Aesthetic bio Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣻⣿⠶⣄⠀⠀⠛⠢⡀⠀⠀⠀⡄⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣄⣻⠿⢡⠄⣈⢷⣦⡄⢀⠐⢶⣄⢼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠙⣾⡉⢶⣟⣾⣿⣥⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣡⣧⡀⠂⣨⡭⣥⠤⠐⢃⣀⠀⠀⣀⣀⡀⠀⠉⢻⣦⣻⣿⣮⠝⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣷⣿⣿⠁⣼⠋⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⢀⠀⠀⠀⠀⣷⡌⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣩⢿⣿⣿⡆⠀⠀⠉⠁⣤⡄⡀⠀⠀⠴⠦⠘⠛⠁⠀⠀⠐⠚⠻⢧⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⣽⠟⢾⡟⣿⠁⠀⠀⠀⠀⠉⠣⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠾⠀⢻⣶⡰⠶⠦⣄⠠⠀⠠⣦⣄⠀⠤⢶⣐⣛⠠⠖⠀⠀⣬⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣏⠀⠀⠀⠀⠀⠈⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠤⢊⣽⡿⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⣶⡄⠀⠚⠂⠐⠦⠠⠙⢮⠀⢛⣀⠀⣴⣶⠖⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠙⣶⣆⡀⠀⠀⣀⡀⠶⠿⠂⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⠉⠀⣀⡠⠛⠻⢧⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠻⣆⡀⠛⠃⢦⡄⠁⠀⠀⠀⠀⠀⠀⢠⣆⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠿⣿⣦⡀⢢⣄⢠⣤⠂⠀⣆⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠢⣄⣲⠀⢀⣠⠆⠰⠒⣏⣤⣭⢯⣙⠠⠉⠑⢾⣿⣡⣴⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⡄⠀⠀⢀⠘⠛⢷⡾⠋⢠⡿⢿⣿⣶⠆⠀⠛⠷⣤⣀⠀⠉⢿⣤⣟⠋⠃⠀⣄⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⡀⠀⠉⣽⣿⣧⣶⠁⠀⠈⠙⠁⠀⢀⡞⠀⠀⠉⣳⠴⠀⢠⡝⠳⣿⣄⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⣸⣿⠋⣿⡿⠛⠛⠡⣤⣴⠤⠭⠷⠖⢋⣉⡁⠐⠛⢻⣦⣀⠈⢻⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣧⠋⠀⠀⠀⠀⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣷⣼⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⡤⠀⣠⡄⠴⠛⢿⡟⢧⠄⠺⠿⠿⠶⠤⠤⠀⠂⣩⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⢻⠇⠀⢀⣀⠀⠀⠀⠁⠀⠀⠀⢀⣀⡀⠀⠤⢠⣾⠋⣸⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⢠⣶⣶⣿⡿⠂⠀⠀⠻⠇⠐⠒⠉⠀⠀⠰⠟⠃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣧⠀⣹⡏⠀⠀⠀⠀⠀⠀⡀⣀⣠⡄⠀⢀⣤⣤⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⣿⠉⠀⢦⡄⠄⠀⣠⡄⠴⠿⣬⡛⠚⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡄⠈⠑⣀⣸⠋⠀⡴⠞⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢀⣿⣧⣠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣟⠁⠀⠻⣴⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⣿⣧⣰⣄⠈⢦⣞⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣇⠀⠀⠀⣠⡃⣀⡀⠋⠛⢹⣧⣀⠀⠉⠃⠈⢰⣦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠋⠀⠀⢐⡿⢫⣿⣿⠣⣤⠜⠋⠹⠓⠀⣾⠒⣯⡉⠃⣄⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢁⠠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⡼⣿⣄⣄⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠶⠀⠤⣴⠟⣰⠋⢉⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢀⠀⠣⠄⣀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠧⠀⠀⠀⢀⡄⠀⠂⠀⣄⣨⡟⣷⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⡀⠚⠁⠀⠀⠀⡰⡿⠿⠁⠀⠀⠀⢘⠫⡞⠻⡦⠄⠀⠀⠀⠀⡠⠂⠀⠤⣀⠀⠀⢀⣠⡶⣾⢿⡟⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠻⣾⣷⠃⠸⢇⡀⠀⠀⡰⠋⠀⠀⠀⢀⡬⠇⠙⠟⠛⣷⡟⢘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⣰⡿⢿⠀⣌⠀⠘⢷⡆⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣜⠀⠈⠱⢦⣄⠀⠈⠑⠺⣤⢀⡀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢩⡦⣄⠴⠾⠛⠛⠀⠀⠀⠀⠙⠳⢧⣄⠀⠀⠈⢳⣆⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠈⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⣧⠀⠀⠱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⢻⣧⣀⣀⣀⢀⣀⣠⣤⣄⢀⣀⣤⣾⠟⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⠿⣿⡟⣻⣏⣿⡟⢻⣿⣩⣿⠿⣿⣋⣿⡿⣿⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣼⡿⢿⣿⢻⣿⠛⠛⠿⠿⠛⠛⠻⢿⣿⢛⣿⣷⣾⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⣷⡾⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣤⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣻⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣿⣷⣄
⠀⠀⠀⠀⢀⣠⣰⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣼⣿⣤⣀
⠀⠀⠀⠀⣿⣋⣿⣷⣾⡿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣧⣼⣿⣉⣿⠆
⠀⠀⠀⢰⣟⢻⣿⣤⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⢿⣧⣼⡟⣻⡇
⠀⠀⠀⠈⠻⢿⣏⣹⣿⣼⡿⢿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣷⣾⣿⣹⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠋⠁⠻⢷⣾⣯⣹⣿⣄⢀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣀⣿⠿⠟⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣉⣿⣿⣿⣶⡿⢿⣷⣴⡾⣿⣏⣽⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣯⣽⡿⢿⣟⢻⣷⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣩⣿⣷⣾⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣼⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⡼⠀⠏⠀⠀⠀⢠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣷⡈⢧⠀⠀⠀⠀⢰⠃⡜⠀⢀⣤⠖⠋⣈⣠⣀⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣉⣥⣀⣄⣉⠀⠈⢦⠀⠀⠀⡿⢠⣇⣴⠋⢀⣴⣋⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠳⣄⠀⣦⣠⡼⠋⠈⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⣽⡟⠉⠀⠀⠀⣀⣤⣶⣶⣤⣄⣀⡤⠿⠶⠚⠋⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡿⠀⠀⠀⣠⡶⠋⠉⠉⠉⠉⠉⠀⠀⠀⣀⡤⠴⠒⠉⢚⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⣷⣾⠋⠀⠀⠀⢰⣿⡃⠀⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢹⡟⠀⠀⠀⢀⣿⢩⡗⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠾⠃⠀⠀⠀⠀⣿⡅⠀⠀⠀⣀⠁⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣾⡁⢀⡀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⢰⣇⠀⠀⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣶⣴⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠛⢋⡛⠟⠛⠉⠛⠿⠿⠿⠿⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠛⣡⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢢⠄⠈⢙⢦⡀⢰⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡷⠀⠀⠀⠀⠀⠀⠙⠋⢿⣿⠟⠙⠀⠈⠓⠠⠂⠈⠉⠴⣤⠄⢠⡀⠀⠀⠀⠀⣧⣾⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⠀⠀⠀⠀⢀⣼⣤⣤⣷⠶⠀⠶⠆⠀⠀⠀⣴⢀⣴⣄⣀⣂⡁⠀⠀⠁⢀⣿⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠗⠠⣤⠒⠋⠀⠀⠀⠐⢀⣽⠊⢑⣋⠉⢁⣿⠄⠀⠀⠉⠈⠉⠛⠁⣴⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡐⠉⠉⣃⣀⡀⠀⢀⠈⠀⠀⠀⠀⠀⠙⢁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢠⣄⡀⠰⣶⠁⠀⠀⠀⠀⠀⠐⡲⠋⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢳⣼⣭⣥⣉⡲⢄⠀⠀⠾⣵⠀⠈⣇⠀⠀⠶⢶⡶⢾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠉⣙⣳⡷⣄⠐⢽⡇⠀⠘⡆⠀⢀⡏⠀⢠⡿⡄⠀⠀⠀⠤⢚⣉⣭⠽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣑⣮⣇⣀⣀⣱⣀⣨⣀⢀⡇⡇⡇⠀⠀⣨⠞⠋⣡⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣟⡟⠛⠛⣛⣿⣿⣿⣿⣿⣿⣽⣿⣿⣦⣿⣿⣿⣿⣿⣶⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠉⠙⢿⡿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣿⢠⠞⠉⠀⠀⠀⠉⠉⣭⣥⠉⠀⢉⣴⠟⠻⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣠⡊⣠⡆⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠺⠿⠷⣤⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡶⠋⢹⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⢤⣷⠀⣀⠤⠀⣾⠁⠀⠸⠷⡀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡄⠀⠀⠀⡠⢴⠞⠁⠀⠀⢀⡴⣧⠀⠀⠀⠀⠚⣦⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠴⠋⠀⠀⠀⡴⠋⠐⠛⢦⠀⠀⠀⡿⠆⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⡀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⡀⠀⠀⠀⢠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣾⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠆⠀⠀⠁⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣤⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⣄
⣠⡷⠀⠀⠀⠋⠁⢀⡤⠀⠀⢸⠁⠀⠀⠾⢥⠙⡄⠀⠀⠈⡀⠀⠀⠀⠠⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡿⠇⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠤⠘⢀⡆⠁⠀⠀⣼⡇⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠀⠀⠀⠀⠀⠀⢠⣔⠷⠀⡖⠀⠀⠀⠀⠀⠑⠀⠀⣼⣿⠁⠠⡇⠀⡠⢄⠀⠀⠀⠀⠁⠁⠄
⡄⠀⠀⠀⠀⠐⠺⢦⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⢠⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡹⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠘⠀⡀⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠡⢆⠽⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠸⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⡀⠀⠔
⢹⣽⡲⠤⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠃
⠁⠓⠁⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠰⠀⠀⢤⠀⠀⠈⠁
⠀⠄⣤⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⡀⠀⠀⠀⠀⠘⠂⢠⠆⠘⢆⡀⠈⠓⢀⢰⠀⢀⠀⢀⠀⢀⣰⣦⢄⣀⠃
⢁⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠀⠀⠀⠀⠈⠃⠀⠸⠑⠦⡀⠀⢸⠀⠉⠀⠀⠀⠀⡼⠁⠀⠀⠧
⠀⠀⣀⢰⠀⠀⠤⠤⠒⠀⠀⠀⠀⠈⠀⢀⡐⡾⠆⠒⠀⠀⠀⠑⠄⠀⠀⠈⢢⠇⠀⠀⠀⠀⠁⠀⠙⠆⠀⠄
⠀⠀⠀⡟⠐⠒⠲⡄⠀⠐⠦⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⡠⠀⠱⣀⠀⠀⣹⠀⠀⠀⠉⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢁⠐⠀⠀⠀⠀⠙⡀⢀⡀
⢠⠃⠀⠀⠈⠉⢒⡟⠒⠲⡄⠀⠀⠈⠇⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢆⣏⡏
⠈⠀⠀⢀⠄⠀⠈⢧⣀⣀⡆⠀⠀⠀⠈⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢠⡖⠲⠼⣾⡇⠀⠀⠀⡽⠉⠛⡆⠀⠀⠀⠆⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣆
⠈⠀⠀⠀⢨⠑⠀⠀⠀⠙⠤⣀⡇⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏
⠘⠀⠀⠢⠞⠀⠀⢐⡀⠀⠀⠀⢯⣹⠀⠀⠀⠶⣶⣶⣦⣀⡀⠀⠀⠈⠁⠐⠒⠂⠤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠙⠓⠀⠀⠀⠀⠀⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⢤
⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣠⣀⣅⣀⣀⣈⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣏⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠁⠀⠀⠀⢳⣤⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣾⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣯⣽⣾⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣦⣼⣿⣿⣿⣿⡿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠻⣿⣿⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢁⠈⠢⠀⠐⠀⠀⠀⠀⢘⣻⣇⣾⡿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⡿⠋⠀⠀⠀⠀⣆⠀⢰⡟⢀⡴⠁⠀⠀⠀⠁⠳⣆⡄⠀⠀⠀⠀⠻⢿⣟⢫⣾⡿
⠀⠀⣠⣀⡀⠈⢻⡟⢀⣤⣴⠆⠀⠀⠀⠘⠻⢿⠷⣿⡇⠀⠀⠀⣿⡀⢈⣠⠞⢁⠀⠀⠀⠀⠂⠀⠘⡄⠀⢀⠀⠀⠀⠻⢿⠟⢏⣴⣶⠄
⠀⠀⠘⠿⣿⣷⣄⣧⣿⡿⠋⠀⠀⠀⠸⢿⣷⣾⣶⣿⡆⠀⠀⢦⠘⢧⡍⠀⣰⠏⠀⠀⠀⠀⠀⠀⠓⢧⠀⠀⠀⠀⠀⠀⠿⠿⢿⡛⣩⣄
⠀⠀⢤⣤⣀⡉⠉⡍⢠⣴⣶⠇⠀⠀⣴⣶⣦⣇⣤⣥⡄⠀⠀⢸⣆⠀⣤⠾⠋⡠⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁⠀⠀⠀⠀⢾⣿⣷⡿⠟⠋
⠀⠀⠈⠻⢿⣿⣦⣵⡿⠟⠋⠀⠀⠀⢈⣉⡉⡏⠙⠛⠁⠀⠈⣦⠙⢦⠀⢠⡾⢁⠀⠀⠀⠀⠈⠠⢤⣸⠀⠀⢀⠀⠀⠀⢰⣶⣶⣷⣿⠿
⠀⠀⢤⣶⣤⣄⡀⡇⣠⣴⣶⣶⠂⠀⢿⣿⡿⢷⣿⣿⡷⠀⠀⠘⢧⣀⠚⠋⣴⠋⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠀⠀⠀⠀⣤⣦⣤⣧⣶⣷
⠀⠀⠀⠙⠛⠛⠛⠻⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠉⡤⠞⠁⠀⠀⠀⠀⠀⠈⠀⣤⡘⠁⠀⠀⠀⠀⠀⣨⣭⣉⣥⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠛⢻⡍⠉⠁



⠀⠀⠀⠀⠀⠀⢰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢷⢸⡇⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡰⢩⢪⠀⠀⠀⠀⠀⢸⠁⡰⠛⠀⠀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢦⢸⣿⣷⣋⣤⠶⠂⠀⠀⠀⣓⣖⢢⢣⣧⣎⣀⠀⠀⢲⠆⠈⡞⢁⣦⠀⠀⠀⠀⡈⢿⣿⠂⣠⣶⠀⠀⠀⠉⠀⠀⠀⢀⠈⠁
⠀⠀⠀⢲⡘⣿⣿⢛⣫⣥⠶⠂⠀⠀⣀⢈⡢⠹⠿⣒⠪⠟⠁⠀⢸⡀⡸⠖⠋⠀⠀⠀⠀⠀⣿⣦⣹⣾⡿⠃⠀⠀⠀⠳⢀⠈⡇⠂⠀⠊
⠀⠀⠐⣦⡻⣟⣮⣽⠿⣷⣚⡀⠀⠀⠹⡗⠮⣷⣖⠩⢗⠆⠀⠀⠀⠱⡇⠀⣰⠟⠀⠀⠀⣤⡈⠻⢿⣠⣴⡾⠀⠀⠀⣄⠀⠈⠟⠀⠀⡤
⠀⠀⠲⣬⡻⣎⠳⢚⣛⣋⣭⠀⠀⢴⢖⡒⢬⣈⡡⠯⣥⣄⠐⢷⠀⢸⠤⠚⠁⠀⠀⠀⠀⠹⣿⣷⡼⠿⣛⣥⠀⠀⠀⠈⠀⠰⣾⠄
⠀⠀⠓⢮⣟⢮⣉⣩⣭⡷⠶⠂⠀⢈⣓⣭⠵⡿⢖⡭⠕⠁⠀⠈⠣⣼⠀⠀⣤⠀⠀⠀⠀⢷⣦⣍⣷⣾⡿⠃⠀⠀⠈⠇⡀⢀⣠⣀⠖⠃
⠀⠀⠙⢶⣽⡳⠤⠴⢶⣟⣛⡁⠀⠹⣓⠬⣳⣷⡫⠭⣿⠇⠘⢇⠀⢸⠤⠚⠁⠀⠀⠀⠀⣌⡛⠛⣧⣴⣶⡿⠁⠀⠀⠀⠀⠄⢛
⠀⠀⠀⠀⠈⠉⠓⠛⠋⠉⠁⠀⠀⠀⠈⠉⠈⡏⠈⠉⠀⠀⠀⠀⠑⢾⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣀⣀⣠⢒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⢀⡀⠁⢀⠌⠀⣤⣀⣀⠗⣠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⣴⠾⢤⣿⢀⠇⠀⠀⠀⠀⠀⡀⠀⠀⣠⢀⡤⣄⠀⠀⠀⠀⡰⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⢣⣤⠆⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣁⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡟⠀⢶⡇⠰⠆⢈⡟⠂⠀⡀⠠⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⣶⠶⣦⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠆⠀⠠⣋⣹⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⠿⣄⢨⡅⣤⣄⢸⠇⣤⣄⢀⡤⣄⣄⢀⡄⠀⠀⠀⠀⠀⠀⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⡾⢸⡿⣇⣸⢿⣿⣇⣹⢾⣡⣇⢿⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀⠘⠳⡒⠌⠉⠉⡁⠀⠈⠁⠉⠁⠈⠀⠉⠁⢀⡽⡟⠈⠁⡟⠀⠀⠀⠀⠀⠠⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣱⠶⠞⠳⠀⠈⠁⠀⠑⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠐⠞⠁⠀⠀⠀⠀⢀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡾⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⢯⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠈⣷⢆⠀⠀⢠⡆⠀⠀⠀⠀⠀⢀⡿⢦⢴⣾⣷⢠⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠬⠆⠈⠉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠃⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣀⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠄⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣈⢉⣁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠓⠒⠛⠋

