⡿⡻⠇
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠉⠛⢿⡄⠀⡀⢿⠋⠙⠋⢩⢻⠆⠛⠛⠑⠚⠒⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠌⢠⣿⠀⢻⢖⣿⣿⣷⣶⣟
⠀⠀⠀⠀⣼⣷⣦⣀⠀⠀⠀⣠⢶⣾⣿⣇⠀⠤⠦⠆⠉⠀⠀⠀⠀⠀⠀⠄⣀⣀⣴⠁⢱⣶⣶⣶⠔⣤⡶⢅⠴⣮⡝⢿⣾⣿⣿⣿⡷
⠀⠀⠀⠘⠿⠟⠉⢙⠱⣞⠿⣇⣂⠟⠉⢻⡆⠸⢿⠇⠀⢠⠀⠀⢀⡀⠀⢰⣮⣍⡛⠂⣚⣛⡻⢋⠸⣄⣀⣴⣄⣀⣽⡌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠸⢿⣧⡄⣧⡝⢶⣶⣦⣤⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⣩⣭⣴⣿⣿⣿⣟⣿⣿⠆⢿⣿⣿⣿⣿⢏⣫⣶⣶⣦
⠀⠀⠀⠀⢀⣀⡤⠷⣶⠄⣉⣥⣵⣶⣮⡻⣿⠋⡤⠉⠁⢀⡀⠀⢀⣄⠀⠈⣿⣿⣿⢿⡙⠋⣿⣈⣀⠀⠸⠿⣿⡿⣰⠛⢳⣿⡎⢿⡁
⠀⠀⠀⠀⣈⣁⣐⠒⠰⣿⣿⣿⣿⣿⣿⣧⠛⠀⠀⠀⠀⠄⠀⠀⠻⠿⢷⡷⠘⠙⠷⠀⠁⡔⠈⠋⠋⠄⢰⣶⡴⠘⡏⣴⣿⣿⣿⣨⣿⡄
⠀⠀⠀⠀⣿⡿⠿⠇⣤⣌⢟⣛⡏⠲⡸⠃⠀⠄⠐⠀⠐⡄⠀⠀⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⠐⢖⡹⣿⣿⣦⡟⠁
⠀⠀⠀⢰⡟⡓⠉⣼⣿⣿⣿⣿⢋⡅⠀⠀⠀⠀⠀⢀⣠⡍⣶⣤⣦⣴⣶⠁⠸⣿⣿⠂⠀⠀⠄⠀⠀⠀⠐⠀⢀⠼⠠⣮⣵⡏⠉⠙⠃
⠀⠀⠀⠀⣵⠦⡌⠋⡀⣀⢟⣿⣌⠷⢦⣶⡰⣷⢎⣭⣛⣁⡽⢶⣛⡶⡴⣤⠀⢛⣀⣤⣤⡠⣦⣷⣿⣛⣏⣿⢮⡠⢴⢲⡝⢆⡐⠂
⠀⠀⠀⠀⢸⣶⣶⣶⣶⣆⠈⣙⠋⣴⠾⣿⣿⡲⢾⣿⣿⣿⣼⢯⣤⣛⣟⡁⣿⣿⣿⣿⣿⣿⡘⣿⠋⠉⠻⣾⣟⣷⡌⣛⣂⣼⣿⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣹⡇⢉⣀⢀⢚⡿⠀⠛⠉⣿⠏⢤⣀⣿⣿⢠⣾⣿⣿⣿⣿⣿⠿⢂⠀⠀⠀⣿⣿⣿⡆⠿⢿⡛⠛⣋
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠶⠋⣦⡀⠀⠉⠉⠁⢀⠹⢦⠌⠁⠀⠈⠿⠃⠨⣿⣿⣿⣿⣿⡷⢶⡼⠷⢶⣬⢈⣵⣶⠅⣲⣾⣀⣽⣿⡆
⠀⠀⠀⠀⠘⠽⠉⠉⠋⠀⠘⠓⠈⠁⠆⠀⠀⠀⠉⠀⠀⠀⠉⠁⠈⠁⠀⠀⠈⠛⠋⠙⠛⠁⠀⠐⠂⠀⠀⠙⠻⢿⠿⢿⡿⠿⠿⠛⠁

⢀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉
⡇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢄⡀⣀
⠈⠂⠠⠎⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⢠
⠀⠀⠀⡇⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⠀⢸
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠸⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠐⡤⠖⠈
⠀⠀⠀⠀⠀⠀⠈⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠊⡆⠀⠀⠐⠙⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⢀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠢⠤⢀⡀⠀⠈⠈⠀⠀⢀⡠⠒⠀⠀⠤⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠖⢁⡀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠋⠁⠀⠀⠀⢀⠀⠈⢆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠐⠁⠀⢰⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠆⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠂⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢨⣁⡑⠀⠀⠀⠀⠀⠉⠁⠀⠀⡐⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⣖⣠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⡀⠤⠙⠐⠂⠤⠤⠀⣄⠀⣀⢈⠁⢠⡀⠀⠤⠀⠲⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠉⠐⠒⢢⠤⠄⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠂⠄⢀⣀⢀⣁⡲⢎⢈⡆
⠀⠀⠀⠀⠀⠈⠁⠒⠂⣄⠀⠀⠀⣠⢁⡜⢀⡀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⢣
⠀⠀⠀⠀⠀⡇⠀⠀⠐⣅⠀⠀⡸⠄⠴⢡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠉⠉⠲⢄⠀⠑⠲⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠐⣄⠀⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠘⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢀⣠⣎⣀⡀⠴⠃

⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀
⠀⣏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⣀⡼⠃
⠀⠈⠓⡎⠙⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠊⠁⢸⡇
⠀⠀⠀⢿⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣤⠤⠤⠤⠤⠤⠤⢤⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⢻⣀⡀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣠⠴⠚
⠀⠀⠀⠀⠀⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠀⠀⠀⠀⠔⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠞⠉⠁⠀⠉⠑⠺⢭⠭⠞⠋⠉⠀⠀⠙⢦⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡼⠁⢬⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡅⠹⡄⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⡇⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⡇⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡇⠀⠀⠉⠁⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⠈⠉⠀⢀⠇⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⢿⣄⡀⠀⠀⠀⠀⠀⠰⠲⡆⠀⠀⠀⠀⠀⠀⢀⣮⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡴⡈⠉⠛⠒⠲⠶⢴⠃⠀⢳⡤⠤⠴⠒⠒⠋⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡄⠁⠀⡀⠄⠊⠀⢀⣰⠀⠀⠀⢠⠀⠀⠰⣦⢎⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⢼⠿⢅⣀⣀⣀⠴⠃⠀⠑⢢⠤⠦⡀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃⣿⠀⢀⡧⡤⠄⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠒⠋⢀⠌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢈⡇⠉⠊⠁⣀⠔⠉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⡂⡇⠀⠀⠀⠀⠀⠙⣏⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣀⣠⠴⠃⠳⢤⣀⣀⣀⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⣀⠠⠃
⠀⢀⠄⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁
⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁
⠀⠈⠂⠄⢺⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠉⠑⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠐⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⡠⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠤⠊
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣁⠇⠀⠀⠀⠣⠸⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢤⣠⣤⣆⢰⠊⠁⣀⠤⠒⠂⠤⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠒⠀⠤⠄⣀⣀⠈⡀⠔⠊⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣠⣴⣧⠀⠀⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢼⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠇⠀⠀⡅⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠀⠀⠀⠀⢰⡉⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⠄⣈⡑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠒⠛⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⢤⠔⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠉⠉⠉⠉⠀⠀⡖⠒⠲⡆⠀⢵⠊⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠜⠉⡝⠉⠉⠉⠉⢂⠀⠀⠀⠀⠀⢀⠤⠤⣀⠀⠀⠀⢠⢇⠀⠀⢰⠀⠀⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣆⡀⢀⠤⠊⠁⠀⠀⠀⠉⠑⡶⠚⢺⡆⠀⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⢀⣀⠀⡤⠓⠄⠀⠀⢠⢐⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠜⢃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡔⠁⠱⡀⠀⠀⢀⡜⠈⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠑⢄⡀⠈⠉⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠢⢄⡀⠈⠁⢋⡉⡇⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠔⠁⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠉⠒⠂⠤⠤⠄⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⡠⣄⢦
⣥⠀⠙⠌⢆
⠀⢀⣤⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡾⢷⡀
⡞⠋⢉⣠⣴⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢺⡇⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠱⠶⠛⠉⠀⠁⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢸⠇⠀⠀⠀⠀⠀⢀⡾⠻⣆⠀⡆⢀⠆⣲⠒⡴⠛⣇⡸⢻⠶⡆⢀
⠀⠀⠀⠀⠀⠀⠔⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⡀⠀⠀⠘⠒⠋⠁⠀⠙⠚⠛⠛⠒⠃⠀⠱⠾⠛⠁⠈⠀⠳⠿⠄
⠀⠀⠀⠀⠀⠀⠀⡴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠑⣬⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣠⡆
⠀⠀⢀⠄⣀⡀⠀⠀⢻⡋⣿⣦⣄⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⠛⠉
⠀⠀⠈⠃⢡⠁⠄⠀⢸⡆⣿⡟⣷⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣟⣿⣿⣄⠀⠀⠠⣄⡴⠟⡋⠉⠙⠉⠉⠛⢷⣶⣤⣀⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⢉⠿⠟⠻⣇⣦⣴⡿⢀⣰⠇⢰⡝⠙⠲⠊⣄⠀⠻⠋⠀⠀⠀⢀⡀⢠⣴⣾⣿⠃
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⣿⣿⣿⣎⣧⡀⠀⠃⠰⠏⡸⠀⠘⢃⣶⣶⣶⣬⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⣽⣿⡿⠋⠀⠘⡄⠁⢀⡀⡠⠶⢶⣾⣿⣿⣿⣿⣿⡿⠿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠹⣿⠁⢸⣷⡄⠑⠚⠉⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠰⡥⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠦⣿⣇⠀⠛⠁⢀⠀⠀⠀⢸⣿⡿⠁⣽⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣀⠰⣶⠶⠀⠀⠉⢀⣰⠿⠛⠁⠐⣷⣤⣀⣀⣠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢀⣠⣖⣉⣨⣽⣿⣿⡷⢶⣿⣿⠟⠁⣀⡤⠐⠛⠉⠉⡉⠉⠁
⠸⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⠀⠂⠀⠀⠀⠀⠤⠤⣬⡇⠈⠑⢄⣿⠁⠐⠿⠛⢣⠀⠀⢠⣞⡅⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠀⠀⠈⠽⠀⠀⠀⠀⠀⣷⠶⠋⠉⠁⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠛⢿⠂⠠⢄⡀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠀⠀⠀⠘⡆⠀⠀⠀⠁⠂⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢠⣦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠁⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠸⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⡿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠠⣦⠀⢶⣼⣿⡿⠛⠛⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠛⠻⠷⣦⣦⣾⠟⠋⠀⠀⣠⣄⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⡇⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⢀⣤⠴⠤⡄⠀⠀⣀⠀⠀⠀⠈⠛⠛⠁⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠺⠟⠁⠀⠀⠀⠀⢠⣾⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣽⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢷⣶⠟⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢄⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠑⢦⣤⣿⠟⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⢿⢀⡤⠒⠉⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⠿⠿⣷⣷⣤⡀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡦⠠⣦⣻⣿⣿⠋⠙⡎⢙⡂⢻⠀⠀⠀⢀⡠⠚⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣈⢙⣿⣿⣿⣿⣿⠿⠛⢿⣿⣤⣤⣴⣿⢿⣿⢇⣈⣞⣀⠇⠀⠉⠒⠚⠉⠀⠀⠀⢜⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⣿⣿⣿⣿⢸⣿⣿⣿⠟⠁⢄⠀⠀⠈⠛⠿⠋⠀⢠⠃⠀⣀⠤⠤⠶⠤⠤⠄⣀⠀⠀⠀⠀⠈⢦⡠⠊⠀⣹⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠛⠛⣿⣿⣷⣿⣿⡏⠀⠈⢿⡿⠂⠀⠀⠀⠀⠠⣾⢠⢊⡁⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⢸⠀⡀⠐⠁⣨⡂
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣦⣤⢔⣛⣽⣿⣿⣷⡀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠘⡇⠀⠈⠓⠀⠴⠆⠀⠀⣾⡇⠀⠱⠀⠀⡜⠀⢁⠀⠈⢁⡝
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⢒⠏⠘⠋⠉⠉⣹⣿⣿⣦⣄⣀⣀⣀⣀⣊⣀⣠⠔⠙⢦⣀⠀⠀⠢⠤⠀⠀⠈⠀⠀⡼⣠⠞⠛⠛⠲⢄⣀⣩⠂
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠈⠀⠀⠀⠀⠻⠋⠉⠁⠀⠀⠀⠉⠉⠉⠉⠛⠻⠤⣀⠤⠾⠭⢭⠴⠖⠶⡒⠒⠒⠛⣩⠝⠀⠀⠀⠀⠈⠟⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠳⣅⣀⣀⡠⠞⢀⠀⠀⠀⠀⢉⠖⠤⢄⣀⣀⡀⠀⣀⡇⠀⢀⡀⠀⠀⠀⠀⠈⠓⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣦⣀⠀⠀⣉⠗⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠤⠲⢮⣁⣀⣀⣠⠔⠈⠲⠤⣬⢥⡎⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠔⠦⣀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⢀⣀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣤⠀⣴⣿⣾⣷⣿⣟⣯⡝⣸⣿⣿⣿⣾⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠉⠒⠺⣿⣿⣀⡙⣿⣿⠛⠋⠛⢸⣿⠩⢝⡿⠿⢿⡿⠰⠿⠉⠻⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⠟⡈⠀⠀⠀⠉⠙⠻⢂⢉⠃⠀⠀⠀⢈⣿⣿⣿⣧⣥⣼⠇⢀⠒⠀⠀⠘⣿⣿⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡡⡆⣤⠹⠿⠄⣋⡀⠁⠠⠆⢈⠀⢘⠘⢦⠄⣀⡀⠻⠛⠛⢻⣿⣿⡩⣮⡄⠀⠃⠀⣰⣿⣿⠇⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠠⣉⠡⣁⣐⣿⢻⣿⣇⠻⣶⣶⢸⣷⡬⣶⣶⣶⣯⣤⣿⠄⠀⢘⣋⡀⠀⠉⣀⣓⣂⠺⠻⣿⠏⠀⠀⠻⣿⣟⠢
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢉⠽⣿⣾⣿⣚⣾⠿⠃⠿⣫⣀⣯⣼⣿⣟⣿⣿⣿⡿⣸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣦⠁⠐⠂⠀⠀⠹⣿⠟
⠀⠀⠀⠀⠀⣤⠳⣤⣦⡔⠟⠁⣹⡟⡾⠛⠋⠀⠈⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⠟⣭⡙⢿⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⠏⣴⠣⣟⣽⣷⠾⠋⠠⠀⡀⢀⣐⢀⡀⠀⠀⠋⠙⠛⠛⠛⠻⢿⢣⣾⣿⣿⡗⠛⠩⠙⠳⠆⢿⣿⣆⣠⣤⠀⠐⡀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⣶⡏⠁⢙⣛⢾⡋⠴⢧⣈⠡⠰⣟⣿⣿⣿⡃⠀⢰⣇⠀⠀⠄⠀⠀⠀⠈⠛⠛⠉⠀⠀⠐⢛⣤⣤⣄⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⡷
⠀⠀⠀⢿⣀⣠⣾⣿⣷⣭⣓⠛⡠⣴⣾⣿⣻⣿⣟⡯⢀⣿⣿⣿⣦⣀⡀⠀⠒⠀⣤⣶⣾⣿⣷⣶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣆⠀⠂⠐⠠⠀⠂⠀⠄
⠀⠀⠀⠀⠻⣿⣿⣿⠟⠻⠿⠇⠳⣿⣿⣿⣿⣿⣿⣧⣛⢿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⡿⠿⠉⢙⣿⡟⠋⢻⣧⠀⠀⣘⡻⠏⠀⠀⠀⠀⠀⠀⠠⠀⣤⣄
⠀⠀⠀⠀⠀⠀⣈⡡⠒⠀⠐⡄⠀⠉⠛⠿⠫⡍⠻⠟⢋⣀⠰⣬⣉⣙⢿⡿⠿⠟⣢⣥⣾⢡⣴⡾⠿⠿⣶⣭⣿⣧⠉⠉⠁⠀⠀⠀⠺⠆⣸⣿⠃⠀⣽⣿⡆
⠀⠀⠀⠀⠀⡌⠀⠀⠈⠳⣤⠃⢤⣤⡴⢃⣀⣠⣶⣿⣿⣿⣧⠹⣿⣿⣷⡾⠿⣿⣿⣿⡿⠈⡷⠀⢀⠀⠰⠎⣿⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡽⠿⠃
⠀⠠⢿⠇⡘⠀⠀⠈⡀⠀⣋⣀⣛⠁⠀⠛⢿⣿⣿⣿⣿⣿⣿⣷⢩⣿⠟⠻⠾⠟⢛⢿⡇⠓⢂⣀⡨⠤⠤⠒⠋⠀⢀⡸⠀⠀⠀⠀⠷⢄⠀⠀⠀⠀⢀⣀
⢀⣴⣤⡄⢇⠀⠀⠀⠚⠈⠿⣿⣿⣦⣄⠺⡿⠛⠻⣿⣿⣿⣿⣛⣆⠻⡈⠀⠀⡀⠙⢸⠣⣼⣿⣿⣿⣿⣩⣄⠀⠈⠀⠅⠀⠀⠀⠲⣈⠀⠀⠀⠀⣤⣥⣤⡇
⠈⠿⠿⣂⢨⡂⠀⣤⣶⡆⠀⢀⣽⣿⣿⣿⣦⣤⡈⡛⠿⠿⠁⠙⣛⡓⠀⠛⠋⠙⠃⢌⣰⢈⣿⣿⣿⣿⣿⣿⣷⣤⣈⠀⠀⠀⠀⠀⠀⢃⣤⣤⠀⣿⣏⣹⡇
⠀⠀⠘⠿⠇⣁⡀⢈⠛⠀⠀⢸⣿⡛⠋⠙⠻⣿⡇⡳⠀⠀⠀⠈⠛⠓⠀⠈⠀⠀⠀⢈⡰⠿⢿⣿⣿⡿⣿⡛⢿⣿⠿⠐⠀⡂⠀⣠⣴⣿⣿⣿⣥⣮⡍⢛⠅
⠀⠀⠀⣴⣿⣿⣿⣦⣍⣒⡄⣌⠋⢁⠒⠀⣹⡟⢃⣇⠀⢀⡀⠀⠀⠀⣠⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠀⢀⠀⠁⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠿⣿⣿⣿⣿⡿⠿⢃⣶⡄⠰⢆⣰⠟⠇⠤⠇⠱⢌⡀⠐⠓⠀⠀⠀⠞⡀⣄⠀⠀⠀⢠⣤⣤⡤⠤⠤⣤⣄⣀⣀⣀⡀⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣹⣿⣿⢋⣾⣿⡜⢛⣛⣉⣋⠀⣂⣁⠈⠢⠐⢃⠞⠀⠀⢀⠅⠒⠚⢿⡿⠋⠉⠙⢮⢻⡙⠂⢀⠀⠋⠏⠿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⠉⣉⣰⠏⠀⠀⣤⡻⣿⠋⡸⢿⣿⠃⠀⠈⠀⠐⠀⠀⣿⠀⠀⡖⠾⣇⠚⠆⠀⣸⠀⣙⣛⣀⣛⠀⢀⣿⣿⣿⣿⣿⣿⡆⠿⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠹⠧⠀⠀⠀⠘⠃⢠⣶⣮⣿⢋⣴⡀⠀⠛⢠⡏⡄⣸⣦⣤⣤⣾⣿⣶⣶⣶⣿⡄⠻⠿⠟⠾⢃⣿⣯⣾⣿⣿⣿⣿⣷⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣠⢘⣏⠘⠀⣿⣿⣿⣷⣾⣿⣆⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠈⠙⠷⢢⣿⣿⣿⣿⣀⣿⣿⢏⣶⣝⣛⡿⠿⢿⣛⣛⡋⠁⠀⡀⠀⢀⠆⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡰⣄⣠⣄⣴⣿⣖⠆⡀⠉⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⢣⣾⣿⠿⢏⣩⣭⣭⣉⠿⣿⣿⣦⠻⡿⠛⠛⢫⣠⡬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠘⢿⣇⠀⢸⣿⣿⣿⡿⠀⢀⡿⠟⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⡀
⠀⣰⠟⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠈⡷
⠀⠹⣄⡀⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢦⠴⠞⠁
⠀⠀⠈⠉⣿⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⣼
