Thanks Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢠⣤⡀⠰⠶⠆⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠂⠀⠀⢠⣶⣦⠀⣿⠛⠁⠀⠀⠸⣿⣿⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠁⠀⠀⠀⠀⠀⠀⠉⠉⠂⢀⠄⠀⡀⠀⠀⢩⡝⠁⠀⠈⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⠀⠀⠀⡀⠴⠀⠀⣿⣶⡄⢹⡄⠀⠘⠟⠀⣀⠤⠀⠀⣈⠘⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠉⠀⣸⠃⢠⣤⡀⠰⡇⢀⣀⠀⠈⠳⡄⠓⡆
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⣿⡟⣿⡧⢄⣀⣠⣤⣄⣀⣀⣀⣠⠞⠁⠀⢸⣿⠃⠀⠙⠻⠿⠃⠀⠀⢹⡀⠑
⠀⠀⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⣿⠁⣿⠃⠀⠀⠀⢉⣉⣉⡁⢀⠀⡀⢀⠀⢸⡏⢰⡄⠀⠀⡀⠠⠆⠀⢸⡇⣴⣶
⠀⠀⠀⠀⠀⠀⠙⢋⡀⠀⠀⠀⣿⠀⣿⠟⠁⢱⡔⠿⠋⠀⣷⢸⡏⠀⠈⣷⢸⣧⠞⠀⠀⢸⣿⣿⡀⠀⣸⠃⠛⣩⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇⣀⠤⠐⣿⣿⡇⠀⢠⣿⢸⡟⣄⠀⠀⠀⠉⠁⢙⡴⠋⠀⣤⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠿⠀⠿⠀⠀⠸⠇⠻⠀⠠⠿⠸⠇⠀⠸⠿⠸⠇⠈⠑⠀⠀⠀⠒⣿⠄⠀⠀⠻⣿⠆
⠀⠀⠀⠀⠀⣠⣴⡤⠄⠀⢀⡀⠀⠀⠀⣾⣶⠀⠀⢠⡄⣠⠎⠁⠈⢷⡀⣾⡶⠀⠀⣶⠀⠀⠀⢸⣿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠛⠋⠀⣴⣿⡟⠀⣤⠀⠀⢹⡏⠀⠀⢸⡇⣿⠀⠀⠀⢸⡷⣿⠃⠀⢀⣿⠀⠀⢀⣿⠇⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢈⣛⠀⡀⠈⠳⠤⠴⢻⡇⠘⢆⠀⢀⡾⠃⢿⣀⣀⡼⣿⠀⠀⣸⡿⠀⠀⢈⠀⠠⠾
⠀⠀⠀⠀⠀⠈⠇⠀⢀⣤⣄⢿⣿⡇⠰⠶⠄⠀⠀⢸⡇⠀⠀⠀⠁⣀⠀⠀⠉⠁⠀⠀⠀⢀⡉⠁⠀⣠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠉⠁⠈⠁⣠⡶⠒⠀⠀⠀⢼⢄⡀⠀⠀⠀⠉⠀⠒⠀⣄⠀⠀⠰⣿⣷⠀⠀⢻⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⡿⠀⠉⠢⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣽⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣸⠃⣠⣦⣶⡈⠙⠛⠋⠀⢤⣶⣶⡄⠀⠀⢀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠠⠤⠚⠁⠀⠻⡿⣿⠃⠀⢀⣿⡆⠠⡉⠉⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠄⠠⠀⡀⠀⠀⢠⣎⠤⠤⠤⢄⡀⠀⠀⢀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⡀⠀⠀⠱⡄⢀⠀⠢⡀⠀⣸⡴⠚⣿⠀⠀⠀⠀⠀⢹⡆⠀⠈⠢⠀⠀⠌⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣠⠴⠛⠫⡙⠉⠉⠉⠉⠉⠉⠋⠉⠛⠳⢺⡷⠤⢤⣴⣋⣀⡀⠸⡄⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡀⠀⠀⠀⠀⣼⠃
⡀⠀⠀⠀⣼⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢠⣦⡿⠀⣰⡟⠁⠀⠀⠀⠀⠈⠓⠒⠖⠛⠁⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠙⠒⠲⠒⠋⠁
⢧⡀⠀⠀⠹⣄⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⡴⣻⡟⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠙⠲⠤⣀⣈⣉⣉⡉⣁⣀⣀⣀⠤⠔⠚⠁⣰⠟⢀⣾⠟⢲⡆⠀⠀⢠⠔⠒⣦⡴⠀⢰⣤⠖⢲⡄⠀⢠⣾⠗⢲⣶⠀⠀⠀⣀⠤⠖⠒⠢⠤⣀
⠀⠀⠀⠀⠀⣀⡤⠿⠿⠯⣄⠀⠀⠀⠀⢀⣼⠋⠀⣾⠃⠀⣸⠇⠀⣰⠏⠀⢠⡿⠁⢠⡾⠁⢀⣾⠃⢀⣿⡡⠔⠋⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⢳⠀⠀⣰⠟⠁⠀⢸⡏⠀⠀⣿⠀⡠⣿⠀⣠⣿⠁⡰⡿⠁⠀⢸⠏⡠⢾⡏⢧⣀⠀⢀⡴⠁⠀⠀⣠⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⢀⣨⣷⠛⠁⠀⠀⣀⣠⡥⣤⣄⣈⣈⡀⠈⠉⠀⠈⠉⠀⠁⠀⠀⠈⠉⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⢀⡻⡀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠉⠐⠛⠉⠉⠀⠸⡄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢉⣴⠄⠀⠀⠀⢀⡴⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠙⢎⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡇⠀⠘⣇⠀⠀⠀⠀⠀⡀⢀⣶⠟⠁⠀⠀⠀⣠⡞⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢏⠀⠀⠀⠁⠂⠾⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣇⠀⠀⠈⠳⠶⠶⠶⠊⣴⡟⠁⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠑⢄⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⡠⣿⠏⠀⣠⢴⢶⡄⣀⣰⠆⠀⣰⡄⠀⠀⡠⠐⠀⠀⢢⠀⠘⡄
⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣼⡏⠀⠀⠀⢀⠞⣱⠏⠀⣼⠃⠈⣻⠏⣡⠏⠀⢰⡟⠀⡠⠊⠀⠀⢀⣀⠼⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣿⠀⠀⠀⡴⠋⣰⠏⠀⢸⡇⠀⣰⠏⠀⣟⠀⣠⡟⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⠤⠔⠊⠀⣼⠏⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠑⠠⠀⠘⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠑⠦⡀⠀⠀⠀⣠⠟⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⢦⣠⠞⠋⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠠⢀⡇⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⣀⣀⣤⠴⠛⠉⠐⠠⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡘

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢀⡀⠀⣿⠇⠀⠀⠀⠀⠀⠤⠄⣼⠇
⠀⣶⣶⡾⠿⠿⣿⣿⠀⠀⠀⣿⡿⢷⣶⣶⣤⣤⠀⢀⣀⣠⣤⣤⣶⣧⠀⠀⠀⣿⣿⠿⣿⡿⣿⣷⠀⠀⢰⣿⣿⣶⣦⣤⣀
⠀⣿⣿⣶⡆⣿⣿⣿⡆⠀⢸⣿⠇⠺⠿⢩⣿⡇⠀⢻⣿⣿⢉⡹⣿⣿⡄⠀⠀⣿⣿⢠⡹⡇⣿⣿⠀⢀⣾⣿⠁⠟⣋⣽⣿
⠀⣿⣿⣿⣇⣿⣿⣿⡇⠀⣿⣿⣰⣿⡇⣼⣿⠀⠀⠘⣿⣿⢈⣣⣌⣿⣧⠀⠀⣿⣿⣸⣷⣀⣿⣿⠀⣸⣿⣇⣶⡆⢻⣿⡏
⠀⠸⠿⠿⠿⠿⠟⠛⠃⠀⠉⠙⠛⠛⠿⠿⡏⠀⠀⠀⢿⣿⣾⣿⠿⠿⠛⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠈⠉⠛⠻⠿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⠀⠀⠀⠀⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣾⡅⠀⢠⣤⣤⣤⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⢻⠟⣻⣿⠀⢸⣿⠏⣥⣬⠙⣿⡇⠀⠀⣿⡏⢻⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⠀⠘⣿⣆⠻⠿⢃⣿⡇⠀⠀⣿⣷⠸⠿⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠾⢿⣿⡏⠀⠀⣿⣿⣷⡾⠿⠿⠷⠀⠀⢿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⣠⡀⠀⠀⢠⣶⠶⣦⡀⠀⣧⣿⠀⠀⣠⡴⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⢀⣾⣿⠃⠀⠀⠀⠻⣷⣜⣷⣄⢻⣿⢠⡾⢋⣠⠞⠃⠀⠀⠀⠀⣠⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣿⡿⠷⣶⣾⣿⣧⣤⣀⣀⠀⠀⠀⠉⠛⠛⠀⠁⠙⠛⠉⠀⠀⠀⢀⣠⡴⠟⠁⢠⣿⡿⠁⠀⢳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢰⣿⡇⠀⢸⣿⠏⠉⠛⠛⠿⠿⠿⣷⣶⣶⣶⣦⣤⣤⣴⣶⣶⡶⠿⠛⠉⠀⠀⠀⣼⣿⠃⠀⠀⣾⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣼⣿⠀⠀⣾⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⢉⣭⠁⠀⠀⠀⠀⠀⣠⣤⠀⢠⣿⡇⣀⣤⣄⠀⠙⠻
⠀⠀⠀⠀⠀⣤⡤⠚⣿⠀⠀⣿⣿⠀⠀⣿⢇⡞⢹⣿⡄⠀⠀⢀⣴⡾⠟⠉⣳⣾⡟⠀⠀⣴⡖⣠⢋⣾⡿⠀⢸⣿⡴⠁⢀⣿⡇
⠀⠀⠀⠀⠀⢹⣆⣠⠾⠂⠀⣿⣇⠀⠀⢿⡟⠀⢸⣿⠁⠀⢀⣾⠟⠁⣠⠞⣿⡟⠀⠀⣼⣿⣵⠃⣾⡿⠁⢲⣿⣿⣧⣴⠿⠛⠀⠀⢠⠆
⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⢿⣿⠀⠀⠈⠁⠀⣿⡿⠀⢀⣼⣯⡤⠞⠁⠀⣿⠁⣠⠞⢿⣿⠃⠀⣿⠁⣠⠞⠁⠛⢻⣇⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡄⠈⠻⣷⣄⠀⠀⠀⠻⣧⡤⠊⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠘⠋⠀⠀⠈⠋⠁⠀⠀⠀⠀⠻⢧⣤⡴⠋⠀⢶⡀
⠀⠀⠀⠀⠛⠷⢦⣌⠳⣤⡙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⢘⡿⢻⡿⠛⣻⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶
⠀⠀⠀⠀⢠⡶⠋⠰⠟⢁⣾⠏⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢀⣴⣶⣿⣿⡿⠗⣀⣀⣀⣀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⣰⣿⣿⣿⡿⠃⠀⠸⣿⣟⠉⠉⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣠⣾⠟⢸⣿⣿⣿⢀⣿⣿⣿⣿⠁⠀⠀⠀⢻⣿⡆⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⢸⠀⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠾⠛⠁⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣧⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠈⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠸⣿⣿⣿⣧⠀⠀⠀⢀⣾⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠛⠻⠿⣶⣾⣿⣿⣿⠁⠀⠹⣿⣿⣿⣧⠀⠀⣠⣿⣿⠁⠀⠀⠹⣿⣿⣿⣆⠀⣠⣾⠏⠈⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣼⣿⣿⡿⢷⠦⠀⠀⠈⠛⠿⠿⣿⣾⠿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠘⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣀⣀⠀⣀⡠⠤⠤⠤⠤⠤⢄⡀
⠀⣠⠖⢀⠔⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⡴⠉⢠⠞⣹⠀⠀⢀⣤⠶⠖⣒⡲⠶⠀⠀⡔
⠀⣴⢣⢏⡎⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⡞⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠈⠁⠚⠁⠀⣸⠀⠀⠒
⠐⠃⣿⢸⡄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀
⠀⠀⠁⠈⠻⣦⣤⣭⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠠⡄⠀⠀⠀⠀⢸⠀⠀⡤⠖⠒⠂⠀⢸⠁⢀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠇⠀⠀⡇⢀⡴⠋⠉⡿⠀⢀⡇⠀⢠⠃⠀⡞⠈⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠀⠀⣸⠃⠸⣇⠀⢰⡇⢀⡜⠀⠀⡎⠀⡰⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡠⠃⠀⠀⠻⠄⠀⠈⠀⠀⠉⠉⠀⠀⠀⠈⡰⢓⠒⢢⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠼⢃⠰⠀⢸
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣠⡤⠤⡀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡴⠋⠁⠀⠀⡿⠀⠀⠀⢠⠖⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⠤⢃
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⡴⠂⠀⢰⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠉⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⡴⠁⠀⠁⠀⠀⢠⠏⠀⠀⠀⡾⠀⢀⠄⠀⢱⡄⠀⠀⡇⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠲⠦⠤⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠃⢀⡏⠀⠀⢸⠃⠀⣸⠁⠀⢸⠇⠀⠀⠀⠀⠀⠐⣦⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⡟⠀⠘⣧⣀⣠⠟⠀⠀⣏⠀⠀⠸⡆⠀⠀⠀⠒⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠂⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡠⠤⠖⠒⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣏⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡈⢉⣉⣤⣴⣞⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠭⠍⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⢳⡄⠀⠀⠀⠀⡇⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠠⠀⠲⠀⠀⠀⠈⠐⣄⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⣀⠀⠀⢸⠎⠁⠀⠀⠙⢦⡀⠀⠀⣀⣄⠀⠀⠘⡣⡀⠀⠀⠀⢀⡀⠀⠀⢸⣤⠀⠀⡸⠅⠀⠀⠀⠀⡠⢋⠜
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⣿⠀⠀⠀⠀⠀⠀⠀⠀⠎⣦⡢⣶⣥⡋⠀⠀⠀⡇⢺⠀⠀⠀⠣⠛⠀⠀⢸⡋⠀⠀⣧⠆⠀⠀⢀⠜⡠⠊
⠀⠈⢧⣹⠀⠀⠀⡏⡮⠼⠇⠀⠀⢀⠞⡝⠂⠀⢀⠃⣇⠨⠙⠉⢀⠀⠀⠀⡇⣼⠀⠀⠀⢸⠀⠀⠀⢸⡃⠀⠸⠁⠀⠀⠀⣃⠖⠁
⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⠀⠀⣌⠸⠀⠀⠀⠘⠰⠃⠀⢀⠀⣺⠀⠀⠀⠃⡇⠀⠀⠀⢨⠀⠀⠀⣿⠀⠀⠀⠀⠴⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⡅⡇⠀⠰⠄⠤⣆⡇⠀⠀⠀⡀⡇⠀⠀⠘⠚⠀⠀⠀⠀⢦⠁⠀⠀⠨⡃⠀⠀⠀⡟⠀⠀⡔⢁⠔⠃⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⢹⡉⠉⠁⠀⠀⠀⡃⡶⢄⡀⢀⡀⠔⢳⣀⡰⠋⢠⡤⢠⣼⡇⢄⣀⣸⣇⠀⣠⢻⠁⠀⠀⠀⠀⣀⠽⢦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠴⠁⡸⠀⠀⠺⡤⠴⢅⡇⠀⠑⠂⠐⠒⠁⠈⠓⠔⠋⠑⠒⠒⠃⠈⠦⠼⠏⢧⣉⡆⠀⠀⠀⠀⠐⢂⡤⠋
⠀⠀⠀⠀⠀⠈⠛⠿⠽⠽⠂⠀⡓⣄⠀⠀⠉⠉⣿⡅⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠐⠀⠀⠰⡄⠀⠀⠀⠀⢸⢆⠀⠀⠀⠁⠀⣼⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠋⠀⠀⠀⠀⢰⢹⠀⠀⠀⢀⠠⣀⠀⠀⠀⢻⣆⠀⠀⢸⢿⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠀⠀⠀⠀⠀⢠⡈⠀⠀⢠⢁⠔⠚⠃⠀⠀⠈⣼⠀⠀⢰⠘⠀⠀⠀⠀⢐⠸⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⡎⠀⠀⠀⠀⠀⢀⡸⠀⠀⡘⠀⠀⠀⠀⠀⠸⡄⠀⠀⢤⢀⣀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⢀⠀⠀⠀⢸⡸⣆⠀⠀⠀⠀⠀⠀⠀⡠⢠⠇⠀⠀⠙⠃⠀⠀⠀⠀⢰⢥⠀⠀⠀⠋⠓⢶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠍⠩⡽⠀⠀⠀⠄⠇⠈⢶⠠⠤⠤⠄⠐⢈⠔⠁⠘⢦⣀⠀⠀⣀⢴⣀⡀⢸⣾⠀⠀⡀⠀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡜⠀⠀⠀⠈⠉⠁⠉⠈⠀⠀⠀⠀⠈⠓⠢⠤⠄⠚⠁⠠⠤⠽⠀⠀⠙⠶⡔⠤⠂
⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠔⠒⠪⠥⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⡁⠱⠎⢘⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡤⠊
⠀⠀⠑⣄⢈⡖⢢⠋⣠⠊⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠤⠐⠒⠢⡀⠀⠈⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⢄⡈⡠⠤⡄⢉⡩⠂
⠀⠀⠮⠤⠌⢁⠛⢤⠀⠥⠀⠀⠀⠀⡠⠄⠈⠑⢲⠀⠀⠀⠀⠀⣈⣀⡀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠖⠒⠁⠑⢄⡀⠀⠀⠀⠀⢠⣊⣁⠛⠲⢁⡈⢑
⠀⠀⠀⠀⠤⠊⠁⠜⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠂⠀⠀⠀⠠⠊⠀⠀⠈⠲⠒⢃⣀⠠⠂⠈⢂⠀⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠠⠓⢄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠈⠈⠁⠀⠀⠀⠀⡀⢀⠄⠀⠃⡤⠴⡀⠀⠀⢀⠌⠀⠀⠀⠈⠁⠈⠚
⠀⠀⡠⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠠⡄⠀⠈⢉⠀⠀⠀⠀⠀⠈⠐⠒⠁⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠰⠆
⠀⠀⠁⠤⠊⠀⠀⠀⠀⠀⡐⠰⠅⠱⠂⠀⠄⠒⡀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠶⠄⣀⠠⠂⠀⠀⠀⠀⡀⠰⠀⠀⠨⡄⠀⢨⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢑⡀⠀⢓⠀⢰⠖⠈⠁⡇⠀⠀⠀⠀⢠⢀⠀⠀⢘⠀⠀⠀⢠⠀⠀⠀⢘⠀⠀⠜⢁⠤⢄⠈⠃⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠐⢄⠈⠀⠉⡒⠉⠈⢣⡠⠴⢀⡠⠃⠀⠀⠀⠀⠄⠈⠠⣈⡐⠀⢀⠂⠀⠀⠀⢠⠃⠙⡀⠃⠀⢀⠃⠐⠒⠂⠀⠀⠀⡀⢀⢤
⠀⢀⡤⠋⠀⠰⠀⠀⡎⢁⠖⠂⠣⣀⣀⠴⢄⣀⢀⡈⠄⠀⠀⠀⠀⠀⠁⠂⢻⠒⠂⠁⠀⠀⠀⠀⠐⢀⣠⠓⠠⢀⠠⠔⢅⣀⠀⠀⠀⠀⢡⠀⠀⡰⠂
⠀⠀⠁⡄⠀⠥⠀⠀⢨⣁⡀⠤⡎⠀⢰⡀⠀⢡⣈⡆⠀⠀⠀⠀⢠⠀⡀⠀⢸⠀⠀⡀⢤⠀⠀⠀⠀⠀⠐⢄⣀⣼⢄⣀⠘⠀⠀⠀⠀⠀⠁⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠇⠀⣦⢈⠦⠠⠀⠀⠀⠀⠀⠀⠀⢄⠈⠢⣘⡐⠁⢀⠂⠀⠀⠀⣀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⡆⠈⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠁⠂⠻⠄⠂⠁⠀⠀⠀⠀⢆⠈⠐⢄⠀⢸⠀⡀⠔⠈⢀⠅⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠰⠀⠀⠀⠀⢀⠀⠈⠐⢄⠀⡇⢀⠀⠊⠀⢀⠂⠀⠀⠀⢠⢀⠀⠀⠸⠀⠀⡀⢀⠀⠀⠀⠈⠂⡀⠀⠑⣌⡐⠀⠀⡠⠊⠀⠀⠠⠊⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢂⣇⠊⠀⢀⠠⠃⠀⠀⠀⠀⠀⢄⠈⠢⣘⡐⠈⢀⠂⠀⠀⠀⠀⠀⠈⠁⠂⢺⠐⠂⠁⠀⠀⠀⠀⢀⠀⠄⠀⢀⠀⠘
⠀⠀⠀⢀⠠⠤⢀⠀⠀⠀⠀⠁⠀⠐⡟⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⠒⠂⠁⠀⠀⠀⠓⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⢀⠡⠔⠁
⠀⠀⡔⠁⠀⠀⠀⠑⡄⠀⢠⠀⠀⠀⠇⠀⠀⢤⣀⣀⠀⠀⣀⣀⡀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠄⠲⠄⠀⡠⠢⡀
⠀⠀⠆⠀⡀⠀⡀⠀⠄⠀⠠⠀⢀⠀⠀⢀⠀⡔⠀⢸⠀⢨⠀⠀⠁⠀⠌⠀⠀⠀⢠⠀⡇⠀⠁⢄⠀⠀⠀⠀⡁⠀⡄⠂⠀⢀⠜⠀⠀⢀⠀⡀
⠀⠀⠈⠢⠄⠁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠂⠐⠀⠀⢀⠘⠀⢀⠉⡄⠀⢂⠁⠀⠀⠀⠑⢤⠀⠀⠂⠀⠀⠀⠰⡁⠀⠀⠀⢇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⢀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠘⠇⠀⠠⣄⠀⠀⠀⠀⠇⠀⡔⢄⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡇⠀⢸⠀⢸⠀⠀⢀⢀⠎⠈⠀⠉⡀⠀⣠⠀⠀⠀⠱⠀⠀⠀⣆⣀⠀⠀⠢⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠑⠒⠋⠀⠈⠀⠐⠁⠀⠀⠀⠀⠀⠈⠀⠑⠒⠊⠀⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀
⠀⠀⠀⠲⢀⡠⠌⠆⠤⡍⠀⢀⡀⠀⠀⠰⠀⠀⠀⢀⢀⠀⠀⢀⣀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠘⠀⠀⠀⠲⠄⡀⠌⠤⠴⡉⠀⢀⡀
⠀⢠⡀⣠⠋⠠⠀⠒⠠⡀⠡⠊⠀⠀⠀⠀⠀⠀⢋⠀⠀⢂⢀⠂⠀⠨⠃⠀⢀⠀⠀⠈⠺⠀⠉⠀⠀⡁⠀⠈⠀⠀⠀⠀⢀⡀⢀⠊⢀⠄⠀⠠⡈⠫⡈⠁
⠀⠀⠀⡃⠀⡁⢌⢈⠄⠰⠀⡉⠴⠂⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⡰⠁⢀⠊⠀⠈⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣚⠀⡁⢐⠉⠆⠀⠀⣃⠲⠆
⠀⠐⠂⠐⣄⠐⠠⠠⠔⠁⣴⢢⡀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠐⢃⠀⠀⠢⠀⠠⠂⠀⢀⠃⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠑⣀⠑⠄⠀⠤⠃⣠⢢⡀
⠀⠀⠀⢠⠄⠃⠖⡔⠲⣙⠀⠀⠁⠀⠀⠐⠄⠠⠀⠀⠀⠀⠀⡀⠀⠒⢄⠀⠀⠀⢀⠠⠈⢂⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⠍⠓⠆⡤⠲⣘⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⢹⣿⣿⠉⠉⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣰⢋⣿⣿⠁⢀⣴⣤⠀⠀⢀⣾⣿⡶⣄⣠⣶⣦⠀⢰⣶⣶⠀⢠⣶⣦⠀⢠⣿⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢀⡼⠁⢸⣿⡟⢠⣾⣿⣿⠀⠀⣼⣿⡟⠀⣿⣿⣿⠃⠀⣾⣿⡟⢠⣿⣿⡿⠀⣸⣿⡇⣠⠟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⣸⣿⣇⡟⢸⣿⡟⠀⢠⣿⣿⠀⠈⣿⣿⠇⠀⣼⢹⣿⣇⡞⣸⣿⡇⢀⣿⣿⣴⠃⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡟⠀⠀⠀⣿⣿⣿⠁⠈⣿⣿⢀⡾⣿⡏⠀⣰⣿⣿⢀⡼⠁⣸⣿⣿⠃⢻⣿⡇⢸⣿⣿⠿⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣼⡿⠃⠀⠀⠀⢿⣿⡇⠀⠀⠻⡿⠋⠀⠻⢿⠾⠁⠹⠿⠟⠁⠀⠻⢿⡿⠀⠘⣿⡇⠹⢿⡟⠀⠈⢿⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⣀⣠⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⣤⣀⠀⣠⣾⠛⠉⠉⠻⣦⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⢿⣿⡇⠀⢸⣿⣿⠃⣿⣧⣶⡶⠀⠀⢸⡇⠀⣼⣿⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠋⠀⠀⢸⣿⡇⢀⣿⣿⡿⠀⣹⣿⣿⠇⠀⠀⢰⣧⠞⣿⣿⠁⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡞⢸⣿⠃⠀⢿⣿⣿⠛⠛⠛⡿⠁⢰⣿⡏⠀⣼⣿⡇⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠁⣿⡟⠀⠀⠘⣿⣿⣦⣤⠞⠁⠀⢸⣿⣧⠞⠹⣿⡀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⣿⣁⣀⣀⣀⣀⣉⣉⣀⣀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠛⠛⣻⡟⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⣰⡟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣦⣶⠀⠀⢠⣿⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠃
⠀⠀⠀⠀⠀⠀⠛⢻⣿⠉⠉⠀⠀⣸⣇⡜⠁⠀⠀⠀⢀⣴⣦⣴⣶⠀⢠⣶⣄⣤⣤⡄⠀⢠⡿⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣿⠿⠛⢿⡇⠀⣰⡟⠁⢈⣿⣿⠀⢸⣿⡏⠁⢹⡟⠀⢸⣿⠲⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣾⣿⠀⠀⢸⡇⢀⣿⠀⢀⣼⣿⣿⠀⣸⣿⡇⠀⢸⡇⢠⣾⡇⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⡿⠋⣼⣿⠀⠀⣿⣷⡟⠻⣷⡿⠋⠸⣿⡿⠋⣿⡇⠀⢸⣿⢿⣿⡇⠀⠸⣿⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⣴⠋⠿⢿⡗⠀⣿⡇⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⣿⠀⣸⡇⠀⠀⣸⡇⢠⡿⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣿⣇⣴⢿⣧⣠⣼⠟⠀⢸⣧⣴⠏⢸⣧⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⠟⠁⠀⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⠀⣤⣤⠀⣀⣀⠀⠀⢠⣤⡀⣤⣄⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣴⣶⠀⠀⣿⣿⠀⣿⣿⡌⣿⣿⣧⠀⣼⣿⢁⣿⣏⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⣿⣿⢸⣿⣿⣇⢿⣿⣿⣇⣿⣿⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣷⣶⣿⣿⣸⣿⣿⣿⣸⣿⣿⣿⣿⣿⢸⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡏⢉⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⢸⣿⡇⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠸⠿⠇⠘⠟⠟⠿⠟⠛⠿⠿⠿⠛⠈⠛⠛⠸⠿⠃⠀⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⢠⣿⡷⣠⣶⣿⣶⣄⢰⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⣾⣿⢱⣿⡟⠉⢿⣿⡜⣿⣇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢸⣿⡇⠀⢸⣿⡇⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢸⣿⣇⠀⣸⣿⠃⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⢿⣿⣦⣿⡿⠀⢿⣿⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣉⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢻⣿⠉⠀⠀⣿⡇⠀⢸⠀⠀⠀⢸⣿⡆⠀⠀⢸⣿⡆⠀⡇⠀⢸⣿⠀⠀⡖⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⡜⣿⡇⠀⠀⢸⣿⣧⠀⡇⠀⢸⣿⠀⡜⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡷⠤⢼⠀⠀⠀⡧⢾⣿⠀⠀⢸⢻⣿⠀⡇⠀⢸⣿⣼⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣿⡇⠀⢸⠀⠀⢠⡇⢸⣿⠀⠀⢸⠈⣿⡇⡇⠀⢸⣿⠘⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⡇⠀⢸⠀⠀⢸⠀⠈⣿⡇⠀⢸⠀⢿⣷⡇⠀⢸⣿⠀⢱⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⣟⠀⠀⠀⣿⡇⠀⢸⠀⠀⡜⠀⠀⣿⣇⠀⢸⠀⢸⣿⡇⠀⢸⣿⠀⠈⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠈⠀⠀⠁⠀⠀⠉⠉⠀⠈⠀⠀⠉⠁⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣴⣷⡄⠀⠀⣿⣿⠀⠀⠀⢀⣴⡿⠋⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠁⢸⣿⡇⠀⢠⢿⣿⡆⠀⠀⣼⣿⠁⠀⠈⠃⠀⠀⣿⡆⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠎⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⣇⠌⠀⢸⣿⡇⠀⠀⣸⣿⣄⡠⠀⠀⠀⠀⢻⣇⡠⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⠀⢀⠆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢉⠑⡄⠀⠀⠀⠀⠀⢀⠔⠊⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠁⠀⠀⠀⢠⠊⣁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠒⠒⠒⢒⠖⠓⠒⢒⣿⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠁⠀⠀⠀⠀⠐⠐⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠁⠀⠆⠀⡄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠠⠐⠀⠐