⠀⠀⠀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⢀⡀⠤
⠆⠐⠦⢀⠒⠈⠁⢀⠤⠒⠂⠚⢳⠞⠒⠒⠒⠒⠃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠐⠁⠠
⠐⠀⠀⣀⣀⣀⣠⣎⠀⡀⢀⡀⠌⠀⠀⣤⣀⣀⣀⡼⢋⣠⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠌⠀⡀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠊⠁⠀⢠⠇⠀⠀⠈⠁⠀⠀⢠⣤⣾⡉⢉⣿⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠘⠉⠐⢢
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⠿⣿⡆⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠐⠚⠉⠻⠆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⣄⣴⡾⠂⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠲⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢈⡀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⣿⡟⠀⢀⣠⡀⠀⢀⡀⠀⠰⣴⣇⡀⠀⠀⠀⠀⢀⣹⣎⠁
⠀⠀⠂⠀⠀⠀⣀⠄⠘⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠘⠿⠃⠀⠙⠁⠀⣀⣏⠀⠀⠀⠠⠀⠀⡎⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⣾⠉⠉⣳⠀⠀⠀⠀⠀⠑⢄⣀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢈⣿⠀⠀⠰⠆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⠀⠀⠐⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡷⠿⠿⠿⣶⡄⢰⡆⢠⣶⢶⣄⢸⣿⢀⣶⢶⣆⠀⣰⡾⣷⡄⣷⠀⢰⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⠀⣸⡇⢸⡇⣾⡇⠈⣿⣼⣿⣾⡇⠀⣿⣶⣿⢀⣿⡇⣿⡀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠠⣿⣧⣶⡶⠿⠋⠀⢸⡇⠹⣷⣾⠟⢸⣿⠹⢷⣶⠿⠈⢿⡾⣿⡇⠻⣷⠟⣿⡃
⠀⠀⠀⡀⠀⢀⡴⢢⣤⣀⠀⠋⠙⣎⠙⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⠇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢄⠄⠀⣀⡤
⠀⠀⠀⠁⣰⠋⣤⣤⣤⣾⣷⡄⠀⠈⠒⠁⠀⠐⢦⣀⣣⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⠏⠀⢠⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣶⠟⠋
⠀⠀⣠⠞⢁⣼⡿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⢸⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠞⠛⠉⠠
⠀⠀⠀⣴⣿⠋⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢳
⠀⠐⣾⡟⠁⠀⠀⠀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⢹⡶⢤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⣤⡀⢀⣠⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣻⡄⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⣹⡛⠻⢿⣿⠇⠘⠛⠓⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠻⡇⠀⠀⠙⢀⣄⣀⣄⡀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠙⠙⠱⣦
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⡦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⡾⠃⠀⠄⠠⠀⠉⢷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡿
⠀⢄⣤⡶⠇⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠌⠀⠀⠈⠼⡽⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠘⠂⠁⠀⠐⠀⠠⠰⠗⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣷⣶⣶⡶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠄⠀⠀⠅⠤⡀⠠⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢣⡶⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⢀⠰⠀⠁
⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠤⠋⠳⠁⠀⠀⡐⠀⠀⢀⠄⠀⠀⠈⠀⠄⠐
⠀⠀⠀⡔⠴⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡲⠀⠀⠀⠈⠀⠀⠒⠁⠀⠀⠀⠄⠀⠀⡆
⠀⠀⠀⠈⠀⢰⡂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠁
⠀⠀⠀⠀⠀⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠕⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⡀⠀⢀⠀⠃⠠⠤⢢⡉⠀⠀⠀⠀⠀⡀⠁⠀⠰⠀⠀⠁⠐⠂⡈⠁⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⣎⠀⠀⠀⢐⠀⠀⡾⠇⠈⠕⠀⠀⠀⠴⠁⠦⠀⠰⠂⡘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠰⠀⠀⡤⡀⡼⠀⢀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠹⣆⢱⠄⢸⠀⠞⠁⠻⡁⠠⠎⠀⠘⢰⠁⠊⢰⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢾⢽⣞⡇
⠀⠃⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠀⠀⠈
⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠀⡠⠄⠀⡄⠀⠂⠠⠀⠔⠀⠈⢠
⠀⢀⢠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⡀⠀⠶⠀⠀⠀⠀⠀⠰⡟⠋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠂⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⢆⡀⠀⣰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢡⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠻⣥⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⢤⣀
⠀⠀⢀⠊⠃⠀⠀⠀⠑⠲⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠛⠁⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠓⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣟⡀
⠀⠀⠀⠀⠀⠃⢘⠀⠀⠀⠈⠣⣄⠈⢁⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⢰⡟⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡰⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠆⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠿⣄⠀⠈⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠐⠂⠄⠀⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢷⣄⣠⣤⣤⣴⠾⠋
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⣷⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⠉⠐⠾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣤⣄⠀⣀⣀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠁⠉⠉⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣤⣤⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣤⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⢻⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⣴⣿⣿⣉⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⠘⢃⣴⣿⣿⣿⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣀⣀⣰⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⢿⣟⢿⣿⢟⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠙⢷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣴⡾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣴⣿⡧⠀⠀⠀⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡧⠀⠦⢤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⣿⣿⣿⣿⣷⣶⣾⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣟⠉⠉⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣿⡿⢿⣦⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣰⡿⠃⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⡿⠿⢿⡿⠿⠿⠿⠿⠿⠿⣿⠁
⣿⡇⣤⢸⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⡀⠀⠀⠀⠀⠀⢠⣿
⢸⡟⢻⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⣾⡿
⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⢠⡿⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣿⠃⢰⣿⠀⣿⠁⠀⠀⠀⠀⠀⣾⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⠀⠀⠀⠀⠀⢀⣾⠇⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣾⠁⠀⠀⠀⠀⢠⣾⡿⠀⠀⢸⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢰⡏⠀⠀⠀⠀⠀⢈⣿⠃⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢰⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡸⠃⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠁⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢸⣿⠀⠇⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡘⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⢠⡿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣼⡇
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡏⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠇⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⡟
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡏⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⣠⣿⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣇⣀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠷⠖⠚⣋⡙
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣧⣤⠴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⡿⠛⠛⣹⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⢏⣙⠻⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠤⠈⢀⣼⠟⠁
⠀⢿⠇⠀⠀⠀⠀⠀⢀⣾⠁⠀⢉⣴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣴⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣷⡶⠀⠀⠈⢻⣿⡿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠉⠀⢻⡆⣹⣿⣿⣿⠇⢀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢉⡡⠋⢹⣿⠏⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠏⠛⠀⠀⠀⣀⡤⠂⣀⣤⠆⠔⠓⢚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠘⠿⠶⠗⠒⠺⣿⡯⠐⠋⠁⠀⣀⣤⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠐⢀⠄⠀⠀⠀⠀⣦⣀⣩⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⠀⠄⠀⠀⠈⢿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢿⡿⠋⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣾⣦⣾⣷⢹⡆⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣫⠿⠛⠭⠴⠶⢾⣛⣛⣵⣫⣿⣼⣷⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢟⣵⣶⣿⠿⠭⣽⣭⣝⣛⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡿⠙⠂⠀⢀⣢⣤⣤⣾⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⡇⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠷⣀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠿⣿⣿⣿⣿⣿⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠭⡛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢳⣦⣀⣀⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣆⢻⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⡟⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆⠋⠥⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠙⣰⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣟⣛⣛⣛⡿⢯⡿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⣽⣷⣶⣶⣶⣶⣾⣷⣯⣭⣍⡙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠊⣬⡍⠻⣿⣿⠀⠀⠀⠀⠠⠖⠿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣿⣿⣿⣿⣿⡏⢰⠈⣼⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⠀⢧⡘⠟⠷⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢤⡴⠟⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠙⠳⠂⠀⠀⠠⡀⠀⠀⠀⠀⢀⣴⣿⠷⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠒⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣀⣀⠀⠀⠀⠀⠈⢷⣦⣄⣀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⢲⣤⣀⠀⠈⠙⠓⠲⠦⣤⣀⡙⣿⣿⣿⠃⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠚⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⢁⣤⣉⠛⠃⠀⢀⣾⠀⠈⠻⢿⣶⣤⣄⣀⠀⠀⠀⠉⠛⢿⣧⣶⣇⠀⠀⠀⠀⠀⠀⠸⢟⣉⣤⡈⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠃⣾⣿⣿⣿⣷⣦⣼⣿⡀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣠⣦⡀⢻⡇⢈⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡸⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣌⣙⣿⣿⣿⣿⣿⣮⣷⣿⣾⣛⣛⣻⣿⣟⣛⣻⣛⣛⣛⣃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⣢⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⡇⠀⠶⠶⢤⡀⠀⢀⡤⠶⠖⠀⢀⠹⣿⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠇⠸⣶⣤⡤⠀⠀⠀⢤⣤⣶⠇⢸⣷⣿⣼⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⢏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠜⣡⣫⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠃⠀⣠⣴⣾⣿⣿⡟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠀⠀⣸⣿⣿⣿⢟⣵⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⡿⣫⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣨⣾⣿⢟⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⣿⣿⣿⣿⣿⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠾⠃⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⢧⠀⡠⠒⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⢈⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣛⣽⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡛⠛⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡏⠀⠀⠛⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⡿⠀⠀⢸⣿⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡾⠁⠀⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⠃⠀⠀⠀⠚⠻⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⢠⣿⣷⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠚⠻⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣶⣦⣤⣤⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⠶⠶⠦⠬⠄⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠚⠛⣻⣿⠿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⡁⢰⣿⣶⣭⣛⠿⣿⣳⣾⣟⢿⣿⣿⣷⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢣⡾⠃⣀⠻⣿⣿⣿⣽⣲⣭⡳⢯⡻⣮⣻⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠟⠀⠀⠸⣿⣮⣙⠿⣿⣿⣟⣷⠦⣝⣮⣳⣝⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠙⠻⣽⣛⡶⢍⠙⢻⣿⣶⣮⣝⡻⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⣛⡲⠀⠀⠀⠴⣛⡛⠛⠳⢦⣝⠻⣿⣿⣿⣷⣌⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⠛⢿⠆⠀⠀⠐⠛⠿⠻⠖⠀⠉⢷⡈⢻⣟⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⡄⠙⣯⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣄⢱⠰⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⣴⣿⣇⠇⠁⣧⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⠐⠒⠀⠀⠀⠀⣀⡴⢺⣿⣿⢇⡶⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣤⣤⡔⠊⠁⠀⠸⠟⢁⡾⠁⢰⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣾⣿⣿⣿⣿⣿⡇⠀⠀⢀⡴⣻⠟⢡⠋⢛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣾⣿⣿⣿⣿⡿⠟⣀⡤⢞⣽⢞⣵⢯⣾⣌⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣾⣿⣿⢿⣫⠭⣖⣫⣵⣾⣿⣿⢟⣵⢟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣫⣵⣿⣿⣿⣿⣿⢟⡵⢟⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⢰⣿⣿⣿⠟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠿⠃⠀⠙⠛

⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄⠀⠀⠀⠀⢰⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⣃⣀
⠀⠀⠀⠀⠺⠻⣿⣿⡿⠉⠀⠀⠀⠀⢠⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠘⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡃
⠀⠀⠀⣴⣶⣴⣿⣷⣤⣄⠀⠀⠀⠀⠀⣀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣁⣻⣷⣤⣤⣤⠀⠀⠀⠀⢀⣀⣴⣿⣷⣶⣶⣦⣄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⢁⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⢿⣧⡀
⠀⠀⣸⡟⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠏⠙⠉⠉⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣿⣿⣿⣿⣿⢿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⠀⠀⠙⣿⣦
⠀⠀⣿⡃⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠆⢿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣥⣤⣽⣿⣿⣿⠇⠀⣿⡀⠀⣸⡟⠈⡿⠿⠿⠟⠀⠀⢀⣾⠿⠋
⠀⠀⢹⣇⠀⠈⣩⣽⣿⣿⡆⠀⠀⠀⣰⠃⡀⠓⠀⣼⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢈⣻⣿⣿⠋⠀⣠⣿⠇⢰⣿⠃⠀⠋⠹⠿⣷⣶⠞⠋
⠀⠀⠀⢻⡆⠐⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠢⢄⡀⠀⠀⠀⣠⣿⣿⣿⣿⣶⡟⠋⠁⢀⣿⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣺⣧⠀⢨⣿⣿⣿⣿⠀⠀⠀⠃⠁⠀⠀⠀⣿⠀⢄⠀⠀⠀⠉⢻⣶⢰⣿⣿⣿⣿⣿⣿⠁⠀⢀⣾⠁
⠀⠀⠀⠀⠉⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠉⠻⠆⠀⠢⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠆⠀⢳⡀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⠀⠀⠀⠀⠂⠀⢘⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠁⠀⠀⠀⢀⣿⣷⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠘⣠⣾⠿⢿⡂⠤⠶⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠘⣿⠟⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⡿⣿⡏⠀⠀⠀⠀⠀⣸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢀⡿⢁⣿⠁⠀⠀⠀⠀⢀⡿⠁⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⡇
⠀⠀⠀⢀⣾⠁⣸⡏⠀⠀⠀⠀⠀⣼⠃⢰⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇
⠀⠀⢀⣿⠇⡸⣿⡇⠀⠀⠀⠀⠀⣿⠱⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷
⠀⠰⣿⠏⠀⠀⣾⡁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿
⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣈⣙⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⠀⠀⠄⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣿⣷⣶⣿⡳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣤⣷⣤⣬⠛⠿⠀⠀⣿⣽⣿⣷⣶⣄⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡾⣿⣿⠻⣿⡟⢠⡀⠀⢠⡟⢛⣿⠟⠿⣟⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣯⣿⣀⣈⣿⣿⣿⣿⣿⠿⠛⠃⠀⠘⢙⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣹⣶⣼⣿⣿⣇⣠⡴⣤⠀⠀⠺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⢻⠏⣿⣿⡝⣿⣿⣿⣟⣻⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⡞⠀⠹⣿⣾⣿⣿⡙⣹⣿⣶⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠇⠀⢀⣿⣿⢜⣿⣼⣏⣹⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠇⣠⣾⣿⣿⣿⡟⣿⡛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣿⣿⣿⣹⠿⣿⣿⣿⡿⢿⣿⣿⣶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⡿⢻⣿⣿⣿⣆⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣇⣛⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠏⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢿⣿⣿⣿⣿⣿⡄⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣯⡧⠘⣧⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⡿⣿⣴⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣧⢻⡏⡀⢻⣀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇⠀⠀⡾⣿⣿⣿⣿⣿⣌⠇⣥⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠁⣹⣿⣿⣿⣿⡇⠰⢸⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠘⣿⣷⡀⠸⡇⣿⣿⣿⣿⡇⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣷⣿⣿⠀⢄⣠⣿⣿⣿⣿⣇⠀⡇⠀⢰⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣨⣿⣿⣿⣿⣿⣿⣆⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣹⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠛⣿⡿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⡟⢺⣿⣿⣿⣿⣯⣭⠛⠿⠛⢿⣯⣹⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣛⣛⣋⣁⣀⣛⣛⣈⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡟⢰⢯⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡇⣿⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⡿⠋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣷⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢟⡽⣻⣯⣾⠙⣿⣿⣮⡙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⢁⡾⣱⣿⣿⠃⠀⠈⠻⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡟⢀⡿⣱⣿⠟⠁⣀⡀⠀⠀⠘⢿⣟⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⢀⣾⣿⡿⠃⠰⠋⠁⠀⢀⠀⠀⠸⢃⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇⣼⣿⡟⠀⠀⠀⠠⠶⠞⠃⠀⠀⠀⠘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⢀⣤⣤⣄⠀⠀⠀⢠⠎⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⡀⠈⠛⠛⠃⠀⢀⠔⠁⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⢌⠢⣄⣀⠠⠐⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⡦⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣷⣲⢱⠃⠀⠀⠀⠀⠀⡀⢀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠘⠡⠖⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠡⠏⠰⠆⠂⣰⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣏⣷⠀⡄⣀⢸⠀⡀⢀⡀⡁⠀⠃⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⠗⡿⣉⢿⢾⣹⢾⣿⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢒⣢⡌⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢴⡟⢀⡜⠀⠀⠀⠀⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠻⢆⠀⠰⠀⠀⠀⠀⠘⢲⠺⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠤⠬⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠦⠦⠶⠰⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⡿⢿⣧⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠙⠛⠛⢫⡄⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⣠⠴⠤⣄⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⡠⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⣶⡋⠀⠀⠀⠀⠀⢸⠁⣸⣇⠈⡇⠀⠀⠀⠀⢸⣿⣿⣿⠄⠀⠀⠀⠀⠀⠳⣬⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠸⠤⠼⠀⠀⠀⠀⠀⠈⠳⢤⡤⠞⠁⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠾


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⣾⣿⣿⠂⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⣴⣾⣿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣿⡀⡀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣧⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠰⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣴⣤⡄⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⢠⣶⡆⢰⣶⡄⠀⠀⠀⠀⠀⣿⣿⡗⠈⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠘⠿⠇⠸⠿⠃⠀⠀⠀⠐⠂⠈⠛⠁⠁⠀⠀⠀⠀⠘⠿⠿⠿⠋



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⠶⠶⠶⢶⣾⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⢀⡀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡠⠮⠤⠛⠉⠑⠒⠈⠀⠈⠉⠁⠀⠀⠂⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠸⠃⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⣸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢀⡬⠤⠦⠤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣄⠀⠀⢀⠶⠀⠀⠁⠄⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠖⡋⣶⣷⡄⠁⠁⠀⠀⠀⠈⠃⠘⠠⠄⠀⠀⠀⠀⠀⢹⡐⠃⠀⠊⠀⠀⠀⢀⣄⠀⠀⠀⠰⠀⠈⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠋⠘⠉⠀⠀⠁⠀⡀⠀⠀⠘⠃⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠂⠀⠀⡎⡇⠱⠀⠀⠀⠀⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⢀⠃⠀⠀⣀⠀⠈⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣤⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠠⠀⢀⡀⠐⠁⠀⠀⢠⠀⡄⠀⠀⠀⡋⠀⢠⠂⠄⢠⠟⠐⣆⠈⠁⠀⠀⠐⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠆⠹⣦⡄
⠀⠀⠀⠀⢀⠀⠀⠀⢦⡄⠀⠀⠀⠀⠈⠁⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⢃⠈⢀⢀⡴⢻⡄⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢴⡽⠛⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠅⠀⠘⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠋⠀⠁⠠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠠⠀⠁
⠀⠀⠀⢸⠀⠀⠀⠄⠈⢻⠀⠀⠁⠀⢠⡤⠶⠂⠀⠀⠀⠀⠀⠶⠀⠀⠀⠈⠃⢀⠂⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠆⠀⡃⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠸⠀⠀⠀⠐⠐⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠁⠀⠀⠀⠄⠂⠀⠀⠄⠠⠒⠚⠋⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠀⡀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡜⠁⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⣴⠁⠀⠈⠀⠀⠀⠠⠊⠀⡀
⠀⠀⠀⠀⠀⢄⣀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠈⠀⠀⠀⣴⣾⣤⠤⣄⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠈⢠⡀⡀⠀⠀⠀⢀⡂⢄⠀⢀⠀⠀⠀⠂⢛⡉⠀⠀⠘⠃⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠃⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡓⠀⠀⠀⠀⠀⠼⠀⠁⢤⠀⠀⠀⠀⠀⠀⠘⠄⡄⠀⠠⠅⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⢠⡀⢤⠂⠀⠄⠠⢄⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠂⠀⠀⠀⠀⠀⣀⡀⢴⣉⠰⠊⠃⠊⠀⠀⠉⡥⠄⠀⠀⠀⠀⠀⠈⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢐⡀⢆⡴⣠⠃⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠡⢛⠙⡆⠔⠀⠀⠀⠀⠄⠀⢤⠀⠀⢀⠀⠂⠀⠀⠀⠈⠀⣀⣾⠀⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠖⠊⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣏⡋⠀⠀⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⠀⠀⢄⠠⠄⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠗⣶⢿⡄⠷⣿⡄⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠧⣘⣃⣀⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠋⠩⠷⠉⠶

⠀⠀⢀⢠⢀⠀⢀⠀⢀⠀⠀⣀⣀⠀⠀⡤⡀⠀⡀⢀⣀⢀⢀⢀⣀⣀⣀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣤⡀⠠
⠀⠊⠄⣜⠀⠁⠘⠁⠘⠢⠀⣉⣉⡇⢂⠀⠈⠀⢢⡊⣓⢻⡌⠀⢿⠿⡿⢠⡈⢰⣿⣦⣶⠀⢰⣿⢸⣿⡆
⠀⠀⢀⠀⠀⠀⡀⠀⢀⠀⠀⠉⠉⠀⠐⠈⠁⠀⠶⠺⠜⠺⠧⠀⣤⠀⡤⠀⠀⠀⠈⠙⠋⠀⣈⣻⣿⣟⣙