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣤⡤⠴⠶⠶⠶⠶⠲⠶⠶⠦⣤⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠈⠁⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣇⣀⣀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⠀⠀⢀⠀⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠑⠊⠀⠀⠀⠈⠒⢂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠤⣤⣐⠒⣄⣀⣠⣀⣈⣁⡤⠴⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠙⠳⠤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⠁⢠⣱⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⠌⢳⡀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⠃⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⣧⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⣿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⢨⠽⠷⠶⢤⣤⣀⣀⣀⢀⡤⠀⠀⢇⠀⠀⠀⣀⣀⣀⣤⠤⠶⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠓⠤⠤⢤⠀⠀⠀⠉⠉⢹⠁⠀⠀⠈⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⢢⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢤⠊⠀⠀⣀⠤⠂⠉⠀⠀⢀⣠⡇⠀⠀⠀⠀⡔⠒⠒⠒⠲⣖⠣⣀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡠⢳⢾⡉⠀⠀⠀⠀⣀⡴⠁⠀⠙⠲⢤⣀⢰⡁⠀⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡟⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠸⡁⠀⢸⠀⢻⠀⠀⠛⡆⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⣸⠀⢀⣰⠏⠫⡉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡉⠉⠀⢀⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠉⠓⠚⠉⠀⢀⡴⠊⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢀⠋⡇⠀⠀⠀⠀⠀⠀⠘⠳⣦⠤⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣀⣀⣠⠴⠋⠀⠘⠦⣤⣀⣀⣀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⡔⠁
⠀⠀⠀⠀⢠⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⢰
⠀⠀⠀⠀⠘⠦⠄⠉⠓⠒⠂⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⡠⠤⠔⠒⠒⠒⠒⠒⠒⠼⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠊⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣶⠆⠀⢰⡶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢓⠂⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣤⣠⠊⠁⢀⡤⢀⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⠒⠉⠉⠢⢄⡀⠀⢀⠤⠊⠁⠀⠀⢈⡱⢦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠋⠀⣀⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⣾⣿⣿⠀⢱⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠤⠆⠛⠀⠀⠀⡤⢄⠀⠀⠀⠀⠹⢟⣿⠀⢰⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⢆⠀⠀⠀⠀⠀⠀⠤⣉⣁⡠⡆⠀⠀⢀⠜⠀⠑⢪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣑⠢⢀⣀⠀⠀⠀⡇⠸⠀⣇⣀⣀⠸⡀⠀⠀⣰⢲⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣻⠆⢉⠉⠉⢒⡾⢅⡀⠀⣠⣛⠀⠀⢸⠀⠈⣀⠴⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⡳⠋⢏⣱⠊⠁⠀⠀⠀⠉⠓⠳⢄⣀⡜⠀⠈⡳⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠃⠀⢀⠼⡀⠀⠀⡤⠒⠀⠒⠀⠀⠈⣧⠠⠔⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⢯⠊⠑⠀⡘⠀⠀⠀⠀⠀⠀⢀⠗⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⣡⠞⠀⠀⡠⠋⠀⠀⠀⠀⠀⢲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠄⠚⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣤⣠⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⣿⣤⠛⢿⣿⠿⠿⣰⡟⠿⣿⣿⡶⣠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⢇⠀⠀⠈⠙⠻⡈⠃⠀⠀⣸⣷⣾⣦⣼⠇⠀⠀⠀⣿⣿⠠
⠀⠀⠀⠀⠀⠀⠀⠦⠶⠊⣿⣨⡄⣀⣀⢠⡀⢄⣓⣠⠄⣍⠉⢹⡿⠇⠾⢂⡀⣠⣿⡿⠀⢨⣦⡄
⠀⠀⠀⠀⢀⠘⠁⠐⣶⣶⣿⢿⡿⠞⠟⡊⣏⣿⣿⣿⣿⡏⣴⣶⣶⣦⣴⣶⣶⣦⡈⠡⠀⠈⢻⣷⠁
⠀⠀⠀⢠⡜⡶⣶⡚⢈⡿⠋⠛⠁⠀⠘⠛⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⠟⣭⣝⠇⠀⠀⠀⠘⠀⠀⣠⣤
⠀⠀⣠⡽⠸⣽⡿⢛⡋⠄⢠⣤⣶⣴⠀⢀⡀⠉⠉⠉⠙⠸⢿⠟⠉⠁⠬⢉⠘⣿⣶⣶⡀⠁⠀⠀⠀⢹⣯
⠀⠀⢿⣀⣼⣷⣭⣒⠊⢠⣼⣿⣿⣿⠆⣼⣷⣦⣀⠀⠀⢀⣤⣶⣶⣶⣾⣿⣿⣼⣿⣿⣧⡀⠀⠠⠠⠈⠁
⠀⠀⠀⠙⠿⠿⠋⠛⠃⠻⣿⣿⢿⣿⣮⡿⠿⠿⣿⣿⣿⣜⠿⣟⠋⢩⣿⣏⣹⣧⢠⣬⠟⠀⠀⣀⢀⣀⠂⢴⡄
⠀⠀⠀⢀⠀⠈⠠⣀⠇⣀⣠⢅⢁⣩⣴⣶⡜⣷⣶⣭⣭⣥⣿⡿⢺⡋⠉⠉⡿⣿⠇⠀⠀⠀⠀⠉⠈⠋⠀⣿⡿
⠀⠰⠆⡀⠀⠀⡀⣉⣐⡋⠁⢻⣿⣿⣿⣿⣷⡌⡿⠛⠶⠛⣻⡇⢂⣀⣈⠤⠐⠋⠀⡔⠀⠀⠘⠦
⠰⣷⡆⢃⠀⠀⣁⠙⠻⣿⣦⣟⠉⠹⣿⣿⠻⣷⡕⢠⣤⢤⠉⢁⢿⣿⣿⣷⣧⡀⠀⠀⠀⠀⠁⠠⠀⠀⢰⡷⣾
⠀⠁⣶⢈⡀⠙⠋⠀⣼⡿⠛⠻⢿⡇⡄⠀⠀⠺⠇⠀⠀⠀⠐⡫⢾⣿⣿⣿⡿⣿⣷⠄⠀⠀⣀⣴⣿⣷⣼⡓⠟
⠀⠀⣴⣿⣿⣶⣤⡄⠜⠁⠀⢨⡟⢡⡄⠠⠀⣀⠀⠄⢀⠠⠊⠀⠀⠀⠀⠀⠑⠄⠀⠀⠈⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠉⣿⣿⡟⣭⡱⡿⠤⠵⠊⠁⠈⠀⠕⡄⠀⠀⡀⠨⣶⡦⠄⢄⢿⢿⠋⠉⢿⣶⣶⣶⣭⡻⣿⣿⣿⡿⠁
⠀⠀⠀⠙⢛⡸⠋⠁⣼⢿⠟⣼⣿⠆⠀⢈⠀⠀⡎⠀⠴⢼⡔⠆⢀⡏⣓⣆⣐⠀⣸⣿⣿⣿⣷⠹⠻⡿⠃
⠀⠀⠀⠀⠈⠓⠀⠀⠈⠀⡿⡾⢫⣶⣄⣉⣼⠀⣼⣶⣶⣾⣿⣶⣿⡇⠙⠋⠙⣼⣿⣾⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠷⠔⠑⠳⡤⣸⣿⣿⢋⣿⡶⣜⡿⢿⣿⡿⠿⠟⠁⠀⠀⡘⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⡠⣤⣄⣾⣷⠄⠉⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⢼⡿⠛⣴⣷⣷⡞⠛⣿⠎⣋⡩⠵⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠛⠃⠈⠁⠈

⠀⠀⠀⢀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢷
⠀⠀⠀⣞⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣀⣠⡞
⠀⠀⠀⠘⠲⠶⡟⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⡟⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠷⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣀⠀⠀⠀⠀⣀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡎⢠⠀⠀⠀⠀⡀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠳⢬⣁⠀⠀⠀⠀⣩⡴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠉⠳⢦⣜⡓⠳⠷⠞⣃⣤⠶⠛⠉⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⠙⣦⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠃⠀⢾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⠃⠈⢷⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⠇⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⣇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⡟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠙⠛⠶⠶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢤⣐⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣙⢲⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠜⠀⠀⠀⠈⠉⠉⠳⡄⠀⠀⠀⠀⣀⠤⠒⠦⣄⡀⠀⠀⠀⡴⠉⠉⠉⠉⠉⠈⠧⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⡤⠒⠉⠀⠀⠀⠀⠀⠉⠓⣤⠼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢯⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣀⡤⠷⠒⠒⠛⠛⠓⠒⠲⠾⣄⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠳⡄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡞⠁⠀⠀⠀⠀⢠⠞⠉⠈⠉⠳⡄⠀⠀⠀⠀⠘⣦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠸⣼⠃⠀⢘⣣⠃⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣀⣈⣙⣛⣋⣀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠚⠉⠉⠀⠀⠀⠀⠉⠉⠓⢦⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢠⠏⠀⠀⢰⣦⡄⠀⠀⠀⣀⣤⡄⠀⠈⢳⡀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡸⡆⠀⠀⠈⠙⠁⠀⠀⠀⠻⠛⠃⠀⠀⢈⡇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⠦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⡀⠀⠀⠀⢀⡟⠉⠉⢫⠉⠉⠉⠀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣉⠒⠒⠞⠀⠀⠀⠈⠧⠤⠒⠋⢀⠈⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠢⢄⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠲⣿⡟⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢀⣀⣀⣤⣄⣘⣄⠀⠀⠀⠀⠀⠀⠈⢣⠔⠉⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠉⡛⢿⣷⣦⡤⢄⡤⢄⠀⡇⠀⠀⠀⠀⣠⠆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣗⡛⠀⢛⣾⣿⣿⡄⢰⡦⠸⡄⠳⠄⣀⠤⠚⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣶⣶⢰⣿⣿⣿⡟⠉⠀⠈⠙⠿⣿⡿⠋⠉⠏⠑⠊⢁⣈⣀⣀⡀⠀⠀⠀⠀⠀⠙⡄⢀⡔⠒⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠀⡽⠿⣿⣿⣾⣿⣿⡏⠀⢢⣷⡤⠀⠀⠀⠀⢀⣸⢀⠔⠋⠁⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⢸⠉⠀⠰⠋⠣
⠀⠀⠀⠀⠀⠀⠙⠿⢷⣤⣀⡩⠟⣻⣿⣿⣇⠀⠀⠉⠀⠀⠀⠶⠂⠀⠙⡎⠈⠑⠆⠀⣤⠀⠀⢠⣶⠀⠱⠀⠀⡸⠀⢃⠀⠐⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⡟⠘⠛⠉⠉⣹⣿⣷⣄⣀⠀⠀⠀⠐⢀⣀⠤⠛⢄⡀⠀⠀⠄⠄⠀⠈⠁⠀⢠⢁⡔⠓⠒⠬⣀⢀⡻
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠁⠀⠀⠀⠾⠋⠉⠉⠀⠈⠉⠉⠙⠛⠛⠻⢄⣀⡤⠿⢍⣒⠶⠶⣖⠒⠒⠚⣋⠽⠀⠀⠀⠀⠘⠏⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠈⠓⠓⣴⣀⣀⡠⠞⢀⡀⠀⠀⠀⡱⠢⠤⣀⣀⣀⣀⣀⣇⠀⢀⡀⠀⠀⠀⠀⠑⠂⠉⠁⠀⠀⡀⠀⠀⠀⡀⢀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⡰⢄⡀⢀⡩⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠲⢅⣀⣀⡠⠖⠈⠒⠤⠭⢴⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠚⠢⠤⠊

⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉
⡔⠁⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢧⡀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⢳⠒⠒
⠀⠉⠁⠘⡁⠀⠈⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣀⣠⡤⠤⠴⠶⠶⠶⠶⠶⠦⠤⢤⣄⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠉⠈⠉⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⣠⠜
⠀⠀⠀⠀⠀⠈⠦⣀⡀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡔⠉⡆⠀⠀⠀⠔⠉⠆⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⠐⠁⠀⠀⠀⠈⠂⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠐⢂⡠⠆⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠉⠙⠒⠢⠤⣀⣀⠀⠀⠀⠀⢀⣠⠴⠊⠉⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠋⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠳⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡰⠁⠀⠐⣾⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣷⠂⠀⠀⢱⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⢸⠀⠀⠀⠀⠈⠿⠿⠟⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠀⠀⠀⠼⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢤⡰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢉⡳⠦⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⣀⠀⣤⠔⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⣷⠖⣋⡁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⣰⠂⠐⠉⠉⠛⠋⡁⠀⠀⠀⠈⠓⠤⣀⣀⣀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠈⠁⠀⠉⠀⢉⠟⣻⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠈⢓⠲⠤⣄⣀⣠⡤⠜⠻⣄⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⡄⠀⠀⠀⠀⣠⡏⣠⠋⢀⣀⡠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢼⠁⠀⠀⠀⣼⠁⢀⡝⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠸⠦⠲⣄⠀⠈⠓⢄⣀⣰⠃⠈⠉⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⣸⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡃⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠋⠀⠱⣄⡀⢀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠾⠤⠤⠤⠒⠋

⠀⠴⡎⢆
⠖⠀⠘⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀
⣀⡴⠾⠀⣘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠋⢹⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⢏⣀⡤⠞⠿⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣿⠀⠀⠀⠀⠀⠀⣼⣟⠁⢀⠀⢀⢀⣀⠀⣠⡀⢠⣤⣀⠀⠀⠁
⠈⠉⠀⠀⠀⠀⡸⢆⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⡄⠀⠀⠠⣄⡴⠋⠘⢧⣼⣤⢎⡼⠁⢹⣠⣷⠏⠈⢹⣠⣆
⠀⠀⠀⠀⠀⠈⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠸⣌⣻⡄⠀⠀⠀⠀⠀⠀⠀⢘⣍⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⠀⢸⡟⣿⣦⡀⠀⠀⠀⠀⠀⠘⠀⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢘⣷⠾⠛⠁
⠀⠀⠘⠎⠉⢠⠀⠘⡇⣸⡟⣯⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣟⣳⣿⡄⠀⠀⢠⣠⠾⢟⠋⠙⠋⠙⠛⢶⣦⣤⣠⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⢸⠀⠀⠀⠀⠇⠀⠀⠀⢈⠿⠟⠻⣆⣦⣾⡇⣀⠞⠰⡌⠉⠒⢊⡄⢐⠛⠁⠀⠀⠀⡠⢄⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠹⣿⣿⣿⡽⢤⠂⠉⠐⠋⠜⠀⠙⣰⣶⣶⣶⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣤⣿⣿⠋⣀⠀⢆⠀⣀⠄⠔⠛⠾⣿⣿⣿⣿⡿⠟⠻⠿⠃
⠀⠀⠄⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⣀⠹⣿⠀⢿⣧⠈⠉⠀⠀⣀⣀⣀⠹⣿⣿⣿⠇
⠀⠙⠊⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠑⠿⣆⠀⠀⠐⠀⣀⠀⠘⠿⠋⢠⣿⡿⠟⣃
⡄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠶⣤⣿⣏⣀⣤⣤⡴⠛⠁⠀⣠⡼⠷⠶⠶⠖⠁
⣣⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⠀⡀⠀⠒⠋⠑⠒⠙⡟⠛⢿⠙⣿⠟⢠⣴⣶⠁⠀⠀⣠⠖⠀⠀⠀⠀⠀⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠘⣻⠀⠀⠈⠀⢱⣤⡼⠻⠃⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠁⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⠀⠀⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠙⠏⠀⠀⠘⡆⠀⠀⠈⠐⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢠⣴⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⢀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢷
⠀⠀⠀⣞⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣀⣠⡞
⠀⠀⠀⠘⠲⠶⡟⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⡟⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠷⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣀⠀⠀⠀⠀⣀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡎⢠⠀⠀⠀⠀⡀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠳⢬⣁⠀⠀⠀⠀⣩⡴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠉⠳⢦⣜⡓⠳⠷⠞⣃⣤⠶⠛⠉⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⠙⣦⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠃⠀⢾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⠃⠈⢷⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⠇⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⣇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⡟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠙⠛⠶⠶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢤⣐⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣙⢲⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠜⠀⠀⠀⠈⠉⠉⠳⡄⠀⠀⠀⠀⣀⠤⠒⠦⣄⡀⠀⠀⠀⡴⠉⠉⠉⠉⠉⠈⠧⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⡤⠒⠉⠀⠀⠀⠀⠀⠉⠓⣤⠼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡐⠿⠀⠸⣷⣿⡿⠟⠁⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠉⠙⠛⠿⢧⣷⣷⠟⠋⠀⠀⣀⣤⣆⡀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢠⣤⠤⢤⡄⠀⠀⣀⡀⠀⠀⠀⠙⠛⠛⠁⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠛⠁⠀⠀⠀⠀⢀⣴⡫⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡸⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⢀⠀⠀⠀⢠⣶⣶⡶⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠉⠀⠀⠉⢻⣣⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠙⠿⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠈⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠈⠢⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁

⠀⠀⠀⣰⣶⣶⣦⣴⣶⣶⣦⢰⣄⡄⠀⠀⢶⣦⣠⣶⡿⣿⢿⣶⣄⢀⣤⣠⡄⣀⣤⣴⣶⣤⡼⠄⠀⠀⢀⡀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠚⠛⣧⢀⢀⠀⢱⣿⣿⣾⡿⡿⣝⢿⡾⢿⠟⣾⣿⣿⣿⣿⣿⣿⣦⣀⣦⣈⣵⣧⣄⣤⡄
⠀⠀⠐⠿⠟⠁⠀⠴⠉⠩⣷⣿⣿⡏⠆⣠⣤⡌⠻⣭⡯⠀⠀⠻⡸⣹⣿⣫⣿⠛⡿⠿⠏⣛⡻⣿⣟⢿⣿⣯⣼⣿⣿⠃
⠀⠀⠀⠆⣑⣀⣀⣀⡀⠂⢠⠀⠻⠄⠀⠀⠈⠁⠼⠗⠄⠀⠀⠀⠀⢴⣶⣦⣥⣼⣦⣠⣾⡿⢣⣶⣦⣹⡿⣿⣿⢿⣿
⠀⠀⠀⣼⡿⢻⡿⢿⣿⠚⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠶⣶⣤⢠⣶⠀⣈⢿⣿⣿⣿⡜⠛⠻⡎⣽⣭⣴⣿⣿⣿⣿⣶⡄
⠀⠀⠀⢻⠀⠈⠉⠀⠿⠛⠀⢰⡖⠀⢰⣶⣦⣾⡿⠟⠀⠈⠉⠲⠠⠀⢿⣧⣠⡌⢿⠏⣤⠀⠘⢿⣿⣿⣿⠛⠻⣿⣿⡇
⠀⠀⠀⠐⠀⠂⢰⣿⣮⣥⠀⠺⡿⠒⠢⡹⠿⠿⣷⠀⡀⠀⠀⣄⠀⠀⠀⢿⣿⣿⡔⠀⠀⣤⣆⢸⡄⠲⠷⠛⠲⡿⠛
⠀⠀⠀⡀⠀⠀⠀⠉⠉⠛⠄⢙⣇⠁⢀⡅⡄⢤⠉⠈⠀⠀⠀⠈⠀⠀⠀⠈⢉⣻⠏⠀⢠⣶⣿⣎⠘⡲⣔⣀⣐⣤⣤⡀
⠀⠀⠀⠋⠀⠂⢐⣠⣤⣐⠁⠊⢭⣟⡻⢇⣐⣨⣶⢣⣿⣿⣷⣤⡄⢤⠀⠀⠀⠁⢰⣦⣤⣹⣿⣿⣿⢡⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠖⠺⠿⠿⢿⣿⡗⠊⠘⢿⣿⣯⣿⢿⣏⢺⣿⡿⣿⡋⠀⠈⠃⠀⠀⠀⠀⠈⠉⡈⢩⡝⢭⣓⣙⣟⣛⣫⡝
⠀⠀⠀⠀⣦⣀⠀⠀⠀⠈⢙⣀⣴⣜⠁⢀⣀⡛⠏⠀⠈⠀⠁⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⣀⡛⢃⣘⢷⣿⣻⣿⣿⣧⡀
⠀⠀⠀⢾⣿⠟⠷⢆⣤⣶⠋⢹⠟⢿⡄⢩⣬⠅⠀⠀⠀⠀⠀⠀⢡⣓⠾⡇⠠⠿⣿⠟⣼⠉⢉⡍⠉⣳⡹⣿⣿⣿⠋
⠀⠀⠀⠀⠁⠀⠀⠰⣷⣈⢹⣕⢦⣬⣧⣀⣉⣀⣄⠁⠀⠀⠁⠀⠈⣛⣋⣴⣾⣿⣶⣾⣾⢹⣿⣿⣿⣿⡗⣛⣭⣭⡄
⠀⠀⠀⠀⣀⣠⠶⣤⠈⣋⣬⣽⣦⡻⣿⠟⡍⠙⠉⡀⠀⢀⣀⠀⠸⣿⣿⡿⡟⠛⣿⣉⣁⠀⣿⣿⣿⢱⠾⢻⣾⡻⣇