⠀⠀⠀⠀⠂⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠑
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁⠀⠣⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢱⠢⠈⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠁⠀⠀⠀⠂⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠍⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⢠
⠀⠀⠀⠠⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⠄⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⢀⠅⠄
⠀⠀⠠⠀⠀⣀⠂⠁⡄⠠⠄⡁⠂⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⣀⠃⠰⠁⠀⠈⡀
⠀⠈⠐⠈⠀⠀⠂⠐⢄⠀⠠⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⡀⠂⠀⡀⠀⠀⠀⠈⡀⠀⢈⠄⠤⠁⠈⠃⠄⠂
⠀⠀⠀⠀⠠⢘⠀⡀⡈⠁⡂⠀⠀⠀⠀⠄⠁⡄⠈⠀⠀⠁⢠⠀⠠⠀⠀⠀⠀⠠⢐⠃⠀⠀⠆⡀⠂
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⢀⠀⠀⡀⠀⡀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠐⠂
⠀⠁⠀⠀⠀⠀⡀⠄⡐⢀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⢁⡀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠡⠊⠜⠠⠈⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠈⠌⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣾⠃⠀⠜⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣸⡏⢠⠊⢀⡾⠋⠀⠀⠀⠀⠀⠀⣸⡏⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⣿⠀⠀⠀⣿⠇⠁⠀⣾⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⣿⠀⡀⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣸⡏⠠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠀⢸⡟⠀⠀⢸⣿⢀⡀⠀⣀⡀⠀⢠⣄⢠⡄⠀⣿⡇⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⢾⡇⢰⡇⣾⠇⠀⡿⢸⡇⢠⣿⣣⠟⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢺⣿⢸⣧⠜⠓⢿⡄⠘⠃⣼⡇⢾⣿⣿⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠜⠈⠋⠀⠉⠀⠀⠀⠀⠀⠀⠹⠟⢸⡿⠘⢧⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢶⡄⠀⣄⢰⠁⣠⡔⡄⠀⣴⡄⢠⡄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⠈⣿⡀⢱⣿⠀⠖⢠⣿⠀⣿⠇⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣿⣰⢻⡧⠈⣿⢀⠃⠸⣿⡰⣿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠼⣿⠀⠈⠁⠀⠀⠈⠀⢻⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⢠⠤⡀⢠⠈⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣼⠀⢰⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⠀⡠⠚⣆
⡇⠸⠇⠀⢿⡄⠘⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡘⠆⠁⣠⠋
⠃⠀⠈⠀⢀⣀⣀⣀⠛⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⠀⠘⠛⠀⣼⠁⠀⢀
⠀⢃⠀⠀⠀⠀⠀⠈⠛⢀⣤⢠⡇⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠉⠘⠂⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠘⠧⣄⠀⠹⠀⠀⠀⠀⠀⠠⠄
⠀⠈⢦⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠐⠶⠀⠀⠛⠀⠉⢈⡵⠈⠹⡆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠳⣤⡀⠀⢀⡤⣤⠁
⠀⠀⠀⠉⠲⣄⠀⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⣀⣀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡼⠃⠀⠈
⠀⠀⠀⠀⠀⠈⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠉⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠛⠀⠀⠀⢀⡛⠀⢠⡤⠚⠁
⠀⠀⠀⠀⠀⠀⣗⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠋⠉⠉⠉⠉⠁⠒⠲⣤⡀⠀⠀⠀⠀⠲⡅⠀⠀⠀⠀⣴⠋⠀⢀⡎
⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⠆⠀⢀⡴⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⢣⠀⢠⠞⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠹⡀⠀⠀⡀⠀⢀⡞⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⣼⠦⠶⠛⢻⠇⠀⠀⠀⠀⠀⠀⡠⠒⠲⡄⠀⠀⠀⢀⣠⠾⠀⠀⠀⢫⡀⠀⢹⣦⠛⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⢁⣼⠀⠀⠀⠘⣇⡀⢸⡟⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⠀⠀⠀⠀⢸⣿⡗⠙⠛⠛⠛⠉⠀⢀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠙⡿⢦⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠀⠸⣿⣷⣤⣤⣤⣤⡶⠛⠋⠉⢹⡟⠁⠀⠀⠀⢷⡤⠛⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣷⠀⠀⠀⠀⠹⣿⣿⣿⠿⠟⠐⠒⠒⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣆⠀⠀⠀⠀⠈⠻⠇⣀⣀⡠⠶⠞⠁⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠙⠶⢦⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠳⠾⠙⠛⠛⠛⠋⠁⠰⠶⠶⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠈⠙⠛⠛⠛⠲⠦⢤⣤⡤⠤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠐⠛⠋⣉⣁⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠳⣤⡤⠔⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡤⠴⠶⢶⣶⠶⠖⠛⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠈⠉⠉⣿⠉⠙⠛⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⠤⣼⡁⠀⠀⠀⠀⠀⠀⡰⠓⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢘⡇⠀⠀⠀⢀⡤⠚⠀⠀⠀⢠⡇
⠀⠀⣀⠀⡀⣀⣠⡴⠖⠉⠀⠀⠀⠀⡀⠘⣧⣀⠰⠊⠉⠀⠀⠀⠀⠠⠺⡇
⠀⢘⠋⠤⠀⣀⣈⣡⡀⣀⡀⠀⢀⡠⠚⠛⠁⠉⢠⡀⠀⠀⠀⠀⡜⠁⣀⠃
⠀⠈⠑⠂⢤⣤⣤⣤⣤⣤⠀⠠⣏⠑⠂⠠⠀⠀⠀⣴⠄⢤⣤⠘⠓⢠⠌⠁
⣤⣤⣀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣉⣤⣤⣶⣶⣶⣤⣶⣤⣤⣌⣁⣠⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣠⣤⣤⣤⣄⣠
⣿⣿⣿⣿⠿⠛⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⣤⣌⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢛⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣿⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠀⠉⠁⠀⠀⠀⠀⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⠆⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⠱⠀⠈⠔⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠃⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⢣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢦⢀⣀⢄⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡆⠀⠀⠀⠀⠀⢴⠒⠚⠉⢀⡠⡾⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⢶⠒⠚⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⠲⠒⠁⠀⠀⠀⢀⣴⠏⠀⢀⡤⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⢀⣀⣠⠴⡟⠴⠒⠊⣁⡠⠟
⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠝⠋⢅⣀⡠⠃⡖⠂⠉⠁
⠀⠀⠀⠀⠠⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⡏⠀⠉⢉⣠⢴⠆⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠞⠁⠒⠒⠒⠒⠉⠑⠒⠉⠉⢀⡞⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⠬⣉⠑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠉⠓⠢⠤⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠺⣆⠀⠀⠀⠀⠁⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣯⠙⡖⠒⠒⠃⠤⠤⠔
⠀⠠⢄⠀⠀⠀⠀⠀⢀⡔⢉⡴⠊⠉⠹⠦⠖⠒⠒⠓⠓⠂⢄⡈⠉⠉⠛⠒⠒⠒⣃⣀⠀⠀⠀⢀⠏⣸⠁
⠠⡑⢄⠀⠒⠤⣀⢠⠎⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢇⠀⠀⠀⢀⡏⣀⠠⠔⣚⠖⠁
⠀⠘⢄⠳⣀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⡼⠒⠒⠒⠉⠁
⠀⠀⠀⠙⠢⡑⢤⡀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣤⣼⣿⣿⣿⣷⣾⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠟⣁⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠁⠚⠉⠀⠀⠀⠀⠀⠀⠉⠃⠀⠸⣿⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⠋⠀⢠⣤⡀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠙⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣸⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠀⠀⢀⢆⠠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⡶⠖⠻⠁⢀⡀⠔⠈⠀⠀⡠⠂⠀⠈⢞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣠⠄⢴⡈⠠⠠⠠⣦⠒⠁⣀⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⢿⣧⣾⡇⠀⠀⠀⢹⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⠄⠺⣇⠀⠀⠀⠀⠋⠈⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠃⠈⠁⣀⠠⠄⠛⠒⢺⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠢⠐⠉⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣤⣤⣤⣤⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⠛⠈⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⢰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣆⣀⡇⠀⠀⠀⡀⣠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢯⣷⣿⠟⠀⠀⠀⠀⢿⣿⣴⣛⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡨⠊⠀⠀⠀⠀⠀⠈⢛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⣿⣿⣻⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿⣻⣛⣿⣟⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣀⣤⡀⠀⣟⣷⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣴⡿⠀⠀⠙⢧⣽⣆⢻⣇⣴⣋⡼⠃⠀⠀⢀⠄⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⢻⡿⠛⢿⣿⢷⣦⣤⣤⣀⣈⣉⣁⠀⢉⣁⣀⣠⡤⠞⠉⢀⣿⠏⠀⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣾⡇⠀⣿⠃⠀⠀⠀⠉⠉⠙⠛⠛⢛⡛⠋⠉⠁⢀⣀⠀⢸⡏⣀⣀⠙⠲
