Cough Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⠖⠉⡙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠎⠁⠀⣰⡟⠁⢀⣼⠔⠊⠉⢉⡹⠆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡎⣿⠁⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢷⡇⠀⠀⠀⢰⠀⠀⠠⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠈⠉⢩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⡀⠸⠆⡀⢴⣦⠔⠈⠏⢠⠄⢐⣆⡄⠀⠀⠶⠃⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠄⠀⠛⠚⠀⠋⠈⠉⠀⠘⠋⠀⠀⠈⠙⠀⠀⠀⠀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⣤⠖⠓⠂⠀⢰⠇⠀⠀⠀⠈⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠠⠞⠻⠶⠀⠀⠀⠁⠀⢀⠀⠀⠈⠻⡆⠀⠀⣀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⢣⠀⢀⣀⡀⠀⠀⠀⠀⠞⠋⠉⠀⠀⠀⠀⠁⠚⠡⠄⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢾⣄⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢂⠀⠀⠦⠀⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠆
⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⣿⠀⠀⠞⠓⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠟⠛⠶⠀⠀⠀⠀⠀⣤⠖⠀⠈⠉⢦⡘⣄⠀⠀⠈⠲⠆⠀⡨⠃⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣹⠉⠲⣤⣀⠀⠁⠈⠀⠀⣠⠔⢶⠟⡗⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠘⢄⡀⠀⣀⣐⠒⠉⠙⠀⢀⠴⠛⡿⢶⠀⠐⠊⠀⠀⢀⠀⡇⠀⠀⠀⠁⠲⡄
⠀⠀⠀⠀⠀⠀⠾⠟⠉⠉⠁⠀⠀⠀⠈⢳⠒⠉⠉⠒⣤⠛⡰⠃⠀⠀⣱⢄⣢⡄⢤⠒⡖⠠⠔⠀⢀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⡓⠒⠒⠀⣀⣠⠁⢀⡀⠄⠳⣤⣁⠘⠻⠆⢹⠁⠀⠀⢸⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠀⠀⠀⠀⡌⠁⠚⠓⠀⠙⠇⠀⠸⠋⢠⣤⠄⠀⠀⠀⠀⠀⣦⠠⡤⢤⣇⠀⠀⢀⠄⠀⠸
⠀⠀⠀⠀⠀⠠⠐⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⡿⠠⣴⠦⠀⠀⠀⠀⢀⡷⠀⡀⠀⠙⠳⠶⠦⣄⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠂⠀⡇⠀⠉⠘⠃⠈⠛⠛⠛⢄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠴⠊⠉⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠆⢀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠙⣒⣢⣤⡤⠤⠴⠒⠒⠛⢓⣂⡈⣭⡥⠒⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠉⠁⠐⠒⠒⠒⠒⠒⠉⠉⠉⠁⠘⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠲⢄⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣤⠤⠴⠒⠫⠝⠷⢦⣤⣤⡤⠖⠺⠟⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⡶⠖⠀⢠⠐⠲⠆⢸⠒⢤⠃⢀⣀⡟⣉⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠴⠞⡉⢧⣟⣻⠇⠀⠀⠀⠸⣧⣷⡾⢏⣉⣁⣸⡟⡞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⢶⠩⠤⠀⠀⢀⣀⢰⡮⠟⠋⠈⠉⠉⠉⠛⠃⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣜⣀⡤⣤⣤⣼⣠⣀⣙⡃⠴⠾⢿⡁⠀⠀⠀⢀⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⢎⡁⠀⠋⠀⠀⠈⢋⣡⠞
⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣦⣶⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣤⣤⣴⣦⣤
⣿⣿⣿⡟⡏⠛⢿⡟⠙⡿⠛⢿⡛⠟⣻⣻⡿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣷⣶⣷⣶⣷⣶⣾⣿⣶⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡤⠖⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢳⣀⠴⠋⠈⠋⠁⠀⠀⠀⠀⢧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣠⠚⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠛⢹⣆⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⣇⠀⠀⠀⠀⠀⠉⢹⡀⠀⡞⣩⠉⢦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⠀⠀⠀⠈⢳⡀⠀⠹⡄⢠⢆⡀⠀⠀⠀⢹⣘⡇⡟⠂⡼⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠈⠓⢦⡀⠀⠀⠑⠀⠀⡇⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⢻⢧⡀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢀⣀⣀⣴⠿⢆⡀
⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠈⠛⠳⣤⠴⣲⠆⠀⠀⠀⠀⡼⠁⠀⠙⣫⠴⢋⣠⠖⠋⠉⠓⢤
⠀⠀⠀⠀⠀⠉⠙⠂⠀⢀⡤⣾⢓⣦⡀⠾⣥⠀⠀⠀⠀⡼⠁⣀⠴⠋⢁⡴⠋⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣸⠋⡀⠙⢆⠸⣄⠀⣠⠞⢻⠋⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡘⢻⡿⠁⠀⢸⠀⠈⠉⠁⠀⢸⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⣹⢧⠀⠀⢸⣆⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠓⢠⠏⢸⡀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠒⠋⠀⣸⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⠖⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⠀⠀⡔⡞⢦⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠻⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⡟⠻⣿⣧⡄⠷⡘⠘⠀⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢛⢁⡤⠀⠹⡿⣇
⠀⠀⠀⠀⠀⣿⣿⠛⢛⡁⡀⢴⡀⢸⠉⡿⡄⠀⢢⠀⢀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⢹⠁⠘⠸⠀⠉⠀⠀⣀⣼⡀
⠀⠀⠀⠀⠀⢻⠿⣀⣀⡿⣚⠭⢭⠉⢾⠀⠈⢢⠤⠐⠚⡀⠒⠒⠚⠉⢉⣉⡩⠍⠀⠈⢆⠀⢎⠉⡣⠀⣠⣿⡿⢏⣀
⠀⠀⠀⠀⠀⠘⢄⣠⣹⠈⣒⡈⠥⢂⠌⠀⠀⠈⠄⠀⠀⠇⠀⠀⠀⠀⠁⠠⠀⠀⠀⡀⢼⠟⠉⢹⠗⠉⡝⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡳⡶⠒⠊⣹⡆⠀⠒⠒⠂⠀⠀⢰⠀⠀⠀⢀⠄⠀⢀⡴⣪⠖⡰⢆⣀⣠⠋⠊⠁⠀⠀⠐⠢⡀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠈⠒⠾⢿⣿⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠉⣡⠔⠁⠚⠁⢀⠇⠀⠈⡷⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡜⠁⠀⣀⡠⠔⠲⡁⠀⢸⠀⠀⡖⠒⠒⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⢄⣀⡼⠁⢃⠀⠀⠀⣠⠊⠉⠉⣢⠞
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣡⠤⠊⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠾⠞⠁
⠀⠀⠀⠀⠀⠘⠢⠄⣀⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠶⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡀
⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡷
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⢠⠄⢀⡤⠀⣠
⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠒⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠐⢤⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠉⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠦⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⡴⠶⠶⠟⠃
⣿⣷⠿⠿⠾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⡤⠒⠋⠉⠉⠑⠒⠂⠐⠲⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣄⣠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠀⠀⠹⡄⠀⠀⡴⢋⣉⡙⡀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⣤⣄⠀⠘⠛⠁⠀⣀⡄⠀⠀⠀⠈⠢⠚⠁⣯⠀⠁⡇⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⡼⣆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢠⡇⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠋⠘⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢘⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⣄⡠⠴⠊⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢠⣞⣷⡶⢤⡄⠀⠀⠀⠀⠀⠀⠀⡸⢀⠞⠀⠀⠀⠀⠀⢱⢄⡀
⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠄⠒⠒⠒⠒⢂⠈⠢⡈⠑⡇⢁⣾⠃⠀⠀⠀⠀⠀⢀⡴⣡⠋⠀⠀⠀⠀⠀⠀⡆⠀⠉⠓⠦⣀
⠀⠀⠀⠀⢀⡠⠔⠒⠒⠤⣀⠀⠀⠐⠉⠁⠀⣠⠈⠲⣍⠋⠁⠀⠀⠀⠀⢀⡰⢋⡜⠁⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⣠⠔⠉⠀⠦⣌⡙⠲⣄⠙⢆⠀⠀⠀⡰⠃⠀⣀⡤⡷⠦⣄⣀⣀⠴⠊⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⡾⠁⠀⠒⠦⢄⡀⠙⢦⠘⡇⢀⠀⢀⠞⠁⡠⠞⠁⢰⠁⠀⠀⠉⡟⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⡇⠀⠀⠠⣄⠀⢹⠀⢸⠀⣅⡈⢠⠏⢠⠞⠀⠀⠀⡏⠀⣀⠤⢄⣡⣹⠤⠒⠒⠢⢄⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢣⠀⠀⠀⠸⣧⢸⣤⠼⠒⢻⠀⠁⢠⠃⠀⠀⠀⠀⣧⠎⠀⠀⠀⠙⢇⠀⠀⠀⠀⠀⠳⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⢆⠀⠀⠀⡇⠀⠀⠀⣴⠃⠀⢀⡎⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢱⠇
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢹⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡰⢆⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⡇⠀⢀⡠⠊⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢠⡏⣡⡴⠋⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⢘⡋⠁⠀⠀⠀⣠⠔⠋⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣨⡇⠀⢸
⠀⠀⠀⠘⣇⣀⣠⠴⠊⠁⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠈⢩⠁⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⡀
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣇⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣄⡤⠚⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣤⣴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠋⠁⠀⠙⠒⠀⠘⣿⣿⣿⠀⠀⠀⠀⣀⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠟⠛⠉⠀⣶⡤⢠⠀⢴⣾⠀⣤⠘⠻⢿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠁⢠⡞⠀⠀⠉⠠⣿⠀⢀⡸⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⡀⠀⢀⡠⠭⠤⣄⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠀⢠⠤⢤⣿⣿⣿⣷⣄⠘⠲⢤⣤⠞⢁⣴⣿⣿⣟⠦⠴⠚⠑⢦⠘⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⡔⠃⠀⠀⠋⣩⡯⠟⠛⡻⠋⠁⠀⣽⠟⠻⠿⢍⡉⡳⠀⠀⠠⡞⠀⠀⠀⠈⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⡀⢀⣀⠇⣠⠖⠙⣄⠀⠀⡠⠇⠀⠀⠀⠀⠈⠳⣤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢉⡴⠋⠀⠀⠀⢈⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠞⠁⢀⣀⣀⣖⠢⣄⣀⣀⡤⠋⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠖⡏⠀⠀⠀⣜⠲⠤⠉⠀⢸⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⠿⠇⢀⣀⣸⣀⣀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠉⠉⠉⠁⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣾⣿⡆⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡅⠘⠟⠀⠀⠀⠈⢻⣿⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠀⠀⠀⠀⠈⠻⠿⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⠛⠛⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠠⡤⠄⢰⣀⡀⠠⠤⠤⠤⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⣠⠔⠋⣱⡀⠄⠀⠀⠒⠐⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠴⢦⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣏⣠⣒⠁⠈⢆⣀⠤⠒⢉⡠⠒⣽⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢤⡀
⠀⠀⠀⠀⡠⠛⠀⠀⠹⣤⡤⠀⠀⢿⣿⣿⣿⣿⣿⠟⡥⠞⢻⣿⠉⠉⠹⢤⠖⢁⡠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠳⣀⡀⠀⢀⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠊⠡⠽⡗⠒⠒⢞⠉⠉⠙⠟⠛⠛⠋⢩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢦⡀⠀⢸⠁⠀⠀⠘⣆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠓⠦⣾⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⠀⠈⢳⣿⣿⡿⢏⡀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⣼⠀⠀⣸⣷⠀⠀⠀⠀⠀⡏⠉⠉⠁⠈⢦⣀⡀⠀⣀⣀⣸⣄⣀⣀⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡤⠤⠴⠁⠀⠀⠀⠀⡿⠻⠴⠚⠁⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠣⠤⠤⠤⠴⠦⠤⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠤⠤⣤⣤⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣄⡀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠉⠻⠿⢿⡿⠿⠟⠛⠉⠀⠀⢀⣠⣴⣶⠀⠀⠀⢠⣾⣿⣿⣶⣬⣿⣿⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣤⣼⣿⣿⣿⣿⣿⠀⠀⢀⣾⣧⡀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⠿⢿⠿⠟⠁⠉⣿⣿⡄⠀⣾⣿⣿⣿⣶⣶⣦⣽⣿⣿⡟⠛⠻⠿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣤⠄⠀⢀⣿⣿⣿⣶⣼⣿⣿⣿⢿⣿⡿⠿⠟⠉⠀⠀⠀⠀⢤⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⢀⣴⣿⣿⣿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⠹⣿⣿⣉⠹⣿⣿⢟⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡏⣴⣿⣿⣿⡿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣴⠞⠛⠋⠀⠀⠀⠀⢿⣿⡟⠀⣿⣿⢣⢫⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⢀⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢻⠀⠀⢰⢃⠞⡵⠋⠀⠀⠀⠀⠀⠀⣿⠏⢀⣴⣿⡟⣴⣙⣭⣿⢿⣷
⠀⠀⠀⠀⠀⠔⠂⠁⠀⠁⣸⣿⣿⣿⣿⣏⠺⣧⠀⠀⢀⣀⡀⠀⠈⡀⠀⠀⠀⠀⡠⢋⠂⠈⠀⠀⠀⠐⠢⡀⠀⠈⠁⠀⢸⣿⣿⣳⡎⡏⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠙⠲⢿⣿⣗⡭⡭⠿⠏⣿⠀⠀⠤⠌⠙⠲⡞⠉⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⣼⠟⣥⣻⡵⠶⠛⠀⠀⠈⢹⡀
⠀⠀⠀⠀⠀⢧⡀⢘⡟⢳⣶⣤⣀⠻⠥⠭⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣴⢟⡡⣟⣀⣤⡴⢾⢿⡃⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⣿⣮⡷⠶⠟⠛⠛⠒⠶⠦⢤⣤⣀⠀⢻⣤⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣫⡥⢮⣭⣴⣻⢶⢷⡍⢺⡽⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣟⠉⠈⠀⠼⠁⠀⠀⢀⣠⡴⠛⠉⠟⠁⠀⠀⠀⠀⠉⠛⢿⡌⡳⢻⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠒⠒⠶⠾⣿⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣽⢵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡳⣤⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⣽⢳⣦⣬⣉⡙⢻⠋⠉⠙⢿⣦⣀⡀⠀⠀⣸⠀⢸⠀⡾⠏⠀⠀⠀⠘⡀⠀⠀⢀⣠⡾⠛⠉⠁⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣜⢸⣿⠿⠛⠛⠟⠀⣆⠀⠀⠘⢻⣄⣀⣠⣟⣠⠿⠚⠁⠀⠀⠀⠀⠀⢡⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡙⡶⣴⡾⠶⣛⢻⡆⠀⠀⠀⠀⢉⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠫⠄⠓⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣼⣾⣒⢻⣏⡾⢿⡄⠀⢠⠖⢻⡲⠤⠌⠳⠦⢄⣀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣟⡻⢾⢸⡇⢰⠃⠀⣠⡤⠆⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣾⣁⠘⢆⡼⠁⠙⠂⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣇⠀⠀⣰⠁⠐⠀⣠⡆⠀⠀⢠⡦⠄⠠⠤⠆⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⣇⠀⠀⣠⡟⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⣄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡄⠀⢀⡄⢀⢴⠀⡠⣦⢀⣤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣗⡰⠉⣇⠔⠁⠷⠁⠘⠊⠀⠛⠋⠘⢻⠀⢳⡤⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⠘⠀⡄⠉⠀⠀⠀⠸⢤⡤⠄⠀⠀⠀⢸⣠⠞⢰⣡⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢠⠴⠁⠀⠀⠀⠀⠀⢀⠔⢀⡀⠀⠀⠘⠃⠀⠋⢰⠁⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⣤⣀⣀⠀⠀⠀⣰⣟⡮⠥⠖⠒⠀⠀⠀⠀⠀⢈⣱⣃⣀⡀⠤⢔⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠘⠋⢉⡝⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠶⠦⠥⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣈⣓⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠒⠶⣇⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠉⠀⠈⠳⣄⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⡠⠛⠉⠉⠉⠻⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡒⠿⠓⠉⠻⣄⠀⠀⠂⡩⠀⠴⠚⠒⠚⠋⠉⠀⠀⣀⡤⢲⠋⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣉⠂⠀⠀⠀⠙⣟⡶⢤⣠⣒⣀⣀⣀⣀⣤⠴⠒⣉⠜⢐⣧⡀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⢀⠔⠒⣆⡀⠀⠀⢘⣦⠀⢹⡀⠀⠀⠀⠀⠉⠳⠶⣿⣿⠉⠉⠀⠀⠉⠉⠉⣡⠔⠉⠀⠙⢦⠀⠀⠀⣠⡇
