Sour Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⣄⡉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⡀⠐⣦⠀⠉⠓⢦⣙⠶⣌⠛⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣀⠀⠀⠀⠀⢀⣀⣤⣤⣍⣉⡉⠐⠦⣤⣄⠈⠳⣤⠈⣧⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠚⠉⠉⠻⠶⠶⠟⠋⠁⠀⠀⠀⠈⠉⠓⠶⠀⠈⠻⣆⠈⠁⠸⣇⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡀⠀⣠⠎⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⣘⣛⡛⠓⠲⠶⠶⠶⠂⠀⠀⠀⠀⢀⣴⠶⠶⢶⣤⡈⠓⠀⠘⠃⠴⢫⡞⠁⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⣠⠀⣼⠀⠀⣀⣤⠤⢤⡄⠈⠻⡦⠀⠀⠀⠀⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⡇⠀⡾⠉⢠⡶⠖⠲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠳⢦⣄⠀⠀⠀⠀⠀⢸⡇⠘⣇⠸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣉⣙⣛⣛⣩⡄⠈⢧⠀⠹⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⢻⣇
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠒⠖⠦⣭⡉⠉⠉⠀⠀⠀⠀⠀⠀⠹⠊⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⣼⢋⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠈⠉⣀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠾⠙⣎⠧⠀⢤⡀⠀⠀⠀⠀⠀⠐⠋⠁⢀⣀⠀⠀⠀⠙⣦⠀⠀⡼⠁⣠⣶⣿⣿⣷⣦⡀⢸⣾⠃
⠀⠀⠀⠀⠀⠀⢻⡀⢏⡈⠃⠀⠠⣍⠓⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣦⣄⣀⣸⡆⠘⢡⣾⣿⣿⣋⣀⣠⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠸⣇⣾⣰⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣀⣠⣿⠟⠿⠟⣻⡟⠉⢻⣿⣈⠛⠛⠋⢁⡀⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣎⠻⣧⡄⠀⠀⠀⠀⠠⢤⣄⣀⠛⠛⢋⣉⣡⣤⠤⠞⠁⣿⠀⠈⣿⠉⠙⠛⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢀⡀⠘⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣠⣿⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠲⠂⠀⠀⠉⢙⣇⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣷⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧⣀⣤⣀⡀⢀⣰⡟⠛⣇⠀⠀⠺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣴⠃⠀⡼⠃⠀⠀⠀⠈⠉⠉⠋⠉⠀⠀⠹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢿⠀⠸⠃⠀⠀⠀⣀⣴⣶⣶⣖⣒⣦⣀⡀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠸⣆⠀⠀⠶⠟⢻⣍⠉⠉⠛⠟⠉⢩⡍⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠙⢳⣦⣤⣤⡴⢟⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠀⠀⠈⠳⠄⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⡶⠶⠾⠛⠛⠙⠛⠛⠉

⠀⠀⠀⣠⣶⡾⠿⠿⣷⣦⡀⠀⠀⠀⣠⡴⢞⣛⡉⣩⣛⡳⢦⣀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⢰⡆⠀⣾⠟⠛⠛⠛⠶⣄
⠀⠀⢰⣿⡏⠀⠀⠀⠀⠛⠃⠀⢠⡞⣡⣦⣹⣿⡧⣿⣿⣿⣢⣍⢷⡀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⢠⡟⣨⣻⠿⣷⠹⣇⢿⢟⣽⣿⣿⡧⢷⡀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠙⠻⢷⣤⣄⡀⠀⢸⢧⣿⣿⣿⣶⠅⠉⠉⠫⢭⣭⣿⣶⢸⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⡟⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠸⡎⣋⣽⣶⣾⡟⣴⢠⡽⣿⣶⡙⡿⣸⠃⢹⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣧
⠀⠀⢀⣤⡀⠀⠀⠀⢀⣿⣿⠀⠻⡜⢿⣿⣿⢹⣿⢼⣷⢿⣿⡿⣰⠏⠀⠘⣿⣿⠀⠀⠀⠀⠀⣰⣿⠃⠀⣿⡇⠀⠀⠀⠘⣿⡀
⠀⠀⠘⢿⣿⣶⣤⣶⣿⡿⠏⠀⠀⠙⠮⣝⡱⠿⠿⠻⠿⢏⣡⠞⠁⠀⠀⠀⠘⠿⣿⣶⣶⣶⣾⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠹⣷⣤
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⠟⠛⠓⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣽⣥⣍⣹⣍⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠄⠈⠄⠀⢀⢠⣴⡾⠟⠛⠉⢾⣿⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⢀⡀⠛⡁⠠⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠈⠒⠂⡗⣿⢛⠖⠂⠀⠀⡀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⠃⠀⠀⠀⠀⠀⠑⢤⣄⣡⣆⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢄⡀⠀⠀⣀⠔⠀⡤⠩⠲⠃⠈⠁⠀⢀
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡠⠈⠀⠀⠀⠑⠲⠀⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⢀⣶⣿⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⠄⣀⠤⠒⠐⠤⡀
⠀⠀⠀⠠⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⠈⣱⠀⠀⠠⢶⠄⣀⠀⡀⠀⠈⢣⡔⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠈⠋⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠃⢾⣿⢆⡐⠀⠀⠣⡀⠀⠀⠀⠐⠀⢣⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⣀⠞⠀⡀⣀⢀⠁⠀⠀⠀⠑⢀⣀⣀⠠⠔⢀⠑⢲⠶⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⠉⠀⠀⠈⠀⣁⠤⠂⠒⠒⢤⠀⠀⠀⣀⢠⠔⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣷⡄⠀⠐⢠⠊⠀⠀⠀⠀⠀⠀⣷⣈⠉⠀⡜⠀⠀⠀⣠⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠙⣿⠀⠁⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠞⠀⠀⠀⠀⠈⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⡄⠘⡈⠀⠐⡀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢀⠀⠀⢿⣶⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠛⠛⢀⡧⢴⣿⣇⠀⠀⠀⠀⠀⠟⠏⠀⢀⣾⣷⣄⡀⠉⠙⡿
⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠐⣾⣿⡇⠀⠙⠿⠆⠀⠀⢀⠼⢄⠀⠀⠸⣿⣿⣯⣿⣷⠞
⠀⠀⠀⠀⠀⠀⠃⢴⣿⡇⠀⠀⠻⣿⣷⡀⠀⠀⠀⢀⡴⠃⠀⠀⠉⠒⠶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠋⠀⣀⠔⠉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠇⢳⢀⡖⠀⡄⢀⢀⣖⡀⢀⡀⡀⠀⠀⠠⣔⣋⣉⠀⣀⠀⡀⠀⢀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⡴⠀⣀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠼⠋⠘⠦⠌⠘⠿⠸⠤⠿⠜⠯⠑⠿⠘⣿⠁⢀⣀⣨⠤⠋⠸⠉⠸⠻⠸⠏⠀⠀⠘⠧⠄⠀⠾⠣⡎⠸⢯⠱⡇⠶⠿⠼⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠘⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢐⢁⡀⢘⠓⢢⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠘⠋⣈⣁⣉⣉⣁⣠⣤⡀⢄⠀⠀⠀⠀⠠⠀⠈⠉⢉⡉⠉⠁⢀⡀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡽⠜⠁⠋⠁⠼⠿⢲⡆⡄⠀⡀⠤⢤⢄⣠⡾⠁⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠂⠀⠁⠒⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⣤⠠⠤⢼⡁⠀⢀⣀⣀⣼⠋⡀⢀⣀⡀⠈⠓⡀⠀⠀⠀⠤⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⢲⡄⡐⣂⡂⠒⢤⣀⠄⠀⠀⠀⠠⠀⢀⣤⣍⠈⠁⠑⠂⢒⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡨⠒⠉⠀⠀⠀⠀⠈⢤⣀⠀⠀⠀⢀⡄⢨⡟⠀⢰⠀⠀⢨⠸⠀⠀⠀⠀⠀⠀⠀⡀⡀⠁⠀⠈⠉⠀⠉⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡈⠤⡀⠐⠀⠀⠀⢔⠈⠉⠉⠙⢷⣼⠶⠋⡠⣺⠉⠙⡄⠈⡆⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠀⠀⠈⠈⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣾⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣾⠰⠷⠦⠀⠁⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⢿⡄⠀⠀⠀⠀⠀⠂⠀⠈⠀⡸⠀⠀⠀⠸⠿⠦⣄⣠⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠿⣄⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠈⠙⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢨⠀⠈⠓⠀⠀⢈⠉⠀⠀⠀⠐⢀⠀⠀⠀⡀⠀⣸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠈⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣷⡀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢡⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡿⠶⠥⠀⢀⣹⣦⣀⠀⠄⠀⠀⠀⠀⣠⣿⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⢠⠀⣀⡀⠀⢠⢄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣖⣀⠀⠀⠉⠛⠛⠛⠛⠉⠛⠁⣀⠨⢴⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢚⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⢀⡀⢀⣠⡀⠀⠀⠀⡄⠀⡀⢠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠹⢀⡟⢸⡇⡏⠗⢰⣶⡞⠣⠋⠀⣾⢹⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠦⢼⠁⣴⢆⣧⡬⢨⡼⠧⠗⠚⠦⠿⠸⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⡀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⡀⣨⡛⡅⢨⠑⡎⣇⢈⡇⢛⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠬⠵⠛⠲⠞⠁⠬⠞⠑⠞⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠠⠠⠴⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣙⠉⠓⠖⢄⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣘⣀⣀⣧⣚⠁⠀⠎
⠀⠀⠀⠀⠀⣀⣀⡄⠀⠤⢄⠆⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⠉⡉⡿⠿⠛⣿⣿⠇⠀⠀⡄
⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠤⠀⠀⠠⠀⠀⠀⠀⢀⢸⠀⢀⢠⣄⡘⠢⠁⢘⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⢀⡉⢡⠆⠈⣿⠀⠀⠋⠀⠐⠐⠘⠀⠀⠀⠟⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠤⢞⠃⢀⠄⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠉⠁⠀⠀⠀⠀⢀⣀⡧⠀⠰⡇⠀⢀⣠⢏⣿⡠⠀⠀⠂⠖⠀⠀⠀⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣶⠖⠚⠒⠛⣱⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⢏⣼⠈⡧⣰⡃⢠⢀⠀⣴⡀⠀⡀⡀⠀⠀⠀⣔⣉⣉⠀⣀⢀⣀⠀⣀⠀⠀⢀⠐⠒⠂⢀⠀⠀⢠⠆⢀⠀⢀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠟⠁⠙⠆⠁⠋⠇⠇⠼⠒⠻⠅⠺⠄⢻⠃⢀⣤⠤⠄⠋⠸⠉⠺⠛⠸⠋⠀⠀⠘⠢⠄⠰⠏⠾⠅⠻⠅⠿⠰⠻⠱⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠁⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢰⡃⣤⠠⠤⠄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⠖⢶⢂⡴⠶⣶⡐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠀⠠⠤⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣿⡯⠛⠀⠀⢠⠐⠉⠉⠛⡑⣄⠄⠒⠒⠒⠲⣏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⣓⡈⠐⢓⠂⠀⡒⠂⠚⠓⠒⠀⠀⠀⠀⠀⠤⠠⠄⢀⣑⣢⢇⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠈⡈⠉⠉⢝⠰⠄⠒⠀⠀⠓⠎⠀⠀⠀⠈⠀⠀⠞⣶⠀⢀⠀⠀⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠄⠰⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠄⠀⠀⡀⠀⠀⠀⢈⠶⠦⢤⣄⠰⣃⡴⠃⣀⠤⢤⠀⡀⠇⠀⠀⠀⠀⠀⠀⠴⠀⠤⠄⠄⠠⠀⢠⠤⠀⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣶⠀⠉⠠⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⣯⣐⢎⣔⣁⠀⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆⠀⠀⠀⠀⠀⠡⠀⡀⠀⠀⡄⠈⠈⡉⣦⣉⠀⠀⡠⠁⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣻⣄⠀⠀⠀⠀⠀⠀⠁⡠⠂⠀⠀⠀⠀⠁⠨⠷⣾⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡟⠙⠦⠄⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠆⠂⠀⠒⠁⠲⠀⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⡄⠀⠀⡤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣿⢦⣀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠸⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣙⣢⠤⠀⠀⠙⠻⠿⠶⠦⠖⠶⠆⠀⢙⣋⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠯⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣠⠼⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠴⠦⠀⠀⠉⠒⠀⠁



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

⠀⠀⠀⠀⢠⣶⣤⣤⣀
⠀⠀⠀⠀⢸⣝⣿⣧⡉⠙⢄⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⠂
⠀⠀⠀⠀⠈⢳⣒⣿⡻⣶⠘⡄⠀⣠⢾⣹⣿⠟⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⡇⠀⣿⣼⠿⢃⣠⠟⠁⠀⠀⠀⣀⣀⣀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣾⡇⠒⠚⠉⠁⠀⠀⢀⡴⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢓⡶⢗⢄⣀⠀⠀⠀⢰⡿⣷⣿⡿⠀⣿⠏
⠀⠀⠀⠀⠀⣠⡶⢋⣩⣽⣿⠋⠀⠈⠺⢝⢧⠀⠀⢈⣷⣿⣋⣀⠟⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⣶⣿⣿⠿⠃⠀⠀⠀⠀⠀⠳⣝⢲⡞⢓⣂⣉⣀⣀⡴⣟⣿⣻⡿⠟⢛⣷⣶⣤⡀
⠀⠀⠀⠀⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣦⡈⠉⠙⠻⢿⣿⣾⣿⣿⣿⡳⣶⣖⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠻⣦⡀⠀⠀⠙⠛⣿⣷⣙⢁⣨⣈⠙⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠳⣄⠀⠀⠀⠀⠉⠉⠛⠛⠟⠛⠛⠚⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣘⣧⣴⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣀⣀⣀⢠⣾⣿⣫⣵⣿⡅⠀⠉⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡳⠿⢿⣛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣴⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢯⣜⣁⣛⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠰⠶⠀⠀⠔⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⢤⣀⣀⣀⣠⠤⠤⠚⠑⣲⡖⢶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠉⠉⠉⠉⠁⠶⡖⠺⡉⠠⢨⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢳⡅⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⣷⡄⠐⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠶⠤⠤⣧⣼⠈⠢⡀⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⠀⡴⠊⠂⠢⠀⣀⣀⡈⠓⢄⠈⠢⢼⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠊⢳⢸⠀⠢⡀⡴⢋⠀⠀⠉⠆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⡄⠈⣋⣗⠀⣴⠄⠠⢀⢀⣠⢆⡀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡜⠁⠆⣝⠻⣼⣶⣷⣢⣊⡭⢌⣀⢠⣱⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠣⣄⡅⠂⠭⣴⡞⡙⣧⠨⡤⣭⣤⣤⠟⠀⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⢀⡂⢂⠎⡀⠳⠷⠴⠖⠤⠑⡆⠀⢰⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⣀⡀⢀⠀⣀⣈⣦⡈⠪⣤⡍⡠⠁⠂⢳⣄⡀⣀⠔⢁⡴⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠪⠶⢾⣟⡀⠺⢛⣛⣛⣛⠻⠽⠢⣀⡈⠑⠂⠒⠁⠀⣁⣠⡴⣿⣃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⠿⠭⠽⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠈⡴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠔⠒⠚⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⢋⠙⠁⢀⠙⠂⠀⢧⠀⠀⠰⡟⠩⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡩⠉⣉⠉⠀⠀⠀⠈⡁⠁⠀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡇⠣⢠⣙⠖⣠⣸⢠⡇⢧⡶⠶⢼⠂⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢓⡀⢉⠩⡈⠉⣙⠀⡀⠊⢀⣀⡀⣬⠇⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⢶⡇⠀⠁⠈⠀⠰⠘⠀⠃⠂⢸⡜⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡰⡄⠀⣀⢀⠀⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢳⢳⣦⣌⣿⢴⠶⣴⠀⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠁⠀⠈⠈⠁⠀⠈⠃⢼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣥⡒⠛⠓⠭⣒⣚⣒⣒⣂⣒⣊⠥⠞⠛⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⣆⢹⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠨⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⣎⢹⠒⡇⡤⢠⢰⡁⢾⡴⠂⣶⣦⣆⢶⢂⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡾⠀⠟⠹⠛⠸⠃⠘⢑⠊⠉⠁⣡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢁⣤⣤⣠⢙⢦⠀⠰⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢯⠛⠈⠉⠁⠸⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠓⣴⠃⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠇⠀⠀⠻⠤⣤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠼⠆⠀⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠡⠒⠉⢇⣏⠉⠉⠑⢪⡙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣀⠣⡀⠀⠀⠸⡟⠀⠀⢠⣪⠞⠑⣜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡎⠀⠑⠺⣦⡀⠀⣷⠀⣰⠟⠁⠀⠀⠈⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠙⣲⠟⢾⣁⣠⣤⣴⣖⣒⣃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⢸⢈⠭⠭⠛⠛⠋⠉⡽⢦⡞⢤⡀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠈⡎⢆⠀⠀⠀⣠⡞⠁⢸⡇⠀⠙⢗⠀⡀⢀⢇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠘⢌⢦⡠⢊⠎⠀⠀⢸⢷⠀⠀⠀⠑⣬⢉⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡆⠀⠀⡴⠊⠳⢄⡣⢄⣀⠀⢈⠸⣀⣀⠤⣊⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠔⠃⠀⠀⠀⠉⠓⠂⠭⠭⠤⠅⠒⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣛⣛⡉⣭⣉⣙⡒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠆⠢⠌⠿⠛⠀⠉⠈⠙⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣄
⠀⠀⣀⣤⣶⡶⣶⢶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣞⣠⡵⠞⠛⣛⣛⣉⣙⣛⣛⡒⣒⣶⠦⠤⣤⣀⠀⠀⠈⢧
⣴⣿⣯⣍⣘⠁⠙⡟⢹⣇⢻⣿⣷⣤⡀⠀⠀⠀⠘⣿⣛⡉⠛⠛⠻⠶⣤⣀⣈⣀⣿⣁⣉⡙⢙⣲⣮⣝⣶⣄⠈⡇
⠻⣎⠙⢦⣉⡓⢦⡀⠀⠃⠎⢿⢻⣿⣿⣦⠀⠀⠀⠀⢸⣭⣍⡙⠒⠒⢻⡟⢟⡿⠁⠈⠉⠉⠉⣻⠋⠛⢻⠛⠳⠾⠤⢤⣤⣀⣀⡀
⠀⠈⢢⡀⢻⣏⠲⣝⢦⡀⠀⠈⠀⢟⣾⣿⣧⠀⠀⠀⣿⠘⣨⠟⠛⠲⢬⣷⣈⢇⠘⣃⡤⠴⠊⢉⠶⣶⢧⠀⠀⠀⠀⠀⠉⣧⠀⣹⢧
⠀⠀⠀⠙⢮⠹⡄⠈⠶⣝⢦⡀⠀⠈⠹⣿⠙⣇⠀⠀⣿⠐⡏⠸⠦⠀⠀⠈⣹⠟⠻⢥⣀⡀⣠⠏⠀⡏⠀⠑⢤⡂⣀⠀⢙⠏⢠⠇⠈
⠀⠀⠀⠀⠀⠑⣷⢤⣰⠟⠲⣝⢆⠀⠀⠁⠀⣿⠀⠀⣿⡀⡟⣛⠛⠋⠉⠉⡹⢦⣠⣞⣉⡙⠳⠦⣀⡷⠀⠀⠀⢝⣮⠑⠋⢠⡎⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠣⣷⡤⣀⠘⢮⢳⡀⠀⢴⣿⡇⠀⣿⠰⡇⢤⡄⢀⠴⠋⠀⡜⠋⢯⢧⣽⡄⠀⠈⢿⠒⢤⣌⣠⠏⠀⣴⣏⢴⣎⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡓⢾⣧⢳⠀⠀⣿⠁⠀⠸⣆⢽⢦⡞⢁⡄⠀⢠⠃⠀⠈⢧⠡⡃⠀⢒⡞⠀⡼⢻⢁⣠⣾⡿⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣍⠲⣌⣿⡇⠀⡟⠀⠀⠀⠹⣿⡟⣧⢫⠆⡀⣼⠀⠀⢀⠨⣷⠥⠖⠋⣠⠞⠀⡼⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⢻⣇⡼⠁⠀⠀⠀⠀⠘⠿⣾⣿⣶⣥⠧⣤⣸⡤⠖⠁⢀⣴⠞⣁⡀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣤⣀⣤⣴⣾⡿⢋⣴⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⢲⣖⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠮⠔⡶⠶⣶⠶⢶⣤⣤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠛⣒⠶⣴⣶⣴⣖⡒⣋⠁⣩⠟⢛⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠁⠀⠀⠀⠒⠂⠈⠉⠘⢧⢰⢿⣏⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⡞⢻⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠱⢻⠳⣄⢻⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠐⠚⠛⠛⠧⣈⠲⣽⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠒⠐⢤⡈⠉⠉⢢⠀⠑⢌⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢈⡀⠀⠀⠀⠱⠡⠊⢸⢡⠊⢂⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡌⠁⠈⣒⠢⡑⣐⡆⣰⢯⠅⢀⡜⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡇⠀⠇⣀⡀⣀⣱⢬⡮⣿⠷⣎⠥⠒⠈⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠂⠀⠀⠀⠀⠀⣸⠀⢣⠀⢀⠊⢀⠄⡅⢨⡤⠘⡟⠀⠀⣤⡤⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⠀⢀⠀⠄⠀⠘⡄⠀⢡⡀⠢⡀⠀⣀⠜⡀⠆⠡⠀⢄⠨⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⣀⣥⠂⢀⣀⣥⣥⢹⠦⣤⣟⣦⣀⠉⠉⠢⠀⠈⠜⠒⠊⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠩⠶⢶⠤⠤⠴⠒⠈



⡟⠉⣿⣛⢻⢛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⢄⠀⠀⠀⠀⠀⡔⠀⠀⢢
⠀⠀⡇⢠⠛⡄⠀⠠⢦⡀⡀⠀⠀⠃
⠞⢇⠀⢈⠀⠘⠄⡀⠁⢸⡟⢀⡠⠠⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠈⠀⠀⢳⡄⠁⠒⡺⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠔⣺⠇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠈⠆⠀⠀⠀⣀
⠀⠀⠀⢀⡀⡀⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⡨⠓⠀⠀⣤⠘⠆⠀⠀⠈
⠀⠀⠀⠸⡜⠁⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⠀⢸⠆⠀⠀⠉⠀⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⠐⠠⠀⠀⠀⠀⠀⠀⠈⠁⠱⡀⢂⠀⠀⠀⠖⠈⢉⣳⡄⢀⣀⣀⢤⣀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠘⠐⠠⠐⡡⠆⠀⢦⠠⠀⡈
⠀⠀⠀⢰⠀⠀⠀⣀⣀⡄⠉⠂⠄⠀⠈⠁⠒⠠⢀⠀⢀⣥⠟⢂⠀⢠⢛⠒⠈⠮⢇⣀⠳⡴⠦⠠⠤⠄⠀⠚⢤⡳⠆⢍⠒⠤⡀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠉
⠀⠀⠀⠀⢫⠉⠉⠉⠉⠣⡀⠀⡆⠀⠀⠀⠀⡠⠂⠀⣿⡆⢀⠀⠉⠈⣠⡄⠀⡨⠃⠘⢰⠳⠤⠤⠄⠀⠠⠤⠤⢷⣮⠌⣸⠋⠀⠀⠀⢤⣴
⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠤⢀⠦⠂⠀⠀⠀⠀⠀⠀⠁⠀⠰⣶⠛⠢⠤⠒⡈⣑⠒⠶⢔⣀⠀⠀⠀⠀⠀⢀⠰⠋⠁⠀⠈⠁⠐⠒⠞⠁⢀⡀
⠀⠀⠀⠀⠈⠢⠀⠀⣀⢀⣀⠀⠈⠊⠀⠀⠀⣠⠄⢀⠠⠤⠒⢈⠛⠖⠠⠤⢀⡈⠓⠚⠛⠒⢾⡝⠋⠻⠤⠀⡌⠁⠀⠀⠀⠀⠀⢠⣤⢴⣶⡬⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠃⠒⠠⠄⢈⡁⠄⣤⣠⣠⣤⡠⢀⣄⢀⠿⠔⣂⣤⣐⡠⠄⠊⡉⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠟⠚⠀⠉⠓⣄⠁⠀⠀⠀⠀⡀⠓⠚⠉⠋⠙⠛⠙⠛⠘⠂⠀⠤⢈⡅⠀⠀⠁⠀⠰⠃
⠀⠀⠀⠀⠀⢀⠀⢲⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠒⠚⠋⠐⠁⠀⠀⠀⠁⠀⢈⣟⠰⣯⠠⡀⠀⠒⠠⠄⡀⣨⡀⠔⢈⠔⠉⠉⠉⠒⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠬⠁⠀⠀⠀⠦⠄⡀⠀⣍⡠⠒⠉⠐⠠⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠤⠀⠀⠉⢉⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡞⠉⠉⠭⣩⣤⠈⠘⢀⡉⠀⠈⠐⡄
⠀⠀⠀⠀⠂⠀⠀⠀⣀⠔⠃⠉⠉⡩⠭⠛⢂⣉⠀⠀⠤⠀⣇⣀⢀⡄⢀⠤⠌⠖⠁⢡⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡠⠀⠀⠀⢀⡴⠎⠀⢱⡀⠀⠀⡟⠘⢤⠴⠞⠓⠼⣀⠡⠣⢤⠂⠀⡏
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⠁⠀⠀⠰⡋⠀⠀⠀⠀⡇⠀⠀⠘⠄⠈⡓⣒⠀⠴⠄⡤⠀⠂⢁⡠
⠀⠀⠐⠀⠀⢸⠀⠀⠐⠘⠶⣦⠴⠀⠀⠀⢀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⢸⡀⠘⠦⠁⠀⠚⠀⠀⠈⠁⠁⠀⢰⠃⠀⢀⡠⠔⠚⣩⠍⢉⠀⢉⡙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠱⣄⠐⠒⡖⡀⠀⠀⢄⠀⠀⣠⠋⢀⠔⢁⢄⠀⠈⠁⠀⠀⠉⠂⡀⠉⠀⠈⠑⢦⡀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠈⠓⠤⣄⣀⠀⣀⣀⠀⠐⠁⡠⠋⣀⡡⠎⣠⠁⠄⠉⠁⠀⠀⠀⠸⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠐⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡏⠀⢠⡹⡀⠀⠀⠀⣠⡤⠂⠀⠐⡈⠂⠀⠣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣠⠐⢈⣭⣉⣁⡲⡄⠀⢰⠃⡤⠂⠀⠀⢀⠽⠦⠠⣤⠶⠷⢤⡀⠀⠀⠀⠀⠐⠂⠀⠈⡄
⠀⠀⠀⠀⠀⠀⡜⣡⣼⡿⣿⣿⣷⡿⠈⠄⢸⠈⢣⠀⠀⠀⢨⡀⠀⠀⢉⠀⠀⠈⣿⣆⢀⣠⠄⠄⠀⠀⡠⡧
⠀⠀⠀⠔⠃⠀⡁⠒⣻⣿⣿⣿⣽⠇⠀⡀⢸⠀⠀⠱⣄⠀⠀⠈⠂⣀⣈⠲⣀⢠⣟⡽⡅⡂⠀⠄⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠱⣸⣿⣿⠿⢿⠟⠁⡔⠀⡈⢧⠀⢠⠈⢦⡀⠀⠀⠀⠈⡝⡁⠿⢅⠊⠉⠓⠀⠴⠈⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠉⠊⠡⠤⠀⢂⢊⡀⠄⡀⠈⢦⡀⠃⡀⠉⠢⣀⠀⠘⢧⠀⠚⡋⡤⠟⠀⠐⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠠⠒⠑⠢⠱⠀⠱⣄⠀⠀⠀⠈⠓⠦⣌⠒⠈⢁⡤⠤⠆⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡧⠁⢰⠫⠹⢉⠀⠀⠀⠈⠙⠢⢤⣀⠀⠂⠈⠳⣄⣈⡠⠔⠊⠁⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣐⠀⠓⠀⠎⠀⠁⠀⠀⠀⠀⡀⢀⣉⣀⡀⠀⠉⢁⣀⣤⠔⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠀⠀⠉⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠒⠶⠶⠖⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠲⠶⠒⠒⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣞⠰⠰⣧⠉⠅⠢⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣗⣂⡐⡒⠒⠚⢵⡫⣤⡈⢋⡀⠀⠀⠙⡖⠂⣀⣀⣠⣶⣶⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡟⢘⣿⡛⠿⢚⣿⣷⣯⠽⠟⣿⣻⣿⣟⡗⢀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣷⢈⡟⠀⠀⠀⣾⡁⠀⠀⢰⠘⠀⠁⠀⠀⢸⡀⠀⠽⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⡇⠀⠀⠀⢹⣧⡁⠀⣸⠀⣄⠀⠀⠀⢃⣧⣘⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣯⣿⣿⣷⣾⣿⣿⣿⠀⠀⠛⢀⣸⡆⢠⠆⠜⣸⣻⣿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣛⠻⠿⠿⣿⣿⣿⣿⣿⣆⣠⣬⣤⣾⣷⣾⣟⣛⡿⠭⠵⡚⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡍⠓⢘⣿⣶⠾⠾⠟⠋⠹⠭⠉⠁⠂⠘⠒⠒⠈⠁⢰⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢿⣷⣤⣤⣄⣤⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣤⣼⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠿⠿⠿⣿⣛⣻⣿⣿⠿⢿⣿⣿⣛⣛⣛⠛⠭⠽⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢠⠀⣀⠀⠀⢠⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢂⠠⠴⠶⠶⠬⢌⣀⣙⣛⣀⣡⡤⠤⠶⠤⠌⡑⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣔⠕⣠⢞⣡⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣦⣍⡑⡄⢀⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⣰⡿⣻⣵⣷⣴⣴⣾⣿⣷⢭⣥⣭⣤⣶⣾⣿⣤⣴⣬⣙⢳⣾⣆⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢰⣿⣤⡿⣛⢫⡥⠭⢛⠴⠒⠲⢩⠵⠚⠒⢝⠉⠭⠻⢟⣻⣷⡌⣿⡄⢸⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠆⣾⣿⡯⠉⡁⠀⡀⠀⣨⠀⠀⠀⢸⠀⠀⠀⢰⡀⠀⢀⠀⠈⠻⣿⣜⣷⠘⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⢰⣿⣿⣹⣀⣷⣤⣷⣶⣿⣶⣶⣶⣾⣷⣶⣶⣾⣷⣶⣿⣴⣇⣠⢘⣿⣿⡆⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⣖⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⢸⣿⣿⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⡇⢨⣭
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠈⡿⣿⢠⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢳⠻⣿⣿⡇⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣷⡀⣇⣿⡖⢒⣻⣿⡿⠋⠉⠀⠉⠻⣿⣿⠟⠃⠁⠀⠉⢻⣿⣗⡒⢷⣿⣿⡇⣾⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠼⠿⠇⢿⣿⡄⢠⠾⣿⠃⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠈⣿⣿⡀⢸⣿⣿⢠⣛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⣿⣷⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢿⠏⣿⣿⣿⡇⠼⣿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡀⢻⣿⣷⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣬⢟⣾⠟⢸⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣴⢆⢹⣷⣿⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢛⣿⣿⠟⢰⣷⣌⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠋⠀⢿⣿⣷⣿⣷⣤⣆⣤⢀⡄⢀⡠⣌⣤⣶⣾⣷⣿⣿⡷⢠⣿⡙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠙⠻⢿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠻⢋⠰⣦⠙⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡰⢀⣌⠁⡻⣿⢿⣿⣿⣿⣿⡿⢁⡀⠄⠂⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⠁⡌⠓⢤⡡⠄⠋⠔⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⢠⠠⢠⠄⡀⡀
⠀⠀⠀⠸⠀⡆⠀⡀
⠀⠀⠀⢀⠉⠙⠚⠛⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⡀
⢠⡈⢹⠀⠑⠊⠐⠀⠈⡅⠋⠁⡇⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠋⠉⣠⠄⠀⠉⠙⠛⠿⣷⣦⣄
⠘⠥⠴⠤⠴⠤⠴⠤⠤⠦⠦⠤⠇⠀⠀⠀⠀⢠⣵⣿⣿⣿⣿⠏⠀⠀⠀⣠⡌⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡏
⠀⠀⡂⡦⠀⡤⠀⡤⠤⠀⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⠁⠀⠀⠴⣿⣿⣿⣶⠀⠁⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⡇⣀⠀⡂⠀⡀⣸⠀⢀⡇⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⠈⠀⠀⠀⠹⢿⣿⣭⡀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠈⡉⡩⢉⣉⢍⠉⣉⢍⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠴⣶⣄⠛⠛⢀⣤⣀⣀
⠀⠀⠀⡇⡄⠀⠉⢨⠀⢽⢸⠀⠀⠐⢴⣿⣿⣿⣿⣏⡄⠀⠀⢀⣴⢆⠀⠀⠀⢀⡀⢀⠀⠀⠀⣠⣿⡿⠓⠾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠳⠶⠶⠶⠶⠦⠾⠞⠀⠀⠀⣾⣿⣿⡯⣿⣿⣗⣦⡄⠐⢶⣿⣷⣤⣶⣿⣷⣴⡄⣠⣾⡿⠏⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠘⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⠻⡿⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠾⠁⠀⠈⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⠾⠷⢶⣄⡀⠀⠀⣀⡴⠞⡓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⠋⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠉⠛⠓⠮⣝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣠⣤⣤⣿⣦⣄⣀⠀⠀⣠⣾⣥⡤⠶⢦⣤⣄⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠉⠉⠙⠛⠛⠉⠉⠀⠀⠀⢀⣾⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣇⣠⣤⣶⣿⣿⣄⠀⣰⣦⣤⣤⣄⣠⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⠟⠉⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠙⢿⣿⣿⠿⠟⠛⠉⠉⠉⠁⠈⠉⠻⣿⣟⠛⠛⠛⠻⢿⡿⠃⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠙⣦⡀⠘⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠈⢳⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⢻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡐⠳⣶⣤⣤⣤⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢬⣭⠥⠖⠋

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⢀⣠⡶⠟⠛⠛⠛⠿⠿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠿⠛⠛⠛⠿⢶⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⠁⢀⣀⢀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠋⢿⡛⠾⢙⡟⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢹⡏⠾⢋⡿⠈⣻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠽⡿⡯⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠽⠿⠯⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡌⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⣶⣢⣄⣉⡍⣭⡭⢉⣉⠂⠲⠤⣄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⡦⠝⣯⠆⠀⢸⣧⠀⠀⠉⠳⠬⠾
⠀⠀⠀⠀⠀⢸⠁⠼⠀⠀⠀⣄⠀⠀⠀⠀⡀⢀⣀⠈⣿⡄⠀⣼⣿⡄⢀⠖⢉⡆⠀⢠⡺
⠀⠀⠀⢀⡠⣄⠑⡄⢰⠲⢗⠂⣺⢀⣞⡄⢹⠁⠛⠀⢸⣣⢠⣿⣿⣇⢸⡀⠀⡀⢐⠟⡢⠀⡴⠶⠐⡠⢆⡄⢠⡀⡘⠢
⠀⠀⠀⠸⣔⠛⣀⠆⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡏⣾⢽⣿⣽⡆⠉⠁⠀⠈⠀⠛⠁⠉⠉⠈⠁⠀⠐⠃⠈⠑⢧⡜⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣛⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⢿⣿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣿⣸⣿⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡿⣯⣿⣧⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣯⢿⣸⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⣿⣏⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣧⣄⣿⣇⣟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣥⡿⢷⡄⠉⠻⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠿⣵⠷⠃⠀⠀⠉⣿⣿⣧⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⢀⡤⠖⠛⠓⢷⣿⡟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⣠⠏⠀⠀⡴⡶⢳⡀⠙⢦⡙⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣟⠀⠀⠀⠳⢭⠽⠃⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣶⣿⡿⢿⡛⠉⢩⠀⠀⠀⡇⠀⢀⠂⠀⣠⠈⠉⠛⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣤⣿⣿⠛⠉⢱⣀⣀⣧⣤⣼⣦⣶⣾⣷⣦⣼⣤⣠⣏⠀⠀⢀⡞⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⣠⣿⣿⣯⣤⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣅⡀⣠⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⠎⣿⣿⣷⡀
⠀⢤⣀⡀⠀⠀⠀⠙⠲⣤⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣻⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢰⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡆⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠠⠬⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣿⠏⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠤⠴⠶⠿⠿⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⢾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠋⣉⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠻⣿⣿⣽⡿⣿⣿⣿⣿⣿⣷⣽⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣁⣀⣀⣹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠋⣉⣽⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⣠⡶⠟⠛⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⢿⣿⠛⠛⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡾⠋⣹⠿⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣟⠻⢯⡌⠙⣆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠞⠁⡴⢋⡴⠛⢉⣿⠟⢻⣿⠿⠿⣿⡿⠿⣿⡿⣟⠻⣯⡉⠳⣌⠙⢷⡀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁⠀⢸⠃⠀⣼⠃⠀⠀⣿⠀⠀⢸⡇⠙⡇⠈⠳⠀⠈⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠛⠀⠀⠸⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠿⢿⣦⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣷⣾⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣷⣿⣷⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠟⢻⡛⢛⣿⡟⢻⡟⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣿⠋⢻⣿⠋⢻⠛⠿⣿⣿⣿⣶⡥⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡀⠘⢧⣄⣁⣠⡾⠁⡀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣀⠹⣦⣤⣀⣴⠟⢀⣀⠴⠋⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⣧⠀⠀⠘⠆⠉⠈⠀⠀⠛⠷⡀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⢐⡈⠣⠍⢉⠛⠳⣦⡀