⠀⠀⠀⢠⣉⣐⡒⠰⣿⣿⣿⣿⢿⣧⠛⠀⠀⠀⠠⠀⠀⠘⠿⢷⠇⠉⠻⠆⠍⠄⠈⠋⠡⠄⣶⣶⢀⠿⢴⣿⣿⡇⣿⡇
⠀⠀⠀⢸⣿⢛⠃⣴⣌⣛⣟⣘⠣⠋⠀⠀⠀⠀⢰⡐⠀⠠⠀⠀⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡀⢣⣹⣿⣿⣮⠃
⠀⠀⠀⠻⠉⠀⣾⠿⢿⣿⡟⣴⡀⠀⠀⣀⡀⣤⣶⢸⠷⣶⡾⠏⠀⢹⡿⠃⠀⠀⠀⡀⣀⣀⣀⡀⡔⠀⡷⡿⠁⢀⠉
⠀⠀⠀⠐⣛⣓⣁⣂⠀⠹⣿⠎⣛⣿⣯⣛⣹⣷⣶⡖⣭⠷⢯⣿⢇⣤⣥⣴⣶⣮⢻⣿⡿⢿⣾⣿⣦⠺⣽⠊⣤⣌
⠀⠀⠀⢠⣿⣿⣿⣿⣇⠐⣦⠸⠏⠙⠻⣮⠹⡿⢿⣿⠏⠛⣷⣷⢈⣿⣿⣿⣿⣿⣦⠋⠀⠀⢹⣿⣿⡇⣶⣶⣿⡿⠇
⠀⠀⠀⠸⣿⣿⣿⣿⣇⣤⢋⡀⠚⠐⠒⠃⠀⢶⣌⠋⠉⠑⣿⠇⢘⣿⣿⣿⣿⣿⣩⣰⣦⣄⡊⣛⣭⡃⢨⡟⠀⣶⣆
⠀⠀⠀⠀⢿⠏⠙⠟⠀⠰⠌⠓⠠⠀⠀⠀⠓⠀⠉⠐⠂⠐⠂⠀⠀⠹⠿⠛⠿⠏⠈⠡⠄⠉⠑⠿⣿⣶⣿⣿⣿⡿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⠷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣠⣤⡤⠤⠤⢤⣤⣤⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣄⣀⡾⠁⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠈⢳⣄⣠⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣶⠀⠀⢰⣶⢸⠇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣠⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢰⣾⣿⡇⠀⠀⠀⠈⣧⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⡿⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠿⢦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠈⠉⠉⣹⠛⠛⠛⢿⠛⠛⠛⠛⠋⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⠦⣤⣀⣀⠀⢠⠇⠀⠀⠀⠘⣇⠀⠀⢀⣠⠴⠚⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠇⠀⡤⠀⠉⠉⠉⠀⠀⠀⠀⠀⠘⠖⠚⠉⠁⠀⣆⠀⠘⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢈⡇

⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠰⣿⣷⣶⣤⣤⣤⣤⣄⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⡟⠛⢉⡙⣿⣿⣿⣿⣿⡛⠉
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡃⢈⢱⡿⠋⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠻⠟⠛⠛⢻⣿⣿⣿⠋⠉⠉⠙⠛⠿⠋⠀⢀⣴⠆⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣶⣤⣤⡄⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢿⣿⠇⠀⢀⣀⠀⠘⠛⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⣀⡀⠀⠀⢁⣀⣤⣴⣿⣿⣷⡶⠶⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⠉⢋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠋⠁⠀⢘⣿⠋⠀⠀⠀⢹⠓⡄⠈⢂⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⠈⠜⠀⠀⠀⠀⠀⠁⠀⠀⠈⡆⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⢸⡗⠂⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢼⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢷⣄⣀⣠⡷⠤⣶⠋⠀⠀⠀⠀⣤⡀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠀⠀⣹⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡼⠁⠀⠑⠦⠤⢄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈
⠀⠀⠁⠈⠈⠁⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠠⠄⡀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⢀⠈⡄⠀⠀⠀⠀⠀⠑⢄⡀⠔⠁
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⠸⠀⠀⠘⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠈⠀⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠈⠐⠒⠓⠂⢁⡠⠐⠈⠁⠐⠄⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠉⠀⠀⠂⠁⠀⠀⠀⢀⣠⡄⠀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡐⠁⢰⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠘⠐⠂⠀⠀⠀⠀⠀⠀⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠖⠠⠕⢦⣄⣀⠀⠀⠀⣀⣀⣀⠠⠄⠐⠒⠙⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠑⠮⢤⠤⠤⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠀⠈⠃⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠰⡀⠀⡠⠒⠁⠈⠁⠂⡤⠼⣆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠠⠓⠀⠀⠠⢈⠈⡇⠀⠀⠀⠀⠀⠀⠁⠠⠋⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠈⠠⢤⠄⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⢄⠈⠐⠀⠔⡣⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠇⠠⢀⡀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠂⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⡔⠁
⠀⠀⠀⠀⢠⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⢰
⠀⠀⠀⠀⠘⠦⠄⠉⠓⠒⠂⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⡠⠤⠔⠒⠒⠒⠒⠒⠒⠼⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠊⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣶⠆⠀⢰⡶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢓⠂⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣤⣠⠊⠁⢀⡤⢀⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⠒⠉⠉⠢⢄⡀⠀⢀⠤⠊⠁⠀⠀⢈⡱⢦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠋⠀⣀⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⣾⣿⣿⠀⢱⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠤⠆⠛⠀⠀⠀⡤⢄⠀⠀⠀⠀⠹⢟⣿⠀⢰⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⢆⠀⠀⠀⠀⠀⠀⠤⣉⣁⡠⡆⠀⠀⢀⠜⠀⠑⢪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣑⠢⢀⣀⠀⠀⠀⡇⠸⠀⣇⣀⣀⠸⡀⠀⠀⣰⢲⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣻⠆⢉⠉⠉⢒⡾⢅⡀⠀⣠⣛⠀⠀⢸⠀⠈⣀⠴⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⡳⠋⢏⣱⠊⠁⠀⠀⠀⠉⠓⠳⢄⣀⡜⠀⠈⡳⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠃⠀⢀⠼⡀⠀⠀⡤⠒⠀⠒⠀⠀⠈⣧⠠⠔⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⢯⠊⠑⠀⡘⠀⠀⠀⠀⠀⠀⢀⠗⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⣡⠞⠀⠀⡠⠋⠀⠀⠀⠀⠀⢲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠄⠚⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣻⣭⡻⢷⣤⣄⣀⣠⣤⣤⣤⣄⡀⠀⣠⣾⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣶⣾⣿⣯⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣜⣿⣿⣿⣿⠿⣿⣷⡄⠀⠀⣠⣾⣿⣿⢺⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣿⣿⣿⣻⣿⣿⣿⣿⣿⣧⣶⠦⠼⠿⣿⣛⣺⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣛⣻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⡿⣿⠿⣿⣿⣿⣿⡿⠿⢻⣿⣿⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢻⣿⡛⢠⣼⣦⣽⣼⣧⣻⠄⠀⢸⣿⣝⣣⡾⠈⣿⡆⠀⠀⠀⢀⡰⣽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣝⣿⣦⣭⣽⣿⠋⠉⠹⢤⣤⣤⠋⠹⣯⣤⣾⡿⠀⠀⠀⠐⣮⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡛⠛⠛⠛⢿⡋⠀⠀⠀⠈⣿⠁⠀⠀⣿⣿⣿⠁⠀⠀⠀⣠⣿⣿⣿⡟⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢻⣿⣷⣶⣄⣼⣛⠳⠶⠖⠛⠛⠓⠦⣾⣿⡿⠃⠀⠀⣠⡾⣿⣼⠏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⣦⣭⣽⡿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⡀⠀⣠⡾⣝⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣰⣿⣿⣿⣛⠿⣿⣷⠉⠙⣿⣿⣿⣿⣿⣶⣬⣬⣳⣾⣯⣻⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣿⣿⣿⣿⣿⣄⡻⣶⣴⡿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣼⣻⣿⣿⣿⣿⣿⣷⣿⡿⣳⣿⢟⣛⣛⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢱⠟⠁⠘⠳⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣄⣀⡼⢣⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢟⢿⣿⣿⣷⣮⣭⣭⣶⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣏⠘⢿⣿⣿⣿⣿⣿⣿⣿⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠛⠺⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⣇⡠⣤⢙⡏⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⠞⡉⠀⠙⡀⣤⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠕⠉⠤⣲⠆⠈⣿⠛⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⠀⠀⠁⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠿⡄⠀⠀⠀⢀⣀⣄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢷⡀⠀⠀⠩⠅⣰⡟⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠑⠂⠀⠐⢿⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠞⠁⠀⠀⠀⠀⠀⠀⢹⣧⢠⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠃⠈⠋⢰⠀⠀⠀⠀⠈⠁⠀⢀⣸⣿⠋⠁⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⡄⠀⠀⢠⡳⣆⠀⠀⠀⠀⠀⠀⠀⠈⠸⣟⠈⠉⠀⢀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⡀⠀⠈⠀⠈⠀⠀⢦⡀⠀⠀⣠⡞⠁⠙⣤⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡈⣷⡀⠀⠀⠀⠁⠀⠀⠙⠷⢰⡟⠀⠀⢀⣿⠃⠁⢸⣴⠄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡛⠿⡦⣀⠀⠀⠀⠀⠀⢀⡛⠀⠀⢠⡟⠁⠀⠀⠸⣏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⢫⣶⡄⠀⠀⠀⠀⠀⠀⣄⡟⠀⠀⠐⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠀⠻⣧⡄⠀⠀⠀⢈⣿⠁⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⢻⣷⠀⢀⣾⢿⡇⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠡⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠐⣿⡠⢎⣡⢾⡃⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⣠⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⣿⣴⡟⠁⠘⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣢⠔⢯⣬⣾⠾⠏⠹⣂⠘⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢂⠀⠀⣰⠀⠀⠐⣿⡾⠃⠀⣀⠿⠄⢠⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢠⣠⠒⠛⠁⠀⠀⠌⠁⠀⠸⡆⠙⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡿⠁⠠⠾⠃⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠠⠀⢨⡦⠴⠹⡛⠽⠣⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡌⠀⠠⣀⠢⠐⢋⡵⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠉⠢⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣠⡛⢯⠞⠚⠎⠔⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣄⣠⣤⠶⠛⡋⠁⣉⣥⣴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢸⣿⡍⠂⠀⠀⠀⠁⠘⠻⠌⢣
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⠥⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠒⢉⣶⢀⣀⣀⣀⣠⡤⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⣇⠠⠾⠋⣀⣼⠈⠈⠓⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠀⠀⢀⣀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠙⠋⠻⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⡠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢒⣎⡀⠡⡀⠀⠄⠀⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡐⠁⢶⡀⣼⡿⢣⣦⢀⣦⣾⡸⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣣⡄⠀⠀⠌⠀⣀⠈⠻⢌⣰⣿⠀⠂⢻⠿⠻⠿⠿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⠰⢦⣀⢉⣛⢶⣼⢏⡴⠊⠀⠄⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣹⠟⡿⣷⠏⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠲⢶⣶⡀⢂⣤⡀⠀⡀⠀⠜⠿⣷⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠛⠀⢠⡙⠿⡆⠩⢻⣆⠈⠳⣶⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠦⠶⠦⠀⣋⣙⣄⡈⣿⣷⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠛⠛⢿⣾⣿⣿⣛⣀⣴⡄⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠐⢿⣦⡀⠀⢰⡄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠿⢾⡲⢀⡀⠀⠀⠀⠐⠦⡀⠀⠁⠀⠰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⢀⢠⡄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⡿⠁⠢⠁⠠⣤⠅⠀⢀⠀⠀⣀⠀⠀⠀⠹⣤⡀⠀⠀⠲⠀⠀⠀⠀⠰⢻⣎⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠉⠉⣥⠁⣔⠰⡀⣸⡌⠀⣾⠀⠤⠿⣦⠀⠀⠀⣿⣷⣭⣤⣠⣮⣵⡄⣀⢶⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣸⠇⠘⠋⠘⣿⣤⡥⠟⢧⠶⣯⣤⣤⠛⠁⠀⠀⢰⣿⠟⠛⣁⣠⡤⠀⠀⢉⣩⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⣀⡠⠴⢿⣿⣳⣤⣄⢰⠘⠁⢀⣆⠀⠀⢀⡟⣽⠿⢛⡫⣅⣀⣤⣾⡿⠧⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⡽⠁⠂⠈⠙⢿⢬⡁⠈⢹⣷⣀⣸⠋⠀⢠⠊⢘⣿⣶⠱⣾⣿⠿⠿⠿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠖⢀⣉⣽⠤⠮⢽⡗⠠⣵⠟⠁⢀⡴⠁⢀⣼⣛⡋⠚⣛⣤⠴⠶⠶⢾⠾⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⠙⠁⢁⣩⣅⠠⠴⡉⠟⠀⠁⢀⠔⠁⠀⣤⣿⠿⠃⣾⢑⣐⣀⡉⢠⣄⠤⠤⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠁⠀⠋⠀⠐⠚⠋⠁⠀⣠⠾⠋⠀⠀⣸⢿⡏⣶⣬⡣⡈⢿⠁⢰⡆⣀⠀⢠⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠃⣀⣠⣴⣾⡋⠈⠀⠙⠿⣷⡜⣞⠳⢌⢿⣄⠀⠑⠋⡁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣯⣿⣭⣴⠶⠶⠛⣿⣿⡈⠡⢁⡌⠀⠀⠀⠙⣻⣾⣓⠀⠀⢈⣀⠀⠻⣤⢁⡀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢃⠐⠊⠀⠀⠀⠀⠀⠀⠉⠙⠷⠒⠤⠍⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠆⠀⢠⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠇⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣟⣼⡇⠀⠀⢠⣄
⣿⣿⣿⣿⣿⡏⣠⣤⣼⣿⣿⠟⢀⣴⣿⣿⡿⢸⣾⣿⣿⠟⢻⣿⡇⠀⠀⠀⠀⠀⠀⡿⠟⠛⡇⠀⢀⣴⣿⠏⠀⠀⢀⣸⠘⠀⠀⣤⡄
⣿⣿⣿⣿⡿⠷⣿⠛⠋⣿⡇⡴⡋⣉⠻⣿⣦⣼⣿⣿⣿⢰⣾⣿⡆⠀⠀⠀⠀⠀⣀⣿⢠⣴⣧⠀⣾⣯⣽⠶⣦⠀⠘⠛⠀⠀⣄⡏⠁
⣿⣿⣿⣿⣧⣤⣿⣷⣾⣿⣧⡹⠿⠿⣠⣿⣿⣿⣿⣿⣷⣼⣿⣿⡇⠀⠀⠀⠀⠀⠛⠋⠀⠉⠁⠀⠹⣯⣈⣇⡼⠃⠀⠀⠀⠀⠈⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⢸⣿⣿⠿⠿⣿⠛⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⡯⣯⠀⠀⣠⣄⠀⣴⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⢠⣴⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠉⠛⠁⠀⠀⡟⠋⠀⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠋⢉⣹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⣶⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣿⣿⡿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⢠⣶⣿⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢻⡟⢿⢹⣿⣬⣇⠹⣿⣷⣄⣰⣿⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⢸⠀⣄⢹⣿⡆⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⣿⣿⣿⣷⡟⠛⢉⣈⢳⡿⣼⢿⣿⣿⣮⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢱⡟⣸⣿⣿⣿⣦⣾⣾⣷⣁⡿⠀⢿⠻⣿⣿⡟⠙⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠏⢸⣿⣿⡿⣿⢿⣿⡿⣷⠀⠈⠀⠟⣿⢠⣿⣽⡟⠩⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⡟⢸⡏⠁⠀⠀⠤⢀⣶⣿⣾⣿⣿⣧⣞⡛⢷⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣠⡅⠸⡇⢀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⠙⢃⠸⠏⠉⠁⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣰⣿⣧⢰⣷⠀⠀⠀⣼⣿⣿⣿⣿⣋⣿⡇⠀⢙⣂⣀⣷⣷⠾⠉⣯⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣼⡟⢻⣿⠈⢿⣾⣷⣾⣷⢟⣿⣿⣿⣿⢻⣿⠶⠾⠉⠻⢽⠧⢄⣠⣿⣾⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣏⣀⣾⠟⠀⢀⣿⣄⣈⣁⣿⣿⢣⣾⠟⢿⣷⠟⠛⠁⠀⠪⡁⠀⢰⡲⠿⢿⣭⣥⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣩⡿⠋⡠⠞⠿⢷⡈⠛⣿⣿⣿⣿⣿⠖⢉⠀⠀⠀⠀⢀⣽⡷⠶⠛⢃⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠄⠃⣾⠀⣀⣽⣿⠟⠀⠀⠀⠀⣿⣷⣾⠏⡿⣿⢛⣤⣝⣎⡀⠀⠀⣻⣿⣶⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠈⠁⣹⣿⣿⠏⣴⣿⣷⣤⡶⠻⡇⢸⣌⣧⡻⣿⣿⢏⠋⠀⢀⡼⠿⠿⠋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠟⠀⢸⣿⠟⠋⢸⣿⠛⣻⠀⠀⠀⣃⡄⣾⣿⣷⣾⠟⠀⠀⣰⣿⠃⢠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠸⠳⣿⣿⡆⢼⣀⣾⣾⠛⡿⣿⡿⣿⣟⣯⣴⡆⣀⣼⣿⡈⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⣿⠃⢠⣽⣿⡟⢀⣷⣿⢿⣿⣿⣇⣾⣷⣿⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⣿⣿⣁⣞⠙⢋⣿⣿⣿⣿⣿⠟⢿⣿⡟⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣶⣿⣿⠿⢿⣫⣿⣿⣿⣿⣿⠋⠀⠛⠁⠀⠈⣿⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⡿⡳⢿⣟⣿⣿⣿⣿⣿⡧⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠋⠉⠉⠁⠅⠀⠀⢸⠟⠛⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢚⡓⡒⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⠀⠒⣾⠑⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⠿⣿⣿⣶⣾⣧⡁⠀⠀⠐⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢛⢇⢀⠈⢤⣾⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⢿⡇⢺⢿⣾⢣⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡄⠑⢄⠀⠀⠀⡀⡆⢰⡀⠚⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠙⠂⡀⢵⣣⣦⡓⡿⣼⡧⣧⣴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢛⡚⢹⣿⡷⠿⠟⠻⠻⠟⠫⣆⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⡿⡉⢿⡿⠓⣾⡾⣿⣿⣦⠛⣅⡦⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⣋⠱⣎⣸⣯⣽⠻⣯⣤⣶⣧⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠳⣨⠉⠻⣿⣿⡧⡹⡿⠛⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠟⠚⡿⣿⣾⢉⢙⠃⠌⠂⠀⡪⣁⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣼⣯⣶⣽⣤⣤⣧⣾⣿⠑⢌⠀⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠟⢀⠁⠄⠀⠀⢀⠀⠀⢈⠆⠀⢁⠈⠉⠁⣍⠀⠁⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⠀⠀⠀⠘⡉⠹⠒⠤⠄⢀⠐⣀⢴⠀⠁⠀⠀⢰⡏⠀⢈⢀⡈⣀⣈⠀⠁⢿⣾⢣⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡠⠤⠀⠲⠋⠁⣀⠤⠐⠒⠒⠛⠻⢦⡀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢀⠤⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠁⢺⠐⣴⠁