⠀⠀⠀⡰⠛⠀⠀⠐⣿⡁⠀⠀⠈⠓⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣄⣀⣀⠞⠁⠀⠀⠀⠀⣠⠷⡄⠔⠁⡇
⠀⠀⠀⠓⠖⢇⣀⠴⠚⠀⠀⢀⡠⠤⣤⣀⠉⠉⠛⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⠄⠀⢛⡩⠂⠀⠀⠀⠀⠀⠈⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢀⡴⠁⠈⠷⣦⡀⠀⠀⠘⠒⠦⠜⠁⠀⠀⠀⠀⠀⠀⠀⠓⡖⠲⠬⡏⠓⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠸⣀⡄⠀⠉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡤⠖⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣦⠴⠋⠀⠉⠁⠀⠀⠀⠸⠧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠏⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠧⢤⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⢦⠠⡏⣖⡀⣧⠀⠀⠀⠀⠻⡀
⠀⠀⠀⢤⣀⠀⠀⠈⠳⡀⠀⢳⢠⡧⠄⠀⠀⠀⠘⠶⠇⠇⣠⡏⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⣸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡀⢀⣀⡀⠀⠀⠀⠀⢠⠏⠀⢧⠴⣲⠾⢋⣩⠦⢤⡀
⠀⠀⠀⠙⠒⠦⣄⠀⠀⠀⢀⣤⡀⠛⣫⠞⠁⠀⠀⠀⣰⠋⠀⢀⡴⠚⣡⠔⠋⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠙⣧⠛⠳⣌⠙⡆⠀⠀⣠⣧⡴⠚⢁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢾⣧⡖⠀⢸⡆⠓⠶⠚⠁⢸⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠲⣿⠀⠀⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠉⢳⠀⣸⢹⡄⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠤⠞⠁⢀⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣭⣙⡛⠛⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⡿⣧⢏⡤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠛⠐⠀⠈⠳⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡶⠀⠀⠀⠀⠀⢷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⣄⡀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠹⡈⠀⠀⠙⠟⢦⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱⡀⠈⠢⣄⢆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠈⢳⡙⢦⡀⠹⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣰⠳⠃⠀⠀⠀⠀⣇⠀⠙⠂⠈⡉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡟⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⡀⡷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡰⠀⣼⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⡠⠇⠹⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⡇⠠⣿⡇⠀⠀⠀⠀⠠⣷⠶⡶⠚⣥⡄⠀⡇⣅
⠀⠀⠀⠀⠀⣼⠀⡇⣿⣇⠀⠀⠀⠀⢲⣶⣟⣓⣭⠾⣡⠾⣇⢹
⠀⠀⠀⠀⠀⡟⠀⢁⣿⡏⠀⠀⠀⠀⢘⠏⠙⠿⢷⣿⡋⠀⢻⣯
⠀⠀⠀⠀⠀⣷⢸⡀⣿⣧⢀⡀⢀⡀⢺⡆⠀⢀⣿⠅⠁⠀⢸⡙
⠀⠀⠀⠀⠀⠹⣿⣧⣸⡟⣠⡬⣁⡉⢣⣥⠀⢸⢿⡷⠄⠠⡾⠃
⠀⠀⠀⠀⠀⢰⠙⢿⣿⣿⣛⠖⠀⠙⢦⡇⠀⢺⠈⠉⠈⠉
⠀⠀⠀⠀⠀⡿⠀⠈⢻⣿⡇⣿⠀⠀⠈⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠳⣜⣶⣼⣿⣿⡴⡄⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡷⠀⠰⣶⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⡀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣀⠘⢿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣘⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⡠⡄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⣿⣿⣿⣤⣘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⣝⠆⠙⠛⠿⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣧⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠲⢇⣀⣀⣠⡀⠉⠻⣦⣀⣨⠀⠀⠀⢴⣀⣉⠋⠐⠒⠤⣀⠀⠊⡀⢠⡾⠗⠈⠢⡄
⠀⠀⠀⠀⠀⠀⡠⠤⠤⢼⣄⣀⣤⡄⠀⠈⢻⣿⣿⣶⡾⠋⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⠁⠀⣾⡿⠋⢀⣦⣾⡷⠾⠖⠒⠢⡀
⠀⠀⠀⠀⢠⠞⢀⣴⣄⡀⠀⢈⣿⣿⣦⣄⣀⣿⢋⠹⠀⠀⠀⠀⠀⠀⠀⢠⠁⣹⡇⢡⠀⠇⠀⣸⣿⠁⠀⣿⣿⠏⠀⣠⣶⣾⣶⠡
⠀⠀⠀⠀⠁⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⡋⢡⠋⣶⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠛⣻⢲⣀⣼⠋⡇⠀⣸⣿⡏⢀⣴⣿⣿⣿⣿⡆⢃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣘⠸⠛⠉⠉⣶⠀⠀⠀⠀⡇⠀⠄⠀⢼⡂⠉⠀⠀⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠘⡄
⠀⠀⠀⢐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣎⣠⡏⠀⢀⣴⠟⠁⠀⠀⠀⠀⣇⡀⠀⢀⣀⡄⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢳
⠀⠀⠀⢰⠀⠀⢀⣡⣾⣿⣿⣿⣿⢫⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣧⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⡆
⠀⠀⠀⠸⠰⣾⣿⣿⣿⣿⢿⡿⢱⣿⣿⣿⣿⣿⠋⡜⠀⠀⠀⠀⠀⠀⠀⠻⠯⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻
⠀⠀⠀⠀⡄⢹⣿⣽⣿⣟⢸⡇⢻⣿⣿⡿⠿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠋⢙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢸⡄
⠀⠂⠀⠀⢵⣦⣼⣿⣿⣿⣌⣷⣈⠻⠉⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣬⣻⡿⣿⣿⣿⣿⣿⣶⣶⡄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡿⣻⣿⣷⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠟⠛⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣿⣦
⠀⠀⠀⠀⠀⢶⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣤⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⣀⣤⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠿⠟⠛⠉
⠀⠀⠀⢸⢇⡼⣿⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣹⣿⣧
⠀⠀⠀⠈⢸⡎⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣹⣽⠇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠊⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠤⢄⣀⣀⣲⡆⠴⠖⠒⡖⠒⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⢙⣛⠛⠿⠿⠏⠙⠋⠉⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇
⠀⠀⠀⢡⠀⢠⠖⠒⠤⠔⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠈⠙⠋⠀⠀⠀⠀⠠⠶⠄⠀⠀⡀⢠⣤⠀⡇
⠀⠀⠀⠀⢇⠘⢄⠀⠀⠀⠀⠀⠀⠀⡠⠠⣆⠀⠠⠂⠈⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⣤⣜⡁⠀⠀⢰⠃⣀
⠀⠀⠀⠀⠀⠳⢎⠀⠀⠀⠀⠀⠀⠐⣇⣀⣘⠈⠑⠒⠤⣀⠞⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠿⣦⠾⣿⣿⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠉⢈⡹⠂⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣲⣤⣤⢤⣤⣴⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣸⡓⠩⣹⣏⢧⠀⢀⡠⠊⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⣿⣿⣧⣙⣺⣿⣿⣿⡿⢟⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠒⡀⢹⠈⠑⠉⠈⠆⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡇⢸⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⠀⠐⢾⣀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣛⣩⣹⢛⣿⣿⡛⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢱⠢⢤⡋⣘⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣃⣞⣛⣛⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡏⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠛⠻⠿⠿⠿⠛⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠂⠀⠀⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣯⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⢧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣿⢙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠎⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⡵⠃⢸⣿⣝⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢓⡒⡴⠋⠀⢨⣑⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣁⣀⣀⣀⣀⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣩⡤⢤⠔⠦⣤⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⡇⠀⠀⠀⣱⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠰⢥⣠⣋⣀⣀⣀⣙⡆⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠸⡯⠩⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠉⠓⠋
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠤⣴⢶⣤⣬⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡁⠴⢶⡄⣀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢡⣎⣿⣿⣿⣿⣟⣻⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⡟⠀⠟⠀⣽⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠔⢯⡿⡧⠟⠉⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠁⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢳⠈⠁⠀⠀⠀⠀⠀⠒⠒⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠒⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠙⠒⠊
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠚⠀⠀⠀⠙⠿⠃⢠⣶⣆⠘⢿⣶⣎⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⣾⠂⠀⠀⢠⣴⡆⠀⠀⡰⠋⠉⢰⡧⠄⠀⠙⠻⢿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⢷⡀⠀⠀⠀⢉⣴⠶⡜⠀⠀⠦⠞⠛⢠⠀⠀⠀⢠⣿⡗⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠁⠀⣿⣄⠀⠀⢄⣀⣾⣧⡀⠀⠉⠀⠀⠐⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⠀⢘⣿⠿⢳⣴⠀⠀⠀⠰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠠⠀⠾⠄⣼⣿⣿⠿⡿⢿⠁⠀⠀⠁⠈⠙⠛⢀⠞⠀⢀⡿⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣶⣼⣆⣀⡀⠐⡋⠀⢀⣿⡉⠛⠂⠀⢸⠁⠀⠀⠀⢀⣤⡴⠃⠀⣠⣿⠁⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠿⣿⠅⠀⠈⠿⢋⣽⡿⠶⠒⠈⠉⠻⠉⢠⣉⡽⠷⡀⢀⣴⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣥⣤⡤⠤⠶⠦⠼⠿⣷⣿⣯⡅⣲⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣶⣼⣿⣷⣾⣷⣶⣆⢀⣀⣀⣼⣿⣏⢀⣹⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢿⣿⣿⣯⠻⠛⠁⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣷⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⣠⣴⣦⣤⣤⣤⣤⣾⣻⣶⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠘⣿⡀⠀⣀⣀⣉⣩⣭⣭⣩⣉⣉⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⢹⣿⠉⣿⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⠾⠿⠿⠷⢶⣤⡀
⠀⠀⣠⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣀
⠀⢸⣿⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⠷⣦⣀
⠀⢸⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠈⠛⠛⠛⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠷⢶⣄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠶⠾⠟⠋⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⠿⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⢶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠹⠘⠃⠛⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢛⠚⡛⢛⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠹⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠛⠙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠻⣧⠀⠀⠀⢀⣠⣤⣄⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⢀⣰⠟⠋⠈⢻⣷⠟⠋⠻⣧⠀⠀⠀⣴⠟⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣄⠀⠀⠀⠀⢀⣴⢟⣡⣤⣤⣴⣿⡁⠀⢀⣴⣯⡀⠀⢸⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⣠⡾⠛⠻⠟⣉⣀⡀⠀⠉⠻⠿⠛⠿⣯⣿⡆⠀⠻⣦⣄⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢠⣴⢿⣤⢶⣤⣿⠋⠉⠻⣦⣀⣴⠶⢶⣄⠹⠿⢷⣤⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣸⣿⣿⠁⣾⡇⠀⠈⠻⣦⡀⠀⠈⠻⣧⡀⠀⠙⢷⣄⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢿⣅⣿⠆⢈⣿⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠈⣿⠁⢰⣿⠁⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣤⣿
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢀⣿⡀⠀⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠺⠋⠻⢶⣤⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡈⠻⣦⡀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣋⣿⣦⣌⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠟⠋⣾⠋⠉⠉⠉⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⣁⣀⣀⣀⣠⣶⠶⠶⣦⣄⠀⠀⠈⠉⠛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣀⣴⠟⠛⠉⠉⠛⠋⠀⠀⠀⠀⠙⠟⠛⠛⠻⢶⡄⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣶⠟⠋⠉⠀⠀⣠⣤⣤⣄⠀⠀⢠⡶⠟⠛⠻⣦⣀⡈⠿⠶⣦⣄⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡁⠀⠀⣠⡶⠾⠟⠁⠀⠁⢀⣠⣤⣤⣄⠀⠀⠈⠉⢿⡄⠀⠀⠙⣷⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⣾⠛⠁⠀⠀⣿⡀⠀⣠⣴⣦⣴⡟⠉⠀⠀⠙⠿⠀⠀⠀⣸⡇⠀⠀⠀⠻⢷⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠸⣧⠀⠀⠀⣾⠏⠁⣾⠋⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⣼⠟⠋⠀⠀⣤⠀⠀⢘⣷⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠙⣿⡆⠀⢿⣄⣠⡿⠂⠀⢠⣄⡀⢀⣴⡟⠛⠛⠛⠋⢀⣀⣀⣴⠟⠀⢀⣼⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣧⣄⣀⣽⡏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⢉⣿⠁⠀⠀⢀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠘⠋⠀⣽⠍⠙⣷⣀⡀⠀⣀⠀⠀⠀⠀⢠⣶⣤⣀⣀⣤⡾⠛⣶⡶⠶⠟⠁⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⢠⡶⠀⠠⡿⠀⠀⠛⠙⢻⡿⠛⢷⣶⣶⣶⣿⠇⠈⠉⢩⣿⠀⢰⡟⠀⢰⡖⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠈⠁⠀⣤⠀⢀⣿⠀⠀⣀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠸⠃⠀⣈⠁⠀⠿⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⠋⠀⣈⠁⠀⢰⡟⠀⢠⣿⣿⣿⣿⣶⣶⠂⢰⡆⠀⢸⡏⠀⢠⡀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠈⠁⠀⠘⠛⠛⣿⣿⡿⠁⠀⠿⠃⢀⣌⠀⠀⠛⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠀⠀⣰⡆⠀⠘⠃⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣀⣀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣷⣶⣤⣤⣤⡤⠆⠘⢆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠀⠀⠄
⠀⠀⠀⠙⣾⣷⣿⣽⣯⣤⣶⠶⠋⠐⠀⠀⠀⠀⠀⠀⢨⣿⣿⡅⢀⣀⣠⣤⣤⣄⠀⠀⠀⠀⣼⢟⣽⣿⣿⣭⣥⡴⠆
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⡶⠦⡄⠢⠀⠀⠀⢸⣿⣿⡿⠛⠋⠉⠉⠹⠿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⢯⣝⡓⠆⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣟⠲⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠙⠳⠈⠉⡠⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠌⠙⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣶⠶⢶⣤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠒⣀⣀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⡀⣠⠊⠁⠀⠀⠀⠀⠀⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⡇⡀⠶⠿⠿⠿⠿⠷⠄⠀⠀⠀⠴⠁⡇⠀⠁⠀⠀⠀⠀⠀⢰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⣤⢠⡄⣤⢠⡄⡌⠀⠀⠀⠄⠀⢣⡀⠂⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠉⠈⠁⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⢀⣴⣤⣤⣤⣴⠶⠖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣤⣤⠆
