Wow Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠀⠀⢻⠀⡼⠁⣠⡄
⠀⢀⣄⠀⠀⢴⣦⠀⠀⠀⠘⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠢⣄⡀⠀⠀⣀⡴⠋⠘⣇⠀⠈⠀⠠⠞⠉⠀⠀⠀⠀⢀⡀
⠀⠈⠙⠛⠶⢤⡈⠑⠀⠀⠀⠀⠳⡌⠓⠦⣄⣀⠀⢀⣀⠜⠁⠀⠀⠈⠉⠙⠉⠁⠀⠀⠀⠈⠲⣄⣀⣀⣀⣀⣠⠤⠖⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣆⠀⠀⣨⠭⢭⠁⠀⣠⠤⠤⣤⡤⠖⠚⠛⠓⠲⢤⡀⢀⣀⣀⠉⠁⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠸⡄⢠⡇⠀⠈⢧⠀⡏⠀⢀⡇⠀⢀⣠⣤⣤⡀⠀⠈⣿⠀⠉⡇⠀⢀⡤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡒⠒⢷⠀⠀⢳⡼⠀⠀⠀⠘⣾⠇⠀⢸⠀⢠⠏⠀⠀⠈⢻⡄⠀⡟⠀⠀⡇⡰⠋⠀⠀⣧⣄⣠⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠘⣇⠀⠘⠃⠀⣰⡆⠀⠘⠀⠀⡟⠀⡏⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠟⠁⠀⠀⠀⡇⡸⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⢹⡄⠀⠀⢀⡏⢻⡄⠀⠀⢠⡇⠀⡇⠀⠀⠀⠀⢠⠇⢰⡇⠀⠀⠀⣠⣶⠀⠀⡿⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⢴⠶⠭⠥⠤⣤⣀⣀⣧⠀⠀⣸⠁⠀⢷⡀⠀⢸⠀⠀⠹⣄⠀⢀⡠⠋⠀⢸⠀⠀⢀⡴⠃⣿⠀⠀⠀⢀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡤⠇⠀⠀⠈⠷⠶⠛⢷⣄⠀⠈⠉⠉⠀⠀⣠⢾⣀⣠⠞⠀⠀⣿⠀⠀⢠⠞⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⠻⠶⠶⠶⠖⠋⠁⠈⠉⠁⠀⠀⢀⣿⣀⣠⠛⠒⠒⠒⠒⠺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢠⠏⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠞⠉⠀⠀⠀⠉⠓⠦⣀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠈⠳⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⠀⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠱⣄⠀⠀⢢⠀⢀⡴⠻⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡶⠦⣤⣀⡘⠦⠀⣈⡷⠋⠀⠀⠙⢿⣄⣠⡏⠀⠀⣠⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡄⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠾⠷⠖⠚⡿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣃⣤⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡤⠶⠋⣿⡆⢀⡄⢀⣾⡿⠀⢀⡀⠀⠀⣤⣤⡄⠀⠀⠀⠀⣀⣿⡞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣿⣧⣾⣧⣾⡿⢡⣾⡿⣿⣷⡀⢸⣿⡇⠀⠀⠀⣰⣿⠛⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⢻⣿⣿⠿⣿⡿⠁⠸⣿⣷⣿⡿⠃⠈⣿⣇⣴⡆⣴⣿⠃⠀⠈⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⢀⣾⠏⠉⠀⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠿⠿⠻⣿⣿⠃⣀⣤⣾⣟⣫⡉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⠁⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣹⠛⠁
⠀⠀⠀⠀⠀⠐⠚⠉⠁⠚⠋⠉⠉⠉⠉⠉⣩⡛⢷⣦⡀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠻⠛⠳⣤⡀⠀⠀⠀⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣴⠏⣰⠗⠙⣷⣠⠞⣋⠭⠡⣶⠈⠹⣍⠃⣍⣉⠀⠀⠉⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠃⠀⠀⢸⢣⣿⠇⠐⠳⠸⡇⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠻⡀⠀⠀⠀⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠂⣘⢸⣶⠄⠀⠀⠀⢀⡄⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠷⣶⡶⠂⠈⠁⠀⡔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢴⣿⡏⣻⢻⣾⠃⠀⠀⠀⠀⠘⠛⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⣤⣴⣍⡿⢳⠼⢼⣧⢠⢀⣀⠀⠀⠀⠀⠰⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⢻⠋⠁⠈⠹⠀⠀⠀⠀⠈⠉⣻⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠛⣂⡄⡀⠀⢀⣄⣀⠀⠀⠀⠀⣴⢿⡆⠀⠀⢀⣠⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠷⣾⡉⠁⢀⣀⠤⣅⠀⠀⠁⠀⠹⣇⣴⠆⠈⢗⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠘⣿⣀⡞⢰⠀⠘⠓⠦⣄⣀⣰⠌⠙⠶⣄⣼⣇⢈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⣧⣀⡷⢋⣿⣗⢼⠀⢰⣂⣠⣿⣿⡁⠀⠀⠀⢻⣶⠝⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⢿⡇⡞⠻⡏⠀⠀⠘⣋⠀⣀⣽⣆⠀⠀⠀⠸⡄⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢲⣾⣞⣻⢾⠉⢀⠀⠰⣍⣯⠥⠭⠝⠂⠀⠀⠀⡃⠀⠀⠀⢀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠙⠈⠉⠁⠀⠘⠀⠀⢃⣴⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⢀⡿⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⢰⣆⠀⢰⣿⢸⣀⠀⠀⠀⡀⡐⠈⡁⠙⠀⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡟⢀⠀⠀⠀⠀⡈⠀⠸⠁⡬⠉⠀⠀⠀⠋⢈⣇⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠈⢉⠄⠀⠀⣀⣿⠀⠀⢀⡇⠤⠄⠀⠀⠀⢻⣷⠛⠛⠋⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠤⢤⣈⣹⠁⠟⠃⠀⠀⠀⢀⡅⠀⠸⠛⢣⣀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⡿⠆⠀⠀⠀⠀⠻⢷⣾⣇⠀⢈⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠕⠁⠀⠀⠠⡌⠀⠀⠀⠀⠙⢿⣶⡦⠐⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠰⡿⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢰⣟⡀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠙⢿⣷⡿⠙⠇⠀⠀⠀⠀⠀⠀⠀⢸⠇⢐⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠙⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣿⣿⡿⠿⠋⠉⠉⠙⠛⠉⠁⢠⠀⢀⡀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⡴⠀⠀⠀⠲⢄⠀⡀⡀⠒⠦⣤⣄⠈⢰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⡄⠀⠀⠀⠀⠘⡿⣯⣛⢳⣄⠈⠙⠻⣄⣿⣿⣿⣷⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠙⢲⡝⣷⡀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⠼⣿⡀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣰⢇⡴⠃⠀⠀⠀⢻⣆⣹⡇⠀⠀⠀⠻⣿⡿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⡇⠀⠀⠀⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⢼⣿⣿⠇⠀⠀⣀⣀⣜⣃⠹⡇⠀⢀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣆⢹⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠾⠟⠻⠿⠿⠛⢻⡿⢿⣿⣿⣿⣿⡿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣯⣤⣤⣤⠀⠀⠀⠀⠀⠀⣾⠁⣾⡟⠉⠈⠀⠀⠠⠁⢀⣠⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣴⡟⢱⡟⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⢐⣿⢘⣿⠃⠀⠀⠀⠠⣿⠟⠛⠫⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠺⣿⠀⠀⠀⠀⠀⢸⡷⢊⣿⠀⠀⠀⠀⢸⡟⠀⠀⠀⠱⠀⠰⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡀⠀⠀⠀⠀⠸⣿⢈⣿⠀⠀⠀⠀⠸⣷⠀⠀⠀⣼⣿⣿⣿⣿⡋⠉⠀⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⠀⢀⢹⣧⠀⠀⠀⠀⠀⠿⣎⠻⣆⠀⠀⠀⠀⢿⣆⠀⠰⠿⠿⢿⣿⣿⣿⣿⣶⣦⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣧⣼⣶⣿⣿⣷⣶⣶⣄⣀⣐⣻⣷⣿⣶⣶⣦⣄⠘⣿⣿⣶⣶⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣛⣙⡛⠛⠛⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡿⠿⡿⠌⠙⠛⠛⠋⠙⣭⠋⡍⠻⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢴⡞⣧⢺⡛⣿⣿⣷⣄⡀⠀⠀⠈⢫⡈⣷⡀⣠⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⢋⢷⣛⡙⠋⠉⠉⠉⠁⠀⠀⠀⠀⢳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⠴⣭⠆⠀⠀⠀⣀⣠⣴⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣴⣿⣿⠟⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢨⠀⢀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠐⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠉⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢳⡀⢀⣴⠏⠘⣿⣿⡇⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠲⢦⣤⣀⣈⠻⠆⠀⣈⡵⠋⠁⠀⠀⠈⢿⣷⣄⣰⡏⠀⠀⢀⣴⠂⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣿⣷⠶⠶⢛⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⠃⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⣴⣿⣧⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢼⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣛⠋⠁⢿⣿⠀⢠⡇⢀⣾⣿⠏⢀⣀⣤⣄⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣤⣼⣿⡸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣸⣿⣧⣿⣿⣾⣿⠏⢠⣿⡿⠿⣿⣿⡄⠘⣿⣿⡇⠀⠀⠀⠀⣼⣿⠏⢻⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠓⢻⣿⣿⣿⢿⣿⣿⡏⠀⢸⣿⣷⣶⣿⣿⠇⠀⢻⣿⡇⣠⣶⢀⣾⣿⠃⠀⠀⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣸⣿⠛⠛⠀⠛⠛⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⢸⣿⣿⣿⣿⣾⣿⠇⠀⢀⣀⣼⣿⠦⠿⠛⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠛⠋⢠⣾⡿⣿⣷⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣿⣿⣋⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣰⠋⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠊⠉⠭⠉⠀⠀⠀⠀⠀⢨⣍⠛⢿⣦⡀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠈⡛⠉⠙⢷⣄⡀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠟⢁⣾⠃⠹⣷⣀⣴⠿⢛⡉⠭⢥⣤⠉⠩⣟⡛⢂⣋⣙⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠁⢀⡾⠁⠀⠀⢹⡿⣡⣾⠏⠀⣤⡈⣿⡀⠀⠙⢯⠈⠉⠛⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠱⣿⡿⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⠉⠉⠐⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣡⠄⠀⢀⠜⠁⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠃⠀⠀⢀⠔⠂⠁⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠋⢀⠏⠀⠘⠃⠉⠀⠄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⡠⠂⠀⠀⠰⠋⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠄⢀⡤⠐⠒⠂⠸⠀⠀⠀⠊⡠⠀⠀⠰⠄⠄⢀⠸⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⡠⠁⢠⠃⠀⠀⠀⠀⢠⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠊⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢀⡔⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠘⠀⠴⠃⢀⡄⠀⢀⡔⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡦⠴⠃⠀⡫⠀⠀⢀⠂⠠⠠⠖⠉⠀⠛⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢊⢁⠔⠁⡠⠃⠀⠈⠈⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢀⠠⠐⠂⠀⠀⠀⢄⠀⠀⠀⢠⠂⠀⠈⡀⠔⠊⠀⢠⠤⠄⠐⠪⠁
⠀⠀⠀⢀⡠⡄⠄⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠸⠁⠌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⢀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠂⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡟
⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠂⠀⢀⠔⠀⠀⠀⠀⡠⠊⠀⡾
⠀⠀⠀⢠⡀⠀⠎⠀⡞⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠂⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠁⣀⠞⠀⢀⡾⠁
⠀⠀⠀⠀⡀⢠⡀⡜⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠰⠀⠀⠇⢀⠞⠃⠀⠀⢠⠊⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠂⠀⠀⣄⠀⠈⢀
⠀⠀⠀⠀⠀⠁⢀⠔⠁⠀⠀⠀⠀⢸⠀⠀⠀⢘⠀⠀⢀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⢸⠁⣀⡀⠀⠀⠀⠀⠀⢻⡀⢀⠀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀⣤⡆⠀⠀⠘⠀⠀⠀⢆⡰⠀⠁⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠚⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⡀⠀⠀⡇⠀⠀⠀⠀⡗⡀⠀⠀⠀⢸⡄⠀⠀⠀⢂
⠀⠀⠀⢀⠀⠀⠸⠀⡄⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⡃⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⡈⡅⡠⠃⡔⠁⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠰⣤⠤⠤⠶⠋⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⡰⠁⠉⠴⠉⡀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣼⠏
⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⢂⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠡⡀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⣿⠿⠿⢿⣿⡿⠟⠋⠃⠀⠉⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠙⣛⠉⠁⠠⡀⠀⠀⠀⣠⣀⠀⠐⠠⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⡁⠀⠀⠀⠈⢷⣖⠦⣄⡈⠛⠷⣄⣿⣿⣿⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⢠⠀⢰⠈⠻⢤⡻⣄⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠸⠀⠀⠘⡧⢹⣇⠀⠀⠀⣀⣤⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣼⣠⠃⠀⠀⠀⢻⣈⣿⠀⠀⠀⠹⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⠀⠀⠀⠀⣸⠂⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⣿⣿⡏⠀⣠⣤⣶⣾⣦⣟⣀⣤⣴⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡋⢉⣁⠈⠀⠀⠉⠀⣼⠃⣽⠟⠛⠛⠉⠀⠀⠀⢈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⡿⢠⡿⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹⠀⠀⠀⠀⠀⣿⠕⡿⠁⠀⠀⠀⣼⠿⠛⠫⠀⠈⢁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠺⣯⠀⠀⠀⠀⠠⣷⢪⡇⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢺⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⣿⡀⠀⠀⠀⠀⣿⢂⣟⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣏⣁⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢄⡀⣸⣧⣀⡀⠀⠀⠝⣧⡙⣄⡀⠀⠀⢸⣧⣄⣘⠛⠛⠛⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣷⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢉⣭⣽⣿⡟⠿⢿⡿⠿⢿⠿⠿⢿⣿⣿⣶⣶⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⡼⡏⠽⢽⣥⣤⣄⠀⠀⠈⠳⡊⢢⠈⣙⣭⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢈⠓⣿⢚⠞⠛⠛⠛⠃⠀⠀⠀⠹⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⢬⣭⠀⠀⠀⢀⣠⣴⣦⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⢠⣾⣿⣟⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢄⠀⣀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⡈⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡦⠻⣿⣿⠂⠀⠀⠀⠀⢦⡑⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⢻⣿⣦⣖⣮⠁⠀⠀⠀⠀⠀⢨⡟⠀⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⣿⣿⡿⠛⠀⠀⠀⠀⠦⠀⣶⣿⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣷⠀⢀⣾⠿⣍⢔⣲⣒⣀⣀⣀⡘⠻⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢿⣷⣼⡇⠀⣈⠉⠛⠛⠛⢛⠿⠿⣦⢙⢀⣾⣿⣷⢲⡎