⠀⠀⠀⠀⠈⠻⢷⣤⠶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠈⠙⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠸⣅⡀⠀⠀⠱⡄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⠀⢹⡀⠟⠀⠀⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⡝⠀⠀⠀⠀⣀⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢀⠤⠴⡒⢢⠙⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⢀⡔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠦⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⠀⣀⣀⣀⢾⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠈⢶⡆⢦
⠀⠀⠀⠀⠀⠰⡀⠘⣶⡄⣀⠉⠉⢃⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣌⣉⡉⠉⠉⢁⡈⠂⠤⠔⠊⣉⠟⠁⠀⠀⠀⣠⠃⡅⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⣿⣧⠴⠃⠀⠀⠀⠄⠁⣄⣀⣀⣀⣠⡤⠶⠛⠋⠉⠉⠁⠀⡀⠤⠐⠈⠁⠀⠀⠀⡠⠖⠁⠐⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⣀⣠⠼⠛⠉⣉⡩⠐⠏⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠒⠢⠾⢿⠃⠀⣠⠞⠉⠀⠀⣶⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡼⠁⠀⢠⠀⣱⣾⡇⠀⠀⠀⢀⣼⣀⠀⠀⠀⢠⠀⠀⠀⠳⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡼⠀⠀⣄⣸⡷⠿⠋⠀⠀⠀⠀⠈⠉⠉⢰⠄⠀⢸⡰⣶⣀⡀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡦⠀⣶⣂⣳⠀⡄⠀⠀⠀⠀⠀⠀⢇⣧⣶⡾⠟⠋⠀⠀⠀⠀⢠⣀⡀⣀⢠⣴⠋⠀⠀⠉⠋⠀⢹⣷⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠮⠟⢡⡊⠉⣹⡟⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣶⣾⠿⠃

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠳⣄⠀⣠⠶⠛⠉⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣶⠋⠁⠀⠀⠉⠳⠦⠤⠶⠚⠉⠁⠀⠉⠛⢦⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⡴⠞⢁⣴⣿⣿⣿⣿⣿⣷⣀⠀⠀⣀⣤⡀⠀⠀⠀⣠⣤⣀⠈⣧⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸
⣿⠀⠀⠐⢿⡀⠰⣿⣿⣿⣿⡿⠟⠉⠀⠉⠙⣻⠿⠛⠙⠲⠶⠚⠁⠀⠉⠉⠙⠳⢦⣤⡄⠀⠀⣹⡆⠀⢸
⣿⠀⠀⠀⠀⠹⣆⠘⣿⡿⠋⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢀⡼⠋⠀⠀⢸
⣿⠀⠀⠀⠀⠀⢸⣶⠋⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡴⠋⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡧⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣯⡛⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⣼⣿⣿⣿⡿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣦⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣷⠹⣿⣿⣿⣷⡄⠀⠙⣿⣿⣿⣧⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠙⢿⣿⣿⡟⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣶⣶⣶⣤⣤⣄⣀⡀⠈⣿⣷
⠀⠀⠀⠀⢰⣿⣯⣴⣿⣿⠿⠿⠛⠛⠛⣉⡉⠉⠀⢸⣿⣧⣶⣶⣶⣶⣶⣾⣿⣇⠀⠀⣠⣬⡉⠉⠛⠛⠛⣿⣿⣷⣽⣿⡆
⢀⣴⣿⣿⣿⣿⣿⠟⠻⣿⣦⣀⠀⠀⠺⣿⡿⠀⢠⣾⡿⠟⠉⠉⠀⠈⠉⠙⢿⣿⣆⡀⠿⠿⠟⢀⣀⣤⣾⡿⠋⠻⣿⣿⣧⣤⣤⣤⡀
⣿⣿⠁⠀⢸⣿⡇⠀⠀⠈⠛⠿⣿⣷⣶⣶⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⢸⣿⡿⠛⠉⢻⣿
⢻⣿⣄⣀⢘⣿⣇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣀⣼⣿
⠀⠙⠿⠿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⠿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⣀⣀⣀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢀⣴⣿⠿⠿⠿⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣧⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠒⠒⠒⠒⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⢀⡰⡀⡄⠠⣠⢀⡢⡠⡀⢀⡄⢈⡉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⢢⣄⣨⣷⣽⣾⣶⣷⣿⣿⣿⣷⣿⣶⣩⣶⣦⣢⠔⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢄⣀⣢⣽⣾⡿⢿⡿⠫⣿⣿⡿⢿⣯⡻⣿⠻⢿⣿⣿⣿⣶⣚⣥⣠⣌⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤⣿⠟⢻⡟⢀⡏⡀⢀⡼⣿⣷⣼⣿⣿⡜⡆⠈⣿⠉⠛⠿⣿⣿⣮⣔⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣲⢿⠉⠀⠀⢸⡇⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⣿⠀⠀⠀⠈⠙⠿⢿⣿⣥⣶⣊
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠉⠀⠀⠀⠀⠈⣷⠀⢣⡻⣿⣿⣿⣿⣿⣿⡿⣱⠃⢠⡟⠀⠀⠀⠀⠀⢀⡤⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣦⠀⠀⠀⠀⠀⠘⢷⣀⠙⠮⣝⣛⣛⣛⡭⠞⠁⣠⠟⠀⠀⠀⢀⡠⠶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣤⣴⠟⠡⣀⣠⠴⠊⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠒⠶⠛⠻⠟⠛⠟⠈⠙⠉⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣻⣽⣾⠃⣿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣿⠿⠁⢠⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠛⠋⠁⣠⠀⢸⠃⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠒⠚⠋⠁⠀⡿⠀⠀⠀⠀⠸⡆⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⡆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣶⣤⣼⣥⣤⣀⠀⣠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⡇⠻⣷⣝⢿⣿⣿⣿⣿⣿⣿⡟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⠙⠻⢿⣿⣿⣿⠿⠟⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⢀⣠⠾⠋⠀⠀⠀⢀⣀⣀⡠⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠾⠟⠛⠛⠛⠻⠶⣦⣄
⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⣀⣤⣴⡶⠶⠶⢶⣦⣤⣀
⠀⠀⠀⢸⣿⡛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠿⠋⠉⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⢻⡏⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠘⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣠⣿⠉⠉⠉⠉⠉⢹⣿⠛⠛⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⢻⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⣿⡇⠀⠀⠀⠀⢨⣿⠻⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⡆
⠀⠀⠀⠀⠀⠘⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣾⣿⣦⣤⣤⣭⣽⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣹⣿⠿⠛⠉⠉⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⠟⠋⠁⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⠋⠁⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⢀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠶⠶⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⣀⡀⠀⠐⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠘⠳⠦⠌⣄⡀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠠⠤⢄⣠⣄⣠⣀⠤⠤⠒⠊⠁⠈⣹⡋⠝⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠋⠉⠉⠉⠉⠉⠉⠙⠛⢾⡳⠲⣾⠋⢀⠄⣉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⣬⡇⡆⢠⠉⠄⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⣇⠈⠄⢀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡵⢻⢢⡀⠘⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣏⣇⣎⡆⠑⢄⠀⢓⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⡀⣀⣉⠛⠲⣄⠑⢦⣁⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠒⠒⠢⣄⡊⠁⠁⠉⢳⡄⠀⠑⢦⡈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠀⠀⠀⡇⠀⠀⠀⠀⡌⢣⠼⣠⠖⠁⡿⢀⡖⢦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠁⠀⢀⠄⠲⢄⡀⡀⢢⠐⡀⣆⠱⠀⣴⢧⠯⠀⠘⡇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡅⠀⢰⡃⠄⠀⢘⣌⡙⢢⠉⡄⣿⢀⡵⢧⡦⠀⢀⡴⠁⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠸⡆⠀⠀⢀⠀⠀⣦⣁⣞⣿⢿⣷⡪⣙⡒⠴⠀⠓⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⣧⠀⠀⠈⣙⠋⠐⠉⠛⡁⠒⢃⢙⣗⣦⢝⠃⢄⠐⡀⠀⠇⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⢆⠀⢸⠁⠀⠖⠀⠎⠀⢼⠞⠀⠘⡎⢄⢈⡀⢿⠜⠋⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡈⢷⡀⠀⢘⢦⡀⠳⣄⠀⠀⢁⡠⢋⡄⢺⠀⢱⠀⠰⡅⢠⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡳⣤⣤⣥⡭⠉⣽⣖⣾⣶⣷⣿⣿⠷⣄⢀⢨⡷⣤⡈⠑⠛⠉⠳⣀⠁⠀⢂⡸⠤⠤⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠂⠐⠒⠒⠒⠒⠒⠀⠉⠉⠙⠻⢶⣾⡿⢳⠤⣀⠀⠈⠉⠉⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠛⠿⠖⠒⠒⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠈⠉⠙⠓⠋⠉⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣀⣠⠴⠒⢿⠉⠉⠁⠙⡏⠀⠉⠉⡿⠒⠢⣄⣀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⢀⡞⠁⠀⢀⡴⠊⠉⢃⠀⠀⢘⣄⢀⣀⣠⣇⣀⡀⣠⠃⠀⠀⡘⠉⠑⢦⡀⠀⠈⢳⡀
⠀⠀⠀⠀⡞⠀⢀⣴⠃⢡⠀⣀⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⡀⠀⡌⠙⣦⡀⠀⢳
⠀⠀⠀⢸⠁⢠⡫⠈⣤⣴⣿⡿⠿⠛⠛⠛⠛⠿⢿⣿⣿⣿⠿⠛⠛⠛⠛⠿⢿⣿⣦⣴⠀⠿⡄⠘⡇
⠀⠀⠀⢸⠀⣿⣱⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣿⠀⡇
⠀⠀⠀⠸⡄⠈⠻⣍⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⠟⠁⢠⠇
⠀⠀⠀⠀⢳⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠱⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⠎
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣤⣀⣀⣀⣀⣤⣤⠴⠞⠋

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⣅⠠⠋⠶⣠⠀⠀⠀⠀⡠⠼⠄⠀⠀⠀⠀⣀⣤⣀⣤⣀⡠⠆⠀⠀⠘⠉⠉⠟⢐⠀⠀⠀⠀⢠⠭⠿⢏⣺
⠀⠀⠀⠀⠀⠧⢴⠴⠀⠏⠀⠀⠀⠰⠁⠀⠀⢡⠀⠀⠠⢵⠋⢱⠧⡟⣏⡂⠀⠀⡘⠉⣭⢃⢸⠀⠀⢠⠊⢴⢔⣶⢲⣶
⠀⠀⠀⠀⠀⠀⡶⠓⠆⠀⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⠀⠒⠂⠤⠓⠁⠀⠀⢀⢁⡼⢷⠀⠊⠀⠀⡀⡀⠼⣚⠛⡽⠎
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡆⠀⠀⠀⠀⣴⢶⠙⢉⠶⣤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠒⠒⢊⠹⣟⠒⠒⠂
⠀⠀⠀⢀⢢⣿⠑⣦⣤⡤⡀⠀⠀⠀⠘⠃⠣⠀⠀⠀⠘⡍⠂⣴⣦⠒⢡⠃⠀⡠⠚⢳⠋⣲⠓⣄⠀⠀⠀⣧⣦⠈
⠀⠀⠀⠘⠊⡎⡛⢲⢟⣶⠷⠀⠀⠀⢰⠩⢾⠀⠀⠀⠘⡰⣂⠌⠉⣐⢆⠂⠀⡁⠀⠀⠀⠀⠀⠸⠀⠀⢤⢽⣷⣼⣄
⠀⠀⠀⠀⠑⠀⠐⠚⠋⠁⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠈⠀⠁⠈⠐⠁⠀⠀⠈⠒⠒⠒⠒⠒⠁⠀⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⢰⣶⣒⢠⣶⣀⣶⢄⣴⠲⣦⢰⣦⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠤⠸⠯⠭⠘⠏⠿⠹⠏⠷⠴⠟⠸⠏⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠠⠀⢄⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⣀⠀⢀⠼⠸⠼⡄⠀⠠⠊⠀⠈⣨⠋⡙⠀⠀⢰⠖⠈⠀⠀⠸⠀⠀⠀⢠⡜⠈⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡏⡙⣥⡀⠀⠀⠸⠀⠆⠈⡄⠈⢙⠀⡇⠀⠀⡇⠀⠀⠀⠀⠐⠀⠀⠀⡂⢖⠀⠐⠲⠀⠀⠀⠠⡇⠛⡤⠤⡀
⠀⠀⠀⠀⠻⠤⣗⣾⡃⠠⠂⠀⢘⡠⠁⠀⠄⠓⠓⠀⠀⢃⣀⣀⠀⠄⠀⠀⠀⠀⣕⣊⣈⣉⡠⠀⠀⠨⣀⡱⣬⣞⣀⠼
⠀⠀⠀⠀⠤⠄⠀⠀⡤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⢀⢃⢀⢁⡀⢸⠀⢀⠄⡌⢭⠐⡴⢄⠀⠀⠈⠻⠏⠤⠴⠁⠀⠀⠀⢰⠒⠛⢑⠄⠀⠀⢀⠠⠄⡭⢯⠀⢄
⠀⠀⠀⠐⠉⠐⠼⡈⠬⡁⠓⠀⢺⠚⠓⠺⠻⠲⢺⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⠐⡰⠀⠀⠀⠀⡌⠆⠀⠀⠀⠃⠡
⠀⠀⠀⠀⠀⠘⡀⠀⢀⠜⠀⠀⠘⡂⡂⢐⢐⢐⡆⠀⠀⠀⠀⣘⡲⡀⠀⠀⠀⠀⢸⣐⣀⠀⠇⠀⠀⠰⡀⠆⡄⣀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠤⣶⣆⡀⠀⠀⠈⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣽⣷⣤⡀⠀⡀⠁⠶⢀⣀⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⣤⠟⠛⠋⠈⠙⠻⠷⣤⡖⢀⣠⠀⠀⢨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⣤⣀⣀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣉⡛⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠷⣄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣾⠋⣦⠀⠀⠒⠒⠒⠒⠒⠀⠀⣴⣿⣷⣤⡌⠗⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠘⢿⣦⡈⠳⣄⠀⠠⠤⠤⠄⣠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠙⠿⣦⡈⢣⡀⠀⠀⣠⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣤⣴⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢷⡿⠿⠿⠿⢿⣿⣿⠿⢿⣿⡿⣿⠻⠿⠿⢿⣷⣾⡿⠿⣶⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹⣦⣀⡀⠀⠙⠛⣣⡿⠃⠀⠻⣦⣀⣀⠀⠙⢛⣳⣴⠟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠙⠋⠉⠋⣻⠟⠁⠀⠀⠀⠀⠉⠛⠛⠚⠛⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣠⠶⠶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣄
⠀⠀⠀⠀⢰⡇⠀⠐⣿⠁⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠙⡷
⠀⠀⠀⠀⠈⢷⣦⣴⣿⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠶⠶⠦⠴⣶⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⢿⡶⠶⠶⠦⢶⣶⠶⠲⣶⡞⠛⠟⢻⣿⠛⠛⠛⠋⠉⠁⠀⠀⣽⡟⣧
⠀⠀⠀⠀⠀⠀⡾⠀⣷⠀⠀⠀⣼⠆⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿
⠀⠀⠀⠀⠀⣾⠂⢀⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⣧⠀⣸⡇⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢸⡏⠀⠸⣇
⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⢸⣧⣴⠛⠀⠀⠀⠉⢀⡿⠃⠀⠀⠀⢿⡶⠿⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢠⡄⠀⣀⡉⠀⠀⠀⠘⠁⠻⣇⠀⠀⠀⢀⣾⡇⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿
⠀⣀⠰⢾⡷⠞⠉⣡⣴⣆⠀⠀⠀⠀⠻⣤⠀⠀⢸⡿⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢉⣁⣨⣿⠇⠀⠀⣿⡀
⠀⢻⣴⠎⠁⢠⡾⠋⢁⣼⡇⠀⠀⠀⠀⠙⣦⠀⣸⣇⢀⣀⣤⣯⣤⣤⣤⣤⣤⣤⡞⠛⠉⠉⣿⡇⠀⠀⢸⣿⠀⠒⢀
⢀⡾⠁⠀⣰⠏⠀⣠⠟⠙⢻⣦⢠⣤⡤⠚⠛⠛⠛⠉⠉⠀⠉⠁⠉⢉⣀⣤⣼⠟⠛⠛⠛⠛⣿⡇⠀⠀⢸⣿⠛⠛⠛⢦⡄
⢿⣀⢠⡾⠃⠀⣴⠉⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣶⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⠈⠳⠦⣾⡀
⠀⢹⡿⠁⢀⡼⠇⢀⣰⣯⣄⣤⣤⣤⣤⣴⠶⠶⢾⡟⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣤⣴⣿⠁⢂⣀⠈⢹
⠀⢸⣧⡀⠻⣧⣤⣾⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⣄⡹⠿⠋
⠀⠀⠘⠛⣿⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⠁
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⡿
⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣧⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⠟⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠋⠉⢹⡏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠓⠊⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠏⠁⠀⣠⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠠⣶⣄⠀⠀⠀⠉⠙⠿⠷⣤⣤⡀
⠀⢀⣀⠀⠤⠤⠚⠋⠀⣀⣀⣤⣴⠟⠛⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣶⠏⠈⠛⠷⣤⣤⣄⣀⣀⣀⣀⠙⢿⣿
⣀⣘⣿⣷⣶⣶⣶⣶⣎⣉⣋⣁⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠉⠀⠁⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣶⣶⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣁⣢⡀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠈⠉⠲⢄
⠀⠀⠀⠰⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⣨⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⡤⠰⠇⠐⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠁⠀⠀⠈⡇⠀⠀⠀⠃⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⡋⠈⣆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢄⡰⠤⠤⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠉⣦⠈⠢⡀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢸⠀⠀⠘⢄⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠔⠁⠀⠀⠀⠀⡠⠂
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠀⠀⠈⢂⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠔⠈⠁⠒⠒⠒⠒⠈⠁⠘⡀
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢃⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠙⠲⠤⣀⣀⣀⡀⠠⠐⠊⠑⡀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⡀⠀⢤⡀⠀⢠⠄⠈⠁⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠐⠒⠒⠒⠀⠀⠠⠤⠤⠤⠄⠀⢀⣉⣉⣀⣀⣀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢘⢇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠀⠀⠸⠈⡀⢨⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⢡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡇⠀⠀⠀⠨⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢫⣠⣔⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒
⣿⣿⡿⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⣛⣩⡩⣭⡉⢙⡒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠂⠐⠈⠋⠁⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⣀⣤⣶⢶⢾⣻⣻⣷⢶⣤⣀⠀⠀⠀⠀⠀⠀⣾⣤⣖⣛⣉⣭⠭⠭⢭⣭⢭⣉⣹⣗⣒⡶⢤⣀⠀⠀⠹⡄
⣾⣿⣷⢦⣌⡀⠀⠁⠀⠈⣿⣿⣷⣄⠀⠀⠀⠀⠻⢯⣄⣉⣉⡙⠛⣦⣤⡤⣤⡯⣄⣠⣉⣩⣽⣮⣽⣶⣄⢡
⠙⢦⠈⠳⣤⣝⠦⣄⠀⠘⠙⠏⢿⣿⣷⣄⠀⠀⠀⢰⠶⠦⣬⣉⠉⢹⡆⢠⡏⣄⡀⠀⠀⣈⡏⠈⢉⠏⠉⠛⠓⠒⢦⡤⠤⢤⡀
⠀⠀⠱⣄⠹⡎⠳⣌⠳⣄⠀⠀⠈⠻⣿⣿⡆⠀⠀⢸⠀⣴⠋⠈⠙⠚⠷⣤⣧⡩⠴⠒⠉⢀⡜⢹⡟⢦⡀⠀⠀⠀⢠⣿⠀⡸⠳
⠀⠀⠀⠈⠣⣸⡄⠈⢱⣌⠳⣄⠀⠀⠸⠀⢻⡀⠀⣾⠀⢧⣘⣁⣀⣠⠤⢼⠉⠙⣷⢤⣠⠎⠀⢸⠀⠀⠙⢦⡠⠄⣤⠋⢠⠃
⠀⠀⠀⠀⠀⠈⢻⠳⡏⠀⠙⣝⢦⠀⠀⠀⣸⡇⠀⣿⡀⡏⠐⠀⠀⢀⡠⠊⣱⢾⣧⠠⣌⠉⠳⣼⣄⡀⠀⡐⢽⠋⠀⣰⠃⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣗⢦⣈⣧⢳⠀⠈⢻⡇⠀⢸⡈⢷⡘⣃⠔⠋⠀⢠⠃⠀⢣⡻⠻⠀⢀⠈⡇⠉⣷⡴⠊⢀⣾⡷⢸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⢌⠛⣮⣇⠀⢸⠇⠀⠀⢻⡵⡹⡇⡴⡂⠀⡜⠀⠀⠀⢳⣱⢀⣠⠞⢀⠞⢹⣤⣶⣿⣯⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢾⣿⢀⡞⠀⠀⠀⠀⠻⣧⡙⣮⣁⢂⡇⠀⢠⢀⡹⠋⠀⣀⡔⠉⠀⣸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠿⠎⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣽⣉⠙⠋⠉⣀⣴⣾⠏⣰⣦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣯⡵⠾⠟⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣓⣷⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⡏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠿⢛⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣽⣕⢄⠀⠀⢠⣾⣿⣿⣟⣫⣉⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢏⠊⡏⠙⠛⢿⣟⣿⣭⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢠⣴⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣿⡏⠈⠳⣤⣴⣾⣿⣿⣾⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣾⣽⣻⣿⣿⣧⣠⠀⠀⠀⣸⣿⣇⠀⠀⣿⣿⠯⣟⣛⡻⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⣽⣿⣿⣿⣄⣠⠞⣿⠃⣿⠀⠀⠻⣿⣧⣹⣷⣝⠿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⡿⠟⣃⡴⠋⢠⠏⠀⢿⡇⠀⠀⠉⠻⠿⣿⣿⣿⡾⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⣀⣠⣴⣚⠁⠀⢀⡞⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠛⢈⡏⣵⣿⠋⠉⢹⣦⣼⣁⣀⣀⣸⡼⣄⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣶⣾⣾⡟⣸⣿⣇⠀⠀⠀⠙⣻⣿⠟⠛⠿⢧⡿⠟⠋⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠿⠛⠋⠉⠀⣿⣿⣿⣷⣤⣤⣾⣿⠃⣠⠀⠀⠀⠀⠀⣠⣾⣷⣼⣷
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢄⣀⠀⠀⠀⠀⣀⡠⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠈⠑⠦⠔⠊⠁⠀⠀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠉⠀⠈⠑⠒⠤⢄⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢻⠻⣿⣿⣿⣿⣭⣁⣈⣻⠉⠉⠉⠉⡏⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠈⠉⠉⢉⣋⣉⣭⣽⣿⣿⣿⠟⢩⠏
⠀⠀⠀⠀⠀⠀⠀⢳⡈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠻⣿⣿⣿⠟⠋⠉⠉⠉⠙⢻⡿⠛⠋⠉⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⠟⠁⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢨⠏⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢬⡿⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⡰⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡠⠊⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠶⢤⣀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠂⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣠⡴⠒⠒⠚⡏⠉⠉⠒⠒⠒⠲⠒⠒⠋⠉⠉⢹⠉⠑⠲⢇⣀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⣤⣾⣿⣿⣧⣤⣄⣠⡇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⣀⣀⣠⣾⣾⣷⣶⣄⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⣠⣶⣿⣿⣿⡿⠟⠛⠛⠛⠿⠿⢷⣶⣤⣤⣤⣾⣧⣤⣤⣴⣶⡿⠿⠛⠛⠉⠉⠉⠛⠿⣿⣶⣦⣉⠄⠀⠙⠷⣄
⠀⠀⠀⠀⠘⣇⠀⠠⢴⣾⣿⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣿⣿⣶⠄⡀⣹⠆
⠀⠀⠀⠀⠀⠸⣆⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠁⢠⡟
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣾⣿⡿⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠁⠂⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢰⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢦⣤⣤⣤⣤⣤⣤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠠⣤⣴⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⡈⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠐⡈⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣀⡀
⣰⣿⡿⣿⣿⣿⣿⣆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣷⣿⣿⣿⣿⣿
⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏
⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠁










⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⡬⡄⠨⢍⡙⠒⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣾⢀⠉⠉⠀⠀⢀⠉⠀⠀⠈⠂⠐⠠⠤⣉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⢋⡅⠀⠶⢂⢀⠀⡬⠵⣆
⠀⠀⠀⠀⠀⣸⠹⡌⠓⠀⠀⠀⠀⠀⠂⠀⣄⡀⡀⠀⠀⣫⠀⠀⠈⣳⣤⡀⠀⠀⠀⠀⣀⡴⠛⡁⠐⠎⢡⠀⠀⢈⡴⠃⠀⠀⠠⠚⢿⡆
⠀⠀⠀⠀⠀⡏⠀⠹⣆⠠⡀⠀⠀⠀⠀⠤⠴⣿⡞⠳⣞⣋⣀⠀⠀⠘⢣⠙⣆⠀⣠⠞⢋⠠⠸⣤⣀⡀⣸⠦⣶⣏⣀⡀⠀⠀⠈⢁⡞⡧
⠀⠀⠀⠀⠀⡇⠀⠀⠈⠃⢈⠁⠀⠀⠀⠀⠀⠀⠶⠒⢟⠻⢭⡉⠁⠒⠀⠠⡈⢳⡁⠶⠄⠀⣀⣨⣭⣿⣄⠎⠿⡁⠀⠀⠀⠐⢀⠘⢁⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⢀⠀⠀⠀⠈⣇⠀⠈⠲⣄⡀⠠⢨⡄⡇⠐⠊⠉⢉⡴⠋⠰⡏⠁⠀⠑⡈⠀⢀⡴⠋⠀⣸
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣙⡀⠤⠀⠘⢦⡀⠀⢀⠙⠀⠀⣱⠃⡀⢀⡴⠋⠀⠀⠀⠁⣠⠠⣂⣠⠖⠉⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠬⠤⠤⠀⠒⢈⣿⣛⠀⠈⣴⠀⠄⢀⣀⡤⢴⠛⠉⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣼⠫⣝⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣴⡿⠃⠠⠉⠲⣶⣦⣤⣤⣤⣤⠤⠤⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠛⠋⠉⡱⢹⡇⠀⢀⠀⠀⠀⠀⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⡹⠁⠀⠀⠀⠀⢰⠁⢈⡇⠀⠈⠀⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡼⠀⠘⣇⡀⠀⠀⠙⢄⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠜⠀⠀⢸⠁⠳⡄⠀⠃⠀⠹⡀⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⡀⠚⠀⢀⡠⠃⠀⠀⠈⠓⢄⠀⠀⠀⢦⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠴⠁⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⢸⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⠃⠘⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⡀
⠀⢀⣴⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣷⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⢷⣄
⠐⠋⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣦⣄⡉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⣁⣤⣶⣶⣶⣤⣤⣤⣄⣀⡀⠀⠀⠈⠓
⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⣶⣿⣿⣿⠿⠋⠁⡟⣿⣿⣿⣿⡟⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⣿⣿⣿⡟⡏⠉⠛⢿⣿⣿⣷⡄
⠀⠀⠈⠹⣯⣤⡀⢀⠀⠻⠑⢜⡋⠝⠁⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡀⠙⠍⢛⠔⠹⠃⠀⢀⣠⣬⠯⠉
⠀⠀⠀⠀⠀⠘⠛⠷⠶⢷⣶⡾⠿⠿⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠘⠻⠿⠿⣶⣶⠶⠶⠟⠛






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⡴⠚⠉⠁⠀⠈⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⠉⠁⠀⠀⠈⠉⠉⠓⢦⡀⠀⠀⠰⠋
⠀⣰⠋⠀⠀⠀⢤⡀⣤⡠⣄⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⣸⠃⠀⠀⣈⢳⡈⢳⣌⢿⢿⡗⠄⠀⢧⠀⠀⠀⠀⠀⢀⡞⣀⠀⠀⠀⡀⢀⣤⠀⡴⣠⠞⢧⠀⠀⢸⡇
⡏⠀⠐⣦⡘⢷⣽⣦⣻⡻⣧⠉⠀⠀⢸⠀⠀⠀⠀⠀⢸⡏⠁⣀⣴⡯⠔⠋⠿⠟⠋⠁⠀⠈⢷⢛⠚⠧⣀
⣇⠀⠻⣮⣿⢮⣿⡻⣝⢷⡌⠃⠀⠈⠳⡀⠀⠀⠀⠀⠈⣗⠋⢉⣥⣀⡀⠀⠀⠀⠀⣀⣤⡤⢸⢸⠀⠀⢸⠀⠀⠠⠖⠶⠴
⠹⡄⠐⢽⡝⢷⡙⠷⠈⠣⠀⠀⠠⣄⡀⠙⢦⡀⠀⠀⠀⠻⡆⠀⢠⣤⡅⢀⡇⠀⠈⢉⣀⠀⡆⢸⡠⢄⣸⠀⠀⠀⡀⢀⣀
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠉⠓⠢⠳⠀⠀⠀⠀⡇⠀⠈⠉⠁⣸⠀⠀⠀⠙⠛⢠⠋⠉⠒⠀⠻⣄⠀⠀⠈⠉
⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠘⢦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⠴⢛⡍⠛⠦⣠⠘⣄⣠⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡑⠂⠀⠀⠀⠀⠀⣁⠜⠁⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡓⢲⠒⠒⡖⠋⠁⠀⠀⠈⡆⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡇⠀⠀⠀⢸⡤⢄⡀⠀⠀⡋⣉⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠉⢇⠸⣄⠀⠀⠀⣀⠇⢰⠋⠑⠲⡧⠔⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢦⣀⢽⡒⣛⢥⡴⠃⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠇⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⠀⠀⠀⢸⣘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣀⠤⠤⠤⣾⠀⠀⢧⠤⠄⠀⠀⠈⡏⠓⠤⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⢇⠀⠀⢨⠆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⣿⠀⠀⠀⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣄⣀⣀⣀⣀⣆⣀⣠⠤⠤⠔⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣷⣶⡆⠓⠀⠀⠀⠀⢶⣶⣶⣦⠀⢠⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠈⠉⠈⠁⠀⠈⠻⣧⣄⣈⠛⠛⠁⠀⠈⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢁⣠⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⣶⣿⣿⡿⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⡀⣠⣄⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⡟⠛⠛⠛⠿⠷⠶⠦⣄⢀⠀⠀⠀⠀⢀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣅⣠⣇⠀⠀⢀⠀⠀⠀⢠⡀⠀⢉⣲⣦⣄⡈⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠋⠛⠛⠛⠛⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣆⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠛⠿⠛⠛⠿⠿⠋⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⣠⣶⣿⣟⡛⣋⣀⣀⣀⣤⣤⣴⣶⣾⣾⣿⢿⣿⡿⠁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠉⠉⠉⠉⠉⠉⠁⡹⠋⢀⣼⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣦⣴⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣠⣤⣤⣀⣠
⠀⠀⠀⠀⠀⣴⣿⢿⡿⢿⣿⣿⣿⣿⠻⢿⣷⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⢀⡞⠋⠀⣾⠀⠀⣾⣿⣿⣿⡇⠀⠈⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠋⣿⠿⢿⣿⣏⠉⢿⠉⠙⠻⣿⣿⡮⠄
⠀⠀⠀⢘⣇⡀⠀⠘⢄⠈⠈⠃⢙⡿⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⠀⠀⠘⢦⣈⡉⢁⣴⠏⠀⠀⣠⣼⠻⠁
⠀⠀⠀⠀⠀⠉⠙⠒⠠⢭⣞⣛⣭⡤⠶⠛⠃⠀⠀⠀⠀⠀⠀⠀⣀⢠⣴⡄⡆⣤⢤⣠⡄⠀⠀⠀⠀⠀⠈⠑⠲⠦⣤⢉⡉⣭⣤⠴⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⠿⢻⣿⢿⣿⣿⣿⣿⡾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡟⠁⠀⢸⡇⢸⣿⣿⡿⢿⡇⠈⢻⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠈⠱⢄⣀⠘⢷⣄⡉⢉⣠⡿⠁⣀⣤⣿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⠛⠛⠋⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡴⣲⢄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣦⣤⣤⣴⣦⣀
⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⡥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣽⣿⡿⠛⣻⣿⣿⢛⣿⣷⣷⣄
⠀⠀⠀⠀⣶⠟⠉⠀⡿⣿⣿⣿⡟⣿⡆⠙⢿⣾⡭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⠀⢸⡏⠍⢠⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⡍⣀⠀⠀⠹⡉⠙⠛⠃⡹⠁⣀⣼⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠈⢿⡸⣌⢙⣿⣿⢓⣾⣿⡷⣄
⠀⠀⠀⠀⠉⠉⠉⠛⠓⠛⠓⠒⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⢬⡿⠾⠿⠿⠿⠿⠟⠿⣀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣾⣝⣿⣿⡇⡶⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⠋⡟⠻⣿⣿⠟⣿⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⢯⡀⠉⠋⣠⡟⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢿⣦⣬⣭⣼⠿⠿⠶⠒⠒⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣄⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣘⣀⣽⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⡦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠩⠍⠉⠉⠩⢍⡭⢽⠁⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢸⣕⠈⢀⠀⡌⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣀⣨⠛⠉⢙⣇⠙⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⢀⠟⠓⡋⠀⡇⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡸⠦⣴⠟⢥⢸⠇⠀⠀⢠⠤⠤⢼⣧⢤⠴⠧⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠇⣀⡸⠤⠘⣸⠀⠀⠀⢸⠖⠀⣺⣴⠏⠀⡀⡧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⡏⠀⠀⠀⢸⡆⣼⠁⠘⡖⣿⣏⣁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢳⣄⣀⠟⠁⣼⢩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⠿⣷⢦⡀⠀⢸⡇⣠⢿⣶⠞⠃⣼
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠳⡜⠤⠧⣣⠀⠈⡇⣿⣀⣈⠷⡄⡟⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠐⠁⢐⡀⠀⠂⠒⡀⣿⠒⣎⡏⠀⠀⣏⠸⣁⣀⡼⠠⡇⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢲⠁⠈⢣⣀⡰⢰⡟⠛⠉⠀⠀⠀⠯⠥⠤⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⡞⠁⢲⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⢠⢁⠄⠺⡄⢸⠀⠀⠀⠀⠀⣠⠤⠖⣦⠀⠀⠀⠀⡏⠋⠛⠛⢛⣻⣿⡛⢳
⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⠋⢁⡀⢀⡵⣼⠀⠀⠀⢀⡞⠀⠀⠀⠈⡇⠀⠀⠀⡇⣰⠋⠸⠋⢁⣈⡵⣼
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡳⠒⠂⡄⠀⣿⠀⠀⠀⠈⢧⠀⠀⠀⣰⠃⠀⠀⠀⣷⠹⣄⠾⢽⠉⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢌⠙⠀⠀⠋⠀⣿⠀⠀⠀⠀⠘⠷⠒⠛⠁⠀⠀⠀⠀⢸⠀⠀⢀⣠⠗⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠤⠤⠟⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠑⣄⡠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠉⠉⠙⠗⢦⡀⠀⠀⠀⠀⠀⠀⠘⡇⢐⡖⠚⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠂⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣇⠈⠃⣤⠎⡱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠛⠶⠖⠚⠛⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⢦⣄⡀⢀⣠⣴⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⣤⠶⠟⠛⠉⠉⠉⠉⠛⠳⠶⣤⣄⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣴⠟⣁⣠⣤⡶⠟⠋⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠉⠛⠳⢦⣤⣀⡈⢳⣄⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⣿⣿⣏⣩⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣍⣹⣿⣿⠏⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠻⣿⣿⣿⣿⠿⠟⠛⠛⠻⠿⢿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠹⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⢿⣟⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠙⢷⣌⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠙⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠶⠖⠛⠛⠋⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠲⢦⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠻⣄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣹⣄⠀⠀⠀⠀⠹⣆⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠋⠉⠉⠉⠀⠉⠉⠉⠙⠓⠶⣤⡴⠋⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢷⠀⠀⠀⠀⢀⣤⠟⠁⠀⣿⠀⠀⠙⢦⡀⠀⠀⠀⠀⣼⠁⠀⢠⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢻⡄⠀⠈⢷⡀⠀⣠⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠙⢦⡀⢀⡼⠃⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠹⢿⡁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣹⠞⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠳⢤⣄⣀⣀⣿⣀⣀⣀⡤⠶⠋⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⡴⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠟⠛⠲⠦⠤⣤⣤⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠉⠀⠀⠀⠉⠉⠛⠓⠒⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠛⠻⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣿⣷⣶⣦⡀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢀⣠⡤⠤⣤⣄⣀⠀⣀⣤⠴⡖⠒⠶⢤⣀