⠀⠀⠀⠀⠀⠀⠠⡴⠚⡇⠀⣿⠇⠀⣿⠜⢹⡇⠀⠀⣴⠞⠋⣵⡿⠁⢠⣶⡰⣻⡿⢀⣿⡗⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⡷⠚⠉⠀⣿⡄⠀⠿⠀⣾⡇⠀⣼⢃⡤⠊⣿⠁⣠⣾⡿⠁⡿⠁⡹⠻⢷⡟⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⣄⠘⢷⣄⠀⠀⢿⡤⠞⠉⠁⠀⠀⠙⠊⠁⠙⠃⠀⠙⠊⠀⠀⠈⠳⣤⠴⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠓⣮⣷⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣙⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠴⢡⠟⣡⠟⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⠀⠀⠀⠀⣼⣿⣿⠀⢀⣴⣾⣿⠿⣂⣤⣤⣤⣾⣿⡇⠀⠀⠀⠀⣾⣿⣧⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢀⣴⢿⣿⣿⢆⣾⣿⣿⠁⠀⠹⣿⡀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡟⢠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠧⠴⠞⠁⢸⣿⣿⢸⣿⣿⡏⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⠀⠀⣸⣿⣿⡇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⣼⣿⡿⠈⣿⣿⣷⠀⠀⢠⣿⠇⠀⢿⣿⣿⠀⠀⢠⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⠉⣻⣿⣿⣅⠀⠘⠻⣿⣷⣤⡾⠟⠀⠀⠈⠻⠿⣷⡶⠟⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠤⠷⣾⣈⣉⢉⡁⣿⢥⣅⠀⢸⣧⣤⠁⠀⠀⠀⣋⣚⢛⡂⠀⠀⠀⢀⣛⣛⡛⣀⠀⠀⠀⠀⠀⣿⢾⣦⡄⠀⠀⣼⣾⡿⡇⠀⢠⢶⣤⡄⠀⠀⢴⢳⢶⡼⠂
⠀⠉⠉⠈⢉⣿⣛⡏⠉⠈⠉⠀⢨⡟⣛⡂⠀⠀⠀⠇⢹⠻⠇⠀⠀⠀⠿⠿⠻⠄⠩⣅⠀⠀⠀⠐⣁⣰⣋⣐⢀⠀⣨⣤⣤⡀⠀⢨⣿⣭⠃⠀⢨⣭⠿
⠀⠀⠀⠀⠀⣭⣭⡅⠀⠀⠀⠀⣈⣦⣾⣿⡂⢉⣭⣉⢿⣤⠀⠀⠀⡀⣼⡏⠁⠁⡾⣾⡇⠀⠀⠀⠹⡏⠻⠼⠻⢧⠟⠀⣀⡀⠀⢀⡽⠾⠶⠦⠩⠍⠁
⠀⠀⠀⠀⢰⣰⣶⡆⠀⠀⠀⠀⢈⣱⡯⠄⠁⠀⠀⠒⠚⣧⠀⠀⣈⣷⣏⡱⢤⢶⣦⣼⣳⠀⠀⢘⣛⢿⠄⠀⢲⣤⣆⣶⣸⡆⠀⢸⡃⠿⠽⠄⠀⡹⡭⣁
⠀⠀⠀⠀⢈⣋⢹⡅⠀⠀⠀⠀⢨⢻⣥⡀⠀⠀⠀⠥⠤⠬⠀⢀⣉⣉⡿⠉⠈⠈⠩⣮⢹⣄⡀⢠⡷⠘⠇⠀⠀⠁⢉⣉⢉⠁⠀⢠⣅⣬⡁⠀⠀⠈⠷⢟⠾⣀
⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣤⣤⣀⠀⠀⠀⣤⣤⠠⠄⠀⠀⢠⠀⡶⣦⢦⣤⠀⠀⠀⠀⢠⡤⠄⠀⠀⠀⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⣉⠀⠀⣨⡭⣬⠄⠀⣤⣧⣤⠀⠀⠠⢦⣤⣤⠀⠀⠀⣦⣿⣴⡆⠀⠀⠐⢺⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠟⢧⡟⠑⠁⠀⠀⣏⡻⠿⠀⠀⠀⠀⣈⠛⣒⡀⠀⣠⣄⣠⠀⠀⠀⢠⣮⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠛⣿⠁⠀⠀⢠⡛⣇⣥⠀⠀⠀⠀⣀⣒⣚⠀⠀⢼⣿⣽⡆⠀⠀⢀⣦⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣙⣝⠀⠀⠀⠀⠛⢫⡿⢧⣤⣤⣤⣦⡄⠂⠀⠀⠠⠤⡤⡶⡄⣴⣶⣾⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠀⠀⠀⠀⠀⠀⠃⠀⠰⠶⠖⠘⠁⠀⠀⠀⠀⠀⠘⠺⠾⠌⠇⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⢀⢄⡸⠀⠐⢅⡤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠄⠀⠀⠀⠀⠄⡻⠦⠙⠁⡠⠚⠒⡄⠈⠃⢖⢠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⣔⠦⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠪⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢕⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠉⡄⠀⢀⠀⠀⠂⠀⢢⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠆⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡂⢠⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠙⠛⠶⣄⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠀⠀⠀⠀⠠⠁⢈⠀⠀⠀⠀⠛⢈⠇⠀⠀⢄⠄⢀⠏⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⡀⡀⠁⠐⠚⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡰⠂⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄⢴⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠈⠁⠄⠠⠀⠠⠀⠠⠤⠡⠈⢢⠈⠢⠀⠀⠄⠁⠀⠀⠊⠀⠁⠌⠑⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠤⠀⠀⠢⡀⠀⠀⠀⠔⠁⡀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠈⠲⠤⡤⠤⠴⡊⠀⠐⠂⢬⢶⢰⠎⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡠⠁⠀⠀⠀⠀⠙⠒⡒⠂⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠄⠀⠀⠤⠀⠈⠧⠀⠚⠏⢈⣟⠚⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠸⠁⠀⠁⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠄⣠⠐⠓⡂⠀⠀⢐⠀⢀⠂⠀⠀⢄⠀⠙⠃⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠃⠀⠘⠲⠄⠀⠀⠀⢀⠀⠀⢀⡤⠊⠁⠀⢀⠂⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠨⠀⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠐⠀⠃⠀⠀⠀⠀⠀⣆⠀⢀⡆⠀⠀⠀⠀⠀⠄⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⣀⠃⠀⠀⠀⠀⠀⡂⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠈⠳⠀⠀⠀⠀⠀⠶⠂⠀⠰⠤⠀⠀⠀⠀⠠⠬⠉⠀⠉⢥⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡇⠀⠀⠀⠀⢀⠎⠀⡜⠀⠀⢀⡆
⠀⠀⠰⠒⠉⠉⠉⠉⠉⠉⠉⠑⠒⢢⠤⢤⡧⠔⠊⠀⠒⠒⠤⠤⡾⠔⠊⢀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣸⠖⡧⣖⣆⣴⠖⣦⢾⢔⠃⠀⣇⡠⠞⡇⣜⡷⣇⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠚⢒⠞⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠐⠀⠀⠀⠀⢃⠀⠀⠀⣠⠀⡀⣤⡀⠀⠠⡆⠀⣀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣀⡀⠀⠀⢳⢦⢸⠒⠽⠘⡅⡉⠁⠀⠀⠀⠀⡏⠀⡠⠁⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠣⠉⠀⠀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⡁⠈⠀⠀⠋⠀⠁⠀⠀⠐⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠠⠐⠂⠀⠉⠀⠀⠀⠀⠀⢸⠀⣇⢀⡇⡘⡆⢸⠀⡇⣧⠎⠀⠀⠀⠀⢀⣠⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠸⠀⠏⠀⠇⠁⠹⠼⠈⠇⠏⠣⠀⠀⠀⠀⠈⠉⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⠐⠆⠀⠀⠀⠀⠀⠀⠀⢦⣠⢂⠒⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢘⡀⣠⢀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢄⡛⠲⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⠀⠀⠄⠀⢄⣀⠀⠀⠀⠃⠈⠂⠂⠘⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢸⠐⠆⠀⠂⠀⠠⠀⠄⠀⠀⠰⠠⠀⠀⠀⠀⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣛⠈⠀⡤⠘⠃⠈⠋⠐⢤⠀⠀⠘⠃⠆⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢂⡀⠛⠀⠀⠤⠀⠀⠀⠀⣠⠘⠀⠠⠆⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠨⡄⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⣏⠉⣹⣛⣛⣛⣛⣛⣛⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⢀⣤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢠⣿⠃⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣁⡀⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⡄⠀⠠⢼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣧⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⢠⡀⠀⠂⢸⣿⠇⠀⠀⣸⡿⠠⠂⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢠⣿⠃⠀⣼⣷⠀⠀⢸⣿⠈⠀⣾⣿⠀⠀⠀⣿⣧⢰⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⣿⡏⠀⠀⡸⣿⡀⠀⢿⣿⠀⠀⠼⣿⠂⠀⣿⡏⠀⠀⡐⢿⣿⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠔⠀⠈⠁⠀⢿⡇⢀⠌⠀⠈⠉⠀⠸⣿⡠⠂⠀⠉⠀⠀⣿⠇⠀⠀⠀⠈⠁⠀⠻⣧⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⡀⠀⠀⢀⣾⠟⠠⣄⠀⠀⣀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢸⣿⡇⠀⠀⠉⠃⠀⠀⠀⠀⠿⣇⠈⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⡀⠤⢀⠀⡀⠴⠀⢄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⡿⠁⠀⢸⡄⠄⠀⠉⠀⠨⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠟⠁⠀⠀⠀⠳⣄⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡴⠛⠁



⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣻⣿⣿⣿⣿

⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⠋⠙⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀
⣼⣿⠃⠀⠀⠀⠀⠀⢀⣴⡆⣠⠞⠉⠀⠀⠀⠀⠀⠼⠿⠃⠀⠀⣀⡤⠤⠄⠤⢤⣀⠀⠀⣠⡾⠋⠀⠀⠀⠈⢣
⠻⣿⣄⠀⠀⠀⠀⢠⣿⡿⠋⠁⢀⣴⠞⠉⠉⢳⡀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠉⣾⣿⣅⣀⣀⣀⣠⠴⠋
⠀⠀⠉⠙⠛⠛⠉⣿⡿⠁⠀⢰⣿⠃⠀⠀⠀⡸⠀⠀⠀⠀⠙⠛⢀⣤⡄⣠⢴⣶⡀⢰⣿⡏⠀⢀⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣿⣿⠀⠀⠀⠊⠀⣠⣶⠞⢛⠆⠀⣾⣿⣷⠏⣼⣿⠃⣼⣿⣧⠞⢩⣿⡇⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⣿⢀⡴⣿⡆⠀⣿⡿⢸⣿⡇⢀⣿⣿⠏⣼⣿⡟⠀⣿⣿⣯⡴⠟⠉⠀⡴⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⣿⠎⢰⣿⠇⠀⡿⡧⠚⠻⠧⠞⠛⠋⢰⣿⡿⠀⡼⠛⠛⠻⠷⣤⡤⠞⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⠞⠈⠋⠀⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢁⡼⠁⠀⠀⠀⢀⡾⠋⣿⣷⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠅⠈⠀⠀⠀⠀⢀⣀⣤⣀⡀⠘⠧⠚⠀⠀⣠⣄⠀⠘⣧⡀⠉⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠞⠲⠶⣤⡀⢰⣿⣥⣶⠌⠻⣦⡀⠀⠀⣼⣿⡏⠀⠀⠈⢉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⣸⠇⢸⣿⣿⡃⠀⠀⠘⣷⣀⣴⣿⣿⠃⢠⣶⡿⠛⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠁⢀⣿⣿⡟⠛⠶⠶⠶⢿⡟⢹⣿⣿⣰⣿⠏⠀⠀⠀⠘⣿⣿⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠠⠤⠴⢶⣦⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⡿⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⣿⠁⢸⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⢀⣾⠇⠀⠀⠈⢿⣿⣇⠀⠀⠀⣼⠏⠀⠀⣿⣿⠁⠀⠀⠀⠀⢠⡿⠃⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠈⠛⠿⠦⠴⠞⠁⠀⠀⠀⢈⣉⡤⠤⠤⢤⡤⠿⠤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠴⠚⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠤⠀⠆⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠿⠛⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣍⠀⠀⠀⠀⠀⠀⠀⣦⣰⡆⢸⣇⡌⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠸⡇⢻⠀⠀⠀⢠⣴⣷⢀⣿⡇⢧⠟⢿⢷⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⡞⣧⢸⠏⢹⣼⠉⠁⠀⡀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⡇⠘⠇⠀⢠⠖⠛⠲⣄⢹⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠥⣀⡸⠀⢧⠀⠀⠀⣸⠀⠑⠒⠋⢀⣤⣄