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠙⢿⣥⠶⣚⠀⠀⠀⠀⠔⣲⢶⣼⡿⠿⠛⠁⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡀⠀⢀⣸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠷⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠷⣷⣿⣿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⠀⠈⢉⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⢠⣤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⠸⠿⠟⠀⠰⡿⠿⠶⠂⠀⠀⣿⠿⠟⠿⣿⣿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⡦⠀⣀⣾⣄⠀⠀⡀⠀⠀⢀⣼⡷⣤⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡏⠀⠀⠀⠤⠖⠋⠉⣷⢄⣠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⠀⠀⠀⠘⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠒⠦⢄⣀⣀⡀⢀⣀
⠀⠀⠀⠀⡠⠄⠠⠴⠲⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠔⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⣒⠒⠒⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠂⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡰⠋⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠂⠲⠃⢠⠎⡄⢀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⢀⠊⠀⡠⠄⠀⢀⠖⠀⠀⠀⢀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠀⡄⢀⡠⠄⠀⠠⠅⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠀⠀⠀⠂⠀⠀⠀⠀⡰⠀⠀⠀⠐⠁⠔⠁⣠⠀⠀⠄⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇⠒⠀⠪⠁⠀⠠⠂⠀⠰⠈⠀⠘⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⢀⠔⠁⠀⠁⡁⡤
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠐⠁⠐⠀⠁⠀⠀⠀⠀⡀⠀⠰⠁⠀⠤⠂⠁⠀⠐⠀⠀⠁
⠀⠀⠈⡴⠐⡉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠜⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢀⢴⠄
⠀⠀⠀⠀⠀⠀⡔⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⢀⠔⠈⠀⠁⠀⠀⡠⠂⠀⠂⠀⠀⠤⠀⠀⠀⢀⠔⢁⡞
⠀⠀⠀⡄⠀⠈⠀⠎⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠊⠀⠀⢀⡔⠁⢀⡾
⠀⠀⠀⠀⠀⡅⡌⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⢃⠽⠉⠀⠀⠰⠁⠀⠀⡠⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢰⠁⠀⢲
⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⣇⢠⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠛⠃⠀⠀⢁⠀⠀⠰⠂⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⢹⡆⠀⡀⠀⠀⢸⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠂⠀⠘⠀⠀⠈⠀⠀⠁⠀⠀⢸⡄⠀⠀⠅
⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠸⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠠⢁⡠⢁⠜⠀⠀⠀⢠⠟⠀⠀⠀⠀⠈⠦⠖⠒⠚⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠋
⠀⠀⠆⠀⠈⠠⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠁⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⢀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢶⣿⣿⠃⠙⢿⣿⠋⠀⠀⠀⠀⠀⠈⢳⣄⠑⠢⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠐⠻⣿⣿⣰⣯⣷⠌⠀⠀⠀⠀⠀⠀⠀⢬⣟⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⣿⣿⡿⠛⠃⠀⠀⠀⠈⠦⡀⠀⣶⣿⠆⠀⠀⠐⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⡆⠀⠀⢨⣿⣿⠳⡠⢤⣤⡤⠀⠀⠀⠠⠾⣿⡿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⢻⣿⣆⠀⣿⠋⠈⠓⠬⣵⣶⣶⣶⣶⣿⣶⣦⡈⢷⠀⢀⣴⣤⣄⡀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠻⣿⣷⡿⠐⠒⠦⠀⠀⠀⠀⠀⠔⠛⠛⠛⢷⣬⣄⣿⣿⡿⡿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣧⡾⠛⣭⠂⠀⠀⠀⠀⠚⢩⠟⢷⣼⡿⠿⠛⠁⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⢸⡇⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢼⡀⠀⠀⠀⠀⣀⠀⠀⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⣷⣧⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⢶⣴⣶⣷⣿⣿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⣄⠈⠻⠿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠙⠙⠉⠉⠉⣉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠁⠀⡀⢀⣦⣿⡄⠀⠀⣠⣀⣤⠀⠀⠀⠀⠘⣿⣤⣼⣴⣴⣦⣸⡿⢋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠛⠛⠘⠀⠘⠿⠛⠛⠛⠁⠀⠀⢀⡟⠛⠛⠛⠻⢿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣿⡶⠀⢀⣰⣿⡄⠀⠀⢉⡁⠀⠀⠀⣠⣿⣷⣤⡀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⡴⠞⠋⠹⣧⡀⠀⣀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠇⠀⠀⠀⠈⢡⠀⠀⠀⠀⢻⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠉⠓⠲⢤⠤⢤⠀⠠⠤⢀⣀⣀⡀
⠀⠁⠀⠀⢀⡠⠄⠤⠤⢶⠲⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠂⠀⠀⠀⠀⠀⠀⠍⢳⡄
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠁⠀⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠐⣂⡀⠤⠀⠀⣀⣀⣀⠀⠀⠤⠄⠐⡂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠒⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠄⠂⠉⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠤⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⢀⠎⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠁⠀⡠⠰⡖⠀⣠⠪⠀⠀⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠂⠀⠀⡔⠁⢀⠀⠀⠀⡠⠂⠀⠀⠀⢀⠤⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⡈⢀⠃⠀⠄⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠸⠀⢠⠃⠀⢀⡀⠀⡀⠠⠄⢰⠁⠀⠀⠄⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠠⠊⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠚⢀⠔⠀⡀⠀⠀⡠⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢄⠌⠀⣒⠀⠀⠰⠀⠀⠤⠐⠁⠲⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠔⢃⠤⠊⡠⠁⠀⠂⠀⣀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡠⠃⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⡔⠀⠈⡁⠔⠊⠀⠤⠠⠀⢂⠃
⠀⠀⠈⣀⠤⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠋⠀⣑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⢀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠁⢀⠔⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠊⡼
⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⣠⠊⠀⢀⠀⢠⠊⠀⡼⠁
⠀⠀⠀⠑⠀⠌⢠⠃⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣼⠃
⠀⠀⠀⠀⡀⠃⡤⠠⡤⠄⠀⠀⡠⠊⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡁⠀⠀⢰⠈⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠀⠈⣧⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⣀⣤⠀⠀⠘⠀⠀⠈⢄⠌⠀⢠⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢸⡄⠋⠀⠁⠀⠀⣿⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠟⠀⢰⠀⠀⠀⢸⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⢇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠸⡼⠀⠀⠁⢀⡀⠀⠸⠁⠀⠀⠀⠸
⠀⠀⠀⡤⠀⠠⠁⠆⠀⠀⠀⢀⡾⠃⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠰⠰⠔⡡⠊⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠃⠀⠈⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠂⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠊⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢳⠤⣄⣀⣀⣤⠞⠀⠳⢤⣀⣀⣠⠴⢛⡇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠙⠢⠀⢸⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣄⠀⠀⢰⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢠⡄⠀⠀⢸⡇⠀⡏⠈⢳⠀⠹⣆⣶⣬⡇⠀⢠⡟
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⡆⠻⡄⣶⣼⡇⠀⢷⣀⣼⢀⠀⠹⠇⠙⠃⠁⠸⡇⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠀⠖⠂⠀⠀⡇⠀⠹⡏⠻⠃⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀
⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⡀⠀⠈⠳⣄⠀⠀⢀⣠⡤⠴⠦⢤⣰⠃⠀⠰⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠎⠀⠀⢸⣆⡴⠋⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠒⠒⠒⠒⠒⠀⠀⠄⡀⠀⠀⠀⠀⡠⢄⣀⡉⠐⠈⠀⠤⣤⣍⠙⠀⠀⠑⢄
⠀⠀⢀⠀⠒⠉⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠀⢀⠴⠛⠁⡀⠈⢙⡁⠀⠒⠶⠌⢻⡀⠀⣀⠀⠀⠀⠀⡀
⠀⠐⠁⠘⣿⡀⣄⢻⠀⣴⣦⢸⡇⡄⣼⢀⣿⠃⠀⠀⠀⢠⠤⡂⠚⠀⠀⠈⢿⣶⣦⠤⠈⠂⠹⣦⡘⢿⣶⣶⡀⠈⢄
⠀⠀⠀⠀⠘⢷⣿⣾⢸⡇⢸⡏⣿⣿⠇⡼⠃⠀⠀⠀⢀⡊⢰⣿⠶⢔⢦⠶⡆⢻⡛⠁⠀⠇⢀⡹⣿⡛⢊⠻⠓⣤⣈
⠀⠀⠀⠀⠀⠈⠿⠹⠀⠛⠛⠀⠉⠘⢠⠁⠀⠀⢀⣴⡶⠀⠈⠁⠀⠀⠳⣤⣄⣤⢙⡤⣶⣷⣼⣟⣛⠛⠦⣑⢤⣚⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠠⠀⡆⠺⣻⣿⡀⠀⠀⠐⠁⠀⢀⣠⠴⠶⠶⠤⠙⣝⢯⡉⠉⣛⣻⣶⣬⡛⣯⣻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡵⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠳⠯⠛⠉⠋⠑⣦⣙⣎⠛⠷⠿⠁⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏⠘⢿⡹⣿⣴⣆⠐⠀⢸⡆⢍⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡥⠄⡀⠀⠀⠀⠀⠂⣉⠉⠻⠆⠀⠀⠀⠀⠘⠘⢸⡇⠘⣿⣍⣁⡠⢞⣣⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡿⠥⠴⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠖⠆⣼⡃⡀⣿⣧⣄⡻⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣄⣄⣼⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⢳⢧⣧⡿⠋⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⠘⣽⣿⣦⠀⠀⠐⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⣇⢠⠟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠾⢣⠶⢿⠛⠹⣷⣦⡀⢶⣀⣠⡤⡾⢩⠀⠀⠀⠀⠀⣴⠃⣸⣟⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣽⣴⠉⢀⣼⡦⣳⣼⣿⣿⣦⠹⣞⡇⣀⠘⠀⠀⠀⢀⡾⠁⢈⣿⣿⠘⠂⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠋⡿⣷⡄⠉⢠⣜⡊⣙⣼⢟⣷⡄⠬⠥⠚⠀⠀⡐⠛⠁⠀⠈⠛⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠓⠀⠀⢬⠿⠞⠉⠞⠹⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡓⣄⣀⣈⣴⣦⡒⠼⢻⡄⡀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⡿⠋⠓⠋⠑⢈⣭⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠘⣇⡀⣿⣿⢸⡇⢶⣀⣀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⠄⢀⠠⠀⠀⣿⣿⢸⡿⠋⠁⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣳⠦⠡⠲⠠⠉⠀⡩⠀⢼⣿⠜⠃⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠋⠀⡀⡄⠀⢄⠀⠘⠀⡹⠘⠃⠀⢠⡠⠂⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠤⠀⢀⠐⠄⠀⠐⠄⢀⡀⠵⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠢⣈⠀⠀⠀⠐⣀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠉⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣀⠀⠐⠢⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠈⠉⢢⡠⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠒⠐⠢⣠⠊⣠⢀⣈
⠀⠀⠀⠀⠀⠄⠀⠂⠈⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⢀⠴⠊⢉⠭⠍⠐⠀⠀⠀⢀⠤⠤⠤⠤⠄⠤⠔⢂⠉⠩⢁⡒⠤⠤⠤⠤⠤⣀⠀⠀⠀⠢⠤⢄⠠⠒⠂⠀⠐⠲⢄
⠀⠀⠀⠀⢀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⢹⡷⠊⠀⠀⠀⠀⠈⠻⡀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡠⠜⢀⡠⢚⣍⠢⣄⠀⠀⠀⠀⡇⠀⠀⠀⢀⠊⠀⠀⣀⣤⣤⣄⠀⠀⠘⡀⠀⢸⢸⠀⠀⠀⠀⣀⠔⣊⣑⠦⡀⠀⠀⢀⠆
⠀⠀⢸⠀⢐⢡⠒⠁⠘⣿⣆⢃⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⣴⣿⠟⠉⠙⠆⠀⠀⣷⠀⠈⢸⠀⠀⠀⢰⢡⣾⠏⠀⠑⡌⠓⢐⠪⡐
⠀⢀⡬⠓⠈⢦⢣⠀⠀⠸⣿⡌⢒⣐⠢⠇⡀⠀⢠⠀⠀⠀⣿⠁⠀⠀⢠⠂⠀⠀⣿⠀⠀⠘⠤⢒⡐⢃⣿⡟⠀⠀⡰⣱⣿⠟⠑⡘⢆
⠀⡞⠀⠀⠀⠀⢣⠣⡀⠀⢹⣿⠃⠀⠑⢦⡅⠀⠘⠀⠀⠀⠙⠄⠀⠔⠁⠀⠀⢀⣿⠀⠀⣦⠊⠀⠈⣾⡟⠀⠀⡐⣴⣿⠋⠀⢀⢊⠜
⠀⡇⠀⠀⠀⠀⠀⢣⡱⡀⠀⠻⠀⠀⡀⠈⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠁⠀⡀⠀⠸⠁⠀⢄⣼⡿⠁⢀⠔⡱⠋⡄
⠀⠙⢤⣀⣀⠀⠀⠀⠱⡑⡀⠀⠀⢸⣷⡄⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⢀⣾⡇⠀⠀⠀⢌⣾⡟⠁⡠⣡⠎⢀⠜
⠀⠀⠀⠀⠈⢣⠤⠀⠀⠱⡐⢄⣀⣸⣿⡿⠲⣶⣾⣿⡿⠻⣷⣶⣶⣾⣿⠟⢻⣿⣿⣶⠶⢻⣿⣧⣀⣀⣈⣹⠟⠟⢋⠜⠁⠀⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⠪⣙⠛⣛⣃⠞⠢⠭⠤⠒⠒⠤⢭⣭⠥⠤⠒⠒⠠⠭⠥⠺⣜⣛⡻⢛⡁⣿⡷⠤⣂⠜⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠒⠒⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠃⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⢑⡦⠤⠤⠀⢱⡀⠀⠀⠀⠀⠀⠒⠒⠄⠀⠀⠀⠀⡐⠤⠤⠂⠀⠑⠲⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠠⠆⠀⠀⠀⠀⠙⠢⠄⣀⣀⠠⠄⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢏⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢸⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡇⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣿⣦⣄⣘⠀⣂⣀⣴⣿⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣆⣼⠿⠷⠺⣦⣿⡇⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⢀⣀⣀⣀⡀⢀⣤⠖⠛⠋⠛⠺⢿⣧⠀⠘⣿⣿⠀⠀⠀⢹⣿⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣘⣿⣤⣿⣟⠉⠉⠉⣷⠟⠀⠀⣀⣤⣄⡀⠀⠙⣦⠀⣿⡿⠀⢀⡀⠀⢿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠿⣷⣾⡟⠉⠉⠙⣿⣿⠀⠀⠀⣿⠀⢀⡼⠛⠻⣿⣿⡄⠀⠸⡇⢸⡇⠀⣸⣷⡀⠈⠇⠀⢸⡧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⢾⣿⣯⠀⠀⠸⣿⡇⠀⠀⠀⠘⢿⡆⠀⠀⣿⠀⣸⠃⠀⠀⠈⢿⣿⠀⠀⣿⠀⠁⢀⡿⣿⣧⡀⠀⠀⢸⣄⡀
⠀⠀⢈⡏⠀⣣⠀⠈⢿⣿⣆⠀⠀⢻⡇⠀⠀⡀⠀⠈⠇⠀⠀⢹⡂⢿⠀⠀⠀⠀⢸⣿⠀⢀⣿⡀⠀⢸⠇⠹⣿⣧⠀⠀⣼⠿⠿⠛⠓⠒⠂