⠀⢒⡚⢀⣁⠀⠀⠀⠀⡀⠀⠀⣤⣆⡀⡳⠇⠖⠀⠂⠀⠀⠀⠄⢀⣤⣀⠀⠂⡀⠀⣠⠀⠀⣿⣿⣿⣿⣿
⠀⢇⣼⣷⣠⢀⠀⠁⠀⠠⠀⠀⠀⠈⠁⡁⡄⡇⣾⣿⣿⣷⠄⠀⣾⠶⢿⠄⢠⣿⠛⣿⣧⠀⠀⠂⠀⠒
⠀⠀⡙⡋⠁⠀⠀⠀⠠⠀⢀⣀⠀⠀⣀⣀⣀⢑⠐⠭⠭⠕⠊⠀⡯⠤⢼⡀⠤⠀⠀⠡⠄⠀⠀⠀⡄
⠀⠐⢀⠄⢒⠂⠄⠀⣤⡄⠀⠋⠀⣊⠉⠉⢹⠈⠀⠁⡀⠉⠄⠀⠂⠀⠀⠀⠀⠘⠛⠏⠀⠀⠀⠐⡇⠀⠘
⠀⠀⠈⠁⠙⠀⠀⠀⠀⢀⣼⠈⠁⠈⡀⠤⠘⠀⠡⠚⠁⠐⠀⠁⠁⢸⡇⠈⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⢸⠆⢾⠭⢿⠀⠀⠈⢿⣇⠀⠀⡀⢀⠀⠀⠤⠴⠒⠂⠄⠂⠄⣤⣤⡄⢻⠓⣶⣀⡀⠓⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⡘⠋⡀⠀⠀⠈⠀⠀⢀⣠⣤⣀⣤⡀⠀⡃⠉⡇⠀⠀⠠⠦⠀⠀⠀⠀⢸⣿⡇
⠀⣠⣤⣶⣤⣄⠐⣠⣴⣶⣶⡀⠀⠀⠀⠈⠀⢰⠉⠹⠿⠇⠉⠹⠀⠋⠉⠃⠓⠆⢀⣶⣦⣤⠀⠀⢸⣿⡇
⠀⡷⣄⠀⠀⢸⠐⠻⣿⣿⣿⣷⢦⢰⣄⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠀⠀⠀⠸⢿⣿⡟⠀⠀⣸⣿⡇
⠀⡇⠀⠀⡀⣸⠀⠀⠨⠈⠄⠀⢀⡈⢿⠋⠉⠓⠀⠀⢠⣒⠀⠐⢄⡂⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡛⠛⡃
⠀⢧⡀⠀⠈⠵⡀⢬⠇⠁⠀⠐⣬⡓⠐⠰⠟⠋⠀⠠⣟⣛⣉⣂⣘⡋⠀⠀⠀⠘⠀⠀⠀⣶⣀⡄
⠀⣤⣤⣤⣤⢠⠀⠈⠀⠀⠀⠀⠙⠃⠀⡀⠀⢉⡉⠀⢽⣯⣭⣩⣿⠀⠀⠀⠀⡀⠀⠄⠀⢉⠙⠓
⠀⠿⠻⠿⠿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠴⡧⠶⢦⡀⠋⣉⢉⠂⣠⠤⣤⠤⣤⠀⠀⠀⡠⠀⠀⠀⡠
⠀⣬⣤⣤⣤⠄⠉⠁⠈⠁⠀⠀⠀⠀⠐⣿⢹⣟⣿⣻⠿⠄⢐⡨⠀⣙⣛⢃⢐⡁⠀⢀⠐⠄⠐⠨⠄⠐
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢸⠀⢀⣿⠀⠀⢀⠤⠀⣀⠔⢃⠤⠀⠆⡀⠂⠂⠸⠐⠀⠀⠐
⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⢸⠀⠀⣿⠀⢠⠄⠈⠀⣠⠄⠛⠂⠀⠄⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠤⠀⠄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢽⡀⣿⠁⣞⠙⣧⠀⢰⡏⠙⢰⠋⢻⡔⢯⣍⢸⣷⣼⡇⢸⡭⠍⢹⡏⠈⡟⢰⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⠛⠂⠙⠚⠁⠀⠈⠛⠛⠈⠓⠋⠈⠓⠋⠘⠃⠀⠃⠘⠓⠂⠘⠃⠐⠛⠈⠛⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢴⠄⠘⠃⢀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠶⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣋⠭⠭⣍⡓⢒⠲⡄⠻⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡵⠋⠀⢀⡠⢤⠌⠹⡇⣇⠀⣯⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⢸⠁⣸⣏⣻⡴⠟⢲⠆⢱⢹⠀⣿⣸⡃⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⢸⠸⡌⠓⠦⢽⣾⡥⠋⠀⡜⣸⠀⡏⠈⡇⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⠀⣷⡙⢦⣀⠀⠃⢀⣠⢞⡵⠃⢸⣁⡀⢸⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⠦⠛⠒⠮⠭⠭⠭⠖⠋⠀⠀⠈⠓⠒⠋⠀⠀⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠈⠳⠀⢀⢰⣆⠀⠀⢀⡴⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡾⣛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢦⡼⣰⡞⢢⡝⣾⠥⠀⢾⣙⡶⢻⠂⠀⠀⠀⢸⡇⠀⣶⣒⡦⠀⣷⠿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢚⣻⣉⠛⠁⢐⣿⣂⠀⠸⣅⣳⣻⠀⠀⠀⠀⢸⣇⣒⣒⣒⣒⣀⠿⣭⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⠀⠀⣿⢹⢽⠀⠀⣿⡟⣗⠀⠀⠀⠀⢸⡇⢀⡤⠤⣽⠁⣀⣋⣀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⢻⠀⠀⣾⢻⢻⠀⠀⣿⠇⢿⠀⠀⠀⢀⢸⡇⠸⣄⣀⣿⢈⣓⣒⣚⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣾⢸⢸⣒⣒⣿⢸⢽⣒⣒⣿⡟⣷⣲⡆⡴⠋⢸⡇⠤⠈⠉⣿⠀⢀⣂⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠘⢸⠀⠀⣿⠘⢽⠀⠀⣿⠃⣯⢸⠁⠀⠀⢸⡧⠭⠤⠤⣿⠀⠀⣦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣓⣋⣉⣉⣙⣒⣋⣉⣉⣙⣒⣋⣙⡇⠀⠀⠈⠓⠒⠒⠒⢿⣀⣀⣋⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡒⣁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣯⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⢀⢾⢹⣹⣽⣿⣵⢞⡃⠀⠀⠀⠀⠀⠀⡰⣷⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠠⡎⠀⢘⣿⣿⣿⠇⠙⣆⠀⠀⠀⠀⠀⡜⠀⣯⣿⣿⣟⣵⡿⠀⠀⠀⠀⠀⠀⡐⠀⢹⣿⣿⠇⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣾⣿⣿⣿⠀⠀⠹⠀⠀⠀⠀⠀⡁⢀⣾⣿⣿⣿⡏⠡⠀⠀⠀⠀⠀⠰⡁⢠⣿⣿⣿⣦⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⢳⠀⠀⠀⠀⠀⠀⠀⠑⣾⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠘⡎⢇⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣯⢹⣿⡿⡄⠀⠀⠀⠀⠀⣰⡿⣿⠋⠻⣾⣿⣆⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠘⠘⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠃⠀⣀⡜⠀⠀⠀⠀⠀⢰⡟⢱⡏⠀⠀⠈⢿⣿⡄⠘⠷
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⡔⣩⢼⡇⠀⠀⠀⠀⠀⣿⣡⠈⢹⡏⠐⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⢻⠀⡀⢐⠀⠀⠸⡇⠀⠀⠀⠀⠘⢛⡟⠀⢸⡇⠐⠊⢱⡸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⠀⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⣼⢘⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢳⠀⢸⡇⠀⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠨⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣷⢻⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡀⠀⣇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠢⠀⠀⠃⠄⠀⠀⠀⠀⠀⠏⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⢁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠟⢛⣉⣭⠥⣤⣶⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠟⣡⠴⢚⣩⠴⠒⠉⣁⢤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢟⣵⢞⣡⠖⠋⠀⢀⠔⣨⠶⣛⣭⡽⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣵⣟⣵⣟⡅⠀⡠⠂⢠⢞⡵⡿⠋⣁⢤⣭⣭⣿⣿⣿⣿⡍⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⡴⠊⠀⣴⣿⢟⣠⡶⠋⢀⣴⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣥⢖⣡⣾⣿⣿⣿⡟⣡⣾⣿⣿⣿⡿⠿⠟⠁⠰⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡼⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⢰⣮⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠙⣽⢏⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⢟⣯⣿⣿⣿⣿⣿⣷⣝⣿⡿⣻⣯⡤⣶⣦⣤⣄⡀⠀⠀⠀⣀⣼⣿⡏⠠⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢿⣿⣿⣿⣿⣿⣿⣿⣫⠞⢽⣷⣶⣿⣶⣯⠛⠃⠀⢠⣾⣿⣿⣿⣇⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⠀⡄⠐⠿⡯⠤⠁⠀⠀⠀⣿⣿⣽⣿⣿⣿⣬⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡟⠿⣿⣿⣿⠛⢾⣿⣧⠆⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⣇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠿⣿⣿⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⢄⣠⣿⣿⣿⣏⢷⣬⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠛⣻⣿⣿⣛⣛⣛⣽⣿⣿⣧⠀⢀⡾⠁⠀⠀⠀⠀⢀⣀⣈⣿⣿⣿⣿⡌⢀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣌⣛⣿⣿⣿⣿⣵⣾⣿⣿⣷⣏⡀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣱⣎⣉⣡⢟⣿⣿⠟⠉⠉⠻⣿⣿⣶⣄⡀⠀⠀⠉⢙⣿⣥⣥⣭⣥⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⡿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣧⣸⣿⣾⣶⣼⣷⣾⣶⣷⣾⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣥⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣭⣿⣿⣷⣽⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠐⣶⡄⠀⠀⠸⡿⣿⡿⠀⠀⠀⠟⠻⡏⠀⢸⣿⣿⡆⠀⠀⡀⠀⠀⠸⣿⣿⡟⠀⢀⣿⣿⡿⠀⠀⠐⣻⠷⢄
⠀⠸⠿⣿⣿⠃⠀⠀⠀⠀⠀⣄⡀⠀⠀⡀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠁⡎⠀⠀⠁⢈⠀⠀⠀⠛⠉⠁⠀⠀⠀⠘
⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⡀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⢢⣴⠄
⠀⠀⠀⠀⠛⣿⣿⣿⠀⠀⠁⢻⣿⣿⡀⣾⣷⣿⡆⠀⠀⣷⡶⠀⠀⣤⢤⠀⠀⠀⠀⠀⠀⠀⣷⣶⣼⠁⠀⠀⠸⡿
⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⢸⣿⣿⡗⣿⣿⣿⡃⠀⢸⣿⠿⠀⠘⢹⠀⠀⠀⠀⠀⠀⠃⣠⣿⣿⡏⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠄⢛⣿⡄⠀⠀⠀⠀⠛⣿⠀⠻⣿⣿⡿⠀⠀⣉⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⣙⡇
⠀⠀⠀⠀⠀⠀⢻⡇⢸⠀⠀⠀⠀⣿⡆⠀⠹⣿⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢳⠉⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠠⠀⠀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡞⠀⠀⠀⠐⢀⣄⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠛⠲⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠤⠤⢤⠤⠠⣤⡤⣤⠄
⣤⣤⣤⣴⣴⣶⣶⣴⣦⣶⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣗⣒⠭⠉⡝⢿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠠⠀⠁⠀⢠⣽⣼⡇⠀⣠⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠁⠀⠀⡀⢠⣾⠽⠾⠙⠪⠛⠹⠧⠶⣄⡘⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⢀⣠⠖⠋⠁⠩⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⢘⢆⠈⣃⣀⣴⠆⠀⠀⠀⠤⠤⣤⣄⣀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⢕⡅⣈⣭⡽⠁⠀⠀⠀⢤⢶⡿⠿⠉⠀⢰⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡑⠂⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⢢⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠱⣄⡀⠀⠀⠀⠈⠙⠿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣶⣤⣀⣀⣀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠻⢿⣿⣿⢿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⢋⣪⣤⣤⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⠛⠇⠀⠀⠀⠠⢐⣺⣽⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣴⡷⡲⣳⣶⣶⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣼⣾⣿⣿⣻⡿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣠⣶⣶⣦⣼⣿⣯⣤⣤⣄⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⣽⣿⣿⣿⣿⣿⡿⢿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣩⣿⣿⣿⣿⣿⣿⣧⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⠀⠀⢷⠀⠈⢿⠟⠟⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣯⠀⠀⢸⡆⠀⢸⠰⢸⢻⣾⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⡖⠀⣼⠇⠀⠀⠆⣸⣾⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣼⣿⣿⠇⣸⡟⢀⡆⠀⢿⣿⠿⠿⣿⢿⣦⡀
⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⡴⡏⣿⣿⣴⣿⣿⣿⣷⡙⣜⡇⠀⠀⠈⠨⣯⣻⡗⠲⣶⠤⣄⡀
⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠈⠸⢰⣿⣿⣿⣿⣿⣿⣽⣧⢹⣶⢰⣷⣶⣾⣿⣮⣿⣄⠀⠉⠐⠻⢦⣀
⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠴⢾⡿⣿⡟⣿⣿⣿⣿⣿⣦⢻⣼⣿⣿⣿⣿⣿⣿⡟⢧⠀⠠⠀⠀⠉⠳⢤⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠗⣿⣇⣿⣿⣧⣏⣿⣿⣦⣳⣿⣿⣿⣿⠿⣿⣿⣾⡧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⡉⠉⠀⢻⣿⡯⠉⠉⠉⠉⠙⢿⣿⣧⠀⠀⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⠖⠚⠒⢀⠀⠀⠀⠀⢀⣀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠇⠀⠶⠀⠀⢰⡞⡂⠀⠀⠉⠉⠁⣀⣨⠉⣯⠒⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠘⢦⠄⠊⠁⢀⡴⢋⣱⠿⠛⠚⠒⣒⣾⣿⣿⣶⡏⠙⠢⠔⠺⠟⠑⠒⢦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣟⡰⠀⢀⣴⡿⣡⣾⠁⠀⣠⣶⣉⣼⣿⡋⠃⠀⠉⠳⣤⡇⣀⢀⡀⠀⣊⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣦⣴⠋⢀⣴⣿⣿⢰⣿⠿⠗⠉⠀⠀⠀⢉⣄⠁⠀⠀⠀⠀⢹⣿⣿⣶⠀⢀⡾⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⣿⣿⡏⠃⢸⡁⠀⠀⠀⠀⠀⢠⣴⣿⣗⣴⣦⡄⠀⣾⠙⠿⠛⢁⠈⠓⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡏⠰⠀⠘⣿⠏⠀⠒⠰⠎⣧⠈⠻⣿⢰⣿⣿⣶⣄⠁⢀⣷⣦⣤⠾⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⡄⠀⢿⡁⣶⠈⣰⣿⡦⠀⠀⠀⠀⣿⡗⣸⢃⣼⣿⣿⣿⣿⣧⢼⣿⡅⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠋⠉⠉⠀⠘⢿⢿⣆⠻⠟⠁⢀⣀⣠⣤⣿⠗⣱⣾⣿⣿⣿⣿⡿⢉⣈⣉⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢶⠀⠈⠃⢻⡀⠀⢀⣌⡁⠀⠀⠙⠛⠛⠛⠟⠿⢿⣿⣧⣿⣿⣿⣧⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣇⠀⠀⠀⠀⢀⣽⠖⢲⡌⠙⠢⠉⠻⢿⣷⡀⠀⠀⠀⠀⠀⠂⣈⣥⣴⣶⣾⣿⡟⠁⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣾⣇⡀⠀⠠⠄⣿⠁⠀⠀⢿⣖⣄⣀⠀⣠⣭⠙⢀⣀⣰⣤⣶⣿⣿⣿⡿⠟⠿⣿⣿⠀⠶⠀⠀⢰⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡶⠀⠐⢻⡆⠀⠀⠀⢻⣯⣉⣉⣀⣀⡑⠾⣿⠿⠿⡟⠛⠛⠋⠀⠀⠀⢹⣿⠀⢢⣴⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠓⠀⠀⠀⠀⠉⠉⠹⣿⡁⠀⠀⢀⠈⠀⠀⣂⣁⣤⣤⣀⠠⠾⠃⢀⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠠⢦⡾⢶⡄⠀⠀⠀⠀⠀⠈⠑⠀⠀⢹⣿⣿⣯⡉⠀⠁⠉⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠛⢡⣿⣷⣶⣤⣤⣄⠀⠻⣿⣿⣿⣧⡈⡀⠀⢀⣀⡠⢾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠉⠹⠛⠉⠁⣛⠆⠸⣧⣹⣿⠻⣇⠉⠋⠙⠉⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠀⠤⠓⠀⠰⣿⠋⠱⠍⣷⣤⠠⠾⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠜⠃⠉⠙⠀⢄⡃⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢰⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⢠⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⣾⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣧⣟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠟⠋⠋⡿⢸⣿⡏⢻⣿⡈⣿⣯⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣹⢿⣿⠟⡥⠀⠀⢠⡇⠀⢿⡇⠈⣿⡇⠘⣿⣇⠈⢻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣣⠏⠀⡼⠁⠀⠀⣼⠐⠀⠸⡇⠀⢸⣿⠀⠘⣿⡆⢻⡷⠈⠻⣯⡕⠢⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡾⡿⢣⣤⠎⠀⠀⠀⢀⢇⡇⠀⠀⣧⠀⠘⢿⡇⠀⠈⠙⠀⠙⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠜⠁⠘⠃⠀⠀⠀⠀⡈⣼⠃⠀⠀⣿⠀⢀⡈⠃⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠁⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠈⠂⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣁⠀⡙⣻⡷⢒⡶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠀⠁⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣷⣤⣀⣤⣄⣀⣤⣤⣶⣤⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⣦⣉⣻⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢻⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠇⠸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠦⣀⠲⣮⣿⣿⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣄⠀⠀⠀⠀⣠⣴⠷⢬⣙⣿⣿⣿⣦⠠⣇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢷⣄⠀⣴⣿⣧⡀⠀⠙⠋⡟⠿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣻⣷⣦⣤⣈⡑⠪⢽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⣿⢻⡟⢻⡟⢻⣿⣿⣿⣷⣦⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣸⣇⣾⠁⠘⠇⣠⠂⠘⢉⣿⣭⡻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⡟⣽⠇⠠⠇⠀⠉⠀⢂⠸⠀⠉⢳⣌⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡟⢰⠏⠀⣠⣤⣴⣶⣦⣤⣇⡀⠈⡈⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢀⠞⢁⡟⠀⣴⡿⣿⣿⣿⡿⢿⣿⣿⣦⢡⠀⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⣠⠌⠀⣤⠀⣼⣿⡅⢸⣿⣿⠀⠀⣿⣿⣿⣯⢳⡘⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⡴⠊⠀⠀⠐⠁⠀⣿⣿⣧⠀⢻⣿⡀⢠⣿⣿⣿⣿⡇⢱⣼⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⡞⠁⠀⠀⠀⠇⠀⢠⣿⣿⣿⡆⠈⢿⡇⢸⣿⡘⣿⣿⣿⡌⠘⣿⣿⣿⡙⣿⡙⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠘⡃⠀⠀⠀⡼⠀⠀⣼⣿⣿⣿⣿⠀⢸⡇⣸⡿⣿⢀⢿⣿⣷⠀⠐⢻⣿⣷⣜⣿⣄⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡗⠀⢀⡼⠃⠀⣼⣿⣿⣿⣿⡿⠷⢾⣷⣿⡇⣧⠸⢸⡟⣿⡄⠀⠈⣿⣿⢿⠿⣿⣆⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠇⠀⠎⠀⢠⡞⣱⣿⣿⣿⣿⠇⠀⣸⣷⣿⡇⣿⠀⠚⣇⢿⣷⡑⠀⣿⡇⠈⢇⡈⢿⣷⡀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠨⠀⠀⢼⣧⣿⣿⣿⣿⣿⠀⠀⣹⣷⣿⡇⢿⡰⠀⣿⢸⡏⣿⣄⠹⣷⡀⠀⠹⡌⢿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡿⠘⢾⡇⠀⣿⣺⡇⡀⢹⣦⣿⣯⡁⠠⡘⡇⢹⣿⠏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠆⠀⠀⢸⣠⡿⢳⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⠃⡀⢸⡇⢀⣿⠸⣇⠱⡀⢿⠀⠙⢿⣦⡘⠈⢻⣿⡇⠀⢻⣇