⠀⠀⠀⠾⣳⣿⣷⣶⣶⣿⣛⠁⠀⠀⠄⠀⠈⠀⠀⠀⠀⢠⣿⠟⠛⠛⢿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⣧⣤⣄⣀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣟⡓⠶⠄⢀⠀⠀⠀⠀⠀⢸⡿⢠⣴⣤⠘⣿⡇⠀⠀⠀⠀⠀⠀⠙⣻⡻⣿⣿⣿⡿⢯⣍⡛⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡻⢮⣍⡓⠂⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⠁⠀⣿⡇⠀⠀⠀⠀⣤⣶⠟⠋⠀⠘⠻⢯⡙⠷⣌⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠁⠀⠁⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠋⠉⠉⠉⠉⠙⠓⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⠀⡠⡇⠀⢀⢴⣆⡴⢻⣀⡴⢺⣇⡤⢿⣤⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼⣇⠞⠀⣧⠔⠁⠈⠋⢀⠈⠋⠀⠈⠋⠀⠈⢹⠀⠈⡷⢚⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠁⠋⠀⡄⠀⠀⠀⠀⠀⠙⠦⡤⠴⠀⠀⠀⠀⢸⣄⠾⠁⣞⣤⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠘⠋⠀⠘⠋⡼⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣄⡀⠀⠀⠀⠀⠀⠀⢠⣾⢔⣛⣩⠤⠄⠀⠀⠀⠀⠀⠀⠙⢂⠞⠀⠀⠀⣀⡠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣰⠿⠿⢷⠆⠀⠀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠉⢉⣉⡤⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠰⣏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣀⣈⣉⣉⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣤⠤⢶⡖⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣤⣄⣤⣍⣓⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣘⣇⠀⠀⠀⠀⠀⢀⡀⠈⠳⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠉⢻⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣒⠛⠯⠓⠈⠻⣦⡀⠀⠒⠊⡡⠆⠴⠒⠛⠋⠉⠉⠉⠁⠀⢀⣀⡤⢲⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠀⠈⠛⠀⠀⠀⠀⠘⢿⠳⣶⣉⣀⠔⠀⠀⠀⠀⣀⣀⠤⠴⠚⢉⠟⠰⣇⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⢀⡠⠴⢤⠀⠀⠀⠀⠸⣄⠀⠸⡧⠀⠀⠀⠀⠀⠘⠻⣤⣬⣟⣿⠟⠉⠉⠉⠉⠙⠒⠒⠚⢁⡴⠊⠁⠙⣄⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⣠⣾⠀⠀⠛⣷⡦⠀⠀⠈⠙⢦⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣤⣀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠈⢳⡄⠀⣠⠞⢹
⠀⠀⠀⢰⡁⢀⡄⠀⠀⣘⡗⠉⠀⠀⠀⠀⠀⠀⠳⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠋⠀⠀⠀⠀⠀⢀⡴⠋⠙⠚⠁⠀⢸
⠀⠀⠀⠀⠉⠉⠳⠤⠚⠉⠀⠀⢀⡠⠶⠉⢩⣷⣤⠂⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠸⢤⠄⠀⠉⣅⠜⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⠳⢦⣤⡀⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⣠⠾⠀⠀⠷⢮⣅⠀⠀⠀⠈⠓⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠒⠶⢼⠍⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⣀⠀⠐⠊⣡⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠙⢦⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⣤⠤⠤⠤⢤⠈⠱⡄⠀⠀⠀⠀⠀⠀⢠⠞⡵⠒⡲⠊⠲⢢⡉⢦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢿⣁⣀⣀⡴⠃⠀⢱⠀⠀⠀⠀⠀⠀⡏⠈⢓⠚⢦⠤⠤⠤⠟⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠈⠁⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡖⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⣠⠤⠊⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢋⣐⣢⣞⣷⣶⣬⠳⡀⠀⠀⠀⠀⠀⠀⢀⠞⣁⣀⠚⣳⢀⣀⠙⢆
⠀⠀⠀⠀⠀⠀⢰⠁⣷⢿⡿⡿⡟⠿⠷⠀⢡⠀⠀⠀⠀⠀⠀⡎⠀⠀⢏⠈⡃⢰⠋⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠈⠓⠛⠛⠑⠁⢀⣀⡀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠶⡏⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡔⠒⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠒⠢⠤⠃
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⠛⠉⠉⠉⠉⠙⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⣀⣤⡴⠶⠛⠛⠛⠛⠷⠶⢦⣤⣀⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⣠⣾⣋⠀⠀⣠⠟⠋⠛⣷⣦⡶⠶⣦⡉⠛⢶⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠾⢻⡟⠉⠉⠛⢻⡿⢦⣴⡟⠉⠈⢻⣦⣸⣇⣀⠀⠙⣧⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣼⠃⠀⢸⡇⠀⠀⠀⣾⠁⣸⠏⠙⣷⣾⠋⢩⣿⡍⠉⠛⢦⣸⡇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠀⠀⣨⣷⡟⠛⠛⣿⢶⣿⣤⣼⣏⡙⣷⡟⠀⢻⡀⠀⢀⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠸⣧⢠⡟⠈⠻⢦⣴⣇⣼⠃⢨⡏⣉⡿⢿⡷⠶⢾⡷⠶⠛⠁⢻⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠈⢻⣧⣀⡀⠀⣸⣏⣿⣻⣿⡿⠿⠶⢾⣧⣤⠀⢻⣄⠀⢀⣾⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⣷⡉⠙⢻⡟⠋⣿⠁⢹⣇⠀⢀⣼⠃⢸⡇⠀⠉⠛⠋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠈⠛⠾⠋⠀⠀⠹⣦⡀⠙⠛⠛⢁⣠⡾⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣶⣶⠶⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⢀⠀⢀⠀⢀⡀⢀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢰⠆⠈⣇⠔⠹⠆⠈⠆⠁⠗⠉⢷⠈⢧⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠃⠠⠈⠀⠀⠀⠢⠤⠀⠀⠀⢸⡠⠃⡮⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠐⠁⠀⠀⠀⠀⣠⣊⠤⢀⠀⠈⠁⠀⠁⠇⡸⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⣵⠶⢦⠀⢀⣴⠟⠒⠉⠉⠀⠀⠀⠀⠀⠖⠃⠀⣂⡀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠐⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠤⣄⠁⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠈⠉⠙⠫⡉⠉⠢⡄⠀⠀⠀⠀⢀⡠⠛⠂⠐⠳⡦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡚⠟⠋⠻⣄⡀⠀⠀⠀⠖⠛⠀⠁⠉⠀⣀⣤⠰⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡀⠀⣌⠀⠀⠀⠘⣭⡓⣤⣬⡤⠤⠤⢴⣂⡉⠔⡁⠷⣄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠳⣤⠀⠀⠛⢄⣹⠀⠀⠀⠀⠀⠉⠛⠻⢄⣀⠀⠀⠀⡠⠊⠀⠀⠈⢳⡀⢀⡜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢤⠀⡠⠝⠁⠀⠀⢀⣀⠑⠲⠤⢤⣀⡀⠀⠀⠀⠀⠉⠑⠊⠀⠀⠀⢀⡴⠋⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢄⡁⠀⢛⡃⠀⠀⠀⠀⠉⢳⠢⢄⣀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠵⠦⠀⠀⠀⠓⠂⠊⠀⠀⠀⠀⠀⠀⠘⡖⠲⠼⠙⠒⠒⠙⠉⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⣀⣰⣾⠛⣷⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠈⣿
⠀⠀⠈⢿⣤⣤⣤⡶⠶⠶⠾⢿⡟⠛⢿⡿⠶⠶⠶⣦⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⢀⣀⣀⣀⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⣿⠋⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠻⠶⢶⣤⣀
⣿⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⣦
⣿⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠛⠛⠛⠛⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠻⠶⣦⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠙⠳⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠟⠁⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠉⠉⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡶⢶⡶⢶⠶⣶⠶⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠷⣦⡘⠟⠘⠇⠸⠃⠻⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⣿⡶⠰⡆⢸⠆⢶⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⣀⣴⢿⣤⢠⡄⢠⡄⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⡁⢈⡁⢈⠁⣉⠀⣉⠀⠀⠀⣠⡶⠛⠷⣦
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣾⣧⣜⡁⠛⠀⠛⠀⠀⠀⣿⡀⠀⠀⣿⠇
⠀⠀⠀⠀⣼⠏⠁⠀⠀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠷⣶⡾⠋⠀⠀⠀⠀⠉⠻⣿⠿⠿⢶⣄⠀⠈⠻⠶⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣠⡿⠀⠀⢠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠙⠻⠷⠾⠟⠋⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢿⣄⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⣿⣥⣀⡀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣶⣾⠋⠉⠙⢷⣄⠀⠀⠀⠀⠈⠻⠶⢶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⣠⡶⠶⠶⠟⠋⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣶⠟⠋⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢶⣤⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠈⠉⠉⠉⠉⠉⠉⠁⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠈⠉⠉⠀⠉⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⠜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠱⡄⠀⠀⢀⡖⢉⡉⠱⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣤⣀⠀⠀⠀⠿⠞⠁⠀⠀⡀⠀⠀⠀⠙⠦⡠⠞⠠⡋⠉⠁⡇⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⢳⡀⠀⠀⠀⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⢀⡇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⣀⡼⢣⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻⣯⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣼⠀⠀⠀⢀⡴⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⠦⠴⠖⡋⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠟⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢰⣮⢭⣿⠖⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⠟⠀⠀⠀⠀⠀⠀⡿⢤⣀
⣀⣀⣀⣀⣀⠠⠤⠤⠤⠶⠤⠶⠒⠒⠒⠒⠒⢂⠈⠳⣄⠉⢻⠈⣠⡿⠁⠀⠀⠀⠀⠀⠀⣠⠞⣰⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣀⠀⠀⠀⠖⠉⠁⠀⢀⠈⠳⣌⡛⠋⠀⠀⠀⠀⠀⠀⢀⡼⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⣠⠖⠉⠠⣄⡈⠒⠤⣌⠙⢦⠀⠀⠀⠀⣠⠏⠀⠀⣀⣹⡦⣀⡀⠀⠀⣠⠔⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣠⠞⠁⠠⢄⣀⠀⠉⠳⣄⠈⣧⠠⠁⠀⠀⡴⠃⢀⡤⠞⠉⡜⠀⠈⠉⠛⡿⡁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⣿⠀⠀⠀⠀⠀⠙⢦⠀⢨⠄⠱⠀⡇⢀⡞⠀⣴⠋⠀⠀⢰⠃⠀⠀⠀⠀⢇⢻⡎⢀⣠⣄⣀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢿⠀⠀⠀⠙⣷⡀⢸⠀⢸⣀⣸⡖⠡⠋⠀⡞⠁⠀⠀⠀⢸⢀⡴⠊⠉⠓⢾⣾⠉⠁⠀⠀⠉⠳⣄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠈⡆⠀⠀⠀⢸⡗⠘⠓⠋⠉⣸⠇⠀⠀⡼⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⣆⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⣄⠀⠀⠸⠇⠀⠀⠀⢰⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡰⢷⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⠞⠁⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⣴⠛⣀⡤⠞⠁⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠿⠚⠋⠀⠀⠀⠀⣠⠴⠋⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⣀⡴⠛⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⠘⠶⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⡀⠀⠀⠀⣴⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⠴⠋⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣛⡛⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠷⢣⡖⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠏⠉⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡍⠿⣶⣅⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠈⢢⡀⠁⠈⠉⠻⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢢⡀⠉⢲⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠈⣇⠙⢦⡀⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢱⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠁⣿⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⢯⠸⡀
⠀⠀⠀⠀⠀⢠⠃⢸⣿⠀⠀⠀⠀⠀⣸⣠⣀⣤⠖⡀⠈⡇⡇
⠀⠀⠀⠀⠀⡜⢠⣿⣿⠀⠀⠀⠀⢠⣭⣽⡧⠔⣹⠇⣠⡇⢇
⠀⠀⠀⠀⢀⡇⠘⢻⣿⠃⠀⠀⠀⠀⡿⠛⣞⣿⣵⡿⠁⢿⣾
⠀⠀⠀⠀⠘⡇⣦⢸⣿⡀⠀⠀⠀⢠⣇⠀⠀⣼⡯⠃⠀⢸⢻
⠀⠀⠀⠀⠀⣷⣿⡄⣿⠗⣉⡑⠛⢦⣹⠀⢠⣿⣷⡀⢀⣼⠆
⠀⠀⠀⠀⠀⡘⢻⣿⣿⣾⣯⠍⠙⢦⡻⠀⢸⡈⠓⠞⠒
⠀⠀⠀⠀⢀⡇⠀⠹⣿⣿⢹⡆⠀⠈⠁⠀⠈⡇
⠀⠀⠀⠀⠈⢧⣳⣤⣹⣿⣆⣇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⣶⣾⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢂⠈⠻⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⠆⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡻⢿⣿⣿⣧⣤⡙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡳⣍⠀⠉⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⢷⣦⠀⣿⡿⠿⠿⠀⢸⡿⠿⢿⣦⠀⣴⡾⠛⠻⠇⠀⣴⡾⠛⢿⣦⡀⢰⣿⣧⡀⢸⣷⠀⠀⣴⣿⣧⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣾⠟⠀⣿⡿⠿⠗⠀⢹⣿⣶⣾⠋⠀⠙⠿⢷⣶⣤⢸⣿⠀⠀⠀⣿⡇⢸⣿⠹⣷⣼⣿⠀⢰⣿⣉⣿⣆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣷⣴⡦⠀⢺⡿⠀⠻⣷⡄⢶⣦⣤⡾⠟⠀⠻⢷⣤⣾⠟⠁⠸⣿⠀⠘⢿⡿⠀⣿⠏⠉⠹⣿⠄⢿⣿⣦⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⣀⡀⠀⢀⣀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⣀⡀⠀⠀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣿⣏⠀⠻⣷⣠⣿⠋⢀⣾⡟⠉⠉⠻⠀⣿⣟⣉⣉⠁⠈⢻⣿⠁⠀⣿⣿⣦⠘⣿⠀⢸⣿⣋⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⣿⡿⠀⠀⢹⣿⠃⠀⠘⣿⡄⠘⢿⣿⠀⣿⡟⠛⠋⠀⠀⢸⣿⠀⠀⣿⡇⢻⣿⣿⠀⢸⣿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠋⠀⠀⠘⠛⠀⠀⠀⠙⠻⠿⠿⠛⠀⠛⠛⠛⠛⠂⠘⠟⠻⠓⠀⠛⠃⠀⠙⠛⠀⠘⠻⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣤⣤⣶⡶⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⠈⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣤⣾⠋⠀⣾⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢹⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠐⢿⡅⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⢀⡿⠐⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⡶⠟⠁⠀⠀⠈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⣀⠀⠀⠀⠀⣿⠀⣿⠛⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⢰⡄⠀⠈⠙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⣠⡾⣿⡟⠀⠀⠀⠀⢿⣄⠙⢷⡀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢸⡇⠀⠀⢀⣴⠄⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⣰⠟⢠⡟⠀⠀⠀⠀⠀⠀⠹⣧⠀⠻⣦⠀⠀⠀⠀⠙⠷⠄⣀⣬⣄⡀⠰⠟⠁⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡆⢀⣀⣀⣀⠀⣼⠏⠉⠉⠻⣆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⢾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⡇⠈⠛⠛⠋⠀⢿⣄⠀⠀⣰⡟⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢀⣴⠄⠙⠛⠛⠋⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⣀⣤⡶⠟⠛⠛⠛⠛⠛⠛⢻⣦⠀⠀⠀⠀⠀⠈⠛⠷⣦⣽⣷⣶⡄⠀⣼⡇⠀⠀⣿⡇⠀⠀⠰⠟⠁⠀⠀⢸⡇⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠘⢿⣥⣤⡾⠛⠻⣿⠛⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠈⠁⠀⠀⢀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠛⠷⢦⣄
⠀⠀⠀⣨⣿⣀⣀⣀⣀⣀⣀⣀⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠙⣷
⠀⢀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⣿⡟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣤⣄⠀⠀⢹⡇
⠀⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⣷⠀⠀⣀⣤⣤⡾⠛⠛⠛⠛⠳⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢸⡏⠀⢹⡇⠀⢸⣇