⠀⠀⠈⠉⠙⠃⠀⠀⠈⢿⣿⡆⠀⠘⠇⠀⢠⣷⡄⠀⠀⠀⠀⢸⡇⠘⣧⠀⠀⠀⣼⠟⠀⣼⣿⣇⣀⡿⠀⠀⣹⣿⣷⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡄⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⢸⣿⣄⠈⠛⠲⠚⠋⣠⡾⠁⠻⠛⢟⠁⠀⠀⢻⡉⠉⠀⠀⠀⠀⡤⠤⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠙⣿⣷⡀⠀⠀⢸⡁⠻⣿⣧⡀⠀⢈⡿⣿⣿⣶⣶⣶⠿⠋⠀⠀⠀⠙⢂⠳⣶⡶⠶⠿⠦⠀⠀⠀⠙⢷⡼⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⣿⣷⠀⡀⣾⣷⠿⠿⣿⣷⣾⣿⡁⠀⠉⠉⠉⠀⢀⣀⣀⡀⠀⣉⢉⢣⣈⠫⣄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠹⡿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠐⠣⠬⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠑⠟⠀⠀⠀⣿⣦⡀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⣿⠙⢿⣶⣄⣈⣀⣀⣴⣇⠀⠀⢀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠈⠙⠛⠛⠋⠉⠛⠿⠟⠛⣿⣏⣀⣠⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣶⡿⠛⠁⠀⠀⠀⢀⣠⠤⠤⢤⣶⡛⠹⣿⡏⠉⠹⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢀⣸⣧⣾⡟⠉⠉⣷⠋⠀⣠⣤⡀⠈⢳⡀⣿⠇⢀⠀⢹⡇⠀⡇
⠀⠀⠀⠀⠀⢠⣴⠾⠛⣿⡏⠁⠀⠻⣿⠀⠀⣿⢀⡞⠉⠛⣿⡆⠀⣷⠹⠀⣼⣧⠀⠃⠀⡗⠂
⠀⠀⡖⠒⡀⠘⣿⣧⠀⠸⡇⠀⠀⠀⠹⡄⠀⢿⢸⡇⠀⠀⢸⣿⠀⣿⠀⢠⡏⢿⣧⠀⢀⣷⡶⠤⠤
⠀⠀⠉⠚⠀⠀⠸⣿⣆⠀⠃⠀⢸⣆⠀⠀⠀⢸⡈⢧⡀⠀⣸⠇⣰⣿⣦⣼⠀⢈⣿⣧⡾⠃
⠀⠀⠀⢀⣠⣴⣾⢿⣿⡆⠀⠀⣿⢿⣧⡀⠀⢸⣷⣤⣉⣉⣡⡾⠁⠉⣉⢅⣀⣀⣧⡀⠀⠀⢠⣂⣆
⠀⠀⠀⠀⠀⠀⢀⡠⢻⣿⡄⠀⣿⣴⢿⣷⣤⣞⠉⠛⠛⠛⠁⠀⡀⠀⡒⢒⡙⢆⡈⠉⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠋⢠⠞⠻⠿⠟⠉⠀⠀⠉⠀⠙⣆⠀⢀⡴⠋⢉⡉⠓⠇⠀⠙⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⢹⢠⠏⢠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⣀⣠⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠢⠀⢀⣄⣤
⠀⠀⠀⠀⠀⡖⠀⠒⠠⠀⠀⠀⠀⠀⠇⠀⢠⠀⠀⠈⠻⠟⠀⠀⠀⣀⠠⠄⠐⡄⠀⢀⢠⠤⠀⠀⢀⠀⠀⠀⠀⠑⡄⢑⠀⠀⢀⢀⢴
⠀⠀⠀⠀⠀⠡⠀⠀⢂⢣⠀⠀⠀⢠⠀⠀⢸⠰⠀⠀⠀⠀⠀⠀⡊⢸⠀⠀⠀⡇⠀⡇⢠⠀⠀⠀⢸⠀⠀⠀⢠⠞⠒⠊⠀⠀⢈⣁⠷
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠆⢳⠱⡑⡼⠀⠀⢸⠀⡤⠀⠈⠁⠀⠢⢇⢸⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⡌⠀⠀⠠⣢
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠆⠀⠐⠆⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠅⡜⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⢀⠄⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⢠⡏⠀⠈⠳⠀⠀⢠⠀⢸⡌⢡⡄⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡔⠁⠘
⠀⠀⠀⢈⠀⢸⡄⠀⠑⡀⠀⠀⢸⡄⠀⠀⢸⠀⠃⠰⡘⢄⣀⡔⠀⠀⢸⠀⠘⠁⠸⠇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⣀⠤
⠀⠀⠀⠈⡄⢸⡇⠀⠀⠀⠀⠀⡠⠉⣠⢄⡀⢤⠂⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠱⠎⠤⠊⠀⠀⠣⣀⠑⠄⠀⠀⢀⣀⡄⠸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⢠⠠⢲⠀⠀⠀⢁⠔⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠱⠀⠀⠀⠀⡀⠀⠈⠒⠊⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠄⠐⡄⠀⠀⠀⡇⠀⠂⢀⠞⢡⣀⣉⣀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢡⠇⠀⠀⠀⡃⠈⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⡡⠃⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠐⠉⠳⠄⠀⠀⠈⠀⠐⠊⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣠⡾⠃⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠰⠠⠴⠁⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠈⣻⢯⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠤⠊⠀⠀⢀⡜⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⢀⠈⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⡠⠊⢀⠂
⠀⠀⠀⡜⠀⡀⠀⠀⠀⠀⠈⠀⠀⡠⡊⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠠⠂⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠾⠊⠀⠀⠀⠀⠀⢁⡔⠴⠊⠀⠀⠀⠀⠀⡰⢻⠇⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⢀⠔⠀⠀⠀⣀⠜⡱
⠀⠀⠀⠀⣡⣀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⢀⠊⡦⢁⡾⠊⢉⠽⠀⠀⠀⠀⠀⠐⠠⠔⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠊⠑⠊
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⡠⢄⢀⠎⠸⠞⣸⣷⠿⠒⠋⢁⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⢸⠘⢀⢇⣡⣿⠷⠶⠶⠖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠀⡇⡘⣃⣮⣾⡏⠀⠀⠀⡀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠐⣦⠀⠀⠀⠀⠱⠀⡇⣷⣿⡿⠛⠡⡀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠠⠔⠂⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣱⡿⠿⢁⠀⠀⠀⠈⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐⠒⠚⠛⠛⠚⠙⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠀⠀⢂⠁⠀⠠⠃⠀⠀⠀⠀⠀⠐⢢⢠⡄⠀⣔⠀⣰⡆⢀⣖⠀⣰⠀⢀⡂⠀⣐⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠉⠉⠉⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢠⡄⠀⢀⠘⡀⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠫⠄⠀⠀⠀⠀⠈⠀⡐⢄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⢠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⢀⠂⠀⠢⡠⠃⠀⠀⠀⠘⡀⠴⠀⢀⠄⠊⡐⠀⢀⠶⠉⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠰⡐⠂⠠⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⢀⠁⠀⠁⠀⠀⠀⠀⠠⠗⠊
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠐⡄⠀⠀⢀⠀⠀⠀⠀⡀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠤
⠀⠀⠀⠀⠀⠀⠤⡐⠀⠈⠁⠀⠀⠀⢸⡇⣰⣆⢸⡇⣾⠋⢻⡄⣿⣠⡄⢰⡟⠀⠀⠀⠁⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠁⠢⢄⠀⠀⠀⠸⣷⡟⢻⣿⠛⣿⣀⣿⢱⣿⠟⣿⣾⠁⠀⠀⡀⠀⠠⠄⠒⠀⠉
⠀⢀⢀⡀⣀⣀⡈⠀⠠⠂⠁⠀⡀⠀⠀⠿⠁⠀⠛⠀⠉⠋⠁⠘⠋⠀⠹⠃⠀⠀⠀⣐⡀⠀⠀⠀⡀
⠀⠀⠉⠈⠀⠀⢀⠀⠀⠀⠀⠴⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠅⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠝⠂⠈⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠤⠀⠀⡠⠀⠀⡠⢀⡔⢄⠀⠀⠀⠆⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⡐⢀⠤⠊⠀⠀⠄⠊⠀⡅⠠⠁⠈⠀⠀⠑⢄⠀⢀⠀⠀⠀⠉⠒⠤
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⢀⠀⠀⠢⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⣔⠀⠀⠀⠀⡄⠀⠀⠀⠈⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⢀⣴⣤⣿⣯⡀⣀⣀⡀⠀⡿⠀⢠⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡟⣿⣟⣉⣙⡿⠋⠀⠀⠀⠀⣼⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠠⣺⣿⣿⢸⡏⣿⡟⠀⠀⠀⠀⠀⠀⠁⠉⢁⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⢦⡚⢧⡜⠙⠖⠃⠿⠶⢲⣠⣤⡀⠀⠀⠀⠀⠃⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⡿⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠛⢉⣠⢀⡀⠀⢀⣤⣀⡀⠀⠀⠀⢀⡼⢻⡇⠀⠀⠀⣀⣤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠾⣧⣾⡉⠁⠀⣀⣠⣹⠀⠀⠈⠀⠀⠹⣇⣠⡆⠀⠹⡃⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠹⣧⢀⣾⠱⡆⠈⠟⠢⣄⡀⠀⢰⡍⠛⢷⣄⣠⣿⣀⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⢋⣷⡀⠀⣦⠟⣿⡿⢄⠃⠀⡴⠀⣰⣿⣿⠋⠀⠀⠀⢸⣯⣍⡏⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡉⢻⣺⠁⣼⣿⣿⠈⠁⠐⠿⠋⠉⠉⢻⡅⠀⠀⠀⠸⡏⠁⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⣄⣿⣰⣦⣤⠿⠀⠀⠀⡴⢳⠶⠺⠿⢛⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡻⡏⠳⠦⠽⠀⢰⠆⠀⢸⠊⡟⠉⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⡁⣘⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⢀⡀⠀⠀⣨⠞⡇⠀⠀⠀⠀⠀⠀⠀⠾⠦⣄⠀⣤⡫⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⡄⠀⠀⠀⠀⠸⠳⠀⠀⣿⠇⢳⡄⠀⠀⠀⣠⡤⣅⠠⠀⠀⠀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣉⣄⠀⠀⠀⠀⣀⠀⠀⠃⢠⠆⠀⠀⠀⠀⠁⢠⣾⣄⣀⢀⣠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡆⠀⠈⠠⠄⠀⠀⣀⣽⡄⢀⠀⡸⠖⠲⠒⠀⣀⠀⠊⡿⡟⠙⠋⠉⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠶⢤⣄⣸⡁⠸⠛⠀⠀⠀⠀⢀⡅⠀⠐⠛⠛⣤⣀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⣸⠶⠀⠀⠀⠀⠀⠻⣷⣾⣧⡀⠀⣹⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡯⠈⠁⠀⠀⣀⠋⠀⠀⠀⠈⠙⢿⣧⣤⡈⠒⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢶⣶⣄⣸⡷⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢄⡤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢹⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⢀⠀⠀⠒⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⠄⠂⢤⡄⠀⣀⡒⢶
⠀⠀⠀⣤⠀⠠⡆⢀⣀⣶⠀⢠⢀⡿⠂⠀⠀⡊⡁⠠⠀⠀⢦⣤⣈⡁⠌⢢⡀⢶⣄⣀
⠀⠀⠀⠘⣧⣷⡇⡏⢹⢻⣿⡏⡼⠁⠀⠀⢀⠃⣾⡦⠠⡤⠌⢻⡛⠀⠐⢀⣻⣶⠏⠻⠃⣄⡀
⠀⠀⠀⠀⠘⠏⠃⠙⠋⠈⠈⠠⠀⠀⣀⣴⠗⠀⠁⠀⠈⠲⠤⠤⠉⢶⣿⣾⣯⣛⣳⠕⢤⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⢿⣀⣀⠈⠁⠀⡴⠚⠋⠁⠀⠻⣕⣤⡬⢽⣟⡲⣝⣯⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠰⠠⣿⣾⣄⡍⠁⠘⣆⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠄⡀⠀⠀⠀⠔⡀⠒⠆⠀⠀⠀⠉⠐⣿⠈⣿⣟⣀⠰⣟⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡯⠰⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢆⣿⢀⣸⣯⡬⠛⢉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⢦⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢙⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣠⣿⠿⣦⣀⢀⡀⠀⣀⡴⠆⠀⠀⠀⢀⡼⢱⡞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣧⡜⢀⣼⢔⣼⣿⣷⡝⣿⡏⠁⠁⠀⠀⢠⡞⠐⣿⣗⠂⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⣩⢿⡄⠁⢴⣋⣩⡼⣿⣦⠨⠔⠀⠀⠐⠋⠀⠀⠙⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⡁⠀⠐⢟⡋⣀⣥⡁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⡶⢿⠻⠎⢝⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣇⠐⣿⢸⡇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⢀⡤⢀⠀⣿⣿⡿⠋⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠛⠐⠁⠀⢀⠔⠀⣿⠏⠁⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⡀⠀⠁⠀⠂⠀⠀⠃⡄⠀⠘⠂⡀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠄⡀⠀⠀⠀⠁⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣊⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠒⣒⡶⠴⠶⠒⠒⠒⠒⠒⠲⠶⠶⣶⣒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿⣿⣿⣿⣿⣿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠾⣭⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣤⢉⠭⠭⠭⠭⠭⠭⠭⢭⣍⣭⣭⣭⣭⣍⣩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⣭⢽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣤⣬⢷⣒⣒⣒⣒⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣿⣿⣿⣿
⢰⣿⡇⣿⡇⣾⡿⣰⣾⣿⣷⣸⣿⣴⣿⣇⣿⡏⣿⡏⠘⣗⣒⣒⣾⣿⣿⡟⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣺⣿⣿⣿
⠈⣿⣷⣿⣷⣿⡇⣿⡇⣸⣿⠛⣿⣿⢿⣿⣿⠁⣛⠁⢀⣧⡾⠋⠙⢿⣿⡇⢻⢡⡏⣸⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠤⠦⠶⠶⠦⢿⣿⣿
⠀⣿⣿⡏⣿⣿⠃⠻⠿⠿⠋⠀⠟⠛⠘⠛⠋⠈⠛⠀⣼⣿⣷⣫⣤⡴⠞⠿⣆⠀⠀⠠⠶⠶⣷⣶⣤⡙⢿⣿⣿⣿⣟⢿⣿⣿⡭⠭⠭⠭⢽⣿⣿
⠀⠙⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠋⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⢻⣷⣌⠙⣧⢻⣿⣿⣹⣽⣫⣩⣿⣿
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⣹⡇⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⡆⠘⠆⢿⣿⣗⣒⣒⣒⣿⣿
⣛⣛⣒⡦⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⢳⣒⣒⣿⠃⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣒⣒⣂⣿⣿
⠭⠤⠤⠵⠤⠤⠤⠤⠤⠤⠤⠭⠷⠦⣤⢀⠹⡤⢼⣾⣾⣿⠁⠀⠀⠀⠐⠦⣤⡀⠀⠀⠀⢀⣀⣤⠖⠒⢾⣿⣿⣿⣿⡆⠀⢸⣿⣿⠤⠤⠤⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⣽⣿⣿⠇⠀⢀⣴⣞⣓⣀⠀⠁⠀⠀⠀⠉⠉⢀⣐⣲⣾⣿⣿⣿⠉⠁⠀⢸⣿⣿⠬⠭⠭⣿⣿
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣿⣿⣿⣿⠀⠀⠀⠻⠍⠻⠟⠁⠀⠀⠀⠀⠀⠘⠉⢽⣟⣽⣿⡿⢁⠀⠀⠀⣼⣿⣇⣙⣉⣁⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠁⣼⠀⠀⢠⣿⣿⣖⣒⣒⣺⣿⣿
⠶⠶⠶⠶⠶⠶⠒⠶⠶⠖⠖⠶⠲⠲⠖⣿⣿⣿⣿⣇⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣰⡿⠀⠀⣼⠏⣿⣿⣷⡖⣾⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣿⣿⣿⢳⠹⣆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⣿⣷⣼⣿⡇⠀⢠⡟⠸⡇⠙⢿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣿⣎⢧⡁⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⣿⣿⣿⢿⠃⠀⢸⠃⠀⠁⠀⠘⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣿⣦⡁⠀⢻⡆⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡀⠀⠀⣿⣿⠏⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛⣷⡄⠈⣿⣆⠀⠀⠀⠀⢀⡈⠛⠛⠛⢉⠄⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠤⠤⠤⠶⠶⠶⠤⠤⠤⠴⠶⠶⠦⠤⠤⠤⠤⠤⠼⠽⣿⠀⢹⡟⣧⡀⠀⠀⠀⠈⠑⠒⠚⠁⠀⠀⣰⣿⠁⠀⠀⠀⠀⣸⡄⠀⣴⣿⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⡿⠀⢸⡇⢿⣿⣦⣄⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠀⠀⠀⠀⣠⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿
⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠍⠋⠀⠀⢸⡇⢸⣿⠿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠂⠀⠀⠀⠀⠈⠀⠘⡟⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣒⣒⣒⣒⠒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⠃⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡭⠭⠭⠤⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣉⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣮⣧⣤⣥⣭⣭⣭⣭⡭⠭⠭⠭⠭⠅⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⢇⠀⠀⢀⡠⠤⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⣀⠀⠀⠀⠀⡔⠁⠀⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣖⠉⠀⠀⠉⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠔⠊⠉⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡄⠀⠀⢇⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠈⢣⠀⠀⡠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣾⡅⠀⠀⢸⠀⠀⠈⢆⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠉⢹⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢄⠀⠀⠀⣶⠁⠀⠀⠣⡀⠀⢸⣷⠀⠀⠀⢃⠀⠀⠈⠢⣀⢀⡔⡁
⠀⠀⠀⠀⠀⢀⡝⠁⠀⠀⠀⣀⡀⣄⠀⠀⣼⡇⠀⠀⢸⠀⠀⠀⣿⡆⠀⠀⠀⠑⣴⡦⣿⡆⠀⠀⠸⠀⠀⠀⠀⡠⠋⠁⠁