⠀⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⠑⠒⠚⠁⠀⢀⣠⣶⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣶⣾⣿⡿⠿⠟⠿⠛⣟⣿⣟⣛⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡀⠚⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠈⢿⣿⣿⣿⡍⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠂⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣂⣠⣤⣶⣶⣶⡆⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠤⠤⠖⢒⣛⣛⣉⣭⣭⣭⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢄⠀⠀⢠⠀⣿⢿⠲⣤⣀⡀⠀⠀⠀⢀⢶⠀⠀⡶⡀⠀⠀⠀⢀⣀⣤⠖⡿⣿⠀⡄⠀⠀⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⡀⠀⠠⠘⣮⠿⢧⣘⡍⠳⣦⡀⣞⡟⠄⢨⢻⢡⢀⣴⠟⢩⣁⡾⠿⣴⠃⠄⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠂⠈⠀⠀⠀⠁⠈⠳⣦⠘⠛⢲⣯⣷⠳⣥⠞⠻⣯⠞⣞⣽⡖⠛⠃⣴⠞⠁⠈⠀⠀⠀⠉⠐⠂⠀⠈⠈
⠀⠀⠀⠀⠀⠈⠷⣄⣀⠀⡀⠀⣄⢲⡓⠀⠒⠀⠓⠌⠛⠻⠷⠶⠿⢧⠽⠀⡀⠋⠸⠿⠖⠾⠟⠛⠁⠊⠀⠓⠀⢒⡖⣠⠀⣀⠀⣀⢠⠾⠁
⠀⠀⠀⠀⠀⠒⠒⠊⠉⣥⣍⠂⠛⠃⠙⠧⣀⣂⡶⠛⠁⢀⠄⠀⢠⡧⣶⣶⣿⠷⡱⡀⠐⠦⡐⢬⠛⢶⣐⣀⠼⠋⠘⠓⠰⣭⣬⠉⠐⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠉⢈⠁⠀⠀⠀⠀⢀⣴⠞⡙⠀⠀⠐⢣⡤⠀⡎⠹⣿⣔⣶⡿⠇⠸⠌⢥⠀⠀⠀⠀⢉⠳⣦⡀⠀⠀⠀⠀⠨⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠡⠖⣃⢀⣤⣾⠿⠃⠀⠀⠁⢀⠜⣁⠄⠀⠣⠤⢙⣿⡋⢀⡰⠀⠈⠀⠀⠵⠀⠈⠀⠀⠘⢿⣷⣤⡀⡘⠲⠌⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠘⠟⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠃⠈⠁⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⠀⠀⠖⣠⡄⠄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⠀⠀⠀⢀⡄⢠⠏⡀⢀⡀⠀⢀⠀⠀⢠⠎⣀⠀⠀⠀⡀⠀⡀⠀⣀⠀⢀⡀⢀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⡞⢠⠏⡜⢠⠋⡴⢡⠏⡸⢡⠟⡀⠉⠀⠀⣰⢃⡼⢁⡞⣀⢀⡞⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠖⠋⠀⠉⠀⠁⠈⠀⠁⠉⠀⠁⠈⠀⠁⠀⠀⢀⠀⡴⠃⠀⠁⠁⠈⠁⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠞⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⡤⢒⡐⠢⠹⣿⣷⣀⠀⠀⠀⠀⠰⡒⠄⠀⠀⢀⣀⠤⣤⠄⠠⢄⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⣠⣾⣿⡏⠔⢂⡒⢤
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠃⠀⠀⠙⠻⣧⡀⠄⠂⠀⠈⢢⣤⠀⡇⢀⣾⡿⣷⣄⡀⢀⡰⣠⠀⠈⠀⠐⠠⢀⣼⠟⠋⠀⠀⠘⠉⠐⠁
⠀⠀⠀⠀⠀⡀⠀⠒⠀⣦⣤⡀⠀⠀⠀⣈⠽⠶⣄⠀⠈⠣⠄⠀⢱⣞⣿⣬⣭⣛⢇⠌⠐⣀⠄⣀⠀⣠⠷⠯⣅⠀⠀⠀⢀⢤⣴⠀⠒
⠀⠀⠀⠀⠀⠉⣉⡅⠶⠂⠶⠁⡿⢣⠞⠃⠀⠁⠉⠛⠂⢄⣀⢀⣀⡣⣈⠉⠝⠋⢊⣀⢈⣁⡂⠐⠛⠉⠈⠐⠈⠳⡜⢿⠉⠶⠐⠶⢨⣁⠉
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠉⠋⠀⠙⠀⢋⣴⠛⠛⢫⣭⢿⢃⣾⡀⢀⣷⡘⡿⣭⡝⠛⠛⣦⡑⠀⠃⠀⠉⠉⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠈⠁⠰⠄⠀⢀⠂⣴⣋⣡⠜⡟⢉⣷⠏⡯⡦⡝⢻⢱⢽⠹⣾⡉⢻⠧⣌⣙⣦⠑⡀⠀⠠⠆⠈⠁⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⢠⠀⣾⣱⢛⣧⠼⠖⠛⠁⠀⠹⣻⠀⠀⣏⠆⠀⠈⠛⠲⢦⣽⡛⣌⣧⠀⡄⠀⠠⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠠⡷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠄⠀⠀⠀⠀⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⢀⠠⠄⠀⠀⢤⠄⡀⠀⡀⢄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢅⡂⣡⠔⢋⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⡙⠦⣈⢰⡨
⠀⠀⠀⠀⠀⠀⠀⡠⣫⣞⠥⠚⠑⠈⠀⠀⢀⡶⠛⢻⠄⠀⠀⠀⠀⠀⠁⠊⠓⠬⣳⣝⢄
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⣿⢸⣷⣼⡇⠀⣸⣿⡄⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⡀⠀⢀⠎⡤⡀⠀⠀⠀⠀⣿⢸⡟⣿⡇⢀⣿⢻⣇⣿⣤⡤⣿⣤⠄⠀⠀⠀⢀⢬⠱⡄⠀⢀
⠀⠄⢔⠠⡀⡎⡜⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠻⣤⡆⠀⠀⠀⠀⠀⠀⠀⢂⠂⢣⢳⢀⠄⡢⠠
⠀⠀⢸⢰⢸⢡⡿⢑⣀⣀⢀⡀⣀⢀⡀⣀⢠⡞⣉⣟⢀⣀⡀⠀⣀⣘⢛⣷⡀⣀⣀⣀⡊⢿⡌⡇⣆⡇
⠀⠀⠘⣿⣸⠞⠀⠈⢹⡋⢸⣧⣿⢸⡇⣿⣾⣷⣿⢛⣘⣿⡇⢰⣿⢛⢸⡏⣷⡟⢸⡷⠄⠀⠳⣇⣿⠃
⠀⠀⠀⢸⣿⡀⠀⠀⠘⠃⠘⠃⣿⢘⡋⠛⠙⠃⠛⠛⠉⠛⠃⠀⠙⠛⠛⠃⠛⠃⠘⠛⠃⠀⢀⣿⡇
⠀⠀⠀⠀⢻⡨⡖⠀⠂⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢀⡶⠛⡷⠀⠀⠀⠀⠀⠀⠐⠀⢲⢅⡟
⠀⠀⠀⢀⢚⢧⢇⡏⠀⠀⠀⢿⡿⠒⡗⣸⠂⣾⣷⢰⣷⣼⢻⣷⡿⢸⣿⠆⠀⠀⠀⠀⢹⡸⡼⡓⡀
⠀⠀⠀⠀⠈⠻⣞⢇⠀⠀⠀⢸⡇⢠⣟⢻⢠⡿⢿⣸⡟⣿⣸⣿⣧⢰⣿⠇⠀⠀⠀⠀⡸⡳⠟⠁
⠀⠀⠀⠀⠐⡸⢞⢫⡧⢬⡢⢒⡀⠀⠀⠸⣧⣶⠘⢧⣴⠀⠀⠀⠘⢦⡶⢀⡒⢔⡥⢸⡕⡳⣇⠆
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡑⢤⣳⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡠⢊⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⣦⣌⡑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢊⣡⣴⠭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⠤⠤⠤⠤⠤⠤⠴⠓⠊⠁

⠀⠀⣠⠖⠒⠒⢦⣀⣀⠀⠀⠀⢀⣠⣀
⠀⢰⠁⠀⠀⡞⠁⠀⠈⠱⡄⡴⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡿⡇⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠉⠲⡄
⠀⠀⠈⠲⣄⣠⠟⠢⣤⠞⢀⣧⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⡆⢸⡄
⠀⠀⠀⠀⠀⠻⡇⠘⡟⠀⡽⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡴⠉⠀⠀⠀⠀⠀⠉⠲⡄⠀⡇⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣼⣣⠞⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⣾⠁⠀⠀⣀⠔⢁⣠⠔⠀⣹⠀⠸⡄⠀⠀⠀⠀⠆⢀⡞
⠀⠀⠀⠀⠀⡔⠲⡿⠋⢠⠔⠋⠉⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢴⣮⠭⣿⡖⣊⣉⡀⣈⡩⠤⠔⠊⠟⣆⠀⠘⢄⡀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢇⢀⣽⠀⡏⣠⠴⠖⠒⠒⠒⠒⠦⡀⢷⠀⠀⠀⢀⠀⠀⠀⠀⢯⡇⠠⠔⠲⠄⠀⠘⠉⠉⢸⠃⠀⠀⠀⢹⣻
⠀⠀⠀⣠⠔⣿⣿⠀⠀⣷⣟⠒⠒⠒⠒⠒⠒⠒⣇⣾⠀⠀⠀⠻⣉⡗⠀⠀⠀⠙⢄⠀⠀⠢⠤⠔⠂⣠⠏⠀⠀⠀⠀⢀⠧⣀
⠀⠀⠀⠁⠚⢹⢻⠀⠀⣿⢉⡭⡛⠒⠒⠚⠻⠥⠈⣽⡄⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⢑⡄⢤⠄⠐⠞⢹⣦⣀⣀⡠⠤⠺⣯⡼⠃
⠀⠀⠀⠀⠀⠛⠈⢧⠠⡻⡄⠀⢀⣀⠁⠀⣠⠀⣰⠃⠙⡄⠀⣠⡤⢤⠀⠙⠲⠤⣤⠚⠙⠪⡖⠐⡖⠁⠀⠙⣄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣇⠈⠒⣤⣬⣭⠥⣴⣊⣁⣀⣤⡿⠔⠓⠤⠼⠃⠀⢀⠔⢧⠀⢀⠀⡇⠀⠱⣀⡤⠴⠉⣆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠉⠉⡇⠈⠉⠀⠈⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡉⠉⢹⠧⠤⠔⢻⣀⡴⠚⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢹⠇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠊⠀⡀⠀⠀⢻⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⠛⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⢧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⠖⠋⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡦⠇⠘⠦⡤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢀⣀⣀⣀⣀⣤⣴⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣇⠀⠉⠉⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣵⡇⠀⠀⠀⠾⣩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣰⣶⡄⠈⠛⠓⣰⣦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠞⠓⠀⠀⠀⢸⣿⣦⠀⠿⠋⠁⠀⠀⠈⣻⣿⣿⡷⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠄⠠⣄⠀⠀⠈⣭⠉⠐⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠶⠀⠀⣿⣶⡄⠘⡇⠀⠀⠛⠃⢀⡤⠄⠀⠀⡀⠘⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠉⠀⢰⠇⠀⣤⡄⠀⢾⠀⣀⡀⠀⠈⢣⡀⠓⡆
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⣿⣿⢺⣿⣄⣀⣀⣠⣤⣀⠀⠀⢀⣠⠴⠋⠀⠀⣿⡇⠀⠈⠻⢿⠿⠀⠀⠀⢳⠀⠑
⠀⠀⠀⠀⠠⣿⣄⠀⠀⠀⠀⣿⠇⢸⡟⠀⠀⠈⠉⠉⣉⣉⠉⠁⠀⠀⠀⠀⠀⡿⠐⡄⠀⠀⠀⠀⢠⠆⠀⢸⡇⣠⣦⡀
⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⣿⠀⢸⡷⠂⠐⣦⠰⠿⠏⠉⣷⢹⡟⠁⠈⣷⠀⣇⣰⠃⠀⠀⣾⣿⣗⠀⠀⣸⠃⠿⢛⣄
⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢀⣀⠤⣿⣾⡇⠀⠀⣿⡆⡿⣄⠀⠀⠀⠙⠛⠉⠖⣰⠏⠀⣀⠛⠛
⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⣿⠀⢸⡇⠀⠀⣿⡇⣏⠀⢀⣿⢹⡇⠀⢸⣿⠁⣿⠈⠓⠄⠀⣀⡠⢴⣞⠁⠀⠀⢿⣿⡄
⠀⠀⠀⠀⢀⣀⣀⡈⠉⠂⠀⠈⠀⠀⠀⣤⡀⠈⠀⢀⠀⢁⡤⠂⠢⣤⠀⣬⣤⠀⠀⣠⡄⠀⠀⠀⣼⡿⠀⠀⠀⠈⡉
⠀⠀⠀⠐⢿⡿⠋⢀⣤⣶⠒⠀⠀⠀⠸⣿⠇⠀⠀⢸⠇⡟⠀⠀⠀⢸⣇⣿⡟⠀⠀⣿⡇⠀⠀⢠⣿⠇⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⣾⠂⠀⠀⢿⣄⠀⣠⣾⠀⢷⠀⠀⠀⣸⡟⣿⠁⠀⢀⣿⡇⠀⠀⣼⡿⠀⠀⠆⠉⠀⣠⠄
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣰⣶⣦⠂⣀⡀⠈⠉⠁⢸⠀⠀⠑⠀⠴⠛⠀⠘⠳⠴⠚⠙⠃⠀⠠⠿⠃⠀⢀⣴⣄⣬⡉
⠀⠀⠀⠀⠀⠣⠀⠰⠾⠷⠙⠿⠟⠈⣉⣉⠀⠀⠀⣾⠀⠀⠀⠀⠀⢴⠀⣀⠀⠀⠀⠀⠀⣠⣴⡀⠀⢠⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⡿⠢⢄⠀⠀⠀⠀⠀⠈⠀⠒⠄⠀⠀⠛⠿⠃⠀⠀⠹⡿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠇⠀⠀⠙⠲⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣠⠟⢀⣼⣷⣿⡀⠈⠉⢁⠀⠲⢾⣿⡷⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠚⠁⠀⠀⠹⠿⠿⠁⠀⢀⣿⣧⠀⣦⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠒⠒⠒

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣤⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿
⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡻⣦⣀⠀⢰⡆
⠀⠀⣿⠀⣰⡿⢃⣴⡿⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⣮⣿⣧⣼⣧
⠀⢰⣿⣾⣿⣷⣿⣿⣿⣿⢃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⡿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⠟⣹⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣼⢇⣾⡆⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣠⡀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⣿
⣿⡿⠃⣰⣿⣿⣿⣿⠏⠀⠀⠰⣦⠀⣼⣟⣾⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⢾⡆⣿⣷⣿⣿⡆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⠀⠙⢿⣿⣿
⠉⢀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⢻⣼⣷⣿⣿⣿⣘⣿⡀⣠⡄⠀⠀⠹⣿⣿⣿⣿⣷⡀⠀⠙⢿
⣠⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣿⠇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⣁⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣄⣻⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣘⣿
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠏⠀⠈⠉⠘⡀⢰⠂⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠐⠂⠈⠀⠃⠀⠀⠀⢀⢡⣀⡄⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⢈⠀⠀⠀⡑
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡘⠀⠀⠀⢹⣆⠀⠀⠀⢀⣮⢬⠇⠀⠀⣾⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠂⠀⠀⠀⢀⠀⠠⠐⠈⠀⠀⠐
⢴⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣤⡴⠊
⡄⠂⢀⣠⡄⠀⠀⠘⣏⡐⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣖⠲⠀⠀⠀⠆⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠀⠀⠀⢰⠿⠛⠉
⠀⣨⢁⣱⡆⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢸⠀⠀⢀⠇⠄⢡⣀⣀⣸⠀⣄⣀⣀⡇⢠⠀⠀⠀⡞⠀⠀⠀⢠⠋
⠀⠉⠁⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣤⣿⡵⠔⢖⠆⣼⣉⠉⠉⠀⢀⠮⠦⢄⡄⣶⣀⣔⡍⢁⣀⠉⠉⣴⡃⠦⢄⣀⠟
⠀⠀⠀⠀⠀⠄⢀⣀⠤⢼⠀⠀⣢⡽⠴⡜⠸⠟⠈⠉⠉⠁⠉⠉⢀⠄⠁⠀⠀⠀⠈⠢⠀⠀⠠⠇⠀⠈⠉⠛⢢⡤⢔⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠦⠥⠖⠃⠀⡸⠢⡀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢀⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠐⢆⣉⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣇⡀⠁⠀⠀⠀⡆⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠑⠞⠑⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⡅⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡇⠉⠀⠀⠀⣻⡟⠀⠀⠀⢸⢃⡞⣱⠃⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠒⠂⠀⠀⠀⢠⢡⠁⠀⠀⠀⡽⠁⠀⠀⢀⡈⡈⢠⠇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢆⠘⠦⣀⣀⣀⣀⣠⣴⠋⠯⠈⢆⠀⠀⠀⠀⠀⢀⡔⠀⣯⠖⠒⢬⡽⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⢉⣉⣉⠍⠀⠀⠈⠲⢔⣀⣀⡠⠴⠚⠁⠀⠘⢄⡀⠉⠒⠒⢒⡶⠋⡠⠪⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣅⠀⠑⠤⣀⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡀⠀⠀⠀⠀⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⢀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠔⠆
⢀⡀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠤⠀⠖⠂⠀⠀⠈⠉⠁⠒⠒⠊⠋⠁⠀⠀⠁⢸⠠⠤⠐⠒⠈⠁⠀⠁⠈
⠆⠁⠔⠀⠘⠀⢀⠠⠀⠤⠤⠤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠴
⠀⠐⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡔⡈⠰⠄⠀⠁⠌⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠤⠤⠐⠒⢒⡔⠃⠀⠈⠐⢲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢈⠀⠸⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢈⡀⠀⡆⠀⢀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠉⡋⠉⠀⠀⠈⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⣃⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠹⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⢘⡆

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⡗⠀⠀⠀⠈⢉⠃⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣠⣿⢇⣼⠃⣿⡀⣾⠀⣿⢸⣧⢸⠙⣴⠏⠀⣴⡟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣻⡿⢩⣿⢃⡼⠃⠀⣿⠛⢸⣸⣿⣼⡏⣿⠀⣿⠀⣼⡟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣾⣿⠞⣁⡀⠀⠉⠀⠈⠈⠀⢉⣁⠈⣀⠉⢰⣿⡴⢃⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢸⡿⢡⢞⣿⠃⢠⣶⠻⣤⣦⠀⣼⢃⣾⡿⠀⣿⡟⣵⣫⠾⠃⣸⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⡷⠃⣾⠇⣠⣿⢃⡼⣿⠇⣠⡿⠋⣿⣠⣾⣿⣾⣏⠁⣠⠜⠉⣿⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣰⡟⠀⠀⠛⠀⠀⢿⡾⠋⠻⠋⠀⢿⡴⠃⠀⠀⠉⠉⠘⠟⠀⠛⠛⠁⡴⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⡳⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡼⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠿⠟⠛⠋⠉⠉⠉⠠⡀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠙⠛⠻⠿⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⢀⠀⠀⠠⠀⠀⠈⡃⠀⡀⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠀⠈⠁⠠⠞⠛⠳⠂⠀⠐⠚⠀⠈⠖⠛⠂⠀⠐⠆⠀⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠐⠣⠠⠜⠉⠰⠤⠤⠁⠈⠀⠀⠀⠀⠠






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠒⠒⠊⠁⠀⠀⠀⠀⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠄⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠂⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠖⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠀⠠⠁⠀⠀⠀⢠⠦⡀⠀⠐⠠⠀⡀⠀⠈⠉⠁⠀⠀⡩⠀⠀⠀⠀⠀⠰⠢⠔⠘⠐⢃⡮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠑⠂⢄⣄⣀⣀⢀⣀⣠⡔⢺⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⣶⠤⢤⢀⠀⣀⣠⡛⢡⡌⢊⡷⠊⠀⣋⠏⠀⠀⠀⠀⠀⢀⠎⠀⠈⠁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢒⢢⠤⡴⣽⠀⠶⠀⡩⠗⢀⠉⠶⡤⠤⣢⠃⠀⠀⡇⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠆⠀⠈⢓⡒⠚⠋⠀⠠⠴⠀⠀⠀⠮⠹⢀⡜⠀⠀⣀⠤⠒⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠧⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠪⠗⠲⡍⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣠⡜⠢⠤⣄⣀⣀⣀⡀⠤⠴⠊⢡⠃⠀⢀⡁⠘⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠸⢀⡀⠀⠀⠑⢄⠈⢢⠄⠲⠀⠃⡤⠔⠉⢡⠀⢃⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠆⠀⠈⡗⠂⠐⠁⠁⠺⡀⠀⡏⢠⠂⠀⠀⠀⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠀⠀⠀⠀⠀⠀⠙⠢⡝⠞⡐⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⡇⠀⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡌⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠇⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⣮⢀⠀⠠⠤⠤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠰⡘⠾⠁⠀⠀⠀⠉⠀⣀⣀⡀⠷⠠⠤⠤⠀⠒⠠⠜⠒⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠄⠀⣸⠀⠀⠈⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢢⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠐⠞⣰⢠⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢱⡋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣠⠟

⠀⠀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠑⠀⠄⠀⠀⡀⠀⠀⠀⠘⠠⠈⢀⠀⢠⠀⠠⠀⠀⠠⠄⠀⠐⢀⢀⡁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⢠⠈⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠙⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣠⠀⠀⠀⡁⠀⠀⠀⠉⠉⡁⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⠀⠈⠀⢀⠁⠀⠀⠀⠀⢉⠀⠀⠠⠁
⠀⠀⡁⠀⠔⠀⠀⠄⠀⠀⠀⢀⢐⣈⡨⢀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠅⣁⡒⡐⠀⠀⠀⠠
⠀⠀⠀⠈⢀⠀⠀⠓⠀⠀⠀⠁⢊⡒⡄⠂⠀⠀⠀⠀⢀⠐⠀⠀⠀⠠⠀⠀⠀⠄⠀⠠⢂⠒⡃⠄⠀⠀⠀⠐⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⠈
⠀⠠⠀⡑⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠄⠔⠀⠀⠀⠢⠰⡊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠘⠀⠀⠈⠀⠀⢀⠀⠀⠀⠈⠉
⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠖⠀⡀⢀⠁⠀⠄⠂⠀⠀⢤⠀⠀⠀⠠⠀⢀⠀⠄⠀⢩⠀⠀⠀⠊⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢀⠀⠃⠀⠀⠐⠀⠤⠀⠂⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠉⠐⠀⠠⠂⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⢀⠠⠄⡀⠀⡀⠤⢀⠀⢀⠀⠈⠀⠀⡀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠠⠄
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠁⣀⠈⠂⠄⠀⠄⠪⠇⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠪⠇⡀⠀⠀⠄⠈⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⡞⠁⡹⠀⠀⢀⡤⠖⠦⣄
⠀⠀⠀⢀⠴⠒⢉⣩⠶⢾⡏⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⢀⣈⠉⠉⣿⠉⢉⣀⠀⠀⣟⠀⠀⠀⠈⡆
⠀⠀⠀⠘⠶⠚⠋⠁⠀⢸⡇⠀⣿⠀⢀⣀⠀⠀⣴⠢⣼⠇⠁⢠⢀⡎⣼⠃⢀⣿⡴⢍⠈⠁⠀⠈⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣰⠋⢸⠃⢸⣇⡼⣿⠀⢠⠿⡿⠀⣿⠀⡜⢻⠁⠈⠳⣄⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠸⡏⠀⣿⠀⡜⠉⠁⠘⠦⠊⠀⠀⠀⠙⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠴⠦⣄⠈⠚⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⠀⠀⣾⠀⠀⣰⣿⠀⢠⡤⡀⠀⣠⣶⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡼⠀⢠⡿⠀⣰⠃⢿⠀⢹⡧⢼⠞⢹⡇⢠⠟⣇⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣀⠜⠁⠀⠀⠳⠞⠁⠀⢸⠀⠈⢧⠞⠀⠘⣧⠎⠀⠹⣄⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⡿⠛⢋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠘⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⠀⣿⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⡇⠀⣿⢀⣤⠀⣠⢴⡄⠀⣿⣠⠎⣿⠇⢻⡔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠊⣿⢰⣇⡀⣿⡀⠿⠁⢰⡟⠀⢸⣇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠛⠀⠛⠈⠉⠀⣀⠀⠀⠀⠘⠓⠀⠈⣿⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⡆⢰⡟⢯⣀⢰⡿⠀⣾⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⢼⡇⠘⣧⣀⠍⠸⣇⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠠⠒⠂⣶⠀⢠⠊⠉⣩⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⡎⠉⠒⢄⠀⠀⢨⠈⠀⣿⠀⡆⠀⢰⡇⡠⠁⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡄⠀⠀⠂⠉⣼⠁⠀⠀⠀⠀⠀⢈⢻⣆⠀⠀⡇⠀⠀⠈⢳⡄⢨⠀⢠⡿⠀⠀⠀⢸⠟⠀⣰⡟⠁
⠀⠀⠀⡀⠀⠤⠐⠂⠀⢈⡭⠈⠀⢰⡇⠀⠈⠀⠀⡿⠀⠀⠂⠀⣴⢄⠀⠀⢻⡆⠀⠁⠀⣄⠀⠀⢻⣼⠀⢸⡇⠀⠀⠀⠀⠀⠰⣯
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⢸⠀⠀⡀⠀⢸⡇⠀⡀⠀⠠⠇⠸⠀⠀⠈⣿⠀⠀⠀⣿⠣⠀⠀⠟⠀⢸⡇⢸⠀⢀⣷⡄⠀⢻⡆
⠀⠀⢠⡴⡄⠀⠀⣶⠚⠛⠁⠀⠀⠈⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠇⠀⠀⠀⣾⠁⠀⠀⢸⡇⢀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡤⣶⠶⠀⠀⣾⠁⡀⠀⢠⡻⠛⠛⠆⠀⠀⢸⡇⠀⠀⣿⠀⠘⡀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠆⢐⣸⣧
⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⡏⠀⢰⠀⠀⣿⠀⠃⠀⣸⠃⠀⠀⠀⠀⣀⣸⠁⣀⣤⣿⠀⠀⠵⠶⠶⠿⠀⠐⠛⠛⠃⠀⠈⠉⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⡇⠀⢸⢀⣀⣟⠀⣠⡤⠿⠀⠀⠀⠐⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⡄⠀⠀⠂⡌⠉⠀⣾
⠀⠀⠀⠀⢀⣀⣠⡇⠀⠀⠤⠶⠛⠃⠀⠈⠋⠉⠁⠀⣀⣀⠀⠤⡄⠀⠀⡠⠄⠒⣤⣀⠀⠀⢀⠂⠁⢰⡆⠀⠀⡄⠀⠀⣿⠀⠀⠘⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⠉⣵⠀⠀⠘⠀⠀⠀⣼⠁⡠⠊⠀⠀⠀⠈⠹⣦⠀⡈⠀⠀⣼⠁⠀⠀⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⢻⣆⠀⠆⠀⠀⢸⡏⡐⠁⢠⣾⠛⠢⡀⠀⢸⣇⠃⠀⢠⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠹⠟⠀⠀⠀⣾⢠⠀⠀⣿⠁⠀⠀⢱⠀⠀⣿⠄⠀⢸⡧⠀⠀⡄⠀⠀⣾⠅⠀⠀⣵⣶⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢸⠇⡄⠀⢸⡇⠀⠀⠀⡈⠀⢰⣿⠀⠀⠤⠄⠀⠸⠁⠀⠸⠟⠀⠀⠀⡤⠖⠓⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⡿⠀⡇⠀⠸⡇⠀⠀⡐⠀⠀⣼⢿⠀⠀⠐⠲⠛⠀⠀⢲⡶⠂⠀⠰⠂⠀⠀⠀⠆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣸⠃⠀⢃⠠⠀⠀⠒⠊⠀⠀⣼⠏⠘⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢀⢀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢰⡟⠀⠀⠀⠢⣤⣀⠀⠀⣠⡾⠋⠀⠀⠳⢦⣤⣤⣤⠾⠋⠀⠀⠀⠀⠀⠙⠛⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣤⣥⡾⠁⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠄⠤⠐⠤⠀⠄⠂⠔⠂⠄⠂⠔⠂⠐⠉⠀⠀⠚⠃⠑⠀⠓⠀⠑⠀⠑⠈⠑⠀⠁⠀⠁⠀⠀⠀⠈⠒⠈⠀⠀⠁⠀⠁⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⠙⠓⠛⠋⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠶⠲⠦⣤⠴⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣤⣤⠴⠖⠋⠀⠈⠙⠶⢤⣤⣄⠀⠀⠈⢻⠶⣤⠖⠋⠁
⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⡇⠀⠀⠦⠀⠀⠀⠠⠆⠀⠀⡿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠹⡄⠀⠀⣀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⢦⡀⠘⠓⠂⠀⣠⠼⠛⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡟⠙⠓⠛⠛⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡋⠀⠉⠓⣦⡶⠋⠁⠈⣳⣀⡀
⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠈⠙⢦⡀⣴⠁⠱⡀⣠⠞⠉⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⡀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣿
⠀⠀⣀⠴⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠃⠀⢰⠋⠉⠙⢷⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⢤⡀
⠴⠋⠁⠀⣠⠖⠒⠶⣤⠼⠒⠲⢦⡀⠀⢸⡀⠀⠹⣄⠀⢀⡼⠁⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢷⡀⠀⠉⠉⠉⠀⠀⣰⠋⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣽⠳⢦⡤⣤⠶⢻⡅
⠀⠀⠀⠀⠀⠙⠳⢦⣀⡤⠖⠋⠁⠀⠀⠀⢰⠇⠀⡾⠀⠸⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣟⡀⣸⠃⠀⠀⢻⡀⣘⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣉⣹⣏⣀⣀⣀⣈⣿⣉⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈⠉⠉⠉⠀⠈⠴⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠀⣤⣀⠐⠆⢄⠂⠀⠀⠀⢀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⠉⠁⠀⠑⡐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠂⠀⠈⢦⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰⣧⠀⠀⠀⢰⣦⡄⠀⠀⠀⠑⡦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠀⠀⠀⡀⠀⠀⢀⠀⢨⠀⠁⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣦⡀⠀⠀⠉⠁⢠⡄⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠙⢥⠀⠺⣶⡶⠟⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⣀⠈⠉⠁⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠘⠡⠀⠤⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⠤⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠄⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⡀
⠀⠀⠀⢤⡀⠀⢀⠠⠘⠈⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠃
⠀⠂⠐⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠂
⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠲⡄⠀⠀⢨⠁⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠰⠶⠇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠇⠀⠀⠀⡇⠘⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢀⣠⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⢲⠂⠋⠀⠀⠈⠳⠒⠲⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠈⠁⡇⠀⠀⠀⠀⠉⠉⢋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠎⠀⠀⢀⠀⠀⠀⣰⠀⠀⠀⡠⠀⠀⠀⠀⠈⢑⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣀⣀⣀⡀⢀⡀⣀⣠⠇⠀⢰⡋⠀⠀⠀⠀⠉⠀⣀⡠⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣾⣿⣯⣶⣶⣶⣶⣷⣷⣷⣶⣿⣿⣶⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢀⠄⢸⡀⡀⢀⢀⠄⡀⠀⢀⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⣀⢄⢠⢰⣉⠀⡇⢁⢘⠀⣙⢈⣢⢸⠁⢸⢸⢸⠒⠁⠀⢸⠃⠀⡇⡈⢁⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢸⠀⠘⠌⠠⠀⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠊⠈⠤⠌⠘⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡤⠤⡤⠜⠙⠉⠓⢲⣀⠀⠀⠀⠀⠰⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⢀⠔⠃⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠀⠋⠙⠊⠳⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⠔⠋⠁⠀⠀⠀⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⡈⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢐⠉⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠠⢌⣒⡲⠄⢺⠁⠀⠙⠀⠀⠀⣠⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣄⣀⠦⠬⠭⠹⣇⠀⠀⠀⢀⡠⠟⠁⠘⠤⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣸⢸⠀⡝⣿⢿⠀⠀⠀⠸⢤⣀⡶⠞⠀⠀⠀⠀⠀⠈⠉⠓⠋⠒⠩⡃⠀⠀⠀⢦⣀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⡏⡼⢠⠃⠉⠀⠀⠀⠀⠀⢸⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢅⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢰⠃⡾⠀⠀⠀⠀⠀⠀⠤⡄⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡆⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠜⠀⠀⢴⠆⠀⠀⠀⠀⢁⠡⡀⠀⠀⠀⠠⣞⡀⠀⠀⠀⠀⣤⠀⠀⠈⢆⣀⠀⡀⢀⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⢷⢤⡀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⢀⠴⠩⢋⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⣦⠤⠤⠤⢤⢴⣖⠉⠁⠴⡄⠸⣗⡲⠤⠄⢤⡤⠤⠤⠤⢶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠡⠀⢻⡇⠀⠀⡾⠞⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠧⠴⠟⠁⠀⠀⠟⠃⠰⢷⠄⢀⡀⠀⢤⠄⢢⢄⡘⠚⠷⡀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠦⠀⣀⡀⠈⠙⠀⢸⠀⠀⠉⠉⠂⠼⠟⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⣠⠄⠀⠉⠀⡒⠦⢼⣀⢀⠀⡀⢀⠀⡀⠠⣐⠢⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠩⠧⠤⠾⠦⡤⠴⠾⠳⠈⠀⠑⠚⠂⠈⠁⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠀⠈⡀⠀⠀⠈⢂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡄⠌⠸⡄⡤⠂⠀⠀⠐⠤⢠⠃⠈⠢⡄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⠀⢠⣀⠧⡀⠀⠀⢠⠲⠴⠀⠀⡠⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠐⠀⣇⢯⣳⠲⣤⣄⡀⠀⢠⣺⠀⢀⣓⡄⠀⢀⣠⣤⠶⣞⡽⣹⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠄⠀⠀⠈⢷⡛⠳⣴⣌⠻⣆⡇⢡⣂⣐⠍⢲⣰⢟⢡⣦⠞⢛⡾⢁⠂⠀⠠⠄⢀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠠⡙⢯⣤⣤⡛⢺⡌⢿⠁⠈⡯⢡⣗⢛⣤⣤⡼⢋⠄⡀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⢀⡈⣙⠐⠂⠰⠀⣱⡜⢦⡀⠀⠀⣠⡤⠀⠉⠉⠈⡝⢉⣀⣤⣄⢌⠁⢀⠉⢀⢤⣄⠀⠀⢀⡴⢣⣎⠀⠆⠰⠆⣋⢁⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠘⠋⠀⠀⠀⠀⣈⡶⠞⠁⠀⣰⠉⡀⡨⢻⣿⣫⣯⣼⡆⢂⠂⠁⠈⠁⠈⠳⢶⣁⠀⠀⠀⠀⠙⠃⠠
⠀⠀⠀⠀⠀⠀⢀⠔⣠⠴⠀⢀⣠⣾⠏⠀⠀⠀⢀⠔⠉⠁⠣⣀⠻⣟⠿⠉⢀⠈⠀⠁⠀⡀⠀⠀⠂⠹⣷⣄⣀⠀⠦⣅⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠊⠸⡿⡟⠀⠀⠀⠀⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠆⠑⠂⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠈⠁⠐⠂⡤⠄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⢠⠁⡜⣁⢀⡀⠀⢀⠀⡀⢠⠋⡄⠀⠀⢀⡀⣠⠀⡠⠀⢀⡄⣠⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠃⡜⢱⠃⡎⢰⢣⠞⠸⢡⠏⡄⠀⠀⠀⠟⢰⠃⢼⠡⠀⡞⠰⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⢰⣷⣧⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡆⡠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠢⠙⠒⠀⠈⠙⢿⣆⠁⠀⠀⠈⠢⣀⡀⡔⠉⣴⣯⣶⣀⠈⢀⠀⡀⠀⠁⠀⠀⠈⣰⡿⠋⠁⠀⠖⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠂⢠⣄⠀⠀⠀⠀⢉⠷⢶⡀⠀⠺⣀⠁⢑⣼⣿⣙⣛⢿⠆⠌⠄⡀⢀⡀⢀⡶⠞⡉⠀⠀⠀⠀⣠⡄⠐⠂
⠀⠀⠀⠀⠈⢁⣩⠰⠆⠰⠀⡹⢢⠞⠁⠀⠀⠙⠓⠀⣀⡀⣀⣣⢈⠉⠛⠋⢂⡀⢈⣀⠀⠚⠋⠀⠀⠈⠳⡜⢏⠀⠆⠰⠄⢍⡈⠁
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⣡⣞⠛⢛⣥⢼⢃⣼⡀⢀⣷⣘⡧⣬⠛⠛⢳⣌⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠂⠀⠠⢀⡾⣥⡴⠺⢃⣴⠏⠧⣒⠍⠩⣂⢼⠹⣦⡙⠟⢦⣬⢷⡀⠄⠀⠐⠂⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⣎⣞⡽⠴⠛⠋⠁⠀⠘⢽⠁⠀⡯⠃⠀⠈⠙⠛⠦⢯⣣⣱⠀⠀⠀⠐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠲⠂⠈⠉⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡁⠀⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠸⠇⢀⠀⢀⡠⠄⢲⡾⠉⠉⠉⠉⠉⠉⠉⠀⠒⠒⠂⠠⠤⢈⣳⠀⠀⠀⠀⢄⠀⠀⠃
⠀⠀⠀⠀⠆⠀⣾⠞⠁⠀⠀⢈⡴⢂⣈⠁⠢⡀⠀⠀⠀⠀⡠⢊⣡⡌⠑⢤⡈⠑⠤⣀⠀⠀⠀⢸
⠀⢀⡤⠞⠂⢴⡇⠀⠀⠀⠰⠋⣰⣿⣯⣴⣦⣸⡆⠀⠀⢠⣧⠼⠿⠷⠷⠤⣵⠀⠀⠘⡷⠀⠀⢸
⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠙⡗⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠸
⠀⠀⠀⢇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⢀⡏⠀⠉⢿
⠀⠀⠀⠘⢧⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⣏⠃⠠⠴⠦⠖⠘⢋⡼⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠳⢤⣀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⠴⠋⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣐⠂⠲⡄⠀⢀⠔⠁⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠳⢦⣀⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢈⣤⡁⠀⠘⠣⠚⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⡶⠶⠶⣶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠀⠀⠰⣉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠇⠀⠀⢷⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠈⠯⣀⠀⠀⠀⢀⣨⠋⠈⠁⠀⠙⠛⠒⢶⡆⠠⠤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣤⡤⣤⠞⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠤⠤⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸⡧⣤⣤⠤⠤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠘⢄⢸⣧⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢶⡿
⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⢃⠀⠀⠀⠀⠀⢀⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢤⣀⡘⠀⣴⠀⠘⣧⣈⡄⠀⠀⠀⢀⡏⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠉⠛⠦⠻⡤⠚⠀⠈⠀⠀⠀⠀⣘⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠶⠒⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠠⠴⠶⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⣀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣰⠇⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⢀⠴⠚⠓⠠⣄⠀⠀⠀⠀⠀⠀⠀⡌⠉⢻⡄⠀⠀⠀⠀⠀⢸⠇⠀⣠⣤⠤⠴⠦⠄⢈⣦
⡰⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⢒⠦⣷⠀⠀⠀⠀⠀⢸⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣧⣀⣀⣀⣀⣀⣀⣀⣘⣁⢀⠀⠀⠀⢘⠋⠀⣼⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣶⣶⣶⣶⣶⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣧⣱⠀⣾⠁⡴⠄⣾⠡⠤⠃⠶⠆⠰⠢⠜⠤⠉⣇⠰⡄⠘⣧⠐⣇⡅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠈⠛⢀⣉⣠⣼⣤⡴⠤⣏⠀⣸⠉⠟⠀⠘⠟⠙⣆⢠⣙⠤⣤⣼⣥⣄⣉⠐⠛⢀⣶⠀⡀
⠀⠀⠀⠀⠀⣤⣀⠀⠺⣀⣈⡡⢼⡷⠚⡋⠉⡤⢠⡎⢠⠞⠁⣐⡉⠁⠌⠀⠀⠠⠊⢉⣆⠈⢲⡀⣷⠀⡤⠉⣙⠓⢾⡥⣈⣁⣀⠇⠀⣠⢤
⠀⠀⠀⡀⢀⠉⢉⣀⡲⠀⢸⠀⢠⡤⠰⠃⠠⠖⠈⠃⠘⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠁⠙⠀⠰⠆⠹⠆⣤⠄⠸⡆⠐⢖⣀⠉⠉⠀⣀
⠀⠀⠀⠉⠉⠃⠀⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠁⠉
⠀⠀⠀⢼⠿⠿⣿⣿⣿⣿⠛⠿⡿⢸⣿⣷⣾⡆⠐⣿⣿⣿⡿⠀⠀⢀⣾⠿⣿⡇⠀⠀⠘⣿⣿⣧⠀⠀⠀⠸⣿⣿⠐⣿⣿⣿⡗⠀⠀⣰⣷⡗
⠀⠀⠀⠀⠀⠀⣿⠿⢿⣿⠀⠀⠀⠸⣿⠿⣿⡇⠀⣿⠿⣿⣇⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣇⠀⠀⠐⣿⡏⠀⣿⣿⣿⡇⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⣿⠻⢻⣿⠀⠀⠀⢠⣿⣿⣿⣧⣴⣯⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣿⡆⠀⢀⣿⡇⠀⣿⣿⣿⣷⣾⣿⣯
⠀⠀⠀⠀⠀⠀⣿⠻⢻⣿⠀⠀⠀⠀⣿⣿⣿⡟⠉⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⡿⠿⣿⡄⠸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠐⢸⣿⠀⠀⠀⠀⣿⣿⣾⡇⠀⣿⣿⣿⣿⠀⠀⣿⡿⣿⣽⣿⡇⠀⠀⣿⣿⢻⡟⢛⣷⠸⣿⡇⠀⣿⣿⣿⡿⣿⡟⢿⡇
⠀⠀⠀⠀⠀⠀⣿⣴⣾⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣯⣿⠀⢸⣿⡇⣿⡛⣿⡇⠀⠀⣿⣿⠈⢿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠾⢿⣿⠀⠀⠀⠀⣿⣍⣻⡇⠀⣿⣭⣽⣿⠀⣸⣿⠀⢻⣿⣻⣷⠀⠀⣿⣿⠀⠘⣷⡿⣿⣿⡇⠀⣿⣻⣿⡇⠈⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠶⢸⣿⠀⠀⠀⠀⣿⣀⣹⡇⠀⣿⣛⣿⣿⠀⣿⣿⣤⣼⣿⣿⣿⠀⠀⣿⣿⠀⠀⢹⣧⣀⣹⡇⠀⣿⣷⣾⡇⠀⢹⣖⣺⣧
⠀⠀⠀⠀⠀⠀⣿⣴⣶⣿⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣙⣻⣿⢀⣿⡿⠉⠉⣿⣿⣿⡆⠀⣿⣿⠀⠀⠀⢻⣦⣿⡇⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠰⣿⠿⡿⣿⠀⠀⠀⠰⡿⠿⠿⠇⠀⠿⠿⠿⡿⠸⠿⠷⠀⠀⣿⣷⣿⣷⠀⣿⣿⠀⠀⠀⠈⠿⠿⠇⠀⣿⣿⠿⡇⠀⢸⣿⣶⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⢰⡜⠂⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⠀⠀⢠⡶⠶⣶⠀⢀⣠⣶⣶⣦⡀⠀⠀⣴⡦⠀⠀⠀⢠⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⠃⡆
⠀⠀⠀⠀⠀⠈⣗⢡⣤⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⣾⡇⠠⣿⢀⣾⣿⣿⣧⣹⣿⣆⠀⣿⡇⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⣤⡌⢾
⠀⠀⠀⢻⠦⣀⣿⠟⠋⠀⠀⠀⠀⠀⠀⣗⠰⡏⠀⢰⣿⡃⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⢰⣿⡇⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠾⣿⣀⠴⡟
⠀⠀⠀⠀⠘⠻⡇⠀⡰⠛⡆⠀⠀⠀⠀⢻⡀⣇⣠⡟⣹⠇⣼⠃⢹⣿⣮⣿⣿⣿⣿⣿⠸⣿⡇⠀⠀⣸⢿⣿⣿⣿⡇⠀⠀⢰⠟⢄⠀⢸⠗⠂
⠀⠀⠀⠀⠀⠀⢀⣆⡐⢀⠇⠀⠀⠀⠀⠈⠙⠛⠋⢀⣿⣼⠏⠀⠘⣿⣿⣿⣿⣛⣿⠏⠀⣿⡇⠀⣠⡏⢸⣿⣿⣿⡇⠀⠀⠸⡀⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠈⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠶⠿⠟⠁⠀⠀⠀⠈⠻⢿⣿⠿⠋⠀⠀⠘⠿⠾⠋⠀⠸⡿⠿⠿⠃⠀⠀⠀⠈⠈⢻⠁

⠀⠀⠸⣿⣿⡆⢰⣾⠆⢰⣶⣾⠿⠷⠰⠟⢿⣿⣿⠟⠻⠂⣶⢶⡾⠿⠆⢰⣶⣿⡿⢷⣆⠀⠀⠀⣾⣿⡇⠀⠘⣿⣷⡀⠀⢸⣿⠀⣴⣿⢷⣆
⠀⠀⠀⣿⣿⡇⢸⡿⠀⢸⣿⣿⣄⣀⡀⠀⣼⣿⣿⠀⠀⠀⣿⣻⣧⣀⣀⢸⣿⣿⡇⢸⣿⣇⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⠀⢸⣟⢼⣿⣇⠸⠟⠃
⠀⠀⠀⢻⣿⡇⣼⡇⠀⢸⣿⣿⠛⠛⠁⠀⢹⡿⣿⠀⠀⠀⣿⣿⡟⠉⠋⢸⣯⣽⡇⢸⣿⣿⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡆⢸⣷⠘⣿⣿⣦
⠀⠀⠀⢸⣟⣿⣿⠇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣿⡇⢸⣿⡏⠀⣼⡿⣿⣿⡀⠀⣿⠻⣷⣿⣼⣿⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⣿⣻⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣾⣷⡿⠟⠁⠀⣿⠀⣿⣿⡇⠀⣿⠀⢹⣿⣻⣿⠀⠀⠀⠻⣟⣿
⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣟⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣽⣿⣷⣆⣦⢠⣿⠶⢿⣿⡇⠀⣿⠂⠀⢿⣴⣇⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠸⢿⠇⠀⠀⢸⣿⣿⣷⣶⠂⠀⢾⣿⣿⠆⠀⠀⣿⣼⣷⣶⠖⠸⣿⣾⡇⠈⠻⣿⠸⠿⠀⢸⣿⣿⠀⣿⠇⠀⠈⠿⠟⠀⠻⣿⠶⠿⠃
⠀⠀⠀⣀⢀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡀⣄
⠀⠀⠀⠁⠈⣀⣈⠡⠷⠀⠸⡀⠘⠓⠰⡆⠐⠆⢀⡄⢠⠆⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⡀⣠⠀⠰⠂⣴⠀⠛⠃⢰⠇⠠⠦⠉⣀⣀⠁⠈
⠀⠀⠀⠀⠀⠚⠉⠀⢶⠉⢉⡝⣺⡷⢤⣅⣀⠛⠘⢇⠈⠦⠀⠩⣁⡤⠂⠀⢀⠐⢄⣨⠟⢀⠾⠀⡿⠀⠛⣀⣩⡤⢾⡓⢩⡉⢉⡦⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢀⣤⠈⣉⠛⢛⠛⠛⠚⣏⠁⢹⡄⣶⠀⢠⣦⣨⠏⠘⣉⠒⠛⢻⡛⠋⣉⠠⣤⠈⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠏⡹⠂⢿⡀⠱⠂⢿⣑⠒⡔⠲⠆⠰⠖⣪⠒⣈⡟⠰⠃⢠⡟⠀⡏⠃⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣰⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠠⡟⠛⣦⣿⠀⢹⡴⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠐⠦⣄⡀⠀⠀⠀⠀⠀⢀⣀⣹⣄⠈⢿⠀⡾⠁⣠⠇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢾⡀⠈⠙⢷⣼⣾⣧⠾⣻⠟⢻⠟⠿⣷⡺⢧⣀⢀⡤⠀⠀⠀⠀⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠉⣻⠿⢿⡿⠉⠀⠀⠁⠀⠘⠀⠀⠀⠙⠦⠙⢿
⠀⠀⠀⠀⠀⠸⡤⠞⠓⢦⣤⠀⢾⣁⣴⡟⢀⣤⠶⠖⠒⠒⠒⠒⠀⠀⠀⠀⠀⠈⣧⡤⠀⠀⢹⡴⢦⣤⠆
⠀⠀⠀⠀⣤⡾⠁⠀⠀⠀⢹⡄⠀⢁⣼⠐⢫⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⢸⡀⢀⣰⠏⢀⠀⢿⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠗⠀⠀⣧⠀⠀⡏⠀⢸⡇⢸⣧⡄⢠⡾⡆⣷⠶⡄⣷⡼⠁⢸⡇⠀⣿⠀⠘⠃⢸⠁
⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⢹⠀⢹⡇⠀⢸⡇⢸⡇⣿⢾⣠⣧⢻⠀⡟⣿⢳⡀⢸⡇⢠⡇⢀⠀⠀⣾⠒
⠀⠀⠀⠠⢾⡄⠘⠁⠀⠸⠀⢸⡆⢸⡇⠀⠀⠁⠀⠀⠈⠀⠁⠈⠁⠀⠁⠀⠸⠃⢸⢁⡾⠀⠘⠀⢰⠇
⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⠀⢧⠈⣧⠀⠀⢸⠀⢸⢰⡟⠹⡆⡿⠀⣶⠀⠀⠀⣾⣋⠀⠀⣀⣠⠏
⠀⠀⠀⠀⠀⠈⢷⣀⠈⠀⠀⠀⠈⠳⣿⠀⠀⢸⡄⢸⠘⢧⡴⠃⣧⣠⣇⠀⠀⠀⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠶⢾⡄⠀⠀⠙⢻⠀⠀⠴⡴⢾⣅⡈⢁⠀⢸⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣀⡼⠀⠀⢰⡇⢘⣉⡽⠘⠀⣼
⠀⠀⠀⠖⠋⠉⠀⠀⣰⠆⠀⠀⠀⠀⠀⢻⡀⠈⠉⠀⠀⠀⠈⠟⠋⠁⠀⠀⢠⡇⠀⠀⠀⠀⣀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣴⠀⠀⠀⢹⡏⠉⠉⠉⢉⣙⣛⣛⣛⣛⡛⠛⠛⠛⠙⡏⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⢰⠃⠀⠐⠒⢺⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⠛⠓⠂⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄⠀⠀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣾⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⣠⠞⠛⠉⠉⠛⠛⠾⣿⠤⠤⠚⠀⣾⠃⠀⠀⠆⠀⠀⠀⠀⠀⢠⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡄⠀⡘⠀⠀⠀⠀⠀⠀⢹⣀⠀⡠⠀⠀⠀⠀⠹⡄⠀⠀⢠⣾⠟⠁⠀⠀⠀⡸
⠀⠀⠈⣇⠀⠀⠠⢄⠀⠀⠀⢹⣿⡀⠀⠀⢻⣇⠈⠀⠀⠀⠀⢠⡾⠓⡄⠈⠉⢁⢤⡀⠀⠀⠀⠈⠢⣴⡿⠃⠀⠀⣀⡠⠊
⠀⠀⠀⠈⠢⠄⠠⠜⠀⠀⠀⠀⢻⣷⡀⠀⢸⣿⠐⠙⣿⠀⢠⣿⠁⠀⢸⠀⡰⢁⣾⣣⠔⣶⡄⢀⣼⠟⣉⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⣿⡆⠀⣿⡀⢸⡏⠀⣴⠆⡰⢁⣾⠟⠁⢠⣿⢃⣾⡟⠉⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⣷⠀⢿⣆⠼⣧⠞⢿⠔⠁⢾⠏⠀⢀⣾⠇⣾⡟⠉⣯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⢿⡀⠀⢿⠇⠈⠉⡠⠤⣄⠀⠀⠀⠀⠀⠀⠘⠿⠺⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠈⡆⠀⢸⠇⠀⠈⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠘⢆⠀⢸⡇⠀⣰⡆⠀⣾⠓⠀⠀⠀⠀⠸⣇⠀⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⠋⠈⠀⠀⠀⠀⠀⠘⣧⡴⣿⡇⣤⡜⢧⠠⣾⠂⣴⠆⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⢾⡇⣿⠁⡜⠀⣿⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⢠⡖⡄⢷⣇⡀⠀⢀⠂⢸⡇⠈⠋⠀⠀⠀⠀⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠰⣿⠛⠖⢡⣿⠃⠀⢸⣀⣼⠃⠀⠀⢰⡟⢹⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠻⡤⠃⢸⡇⠀⠀⠀⠉⠁⠀⠀⢀⣿⢃⢦⡾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠈⠓⠒⠁⠀⣠⣶⠒⠀⣸⣿⢁⣿⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡀⣿⡇⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢿⣇⣾⠀⣿⡇⣾⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠓⠹⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⡀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠀⣸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⢀⠠⠀⠀⢰⠁
⠀⠀⠀⠀⡲⠀⢰⠲⠞⠀⠀⠀⡴⠀⠀⠀⢀⠠⠰⠲⠆⠀⢧⠀⠀⢠⡴⠄⠀⣆⠀⠀⠸⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡠⠄⠀⢰⡀⢀⣠⣄⠀⠸⠀⠀⠀⡂⠀⠀⡏⠀⠀⠋⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⢸⠁⠀⠀⠸⢀⡈⠃⢀⠀⢰⠀⠀⢰⠆⠀⠀⡇⠀⠀⢺⠉⠳⡄⠀⢀⠌
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠈⠀⠀⠀⠆⠀⠉⠉⠉⠑⠋⠀⠙⠙⠀⠀⠒⠃⠀⠒⠸⠆⠀⠈⠛⠜
⠀⠀⠀⠀⠀⠐⠈⠁⠀⠉⠀⠀⢀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⡠⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⡠⠊⠂⠀⢹⡄⠀⢸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⡀⠃⠀⠼⠀⠀⠀⠤⠄⠀⠠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣴⡄⠀⠀⡇⢈⠒⠤⠀⠚⠁⠀⡀⠀⡀⢀⠀⣰⡆⠀⡠⠊⠉⠷⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠉⠉⠉⠁⠠⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡀⠀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