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠸⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠴⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠁⠀⠀⡀⡀⠀⠀⠀⠉⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣥⣴⣾⣿⣿⣿⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡐⢠⣿⣷⣾⠟⣉⣡⣀⡀⢀⠁⣀⣀⣤⣬⣭⣿⢶⣭⣡⡠
⠀⠀⠀⠀⠀⠀⠀⢇⢀⣿⣿⣼⣦⣼⣿⣿⣿⡕⠖⠺⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢿⡊⢿⣿⣿⣿⠟⣀⣀⣀⠙⢿⣿⣿⠏⣽⣿⠷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡉⠉⢽⡇⠀⣿⣵⣾⡷⢈⡿⠀⢠⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣬⣛⠦⣬⣿⣿⠥⢜⣡⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⡿⣶⣤⣴⣾⢿⣻⣿⣧⠀⠀⠀⠀⢀⣀⣦⣾⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠂⠀⠁⠉⠋⠋⠈⢀⡿⣿⡄⠀⢀⣾⣿⣿⣟⣿⡛⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢀⣀⢀⠔⠋⠹⢻⣿⡖⣻⣿⣿⣟⠻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⣠⠞⢁⣼⠟⠀⣀⣴⠛⣿⣿⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⣧⡦⠀⢀⠔⠁⣰⡿⢁⣠⢾⣯⣿⡿⣋⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠈⣼⣷⡶⠋⠀⠀⣸⣿⣿⣽⣾⣿⡿⠀⠙⢶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢛⡉⢀⣴⣿⣧⠘⣿⣿⡿⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠋

⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⡤⠄⠒⠒⠂⠒⢀⡀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠁⢠⣀⡀⠀⠐⣀⣤⠀⠀⠐⣤⡀⣀⠀⣤⠂⠀⣿⣿⣿
⣿⣿⣿⣿⠀⣀⢰⠴⠆⠀⠀⠀⠀⠀⠄⠁⠀⠆⠄⠀⠀⠀⠀⠊⠟⠁⠀⠀⠙⠿⠀⠀⠀⠋⠋⠉⠙⠙⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠈⠀⢀⠀⠀⠀⠈⠀⠀⢠⠐⠆⠀⢀⢀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⡤⠀⢈⣀⡁⠀⢤⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠐⠀⠀⢰⡚⣥⢠⡀⠋⡀⠀⠀⣦⢤⠴⢠⠢⠶⣐⠀⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠐⠃⡀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⢀⡀⠀⢀⠀⠐⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠰⠆⠆⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠈⠀⠀⢿⠷⠀⠀⠐⢿⠇⠀⠐⢶⡦⠈⠀⠘⠟⠈⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠂⢀⠀⠀⠀⠀⠀⠂⠀⠀⠙⠈⡁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⢀⡀⠀⠠⠀⠀⠀⢤⠀⣿⣿⣿
⣿⣿⣿⣿⠀⣄⡘⠁⠀⠀⠀⠠⠇⠀⠀⠒⠀⠤⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣂⠀⣿⣿⣿
⣿⣿⣿⣿⡄⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢁⢀⣄⠀⠀⢀⢀⠀⠀⣀⣠⡀⠀⢀⠃⢁⢁⡀⡀⣠⣿⣿⣿
⣿⣿⣿⣿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠰⠄⠀⠀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⠀⠀⠀⠖⠀⣿⣿⣿
⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠄⠀⠠⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠈⣉⠙⣢⢀⣀⠀⢂⡀⠀⠈⠤⣠⣤⡄⠶⠗⠀⠀⠀⠁⠀⠀⠀⠀⠘⠃⠀⠨⠈⢀⠀⢀⠁⠘⠂⣿⣿⣿
⣿⣿⣿⣿⠈⡁⠸⠿⣼⡟⠉⣰⡂⠀⠛⠀⠉⠀⣠⡤⠠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠂⡀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⢀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿
⣿⣿⣿⣿⠀⠉⠀⠀⢠⠁⡀⠀⠁⠀⠆⠀⠀⠈⠘⠘⠈⠁⠀⠀⠀⢀⠀⡀⡃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⢀⢀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠶⡄⠐⠆⠀⠀⠀⠶⢠⠀⡇⠀⡆⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⡘⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⢄⠀⣷⠀⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⣿⡇⠙⢟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡀⠀⠈⣾⡆⠘⢷⣦⡀⠀⢹⣆⡀⠀⢻⡒⠚⠙⣇⠀⢈⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢷⡶⠚⢹⠁⠀⠀⢹⣷⠀⠀⠙⣟⠲⡄⠈⠲⣄⠈⠛⠉⣿
⠀⠀⠀⠀⠀⠀⢢⡀⢻⣦⣄⠀⠀⠀⠀⠀⠈⠳⣼⣄⣀⣀⡼⢻⠀⠀⠀⠈⣇⢿⠀⠀⠈⡇⠀⠀⠹⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠪⡝⡄⠀⢸⡀⠀⠀⠈⣧⠉⠁⠀⢸⡀⠀⠀⢀⠏⠘⢦⣀⠴⠁⠀⠠⠆⠆⠆⠞⠓⠒
⠀⠀⠀⣀⠀⠀⠸⡇⠀⠀⠀⢹⣾⠀⠘⠷⠀⠀⠐⠋⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀
⠀⠀⠀⠈⢷⡀⠰⣷⣄⣀⣤⠟⡿⠀⠀⠀⢠⣆⠀⠀⠀⠀⣀⠠⠀⠒⡐⠈⡀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠐⠉⠀⠀⠀⢀⣀⠽⠖⣒⢪⢉⡀⠴⠅⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠒⣘⣾⡧⠅⠚⠋⠁
⠀⠀⠀⠀⠀⠀⢺⣷⣡⣀⡂⠠⠔⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠿⠯⠭⠤⠀⠂⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠉⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣮⣝⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠀⠀⣀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢿⣿⣦⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⢿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣅⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣶⣿⡿⠋
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠉⢹⣿⣿⡿⠋
⠀⢰⣿⣿⠋⠉⠀⠀⠉⠙⣿⣿⡇⠀⠀⣠⣴⣾⣿⣿⣿⣶⣤⣾⣿⠟⢿⣿⣶⣿⡿⠋
⠀⠈⢿⣿⣷⣦⣤⣤⣴⣾⣿⡿⠀⢀⣾⣿⠟⠉⠁⠀⣹⣿⣿⣿⣧⣀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠉⠀⣠⣾⣿⠋⠀⠀⣠⣾⣿⠟⠁⢙⣿⣿⣿⠋⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⠟⠁⣠⣶⣿⣿⣿⣅⢀⣴⣿⡿⢻⣿⣧⠀⠀⣾⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⢁⣀⣀⣿⣿⡇
⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠉⠁⠀⠀⢰⣿⡿⠉⠉⢻⣿⣿⡿⠋⠀⢀⣿⣿⠀⠀⣿⣿⣧⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⣠⣼⣿⡏⠀⠀⢀⣼⣿⠏⠀⠀⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⣿⣿⡇
⠀⢀⣿⣿⠃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⢀⣤⣶⣿⡿⠋⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⣏⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣿⣿⡇
⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⢀⣀⣠⣄⣀⣿⣿⡇⠀⠀⠀⠀⣴⣿⡿⠟⠛⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣷⣤⣤⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⢻⣿⣯⣀⠀⢀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣠⣶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣤⣶⣾⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣗⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠙⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣤⣴⣯⣤⣬⣼⣯⣾⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠙⠶⡤⠶⠶⠶⠒⢲⣶⠟⠉⠙⣷⠖⠛⠉⠉⢻⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣰⠶⣶⣿⣇⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢨⣿⣾⢋⡀⠈⠹⣦⣤⣤⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⢶⣿⡟⣈⠁⠟⣠⡿⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⡽⢿⣿⠷⠞⠁⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠒⠶⢾⡆⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⣿⣄⣀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠃⠈⠙⠻⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⢀⡹⣖⠀⠀⠀⠉⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠉⠀⠀⢀⣹⡟⠓⠒⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠓⢤⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠿⣿⠓⠲⣶⣤⣤⣤⣤⣤⣤⣤⡶⠖⢾⡿⣥⣀
⠀⠀⠀⠀⠀⠀⣴⠛⠉⢻⣏⡁⠀⣼⠏⠀⠀⠘⣏⣀⠀⠀⠀⣀⡟⠁⠀⠈⣷⡄⠉⢻⣶⠶⢤⡀
⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⢿⣽⣶⠏⠀⣠⠤⣄⠈⠙⠒⠒⠛⠁⠀⣀⣀⠀⠸⣆⣾⣾⡛⠀⠠⣯
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡻⣽⠟⠀⠀⢧⣀⡼⠀⠀⠀⠀⠀⠀⢸⡁⢈⡗⠀⢻⣿⠃⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢷⡀⠀⢀⠄⣠⠏⢠⡴⠛⠛⠛⠳⢦⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⢿⡄⣀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠈⢻⣶⣷⣴⣏⣰⠏⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⣀⡤⠟
⠀⠀⠀⠀⠂⠈⠀⠀⠘⡩⢛⡏⠀⠀⢀⠀⠀⠀⠀⠸⡇⢀⣤⠀⠀⢀⢀⣼⡇⣀⣤⣼⣏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠿⣻⣶⠿⠿⠛⠛⠛⣿⠀⠁
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠘⣿⣤⡀⣠⠀⠀⠀⠀⣸⣟⠃⠀⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢸⠈⠻⣯⣅⣆⣀⣤⡾⠋⠻⢦⣄⣀⣀⣀⣀⣤⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠳⣦⠀⠀⠻⣤⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢸⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡴⠟⠀⠀⠀⠈⠙⢶⣄⣀⣀⣻⠄⠀⠀⠀⠀⠀⠈⠻⣄⠀⣀⡴⠏⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣦⡀⢀⣹⣤⣦
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢤⠸⠋⢠⡴⢦⡈⠉⢻⡁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⣤⠞⠁⣠⢦⡄⠀⠀⠘⠷⠾⠃⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⡼⠋⠀⠀⠀⣠⣤⣄⠀⠳⠴⠏⢠⣤⣄⠀⢀⣤⣄⢸⡇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠙⠉⢠⡶⢶⡄⠀⠻⣤⠟⠀⣀⣀⠀⠹⢦⠾⠀⠻⣤⡿⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠀⢀⡶⠲⡆⠈⠳⠞⠃⣠⣤⡄⠀⢸⣏⣹⠇⠀⠀⣤⣤⡀⠀⢀⡿⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢿⡉⠁⠻⣆⠀⠛⠚⠃⠀⠀⠀⠀⠻⣤⠟⠀⠀⠈⠁⠀⠀⠘⢧⡼⠇⢀⡾⠁⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⣄⡀⠀⠀⠀⠈⠻⣄⠀⠈⠳⣄⠀⠀⢰⡟⢻⡆⠀⢀⣤⣄⠀⠀⣼⠛⣷⠀⠀⢀⣴⠟⠁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⠀⠈⠳⣄⠀⠈⠻⢦⣄⠙⠋⠀⠀⠸⣧⡾⠁⠀⠈⠛⠉⣀⡴⠟⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠾⠿⢦⣄⠀⠁⠀⠲⠶⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠢⡀⠀⢀⡜⠁⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠓⠌⣲⠿⠀⢀⠄⠀⠀⠀⠈⠣⡀⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣆⠞⢀⡼⠁⢠⣾⠃⠀⠀⠀⠀⠀⠀⣀⡄⣀⣸⠿⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠙⡶⠋⠀⣠⣟⣡⣀⣀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⢈⣳⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠉⠉⠃⢀⣀⣀⠀⠀⠀⠀⠀⡀⠀⣠⡄⡄⠀⢄⡄⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⢀⣴⠟⠁⣼⡇⠀⠀⠀⣼⠇⠀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣠⣾⢀⣠⠄⠀⢀⣀⣌⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣾⠃⠀⣰⣿⢁⣤⣤⣼⣿⣤⠀⡩⣀⣀⠀⣠⡀⢀⣤⡄⣸⡿⠁⣤⠀⣘⢻⡉⣰⡏⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⣀⣤⣿⢳⡿⢃⣼⣿⣿⡏⡰⣿⡟⡹⢣⣿⡴⢫⡿⢁⡿⠁⣼⠏⣰⡏⣸⣿⠏⣾⢋⡼⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⢸⡏⠁⣼⣿⢾⣣⠏⢸⡏⢿⣣⠁⢿⣉⡔⠹⡿⠁⢿⣁⡼⣇⡴⣯⠜⢿⡼⣿⠏⡸⡏⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠘⠇⠀⢿⡟⠀⠁⠀⠀⠀⠀⣁⠀⣀⠀⣀⢀⣀⠀⢀⣉⠀⠀⠀⠀⠀⠀⠀⡯⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠂⣿⢢⣿⠸⣧⣼⣿⡇⠈⣧⠐⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⢰⠄⠰⣓⠂⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⡆⠘⣯⠅⠀⠀⠘⠁⠙⠚⠁⠀⠀⢯⣀⡿⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢳⡈⠋⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠀⠸⣤⠞⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢇⡀⠙⠋⠀⢀⡴⣾⠟⠒⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠀⠠⠀⠀⠀⠴⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡀⠁⢀⡀⠀⣾⠸⡇⠀⠀⢀⠀⠀⠀⠀⠐⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠉⠀⠀⠀⠁⠀⠘⢦⣉⣒⡶⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠁⢀⣦⡀⠁⠀⠀⠋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣆⡆⠀⠀⣼⡤⠀⠁⠀⠀⠀⠀⠀⠤⠀⣰⢿⣦⠀⠀⣀⡀⠀⠀⠀⠈⠟⠁⠀⠰⠂⠀⠉⣻⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⡈⠀⢀⠀⠘⠀⣠⣿⣄⠀⠂⠀⠀⠀⢰⠇⢹⡘⣇⠘⠃⢹⠀⠀⠉⠀⢠⣄⠀⠀⠀⠀⠈⣡⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⡀⠀⢀⡼⠃⣼⠸⡆⠰⠆⠀⣰⠋⢀⡞⠀⠈⢁⡴⠋⠀⠆⠀⡴⠻⡏⢷⠀⠀⠀⠀⠓⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢹⡆⠀⢁⡴⠋⠀⢰⡇⠀⢹⣄⣠⠞⠁⠀⣾⣁⡠⢀⡞⣠⠶⡄⢠⠾⣇⢀⡟⠈⠳⣄⠛⠀⢘⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠄⠀⠈⢀⡄⠀⠈⢳⣴⠋⠈⠻⡄⠀⠈⠋⠉⠀⠸⡇⢻⡀⠡⡀⠀⠙⣾⠁⠀⣤⠈⠃⠀⢨⣄⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠄⠀⢀⠾⡿⣆⠀⠀⠉⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠹⣄⢳⡀⠙⠀⠀⠀⠀⡴⣿⢧⠀⠀⠛⠚⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⢦⠀⠀⢠⣷⡀⠀⠀⢀⣤⣀⣀⣀⣀⡀⠹⢸⠇⢀⣿⡀⠀⢀⡴⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠆⣠⢞⣟⢦⡀⠐⢫⣯⡓⠀⢀⡾⡴⠀⠀⢠⢠⣿⣆⠋⠀⢋⣿⡙⠀⢀⡾⣿⠳⡄⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠀⡴⢛⣟⠢⡀⠐⣫⣯⡛⢀⣞⡁⠙⠓⠒⠋⡼⣻⣘⣦⠐⢫⣿⣝⠂⢠⠞⣿⡳⡄⠘⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⢛⣟⠳⡄⠜⣩⣯⡙⠆⢉⣉⣙⠛⠒⠞⠛⢋⣉⡀⠰⣫⣿⣝⠆⡠⢞⣿⡛⢦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣫⣏⠳⢦⢚⣵⣷⣍⠂⠰⠆⣄⠰⠶⠶⠀⣀⠰⠆⠘⣡⣿⣌⠃⡴⢋⣿⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠑⠀⢋⣴⣷⣌⠓⠈⣼⣿⣄⢉⡉⣴⣿⣆⠁⢚⣡⣿⣮⡓⠀⠋⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⡷⣌⠓⢈⡴⡿⣆⠐⢀⡴⣿⢦⠀⠋⡴⣿⢦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠾⡿⢮⠀⠈⡼⡿⠦⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⣻⣿⠙⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡿⣿⣿⡿⠟⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣽⣿⣿⣿⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⢸⣿⣿⡏⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠈⢻⣿⣿⡇⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⣀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠊⠀⠈⠻⢿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠋⠀⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⢀⣴⣾⣿⣿⣿⣿⣇⠀⢰⠀⠀⡆⠀⢰⠀⣠⣿⣿⣿⣿⣿⣶⣄⠀⠉⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣯⣈⣋⣭⣭⣴⣶⠀⠀⢶⣶⣦⣭⣭⣭⣬⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⠛⣛⣋⣩⣭⣤⠀⠀⣠⣬⣭⣉⣙⣛⣛⣸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠿⠿⠿⠛⣛⣋⣩⣤⣤⣭⣉⣛⣛⠛⠛⠿⠏⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠙⠛⠻⠿⠷⠶⠿⠿⠛⠋⠉⠀⠀⣸⣿⣿⡙⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⢋⣾⣿⣿⣿⡶⢤⣀⢤⣶⠄⣠⣤⡀⢶⡶⢀⣴⠶⣿⣿⣿⣿⡇⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡅⣰⣿⣷⣶⣿⣿⣿⣿⣷⣶⣿⣷⣀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠧⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣚⢛⡻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣧⣾⣤⣦⣽⣿⣿⣿⣿⣿⣿⣷⣾⣦⣿⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⡜⠀⠀⠀⢀⡤⠆⠀⠻⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠰⠂⣤⣴⠁⢀⡴⠚⢉⡄⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠀⣰⠟⠀⣀⠁⠠⠀⠉⠁⠀⠀⡄⠚⠀⡖⣒⠖⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠡⠀⠻⢄⠘⠋⠀⠀⠾⠀⡀⣒⢀⣄⡀⠤⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠡⠄⢀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠐⠂⢃⡀⠀⠀⠀⠀⢠⢦⣤⣄⡀⠀⠀⠀⢼⠈⣯⡫⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⡌⠁⠀⠀⣤⣤⠃⢀⡀⡨⠃⠀⠀⠀⠈⠳⠦⠚⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠤⢠⢾⢥⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⢀⣠⠀⠾⡐⠂⠘⡚⠊