⠀⠀⠀⠀⡖⠉⠀⠀⠀⢰⣯⠀⠀⠈⡄⠀⢸⣿⠀⠀⠀⡆⢀⡠⠼⠛⠢⣄⠀⠀⠸⡇⠘⢿⠀⠀⠀⡇⠀⠀⡌
⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⣿⣇⠀⠀⢡⣴⡾⣿⡆⠀⠀⢸⠁⠀⠀⠀⠀⠀⠳⡀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠰
⠀⠤⠄⠐⠚⢤⠀⠀⠀⠀⢸⣿⡄⠀⠀⢿⡇⠻⣇⠀⠀⢸⠀⢠⠚⠻⣷⣆⠀⢡⠀⠀⠀⠀⡀⠀⠀⠈⡆⠀⢣⠔⠊⠉⠒
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢿⣷⡀⠀⠈⠁⠀⠉⠀⠀⠘⠀⠇⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⣷⡄⠀⢀⠵⠀⠀⠱⡀
⠀⠀⠀⠠⠊⢧⣀⣀⡀⠀⠀⠸⣿⣇⠀⠀⠀⢰⡄⠀⠀⠀⡇⠈⠒⠚⠋⠁⠀⣸⡀⠀⢀⡠⠏⠛⠊⠁⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢀⣨⡆⠀⠀⢿⣿⡀⠀⠀⢈⡟⠒⠶⣾⣅⠀⠀⠀⠀⣀⠔⠙⠛⠊⠁⠀⠀⠀⢀⡠⠤⠤⢄⠀⠀⠃
⠀⠀⠀⠀⠀⠐⠁⠈⠀⡟⠀⠀⠈⠙⠓⠂⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠑⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣀⠤⢤⠤⠒⠁⠘⠢⢤⠒⠋⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⡔⠢⣀⠀⠀⠀⠀⠀⠀⠣⣀⡠⠤⠒⠂⠉⡏⠁⠂⠊⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠓⡄⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠣⠤⠤⠄⡶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠙⢄⠀⠀⢸⠁⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢀⣿⡇⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⣿⣿⣆⠀⣼⣿⣿⠀⢀⣾⣿⡇⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⣿⣿⣿⣶⣿⣿⣿⣧⣿⣿⣿⠇⣀⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣀⡀⢻⣿⣿⣿⣿⣿⠹⣿⣿⠈⣿⣿⠏⣿⣿⣾⣿⣿⡿⠁⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠈⠛⢿⠀⠘⠃⠀⠹⠁⢀⣿⠟⣉⠿⢿⣥⣶⣾⣿⡿⠋
⠀⠀⠀⠠⣤⣄⡀⢀⣀⣈⣻⣿⣮⡉⢸⣷⣶⢀⠀⣦⣀⣇⣠⡆⢀⡀⢠⣿⣿⡇⣻⣿⡿⠋
⠀⠀⠀⠀⠠⣶⣾⣎⢿⣿⠿⢛⠻⠷⢸⣿⣿⠘⢛⣭⣴⣶⣭⣕⠛⣀⢸⣿⣿⢱⣿⠿⢷⣶⣶⣶⢆⡐⠖
⣤⣾⣿⡄⠀⠹⣿⣿⣎⢿⢸⣿⣿⡄⢸⣿⣿⣰⣿⡿⠟⠻⣿⣿⣧⠁⣾⣿⣿⠀⣠⣿⣶⠙⠟⣱⣿⣿⡆
⠈⠻⣿⣿⣆⢶⡹⣿⣿⣆⢸⣿⡿⣿⣼⣿⣿⢿⣿⡅⠀⢀⣼⣿⣿⢠⣿⣿⣧⣾⣿⣿⣿⢀⣾⣿⡿⢋⣾⣶⣤
⠀⠀⠘⢿⣿⣦⠀⠹⣿⣿⣿⣿⠁⠹⣿⣿⣿⢈⠻⣿⣿⣿⣿⡿⡃⣼⣿⣿⣿⢟⡁⣿⣷⣿⣿⠟⣠⣿⣿⣿⠟
⠀⠀⠀⠀⠉⣥⣶⡀⢻⣿⡿⢏⣴⣶⠙⠉⠉⢀⡷⢠⡍⣭⠱⣦⠁⠀⠀⢮⣤⣬⢱⣿⣿⡿⢃⠐⢿⣿⠟⠁
⠀⠀⠀⠀⠐⠙⣿⡿⠀⠉⣸⣿⣿⣡⣴⣾⠃⠀⡀⠘⠀⠘⠀⡈⠀⢲⣤⡈⢻⣿⣷⠮⠛⢰⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⣿⣏⣴⣾⡇⢀⣾⡄⢀⣿⣦⠸⣿⣿⣿⣿⣿⣆⠀⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠉⠀⣾⣿⣿⡿⣿⣧⣾⣿⣿⣼⣿⣿⣷⣿⣿⠉⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⠋⠀⣿⣿⣿⠋⣿⣿⣿⠃⠻⣿⣿⡇⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⣿⣿⠃⠀⠘⣿⣿⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠹⡏⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠙⡀⢠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢈⠀⠀⠡⠀⠀⠀⠀⠀⠀⡄⡇⠀⠉⠁⠀⠀⠰⠀⠀⠘⠀⢠⠀⠃⠀⠀⠂⠀⠀⠀⣠⣀⠄⠀⠰⢔⣏
⠀⠀⠀⠀⠀⠄⠀⠠⢠⡤⠠⡰⠀⠀⡇⢃⠀⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⡃⠀⠀⠌⠁⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠲⠁⠁⡅⠀⠀⡀⠈⠀⠀⠀⠀⠀⠸⡀⠀⢸⣀⠀⠅⠀⠀⢀⠁⠀⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⡎⠀⠐⠄⠀⢀⠀⢠⠌⡠⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡠⠂⠘
⠀⠀⠀⠀⢸⡆⠀⠡⠀⠀⠘⡄⠀⠀⠃⠀⠀⠌⠠⠤⠂⠀⠀⠀⠈⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠖⠠⠐⢈⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⡠⠀⠈
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⠂⠀⠀⠀⠑⠀⠁⠀⠠⠄⠂⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢅⡩⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⡴⠂⠑⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡰⡈⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡅⠀⠀⢰⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠢⠆⠊⠑⠄⠀⠀⠁⠀⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣀⣠⠞⠁⠀⠀⠀⠀⡠⠊⠀⡜
⠀⠀⠀⠀⠀⠐⠐⠊⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⣹⠿⠄⠀⠀⠀⠀⠉⠁⠀⠀⢀⠠⠔⠉⠀⠀⡰⠀⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡠⠊⢀⠀⠀⠀⢀
⠀⠀⠘⢀⡲⠀⠀⠀⠀⠡⠀⢈⠐⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⡠⢋
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡐⣄⢂⡾⠔⢒⠄⠀⠀⠀⠐⠀⣠⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠠⠊⠔⠁
⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢀⢄⢀⠋⡧⢁⣿⡾⠔⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⢀⡈⡰⣤⣾⠿⠖⠲⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣈⣵⣽⠟⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣹⠆⠀⠀⠀⠡⣸⣾⠟⠋⠁⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⣀⠀⡀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠋⠉⠀⡀⠀⠀⡠⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⡀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠐⠚⠘⠃⠒⠃⠈⠃⠀⠁⠀⠓⠀⠁⠀⠛⠐⠛⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠀⣠⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⢹⡷⢤⣄⣀⣀⣀⣤⡾⠁⠀⠙⢶⣤⣀⣀⣀⣤⠶⠋⣿⠁⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⣧⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣿⠀⠀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠛⠶⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢀⡀⠀⠀⠀⠀⣶⠀⠀⣼⠏⠙⣷⡀⠈⢿⡄⢠⣦⡀⣿⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢷⡄⠀⢀⠀⢿⠀⠀⢿⡀⠀⢸⡇⠀⠈⢿⣼⠟⢿⣿⢀⡀⢸⡇⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠄⠀⠀⠸⣧⠈⢿⡄⣿⣧⣽⠀⠀⠘⣷⣄⣾⠃⣄⠀⠈⠛⠀⠈⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠂⠀⠀⢀⡿⠀⠈⢿⣿⠘⠿⠀⠚⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠛⠛⠛⠲⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⡀⠀⠈⠙⢷⡀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⢠⡏⠀⠀⢳⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠞⢀⡖⠀⠀⠀⢻⡀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠋⠀⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠂⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡰⠋⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠂⠲⠃⢠⠎⡄⢀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⢀⠊⠀⡠⠄⠀⢀⠖⠀⠀⠀⢀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠀⡄⢀⡠⠄⠀⠠⠅⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠀⠀⠀⠂⠀⠀⠀⠀⡰⠀⠀⠀⠐⠁⠔⠁⣠⠀⠀⠄⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇⠒⠀⠪⠁⠀⠠⠂⠀⠰⠈⠀⠘⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⢀⠔⠁⠀⠁⡁⡤
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠐⠁⠐⠀⠁⠀⠀⠀⠀⡀⠀⠰⠁⠀⠤⠂⠁⠀⠐⠀⠀⠁
⠀⠀⠈⡴⠐⡉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠜⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢀⢴⠄
⠀⠀⠀⠀⠀⠀⡔⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⢀⠔⠈⠀⠁⠀⠀⡠⠂⠀⠂⠀⠀⠤⠀⠀⠀⢀⠔⢁⡞
⠀⠀⠀⡄⠀⠈⠀⠎⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠊⠀⠀⢀⡔⠁⢀⡾
⠀⠀⠀⠀⠀⡅⡌⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⢃⠽⠉⠀⠀⠰⠁⠀⠀⡠⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢰⠁⠀⢲
⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⣇⢠⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠛⠃⠀⠀⢁⠀⠀⠰⠂⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⢹⡆⠀⡀⠀⠀⢸⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠂⠀⠘⠀⠀⠈⠀⠀⠁⠀⠀⢸⡄⠀⠀⠅
⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠸⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠠⢁⡠⢁⠜⠀⠀⠀⢠⠟⠀⠀⠀⠀⠈⠦⠖⠒⠚⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠋
⠀⠀⠆⠀⠈⠠⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠁⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⢀⣴⢦⠈⠁⣿⡍⠿⣅⢺⡿⠡⢀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠴⠴⣄⡤⠤⠶⣾⡋⠀⢨⣠⣄⣿⠃⠀⢸⣿⠇⠀⣼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⣼⠛⢏⢀⣘⣿⡀⠀⣿⡇⣤⣴⡙⢿⡄⠀⣛⠁⢙⡁⣸⠇⢿⣦⣮⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⢼⣏⡀⡀⣻⡟⠛⣃⠀⣿⣄⢪⣿⡃⣼⣿⡄⠉⢠⡈⠀⡜⣰⡞⠒⣦⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢷⣄⠀⠓⢐⣄⠀⠝⢿⡄⠉⢁⡏⠘⢷⡶⠞⠷⠶⠃⠘⢷⡶⠏⢻⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢷⣆⣰⣿⣿⠟⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⠙⠛⠁⢉⠙⠓⠲⠶⠦⠤⠴⠶⠶⠖⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠈⠓⠶⠶⠞
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣶⡞
⠀⠀⣾⣿⣿⣿⣿⣿⡿⠞⠉⠉⠛⠳⠶⠄⠀⠀⠀⠀⡴⠶⠛⠋⠉⠻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⣿⣿⣿⣿⣿⣿⡇⣀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣏⠙⠃⠈⠻⣍⣼⡧⠀⠀⠀⠀⠀⠀⠀⢰⣾⣌⠝⠋⢸⣿⣿⣿⣿⡟
⠀⠀⢻⣿⣿⣿⣿⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⠛⠿⠟⠋⠀⠀⠀⠀⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠈⠢⢄⠀⠀⠀⠀⠀⣀⠔⠋⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⣿⡄⡀
⣀⡤⠔⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢿⣧⠸⣿⣧⡀⢉⠒⠠⠄⣀
⠀⠂⠄⠀⡀⠀⠉⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡜⣿⣆⢙⠻⣿⣦⣀⡀⠀⠀⠁⠀⠑⠂⠄
⠀⡁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⡄
⠀⠃⠀⡀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠙⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠢⣄⠀⡀⢀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⣀⡀⢀⠔⠉⠀⠐⠈
⠀⠀⠀⠀⠀⢀⠐⢀⠀⠤⢀⡞⠁⠀⠀⠀⠀⠀⠈⠊⠀⠀⠑⡤⠤⡤⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣧⠔⠋⠐⠢⣄
⠀⠀⠀⠀⠀⠈⠐⣀⡠⠤⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣂⣀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠁⢀⠖⠉⠀⠀⠀⠀⠀⠰⢚⢛⠉⠉⠩⣝⢚⡡⠐⠀⠀⠐⠨⣩⠍⠉⢙⣛⠳⡀⠀⠀⠀⠙⠉⠢⠈⠁⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⡏⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠡
⠀⠀⢀⡴⠊⢁⠴⢊⠴⣮⡑⢦⠀⠀⠀⠀⡇⠸⠀⠀⢀⠎⠀⠀⢀⣴⣾⣷⣦⠀⠀⠀⠆⠀⠀⠇⡇⠀⠀⠀⢀⠔⣋⡴⠌⡓⢄⡀⠀⠀⡄
⠀⠀⠸⡄⠰⡁⡎⠁⠀⠸⣿⡆⢇⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⢠⣿⠟⠉⠀⠙⠃⠀⠀⢸⡀⠀⠀⡇⠀⠀⢀⡎⣼⣿⠁⠀⠈⡢⣙⣂⡐⢎⠂
⠀⢀⡴⠙⠀⠱⡌⠄⠀⠀⢹⣿⡌⢋⢍⡲⠇⡄⠀⠀⡄⠀⠀⢸⡏⠀⠀⠀⡠⠀⠀⠀⣸⡇⠀⠀⠷⢒⠩⡛⣰⣿⠃⠀⠀⡐⣰⣿⠟⠈⢂⠻⡀
⠀⡞⠀⠀⠀⠀⠘⡌⢄⠀⠀⢻⣿⠁⠀⠙⢶⡇⠀⠀⡁⠀⠀⠈⠣⠄⠠⠚⠀⠀⠀⢀⣿⠃⠀⣧⠞⠁⠀⠸⣿⠏⠀⠀⡜⣼⣿⠋⠀⢀⠜⡰⠃
⠀⡇⠀⠀⠀⠀⠀⠘⣌⢂⠀⠈⠇⠀⠀⡀⠀⠁⠀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠉⠀⢀⠀⠀⠟⠀⠀⢌⣾⡿⠃⠀⡐⣡⠎⠡
⠀⠹⢄⡀⠀⠀⠀⠀⠈⢆⢢⠀⠀⠀⢸⣷⡄⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⠀⠀⠀⣰⣿⠀⠀⠀⢀⢪⣾⡟⠁⡠⢊⠜⠁⡠⠃
⠀⠀⠀⠉⠀⠳⣄⠀⠀⠈⢆⠡⠀⢀⣸⣿⣿⢦⣤⣴⣿⣿⠿⣷⣦⣤⣴⣶⣿⡿⢻⣿⣿⣤⣤⠼⣿⣿⣀⡀⢀⠂⣿⠿⡷⢊⡴⠃⠀⢫
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠣⣙⠻⠿⠿⠟⣰⢬⣙⣉⡩⠤⢄⣉⣙⣛⣉⣡⠔⠬⢉⣙⣋⠴⡄⠿⠿⡿⠟⢱⣿⣧⣀⠆⡼⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠙⡖⠀⠀⠀⠀⠈⠙⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠙⠢⠭⠥⠒⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⢀⡎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢠⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡙⢄⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈⠢⣀⠀⠀⠀⣣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⠉⠚⠋⠉⠉⠁⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⢀⣠⠎⠉⠉⠁⠀⠀⠀⠉⢉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠁⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡴⢧⠀⠀⡇⠰⠃⣠⡦
⠀⢠⣦⣀⠀⠻⢦⡀⠀⠀⠙⢿⠦⣀⠀⠀⠀⠀⠀⠀⢀⡞⠈⠑⠦⣄⣀⣀⡴⠊⠀⠘⣆⠀⠀⠀⠊⠁⠀⠀⠀⢀⣠⣤
⠀⠀⠈⠉⠛⠲⠄⠉⠀⠀⠀⠀⠳⡌⠙⠲⠤⣤⣤⠤⠊⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠓⠦⠤⠤⠤⠶⠒⢋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠿⡄⠀⣰⠒⠲⡄⠀⣸⠋⠉⡷⠚⠉⠁⠀⠉⠉⠲⣄⡶⠦⢤⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣻⡀⠀⢳⢀⡇⠀⠀⠹⡄⡏⠀⢰⠇⠀⡠⠖⠛⢶⣄⠀⢈⡇⠀⢸⠀⢠⠞⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣍⠉⣧⠀⠈⣿⠀⠀⡀⠀⢻⠃⠀⣼⠀⡼⠁⠀⠀⠀⢹⡄⢸⠇⠀⣼⡴⠃⠀⠀⢸⠓⡶⠋⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠸⡆⠀⠀⠀⢸⢷⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢸⠇⢸⠀⠀⠛⠀⢀⡀⠀⢸⡜⠁⠀⡴⠁
⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⢿⠀⠀⢀⡏⠈⣧⠀⠀⢸⠃⠀⢧⠀⠀⠀⢀⡞⠀⣼⠀⠀⠀⡰⢻⡇⠀⠘⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠺⣇⠀⣸⠁⠀⠘⣇⣀⣼⣆⠀⠈⠓⠦⠖⠋⠀⢀⡏⠀⢀⠞⠁⢸⡇⠀⠀⣠⠟⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠉⠉⠀⠘⠳⣦⣄⣀⣀⣤⠔⠋⠛⠗⠋⠀⠀⢸⡃⠀⣰⡧⠤⣤⣌⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠉⠙⠒⠦⣄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡴⠚⠉⠉⠙⠲⢤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠸⡄⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡴⠄
⠀⣀⠀⠀⣁⣀⠀⠀⠀⠉⠉⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣝⣿⣶⣟⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⠀⠘⢿⡷⣤⣀⠀⠀⣀⣴⣿⣿⣿⣟⢻⣽⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⡰
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠈⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠄⠀⠈⠀⠀⠀⣀⣠⡾⠃⠀⠀⠀⠀⣠
⠀⠀⠙⠲⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⢁⣷⣶⣶⣶⣶⣾⣿⡿⠛⠉⢀⣠⣤⣴⣶⠎⠁
⠀⠀⠀⠀⠘⢿⣿⣿⠿⣤⡄⢀⡀⠀⠀⠀⢘⡿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠙⢿⣿⣶⣿⣿⣿⣿⣿⣿⠟⠁⣀⢰⣶⠟⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠛⣳⣾⣿⣿⡟⠀⠀⠀⠙⢲⣦⣝⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⡿⣿⡿⠃⠀⠘⣿⣿⣿⣞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⣿⡟⠛⢦⣤⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣾⢅⣤⠞⠋⠹⣯⣯⣅⡀