⣿⠁⠀⠀⠀⠀⠀⡀⠘⢷⣦⡙⢿⠻⣦⡙⠃⠀⠀⠀⢸⡇⠀⠀⠀⣰⠋⠀⠀⠀⠀⡏⠉⠉⠉⠀⠀⡇⠀⠀⠀⠉⢳⣤⡀
⣿⠀⠀⠀⢠⣤⡈⢻⣷⣬⡻⣝⠻⣷⣌⠃⠀⠀⠀⠀⢸⡗⠀⠀⠀⣏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢸⠀⠻⠛⠓⢦⡀
⢻⡄⠀⠀⢤⡙⢿⠳⣽⣧⡙⢿⣷⡈⠛⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⣸⢻⡄
⠈⢷⡀⠀⠀⠉⠻⢷⠈⠙⠛⠀⠈⠁⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣇⠀⠠⣄⣤⠶⢦⣄⣠⠴⠚⠉⠉⠛⢦⣄⣀⣠⣤⣀⡀⠀⠃⣼⠁⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢦⡰⠋⠀⠀⠀⠀⠀⠹⣆⠀⢈⡇⢀⣀⠀⠀⠘⠛⢋⡙⠛⠀⠀⠀⣀⡀⠈⢻⠶⡦⣼⣧⠀⠀⠀⠀⠀⠠⠞⠋
⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⣀⣈⣹⠷⢦⣄⡸⣧⠀⠀⠀⠀⠀⠀⠀⡟⠢⣼⠀⠈⠙⠛⠶⠀⠀⠀⠀⠀⠰⠾⠛⠛⠁⠀⡿⢸⡇⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠻⠿⠆⠀⠀⠀⠀⠀⠸⡇⠀⢸⣀⣀⣴⣶⣶⣀⣀⣆⡀⠀⣠⣶⣶⡆⠀⠀⡇⢸⠁⠀⠀⢸⠇⠀⠀⠠⣄⣀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣸⠉⢏⡀⠙⢉⡏⢹⠈⠉⢻⠁⠙⠋⣹⠖⢸⡇⢸⣀⣤⡀⢸⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⠀⠀⠉⠉⠉⠀⠣⣄⡀⠈⠑⠒⠚⠁⠀⣸⠗⠛⠧⠄⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⣠⡖⢺⠉⡏⠙⡋⠙⠒⣄⠀⠀⡿⠀⠀⠀⠀⠀⠀⢹⠆⠀⠈⠓⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠾⠴⠦⢚⣶⣷⣶⣷⣾⠶⢾⠧⠘⣇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠰⠄⠋⠁⠤⠶⠀⠉⠁⠛⠀⢀⣧⣀⣠⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠈⢻⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⠶⣶⠶⠖⢲⡞⠉⠀⠀⠀⠀⢸⣆⣀⣠⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡆⠀⠘⡇⠀⠀⠀⠀⠀⢸⣈⡁⢣⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⡀⠀⠀⠀⢀⡷⢤⡀⠀⠀⠀⠀⡏⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⢺⠧⡿⣄⠀⢙⡶⠤⣶⡋⠀⣼⢿⠋⠒⠦⢤⣇⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⢸⡋⣧⠈⠙⠉⡇⣠⠀⠈⢹⣏⣿⠀⠀⠀⠀⣿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠹⡆⠀⠀⠀⠀⢯⡽⣆⠀⠀⣇⠈⠀⣠⠿⣼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣉⢷⣄⠈⣡⠾⣩⠿⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠾⣭⣯⠽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⣧⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⠳⢤⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⢤⣤⠤⠄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⣰⢿⠃⠀⡴⠒⠚⠄⠀⠀⠈⠒⠐⠠⣈⠙⢦⣀
⠀⠀⣸⠇⡇⢀⠼⢥⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢱⡀⠙⢦⡀
⠀⢀⡟⢰⠀⣜⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⢳⡀
⠀⢸⡇⠘⡆⠹⣤⣤⣤⣤⡀⠈⠑⠂⠀⠔⠁⠀⠀⠀⠀⠐⠈⢧
⠀⠘⡇⠀⢧⠀⡆⠀⠀⠀⠀⢀⠔⠁⠦⢆⠀⠀⠀⠀⠀⠈⠀⢸⡄
⠀⠀⢷⠀⠈⢧⠘⢄⠀⢀⡔⠁⠀⠀⡇⠀⠢⡀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠒⠒⠒⠶⢤⣄⡀
⠀⠀⠈⢆⠀⠈⠳⡀⠉⢧⠀⠀⠀⠀⡇⠀⠀⠈⢄⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠈⢂⠀⠀⠈⠢⢄⠑⠢⠄⣀⡇⠀⠀⠀⢀⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠉⠒⠠⣀⣀⡈⠀⢉⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⠀⠈⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣠⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠚⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠠⠤⢒⠖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⡀⣀⡀⣂⣄⣤⠆⠶⠒⠛⠁⠉⠉⠉⠉⠀⠉⠉⠁⠤⣄⡀
⠀⠀⠀⠀⣴⣴⣀⣔⣵⣢⣴⣶⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠧⣀
⠀⠀⠀⠀⠿⡿⠿⠿⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⣢⢤⣤⣴⡿⢿⠿⠿⢿⣿⣷⣖⣶⣒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣠⣶⡾⣟⣾⣧⣤⣶⣷⣾⣾⣾⣿⣿⣿⣿⡟⣶⣿⡵⣂
⠀⠀⠀⠀⠀⠄⢀⠄⢤⣜⣿⡿⣯⣷⣿⣿⣿⠿⠛⣻⣟⣽⣿⣿⣛⠿⣿⣿⣾⣿⣿⣿⣿⣝⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣵⢍⣿⡿⠛⣿⣿⣻⢠⣷⣾⣿⣿⣿⠁⠀⢸⣇⡉⢿⣿⣿⣿⣧⡟⠿⣆⡠
⠀⠀⠀⠀⠄⠠⠆⢄⣼⣉⣾⠁⠀⢀⣿⣿⢟⣿⣿⣿⣿⣿⣿⡦⢤⣾⣿⠄⢸⠈⠹⢿⣿⣿⣿⣯⠖
⠀⠀⠀⠀⠀⡄⠐⣻⣵⣿⣿⠀⢀⠀⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣶⣿⡟⢀⡼⠀⠀⣻⣿⣿⣿⣟⢋⠁
⠀⠀⠀⠀⠠⠀⠲⣿⣿⣿⣿⣷⣜⠄⠹⣿⣯⣜⣿⣿⣿⣽⣿⣿⣿⡿⢃⣼⠏⣀⣴⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠛⢿⣶⣥⣀⠈⠻⢿⣿⣿⣍⣿⣿⣿⣭⣾⣿⣵⣿⡿⠿⠛⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠿⢿⣿⣿⠿⠿⣿⣿⠻⠍⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⢳⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡩⠀⠀⠀⠀⠀⠁⠀⠃⠀⠘⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣤⡄⠀⡀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡠⠊⠁⠀⠀⣠⠔⢀⣴⣿⢀⣴⣫⡴⠛⠁⢻⡄⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣀⣴⣊⣥⠶⠋⠁⡿⠟⠋⠁⠀⠀⠀⠈⢷⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣿⠧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡆⠀⠀⠉⠙⠳⢤⡠⠀⡄⠀⠀⣀⡴⠛⠉⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⢠⣿⣷⡶⠀⢸⠃⠀⠀⠀⣴⣶⣶⠀⠀⢸⠀⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠠⣙⣛⡅⢀⡟⠀⠀⠀⠀⢙⠿⣋⡄⠀⢸⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠧⣤⡀⠀⠀⠀⠉⠁⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡄⠤⠤⠶⠶⠶⠶⠶⠀⠆⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡿⠉⡟⠛⠛⠉⠉⠻⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠛⡿⠁⢸⡄⠀⠙⠀⠀⠀⠀⢠⡇⠈⣿⠓⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⢰⡇⠀⠈⢷⡀⠀⠀⠀⠀⢀⡾⠁⠀⢹⠀⠀⠀⠀⣀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠐⡄⠀⠀⠀⠸⡇⠀⠀⠀⠻⣄⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⡄⠀⠀⠀⠷⠖⠛⡇⠀⠈⢳⠾⠋⠀⢸⠛⠺⠃⠀⠀⠀⠌⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠓⠦⣼⣤⡴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠌⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⠀⠸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣈⠙⠲⢤⣄⣰⡆⠘⣇⠀⠀⠀⠀⣠⠇⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠴⠚⠉⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠛⠶⠶⣤⣽⡇⠀⠈⠙⠲⠖⠋⠁⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣅⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠶⠾⢿⣤⣄⣀⣀⣤⣤⣤⣤⣀⣀⣀⣇⣿⢀⣀⣀⣤⣤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣬⣭⣭⣿⠿⠿⠿⠟⠛⠻⠶⢶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⣻⠄⠀⠀⠀⠀⠀⠀⠀⣬⡻⣧⡀⠀⠀⢀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣄⣀⠀⠀⢀⣒⠶⣝⢮⣻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠶⣯⣿⣾⡷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⢚⣋⠭⠥⠄⠒⠒⠒⠂⠠⠭⣙⡓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⣉⠤⠒⢋⣡⠤⡖⠂⠉⠉⠉⠉⠉⢣⠀⠀⠈⠑⠦⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡡⠖⠋⣠⢔⡪⠍⠊⠁⠀⠀⠀⠀⠀⠠⠀⢸⠯⠍⢕⠢⡀⠈⠳⡄⠙⢦
⠀⠀⠀⠀⠀⠀⢠⡞⡵⠋⠀⣀⣸⠑⠁⢀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⡎⠀⠀⠀⠱⡜⢆⠀⠈⢦⠈⢷
⠀⠀⠀⠀⠀⣰⠏⠌⠀⢠⡫⠄⠉⠳⣄⠈⢢⡀⠀⠀⠀⡀⠈⢠⠃⡸⠀⠀⠀⢀⠀⠘⠈⢇⠀⠀⢧⠘⣇
⠀⠀⠀⠀⢰⡏⡌⠀⠀⣿⠀⠀⠀⠀⠈⠳⡄⠑⢴⡄⠀⢡⡀⢸⢰⠃⠀⠀⠔⠁⠀⠀⠀⢸⠀⠀⠘⡆⣿
⠀⠀⠀⠀⡿⢰⠀⠀⢸⠈⠀⠀⠀⠀⡀⠀⠈⠢⡀⠣⠀⠈⠀⢇⡎⠀⠀⣀⠤⡪⠔⢋⣀⣜⠀⠀⠀⡇⢻
⠀⠀⠀⢸⡇⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠁⠒⠤⠈⠢⢄⠀⠀⡼⠠⠒⠩⣒⠭⠔⠚⠉⠠⡈⡇⠀⠀⡇⣾
⠀⠀⠀⢸⢁⠁⠀⢰⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣽⢺⣥⠴⠚⠉⠀⠀⠀⠀⠀⠀⠁⠹⠀⠀⡇⣿
⠀⠀⠀⢸⡈⠀⠀⠈⢆⡀⠀⠀⣀⣀⡤⠤⠴⠒⠚⠉⣩⡟⠻⡄⠀⠀⡐⠤⡀⠀⠀⠀⠀⡇⡇⠀⢠⢧⡇
⠀⠀⠀⢸⡇⡇⠀⠀⢠⠋⠉⠉⠀⠀⢀⡠⠐⢈⠔⡰⡹⠀⢠⡘⢦⠢⡈⠀⠈⠑⠀⠀⠰⢡⠃⠀⣸⣾⠁
⠀⠀⠀⠀⣷⠸⡀⠀⢸⡀⠀⠀⠠⠂⠁⠀⠀⢂⠌⢰⠃⢰⢀⠱⡀⠳⡈⠢⡀⠀⠀⠀⠀⡎⠀⢠⣿⠇
⠀⠀⠀⠀⠘⣇⠱⡀⠀⢣⡀⠀⠀⠀⠀⠀⡠⠁⢠⠃⢠⢸⠀⢇⢱⡀⠘⢆⠀⠁⠀⠀⡼⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠹⣆⠑⣄⠀⠱⣄⠀⠀⠀⠈⠀⡠⠃⠀⠀⠸⠀⠀⠀⢇⠀⠈⢦⣀⡤⠞⢁⡴⣻⠟
⠀⠀⠀⠀⠀⠀⠙⣦⠈⢢⠀⠈⠳⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⡠⢊⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢦⣀⠀⠀⠱⢌⣂⣒⣒⣀⣤⣤⡤⠠⠤⠖⠋⣀⠔⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣌⡉⠒⠦⢤⣀⣀⣀⣀⣀⣀⡠⠤⠔⠒⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⢤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣠⠤⢤⠄⢰⠒⡩
⠀⠀⢠⣦⠀⠀⠀⠀⠐⠟⠀⢲⠤⠀⠀⢠⠟⠁⣀⡤⠒⠋⡐⠁
⠀⠀⠀⠈⠙⢖⣤⡀⠀⡠⠂⠀⠀⠀⡴⣣⡴⠋⠉⠉⣠⠞
⠀⠀⠀⠀⠀⠈⣇⡉⠙⠳⣦⣤⣤⡴⠿⣿⣓⣈⣉⣁⡀⣤⣖⣚⡉⠛⠳⠶⣄
⠀⠀⠀⠀⠀⠘⣿⠙⢦⡀⠀⠙⢦⠀⠀⠀⠉⠑⢤⣀⢀⣸⡀⠁⠙⠲⢄⡈⠉⠳⣤⣄⡀
⠀⠀⠀⠀⠀⢰⠉⢷⡈⠳⣀⡴⠾⣶⠦⣄⠀⢀⠞⠾⠋⠀⠉⠛⠦⣄⠀⠈⠙⠺⠠⠤⠭⠟⡚
⠀⠀⠀⠀⠀⠀⡄⠀⠹⡄⢹⡇⠀⠀⠁⠫⡹⣞⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁⠀⣇⠀⠀⠀⠀⠀⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⣠⠇⠀⠀⠀⠀⠀⠐⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠀⠀⠀⠀⠀⠀⠀⡿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠳⢄⣀⣀⣀⣠⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠢⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡇⠀⣸⠀⠀⣠⠤⠶⠤⠤⣄⣀⠀⠀⠀⠀⢀⡠⡏⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠃⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣇⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠄⠀⠈⢾⠀⠀⠀⡠⠄⠀⠀⠀⠀⢤⡀⠀⠀⣿⠁⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⠤⠤⠤⣄⠀⠀⠀⢀⠇⠀⠀⠘⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠺⣱⣄⠀⠀⠀⠀⠱⡄⠀⠘⢇⡎⠹⢱⡨⠃⠀⢀⠞⠀⠀⠀⢠⠇⡸
⠀⠀⠀⠀⠀⠀⠀⢾⠵⠫⠵⠀⠀⠀⠀⠈⠢⣄⡀⠳⠤⠞⢀⣀⠔⠋⠀⠀⠀⠀⢸⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠉⠀⠀⠹⡄⠈⠈⠀⠀⠁⣨⠋⠀⠀⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘





⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠇⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⢠⠄⢠⠤⡀⡀⢤⠄⢠⠀⠠⢄⠀⠀⠠⠄⠀⡠⠀⢤⠄⡤⠄⡠⠤⡀⡀⠄⡠
⠀⠀⠀⠀⠀⠀⢘⣁⣸⣀⡄⡇⢸⢀⠃⣃⣉⣱⣸⣀⣈⣀⠀⣈⡆⢸⠀⡇⣃⢣⣀⢇⡛⣄⣋⣀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣶⣄⠀⣠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣩⣷⣦⣿⣿⣿⡇⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣿⡷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣠⣤⣿⣧⣭⣿⣶⣷⣶⣤⣀⠉⠙⠻⣿⣷⠁⠀⢀⠂⣠⣾⣿⣾⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⡶⠄⠛⠉⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤⣀⠈⠄⠀⡬⣀⣿⣛⣫⣭⣥⣀⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⢀⠞⣴⣿⡿⠿⠿⠿⠿⠛⠋⣴⡶⠶⣧⣄⠲⣼⣇⠉⠉⠉⠙⣿⣿⣿⡍⠂
⠀⠀⠀⠀⠀⢰⠟⠛⠛⠁⠀⠀⣛⣉⣁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⣿⣿⠶⣿⠿⠿⠿⠶⢶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣿⣿⣷⠀⠀⠀⢀⣠⡴⠾⠻⣿⣿⠿⠁⠀⠀⢀⠀⠀⠀⠀⢻⣿⣿⠙⠢⠑
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠉⠙⠻⠿⣿⣄⣠⣾⠟⠉⠀⠀⠀⠀⠀⣠⣄⣀⠀⢾⣧⡀⢰⣦⡈⣿⠁⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠚⠙⠈⠂⠀⠀⠀⠀⠀⣰⣷⣿⣿⣷⣾⣿⣷⠸⣿⣯⣝⢃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢿⣻⡾⢡⠂⠀⠀⠀⠀⡀⣀⡀⠀⠀⡏⣿⣏⢉⢘⣿⡿⢹⣇⠿⠷⣽
⠀⠀⠀⠀⢀⣠⣚⣩⡟⠋⢡⢘⣿⣤⣾⣤⣄⣷⡀⢠⣿⠥⠹⣆⠀⢻⣿⣿⣿⣿⡿⠁⡀⠈
⠀⠀⢀⡴⢋⣩⡟⠁⢻⡄⣰⣾⣿⣿⣙⣿⣿⣿⣿⣜⣿⣷⣾⣿⠀⠐⢽⣿⣿⣿⣶⣿⡄⠀⢀
⠀⠀⣾⣟⣿⠁⢹⣄⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⣿⡄⠀⠀⠉⠻⣍⣿⣿⣿⣆⠀⠰⡀
⠀⣸⣻⣿⣷⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣅⡘⣿⣿⣿⣿⣿⣷⣿⠇⠀⠀⠀⠀⠘⢿⡏⣿⣿⠀⠀⠁
⠀⢹⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⠿⢄⠀⠀⠀⠀⠀⠘⢿⢿⡇
⠀⡾⠛⠉⠀⣾⣿⣭⣭⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡟⠛⠋⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢠⣿⣿⣥⣾⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⡷⡦⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⣠⣾⠿⠿⠿⠟⠛⠉⠀⠀⠀⢀⣾⡿⠿⣿⡟⣣⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠘⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⢀⣀⣤⣴⣶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣄⣀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⣠⣴⣾⣟⣛⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣻⣿⣶⣤⣀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠷⣶⣶⣶⣶⣶⣶⣶⡶⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠹⣿⡄⠉⠙⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠉⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣶⣶⣶⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠗⠒⡇⠉⣇⢸⢁⡌⢠⢉⡿⢳⠗⢠⡶⢀⢀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠯⠉⢀⠑⢤⣈⣷⣽⣦⣽⣾⣿⣴⣷⣿⣣⣿⡴⣯⣀⣠⢜⣫⡿⢴⣁⡤⠊⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⡘⢧⡈⡐⣦⣨⣶⣾⢿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣟⣽⣿⡦⠶⢰⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣶⠿⠋⠉⢡⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣯⣿⠻⢿⣿⣿⣿⣿⣓⣲⡽⠷⠖⣋
⠀⠀⠀⠀⠀⣒⣺⡿⠛⠁⠀⠀⠈⠀⠈⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⡟⠈⠀⠁⠈⠟⣿⣿⣿⣿⠶⠶⠮⠥⢀⠰⣂⣠⠴⠂
⠀⠀⠀⢴⡾⠿⢟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡌⠈⢻⣿⣿⣿⣿⣿⡿⠛⢳⣾⠇⠀⠀⠀⣰⡾⠏⣠⣭⣙⣛⠢⣾⠿⠭⠥⠄
⠀⠀⠰⣄⣀⣀⡀⢀⣂⡤⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠛⠛⠛⢋⣥⣴⡿⠃⣀⡠⠴⣚⣡⣔⣛⣛⠻⠯⡍⠁⠈⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠀⠗⡢⢤⡄⡠⣠⢰⢈⡁⠖⡶⠖⢛⣛⣓⣚⣩⡡⠶⢿⡺⣿⢯⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠈⡰⢋⠛⠃⣿⡮⢲⢿⡻⡙⠢⡹⠙⢆⠑⠌⠱⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠘⠀⠑⠄⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⠉⠓⠲⠶⠖⠚⠋⠁⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣠⡄⠀⠱⠦⢄⡀⠀⠀⠀⣀⣠⣴⠂⠰⠶⢀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣠⡔⠈⠀⠁⠀⢠⡀⠀⠀⠀⠙⠁⠀⠀⠀⠳⠈⠀⢢⠑⢦⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣴⣿⣿⣶⣾⣀⣀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣨⣷⣶⣿⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣴⣿⣿⣿⠿⠟⠛⠻⠿⠿⢿⣷⣶⣶⣿⣿⣷⣶⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⠄⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠞⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡴⠻⢄⡀⠀⠈⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣶⣤⣄⣀⣀⣀⠀⠈⠳⣄⡀⠉⠓⠤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠛⠛⢛⣿⠟⠋⠙⠻⣿⠛⠛⠷⣮⣝⠳⠦⣶⣋
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⣾⠃⠀⠀⠠⣶⢸⡇⠀⠀⠀⠙⢷⡄
⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠈⢿⡆
⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⡀
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠈⠻⣶⣄⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣴⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⣷
⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⣿⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⡴⠋⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠘⠳⢶⣄⠀⢀⣤⡦⠀⠀⠀⠀⣿⠃⠀⠀⠀⠘⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠴⠞⠛⢻⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⣴⠋⠉⢹⡇⠀⠀⠀⣠⡴⣾⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣠⠞⠉⢰⡏⡘⣡⠔⠙⡏⠉⣩⡿⣏⢛⠟⠛⠛⠛⠛⠛⠛⠛⠛⣿⠷⣤⣀