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⣼⡏⠁⠀⠀⢀⣤⣤⣄⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡇⠀⢸⡇⠀⢸⡇⠀⢸⡟⠻⣄
⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣶⡶⢿⡅⢀⣿⠶⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠛⢷⡄⠀⣿⠀⢸⡇⢠⡿⠁⠀⠘⣷⠀⢸⡇⢠⣿
⠀⣿⠀⠀⠀⠀⢰⡟⠉⠙⢷⡄⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠈⠙⠛⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⣀⣼⡇⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⣹⡇⢸⡇⢸⣿
⠀⣿⠀⠀⠀⣤⣼⣧⣤⣤⣾⣥⡄⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠛⠛⠛⠛⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡟⠋⣹⡇⠀⣿⠀⢸⡇⠘⣷⡀⠀⢠⡿⠁⢸⡇⠘⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡁⠈⠛⠛⠋⠀⣿⠁⠀⣿⠀⠀⠀⠀⠀⣀⢸⡿⠟⢹⡇⠀⠻⣦⣼⡇⠀⢸⡇⠀⢸⡇⠀⢸⣧⣼⠟
⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⣿⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⠀⠀⠀⢀⡾⠋⢸⣷⠶⢻⡇⠀⠀⠈⢹⡇⠀⢸⣇⠀⣸⡇⠀⢸⡏
⠀⣿⠀⠀⠀⠀⠛⠛⠁⠘⠛⠛⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁⠀⠸⠿⣶⠾⠇⠀⠀⠀⢸⡇⠀⠈⠛⠛⠋⠀⠀⣸⡇
⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣿⡇⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⡿
⠀⢻⣦⣤⣤⣤⣶⣤⣤⣶⣤⣤⣤⣤⣤⣾⡏⠀⠘⢷⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠋⠀⠀⠀⠀⠀⠀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣦⣶⠞⠋
⣿⣿⣿⡿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⠆⢸⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠠⠴⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⢤⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠂⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣯⠉⠩⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⡄⠀⢿⠀⢤⡀⠀⠀⡿⠑⠀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⢀⠈⠖⢨⣇⢸⠁⠀⠀⡹⠀⠚⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠉⠉⠠⣬⣶⢹⣧⣠⣤⠀⣧⠀⠀⢈⣿⡏⠀⠀⠠⠀⠀⠀⠀⠀⣤⠀⢸⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠉⠉⠉⠁⠉⣶⣿⣿⠇⠀⢠⡂⠀⠀⠛⢿⣷⡀⠀⠀⠀⠐⠀⢰⠂⡟⢀⣿⠇⠀⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⣱⣶⣽⣆⡀⠀⠀⢀⣄⡀⠀⢀⣼⣧⣀⡴⠟⢀⣠⣤⡤⠈⠹⣿⣦⣄⠀⠀⠀⠸⣷⡿⠀⡿⣀⣶⡏⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⡾⣏⣴⡿⠛⠛⡿⠿⠋⠀⣤⣾⣿⡿⠿⣦⣄⡄⠈⠛⣻⠶⣦⣄⡀⠄⠚⣿⣿⣇⠤⠔⠛⠁⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣶⣤⠴⠂⠈⠉⠁⠀⢠⡿⠄⣤⣾⡿⠋⠁⠀⠀⠀⠉⠛⠿⢦⣅⡄⠈⢻⣿⣶⣿⣿⣿⠇⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠼⠷⣤⣲⣦⣤⠔⠒⢉⣰⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠿⣤⠝⢿⣿⠛⡷⠶⢄⡙⠋⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀⣤⠶⠶⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠒⣿⣆⢁⠀⠈⠙⠓⠦⣴⡀
⠀⠀⠀⠀⠀⠀⠡⠶⠄⢀⣠⠀⠀⢠⣼⠁⣠⠖⢻⢿⣿⣅⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⠀⢿⣀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠋⠀⠠⠋⣥⡔⣡⣼⣿⢲⡿⢿⠉⠙⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⡿⠦⢻⡟⢛⠓⣦⣤⣤⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⣠⡟⣰⠁⣹⡯⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠇⠀⣸⣷⣾⣿⣄⣿⣿⣇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣠⠄⠸⠈⢁⣟⣿⣿⣹⣿⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⢿⡀⣸⡇⣿⠃⣿⡏⠸⡍⠙⣢⣈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⠈⡶⠁⢃⠋⡍⢡⣟⠻⡟⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣸⠟⢹⡗⢻⡈⢻⣿⡀⢱⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠘⢣⠞⡽⠙⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⠀⠟⠣⠘⣧⠘⠉⠷⠀⢲⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⢠⡿⠀⠀⡆⠺⡙⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠜⠀⠀⣰⠃⠈⡁⠁⠀⠀⢠⠌⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠈⠇⠀⠀⠀⠀⠷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢌⠀⠁⢈⣵⡄⠀⠐⠀⠀⣄⠘⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣷⡿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿
⣿⣿⣿⣀⣀⣤⣤⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⠀⢲⣿⣿⡧⠀⠁⠉⠛⠀⠀⠒⠀⠀⠀⠚
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠉⣾⣿⠇⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⣀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣠⠋⠀⠀⠀⠀⠄⠀⠀⠀⠐⠻⠍⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠁⠢⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣿⡿
⠄⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⠁⠀⠀⠀⠑⡀⠀⠀⠀⠠⣀⣀⡈⠁⠀⠀⠀⠀⠀⠤⠀⣰⣦⣤⣷⣶⣤⣾⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⢳⠎⠈⠀⠁⠀⠀⠄⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣿⣿⣿⣶⣶
⠧⠀⠀⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠙⠂⠀⠀⢀⠴⠀⠀⠐⢻⣿⠧⠉⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⣘⣧⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠒⣤⡀⠀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠸⡌⠁⠉⠀⢀⠱⢤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣼⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⡀⠀⠈⠉⠛⣿⣿⣷⢠⣿⣶⣾⣧⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⣿⣿⠿⠿⠯⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠐⢃⠀⢀⠆⢀⠠⡤
⠀⠄⠀⡀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣶⠿⣷⡆⠠⡀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⡀⠀⠀⠀⠠⠀⠉
⠀⠀⠀⠻⣄⠀⠀⡀⠀⠘⠀⠂⠀⠀⠀⠀⣀⡈⠉⡤⠶⢋⠄⡀⠈⠳⡄⠀⠀⠀⠉⠉⠂⠀⠂⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠒⠃
⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠁⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠒⠻⢷⣶⣆⣀⠀⠀⠀⠀⠀⠠
⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⣀⡀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠀⠀⢀⠀⡀⠀⠀⣠
⠀⠀⠀⠰⠠⢄⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣷⡀⣸⣧⣥⣴⡿⠛
⠤⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢳⣿⡀⠀⠀⡰⠃⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⢿⣿⣆⠐⢀⠀⢀⣰⣿⡿⠋⠀⠀⠀⠀⢀⡀⠀⠾⠛⠝⢶⡠⠀⢀⣀
⠤⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠉⠑⠢⡀⠀⠘⠿⣿⣿⠾⠀⣼⣿⣟⡁⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠓⠦⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⢘⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣦⣬⣋⡰⠤⢙⣿⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠒⠉⠉⠩⠝⠛⠛⢻⡿⢿⠁⣰⡔⢩⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢰⣲⣀⠠⠜⠣⢄⣀⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣀⢤⠐⣓⠠⢄⡰⠆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢰⣡⢘⡡⡯⢘⣣⣵⡀
⠀⣏⠶⡁⠀⠀⠀⠀⠀⠀⢈⠶⡱⠀⠐⢷⠔⢺⣏⠉⠁⡢⣾⠂⠀⠰⠒⢇⣬⣨⡑⣖⣬⠱⠆⡗⠺
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⠃⣴⡤⠀⠀⠈⠋⠈⡄⠀⠀⠀⠘⣤⠾⢠⡝⣤⣛⡜
⠀⠀⠸⠽⠉⠒⢢⡔⠒⠉⠯⠇⠀⠘⢻⣤⡂⠁⠐⠽⠀⢢⣄⡟⠛⠀⠀⠛⠀⠉⠚⣖⠊⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠓⢋⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⣦⠆⠀⢈⣝⠀⠀⠀⠀⠀⣠⢤⠀⠉⢸⡽⠂
⠀⠀⠀⠀⢦⠀⢀⣸⣀⠀⢀⠆⠀⠀⠈⠋⠈⠑⢻⡖⠊⠁⠙⠁⠀⠀⠀⠀⠈⡽⠉⢝⣱⠈⠑⠤⡮⠧
⠀⠀⡀⠀⡠⠋⡧⢰⠆⣭⢣⡀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣔⣢⠜⠀⠀⠈⠈⠀⢀⠀⠇
⠀⠀⠉⢹⠙⠃⠶⣙⡁⠶⣛⢻⠉⠁⠀⠀⢀⡀⠀⣀⡠⠻⢄⣀⠀⢀⠈⠀⢇⠀⢀⣄⠀⠀⠺⣋⡯⢞⡇
⠀⠀⡤⠼⡟⢑⡗⠨⠜⢣⡍⡸⠤⡄⠀⠀⠘⡏⠉⠀⠀⠀⠀⠀⠙⢹⠁⢔⡾⠤⣑⠛⠀⠀⣀⡜⠁
⠀⠀⠀⠀⢈⠮⢅⣗⣘⠧⢎⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠁⠀⢀⠵⡏⠉⠀⠯⠃
⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢀⡃⠀⠀⠘⠟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠻⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡔⡆⢀⣀⠧⡏⠀⠀⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠄⢤⠀⡄
⠀⠀⠀⠀⠀⠀⡨⠋⠱⢭⡆⠈⠑⢺⠬⠂⠰⠕⠢⢄⣀⠀⣀⡠⠔⢾⠆⠀⣄⡹⢚⡉⢭⠍⣓⢎⣠⡀
⠀⠀⠀⠠⣋⢖⠁⠀⠀⠀⠈⠀⡀⠈⡄⢀⠀⠀⠀⠀⠈⠾⠀⠀⠀⠀⠀⠖⡞⣤⢈⡁⠶⢀⣉⠴⢧⠴
⠀⠀⠀⠀⠀⠈⡄⢀⣴⡄⠀⠈⠺⢽⠘⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⡤⣨⡁⢟⢠⡍⠶⢸⠒
⠀⠀⠀⠀⠰⣊⠗⠪⣒⠁⢀⣀⣠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⣤⡾⢠⡝⢡⡘⣣⢎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢱⠚⡥⢺⡋⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⠀⢰⠆⢃⠀⢀⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠘⡚⣏⡄⠈⣇⡇⠀⣜⡟⡚⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣰⣼⣦⣤⣽⡀⠀⢹⠀⠀⣸⣥⣤⢾⣴⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⢿⣿⣿⣷⣿⣿⣶⠞⢶⣿⣿⣷⣿⣿⡿⠧⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣩⣽⣿⣿⠁⠀⠀⢹⣿⣿⣭⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠿⢛⣯⣿⣿⣿⡿⢂⠀⢀⡺⣿⣿⣿⣯⣟⠻⠿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠰⢀⣰⣯⣿⣿⣼⣧⠛⠀⠘⢡⣿⣼⣿⣏⣷⣄⠰⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣾⣿⣿⣿⡻⣿⠟⠀⠀⠀⠘⢿⡿⣻⣿⣿⣿⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡍⣍⣽⣇⠂⠹⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠀⠊⣿⣍⡌⣅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣛⠃⠈⠂⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠊⠀⢛⣧⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓

⣷⣄⠀⠀⠈⠻⣿⣿⣿⠆
⣿⣿⠗⠀⠀⠀⠙⠛⠁
⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⢀⣴⡦⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⢀⣴⣌⢉⣴⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣆⡀
⣿⣿⣿⣿⣿⣿⠟⠋⢿⣿⣦⠀⠀⠀⠙⢿⣿⡿⠃
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠁⠀⠀⢀⣴⡿⠋⣠⣄
⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⢋⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠈⠋⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣷⣄
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⣴⣿⣿⠿⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠙⢿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡤⠖⠚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄⢀⡴⠛⠙⠞⠋⠁⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠏⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡤⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⠋⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⢻⣠⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡞⣡⠀⢻⡄⠀⠀⠀⠀⢸⡁
⠀⠀⠀⢀⠀⠀⠀⠀⠘⢧⡀⠀⠸⡆⢀⡖⣀⠀⠀⠀⠀⢹⣸⡇⣿⠃⣸⠇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠓⠀⠀⣿⠸⠋⠁⠀⠀⠀⠀⠀⠉⠀⠈⢸⡷⡄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢻⠀⠀⠀⢀⣼⡷⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⡄⣀⡤⣤⠀⠀⠀⠀⠀⢠⠏⠀⠘⠶⢛⡭⠟⢁⡤⠾⠓⠦⣄
⠀⠀⠀⠈⠉⠓⠶⢤⠀⠀⠀⠀⣠⢤⣀⠉⢩⠖⠃⠀⠀⠀⠀⣰⠏⠀⢀⣠⠞⠋⣠⠞⠉⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢹⡴⠋⠳⣌⠉⣷⠀⠀⠀⣰⣯⡤⠖⠋⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⣸⣥⡶⠀⠘⡆⠙⠦⠤⠞⠁⢸⡆⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦⣽⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡼⠻⡄⠀⢠⡿⡄⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⢀⡾⠀⣷⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠚⠋⠀⢀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠓⠲⣄⠀⠀⣸⣿⣿⡀⠀⢀⡴⠚⠛⠛⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⡀⠀⠀⢸⣷⣾⣿⡿⣿⣿⣶⣿⠀⠀⠀⡀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⣠⠈⠙⢛⣶⣶⡿⡿⠋⠁⠀⠀⠉⠻⡿⣷⣶⡞⠛⠉⢠⣀⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⠀⣤⣬⣿⡶⠚⣹⣿⡟⠀⡇⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⡛⠲⣾⣯⣤⡄⠀⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡽⠛⠉⣴⡏⠀⣰⠟⣰⣇⠀⡇⠀⠀⠀⠀⠀⠀⡇⢀⣿⡘⢷⡄⠈⣷⡌⠙⠻⣅⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣀⠀⠘⠁⠃⣰⢿⡄⠉⡟⠂⡇⠀⠀⠀⠀⠀⠀⡇⠚⡏⠁⣼⢿⡄⠃⠙⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢶⣄⣀⣭⡷⠶⠶⣺⠏⢰⣷⣄⡙⠀⡇⠀⠀⠀⠀⠀⠀⡇⠘⣁⣴⣷⠈⢿⡒⠶⠶⣿⣅⣀⣶⠆⠈⢧
⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⣴⠋⢡⠿⠄⠀⣰⡇⠀⠞⠈⣧⡉⠀⡇⠀⠀⠀⠀⠀⠀⡇⠈⣡⡏⠘⠆⠀⣷⡀⠀⠼⢧⠈⢳⣤⠀⠘⣇
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣀⠈⠀⢀⣴⡇⢹⣄⠀⠀⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠉⠁⠀⢀⣼⠁⣷⡄⠀⠈⢀⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠲⠦⣽⡦⢶⡟⢁⡇⠀⢻⡛⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠚⣻⠃⠀⣇⠙⣶⠤⣾⡵⠶⠂⠀⠀⢷
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠶⠞⠁⣠⡏⠀⣼⣷⡀⢰⡟⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢠⠞⣿⠀⣰⣿⡄⠈⣧⡀⠙⠶⠆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠴⣾⠋⣇⠀⢁⡏⠁⠈⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣈⠀⠈⠀⠉⣇⠁⢀⡏⢻⡶⠤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠾⠃⠀⠈⣀⣨⡥⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⣭⣀⡈⠀⠀⠻⠆⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠶⠚⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠠⠤⠤⠀⢀⣀⣀⠀⠀⠀⠀⠀⠸⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⣀⣠⡄⠀⠒⠚⠃⠀⠁⠀⢀⠀⠀⠸⣅⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠉⠉⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⠀⠐⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠠⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠋⠀⠀⠢⣦⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠉⠢⡈⠲⣌⣦⡖⠠⢥⡒⠷⠠⠀⠀⠈⣶⠏⠀⠀⠀⠀⢁⠀⠁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠀⣻⣋⠐⠓⢠⣐⠁⠈⠆⠀⠂⠀⡀⠻⠀⠀⠀⠀⠀⠘⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⡁⠑⢌⡀⣈⠬⠘⠂⢬⠀⢀⡼⠀⠐⣇⠀⠀⠀⢸⡔⡀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠌⢀⠄⡞⠀⠒⠀⠠⠤⠂⠀⠀⠊⠉⢀⠛⠀⠀⣤⠆⢀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠈⡔⢀⡀⠛⠀⠀⠀⠀⠀⠀⢀⠒⠂⠐⠁⠀⠡⡐⠈⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢷⠊⠉⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠐⠒⠒⣲⠚⣯⡀⠁⠱⠒⠁⠁⠺⠀⠂⠀⠀⠠⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠭⠧⡐⠀⠀⢃⣘⡀⢬⠈⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⢪⠁⠀⠔⠢⠤⠅⠀⠈⠙⡛⠁⢘⣀⠤⡐⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⠚⠦⠤⠄⠓⠶⠤⣲⡄⢠⠄⠀⠄⠠⡤⣀⡉⠀⣀⣡⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠄⣰⡃⠠⠀⠉⠐⠺⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡀⠀⠀⠈⠸⡁⠈⠈⠀⠤⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠈⢄⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠈

⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄
⢀⣰⣿⣷⣷⣤⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣠⡆⠀⣴⣶⣧⣤⣶⣾⣷⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣴⣶⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣶⢀⣰⣤⣤⣾⣿⣷⣿⣴⣶⣶⣦⣷⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄⠀⢸⠃⢀⡤⢐⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⣛⣿⣿⣿⣷⣄⣰⣯⣾⣿⣛⣉⣙⡆⠀⠒⢤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡩⠒⠂⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⣦⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣿⣿⣿⡿⠟⣏⢈⠀⠙⢿⣿⣿⣿⣿⠻⠿⣿⣿⣶⣄⠙⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠊⢩⣿⢛⣽⣿⡿⢻⣠⠞⡏⢸⠀⠀⠘⣿⣿⠹⣿⡄⠘⠾⡙⢿⣿⣷⣤⣄⣀
⠀⢇⠈⠉⠐⠒⠈⠁⠀⠀⠀⠀⠀⡟⢁⣾⣿⡿⢁⢸⠫⠔⠁⣀⣤⠀⠁⢸⠏⠄⡿⢘⣀⠘⢹⠀⢻⣿⣿⣆⠈⠙⠒⠤⣀⡀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⡇⠀⠃⠀⣴⡿⢟⣋⣁⡀⠀⠀⠀⠀⠹⣿⣷⣤⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠄
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢠⡞⠍⠎⠂⠍⠀⠀⠀⠀⠀⠸⡹⠏⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⡤⠀⣿⣿⣿⠇⠀⠀⠀⣾⠁⠀⠈⣈⣠⠄⠀⠠⣦⠀⠀⡇⠀⠀⠀⣿⣿⣿⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠠⣃⠎⡠⢶⡿⠿⣟⠀⠀⠀⠀⠸⣆⠀⠁⠀⠁⠀⣀⠀⠀⢀⡴⠁⠀⠀⣸⣿⣿⡟⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⡾⠀⢀⣃⠀⠀⠀⠀⠀⠀⠐⠬⣭⠶⠋⠀⠀⠈⣳⠾⠯⠔⠁⠀⠀⢻⡙⡛⡙⢳⠀⠷⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣇
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⡗⢹⢸⡇⠒⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⢀⣼⠃⠀⢉⡼⠁⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣴⡿⣄⠀⠀⠀⠀⣀⣴⡶⠖⠲⠴⠚⠁⠀⠀⣀⣤⠶⣿⣥⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⠀⢀⣀⠤⢜⣫⠭⢑⣰⠋⢀⣀⡀⠀⢀⣤⣶⣾⣿⣿⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠃⠀⠀⠀⣼⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⢀⡠⢴⣮⡥⠖⢊⡡⡴⠾⠛⠛⠛⠻⡟⠀⠀⡾⠁⠀⠉⠻⣿⠀⠀⠀⠈⠑⠲⢄⡀⠀⠀⠀⠀⢠⢣⠇⠀⠀⠀⢀⡿
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠆⠱⠶⠞⣡⠴⠊⠉⠀⣳⠟⠈⣹⣷⣄⠃⢀⣎⣴⣿⠉⠓⠚⢃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣞⡞⠀⠀⠀⠀⣼⠇
⠀⠸⣷⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⢠⠾⣫⠊⣴⣿⡟⢙⣧⣾⠁⢿⣿⡆⠀⣄⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣸⡏
⠀⠀⣿⡄⠀⠀⠀⢀⣀⣀⡠⠒⠒⢄⠀⡔⠁⡼⢁⣼⣿⡿⠁⢸⣿⣿⠀⠸⣿⣿⡀⠘⣆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣰⡟
⠀⠀⠸⣧⠀⠀⢰⠁⠀⣼⠁⠀⢀⠀⠀⢌⣸⠁⣼⣿⣿⠇⠀⣾⣿⣿⠀⠀⢻⣿⣷⡀⢘⣇⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⡟
⠀⠀⢀⣿⡦⠶⢾⠀⠀⢸⡄⠀⢸⠀⠀⢸⠉⠉⠉⠉⠙⠒⠦⢿⣿⡿⠶⠚⠋⠉⠉⠉⠉⠉⠙⠓⠚⠷⢤⣀⠀⠀⠀⠀⠃⠀⢀⣾⠏
⠠⠖⠉⠀⠀⠀⠀⠳⣄⡠⠷⢄⡼⠓⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⠁
⣿⠿⠿⠿⣿⢿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣻⣟⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⠻⢳⣤⡿⠟⠟⠻⠛⠛⠛⠛⠻⠛⠛⠻⠛⠛⠛⠛⠛⠻⠻⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠨⠀⣤⠭⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⢄
⠀⢠⠄⠀⠀⠀⠤⠊⠉⠑⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡁⠀⠐⠠⠤⠐⠀⣀⠤⠐⠀⠈
⠈⠘⠀⠀⠀⠀⠢⣀⣀⡠⠓⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠐⢖⢒⠒⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠀⠀⠀⠀⠀⠀⠀⡀⠂⠄⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠄⠀⠀⠀⠀⠙⠲⠤⣤⣀⣸⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠄⠀⢀⠄⠀⠐⠀⠈⠓⠎⣀⡀⠀⠈⠀⣠⣈⠀⠀⡀
⠀⠀⠀⢀⠀⠆⠀⠐⠂⠀⠀⠀⠈⠀⠀⠉⠀⡀⠂⠀⠀⠘⠁⢄⣰⠁⠀⠉⠒⠤⡁⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⠂⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄⡀⠉⠒⠂⠍⣙⠒⠲⠤⠤⣄
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠆⠀⢀⡀⠤⠀⠒⠂⠁⠀⠀⡀⠈⠁⠒⠤⡀⠀⠈⠉⠐⠂⠊
⠀⠀⠀⠀⠀⠘⠊⠁⡀⢀⠀⢀⡠⠄⠠⠤⠄⠁⠀⠀⡅⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⡆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⢢⠒⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠔⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡔⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠉⡀
⠀⡤⠄⠤⠒⢪⠡⠉⡠⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁
⠀⠀⠂⣀⡹⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡰⠀⠂⠰⠀⠀⠰⠀⠚⠀⠂⠰⠀⠆⠰⢀⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⣧⣶⣴⣶⣾⣴⣧⣼⣶⣷⣴⢳⣿⣶⣧⣾⢿⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠤⢄⣀⣀⣲⡆⠴⠖⠒⡖⠒⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⢙⣻⠻⠿⠿⠏⠙⠋⠉⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇
⠀⠀⠀⢡⠀⢠⠖⠒⢤⠜⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡋⠈⠙⠋⠀⠀⠀⠀⠠⠶⠀⠀⠀⡀⢠⣄⠀⡇
⠀⠀⠀⠀⢆⠘⢄⠀⠀⠀⠀⠀⠀⠀⡠⠠⣆⠀⠠⠂⠈⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⣤⣜⡁⠀⠀⢰⡃⢀
⠀⠀⠀⠀⠀⠳⢎⠀⠀⠀⠀⠀⠀⠐⣇⡀⣘⠉⠑⠒⠤⣀⠞⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠿⣦⠾⣿⣿⣿⣷⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠉⢈⡹⠂⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣲⣤⡤⢤⣤⣤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣸⡓⠫⣹⣏⢧⠀⢀⡤⠊⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⣿⣿⣧⣙⣺⣿⣿⣿⡿⢟⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠖⡀⢹⠈⠑⠉⠈⠆⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡇⢸⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⣀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠢⢤⡋⢐⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣧⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡏⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠛⠻⠿⠿⠟⠛⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠂⠀⠀⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⢣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢟⣿⢙⡟⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠎⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⣿⡷⠃⢸⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢓⡒⡴⠋⠀⢨⣑⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⠏
⠉⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⡄⠀⠀⢿⣿⣿⠀⠀⢠⣿⣿⡿⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿
⣦⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠘⢿⣿⣷⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁⠀⢀⣾⣿⣿⠟⠁⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣶⣄⡀⠀⠀⠀⠈⠙⢿⣿⣿⣷⡀⠀⠈⢿⣿⣧⠀⠈⣿⣿⠀⠀⣿⣿⠇⠀⢠⣾⣿⡿⠋⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠈⢻⣿⡆⠀⢻⣿⠀⢠⣿⡟⠀⢠⣿⣿⠋⠀⠀⣠⣾⣿⣿⣿⠿⠛⠁
⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠻⢿⣷⣄⠀⠻⣿⡀⢸⣿⠀⣸⡿⠁⣰⣿⠟⠁⢀⣤⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣷⣦⣀⠀⠙⢿⣷⡀⠹⣷⠀⣿⠀⣿⠃⣴⡿⠃⣀⣴⣿⠿⠛⠉⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿
⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣦⣄⡈⠻⣦⡙⣇⢻⢰⡏⣼⠋⣠⡾⠟⠋⢁⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⣈⠙⠛⠷⢬⣑⢜⢼⢺⣞⠵⢛⣩⣤⣶⠾⠿⠿⠛⠛⠛⠉⠉⠉
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣒⣲⣗⢵⣷⢽⡯⠭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⡾⠿⠛⣋⣭⢖⡝⡧⡷⡕⢍⡓⢶⣤⣄⡉⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⢀⣠⣴⡿⠋⣠⡟⣰⠇⣧⢹⣌⠻⣦⡈⠙⠻⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛
⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⣀⣤⣶⣿⠟⠉⢠⣾⠟⢠⣿⠀⣿⠀⢿⣆⠈⢿⣷⣄⠀⠉⠛⢿⣿⣿⣷⣶⣤⣄⡀
⣿⠿⠟⠛⠉⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠁⢀⣴⣿⠏⢀⣾⡇⠀⣿⡇⠈⣿⣦⠀⠙⢿⣷⣦⡀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠋⠀⠀⣠⣿⣿⠃⠀⣼⣿⠃⠀⣿⣧⠀⠸⣿⣧⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠟⠁⠀⠀⣠⣾⣿⡿⠃⠀⢰⣿⣿⠀⠀⣿⣿⡀⠀⢻⣿⣷⡀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠛⠿⣿⣿
⣤⣶⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⢀⣴⣿⣿⡿⠁⠀⢀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⢿⣿⣷⡄⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⣾⣿⣿⠃⠀⠀⣿⣿⣷⠀⠀⠘⣿⣿⣿⡄⠀⠀⠈⠻⣿⣿⣿⣿⣶⣄
⣿⣿⡿⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣤⡀
⡿⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⢻⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⣄⠀⠀⠀⠀⠀⢀⣤⢶⣶⣿⢶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠓⢶⣤⡀
⠀⠀⠀⣰⣿⡏⣻⣿⣿⡮⣿⣿⣧⠀⠀⠀⢰⣿⣿⣟⡿⣿⣿⣯⣿⣷⠀⠀⠀⢠⣾⡟⠋⢌⢤⣷⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠛⡃⠀⠀⠀⣿⣿⣿⣿⣧⣿⠿⠿⠟⠻⠀⠀⠀⣼⣿⡶⣀⣷⣿⠿⠿⠟⢻
⠀⠀⠀⢿⣿⡟⠛⠿⠁⠀⢠⠄⢡⠀⠀⠀⢻⣿⣿⠛⠻⠃⠀⢠⠄⠰⡀⠀⠀⢸⣿⣿⠛⠻⠏⠀⠀⡄⠘⡄
⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⡏⠀⠀⡤⠤⠠⠜⠀⠀⠀⠀⠀⠀⡉⠁⠀⢠⠤⠤⠴⠁⠀⠀⠀⠀⠀⢹⠁⠀⢠⠤⠄⠤⠃
⠀⠀⠀⣀⣴⣇⣀⣀⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣇⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⢀⣠⣮⣀⣀⣸⣄⡀
⠀⠀⠚⠿⠿⠿⣿⠿⠿⠿⠛⠀⠀⠀⠀⠘⠿⠿⠿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠐⠻⠿⠿⢿⡿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⠲⠲⣂⡦⠽⠍⠚
⠀⠀⠀⠠⢀⠀⣀⡀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠇⡀
⠀⠀⠀⠘⠂⠚⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠈⠂⠈⠉⠉⠁⠉⠉⠁⠢
⠀⠀⠀⠀⢀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣀⡀⠀⠤⠀⣀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡔⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢄
⠀⠀⠀⠀⡆⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⢀⠠⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠑⠐⠐
⠀⠀⠀⠀⡇⠠⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⣇⢂⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡖⢉⡁⠀⠃⠀⣲⡀⠀⠀⠀⠀⠀⠀⢄⡐⠠⠂⠀⠤
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠐⡀⢁⣰⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠂⠀⠀⠀⠀⡄⢸⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠄⠠⠀⠐⠈⡄⠀⠈
⠀⠀⠀⠀⢄⣀⠀⠈⣀⢖⠁⠀⠀⢀⠎⢁⢀⠁⣐⡂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡭⠀⠀⢈⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠈⠙⠉⠀⠀⠀⠀⣠⠧⠆⠀⠀⠀⠀⠀⠀⠑⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢀⠁⢀⠀⣀⢀⡀⣀⠠⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⠃⠁⠁⠀⠁⠈⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠁⠀⠀⠀⠀⠔⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⠌⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠒⠀⢁⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣯⣶⣶⣮⣝⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⠉⠛⠿⣿⣻⣷⣮⣭⣑⣯⣿⣿⠿⣿⣿⣿⣾⣿⣿⣿⣯⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠊⠉⠉⠉⠙⠛⠛⠛⠓⠋⠉⠿⠽⠯⠷⢾⣿⠿⠿⠿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠖⠀⠀⣀⣉⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⣾⠿⢿⠿⠿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣦⣤⣭⣍⣁⣀⣀⣚⣛⣛⣛⣚⠲⠦⠤⣤⡤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠠⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⣤⣄⣀⠀⠁⠉⠉⠙⠋⠉
⣿⣿⣿⣿⣿⠿⢛⣻⣿⣭⣭⣭⣭⣽⣿⣷⣶⣤⣤⣴⣒⣒⣒⠲⠶⢶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣍⣛⣛⡛⠛⠦⠤⠤⠿⠿⠿⢿⣿⢶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⢸⡇⠀⡇⠀⢀⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣠⢀⣀⠹⡄⢸⡇⠀⡇⢠⠏⢀⡀⣄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⠹⣼⡇⠀⣧⡼⠀⠀⢳⣼⠀⠸⣧⠏⣀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⢻⣛⣻⣄⠀⣸⣀⡤⠤⣄⣇⠀⢠⣟⣛⡟⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣤⠀⠈⠁⢀⡼⠛⠹⡏⠀⠀⢸⠏⠙⢯⡀⠈⠁⠀⣤⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⡦⣭⣹⣦⡴⠋⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠙⢦⣴⣏⣭⢴⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣦⠴⢋⡴⢻⠁⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⠈⣟⢦⡙⠦⣴⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡞⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡟⠀⢳⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡼⣇⠀⠘⡟⠢⣄⢀⠀⠸⡇⢸⠇⠀⡀⣠⠴⢻⠇⠀⣸⢷⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⡤⣾⠁⠸⣆⠀⢹⡀⠈⠯⠀⢀⡇⢸⠀⠀⠹⠁⠀⡟⠀⢰⡇⠈⣷⢤⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢠⢷⡀⢸⠿⡆⠈⣇⠀⠀⠀⣸⠁⠈⣇⠀⠀⠀⣸⠁⢠⠟⡇⢀⡾⡆⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡾⠀⣷⠀⢀⣿⡀⣿⠦⢀⡴⠃⠀⠀⠘⢦⡀⠰⢿⢀⣿⡀⠀⣼⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣇⠀⠟⠦⠈⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠁⠰⠻⠀⣸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠉⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢤⡀⠀⣧⠀⠈⡇⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⠀⠀⠹⡀⡗⠀⠈⡇⣾⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠂⠀⠰⡇⣿⠀⠠⡇⢿⠀⠄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⠀⠀⠀⢤⢀⠐⠀⢸⠁⣿⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡀⠂⢸⠃⠀⠀⢸⠀⡇⠀⢸⡇⢸⡇⡀⠀⠀⠀⠢⠀⠊⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠐⠀⠀⠀⠀⠀⢸⠀⠁⠀⠸⡼⠁⠀⠀⠳⣼⠇⠀⡀⠀⠀⠀⢀⠘⠀⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠉⠒⠢⣄⠀⠈⢆⢀⣀⡄⡴⠊⠉⠑⢦⡀⢀⠠⠀⠄⠀⢠⠌⠀⢀⠠⠅⠀⠈⢣
⠀⠀⠀⠀⠀⡼⠁⠂⠀⠀⠈⠈⠀⠀⠈⠙⠒⡺⠋⣸⠁⡇⠀⠀⠀⢠⡇⡆⠉⠒⠒⠚⠋⠂⠒⠢⠤⡀⠢⡀⠈⢣
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣁⠔⠀⢀⡇⠀⡇⠀⠀⠀⠀⡇⢸⡀⠂⠄⢇⠀⠌⠁⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⢀⣀⡼⠀⢀⡇⠀⠀⠀⠀⢷⠈⡇⠈⠀⡀⠀⠠⠀⠐⠀⠈⠀⠀⠀⠀⠘⣆
⠀⠀⠀⡞⠀⠠⠀⠀⠂⠀⠈⠀⠀⠄⠂⠀⠀⣰⠃⠀⣼⠀⠀⠀⠀⠀⢸⡄⠹⣄⠀⢀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠠⠄⠸⡄
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡴⠁⠀⠠⡇⠀⠀⠀⠀⠀⠈⡇⠀⠈⠦⡄⠈⠀⠀⡀⠀⠀⠀⠌⠀⠁⠀⠀⢹⡀
⠀⢠⠁⠀⠀⠀⢀⡠⠤⠶⠒⠂⠖⠚⣹⠋⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠹⡗⠒⠠⠤⠤⠔⠦⡀⠀⠀⠂⠀⠀⣇
⠀⡏⠀⠀⠀⡚⠉⠀⠀⠀⠀⠈⢀⠀⠃⢀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡇⠀⠁⠠⠂⠱⠀⠀⠀⣀⡀⠀⠈⢀⠀⠠⠀⠀⠸⡄
⢸⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠂⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠲⠄⠀⠄⠀⢷
⡿⠀⠘⠀⠀⠀⡀⠀⢀⡀⠀⢀⠀⠀⠀⢀⠠⠀⡀⢠⠃⠀⠀⠀⠀⠀⠀⢹⡀⠀⠐⠀⢈⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸
⡇⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠂
⣧⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠲⠤⢄⣀⠀⠀⠀⠀⠀⢂
⣬⣷⣶⣶⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣶⣶⣾⣭
⣿⣿⣁⣍⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠟⠋
⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢢⣴⣾⣿⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠁⠀⢀⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡜⡡⠂⢀⠀⠀⣿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠘⠃⢠⠆⢀⠰⠈⠇⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢧⡄⠀⢀⠘⠀⠀⠀⢄⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣧⠀⡄⠀⠐⠀⠣⡀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠀⠀⣠⣾⣤⢴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣾⠶⣿⠿⠉⢿⡄⠙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠘⢦⡀⠀⢁⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠤⣦⣤⡴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣶⡟
⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣁⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠎⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⣠⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⣤⣿⠃⠀⠜⠁⠀⠘⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⣿⠋
⠀⠀⠀⢀⣄⠀⠀⠀⠛⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢳⡶⣤⡀
⠀⠀⢸⡏⠀⢰⣤⣤⣤⣼⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⢷⣴⣅⠀⠉⢀⣿⡇⢸⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⢹⡉⠀⠀⢸⡿⠃⣼⡇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣠⠟⠀⠀⣠⣶⡿⠟⠋⠁
⠀⠀⢀⣤⣤⣴⡾⠃⣿⠃⢀⣿⣿⠀⠀⣿⡆⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⣿⡇⠀⠀⣤⣄⠀⠀⢀⣼⡿⠋
⠀⠀⣾⠃⠀⠀⠀⠀⢻⡆⠘⣿⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣿⠀⡀⠀⢀⣿⠃⠀⣼⣿⠃
⠀⠀⠘⣷⡄⠀⠀⠀⠀⢿⡄⠹⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⣼⡟⢡⡀⠀⠀⢰⣿⠀⠙⢿⠛⠁⠀⣰⣿⡏
⠀⠀⠀⢿⣄⣴⠟⠻⡆⠈⠻⣦⡘⣧⠀⢿⡄⠀⢰⣿⠀⠀⠀⢰⣿⡁⣿⡇⠀⠀⣾⡟⢠⣤⡿⠁⠀⣴⣿⠿⣶⡀
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣼⡟⠛⢻⡇⠸⣧⠀⢸⣿⠀⠀⠀⢿⡏⠀⢹⡇⠀⢸⣿⣧⣼⠏⠀⣠⣾⠿⠋⠀⣿⡇
⠀⠀⠀⠀⣿⣷⣶⡟⠀⣼⡟⠀⠀⡈⢿⣄⣿⡄⠈⣿⡆⠀⠀⠸⣧⠀⠈⢿⣄⣈⡿⠀⠀⣠⣾⠟⠁⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⢀⣾⠃⠀⠉⠻⣧⡀⢿⣷⠀⠀⠀⣿⡆⠀⠀⠉⠉⣡⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⢆⣀⠀⠀⣿⣇⣾⢃⣼⠃⠀⢀⣼⡿⢾⣿⣧⡀⠀⢻⣧⠀⠀⢀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⣄⣠⡟
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿⠃⣾⠃⠀⢰⣿⡟⠀⠀⠀⠹⣷⡀⠘⣿⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⢰⡿⠋
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠈⠿⣾⡏⠀⣠⣿⠏⠀⠀⣴⠇⠀⢙⣷⣴⠟⠉⠀⠀⠀⠀⣤⠀⠀⠀⢀⣼⣷⠟⠁
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠀⠸⣧⣤⣶⠿⢛⠁⠀⠀⠀⠲⣤⡴⠏⠀⠀⢀⣼⠟
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣾⡇⠀⠒⠶⣿⠋
⠀⠀⠀⠀⠀⠛⣷⠀⠀⣶⠀⠀⠀⠀⠙⢷⣿⣀⣤⠀⣴⣶⡶⠏⠙⢷⣴⡾⠿⠋⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠻⣶⠾⠋⠀⠀⠀⠀⠀⢀⣈⣽⣿⣏⡉⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠛⠋⣉⣉⡛⠿⣷⣤⡀⠀⠀⠈⠙⠳⠶⠖⠋⠁
⠀⠀⠀⠀⢀⣤⢶⣄⢠⣿⣏⠀⠀⠀⠀⢦⣀⣿⠀⣉⣀⣀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠙⠷⠾⠏⢸⣿⠛⢷⣤⡀⠀⠀⠉⠀⢾⣋⣉⡽⠇⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⣷⣤⡀⠀⠀⠉⠹⠶⠾⣿⣻⡿⠀⢀⠀⠙⠻⣿⣶⣤⡀⠀⠀⠀⠀⣧⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠀⠀⠚⢡⡴⣶⣏⣿⠆⠀⠀⠉⠙⢿⣷⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡶⠛⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠉⠁⣀⣀⣠⣤⡶⠛⠉⢻⡆⠀⢀⣀⡀⠀⠀⠀⠀⢀⣴⠖⢶⡀
⠀⠀⠀⠀⢠⡾⠛⢶⣄⣠⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣤⣤⣶⡿⠿⠛⠉⠀⠀⠀⠀⢸⠃⡴⠉⠉⢻⣆⠀⠀⠀⠈⠃⠴⠾⠃
⠀⠀⣠⣤⣤⣿⣄⣀⣿⡷⠶⣦⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠴⢤⣾⡇⠀⢀⣀⣀⡀
⠀⠀⠘⢧⡽⠉⠉⠙⣿⠀⠀⢸⡿⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⠟⠉⠉⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠃⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠋⠁⠀⠀⠀⠀⠀⢿⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⣾⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⡶⠟⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣶⢸⣿⠛⠛⠀⣿⠟⢿⣦⢰⣿⡛⠷⠀⣴⡿⠛⢷⣆⢸⣿⣆⠠⣿⠀⢠⣿⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠋⢸⣿⠛⠛⠀⣿⠿⣿⡋⢈⡛⠛⣿⡜⣿⣄⠀⣸⡿⢸⣿⠹⣿⣿⢀⣾⣧⣼⣷⠀⣿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠛⠛⠛⠀⠛⠀⠙⠃⠈⠛⠛⠋⠀⠈⠛⠛⠋⠀⠘⠋⠀⠘⠛⠈⠋⠀⠀⠛⠃⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⢠⣄⠀⢀⣄⠀⣀⣤⣀⠀⢠⡀⢀⣀⣤⣄⢀⣄⡀⠀⣠⠀⣠⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⣿⠈⢻⣶⡿⠁⣾⠋⢉⣙⡁⢸⡇⢸⣿⣭⣭⢸⣿⣿⣄⣿⠁⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⠀⠀⣿⡇⠀⠻⣧⣬⣿⠇⢸⡇⢸⣿⣤⣤⠸⣿⠈⢻⣿⠇⣿⣧⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢾⣏⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡉⠈⠙⣿⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠈⢻⣦⣴⣿⣦⣄⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣈⡉⠀⠀⢸⡟⠁⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣦⣄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⢀⣄⠀⠀⠻⣷⡀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢰⡿⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⡄⠀⠸⣷⠀⠹⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣤⣀⣼⣿⣶⠾⠟⠛⠃⣼⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⡇⢰⣿⠀⠶⣿⣧⣤⣤⡄
⠀⠀⠀⢠⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢠⣿⡷⠟⠁⠀⠀⠀⠈⠉⠛⠿⣶⣶
⠀⢀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠛⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠋⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤
⠀⢨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⣷⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠛⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣿
⠀⠛⢿⡄⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣄⡀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡷⠀⠀⣸⣿
⠀⠀⢾⢿⣄⠀⠀⠀⠀⢸⣿⣀⣀⣽⡏⠙⢿⣆⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣼⡿⠛⠛⠛⠛⠛⢻⡟⢻⣿⠋⠀⠀⣴⡿⠛
⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠙⢷⣄⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣶⣶⣾⣷⣾⣿⣶⣶⡄⠉
⠀⠀⠀⠀⠀⠀⣹⡿⣶⠀⣾⠇⠘⣿⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣦⣀⣀⣀⣿⣀⣀⣀⣀⡀⢀⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢨⣿⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣾⣟⡉⠉⢉⣉⣩⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⣿⡇⠀⠀⠀⢀⣾⠿⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⣰⡿⠛⠛⠛⠛⢛⣿⡀⠘⠛⠛⠛⠛⠛⠛⠙⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣧⣀⣼⡇⠀⠀⠀⠀⣿⡇
⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣉⣉⣁⣀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠷⠿⠿⣷⡿⠶⠶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣶⣶⣶⣶⣶⣿⣧⣴⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣿⡇⠀⠀⠀⠀⠾⠿⠿⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠻⢷⣤⣄⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣿⠉⠉⠉⠁⠀⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠙⠀⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⣿⡇
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⣿⣶⣄⣀⣀⣀⣀⣀⣤⣾⠟⠛⠃⠀⠀⢿⣷⣤⣴⡄⠠⣤⠀⢴⣴⣤⣴⣤⣤⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣉⣿⣛⣛⣿⣋⣉⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣾⣿⣷⣦
⠀⠀⠀⠀⢀⠈⠄⢸⡇⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡿⠻⡆
⠀⠀⠀⠀⠘⡄⣀⣿⡗⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣾⡇
⠀⠀⠀⠀⠀⢈⠉⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣡⡀⠀⠀⠀⠘⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⡠⠀⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠸⣀⠀⢩⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠈⢆⠀⢧⠀⠀⠀⠀⢈⡆⠸⡄⠀⠀⠀⠀⠀⠃⠙⣀⡌⠀⠀⡤⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠀⠀⠈⠛⠓⢈⣠⠶⠋⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⢸⠟⠛⠛⡄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠠⢖⡖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣂⠠⠔⠛⢦⡀⠀⠘⢦⡀⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠂⢂⡴⠊⢹⠀⠀⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢄⠠⠇⠘⡄⢰⡛⢤⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡰⠁⡇⠈⠀⠠⠊⣺⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠃⠈⠁⠈⠀⠘⠢⠐⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢻⠀⠰⠙⠃⠀⠀⠀⣼⠀⠤⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃⠀⡆⡀⠀⠀⢢⠤⠊⠟⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⢀⠀⠨⠁⠄⠀⠂⢀⡄⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢧⡙⠒⠦⣄⢀⣠⠜⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡏⠀⠉⠀⠀⢠⠔⠛⡁⠐⢄⡀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠀⠀⠈⣦⠌⡉⠉⠒⠲⠦⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡘⠁⠀⠠⠀⠐⠁⢀⡀⣈⠀⠀⠰⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠴⢉⢧⡈⢐⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠠⠀⠀⠚⠁⠉⠀⠀⠙⠲⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣾⡇⠐⠁⠀⠀⠀⠀⠀⠀⠀⣙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⡀⢀⡤⠴⠒⠒⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠠⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⢀⠀⡀⠀⣀⠀⡀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠈⠈⠃⠉⠉⠀⠁⠁⠉⠈⠉⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢧⡀⠀⢠⡆⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠈⠛⠿⠋⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣦⣀⠀⢀⡀⣠⣾⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠉⢉⡡⣿⣯⣸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⡻⣿⠀⠈⣷⣿⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠡⣟⣿⡇⣷⡿⠀⠈⢻⡟⢸⠸⣿⠿⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⠶⠶⣦⡉⠀⣿⣷⡄⣨⡿⠋⢉⣤⡶⠾⠿⢒⢀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⠀⠳⠀⣹⡄⠙⠟⠛⣉⣴⠾⠛⣽⡆⠀⠀⣿⠈⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠰⢰⣿⣷⠀⣠⣞⠟⠁⠀⠀⠘⣷⠀⠐⣯⠄⠁⠀⠁⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣴⢹⡇⢠⣿⠃⢀⠰⠞⡻⠇⣿⣿⠀⣽⡆⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠠⠞⠃⣾⠁⣿⡏⠀⣮⡾⠉⣻⠶⣿⡛⢷⢻⡟⢃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣅⡋⠈⠀⠀⠀⠀⠂⢠⠤⠄⢰⣿⣸⡿⠁⢰⠏⠀⣼⣿⣶⣿⣿⣿⣿⣷⡄⠁⠀⠰⠀⢴⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣥⡛⠃⠀⠀⠀⠀⠀⠐⢠⣦⣴⣿⣿⢻⣇⠀⢾⠀⠀⢩⣿⣿⣬⣟⡙⠛⣿⡙⢶⠀⠀⠀⠤⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠠⠘⠿⠤⠁⠀⠀⠀⠈⠀⣴⢶⣿⢻⣿⠿⠁⢠⣾⡄⢸⠀⡄⢈⣿⣿⣿⣿⣿⣶⣌⣷⠄⢠⡀⠀⠴⢮⠁⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣷⣆⣠⣾⠁⠀⠀⠀⣤⡇⠘⠀⠈⠏⠀⠿⢿⡿⣿⣿⣿⣇⠘⢠⡄⠀⠀⠠⠖⠏⢰⣹⡇
⠀⠀⠀⠀⣰⠎⠀⠐⠀⠘⠇⠀⠀⠀⢀⣠⡿⣿⣿⣿⣦⣀⣛⣠⡾⠃⡀⠀⠀⣠⠀⠀⠈⠛⠊⢱⣙⣿⡿⢲⣇⣀⢁⣀⡆⣀⣐⣀⣷
⠀⠀⠀⠀⣼⣂⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⣿⠏⣩⡟⠛⣡⠄⠀⠀⢈⣠⣤⡆⠀⠂⠆⢸⡛⢻⣋⣻⠛⣉⣇⠈⢡⡈⠍⠹⠋⢡
⠀⠀⠀⢸⣯⡅⠒⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣧⣲⡽⣧⡂⠶⠖⣀⠈⠰⣯⠚⣟⣿⠇⠀⠀⠀⠀⠉⢻⣿⣿⣿⣼⣏⣟⣠⠛⠂⢄⠒⠫⣦
⠀⠀⢀⣾⡿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣤⣽⣿⣿⣟⣿⣿⣧⡀⢀⡘⠂⠀⠃⠁⢘⣿⠂⠀⠀⠀⠀⠚⣿⣿⣿⣽⣟⣿⣍⡿⢳⣶⡟⢳⣾⠿
⠀⠀⣿⣿⠇⠻⣆⠀⠀⠀⠀⠀⠀⢉⣹⣿⣿⣟⣭⣿⣧⣿⣿⣤⠉⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣹⣿⠟⢿⣿⣿⣿⡃⢸⠿⠞⠉⢀⣾⡄
⠀⢠⣿⣿⡰⠈⢿⡄⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡓⡾⠿⣟⣿⣿⣆⡆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣹⡟⠀⠀⠀⢬⣿⡇⠀⠀⠀⠀⣾⠻⡇
⠀⢸⣿⡿⠁⠠⠘⣷⡀⠀⠀⢀⠀⠘⢿⣿⣿⣿⡹⠅⣤⣹⡟⠙⠻⣿⣿⣷⣦⣾⣷⣶⣶⣶⡄⢠⣤⣿⠁⠀⠀⠀⢺⠿⠁⠀⠤⣂⣾⡏⣿⣿⡆
⢠⣾⣿⣿⡤⠁⠀⢿⡧⠀⠀⠀⠤⢤⣿⣿⣿⣿⣓⣾⣿⡿⢡⣀⣄⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢩⠀⠄⢹⢵⣿⠆⠀⠀⢀⣿⡟⢂⣼⢻⡇
⠈⣿⣿⡞⠀⡄⠀⠘⢣⡛⠀⣀⠀⢀⣿⣿⣿⣯⣼⣿⠉⢀⣴⣿⣿⡔⠂⠀⠠⣾⡩⣛⠿⣿⣿⣿⢿⣦⠴⣖⣌⣿⡏⠀⠀⣿⡿⠋⠉⠁⠉⣳⠄
⢠⣿⡿⣷⠟⠀⠀⣀⠘⣇⠀⣤⣴⣿⣿⣿⢟⡋⠛⠀⣴⡟⠁⣠⠄⠀⠀⠀⢠⢼⣿⣿⣤⡀⢀⠘⢣⣽⣷⣿⣼⣿⣷⡾⠟⠛⢂⣀⠀⠙⣳⢿⣇
⣺⣿⣿⣦⠋⠀⠒⠀⠀⠹⣦⠆⣿⡿⠿⠉⠙⠁⠀⠀⠉⠁⠞⠁⠀⠀⠀⠀⠀⠈⢿⣆⠙⠿⣿⣷⣼⣿⣿⣿⣿⣿⣿⠔⠀⠀⠀⠀⢀⣈⠍⠳⣿
⣿⣿⡿⢿⡏⠃⠸⡋⠀⡄⠹⣾⡿⠁⢀⣴⠃⣴⣤⣶⡔⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣷⣶⣿⡟⢿⡏⢹⣿⣿⣿⡯⠛⠲⡀⠀⠀⠄⠁⠰⣦⡿
⢻⣿⣷⡟⣗⣠⡾⠡⡔⢄⣰⡟⢁⠴⠟⠁⢠⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡆⠀⠹⣿⠿⣿⣿⣄⣀⣀⠷⣲⣆⣠⣟⣊⣹⡷
⢺⣿⣿⣯⣿⣷⣾⣿⣿⣛⡿⢱⠊⠀⠀⠀⣰⠄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣹⣄⢀⠀⠀⣸⢿⣿⣿⣿⡆⠸⠤⠞⢿⢿⣿⣿
⢸⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠠⡄⠀⠀⠁⠀⢂⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣼⡉⠛⠿⣧⣿⡿⣧⣭⣿⠿⣿
⢸⣿⣿⣿⠙⠋⠁⠀⣀⡀⣰⣿⡁⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠻⠿⡿⣿⣿⣶⣀⣹⣿⣿⣿⣷⣯⣿⣿
⠀⣿⣿⣉⣥⣶⣤⣤⣿⣿⣿⡿⢃⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⡏
⠀⢹⡟⠻⢿⣿⣿⣿⠿⠛⠩⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠻⣿⣿⣤⣿⣿⠟⠋⢰⡿
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠢⠉⡉⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄

⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿
⠙⢿⣿⣷⣄⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠙⢿⣿⣷⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⢀⣶⣦⡀⠀⡀⠋⠁
⠀⠀⠀⠈⠙⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⣀⣴⡿⠀⠀⠀⠈⠻⣿⣿⠟⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠐⠟⠁⠀⠀⣀⠀⠀⠀⠈⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠀⠀⠀⢠⡾⠋⠀⢀⣠⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⣷⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⠠⡀⠀⠀⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⡿⠃⠀⠀⠀⢀⣦⡀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡽⠂⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠉⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣴⣖⡂⣖⣶⢴⣶⢠⡖⢶⣰⣦⣰⢀⡶⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠓⠂⠋⠓⠐⠞⠋⠓⠚⠙⠃⠛⠘⠛⠓⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡖⢦⡖⣴⢒⡂⡆⣶⣒⢰⣆⣶⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠀⠙⠚⠃⠃⠛⠒⠙⠈⠋⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣏⢙⣦⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣈⣋⠉⠟⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⡤⠤⠤⠤⠤⢀⡉⠻⣆⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⢸⠃⢸⠃⠀⠀⠀⠀⠀⠀⢰⡆⢸⠆⡷⢠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠊⠁⠀⠀⠀⠘⣆⠘⢦⣀⣀⣀⣀⣀⣠⠼⠁⣼⠋⠁⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠛⠛⠛⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢤⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠹⣯⡽⠙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠸⠷⠟⠛⢛⣻⣻⣟⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠶⢠⡆⣶⠀⠀⠙⢦⡀⠙⠂⠀⠀⠀⢠⠾⣆⣀⣿⣉⣉⣉⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣈⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣟⠀⣹⠄⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠿⣆⠰⠾⠷⠶⢶⡄⠀⠈⠁⠀⠀⠀⠈⠉⣿⠀⠀⣞⣹⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⢉⣭⣭⣭⣿⣭⣭⣽⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⠥⠤⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠒⠒⣶⡗⣶⠋⠀⠀⠀⠀⠀⠀⢀⡆⣿⠀⠀⠘⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢤⣤⣬⣭⣭⡽⠏⠉⠁⠶⣄⠀⠀⠀⠀⣠⡞⠃⣿⣒⠒⠒⢒⣒⣒⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠛⠃⠀⠀⠈⠉⠁⠁⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣀⣀⣀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡏⣿⣿⣿⣿⠛⠛⣡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⣤⣤⣽⣾⣧⣈⠘⠿⠾⠂⢠⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣧⠉⢻⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⢷⣀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠿⣿⣷⡄⠀⠈⠛⠻⠿⠿⣿⢻⣷⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣀⠾⠿⣿⣿⣿⣿⣦⣾⣿⣿⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢸⢿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⢄⣨⡄⠴⠟⠁⠈⠓⢄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⣿⣿⡶⣾⣿⣷⠠⢄⣀⠀⠀⠀⠈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠰⠁⠒⠄⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣤⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢣⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⡿⣿⣿⣿⡿⠻⠋⠉⠀⠀⡃⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣶⣧⡿⠁⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣟⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⣤⡶⠂⣤⣤⠀⡀⠀⠤⣤⣘⣻⣄⠀⡀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣀⡀⠀⠸⠏⢤⠀⢰⣯⣿⣿⣿⣆⢁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⣄⠋⠉⠉⠛⠛⠛⠛⠓⠀⠀⠘⠛⠚⠛⠛⠻⠿⢿⣆⠳⡄⠀⠀⠀⢀⣧⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠄⠠⠾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣋⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣶⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⢠⡶⠾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⣾⢁⣼⢿⣿⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠃⢻⡇⢿⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣉⡴⠟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠁

⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⣀⣀⠀⡀⡀⠀⡀⠀⠀⠀⣀⣀⡠⠤⠤⠴⠖⠒⠒⠒⠒⠛⠛⠋⠉⠙⠛⠛⠒⠒⠒⢶⠤⣤⣀⣀⣤⠞⠉⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⣀⡠⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠔⡶⠀⠀⢀⣠⠤⠶⠒⠚⠛⠉⠉⠛⢒⣢⣤⠈⠉⠀⡀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡴⠚⠁⠀⣼⡡⠖⠛⠉⠀⠀⠀⠀⠀⢀⣤⣖⣛⣉⣁⣀⠀⠀⠀⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⡴⠋⡽⢀⡴⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⣴⠚⠋⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠋⣇⠀⠀⠀⠀⢸⠃⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⡠⠽⠆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⣿⣶⣿⠀⠀⡗⢄⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠒⠒⠢⠤⠞⠉⠉⢤⣀⠁⠙⣇⡠⠜⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠇⡏⣈⣉⣹⡆⠀⠹⣄⠁⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣤⡤⣍⡳⢄⠈⢂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣸⣿⠿⠿⢿⡇⢰⠖⢉⡀⠀⠀⠀⠉⣐⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠙⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣯⣤⣤⣤⣼⡇⢸⡴⢹⠃⠀⠀⠀⠀⠙⣦⠀⠀⠀⠠⠚⣡⢤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠛⡇⠀⠀⣿⣀⣀⣀⣤⡀⠀⢸⡇⠀⠀⠀⡞⠁⠀⠈⢻⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢼⣿⣿⣿⣿⣿⡇⠀⠀⢈⣩⠤⠤⠤⠷⠤⣼⠁⠀⠀⠰⡧⠖⠚⠿⠁⡏⠹⡿⠒⠶⠤⣄⣸⠃⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣸⡁⠀⠀⣀⢀⣧⠀⢠⠏⠀⡠⣖⣒⣦⠄⠀⠀⠀⠀⠀⠳⣄⠀⣀⣴⢷⣄⣀⣀⣀⣠⡼⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡿⠿⣿⠀⢸⣄⠀⠀⠈⠉⣀⣀⢤⡀⠀⠀⠀⠀⠘⠛⠛⠉⠀⢉⠙⠉⡉⠥⠤⠦⣄⠀⠀⡟⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣼⣇⣀⣀⣄⣀⣿⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢁⣝⠐⣠⡀⠀⣀⣠⠟⠀⢰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡏⡿⠟⠿⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢷⢿⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠐⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠾⣿⣿⡶⣶⣤⣤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡼⣏⣉⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠈⠓⢦⣀⣀⠤⢤⣴⠒⠚⠋⠉⠁⠀⠉⠉⠁⠀⠈⠈⠉⠛⠻⢿⣿⣟⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢻⡿⠿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠳⣤⠖⠁⢹⣤⠊⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢳⣾⣿⡇⠀⠀⣀⣀⣀⣠⠖⣶⢻⠗⣲⣇⣀⣀⣀⠈⠛⠃⢰⢸⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠉⠉⠉⢉⠽⡍⠉⠙⠒⠓⠼⠲⠋⠀⠀⠀⠉⠉⠉⠉⠛⠛⢶⣀⣇⣸⣉⣦⣀⣀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠰⣦⠒⡒⡤⠊⠀⣧⡤⢤⠤⢲⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣞
⠀⠀⠀⠐⠉⠀⠀⣀⠤⠖⠛⠉⠣⡙⡄⠀⠈⠖⠁⠂⠀⢀⡥⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⢃⠈⠀⠈⠀⠀⡠⠚⠁⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⡀⠀⠀⢀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⣀⠀⠉⠘⠒⠀⠤⠦⠚⠒⡠⠖⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣈⠒⠠⢽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠉⠉⠓⠒⠒⠢⡔⠋⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⣄⣀⡀⠀⠀⠃⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠐⠂⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⣾⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣐⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣉⣡⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢹⡇⠀⠀⠀⣠⣶⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠅⢸⡇⠀⠀⠈⠀⠈⢩⣟⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⠀⠀⣇⣽⡇⠀⠀⠀⠀⠀⠈⢉⣟⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢌⣹⣄⣣⠀⢸⡄⢀⡧⢿⣇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⣲⡷⢬⣏⡱⣀⣸⠇⣚⣩⣭⣹⣻⡆⠀⠀⠀⠀⠀⠀⢭⡻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠊⠛⠁⡠⠚⢉⣽⣿⣿⡿⠛⠁⠈⠉⠻⣷⡄⠀⠀⠀⠀⠀⠤⣛⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠚⢻⣿⡿⠋⣿⡇⠀⠀⠀⠀⠀⣿⣿⣧⣄⠀⠀⠀⢀⣵⠟⠁⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡰⣦⡄⣄⢰⡦⣶⣿⡿⣧⡀⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⡞⠋⠀⢀⠀⡋⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⡰⢚⣹⢿⣿⣿⣇⠀⠁⣿⡇⠀⠀⠀⠀⠘⠿⠿⣿⠿⣿⣿⣷⣤⣀⣠⡞⡵⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠰⢂⡽⠃⠼⣩⡾⣷⣯⣿⡀⠀⣿⡇⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠹⣐⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⠀⢾⣿⡇⠛⢀⣦⠀⢀⣿⡇⠀⠀⠀⠀⢀⣿⠽⣿⣿⠀⠑⣻⣿⣿⣿⡆⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⠀⠹⠛⠃⠀⠝⠋⠀⢺⣿⡇⠀⠀⠀⠀⢸⡇⢸⣿⣿⣶⣄⣴⣿⡿⠛⠁⠀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣁⣠⣤⣀⣠⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⡿⣿⢿⣿⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣾⠷⠾⠛⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉

⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣧⡐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⡀⠈⠻⣿⣿⣦⡀⠈⠳⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⣿⣿⣶⣤⣀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣷⣦
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣿⣿
⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿
⣿⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿
⣿⣿⣿⣿⣦⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⠻⢿⣿⣿⣿⣷⣄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠙⢿⣿⣿⣟⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⢿⣿⣿⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⣛⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢙⠿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⡰⠊⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠐⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠐⠀⠀⠈⠻⡿⠋⠀⠀⠀⠤⠶⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠸⡛⠛⠋⠋⠀⢳⠀⠈⠀⢰⣇⠀⠀⠀⠀⠀⠸⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠺⢷⣄⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠐⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⢰⠚⠁⠀⠀⠀⠋⠉⡀⠀⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠸⡄⠀⠀⠀⠠⢴⠈⠉⠡⠀⠈⠀⠀⠀⡠⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠀⠀⠀⡶⠀⠀⠀⠀⡠⠀⠀⠀⠀⠖⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡼⠒⠦⠮⢀⠤⠃⠀⠂⠀⠁⠰⣀⠀⡠⠀⢀⠌⠀⢀⠎⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰⠈⠀⠀⠸⠂⠉⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡰⠦⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠐⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⡿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣿⣿⣿⣦
⠀⠀⠀⠀⢀⡀⠀⣄⢸⣿⣿⡇⠀⠀⠀⠀⠠⠀⢸⣇⢀⠀⠀⣀⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢠⡀⢸⡇
⠀⠀⠀⠀⠀⢆⠀⠶⠀⠛⣹⠁⠀⠀⠀⠀⠀⡀⠈⠓⠴⠄⠀⠶⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠂⠘⢋⡟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠶⠀⠰⠄⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠁
⠀⠀⣤⡀⠀⢀⠚⡣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢛⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⢱⠄
⠀⠀⠠⠛⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠗⠤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡄
⠀⠀⠀⢠⣿⢷⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠈⡏⢀⡄⠀⢤⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⢘⡅⠀⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣉⠀⢘⡃⣸⠁⠀⠀⠀⠀⠀⠀⠀⢿⠐⣏⠀⢐⠂⢸⠁
⠀⠀⠀⠀⠓⠘⠂⠄⠘⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠂⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⣀⠀⠀⠖
⠀⠀⠀⠀⡀⠁⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠤⡤⡀
⠀⠀⠀⠀⠀⠇⢀⣀⣀⡸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠦⢐⠀⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠂⠀⠂
⠀⠀⠀⠂⠀⠲⠤⠽⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠞⠛⠛⠛⢻⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠠⠆⠀⠶⠜⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣉⠉⢉⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⣿⠐⡂⠀⢐⠂⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠐⡒⠀⢘⠂⢸⠃
⠀⠀⠀⠀⠀⠣⠀⠛⠀⢋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠫⠈⠧⠀⠰⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠋⠄⠘⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⠛⠂⠘⠅⠯
⠀⠀⠀⠂⠀⢐⠒⣀⠠⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⣀⠘
⠀⠀⠀⠀⠁⠀⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠔⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠊⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢖⠀⠀⠀⠀⠒⠂⠤⠄⡀⢋⠒⢄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠊⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠲⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢆⠑⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠀⠀⠀⠈⡆⢁
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⠖⠉⠁⠢⡀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡇⠸
⢀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⢋⡠⠊⠁⠸⡔⠁⠀⠀⠀⠀⠈⢂⣠⠀⠀⠀⠀⢡⠀⠀⠁
⡆⢰⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⡄⢲⡠⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠰⠀⠀⠠⡤⠆⠀⠀⠀⠀⡀⢀⠀⣀⢀
⡇⠸⠀⠀⠈⡄⠀⠀⠀⠀⢀⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⢓⠀⠀⢠⠀⠀⠀⠀⠰⠀⠈⠁⠀⠀⡁
⢡⠀⢣⠀⠀⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠷⠂⢤⠀⠀⠀⣠⢂⡁⢀⠀⠀⠀⠉⣄
⠀⢢⡀⠁⢠⠀⢀⢤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠁⠀⠀⠈⣄⢀⠖⠐⠁
⠀⠀⠁⠀⠘⠀⠁⠀⠠⠀⠁⠀⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢳⡀⠶⠀⠀⠀⠀⠀⠀⣀⣀⣈⣀⡀
⠀⠀⠀⠀⠀⠲⣦⡀⠀⢶⠀⠀⠀⠤⠠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠈⠲⡄⠀⠀⠀⠀⠓⠤⡎⢰⠞⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠠⠄⠉⠉⢆⠀⠈⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⠤⠄⠤⣸⠀⠀⠀⠠⠔⠈⠁⠙⢤⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠧⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠐⠠⠔⠁⠀⠀⠀⡃⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠂⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠓⢤⡀⠠⠀⠀⠉⠐⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢄⠀⣀⠀⠀⠀⠀⣀⠀⠀⠤⠐⢲⠋⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠈⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠉⢣⡄⠄⠗⠀⠀⠀⠸⠀⠀⠰⡦⣄⠀⢀⢀⠀⠀⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⢀⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠄⠁⠁⠀⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠃⡰⠁⣀⠀⢀⠀⠀⠀⡇⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢇⠀⡀⠀⢀⠀⡨
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠐⠒⠈⢇⣀⣀⣀⣀⢠⣬⡎⠤⠊⠀⠀⠀⢡⠀⠀⡧⠤⠀⠞⠒⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠈⠣⠠⠊⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣀⢀⠠⠒⠁⢇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠐⠈⠀⠀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⢖⠉⠀⠀⠀⠀⠈⡲⠀⠄⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡠⠔⢛⠁⠀⠀⠀⣀⣀⠀⠀⢤⡤⠄⠤⠄⠀⣀⣀⡄⠀⢃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⡀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠉⢆⠀⠈⢢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠦⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠠⢈⠀⠂⠠⠄⠊⠀⠀⠀⠹⠀⠂⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⠁⠱⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⡁⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣶⣶⣶⠀⠀⠀⠀⣀⣀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠸⣿⡿⠀⢸⣿⣿⣿⠀⠀⠀⠐⣿⣿⠂⣿⣿⣿⣿⠀⠀⢠⣤⣄
⠀⠀⢠⡄⠰⣦⣀⣾⣷⡀⠀⠀⢠⡄⠀⠀⠀⣰⣷⡆⠀⠀⠀⣠⣶⣶⣶⣦⣄⣠⡦⠀⠀⠀⠀⢰⣶⣶⣦⣤⣤⡀⠀⠀⠀⠘⠿⣟⡀
⠀⣀⣼⣄⣀⣈⣉⢻⣿⡇⠀⢀⣸⣇⣤⣴⣾⣻⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠉⠉⠁⠀⠀⠀⠀⢀⣿⣿⡀
⠀⢸⣿⣿⣿⣿⠇⢸⣿⠃⠀⠀⣿⣿⣿⣿⡿⢸⣿⠏⠀⠀⠀⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⣟⣿⣿⡇⠄⢠
⠀⠠⡄⠈⠁⠀⠀⢸⣿⠀⠀⠀⣤⠀⡍⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠀⠀⢸
⠀⠀⠑⠊⠀⠀⠀⢸⣿⠀⠀⠀⠈⠒⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆⡀⠀⠀⠓⣼
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠙⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠾⠿⠷⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣛⣋⡉⠉⠳⣄⠀⠀⠀⠀⣰⣶⣿⣯⣤⡶⣀
⠀⠀⠀⠹⣦⡀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠉⠉⠁⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢉⢉⣿⣦⡀⢿⠃⠀⠀⠀⣿⣿⡀⠀⠀⣠⠾
⠀⠀⠀⠀⠘⢷⣝⢿⠟⣙⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⠀⠸⣿⣇⠀⠠⡚⣿⣿⠇⢸⠀⠀⠀⠀⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⠶⠶⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⠀⠰⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡟⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⡏⣿⠀⠀⠀⠀⠀⢸⣿⠀⢐⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⠀⠀⠀⢸⣿⣿⡇⠿⠀⠀⠀⠀⠀⠻⠇⠀⠐⠶⠶⠶⠶⠾⠀⠀⠀⠀⠘⠿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⢀⣴⣶⣄⣠⣶⣶⣀⠀⠀⠀⠀⠀⣰⣶⣦⣠⣶⢶⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣙⣟⣽⣷⣛⣏⣹⣦⠁⠀⠀⢀⣈⣻⣋⣻⣟⣞⣙⣧
⠀⠀⠀⠀⠀⠀⣸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠻⠿⠿⠏⠫⠄⠀⢀⣿⣿⣿⣿⠿⠿⠿⠏⠋⠄
⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢳⣦⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣉⣡⣀⣀⡀⠀⢸⣿⣿⣿⣿⣀⣈⣁⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⣉⠉⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠿⠿⠿⠿⠿⠇⠀⠀⠁⣿⣿⣿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⢀⣀⣙⡛⠛⢻⣷⠀⠀⠀⠀⠀⠀⢀⣙⣛⡛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⠇⡀⠀⠀⠀⠀⣼⣿⣿⣧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠈⠀⠀⠀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠶⡇⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠹⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠓⠶⠈
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢤⣀⠁⠲⠉⠤⡀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣠⠞⠉⠀⠀⠀⠹⣄⠋⠀⡴⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⣿⠀⢺⠃⠀⠀⠀⠀⠀⠀⠈⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⡀⠀⠀⠀⠀⣸⣿⡀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠒⠂⠐⠐⠒⠂⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣹⠣⣄⠀⠈⣟⡿⠀⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠉⠁⠘⠦⠦⠉⠘⣿⠁⠘⠀⣿⠀⠀⠀⠰⣦⣆⠀⠀⠀⠀⠆⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣸⠀⠀⠸⠻⣷⣄⣀⣴⣿⡉⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠁⠀⠀⠀⠀⠀⣠⠔⢁⠎⢈⡆⠀⠀⠀⠈⠙⠇⣼⣻⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠤⠀⠠⠀⠤⠤⠀⠀⠀⠀⢸⡏⠐⠀⠄⠀⠀⣀⣀⣠⡄⣸⠔⠈⡇⠀⢠⠀⠀⠀⠀⢹⣿⡅⣀⡀
⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⣀⣄⠈⢏⠀⠀⠀⠈⠉⠉⠈⠁⠇⠀⠀⡇⠀⠈⡄⠀⠀⠀⠸⢿⣷⠈⠁⠀⠀⠀⠀⢰
⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠧⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠙⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡀⠀⠀⠀⡀⠀⠀⣀⠀⢀⡗⢀⣄⡀⠓⠄⠀⠈⠀⠀⠀⠀⠀⢰⠃⠀⢰⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣁⣀⣀⠀⣁⣀⠀⠈⠀⢸⠁⠙⠙⠉⠁⠘⠦⠀⠀⠀⠀⢀⣠⠞⠀⠀⠈⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣴⢷⠞⠛⠛⠛⠛⠻⢤⣤⣀⠀⠀⡉⠀⣀⠈⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⠤⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⠟⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠙⠛⠛⠛⠛⠛⠲⠦⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⣹⠋⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠠⠇⡀⣤⡶⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⣀⠐⣥⡟⣧⠀⠀⣿⠃⠀⠀⠠⠀⠀⠀⠀⠀⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⢓⣳⣼⢷⣄⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣾⡁⢸⣇⣿⣴⣽⣿⣿⣶⣀⡄⡀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣷⣿⡟⢻⣧⣿⣿⠈⢿⣯⣿⢿⣷⡉⣦⠀⠀⠀⠀⠀⠀⠀⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡤⣽⢿⣿⢿⣹⡟⢻⠃⣾⢿⣟⣶⢬⣿⣿⠆⠀⠀⠀⠀⠀⢰⡀⢻⠀⢀⣤⣴⣶⣶⣶⣤⡉⠀⠀⠀⠊⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣯⣧⡜⣻⣻⣿⣯⣸⠇⠀⠘⢿⢭⣷⡚⢿⠟⠀⠀⠀⠀⠀⢠⣎⡔⢁⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⠀⠏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣹⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠛⠁⠀⠀⠠⠀⠀⠀⠀⣦⣼⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⡼⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣿⣯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⡞⣩⣶⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣽⣋⣄⠹⣿⣿⣿⣿⣿⡿⠟⢁⣴⠃⢸⣯⡿⣠⣇⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡿⢋⣷⢄⣟⠉⠋⣹⣧⡴⠟⣴⣷⠈⠉⣀⡍⠉⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣿⣿⣻⣾⣻⡿⢶⣿⣠⣏⣹⢀⣼⡿⢡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡛⣯⣥⣤⣽⣿⠻⣿⣍⣉⣅⠈⠃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⢉⣡⡘⠻⠟⠛⠁⠈

⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢶⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢹⠄
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⣠⡴⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⡷⠟⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠹⣦⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⣉⡽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⣴⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⣤⣾⣯⡤⠴⠶⠾⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠛⣿⠀⠀⠀⠀⠀⠀⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⡶⠋
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠛⠛⠛⠲⠶⠶⠶⣿⣥⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿⣴⠞⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠳⠶⠶⠤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠳⠲⠾⣧⠀⠀⠀⢀⡟
⠀⡇⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣠⡇
⠀⡇⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠿⠚⠛⢹⡇
⠈⡇⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⢈⡇⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠚⠛⢋⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⢈⡇⢀⣴⠋⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠛⠛⠋⠉⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢨⣧⡾⠃⠀⠀⠀⠀⠀⠈⠙⠳⢦⣶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⠉⠉⠙⠛⠓⠶⠶⢤⣤⣄⣀⣀⡀⠀⠀⠀⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠙⠛⠛⠾⠷⢦⣤⣤⣀⣀⣀
⠛⠳⠶⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠛⠛⠲⠶⠦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠦⣤⣤⣄⣀⣉⡛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⠛⠻⠻⠷⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣈⣷⣄⣀⣀⣠
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣭⣯⣭⣿⣯⣿⣅⣽⣬⣭⣭⣭⣽⣯⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣅⣈⠁⠀⠀⠀⠀⠀⣱⠀⢀⠤⡀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⢿⢿⣾⣿⣿⣿⣿⣿⡎⣿⡿⣛⢿⣿⣿⣿
⠀⠀⠀⠀⠙⢿⣿⣿⠱⣄⠀⠀⠀⠀⡼⠃⠤⠤⠀⠓⡄⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⣿⢻⣿⣿⣿⣿⢋⣼⣛⣛⣿⣭⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢹⠋⠀⠈⠳⣄⣀⢀⡜⠀⠉⢓⠀⡖⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⣾⣿⣷⣝⠻⢿⣿⢣⣿⣶⡿⣿⢭⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⡀
⠀⠀⣰⠄
⠀⠀⢸⡇⠀⠀⠀⠘⠁
⠀⠀⠐⣿⣆
⠀⠀⠀⠈⠙⠕

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠐⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣧
⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⣿⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣶⣶⡄
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⠇
⠀⠉⠉⠓⠂⠤⢤⡈⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⣶⠀⠀⠀⠀⢠⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⣠⣟⡼⠃⠀⠀⠀⠀⠀⠠⠐⣀⠠⡔⠁⠀⡰⢿⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⣿⣿⠟⠋⠁
⠀⠀⣾⠇⠀⠀⢀⣯⠟⠁⠀⠀⠀⠀⠀⠀⠁⠈⢁⠾⠀⢀⡜⠁⢸⠀⠀⠀⠀⠀⠀⠀⠉⠋⢻⣏
⠀⠀⣿⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠉⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣾⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡄⠀⠀⠀⣀
⠀⠀⠀⠀⢰⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡼⠁⠀⣠⡾⠁
⠀⠀⠀⠀⣼⣿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃⢀⢼⠟
⠀⠀⠀⠐⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⣴⠃⠀⠘⠋
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣟⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠴⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣀⠴⠋⠈⠋⠁⠀⠀⠀⠀⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠓⢻⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⢹⡀⠀⡴⣩⠙⣦⠀⠀⠀⠀⢰⡇
⠀⠀⠀⡀⠀⠀⠀⠈⢧⡀⠀⢹⡄⢠⠆⡀⠀⠀⠀⢹⣸⡇⡟⠂⣼⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠉⠳⢄⡀⠀⠀⠓⠀⠀⡇⠟⠋⠁⠀⠀⠀⠀⠉⠁⠁⢺⢧⡀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⣀⣀⣴⠷⣄
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣀⡴⢲⠄⠀⠀⠀⠀⡴⠃⠀⠙⢚⡵⠛⣡⠴⠚⠛⠲⣄
⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⣠⣴⠲⣄⡀⣾⣁⠀⠀⠀⠀⡼⠁⢀⡤⠚⢁⡴⠚⠁⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⣸⠋⠀⠙⣄⢸⡄⠀⢀⡜⢷⠞⠁⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡙⢿⡾⠁⠀⢸⠀⠉⠛⠉⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⢻⣧⠀⠀⢸⡄⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠈⠧⢀⡏⢹⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠴⠋⠀⢸⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠴⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⡀⠄⣀⣃⣀⣀⡀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠤⠒⠡⠀⠀⠀⠀⠀⠀⠰⡿⠉⠙⠁
⠀⠀⠀⠀⡈⠢⣜⡛⣛⣹⣿⣧⡆⠐⠃⠀⠀⠀⠀⠀⠀⠘⠐⠀⠈⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢠⠋⢁⠈⠉⢉⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠠⠉⠀⠀⠩⠄⡌⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠄⠀⡢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣬⣂⣂⣨⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣿⣶⣦⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠘⠇⠿⠿⠿⠷⠟

⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠛⡉⠛⡛⠛⠀⠀⠀⠀⠀⠀⠈⠈⠨⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⡆⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⢀⣤⡤⢤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢠⣾⣿⣽⣿⣶⠀⠀⠀⠀⠰⣶⣷⣦⣴⣷⡶⠀⠀⠀⠀⢀⣴⣵⣴⣶⡄⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠃⠀⠀⠀⠀⠤⠤⠤

⠀⠀⠀⠀⢀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦
⠀⠀⠀⠀⠘⠋⠀⠠⠀⠀⠀⠀⠀⣴⡆⠠⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠉⠀⠀⠀⠀⠀⠀⠉⠡⠉⠩⢉
⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠰⣯⣽⠀⢠⣄⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠁⠀⠀⠀⠈⠉⠙⠉⢀⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