⠀⠀⠀⠀⠀⠀⠐⠩⠋⢸⣿⣿⡀⢈⠿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠿⠿⠟⠁⢀⣾⣷⣿⣿⣿⣿⣋⣁⠀⢠⡴⣿⡗⣯⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⡼⠾⣿⣿⣿⣷⣿⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⢿⣿⣿⣿⣿⣿⠙⠂⠀⠰⣿⣿⠻⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⡏⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⠂⠀⢀⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠁⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⣿⣿⣿⣿⣷⣿⡿⢿⠃⠀⣸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠐⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣌⡃⠀⠀⠀⠃⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⠀⣠⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⠿⣿⣿⣻⣾⣷⣤⣀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠘⢿⣿⣿⣯⡿⢿⡿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⡋⢘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢣⣸⡜⠋⠿⠛⠁⠀⢻⣷⣿⣿⣿⣿⣿⡟⠀⠀⣸⣿⣿⣿⣿⣧⠻⠀⠀⠉⠻⠀⠁⣿⡟⢠⡹
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⡏⠛⠀⠀⠀⠀⠀⠀⠀⢼⣧⢾⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣧⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠿⠃⢀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡧⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣤⡀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣮⣉⠋⠓⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⢿⣿⣿⣿⣿⡿⢻⣿⣷⣶⣦⣍⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⠿⢿⣿⡿⠟⠹⠋⡤⠰⠁⠸⡟⠋⠛⠛⠿⠻⢶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠐⠁⠐⠟⡩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⣷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣴⣷⢇⣶⣾⢰⡄⠀⢱⣠⡆⢠⢰⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣠⠀⠀⠀⡄⢰⣿⣿⣿⣿⣿⣿⣠⣿⣇⣾⣿⠃⢠⣾⠘⣿⡇⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⢧⠆⡾⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡏⣾⢸⡟⢀⢿⣇⡄⠀⢹⡏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⡟⣸⠃⢸⡏⣼⣿⣿⣿⣿⣿⣿⡿⣷⠟⣸⠃⡾⠁⡟⢻⡿⠛⠓⠈⠇⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣸⢃⡟⠀⣿⢱⣿⢟⣿⠿⠛⠁⠋⠀⠁⠀⠁⠈⠁⠀⠀⠀⠀⣀⣀⠀⠀⠐⢿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠇⣟⢸⠃⢰⡟⢸⠇⠛⠀⠀⠠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠟⠘⠂⠀⢸⣿⡇⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢰⡏⠀⠐⠙⢃⠎⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠞⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠁⠀⣾⣄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟⠀⠀⣿⢁⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠌⣸⠇⠀⠀⣿⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡾⠀⠀⠈⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⡾⢁⡆⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿
⠀⠀⠀⠀⠀⠀⣾⠃⣸⢣⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⢀⡖⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠈⠢⡍
⠀⠀⠀⠀⠀⢠⡟⢰⣿⣿⣿⣿⣿⠿⠟⠋⠉⡟⠀⠀⠀⠀⠀⠀⠀⠞⠑⠤⣀⠀⠉⠒⠤⠤⠤⠤⠴⠞⠁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢸⠇⣾⣿⠿⠛⠉⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠢⣄⡀
⠀⠀⠀⠀⠀⢸⣸⣏⠁⠐⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠖⠃⠈⠛⠓⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣁⡟
⠀⠀⠀⠀⠀⢸⠏⠀⠑⠀⠀⠀⠄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⠁⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⡶⣄⢰⡏⠑⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣶⠒⢢⡤⠐⢶⡍⠈⣤⢼⡇⢨⣿⠃⡸⢇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⣤⣀⢠⡏⠑⢶⠿⡆⢸⡃⣿⡇⣿⡄⠒⡀⢂⣏⡽⢟⠃⣰
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡘⠂⣄⣸⢳⣌⣁⡇⠻⠶⠛⠾⠈⠷⠞⢚⠁
⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢶⠟⠯⢄⣉⣁⣀⣀⣀⡤⠤⠤⠤⠴⠚
⠀⠀⣼⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⢐⠢⠴⠂
⠀⢰⣿⣿⣿⣿⣣⠤⢤⣀⡀⠀⠀⠀⣀⣀⠤⢿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⢀⣀⣤⡄⠁⠀⠀⠈⢁⣤⣀⣀⣿⣿⣿⣿
⠀⢸⣿⣿⣇⠁⠈⠋⠼⠃⠀⠀⠀⠀⠐⠿⠘⠉⣿⣿⣿⡿
⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠠⢶⡶⠄⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡏
⠀⠀⠀⠀⢻⣿⣿⣿⠀⠁⠀⠀⠀⠀⠊⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⡀
⢀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⣧⢻⣧⡀⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠻⣆⠙⠻⠶
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣿⠙⢷⡄⠀⠀⠀⠀⢠⡟⠙⣧⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠳⣄⡀⠀⠀⠀⡟⠀⠀⠙⢦⡀⠀⢀⡞⠀⠀⠘⣧⠀⠀⠀⣠⠞⠋⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⠙⢦⣄⢰⡇⠀⠀⠀⠈⠻⣤⡾⠀⠀⠀⠀⠘⣧⣠⠞⠁⠀⠀⠀⢻
⠀⠀⠀⢿⠶⣤⣀⠀⠀⣧⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⠶⢶
⠀⠀⠀⠈⢧⠀⠉⠛⠲⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⠶⠛⠉⠀⠀⣼⣀⡴⠖⣷
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢀⡟
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⣷⡤⢶⡀⠀⣰⢦⡀⢠⠇⠀⠉⠉⡇⠀⠀⠀⣠⣄⠀⠀⠀⢳⠀⠈⢹⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠛⠲⢦⡄
⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠳⣴⠃⠀⠱⡼⠀⠀⠀⢀⡇⠀⠀⠈⠓⠋⠀⠀⠀⢸⠀⠀⠼⠖⠋⢹⣀⡠⠞⠛⠲⣤⡀⠀⣸⠁⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡼⠃⢠⡇⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⢤⣄⣀⣀⣀⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁⠀⠀⣖⠀⠀⢀⣀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣀⣽⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣀⣠⠼⠀⠳⠶⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠊⢹⣄⠀⠀⣠⠞⠁⠀⠀⠀⣾⠁⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣧⠀⠀⠀⢀⣤⣄⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⠀⠀⠀⣨⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡴⠋⠁⠈⢷⡴⠋⠉⠛⢦⣠⡴⠛⠙⢷⣤⡴⠞⠛⣦⣀⣠⠶⠛⢳⣄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⢸⡏⠳⣄⠀⠀⠀⠀⡾⠙⣆⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣏⠙⠦⣄⠀⠀⢸⠃⠀⠘⢦⡀⠀⡼⠁⠀⠘⣆⠀⢀⡤⠞⠁⢸⡄
⠀⠀⢀⣀⠀⠀⠀⢸⠀⠀⠈⠳⢤⣼⠀⠀⠀⠀⠙⡾⠁⠀⠀⠀⠘⣶⠏⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⢯⠓⠶⢤⣘⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣀⡤⠖⠋⢩⡇⢀⣠⣄
⠀⠀⠀⠈⢇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠸⠟⠋⢁⡏
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠎⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠁
⠀⠀⠀⠀⠀⢈⣧⠴⢆⠀⣰⢦⡀⠸⠉⠉⠉⡇⠀⠀⣠⢄⠀⠀⠈⡇⠈⢹⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠲⣤
⠀⠀⠀⠀⣾⠉⠀⠀⠈⢦⠃⠀⠱⠇⠀⠀⢀⡇⠀⠀⠉⠋⠀⠀⠀⡇⠀⠸⠚⠁⢻⡤⠚⠉⠓⢦⡀⢠⠃⠀⠀⡴⠋
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡏⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠈⠓⠒⠒⠒⠋⡅⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠳⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡸⠑⣤⠤⠴⠾⠀⠀⠀⠀⠀⠀⠀⠳⢤⡴⠛⡀⠀⠀⣠⠎⠀⠀⠀⡞⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢧⠀⠀⠀⣀⣄⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠓⠞⠁⠀⠀⠀⠀⡽⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠞⠉⠈⠳⠴⠋⠙⠳⣤⠴⠋⠙⢦⡴⠞⠙⢦⣠⡴⠛⠳⣄⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡤
⠀⢀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡛⢿⣭⡿⣻⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⡀
⠀⠉⠀⠀⠈⠉⠁⠒⠀⢀⡀⠀⠀⠈⢿⣷⣦⣀⠀⠀⠀⢀⣠⣿⣿⣿⡟⢿⣻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠻⣮⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣄⠠⠐⠊⠁⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢀
⠀⠀⠻⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣱⣤⣤⣤⣤⣤⣴⣶⣾⠿⠋⠁⠀⣀⣠⣤⡔⠋
⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠻⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⠟⠉⠀⣤⣤⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢹⡿⢧⣹⣿⣶⣿⡆⠀⠀⠀⠾⢭⣙⠻⣿⣿⣿⣿⣿⣿⣿⣦⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢼⣷⣿⣧⡴⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣏⣿⣿⢿⣄⡀⣀⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣯⣟⠁⢀⡴⠞⣿⡿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠴⡿⠋⣿⣿⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣸⣿⣽⣿⣿⣿⡯⠚⠁⠀⠀⣸⣿⢛⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⣦⣼⣿⣿⣷⣟⣿⣿⣿⣆⡈⠉⠉⠀⣀⣴⣿⣿⣿⣿⣻⣿⣿⡿⢆⠀⠉⢹⣿⡞⣷⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⠉⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣂⣀⠀⠀⠈⣯⣷⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⡇⠹⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠀⠀⠀⠘⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢨⣿⣿⣿⣿⣿⡟⢿⠉⠂⠀⢠⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡇⠀⠀⠀⡀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⡷⣦⣄⣀⡀⠀⠠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠇⣴⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢐⣿⣿⣿⣿⣏⣿⣿⣷⣿⣿⣿⣷⡄⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠳⡄⣭⢿⣿⡿⠾⠛⣿⣟⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⢿⡏⠙⠿⣿⡝⢷⢄⣼⡿⠛⡷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣮⡶⣷⠀⠘⠋⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⡏⠀⠀⢼⣿⣿⣿⣿⢿⠘⠀⠀⠀⠈⠁⠀⠘⣿⠇⢷⡛
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡷⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣦⣄⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⠈⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⣿⣿⣦⠀⠀⣸⣿⣿⡀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⣿⣿⣿⣧⣠⣿⣿⣿⣇⣠⣾⣿⣿⠇⠀⠀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣿⣿⡿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣤⣀⣻⣿⣿⢿⣿⣿⡇⠙⣿⡟⠀⢻⣿⠏⢸⣿⣿⣿⣿⣿⣿⠁⠀⢀⣀⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⠆⠈⠻⡇⠀⠈⠃⠀⠘⠁⠀⢸⠿⠋⣁⡛⠿⣷⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢤⣤⣤⠀⢀⣀⣀⣙⣿⣿⣦⠉⢸⣿⣿⡆⡀⠘⣦⢀⣷⢀⣾⠀⢀⠀⢸⣿⣿⡗⣸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢠⣶⣾⣎⢻⣿⣿⠿⠿⠿⠷⢸⣿⣿⡇⠻⢃⣩⣥⣭⣭⣁⠺⠏⡀⣸⣿⣿⡇⣿⣿⣿⣶⣶⣶⣶⡆⡀⠤⠄
⣀⣤⣶⣄⠀⠈⢻⣿⣿⣆⢻⡇⣾⣿⣷⡀⢸⣿⣿⡇⣴⣿⣿⡿⢿⣿⣿⣷⡜⠃⣿⣿⣿⠀⠉⣱⣶⣌⢻⡿⢋⣾⣿⣦⡀
⠙⢿⣿⣿⣦⢀⣄⢻⣿⣿⣆⠃⣿⣿⣿⣷⣸⣿⣿⣷⣿⣿⠃⠀⠀⢸⣿⣿⡇⢸⣿⣿⣟⣠⣾⣿⣿⡿⠀⣰⣿⣿⡿⢋⣤⣄⡀
⠀⠀⠻⣿⣿⣧⡙⠊⢿⣿⣿⣦⣿⣿⠙⢿⣿⣿⣿⠃⢿⣿⣦⣤⣤⣾⣿⣿⠃⣾⣿⣿⣿⣿⡿⢻⣿⣇⣼⣿⣿⠟⢡⣾⣿⣿⣿
⠀⠀⠀⠈⢻⡿⠗⠀⠈⢿⣿⣿⣿⡏⢀⡈⢿⣿⣿⡄⢆⣙⠿⠿⠿⠿⢛⢡⠠⠿⠿⠿⠟⠉⠖⣸⣿⣿⣿⡿⠃⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠠⣾⣿⣆⠘⣿⡿⣛⣡⣿⡿⠀⠀⣀⠀⠼⠋⢸⠆⢶⡎⠻⡄⠀⠀⠀⢳⣶⣶⡆⠿⣿⣿⠟⣠⣈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠂⠙⣿⠿⠃⠈⢀⣿⣿⣟⣤⣶⣿⠇⠀⣠⡀⠈⢀⠈⠀⣀⠀⠀⣷⣦⣀⠻⣿⣿⡷⠮⠉⢼⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⣠⣾⣿⡆⢀⣾⡆⠀⣿⣷⣄⢹⣿⣿⣿⣿⣿⣷⡄⠀⠀⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠛⠉⠀⣾⣿⣿⣿⣿⣿⣇⣾⣿⣿⣄⣿⣿⣿⣾⣿⣿⠉⠻⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⢸⣿⣿⡿⠋⢸⣿⣿⣿⠟⣿⣿⣿⡿⠙⢿⣿⣿⡇⠀⠀⠈⠛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠀⠀⢸⣿⣿⠏⠀⠹⣿⣿⡇⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢻⣿⠃⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣮⣉⠋⠓⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⢿⣿⣿⣿⣿⡿⢻⣿⣷⣶⣦⣍⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⠿⢿⣿⡿⠟⠹⠋⡤⠰⠁⠸⡟⠋⠛⠛⠿⠻⢶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠐⠁⠐⠟⡩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⣷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣴⣷⢇⣶⣾⢰⡄⠀⢱⣠⡆⢠⢰⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣠⠀⠀⠀⡄⢰⣿⣿⣿⣿⣿⣿⣠⣿⣇⣾⣿⠃⢠⣾⠘⣿⡇⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⢧⠆⡾⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡏⣾⢸⡟⢀⢿⣇⡄⠀⢹⡏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⡟⣸⠃⢸⡏⣼⣿⣿⣿⣿⣿⣿⡿⣷⠟⣸⠃⡾⠁⡟⢻⡿⠛⠓⠈⠇⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣸⢃⡟⠀⣿⢱⣿⢟⣿⠿⠛⠁⠋⠀⠁⠀⠁⠈⠁⠀⠀⠀⠀⣀⣀⠀⠀⠐⢿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠇⣟⢸⠃⢰⡟⢸⠇⠛⠀⠀⠠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠟⠘⠂⠀⢸⣿⡇⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢰⡏⠀⠐⠙⢃⠎⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠞⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠁⠀⣾⣄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟⠀⠀⣿⢁⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠌⣸⠇⠀⠀⣿⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡾⠀⠀⠈⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⡾⢁⡆⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿
⠀⠀⠀⠀⠀⠀⣾⠃⣸⢣⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⢀⡖⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠈⠢⡍
⠀⠀⠀⠀⠀⢠⡟⢰⣿⣿⣿⣿⣿⠿⠟⠋⠉⡟⠀⠀⠀⠀⠀⠀⠀⠞⠑⠤⣀⠀⠉⠒⠤⠤⠤⠤⠴⠞⠁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢸⠇⣾⣿⠿⠛⠉⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠢⣄⡀
⠀⠀⠀⠀⠀⢸⣸⣏⠁⠐⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠖⠃⠈⠛⠓⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣁⡟
⠀⠀⠀⠀⠀⢸⠏⠀⠑⠀⠀⠀⠄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⠁⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠢⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡠⠊⠢⠀⠔⠂⠉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⠐⠤⣀⡤⠒⠓⠒⠤⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣀⣀⠤⠀⠤⠤⠊⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢡⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢣⠀⠀⡠⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡏⠁⠀⢃⠀⠀⢢⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠙⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡀⠀⠀⣴⠁⠀⠈⢆⠀⠘⣷⠀⠀⠈⡄⠀⠀⠳⢄⣀⣎⡀
⠀⠀⠀⠀⢀⠜⠁⠀⢀⡀⠤⠤⡀⠀⢾⡇⠀⠀⠇⠀⠀⢿⡆⠀⠀⠀⢲⡶⢿⡇⠀⠀⢁⠀⠀⢀⡴⠁
⠀⠀⠀⢨⡁⠀⠀⠀⢺⣧⠀⠀⢣⠀⣸⣿⠀⠀⢸⡠⠔⠚⠉⠐⢄⠀⠀⠿⠀⠛⠀⠀⠸⠀⠀⡎
⠀⠀⠀⢀⡕⠀⠀⠀⠈⣿⡆⠀⠀⣿⡟⣿⡆⠀⠀⠁⠀⣀⣤⣄⠀⠱⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⢀⡀⢀⡀
⠀⠉⠀⠀⠈⡆⠀⠀⠀⢸⣿⡄⠀⠸⠇⠈⠇⠀⠀⡇⢸⠀⠈⣿⡇⠀⡇⠀⠀⢰⣆⠀⠀⢰⠀⠈⢇
⠀⠀⠀⡠⢖⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⢠⠀⠀⠀⢠⠘⠤⠴⠟⠁⢀⡇⠀⠀⢈⠟⠦⠐⠉⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠉⠉⢱⡄⠀⠸⣿⡆⠀⠀⢸⠣⠤⣴⣎⠀⠀⠀⠀⢀⠜⠛⠒⠈⠁⠀⠀⢀⠀⠀⢀⠀⠈⡄
⠀⠀⠀⠀⠀⠊⠉⠁⡇⠀⠀⠙⠓⠒⠈⠁⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠑⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠐⠒⠖⠊⠀⠈⠒⡂⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⡀⢤⠄⣀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠚⠉⠑⢄⠀⠀⠀⠀⠀⡗⠂⠉⠀⠀⡰⠀⠀⠀⠀⠀⡇⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⡴⢄⠀⠀⠠⠊⠣⡀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠰⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡤⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⡠⢳⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡞⠂⠀⠀⠀⠀⠀⠀⠃⠀⠰⡀⠀⠀⠉⠁⢠⠊⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⢆⠀⢀⠔⠁⠀⠀⠀⢡⠀⠀⡀⠀⠀⠀⡠⠊⡹⠀⠀⠀⡰⠶⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠀⠀⠖⢄⡈⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠈⠖⠊⠹⢀⠔⠋⠀⠰⠁⠀⡰⠉⠀⠀⠀⠀⠀⢀⠄⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠈⢇⠉⠁⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋
⠀⠀⠀⠀⠀⠀⠙⠖⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠀⢔⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠈⠉⠀⠀⠀⠀⣿⠀⢠⣄⠀⣿⢃⣾⠟⢻⣇⢰⣿⢀⣀⠀⣰⡆⠀⠀⠀⠈⠑⠢⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⣿⣆⣿⢿⣦⣿⢸⡏⠀⢸⡟⢸⣿⣿⣿⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡈⠡⠶
⠀⠀⠀⠀⠀⠀⠀⠰⡲⠀⠀⠀⡁⠖⠀⠀⠀⠀⢹⣿⠇⠈⣿⡇⠘⣿⣤⡿⠃⣼⡿⠁⢹⣿⠇⠀⠀⠀⠰⡒⠂⠈⠁
⠀⠀⡀⣠⣀⡄⣠⢀⡀⠀⠒⠃⠤⠤⠄⡀⠀⠀⠘⠟⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠁⠀⠈⠋⠀⠀⠀⠀⢀⣱⠄⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠮⠤⠠⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠬⠆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⡺⠭⠐⠃⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡄⠀⠀⢀⠀⠀⠀⣠⠀⡠⢄⠀⠀⠀⠀⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⢠⠊⠀⣀⠤⠊⠁⠂⣠⠔⠁⡇⠀⡰⠀⠓⠁⠀⠱⡄⠀⠀⢰⠀⠀⠀⠈⠒⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⢠⢃⠴⠊⠁⠀⠀⠀⠋⠀⠀⠀⠃⠰⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⡄⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡀⠀⠀⠘⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡀⠀⠀⠀⠀⣔⠂⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⢰⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⢺⢻⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢐⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢿⣿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⡀⠀⠀⢸⡄⠀⢸⢧⢸⣿⣸⣇⠀⠀⢀⣼⠃⠀⠀⢀⡀⠀⣼⡿⠃⠀⠀⡠⠊
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⣷⡄⣿⣆⠀⡼⠙⠶⠋⠈⢻⣿⠃⠘⠦⠴⠋⢸⡄⢀⡴⣿⠃⣼⡟⠁
⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠈⣿⣏⠈⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠋⠀⣿⡼⠋
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⢀⣀⡗⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠷⠴⠞⢁⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⣤⣶⣦⠄⠀⠀⢀⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⣾⣿⣿⠀⠈⣿⣿⣿⠀⠀⡀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠈⢷⣤⣤⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣏⠀⠀⠀⢻⣿⣿⣰⣧⣿⣿⣿⣠⣴⣷⣿⣿⣿⢠⣶⣿⣿⣿⠀⠀⠀⠴⢾⣿⣿⣿⠟⠛⠋
⠀⢠⣤⣤⣤⣤⣀⡒⠾⣅⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⠟⠉⠁
⠀⠘⠻⠿⠿⠿⠿⠿⠿⣿⠇⠀⠀⠸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⢷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⠀⠀⠀⠂⠉⢉⠉⠄⠠⠀⠛⠿⣿⣿⠟⣿⣿⣿⠿⣿⣿⡿⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛
⠠⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠛⠻⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠀⠀⠀⣷⢖⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⣰⠋⠉⠻⣦⠀⠈⢻⣿⣆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢻⣤⠖⠳⣄⣾⠁⠀⠀⢀⣀⣄⡀⢠⠟⣏⠙⢿⡀⠀⠀⠈⠁⠀⠀⠙⠿⠇
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢸⠃⠀⢀⣿⡿⠋⢳⣀⡞⠀⠈⢳⣿⠀⢹⣧⡀⠛
⠀⠀⠀⠀⡀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠠⢠⣿⣿⠃⠀⠘⣿⠀⠀⠀⠀⢿⡀⠀⢻⣿⡄
⠀⠀⠀⠊⠀⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⢠⣿⠏⠈⠀⠀⠀⣿⠀⠀⠀⠀⠈⠁⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠉⠢⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠢⣀⠴⠖⠶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠒⠒⢤⡠⠊⣀⣔⣈⠁
⠀⠀⠀⠀⠀⠀⡍⠉⠐⠀⠒⠒⡾⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊⠁⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢈⡠⠴⠖⠚⡓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡠⠴⠖⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢈⡤⠤⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⠊⠉⠁⠀⠀⠀⠀⠀⡴⢉⠉⠉⠩⠤⢤⣍⣩⠔⠒⠉⠉⠑⠢⢤⡭⠭⠍⠉⢍⠹⡆⠀⠀⠀⠀⠉⠉⠖⠁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣠⠖⠊⣀⡴⢊⠥⣌⡓⢦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠁⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⢰⡀⠀⠀⡇⡇⠀⠀⠀⠀⣠⠔⣋⡤⢍⡳⢄⡀⠀⠀⢠⠃
⠀⠀⠀⡇⠀⣾⢡⠊⠁⠀⠹⣿⣆⢳⠀⠀⠀⠀⡇⡇⠀⠀⢰⠁⠀⠀⣰⣿⡿⠋⠉⠙⠇⠀⠀⠀⣧⠀⠀⡇⣧⠀⠀⠀⢠⠃⣾⣿⠁⠀⠉⢢⠙⠖⢒⠢⡑⠄
⠀⠀⢀⠽⠆⠘⢆⢣⡀⠀⠀⢻⣿⡄⠳⠒⠦⣤⡇⠇⠀⠀⠘⠀⠀⠀⣿⠏⠀⠀⠀⢀⠇⠀⠀⠀⣿⠀⠀⡇⣿⡤⠖⠲⠏⣼⣿⠃⠀⠀⢠⢃⣾⣿⠏⠣⡙⣮
⠀⣰⠋⠀⠀⠀⠈⢦⠱⡀⠀⠀⢿⣿⡴⠁⠙⢦⣄⡇⠀⠀⡖⠀⠀⠀⠻⣄⠀⣀⡠⠊⠀⠀⠀⢸⣿⠀⠀⣇⣠⠖⠁⠹⣴⣿⠏⠀⠀⡠⢣⣿⡿⠃⠀⠀⡵⢸⠆
⠀⡏⠀⠀⠀⠀⠀⠈⢧⠱⡀⠀⠈⣿⠃⠀⠀⠀⠹⠇⠀⠀⢙⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣾⡟⠀⠀⠿⠁⠀⠀⠀⢻⡟⠀⠀⡰⣱⣿⡟⠁⠀⡠⢊⡴⡇
⠀⢷⡀⠀⠀⠀⠀⠀⠀⢳⡘⡄⠀⠘⠀⠀⢰⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁⠀⠀⠀⠀⣴⡄⠀⠈⠀⠀⡔⣴⣿⠏⠀⢠⠞⡱⠋⢀⡇
⠀⠀⠓⠦⠤⢤⡀⠀⠀⠀⠳⡘⡄⠀⠀⠀⣾⣿⣆⠀⠀⠀⣾⣿⣷⣄⡀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⠀⠀⢀⣼⣿⡇⠀⠀⠀⡜⣼⣿⡋⢀⠔⣡⠞⢀⡤⠊
⠀⠀⠀⠀⠀⠀⢳⠤⠀⠀⠀⠱⡜⢄⣀⣠⣿⣿⣿⠳⢶⣿⣿⣿⡟⠛⢿⣿⣶⣿⣿⣿⠟⠋⣿⣿⣿⣷⠶⠚⣿⣿⣧⣄⣀⢜⣨⡿⠛⢟⢁⠜⠁⠀⢈⡀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⠙⠦⣙⠻⠿⠛⢃⡼⠢⠭⠭⠤⠒⠒⠤⢌⣉⣉⣡⠤⠖⠓⠦⠤⠭⠥⠺⣄⠛⠛⠿⢛⡁⣿⣿⣧⠤⢂⡞⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠑⠦⠤⠖⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠐⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠈⢇⠀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠂⠘⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠦⢿⠒⠖⠒⠂⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⠈⢆⠀⠀⠀⠀⢀⠖⠒⠒⠚⠁⠀⠉⠛⠒⠖⠺⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠤⠴⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⠠⠔⠀⠀⠙⠲⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠸

⣒⣲⡶⠶⠒⠒⠒⠒⠒⠒⠶⠶⣶⣒⡒⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢻⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣆⡒⠒⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⣀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠴⠤⠤⠤⠤⠤⣤⣣⣭⣶⣶⣶⣦⣥⣄⡤⠤⠤⠤⠤⠤⠴⠶⠶⠤⠮⣿⣿⣿⣿
⢀⣀⠀⣀⠀⣀⡀⢀⣀⡀⢠⣤⠀⣤⡄⣤⣤⣶⡞⣯⠭⠭⢭⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣿⣿⣿
⢸⣿⣸⣿⣷⣿⢱⣿⠻⣿⡾⣿⣿⣿⣿⣿⢹⣿⠀⢻⢭⣭⢿⣿⣿⡇⣼⢟⡿⣻⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⡭⠭⠭⠭⠭⠥⢿⣿⣿
⠀⣿⣿⣿⣿⣿⢸⣿⣴⣿⠃⣿⣿⢹⣿⡟⢰⡦⠀⣿⣿⣡⠀⠙⣻⣧⡙⠘⠁⠏⣉⣉⣉⡛⢿⣿⣿⣿⣿⣿⣿⣯⡭⠭⠭⠭⠽⣿⣿
⠀⢿⠿⠸⠿⠏⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⢀⣼⣿⢿⣷⡿⠋⠁⠉⠓⠀⠀⠉⠉⠙⠻⣿⣶⡝⢿⣟⠻⢿⡹⣿⣿⡭⠭⠭⠭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠁⠁⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⣷⡀⢳⢸⣿⣿⠭⠭⠭⣿⣿
⠳⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠥⢤⣿⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠈⠀⣿⣿⡧⢥⠅⢸⣿
⣿⣯⣑⣶⣀⣭⣭⣭⣽⣯⣶⣦⣄⣀⠈⢿⣟⣿⣤⣿⡇⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⣀⣠⠤⢼⣿⣿⣿⣿⠀⠀⢹⣿⡧⣁⣀⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛⣛⣺⣗⣻⣿⡿⠀⠀⣠⣴⣒⡀⠉⠁⠀⠀⠀⠛⠉⢀⣠⣼⣿⣿⣿⠹⠀⠀⣼⣿⣗⣒⣒⣺⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣾⣿⣿⡇⠀⠀⠻⠍⠹⠟⠀⠀⠀⠀⠀⠀⠚⢻⣿⣹⣿⡿⠃⠀⠀⢀⣿⣿⣐⣒⣒⣾⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡟⠁⣾⠀⠀⣸⣿⣷⣒⣒⣂⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⡌⢻⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⣸⣿⠁⣼⡟⠀⢀⣿⢹⣿⣿⣶⣺⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⣿⣿⣿⢏⢷⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣾⣿⡇⠀⢸⠇⢸⠈⠻⣿⣿⣿⣿
⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠙⣿⣞⠧⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⣿⣿⣿⠻⠀⠀⢸⠀⠀⠀⠀⣿⣿⣿⣿
⠭⠭⠭⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠯⢬⠿⣧⡀⠈⣷⡀⠀⠀⠀⠀⠰⢶⣶⣶⣾⠦⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣷⡄⢹⣿⣄⠀⠀⠀⠐⢤⣀⣉⡠⠆⠀⠀⣿⡏⠀⠀⠀⠀⣄⠀⠀⣰⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢽⡇⠈⡟⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⢀⣿⣀⣴⣿⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⠟⠀⠀⡇⢸⣿⣿⣷⣦⣄⣀⣀⣤⣾⣿⣿⣿⣄⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠉⠀⠀⠀⠀⡇⠀⣿⠀⠙⢿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠁⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠻⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡉⠉⢉⠉⠉⠉⢉⣉⡉⠉⣉⡉⠉⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣤⣤⣀⣀⣚⣀⣊⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣤⣬⣍⣙⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⡻⢟⢙⠏⢸⡿⠿⡿⣿⣶⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠫⠚⠩⠿⢋⠁⠀⠀⠈⠀⠀⠈⠁⠀⠀⢀⣠⣮⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣴⣷⣢⣴⡆⡄⠀⢰⣠⡀⡀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡤⠀⠀⡀⢠⣿⣿⣷⣿⣿⣇⣿⣇⣼⣿⠁⢡⡇⢻⣿⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⡖⣼⠁⣿⢿⣿⣿⣿⣿⣿⣿⡿⣹⢏⡾⣽⠇⣘⣿⣰⠀⢹⡏⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣹⢱⡇⢰⡏⣾⣿⣿⣿⣿⢿⡿⣷⠏⡼⢡⠏⠸⠋⡿⠋⠛⠀⠇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⡏⣾⠀⣾⢹⡟⣿⠟⠋⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⢄⡀⠈⣿⣧⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢸⠗⠃⣸⡟⣸⠁⠀⠀⠀⢘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢻⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠿⣀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢀⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⢹⠃⠀⣦⡀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡄⠀⠀⢀⠁⠀⠀⢷⣦
⠀⠀⠀⠀⠀⠀⠀⡎⡞⠀⠀⡿⢸⢂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠃⠀⠀⠋⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠰⢰⠇⠀⢰⡇⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠙⣿⣿⣿⣿⣿⡁⠀⡈⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢢⡟⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⢀⣿⢁⡇⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠢⡘⢿⡇
⠀⠀⠀⠀⠀⣸⠇⣼⣿⣿⣿⣿⣿⣿⡿⢿⠏⠀⠀⠀⠀⠀⢀⡾⠉⠐⠀⠤⡀⠀⠀⠀⠀⣀⡔⠻⡄⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⠿⠋⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠤⢀⡀⠉⠉⠉⠉⠉⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀
⠀⠀⠀⠀⠀⣏⣾⠛⠩⡀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡩⠟⠢⠤⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⣬⢓⡆
⠀⠀⠀⠀⠀⡿⠋⠢⡀⠀⠠⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡸⢄⠀⠀⠀⠀⠁⠀⠀⠀⢀⠡⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣎
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡤⠁⠀⠉⠀⠀⠀⠀⠈⠻⢿⡿⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠊⡀⠀⠀⠀⠀⢀⡤⠂⣀⣄⠘⡗⢀⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⠐⠁⠀⠀⠀⠐⠉⠀⠐⠛⠙⠃⠛⠉⢹
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠂⠠⠀⠀⠄⠚⣠⢄⡀⠀⠀⢀⣠⣂⡟⣀⠠⠀⠀⠀⠀⠀⢀⡀⠂⠀⠈⠀⠅⡤⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶⡀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠀⠀⠘⠋⢛⡇⠛⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢦⠀⠙⣤⠁
⠀⠀⠀⠀⠀⠀⣔⠀⠀⢸⠙⠧⣇⠀⠀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠒⠢⣆⠞⠁
⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠀⡐⠛⡀⠀⠩⣷⠀⠀⠀⠀⠀⠈⠠⠄⠎⠁⢠⡆⣠⠄⠀⠀⠀⠀⠒⠒⠈⠁⣄⣰⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⡀⢐⠈⠓⠤⣀⠀⠀⠀⢀⠼⣄⠀⢸⡟⠁⠰⠠⠀⠆⠁⠀⠀⠀⠀⢘⡏⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠈⢠⡶⠀⠉⠛⠆⠐⠈⠴⠈⠲⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⡄⠀⠀⠀⡰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠂⢀⠠⠋⠁⠄⠀⠀⠀⠀⠀⠃⠀⠀⠈⡟⠀⠐⠁⠀⠀⠀⠀⠀⠀⠺⣀⡈⢲⣄⠘⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠂⠁⠀⡀⠛⢦⡀⠀⡀⠀⠀⠀⢀⠈⠀⢀⣾⠀⠀⢠⠀⠀⠀⠀⠀⠀⡔⠀⠀⣈⡅⠸⠋⠀⠁⠠⡈⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠶⣄⡇⠀⠈⠿⠀⠃⠀⠀⢀⢠⠄⠀⠃⡎⠀⠀⠀⢀⡀⠀⠀⠀⣼⠁⡰⠀⠸⣦⡰⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠄⠀⠀⠀⢾⠀⠀⡄⠊⠀⠠⡿⠀⡇⠀⠰⠈⠀⢑⠀⠀⢠⠏⠸⣤⠀⠀⣨⢁⡄⢰⠀⣀⠀⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡇⠀⠄⠀⣢⠀⠁⠀⡠⠄⠃⠣⠄⠇⠘⢤⣀⡡⡎⠀⢠⠟⠀⠘⠋⠀⠘⠁⢨⠃⠠⠈⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡠⠄⠀⠀⠌⠉⠀⡶⠀⠀⠀⠘⠀⠰⠀⠀⠠⠀⠀⠇⠀⢬⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠲⡌⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⠀⠘⠀⠀⢀⠈⠆⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠠⣈⠀⠄⠀⢄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠉⠀⠀⠀⠀⣰⠏⠃⠘⠢⠄⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⢁⠐⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠠⠙⠦⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠀⢀⣀


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⢀⣀⣀⢀⣀⠶⣦⠄⠀⠔⠸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⢠⣄⣐⣠⡾⢧⠀⠙⣻⢆⣇⡀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣯⣽⣇⠀⠰⠁⠀⢱⣄⠻⣦⢛⣄⠈⠙⠒⢓⣯⡡⠾⣅⠙⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣼⡿⠛⢋⣦⠇⠀⠀⢈⢷⠛⠇⣼⣻⣿⣿⣷⣟⣇⣻⣦⣄⡰⢄⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⣿⢳⡴⠟⡱⣇⠉⠀⠁⡿⢎⣾⣿⣿⣻⣿⣿⡟⠋⠫⡉⠀⠀⢹⠏⡉⠙⠢⠉⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡓⢳⣄⡲⣼⠟⣹⣮⣭⢽⢿⣾⣿⣿⣿⡇⣿⣯⣀⡀⠀⠠⢶⠐⠀⠀⠘⠂⠄⠠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⣬⣋⣿⠟⠋⠉⣐⡎⠻⠉⠧⠊⠱⠻⣿⣛⠛⠛⠿⠶⠦⣤⣿⡂⠨⢷⠄⠠⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡟⢛⠟⠉⠈⣀⣤⣴⢿⡁⠀⠃⠀⠈⠀⠐⡏⠓⠀⠀⠀⠀⠀⠀⠉⠙⠓⣮⡾⠃⠂⠿⠂⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠙⠘⠀⠆⠌⠀⠀⠀⠩⡤⠟⠀⠛⣦⣤⣦⣴⣦⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠀⢠⣿⠿⠿⠻⣿⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢾⡷⣄⠐⠉⣯⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⣷⣄⣠⡾⠹⡜⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠺⠀⢀⠁⠀⠀⢻⠬⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠊⡀⠀⢸⠀⠀⠀⠈⡆⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠀⠀⠀⣸⡿⠖⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠐⠂⡴⢙⠅⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢹⡙⢶⣤⠞⠑⢘⣴⣿⡤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⢸⣟⣄⠉⠔⣰⢿⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⢽⣿⣷⣿⣿⣿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⡁⣷⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢲⣾⣿⣿⣿⣯⣿⣿⣷⣶⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠔⢀⣿⠿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⣠⣴⣿⣿⡟⠉⠉⠛⢶⣦⡏⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⣿⣿⡀⢀⣼⣿⣿⠛⣿⣇⠀⠀⠀⠀⣹⡅⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠹⣿⣷⣦⣤⣿⣿⣴⣿⣿⣇⠀⠀⠀⣧⡴⢹⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⡀⠀⣀⣸⣿⣿⣿⣯⣿⣿⣼⠍⣻⡄⠄⢆⠙⠹⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣵⣤⣿⣿⣿⣿⣟⢿⣹⣿⠿⢧⡈⠉⠳⣮⡀⠔⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡿⠋⠉⠛⠫⢍⠻⡟⣿⣶⣿⣿⣾⣜⠿⢷⣦⣀⢟⣥⡾⣿⣿⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣶⡿⠋⠀⠀⠀⠀⠀⠀⠡⠌⢺⣿⣯⣿⡿⣿⣷⣶⡶⠯⡡⣿⣶⣿⣿⣿⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣿⣟⡁⠀⢀⡀⠀⠀⠀⢀⠀⠹⡀⢻⣿⣿⡁⢿⣿⣿⢆⠀⠐⣾⣹⣿⣿⡿⡆⠀⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣻⠽⢦⣿⣆⡐⠀⠐⠃⠀⠀⠀⠀⡇⢸⢿⣿⣿⣨⣿⣧⠈⣼⡀⣯⢹⣿⣿⣆⢷⣅⠗⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⣴⣾⣿⣿⣿⣿⣿⡶⣽⣷⣾⣶⣼⣧⣼⣿⣿⠽⣿⣿⣿⣤⣿⣵⣿⣼⣿⣿⣟⣫⣴⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠿⣿⣿⣿⡯⢇⠀⣀⠈⠙⢻⡿⣿⣿⡿⣿⣧⡀⠙⠿⠻⣿⣿⣿⣿⠍⣿⣿⣿⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣷⡿⠋⢑⣷⣾⣿⣿⣦⡀⢄⣻⣿⣯⣽⣿⣩⠛⠂⠄⣠⢄⠘⠷⣃⡐⠻⣿⣦⣆⣴⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣿⣿⣭⣿⣍⡙⠟⠛⠉⢻⣿⣿⢿⣿⣀⠀⠀⠀⠂⠀⠀⠣⠀⢀⣿⣿⣿⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⡿⠿⠛⠿⢿⣾⣿⣿⣿⣧⠀⠹⣿⣷⣦⡀⠀⡇⠀⠀⣸⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣷⣲⠀⠀⢀⠀⠀⠘⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣦⣧⢠⣶⢿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣍⡒⢬⣬⣅⡠⣿⣿⣿⣿⣿⣿⣉⠀⣿⡏⠛⢿⡿⢁⣾⣾⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⢌⠹⢆⡛⣿⣿⣿⣿⣿⣿⢉⠛⢶⣻⣇⠀⠘⢁⡼⠉⠘⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣻⣿⠿⠷⣄⡈⠺⣷⣤⠾⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠀⠀⠀⠈⠉⠐⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡎⠀⠀⠀⠘⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠐⢴⢿⡀⠀⠀⠀⠀⠀⢣⠀⠀⡀⠀⠀⠮⠆⠀⡠⠊⢱⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⠀⠀⢰⠑⡄⠀⠀⢠⠞⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡠⢶⠀⠀⠀⣀⡠⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⠀⠇⠀⠈⢆⠔⠁⠀⠀⠀⠀⢣⡠⠲⠀⢀⡴⠊⢀⠃⠀⢠⠖⠉⠁⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠐⣖⠂⠠⠤⠼⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⠀⣰⠷⠃
⠀⠀⠀⠀⠀⠐⢿⠟⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡰⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⠒⠈⠀⠀⠀⢸⡇⠀⣤⠀⣼⡇⣼⠟⢿⡆⢸⡇⣀⠀⢠⣶⠀⠀⠀⠘⠢⠤⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣧⣿⢸⡟⠀⣸⡇⣾⣷⣿⣇⣾⠇⠀⠀⠀⠀⠀⠀⢀⣈⣉⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⢀⡩⠆⠀⠀⠀⠀⣿⡿⠀⢻⡿⠘⢿⣴⡿⠁⣿⠟⠈⣿⡟⠀⠀⠀⠰⡒⠊⠉⠁
⠀⠠⣄⠤⣠⠤⠤⠄⠀⠘⠃⠒⠂⢤⠄⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⢠⠵⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠈⠀⠀⣀⡀⢤⠀⠀⠀⠀⠀⠃⠒⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠺⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⡠⢆⠀⢀⡠⡀⠀⢠⢦⣠⠒⢄⠀⠀⠀⢁⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⡜⢀⡠⠖⠉⠀⢸⠔⠋⠀⡇⢀⠆⠀⠁⠀⠀⠱⣄⠀⠸⠀⠀⠀⠈⠉⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⡼⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢡⠎⠀⠀⠀⠀⠀⠀⠈⠢⡀⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠂⠀⠀⠀⣼⠶⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠅⠀⠀⠀⠀⠀⠀⠠⣴⣦⡀⠘⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠉⠀⢀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢳⡀⢀⡴⠋⠘⣿⣿⡇⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠲⢦⣤⣀⣈⠻⠆⠀⣈⡵⠋⠁⠀⠀⠈⢿⣶⣄⣰⡟⠀⠀⢀⣴⠂⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣿⣷⠶⠶⢛⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢰⠃⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⣴⣿⣇⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢼⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣚⠋⠁⢿⣿⠀⢠⣆⢀⣾⣿⠏⠀⣠⣤⣄⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣤⣽⣿⡸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣸⣿⣧⣿⣿⣾⣿⠏⢠⣿⡿⠿⣿⣿⡄⠘⣿⣿⡇⠀⠀⠀⠀⣼⣿⠏⢻⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠓⢻⣿⣿⣿⢿⣿⣿⠏⠀⢸⣿⣷⣶⣿⣿⠃⠀⢻⣿⡇⣠⣶⢀⣼⣿⠇⠀⠀⢙⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣸⣿⠛⠛⠈⠛⠛⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⢸⣿⣿⣿⣿⣾⣿⠃⠀⢀⣀⣼⣿⠦⠿⠛⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠛⠋⢀⣾⡿⣿⣶⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣿⣿⣋⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣰⠋
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠊⠉⠭⠉⠀⠀⠀⠀⠀⢨⣍⠛⢿⣦⡀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⣿⢠⠛⠶⣄⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠟⢁⣾⠃⠹⣷⣀⣴⠿⢛⡩⠭⢥⣬⠉⠉⣝⠻⢦⣿⢸⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠁⢀⡾⠁⠀⠀⢹⡿⣡⣾⡏⠀⣤⡈⣿⡀⠀⠈⠧⠀⠉⠚⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠵⣿⡿⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿
⣿⣿⣿⣿⢏⠀⠀⣿⣿⣿⣿⣿⣿⡛⣿⣿⠁⠀⠀⠀⠀⠀⠀⣀⣤⡄⢰⣤⣦⠀⠀⠀⠀⠀⠀⠸⡿⢛⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿
⣿⣿⣿⣿⡎⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⡞⠉⠀⠃⠀⠀⠉⠁⡄⠀⠀⠀⠀⠀⠱⠻⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣯⡟⣿⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠘⠉⣿⣿⣿⣿⠏⠀⠀⠀⣿⣯⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠘⠀⠀⠀⠀⠈⡟⣠⠀⠀⠀⠀⠀⢠⣄⣿⣿⠟⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⠇⢹⡷⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢷⡄⠀⠀⠀⠀⣥⣸⠏⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠞⡿⠀⣶⣤⣄⠀⠀⠀⢀⣤⡴⡇⠀⣮⠓⢀⠀⠀⠀⡹⠃⠀⠀⠀⠀⢀⣿⣿⢿⣷
⣿⣷⣽⣤⣙⢿⡄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡇⣴⡇⠀⣇⠈⠛⠁⠀⠀⠉⠙⠁⠇⠀⢿⣤⣾⠀⠀⠀⠁⠀⠀⠀⠀⠀⡼⠛⣽⣿⣷
⣿⣿⡎⢿⣿⣦⢻⣄⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⠀⣿⡄⠀⠀⠀⠀⠀⠀⣼⣆⠀⠸⣿⡏⠀⠀⠀⠀⠀⣶⠀⢀⡜⢠⡄⢿⡟⣿
⣿⣿⣿⡌⠻⣿⣿⡿⠀⠘⢹⣷⡄⠀⠀⢄⠀⠀⠀⣿⠃⡄⣯⣽⣄⠀⠀⠀⢀⣼⣷⡟⢰⡄⠻⠀⠀⠀⠀⠀⢨⠋⠀⣼⢃⣿⣾⠎⣼⣿
⡿⠿⣿⣿⣄⠙⣿⣿⡄⠀⠀⠙⠿⡄⢀⡀⠀⠀⢠⣾⣷⠀⢻⣿⣥⢑⢒⠂⣹⣿⣿⡃⠂⣿⠀⠀⢀⣽⣦⠚⠁⠀⠀⣿⣿⡿⠋⣼⣿⣿
⡇⢶⣌⠙⠻⣦⠈⢻⣷⡀⠀⠀⠀⢠⡿⢷⣄⠀⢸⡆⣿⣷⡞⢿⣿⡿⠿⣿⣿⣿⣯⣿⣧⡙⡠⠾⢟⣡⣿⣧⠀⠀⣰⡿⠙⠁⣼⠿⢋⣴
⣿⣄⠻⣷⣤⣈⠁⠀⠙⠷⠀⠀⠀⣾⣥⢿⠿⣗⣼⣿⣿⣿⣿⠛⢿⣿⣶⣿⠿⢋⣽⣿⣽⣿⠃⠈⢫⣙⣻⣿⠀⣰⠟⠁⠀⢈⣡⣾⣿⠟
⣿⣿⣦⠈⠛⠻⠿⣦⣄⠀⠁⠀⠰⣿⣿⣻⣮⡁⢫⡿⠟⠛⠻⣷⠀⠈⡟⠁⢠⡿⠉⠉⠉⢻⡄⠼⠋⠀⠘⠟⠀⢎⠀⢀⣴⡿⠛⣿⡿⢀
⣿⣿⣿⣷⣄⠹⣦⣀⣽⣦⡀⠀⡄⠙⠿⠅⣼⠣⠏⠀⠀⠀⠀⠹⣧⠀⠇⢠⡾⠁⠀⠀⠀⠀⠀⢿⡄⣠⠀⢺⡇⠈⠀⢺⣥⣴⣿⠟⢡⣾
⣿⣿⣿⣿⣿⣦⠘⢿⣿⣿⣷⣄⠈⠀⢊⢩⣿⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⠚⠀⠀⠀⠀⠀⠀⠀⢨⣿⣴⠞⣻⣿⣄⣪⣿⣿⣿⠋⣰⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣄⠠⡂⢸⣿⠀⠙⠄⠀⠀⠀⠀⠀⠸⡿⢠⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠘⠿⣿⡿⣷⣍⣿⢹⣧⡀⠀⠀⠀⠀⣀⣴⡄⠁⣰⣦⣀⠀⠐⠊⠀⣰⣿⠟⠋⠁⠀⠀⢛⣯⠝⠃⣸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣟⠛⢿⣶⡟⠛⢛⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⣸⣿⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢧⠘⡏⠉⠉⠛⠇⢸⣿⣿⣿⣛⠝⠁⠀⠀⠉⠀⠀⠀⠐⢞⣷⣿⠟⠀⠐⠀⠀⠀⠀⠸⠆⢠⠟⣩⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⠙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠰⡇⠀⣿⠀⠀⠀⠀⢀⠀⠠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⠌⣠⡿⡄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣷⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⠂⠀⠙⠄⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣻⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣿⢦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠰⣷⡀⠀⠀⠀⠀⣿⠛⠛⠉⢉
⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣶⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣏⡆⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⣯⠛⢿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⢟⣿⣧⠸⣿⢿⣦⣄⠀⠀⠀⠀⣀⣴⣾⠟⣛⠋⢡⣿⣷⡀⠀⠈⢿⣷⠀⠀⠀⠀⠀⡜⣻⣿
⣿⣿⣷⣄⠙⠻⣀⠀⠀⠀⠀⠋⠀⠀⣿⣿⣿⣿⢣⣿⣿⣿⣾⣝⡿⣶⡶⣾⣋⣵⣾⣿⣿⣯⠀⣓⣿⣧⠀⠀⠈⣿⣤⠀⠀⠀⠘⣱⣿⣿
⣿⣿⣿⣿⣆⠀⠈⠀⠀⠀⠀⠀⠀⢀⣿⡻⢿⣯⣾⡇⠀⢻⣿⠿⣿⣿⣾⠿⣿⣿⡽⠋⠹⣿⣧⠹⣿⣿⠀⠀⠀⠸⣿⡀⠀⠀⢠⡿⢹⣿
⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢃⣾⠏⠀⠀⠀⠈⠉⠛⠗⠾⠟⠋⠁⠀⠀⠀⠹⣿⡑⣿⣿⡇⠀⠀⠀⢻⣿⠀⠀⠀⢁⣾⣿
⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣿⣷⢾⣿⡇⠀⠀⠀⡿⢋⠀⠀⠀⠼⣿⣿