⠀⠀⠀⠀⠀⠀⡀⠀⡴⠓⣀⠃⢄⠘⠋⠀⠀⠉⠀⠀⠁
⠀⠀⠀⢨⠉⡍⠉⠉⢑⡀⠋⠀⠀⠷⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢵⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠘⢧⣅⡑⠀⠀⠀⠀⢠⠶⠚⠛⠃⠀⠀⠀⢄⢀⡀⠀⠠⠄⠀⠀⠆⠀⢀⠞⠐⠂⢀
⠀⠀⠀⠀⠠⠄⠀⢀⣀⡀⠈⠳⠶⠊⠡⡦⠀⠀⠀⣀⡄⠀⣴⠆⠀⠀⠀⠸⣌⡢⡀⠶⠘⣠⣤⠀⠰⠋⣄⠃⠴⢠⡅⠀⠑⣀
⠀⠀⠀⠀⠀⠀⣀⣫⢔⡉⠁⠀⠀⠀⠠⢄⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⢂⠈⠁⠀⠀⠂⠀⠀⠀⠀⠈⠙⠲⠶⠆
⠀⠀⢈⠃⢀⠀⠉⢀⣩⠶⠋⣀⠀⠀⠙⠀⠠⡶⠀⠀⠀⢠⡆⠀⠀⠀⠀⡄⡄⠠⣄⠀⠋⠀⠄⠉
⠀⠀⠘⠀⠼⠀⠰⠋⣀⠘⠂⠁⠀⢀⣤⣤⠀⠂⠀⢰⠖⠀⢢⡀⠀⠘⢁⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡴⠃⠈⢁⡤⠀⠖⠀⢢⡴⠛⢡⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠶⠒⠀⠀⠀⠀⠀⠈⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⣴⡄⠀⠀⠁⠐⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠁⠐⠀⠀⠠⠇⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠐⣻⡀⠀⠀⠀⣀⢀⠬⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠥⢷⠀⠀⠀⠸⣁⣠⢔⠗⠀⠀⠀⠐⠒⠒⠁⠂⠈⢠⠀⠀⠀⠀⢰⣃⣀⠀⢄⠀⠄⠠⠀⠁
⠀⠀⠀⠀⠀⠸⠽⡇⠀⠀⠀⠈⠃⠤⣂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⢒⣻⣤⡠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⡀⠺⡜⣏⠀⠉⠢⠐⠀⠀⠀⢀⠀⠀⠀⠀⢠⡁⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠧⢻⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣨⠈⡃⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢁⣀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⡴⠧⣀⠀⢁⣆⠀⠀⠀⠀⠀⠀⠠⡅⣈⠃⠀⠀⠀⡀⠀⢙⠛⡄⠀⠀⠈⠥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠀⣠⡜⠀⠈⠣⡀⠀⠀⠀⠀⠀⢠⠀⣸⠀⠀⠀⠂⠀⠀⠀⠣⣘⡄⠀⠀⠒⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠤⡌⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣥⠀⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡤⡄⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢄⠀⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠈⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡁⡆⠀⠀⠀⠁⣠
⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠀⠀⠀⠀⠀⡀⠤⠂⣷⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠇⠀⠑⢀
⠀⠀⠀⠀⠀⡲⠦⡀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⣶⡂
⠀⠀⠀⠈⢞⢀⣐⠍⡀⠀⠀⠀⢲⠞⠑⠢⠄⣀⠀⣀⠀⠀⠈⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⠀⠀⠀⠀⠈⠀⢒⠈⠭⠉⠭⢙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁
⠀⠀⢀⣀⢀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠄⠀⠀⠀⠬⠀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⡠
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢀⠀⠈⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⣉⡈⠁⠐⠈⠀⣠⣤⠱⠀⡁
⠈⠉⠉⠻⣿⣿⣿⠿⠿⠿⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢿⠇⠀⠀⠀⠀⠘⠛⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠱⣀⠀⠀⠀⠀⠀⢀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⡬⠀⠌⠉⡉⠄⡀⣨⠐⠊⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⡃⡘⠸⠏⠀⠀⠈⠉⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠑⠀⣇⠀⠀⣠⠀⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠇⠠⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣄⢦⣨⡿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⡄⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢠⣤⣶⣶⣶⣶⣶⣶⣶⣄⠐⠲⠦⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⢘⣿⠀⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠷⠀⠀⠀⠀⠘⣆⠀⠀⠘⣷⣤⣾⣿⣧⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢟⡿⠟⠛⢿⣿⣿⠛⠛⠿⣟⠀⠀⠀⢠⣄⡀⠀⠙⢷⣤⣄⡀⠙⠛⠿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⠟⠃⢸⠀⠀⠀⣸⡿⢿⡀⠀⠀⣸⠀⠈⣿⡌⠛⣷⡄⠀⠀⠙⢿⣿⣦⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⢀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣷⣶⣾⣿⠃⠘⣿⣶⣶⡿⠁⠀⢸⢳⡀⣸⣿⠀⠀⠀⠀⢹⣿⣆⣀⣠⣿⣿⡿⢻⣽⣯⣷⣿⡿⠟
⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⡀⠀⠲⢹⣟⢻⣿⣶⣶⣿⣏⣼⡏⠀⢀⠀⠀⠀⠙⢿⣧⡀⠀⢀⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠘⣦⠀⠴⠟⠀⠚⣁⠀⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠿⣦⠀⣀⣀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉
⠀⠀⠢⡀⢸⠃⣄⣤⠞⠑⠋⠀⠁⠀⠈⢻⣿⣿⣿⣿⣿⠟⠱⠇⢸⠏⢀⡻⠇⢨⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⢹⡆⠀⠀⠁⠷⠀⠀⣰⡇⢰⠆⠀⠉⠉⠉⠉⠀⢀⡀⠀⠀⠀⢾⡿⠀⢸⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠰⠿⠤⠤⢀⠀⠀⠀⢀⣴⣿⡿⣋⡀⠀⠀⠘⠷⣠⣾⣿⣿⠿⠋⠁
⠐⢦⣄⠈⠻⣷⣤⠄⣴⣷⣤⡄⠀⠀⠀⠀⠀⣴⣿⣿⡟⣷⡿⠟⢀⣠⣤⣿⣿⠿⠋⠁
⠀⠀⢹⣧⠀⠈⠁⠞⠙⠋⠉⣠⢆⡴⣠⢀⣾⣿⣿⣿⣷⢾⣠⣶⣟⡿⡿⠋⢸⡄
⠀⠀⣼⣿⠀⠀⠀⠀⣀⣠⣤⣤⣼⣤⣵⣿⣿⣿⣿⣿⣽⢿⣽⣿⣻⣿⣿⠀⠸⠇
⠀⠘⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠕⠋⠀⠉⠀⠀⠀⠂
⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣦⣀⣀⣤⡄⠀⠣
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡕⠎⠳⡽⡿⣫⣿⡿⣯⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣎⢢⣐⣺⣯⣵⣿⣿⣿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⣿⣿⡿⠋
⣶⣶⣶⣦⣶⣴⣶⣶⣶⣤⣶⣬⣿⣯⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣴⣦⣤⣴⣦⣤⣤⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠁⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣰⣴⣬⣧⠀⠀⣠⣤⣀⡀⣀⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠐⠒⠲⣾⠫⣴⣶⣿⡏⠐⠲⣞⢠⡫⠁⠙⢷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⡀⠀⠀⠀⠌⢸⡿⠋⠉⠉⠀⠱⣸⣆⡁⠀⢈⠋⢳⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠃⠀⣼⠿⠶⢀⢞⠁⡀⠀⠀⠀⢠⣿⣿⡇⠤⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣨⢴⡞⠉⠁⠀⢋⠂⡸⠃⠀⠙⣦⠀⠀⠸⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠂⠜⠀⣿⠇⡀⠀⠀⣸⣴⣥⠄⡠⢶⡻⡧⠀⢠⣴⠎
⠀⠀⠀⠀⠀⠀⣇⡘⠀⢰⣿⢋⣀⠀⣤⡿⣿⠁⠊⠀⠈⠉⠀⢦⡀⠷⠸⡀
⠀⠀⠀⠀⠀⠘⠿⠇⠀⣼⠟⠋⡀⠀⠈⠻⣾⣆⠀⠀⢰⡷⠁⠀⢷⣀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⣤⣾⡄⠠⠀⡠⠂⠁⠀⢀⢸⣥⡀⠀⠈⠻⣄⠀⠀⠙⠳⣤⣀⡀⢄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢙⠿⣿⣣⣾⡁⠀⠀⠀⣼⡿⠉⠻⢗⢦⣄⣀⠉⠒⠠⠤⢄⣉⠳⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⠉⠉⠉⢹⣷⣭⡴⠊⠀⠀⢰⠋⠀⠈⣿⡄⠀⠀⠀⠈⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠽⣿⡸⣿⣄⣠⢶⣯⣤⡴⠤⢉⡇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢘⢡⠉⠻⢿⠘⠁⢉⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢸⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⡶⣞⠁⠀⠀⠀⠀⠀⠈⢧⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡼⢇⡟⠀⠀⠀⠀⠀⠀⠀⣾⠃⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⡾⢇⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸⣷⡤⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⠀⠀⣀⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣶⡏⡏⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠁⡁⠀⣀⣀⡄⣓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠛⠛⠛⠋⠉⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠛⠛⠛⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠤⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣶⣤⣄⣀⠀⠀⢀⣀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠿⣿⣿⣿⣿⣡⠀⢌⣿⣿⣿⣿⣷⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡆⠀⠀⠀⠀⢸⣿⡿⠛⠳⠶⠛⠁⣠⡍⢻⣿⠃⠀⠀⠀⢹⣷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠇
⣠⣀⣀⣀⣀⣀⡀⠀⣀⣠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⢿⡇⠳⠆⠀⠀⠀⠈⠃⠸⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣴⣶⣶⣦⣾⣿⣿⣿⣿
⠈⠹⣿⣿⣿⣿⣿⣿⣿⡟⠋⢙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠻⡄⠠⣤⣤⣶⣾⠀⠐⢀⣀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠟⠋⠻⣿⣿⣿⣿⠟
⠀⠀⠘⢿⣿⣿⣿⣿⡿⠷⣥⣴⠟⣩⡙⡷⠀⠀⠀⠀⠀⠀⠀⠠⡛⠁⠈⠉⠈⠦⠾⠛⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⣮⣀⣱⣿⢿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⠋⢴⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣤⡄⠀⠉⠁⢠⡆⢹⡿
⠀⠀⠀⠀⠈⠻⣿⡀⠤⠀⠀⠃⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⢈⠤⡀⠀⠀⠈⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠨⠟⣿⠀⠀⠀⢯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⢛⠟⠀⠼⣿⡿⠿⠖
⠀⠀⠀⠀⠀⢤⣄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠦⠎⠀⠀⠀⠈⢡⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⢿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡇⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡧⣀⢳⡜⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣉⣙⣿⣿⣿⣿⣌⡲⣯⡳⣄⠙⠛⠿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⠟⠟⡁⠀⠑⠑⠄⠙⢿⣿⣕⣿⣿⣌⠑⠴⠿⠦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣭⣉⣉⣠⣴⣿⣧⣷⠶⠛⢶⣤⣀⡈⡙⠛⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣷⣤⣭⣻⣿⣿⣷⣶⣷⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠰⠖⠀⠀⢻⣆⢸⣿⡄⠀⠀⠀⠀⠀⢰⡟⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⢠⠴⣦⣄
⠀⠀⠀⠀⠀⠀⣠⠴⣲⠆⡀⠘⠓⠋⠀⠀⠀⠀⠀⠻⢾⡟⢿⣾⣧⠀⠀⠀⣿⠁⠘⠟⠁⠀⠀⣞⠛⣉⣉⣩⡿⠓⠾⠟⠀⣤⢦⣤⡀
⠀⠀⠀⢀⣴⣏⣀⠐⠛⢩⠟⣀⣾⣷⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢮⡐⠻⡆
⠀⢠⠞⠉⠀⠀⠈⠻⣦⡞⣥⣿⠉⠈⠀⢩⡟⠀⣀⣀⡀⠀⢀⣠⡤⠶⠶⣖⠒⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠶⠃
⠀⣿⠀⠀⠀⣀⡀⠀⣹⠀⠈⠉⢻⣤⠤⢿⠀⠀⢯⡉⠉⠙⠻⣅⡀⢀⡀⠈⢳⡀⡀⠀⠈⠙⠳⣄⠀⠀⠀⠀⢰⡶⣤⠏⣷⢾⣆
⠀⠹⣄⠀⠸⠋⣠⣠⠏⠀⠀⠀⢸⡆⠀⢸⡇⠀⢠⣿⣶⡄⠀⠈⠋⠀⣙⣦⡀⠉⠁⠰⠋⣴⣀⣈⢷⡀⠀⠀⠈⢳⣄⠀⠀⠹⣿⡄
⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⣷⢰⡯⠤⢴⡶⣠⠀⣈⣭⣭⡉⠁⠀⠀⠀⠀⣀⣠⣀⠀⢿⡀⠀⢀⡾⢧⣄⣴⠚⠉⠁⠀⠀⣠
⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠹⣿⠀⠀⣾⠞⢁⡼⠋⠀⠀⠙⠶⠚⠉⢻⡄⠈⠳⣄⠘⠛⠉⠛⠻⢦⣠⡟⠁⠀⠀⠀⣴⣾⡋
⠉⠉⠉⣩⡿⢟⣡⣤⣴⣄⣀⣤⠄⣿⠀⠀⠀⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⢳⠀⢀⢲⣤⣄⣴⠟⣠⡾⣿⣦⣴⠾⠟⠋
⠀⠀⠀⠿⠟⠛⠉⠁⠛⠛⠉⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⣿⣦⡧⣰⠏⠘⠉⠀⠉⢁⣠⣤⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡤⠸⡇⠀⠀⠹⣄⠀⠀⠘⢆⣀⣀⡤⠴⣄⠀⠀⣀⣀⠀⠀⡼⠁⢀⡿⠁⣰⠇⢀⡤⠖⠛⢻⣋⣤⡴⠏
⠀⠀⣀⣴⠾⢻⣿⣴⠾⣷⠾⠋⠀⠀⢷⠀⠀⠀⠙⣦⡀⠀⠀⠉⠁⠀⠀⠈⠛⠉⠁⠸⠦⠾⠁⢀⡾⠁⢠⠏⢀⡞⠀⡴⠂⠀⠉⠀⠹⡄
⠀⠾⠋⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢠⡟⠀⠸⡇⠸⡇⢰⣷⣄⢠⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠴⢤⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠖⠖⠒⠛⠋⠁⠀⠀⢠⡟⠀⠀⠀⠳⡄⠁⢻⠀⣼⣿⢠⠏
⠀⠀⠀⠀⠀⠀⠻⣤⣼⡀⠀⠀⠀⠀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣤⣄⡀⠈⢳⠏⠀⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠹⣄⠀⠀⣾⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⡷⠀⠈⠓⠛⢡⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠹⣆⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣷⠀⠀⣀⡞⢻⠷⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⣿⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⡀⠀⠀⢻⠀⠀⠀⠀⠀⡇⠀⠘⢧⣤⠀⢠⠶⠞
⠀⢀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣈⣷⣄⣀⣀⡶⠋⠁⠀⠀⠀⠀⠀⠉⠳⣄⣸⣄⣀⣀⣀⣠⣧⣀⣀⣀⣈⣉⣉⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⢛⡏⢀⣸⣧⡀⢠⣠⣀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣬⣥⣭⣤⣤⣅⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣸⠋⠉⢉⡽⠗⠈⢧⣀⠀⠈⢯⠉⠁⢹⡙⢀⡁⠉⣡⢶⡖⠛⠛⢛⡷⠶⣄⠀⠀⠀⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠰⣏⣉⣁⣀⣀⣀⡼⠃⠀⠀⠀⠈⠙⠲⢬⣳⡄⠀⠀⠛⠋⠀⠙⠒⣃⠀⠀⡼⢦⣀⡘⣇⠀⠀⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠖⠚⠋⠉⠙⠛⠁⠀⠈⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣀⠀⠀⡲⣄⠀⠀⠀⢀⡀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠁⡤⠀⠀⠄⠀⠀⠰⠂⠀⠀⣴⠄⢀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠸⢊⠆⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⣐⠀⠨⠥⠤⣀⣈⣓⣀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠈⢇⣒⠀⠀⠀⠋⢐⡖⠂⠬⠭⠭⣝⣚⣒⣒⣒⣶⣒⠒⢲⡖⠒⠒⠒⡾⠉⠉⠀⣀⠄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠒⠒⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢀⣠⣧⣤⣬⣤⣴⣶⠮⠍⣉⣽⡋⠉⠉⠒⠒⠢⠤⠾⣀⣀⠐⠀⠀⠀⠀⠐⣒⠚⠉⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠉⠑⠒⠢⠭⢍⣓⡒⠤⢄⣐⡀⠀⠀⣨⠉⠁⠀⠰⠋⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠛⠛⠓⠒⠒⠂⠀⢠⠀⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠑⠢⢄⢀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠙⢿⠃⣀⠤⠒⠈⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⡶⣶⣷⣄⡀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⢀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠆⢶⣽⣿⣿⠊⢱⡉⣇⠸⡀⠀⠀⣀⠔⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣤⢩⣿⣿⣿⡿⠟⠛⠻⢷⣶⣶⡿⠛⡿⠣⠼⠥⠜⠀⠈⠉⠉⠀⠀⠀⠸⡀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠚⠛⢛⣿⣿⣿⣼⣿⣿⡟⠀⣰⣄⡀⠀⠉⠉⠀⢠⠁⣠⠔⠂⠉⠉⠐⠒⠤⣀⠀⠀⠀⢱⠚⠁⡠⠧⡄
⠀⠀⠀⠀⠀⠐⢶⣶⣌⡀⢚⡿⢿⣿⣿⡅⠀⠘⠛⠁⠀⢠⠄⠀⠻⡞⠐⠦⡀⢀⣀⠀⠀⣠⡀⠱⡀⠀⢸⠀⠂⠀⠔⢣
⠀⠀⠀⠀⠀⠀⢀⠤⠭⢽⠋⠶⠛⠛⣻⣿⣦⣀⠀⠀⠀⠠⠄⢀⡠⠷⣀⠀⠀⠈⣁⠀⠀⠛⠁⢀⠇⡠⠧⠤⢌⠀⠐⡇
⠀⠀⠀⠀⠀⠀⡇⢀⠀⠃⠀⠀⠀⠶⠛⠋⠉⠉⠉⠙⠛⠛⠛⠿⢆⣀⡨⠿⣐⣲⠤⢤⡤⠤⠴⢚⡸⠁⠀⠀⠀⠹⠋⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠙⠚⢪⢀⡀⡠⠞⢀⡀⠀⠀⢈⠖⠤⢀⣀⣀⣀⣀⣇⠀⢀⠀⠀⠀⠀⠉⠒⠈⠁⠀⠀⣀⠀⠀⠀⠀⣀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡰⠤⣀⣀⠽⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠒⠯⢤⡄⡤⠒⠈⠓⠠⠍⡎⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⠒⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣹⠢⢄⡀
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⠃⠭⣁⠀⢿⡲⢦⣄⣀
⠀⠀⠀⠀⠀⢠⣿⠀⡦⣿⠃⡶⠶⣂⣿⣿⠑⠊⠀⠀⣄⠀⠍⠀⢈⢁⠠⢠⣴⣛⢺⣿⢦⡀
⠀⠀⠀⠀⠀⢸⣟⣠⣏⣿⠄⣠⡄⢹⣻⣿⡇⠀⢔⣦⣟⡥⠀⠀⢸⣙⠺⢶⣚⡋⣸⣿⡇⢩⣭⡄
⠀⠀⠀⠀⠀⢸⡏⠿⢿⣿⡄⢽⠇⢾⣿⣿⣇⣰⣶⣖⣤⡥⠀⣶⣿⣶⣾⣤⣤⣥⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣷⢲⣾⡇⣒⣀⣤⣼⡿⣿⡿⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠸⣿⣄⣿⠟⠛⠉⠁⠀⠀⣿⡇⠀⠀⠀⣤⡿⣿⣿⣿⣻⠿⣛⠋⠉⠉⠛⡋⢹⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣼⡿⢱⣹⣿⡿⢿⡟⠁⠀⠀⠘⣆⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠟⠁⡾⠿⣿⢃⡟⢀⣠⡤⠶⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⣿⣿⠇⡼⠱⣿⣥⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠟⢸⣷⠶⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠹⢿⡦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏⠉⠻⣿⡀⠀⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⣾⣿⡟⠃⠈⢿⣿⣆
⣷⡀⠀⠀⠀⢀⣠⣤⣿⣿⠟⠁⠀⠀⠀⠛⠟⠛⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠹⣿⣷⡀