⠀⠀⠀⠀⠀⠸⣧⡾⠁⠀⠀⣾⠷⢂⠃⠀⠀⣹⡞⠁⠀⠈⢿⠸⣀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣸⠀⢀⡼⠃⢻⡀⠀⠀⢸⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡟⢠⡞⠄⠠⡘⣇⠀⠀⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠈⠳⣬⣦⣬⠿⠀⠀⢸⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠻⣿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⢤⣼⣇⠀⠀⢀⣠⠤⠶⠶⠶⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣤⣶⣈⣧⠀⢸⣅⣀⣀⣠⡴⣆⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠦⠄⣀⠀⣀⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⠁⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣀⡴⠚⠉⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⡀⠀⠀⡆⢸⣸⣦⣶⣾⣟⣋⠀⠀⠀⠀⠀⠀⢢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣧⣀⣰⣡⣿⡿⠟⠛⠛⠙⠋⠀⢀⢉⡑⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢇⠀⠀⠀⠀⠁⠀⠉⠃⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠢⢤⠤⢤⣴⠾⠃⠀⢸⣿⣉⣉⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⢾⠟⢋⠟⠀⠀⠀⢘⣹⢿⡅⠀⠀⢀⣈⣑⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠉⠉⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⢸⡇⠐⢳⡄⠀⠀⠀⠈⠉⠙⠦⡀
⠀⠀⠀⠀⠀⡴⠁⠐⠀⠀⣤⡦⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠟⠖⠀⠀⠀⠐⠶⠶⢿⣦
⠀⠀⠀⠀⢰⠁⠀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⠀⠀⠀⠀⠀⠤⠶⢾⣷
⠀⠀⠀⠀⣼⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣹⣄⡸⠀⠀⢳⣄⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠿⢅⡀⢀⠀⣀⢀⣀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⢁⡀⡀⠀⠉⠉⠙⠧⣤⣀⠈⠱⡄⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⣇⠀⠀⡴⠻⠿⣆⠀⠘⠳⣄⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠖⠈⠀⠀⡈⠀⠂⠀⠠⠀⢀⠁⣐⡡⠀⠀⠈⢷⣄⣢⡙⣿⣶⣞⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠰⢇⡀⢒⠀⠚⠀⠀⢡⠀⠀⠀⢈⠠⠁⠁⠀⠀⠀⠀⣀⠀⠙⢿⠍⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠃⠀⠁⠀⠀⠉⠘⠆⠰⡀⢀⣀⣾⣥⠆⢀⠀⠒⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣈⢓⣐⠀⠄⢤⡀⠀⢀⢳⣀⣿⡟⠁⣀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⠶⠺⠦⣤⠶⠦⠢⢬⡅⠉⠈⠿⠯⡶⠛⠋⠁⠐⠚⠠⢌⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠎⠉⠃⣻⠡⣄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⠁⡀⣸⠆⠀⢀⠈⣰⠆⡇⠀⠀⠄⡀⢔⠢⠀⠀⠀⠀⣴⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠒⢊⡠⠾⠒⠀⠀⠠⡛⠀⠀⠀⠀⠠⠙⢄⠓⠂⠀⠲⡴⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠹⡀⠀⠀⢀⠄⠐⠋⠁⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡈⠐⠄⣀⣀⣠⠎⠀⢀⡘⡀⠀⠀⠀⠀⠀⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⠀⠀⠀⠚⠛⠁⢀⣀⣠⡶⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠃⠀⠠⠶⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⠄
⠀⠀⣠⠖⠋⠁⠀⠀⠀⠉⠳⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⡀
⠀⢰⠃⠀⠀⠀⠶⣷⣶⠄⠀⠸⡄⠀⠀⠀⠀⢀⡴⠊⠉⠀⠈⠉⠉⠢⡀
⠀⣿⠀⠀⡀⣬⣻⣟⢯⠁⠀⠀⡇⠀⠀⠀⣰⠋⠀⠀⠀⣀⠀⢀⣤⠀⠈⡆
⠀⠸⡄⠀⠹⣟⠳⠉⠃⠀⠀⡸⠁⠀⠀⠀⡗⣀⣤⠦⠒⠳⠛⠁⠀⠳⠒⠳⠀⠀⠀⠠⠤
⠀⠀⠙⠢⢄⣀⣀⣀⡐⠖⠦⣷⠀⠀⠀⠀⢻⠁⠐⡒⠀⡀⠀⠤⠤⠀⠀⠀⠈⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠛⢠⡃⠀⠸⠇⠔⠛⠈⠣⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣈⣀⡀⠀⡀⠀⠀⢐⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡐⠈⠀⠁⠁⡡⠊⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠙⠉⠀⠀⢨⡍⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢖⢸⠀⠀⠀⡋⡰⠂⠘⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠢⠜⠂⠃⠔⠁⠀⢈⡀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⣀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⡾⠿⠿⠛⠛⠛⠛⠛⠛⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⡞⠟⠛⠛⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣋⣡⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠉⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⠟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣤⣄⠄⠉⠙⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠉⠀⠀⣾⣿⣿⡟⠉⢹⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⢰⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢣⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⠁⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣈⣦⣀⣀⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠙⠻⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⠈⠀⠀⠀⠈⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠸⡄⠀⠀⠀⢤⣒⣾⡤⠄⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠱⡀⠀⠀⠀⠀⠈⠐⠚⠆⠀⠀⣠⣔⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⢀⠎⠀⢠⠇⠀⠙⣄⠀⠀⠀⢂⠀⢀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠏⠀⠀⠀⠈⠢⡀⠀⠀⠁⢺⠗⠟⠀⠀⠀⠀⠀⠀⠀⡄⣀⣀⡄⠙⠄
⠀⠀⠀⠀⢀⡆⠀⠸⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠂⠀⠀⢳⡀
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢀⠀⠀⠀⠀⠙⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⠤⣀⣀⣀⠀⡀⠀⠀⣀⣀⡀⢸⠆
⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁⠀⡼
⠀⠀⠀⠀⠀⠘⡄⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⡄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠠⢀⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠘⢄⡬⣄⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⣀⣀⠀⢠⠀⠀⣀⠀⣀⢍⣠⡾⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠙⠛⠓⠒⠓⠓⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠓⠎⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⣹⠦⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠉⠉⢉⣱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠶⢾⣦⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠓⣶⠶⠒⠉⠉⠉⠻⠶⠶⠒⠊⠙⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⣄
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠸⡇⠸⠃⠀⠀⠀⠀⠀⠀⠈⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⡀⠀⣿⡆
⠀⠀⠀⠀⠀⠈⣿⣧⠀⢰⡟⣠⠇⢠⣤⠤⠀⠀⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⡿⡷⠀⣹⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠂⠄⠀⠼⢯⣤⣄⠐⠀⠀⣦⡀⠃⠀⠁⠀⠈⠀⠀⠀⠸⠋⣀⢀⣠⣶⠞⠋⠁⣧⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢿⡁⠀⣀⢤⣐⠉⣡⡼⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⡟⠙⠋⢡⡄⠀⣴⣷⠈⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠉⠁⢀⣀⠉⠁⠉⠒⠒⠙⠦⡀⠀⠀⠀⢰⣤⠞⢉⣥⡁⠰⠇⣀⣤⠶⢺⠿⣿⠀⣿⣯⣿⠘
⠀⠀⠀⠀⠀⠀⣿⣿⠃⣤⣤⡴⢦⣉⠀⠾⠇⢸⣿⠇⡀⢹⡇⠀⠀⠘⣿⠀⣈⣟⣁⣤⣼⣿⣿⡴⣯⡼⠃⢠⣿⢰⡟⢰
⠀⠀⠀⠀⠀⠀⡇⣿⡄⢿⣿⣷⣿⣿⣷⣤⣤⣴⣾⠃⠀⣾⡇⠀⠀⠀⣿⢻⣿⡈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⢠⢸⣧⠀⠹⣿⡷⣿⣿⣿⣿⣿⣧⣾⣿⡿⠀⠀⠈⠀⣿⣾⣿⣿⡶⢮⣿⣶⡿⢯⣿⠀⠀⠀⠀⣿⣡⡌⣦
⠀⠀⠀⠀⠀⠀⠈⢸⣿⡀⠀⢻⣿⣾⣝⣋⣙⣽⡟⣱⣿⠃⠚⠀⠀⠀⢻⡘⢿⣿⡄⠲⣬⣭⣵⠾⠃⠀⠀⠀⠀⠈⣿⡟⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠈⠙⠛⠋⠉⠁⢰⣿⡏⡨⠇⠀⠀⠀⠈⠳⠾⢿⣷⡄⠀⠀⠀⣠⣾⣷⡆⠀⠀⠀⢸⡧⢸⡇
⠀⠀⠀⠀⠀⠀⠸⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⢀⣀⠀⢈⡿⠁⠀⠀⣸⣿⣿⣿⡇⡆⠀⣠⣟⡇⢨⠃
⠀⠀⠀⠀⠀⠀⠂⢸⣿⠀⠀⣰⣿⣷⣦⡀⠀⠀⠀⠣⠄⣤⣾⣷⠀⠀⣾⠿⠛⠋⠀⠀⠀⢘⣿⣿⣿⣿⠯⠁⣠⣿⣽⠃⡟
⠀⠀⠀⠀⠀⠀⢸⡼⣿⠀⠀⠹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢰⣷⡀⠀⣽⣷⡀⠀⢀⣠⣤⣾⠟⣹⡿⠟⠁⢰⡏⣿⠏⡜
⠀⠀⠀⠀⠀⠀⠀⢴⢿⣷⡄⠀⠸⢿⣿⣹⢯⣷⣦⡤⣄⣀⣘⣿⣷⣛⣻⣿⣧⣾⣿⣿⡏⠁⢰⠋⠀⠀⢠⣿⢱⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⣿⠤⢀⠀⠉⠻⣆⢹⣿⡛⢿⠿⣭⣋⠛⠛⡛⣿⣿⡿⠏⣀⣷⣿⠃⠀⠀⢀⣼⢿⠘⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⡇⠘⣷⡄⠀⠙⠋⠙⣿⣷⣤⣉⠋⠛⢿⠟⠛⢩⣴⠿⣿⠿⠁⠀⠀⠀⢠⡏⠸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⣿⣿⣄⠀⠀⠀⠘⢷⣌⠙⠷⣦⣴⣤⣶⣿⣥⡿⠋⠀⠀⠀⣠⣶⡿⢧⠀⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡁⠙⠉⣉⠻⢶⣄⡀⠀⠛⢷⣴⣫⣤⣇⣰⣾⡿⠁⣀⣤⣶⣿⡿⢋⡔⠁⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣅⠀⠂⢉⠀⠈⢻⣿⣿⣶⡄⠈⠙⠛⠛⠋⠁⠀⢰⣿⣿⡿⠏⢠⣾⡁⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢼⡄⠀⠀⠀⠘⠻⣿⠇⠀⠀⠀⠀⠀⠀⣀⠈⢿⡏⠀⠀⠈⠻⣿⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣼⣿⣦⡀⠀⠀⢠⡎⠀⠻⠇⢀⠀⠀⠸⠿⠀⣸⠃⠀⠀⠀⠀⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄⣼⢿⣿⣿⣿⣦⣄⣀⠈⠦⣤⣤⡰⠶⠾⠶⠶⠖⣿⠀⠀⠀⠀⠀⣠⣤⠞⠡⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣧⡟⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⣀⠀⠻⣤⣀⠀⣴⣿⣟⡁⠈⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⢻⣴⣴⣶⣅⠀⠀⣠⠄⠀⠀⠉⠉⠛⠿⢿⣿⣿⣶⣤⡙⢠⠻⣿⠉⠁⠒⠉⠉⠀⠀⠈⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠉⢿⣿⣇⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠙⣿⡄⠀⠹⣷⡀⠃⠀⠀⠀⠀⠀⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⣿⠿⠋⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢺⣇⠀⠀⠘⣷⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣀⢀⣀⣠⣤⣴⣶⣾⡿⠿⢿⣿⣿⣶⣶⣤⣸⣿⣆⠀⣠⡌⠀⣐⠀⠤⠀⠀⢘
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠀⠀⠀⣀⡀⠀⠈⢛⣟⣿⡿⠛⠛⠷⠟⠀⠀⠚⡉⠀⣠⡿⡿⢻⣿⣿⠟⠋⠀⠀⠉⠙⣀⠠⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠂⠀⠀⠈⠉⠀⠀⡀⠃⠈⡟⠁⠲⠄⠀⠀⠀⠀⠤⠀⠀⠙⠧⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡆⢀⢄⣸⡿⠖⢁⡀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣘⣵⣾⡿⠛⠙⢛⡛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢻⣿⠀⣰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠄

⠀⠀⠀⢀⣠⠴⠒⠉⠉⠉⠉⠉⠑⠒⠦⣄⡀
⠀⢀⡔⡟⠁⢀⡤⠔⢆⠉⠀⠉⠐⠄⠤⢀⠙⠳⣄
⢀⡾⢸⠀⢀⣏⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠙⢦⠈⠳⣄
⣼⠁⡆⢀⠇⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠘⣧
⡟⠀⡇⠘⡄⠀⠀⠀⠀⠈⠒⢄⠀⠀⡠⠊⠀⠀⠀⠀⢠⠘⣇
⣇⠀⢻⠀⢈⠉⠉⠉⠉⠁⠀⡠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢻⠀⠈⢧⠘⣄⠀⠀⠀⡠⠊⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⠀⠈⡇⠀⠀⠀⠀⠀⠤⠤⣤⣀⣀
⠘⣆⠀⠈⢧⡈⠂⠤⠋⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢀⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣦
⠀⠘⢄⠀⠀⠑⢄⡀⠳⢄⠀⠀⢸⠀⠀⠀⠀⠑⢄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠈⠢⡀⠀⠀⠉⠢⢄⡉⠀⠚⠀⠀⠠⠤⢰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠒⠦⠤⠤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠴⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣷⣶⣶⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⣀⡤⠴⠦⣄⣀⣀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠁
⠀⠀⠀⡞⠁⠀⢠⡀⢢⡘⣦⣲⣄⠀⠙⡄⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡄
⠀⠀⢸⠁⢀⡀⢦⡹⣆⢻⣮⣏⠷⠀⠀⢱⠀⠀⠀⠀⠀⣼⠐⠀⠀⡠⣂⡴⣟⣴⠵⠋⠘⣧⣄⣸⡇
⠀⠀⣿⠀⢦⡹⣦⡻⣿⣗⢷⡙⢦⠀⠀⣌⠀⠀⠀⠀⠀⢻⠤⠶⠛⠋⠁⠀⠉⠀⠀⠀⠀⢈⠁⡆⠈⠳⠀⠀⢀⣤⣀⡀
⠀⠀⢸⡀⠈⣿⠳⣝⢷⡝⢦⠁⠀⢀⠀⠈⠳⡄⠀⠀⠀⠸⡇⠀⢛⣛⠃⢀⡆⠀⠐⠛⠉⠸⠀⠁⡀⢸⠀⠀⠀⠀⠉
⠀⠀⠀⠹⢄⡈⠁⠈⠀⠀⠀⠀⢀⡤⠉⠓⠦⠙⣆⠀⠀⠀⠃⠀⠈⠛⠃⣸⠁⠀⠀⠿⠃⡴⠚⠧⠈⠻⡀⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠉⠒⠶⠤⠴⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⡃⠀⠀⠀⠀⡇⠀⠰⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⠴⢚⡛⠲⢄⡀⢁⣀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡐⠀⠀⠀⠀⠀⢈⡴⠋⠈⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠒⠒⡏⠁⠀⠀⠀⣃⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡇⠀⠀⠀⢳⠂⢤⡀⠀⠉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠰⡈⠢⣀⣀⣀⠔⢀⠎⠀⠉⠛⠈⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⠖⠛⠤⡜⠒⠁⠀⠀⠀⠘⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⡼⠀⠀⣆⡀⠀⠀⠀⠗⠦⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⢀⣀⣀⣀⣄⠠⠤⠤⠐⠂⠉

⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣿⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠛⠳⢦⣄⣀⣤⠶⠛⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀
⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀
⢀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣤⣀
⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣦⣄⣠⣴⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⢿
⢸⡇⠈⠉⠛⠻⣿⡟⠛⠛⠛⠛⣿⠋⠉⠉⠀⠀⠈⣿⠀⠈⠉⠙⠛⠛⣶⠶⠶⠶⠶⠶⢶⣶⠚⠛⢛⣿⠋⣩⡿⠋⠀⠀⣾
⠀⢿⡀⠀⠀⠀⠙⢿⣷⣶⠶⠶⠻⣦⣤⣤⣤⣤⣼⣿⣦⣀⣀⣀⣀⣠⡿⠷⠶⢶⣶⡶⠟⠉⠛⠛⠋⢁⡾⠋⠀⠀⠀⢠⡟
⠀⠘⣷⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠻⢦⣄⠀⣀⡴⠋⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠘⣧⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣠⠴⠛⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⡴⠶⠛⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠿⠿⠿⠟⠻⠟⠛⠛⠟⠿⠿⢿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣦⣄
⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄
⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⡆⠠⢣⢸⠇⣿⠃⡰⠃⠂⡾⢃⡼⢣⢷⠇⡤⣠⡔⡀⠀⣠
⠘⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣦⣷⠇⢅⣰⣠⣬⠼⣶⣿⣶⣷⣿⣿⣿⣿⣿⣷⣶⣶⣤⣵⣿⣯⣿⣷⡧⣤⣀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⢀⡈⣷⡿⣿⡬⠚⠋⠉⠀⠀⣴⣿⣿⢿⣿⣿⣿⣿⠉⠉⠉⢹⣿⣷⡈⠙⠻⠿⣿⣾⣿⣿⢾⢦⣀⡄
⠀⠀⠀⠀⠀⡀⠐⠶⢝⣥⠞⠉⠀⠀⠀⠀⠀⢸⣿⠏⣴⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠉⠛⠶⣥⡞⠛⠁
⠀⠀⠀⠀⠢⢬⣽⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢉⡽⠟⢷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡅⣿⡿⠿⣿⣿⡿⣿⢿⣫⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠘⣿⣿⣷⣴⣷⣤⣷⣤⣴⣿⣿⣿⣿⠟⠀⠀⢀⣀⣠⣴⣶⠿⢿⡿⣼⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣄⣀⠈⠛⠿⢿⣿⣿⣿⠿⠿⠟⠛⢉⣠⣤⣶⡾⡟⠉⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠿⠶⢶⢶⠶⢶⠶⠶⠶⠾⠛⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⡳⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⢶⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⣤⡆⠀⠈⢓⡄⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣂⣋⣁⣰⠷⠐⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⡈⢡⡆⠀⠀⠀⠀⠀⠙⣶⣤⣶⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⠽⢷⣄⣧⠀⢀⠀⢠⡴⠟⢻⣅⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠐⢤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⢹⠿⠶⠎⠀⠀⠀⡯⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⣠⠶⢶⣤⣤⢶⣄⣤⡤⠤⠤⠖⠛⠋⢠⠇⢠⠃⠀⠀⠀⠀⢷⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣤⣤⣤⣬⣭⡭⠛⣅⠸⡅⠀⠀⡇⠀⠀⡿⠀⢸⡄⠀⠀⠀⠀⢚⡦⡈⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠄⠼⠿⣿⠟⠩⠂⠀⠸⠋⠁⠀⢸⡇⠀⢸⠃⠀⡌⢷⡀⢀⡀⣀⢻⣵⠃⠀⠀⠀⣠⣤⠉⢻⣧⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⣷⣾⠋⣀⠰⢀⡀⠀⠀⠐⡀⠀⠀⣄⡁⠀⣽⠀⠀⡅⠸⡗⢻⣧⠹⢣⣽⣷⣄⣀⣄⠻⠿⣷⣾⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣿⣷⣿⣿⣴⣾⣤⣾⡿⣦⣿⢰⣶⡿⡱⣀⢿⠀⠀⢱⡄⠀⢻⣿⣧⣼⣿⣿⣿⣟⠘⢳⣾⣟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠹⣿⣿⠟⠿⠿⠋⣿⡟⠛⢟⢸⣿⠋⢃⣉⣿⣿⣆⠈⢯⣁⠘⢿⣿⣿⣿⣻⣾⣿⣷⣄⣿⣇⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠓⣤⡈⢱⢧⡟⢲⣶⣾⡇⣾⣀⣹⣿⡿⢻⡇⠀⠙⢷⣌⠻⣏⠙⠿⣿⣏⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣧⣾⣎⢷⣮⣿⣿⣿⡜⢿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠳⣝⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡿⠇⠉⠉⠙⢉⣀⡄⢹⡟⣺⢿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠰⠊⠉⠁⠈⣨⣿⣾⣅⠈⠈⠙⣿⣶⣭⣄⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠊⠀⠀⠀⠀⣚⣻⣿⠏⡎⡈⠢⣀⡀⠈⠈⢿⣿⣷⣶⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣠⣴⠀⠀⢒⣾⡿⣷⠻⠄⠀⣌⢷⣬⣿⣶⡀⠀⢉⣈⣛⢛⠋⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠃⠈⠀⠀⠰⣿⠟⠚⠃⡇⣤⢼⣿⣷⣻⣿⡿⡷⠆⣷⣆⣼⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⡾⠯⡄⠀⣿⣦⣿⠟⠉⢉⣉⣿⡿⣿⣿⣿⣿⣿⡇⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠁⠀⠀⢃⢨⣽⣿⣏⢰⡏⣠⣴⣿⣷⣼⣿⣿⣿⣿⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠓⠛⢾⣷⣽⣿⣿⣿⡛⢟⡿⣿⡃⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠓⢄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⢳⣇⠀⢀⡄⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢻⠀⡜⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣷⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⡇⠀⢰⠏⠀⠀⠀⠀⠀⣠⠤⠀⠢⣤⠤⢄⡀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠉⠉⢔⡢⢄⠀⢀⣼⠁⠀⠀⠀⠁⠈⠁⠚⠍⡲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢗⣼⣿⠈⠀⠀⠀⠀⢠⠀⠀⠀⠘⠢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢆⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⡴
⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠣⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠸⡀⠀⠀⢘⠨⡀⠀⠀⠀⠀⢀⠰⢢⠲⡶⠶⠘⠂⠠⠤⢈⢶⠄⣀⠀⠠⠀⠀⠴⣾⡝⢁⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⡄⢣⠀⠀⠀⠂⠁⠀⠆⠀⠁⠀⠈⠀⠀⠀⡜⣸⠀⠀⣉⣐⣶⠶⢏⠁⠀⠀⠀⠀⠀⠬⢔⣦
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⠱⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠛⢿⠛⠋⠛⠀⠀⠀⠈⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⡦⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⢀⣶⠀⠀⠀⠀⡀⠀⠀⠀⣄⡀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣶⣿⡏⠀⠀⢀⣴⣷⠀⠀⣰⣿⣷⡄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣀⣴⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠟⠛⠻⠿⠷⣶⣤⡀⠀⠀⠀⣿⣿⣿⡿⠛⠉⣁⣀⣤⣤⣬⣉⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣤⣀⠉⠋⠀⠀⠀⣿⣿⣿⣶⣾⠿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡟⣉⠙⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⣴⢶⡄⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣬⡴⠛⠋⠁⠀⠀⠁⠀⣿⣿⣿⣿⣷⣦⣤⣙⢚⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠢⣤⣠⠟⠛⣻⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣴⠛⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠒⠫⠭⠶⠶⡶⣾⣉⣙⣛⣛⣓⣦⣽⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠉⠉⣶⣾⣿⣿⣿⣿⣿⣿⠿⢛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠠⣿⣿⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠄⠀⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠶⠶⠶⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⣧⠴⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣋⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣳⠆
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣟⡉⠉
⠀⠀⠀⠀⠀⣀⡤⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⠀⠐⢦⣤⢤⡤⠤⠤⠤⠴⠚⠁
⠀⠀⠀⠀⣼⢁⣠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠖⠲⠶⠶⠖⠚⠋⠉⠀⠰⡀
⠀⠀⠀⠀⠙⠉⡠⠂⠀⠀⠀⠀⠀⠐⢲⡄⠠⠄⠀⠀⠀⠒⠒⠉⠁⠀⠀⣠⠖⠋⠉⠙⣄⠘⢦
⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⣠⠖⠚⠛⠓⢦⣰⠋⠀⠀⠀⠀⢹⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠸⣄⣼⠇⠀⣀⣠⣤⡄⢸⡏⠀⠀⠀⠀⡏⠀⢀⣀⣤⠤⢿⠒⠂⠒⠒⠢⡾⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠙⠀⠓⠋⠁⠀⢸⡇⢸⡇⠀⠀⠀⠀⣷⠞⠋⠁⢀⣤⣸⠀⣤⠀⠀⣼⠃⠀⠀⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠁⠀⠀⠀⠀⠘⠷⣄⣀⣈⣻⠟⠒⠧⠶⠚⠁⠔⠋⠉⠉⠉⣳⠈⣷
⠀⠀⠀⠀⠀⠀⢠⠔⠚⣉⡉⠻⣯⠞⠁⠀⠀⠀⠀⠀⠀⢤⣤⡭⠟⠁⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⠴⠋⠀⠘⣇
⠀⠀⠀⠀⠀⢠⠃⠀⠈⠭⠟⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⠠⠴⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠠⡶⡄⠠⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⡌⠀⠃⠘⡅⠈⠙⠲⠇⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢁⢠⢠⡃⢀⠨⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠲⠤⢄⣀⠘⠤⠤⠔⠁⠀⠀⠀⠀⣀⣀⠤⠴⠚⠉⠀⠀⠀⠀⢀⣀⣀⠨⡈⠋⠐⠳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⢀⠀⠀⠉⠉⠛⠋⢻⡏⠉⠋⢄⡀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⣳⡄⠀⠀⠀⣰⠁⠀⠀⠀⠲⠛⣲⠾⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⣼⣷⡇⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠛⢺⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⣯⢳⣼⣿⣏⣉⠉⠉⣁⡹⣤⠤⢄⣀⡀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣀⣀⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠀⠀⠈⠁⠈⠈⠳⣁⣠⠞⢁⡟⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠧⠠⠍⠉⠀⢱⠀⠀⠀⠀⠀⠀⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡤⠀⠀⢀⡠⠯⠤⠤⠴⡶⢲⡋⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠉⠁⠀⠀⠀⠀⢠⡏⣸⣀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠒⣲⢾⡟⠛⠛⣀⡼⣻⠗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣻⣝⢶⣤⣄⣀⣀⣀⣠⣤⣾⠶⠚⠛⠙⠛⣉⣇⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣏⠈⢉⣀⣠⠤⣛⠽⠋⠉⠚⠭⠭⠉⢉⠭⠭⠥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠭⠭⠿⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⣘⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣼⣿⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠀⠐⠂
⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⢾⡟⠸⡿⠇⠚⠛⠒⠊⠤⣤⠀⠀⠀⠀⠀⠐⠶⠦⠄⠀⠀⢀⠘⡟⢀⣈⣀
⠀⠀⠀⠀⡀⠛⠷⣦⣤⣴⠾⡟⠛⠧⠶⠖⠁⠀⠀⠀⠈⠳⡦⠀⡀⠀⠀⠀⠀⠀⠀⠐⢤⣰⠿⠏⠉⠛⠳⠶⠶
⠀⠀⠀⠀⢙⣗⣆⣽⡄⠀⠻⠓⣤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡆⡀⠀⡀⢹⡷⠀⡄⠀⠀⡰⢎
⠀⠀⠠⠞⠛⠛⠉⣿⣥⡔⠀⡼⣿⣿⣄⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠁⠈⣿⠁⠚⠁⡿⠀⣠⡅⠀⠶⠛⢁⡴⢂
⠀⠀⠀⠀⠀⠀⠖⠛⠓⡞⠀⠀⠙⣿⣿⣆⢸⣧⠀⠀⠀⣠⠤⠄⣀⠀⠈⣃⠀⣍⠀⠀⠸⣥⣾⣿⣖⣂⡠⠴⠃⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⣻⣟⣿⡘⠇⠀⠀⠀⠈⠛⠋⠀⡀⠀⠈⠳⣿⡆⢀⣾⠙⠋⠙⣯⡀⢀⠲⢴⢞⣉
⠀⠀⠀⣀⣤⣤⣀⣠⣶⢿⣷⣶⣿⡛⣿⣾⠁⢀⠀⠀⠀⠀⠀⠰⢄⣡⣆⠀⠀⢾⣯⣸⠇⠀⠄⠀⠀⢉⠠⣧⢤⠈
⠀⠀⢰⣿⣿⣋⣹⣿⣿⣦⣿⣿⢿⣷⣿⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⠿⠀⠀⠀⢹⠇⠀⣠⡀⠀⠀⢼⠀⠉
⠀⠀⠈⠻⣿⣿⡟⠉⢻⣿⣿⠛⢻⣦⠰⣶⣶⣤⠀⠀⠀⢳⣿⡞⠀⡈⠀⠀⠀⢀⡟⠀⢀⣿⠇⠀⠀⠀⠆⠀⢦⣀⠀⡀
⠀⠀⠀⠀⢿⣿⣧⣤⣿⣿⣿⣦⣼⣿⠀⠉⢐⣿⡟⠈⠳⠛⣛⠿⠀⠁⠀⡀⢀⣾⣄⠀⣾⠕⠀⠀⠀⠀⠀⠀⠀⡙⢈⣷⡄
⠀⠀⠀⠀⠀⠉⠛⠋⣡⠈⠛⠛⠛⠁⠀⠀⠙⠟⠀⠀⠀⠀⠀⣷⣦⣰⣟⢋⣾⠋⠛⠿⠿⡶⠀⠤⠀⢀⡀⠤⠚⠿⠖⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠸⠉⢿⣿⠃⠀⠀⠀⠀⠈⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣀⡀⠀⠀⠀⠀⠀⠐⣶⣾⡀⠀⠀⠀⠀⣼⡃⠀⠠⠒⠀⠀⠀⢱⣀⠣⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⣿⣿⡆⠀⠀⠀⠀⠀⠛⢻⠃⠀⠀⢠⣼⡟⠁⠀⠀⠀⠀⣴⣧⣾⣿⡤⠹⡿⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡶⢿⣽⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠎⠉⣷⠀⢀⣆⠀⠀⠈⠿⡟⠀⠋⠘⠁⠜⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣩⣿⡷⣷⣶⣏⠹⣿⣿⣧⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢹⣿⡿⠀⢋⠙⠻⢦⣴⣿⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠐⢻⡇⠑⠆⢀⠀⠀⠉⠉⠉⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡼⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⢀⣴⠏⠀⠀⣂⢂⠉⠀⠂⠠⡴⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠁⠀⠀⠀⠘⢻⣿⣷⡀⠀⠀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢿⡏⠣⠈⠀⠀⠀⠀⠀⣶⣶⡒⣳⣴⡶⠀⠀⠀⠀⠀⠤⠄⡀⣀⣰⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣶⣶⣿⣷⣤⠾⠀⠂⠀⠀⠀⠈⠛⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠾⠛⠛⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⠇⠀⠀⠀⠀⠄⠀⢸⡏⡄⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣠⣤⠶⠶⠶⠶⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⢀⡞⠁⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣾⠁⠀⠀⡀⢀⣄⠰⣆⢸⣿⣿⣤⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠉⠉⡏⠓⠦⠤⢶⠒⠛⠓⠦⡀
⣿⠀⠀⡌⢿⣄⢿⢦⢻⢳⣯⢻⡌⠃⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⠤⠤⠞⢹⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠀⠀⢹
⢻⡄⠀⠈⢾⡌⢿⡎⢻⣇⠹⡆⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⢎
⠀⠻⣄⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⠁⣀⠤⠤⠤⠤⠞⠉⠉⠙⠲⠤⠴⠒⣦⠤⠀⣸
⠀⠀⠈⠳⣤⣀⠀⠀⢤⡀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⡿⡀⠀⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⣰⠃⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠉⠉⠛⠊⠻⣦⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⢰⠇⠀⠉⠉⠛⠀⢀⠀⠀⠘⠉⠉⠉⢸⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⣄⣀⣰⣿⡿⣤⡾⠤⣤⠴⣶⣶⣀⣨⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡇⠘⢧⣀⠤⢾⣇⣀⡘⢦⣍⡡⠇⠀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡖⡋⢉⣡⣄⣈⣉⢙⢲⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⣤⣄⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡆⠸⢚⢋⠭⢿⣿⠭⣍⢙⠓⠗⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠐⠒⠂⠼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠀⠀⠈⠁⠀⠀⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⢸⢸⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⡖⠒⠒⢺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣤⣄⣀⡀⠀⠈⡄⡇⠀⠀⠀⣠⠤⠶⠒⠳⢦⠲⢶⡶⣞⠁⠀⠀⣨⠗⢲⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠒⠚⠛⢦⠀⠇⢷⢀⡴⠛⢁⣠⣤⠀⠀⠈⠻⣧⢣⣈⣳⣴⣊⠁⣠⠞⣼⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢉⣉⣉⣽⠂⢸⢸⠘⠓⠛⢉⡾⢧⡀⠀⠀⠀⠈⢻⣧⣼⡙⠈⠙⠁⣰⠃⠀⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡟⠛⠋⠁⠀⢸⢸⡇⠀⣠⠏⠀⠈⠷⢤⠾⠷⣤⢾⣹⠇⠉⠲⢤⡞⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣧⣤⣄⣀⣀⣈⣄⡇⣴⠋⠀⠀⠀⠀⣸⡆⠀⠘⢶⠋⠀⠀⠀⠀⠙⢦⣀⠐⢼⡄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠚⠋⠉⢻⡀⠀⡼⠁⠀⠀⠀⢀⡾⢹⠇⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠉⠳⢾⣇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢳⡾⠁⠀⠀⠀⣠⠟⠀⢸⠀⠀⠀⠀⠀⠀⢨⠳⣄⡀⠀⠀⠀⠀⠉⠳⠄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⠃⠀⠀⢀⡴⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠳⣦⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠋⠀⠀⠀⠀⠀⠀⣿⠤⠭⣭⠶⠀⠀⢸⡄⠀⠀⢀⡴⣖⣒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⠀⠀⠀⠸⡇⠀⠀⠀⠉⣀⣠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⡉⠛⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⣿⠞⣧⣬⣿⣿⣿⣿⣿⡟⣥⣾⣿⣿⣿⡏⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣌⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣟⠝⢿⣿⣿⣿⣿
⠀⠀⠀⠉⠹⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⣿⣾⣿⣿⣿⣇⠈⢿⣿⣿⣿
⠀⠀⠀⠀⣠⠞⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣈⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⢀⠀⡀⢤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢻⣿
⠀⠀⠀⢸⠋⡰⠋⠰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣈⡈⢁⠀⣀⢀⠀⣀⣀⣪⣿⣿⣷⣾⣿⣿⣿⣿⣿⣯⣟⣯⣿⣿⣫⣍⡛⢽⣿⡤⠄⢸⣿
⠀⠀⠀⠀⠀⠅⠀⠰⢣⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⣿⣿⠿⠿⠩⠷⠛⠋⠉⠁⠀⠐⠃⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⣀⢨⠁⠈⠀⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠿⣿⡎⠛⢹⡜⢯⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣧⢘⠃⢿⠛⣽⠯⣿⡽⡟⠿⣿⢿⣿⣿⡋⣿⠙⣿⡏⠈⠳⠰⠇⠀⠻⡀⠀⠃⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠋⠀⢹⡆⠇⠀⢻⡌⣿⢹⡆⢩⠀⢻⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠉⠐⠒⢤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠤⢄
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠒⠆⠀⢶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⠰⠂⠀⠀⠀⠀⢰⠒⠂⢁⣠⡴⠛⠉⠙⠓⠀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⢀⠂⠀⠀⢀⠔⠉⠑⡄⠀⠀⠀⠀⠀⢀⡔⠁⠉⠐⢈⠁⠀⠑⠤⠄⠤⠒⠁⠙⠛⠛⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⡆⠀⠀⠀⠰⡀⠀⠀⠁⠀⠀⠀⣀⣴⡧⠶⢶⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠒⠠⠀⠀⠣⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠂⠄⠀⠀⠈⠁⠒⠒⠒⡎⠉⠡⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠒⢲⣶⣤⣄⣀⣰⠁⠀⠀⠀⠀⠀⠀⠠⠔⠁
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣵⢤⠶⠚⠋⣹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣿⣿⣿⠞⡉⢩⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⢸⠀⠀⠀⠘⠻⠟⢻⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠇⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠢⢀⡀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠓⠒⠂⠀⠒⠒⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠐⠒⠚⠀⠀⡦⡀⠀⠀⠀⠀⠀⢀⡀⠴
⠀⠀⠀⠎⠀⠀⠈⠁⠐⢏⠀⠉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣈⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢉⠀⡹⠂⠀⠀⠀⠀⠱
⠀⠀⠈⠂⠀⢒⠆⠀⠀⠀⠙⠈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⣈⠠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠁⠛⠀⠀⠀⠰⡒⠀⠐⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣀⠄⠘⡄⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣁⣀⠀⢀⣰⡆⠀⠀⠀⠀⠀⠀⢠⠃⠨⣀⠄⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠢⠠⠔⡏⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠠⠿⠔⠊⠉⣀⠀⠀⠀⠈⢁⠄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢹⠢⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠉⢂⠀⠀⠀⠀⠀⠈⠑⠒⢤⠀⠀⠄⠠⠀⠀⡤⠐⠊⠁⠀⠀⠀⠀⠀⡐⠉⠒⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⡀⠤⠐⠂⠐⠺⠀⠀⠀⠀⠀⠀⠗⠒⠐⠂⠤⢀⣀⡠⠜⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢁⣴⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡴⢋⣴⣞⣃⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣀⣴⣿⣿⣿⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⠀⠉⠉⠉⣉⣀⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⣧⣸⠏⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⣡⣾⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⣶⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣴⣿⠋⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣴⣿⣱⡿⠟⠘⢴⣀⣀⠤⠐⠀⠂⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⢀⠜⠉⠤⠀⠈⠀⠈⠀⠁⠠⠈⠂⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⠏⠰⠁⠀⠀⠀⠃⢀⠐⠀⠀⠀⠀⠐⡀⠀⠠⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠡⠀⠀⠀⠀⠄⠠⢀⠀⠈⠀⠀⠂⢸⠶⢶⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠀⠤⠀⠐⠣⠠⢄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⣀⣈⣡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠒⠂⠈⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⠯⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣦⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠖⠛⠉⠉⠉⠀⠀⠉⠉⠉⠙⠻⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢁⣠⠴⣒⣪⣭⣭⣤⣭⣭⣭⣙⡳⠶⣦⣌⣿⣿⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⣴⣿⣿⣤⠶⣫⠴⠋⠿⠿⢿⣿⣿⣿⡿⠿⡟⠟⢷⣮⡛⢿⣿⣿⣿⠭⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣟⣥⠾⠁⠀⠀⠀⠀⢸⣿⣿⣟⠀⢀⠇⠀⠁⠉⠻⣦⡙⠟⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠽⣟⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠊⠀⠀⠀⡁⠴⠯⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡄
⠀⣰⠟⠁⠀⠀⠀⠀⠀⡀⢠⣄⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠐⠋
⢰⡏⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⠾⣷⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠀⠀⠈⠙⠛⠛⠳⢦⡀
⣿⠀⠀⠀⠀⠀⢀⠀⠻⣶⣌⡻⣝⠷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⡯⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⠀⠀⠀⢠⣄⡙⢿⡶⣌⣳⣝⠻⢷⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠹⣆
⢻⡆⠀⠀⢄⡙⢿⠲⣽⣦⡙⠻⢷⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⡟⣠⠔⠀⠀⠀⡠⠂⣠⣾⠀⣠⣊⡴⠋⠹⡆⠂⠀⢸⣿
⠈⢿⣄⠀⠀⠉⠛⠷⠀⠉⠋⠀⠀⠀⠀⠀⠠⡀⣼⠃⠀⠀⠀⠀⠀⢸⡟⠁⠀⣀⣴⣫⠴⠞⠁⠾⠟⠛⠁⠀⠀⠀⠹⣦⢀⣤⣿⣀
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠘⣷⠶⠛⠋⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢻⡿⠀⡇⠈⠑⢦⠀⠀⠀⢠⠾⠻⠞
⠀⠀⠀⠀⠈⠙⠳⠶⠦⣤⠤⠶⠖⠛⠁⠉⠛⠾⣿⠄⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⢉⣉⡙⠃⠀⡤⠀⠀⠀⠋⠉⠀⠀⢸⡇⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⣿⡿⠃⢸⠇⠀⠀⠀⣾⣿⡇⠀⢸⡇⢸⣀⣤⡀⢸⠀⠀⠀⢤⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠐⠆⠀⠀⣾⠀⠀⠀⠀⠨⠍⠀⠀⣸⠗⠛⠷⠄⠙⢯⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣠⡴⠖⢒⡛⠛⠿⠿⢷⣶⡶⢤⣄⡀⣽⠀⠀⠀⠀⠀⠀⣹⠀⠰⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠛⠡⠃⣴⠁⣠⡴⠶⠶⠦⣄⢙⡆⠙⠙⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠻⣶⠁⠐⠛⠋⠀⢀⡽⠇⠀⠀⡻⢤⣴⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⠀⠈⠙⠛⠒⠚⢛⠉⢀⠀⠰⣀⣹⠏⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠤⣼⣀⣀⣇⣀⣸⣠⣼⣶⠞⠋⠁⠀⢠⠷⡶⡖⢚⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢨⠉⡏⠈⡄⠀⢸⠀⠀⠀⠀⠘⡶⠧⠵⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣇⡎⢰⠀⢸⠀⣧⢘⡴⠳⠤⣀⡀⠀⡇⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠈⠑⠻⠦⠾⠴⠞⠋⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠇⢹⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⣷⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠦⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠤⣔⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣉⣀⡀⣄⣀⣉⣛⡐⠂⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣫⣍⣉⣬⠋⠚⠻⠗⠙⠛⠻⠟⢖⣾⡦⢀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⡱⠶⠷⠶⢶⣛⣛⣉⣁⡀⣠⡾⠓⢹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠈⠀⠐⠀⠈⠓⠒⠤⠀⠉⠁⠈⢻⡀⣼⢻⣟⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣏⡏⣿⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠒⢎⢿⣿⣅⠈⢷⣲⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠜⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⢷⠙⢧⡈⢻⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣙⢸⢸⠀⠀⠙⢦⣭⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢸⢸⠀⠀⠀⠀⠉⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡌⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⢐⠜⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠌⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠒⠀⡀⠀⡦⢊⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡄⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠒⠀⠀⠄⠠⡖⢋⠄⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠋⠀⠁⡠⠴⡊⠰⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠒⠦⢄⠐⠂⠈⠉⠤⠤⠠⠔⠒⢊⠡⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠤⠤⠤⠬⠍⢁⣤⡭⠭⠥⠦⠑⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠂⠉⠉⠉⠉⠙⢒⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠴⠖⠛⠛⠒⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⠍⡉⠑⠒⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠷⠮⠝⠓⠒⠒⠂⠤⢤⣀⣀⠈⠛⠶⡤⢄⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠎⠀⠀⠉⠙⠒⠺⠆⠈⠑⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡞⢁⡆⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⠞⠁⠑⡄⠀⡼⠂⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⡀⡾⠀⢀⡠⢀⠀⠇⠀⠀⠀⠀⡆⠀⠀⠀⠇⣾⠄⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⡇⢽⠀⠀⠀⠀⠑⠼⠤⠊⠀⠈⣏⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠒⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⠖⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⢀⡀⢆⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠸⡄⠀⢸⡰⠽⢆⢀⠊⠑⢄⡔⠉⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠉⠁⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡮⠛⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠦⡀⡬⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⣠⣊⣑⠄⠈⠶⡲⣄⣀⡀⠀⢀⣀⣀⠀⡿⠷⣒⢲⠶⣖⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢉⡿⠤⠠⠔⢢⢴⠀⠀⠀⠈⠉⠛⠒⠊⠝⠷⣔⡫⢛⠔⠃⠤⠄⠐⠂⠈⢱⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⢠⣔⡊⠁⢀⠔⠁⠱⣄⠀⠀⠈⠉⠈⠁⠒⠠⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠤⢴⠀⠀⢘⣤⡔⠁⡽⢀⠷⡀
⠀⠀⠀⠀⠀⠀⠺⡀⢸⣂⠏⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⡆⠀⠉⠦⡇⠈⠀⠈⢐⡡⠈
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠁⠀⡐⢀⡄⠀⠀⠀⢈⠆⠀⠀⠀⠀⡳⠃⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠘⢆⢠⠊⢠⠖⠲⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡑⠰⠄⠀⠀⠘⢰⠗⠊⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠈⠀⣀⠤⠤⠴⠆⠤⠤⠾⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⠂⠀⠈⠉⠉⢹⢹⡋⠁⠀⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣸⢸⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡀⡀⡄⣄⣄⡀⣹⢉⣉⣉⣄⣌⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠟⠻⣟⡛⠿⢛⣟⠻⠿⠛⣛⣻⠉⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⣚⠩⠭⠀⠀⠈⠀⠀⠈⠉⠐⠂⠐⠈⠭⠍⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠠⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠈⠑⢤⡀⠀⠀⠀⣀⠔⠈⣀⠔⠊
⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠔⠀⠀⠀⠀⠀⣇⡠⠔⠉⣀⠔⠊
⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⡠⠔⠁⡠⠔⠉⠑⠖⠊
⡠⠂⠀⠀⠠⠶⠀⠀⠀⡠⠔⠉⢀⠔⠊
⡇⠑⠤⡀⠀⠀⠀⠐⠉⢀⠤⠊⠁
⠑⣄⠀⠈⡆⠀⢀⠤⠊⠁
⣤⣬⣵⣶⣶⣪⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣳⣦⡀⠀⢀⣤⣴⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣭⣿⣿⣿⣿⣾⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣠⣴⠿⠋⠉⣿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⡿⣿⣿⢟⣯⣿⣷⠀⠀⠀⠈⣧⣿⠻⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣁⡀⢀⢟⣵⠿⣽⣿⣿⠀⠀⠀⠀⢸⣿⠀⠙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⢩⣿⣵⡟⢧⣠⣿⣿⣿⠀⠀⠀⠀⢸⣿⡄⠀⠀⠻⢯⡢⣀⣤⣴⠶⠒⠒⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠶⣲⣿⠟⠻⣇⣾⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⠇⠀⠀⢀⣴⣿⢮⣿⢶⣖⣛⣀⣀⠀⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢠⣿⢿⣄⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⡇⣼⠻⡷⣽⡻⣷⣂⣀⣠⣄⣀⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣯⣿⣤⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠠⣿⣿⡇⣿⠀⠀⣿⡹⣾⣝⢿⣷⠀⣈⣉⣻⣿⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡷⠿⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠻⣿⣿⣿⣧⡀⣿⣆⠹⢿⢷⣿⣿⣟⠛⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣼⣆⠈⢻⢿⣷⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⣀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣷⣾⣧⡙⣿⣷⣙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⢠⣿⣿⣿⣿⡿⠁⣾⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⢿⣿⣿⣿⣮⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠈⢿⣿⣜⣿⣓⣶⢿⣿⣛⣭⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣄⡀⠀⠙⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠛⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⢀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠚⠋⢩⠉⡭⠉⣳⡄⠀⠀⠈⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⣠⠒⠨⠅⠀⠤⣒⣒⣉⠩⠭⠔⠒⣉⡴⠃⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠸⣭⣉⣉⣉⣁⣁⣠⡤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⢀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⢸⠀⣠⡞⠁⠀⢀⡀⠈⠳⡄⣰⢺⠋⠀⠀⠀⠙⢦⡟⢰⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⣴⠃⠀⢀⣴⠮⠴⠋⡴⡛⠀⢠⡾⣅⠉⡆⠀⢹⠁⡇⠀⣰⡋⠳⡄⠀⣧⢸⠀⢠⡙⣆⠀⢹⡀
⠀⠀⠀⣸⠉⠀⢀⡏⣀⣤⣤⣴⡇⡇⠀⢸⠀⠘⡇⡇⠀⢸⢸⠀⠀⣏⢹⠀⡇⠀⣹⢸⠀⢠⣷⢸⠀⠘⡇
⠀⠀⠀⣿⢃⠀⠈⣿⢻⣀⠀⠀⡇⡇⠀⠘⣆⢸⣧⠃⠀⡜⢸⠀⠀⢿⢸⢠⠇⠀⣿⢸⠀⠘⣿⡞⠀⢰⡇
⠀⠀⠀⢿⠘⣄⠀⠈⠑⠚⠀⠀⡇⠸⡄⠀⠈⠛⠃⠀⣰⣧⠀⣇⠀⠈⠛⠊⠀⢠⣿⢸⠀⠀⠉⠀⣠⡟
⠀⠀⠀⠈⢧⡈⠓⠤⢄⣀⣀⣠⡿⣄⠈⠲⢤⣄⣤⣴⠏⠘⢧⡈⠳⢤⣀⣀⣤⠟⣿⢸⡤⠤⢒⣿⠏
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠋⠀⠈⠓⠶⠶⠶⠛⠁⠀⠀⠀⠉⠓⠒⠒⠛⠁⠀⠻⠶⠖⠚⠋⢀
⠀⠀⠀⢠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡳⣄
⠀⠀⠀⡏⠀⠳⣄⡀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⣀⡼⠀⢹
⠀⠀⠀⢷⡀⠀⠈⠉⠙⡅⠀⠀⣀⠀⠈⠳⡄⠀⡴⢻⠁⠀⠀⠘⡇⢠⡎⡇⠀⠀⠀⠉⠓⢏⠁⠀⠀⡼
⠀⠀⠀⠀⠙⠦⣄⣠⠂⡇⠀⢰⣦⠕⠀⠀⡇⢰⢇⡟⠀⢰⡆⠀⢿⢸⡇⡇⠀⢰⠒⢤⡀⠀⢳⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠠⣾⢃⡟⢸⠁⠀⠺⠇⠀⢸⣾⡇⡇⠀⠘⡷⡄⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠠⡇⠀⢐⠒⢄⠀⠈⣿⢁⡇⠀⠀⣀⡀⠀⠀⣿⡇⡇⠀⠀⣧⡧⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠇⠀⠿⠷⠈⠀⠀⡸⡼⠀⠀⢸⡇⢳⠀⠀⢹⡇⡇⠀⠀⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣅⣀⣀⣀⣀⣤⣾⢱⣁⡀⢀⡟⢷⠸⣄⣀⣈⡇⠧⡠⠤⠤⢴⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠴⠖⠋⠙⠶⠶⠤⠟⠁⠘⠒⠒⠲⠞⠓⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠦⣄⡀⠀⠀⣀⡤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠙⢦⠞⠉⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣤⠶⣿⠉⠉⠉⠉⠉⢻⡏⠉⠁⠉⠉⠙⡷⢦⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⣀⣠⡴⠚⠉⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣟⠀⠈⠙⠲⢤⣀⡀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣴⣶⣶⣿⣿⣿⣷⣶⣦⣤⣤⣿⣤⣤⣄⣀⣀⣼⣧⣀⣀⣤⣤⣤⣿⣤⣤⣶⣶⣾⣿⣿⣿⣶⣶⣬⣽⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠿⢿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠛⠁⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣶⣶⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡠
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢰⠋⠀⠀⠀⢠⣶⣤⡀⠀⠹⡄⠀⠀⣀⢀⣀⠀⣀⡤⠤⣀⠀⠀⠀⠀⠁
⠀⠀⡟⠀⢀⡠⣌⣳⣿⣟⠅⠀⠀⡇⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠓⠤⢄⡀
⠀⠀⢹⡀⠀⢿⡻⠯⠳⠈⠀⠀⣰⠃⠀⡔⠀⣀⣀⣀⡠⠤⢄⣀⠀⠀⠀⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠙⠦⣀⠀⠀⠀⣤⡀⢶⠁⠀⠀⢱⢄⡅⠤⣄⠀⠀⢀⡤⠄⠘⠒⠺⠄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠛⠀⠀⠀⠸⣀⠦⠰⠶⠠⠠⡰⠶⢄⡀⠀⠀⠈⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⣀⠠⠤⠀⡀⠀⠔⠛⠈⠣⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠚⡃⠶⠶⠶⢺⠂⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣀⣀⡠⠔⠁⠃⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠃⠈⡀⠀⠀⢨⡍⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣖⠏⣱⠤⠆⣹⡖⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⠀⢳⢄⠀⡈⣠⡿⠃⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣷⠾⠋⠁⠀⠀⠀⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠐⠀⠉⠉⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢁⠁⢀⠈⠁⢟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠐⠀⠀⠠⠊⠀⠀⠘⠂⠀⣄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣁⠐⠁⢀⢄⠔⠁⠀⠀⠀⠀⠀⠈⠀⡚⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⠈⣤⠃⠂⢞⠅⠀⠔⠀⠀⠀⠀⠀⠠⠃⡰⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠠⠀⠀⡱⠁⠀⠔⠀⠀⠄⠀⠀⠀⠀⠀⠒⡀⢀⠁⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠀⢐⠘⠀⠈⠀⠄⠀⣀⠄⠂⠀⠠⠀⠀⣠⠌⢤⠃⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠤⠠⣀⣬⠤⢯⡤⠶⠚⠉⠁⠀⣀⠀⠀⢀⠔⢁⡮⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠴⠉⢀⡀⠶⢒⠿⠲⡞⣍⠁⠀⠂⠐⠂⠀⠲⠈⠁⣔⠎⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠄⠀⠀⠁⢀⡴⠏⠈⠀⣆⠁⠘⠂⠀⠀⠀⡤⠊⢀⠓⠁⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡜⡵⠀⠀⠀⢄⠔⠉⠀⡈⠄⠁⠘⡄⠠⠀⠀⠅⢀⣰⠜⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡞⡈⠀⠀⠘⠕⡡⠠⡂⠚⠀⠀⠂⠀⡧⠀⠐⠁⣔⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡸⠀⠂⢀⡴⠁⠀⠀⠈⠁⠈⠀⠀⡀⠔⠀⡀⠦⠓⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢈⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⡁⠀⠤⠂⠀⠀⠂⠀⣠⡃⠠⠔⠉⡠⡰⠒⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⡗⡀⠐⠂⠤⠤⠂⠊⢀⠠⣄⠦⠓⠋⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠘⢮⠱⢦⣄⣀⡤⠦⠐⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠁⠀⡁⠀⠀⠄⠁⠈⠀⣠⠜⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠞⠁⠀⠀⠀⠀⣀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠓⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⡤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣷⣶⣶⣶⣿⢟⡻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣯⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠛⠛⠹⠋⠉⠉⠛⠛⣿⠛⠉⠉⠉⢹⠉⠙⢻⣿⣿⣿⣟⣽⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣅⡀⠀⠀⠀⣡⠀⠀⠀⢀⣸⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⠿⠛⠊⠉⠛⠉⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠈⠀⣰⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⣷⣦⣄⣄⣴⣂⠈⣀⣠⡴⣞⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣻⣿⣿⣟⣫⡴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠒⠒⠒⠒⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠴⠚⠛⠃⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢠⠊⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣰⢇⡠⠄⢤⠘⠀⠀⠀⠀⠀⡀⢀⠃⢯⡀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣆⠀⠀⠀⠀⠀⠑⠁⠀⡠⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⢠⡈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠴⠀⠀⠑⠄⠔⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢴⠀⠀⣠⡀⡀⠀⠒⠠⠤⡖⠒⠤⣏⠹⣮⢏⡄⡀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠚⠀⠐⠫⠈⠀⠀⠙⠒⠀⠀⢄⣀⡀⠀⠀⠉⠀⠀⠀⠠⡀⢀⡄⠀⢸⡀⠔⠐⠂⣆
⠀⠀⠀⠀⠀⠐⡇⠀⣀⠃⠀⠀⠀⠒⠄⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⢀⠙⠀⠁⠲⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠦⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠂⠀⠐⠢⠎
⠀⠀⠀⠀⠀⠀⠀⢈⠐⠀⠐⠀⡀⠋⠁⠀⠀⠀⠀⢠⡀⠀⢀⠀⠠⠄⡰⡎⠀⠆⠐⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠆⡇⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡂⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⢀⡀⠚⡖⡓⠂⡀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠻⠻⣟⡛⠿⢿⡷⠿⢛⣻⣏⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠈⠉⠀⠀⠀⠀⠈⠁⠀⠐⠊⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠐