Joyous Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠾⠟⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⢳⡶⠰⣋⠀⢲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠀⠀⠀⣠⡀⠄⠒⠉⠉⣀⠈⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢓⣊⣘⠉⡘⠊⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢻⣋⡀⢳⡗⢺⣧⡄⣀⠲⢤⡆⠉⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢶⢿⡆⡇⢀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠁⡶⡼⠷⢾⢱⠓⠃⢢⡟⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⣈⠀⣃⣩⣭⠠⠰⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡘⠈⠂⠀⠀⠓⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠈⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠴⠲⠄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣤⣠⣤⣠⡄⣀
⠀⠀⠀⠀⠀⣶⣤⣶⠀⠠⡆⠀⢸⢹⠀⢰⠄⠘⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢈⠁⢁⠉⡀⠋⣀⠰⡀
⠀⠀⠀⠀⠀⠈⢡⡿⠄⡀⠣⡀⢸⣧⠸⡄⡎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⣖⢂⣦⡈⡏⢾⢺⣸⣍⢸⢿⢘⣧
⠀⠀⠀⠀⠀⠀⠈⠇⠘⠉⣠⠬⡁⣉⠃⣠⡈⠀⠃⢀⠉⡀⠀⠀⠀⠀⠀⠋⠘⠁⠉⡤⣄⣠⡨⣭⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠴⠃⠋⠊⢁⡟⠁⠂⠑⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠋⠃⠻

⠀⠀⠀⠀⠀⠀⢀⣴⣞⣛⣛⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣛⣛⣷⣶⡀
⠀⠀⠀⠀⠀⠠⠞⠉⠉⠀⠀⠉⠙⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠉⠁⠀⠀⠉⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠻⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣓⣒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣒⣾⣶⣤⠀⠀⠀⠀⠀⠛⠛⠳⣄⠀⠀⣠⠽⠛⠃
⠀⠀⠀⠀⠀⠴⣹⣿⡟⠻⣿⣷⢮⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣫⢾⣿⡿⢻⣿⣧⠑⠀⠀⠀⠀⠀⠀⠀⠘⣆⡼⠁
⠀⠀⠀⠀⠀⣠⣯⣿⣿⣞⣿⣿⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣿⢷⣾⣻⣟⣯⣽⣽⣇⡀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠉⠉⠛⠳⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠖⠛⠋⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠴⠖⠚⠛⠛⠁

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣾⢻⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⡆
⠀⠀⠉⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⠀⠀⢀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣤⡴⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⣩⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣤⣀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢉⠀⣠⠴⢿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠲⠛⠲⠶⣆⣀⣀⣄⣀⣠⠟⠛⠋⠉⠁⠀⠀⢻⠟⠀⠀⠀⠀⣀⣤⡾⠁⣠⠶⠾⠷⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠃⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡼⢉⡞⠁⡼⠁⣤⠤⠾⠧⠤⢬⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⡃⡞⢀⡼⠁⣼⠃⢀⣀⣀⣤⣄⣳⣼⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠈⡇⢹⠋⠛⢻⣃⡴⠋⠙⠋⠀⡀⠉⠙⢿
⠀⠀⣀⡴⠋⠧⣄⡀⠀⠀⠀⠀⠀⠀⠘⠷⣖⠶⠲⣦⠶⢤⠴⡴⠲⠚⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠉⠀⠀⠀⣠⠞⠙⠳⢦⣈⡇
⠀⠀⠉⠳⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⠀⠀⠀⠀⣀⣤⣴⣋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠉⠙⠓⠦⠴⠛⠁

⠀⢀⣴⠟⣿⣻⡆
⡜⢻⢏⣴⠯⡄⠙⣦
⠙⢿⠟⠁⠀⠀⠀⣿
⠀⠈⠳⣄⠀⣀⣤⡿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⡀
⠀⠀⠀⣼⠟⠉⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠷⣾⠁⣿⢦
⠀⠀⢨⠿⢂⡴⠞⠛⠛⢻⠀⠀⠀⠀⠀⣠⠾⠛⠛⠉⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠀⠛⠛⠿⣥⠎
⠀⠀⠀⣴⡏⠃⠀⠀⠀⠙⣦⠀⠀⠀⣴⠋⢀⡤⠞⢷⣀⠀⠘⢆⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⢷⣄⠀⠀⢀⣨⠟
⠀⠀⠀⠙⣧⠁⠀⠠⠀⠀⢩⡄⠀⢠⡏⢠⡟⠁⠀⠀⠙⢷⣄⠀⠄⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢶⡄⠀⠙⢧⡴⠋⠁
⠀⠀⠀⠀⣿⠀⠸⠀⠐⠀⠘⣇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠹⣆⠀⣀⣷
⠀⠀⠀⠀⠹⡀⠀⠈⠀⠀⠀⢹⠀⣾⠀⢸⡇⣾⠀⠀⢰⡆⠀⠸⣇⠀⠀⠈⡇⠀⠀⠀⠀⢀⡴⠋⠀⡀⠀⠀⠀⠀⢛⣀⣹⠋
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⣷⡇⠀⣾⠀⠀⠐⠂⢈⡁⠀⠀⠙⢳⡆⠀⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠃⠀⢀⡶⠛⠟⠁
⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠸⣧⡀⢻⡄⠀⢶⡂⣈⡁⠀⢠⡐⣾⡇⠀⣇⣠⠾⢋⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢹⡇⠀⢀⠀⠀⠀⠀⠈⠻⣟⣻⣦⣀⣀⣀⣠⣴⣛⡁⠘⠋⣴⠟⠉⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠐⠀⠀⠀⠀⠈⠉⢿⣈⠉⠉⣉⡟⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⠋
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⠋⠉⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁⠄⢀⡰⠆⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⢹⡄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⡾⠁
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⢘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⡄⣰⠃
⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⢰⡇
⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⣼⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣽⡀
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠈⢳⡄
⡎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠁⠀⠀⠀⠀⠀⢻⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣄⣼⠇
⢸⡷⠀⠀⠀⠀⠀⠠⠀⠄⠀⢀⣀⡴⣶⠖⠛⠛⠉⠁⠈⢫⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠘⡇⠀⠀⠀⠀⠀⢀⣠⠴⢿⡋⠁⠀⠘⢦⡀⠀⠀⠀⠀⠈⣷⣀⣀⣠⠴⠶⠞⠛⠁
⠀⢱⠀⠀⠀⣠⡾⠋⠁⠀⣀⣙⣷⣄⡀⠀⠹⣄⠀⠀⠀⠀⠘⢿⣥⣤⣀⣀
⠀⠘⢦⣴⣮⣥⠔⠲⠚⠉⠉⠉⠁⠈⠓⢦⣄⠙⢦⡀⠀⠀⠀⠘⣧⠀⠈⠉⠙⠲⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⣷⡄⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠘⢳⣄⡀⠀⠀⠀⠀⠂⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢻⡝⢧⡀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠛⣦⢻⡄⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠈⣆⠹⣆⠀⠐⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠙⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⣿⠀⠀⠘⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠒⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⣤⠾⠁

⠀⠀⠀⠀⠀⠀⣰⣆⠄⢤⠀⣂⠰⠆⢰⣄⠀⠀⠀⠀⠀⠀⣼⠣⣠⡷⠀⣤⣄⠀⣴⣔⣶
⠀⠀⠀⠀⠀⠀⠋⠙⠀⠈⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠉⠙⠀⠘⠓⠋
⠀⠀⠀⠀⠀⠀⢠⣶⡀⢀⢠⣀⣶⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⢠⡃⢠⢠⡜⠀⢸⢓⡷
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠈⠀⠈⠛⠀⠒⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⣶⣏⠀⠾⠷⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⡾⢣⠀⠘⢼⠁⠈⠁⠨⠖⠄
⠀⠀⠀⠀⠀⠀⣠⣀⠀⡀⣀⢀⣀⣔⠂⣠⠀⡂⠀⠀⠀⠀⢠⢀⠀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⣾⡿⠃⠛⠟⠸⠛⠉⠀⠳⠀⠟⠀⠀⠀⠀⣜⡶⠄⠀⠄⠀⣗⡎⠀⢠⠂⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃

⠀⠀⠀⠀⠀⠀⣾⠛⢿⣄⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡄⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⢹⡄⢸⣿⣦⠀⠀⠀⠀⢸⣿⡧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⣴⣾⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣯⡀
⠀⠀⠀⠀⠀⠀⢠⣦⣴⡿⠁⠀⠀⠀⠀⠙⠛⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠷⠀⠀⠀⢼⣿⣏⣰⡆⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⣴⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⢰⣷⣽⣿⡆
⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠏⠀⠀⠘⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀



⣿⣾⡾⠿⠾⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣿⣿⣾
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢿⣦⡀⠀⠀⠀⠀⡴⠀⠀⣴⠏⠁⠀⠿⠃⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣷⠀⠀⠀⢸⡇⠈⣼⣿⣷⣤⠶⢚⣻⣿⣿⠿⣧⠀⠀⢀⣴⣿⠿⠳⢦
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⢿⢿⡀⠀⠀⣸⣿⣴⡟⠻⣿⣶⣾⡉⢈⣙⣷⣞⣁⣤⣤⣸⠹⣿⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣶⣿⡿⣗⡶⣞⣿⣿⣧⣿⢶⣿⠛⠿⠿⠟⠛⣿⣿⢫⣿⣛⣿⣷⣭⣽⣿⣦⡀
⠀⣀⣤⣀⡴⣲⣶⠀⠀⠀⢀⡀⣾⣿⣿⣿⣿⣿⣿⣮⣿⣿⡿⣋⣤⣿⣼⠟⠀⠀⠀⠀⠀⠸⣧⣾⣿⠟⢻⣿⡿⡏⠻⣿⣿
⢰⣥⡈⣿⡇⢻⡻⣄⣀⡤⣺⣿⣿⣿⡿⠋⣩⣽⣿⣿⣻⣿⣿⣿⠋⢉⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣬⣿⣷⢇⣼⣿⠋
⠀⢹⡟⣿⣿⣦⣽⣶⣶⣿⣿⣿⣯⣹⢠⣾⢟⣥⣶⣾⣿⣿⣿⣿⠰⠿⠻⢿⣿⡀⠀⠀⠀⠠⠺⠑⠉⠉⠛⣿⣿⣿⣻⣇
⢀⡾⣱⣿⣿⣍⡻⣟⡦⣀⠉⢻⣿⣿⡟⣱⣿⣿⡿⠟⢻⣿⣿⢿⡀⠀⠀⠀⠙⣿⣆⠀⠀⢀⣤⣦⣴⣿⣳⣬⠙⣿⣿⣿⣷
⣨⡾⢿⣏⢿⣿⣿⣿⣯⣿⣷⣌⢻⣿⢠⡿⠋⢀⣀⣀⣼⣿⣏⠀⠻⢿⣿⣿⣷⡜⢿⠀⢰⠟⠋⣉⠉⠉⠙⠉⣰⣿⠿⠋
⠃⣀⣉⡻⣿⣯⠛⠛⠛⠻⢿⣿⣿⣿⣿⠃⣠⣴⣾⣿⢿⣿⣏⠀⠀⠀⠉⠙⠛⠻⡎⠀⣡⣾⣿⣿⣷⣦⣤⣴⣿⢁⡄
⡞⣿⠋⠙⡞⣿⣦⡠⣴⣒⣤⣘⢿⡟⣿⣴⣿⠿⣛⣵⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⠃⠈⠇⠀⠈⠉⠉⠉⠀⣻⣿⣾⡁⢀⡀
⠹⢿⣄⠛⠁⢻⣿⣿⡿⠿⠿⢿⣿⣧⠹⣿⣿⣿⠋⠀⠉⣿⣿⣿⢳⠦⠄⠠⣤⡀⠀⠘⠓⡄⠀⠀⠀⢀⡼⠋⠻⠛⠛⠋
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠙⡿⢿⡄⠀⢶⣿⢿⣿⠀⠀⣤⣤⣠⣱⣶⠛⠉⠁⠠⡴⣾⠏
⠀⠀⠀⠀⠁⠀⠘⢿⡏⢍⠁⠈⣽⡻⠀⠀⢡⣾⣿⣿⣶⣶⣿⣿⡀⠀⠈⠻⠷⢬⣽⡿⠖⠀⠀⣰⠋⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣠⣿⣿⣶⣤⣿⠟⢩⣽⣿⣿⣿⡏⢳⡀⠀⠈⠀⠂⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⢻⣿⣿⣷⣄⣠⣤⢾⣿⡟⣿⣿⡇⠀⠙⢦⡀⠀⠀⢀⣴⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣽⣿⣋⣡⣤⡿⠿⣿⣿⣿⠀⠀⠀⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣿⣿⣿⣛⣀⣠⡿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣻⣿⣿⣷⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣩⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⡾⢻⣯⣦
⣞⣻⣱⠟⠦⠈⣷
⠈⢿⡁⠀⠀⠀⣿
⠀⠀⠙⣦⠴⠚⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡿⢳⣄
⠀⠀⢸⡯⢀⣠⢴⠶⣧⠀⠀⠀⠀⢀⣠⠶⠶⠚⠛⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠡⠼⢷⡜⣝
⠀⠀⢣⣴⠛⠀⠀⠀⢺⡄⠀⠀⢀⡟⠁⣠⠶⣄⠀⠀⠆⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣧⡀⠀⠀⢙⣷
⠀⠀⠈⢻⡜⠀⠀⠀⠈⢣⠀⠀⡼⢁⡞⠁⠀⠈⠳⣄⠀⠁⠂⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⡀⠀⠹⣤⠶⠋
⠀⠀⠀⢸⡇⠈⠀⢐⠀⠸⡄⠀⡇⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠹⣄⢀⣸⡆
⠀⠀⠀⠈⢇⠀⠈⠀⠀⠀⣿⣸⠃⢸⡇⠿⠀⠀⣾⠀⠈⣇⡀⠀⢸⡁⠀⠀⠀⣠⠞⠁⠠⠀⠀⠀⠄⣩⣠⠟⠁
⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠘⣿⠀⣾⠀⠀⣈⠀⢤⠀⠀⠈⢹⡆⢸⡇⠀⣠⠾⠁⠀⠀⠀⠈⠀⣠⠟⠉⠁
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠹⣦⣸⣦⠀⠛⠑⠆⠁⣰⠆⢻⡇⣸⡶⠞⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠐⢀⠀⠀⠀⠈⠻⠺⣿⠲⠶⠶⢻⠷⠆⠀⠘⠋⠀⠀⠀⠀⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡦⠖⠋⠀⠈⠀⠀⠒⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢿⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠄⠀⠁⢀⡤⠞⠁
⠀⠀⠀⠀⠀⢸⡄⠁⠀⠔⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠎
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠈⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⣼⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡸
⠀⠀⠀⢸⠇⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠐⠒⠀⠀⠀⢸⡇
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⠀⠀⣾
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣾⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇
⢠⡧⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠹⣆
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠰⠀⠀⠀⠀⠙⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠘⢧⡀
⢸⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣤⡤⠤⠴⠤⠤⠤⠤⠤⠤⣼⠃
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣄⠤⠖⢻⡋⠉⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⡠⠞⠃
⠘⡇⠀⠀⠀⢀⣤⠞⠛⠳⣄⠀⠀⠙⢦⠀⠀⠀⠀⠘⣦⡶⠚⠛⠋⠁
⠀⢿⣀⣠⡖⢋⣤⡤⠶⠶⠾⠛⢦⣄⠈⢷⡄⠀⠀⠀⠹⡏⠋⠲⠶⣤⣄⡀
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣽⣄⠀⠀⠀⠘⣄⠀⠀⠀⠀⠙⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠘⣦⣀⠀⠀⠀⠀⠄⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢿⡳⣄⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠈⢳⡜⣆⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢱⠘⣷⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠈⢣⣄⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢸⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣀⡰⠎

⠀⠀⠀⠀⠀⣠⠞⢹⡆
⠀⠀⠀⠀⡴⠁⠀⢸⡗
⠀⠀⠀⡜⠁⠀⠀⣿⡇
⠀⢀⡜⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡴⣲
⠀⠈⠀⠀⠀⠀⢸⣿⠁⢠⣞⡍⠳⡄⠀⠀⠀⠀⣤⡀⠀⠀⣴⣦⠀⠀⠀⠀⢀⣞⡍⠑⡄⠀⠀⠀⢠⣿⡇⠀⢠⢿⣿⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⢀⠴⠊⠀⠀⣿⡏⠀⣼⡏⠀⢀⠇⠀⠀⠀⣸⣿⠁⠀⣸⣿⡟⠀⠀⠀⠀⣸⡟⠀⠀⡇⠀⠀⠀⣸⣿⠃⢀⡏⣿⡿⠀⠀⠀⢀⠔⠁⢸⣿⡄⠀⠀⢀
⣰⠃⠀⠀⠀⢸⣿⠁⢰⡟⠑⢤⣼⣀⣠⠤⠊⢸⣿⠀⣰⢫⣿⡇⢀⣠⠴⠋⣿⠙⢦⣼⣁⣠⠤⠚⢻⣿⠀⡼⢸⣿⠃⠀⠀⡠⠋⠀⠀⣸⣿⡧⠔⠚⠁
⡇⠀⠀⠀⢀⣿⠃⠀⠀⠛⠒⠊⠀⠀⠀⠀⠀⠀⠻⠶⠃⣸⣿⠖⠉⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠘⠿⠞⠁⢸⣿⠀⣠⠞⠁⢠⡖⠋⣹⣿⠃
⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠈⠒⠶⠛⠁
⠘⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡿⠁
⣿⣿⣷⢶⣿⠾⣿⣿⣿⢾⣿⡷⣿⣿⣶⣶⣾⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷
⣿⣿⣤⣷⣵⣶⣦⣾⣼⣾⣧⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⠀⠀⢀⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣾⣿⣿⣇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣶⣤⣀⣠⣤⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠈⠉⣿⣿⡿⠀⠀⠈⠙⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠟⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⣀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⡿⠋⠀⠀⢠⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠋⠈⠻⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠟⠁⠀⣠⣄⠀⠈⠛⢿⡿⠋⠁⠀⣠⣾⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠿⠋⠀⢀⣤⣾⣿⣿⣿⣦⠀⠀⠀⢀⣴⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣾⡇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠸⠿⠿⠿⠿⠿⠿


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

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

⠀⠀⠀⠀⠀⢀⣠⡤⣴⣶⣦⣤⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠈⠑⢾⣿⣿⣷⣾⣔⣤⣈⣀⣉⣿⣯⣯⣤⣬⣍⣉⠙⠒⢢⡤⠂⠀⠀⠘⢶⣿⣏⣋⠁⢸⠉⠉⠛⣲⣶⠃⠀⠀⠀⣤⡐⠀⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠹⣿⣩⡟⠛⠛⠛⠿⠻⠛⠛⠛⠛⠋⠋⢛⣟⡽⠋⠀⠀⠀⠀⠀⠀⠙⣯⣿⡲⡾⠶⣶⣿⠿⠃⠀⠀⠀⠀⠈⠛⠓⠲⠾⠛⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣴⣦⣤⣤⣀⣴⣤⣾⣶⣿⣲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⠤⡤⠤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⡶⠶⠋⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠰⣶⣶⣾⡿⢿⣿⣿⣥⣬⣽⣭⣭⣭⣬⣴⡾⠻⣻⡿⠀⠀⠀⠀⠀⠀⠀⣎
⠀⠀⠀⢿⣿⣏⡀⡰⠁⢀⠟⠉⢹⡏⠉⠙⡟⠀⢣⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠂⠀⠂
⠀⠀⠀⠘⣿⣿⣿⣿⣶⡞⠀⠀⠘⠁⠀⠀⣳⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⡀
⠀⠀⠀⠀⠘⣿⢿⣿⣾⣿⣿⠷⣿⡶⡶⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⣏⣳⣢⣤⣀⡄
⠀⠀⠀⠀⠀⠈⠻⣍⠻⢿⢿⣷⣿⣾⣿⡿⣯⣱⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣧⣕⣯⣷⣻⠟⠁⠀⠀⠀⠀⢀⢠⠶⠶⠂⠀⠀⠀⠀⠙⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠭⣭⣭⣭⠬⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⣀⣀⣴⡤⠖⠛⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣫⣥⣽⣿⣿⣴⣿⣶⣶⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⢠⣀⣀⣴⣿⡷⣿⠿⠿⠿⠛⢿⠛⠛⠛⠻⡟⠛⢿⣿⣷⠶⠃⠀⠀⠀⢀⣀⣠⣄⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠈⠛⢿⣿⣷⣤⣼⣄⣀⣀⣀⣸⣀⣀⣠⣤⢷⣶⣛⣿⠃⠀⠀⠀⠀⠀⠘⣏⡛⡖⠲⠖⠞⢛⣙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣤⡤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⣿⢃⠟⣿⣿⠟⣸⣫⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣛⣛⡫⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡔⣩⣴⣴⣭⠓⡄⠀⠀⠀⠀⠀⢀⠀⠀⣺⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⠛⡳⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡔⢛⣷⡆
⠀⠀⡜⢸⣿⣿⢛⣽⣷⢸⠀⠀⠀⠀⠀⠸⠅⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠁⢸⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⣿⣄⠀⢀⡎
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠈⠱⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡤⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⣀⣼⣏⣹⡆⠀⠀⠀⠀⢘⡏⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠛⠃⠀⠀⠀⠀⠀⠀⠀⡀⣴⣿⣿⡄⡀
⠀⠀⠀⠀⣠⣾⣿⡿⠯⠟⠁⠀⠀⠀⠀⠈⢙⣛⡛⡜⠉⢻⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢴⣿⠄⣿⣿⢴⣿⡄
⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⡽⠛⠶⠛⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠖⡛⠓⣮⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣴⠚⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⠿⠟
⠀⠀⢀⣼⡞⠻⠞⡿⠀⠀⠀⠀⠀⠀⠀⠠⠟⡿⢟⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⣡⡀⣿⡇
⠀⠀⠈⠁⠉⠛⠋⠙⣦⡀⠀⠀⠀⠀⠀⢀⣼⠷⠞⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣦⢴⣶⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃


⠀⠀⠀⠀⣀⣰⢦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠠⢤⡛⢠⣶⡀⢓⠆⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠐⢧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠁
⠀⠀⠀⠰⢥⠈⠏⣠⡔⠓⠀⠀⠀⠀⠀⠀⣿⡏⠈⠋⢿⡷⠀⠀⠀⠀⠀⠀⠀⠉⡗⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⢻⡇
⠀⠀⠀⠀⠸⠖⠙⠋⠓⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠿⠽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠳⡼⣟⢿⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠁⣠⣄⣀⡤⠬⡟⠳⣄⣹⡈⠩⣿⠗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠆⠀⠉⣤⡀⠘⠁⣹⠀⠱⣄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⡄⠈⠛⣇⠐⠖⠀⢀⣠⢄⠀⢰⠁⢀⡠⠚⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⠏⡇⠀⠀⠘⣆⠀⠘⢿⣀⡼⢠⡾⠚⡏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠶⠿⢧⠀⣀⡴⢾⢓⡶⣶⣲⢶⢋⠿⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡋⠁⣀⡤⣼⣿⠿⢳⡿⢯⡠⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⣈⡧⣄⣀⣠⣴⠟⣠⠴⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⠛⠻⡖⠚⠁⠀⢨⠿⠟⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠋⠂⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠦⢄⣀⠀⠀⠈⠳⢶⠇⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣼⠟⠒⣶⠦⣄⠙⠲⠤⣄⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠞⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⡴⠟⢦⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢳⠀⠀⠈⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⣆⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣷⣤⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⣿⣿⣿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿
⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⣠⣴⣶⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡄
⠈⠻⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠲⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡖⠙⢶⡐⣷⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢧⣠⠾⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢬⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠴⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⡿⠿⠿⣿⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣿⢰⣶⣶⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣩⣭⡍⣿⢸⣿⣿⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠛⠃⠛⠘⠛⠛⠘⠃⠘⢻⣷⡼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣋⣥⣶⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡂⠠⣶⣿⣿⣿⠁⡦⢙⣛⣛⣓⣦⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⢠⡄⠀⠀⣍⡛⠿⣷⣶⣾⠿⢛⣉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢸⡇⠀⠀⣿⣿⣷⣦⣭⣴⣾⣿⣿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆⢀⣾⡿⠛⠟⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡤⠘⠻⢷⣶⣾⣿⣿⣿⣦⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣠⣴⣦⠀⢀⣴⣤⣄⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠛⠛⠟⠃⠈⠿⠿⠃⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣶⣶⣶⡶⠄⠀⠀⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⡀⠀⠀⠈⠉⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⡀⠀⠀⣠⡾⢀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠰⠿⠂⢹⣿⠛⣿⣶⣶⣟⡿⠿⢿⣿⠀⠸⢦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠃⡀⠀⣶⣄⡀⢸⣿⣷⡿⠛⠉⢻⣿⠀⠀⣿⣀⣀⣼⡏⠈⠉⠲⠆⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⠀⣿⡀⠸⠟⠁⣸⣿⡀⠀⠀⠀⢸⣿⠀⢸⣿⠿⢿⣟⠃⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣿⡿⣷⣶⡆⠙⠋⠀⠀⠀⠀⠘⡏⠀⣸⣿⠀⠈⠻⡇⠀⠀⣦⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠀⠀⠀⠀⠀⠀⢠⡄⠘⠋⠀⠀⠀⢠⡄⠀⣴⣦⠀⠀⠀⠙⠛⠿⠋⠀⡀⠴⡄⠀⢀⣿⣤⠀⠀⠀⠀⠀⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠁⠐⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠉⠁⠘⢷⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⢀⣀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠸⣿⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡿⢇⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠎⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⠎⠉⠛⠀⣰⠀⠘⠇⠀⠀⠀⠀⠿⠁⣶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⣀⣤⡄⠘⠻⡟⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠁⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠁⠀⢤⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡿⠀⠈⢻⣿⠛⠀⡀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠓⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡗⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢀⡷⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⡇⠀⠀⠈⣷⠦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣿⡇⠀⠀⠀⣣⡄⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣶⣿⠿⠿⠿⣷⣤⠀⢀⡏⠁⠀⠀⣾⣷⣶⣶⡷⣦⣤⣦⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡘⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⢻⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠃⠀⠀⠀⠀⠀⢰⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠻⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢠⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠹⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⢿⣿⠻⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠳⣤⣾⣿⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣷⣄⣀⣀⠀⠀⢸⣿⣷⡀⠀⠀⣀⣀⣀⣽⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠉⣉⣈⣹⣭⣿⣿⣿⣿⣿⣿⣶⣿⡿⠛⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⣿⣿⣶⣶⡾⠟⠉⠙⠛⠛⠙⠉⠁⠀⠈⢿⣋⡀⠀⣀⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠟⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠿⠟⠛⠻⠟⠛⣿⠟⠛⢿⡦⠙⠛⠛⠻⣿⠋⠉⠉⠉⠉⠉⣉⣉⣁⣀⣉⣉⣩⣥⣼⣿⣿⣿⣿⣉⣉⣉⣉⣉⣉⣽⣉⣉⣉⣉⣉⣉⢉⣉⣙⣿⣿
⣿⣿⣷⣾⣿⣷⣶⣾⣷⣾⣿⣿⣶⣿⣿⣶⣾⣷⣾⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣠⣤⣤⣤⡸⣷⡀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠁⠀⢹⣷⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡏⢻⣆⠀⢀⣾⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠙⠛⠛⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢁⣴⠶⢶⣦⠀⠀⠀⠀⠀⠀⢹⠉⣭⠀⠀⣶⠄⣴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⢸⡄⣿⣤⣶⣿⡀⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⣿⠀⢠⣄⠀⣼⡇⣠⡾⠛⠻⣦⠀⣟⢀⣀⣼⡟⢰⡇⠀⠀⠀⣿⠀⠀⠹⣿⡄⣾
⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣷⣿⢡⡟⠀⠀⠀⣿⠃⣿⣿⣟⠉⠀⢸⣿⠀⠀⠀⣿⠀⠀⣠⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⡿⠈⢿⣤⣤⣾⠏⠀⣿⠀⠻⣷⠄⢸⣿⣤⣤⡀⣿⣴⡾⠟⠃⠀⡛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡦⠀⠈⡰⠀⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠹⡠⣤⣼⡏⠔⣂⣴⡿⠇⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠉⣀⠀⠀⡃⣉⢪⡛⢍⣯⡤⠤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡁⡔⠁⢀⡜⢳⡘⣡⢪⡁⠟⠙⠋⠁⠀⠉⠉⠉⠙⠛⠒⠤⢄⡀⣴⣲⠄
⠀⠀⠀⠀⣀⣀⡀⠀⡜⡡⣨⣿⣳⠖⢲⣹⣮⣿⠕⡡⠒⠁⠀⠀⠀⠀⡀⠐⠃⠀⠀⠀⠀⠀⠀⠉⠫⣅⣠
⠀⠀⠀⠙⠄⡀⠨⢱⣆⢿⣾⣿⢃⣦⠛⢋⠄⠀⠉⠀⠀⠀⠀⠀⠀⠐⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣶⠟
⠀⠀⠀⠀⠀⠀⣠⣿⣯⣷⡿⢖⠁⠷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠄⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠸⢷⣶⡲
⠀⠀⠀⢠⢴⠛⠉⢸⣽⢧⣿⠬⠀⡿⠄⠀⠀⠀⠀⢾⠁⠀⠒⣰⠀⡧⠖⠋⠀⠀⠀⠀⢸⠃⣤⡀⠐⠂⠀⠀⠈⠫⢶
⠀⣠⡿⠙⠀⠀⢠⣿⡥⢸⣿⠀⠀⠛⠀⠰⠄⠀⢠⠋⠀⠀⠀⢿⠀⢇⢂⠀⠀⠀⠀⠀⠘⣄⠓⠈⠙⠀⠀⠀⠀⠀⠻⢇
⠀⠀⠁⠀⠀⣰⣿⣍⡀⢸⣿⡆⠀⠀⠀⠀⠀⠊⡀⡀⠀⠀⠀⢸⠀⠋⠀⠑⠒⠈⠀⠀⠀⠀⡄⢰⠂⠀⠀⠀⠀⠀⠘⠘⣀
⠀⠀⠀⠀⢀⣿⣿⣿⠷⠈⣿⣇⠀⠀⠀⠀⠁⠠⠈⠈⠈⠁⣀⠀⠀⢠⠀⠀⠀⠀⠤⠀⢶⠠⡟⢰⡖⠄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢸⡟⣿⠃⠀⠀⢻⣿⡄⠀⠀⠀⠤⡇⠀⠀⡶⠤⠛⠂⡀⠘⡀⠀⢸⠁⠀⠀⢸⠀⣇⠸⠇⠘⠀⠀⠀⠀⠀⡄⢸⡄
⠀⠀⠀⠀⢸⠁⢻⠀⠀⠀⠈⢿⢳⡀⠀⠀⠀⠛⠀⠰⠧⠄⠀⠀⡖⠘⠇⠀⠀⣧⠓⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠸⠆⠀⠀⠀⠈⠣⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⠀⠠⡄⢠⠀⠀⠀⠀⠘⠀⡘⡍⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢸⠀⢀⠀⠀⣇⢈⡆⠄⡔⠸⣁⡠⠘⢄⠶⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢣⣸⢁⠃⡆⠐⠻⢘⡇⢀⣻⠂⢎⠑⠀⡆⢱⠆⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠘⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⠸⠴⠃⠐⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡷⣀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⡔⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⢿⡛⠺⡇⠀⡀⠀⠀⠀⠸⠈⠒⠀⠈⠀⠑⠀⠀⠉⠉⠀⢀⡀⣠⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⡂⠰⠆⠀⠀⢀⡀⡤⠄⠀⡠⢴⣶⣤⠾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠐⠖⠚⠛⠋⠁⠉⠃⠘⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠀⠐⠒⠒⠀⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡡⠊⢀⠠⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠢⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠈⠄⠈⠀⠀⠀⠀⢀⠠⠀⢀⢄⠀⠀⠀⠀⠈⠢⠀⠱⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠖⠁⠀⠑⢄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡁⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠑⠠⢈⠒⠠⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⢈⡗⣹⡠⡄
⠀⠀⠀⠀⠀⠀⠀⡤⣘⢣⡞⠀⠀⠀⠀⠀⠀⠀⢸⢉⢉⡇⠀⠀⠀⠀⢴⠄⠀⠀⣀⠀⠀⠀⠶⠀⠀⠈⠁⠀⣶⡚⠧⠀⠀⠀⠀⠀⠁⠀⠁⠉⠮⡄
⠀⠀⠀⠀⠀⠀⢠⢴⠘⠈⠁⠰⠘⠆⠀⠀⠀⠀⢳⠈⠒⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠠⠀⠀⡰⠂⢁⠀⠀⠀⡠⠊⠑⢄⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⠏⠑⡈⣆⠀⠈⠢⠋⠀⠀⠈⠛⠋⠀⠀⠀⠘⠐⠀⡰⠇⣀⠀⠈⠶⠊⠀⠀⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠒⠉⠀⠑⠄⡀⠀⠀⡙⠀⠠⠟⠉⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢾⠢⡤⠜⠢⡀⠀⠣⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠈⠁⠚⠠⠀⡇⠐⡎⠸⡁⠈⠁⠲⠂⢤⠠⠀⡔⠒⡼⠁⠀⢀⡆⠈⠀⠀⠘⣄⠐⣣⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⢠⡒⢤⡞⠀⠀⢠⠀⠈⡷⠀⢤⠱⣀⠋⠈⣣⠌⢰⠠⠂⠉⡆⠀⠃⠀⠀⢀⣯⠐⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢠⡄⠉⡘⠁⠀⠀⠸⠀⠀⠇⠀⠘⢄⠙⡄⢠⠋⡠⠊⠀⠀⠀⠃⠀⣤⠠⠊⠁⠀⠢⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠔⠋⠲⠀⠀⡆⠀⢠⠀⠀⠀⠀⠑⡜⢣⠎⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⡄⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠰⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⣠⢃⡀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⢀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡇⠀⠀⠀⠈⠉⠉⠉⢉⡍⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡸⠉⡲⠒⡦⠀⣤⢀⣄⠀⡤⠠⠴⠲⠆⠸⠉⢏⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⡰⠁⡰⠁⢠⠁⠀⡄⠀⡇⠀⡇⠀⡀⠀⠇⠀⢃⠈⢆⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠜⠀⡜⠀⢠⠃⠀⣤⠀⢀⠇⠀⡇⠀⣷⠀⢠⠀⠈⡄⠈⢆⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡰⠋⢀⠞⠀⣠⠃⠀⢰⠆⠀⢸⠀⠀⠁⠀⣿⠀⠘⠀⠀⢡⠀⠈⢳⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣀⡰⠃⠀⡴⠁⠀⢀⠇⠀⠀⡸⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡆⠀⠀⠱⡀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠓⠠⠾⣀⠀⢀⠎⠀⠀⠀⡇⠀⢸⠀⠀⣸⠀⠀⢰⠀⠀⠸⣀⣀⠤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠊⠀⠀⠀⠀⠀⡨⠋⠑⠒⠀⠴⠤⠤⠤⠤⠤⡼⠤⠀⠚⢲⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢄⡄⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠂⠠⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠚⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⢻⣛⣟⣋⡜⣛⣿⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣍⣉⣿⣟⣿⣿⣟⣛⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠛⠛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠸⣿⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠙⢷⣄⠀⠀⠀⣠⡶⠊⠉⠉⠛⠶⢶⣌⠹⢆⠀⠀⠘⢷⣄⡀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠙⢷⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠨⠿⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠛⠿⣄⠀⠀⠀⠀⠀⠾⢇⣼⣧⠶⠶⠞⠛⣶⡀⠆⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣠⣴⡿⠉⠁⠀⠀⠀⢀⣿⣇⠶⠟⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢷⣦⣄⡈⢠⡀⠀⠀⠈⣿⠙⢿⠆⠀⠀⢀⣰⡟⠁⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠉⠙⠻⠿⠆⠀⠀⠻⣦⣤⠴⠶⠒⠉⠉⠀⠀⠀⠀⢀⣴⠶⠿⠛⠻⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣥⣄⠀⡧⠀⠈⠛⠀⠰⣦⡤⠿
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⡶⠒⠀⠀⠀⠈⠉⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⣠⠶⢿⡋⠙⢧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣼⠇⠀⠀⢠⣤⣴⠾⢶⣤⣤⡴⠋⣴⠾⢷⣦⣠⣄⠀⠀⠻
⣤⠛⢿⡄⠘⣷⣴⠟⠋⠘⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣁⣸⡏⠀⠀⣰⡿⠁⠙⠶⠚⠉⣠⣴⣶⣏⠀⠸⠃⠈⣿⣄⣀⣤
⢿⡀⠘⢷⣄⣼⡃⣀⡴⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠹⡟⠀⠀⣴⠛⠀⠀⠀⠀⠀⠐⣏⠀⣀⠙⢿⡄⠀⢀⣿⠉⡟⠁
⠀⠻⣤⣼⣿⠉⠁⠉⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣰⠏⠛⣶⠟⠀⢀⡾⠃⣠⣤⠖⠻⠿⠛⠟⠻⢆⠈⢧⣼⣷⣤⡿⠃⣸⡇
⠀⠀⠀⠀⠉⠳⠄⠻⣆⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠋⠀⣰⡟⠁⣾⠋⠀⠀⠀⣶⠆⠄⠀⢈⣷⠀⠉⠀⢈⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣶⡆⠙⣶⡶⠀⠀⣿⠲⣖⣹⡟⢀⣤⠚⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠹⢆⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠈⠛⠛⠁⠀⠀⠀⠈⢷⣘⢿⣄⡾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠳⣄⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢛⣶⣿⣷⡷⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠙⢧⣤⣤⡀⢸⣧⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⣠⡄⠀⣠⡟⢀⡾⠃⠈⠙⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠋⠀⠻⡄⢹⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢀⣼⠟⢡⣶⠏⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠹⣆⣿⡆⠀⠀⠠⣟⣀⣀⠀⠀⣠⡿⠁⢰⣿⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠘⢿⣿⡀⠀⠀⢿⣏⣿⣀⣰⡟⠁⠀⠀⢻⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠻⣷⠀⠀⢸⡏⠈⠻⠋⠀⠀⠀⠀⠈⠱⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄⠀⠀⣿⢃⣴⠏⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡞⠿⠟⠁⠀⠀⠀⠀⢀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠰⠟⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠻⠟⠻⠟⠻⡟⠻⣏⣯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣙⡋⢙⣋⠈⡛⠀⢸⠄⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣀⣿⠀⠻⠃⠿⠓⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡅⠿⠀⠸⠇⠰⣤⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣶⣶⣶⣾⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⡇⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠻⠀⠹⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢻⣄⣰⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠸⣿⠁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠰⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡀⠀⠀⢀⣀⣀⠀⠀⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣶⢶⡆⠛⠿⢿⣯⣭⡉⠑⠾⠿⠛⠙⠛⠿⠛⣿⣿⡿⠿⠿⣿⡿⠶⠾⢿⠷⢶⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠺⠃⠀⠀⢀⠹⠷⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠚⠂⠺⠄⠺⠼⠿⠀⡣⡉⣀⠀⠠⠀⢀⣀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠟⠉⠌⢁⠀⠀⠀⠀⠀⠀⠀⠓⠘⠁⠋⠓⡋⠸⡄⠛⠍⠟⠒⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⣛⣿⣻⢰⣄⢲⠀⢢⡄⠹⠂⠀⠠⣶⠀⢦⠀⡀⢠⡶⣴⣒⠂⢰⣄⢀⣴⠀⣤⣨⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠆⠙⢓⠿⠈⠈⠘⠶⠟⠋⣀⠦⣉⢁⠻⠶⠋⠷⠈⢼⠇⠿⠶⠆⠸⡎⠛⢹⠼⠛⠷⣧⡴⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁⠀⢀⠈⠉⠚⠣⠀⠘⢋⠛⠉⣛⠁⡘⠁⠌⠹⠃⠐⠀⠀⢘⣂⠲⠂⠀⢀⠐⠆⠖⠲⠄⠰⠶⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⣀⣀⢀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠈⠀⠄⠀⢀⣀⠀⠛⠉⠀⠀⠀⠈⠁⣀⢀⠀⡈⣀⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⡞⢛⠃⠋⠉⢀⠁⠛⠛⢛⠀⠓⡻⡄⠚⠂⠐⠚⠋⠒⠓⠘⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠰⠠⡄⢠⠴⢀⡄⠠⣦⣢⡄⣤⣰⠀⢄⣀⡦⣤⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠋⠀⠈⠀⠃⠈⠀⠁⠉⠉⠋⠉⠍⠁⠚⠛⠙⠙⠉⢿⡈⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠛⠋⠘⠷⠻⠉⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⢱⡄⠐⠂⠀⠠⠀⠐⠂⠀⢸⠏⠀⠀⠀⠀⠑⠘⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠁⠀⠂⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢀⠀⠀⠠⠀⢀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠐⠀⠃⠒⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠦⠠⠄⢸⡀⠀⠀⠀⠈⢮⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣀⣂⡤⣜
⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣈⣀⣀⡘⡇⢉⢩⡟⠀⣼
⠀⠀⠀⠀⠸⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡒⢾⡛⣿⣾⢿⣽⣷⠗⠂⠈⡉⣻⠁⠋⠀⠀⢄⣋⡜⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣦⣠⣤⣤⡀⠀⠴⡆⡠⣴⣶⠒⡀⠀⠀⠀⠀⠀⢀⠸⣂⣤⣠⣍⡯⡈⢉⠋⡙⠛⢇⡰⡀⠀⢲⡔⢇⢀⠸⣡⡄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣟⡋⠃⣻⡀⠈⠟⠼⠿⣀⠁⡇⠀⠀⠀⠀⠈⣠⡿⡿⠘⠋⢹⠅⠿⠑⡷⠰⠆⠓⠁⠀⠀⠀⢈⠉⠀⣬⡅⢐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⣾⢅⢻⡷⠢⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣧⠋⠁⠈⠘⢧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠁⠈⣀⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡓⠀⣤⠾⣏⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣱⡦⣸⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢘⡉⠩⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⡽⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢢⣄⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⢡⡟⡦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢰⠆⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⡰⢦⡰⠟⢡⡟⠂⠘⢃⣇⡛⣋⠡⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠛⠀⠀⠀⣀⠀⠀⢀⠀⡀⣠⣤⠆⠀⠀⠀⠀⠀⡐⠉⡡⠝⢋⡙⢧⠈⢠⡡⡷⣟⡿⣤⠸⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠈⠉⠙⠛⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⣳⠷⣴⠟⠴⠲⠇⢸⡇⠸⡴⡿⢠⣭⣜⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢄⠁⡂⠠⠀⠔⣡⠠⠆⠁⡄⠀⠀⣤⠤⠌⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢦⡤⠒⠘⠀⠐⠀⠖⠄⠄⠀⠀⠀⠐⠦⠆⠴⠄⠂⢂⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⡀⠐⠞⠠⠀⠇⠠⠐⠒⠀⠰⠆⠐⡎⣥⠀⠀⣀⢀⡄⠀⡄⣤⣠⢈⣀⠀⠀⢠⠀⠀⣀⣠⣌⠄⣠⡀
⠀⠀⠀⠀⠀⢀⣶⡶⢆⣦⡚⠻⢦⣤⡤⠄⠀⠀⠀⠀⠀⡀⠀⠀⣠⢀⣄⣀⣀⣀⠀⠁⠋⠀⠀⠁⠀⠁⠀⠀⠁⠉⠁⠉⠀⠀⠈⠉⠉⠁⠁⠉⠀⠩
⠀⠀⠀⠀⣴⣿⠈⠀⠺⠀⠁⠶⠨⠁⠄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣣⠀⠟⣀⠳⠚⠀⠐⠒⠘⠠⠐⣂⠀⢀⡀⠆⠀⠀⢸⠒⠆⠀⠄⢶⠂⠰
⠀⠀⠀⣤⡤⣤⣃⣠⡇⠯⣬⠐⠴⠑⠄⡿⠘⠳⠳⠰⠂⠻⢧⠿⡄⢋⠤⢽⠀⠐⠻⠿⠀⣷⢿⣦⣤⠄⣼⢭⡷⠞⠷⣤⠿⠇⣄⣀⣀⡷⢸⣧⣩⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⣶⣾⣿⣿⣷⣾⣍⣽⣿⣿⣿⣯⣽⣿⣷⣶⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣄⢠⠴⠃⣠⢾⣿⢿⣿⡧⣿⣽⢿⣿⣿⣾⣿⢿⣿⣋⠻⢷⣿⣿⣦⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡏⡤⠉⠀⡆⠀⠿⣟⣘⢿⣿⡿⠟⠛⠛⠛⠰⠂⠈⡁⠈⠛⠯⣙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢀⠀⠀⠀⠉⠉⠀⠐⠂⢀⠁⠀⢀⠀⠂⠀⠸⡄⠀⠀⠀⢉⠛⠉⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣀⠴⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠤⠄⡀⡆⠀⢠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢸⠁⠀⠀⠒⡐⠒⠒⠒⠒⠒⠚⢦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠋⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⢀⡆⠀⠀⠀⡠⠒⠚⠳⣦⡀⠀⠀⠀⠀⠀⠀⢡⣦⠀⠀⠔⠐⠠⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⢸⡇⠀⢠⠊⠀⠀⠀⠀⢸⣷⠀⠀⢀⣀⡀⠀⠘⣿⡇⠀⣴⣶⢄⡀⣠⣴⡄⠀⣠⡶⠒⠄⠀⣼⡟⠁⠀⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⣿⠁⡀⠀⠀⢿⡇⠀⢏⠀⠌⢠⡿⠋⠀⢰⣿⠁⠀⠁⣸⡟⠀⠐⠈
⠀⠈⣿⠀⠀⠀⢀⠤⢀⡴⠋⠀⠀⠸⣿⠈⡠⠀⠀⢸⡇⠀⠀⠀⠀⣿⠁⠀⢀⠙⢿⣄⠀⠔⢹⡇⠀⠀⠀⡀
⠀⠀⢹⣧⠀⠀⠈⠀⠉⠠⠀⠀⢀⠂⠈⠁⠀⠀⠀⣾⠁⠀⠀⠀⠀⠻⡀⡀⠂⠀⠀⠀⠀⠀⠈⠧⠀⠄⠊
⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠈⠀⢁⡀⠀⠈⠄⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠂⠀⠀⢀⠠⠀⠀⠂⠀⠀⠐⠠⡀
⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡐⠁⠀⠁⠐⠀⠀⠤⠤⠠⠤⠄⠀⠐⠀⠀⠀⣠⠶⠀⠑⠎⠀⠀⠀⠀⠀⠀⡀⠂⠁
⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⡘⠀⢀⡴⡆⠀⡄⠀⡀⢀⡀⣀⠀⣠⠀⢤⣸⡇⠙⣋⣹⠂⠈⡄⠥⠀⠈⠁⠀⠁⠀⢄
⠀⠀⡠⠐⠈⠀⠉⠀⠪⠐⠁⠀⠚⠓⡧⢸⣇⠀⢿⡞⣧⠏⢀⣯⣷⡀⠉⠃⠀⠈⢀⣠⣾⡇⠀⢀⣀⡤⠶⠒⠚⠚⠁
⠀⡌⠀⠀⠀⠀⣀⠀⠀⠀⠃⠢⠀⣀⠀⠀⠉⠁⠈⠁⠉⠀⠈⠁⠀⠁⢀⡠⠤⠒⠈⠠⣠⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣂⢀⠀⠀⠈⠀⠐⠂⠀⠀⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⡀⠀⠊⣀⣀⡃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢸⡏⠀⢸⣟⣉⡁⠙⢻⡟⠻⠂⠀⠰⠒⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠘⣿⠉⠁⠀⣼⡇⠀⡀⢠⣴⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣮⣭⣭⣄⣛⣛⠛⠀⠿⠀⠀⢉⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣅⣷⠙⣯⢸⡯⢽⡿⢿⣿⢛⢿⠻⣟⢟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣿⣷⣿⣬⣧⣾⣿⣾⣿⡸⣸⣠⣻⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠠⡌⠉⠉⠛⢻⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠩⠁⠀⠀⠀⠀⡏⣳⢸⠛⢻⢸⡏⡇⣾⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⠀⠀⠉⠁⠈⣄⠉⠈⣁⡉⠻⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠁⢹⣇⢀⣾⠟⢷⡄⠻⣦⣼⠟⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠈⢿⣄⣼⠇⢠⡿⡧⠀⢸⡏⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠿⣶⠟⠀⠀⠉⠁⠰⠟⠁⡄⠀⠸⠆⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠁⠐⠂⠀⠀⠐⠀⠈

⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⡙⠻⠿⣿⣿⠿⠛⣡⣌⠛⠿⣿⣿⣿⡿⠟⠁⠀⠉⠻⢿⣿⣿⡿⠟⢋⣴⡌⠙⠻⢿⣿⡿⠿⢋⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣦⣴⣶⠆⣿⣿⣿⣶⣤⣤⣤⠄⠀⠀⠀⠀⠀⢰⣤⣤⣶⣾⣿⣿⣇⢸⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡩⠞⢋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡙⠮⠁
⠀⠀⠀⠀⠀⠀⢠⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠃⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠻⠿⠛⢋⣴⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄⡙⠻⠿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⣙⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣿⠀⠀⠀⢻⠀⢠⣾⠂⠀⢠⣿⣧
⠀⠀⠀⣠⠴⠋⣉⣐⣦⣀⢦⡈⠳⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⣹⣦⣀⣀⣸⣀⣸⣿⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⣶⣄
⠀⠀⠀⠈⠓⠦⣄⣩⡟⠋⣽⣫⠽⠋⠉⠉⠉⠉⠉⢹⣿⡴⢺⣿⠀⠀⠀⡼⠀⠈⣿⠀⠀⡞⠀⣿⣧⠴⠋⠉⠉⠉⠛⠛⠉⠉⢹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠘⣿⣄⣀⡼⠁⠀⠀⠙⠓⠚⢁⠴⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⡀⠘⣧⠀⠀⢿⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢹⡄⠀⠀⠀⣼⣥⡾⠋⠳⠶⠾⠷⠶⠼⠷⢦⣤⣤⣿⠀⠀⠀⠀⡾⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠻⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠸⠃⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣄⠈⠀⠀⠀⠀⢿⡁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⣹⠆⠀⠀⠀⠈⢀⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⢀⣀⣛⣃⣀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢠⣤⡀⠀⠀⠀⠈⢷⣄⠀⠈⠣⣤⡴⠋⠀⢀⣴⠏⠀⠀⠀⠀⣠⣤⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣷⡹⣿⠈⡇⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⣤⠶⠋⠁⠀⠀⠀⠀⢀⡟⣸⡿⣱⣏⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣮⡻⣧⠹⠇⢷⣴⠖⣦⠀⠀⣀⣠⡾⠷⣶⡶⠿⣦⣀⡀⠀⢠⡖⢶⣼⠇⠿⢱⡿⣫⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣯⣟⡛⠀⠀⠀⠈⣡⡞⢁⡴⠟⠉⠻⢦⣴⡟⣷⣤⠾⠋⠛⢶⣄⠻⣦⠁⠀⠀⠀⠘⣛⣭⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⣠⣟⣰⠟⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠙⢷⣸⣇⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠟⠛⠋⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠠⠂⠒⠀⡄⠀⠁⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⢰⠒⠂⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢐⠉⠐⠀⠀⠀⠀⠠⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠠⠄⣄⠀⠀⠀⠤⠐⠀⠠⠀⠠⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⠃⠀⡐⠁⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠃
⠀⠀⠀⠀⡀⠀⣀⠁⠄⠀⠀⠉⠀⢀⠄⠀⠀⠰⠀⠀⠘⡀⠀⢀⡀⠤⠒⠇⠀⠀⡄⣀⠈⠀⠀⠀⢡⠀⠀⡰⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠑⠀⠂⠁⠀⡄⠀⠀⠀⠑⠂⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠃⠒⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⢀⠀⡄⠒⢤⢀⠀⠀⠀⠀⠀⢐⣰
⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⢉⠣⡀⠇
⠀⠐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠐⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠐⡄⠂⡀⠀⡡⢄⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠈⠀⠐⠁⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠑⠰⢉⡈⠄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⢀
⠀⠀⠀⠀⠒⣤⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡎⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠐⠀⠒⠁
⠀⠀⢄⠈⢢⠦⣔⠁⡰⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⠄
⠀⠠⠌⣠⠝⠰⢂⣈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⢀⡠⠣⠀⡀
⠀⠀⠀⠠⠄⠓⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⢢⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⢁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⠐⢄⠀⠁⠀⠀⠀⠀⠢⠀⠀⠠⠊⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠆⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠡⠀⠐⠈⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⠀⠀⠐⢠⠉⢒⢀⠆⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠔⠠⠰⣅⠀⠀⠀⠀⡐⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⡀⠄
⠀⠀⣀⡀⠁⢠⠄⠐⠢⢀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡡⠊⠀⠀⠀⠀⠀⠈⠁⢀⠀⠒⠀⠀⡐⠀⠀⠀⠀⠀⡠⡅⠙⢠⠁⠀⡃⠜⠠
⠀⠈⠄⠈⠖⠀⠀⠀⠀⠈⢃⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⢀⠂⠉⠁⠢⠀⠀⠈⠀⠀⠀⠜⠀⠀⠀⠀⠀⡀⠴⠐⠦⡀⠆⠬⠁⣀⠈⠰
⠀⢀⠦⠠⢆⠀⠤⠀⠴⠀⡈⠀⠢⠈⠀⠀⠔⠁⠀⠂⠁⠀⢀⠃⠀⠐⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠆⠠⠤⢄⡀⡀⢅⡀⠈⠱⠄
⠀⠈⠰⠁⢀⠱⠤⢤⡄⠊⢋⠐⡆⠀⠀⠀⠈⠢⠀⣀⡀⠀⠊⠡⡀⠀⠀⠈⠁⠀⠀⢀⠴⣀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠚⢀⡠⠎⠀⣧⠀⠑⡈⠄
⠀⠀⠀⠀⠁⠼⡀⣤⠈⠦⠜⠀⠀⠀⠀⠥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⢂⡠⠂⡐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣻⣟⣿⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢛⣉⣥⣤⣴⡆⣴⣤⣤⣍⣙⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣁⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⣉⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⣠⣿⣿⣿⡿⠛⣉⣥⣤⣶⣦⣤⣍⣙⠻⣿⣿⣿⣷⡈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣾⣿⠋⣼⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣌⠻⣿⣿⣿⡄⢻⣿⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡏⣈⣻⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⣧⠹⣿⣿⣋⡈⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡿⢛⠋⣰⣿⣿⣿⣿⣿⡇⢻⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡄⣿⣿⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡀⠿⠇⣸⣿⣿⣿⣿⣿⡇⢸⣿⣿⡟⢸⣿⡇⣰⣤⣤⣤⣀
⠀⠀⠀⣠⣾⠿⢛⣉⣭⣭⡄⢿⣿⡄⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⡇⣼⣿⡇⣬⣭⣍⣙⠻⢿⣦⡀
⠀⢀⣾⡟⣡⣾⡿⣿⣿⣿⣷⠸⣿⣧⠸⣿⣿⣿⣌⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⣼⣿⣿⡿⢠⣿⡿⢰⣿⣿⣿⡿⢿⣦⡙⣿⣦
⠀⣾⡟⢰⣿⣿⣷⡌⢻⣿⣿⣇⠹⣿⣧⠙⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⢡⣾⡿⢁⣿⣿⣿⠏⣴⣿⣿⣷⡘⣿⡆
⠀⣿⡇⣿⣿⣿⣿⣟⠠⡦⢸⣿⣧⠙⣿⣷⣌⠻⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⡿⢋⣴⣿⡟⢡⣾⣿⠡⣦⠘⣿⣿⣿⣿⡇⣿⣷
⠀⣿⣇⢹⣿⣿⣿⣿⣶⣶⣾⣿⣿⣷⡌⢻⣿⣷⣌⡛⠿⣿⣿⣷⣶⣶⣶⣾⣿⣿⡿⠟⣋⣴⣿⣿⠋⣴⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⢃⣿⡇
⠀⠘⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⡿⣿⣷⣦⣬⣍⣉⣉⣉⣉⣭⣤⣶⣿⡿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣼⡿⠁
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠛⢿⡿⠋⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠛⠛⠉⠁⠀⠈⠻⣿⠟⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⢀⣀⡀⢀⣀⡀⢀⣀⡀⢀⣀⡀⢀⡀⢀⡀⠀⣀⠀⢀⡀⢀⡀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡬⠇⣿⠦⢸⡧⣟⢸⡧⠄⣟⠀⣿⢸⡧⡟⢸⢿⡼⣿⢰⣯⣇⢸⡿⣼⡧⣏⠀⢸⡷⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣿⣿⣾⣧⣿⣾⣧⣤⣽⣿⣧⣼⣦⣿⣾⣼⣧⣿⣾⣤⣿⣾⣧⣿⣧⣽⣿⣶⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠠⠦⣤⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠈⠻⣿⡿⠋⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⣀⣤⠄⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣾⡿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠈⠻⠿⠷⠆⠀⠀⠤⠤⠤⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠒⠋⠁
⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⣀⣴⠾⠋
⣠⣞⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠰⡃⠀⠀⠀⡣⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢣⡀⠑⠤⠤⠔⠁⣠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⡄⠀⠀⠀⠘⢄⠀⠙⠢⠤⠤⠤⠚⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⡇⠀⠀⠀⡜⠀⡜⢦⡀⠀⠑⡄⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⣠⠔⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠙⢄⠈⠓⠒⠊⢀⡴⠁⢀⠇⠀⠀⡇⠀⢀⣀⠀⠀⡇⠀⢀⡴⡆⠰⡇⠀⠀⠀⡱⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠉⠒⠒⠊⠁⢀⡠⠋⠀⠀⠀⡇⠀⠁⠈⠀⠀⡇⠀⡏⠀⢣⡀⠑⠤⠤⠔⠁⣠⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⠘⢄⠀⠙⠲⠤⠤⠴⠚⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠤⡀⠀⡇⠀⠀⠀⠀⠀⠧⠤⠇⠀⠧⠤⠇⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠄⠀⠦⠤⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⠀⠈⣿⣿⣿⣷⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⡄⣤⣿⣿⡟⠉⠁⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⢀⣿⣿⣿⠷⠄⢼⣿⣿⡏⠀⠀⠀⠈⢻⣿⣇⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠾⢿⣿⣁⠀⠀⠀⣀⣿⣿⣇⣀⠀⣿⣿⡧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⣠⡿⠟⠳⡄⠀⣿⡿⠟⠉⠈⠣⠛⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡤⠞⠉⠀⠀⠀⠘⠚⠁⠀⠀⣀⠀⠀⠀⣠⣴⡏⠱⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⣠⣴⣶⡀⠀⢀⣠⣴⣿⡏⠦⠖⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⣖⣉⣿⣿⣿⣟⣒⣉⣿⣿⣿⣇⣀⣀⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉


⡿⢿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢀⣤⡾⠿⡿⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠈⣿⣦⣤⣤⣶⣷⣄⡙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠈⠀⠈⠉⠉⠙⠛⢻⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⢀⣤⣄⠀⢀⣀⣀⡀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠘⠿⠿⠀⠸⢿⡿⠁⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣤⣤⣤⣴⡄⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⠀⠀⠙⠋⠀⠀⠀⢰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡦⠤⡄⠀⠠⢀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⣷⡀⠀⠀⣠⣘⣁⣼⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠰⠾⠃⢸⣿⡉⣿⣶⣾⡉⢻⠏⢻⡇⠀⠻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⢠⡄⠀⣶⣦⡀⢸⡿⠿⠟⠉⠉⢻⡟⠀⢸⡇⣀⣠⣿⡁⠀⠙⠛⠀⠚⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⢹⣆⣤⡟⠁⣹⣿⡀⠀⠀⠀⢸⡿⠀⢸⡇⠛⣷⡟⠁⠀⠀⡔⠃⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⣶⠛⠻⠀⠙⠀⠀⠀⠸⣧⣀⣿⡇⠀⠀⠻⠃⠀⠀⠇⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡀⠀⠀⠀⠀⠠⠶⠈⠁⠀⠀⠀⢹⠀⢰⠴⣷⣀⠀⠀⠘⠛⠋⠁⠠⠴⠾⢀⣀⣠⣶⡆⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠇⠀⠈⠉⣷⣶⠆⢸⠆⠀⠀⠀⠀⠀⠈⠉⠉⠰⢶⡄⢀⣀⠀⢻⣇
⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠄⠀⠀⠀⠸⣿⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⠆⠀⠿
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠏⠉⢠⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠜⠛⠋⠀⠀⠀⠀⣀⠀⠿⠀⠀⠀⠀⠈⢹⡟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠘⢷⣤⣤⡀⢀⣤⡶⠛⠁⠀⡏⠀⠀⠀⠀⠀⠀⣼⠛⢳⣤⣛⡤⠀⠁⣰⡿⠁⠐⠀⠀⠀⠀⠙⠓⣤⣦⠄⣀⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠈⠉⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢰⣿⠀⠀⠀⢷⣄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣼⡿⠀⠀⠀⣿⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡟⠀⠀⠀⣼⠃⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⣤⣶⣶⣾⣇⡀⠀⠀⠹⠃⠀⠀⣹⣧⣀⣤⣤⣀⣀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡿⠛⠁⠀⠈⠙⠛⠶⣶⠦⠠⠤⠀⠙⠛⠛⠋⠈⠙⠛⠻⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣶⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣾⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⢰⡄⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡇⠀⠀⠀⠀⢠⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡆⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⣹⣀⣠⣶⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠘⠃⠀⣿⡟⢻⣿⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⢦⡄⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣣⡀⠀⣀⠀⠀⢀⣿⣷⠀⠀⠀⠀⣈⠁⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⢉⣙⣶⣿⣿⣶⣿⣿⣿⣶⣴⣿⠿⠛⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣴⣶⣾⠟⠋⠉⠛⠛⠛⠉⠉⠀⠈⣿⣋⠀⠀⣀⣈⣿⡄

⠀⠀⠠⡄⠠⢤⡄⢠⠄⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⣄
⠀⠀⢸⣸⣧⣎⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠾⢿⡾⢿⡀⡀⠀⣄
⠀⠀⢸⣹⣏⣽⣿⣿⣿⣿⡿⠿⠟⠻⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⣠⠀⡛⢿⣏⣿⡿⠟
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⢸⣿⣷⠆⢀⣀⣠⠀⠀⠈⠉⠁⠀⠹⣟⣿⢿⡿
⠀⠀⢸⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠞⣯⡅⠀⠈⣻⣿⡄⠐⠶⠀⠀⠀⠀⠙⠻⣷⡤
⠀⠀⢸⣻⣿⣿⣿⣿⡧⠤⠀⠀⠀⠀⠀⢐⠒⠚⣿⣿⠀⠀⠀⢀⣼⠆⢿⢫⣿⠶⣤⣄⣠⣴⣶⠀⠀⠠
⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠉⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⢎⣧⠀⠀⠘⠃⢸⣟⢹⣿⡈⠿⠷
⠀⠀⢠⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⢈⠴⢃⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢈⢻⡿⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⢀⡀⠁
⠀⠀⣸⢉⣵⣿⣿⣿⡧⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣤⣤
⠀⠀⢉⣼⣿⣿⡿⠛⢆⠈⠢⣀⠀⠀⠀⣠⠊⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣿⠟⠻⣿⠁⠀⢰⣄⡀⠀⠉⠉⠉⠁⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐
⠀⠀⢏⠀⠀⠔⠈⠁⠈⠀⠈⡆⠀⠀⠀⠀⠈⣉⣩⣉⣍⣩⢉⡍⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⣦⠀⠀⠀⠀⠀⢀⡠⠁⢠⣾⣿⣦⣾⣿⣿⠁⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣠⣴⡿⠃
⠀⠀⠸⠁⠀⢄⠀⠀⠀⠃⠀⠀⢸⣿⡟⠋⠉⢻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⢣⣀⣠⠔⢄⠀⠀⣨⣀⣴⣶⣿⠀⠀⠀⢈⣿⣿⡷
⠀⠀⢸⢿⣿⣧⣤⣑⢊⣽⠋⣿⣿⣿⣄⣀⣠⣾⣿⠟⠁⢀⡴⠶⢤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣄⢀⣠⠶⠶⣄
⠀⠀⢘⣿⣿⡈⠙⠻⡟⢸⣤⣝⡛⢛⢻⣿⣿⣿⢈⡇⠀⢸⣀⠀⣀⡗⠀⠆⠀⠀⢡⢸⠡⣿⢟⣼⠈⢇⣀⢀⡿
⠀⠀⠠⢹⣿⡷⣤⣾⢃⡟⠉⠛⠛⡟⠀⠛⠿⠛⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠁⠀⠀⠉⡉
⠀⠀⠘⢿⣿⢻⣿⣿⡟⠙⠒⠒⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⢀⡴⠙⣷
⠀⠀⠀⢠⠎⠀⢀⣿
⠀⠀⢠⠋⠀⠀⢸⣿
⠀⠠⠃⠀⠀⠀⣾⡟⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢠⣄⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣴⡽
⠀⠀⠀⣀⠀⢠⣿⠇⢠⣿⠆⠈⡆⠀⠀⠀⢰⣶⠀⠀⣴⣿⠀⠀⠀⠀⢰⣷⠂⢸⡄⠀⠀⠀⣿⡏⠀⣰⣿⣿⠀⠀⠀⠀⣠⠖⣿⡄
⢀⡔⠉⠀⠀⣸⡿⠀⣼⢯⡀⢰⠃⠀⢀⡠⣿⡟⠀⣰⢿⣿⠀⠀⢀⡠⣾⢇⠀⣸⠀⠀⢀⣼⣿⡇⢠⢇⣿⡇⠀⠀⢀⠞⠁⠀⣿⣇⢀⣀⠴
⡎⠀⠀⠀⢠⣿⠃⠀⢿⣀⡹⠛⠒⠚⠉⠀⢹⣇⣰⠃⣿⣇⠤⠒⠉⠀⢿⣀⡽⠓⠒⠋⠁⠘⣿⣠⠏⢸⣿⠀⠀⡴⠋⢀⣠⠴⣿⡟⠉
⡇⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠿⠴⠊⠀⠀⢾⣀⣼⠟⠁
⠱⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡞⠁⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡰⠶⠖⠒⠚⠀⡏⠀⢈⠱⣇⣸⢀⣾⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡪⠑⠈⠀⢠⡀⠀⠀⠀⠀⠓⠤⣌⣱⡏⠉⣻⠤⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠕⠁⣀⣀⡀⠀⠈⢀⣀⠀⠀⠀⢠⠊⠁⠔⢳⠖⡏⠻⡒⠒⠉⡀
⠀⠀⠀⠀⠀⠀⠀⡔⡑⠁⣠⡞⠁⠀⢸⠀⢠⡟⠀⠅⠀⠀⠸⣄⣀⡠⠛⡆⠈⠀⣹⠀⠀⢪⠧
⠀⠀⠀⠀⠀⠀⢌⠌⠀⢰⡟⠀⠀⣠⠎⠀⣿⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡜⡌⠀⠀⣿⠃⠀⠀⠀⠀⢸⡏⡠⢺⡆⢰⣯⠱⠀⣮⠕⡀⣴⠓⠀⠀⣴⣛⠆⠀⢹⡄
⠀⠀⠀⠀⢀⢱⠀⠀⠰⣿⠀⠀⠀⠀⠀⣼⡷⠀⣸⡇⢸⣏⡸⠉⣿⢀⠋⠙⣷⡤⠚⣿⠀⡠⠃⠀⢳
⠀⠀⠀⠀⠈⡄⠀⠀⠀⢿⡄⠀⠀⣠⠎⠀⠁⠀⣿⢁⠆⠈⠁⠀⠈⠁⠀⣄⣽⠇⠀⠈⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠲⠆⠈⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡇⠀⠸⠂⠀⠀⠀⠀⢀⡠⠖⣿⡇⠀⣧⡄⢱⠀⢠⣿⠆⢀⣾⣷⠀⠀⠀⠀⠁⠀⠀⠤⠁
⠀⣾⡇⠀⠀⢿⡀⠀⠀⠀⣀⡤⠒⠉⠀⢰⣿⠇⢰⣿⠧⢴⠖⢻⣿⠀⡼⣿⡇⠀⠀⠀⢀⡠⠴⢒⠚⠉⠉⠛⠻⣶⡄
⠀⠘⠻⢦⣤⡬⠥⠔⠒⠉⠁⠀⠀⠀⠀⢸⣿⠔⠉⠻⣤⠞⠀⠘⠿⠞⢱⣿⠃⣠⠴⠚⠁⠀⢀⢣⠃⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⠀⠀⠀⡠⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡼⢰⡟⠀⠀⠀⠀⣠⠀⠀⠀⢀⡎⣾⠃⠀⠀⠀⢀⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠳⠟⠀⠀⠀⠀⠈⠻⠁⠀⠀⠘⠶⠃⠀⠀⡠⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⡠⣤⡲⠝⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣩⣴⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢰⣶⡆⠀⢀⠚⠁⠀⠀⠙⣿⣧⠀⠀⠀⣤⣶⣤⠀⢹⣿⣿⠀⢠⣴⣶⠈⠀⢀⣤⡀⠀⢀⣤⣤⡄⠀⢠⣾⣿⠟⠙⠆
⠀⠀⠀⢸⣿⡇⠀⠄⠀⠀⠀⠀⠀⢹⣿⡇⠀⣼⣿⠃⠠⠀⠀⣿⣿⠀⢾⠟⢑⠞⢨⣿⡿⠋⢠⣿⡟⠁⠀⢀⣿⣿⠁⠄⠊
⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣿⠁⠀⡀⠀⣿⣿⠀⠈⠁⠁⢠⣿⡟⠀⠀⠾⣿⡇⠀⢀⢸⣿⠃
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢀⠀⣠⣿⠟⠀⠀⠈⠛⠖⠈⠀⠀⣿⡟⠀⠀⠀⠀⠸⣿⠀⠀⠈⠀⠉⠛⠀⠁⠘⣿⡄⠀⡀⠔
⠀⠀⠀⠀⠘⣿⣇⡄⠀⠀⠰⠶⠛⠣⠀⠀⠀⠀⠀⠐⡀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠙⠂⠀⠀⠠⠤⠤⠤⠄⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣤⣤⠀⠀⠀⠴⣦⠀⣶⣦⠐⠰⣶⠂⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠐⠐⣿⡇⠈⢿⡇⠠⣿⡠⠃⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⣴⡾⠷⢆⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣤⣄⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠠⠀⠀⠀⠀⠈⠂⠄⢀⠀⠀⡠⠞⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⢀⣿⠃⢀⠈⠀⣀⣀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⢀⠔⠁⣴⡿⠋⠉⡁⠀⠀⠀⠀⢸⣿⠀⢠⣦⠐⣿⠐⠁⠀⣾⡏⠀⣁⠀⠀⠀⠀⣿⡇⠀⠠⠁
⠀⠀⠀⠀⢀⡀⣸⣿⡇⠐⠁⠀⢀⣿⡇⠐⠊⠀⠀⠀⠀⠀⠈⠻⠖⠁⣿⡆⢿⡀⠀⡐⠉⠓⣾⠏⠀⣤⡶⠒⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠪⠁⢤⠀⠙⢧⣀⠔⠀⠀⠀⠀⠀⠀⡀⠐⠀⣿⡇⠈⠳⠔⠀⠀⠀⠯⠄⢺⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⢻⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢦⣄⠀⣠⣤⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⠀⠀⣀⣶⣷⡏⢃⠺⣏⣷⠀⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢬⠁⠁⠀⠀⣿⢣⠃⠱⣌⣜⣫⠹⣿⠶⣿⣿⠼
⢠⠈⢀⣧⠀⠀⣄⣰⣄⠈⠀⠀⢰⠠⠀⠀⠀⠀⢈⣷⡀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⢦⡤⠄⠸⠛⠁⠈⣤⣿⣾⣌⣘⠟⠀⠙⢿⣼
⣈⠀⣤⠁⠀⢀⣧⣍⢹⡎⣤⣄⡠⠦⠆⠒⠠⡦⢨⡿⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⡀⠈⠀⠀⠀⠀⢰⣾⣿⠏⠀⢸⣿⣟⡻⡿⢿⣿
⠋⠀⣠⡀⠀⠸⠃⣨⣅⠀⠈⠀⠀⠀⠀⠀⠐⠂⠁⠙⠀⠁⠃⠀⠀⠈⠹⠤⠠⠚⠂⠀⠀⠀⢄⣀⣀⣀⣀⣦⣠⣿⡿⠁⠀⢀⣾⣿⣿⣧⢢⣾⣿
⠚⣉⣁⣀⣀⡤⠔⢏⠀⢁⣴⠀⢀⡜⠀⠀⠠⣄⠀⠀⠀⠀⠀⣀⣠⣆⠀⣀⡀⣀⣶⣦⣠⡀⠠⠀⠛⠹⠅⢬⣉⣀⣀⣀⣤⣾⣿⣿⣿⡿⠍⠙⣿
⠀⠀⢰⡄⢹⡄⠀⠘⡿⣾⠃⢠⡾⠷⠤⣔⣺⣿⡏⠓⡖⠚⠛⠉⠉⠁⠀⠈⠿⢉⠀⡿⢟⡷⣤⣤⣴⡇⠀⠀⠉⠉⡋⠀⠈⣿⡟⠛⣿⣧⣴⠇⣸
⠀⠀⠘⣷⡿⠃⠀⠀⠀⠊⠀⠘⠀⠀⠀⣿⣿⣿⣏⠐⠁⠀⠀⢤⡤⠀⠀⠀⣧⠀⢸⣴⡊⢠⠏⠤⢄⡁⠓⠤⣔⣿⣷⣤⣼⣿⠧⣴⣯⡁⣁⢀⣠
⠀⠀⠘⣿⣇⣀⣄⠀⠀⠀⠀⠃⠀⠀⠀⡿⣿⡿⠋⠀⠀⠀⠁⠔⢀⠀⠀⡆⠀⠀⣨⠴⣮⡄⠀⠀⠈⠉⠢⣄⠈⠁⠈⠉⠻⣄⠀⠀⢻⣿⣦⣌⣿
⠀⠀⢰⣿⢥⣹⣿⣷⠁⠀⣸⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠊⠂⠀⣸⠑⠐⣶⣮⢦⠘⡃⢰⣦⣰⠀⠀⠀⡉⣽⣶⣿⣷⡞⠀⠀⠀⡟⠻⣿⣿
⠁⠀⠠⣿⢀⣿⠀⣿⡀⠠⠃⠀⠀⠄⠀⢸⡿⣿⡇⠈⠀⠀⠂⠀⠂⠀⣯⠀⢠⡟⢿⣿⣧⢧⠈⢹⣿⠀⠀⠀⡘⣮⣿⡦⣾⣷⣄⡒⠛⢡⣾⣿⣟
⠆⠀⠀⢿⢸⠟⠿⠟⣐⣀⠰⠆⠀⠀⢀⡼⢳⣿⠁⠀⠀⠀⢀⣲⣄⠈⣇⠃⣸⣷⠤⠟⢙⣿⡀⣾⣿⠀⠀⠀⣷⣼⠁⠀⠈⠅⠈⠉⠛⠛⢋⢻⣿
⠀⠀⠀⣈⣸⣤⣶⠄⠸⠁⠀⠀⠀⢀⣤⣴⣿⣿⠀⠀⠀⠀⠁⠉⠝⠈⡏⣸⣇⣀⢐⣾⣿⣿⠇⣿⢿⣆⣄⡀⡀⠋⠀⢺⣧⠈⠄⠀⠀⠀⢈⣼⣿
⡤⠤⠤⠤⢹⣿⢯⠠⢁⡀⠀⠀⠀⠩⢟⣯⣾⠋⢀⣀⣠⢤⣕⠍⠀⠘⠉⠻⠁⠉⠋⣋⠋⢴⡞⠛⣛⣋⣩⣿⡝⠀⣠⣾⡷⠟⣶⣤⣤⣾⣿⣿⣿
⣠⠀⠐⠀⢿⡁⠘⠀⢸⣀⣀⡠⠤⠄⠀⣼⣿⡿⠿⠿⠟⠛⠂⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠚⠁⠀⠀⠉⠀⠀⢀⡴⠁⢀⠀⠀⠁⠀⠞⣉⣠⣴⣶
⣼⣶⣴⡷⡿⠉⠀⠈⠐⠛⠁⠀⠀⠀⠀⠀⡠⠀⢀⠀⢠⡤⠀⢐⡀⢀⣀⣀⡄⠀⠠⠀⠀⠀⠠⠴⠀⠀⠀⢀⣼⠷⠛⠉⠉⠀⠳⠖⠚⠛⠛⠉⠻
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⠂⠒⠤⡀⠀⠀⠀⠀⠀⠀⣀⠄⠒⡀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⡀⠐⢄⠈⢆⠀⠀⠀⢀⠎⠀⡠⠜⡬⠠⢄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡠⠛⢲⣤⠟⠁⠘⠀⠀⠀⡈⠀⠀⡃⢐⠉⣂⠰⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢠⠊⠀⣰⠞⠁⠀⠀⢠⠀⠀⠀⢡⠀⢸⠬⡞⠉⢮⠒⡅⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⠛⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠢⡘⠀⠓⠀⠓⠀⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠒⠂⠁
⠀⠀⠀⠀⠀⢀⠤⠐⠒⠠⠄⡀⠀⠋⠑⠛⠚⠛⠓⠐⠚⠙⠙⠚⠃⠋⠛⠀⠀⠙⠊⠋⠘⠛⠚⠓⡿⠰⠀⠀⠀⠀⡀⠄⠐⠂⠠⢀
⠀⠀⠀⢀⠞⠀⡠⢐⣆⡂⢄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⢲⡄⠀⣰⣦⠙⢄
⠀⠀⠀⡌⠀⡌⠌⠔⡓⢪⢢⢡⠀⢃⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⢠⠁⠀⠑⠡⡓⢾⡟⠁⠀⠈⡀
⠀⠀⠀⢃⠀⢏⢍⠢⣁⠜⡉⡙⠀⡌⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡀⠀⢀⠠⠟⢪⠢⡀⠀⢀⠁
⠀⠀⠀⠈⢆⠈⠒⠥⠧⠭⠊⠀⡔⠁⠀⠀⠀⡇⠀⠐⠒⠀⡔⢐⠆⡖⠩⢍⠲⡀⣂⡀⠐⢒⠀⠀⡇⠀⠀⠀⠱⣀⠧⠜⠀⠀⠑⠼⢀⠎
⠀⠀⠀⠀⠀⠉⠒⠤⠤⠄⠒⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠎⢠⠃⠀⠇⠐⠜⢀⠄⠀⠇⠈⠉⠀⠀⡇⠀⠀⠀⠀⠈⠑⠢⠤⠤⠔⠊⠁
⠀⠀⠀⢰⠦⡆⡤⡶⢴⠰⠒⢤⠰⠆⠀⠀⠀⡇⠀⠀⠀⣄⠘⢄⠀⠀⢸⠉⠁⠀⢀⠀⡄⠀⠀⠀⡁⠀⢄⢰⡦⠖⠤⠶⠰⠰⠄⡦⠖⠤⠢⠐
⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠇⠱⠄⠇⠠⠸⠀⠀⠸⠈⡀⠉⠱⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⠤⠤⠄⣀
⠀⠀⠀⠀⠔⠁⣠⡖⠒⠒⠰⠈⢢⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⠊⢀⢤⠀⠀⠀⠀⠑⡄
⠀⠀⠀⡘⠀⠈⠉⠭⠍⠥⢘⠀⠀⠃⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⣚⣚⣒⣒⣒⡒⠒⠒⠒⠒⠚⠒⠊⠀⠀⢀⠃⠀⡬⠒⢄⠀⣁⣀⣸⠘⡀
⠀⠀⠀⠇⠀⠀⠀⣀⡄⢰⣾⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⣷⣀⡟⢂⣅⠀⠀⢀⠃
⠀⠀⠀⠘⢄⠀⡀⠐⠫⠡⢫⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠡⣀⣨⣱⢛⠃⣀⠇⢀⠌
⠀⠀⠀⠀⠀⠑⠤⠀⠀⠠⠄⠊⠀⠀⠀⢀⠔⠊⠀⢀⡀⠈⠑⢄⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠢⡄⠀⠀⠀⠈⠂⠤⠀⠀⠠⠐⠁
⠀⠀⠰⠖⠶⠰⠔⠠⠆⠦⠧⠰⠔⠒⠂⠎⢀⠦⡔⣍⣱⢢⢤⡀⢡⠀⠀⠀⠰⠁⣾⠛⠛⢛⣛⡛⢻⡌⠆⠀⠀⠀⠐⣴⠢⠆⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢋⡁⡡⡈⢈⡉⠀⠀⠀⠀⠀⠆⠀⣿⠀⠀⡆⠀⠈⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠰⡑⠵⠞⠢⠂⠀⡘⠀⠀⠀⠸⡀⠛⠒⠒⡧⣤⢼⠚⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠁⠈⠀⡠⠚⠀⠀⠀⠀⠀⠘⠠⡀⠀⠈⠉⢁⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⡀⡈⢀⣄⢠⢠⡀⢀⡄⣠⣀⢠⣤⣀⢅⡈⣠⣄⢠⣀⢄⡠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠈⠀⠁⠈⠈⠁⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⢰⠛⠱⡀⠀⠀⠀⠀⠀⠔⠀⢢
⠀⢀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⢸⠀⠀⢱⠀⠀⠀⠀⡘⠀⠀⢸
⠀⢸⢁⠀⠠⢌⠳⡄⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠉⠑⠠⡀⠸⡴⣚⠧⢀⠼⣴⢤⣸⡀⠀⠀⢀⠃⠀⠀⡌
⠀⠈⢮⡈⠁⠀⣹⣼⣆⠀⠀⠰⠃⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆⠹⡀⢀⠀⣀⠈⠓⠺⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠑⣤⠀⣩⡅⠈⢆⠀⡅⠀⡀⠀⣠⠔⠢⣄⠀⠈⠀⠘⡀⠈⢦⠀⠀⠀⠉⢀⡇⠀⠀⡧⠂⠀⡎
⠀⠀⠀⠀⠈⠻⡻⣠⣤⣼⡄⡇⠀⠁⣰⠃⠀⠀⠈⡆⠀⠄⠀⡇⠀⠀⢏⠀⠀⠀⠨⠇⠀⢸⣣⡠⣰⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠿⠟⢃⢸⠀⠀⠜⢦⡀⠀⡰⠁⠂⠆⢠⠁⠀⠀⢸⣤⣤⣍⣸⠀⠀⣇⣈⣍⡏
⢀⡤⡀⠀⠀⠀⠀⢸⡄⢒⢻⠀⢃⠀⠠⠀⠈⠉⠀⠠⠂⢀⠎⠀⠀⠀⠤⠸⠿⠿⠇⠀⠘⠯⠿⡾⠁
⣏⠀⠈⢂⠀⠀⠀⡸⠁⠀⠉⠀⠀⠳⢄⠀⠀⠀⠀⣀⠔⠁⠀⣀⠴⠚⠫⢲⡵⠃⠀⢀⣤⢬⡉
⠱⡀⠀⠀⡱⣲⢾⡅⠀⢠⠁⠀⠀⠀⠀⠉⠉⠈⢉⠔⠒⠊⠉⠀⠀⠀⣠⠞⠀⠀⢰⠁⢿⡄⢸
⠀⠑⠤⢀⡇⢘⠂⢉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⠔⠊⠀⠀⠀⠀⠈⠢⠿⠧⠞

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢸⡇⠀⢀⠐⠁⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⢿⣇⠀⢄⣀⡀⠀⠀⣀⠀⠀⢀⣀⡀⠀⢀⣴⠟⠛⠆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣶⠛⠀⠀⢸⣿⠀⣾⠛⠱⢂⣾⠟⠀⣰⡟⠁⠀⢀⣿⠃⠀⠔
⠀⢸⣇⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⡧⠀⡀⠀⠈⣿⠀⠈⠈⠀⣾⠃⠀⠀⢻⡇⠀⠀⢸⡇
⠀⠀⢿⡄⠀⠀⠀⠚⠋⠀⠀⠀⠀⠙⠂⠀⠀⢰⡏⠀⠀⠀⠀⢿⠀⠀⠂⠀⠉⠁⠀⠈⢧⠀⡀
⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠂⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⡤⠤⠈⠒⠁⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠈⠀⢀⠀⡀⠀⠀⠀⠀⢀⡴⡆⢠⠄⢠⠀⡄⢀⠀⣠⡀⠦⣼⠀⠙⣭⠗⠀⡠⠠⠀⠈⠀⠀⠀⠀⡀
⠀⢀⠔⠈⠀⠀⠁⠘⠐⡀⠀⠋⠙⠇⢼⣀⡘⣿⢻⡟⠀⡯⢷⡄⠘⠂⠀⠀⡠⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠂⠁⠀⢠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡤⠖⠒⠲⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⡠⠒⠢⢄⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⣼⡇⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⣅⣠⣄⠀⡆⠀⠀⣀⡀⠀⠀⠀⢀⣠⣄⡀⠀⠀⣠⣾⡿⠛⠉⠙
⠀⠀⠀⠀⣿⡇⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⣠⣶⠶⢆⠀⠀⠀⣿⣿⠀⠀⣿⡿⠿⢤⠧⢐⣾⣿⠏⠀⢀⣾⡿⠋⠀⡑⠀⢰⣿⠏⠀⠀⢀⠜
⠀⠀⠀⠀⢿⣇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢰⣿⠃⠀⡜⠀⠀⠀⢻⣿⠀⠀⠻⣀⡠⠊⢀⣾⡟⠁⠀⠀⣼⣿⠃⠀⠀⠀⢀⣿⡟⠒⠀⠉
⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⢀⣀⡀⢀⣼⠟⠀⠀⠀⢸⣿⡒⠉⢀⠄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢰⠹⣿⡆⠀⠀⡠⠚⣿
⠀⠀⠀⠀⠈⣿⡆⡇⠀⠀⠐⠥⠤⠞⢋⠀⠀⠀⠀⠀⡜⠙⠷⠒⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⢀⠃⠀⠈⠙⠓⠉⠀⠀⢻⡄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⠜⠀⠀⠀⢄⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠣⠤⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠙⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠈⢦⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠑⣄⠀⠀⠀⣀⠤⠐⠒⠈⠉⠉⠉⠉⠐⠒⠤⡀
⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠀⣀⡀⠀⠀⠀⡰⠁⠈⠑⠒⠠⠤⠀⣀⣈⣀⠀⠀⢀⣀⣀⠠⠤⠄⠒⠈⠁⠀⠀⠀⠀⠈⢢⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉
⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠑⡄⢠⠔⡄⠀⠶⠄⠀⠺⠄⡀
⠀⠀⠀⠀⠀⠈⢣⡀⠀⣀⠀⠀⣀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡰⢻⣦⠀⠀⠸⢿⣷⡀⡇⠀⢸⣿⡆⠀⠡⠀⠀⠀⠀⠀⠀⠀⡼⠈⠒⠃⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⢀⠔⠊⠀⠀⠀⠉⠀⠘⠌⠂⠁⠀⠀⠀⠀⠀⠀⠸⠃⢼⣿⡄⠀⢸⠈⢿⣷⡇⠀⠀⢿⣿⡤⠃⠀⠀⠀⠀⠀⡠⡊⠔⠀⠀⠀⠀⢀⡠⠤⠒⠂⠈⠉⠉
⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄⡀⠀⠀⠀⠀⠁⠀⠘⠿⠃⠀⠘⠀⠈⠛⠃⠀⠀⠈⠀⠀⠀⠀⢀⡠⠖⠙⡔⠀⡄⢀⡠⠒⠉⠁
⠀⠘⠀⠉⠉⠉⠉⠉⠐⠒⠠⢄⡀⠀⠘⡄⠀⠉⢰⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠂⠉⠀⠀⠀⠀⠘⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⣄⡡⠃⠀⣀⠤⠄⡀⠀⢁⡉⠉⠉⠁⠀⣠⣴⣶⣄⠀⠀⠀⢠⣾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡎⠀⠀⠀⠁⠀⠙⠃⠀⠀⠀⡜⠁⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⣶⡆⠀⠀⣤⡇⠀⠀⣿⡟⠀⠀⠀⣿⣧⡀⢀⠌⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⢿⣷⣄⠀⠀⢿⡇⠀⢠⠛⠿⠀⠀⣿⡇⠀⠀⠀⡏⠛⠛⠉⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠀⢻⣿⣧⠀⠈⠳⡀⠎⠀⠀⠀⠀⠹⣧⠀⠀⡰⠀⠀⡔⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⡎⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⡀⠀⣀⣠⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣼⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠸⢟⣀
⠀⠀⠀⠀⠀⠀⠘⠻⠖⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡧⠉⠘⠀⠐⠒⠃
⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣶⡂⠀⠘⠒⠀⠀⡄
⠀⠀⠀⠀⠈⠙⠒⠢⣤⠖⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢿⣯⣿⣿⣿⡋⠻⢤⡜⣄⡀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣥⣾⣷⣦⣌⡙⢃⡠⠀⣹⣿⡇⠀⠀⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠉⠻⠛⠉⠉⠉⠙⣀⣾⣿⡿⢿⣷⣤⣷⡀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣤⣽⣿⣿⣷⣾⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣤⣠⣴⣶⣿⡿⠛⠋⠉⠙⠿⣿⣿⣿⣿⣿⣿⡀
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡀⣉⣀⣤⣴⣦⠀⢀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠉⣉⣽⣿⣿⣇⣿⣿⣶⣴⣾⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⣤⢈⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠏⠀⢀⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠻⢿⣤⠴⠶⠖⣿⣿⣿⡗⣭⣥⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡟⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠄
⠀⠠⠾⠧⢶⠒⣿⣬⠽⠛⠛⠛⢃⡽⢋⣴⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⠟⠿⠁⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧⣀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣾⣧⣭⣅⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣶⣤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏⠉⠉⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣇⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡄⢻⡇⠀⠁
⠀⠀⠀⠀⡀⠀⠀⠀⣀⣠⣾⣟⠿⣟⠁⠀⣰⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⡇
⠀⢈⣥⣤⣄⣀⣤⣶⣿⣿⣿⣷⣦⢄⣥⣶⣿⣿⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡾⠁
⠀⢹⣭⣙⠻⠿⣿⠏⠉⠁⠙⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢠⣈⣿⣿⣿⣿⣿⣿⣿⡿⣏
⠀⠈⣿⣿⣷⣦⡤⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣾⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⢀
⠀⠰⣿⣿⣿⣿⣶⣧⡼⣫⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣽⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣷⠦⠀⠆
⠀⠀⠻⣿⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠂
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣴⣶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠷⠒⠒
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠙⠁⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠉⠉
⣄⣠⣄⣠⣀⣀⣀⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⣄⣀⣠⣤⣤
⣿⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣀⢀⣄⢀⣠⡀⢠⣀⢀⠀⢀⣠⠀⠀⣠⡀⠀⠀⢀⠀⣠⣀⢀⣄⠀⣠⡀⠀⠀⢀⠀⡀⢀⠀⡀⢀⡀⠀⣀⠀⡀⠀⠀⡀⢀⡀
⠀⠀⠿⠋⠸⠧⠸⠦⠟⠸⠛⠸⠤⠸⠷⠀⠠⠿⠧⠀⠠⠼⠘⠧⠼⠸⠿⠆⢽⠖⠀⠀⠠⠈⠁⢨⠀⡀⢀⠀⡁⢀⡀⣀⢈⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⡆⠴⡀⡔⡆⠀⣆⡖⠂⡔⢦⢰⣴⠰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠄⠀⠀⠄⠠⠀⠄⠠⠄⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠁⠀⠁⠉⠁⠉⠁⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠀⠆⠰⠀⠆⢐⠂⠂⠰⠂⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡃⢘⠀⠃⢀⠀⠂⠘⠁⠀⢀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡁⢨⠀⡀⢈⠀⡁⢈⠀⡁⠈⠁⣉⢈⡁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠐⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠰⠀⠄⠰⠀⠄⠠⠄⠤⠠⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠒⠀⠂⢘⠀⡂⠘⠀⠂⠘⠀⠂⢘⠂⡃⠀⠂
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⠀⠁⠀⠀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⠐⠒⠲⢦⡄⠀⠀⢠⡴⠖⠒⠂⠀⠀⠞⢳⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⢲⣶⠢⠀⠀⠀⠀⠔⣶⡶⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠙⠒⠒⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠑⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⢉⡉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⣀⠔⢦⡀⠑⡄⠀⠀⠀⠀⢠⠊⠀⠠⠤⢡⡆⠤⣄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣀⠿⣵⣄⡱⠞⠀⢸⠀⠀⠀⢀⠃⠀⠀⠀⢠⠞⠳⠀⣘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠞⠁⢠⡸⠋⠁⠀⠀⢸⠀⠀⠀⠘⡀⠀⢰⣋⣦⠗⠺⣸⣁⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠣⣐⡼⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢣⡀⣔⣉⣫⣀⣀⣏⣀⣱⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⡀⢈⠉⠉⢋⡉⣁⡀⢀⠀⣀⢀⡀⠀⠀⡀⢀⢈⠉⡃⣀⢉⠁⣀
⠀⠀⠀⣀⠤⠔⠒⠒⠢⠤⣀⠀⠈⠁⠘⠋⠑⠛⠙⠚⠐⠙⠘⠉⠘⠙⠉⠀⠙⠀⠀⠀⠛⠘⠈⠐⠛⠛⠋⠉⡼⠙⠇⠀⠀⠀⠀⣀⠤⠐⠒⠒⠢⢤⡀
⠀⢀⠜⠁⢀⠤⠲⡖⠤⡀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣁⠰⡶⡀⠀⣠⣶⡌⠳⡀
⠀⡎⠀⡔⠡⢊⠽⠯⡑⢌⢣⠀⠸⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢠⠃⠀⠫⣓⣣⢦⣴⣿⠟⠁⠀⠱
⠀⠀⠀⡗⡗⢇⠐⠆⠸⢺⠒⠀⠀⡇⠀⠀⠘⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡏⡇⠀⠀⢸⠀⠀⠀⠀⢈⣦⠙⢇⠀⠀⠀⠀⡇
⠀⢇⠀⠱⣘⠮⢑⡄⠄⢀⠌⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⢀⠀⣀⡤⠀⣄⠀⠀⡀⢀⣀⣀⡀⠀⠀⡇⡇⠀⠀⠈⡄⠀⢀⠀⡈⠀⠐⢈⠣⡀⠀⡰
⠀⠈⠢⡀⠈⠑⠒⠓⠊⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠇⡔⠁⡜⠁⡇⢠⠒⡄⠱⠘⠤⡄⠀⣀⡽⠀⠀⡇⡇⠀⠀⠀⠈⢦⡑⠊⠁⠀⠀⠀⠑⢃⠜⠁
⠀⠀⠀⠈⠑⠂⠶⠶⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⢀⠎⠀⠀⡇⠈⠒⠁⡠⠁⠀⡇⢰⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⠑⠒⠶⠶⠒⠊⠁
⠀⣔⢸⣒⣄⣴⠦⡖⠰⣂⢢⠤⣶⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠣⡀⠀⡇⠀⡁⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⡇⡇⠠⣢⢴⣢⢶⣦⠖⠖⡄⠴⣆⣲⢴⢲⢴⡠⠖
⠀⠀⠀⠀⠈⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣁⠱⣄⣘⡄⢃⣀⡇⠀⠀⣘⡘⠁⠘⠐⡆⠀⠀⡇⡇⠀⠀⠈⠁⠀⠀⢀⣀⡀⣀⣀
⠀⠀⡠⠒⠁⣀⣀⣀⣀⣈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡇⡇⠀⠀⠀⠀⡠⠚⢁⠀⠀⠀⠀⠉⠲⣄
⠀⡜⠁⠀⡞⠽⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠠⣐⣒⣒⣒⣒⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣒⣒⣒⣃⠇⠀⠀⠀⡜⠀⠐⢍⡱⠀⠀⡖⠒⠒⡌⢢
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠧⠾⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢔⡉⢑⣢⠀⡧⠤⠤⠇⠈⡄
⠀⠆⠀⠀⠁⠤⢴⣄⣰⡏⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢿⢣⢼⠎⣠⠇⡀⠀⠀⢀⠁
⠀⠀⡀⠀⣀⣒⡚⠬⠡⢂⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣨⣬⣞⣛⣁⣀⡇⠀⣠⠊
⠀⠀⠈⠓⠄⣀⡀⢀⣀⠤⠒⠁⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠈⠲⠄⣀⣀⣀⣀⠤⠚⠁
⣤⡤⡀⢠⣄⢦⢠⢠⣤⠦⣄⡤⣠⢤⠀⡜⠁⢀⣀⢤⠊⢲⠄⣀⡀⠘⣆⠀⠀⠀⠀⡰⠁⣤⠤⠤⠤⠤⠤⠤⢤⡌⢣⠀⠀⠀⠀⠀⡤⣄⡤⣀⢤⠐
⠈⠀⠁⠈⠀⠀⠉⠈⠀⠀⠀⠉⠀⠈⢰⠁⠐⢧⠿⠘⠓⠚⠠⠥⡟⠀⢸⠀⠀⠀⢀⠃⠀⣿⠀⠀⢠⠖⠒⢲⢸⡇⠀⡇⠀⠀⠀⠀⠙⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠲⡠⡒⡢⢠⢲⠀⠀⢸⠀⠀⠀⠘⡀⠀⣿⣀⣀⣸⠀⠀⢸⣸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠳⢝⠛⢓⡡⠂⠀⢠⠏⠀⠀⠀⠀⠱⡀⠉⠉⠉⠹⣓⣶⣺⠉⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⣁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡈⢉⠉⣀⢀⡀⣀⠀⠀⣀⣀⣀⣀⣀⣀⡀⣀⣉⣉⣁⣀⢀⡀⣀⡀⡀⢀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣉⣉⣃⣉⣈⣉⣈⣁⣈⣁⣀⣇⣉⣉⣁⣈⣀⣉⣁⣉⣀⣀⣀⣉⣁⣁⣉⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠉⡉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡤⠀⠀⠀⠀⢳⡿⠀⠸⠿⠀⠀⠀⠀⠐⠒⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠁⠤⠄⠉⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⢤⡀⠀⠀⠀⠀⣠⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠙⣆⠀⠀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠹⡾⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢀⣀⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⠀⠀⠀⠀⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⢸⡄⢸⡇⣿⣾⣿⡀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⢹⠀⠀⠀⠀⣸⠏⠀⣼⣿⢟⣴⡿⣻⠟⢡⣾⢿⣿⣿⣿
⡇⠀⢸⡇⢸⣷⢸⣷⣿⣇⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⣿⣧⠀⠀⣇⠀⠀⣶⣿⡄⣼⣿⣿⣿⣿⣽⢯⣠⣿⣿⣿⣫⣿⣋
⢷⡀⢸⣧⠀⣿⡄⣿⣿⣿⡀⠀⢻⣧⠂⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡏⣿⡆⠀⢹⣇⢺⣿⣿⣾⣿⣿⣿⣿⣿⣿⢷⣿⣿⡛⠃⠛⠛⢿
⡘⣧⡘⣿⠀⢻⣟⢹⣿⣿⣷⡀⠀⢻⣆⠀⠋⠀⠉⠀⠈⠉⠻⣷⠀⢀⡀⠀⠀⠀⠀⢸⣿⠈⠻⡄⠀⢿⣾⣿⣿⣿⣿⣿⣿⠿⡟⣁⣸⣿⣿⣿⣿⡶⣦
⣳⡘⣷⣿⣇⠈⣿⡄⢻⣿⣿⣷⡀⠀⢻⣶⡟⠀⠀⠀⠀⠀⠀⠹⣦⠈⠙⢧⡄⠀⠀⠘⣿⠀⡦⠙⣆⡈⢿⣿⣿⡿⠟⠉⢀⢸⣴⡿⠛⠉⠀⠀⠀⠠⠛
⣿⣷⣹⣿⣿⡄⠻⣶⣈⣽⢿⣿⣿⡄⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠹⡆⠀⠀⢸⠆⣴⠀⠈⢻⣶⠿⣯⡴⠊⣰⣃⠟⠁⠀⠀⠀⠀⠀⠀⠇⠀⡴
⣿⣿⣿⣿⣿⣷⠞⠛⠛⠙⠿⠻⠟⢿⣦⣨⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠁⠀⠀⠈⡆⠈⢷⣀⡾⣿⣜⣿⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢘⢸⡇⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠹⡇⠀⠀⠁⠀⠈⠻⡄⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠈⢷⢀⡀⠀⠀⠀⢠⣟⡀⣿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⣧⠸⣿⠅
⣿⣿⣿⣿⣿⣧⠀⠐⣧⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠆⣤⣀⣄⠀⢿⡇⠀⠀⠀⣼⡇⢱⣿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣤⢰⡇
⣿⣿⣿⣿⣿⣿⢿⣧⣹⣧⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢿⣿⣾⡇⠀⠀⠀⠉⢠⡏⣸⠇⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡞⠛⠏
⡻⣿⣿⣿⣿⣿⠙⠿⢿⣿⠁⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⠀⠀⢰⣆⣸⠙⠃⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣻⠇
⣿⣮⣿⣿⡿⣿⣤⣀⡀⠘⠀⠀⠈⠓⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣝⣿⣧⠀⢨⣿⡇⣠⢾⠀⣼⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡷
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡇⠀⠀⠀⠀⠈⠑⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡌⠘⣿⡿⢱⣦⣰⡿⢉⡇⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢈⣷⣤⡀
⠻⣿⣝⢿⣿⣿⡿⠿⣭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣾⣿⣷⠿⢻⡿⢁⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⠤
⠀⠙⠛⠳⡍⠛⣿⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣥⡴⠟⠋⣩⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠐
⠀⠀⠀⠂⠀⠋⠐⠂⢀⣡⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⠀⠀⢀⡼⠡⣿⣿⣿⣿⣿⣿⣿⣷⣶⠾⠋⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⠾
⠀⠀⠀⠀⠀⠂⠀⠙⢏⣩⣼⠷⠄⠀⠀⠀⠀⠀⠀⢙⡦⣄⢀⣳⡴⡀⠊⠁⣼⣿⣿⣿⣿⣿⣿⣴⣿⠿⠖⠒⠶⠆⠈⠀⠀⠁⠀⠀⠀⠲⠂
⠒⢤⣀⠘⠻⣦⡀⠀⠘⣿⢤⢶⣿⣷⣄⠀⠀⠀⠀⣸⣧⡹⣯⣿⣿⣷⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤⣀⡀
⠀⠀⠈⠙⠲⣌⠛⢷⣤⡙⢦⠘⠙⢻⡿⢷⣄⠀⠀⠻⣽⣿⡘⣿⣿⣿⢿⡙⠻⣿⣿⣿⣿⣿⠻⣧⣤⣠⡌⠙⠲⣤⡀
⣤⣀⣀⠀⠀⠀⠈⠀⠙⠛⠿⣷⣦⣬⡁⢠⣻⣆⠀⢠⣾⣿⣿⣜⣿⣿⣸⣷⠀⠈⢿⣿⣿⣿⡷⠈⠻⣿⡿⠃⠀⠈⠻⣇
⣿⣿⣿⣿⣿⣶⣤⣤⣤⣠⣀⣀⠈⠁⢛⡿⢿⣿⣿⣤⣼⣿⣡⣿⣾⣿⣿⣿⡆⠀⠈⢡⣿⢦⡛⢧⡀⠘⣷⠶⠘⠦⡶⠙⣷⡀⢀
⣿⣿⣽⡿⠛⠋⠉⢉⣤⠤⠾⠿⢷⣆⣤⣄⣉⠻⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢻⣾⡿⣎⠁⠀⢸⣿⣠⣴⣮⡀⣻⣧⠀⠐
⣿⣿⠋⠀⠀⠘⠛⠛⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣷⣶⣾⠿⠿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡈⠀⠈⠳⡀⠘⠟⢟⢻⣿⣿⣿⣿⣷⢌⠐⠄
⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣿⠿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠸⢃⠀⠀⠀⢘⣷⠀⠀⠀⠘⠫⣿⣿⣿⣦⠁⠀⠁
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⢯⣤⣤⣬⣉⣀⡀⠀⠸⠟⣙⣎⠹⣿⣿⣿⣿⣿⡀⡄⠀⠑⣄⠀⠸⠁⠀⠀⠀⠀⠀⣸⣿⣏⠜⢷⠀⠀⠀⠀⠐⠱⢄
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣷⣿⡟⣿⣇⠹⣿⣧⣿⣿⣇⡇⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⢠⡿⠉⣏⡄⠀⠳⡄⠀⠀⠀⠀⠈
⣆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠉⠠⢻⣿⣿⣿⣿⣆⠙⠿⠿⠋⠛⡧⠀⠀⠉⢰⣧⣀⠀⠀⠀⣠⡾⠁⠀⠈⠀⣄⠀⠙⣆
⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⢡⣾⣿⣿⣿⢿⣏⠀⢄⢀⡀⢪⠙⢦⡀⢀⣼⣿⣉⠁⠀⣼⣿⣇⠀⠀⠀⠀⠘⢦⠀⠈⢧
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣟⡒⢈⡽⠋⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠻⡑⠀⠀⠀⠙⠷⡤⠺⣿⢻⣿⡯⢰⡟⢻⡿⢿⡄⠀⠀⠀⠀⠘⣧⡀⠀⠁
⣿⣿⡄⢀⠀⠀⠀⠀⠀⠿⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠙⣄⠀⠀⠀⠀⠐⣶⣿⣿⣏⣿⠀⠁⠈⠀⢸⣧⠀⠀⠀⠀⠀⠘⣷
⣿⣿⡇⠈⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⠀⠈⠁⠈⢲⠀⠀⢹⣿⣇⣿⣿⡆⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠹⣧
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠻⣽⣿⣿⣿⡇⠀⠃⠀⠘⣿⢹⣆⠀⠀⠀⠀⠀⠹⣧
⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⡿⢫⣿⢃⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⣀⠀⠀⠈⢿⣿⣿⠀⠀⢰⠀⠆⠿⡜⣿⣆⠀⠀⠀⠀⠀⢹⣇
⣾⣿⣿⣿⣿⣷⣦⣤⣤⡶⠿⣋⣩⣻⣿⡟⠋⣠⣿⠃⣾⣿⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⣿⠃⠀⠈⣼⣿⡿⠀⠀⢼⠁⣶⠃⡇⢻⡟⡄⠀⠀⠀⠀⠀⢿⡄
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠠⠦⣍⣾⡿⠃⠀⣰⣿⠁⣼⣿⣿⣿⡇⠀⢸⡇⣿⠀⠀⠀⡆⢸⣧⡀⠀⠈⢿⡄⠀⠀⠀⢲⣽⠃⠁⠸⣇⠀⠀⠀⠀⠀⠀⠘⣷
⣿⣿⣿⣿⢿⠋⠀⣠⣾⠃⣴⡿⠋⠀⠀⡴⠋⠀⣰⣿⣿⣿⡿⠃⠀⢸⣷⣿⠀⠀⠀⣏⣿⣿⣿⣦⣄⠘⠃⠀⢀⠀⠘⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢻
⣿⠛⠉⡐⠀⣠⣾⢓⣵⣾⠋⠀⠀⢀⠘⠀⠀⣴⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡄⢀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢼
⠀⡀⢈⣰⣾⡟⢊⣱⠟⠁⠀⠀⠀⠀⠀⠀⣨⡿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣇⠀⡀⢡⣾⣏⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠘
⣶⣿⣿⣿⣁⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣰⣞⣹⣿⣿⣿⣀⣀⣀⣀⣀⣿⣿⣀⣀⣿⣿⣿⣏⣿⣿⣿⣿⣿⣷⣔⣶⣻⣇⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀

⠀⣠⡾⠛⣯⣷⡀
⡞⢻⣳⡾⠷⠈⢳⡄
⠉⢿⡉⠀⠀⠀⢸⡇
⠀⠀⠹⣆⣠⠴⠞⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡶⢦⡀
⠀⠀⢰⣟⠁⢀⣠⣤⣵⠀⠀⠀⠀⠀⠀⣠⣤⣤⠶⠶⠒⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⣹⣧⡜⡹
⠀⠀⠈⣉⠾⠋⠀⠀⠸⣇⠀⠀⠀⢠⡞⠁⢀⣤⡀⠀⠀⡀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⡁⠀⠀⠈⠻⣧
⠀⠀⠘⢿⡀⠀⠀⠀⠀⠻⠀⠀⢀⡟⢀⡾⠉⠈⠳⣦⡀⠁⠒⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡀⠀⠙⢦⣀⣴⠞⠉
⠀⠀⠀⠈⡇⠀⡆⠁⡄⠘⡇⠀⢸⠇⢸⠁⠀⠀⠀⠈⠻⣦⠁⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠙⣦⠀⠀⣻
⠀⠀⠀⠀⣇⠀⠐⠂⠁⠀⢹⡄⣼⠀⢸⢠⡆⠀⠀⣤⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠸⢇⢘⡿
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣷⡇⠀⡞⠈⠁⠦⠀⡋⠀⠀⠛⢢⡄⢸⡇⠀⠀⣠⡴⠋⠀⠀⠀⠂⠀⢈⡴⠿⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣇⠈⣧⠀⠰⡶⢄⡙⠄⢀⡀⣼⡇⢸⣇⣠⠞⡉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣿⠀⠀⡁⠀⠀⠀⠘⢿⣟⣷⣄⣀⣀⣁⣴⣟⡁⠘⢃⡼⠋⠁⠀⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠂⠀⠀⠀⠀⠉⠙⢧⣉⠉⣉⡽⠉⠱⠀⠀⣀⠀⠀⠀⠀⠐⢀⡴⠋
⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢸⡀⠠⠀⠀⢀⠀⠀⠀⠐⠀⠠⠛⡀⠀⠄⠁⠀⠀⠁⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⢠⠞⠁
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠏
⠀⠀⠀⠀⡼⠇⠁⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⢀⠇⠀⠀⠀⠇⠀⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⠇
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡄
⡘⠁⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠀⠙⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⢰⡅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣈⡷
⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣠⣤⠔⠒⠒⠊⠉⢻⡉⠉⠉⠉⠉⠉⠉⠉⣉⡽⠃
⠸⡇⠀⠀⠀⠀⠁⣀⡤⢾⡋⠉⠀⠹⣆⠀⠀⠀⠀⠈⢳⣀⣀⣠⣤⡤⠒⠊⠉
⠀⢷⠀⠀⢀⡴⠟⠉⠀⣀⣙⣦⣄⠀⠈⢷⡀⠀⠀⠀⠈⢿⡭⣤⣀⣀
⠀⠘⠶⡶⠯⠤⠚⠛⠉⠉⠉⠁⠈⠛⢦⣄⠹⣆⠀⠀⠀⠘⢧⠀⠉⠉⠛⠓⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠘⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠳⣦⣄⠀⠀⠀⠀⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠹⣏⢳⡀⠀⠀⠁⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⢎⢷⣀⠀⢀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⡆⠹⡆⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇⠀⠙⢦⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠄⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣀⠀⠀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣨⣦⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⢷⠸⣿⣿⣿⣿⠀⠀⠂⠀⠀⢸⣯⣷⣿⣿⣸⡆
⠀⠀⠀⠀⠀⠀⠀⣧⣾⣿⣿⣿⣿⢟⡾⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⣛⣿⣿⣿⣟⢉⣧⡄
⠀⠀⠀⠀⠀⠀⠀⢣⢿⣿⣿⣿⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⣿⣿⣿⣿⢸⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⣿⣷⣿⣿⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠒⠓⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡟⣄⠘⠫⠉⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⢶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣴⣿⣿⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠉⠉⠉⠁⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣠⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣰⠃⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡀⢻⠀⠀⠀⡇⠀⠀⠀⢧⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣚⣧⠸⡄⠀⣀⢻⡰⠊⠙⠊⠃⠀⠀⠸⠆⠉⠙⢺⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡞⠻⡄⠁⢠⣄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡆⡞⢙⡆⡸⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⣇⠀⠙⢤⡇⠀⠘⠟⠀⠀⠤⠀⠀⠀⠿⠀⠀⢀⢇⣧⠞⡴⠁⢆⠀⢦⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣬⣧⠀⠀⠙⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠀⢀⣜⣜⡠⠊⠀⠀⠈⠳⣄⡙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠾⠟⠋⠀⢧⠀⠀⠀⠓⠤⣀⠀⠀⠀⠀⢀⡠⠔⠛⠛⣩⠴⠶⠶⠶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⢔⠆⠀⢸⡀⠀⠀⠀⠀⠈⠁⡖⠂⠈⠉⢰⠀⠀⠀⢿⢉⠉⠉⣫⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣴⠇⣒⣢⡌⡇⢀⡠⣤⢀⡤⠴⠁⠀⠀⠀⠸⢤⢖⠒⢿⢸⠶⠯⣛⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠰⢟⣋⣛⡦⢧⠀⠀⠈⠣⣕⠶⠤⠤⠤⢶⣺⠵⠋⠀⢸⠴⠓⠒⢬⡁⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⡭⠭⠵⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⠯⠭⡙⠂⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⣸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣰⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣾⢹⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⠞⠈⡎⠧⠤⡤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠰⠋⢀⡠⢿⣠⠤⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡤⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠢⠀⠀⠀⠀⠔⠀⢠⡄⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠔⢧⡴⠂⠠⠀⠀⡄⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣁⠴⠊⠀⠀⡈⠀⠀⢁⠀⠓⡋⢱⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠁⠄⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠀⠀⠖⠒⠒⠒⠒⠖⠆⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡹⣆⣰⡶
⠀⠀⠀⠀⠒⢆⠂⢔⠺⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠂⠂⠒⠒⠂⠐⠀⠐⠂⠀⠐⠀⠀⢈⡮⠋
⠀⠀⠈⢄⠑⠠⠧⠐⢁⠔⠀⠀⠀⠀⠐⠲⡠⢒⠀⢂⢄⠂⠀⡀⢒⡀⠀⠀⠀⠀⠠⡠⠔⠀⠀⠴⡠⠁
⠀⠀⠀⢀⠀⠐⠒⡀⠁⠀⠀⠀⠀⠀⠀⠈⠡⠁⠀⠀⡈⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠒⠂⢀⡀⠀⢀
⠀⠀⠈⠉⠉⠉⠁⠉⠈⠉⠀⠀⠀⠀⢀⢈⠡⠀⠄⠀⠄⠀⠄⡀⠃⡄⠀⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠐⢁⡄⠀⠀⠅⢀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀⠀⠤⠀⢀⠈⡢
⠀⠀⠈⠀⠀⠀⢀⣀⠄⠀⠄⠀⠀⠈⠉⠁⠁⠡⠀⠙⠏⠀⠍⠉⠉⠁⠈⠀⠀⠀⠀⢰⠉⣄⠰⠀⠂
⠀⠀⠐⡀⠀⠠⢗⡋⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢙⣼⠷⠁⡂⢀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⢀⠔⠀⢀⡀⠀⠢⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⡀⠀⠀⠁⠀⠠⠀⠀⠁
⠀⠀⠋⠁⠉⠀⠈⠉⠈⠈⠁⠂⠰⠦⠁⠊⠾⠆⠠⠀⠀⠀⢸⠉⠉⠉⠨⢹⠀⠀⠀⠀⠛⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠬⠤⡒⠀⡈⠀⠀⢁⠸⠤⠤⣀⣰⠼⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠐⠰⠒⠐⠀⠰⠲⠒⠐⠒⠐⠒⠒⠔⠂⠂⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠓⠲⢤⡀⠀⠀⠀⢀⡞⠉⠀⠈⠙⢦⠀⠀⠀⠀⣠⠎⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢧⡀⠀⠀⠀⠀⠈⢇⣰⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⠀⠀⠀⠀⢠⣻⡆⠀⣜⣦⠀⠀⠀⠀⢀⣀⠈⣇⠈⠳⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⡇⢸⠁⠘⡏⠀⠀⠀⠀⠈⠛⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⢰⠋⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠸⠶⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⢳⡀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠉⠩⠭⠉⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣀⣀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣋⢰⡏⢹⡗⢀⣹⠤⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢽⣯⣛⠫⢌⣹⣥⠴⣟⣀⣉⣦⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠈⣳⣾⣤⡙⢆⣠⣄⠀⠀⠘⣽⠛⣷⣴⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣛⢝⡻⣶⢾⡁⠈⠦⣀⠀⠈⠙⢿⢀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣃⠀⢈⣣⠀⡠⢧⣀⢠⠼⠿⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠠⢯⡉⠑⢦⡘⣿⢛⣾⣛⣽⣧⡖⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢞⠡⠀⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣷⢀⢀⠀⠀⠀⠀⠀⡟⢁⡆⢠⠍⠈⢻⣽⡏⡞⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡇⠀⠀⠈⢣⠀⣠⠋⠀⠀⠀⠀⣼⠃⠀⠙⣓⣤⠀⣠⠞⡠⠋⠀⠈⠀⠀⠀⠈⠣⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣲⠚⣌⣇⢀⡧⣆⠀⠀⣸⠜⠁⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠙⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠁⠀⠠⣠⠋⠀⠘⣷⠚⠁⡤⠠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠾⠉⠁⠚⡄⠀⠀⢠⠁⠏⠀⢠⠀⠈⠑⠶⠚⠋⡠⠊⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⣄⠀⠂
⡇⠀⠀⠀⢰⡅⢀⠀⡆⡼⡇⠰⠛⠈⠛⢲⣄⠀⢼⠄⢀⣎⣍⣩⣩⢯⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘
⠈⠣⠀⠴⠋⠙⣽⠚⠖⠁⢹⢁⠀⠀⡴⣫⡦⡄⠘⠀⠘⠻⣿⣆⢱⡁⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣧⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠉⡆⢘⢻⣠⡄⢧⣁⡴⠉⡇⣀⠀⠀⠈⠙⢦⣧⠀⠣⣎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⢀⠔⠤⡀⠀⣬⡍⠀⣏⡀⡄⢀⡅
⠀⠀⠀⠀⠀⠀⠀⡸⠄⠹⡇⠈⢿⠽⠖⠰⠒⠂⠙⠛⣄⠀⠀⠀⠀⠈⠑⠢⠜⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⣠⣔⠝⠉⢹⠃⢠⣿⣧⣤⡺⣩⣯⠱⡈⠃
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡭⣀⡸⠒⠒⠊⠉⢹⠃⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠎⣭⣤⣽⣹⠁⠀⡰⠃⢰⢩⡇⠈⡢⠟⠠⠞⡰
⠀⠀⠀⠀⠀⠠⠹⡀⠀⡞⠉⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⠑⠒⠋⠒⠸⠖⠒⠲⠶⢶⠲⣿⠸⡄⢻⢋⡕⢀⣼⡁⡴⠉⣲⡨⠩⠥⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠴⢊⣹⡆⠉⡳⢌⡉⠘⠴⠫⠖⡹⢠⠋⠀⠑⣄⠐⠈⠀⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⢀⣘⡷⠀⠤⠤⠭⠤⠄⠈⠃⠀⠈⠁⢀⣘⠭⠛⢀⠤⠛⠓⢟⡾⠒⢳⡶⠆⠀⣛⠀⢈⣉⡩⢤⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡟⠀⠀⡏⠈⠓⠒⠒⠒⠲⠶⠶⠶⠒⠒⠒⠊⠉⠀⣀⣴⢁⣠⠖⠋⠉⢟⠟⢉⠗⠒⠊⠉⠉⠁⠀⠀⡎⣠⣄⡟⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢁⣀⠀⡩⢶⣤⡤⢄⣀⣀⣀⣀⠀⠤⠤⠔⢒⠊⠉⠀⠈⠁⠀⠀⠀⢀⠞⣩⠛⠢⢄⡀⠀⠀⠀⠀⠀⣇⠙⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣜⣤⡇⠙⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⠢⢄⠀⢀⠔⣉⠑⢁⡜⠁⠀⠀⠀⠈⠙⠒⠢⠤⠤⠼⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⢁⢤⢇⣄⠇⠀⠀⠀⠀⠀⠀⢤⠤⠤⠚⢡⠀⠀⠀⠀⠀⠈⣋⢰⠁⠐⡇⢸⢆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠋⢣⣤⠀⠀⠀⠀⣎⠐⡹⠑⣚⣁⣲⡌⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣧⣀⠀⡀⢈⣉⠗⢾⡀⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠈⡙⠀⠉⡇⠺⠇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡄⠉⢐⣽⡋⠉⠓⢆⠓⠈⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠛⠻⠿⠆⠒⠊⠐⠚⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠢⠢⣐⡀⠜⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠰⢀⠐⠀⠤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢰⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⢀⠀⡍⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠊⢀⠈⠊⠡⠘⢄⠀⠠⠀⠀⠔⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣶⡡⠔⠊⡠⠖⠁⠀⠀⠀⠑⢄⠳⣔⠀⠀⡀⠀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠄⠀⠀⠁⠁⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠤⠇⠁⠀⠀⠀⠀⠐⠢⠠⢤⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠉⠀⠘⠒⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣏⣛⣟⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠿⠿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⢙⣥⢊⣴⡄⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡶⢚⡵⠋⢿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠁⠹⣿⣿⡄⠀⠀⠀⣿⣿⣿⡿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠋⣠⣾⣧⠀⠀⢿⣿⣿
⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠀⢻⣿⣧⠀⠀⠀⣿⣿⣿⠃⠀⢰⣿⣿⣿⣿⣿⡿⠛⠁⢀⣴⣿⣿⣿⡆⠀⠘⣿⣿
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠻⣿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣄⠀⣠⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢀⣴⡾⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠘⢿⣦⣶⣶⣶⣶⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠀⠀⠀⠀⠀⠀⠉⠉⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡏⠀⢠⣶⣦⠀⣴⣶⡄⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠈⠛⠋⠀⠘⠿⠋⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠙⠛⠿⠟⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠉⠁⠀⢠⡎⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⠖⢘⣿⠟⢶⣀⡐⢛⡿⠿⣿⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠚⠓⢀⠀⢴⣆⠀⠈⣿⡿⠟⠛⠻⣿⠇⠀⣿⡀⠀⣻⠁⠘⠻⠃⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢸⡇⢀⡽⠛⢹⣿⡀⠀⠀⠀⣿⡆⢀⣿⠛⢿⡟⠁⠀⠀⡄⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠀⠀⠀⠀⠀⠀⣸⡷⠾⢷⣼⠛⠀⠙⠀⠀⠀⢿⣀⣨⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠾⢀⡀⠀⠀⠀⠀⠴⠆⠉⠀⠀⠀⠈⠀⣰⠶⣦⣀⠀⠀⠋⠙⠉⠀⠴⠒⠤⢀⣰⣶⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⠉⣷⡖⢠⡀⠀⠀⠀⠀⠀⠈⠉⠉⠰⢦⣄⣀⡀⠙⣧
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠀⠀⠀⠀⠹⣷⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⢣⠀⠙
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠻⠏⠀⠐⠃⣀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⢀⣠⡤⢶⡟⠛⠀⠀⠀⠘⠁⣶⡀⠙⡀⠀⠀⠀⢨⡆⠙⠛⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠈⠋⠀⠶⠶⠶⠛⠁⠀⠀⡇⠀⠀⠀⠀⠘⣷⠀⠉⢰⣷⠆⠀⠤⠛⠃⠠⠀⠀⠀⠀⠀⠘⡿⠂⠶⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠈⣧⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣿⠀⠀⠀⣈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠟⠃⠀⠸⡇⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠸⣿⠀⠀⠀⠐⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣤⣤⣤⣿⠀⠀⠀⠲⠆⠀⢰⣯⣀⣀⣀⣀⣀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡿⠛⠉⠉⠉⠳⠶⠶⠤⠤⠴⠾⠿⠛⠛⠋⠛⠛⢿⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⠿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢠⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⢠⡆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⢰⡆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠘⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢸⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣦⣤⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣾⡟⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠘⠣⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠷⢶⣶⣄⣀⣼⣿⣿⣀⣀⣤⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⣁⣀⣤⡶⠶⢶⣿⡿⠿⠟⠛⢿⣿⠟⠁⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣶⣶⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠉⠹⠟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠒⠒⠊⠁⠀⠀⠀⠀⠠⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠘⡜⢁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⢀⠒⠄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠜⠀⠀⠀⠀⠀⢃⠒⠀⠁⠉⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠈⠁⢢⠤⢤⢤⡠⢤⠴⣊⠹⠀⠀⠀⠀⠀⠀⡸⢀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣠⠞⠗⠢⡢⡀⠤⣴⡍⠱⠆⠡⡟⠁⠀⡵⠁⠀⠀⠀⠀⢀⠔⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⡭⠖⠚⠳⣧⡘⠃⢄⠋⠠⠂⠘⠒⠒⠻⢁⠀⢰⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⡨⠥⠉⠁⠀⠐⠚⠀⠀⠀⠑⠚⣠⣁⠀⡀⠄⠊⠁⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠢⣉⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⡎⠙⠫⠙⢆⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡞⠀⠉⠒⢖⠒⠒⡖⠈⢉⠀⡐⠀⢀⠰⠁⠈⡄⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢆⠁⠂⢤⣀⢀⢁⠢⣘⠁⠉⡤⠀⡃⠁⠀⢡⠀⢠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠈⡄⠀⢸⠀⠀⠀⠢⣈⠢⣠⢃⠌⠀⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠱⡁⠆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠀⢡⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠎⠡⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢀⣀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠀⡄⢦⠇⠄⠀⠀⠒⠒⠂⠀⠈⣭⢉⣀⢀⠀⡀⢀⢤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠈⠲⠖⠒⠒⠒⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠈⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠖⡄⣠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠠⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠄⢴⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠉⠁⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣤⣠⣤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⣿⣿⠃⠀⠀⠈⠹⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠤⠤⠀⠠⡤⢿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⠁⠀⠀⢀⡀⣀⠀⢸⣿⣿⠟
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⡿⠿⢿⠏⠄⠀⠀⠙⠛⠋⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⢀⢀⡞
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠋⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠄⠀⠀⠤⠀⠀⠀⡀
⠀⠀⢠⠂⠀⠀⠀⠀⠀⡀⠠⠖⠁⠀⢀⣤⣶⠤⡤⠄⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠓⠀⠐⠁
⡆⠀⠠⠆⠀⠀⠀⠀⠁⠀⠀⠀⠈⠝⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠈⠀⢠⡶
⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⠐


⠀⢀⠀⠀⠠⣀⡀
⡀⡀⢱⠄⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄
⠇⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼
⣧⡀⠀⠀⠀⠀⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⡀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⡆⠀⠀⠈⠫⠀⠀⠀⠀⠀⡠⠂⠀⠀⠃
⣿⣿⡄⢀⠈⠀⠀⢀⣬⣄⠀⠀⠀⠀⠀⠀⠀⡘⠉⠑⢄⣀⣀⠴⠋⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⢻⠂⠀⠈⠀⠀⡀
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠓⢤⣄⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠄
⣿⣿⣿⣿⣦⣄⠀⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠉⢻⠈⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠒⣂⠁⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠙⠻⠿⠿⠛⠁⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣾⣿⣷⣶⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠙⠛⠁



⣤⣤⣤⣦⣴⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣴⣶⠟⠛⠲⡄⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⢺⣿⡇⠀⠀⠀⠀⢿⣿⣤⠀⠀⣘⣀⡤⠖⢻⣿⡿⠁⠀⠀⣠⣿⣿
⠀⠀⠀⠀⢼⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣀⡤⢾⣿⡟⠛⢻⠋⠁⠀⠀⢸⣿⠁⠀⡠⠘⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣿⡿⠶⠚⠋⠁⠀⠘⢿⣧⠔⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⣼⣿⠏⠀⠀⠀⠀⠀⣀⣤⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠒⣾⣿⠟⠒⠒⠒⠒⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠶⢠⠄⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠚⢁⠴⡄⠀⠀⢀⠠⢀⠀⠀⠄⠤⢄⡀
⠀⢀⠔⢋⠀⠀⠈⠐⢄⠀⠘⢄⢸⠀⠀⢘⠣⡆⠡⠀⠊⠐⠇⠀⠀⠀⠀⠀⢀⠔⠛⠓⠶⡎⠀⠀⠀⠸⣀⠜⠀⠘⠀⠀⠀⠀⠉
⢀⠃⠀⠠⠓⡐⠢⡀⠠⠇⠀⠀⠻⠃⠀⠦⣘⠇⠀⠀⠁⣬⠀⠀⠀⠀⠀⠀⢈⠀⠸⠿⠀⢰⠀⢀⠎⡉⠁⠒⡄⠸⠤⠄⣀
⠘⠀⠈⠃⡘⠚⡠⠁⠀⡈⠀⠀⠀⠀⢸⡆⠙⠐⢡⠠⢐⡂⠀⡤⢶⠒⣄⣀⣀⠉⠐⠒⠂⠂⠀⣂⢾⠀⢠⠉⠪⡇⠀⠀⠀⠁⠈
⠀⠁⡀⠀⠈⠉⠀⠀⡠⠁⠀⠀⠀⠀⠒⠁⠆⠆⣩⢷⢮⠁⠀⣷⠈⢻⣅⡟⡴⡇⠀⠀⠀⠀⠀⠀⠄⠼⢸⠀⢀
⠀⠀⠀⠑⠚⠂⠂⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⣈⣀⠈⢙⠓⢈⣀⢀⣀⣀⣀⡀⡀⡒⠀⠚⠀⠀⠀⢀⣰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡞⠉⡉⠉⣷⡀
⠀⠀⡔⡖⣖⠶⠀⠀⠀⠀⡇⠀⠀⡀⠤⠄⠀⢀⠤⢄⠀⠀⡠⠄⠀⠠⢄⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⡇⠀⠀⠀⣄⠀⠉⠁⣸⠃
⠀⠀⠐⠚⣽⠒⠁⠀⠀⠀⡇⠀⠀⡇⠀⠀⡠⠃⢀⠂⠀⠀⠀⢠⠒⢢⠀⠃⠈⠒⢠⠀⡠⠤⠇⠀⠀⡇⠀⠀⠀⠙⠒⠶⠚⠋
⠀⠄⡤⡏⠍⠱⢠⠠⠀⠀⡇⠀⠀⡇⠀⠘⠁⠠⠃⠀⠀⠀⠀⠈⠀⠈⠀⡄⠀⠀⠈⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠄⠅⠧⠀⠰⠘⠔⠀⠀⡇⠀⠀⡇⠀⢠⠀⠐⢄⠀⠀⠀⡇⠀⠤⠐⠊⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇⠀⣀⣴⠒⢩⣉⢹⡍⡗
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⠑⡄⠈⠢⡀⠀⡇⠀⡇⠀⠀⠀⡔⠒⠂⠐⠂⠠⠀⠀⠀⡇⠘⢯⡛⢎⠣⠸⢰⣾⠁
⠀⠀⣠⠐⢲⢇⠀⠀⠀⠀⡇⠀⠀⠃⠀⠘⠀⠈⠒⠀⠁⠀⠑⠐⠂⠀⠀⠀⠁⠐⠒⠒⠂⠐⠀⠀⠀⡇⠀⠀⠈⠻⠖⢴⡎⠉
⠀⠀⠈⢟⡺⠬⠂⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠀⠀⠀⡼⠾⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⡷⠟⠛⠉
⠀⠀⣴⡒⣶⣤⢄⠀⠀⣀⠤⠠⢀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡐⠀⠀⠀⠀⢀⠄⠠⡀⠀⠈
⣤⠸⣧⣑⣹⠏⠁⢀⠊⡀⢂⡐⡲⡟⠅⠀⢈⠀⢀⠠⠐⠀⠒⠤⠀⠄⡪⠥⠭⠀⠀⠀⠀⡠⢼⣄⢠⡟⠐⡄⠀⠀⢀⠔⡩⡙⡄
⠈⠠⢷⠋⠀⠀⠀⢸⠐⡀⢅⣈⠄⡁⢈⠀⢰⡊⠀⢠⠁⠀⠈⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠕⠺⠀⠀⠟⠙⡄⠀⠀⡎⠠⠖⢁⠇
⠀⠈⠁⠀⠀⠀⠀⠀⠢⡑⠂⠀⠊⡠⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠤⠤⠇⠼⠤⠤⠄⠀⠘⡄⠛⣀⠎
⣶⣶⠶⢶⢶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣾⣷⣶⣾⣷⣶⣾⣷⣶⣶⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠅⢠⡄⢠⡄⣾⠉⠙⣷⡄⠀⠀⠀⠀⠀⣨⠀⣤⠤⢤⡘⣧⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠃⣿⢸⢿⣼⡇⣿⠀⢀⣾⠇⠀⠀⠀⠀⠀⣿⣼⠃⠀⢨⡗⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠿⠼⠈⠿⠇⠿⠶⠟⠋⠀⠀⠀⠀⠸⠿⠇⠘⠦⠴⠟⠁⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⣄⠀⣠⠀⠀⠀⠀⠉⠙⡟⣛⠀⢀⡄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣧⣿⠀⠀⠀⠀⠀⠀⡇⣿⣤⣼⡇⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠹⣿⠀⠀⠀⠀⠀⠀⡿⢿⠀⠸⡇⣿⣠⣄

⠀⠀⠀⠀⠀⣠⡶⠶⣦⠀⢀⣀⣀⠀⡴⠶⢶⣄⢀⢰⣦⢠⣶⠀⠀⣀⠀⢀⡴⠶⢶⡆⢀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⢠⡏⠀⠀⣿⢿⣋⣩⣽⢿⣇⠀⠀⣿⣶⢸⣿⣾⣿⠀⡾⣿⣧⢸⣧⣶⠞⠃⠹⣦⣾⠋⢸⡇
⠀⠀⠀⠀⠘⣧⣀⣴⠟⢸⡟⢿⣄⢸⣿⣤⡾⠏⢸⣸⡇⢿⡟⡼⠛⠛⣿⣾⡏⠻⣦⡀⠀⢿⡇⠀⠸⠇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠈⠈⠀⠈⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡆
⠀⠈⣷⣥⣬⣧⡄⢠⣾⣿⣷⢱⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢸⣆⣤⣵⣤⠄⢠⣤⣤⣌⣿
⠀⠀⢏⠘⠛⠛⣱⣧⠙⠻⠟⢈⠇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⡁⠛⠛⠋⣾⡎⠛⠿⠃⡹
⠀⠀⠀⡟⣧⡀⠉⠉⢀⣰⡏⡇⠀⢸⠀⢧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇⠀⢸⢻⣄⣀⢉⢁⢀⣼⢹
⠀⠀⠀⢧⠈⠙⠆⠒⠘⠉⢀⠃⠀⢸⣆⢘⡗⠀⠒⣛⣓⠒⠤⠀⠀⠔⠒⢒⡒⠂⠀⢾⡖⢸⠇⠀⠸⡌⠋⠉⠉⠈⠙⠛⡸
⠀⠀⠀⠀⠑⢄⣀⣀⣤⠔⠁⠀⠀⠀⢿⣿⢰⣿⣿⣿⣿⣿⣦⠀⠀⣾⣿⣿⣿⣿⣷⡄⣧⡾⠀⠀⠀⠈⠢⣠⣀⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢸⣿⣿⣿⣿⣿⠟⢠⡄⠻⣿⣿⣿⣿⣿⠇⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡆⠀⣉⠛⠛⠛⠁⢰⣿⣿⡄⠈⠉⠿⣿⠽⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⡾⣴⣦⠀⠀⠀⠻⡿⢿⠇⠀⠀⠀⣤⡔⣷⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠰⣄⠄⣐⣄⠀⠀⣀⣀⠀⠆⡄⠀⠀⠀⡇⠘⣿⣦⣀⣀⣀⣀⣀⢀⣀⣀⣴⣿⠁⣿⠀⠀⠀⢃⠠⠠⢄⠀⠀⠀⣀⠄⠠⣐⠇
⠀⠀⡿⢾⣿⣿⠇⡰⣿⣿⣿⢸⡇⠀⠀⠀⡇⠀⠹⡿⡳⠦⡬⣠⣬⢭⠴⣖⢿⠛⠀⣿⠀⠀⠀⢸⡧⣾⣿⣿⢂⢺⣿⣿⡧⣿
⠀⠀⠳⣤⡉⠁⢼⢿⠀⠉⡁⡀⠁⠀⠀⠀⢻⣧⠀⠀⠈⠉⠓⠒⠒⠒⠋⠁⠀⠀⣠⠏⠀⠀⠀⠘⢦⣈⠉⠁⣿⡇⠈⢉⣁⠜
⠀⠀⠀⡏⣿⣤⣀⣀⣠⣾⡇⡇⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢀⠐⣠⡾⠋⠀⠀⠀⠀⠀⢸⢿⣤⣀⣀⣠⣀⣼⢸
⠀⠀⠀⢇⠙⠛⠟⠻⠟⠋⢁⠇⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣆⣀⣀⣴⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⡌⠛⠟⠻⠻⠟⠛⡸
⠀⠀⠀⠈⠳⣄⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣠⡠⠊
⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣩⣙⣻⣋⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠰⠄⠀⠐⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⢀⠂⠈⡉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠉⠂⠻⡆⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠒⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠤⠒⠒⠐⠒⠒⠤⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠊⠁⠀⠐⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢃
⠀⠀⠀⠀⠀⠈⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⡄
⢃⠀⠊⠀⠔⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠜⢧⠘⡇
⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⠀⠎⠁⠙⠢⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⢰⠃⠀⢈⠀⠀⢲⢀
⠀⠀⠀⠀⠀⠑⠢⡂⢤⠀⠀⠆⠀⠀⠀⣈⠕⠂⠀⠀⠀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠠⡁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠉⠠⢈⠢⢄⠄⡀⠀⢀⡤⢢⡀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⠄⠀⠘⠣⡁⠀⠀⠀⠀⠂⠠⠚
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⢀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠒⠆⠹⠤⡧⠒⠀⠉⢐⠄⠄⡀
⠀⠰⠁⠀⠀⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠤⠀⠒⠒⠲⠖⠐⠂⠀⠐⠀⠒⠒⣤⠀⠀⠀⠀⠀⠀⠀⠀⠨⣒⠊⠷⢄⠀⠀⠘⠋⠐⠃⠤⣀
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣜⣀⣀⠔⠚⠉⠀⠀⡀⠤⠀⠒⠒⠦⠄⢀⣀⡤⠚⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡠⠊⠀⠀⠀⢸
⠀⠈⠀⠀⠀⢀⡀⢈⠷⠃⠀⠙⠉⠈⠉⠉⠉⠐⠒⠀⠀⠀⠤⠶⠈⠉⠁⠀⠀⠀⠸⠀⠾⣀⡀⠒⠿⠐⢠⣈⣹⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠰⠀⠀⣨⠁⠌⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠸⠀⠀⠀⢹⠃⠲⠴⡖⠁⡙
⠀⠀⠀⡄⠀⠍⠜⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠃⠀⡇⢀⠇
⠀⢠⠞⠃⠒⠺⢅⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠒⠃⢢⡽⣸⣀
⠀⠛⠢⠤⠄⢀⣀⣹⠏⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡐⠁⠀⠀⢠⠊⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⠰⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠈⠁⢇⣀⣀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠢⢤⣤⣄⣀⣀⡀⠀⠀⠘⠀⡤⠚⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢤⠒⠲⠦⠄⠀⠀⠀⢀⠈⣉⡙⠓⠒⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡤⠤⠄
⠀⠀⠀⠀⢀⣀⡀⠈⠉⡀⠄⠚⠀⠒⠤⠄⢀⣀⡀⠄⠂⢉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠨⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢆⠈⠉⠙⠛⠐⠂⠒⠦⠀⠠⡔⠊⠀⢀⠤⠂⠁⠀⠀⠀⠀⠠⡀⠄⠀⠀⠠⠀⢄⡀⠈⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠒⠒⠀⠤⠤⡇⠐⠋⠀⠀⡠⠒⠀⠒⠲⠤⣀⣸⡀⣀⠀⠠⠀⠀⠈⠢⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠔⠊⠀⠀⢀⠔⠀⠀⠙⠈⠀⠐⢄⠙⠂⠀⠀⢀⢅⠀⠈⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠋⠀⠈⠓⠤⠀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣦⠀⠀⠈⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠤⠤⠾⠷⠀⠀⠀⠀⠀⠀⢋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠒⢀⡀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⣿⠇⠀⠈⠱⠆⠀⠀⠀⠀⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢻⣞⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠘⣿⣿⡇⠀⠈⢱⣤⡀⠈⠀⠀⠀⠀⠀⠀⠀⢒⠁⢾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⣿⣠⡀⠁⠒⣒⣢⡤⠄⠒⠈⠁⠒⠘⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠠⣾⣯⣄⣤⠿⣿⣀⢠⣤⣀⣭⠀⢀⡿⠀⠀⠘⠛
⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠙⠋⠁⣧⣄⠠⠛⠛⠛⠛⠻⠏⠉⠀⠀⠀⠄
⠶⡄⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⡁⠀⠀⠀⠀⠀⠀⠁⢠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠀⠀⠀⠀⠀⠀⠀⠜⠛⠟⠉⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⡀⠴⣦⡄⠀⠀⠘⠋⠀⠀⠀⢤⣿⠟⣿⢷⣦⣀⣀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⡶⠀⠀⢨⠆⠀⠞⣁⣴⣿⡆⠙⠷⠿⠿⠿⠛⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠄⠀⢀⣀⠀⠀⠀⠀⠀⣼⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠆⠀⠀⠀⢸⠇⠀⢠⡴⣄⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠳⠦⣀⣀⠀⣠⡆⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠁⢨⡀⠀⠀⠀⣸⠉⠀⢾⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡍⠀⠀⠀⠀⠀⠀⠍⠻⠙⢳⠖⣴⣿⠀⠀⢸⡯⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠈
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣇⣷⣤⣄⢠⣤⡆⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠀⠈⠡⠐⠄⢈⣵⢻⣥
⠀⠀⠀⠀⠀⢀⣤⢴⣿⣿⣿⣿⣿⠰⠆⠂⠀⠀⠀⠀⠘⣿⠿⠟⣷⣄⠀⠀⠀⠀⠄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠉⣱⠤⡠⣀⠀⢰⣿⣷⣤⣄⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣦⠘⠋⠀⠀⠀⠈⠓⠶⠉⠉⠙⠻⠄⠈⠀⠀⠀⠀⢀⡖⢻⣤⣴⠏⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠐⠉⠁⣠⠐⠐⠂⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠸⢧⠚⠒⠒⠉⠐⠀⠀⠀⡀⢀⡈⠉⠙⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠀⢀⡀⠀⠀⠀⠀⢸⠁⡠⠓⠐⢃⡴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣘⣁⣀⣀⣀⣀⣀⣉⣁⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠶⠶⠶⣦⣴⣶⠶⢶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠛⠛⣿
⠀⠀⠀⠀⠀⢀⣾⠏⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡏⠀⢠⣾⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠁⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⢙⣿⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⢿⣄⠀⠀⠀⠀⢸⣿⠀⠀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⠀⠈⣿⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠙⢻⡷⠶⠶⠶⠾⠛⠋⠀⠀⠈⠻⢷⣄⡀⠀⣼⡇⠀⠀⣿⠉⠉⠉⢹⡇⠀⣿⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⡄⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠸⣿⠀⣿⡇⠀⠀⣀⣿⡶⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⢀⣾⠛⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠀⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⣀⣀⣀⣐⣿⠀⣿⣇⣾⡇⠀⠀⠀⠀⣠⣾⠗⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠙⠛⠛⠛⠛⠋⠀⠙⠛⢿⡇⢴⣦⣠⣾⠟⠁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⣀⠀⣀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⠛⠁⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠛⠛⠛⠉⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠻⢷⣄⣀⣠⣶⠟⠻⣷⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⣴⠿⠛⠙⢿⣦⡀⠀⣹⣿⣯⠀⠀⣠⡾⠋⠛⠻⣶⣄⠀⣿⠀⠀⠀⢸⡇⠀⣿⠋⠉⠉⠙⣿
⠀⠀⢠⣾⠋⠁⠀⠀⠀⠀⠙⢷⣾⠟⠀⠹⣷⡾⠋⠀⠀⠀⠀⠀⠙⢷⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢀⣿⠀⣀⣀⣀⣀⣀⣀
⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠐⣿⠀⢿⡏⠉⠉⠉⣿
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢠⣿⠀⣸⡇⠀⠀⠀⣿
⣼⡇⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⢰⣶⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢐⣿⠀⣿⡇⠀⠀⠀⣿
⣿⠁⠀⠀⠀⢸⣇⠀⠀⠀⠀⢰⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⣄⣀⣀⣨⣿⠀⣿⣇⣀⣀⣀⣿
⣿⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⣸⡇⠀⠀⠉⠉⣿⠋⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉
⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣿
⣸⣇⣀⣀⣀⣀⣿⣄⣀⣀⣠⣿⣀⣀⣀⣀⣀⣿⣆⣀⣀⣀⣿⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠱⣠⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢤⡈⣑⡄⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣷⣆⡸
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢠⣼⡗⠁⠀⠀⠀⠀⠀⠈⢻⡧⠀⠀⠀⠀⠀⠓⣿⠛⠁⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⡟⢿⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⢸⢻⡄
⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠉⠀⢛⠀⠀⠀⠀⠀⠀⠀⡝⠸⠀⠀⠀⠀⠀⢨⠁⢇⠀⠀⠀⠀⠀⠨⠂⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠠⣀⢀⠀⠀⠀⠀⢀⣰⡠⠊⠀⠀⠈⡅⣀⠀⡰
⠀⠀⢤⠀⢠⡄⠀⢀⠀⠀⠑⢴⣦⣀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⣿⠊⠁
⠀⠀⠀⠉⣿⡏⠉⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⢻⡄⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠐⡏⠿⠀⡀⠀⠀⠨⠁⠇⠀⠀⠀⠀⠀⠸⠉⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣧⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠙⣦
⣿⣿⣸⣆⣤⣀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⣄⢱⣾⣿
⣿⣿⣿⣿⡛⡁⡀⢀⣽⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⢄⠩⣳⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣻⣴⣷⣿⣄⢠⠖⠀⢠⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣸⣈⣵⢦⣾⣧⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⠤⣶⣦⢿⢁⡇⠀⣳⣀⠔⠀⣠⡴⠀⢂⢸⡆⠀⡘⣀⡀⠓⣐⢛⣋⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣾⣿⣷⣾⣿⣧⣶⣴⣃⣿⣿⣯⣞⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠐⠄⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⣆⠂
⠀⠀⠀⠀⠀⠀⢠⢀⢂⡔⡀⠀⠀⠀⠀⠀⠈⠖⢰⠈⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢄⠰⡆⠀⠀⠀⠍⠒⠀⠁⢊⣌
⠀⠀⠀⠀⠀⠀⢤⡜⠈⠃⠇⠂⠀⠀⠀⠀⠀⡈⠨⡄⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⢠⠇⢁⠀⠀⠀⠀⠓⢄⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠇⠀⢈⣄⠀⠑⠘⠀⠀⠈⠐⠀⠀⠀⠀⠀⠁⡠⠣⣀⠀⠀⠆⠁⠀⠀⢀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠈⠐⠄⡀⠀⡘⠀⡠⠋⢈⣡⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠺⡓⠶⠚⠢⡀⠘⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢉⠁⢀⠓⠒⡅⠠⢀⠀⠙⠆⢲⡀⢒⠆⠉⠃⠀⢀⡄⠀⠁⠀⠑⢄⡠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠮⢑⡆⠀⠀⠀⠀⢸⠁⠘⠀⢦⠀⠀⡖⠁⡃⠈⠀⢀⠀⠀⠀⡀⠘⢆⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⡤⣀⠀⠀⡀⠀⠘⠀⠀⠑⠀⢠⡌⡠⠊⠀⠀⠀⢸⠀⠐⠈⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⣁⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡁⠀⠀⠀⠈⠉⠉⠁⡍⠉⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⢁⠖⢠⠆⢰⠀⠦⠀⡦⠰⠀⠒⠀⠏⠡⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠊⢠⠊⢀⠆⠀⡆⠀⡄⠀⠇⠀⡀⠀⠀⠐⡀⠡⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⠁⡠⠃⢀⠎⠀⠰⠀⠀⡇⢀⠀⠀⡇⠀⠀⠀⢡⠀⠱⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠜⢀⡔⠁⢠⠊⠀⢠⠁⠀⢰⠀⢸⠀⠀⡇⠀⠂⠀⠀⠆⠀⠑⡄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡮⢀⡔⠁⠀⢀⠆⠀⠀⡘⠀⠀⠀⠀⡇⠀⢰⠀⠀⠰⠀⠀⡈⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⡠⠞⠀⠤⠤⠇⣀⣀⣀⣀⡇⠠⠤⠦⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⡤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠐⣶⠂⣀⠀⠆⣴⣶⣶⣔⣶⣶⡶⠶⠶⠖⠂⠀⣾⠬⣍⡿⠨⢰⡗⡚⣒⠚⡸⢏⡿⣟⣹⣿⣷⣶⣾⣧⡀⢲
⠀⠀⠀⠀⢘⢃⠒⣉⠀⠸⣼⣿⢻⣟⣉⣯⣄⢶⢖⣾⣿⡒⢸⣥⡞⢬⣲⣼⠶⠓⠚⠚⠡⠴⠀⢺⠿⣿⣿⡇⡀⢀⢉⣳⣦⠄
⠀⠀⠀⠀⢀⣸⠒⢀⡀⠰⣹⣿⣇⣽⠿⡿⠛⠡⠿⠟⣋⣤⡟⢡⣦⡮⢋⣴⠂⡈⡁⣉⠠⡈⠎⢴⣧⣾⣿⣧⣿⠿⠙⠉⡙
⠀⠀⠀⠀⠈⠉⣔⢢⠄⠰⣹⣍⡟⣩⣿⡟⣰⣾⣿⣿⣿⣿⢣⣉⣩⡃⢌⣾⢤⣀⣁⣌⣄⠙⠦⣬⣽⣿⣿⡟⠟⠓⢦⣴⠆
⠀⠀⠀⠀⢠⡴⡎⢹⢦⣞⠙⣿⡿⠿⠋⣷⣷⣿⣯⣽⣿⠿⠏⠻⠽⠿⣒⢿⣍⠅⠀⠀⠑⠚⠒⢶⣿⠿⠿⠿⠠⣾⡟⠛⠄
⠀⠀⠀⠀⠈⠁⠀⠸⣿⠿⣏⡀⠀⠀⠀⢀⣼⣿⡟⠉⠀⢀⣀⠀⠀⠀⠈⠫⣷⣶⣄⡀⠀⠀⠀⢸⣿⠄⠀⢀⣤⠖⢦⣧⠄
⠀⠀⠀⠀⢠⣀⡬⢶⣿⣿⣶⣍⠀⠀⠀⢸⣶⡟⠀⠀⠀⡆⡾⣷⡀⠀⠀⠀⢹⣿⣿⣷⣄⠀⠀⠀⠁⢀⣴⣽⣷⣼⡟⠉⡄
⠀⠀⠀⠀⠈⠃⣡⣿⢹⣿⢙⡿⠀⠀⠀⢸⣾⡅⠀⠀⠀⢇⡿⣿⣇⠀⠀⠀⢸⣿⣭⣿⣿⡄⠀⠀⠀⣸⣿⣿⣏⣿⢶⣿⣆⠄
⠀⠀⠀⠀⠰⠟⡿⠋⠈⠉⠌⢿⠀⠀⠀⣸⣿⣇⠀⠀⠀⠈⢽⣿⡧⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⢿⣿⣾⣿⣯⣿⡿⣇⡄
⠀⠀⠀⠀⢀⡸⣷⠀⠀⠀⠒⠀⠀⠀⢀⣿⣿⣿⣆⡀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠘⢛⣿⣿⣭⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⢸⣿⣦⣀⡀⣤⡄⢰⣿⣿⣿⣿⢣⢸⢓⠠⣤⣶⣶⡇⢄⢹⣿⣿⣿⢻⣿⣶⣶⣦⢆⡸⣿⣿⣿⣿⣿⡟⣽⣇
⠀⠀⠀⠀⢠⡀⣬⠛⣏⣿⣿⠟⣷⣾⣿⣿⣿⡏⠬⠐⠠⠰⢩⣧⡾⠩⠤⢸⢯⡿⣿⠟⠿⢣⣿⣻⣚⣟⣿⣿⣿⣿⣿⡱⢸⡇
⠀⠀⠀⠀⠘⠁⠈⠀⣿⣽⣷⣯⣿⣿⣾⣿⡿⠅⠥⠅⠀⢸⠾⣛⡇⠰⠀⢰⠀⢇⡌⠘⣀⣿⡟⠧⣯⣿⠻⣿⣿⣿⢃⡔⢸⡇
⠀⠀⠀⠀⢠⣰⡄⡶⠷⠾⣟⠫⢽⣿⣿⣿⣯⡏⠁⠥⠅⢾⣿⠋⠀⠤⡔⢸⣷⣾⣿⠆⡾⠳⣸⣿⣹⣿⣿⣧⣿⣟⠡⡜⢻⠃
⠀⠀⠀⠀⠈⠁⣤⡧⢀⡠⢿⣿⠿⣿⣾⣿⣿⢱⣀⣠⡤⠠⠴⠿⠎⠱⠁⠌⠿⠿⠋⠼⠙⠛⠳⠿⠷⠛⠛⠾⣿⣯⠰⢘⡏