⠀⠀⠀⠀⠀⢠⣼⡀⠀⠀⠀⠀⠀⠀⢘⣿⣧⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡿⠀⠠⢸⡇⢸⣿⠀⢻⡄⠁⠈⡄⠀⠀⠛⣷⡀⠈⢿⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⢀⣴⠾⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣧⠀⢀⡿⠀⢸⡏⢠⠈⣿⠀⠀⢡⠀⠀⠀⢸⣷⡄⢸⣿⢃⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⣶⣿⣅⣰⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣦⣸⡏⢀⠚⡇⠀⢸⡇⠨⡄⣿⡆⠀⠈⣄⣀⡀⠀⣻⠇⢀⣷⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣿⣿⢸⣿⣿⠃⣿⠙⣧⡌⠀⠀⠀⢠⣷⣤⣴⣿⣿⣄⠀⢿⡿⠿⠟⠉⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠹⢿⣿⣿⠿⣏⢸⣧⠀⢻⣷⣤⣤⣤⣼⡿⠟⠛⠿⠿⣿⢶⠾⠃⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢻⡿⠁⢀⣨⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣈⣤⣉⣧⣤⣌⣠⣽⣌⣠⣿⣋⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⢟⣛⣛⣓⣒⣲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⣾⣿⠟⣫⣵⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⠏⣴⣾⣿⡿⣛⣯⠷⣞⣻⣿⣿⣿⣿⠋⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡿⢣⣾⣿⢟⡵⢟⣩⣶⡿⢟⣻⣿⣿⡿⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⢃⣿⠏⣱⢋⣴⡿⢛⣵⣾⣿⣿⡿⠛⠀⠀⠀⠒⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣼⡟⠐⣱⡿⢋⣴⣿⣿⡟⢉⠁⠀⠀⠀⠀⠰⢿⢧⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⠃⢰⡿⣱⣿⢟⣿⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⠀⣿⣱⡟⣱⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣿⣷⠹⠀⡏⡟⣼⡟⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⣰⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢿⣧⠀⠃⡇⣿⡁⠀⠀⠀⠀⣠⣾⣿⣶⣤⣀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣿⣧⡀⠁⢻⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡝⢄⠀⠻⣧⡀⠀⠸⣿⣿⣿⣿⡿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡁⠀⠙⢷⣄⠀⠀⠉⠉⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡄⡀⠠⠖⣂⡄⢴⡦⢀⡐⠲⠄⢀⢰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⣠⡞⣫⠭⢥⣬⣤⣶⠶⠷⣥⡼⣻⣉⢷⡞⣉⣟⢧⣬⠾⠶⣶⣤⣥⡬⠭⣝⢳⣆
⠀⠀⠀⠀⠀⠀⢻⡸⣽⣧⡾⠫⢬⣽⡿⠿⢯⣥⡭⢞⣼⣧⡳⢯⣬⡽⠷⢿⣯⡥⠝⢷⣼⣦⢇⡟
⠀⠀⠀⠀⠀⠀⢠⡦⢠⡟⢨⠖⠀⠀⠂⠁⠀⠉⠛⠋⠁⠈⠙⠛⠉⠀⠈⠐⠀⠀⠲⡅⢻⡄⢴⡄
⠀⠀⠀⠀⠀⠀⡾⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠇⡇⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠥⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟⠬
⠀⠀⠀⠀⠀⠀⠀⣙⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢩⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⡍
⠀⠀⠀⠀⠀⠀⠀⡒⣼⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣧⢒
⠀⠀⠀⠀⠀⠀⡆⡃⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⢸⢰
⠀⠀⠀⠀⠀⠀⢳⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠘⠗⠘⣧⢘⠦⠀⠀⡄⡀⠀⣀⣤⣄⡀⢀⣠⣤⣀⠀⢀⠠⠀⠀⠴⡃⣼⠃⠺⠃
⠀⠀⠀⠀⠀⠀⣼⢱⠻⡟⢷⣔⢚⣻⣷⢶⣞⠛⣓⢮⢻⡟⡵⣚⢛⣳⡶⣾⣟⡓⣢⡾⢻⣟⡎⣧
⠀⠀⠀⠀⠀⠀⠹⢧⣝⣒⡚⢛⠛⠿⠶⠶⡛⢳⣽⣉⡾⢧⣉⣯⡞⢛⠶⠶⠿⠛⠛⢓⣒⣫⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠇⠁⠐⠦⠌⠁⠺⠗⠊⠡⠴⠂⠈⠸⠖⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠉⠋⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠦⠀⢠⡤⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⡀⣀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢠⡤⡴⢿⣏⣱⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣍⠀⢂⡝⢌⡀⢾⣭⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⠶⣺⠀⢻⣬⡳⣝⣾⢮⣍⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠜⡁⢯⣰⣈⣿⣽⠿⣎⠳⢯⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠘⠒⠾⣽⣿⣳⣦⠈⢧⠻⢿⣷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⣻⣻⣵⡻⠼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡱⢿⣿⣯⣹⣾⡬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢠⣭⣷⣽⡪⢽⣏⢼⠨⣕⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣦⣾⣽⣵⣿⣮⣷⣔⣵⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠶⣥⣭⣍⣿⡟⢿⣫⣟⠿⡻⣷⣚⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡛⢲⠆⢸⣹⣮⢺⣿⡻⣕⢮⡛⢍⠓⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣿⡅⢸⡟⠟⣻⣿⣝⣼⡻⣿⣷⣏⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⣶⢿⠇⣸⣠⣮⣿⣿⣫⣏⣗⡟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣿⠯⠞⣸⠇⣭⣝⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠘⣏⣒⡷⠶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡿⣅⢯⣭⣉⣻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣍⣷⠷⢶⠶⢬⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢻⣿⢿⡷⠒⠺⠖⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢺⣿⣏⣿⢛⣽⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣋⡟⣿⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠧⣾⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠉⠻⢫⡀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠐⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣾⠀⠸⣿⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣟⣛⣛⣛⣛⣻⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⡀⠀⣼⡆⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠹⣿⣿⣿⣿⠟⠉⠘⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⢀⣾⣿⣿⣿⠿⠷⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⣿⡁⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣿⣷⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠛⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⠀⠦⠀⠱⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠈⠀⠀⠀⢰⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡠⠀⠀⠀⠀⠀⠈⠐⠠⣀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣷⡀⠀⠀⠀⠀⠀⠠⢄⣉⣙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⡀⢻⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⢻⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⢿⣿⣿⣿⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠛⠛⠉⢸⣿⣷⣝⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢳⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠋⠀⠀⠀⠰

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡇⠠⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢼⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⢢⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠄⣘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⢀⠀⠐
⠀⠀⠠⠀⠀⠄⣿⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠠⠤⠤⠄⠀⠈⠀⠀⠀⠀⠚⠁⠄
⠀⠀⠀⠑⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠒⠤⠔⠃⠀⠀⠐⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⢠⣄⣀⣀⣠⣀⣤⡄
⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠉⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡆⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠
⠀⠀⠀⠀⠀⠀⡀⠈⢆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⣦⠀⠘⣿⣦⣼⡗
⠀⠀⠀⠀⠀⠀⠉⢳⡎⢁⣀⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⣀⠀⠀⠀⢀⡀⠀⠀⣀⠀⡄⢠⠀⠀⠀⠠⠀⠀⠅⠀⠀⠀⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋⠁⠁⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⡡⠇⠀⡇⢸⠀⠀⡆⠀⠇⠀⡇⡎⡈⠁⠡⠘⡁⠀⠀⣀⣠⣤⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠇⡇⢘⣀⠈⢇⡀⠆⡀⠁⢣⡸⠂⣀⡠⠇⠀⠀⣿⡿⢋⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢳⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠈⠡
⠀⢀⠀⠀⠀⠀⠆⠀⠡⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⢀⠀⠀⠀⠁⠀⢀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⢀⣾⠟⠙⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠃⠀⠄⠸⠀⠀⠀⠀⠀⠀⠰⠛⠁⠈⠁⠀⢄⠀⠀⢸⡅⠀⠀⢀⣇⠀⠀⠀⠀⠠⠀⠄⠀⢆⣀⡀⠀⠀⠀⠁⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⡀⠀⡀⠀⠺⠗⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⢈
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⠞⠃⠀⠀⠄⠀⠀⡀⢀⠀⠀⠀⠀⠃⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⡦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⠀⠀⠆⡶⢴⡡⠄⣠⠔⠁⠀⠀⠀⠇⢀⠀⠛⠁⠀⠀⠀⠈⠀⠀⠀⠀⠂⠁⠑⢄⠀⠀⡀⠀⣠⣤⠤⢤⠀⠀⠀⠀⢀⣀⡀⢓
⠀⠀⢰⠦⣂⠕⠄⠄⠳⡞⠁⠸⠃⠀⠀⢀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠲⠀⠠⠀⠀⢁⠀⢰⡄⠀⠂⠁⡄⠀⠀⠀⠀⢧⢀⠔⠊⠁⠀⠃⠀⠁
⠀⠀⠈⢾⠛⢴⡺⠀⢀⣀⠰⣄⡀⠀⠐⠀⠠⠄⢠⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⢠⣄⠁⠀⡡⠎⠓⡈⡀⠍⠀⣀⠁
⠀⠀⠀⠀⠉⠀⢀⣤⠀⠀⠙⠾⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠼⢢⡆⢀⡞⠀⢤⠙⠉⠁⠎⣷⠞⣡⡄
⠀⠀⢰⣆⠀⠚⠉⠀⠀⡄⣀⢀⡄⠀⠀⢄⠀⠀⠁⢀⣤⣈⡠⠀⠀⢐⣤⠀⠀⠀⠀⠀⠀⠀⠐⢀⠠⠀⠠⠀⠀⠸⠤⠔⠠⣗⣦⠠⠌⢁⣴⢿⢄⡄
⠀⠀⠈⠀⠠⠍⠀⠀⢀⣧⡻⣾⡁⠀⠠⠈⠀⠘⠁⠸⢋⡴⠃⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠒⢰⣿⠆⠀⢀⣀⡀⠀⠦⣥⣀⣠⣄⡀⠀⠈⠂⠀⠈⠁
⠀⠀⠀⠠⣄⡐⠀⠀⠸⢿⣵⡿⠀⠀⠀⠀⠀⠀⠠⢤⡤⠀⠀⠺⠀⠀⠀⠀⠀⠀⠠⣤⣹⣿⠄⣉⠀⢠⢟⣙⠠⡆⠀⠀⠀⠙⠳⠛⠂⠀⢀⡇⠠
⠀⠀⠀⠢⢀⠀⡁⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠐⢈⠡⡌⠠⡤⡴⠀⢀⣀⡀⢠⠀⠀⠊⡀
⠀⠀⠀⠐⠂⠀⠁⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢈⡷⢢⣦⡁
⠀⠀⠐⠒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠇⠷⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠉⢀⠄⠀⠑⠜⠂⠀⠀⢀⠀⡠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁⣒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⣀⠀⠀⠀⠀⠐⣁⠘⠀⠇⠀⠄⢀⠆⠐⠂⠀⠃⢠⡄
⠀⠀⡉⠆⠀⢦⣄⢀⣀⣠⡴⠆⠀⢠⡄⠀⠀⡆⠈⠉⠁⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠁⠐⠁⠐⠖⡀⠂⠈⠀⠀⠂⡀⠂⠀⠀⡇⢬⡶
⠀⠀⡇⠘⢀⣾⣟⣿⡿⠿⣿⡇⠀⠊⠭⠧⠤⠈⠐⠀⠐⠠⠄⠠⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠃⢀⠃⠀⠀⠀⠈⠀⠀⠀⠠⠤⠠⡄⠁⠈⠂⣘⢯⡁
⠀⠀⠀⠈⠀⠈⠛⠶⠖⠛⠉⢡⡄⢀⣄⠀⠀⢢⠄⣀⡐⠀⠒⠀⠀⠀⠈⠁⡆⠤⠄⠀⠀⠤⠄⠠⠀⠀⠀⢠⠀⠀⡰⡀⠀⢀⠠⠀⡆⠐⠁⠻⣦⡕
⠀⠀⠠⣀⣀⠀⢀⠀⠀⠂⠀⡣⠀⠀⡄⠁⠸⠃⠺⣍⡵⠠⠖⡆⠀⠰⠀⠂⡆⠀⠒⠀⢀⡤⠒⢶⣄⠀⠀⠀⠉⠉⠁⠀⠀⢤⣦⡾⠆⠠⠀⣠⡿⡄
⠀⠀⠀⠉⠉⠒⣄⠀⠀⠀⠀⠁⠀⠑⠞⠀⡸⢤⣄⡀⢀⣀⣸⠇⠀⢀⣀⣠⡄⣀⠀⠀⢺⡆⠀⢀⣿⠄⠀⡦⡖⣀⡀⠏⠰⠄⢃⣶⠀⠞⢀⠈⢫⠅
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠉⠣⠀⠀⠀⠀⠋⠃⠀⠉⠀⠈⠓⠾⠟⠋⠠⠥⠤⠾⠯⠀⠀⠀⠀⠠⠉⢠⠔⠃⠘⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣥⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣾⣿⣷⣿⣷⣿⣿

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣧
⠀⠀⠈⠉⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⣠⣀⣼⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⠀⠉⢁⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣷⣶⡆
⠀⣿⠛⣿⣿⣿⣿⡇⠀⠀⠀⢀⡾⠋⠀⠀⠀⡰⣇⠀⣴⠛⠋⠀⠀⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⣿⠛⣿
⠀⣿⠀⠹⣿⣿⣿⡇⠀⠀⠀⢹⣇⠀⠀⠐⠄⣠⢿⡾⠁⠀⠀⠀⠀⣿⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⡿⠀⢹⡆
⠀⠹⣆⠀⣾⣿⣿⣿⠀⠀⠀⠀⠹⡄⠀⠀⠀⣿⢀⡀⠀⠀⠀⠀⠀⣿⡆⢸⣿⣿⣿⣦⠀⠀⠀⠀⠘⢷⡔⠛⠛⢈⠁⣾⠁
⠀⠀⣿⡀⢙⣿⣿⣿⡄⠀⠀⠀⠀⠙⡿⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠁⠐⠀⠀⠁⠁
⠀⠀⠙⠃⢺⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢏⡀⠀⠂⠠⡀⠠
⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⡆⠐⠈
⠀⠀⠀⠀⠀⣺⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡌
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⢰⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⡾⢁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⢀⣾⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⢀⣾⠏⠐⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠁⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣄⢰⡿⠚⠀⠀⠀⠤⠀⠉⠁⠉⠉⠀⠷⣿⠇⠀⠀⠈⠁⠀⠀⠉
⢠⡾⠋⠉⠀⠀⡀⠀⠠⠒⠀⠀⠀⠠⠟⠀⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡷⠀⢀⡄
⠀⠀⠀⠀⠀⠻⡇⠂⠀⠀⢀⡀⢠⡄⠠⠄⠀⠀⠀⠀⠀⠈⠃⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠡⠃⠀⣴⠀⣤⡇
⠀⠀⠀⠀⠸⠃⢀⠀⠀⠀⠈⠁⠛⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡼⠀⣩⡎
⠀⠀⠀⠀⠛⣶⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣀⠤⡾⠇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠊⠁⢀⣠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣼⣷⣆⠀⠀⠀⠀⠀⠀⠀⠔⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠰⠁⢀⣠⠀⠈⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⠛⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢈⣠⠀⠒⠃
⠂⠀⠂⠠⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠆⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠈⢠⠀⠆
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⣀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⠀⠀⢰⠀⣼⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠈⢉⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠂⣋⢺⣤⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⢀⠀⠀⠀⠐⠚⠈⢉⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⡄⠀⠀⢘⣿⣿⣿⣿⣿⠀⠸⡟⠻⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡆⠀⣤⣶⣶⠀⠀⠀⢀⣸⡩⠬⠽⠒⣐⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢞⣤⣸⣝⡁⠀⠆⠃⢰⡏⠟⠂⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣠⠔⠉⠀⢸⣇⡘⣛⣀⠉⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠘⠙⢠⠃⢶⠆⢸⣿⠟⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⡀⠀⠀⠀⠀⠀⣸⣿⣴⡶⢛⡟⢻⣭⠀⣽⠁⣽⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⢳⣼⡷⢿⣠⢠⣞⠶⠄⠀⡆⠀⠀⠀⣧⠀⢀⠠⣤⡶⣼⣿⢿⣿⡆⡇⢸⡏⠀⣿⠀⠿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠐⠳⡀⠘⡿⠀⠀⣿⡤⣿⣦⠔⠀⠰⠂⠀⠀⠂⠒⠂⠀⠉⠀⠈⣁⣀⠀⠄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⢋⢻⣕⠐⠄⠱⣄⠀⠀⠀⠏⣱⠋⢧⢀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢨⠉⣉⠆⠀⣉⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣠⢊⣧⠊⠳⡊⣆⠫⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⣸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢻⢃⣪⡏⢀⡀⠸⠺⣗⠁⠀⠀⠀⠀⠀⠀⠀⠾⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡀⡌⣠⢿⣱⣉⠁⠀⠒⠠⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡰⡿⡜⢡⡖⠉⠉⠀⠀⠈⢷⢻⣦⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⣱⡝⣴⠈⠀⠀⠀⠀⠀⠀⠀⠙⢱⡺⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠺⣏⢰⣠⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⢷⠗⢸⡆
⠀⠀⠀⠀⠀⠀⠐⠰⠋⣼⢣⡾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢦⡑⣴⠅⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠸⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠙⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣤⣤⣤⣴⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠀⠀⠀⢠⡆⢠⡟⠑⡄⣴⠏⠙⡆⠿⢷⡎⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠋⠀⠀⠀⠀⠀⢹⡇⢸⣇⣠⠁⢿⣇⡼⠁⢠⡿⢱⣳⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡟⠁⠀⠀⠀⠀⠀⣀⡈⠁⠀⠉⠁⠀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⣿⣷⡞⠋⠉⠉⠉⠙⣦⡀⠀⠀⢹⣿⣧⠀⠀⠀⢀⣴⠟⠁⢿⣿⣷⡄⠀⠀⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⢸⣿⡇⠀⠀⢀⣀⣴⣿⠃⠀⠀⢸⣿⣿⠀⠀⣰⣿⡏⠀⠀⠀⠀⠙⣿⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠸⣿⣷⣾⣿⣿⡿⢿⣁⠀⠀⠀⢸⣿⣿⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⣿⡟⠋⠉⠁⠀⠀⠈⣿⡄⠀⢸⣿⣿⡇⠈⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⣿⣷⢀⣀⡀⠀⢀⣠⣿⠇⠀⠸⣿⣿⡇⠀⢻⣿⣇⠀⠀⠀⠀⠀⠸⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⢿⡿⢻⣿⣿⣿⡿⠿⢋⠀⠀⠀⠻⠿⠇⠀⠀⠙⢿⣦⣄⣀⣠⠔⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢂⡀⠄⠀⠀⠀⠀⣰⢂⠆⠀⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠄⠀⢸⠁⣩⠤⣤⡶⠐⣶⠆⡏⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠐⣿⢣⠙⠐⠁⠁⠈⠈⠀⡇⢀⠄⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⢁⡜⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣷⣶⣬⣤⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠛⠿⣿⣿⣿⠿⠋⠁




⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⣠⠠⡒⣒⣒⣒⣒⡢⠀⠀⠀⠀⠀⠁⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠤⠴⠡⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠀⠀⢀⠄⣀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠁⡎⠀⢠⠀⠑⠀⠈⠉⠀⠉⠉⠈⠉⢁⠴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⡀⠁⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢷⠶⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⡀⠀⠀⠂⠶⠼⠲⠁⣤⠟⠀⠀⠀⠀⠀⠀⠀⠐⠤⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⠇⠠⠀⠄⡠⣀⢤⣤⡤⡭⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡦⢚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢀⣤⡄⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⠘⠛⠃⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠛⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡶⢀⣬⣿⣶⣄⣀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡿⣧⣿⣿⣿⣭⡉⠉⠀⠀⠀⢠⣄⣠⣤⣤⣤⣤⣤⣤⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⡀⠈⠉⠀⠉⠻⣿⠻⣿⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣟⣧⣸⣿⣿⣿⣿⣿⠇⣶⣦⡀⠰⣾⠛⢣⡄⢻⣷⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡈⢙⣿⣿⣿⣿⣷⣾⡿⠧⠀⠘⠛⠛⠀⠹⠇⠘⠿⣾⣿⣿⣿⡏⠙⠻⡟
⠀⠀⠀⠀⠀⠀⠸⣷⣦⣀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣄⡀⠀⠒⢤⡤⠀⠀⠀⠀⠀⠈⠛⢿⣿⣉⡄
⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣷⣿⣿⣿⣦⡈⣉⣩⣇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠉⠀⠙⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢉⣿⣷⣶⣦⣀⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣹⣿⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠈⠻⣿⡟⢿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠉⠁⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣀⡛⠉⠀⢠⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⣀⣠⠶⣶⡶⢿⠿⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠹⣿⣀⠀⠀⠸⠋⣹⡿⢿⣿⣿⣿⡆⠀⠟⠋⠛⠒⣾⣷⣿⡷⠶⠆⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠈⡉⠀⠀⠀⣼⡿⠇⠀⣽⣿⠛⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⣼⣷⠀⢀⣀⠸⣿⡀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠋⣻⣟⣀⣈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⣿⣿⣤⣽⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⠋⣿⠿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠁⠈⠉⠁⠈⠉⠁⠀⠀⠀⠀⠉



⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣷⣶⣶⣄⠀⠀⠀⠀⠀⣴⡄⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⡀⡶⣤⣄⠀⠀⢀
⣿⣿⣿⣿⠁⠀⣸⣿⠿⠻⣿⣿⠿⠻⣿⣿⣷⠆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣿⣿⣧⡀⢀⣀
⣿⣿⠿⣿⣶⣀⠈⠁⠀⢤⣄⣡⣄⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣶⣿⠟⠻⠋⠐⣶⡾⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⣉⠀⠂⢠⡄⠻⢷⡄⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡏
⣿⣿⣠⡏⢀⣴⡆⠀⢀⣤⣄⣘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣷⡸⠿⠉⠛⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⡀⠻⠏⣸⠁
⣶⣟⣿⣿⣿⣿⠙⣿⣧⣀⡀⠀⣀⣀⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⠟⠀⠈⢻⡄
⡌⠛⢿⣿⣿⣷⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠀⠀⠈⠻⣿⣿⣿⡴⠃⢠⡾
⣀⠀⠀⠙⢿⣿⣿⠋⢿⣿⣷⣶⣿⣿⣿⡿⠡⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇⢀⣾⠇
⠙⠣⡄⠀⠀⢙⣿⣿⠟⠿⠿⠻⢿⠿⠿⢇⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⡀⠀⠀⠂⠀⠈⠹⠟⠀⣠⣤⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠛
⣹⣶⣀⠀⠀⢠⡄⠀⠘⣭⣉⠿⠟⠀⢀⠀⠀⢀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣤⣾⣗⣀⡀
⣿⣹⣿⣧⡀⠈⠋⢀⣈⡉⠟⠀⣴⠀⠀⣠⣶⣾⡿⠿⢿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠻⢿⣿⣿⣿⣍⣿⡿⠷⠄
⣿⣿⣿⣿⣁⡀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠘⠋⢀⣴⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⢻⣿⣦⣾⣶⣶⣶⣿⣏⣀⣿⣿⣁⣤⣤
⣿⣿⣿⣿⣿⣿⠆⠀⠘⣿⣷⣄⠀⣰⠟⠀⠀⣼⠃⠀⠀⣤⣌⢻⣶⣿⣿⣿⣿⣿⠿⠛⠀⢘⣛⠧⠈⠉⠉⠀⠀⠀⣁⣸⣿⣋⢼⣿⣿⣷⠶
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⠿⠂⠁⠀⠀⡆⠀⢰⣾⣿⣿⡟⢀⠉⠀⠹⣿⣟⣿⣶⠿⠇⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⡉⣿⣿⣼⠟⣯⣄⣤
⣿⣿⣿⣿⣿⠿⠟⢰⡆⠀⢀⣠⠀⠀⠀⠀⣀⡀⢾⣿⣿⠟⢁⡼⠀⢸⣿⣿⣿⠋⠉⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⢿⣿⡟
⣿⣿⣿⣿⣿⡖⠒⣿⣷⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⣽⣧⣴⣾⠀⠀⢠⣿⣿⣤⣶⣾⣷⣶⣿⣿⣿⣿⣭⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⢛⣉⡀
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣾⣿⣿⣄⠀⠀⣸⣷⡀⠘⣿⣿⣁⣀⡂⢠⣿⡿⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿
⠛⠋⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠋⠀⠁⠙⠛⠁⠀⠛⠛⠛⠛⠁⠀⠉⠉⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠾⢦⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢧⣶⣿⣶⣤⣄⡀⠉⠀⠻⠿⠃⠀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⡟⠋⠹⢷⣄⣰⣶⣶⣾⣿⣿⣷⢆⣀⠸⠷⣨⣧⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠇⣀⣽⣿⣿⣯⡝⠛⣛⡻⠿⢿⣧⠸⢿⡏⠛
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠛⣿⡛⢿⣁⣠⡿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⢁⡀⠀⠀⠀⠀⠀⠻⢿⣿⣦⣶⣲⣶⣾⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠻⢿⠿⠿⣿⣿⣿⣌⣿⣿⣾⣿⣿⣷⣿⣿⣿⠛⢿⣋⣉⣛⣤⣤⣌⠁⠀⢀⡀⣀⠀⠀⣠⣶⡛⠛⠛⠉⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣷⡄⣈⣿⠋⠀⠀⠉⠙⠳⠀⠀⠀⠀⠀⠹⣿⡟⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⡟⣿⣿⠛⠛⣏⣉⣷⣏⣿⡿⢿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⡿⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣤⠞⢻⣿⠟⠁⡾⠟⢩⣿⣿⣿⣿⣾⡿⠛⠿⣿⣿⡿⠿⠉⣿⣿⣿⣿⡗⠀⠀⠀⣿⣿⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠉⢰⣿⡿⣷⣿⣗⣶⣮⣼⡿⣿⣁⠁⣀⣴⣾⣿⣿⡀⠀⠀⢿⡿⢿⣿⠟⠀⠀⠛⢻⣿⡇⠚⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣦⣤⣀⣀⣵⣿⣿⣿⣿⣿⣟⣿⠿⢿⣿⣿⣛⣿⣿⠿⠿⠿⢀⠀⣠⠀⠀⠸⠗⠀⠀⠈⠙⣷⠀⠀⠙
⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠈⢿⣿⣿⣷⣶⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠙⠛⠒⢾⣿⡆⢠⣦
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣄⡸⣿⣿⣿⣿⣿⡀⠀⠀⣿⡿⠛⠛⢿⣿⠋⠹⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣦⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣷⣤⣀⣀⡀⠀⢠⣤⣤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠙⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠉⠉⠹⣿⠟⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠏⣀⡀⠘⢢⡴⣿⣦⡀⠀⢠⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠈⠁⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠁⣠⣶⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣶⣄⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢐⡅⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣯⣤⣤⣧⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⢀⣰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣾⣿⠁⢸⣿⡧⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠠⣦⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠘⠿⡷⠾⠯⠟⠲⠄⠀⠀⠀⣾⣿⣿⣧⠀⢠⣿⣿⣿⣿⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⠰⠀⡀⠀⢰⣿⣿⣿⡿⠳⢸⣿⣿⣿⣿⣧⠀⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠀⠀⠀⠀⠀⣄⠀⡏⠀⢼⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⠀⢿⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢙⡄⡀⠀⡄⢀⣆⣿⠠⡆⠀⣾⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⡆⢸⣿⡀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠄⣺⣷⣼⣶⣷⣾⣿⡿⣇⢻⣸⣿⡏⡹⠛⠻⠹⠟⢯⣿⣿⣿⣿⣅⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⣴⠏⣠⣶⣾⣿⣿⣿⣿⠹⣆⢹⡿⠓⠓⠀⠀⡄⠀⠘⠻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⠟⠙⢀⣼⣿⣿⣿⣿⣿⣿⡇⠸⣦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⠁⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠃⠸⠋⠀⠐⡃⠀⢀⡷⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣀⠀⢀⠀⡄⡱⠄⠸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠁⠀⠀⠈⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⢿⣽⣿⠀⠁⠀⢸⣆⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⢰⣿⢟⣿⠀⠀⠀⢸⣿⡄⠰⠇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⢀⣿⣿⣿⡀⠀⠀⢸⡏⣷
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠳⠀⠀⢀⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⠀⠀⠀⣷⠸⣇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡾⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⠸⣿⣼⣿⣿⣿⣿⡇⠀⠀⠀⠏⠂⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⣿⣿⣿⣿⡇⠀⢀⣤⣤⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣸⠀⢱⣿⣿⣿⣿⠿⠛⠁⠀⢹⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⢸⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⢀⡞⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠃⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⣉⣷⣖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠘⠟⠀⢬⣾⡋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠔⣰⠇⣰⠒⠛⠀⠀⠽⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡖⠀⠀⠀⠈⠉⠠⡄⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⡻⠁⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⠻⣾⣳⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣷⣖⠃⠀⠰⡿⣏⣊⠀⠀⠌⢻⡧⠨⢻⣷⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣴⠎⢻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢽⣿⢰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠶⠓⡦⠀⠀⠀⠀⠀⠀⠀⠉⣺⡙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢰⡇⠀⠀⠉⠀⠀⠀⠀⢠⡖⣦⢔⣿⡟⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣼⣇⣤⠀⠀⠀⡀⠀⠀⠀⡠⠎⡸⠀⠀⠈⢑⡄⠀⠄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣽⣿⡆⠀⠰⠟⢛⠂⠉⢀⡔⠁⠀⠤⠚⠉⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢛⣻⡟⠁⠀⠀⠠⠇⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠏⡻⠁⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣶⢏⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡾⠟⠀⠀⠐⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⡘⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⣀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢸⠃⠀⠀⠠⡀⠰⡀⠀⠀⢓⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠂⢈⣙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠟⢠⠀⠀⠀⠉⠙⢯⣀⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⡀⢁⣄⠲⣽⣷⠦⣿⢳⣔⠏
⠀⠀⠀⠀⠉⠉⠙⠛⠋⠙⠋⠉⠉⠋⠐⠀⠀⣀⣇⠀⠀⠀⠀⠀⡙⣦⠙⢷⠀⠹⡷⢻⠆⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⠀⠀⠀⠁⣈⣠⣬⣇⣀⠀⠀⣧⢦⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣷⡿⣟⣿⣿⣄⠰⣻⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠹⣯⠙⢿⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⠀⢰⠂⢰⠀⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⠇⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠀⠀⠀⠀⠀⠀⢰

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

⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⡦⠁⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣤⣭⣍⣬⣥
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠠⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⣿⠛⠻⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣤⠐⠀⢶⣿⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣾⡿⠀⠳⠄⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⣩⡇⠀⠀⢀⡍⠈⠛⢷⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢿⣦⣽⣿⣟
⠀⣠⠰⠿⠛⠓⡿⠿⠷⣶⡿⠃⠘⠛⠛⠀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣾⣏⠀⣿⢩⣿⢋⣿⣦⣄⣀⡀
⠰⣿⠂⠀⠀⠀⣿⢘⠻⡍⠁⠀⠀⠀⠀⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⡟⠁⣼⠁⣼⣟⣻⣿⣿⣿⣿⡡
⠐⣿⠑⠢⣤⡀⣿⣻⣾⠁⠀⠀⢀⣤⡾⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡷⣿⣿⠁⠀⠁⢰⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⠀⠀⠘⢿⣿⣿⡯⠀⢀⣴⠟⣿⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣗⣹⡿⠀⣶⠤⣾⣿⣿⣟⣿⣛⣿⣿⣿
⢸⣿⠀⠀⠀⢠⣿⣿⡗⣠⣾⡟⠀⠧⠸⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⠃⢐⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢈⣿⠀⠀⠀⣼⣿⣏⣻⣿⡿⠃⠀⠀⢀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⢨⡅⢸⣿⣿⣿⣿⣿⣏⣿⣿⡟⢡
⠀⠇⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⡀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⡟⣿⣿⠃⠀⢈⡇⠸⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿
⠀⠒⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡳⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⠁⠀⢸⡇⠀⣿⣿⣿⣿⡿⠂⠘⣿⣿⡆
⠀⠀⠀⣾⣿⣿⣟⣿⣿⠀⠈⠁⠀⠀⠙⣵⢪⣿⡷⣿⠀⠀⠀⠀⠀⠀⠈⡏⠀⣿⣟⡀⠀⠀⠘⣇⠀⢸⣿⣿⣿⡋⣷⠀⣿⣿⡷
⢄⣀⣸⣿⣿⣿⣿⢻⣿⣶⣂⣿⣀⣀⣀⣤⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣦⠀⣿⣿⡇
⣸⣿⣯⢿⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠄⢤⣿⡿⣟⣯⡙⠿⣿⣿⡾⠃⠀⢻⣿⣁
⣏⢩⣭⣟⣹⣿⢿⣽⣿⡿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣴⣶⣶⣶⣯⣿⣿⣏⣩⣅⠀⠀⢸⣿⠉
⠻⣖⣉⠉⣉⠋⣹⢾⣿⣿⣷⣿⡿⠋⠉⠛⢷⣾⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣶⣤⣤⣦⣤⣏⢙⣩⡄⠁⠀⠀⠀⢸⣿⠂
⠀⠙⠻⣇⡁⠀⢻⣾⣿⣿⣿⠋⠀⠀⠀⠠⡎⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⣉⠛⢛⣿⣿⣛⣟⡛⠀⠀⠀⠋⠁
⠀⠀⠀⠈⠹⡟⠙⠛⡏⢹⠃⠀⠀⠀⠀⢀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⢛⡋⠙⣷⣼⣿⢿⣿⡿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⢸⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡛⡻⠧⣀⣉⣏⣱⣬⣿⣄⣸⡃⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣛⣶⣾⢶⣿⣿⣿⣿⣿⣾⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⣡⣴⣿⣿⣿⣿⣿⣟⡻⠆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣶⢼⣿⣿⣿⣯⣽⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣤⣤⣽⣿⣿⣟⣻⣟⣿⣿⡃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠛⠉⢿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠐⡓⢰⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣶⣶⣷⠀⠀⠀⠀⠀⠀⠀⣀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠟⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠈⠻⠿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠿⠃⠀⠀⠀⠀⢠⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣦⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣀⠀⠿⢋⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣮⠛⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢨⣭⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣶⠆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣛⣛⣛⣻⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⢇⣴⣿⣿⣿⡟⣛⣻⣿⢿⣿⡿⠥⠁⠀⠠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⣿⣿⣿⠏⢼⣿⣿⡟⠛⠈⣉⠉⢉⠀⠈⢀⣴⣦⣀⣤⡀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣴⣶⣶⣌⢧⣈⠻⢁⡁⣀⣀⣀⣀⡀⠀⠀⠘⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣿⠿⠋⢠⡿⣀⢺⠛⣿⣿⣿⣿⣿⣿⠀⠀⠠⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢠⡇⣤⣀⣴⣾⠿⠸⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠰⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠙⢿⣷⣶⠶⠚⠻⢿⠿⢬⣽⣧⣀⠀⠀⠀⠹⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠋⡤⢈⡿⠁⠠⣤⢀⠁⠀⠀⢈⢙⠻⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⡽⠟⠁⠐⠀⠀⠠⠀⠀⠀⠀⠘⢃⠂⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡎⠃⣀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠐⠀⠀⡀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⡇⠀⣿⡗⠀⠀⠀⠓⢦⣄⣤⠀⢴⣦⡄⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⣿⡧⢦⣠⣀⣀⠀⠀⠀⢰⣾⣿⠙⢀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣴⣿⣷⣶⣿⣯⠀⡷⠶⣾⣿⣶⣿⡖⢀⢾⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡿⢾⣷⠽⠟⣿⡿⣿⠟⠒⠒⣿⣿⣿⣿⣿⠐⠍⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣶⣶⠀⣌⢑⣯⠁⢀⡈⠉⠉⠛⠃⠿⢀⣄⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⣉⠁⠜⠉⠈⠁⠀⠈⠉⣗⠀⠀⠀⢀⡽⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣶⣶⡦⠄⠀⠀⠰⢶⣿⠿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣦⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣒⣶⣶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣩⣿⣯⣉⣿⠊⣿⣿⣿⡿⣿⡵⣾⢟⡻⣦⡿⠛⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣴⡆⠓⢶⡀⢿⣏⢙⡍⢡⣼⠓⣿⣴⣿⠠⣟⢣⡀⣼⢁⣠⡴⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⢻⠆⠀⠁⠈⢻⣾⣿⣦⣱⣮⢸⣛⣆⣰⠉⡦⣉⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠻⣩⡿⣉⠏⣰⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢤⣽⣍⣳⢻⣾⠩⣿⣥⣾⠉⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡜⢿⣗⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⢩⡿⣿⣩⡝⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⣿⢀⣼⡍⠦⣴⣻⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡽⣶⣿⡷⠻⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⠰⢾⣿⠾⡦⢠⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣏⡙⠀⠼⢏⠀⢀⠀⠉⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⠘⣛⡘⣻⣿⣇⠙⡗⠚⣉⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣳⠶⣯⣍⣩⣍⣻⣏⣉⣩⡕⠐⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣯⡝⢤⣴⢠⣬⣽⣧⣈⣥⠶⢆⣉⠁⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡒⢲⠿⢯⣴⣮⣭⣝⣧⠤⠶⠦⠴⠇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠒⢃⢹⣍⣲⡋⢟⣿⣿⡿⡁⠚⡓⠀⠚⠓⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠛⠉⠙⣿⠀⠙⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⢀⡶⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣷⣿⣷⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⢴⣾⣿⣿⣿⣿⣿⣅⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⡿⣿⠻⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡌⣩⣿⡟⢺⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⢸⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣿⣿⣿⣷⣾⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣾⣿⣿⣷⣿⣿
⣿⣿⣿⡏⠀⠉⠁⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣠⡾⡻⣛⣋⠵⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠐⣫⣤⣝⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢠⠈⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠍⣛⠻⠻⣦⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣥⣬⡓⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢰⣦⡀⠀⣀⣤⣤⡺⣿⣿⣿⡘⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢏⣼⡿⣸⣷⣌⠻⣿⣿⣮⡻⣿⣧⣙⠿⣿⢟⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡿⣱⣿⣿⢏⣾⣿⣿⣿⡇⣼⣿⣿⢟⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠿⣿⣷⢸⣿⣿⣿⡟⣼⣿⣿⣿⣌⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠷⠎⠛⠎⣿⠿⠟⠑⠛⠿⡿⢿⣿⡷⠈⠻⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢠⣠⢦⣠⠖⢠⣠⡄⣤⢠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠙⠙⠚⠉⠒⠘⠘⠛⠉⠋⠉⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣥⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠻⢟⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠜⢀⢌⠛⠁⣤⠄⠀⠈⠛⠻⢫⣶⡍⢋⣥⡀⠀⠃⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠡⣎⣹⣧⠀⠀⠉⣷⡄⠀⠀⢠⡦⠀⠀⠀⣡⣄⣠⠈⣀⠀⣿⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢋⡴⣦⢸⣿⣿⣀⣤⣶⠖⠤⠀⠀⠹⣿⣿⣄⣬⡀⠻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡙⠃⠘⠟⠛⣿⣿⠟⢴⡶⠠⠐⠀⠘⢷⠘⠛⠁⣩⣤⠀⡀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣽⡥⠀⠀⣀⡀⠈⠁⠀⠀⠀⠐⠠⠀⠁⢀⣀⡀⢠⣬⣯⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠠⢶⡿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠠⣾⡿⢿⡶⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢷⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠛⠣⡀⠀⠀⠀⠀⢀⣤⣤⡤⢤⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⢻⣄⠀⠀⠀⠙⠏⡴⠂⠿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣆⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠘⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠰⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⣿⣿⠟⠋⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣦⠄⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢛⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⣄
⠀⠀⠀⠚⠛⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠙⠃⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣷⡀
⠀⠀⣴⣦⣴⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣷⣆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣿
⠀⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡟⢿⣿⣿⣿⣿⠁⢿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠈⠻⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⢠⣿⠁⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⠀⠸⣿⣿⣿⡿⠀⠘⣿⠀⠀⠀⠀⠀⣾⡟⠙⣿⣿⣿⣿⠃⠀⠀⢀⣾⡿⠃⠀⠀⠀⣸⣿⣿⣿⢻⣿
⠈⣿⡄⠀⠙⢛⣻⣟⣿⡇⠀⠀⠀⠀⠘⣿⣆⡠⠿⣿⣏⠡⠦⢰⡟⠀⠀⠀⠀⢰⣿⠃⢔⡷⢶⣶⡆⢀⣤⣶⡟⠁⠀⠀⠀⠀⣼⣻⣿⣿⡿⠈⣿⣇
⠀⠘⣿⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⢿⡦⢂⡁⠀⠐⢄⢹⠁⠀⠀⠀⢠⣿⠋⠀⠘⠀⠀⠈⠁⠉⠉⢹⠁⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡄⠘⣿⡄
⠀⠀⣿⡇⠀⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣀⡈⠀⠀⠀⠈⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢠⣦⠀⠏⠉⣀⣀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠲⠽⣿
⠀⠀⠻⡿⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣁⠀⠀⠙⠂⠤⡀⠠⡄⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢸⣿⠃⠀⠀⣾⣿⣿⣿⣿⣯⣿⡄⠀⢻⣧
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢹⡆⠀⠄⠒⠁⠀⠀⠀⠀⠹⠧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⢸⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡝⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣿⠏⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⣾⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⢀⡿⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⢠⣾⠁⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿
⠀⢀⣿⠃⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⢰⣿⠏⠀⠁⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠁⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡠⠊⢠⠀⠺⠇⠀⢀⠈⠖⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⡄⠀⠈⣁
⠀⠀⠀⡰⠁⡀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠄⢀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠿
⠀⠀⠀⡃⠀⣉⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⣠⣄⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠁⠒⠙⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⠀⠀⠴⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠀⠀⡈⠀⠀⠀⢠⣿⣿⣿⡆⠀⡀⠀⠀⠀⣀⠙⠁⠜⣃⣓
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠺⣿⠆⠀⣴⡄⠻⠿⠿⠋⠀⠀⠀⠀⢘⠃⣈⠀⠐⠠⠄⠚
⠀⠀⠀⠀⠠⠊⠹⠆⠀⠀⠉⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠄⠀⠈⣤⡌⠁
⠀⠀⠀⠀⠰⡀⠐⠆⠀⠀⠖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡶⠀⠍⠅⠲
⠀⠀⠀⠀⠀⠘⢄⡀⠰⠄⢰⣦⠀⠀⠀⠎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢴⡆⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠅⣀⠲⠀⠀⠤⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠉⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⢀⠀⠈⢆⠀⠀⠀⠀⠀⢐⠀⠀⣀⠀⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⠂⢀⠈⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠰⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣵⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⢀⠀⣀⣠⣄⠙⠃⠀⠉⠉⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⡀⠤⠀⠐⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠰⣿⣿⣿⣗⠀⠀⠀⠈⢡⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡤⠀⢃
⠀⠀⠀⠀⠙⠻⠟⠃⠀⠙⠀⠀⠈⠛⠁⠠⠄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠐⠶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠓⠚⡈⠀⠤⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⢲⡄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠈⠅⠀⢀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠃⠄⠀⠘⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⠿⢀⣀⢀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠨⣴⡆⠀⠀⡀⠀⠁⣠⠁
⠉⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠈⢀⡋⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣈⡤⢀⠁⡼⡲⠥⠈⠐⠈⢀
⠀⠃⠀⠐⠁⠠⠁⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠈⠁
⠒⢄⡰⠀⢀⣀⠀⢀⡀⠀⠑⡄
⠀⠀⠈⠢⡘⠛⢀⠀⠁⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠘⠦⡉⠀⠀⠖⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘⣂⣤⠙⢈⡁⢏⢖⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢊⠛⠀⠂⠑⠊⠨⢁⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠆⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣠⡦⠄⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠
⠀⠀⠀⠀⠀⠀⡿⠑⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢄⣷⣶⣀⣤⠤⠤⠤⣤⣀⠀⡆⢸
⠀⠀⠀⢀⣤⣤⣴⣦⡀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣾⡟⠉⠉⠀⠀⠀⠀⠀⠀⠙⢧⡊
⠀⠀⠀⣼⣿⣿⣿⣿⣧⣄⣼⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠓⠻⠃⠀⠀⠀⠀⠻⠋⠀⠀⢻⡆
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢂
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⢀⣬⣿⡿⣿⠛⢿⡏⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣶⣶⣦⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠈⣿⣿⢣⡏⢠⣸⡇⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⣿⡏⡾⠀⠘⠁⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⣸⣿⣿⣿⣿⣿⣷⣦⣤⡤
⠀⠀⠀⠀⢠⣿⠑⠣⠀⠀⠀⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⠞
⠀⠀⠀⠀⢸⡏⢀⡇⠀⠀⠀⡇⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⡀
⠀⠀⠀⠀⠀⢁⢸⠇⠀⠀⠀⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠁
⠀⠀⠀⠀⠀⡸⡈⡀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠳⡀⢳⣄⠀⠐⡇⡇⢀⣀⠧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣇⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡙⢻⣿⡟⢿⣿⡿⠛⠿⣟⣿⣿⠀⣿⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾⣿⠁⣸⣿⠃⠀⠀⠀⣿⣿⠀⢹⣿⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢠⣿⡏⠀⠀⣾⠀⣿⣿⠀⢸⣿⢘⣿⣿⠟⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣩⣵⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⡄⠀⣶⣿⠀⠀⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣿⡿⠃⠀⣿⣿⠀⣼⣿⡟⠁⠉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⠛⢻⣿⣦⠀⣿⣿⠀⢿⣿⣧⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⢿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣶⣿⠿⠋⠀⣿⣿⠀⠈⠻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡈⢿⣿⠟⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡷⠀⠙⠷⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⡿⠟⠛⠿⣿⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣦⣤⡀⠀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢠⣤⣴⡿⠃⠀⣠⣤⣤⣿⠃⠀⣠⣤⡀⠈⢻⣿⣿⣿⠀⣨⣿⣦⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢀⣀⣸⣇⠀⠘⣿⣿⣿⣯⠀⠀⣿⣿⠇⠀⢸⣿⣿⣿⠀⢿⣿⣿⣧⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠈⠉⠉⣿⣄⠀⠈⠉⠉⣿⣆⡀⠈⠉⠀⣠⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣩⣍⡉⢩⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣟⣛⣭⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⢟⣛⣭⣭⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢵⣺⣭⣿⣿⣿⣿⣿⣿⣿⡿⢃⡿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⠿⢛⣩⡴⢟⣵⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣭⣭⣭⣥⣶⠶⢟⣛⣭⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠙⠿⣿⣿⢿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣮⣝⡻⢿⣭⣭⣍⣉⣭⣥⣶⢟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠂⠀⠀⠀⠙⢮⡙⢶⣌⠻⣿⣿⣿⣿⣿⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⡀⠀⠂⠰⣶⣶⠆⠀⠀⠀⠈⣻⡄⢹⣷⡌⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠃⢸⣿⣿⡘⣿⣿⣿⠟⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⢸⣧⣿⠟⠁⠀⠐⠐⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣿⣦⠀⠀⠀⠀⢶⣦⡄⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⢸⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠛⢿⣇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⡀⢻⠀⠀⠀⠀⠀⠀⠀⠚⠛⢟⣿⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡟⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢣⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠴⠖⠛⠷⠀⠀⠀⠀⠀⠀⢸⣼⣷⠀⠀⠀⠀⠀⣿⡆⠀⠀⠾⠛⠲⠦⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢴⠓⣦⢤⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⢀⡤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⢸⡌⠛⣘⠃⠀⠀⠀⠀⠀⠀⠉⠁⡴⠋⣶⢾⠄
⠀⢀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠺⣇⣸
⠀⠸⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡠⠤⢈⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁⠀⠀⠀⠀⣤⣶⡆⠀⠀⠀⠀⠀⠀⡀
⠀⡱⠶⠛⣃⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢀⡀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢂⠀⠘⠀⠀⢰⣦
⠀⠈⠉⠉⠀⢀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣽⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢤⣀⡑⢄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣾⣋⣡⠬⠃⢀⣀⠀⠀⠀⣴⣿⣿⡇⠐⠂⠀⠀⠤⠡⠄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⢀⡖⠲⣏⢹⡆⠀⠀⠉⠛⠿⠇⠀⠀⠀⢤⠶⣖⡒⢓
⠀⠰⡻⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣝⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⣙⡯
⠀⠐⡇⢠⣤⣌⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠹⣤⣄⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⢄⠰⢤⣀⠉⠀⠀⠀⠀⢠⠊⣼⠇⡟⠀⠀⢠⡖⢛⣻⡷⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡀⠢⠀⠈⢛⠂⠀⠀⠀⠀⠉⠓⠜⠁⠀⠀⠀⠙⣛⣁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⣀⢤⡀
⠀⠛⠃⠀⠘⠀⠠⠂⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠉⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡴⠶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⡐⠒⠚
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⡏⢹⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁
⠀⠀⠀⠀⠀⡷⠿⣠⠞⢀⡀⠀⠀⠀⠀⠀⠀⢀⡊⢤⡘⣇⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠷⠚⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠈⠳⠾⢧⡟⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣠⢤⣄
⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣼⠶⢦
⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠿⢃⣴⠟
⠀⢀⣀⡈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠞⠋
⠀⠘⡤⠬⠽⢒⡞⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠃⠐⠂⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⣤⣶⡆
⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⢌⠉⠁⢄⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢄⣀⠑⠄
⠀⠀⢀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣶⣟⣁⡼⠃⠀⠀⠀⠀⠀⣴⣾⣿⡆⠐⠠⠤⠀⠈⠁⠀⠐
⠀⠀⠻⣅⠷⠶⣤⠓⠀⠀⠀⠀⠀⠀⠀⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠇
⠀⠟⠃⢈⠓⠦⠟⠀⠀⠀⠀⣠⠖⠒⠒⠶⠤⣀
⠀⠀⠀⣸⢳⠀⠀⠀⠀⠀⠀⠻⣤⡾⢃⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣠⠞⣷⠀⡇⠀⠀⠀⠀⠀⠀⠿⣠⠞⠁⠀⠀⠀⠀⣠⠒⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⢸⡁⣸⠏⢀⠇⠀⠀⠀⢠⣿⣿⣶⡄⠀⠀⠀⠀⠀⢰⣁⣞⣋⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢒⡤⡀
⠀⠉⠈⠓⠊⠀⠀⠀⠀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡹⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⡀⠀⠀⠀⠀⠀⠀⣾⣃⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⡇⠀⠀⠀⠀⠀⠀⣧⣤⠀⠀⠀⠀⠀⠰⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠀⠈⢁⣀⠀⠀⠀⣴⠚⠛⠛⣷⣦⠀⠀⣀⣰⡿⣇⡀
⠀⠀⠀⠀⠀⡠⣶⣯⣿⣿⣶⢶⣿⣿⡷⡄⣼⣿⣿⣷⣾⣿⣿⠀⢸⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠟⠛⢸⣿⡿⠃⢻⠇⠐⣻⡿⢻⣿⣿⣿⢹⣷⣸⣷⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⠂⣰⣾⣿⣯⡐⠃⠀⠸⠟⢻⣏⢿⢿⣿⣟⠻⠛⣿⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣿⣇⠀⠀⠀⠀⠋⠸⣿⣿⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⡯⡿⠀⠀⠀⠀⢀⠀⠈⣷⣿⣧⠀⠀⢹⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣴⡇⠀⠀⠀⠀⠀⢧⠀⢳⣿⣿⠀⢀⠖⢞⣾⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⠀⠀⠘⣷⣸⣿⡇⠀⢸⡿⡿⠓⡝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠲⢾⣿⣿⣿⣥⣷⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠊⢹⣿⡿⣿⠎⠂⠀⠀⠀⠀⡇⠁⢿⠀⠀⠀⣾⢨⣿⣿⣿⢿⡆⣧
⠀⠀⠀⠀⠀⠀⡼⡁⠀⡎⢀⠀⠇⠀⠀⣝⠒⢤⡠⣷⣦⠘⠄⠀⠀⣿⡜⣷⡟⣿⠘⢠⢸⡄
⠀⠀⠀⠀⠀⣼⠁⠀⣸⠁⢸⠀⢰⠀⠀⠈⢧⠹⡗⣾⣿⡀⠀⠀⢠⠃⠃⠛⠀⢹⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠃⠀⢠⢇⠀⠈⠀⢸⡆⠀⠘⢿⣷⣇⠈⣿⢷⡀⡆⠀⠀⠀⠀⠀⠘⠀⠀⡆⢸⠀⠀⠀⢀⡔⢁⠞
⠀⠀⠀⠀⠀⠀⠀⡎⡸⠀⢰⣀⠘⣷⠀⠀⠈⠿⠻⠁⣿⢸⣿⡇⠘⠇⠀⠀⠀⢠⠀⠀⠉⠀⠇⢀⡀⠉
⠀⠀⠀⠈⠚⠄⣾⠁⢧⡆⠸⡿⠄⠻⠀⠀⠀⠁⠀⠀⠋⠈⠿⡓⠀⠀⠀⠀⠀⠈⠰⣆⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣥⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