⡿⠁⠀⠀⣤⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡆
⠀⠀⠀⢰⣿⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠸⠛⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡆⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣴⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣿⣷⣤⣀⢀⡘⠁
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠛⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠛⠿⢿⡿⠃
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠛⢿⣿⣿⣿⣷⡀⠀⠀⠀⣸⣿⣿⣿⠋⠀⠙⣿⣿⣿⣧⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠘⣿⣿⡇⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⡿⠃⠀⠀⠀⣼⣿⡿⠇⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⠟⠀⠀⠀⠀⠀⢹⣿⠂⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⣶⣶⣤⡁⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⢿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠀⠀⠀⠀⠈⠙⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⢰⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⣿⡿⠟⠋⠀⠀⠙⠿⢿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⢻⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣾⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣿⣄⣄⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢴⣤⣀⠀⠈⠹⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⣿⢿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⣰⣿⣷⣤⠈⠿⣿⣿⣦⠀⠈⠻⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⡇⠈⣿⣿⣿⣿⠋⠉⠀⢀⣠⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣧⡀⠀⠈⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⣤⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⣧⣼⣿⣿⡟⢁⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠹⣿⠀⠀⢸⣿⣿⠋⠀⣠⡔⣿⣷⡀⠀⠻⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣿⠀⠀⢸⣿⡇⢀⣼⣿⡇⣿⣿⣿⣆⠀⠈⢿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡿⠀⢸⣿⣿⡇⣿⣿⣿⣿⣷⣦⡀⠈⣿
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢻⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣄⣈
⠀⢸⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠘⠛⠛⠛⠛⠁⠤⠄⠜⠛⠛⠛⠛⠛⠛⠛
⠀⢸⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⢸⣿⡏⠀⢸⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣷⠀⠈⢧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣦⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣀⣼⣿⣿⠿⢿⡄
⠀⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣿⣿⣿⢀⣼⡇
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀⣠⣤⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⡀⠈⠛⠉⣿⣿
⠀⢸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣾⣿⣿
⠀⢸⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢧⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢤⣄⣀⠀⠀⠹⣿⠀⢸⣿⣿⣿⠏⠀⠀⢀⣴⣦⡀⠀⠀⠙⣿⢳⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠈⣿⣿⡧⠀⠀⣿⠀⢸⣿⣿⡿⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠐⢿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⣼⣿⣿⠇⠀⠀⣿⠀⢸⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠹⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠋⠁⠀⠀⣠⣾⣿⣿⣿⠀⢠⣾⣿⠀⢸⡿⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⡀⠀⠈⢿⣿⣿
⠀⢸⣿⣿⣿⡇⢀⣾⣷⣾⣿⣿⣿⣿⣿⣄⣼⣿⣿⠀⢸⡇⠀⣾⣿⣿⣿⣿⣿⣧⡈⠀⠻⣿⣿⣿⣤⡀⠀⠛⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠈⠉⠤⠠⠠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠒⠰⣖⠃⡀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⡠⠂⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⠀⠀⠀⣀⡀⠀⠰⠀⠀⣀⣤⠶⠶⢚⠿⣿⠿⢿⣿⡗⠒⠒⠒⠒⠂⢪⡍⢓⠆⠀⠀⣺⣿⣇⠀⠀⣀⣠
⠸⡀⠀⠈⠁⠀⢀⡏⢀⠰⡇⠱⢤⣀⣘⠢⠕⢲⣤⠤⣤⣤⣤⣶⡲⠾⠓⠊⡁⣤⣀⢰⣟⣛⣻⢶⡿⠿⣿
⠀⠈⠐⡄⠀⠀⠀⠰⡁⢀⠈⠒⠤⢄⣉⣉⡉⠉⠛⠉⠉⣉⣉⠤⠛⠀⣀⡈⠓⠍⠹⠀⠙⠛⠉⠘⠿⢿⠿
⠀⠀⠀⠐⢄⠀⠀⢀⡈⢠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠂⠐⠦⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⡀⠱⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠁⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⣿⣿⣿⣿⡿⠶⠶⠤⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣴⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣉⠇⠀⠀⢸⠿⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢛⣋⣁⣤⣽⡄⠀⠀⠀⠀⠀⢀⣿⣤⣤⣈⣙⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠛⠛⠻⣿⣿⠛⢻⣿⣿⣿⠛⠻⣿⣿⠛⠛⠛⠛⠛⣿⣿⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠶⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠸⡶
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣦⣤⣶⣾⣿⠟⠁⠀⠙⢿⣿⣷⣶⣤⣴⣶⣾⣿⣿⣿⠃⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠞⠋⠉⠳⢼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢶⡂
⠀⠀⠀⠀⢀⣤⣀⣠⣄⠀⢙⣿⡿⢿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⣿⢿⡋⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣸⣶⣿⠆⠈⢻⡉⠀⣇⢼⣁⠀⠹⣿⣿⣿⣿⠟⠉⢀⣈⠇⣰⠃⠀⠹⣄
⠀⠀⠀⠘⣇⠉⠡⣼⡶⠆⠙⣆⠸⣆⡞⠁⠀⠙⣿⡿⠁⠀⠀⢙⣦⡜⠁⠀⠀⠀⢸⡆⠀⠀⠀⠀⣄
⠀⠀⠀⢠⣿⣷⣄⠘⣟⢹⡆⢘⡆⠘⢏⡉⠉⠀⠀⠀⠀⠉⣉⡽⠋⠀⠀⠀⣷⣴⣾⠁⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠙⠛⠛⠳⣌⣩⡶⠋⠀⠀⠀⠙⠦⣄⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⠀⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢾⣁⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣤⡤⠠⣤⣤⣬⠏⢹⣀⣀⣀⣀⣠⣤⣤⣤⠶⠾⠃⠀⠀⠀⠀⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠀⠀⢀⡗⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⢀⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣦⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠋⣰⣾⣿⣶⡄⢰⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠃⠙⢿⣿⠿⠃⠆⣹⣿⣿⠁
⠀⠀⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠂⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠘⠓⠶⢶⣦⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⣀⣾⣿⣿⡿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠛⢿⡟⠙⠁⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠸⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠄⠀⠀⠀⠀⢰⡆⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⢰⡄⣀⣀⣰⣄⣾⣷⡄⢸⣿⡇⢀
⠀⠀⠀⠀⡇⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣴⣧⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣀⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣤⣴⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⣦⣿⣟⣠⡟⢀⣴⣰⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠟⢻⣿⣿⠿⣿⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⠁⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠤⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⣁⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⢀⡀⡀
⠀⠀⠐⠘⣆⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡃⠀⠀⢈⠇⠐⠀⠀⡧⢴⡷⢱⣂⠃⠀⡄⢄⡀
⠢⡄⠀⠀⠈⠀⠀⠀⠀⠜⢡⠁⠀⠐⠻⠿⠟⠀⠀⠀⠀⠀⠙⠁⠀⢠⡏⠀⠀⠀⠀⠳⡮⢻⠞⠿⡾⡟⣱⣈⠨⡦⡀
⠀⠀⠠⠀⢤⣬⣶⣦⣴⡀⠀⠐⠛⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠊⠹⣸⡟⡔⡛⠉⢈⠰⣛⣧⡀
⠀⠀⠀⢀⢿⣟⣷⣟⣿⡿⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠐⠛⠀⣉⠀⢈⠉⢁⠋⠉⠘⢋⡐⠞⢥⣷⡄
⠐⠂⠉⠉⠛⣟⢿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⢘⢻⡞⡊⡥⣈⠉⠀⠺⠟⠓⠚⢴⣻⡇
⠀⠀⠀⠀⠀⠐⠟⠓⣧⠄⠒⠂⠦⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡄⠀⠊⠅⡶⣼⣇⡀⠈⡐⠀⢲⢄⣼⠟
⠀⠀⠀⠀⠐⠀⠔⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⠀⠐⠃⢾⣷⣀⡿⠞⠹⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠐⠀⠀⢠⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣈⡀⠀⠀⠐⠊⢁⣠⠆
⠀⠀⠀⠀⠀⠐⠀⠀⣾⡖⢒⠀⠀⠀⠀⠀⠠⠀⠀⠉⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⢤⡀
⠀⠚⠹⡎⢣
⠞⠀⠀⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣀⣴⡶⠀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠀⠰⠟⠙⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡟⠉⠀⣀⣤⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠹⠶⠞⠋⠁⠉⠠⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠟⣷⡀⢀⡇⢀⡆⢰⡖⢢⡔⠚⡇⣸⠛⡷⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⡟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠆⠀⠀⠀⠳⠤⠞⠃⠀⠘⠷⠾⠷⠿⢤⠟⠀⠀⢧⡼⠷⠃⠘⠀⠺⣴⣯⡄
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠠⣾⣠⣴⣶⠏
⠀⠀⠀⣘⢢⡤⡄⠀⠀⠸⡟⢸⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠘⠂⢀⠀⠆⠀⠐⡷⢸⣿⠛⣷⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠏
⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡛⣷⣿⣤⠀⠀⠈⢢⣀⡾⠟⡻⠉⠁⠛⠉⠙⠛⢿⣶⣦⣄⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡟⠋
⢰⠃⠀⠀⠀⠀⢸⠀⠀⠀⠈⠑⢻⣿⡿⢻⣆⢠⣀⣼⣯⢀⢀⡵⢂⣔⠒⠲⢤⠚⠤⡁⢼⡿⠋⠀⠀⠀⠀⠀⢀⢤⣴⣴⣿⠿
⠘⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠹⣟⣿⣿⣷⡘⡿⠀⡸⠩⠀⢠⡖⢱⠋⠰⡟⢦⣤⣤⣄⣐⣶⣷⣶⣿⣿⣿⠻⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢹⣿⣿⣿⠷⠛⢻⡀⠢⠀⠀⠀⠋⢀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁
⢤⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⠃⢠⡀⠀⢷⡀⣀⡤⠖⠚⠉⠙⠿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠸⣿⣦⠀⠉⠀⠀⠀⢀⣀⣀⣀⠹⣿⣿⣿⣿⠁
⠀⠘⢧⠃⠈⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠛⠦⣿⣆⠀⠉⠁⠠⡄⠀⠀⠀⠹⣿⣿⠃⢀⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀⠀⢶⣶⡗⠀⠀⠀⠀⣀⣾⠿⠛⠁⠐⢻⣦⣀⢀⣀⣠⡔
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⡔⣋⣉⣛⣿⣿⣷⣶⣶⣶⣶⡶⠋⠀⠀⣀⠴⠞⠛⠛⠛⠛⠋⠁
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⠀⡄⠀⠀⠉⠁⠉⠉⣉⡈⣏⠉⠛⢏⠈⢿⠟⠁⣴⣾⠿⡆⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⢠⡄
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠻⡄⠀⠀⠹⢻⠀⠀⠀⠀⠀⠈⣦⣀⣸⠿⠿⠀⠀⠀⠀⠀⣀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢶⡄⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠈⠙⣿⠁⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠛⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠙⡆⠀⠀⠀⠈⠁⠠⡀⠀⠀⠀⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠤⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡄⠀⠄⢠⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⡄⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠉⣛⠀⠀⠁⠠⣄⠈⢢⡀
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢀⡴⠃⠀⠂⢀⠠⠊⠀⠀⠀⠀⠀⢀⠠⠠⠅⢤⡀⠀⠀⠀⢀⢤⡖⠀⣰⡖⠀⠀⠀⠴⠊⡀⠀⠃⠀⠈⠄
⠀⠀⠀⠹⢶⡔⠚⠤⠰⡆⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠊⢙⡇⠀⠀⠘⠀⠀⠒⠉⠀⠀⠀⠴⠋
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠉⠀⠀⣤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠠⢂⠀⠈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢔⠀⠀⠠⠐⠊⠁⠀⢀⠄⠀⠀⠐⠀⠀⠈⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠬⠆⠊
⠀⠀⣤⠀⠀⠀⡇⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠰⠆⠀⠀⠀⠀⠚⢲⠇
⠀⠀⠙⣧⠀⢀⡼⢇⣀⣀⡀⠑⠀⠀⠘⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⠈⠂⠀⢰⢦⡀
⠀⠀⠀⠘⣧⠸⡅⠻⠩⡉⢻⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠤⡞⠙⢃⠃
⠀⠀⠀⠀⠘⣧⠈⠒⢒⣚⡏⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠉⢻⡦
⠀⠀⠀⠀⠀⠘⣧⠀⠈⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠙⠁⠀⠠⡆
⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⡀⠐⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⡤⠤⠔⠀⠀⠈⠛⣋⠉⠀⠀⠀⢀⣀⣠⠀⠐⠁
⠀⠀⠀⠀⠀⢈⠁⠀⠆⠨⠈⠡⠐⢈⠀⠄⠀⠀⠈⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢰⢫⣤⣄⠇⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⢠⣤⠀⠐⠀⠈⡉⠃⠀⠐⠀⠂⡄
⠀⠀⠀⠀⡏⡿⢈⡗⠂⠀⠀⠀⠀⠀⠈⠰⠆⠀⢀⡨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡄⠀⠀⠉
⠀⠀⠀⠐⣧⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⡿⠀⢀⣀
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡶⢦⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠄⠀⠀⠀⣸⢀⡀⠀⢠⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⣰⣤⠚⣀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣷⣧⣾⣥⣶⣷⣾⣷⣼⣯⡴⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⢿⣿⢿⠏⣻⣷⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣷⣿⣿⡿⠿⣿⠟⠻⢿⣿⣿⣶⣭⣍⣙⠻⠿⠿⠿⣿⣿⣿⣿⣿⣶⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡀⣾⠛⣿⣿⣿⠟⠁⡘⢀⡄⠀⠀⡿⠿⠟⠛⢻⣿⣿⣾⣷⠂⠀⠉⠛⠿⣿⣿⣿⣮⣄⡿⠋
⠀⠀⠀⠀⠀⠠⣝⢿⣴⡏⠸⣷⡆⣠⢁⣾⣷⡇⢾⣅⢀⠆⢀⠎⠹⣿⠟⠁⠀⠀⠀⠀⠀⠀⢩⣭⡍⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠠⠌⢻⡉⣧⢱⣼⣷⣿⣸⣿⣿⣷⣽⣿⣿⣶⣮⣼⣹⣯⣀⣤⡀⣀⣤⣤⣄⠀⢸⣿⣃⣤⣤⡀⠀⢀⣠⣤⣤⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⠁⣿⡟⠉⣹⣿⣋⣉⣿⣷⢸⣿⠋⠁⠹⣿⡄⣾⣏⣉⣹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⡏⠉⠀⠀⠀⣿⡇⠀⢹⣿⣍⣉⣭⡍⢸⣿⣆⣀⣠⣿⠁⢿⣏⣉⣩⣭⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⣿⣿⣿⢿⣿⣿⣿⣿⡀⠂⠀⠀⠀⠀⠀⠙⠃⠀⠀⠙⠛⠛⠉⠀⠘⠛⠛⠛⠋⠁⠀⠈⠙⠛⠛⠁⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⠻⣿⣿⡋⠓⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣟⣶⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡆⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣿⣿⣿⣿⣿⢿⣾⣻⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⡀⣄
⠀⠀⠀⠀⠀⢠⡾⣿⣷⣿⣿⣻⢼⣿⣿⣿⣿⣿⣽⣷⡀⠀⠀⠀⢀⣿⣿⣿⣿⣏⢾⣟⣽⣏⣷⡠
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣾⣵⣯⣶⡄⣰⣽⣿⣿⣿⣿⣹⢿⣿⣷⣿⣳⣿⡆
⠀⠀⠀⠠⣼⡿⢿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣠⠠
⠀⠀⠀⢠⣿⣿⣾⣿⣟⣿⢿⣿⣿⣿⣿⣿⢿⣇⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣯⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⡿⠾⠛
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣿⣿⣿⣿⣿⡷⣿⣿⣿⣼⡿⡿⣿⠿⢇
⠀⠀⠀⠀⠁⣹⣿⣿⣿⣾⡿⠟⠻⣻⣿⣿⣿⣿⣿⣯⡞⡾⣿⣿⣿⣿⣝⣿⣿⣿⣿⣶⣦⣿⣤⣠⠤
⠀⠀⠀⠀⠀⠿⠛⠛⢿⡅⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠛⢿⣿⣿⣻⣿⣿⣿⣿⣟⡹⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⢟⣿⣿⡿⣿⡿⠿⠟⠁⠀⢀⣴⣳⣿⣏⣿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡾⠿⠛⠉⠁⠀⠀⠀⢀⣴⣿⣿⠽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠒⢴⠋⠀⠀⠀⠀⢀⠞⠀⠀⢀⠀⠀⠀⠀⢠⠋⢉⢲⡖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⠋⠀⠀⢀⣏⡩⠛⠊⠉⣏⠀⠫⠋⠉⣯⡉⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠠⡀⣄⠀⠀⠀⢠⣇⣀⠜⠁⠀⠀⠀⢸⠋⠀⠀⠀⠀⠈⠛⠉⠙⢺⣉⡱⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⢻⠁⠉⠁⠈⠁⠀⠈⠑⢄⠀⠀⠀⠃⢠⠞⠛⣆⠀⠀⠀⡴⠚⢲⡉⢳⣽⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠗⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠐⢚⡄⠘⢦⣠⢏⣠⣤⡀⢧⣀⡰⠃⢸⣽⠧⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠤⡇⠀⡔⠒⠲⡀⠀⠀⢰⠋⠉⡆⢸⢀⣡⠞⢦⣀⠀⡟⡫⡓⣼⡆⠀⢀⣠⠿⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠀⢧⣀⡠⡗⣺⠯⣾⠦⠴⢃⠎⠺⢤⢧⣤⠞⣏⡿⢴⠼⣻⢓⠟⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⠋⠙⣿⣗⣝⡼⢤⠚⠁⠀⠀⠀⢎⠘⢄⣸⠃⠀⠀⠀⠘⡴⠊⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⣀⡼⠋⢙⠻⣏⡯⡄⠀⠀⠀⠀⠈⠛⠉⢹⠀⠀⠀⠀⠀⠣⢤⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠁⠁⠀⠒⠋⢹⠉⠀⡷⠀⠀⠀⠀⠀⠀⡴⠋⠉⢑⡟⠒⠒⠲⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠦⠤⠤⣤⣀⠜⠒⠒⠁⠀⠀⠀⠀⠀⠀⠳⠤⠤⠞⠣⠄⠀⠤⠊

⠀⠀⠀⠀⠀⡀⠐⠉⠩⠒⣄⠈⡀⢀⡀⠀⠂⠀⠀⠄⢀⠀⠀⠀⣀⠀⠀⠀⠄⠀⣀⠀⠠
⠀⠀⠀⠀⠘⠀⠀⢠⠊⠨⠱⠀⠂⠀⠀⠂⠠⠁⡠⠤⣿⡷⣼⣦⠈⠴⣦⡀⠀⣀⡁⢣⣦⡀⠀⢐⠀⠀⠀⠠⠀⠄⢀
⠀⠀⠀⢰⠀⠀⠀⡈⠧⠐⠈⠀⠢⠀⠀⠀⠀⠘⠃⠄⠈⠳⠈⡙⢤⣤⡿⠃⠀⣸⣿⢋⣭⣶⣤⠤⠈⢢⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠐⠇⠂⠀⠀⠀⣿⣿⣶⣦⣀⠨⡀⠌⠐⠦⠹⠐⢌⠠⠂⠀⡙⠂⣿⡿⠋⠀⠀⠀⠃⠡⠀⠀⠐⡀
⠀⠀⠀⠀⠁⠀⢀⠀⠐⠄⠀⣁⠔⠋⠏⡟⡿⡙⠷⣑⡄⠀⠄⠀⢴⠁⠀⠀⡀⠰⢃⡋⠀⢠⠐⠊⣀⡿⣰⠄⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠐⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⢡⡀⠦⠙⠀⠄⡀⠼⢥⡀⠰⢀⠄⠈⠓⡀⠐⠊⡉⢡⠇⠌⠸⠀⠀⡰⠂⠂⠀⠀⠀⡀
⠀⡄⠀⠠⠀⠘⠀⠤⠀⠀⠀⠈⠀⠀⠀⠨⣄⢀⠰⠀⠄⢅⠘⠖⢄⠂⠐⣦⣈⠜⠀⠀⠀⠀⢰⠁⠈⠀⠀⠀⠛⢀⣉⡠⢀⠀⠠⠄
⠀⠐⠀⢠⡾⠋⡀⠀⣶⠆⢠⠀⡀⠀⠀⠤⠀⢦⠀⠀⡘⠂⠑⢀⣨⠬⠇⡫⢀⠔⠀⠀⠄⠀⢄⠘⠠⠀⢀⣀⣴⠏⣠⡘⠠⡑⠀⠀⠄
⠀⠀⠀⠜⠇⠀⠀⣾⣿⠉⠁⠀⢹⣿⣷⣷⡶⠂⠀⠉⠛⠓⢄⡘⠐⠉⠚⠉⠀⠐⠁⠀⠀⠀⠉⢈⠱⠚⠁⠈⠙⢂⣑⣠⣄⠇⠀⠈
⠀⠀⠀⠈⠈⠻⣶⣷⠎⢠⠒⢀⠀⠈⠉⡏⠀⠠⢁⠀⠠⠠⣨⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠄⠁⠰⠈⠙⣋⠀⢀⡄
⠀⠀⠀⡀⣅⣼⣿⡿⡀⣸⠙⠀⠌⠠⣀⡀⢀⡰⠁⠀⢀⠀⠐⠈⠂⢀⡀⠄⠐⠈⠀⠀⠀⡀⠄⠂⠉⠀⠀⣠⠐⠁⠀⠇⢠⣴⣿⠟
⠀⠀⠀⠀⢈⡿⠋⠀⠉⠀⠀⠀⠀⠀⠆⠀⢤⡄⠀⠀⢈⠁⠠⠀⠀⠀⠀⠀⢀⠡⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⣿⣯⡿⠂
⠀⠀⡀⠘⡄⡰⢀⠂⢀⣤⣤⡔⣈⣬⠃⠈⠁⠁⡀⡠⠰⠀⠀⠀⠐⠄⠂⠉⠀⠐⠀⠀⠀⠙⠁⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠙⠁⠁
⠀⠀⠀⠀⠇⠀⡈⢠⡿⠛⠻⠀⢿⣤⠀⠀⠀⣾⠀⣟⠛⠒⠁⠠⢀⣐⡀⠠⠐⠰⠀⠀⣐⠆⠀⠄⠀⠀⠀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠻⠾⢃⡗⠂⠉⡃⠀⠀⠸⡄⠘⣄⠀⠤⠰⠊⠁⠀⠀⠰⢀⠐⡠⠀⢂⣵⣷⣦⠄⢀⠀⠠⠁⠀⠀⡀⠀⠠
⠀⠀⠀⠁⠀⠀⢀⡰⠀⠠⣿⠁⠁⠈⠁⠀⠀⢀⡎⠀⡀⠑⠒⢀⡶⠀⠀⠀⠄⠀⣼⡿⠟⠁⠀⣉⡣⡄⠊⣼⠃⡀⠠⠈
⠀⠀⠀⠀⠀⠈⠻⣿⣶⣟⡌⠀⠀⠀⠀⡠⣶⣟⣿⣦⣅⡀⠀⠀⠂⠀⣁⣀⡶⣓⡹⣶⣥⣀⡡⡌⠁⠀⢞⣛⣋⣠⣶⠄
⠀⠀⠀⡐⡏⠀⠀⡙⢿⣟⣂⣀⡀⡀⡉⠡⣸⠋⠉⠀⠈⠀⠁⠈⠩⠃⡸⡐⠒⠤⠐⣟⠟⠪⠀⠈⣢⠲⢞⡛⠛⠉⠁⠀⣀⡀⢀⠄
⠀⠀⠀⡆⠌⡠⠉⠀⢀⢻⣿⡟⠟⠋⠁⡩⠀⢀⠔⡠⣆⠘⠇⣀⢠⢹⣙⣤⡠⣤⢎⠑⠄⠻⢀⠖⠂⠁⡀⠀⢖⢂⣤⡾⠿⠓
⠀⠀⠀⡀⢠⠘⠀⠀⠐⠈⠀⠁⠰⢀⣠⣶⣿⡆⠒⣙⠔⢢⣀⢸⠀⢨⠫⠔⠨⠒⡡⢳⣄⣐⠋⠁⠄⣂⢁⡾⠻⠆⡿⠃
⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠟⠛⢿⣷⣄⠈⠀⠐⠉⠉⣠⡖⠡⠊⠕⢊⠀⠀⣸⠆⠐⠸⡟⠁⠛⠿⠛⠰⠁⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠐⠠⠌⠋⠈⠙⠿⣿⣿⣿⣆⠑⢉⢆⠡⡈⢌⡜⠀⠀⣀⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠁⠀⠙⠂⠡⠀⠔⠺⢿⣿⡿⠃⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣦⣀⣴⣶⠀⠀⠀⣠⣴⠶⠶⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢨⣿⠁⠹⣇⠀⣼⠏⠁⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⢀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⡀⣸⡇⠀⣀⣽⣶⣿⣴⡟⠻⣦⠀⠀⠀⠸⣷⣀⣀⣀⡀⢠⡿⠁⢻⣶⣿⠿⠛⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠛⠿⣷⣾⠟⠉⠀⠀⠉⢿⣆⣿⠀⠀⠀⠀⣿⠋⠛⢛⣿⠟⠁⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣴⠾⣿⠀⠀⠀⠀⠀⢸⣿⣧⣄⣀⣠⣾⠏⠀⠀⠙⢿⣤⡀⠀⢸⡿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣀⠙⣷⣦⣤⣤⣶⠿⠁⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠿⠟⠁⠀⠀⠀⣤⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⠙⠛⣹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣶⣤⣤⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢿⡓⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢻⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⢈⣷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠶⠶⢶⣾⣿⣤⣀⡀
⠀⠀⠀⣤⣶⡾⠟⠛⠋⠹⣿⣄⣠⣤⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠶⣤⣤⣾⠟⠀⠉⠉⠛⠛
⢻⣷⢦⣄⡈⠻⣿⣷⣶⣴⡟⢻⣿⣤⡶⣴⡿⠛⠉⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣾⠟⠛⢷⣶⣶⡶⢶⣦⣤⣀
⠀⢉⠛⠛⢿⡶⣿⣟⠙⢎⢻⣶⡿⠛⣿⡿⠁⠀⠀⠀⠀⠀⠿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠛⠛⠿⣿⣶⣾⡏⠙⠁⠀⢀⡭⠿⠷⠄
⠀⠹⣟⢦⠀⠉⠳⣍⠳⣾⣿⡿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⠻⣷⣶⣶⡶⠿⠿⠿⠛⠛⢉⡿⠃⠀⠀⠀⢸⣿⡟⢛⣃⣀⣀⡟
⠀⠀⠘⢦⡑⠦⣤⠞⣨⣙⠻⣦⣀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⣤⣤⣤⣿⣄⣴⠟⠉⠉⠙⠃
⠀⠀⠀⠀⠙⠷⠶⠾⠟⠉⠳⢦⣙⠳⢦⣹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠙⣦⣼⠋⠀⠀⠀⠁⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣍⠻⢿⣷⣤⣄⣀⠀⢀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⣀⣤⣤⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣌⣻⡿⠛⢻⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣛⣋⠉⢿⣏⣽⣿⣷⡄⠀⢶⡶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⣾⠋⠉⠉⠛⢿⣏⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⣈⣙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠙⠛⠛⣶⣤⡀⠙⢷⡄⠀⠀⠀⠀⠀⠘⣧⡿⣯⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣯⣤⣤⣄⡀⠀⢀⣠⣾⠟⠉⢻⣦⡀⠻⣦⡀⠀⠀⠀⠀⠸⠇⢈⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣁⣀⣀⣀⣤⣤⣴⣾⠿⠛⠉⠉⠁⠀⠉⠉⠛⣿⡟⠋⠀⠀⠀⠈⣿⠻⢦⣌⠛⠶⣤⣤⣤⣤⣴⠾⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣸⡿⠀⠀⠉⠛⠶⣦⣤⣬⣤⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣀⣀⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣀⣀⣀⣀⣀⣀⣤⣶⠟⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⣠⣤⣶⣾⡿⠿⠟⠛⠛⠉⠉⠉⢻⣿⣷⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠉⠙⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠿⠋⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⢿⣿⣿⣿
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⣇⣀
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠘⠒⠠⡄⢐⡄⠤⠤⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡎⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡄⠀⠀⠀⠀⡀⠈⠁⠀⠰⢎⣅⠀⠀⠀⡠⠔⠰⢲⢔⠤⢀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⡠⠄⠒⠒⠥⣄⠱⡀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠉⢢⠌⠄⣀⠤⠘⠂⣀⡀⠐⣐⡌⠀⡃
⢰⠉⢁⠎⠀⠀⠀⠀⠀⠀⠱⣿⠁⠀⠀⠀⣀⢈⠀⠀⠀⢈⠒⢎⣨⣁⠔⠒⠔⢂⠀⠁⠀⠑⠈⣈⠖
⠀⢣⢸⠀⠀⠀⠀⠀⠔⢢⠊⠉⠲⡺⠂⠁⠀⡠⠀⠀⠀⡸⠀⠀⠀⠘⠤⠿⡽⠍⡅⠁⠀⠃⠀⠀⠀⠀⠂
⠀⠈⠹⡀⠈⠉⠀⣶⣤⣼⢀⢦⡀⠄⢀⠄⠊⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡈⠈⡷⢒⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠉⠳⠟⠻⣙⣫⠇⠑⠀⠀⠀⠀⢑⠄⠀⠀⡠⠚⠣⡈⠁⠀⠒⠀⠀⠁⡠⠇⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠹⢺⠀⠀⡄⠀⡀⠰⢺⠀⠀⢀⢇⠀⢀⣽⡒⠆⠀⢐⢒⣍⠀⠀⠀⠀⠀⠰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢖⡀⢠⠲⡐⠏⠈⠉⠁⠀⠈⠁⢰⠑⠊⠀⠱⠊⡴⠃⠲⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡃⠀⠀⡀⠂⢎⣲⠎⢶⡶⠡⡖⣩⠤⠀⠤⠤⠴⠒⠢⠤⠜⠛⣇⡀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠴⠉⠇⠀⣷⣀⣠⠴⡑⢺⠁⠑⠒⠉⠒⠊⠠⢀⠀⡀⠜⠂⠀⠀⠀⠈⠂⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⣠⠤⠒⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⠠⡎⠀⣀⡴⠋⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣟⢀⡤⠛⠉⠀⢀⠠⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠁⠀⠀⠀⠀⠀⠈⠀⠋⠀⣠⠦⠤⠀⠀⠤⠦⠤⠬⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⣀⡀⣠⣊⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⡠⠚⠉⠰⠶⢉⣀⢰⣎⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢄⣈⣻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠱⠇⠐⢀⡀⢀⠀⠐⠈⠀⠄⠓⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠋⢃⣤⠈⠑⠈⠀⠐⠃⣎⠀⠘⠉⠱⡝⡄⠀⠀⠀⠀⠀⠀⢺⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢁⠠⠌⢀⠀⠈⠃⠀⠉⢁⠀⠸⠉⢠⢀⢹⡀⠀⠀⠀⣀⣴⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣦⠳⡄⠘⠀⠃⡐⠀⢠⠛⣄⠀⠀⠒⡺⢸⠀⠀⢠⣴⣿⣿⣿⠳⢶⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⡳⢬⡐⠦⣀⠛⠒⢂⠀⢠⣄⡤⢞⣠⣿⣤⣶⣿⣿⠟⠋⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣩⠓⠶⠤⣭⣭⣭⡥⠥⢖⡿⣛⠛⠋⠉⠉⠀⢰⣩⡆⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣦⣴⣤⣤⠴⠖⠊⠁⠀⠈⠉⠑⠒⠒⠒⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣈⠴⡞⠋⠃⡰⡏⡇⠀⠈⠳⠦⢌⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡠⠊⠀⠀⠁⠀⠀⠀⠀⠁⢠⠂⠀⠂⠀⠙⢦⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣼⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⣣⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⠏⣠⣿⠀⢀⠀⡆⠀⡄⡀⠀⠀⠀⠀⡄⠀⢸⢠⠸⣿⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⣾⣻⢟⣿⠸⡀⢧⠰⡇⠹⡄⢀⠄⡄⡇⢰⠈⢺⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠓⡏⠣⠻⠀⠳⡁⠁⠹⡀⠙⠎⠀⠏⠀⡹⠀⠘⡽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠀⠀⢀⠀⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠤⠤⠀⠀⠤⠤⠤⠤⠤⢤⣸⣫⣡⠀⠀⠐⠲⠶⠐⠀⠘⠀⠀⠀⠂⠲⠶⠒⠀⠀⢰⢹⣣⣀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠬⠍⣹⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⢠⠀⢀
⠀⠀⠀⠠⡀⢠⠀⠀⡆⠀⠘⠀⡘⠀⠀⠀⠸⢼⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡯⠄⠀⠀⠀⠸⡀⢸⠀⢀⠀⠀⢠⢀⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⠚⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠑⠘⠂⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠠⣤⠀⠀⣤⠄⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠳⡄⣠⠃⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠚⠛⠛⠛⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡂⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠉⠰⠄⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠐⠂⠢⠤⢀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠱⣀⣀⠀⠀⠠⠤⠤⠤⠄⠈⠒⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠀⠀⣀⡀⢰⣶⡆⢠⣄⣀⣤⠀⠀⣼⠓⠒⠒⡞⣷⡀⠰⢤⣴⡄⢰⣦⠀⢀⢀⣀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⠆⢠⠏⠲⠟⠀⢥⣾⡁⣸⣟⠧⢸⣇⣼⣀⡀⠿⠙⣷⡿⢋⡉⢿⣾⣍⣧⣜⡏⣹⣧⠀⣠⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡴⣆⣤⣄⣤⡄⢤⣤⢼⣿⣸⣿⠈⠳⡏⠻⡟⢩⡌⢠⣸⡷⠻⣇⠘⠛⠀⠈⠙⠿⢿⣽⣿⣿⣏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣉⣛⠿⠉⢿⣯⣿⣿⡏⠀⠻⠿⠟⢓⠋⢠⣦⡏⢻⡇⠹⠟⠀⠄⠀⠀⠀⣐⡀⠀⠀⡤⣄⣈⢙⣫⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣸⣿⣧⣽⣧⢿⡏⢘⠿⣿⣟⡙⢾⣷⢯⣽⣧⠈⣧⠁⢲⡇⢤⣖⠈⣽⢛⣿⣯⣿⠀⢹⣿⢛⢿⣦⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣧⣿⣧⣬⣇⣸⣯⣿⣤⣤⣿⣿⣷⣿⣷⣖⠛⠛⢻⣿⣿⣿⣴⣇⣿⣾⣿⣾⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠻⠛⠲⠿⠿⢿⣿⣿⣿⡿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⠟⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉⠁⠀⠀⠉⠙⠻⣿⣿⣟⣩⣤⣠⣴⠲⢲⣄⢀⠀⣀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡶⠿⠖⠛⠋⠉⠁⠀⠀⠀⠈⠉⣽⠒⣾⡿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡇⠈⠝⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣰⠆⠀⠀⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⡋⠹⠉⠻⠃⣤⡐⠿⠛⠷⢤⣄⣀⡀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⠖⠉⠀⠀⠀⠀⢰⠇⣼⠀⠀⢸⠀⣇⠀⠀⠀⠀⠈⠉⠻⢷⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⣠⡟⠀⣿⠀⠀⠈⣧⠈⢣⡄⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠿⣤⣴⣿⣀⣀⡀⣻⣧⣀⣿⠶⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠤⣄⠀⠀⠘⠑⠒⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⣀⠤⠶⠒⠋⠈⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡜⠋⠉⠀⠀⠀⠀⠀⠀⣧⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠦⣄
⠀⠀⢠⠂⢲⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣨⠇⠀⠀⠀⠀⠀⠀⠀⢸⡥⢄⣉⡉⣱⡄⣀⣀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠑⡎⠀⠀⠀⠀⠀⠀⠀⠘⡄⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⢠⡗⢋⠙⣆⠀⠀⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠧⠜⠀⢹⠏⠉⠁⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠙⣄⣀⡼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠈⠃⠒⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠀⠀⢸⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡤⠀⠠⠀⠀⠀⠀⠀⠋⠀⢀⠂⢀⡞⢆⠀⢋⠐⠀⠀⠀⠀⠀⠀⢀⣖⠰⣄⠀⠀⡜⠀⠉⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠸⡅⠀⠆⠀⡤⡀⠀⠀⠀⠀⣠⠞⠀⠈⢷⡈⠀⠀⠀⠀⣀⡒⠀⠀⣀⠤⠀⣀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⣤⣀⣀⣈⣀⣁⡀⠤⠔⠚⣟⠉⠑⢦⡀⢉⠷⠤⣀⣀⠡⠈⡴⠉⠙⢦⠞⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡀⠀⢝⣤⣤⢀⡤⢄⣠⡟⠻⢅⣠⠔⠒⢃⡳⠤⣤⠀⢀⣄⠉⠳⣀⠀⡸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⣡⠚⠒⠊⡽⠀⠁⠀⠀⠀⠙⣄⠀⠉⠲⣎⠁⠀⠰⠀⠉⠁⠀⠉⠙⣇⠉⣠⠄⠚⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠉⠒⠐⠊⠉⢢⠀⠀⡠⠔⠒⢾⡦⡀⠀⡸⠀⣠⠟⠈⠐⢄⡀⢀⠖⠉⠓⠤⠴⠋⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢠⡞⠁⠈⠓⠊⠀⠀⠀⡜⢠⠝⠛⢻⠋⠈⢢⡀⠀⠀⠙⠶⠊⠉⠶⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡶⢤⣤⡄⠀⠀⢈⣧⣀⠀⣀⣸⡄⠀⠀⠀⠀⠀⠀⡜⠀⠈⠲⡤⠃⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣜⠀⠀⢀⡜⠓⣄
⠀⠀⠀⢀⡇⠀⣘⠁⠀⢠⠋⠀⠈⠉⠉⠀⠈⠢⣀⠀⡀⣠⠞⠓⠉⠙⠂⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⠴⠋⠈⠙⠛⠉⠀⠀⢘⠁
⠀⠀⠀⠈⠁⠈⠉⠀⢀⣸⡀⠀⠀⠀⠀⠀⠄⣀⢈⡉⢩⠍⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠐⠋⠁⠈⠙⠢⠤⠂⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⢀⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠘⢄⣀⢀⣸⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠦⣄⣀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠲⢄⣀⣀⠴⠋⠱⢄⣀⢀⡠⠚⠦⣀⡀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢰⡦⠀⠰⣶⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣷⣦⣶⣶⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢰⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⢻⣿⡿⠃⠀⠀⠀⣤⡄⠀⠀⠀⠸⣿⣿⠏⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣠⠴⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣻⠟⠛⠻⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⠟⠛⢻⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⢻⡏⠀⠀⠀⠈⣿⠛⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠐⠲⠦⠤⢀⢤⣄⡀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⣴⣦⠳⠄⠀⠀⠀⠚⠓⠀⠀⠀⣰⣿⢋⠀⠀⠀⠠⠈⠒
⠀⠀⠀⠀⠀⠀⠙⠤⠄⡼⡶⠀⡄⠉⢻⠛⠻⣿⣿⠇⠀⠘⠛⠀⠀⣠⣴⠿⠛⢉⣀⣠⣤⣶⣶⣤⡀⠀⠀⠐⠶⠿⣡⠍⣿⢠⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠒⠋⠀⠀⠈⠉⠀⠀⠀⢀⠀⠉⠁⢠⣖⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠈⢃⣤⠞⠉⣤⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢷⣄⠉⠻⣿⣿⣿⣿⣿⣿⣟⠻⣶⣄⡀⠀⠈⠉⠛⠁⠀⠉
⠀⠀⠀⠀⠀⢠⣖⣟⣋⣛⣛⡛⠋⠿⢩⠙⢷⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⣤⠀⠀⠀⠝⣡⣞⣀⣼⣿⣿⡇⠀⠀⠀⠀⠆⠚⣆⠀⠀⠙⢷⣄⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠹⣿⣷⡤⣴⣶⣶⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠙⢷⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠉⠉⣩⠉⠛⠻⣿⣿⡿⠀⠀⠉⠀⠀⠀⠂⠐⠀⠀⠐⠂⠀⠙⣷⣄⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢉⣉⣤⣾⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠒⢄⠀⠡⠀⠀⠹⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠉⠁⢠⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⡠⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡠⠎⠀⡀⠂⠀⠀⠒⠂⠀⠈⡠⣄⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡏⠁⠀⢤⠀⢑⡴⠆⠉⠀⠈⠅⠘⠂⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠉⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⣤⡀⣀⠀⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⣠⣶⣛⣩⣅⣀⡀⠀⠀⠀⠖⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀
⠀⠀⠀⠀⣐⡾⠿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⢠⠒⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠛⠛⠻⠿⠷⢿⣯⣽⡉⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⠀⢀⡴⠠
⠀⠀⠀⠉⢻⡒⠶⢷⣖⣦⣤⡄⣸⡟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠁
⠀⠀⠀⠀⠸⠇⠀⠀⢸⠅⠉⠉⢸⡇⢀⡀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡤⠤⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠸⢃⣜⠻⠖⠀⠀⠰⠀⠘⠀⠄⠀⠠⠄⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣯⢠⡿
⠀⠀⠀⠀⠀⠀⠰⢥⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡏⡤⣤⢈⣽⣏⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⡀⠠⠆⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠿⠓⠋⠻⠂⠰⣷⠆⠀⠀⢰⣿⣿⡟⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠘⠿⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢀⣤⣄⣤⣤⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠠⠀⠀⠀⠀⠐⢮⣌⣩⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣻⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⣤⢤⣈⡛⢦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣩⣶⣾⣷⣿⣿⣿⡆⠀⠀⠀⠀⢠⠞⣄⡈⠳⣜⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣀⠒⠆⠈⠙⠲⡌⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢶⡈⠓⠂⡉⠳⠄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡈⠻⣿⣿⣿⣿⣿⣿⡇⠀⢀⣘⢦⣓⠰⠦⣭⠉⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠙⠟⠛⣃⣩⡅⠀⠠⣌⠉⠀⢬⡑⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⠏⢠⡛⠳⣮⡳⢦⣼⠃
⠀⠀⠀⠀⠀⠀⠰⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠴⢮⡝⢦⣄⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣷⡌⠳⣤⡓⠶⠬⠍⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠛⠿⣿⣿⣆⠀⠙⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣀⡤⠤⢤⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣹⠞⢁⣠⢾⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠁⠀⣠⣖⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⢀⡞⠁⠀⢱⠀⠀⠀⠀⠀⠀⠠⠄⠀⢰⣇⠀⠀⠘⣦⠎⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⠉⠉⠷⣀⠀⢣⡀⠀⣠⠴⠒⠚⠛⠛⠛⠓⠲⣤⠴⠳⠦⠔⠚⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣰⠇⢀⠀⠀⠀⠀⠙⢦⡀⠈⠲⣅⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⣀⡤⢲⣮⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⣾⣤⣶⠭⢛⣒⣒⠺⢆⠀⠈⠛⠒⠒⠒⠒⠒⢉⣡⣴⠺⠯⢴⣤⠏
⠀⠀⠀⠀⠀⠀⡖⢱⡀⠀⠀⠀⠀⠙⠃⢸⡀⠈⠉⠀⠁⣑⠀⠀⠀⠀⠀⠀⣀⣔⣊⣵⣋⡠⣀
⠀⠀⠀⠀⠀⠀⢹⠆⠉⠁⢲⣀⣀⣀⣀⡤⠭⠥⢒⣋⡉⣏⠀⠀⠀⣠⠴⠋⠓⡶⠒⠦⣄⠁⠀⢣
⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⢠⠒⠒⢲⡶⠃⣀⣀⣀⡀⠈⢻⠀⠀⢎⡀⠀⢀⡜⠀⠀⠀⠈⠑⠒⠋
⢀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠜⠀⣀⡨⠟⠉⢛⡉⠀⠀⠀⠸⡀⠀⠀⠈⠉⠉⠉⠛⠛⢓⣂⠴⠂
⠀⣢⠞⠙⢿⣿⡻⢿⣦⡴⠒⠉⠁⠀⠀⡠⠊⠁⡊⢂⢄⡠⢧⣀⣀⣀⠤⠤⠲⢶⣉⠉⠀⢀⣀⣀⣀
⠀⠹⠤⢴⠞⠁⠀⠁⠀⠙⠦⣄⠀⠀⠂⢅⠟⠤⣨⠔⠋⠀⠀⠀⠀⠙⢄⣠⣮⢝⣓⡀⠐⠣⣄⡰⠧⡝
⠀⠀⠀⠙⠒⠂⠐⠂⠤⠤⢤⣴⣻⡒⠲⠤⠭⠾⠧⠤⡄⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠉⠹⣤⣟⢒⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⣀⣠⡤⠤⠤⣗⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣄⣀⣠⣴⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⢿⡏⢉⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢴⠄⠈⠉⠤⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⣿⣧⣀⡀⢀⣀⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡔⠋⠁⠙⠋⠉⠃⠀⠁
⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⠀⠀⡤⠤⠀⠀⡀
⠀⡨⠂⠐⠁⠉⠛⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⡋⠀⠀⠐⠢⣀⠀⠀⠀⢀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠇⠀⠀⠀⠀⠊⠀⠐⠂⠘⠒⠁⡆⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣞⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⠮⠷⠉⠒⠢⠀⠠⠀⠤⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠔⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⣇⠀⠀⠀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣿⡄⢀⣾⣿⡏⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⣰⣿⣧⣿⣿⣿⣇⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⣠⣿⣿⣿⣿⣿⢿⣿⣾⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠟⠁⢀⣴⢻⣿⣿⣨⣯⠿⡽⡞⣿⣿⣇⠄⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠠⢺⡛⣻⣷⣿⡕⠱⣟⠔⣧⣹⣻⣿⣄⠀⠀⢸⣿⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡖⣥⡁⠱⢿⠃⠻⢡⠠⣟⢶⢷⢦⡜⡙⢉⡦⠀⠈⢿⣿⣧⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠷⠸⠾⠧⢧⣾⠀⣦⣾⡘⢣⠀⣿⡴⠉⡋⠄⡵⠙⠄⠀⢻⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⢀⠀⠀⠈⠀⣿⠿⠛⠛⠃⠛⠛⠛⠰⢀⣠⡂⢰⣆⠀⠻⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢀⡴⣣⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠻⡍⣬⠀⠐⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡾⠁⣹⣶⣦⣴⠀⠀⠀⠀⠀⠶⢀⣄⣄⢸⡋⢹⣧⠀⠄⢹⣿⠀⣨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣾⠁⠠⡟⢸⡏⣿⠀⠀⢠⣤⡄⢲⡘⣿⢻⡜⠷⠚⢹⡇⠀⠀⡟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢩⢶⠆⢀⡀⢀⠈⡕⠸⠞⣯⠻⡜⠧⢩⡄⠀⠀⠀⢨⡇⠀⠆⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢣⡴⣼⣇⣾⡄⣿⠳⣤⡍⢠⠴⢶⠀⠻⣦⣀⠀⠘⡁⠠⠂⠀⠀⠠⢨⡆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣆⠡⠏⣠⢤⣍⠈⣀⠴⣄⣿⢴⣍⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣇⢀⣫⣦⣻⠶⢃⠿⢀⡙⠓⠶⠐⠀⡩⠆⠀⠀⢠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣏⠀⢨⡁⣾⠙⢷⠸⠧⠚⠃⠐⠀⠀⢠⢖⡽⣿⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡉⣦⣈⠿⣿⠓⠁⡠⠀⠀⠀⠀⠀⠰⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⣦⣤⣄⡀⠀⠀⠀⢀⡾⠛⠻⡟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢈⣉⣿⣿⣷⣶⣾⡓⠀⠀⣅⠀⣿⣷⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢠⡶⠛⠉⠉⠉⠉⠙⢿⣟⠘⠈⡟⠀⠈⢽⣿⠋⠉⠉⠙⠛⠛⠻⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣦⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣀⣥⡀⡄⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢉⣯⠀⠈⣿⣿⠀⠀⢀⣀⠀⠀⢰⣾⣷⣆⢹⡄⣯⠀⠀⠩⠍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⠿⠀⠈⠛⢻⣇⢰⣿⣿⣷⠀⠘⠿⠿⠃⡾⠻⠿⣶⣶⣾⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⠀⠀⣤⡀⢹⣦⡙⠛⠋⠀⢠⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡁⠀⠻⠁⠀⣿⣿⠳⣆⠀⠈⠁⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣛⠀⣸⡿⠃⠀⢹⣤⡴⠴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠿⣿⠿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⢹⣟⠉⠙⠋⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠤⠴⠶⠾⣿⡟⠛⠀⠀⠀⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⡶⠤
⠀⠀⠀⠀⠀⠀⢠⣾⠟⢉⣩⣿⣿⣶⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣀⣀⣴⡿⠿⠿⣧⣄
⠀⠀⠀⠀⠀⠀⣿⠷⠾⠛⠉⠀⠘⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⡿⣦⣀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⣻⠿⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⢿⣏⢳⡀⠙⠻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠞⣿⣯⣷⠀⠀⠈⠙⠻⣿⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣿⡿⠟⠉⠁⠀⠀⢸⣟⣟⡹⣄⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡶⣾⢿⣿⣿⣷⣤⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⣀⣤⣿⣿⣿⣿⡻⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢾⣿⣾⣛⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀⢀⠀⣠⡙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣼⣿⡛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠘⠃⠈⠀⢤⡆⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠛⠉⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⠀⠘⠿⠿⠀⢠⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠰⡧⠰⠶⢿⣼⣿⣿⡿⢿⣿⣟⣛⣛⣛⣻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⣇⣅⣼⣧⣀⢹⣏⣉⣿⣙⢉⣟⢛⣻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣏⠉⠙⣏⠉⢹⡏⠉⣿⠉⠉⡟⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣷⣤⣶⣷⣦⣴⣦⣤⣶⣤⣤⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣛⣛⣛⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠉⠙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⢊⢁⡄⢠⡄⢀⠀⠀⡁⠀⠀⠀⠄⢄⠀⠀⠊⠀⠈⠀⢁⠀⠃⠠⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠐⢁⣶⠀⢠⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠆⠉⡄⠀⠀⠀⠀⠀⠆⠀⣠⣀⠀⠀⠀⠙⢆⠤⣴⣷⣾⣿⣿⣀⡐⠀⡀⠈⠀⠄⠀⠀⠀⠀⠀⡶⠋⠀⡰⢀⠔⠀⠠
⠀⠀⠀⠰⠀⠀⠠⠃⠀⣠⠑⠈⠀⠀⠄⠀⡤⠄⠀⠀⢸⠛⠻⠷⠆⠂⠠⡈⡄⠈⡀⢻⣿⣿⠟⣿⣤⠀⠀⠄⠐⣀⠉⠉⡀⠨⡆⠀⠀⠃⠂⠀⠀⠐⠄
⠀⠀⠀⠢⠀⢀⠄⢂⠀⠠⠀⠐⠀⠀⠰⠀⠐⠒⠀⠓⣤⣀⡃⠀⠮⠁⠀⢠⠈⠀⠀⠀⠠⠄⠀⢹⣿⣤⠀⠀⠊⠀⠤⢀⠀⡳⢈⡱⠀⠀⠰⣅⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢁⠀⠀⠂⠀⠸⠁⠐⢌⠘⢷⣄⣀⡀⡀⢂⢡⠀⢀⣀⡀⠀⣀⣼⣿⡇⠀⠀⠀⠐⠀⢀⠛⠁⡐⠄⠐⠀⠁⠉⠁⠒⠈⠦
⠀⠀⠀⠀⠀⠘⡀⠁⠀⠠⠘⡄⠈⢀⠀⠐⠀⠀⠁⣁⠀⢠⡀⠹⣿⡇⠀⠀⢂⠋⠁⠘⢄⠈⠟⡉⠉⠁⠀⠡⡀⠚⢁⡜⠐⡀⠀⠰⠘⡃⢄⠁
⠀⠀⠀⠀⠢⠀⡈⢄⠓⣼⣶⣷⡀⠀⡄⠀⠉⡀⠈⠁⣀⠈⡑⣄⠈⢷⠀⠀⠈⣆⠀⠀⠀⠙⡀⠀⢀⠜⠉⠢⣀⢉⠈⠀⡀⠀⠀⣐⢠⡅⢀⠁⠀⢈⠐
⠀⠀⠀⠀⠀⠀⠊⡰⣦⡙⣿⣿⣷⡁⠠⠄⠀⠉⠀⠀⠀⠀⢤⣿⠆⣸⣧⠈⠁⠀⠘⠢⠁⠀⠈⠀⡇⠀⡀⣘⢉⡍⠁⠠⢀⠔⠁⠀⠉⠊⠄⠀⠀⠀⡄
⠀⠀⢠⠠⠀⠈⠛⠀⢙⠷⡈⢩⣉⠃⠀⠀⠀⠀⠀⠄⣠⣾⠟⢁⠴⣿⡏⠀⠀⠠⠑⠠⣀⠙⠈⢈⡇⠀⠀⠈⡶⡁⢢⠖⣻⡀⠄⠀⠀⢠⠊⠀⠀⠃
⠀⠀⠀⠄⠀⠴⢀⣻⡏⣰⣿⣦⠙⢷⡈⣷⣶⣶⣾⣈⣋⣁⠾⢷⣄⠈⠻⣶⠿⠃⢈⣢⠀⠉⠐⠀⡇⠀⠀⢰⢅⢀⠗⣈⡴⣧⣀⣀⡐⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣿⠘⣿⣷⣌⣛⣿⣷⡬⢡⣿⣿⣼⠟⠉⠉⠉⠁⠄⠈⡷⠛⡐⠢⡴⠻⣿⠃⠒⢠⡼⠀⠀⠐⠬⣒⠊⠙⠋⠀⢹⣿⠻⢀⠀⠄⡠⠄⠂⢀⠠
⠀⠀⠀⠁⠀⢨⣷⡘⠻⠿⠿⠿⠟⣡⡌⢻⡟⠁⠀⠀⠀⣀⠀⠀⠈⠠⡋⢈⣠⣀⣰⠏⣨⡆⠊⠀⠀⠀⢀⠂⠀⠀⠀⠄⣠⣿⠇⠘⣨⠊⠎⠄⡂⠀⠄
⠀⠀⠀⠀⠢⠀⠹⣿⣷⣶⣶⡶⠿⢿⡿⠀⠀⠐⠁⢠⠀⢀⠂⠀⠀⠀⠀⠀⡈⠁⠀⠚⠀⠆⠐⠄⠀⠠⢀⠁⠀⠀⢠⣾⡿⠁⠀⠐⠁⠘⢂⠀⠔⠂⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠃⢠⣤⡄⣇⠀⠣⠀⠀⠀⠢⠤⠀⠀⠄⠂⠠⠄⠈⠢⠄⠀⠰⡀⠀⢀⡈⠑⠀⠀⡘⢸⣧⡙⡅⠀⢀⢐⠂⡴⡃⠄⠒⢻⣄
⠀⠀⢀⠀⠈⠐⠁⣶⣾⡿⠀⠘⠿⠶⠿⠃⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⡄⠀⠐⠀⠀⠀⠈⠃⠀⠀⠀⢼⣛⡇⢸⠄⣼⣈⣠⠓⠔⠀⠀⣼⣿
⠀⠀⠈⠀⠀⠀⠀⠾⡏⠀⢈⠀⠀⣤⣶⣾⡿⠦⠀⠀⠢⡀⠈⢄⠀⠀⡀⠀⡠⢊⠳⠄⠀⠁⠑⠀⠀⠀⠰⠐⠀⢸⣿⠇⣸⠓⣤⡛⣷⡎⠘⢣⡾⠋⠁
⠀⠠⠂⠀⠀⠀⢀⠂⠀⡐⠌⢂⠄⠸⠋⠁⠀⠀⠈⠂⢀⠀⠐⢄⡈⢁⠠⠌⢉⣤⢐⠨⢳⣄⡈⡂⠀⠠⡀⠀⡰⠿⢋⡰⣏⠀⠘⢝⡌⡷⠞⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠈⢳⣾⡦⢀⣀⠴⠀⠀⠄⢠⣤⡁⠀⠈⡱⠖⠁⠄⠀⠰⣦⠖⠀⢈⠐⠥⢡⠩⢙⢒⠐⠘⠛⠒⠒⠉⠂⠀⠁⢲⡖⠻⢸⠂⠀⠀⣐⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠘⠏⣾⠋⠡⢺⠀⠀⠂⡈⠛⠁⠀⡄⠀⠄⡀⠀⠀⠈⠉⠀⠂⢜⠓⠀⢣⡂⣠⢼⠀⠐⡀⢄⢠⠀⠈⢢⡴⢉⠙⢿⢸⠰
⠀⠀⠈⠀⠐⠀⠀⠀⢱⠁⠬⠑⠈⠣⣀⠐⠀⠀⠀⢀⢀⣷⡟⠁⠀⢀⠐⠂⠀⠀⠠⢤⠐⠀⠖⠁⠀⢈⠲⢇⠈⢱⣀⢔⠛⠘⠠⢠⠆⡌⢀⠂⡘⢠⡑
⠀⠀⠐⠠⢀⠐⣒⠢⡆⠨⠀⠀⣀⠃⣼⣷⣶⣶⢾⣏⢸⣿⠇⠀⠀⠀⠼⡿⠂⠅⠀⠀⠀⣉⠀⠁⠀⡉⣱⠀⠘⢻⠟⣐⣄⣁⡠⢋⡾⠃⢂⡐⡀⠠
⠀⠀⠀⠀⠸⡘⠠⠸⠆⠿⠄⠀⠄⢸⢿⣿⡿⠿⢾⣿⡀⢿⣧⡠⠀⠀⠒⢀⠈⠠⢀⡀⢀⠉⣶⣄⠀⠐⠁⠀⡠⠼⣄⠛⣛⣩⣴⡿⠁⢀⠢⡀⠀⠀⠃
⠀⠀⠠⢀⠆⢈⠀⢀⠶⠸⡀⠀⠁⡈⢞⠛⣀⡦⠈⣿⣷⡈⠋⠁⠦⠀⣀⣀⣠⣴⣾⡿⠃⣴⣿⣿⣿⣦⣀⠀⠣⠃⠄⡰⠙⠿⠏⠀⠀⠀⠠⠁
⠀⠀⠠⡈⠀⠤⠐⢁⠄⠁⠑⣀⠀⠀⠀⠉⠉⠁⠰⢿⠿⢿⣦⣈⠑⠶⠿⠿⠟⠛⢁⠄⠚⢿⡟⡻⠿⣟⣿⣷⣦⣠⡌⠀⠀⠀⠂⠁⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⡀⠄⢀⠀⠀⠈⠓⠒⠰⠀⠂⠈⠀⠀⠠⠼⠿⠛⠛⡋⠁⠐⠂⠀⠀⠀⠀⠠⠋⠁⠀⠈⠙⠻⠿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠈⠠⠂⠐⠀⠀⠀⠈⠉⠀⠐⠄⠁⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣠⣤⣄⡀⣀⣤⣤⣄⡀⢠⡀⠀⠀⣀⠀⢀⣤⡀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣀⠀⡀⡄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣷⣾⡇⠀⠀⠈⠻⠟⣴⣿⡿⠋⣭⣭⣿⣿⣷⡔⣶⣿⣾⠟⣠⣶⣿⣿⣿⣿⣿⣷⣌⠀⡀⠀⠀⢶⠆⡅⠆
⠀⢘⣿⣿⣿⣿⠿⢿⣋⣻⣿⣿⢃⠉⢹⣧⠠⠀⡀⠀⢠⣿⣿⣷⣿⡟⠻⠇⣎⢻⣷⢻⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣷⣦⣤⣾⢿⣦⣤⣶⡆
⠀⠻⢿⡿⠋⠀⠀⣦⠉⠉⢩⣴⣿⣿⣿⡣⠆⢠⣤⢠⡀⠟⣫⣝⣋⠄⠀⠀⠿⢸⠏⣾⣿⢇⣯⡽⠿⢿⠿⢿⡙⢛⡛⣿⣿⣇⢿⣿⣿⣯⣸⣿⣿⣿⠇
⠀⣠⡄⢦⠀⢀⠐⢀⠀⠠⠆⠙⠉⢻⣏⡁⠀⠈⢉⠺⠃⢀⣶⡆⡆⠀⠀⠀⠀⠁⢈⣩⣭⣈⡛⢂⣨⣁⡁⢀⣴⣿⡿⣨⣭⡉⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢰⣶⢾⣶⣶⢶⣶⡀⣀⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢈⢅⠀⠀⠀⠀⢐⠈⠻⠿⣿⣿⣿⣿⣿⡇⣿⣿⣯⡐⡿⠿⣿⠳⣋⣭⣶⣶⣭⣝⣛
⠀⣰⣿⠟⢉⣟⡛⠻⣿⣇⠉⢀⠂⠤⠐⠀⠠⠤⠀⠀⠀⢈⣀⡀⠻⢿⣷⣆⠺⣿⠃⢀⣄⠻⣿⣿⣿⣿⣧⠉⠉⠹⠷⣰⣿⢶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⢿⠀⠀⠉⠁⠀⠿⠛⠃⠈⠰⣾⠆⠈⢰⣿⣷⣤⣾⣿⡿⠟⠀⠈⠁⠀⠶⠆⡄⠘⣿⣷⣀⣤⡠⢿⡿⢁⣤⠀⠘⠿⣞⣯⣿⣟⡟⠛⠻⣿⣿⣿⡇
⠀⠀⡬⠤⣠⡀⣴⣶⣔⢒⡀⠄⣴⣿⣦⠤⣌⠉⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⣿⣿⣇⠠⠀⠈⠁⡀⣀⠀⣙⠉⢢⣹⣤⣶⡄⣾⠻⡿⠁
⠀⠈⡄⠀⠀⠀⠻⠿⠿⣿⡟⢀⣄⣻⢁⡀⠀⢱⠉⠁⠀⠝⡗⣠⠃⠀⠀⠀⠓⠀⠀⠀⠀⠸⠿⢿⣿⣧⠀⠀⠀⠿⠿⢆⠳⣣⢀⠑⡀⠀⠀⡛
⠀⢸⡀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢠⣿⣄⣉⣠⣞⠰⠆⠲⠂⠀⠀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠠⠠⣤⡜⠃⠀⠀⠸⣿⣿⣿⣧⣀⣬⢳⣭⣥⣭⣵⣿⣶⡄
⠀⠘⠈⠀⠲⠀⢋⣠⣤⣄⡚⠀⠂⢠⣍⣛⡻⠿⠀⠋⣩⣴⡏⣴⣿⣿⣿⣷⣤⣄⠠⣄⠂⠀⠀⠐⠀⢰⣷⣦⣠⢹⣿⣿⣿⣿⠩⣾⣿⣿⣿⣿⣿⣟⠃
⠀⠀⢠⣶⢤⣿⣿⣿⣿⣿⣿⣦⠔⠃⢻⣿⣿⣿⠛⣿⣿⣿⠃⣿⣿⣿⣿⣿⠟⠁⠀⠛⢄⠀⠀⠀⠀⠈⠉⠙⠟⡘⠻⣙⠻⣛⢷⡌⠻⠿⢿⡟⣙⠿
⠀⠀⠀⡄⠀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⢿⠋⠉⠛⠋⣽⢻⠂⠛⠛⠋⠒⠛⠋⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠃⢀⠃⢠⣿⠄⢻⡷⣺⣿⣿⣷⣶⣿
⠀⠀⢸⣷⣦⡄⠀⠀⠀⠀⢀⣤⣴⣾⣿⡦⠀⣀⣤⣤⡄⠉⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣠⡜⠈⢶⣤⣠⣴⣤⢆⣤⣬⡉⣠⣤⣝⢿⣿⣿⣿⣿⣿⣷⠄
⠀⠰⢿⣿⡿⡛⢷⢂⣤⣶⣟⠁⢙⡿⠛⢿⡆⢡⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⡓⠿⣧⢀⠼⠿⣿⡿⢃⣌⠉⠁⣸⡀⠉⢙⣄⢻⣿⣿⣿⣿⠁
⠀⠀⠈⠁⠆⠀⠀⢠⣶⡌⠉⢳⡪⢤⣀⣘⣓⠀⠈⠉⠀⡀⠈⠀⠀⠀⠛⠀⠀⠈⠻⠟⠓⣠⣶⣾⣷⣦⣴⣦⣭⢻⣿⣿⣿⣿⣿⣿⠾⡿⣛⣛⣛
⠀⠀⠠⠀⠀⢀⡀⠈⠙⣿⠷⠸⠟⣂⠻⣿⣿⣷⡿⠷⢾⠿⠇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣻⣷⠟⡿⠋⠈⢻⣿⣿⣿⡟⣫⣶⢞⣛⢿⣿⡇
⠀⠀⠠⡤⠴⠛⠛⠿⣃⣠⣶⣶⣿⣿⣷⡍⣿⡏⠻⠃⠀⠀⡤⠠⠀⢤⣴⣦⠐⡄⢻⣿⣯⠟⢫⡬⠀⠿⣤⣤⡤⠀⣚⡛⠿⠿⣡⡏⠉⣸⣿⣧⢹⣧⡀
⠀⠀⢰⣁⣀⣏⠁⠸⣿⣿⣿⣿⣿⠿⣿⣷⠙⠁⠀⠀⠄⠀⠇⠀⠀⠈⠛⠛⢿⡟⠈⠉⠻⠇⠠⠠⠌⠁⠉⠉⠰⠥⢿⣷⡜⠂⢻⠰⣿⣿⣿⣿⢨⣿⣷
⠀⠀⣼⣿⡿⠿⢇⣤⣄⠿⢛⠿⠕⠖⡸⠃⠀⠀⠀⠂⠀⢸⡄⠠⠀⢀⠀⠠⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠛⠻⠿⠀⠻⠆⢿⣿⣿⣶⣝⠋
⠀⠀⣿⠻⠄⢃⣼⣿⣿⣿⣿⡿⢛⠂⠀⠀⠀⠀⠀⠀⠀⢀⡉⣶⣄⣢⣤⣤⡖⠁⠰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠇⣤⣭⣼⠟⠉⠹⢿
⠀⠀⢑⣨⡀⠸⡏⠉⠉⢿⣿⣇⢻⣦⣀⣠⢄⣴⣤⢔⡺⠿⠃⢚⣫⣭⢭⣝⣁⡀⠀⢺⠏⠁⠀⠀⢀⣅⣶⣰⣶⢶⠦⣶⡜⠆⢀⣬⡝⢯⡁⢀⠤
⠀⠀⢛⣭⣉⣁⣘⣀⡒⠁⠾⣿⡇⣉⣹⣿⣧⡝⣫⣾⣿⣟⣿⠯⡽⣚⡷⡳⢿⠏⣤⣶⣶⣾⣿⣿⣮⢻⣿⣿⠿⠾⣟⣻⣽⣦⠙⠶⠎⠂⣶⣶⣄
⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⢴⣦⡰⢿⠛⠛⢿⣷⡝⢻⣿⣾⣿⣿⠛⠛⢾⣷⣛⠄⣻⣿⣿⣿⣿⣿⣿⣦⡙⠃⠀⠀⠈⣿⣷⣿⣇⢠⣤⣴⣿⣿⣿
⠀⠀⣽⣿⣿⣿⣿⣿⣿⠀⢠⡿⠁⢠⣬⣠⣩⡼⠁⢀⣡⠈⢛⡓⠒⠶⣼⣯⡟⢸⣿⣿⣿⣿⣿⣿⣟⠛⠃⣄⠀⠀⡰⢿⡿⠿⢿⠘⠛⣿⠉⣉⣤⡀
⠀⠀⠙⣿⣿⣿⣿⣿⠟⠚⡏⢲⣆⠀⠀⠀⠀⠁⢀⡀⠶⢳⠀⣁⠀⠀⠙⡟⠀⠰⢿⣿⣿⣿⣿⣿⡟⠸⠶⠿⠛⣿⡴⢠⣶⣿⠖⢀⣿⣧⣀⣾⣿⣧
⠀⠀⠐⠯⠏⠀⠈⠏⠀⠐⠛⠂⠉⠠⠆⠀⠀⠀⠈⠉⠀⠀⠀⠈⠋⠈⠋⠀⠀⠀⠀⠛⠟⠉⠛⠟⠁⠀⠀⠒⠁⠀⠀⠛⠻⢿⡿⢿⣿⡿⠿⡿⠛⠁

⠀⠀⠁⣀⠈⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡠⢄⡄
⠀⠀⠄⠀⠀⠀⠑⢷⢈⠀⠀⢠⠖⠉⠉⠛⢦⡀⢀⠀⠀⠀⠀⠁⠀⠐⠛⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠁⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⢀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣀⣨⢚⣿⣶⠀⠀⠀⠀⠀⠀⢄⠀⠰⡄
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡟⠀⠠⣄⣠⠤⠙⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠂
⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣗⡞⣞⣾⣸⣵⣇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⢀⡉⢉⡉⠈⠁⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡡⠔⠂
⠀⠀⠀⢠⣄⣀⠀⠀⠆⠀⠀⠀⠀⢤⣤⣤⠀⠀⠏⢠⣞⡵⡾⡋⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠉⠐⠀⠀⠀⠀⠀⠀⠈⢣⠞⠀⣀⣴⠻⡟⣸⢱⣷⣧⣷⣿⣴⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠘⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢖⠒
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⡠⡄⠀⠠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⢠⠀⢽⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠍⢡⣨⠆⠄⠀⠀⠘⠂⢰⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢠⡇⠘⠄⠀⠀⠀⠀⠀⣝⠃⡞⢉⢹⣦⣐⣫⠵⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⢸⡇⠀⠀⠀⠀⠐⠀⣾⠏⠀⠘⠊⠹⡿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠄⠉⢉