⣿⣿⡇⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⢇⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⠖⠀⠀⠀⠀⠀⠀⠀⢿⣿⣸⣿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⢹⣿
⣿⣿⣇⠀⠐⠀⠈⠀⠀⠀⣀⣀⡀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⣿⣿⡿⠀⠀⠀⠑⠀⠀⠀⠀⠀⢸⡄⣿
⣿⣿⣿⡆⠀⠀⣄⠀⢀⣠⣤⠟⠁⠀⠈⣿⡮⣻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⢸⠃⠀⠀⣄⡀⠀⠀⠀⢀⠀⣸⡇⢸
⣿⣿⠋⠀⠠⠞⣡⣾⡅⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⠖⠀⠀⣠⣿⣿⡟⡄⠀⠀⠈⢻⣅⡀⠀⠀⠀⠀⡙⠃⠸
⣿⠃⠀⠀⠀⠈⣿⣮⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣿⣿⣿⡄⠀⠀⡰⠋⢀⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣼⠁
⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣯⣿⣇⠀⠀⣶⣾⣿⣿⣿⣿⣿⠁
⡅⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡗⠀⢠⢿⣿⣿⣿⣿⣿⠁
⣧⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⣆⣸⣾⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⣦⣄
⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣈⣿⣿⣿⣇⣿⣿⣿⣻⣷⣿⣇⣀⣀⣀⣀⣀⣈⣠⣄⣀⣀⣠⣷⣜⣿⣿⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣏⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⡴⢺⡿⢷⣦⠾⠟⢹⣏⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣏⣼⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣿⣶⣿⠁⠀⠘⣇⠀⠘⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⢰⡟⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⣴⣼⡆⠀⢀⡿⢸⣿⣿⣿⠀⠀⠀⣿⠀⠀⢹⣿⣿⡇⠀⠀⢠⣯⣧⠀⠀⣸⠇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⢠⣿⣿⡇⠀⣸⠃⢸⣿⣿⣿⠀⠀⠀⣿⠀⠀⠈⣿⣿⣿⠀⠀⣼⣿⣿⠀⢀⡟⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⡌⢹⣿⣧⣠⠏⠀⠈⢿⣿⣿⡄⠀⣰⠃⠀⠀⠀⠘⣿⣿⣧⣠⠋⣿⣿⣄⡼⠁⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠙⠛⠋⠀⠀⠀⠀⠙⠻⠿⠚⠁⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠈⠛⠛⠁⠀⠀⠀⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⡤⠘⠁⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣙⡷⠂⠉⠀⠀⣠⡄⠐⠈⠉⢀⣠⣴⣾⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣤⣀⡤⣀⠴⠆⠈⠁⠀⣠⠾⠛⢉⠀⢠⣶⣦⣈⡋⠛⠉⣡⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢪⡟⠀⠀⠤⢰⣶⠄⢠⠀⠀⢤⣴⣾⣿⣷⡾⠋⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⢀⣠⣤⠞⢳⣤⡀⠀⠰⠞⠋⠁⠀⣠⡾⢷⡶⠀⠁⠀⡾⠛⠓⠄⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣿⢹⣧⣀⣠⠶⡶⠿⠁⢀⢠⣤⣤⣴⠈⠿⠟⣉⣠⣄⠀⠈⠀⢠⡆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠀⠀⢶⡿⢿⣿⠋⠀⠀⠀⠀⠀⠈⣿⣿⣴⣿⠆⠀⡺⠛⢁⡀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣃⣤⣴⣶⣷⣶⣶⣾⡿⢶⡇⢀⡤⠈⠀⢒⠒⠀⠀⢹⡟⣿⣦⡴⠀⠴⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⣾⠿⠟⣻⣿⡿⠟⠛⠛⢁⡹⠋⠉⠀⠀⠀⣀⣧⣤⡀⠀⠊⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠷⠊⣾⣿⠯⠀⠀⠀⠀⢸⠏⠠⡖⠛⠓⠀⠉⠀⠘⠁⠀⠈⣶⣷⡄⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡘⠀⠟⠡⢀⣄⡀⠂⠀⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠟⠀⠀⠠⠄⠠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡱⠗⠀⠀⠙⣿⣧⣴⠞⠁⣼⠀⢹⣄⠀⢿⣿⠀⢀⣾⡁⠈⠀⠁⢰⣦⡤⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣦⣦⣴⣼⣿⣷⡆⠀⣻⠀⠀⢿⣷⣤⣤⣴⡾⠛⣧⢀⢀⠀⠙⢁⣠⣀⡴⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⢋⣩⠛⠿⢿⣧⡝⢿⣥⣦⠀⠙⠡⣌⢄⣹⣿⠛⠀⠁⢨⠿⠚⡀⠀⠀⢸⡿⠟⣠⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⡠⣿⣿⣶⠟⠉⠀⠀⠀⠀⠀⢀⡉⠉⠁⠀⠀⠉⡀⠋⠉⠀⠀⠀⠀⠀⠐⣄⢀⣀⡘⠳⣶⠁⠰⠶⠂⠲⠂
⠀⠀⡀⠀⢴⡈⠟⠀⣿⣋⣀⠀⠀⠀⠀⠀⠘⣭⣉⡀⠐⠂⢠⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣟⡥⠀⠒⠻⢿⠖⠉
⠀⢠⠁⠸⠀⠁⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⣫⡿⠛⠁⢶⣖⢋⡉⢰⠇⠠⣤⣤⣶⠈⠁⠀⠀⠀⠈⠿⠿⡁⠀⠀⠀⠀⣄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⡖⠀⠜⠁⣰⡄⠀⠠⢿⣿⠁⠒⠋⠐⠋⠠⠄⠀⣠⡀⠀⠀⣰⡆⠀⢰⠀⡠⢀⣿⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⡸⢿⡞⢋⡤⠀⠀⠘⠿⠂⢈⣠⣾⣆⢷⣾⢺⣿⣿⣶⣾⡿⠇⠀⠈⣿⠟⢿⣿⣯⠁⢺⣤⣀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠃⠀⠀⠀⠠⣻⣯⠃⣸⠎⠁⢉⠓⠰⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠏⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⠀⠀⡛⠀⠀⣨⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢾⠃⠀⠠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢿⠸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣄⢀⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃⠀⠘⢳⣀⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⢉⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠉⠉⠆⠀⠀⠀⢹⣧⠀⣤⠄⣀⠀⠀⠀⠀⠀⠀⡀⠰⠛⠀⠻⣿⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⡀⠀⠸⢄⡄⠂⠘⢉⣀⠉⠀⣹⠖⠀⠀⠤⢠⠞⢁⡀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠮⣀⠈⠀⢾⠁⠀⢰⣿⣿⣶⡾⢻⠟⢀⣄⡴⣾⡁⠈⢳⣆⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣀⣶⠞⠋⠛⢦⡀⠀⢀⣼⣿⣿⣯⠈⢃⣾⠿⠋⠡⣼⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⠀⢈⠀⠀⢸⡿⢿⣿⣿⡿⠛⠁⢀⣴⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣷⣿⣿⣿⣿⣿⣴⠟⠁⣿⢿⣿⠟⠂⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣮⣉⠛⢿⣿⣿⣗⣸⠗⠉⢰⣿⣷⠿⠟⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠟⠉⣀⡀⢠⣶⡶⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁⠀⣤⡞⢹⣇⣼⠋⠀⡤⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⠟⠋⢀⡠⠞⠛⠀⣠⣿⣦⠀⠰⢃⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠴⠁⠀⠀⠤⣾⣿⣿⣿⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢁⠋⣀⠴⠖⠁⠠⠿⠿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡐⢶⣤⣿⣷⣾⣿⣤⣤⢄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣄⣠⣾⠀⢀⡀
⠀⠀⠀⠀⠄⢹⢀⡌⠛⢡⡈⠁⠘⠃⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡟⠻⠟⠛⠟⠻⠛⠟⢠⠖⡆
⠀⠀⠀⠀⠁⠴⢤⣤⣶⣤⣤⣆⣤⣦⣼⠃⠇⠀⠀⠀⠂⠀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⠇⢀⠀⢀⠀⢸⢠⡃
⠀⠀⠀⠀⠀⠀⠼⠟⠛⣿⠋⠛⠧⠀⠀⠀⠀⠀⢀⣰⣦⣤⣿⣶⣿⣧⣤⣶⣀⢀⡀⠀⠀⠈⢹⣿⠷⣶⡿⢿⣟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢿⠀⠙⠀⠁⡤⢼⡄⡿⠀⠀⠀⠉⠀⠀⠿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣤⣦⣼⣤⣦⣤⣦⣠⡜⠃⠗
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⢹⡿⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⢲⣦⣼⣿⣶⣿⣧⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣾⣦⣴⣿⣀⣠
⠀⠀⠀⠀⠻⠉⢡⡏⠀⣾⡁⠠⣿⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⢋⡉⠻⠉⡉⠛⠉⢼⡇⡇
⠀⠀⠀⠀⠺⠇⣸⣧⣤⣼⣬⣤⣿⣸⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⣠⡆⠀⠀⠀⠀⠀⠀⠈⢁⣴⣈⣁⣴⣀⣶⣟⣂⣠⠖⠇
⠀⠀⠀⠀⠂⠠⠟⠋⢹⡟⠉⠻⠆⠀⠀⠀⠀⢀⡀⢸⠟⡾⠿⠟⠟⠷⠞⠻⠠⡄⠀⠀⠀⠀⣼⠟⠛⣿⠟⠻⣯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠁⢸⡆⢰⡇⢀⡽⢠⠃⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣾⣶⣿⣷⢾⣷⠚⠻⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠸⠏⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣿⣷⣿⣷⣤⣴
⠀⠀⠀⡀⠨⡿⢶⡿⠿⢿⠿⡶⡾⢢⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠈⠃⢸⠁⣄⠙⡇⣹⡦⠄
⠀⠀⠀⠀⠀⡁⠘⡇⠨⢿⡆⢰⣇⣸⠃⠀⠀⠀⠀⠐⣶⣤⣾⣷⣾⣿⣤⣴⡤⣄⠀⠀⠀⠐⠧⣼⣤⣼⣤⣡⣼⡇⢿⡄
⠀⠀⠀⠈⠀⢳⣶⣷⣶⣾⣷⡞⠻⠾⠀⠀⠀⠀⠀⣤⠈⠁⠘⠃⢸⠃⠠⠼⡇⡿⠀⠀⠀⠀⠀⠿⢋⠙⡿⠉⠙⠧
⠀⠀⠀⠀⠀⠚⠉⠀⣿⠁⠈⠛⠀⠐⠀⠀⠀⠀⠀⠀⢤⣆⣤⣦⣸⣦⣀⣴⣏⠃⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠉⣿⠋⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣦⠄
⠀⠀⠀⢠⣶⣶⣤⡀⠀⠀⠀⣀⣤⣶⢆⠀⠀⠀⠀⠀⠄⠀⠀⠀⢻⣟⣻⡇
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣦⣤⣅⠀⢀⠀⠀⠀⠀⢸⡿⠁⠈
⠀⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣷⣿⣿⣿⣿⣷⣶⣶⣾⣤⣴⣾⡿⠉⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠇⠀⠀⠀⡀
⠀⣿⣷⣶⣾⣷⣶⣿⣿⣿⣿⣿⣮⣧⣿⣿⣿⣿⣿⡿⣿⣟⣯⡉⣿⣷⣶⠀⢴⣶⣿⣿⣾⣿⣿⣿⣷⣦⣰⣶⣶⠞⣁
⠀⠈⢿⣿⣿⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣦⣤⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣾⡀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡉⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡏⣿⡏⠉⠋⠛⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⠛⠛⠛⠛⠈⢿⢿⡿⠙⢿⠟⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡇⢘⣿⡄⠀⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠘⠲⠶⣶⣌⣛⡿⢿⡟⠙⠻⣿⠀⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠉⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣢⣽⠀⢀⡀⢠⠄⠀⠀⠀⠀⠀⠉⠉⢻⡛⠁⠐⢶⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡿⠊⠁⠀⣶⡖⠀⠀⠀⠀⡀⠾⢿⣦⣄⣀⣙⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⢻⣤⠐⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣶⣾⡅⠀⠀⠙⢿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣶⣿⠗⣙⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠀⠀⢠⡿⢛⠻⣯⣠⣤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢶⣿⣶⣽⡧⠾⣿⢿⣿⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⢀⡙⠳⣶⡷⣯⠚⣿⠶⢨⡙⠿⢦⣯⣤⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⠴⠘⡆⣻⣶⣿⡿⣿⣷⣾⣿⡄⠈⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⢉⣤⣿⣿⣿⡿⣷⣤⣜⢻⣿⣆⣾⡄⠈⣹⠃⠀⡀⣀⡀
⠀⠀⠀⠀⠠⢿⠛⢦⣴⣿⣿⣿⣿⣿⣿⣿⠿⠟⣤⡄⠀⠀⠀⠀⠁⣀⣀⠐⢠⠐⠉⠉⣻⣿⣿⣷⣾⣿⣿⣿⣿⣿⠛⠻⠾⣿⣿⣶⣯⡉
⠀⠀⠀⠀⠀⠠⡘⣿⠟⠋⢈⣿⠟⠙⣿⢣⣠⣼⣷⣿⣦⠀⢀⠀⡀⣨⣏⠀⠀⠀⠀⢀⣿⣿⣿⣯⣿⣤⣬⣿⣁⡌⠀⠀⠀⠈⠙⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢘⣷⣿⣶⣿⣬⣿⣿⣿⣿⣿⣶⣦⣴⡶⠿⠿⠷⣦⣴⣶⣦⢿⣿⣿⡿⣿⣾⣿⣿⣿⠓
⠀⠀⠀⠀⠀⢳⣤⡄⣀⡄⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣮⣿⡿⠛⠛⣷⣤⠁⠙⢷⣟⣷⣿⣿⣿⣿⣿⣿⡿⣿⣽⢁⣀⡀⣀⣄⠆
⠀⠀⠀⠀⠀⠀⠙⠳⠿⠿⠿⣟⣙⣿⢻⣿⣿⣿⣿⣿⣿⣏⠀⠀⢰⣿⢻⣇⠀⠀⢿⣟⣽⣿⣿⣿⣿⣏⣴⣏⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⡿⠿⣿⣿⣿⠿⠻⣿⣽⣿⣿⠛⠁⠀⠙⢷⣶⡌⢿⣿⣿⣿⠟⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣠⠙⠁⠀⠸⣿⠋⠀⠹⣦⣀⠀⢀⠈⢹⡅⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠟⢿⣤⡈⣦⣠⣦⣾⣿⡄⠀⠀⠀⠈⠻⡇⠀⠈⣷⣄⣀⠀⢠⡄⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⣻⣤⠈⠙⠚⠉⠉⢹⢯⣿⣿⣷⣄⣀⡀⠀⣦⣀⣶⠉⠉⢹⣇⠀⢣⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠓⠛⠁⠀⠀⣸⣆⡝⠻⠙⣿⡋⠉⠉⠙⠛⠻⡀⢀⠀⠉⠀⠈⠉⠙⠻⣻⣿⡦⠤⣄
⠀⠀⢀⣀⠈⠃⠀⠀⠀⠛⠀⠀⠀⠀⠀⡀⢠⣜⠀⢉⣿⣷⣿⣿⣷⣔⠀⢀⣀⣠⣽⣧⡀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠆
⠀⠀⠊⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣿⣄⡹⣤⣿⣷⣿⣿⡿⠺⠟⠻⣟⠉⠁⠀⠀⠉⢿⡀⠈⡇⠀⣀
⠀⠀⠀⠀⣄⣤⠤⠖⠃⠀⠀⣀⠀⡄⠀⠀⠀⢶⣽⠟⠛⣿⣟⢳⣶⣔⡂⢿⣷⣶⣶⡶⠊⠉⠁⢀⣷⠄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡈⠿⡿⢿⠷⢶⡾⢋⣤⣶⠟⠻⣿⣿⣹⣟⠋⣉⣬⣿⣿⢻⡿⠶⣿⡁⠀⠀⢀⣇⡀⠐⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠿⣶⣶⣦⣴⣦⣴⣿⣿⣿⡆⣠⣼⣿⣿⣾⣟⣿⣧⣶⣾⣭⣟⢋⣼⠟⠛⠛⠷⠛⡷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡶⠟⠋⣁⡨⢉⣇⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⢙⣷⣤⣴⣶⣶⣤⣾⣆⣀⣀⣄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣬⣿⣯⠄⠀⠀⠀⠀⠀⢻⣷⡖⠂⠰⣿⢿⣿⠛⣿⣿⡿⢿⣆⠊⢠⢶⡿⠿⠿⠍⠉⠉⠠⢹⣄⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣜⣿⠿⠛⢙⠃⢸⣿⣶⣿⣿⡇⢸⡉⢻⣷⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣰⡟⣿⣿⣿⣶⣦⠀⢀⣠⣌⣹⣯⣁⣀⢉⡀⢸⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⣿⣄⡠⠀⠀⣠⣾⣷⠇
⠀⠀⠀⠀⠀⠠⢿⡜⠌⠻⣿⣿⣿⣿⣿⣿⣿⡿⣛⡟⠉⢹⡧⢿⡿⠉⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⠿⠿⢛⣿⣿⣿⡄
⠀⠀⠀⢠⣮⠀⣼⣷⡄⠀⠀⠉⢙⣛⣋⣭⣵⣾⣿⣇⠀⢿⣿⡖⢿⣾⣿⣿⣿⣿⣿⣿⠿⣿⣶⡆⠀⢀⣸⣿⣿⣏⣀⣱
⠀⠀⠀⠀⠻⣶⣿⠟⢁⣀⠀⠀⠀⡉⢉⣿⣿⡇⣼⣿⡆⠀⢻⠓⣿⣿⣿⡿⠛⣿⣿⣿⣶⡀⢸⣿⣿⣄⣀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢈⣟⡾⢿⣿⣄⠀⠀⣠⣿⣼⣿⣿⠿⠟⠃⠀⠈⢆⠙⣝⣿⡂⠀⠀⠙⢻⣿⣷⣿⣿⣿⡿⠛⠛⠿⠿⢧⡀
⠀⠀⠀⣀⠐⠋⠉⠙⠓⠒⢯⣥⣬⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⠁⠀⠀⠀⠀⢸⣿⡿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⣼⡧⠰⣄⣤⠀⠀⠀⠀⠀⠀⡙⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠲⣼⣿⣷⣿⣃⣲⣞⣀⠁⣿⣴⣤⣼⡖
⠀⠐⣿⣷⣶⣿⣧⣴⣼⣶⣴⣴⣶⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠻⠿⠿⡿⠿⠟⠁
⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢿⣆⣧⣠⣼⠠⣤⡀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⢠⣤⢸⣧⣤⣿⣼⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⡟⢿⣿⣿⣕⡙⢢⡄⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⠙⡓⣼⣿⣿⡟⣿⡧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢾⠏⢹⢛⡛⠉⠠⢈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣉⡀⠉⢙⢻⢹⠃⣿⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠸⠿⠿⢃⢀⡀⡃⡚⣿⠟⡱⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁⠀⠰⣝⢿⡿⠁⣀⣀⠀⡻⠿⠿⠁⣴⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣽⠇⢀⠁⠁⠌⠙⠋⠋⠃⡀⣋⢘⡃⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣚⢌⡃⡈⢹⠙⠙⠋⡍⢈⠈⡀⢽⡽⢯
⠀⠀⠀⠀⠀⠀⠀⠸⠛⢛⠆⢫⡎⠙⡀⣀⣠⡆⣈⠡⢨⣏⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢫⠉⠉⣄⢠⣅⠀⢘⠉⣫⠓⢚⡳⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠟⣋⠈⣅⣔⠞⠀⣉⡽⢧⠘⣀⣨⡧⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣢⢄⡉⠻⠤⣍⡉⠀⢇⡠⡝⠈⣛⠱
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠋⠍⠺⢀⣥⣽⣯⡅⣑⣙⣚⡗⠀⡠⠺⣿⠅⢼⡿⠶⡀⠀⣛⣎⣗⡋⣭⣿⣩⣇⠀⡂⠍⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡰⠀⡴⠀⠖⠛⣭⣾⡟⠈⣿⢇⣦⡭⠞⣇⢸⣇⠛⣼⠀⢨⡔⢃⣿⣄⣝⠧⢥⣄⢹⡯⠀⢿⣦⣝⠛⢦⢀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣶⣴⣤⣀⣼⣇⠿⢋⠀⣴⢉⣴⡿⡟⡏⣿⡍⣻⣛⡿⠀⠨⣟⣿⣋⣹⡏⡟⡿⣿⣦⢱⡦⢀⠛⢯⣷⣦⣤⡶⣵⣦
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡯⠟⠏⢁⣀⣿⣷⡜⡏⣇⠇⢳⣽⣮⣣⣳⣻⢗⠀⢐⠾⣛⣫⣻⣷⣧⡃⠇⡏⡿⣤⣿⣇⣀⠈⠓⠫⢿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡍⢀⡀⣾⣷⢺⣣⡞⢇⠱⡩⢞⣴⣿⣿⣿⢿⣿⡷⡼⠀⢸⣳⣿⢘⣿⣿⣿⣿⣼⠦⠁⢇⡟⣶⣛⠲⣿⡆⣀⠀⣽
⠀⠀⠀⠀⠀⠀⢸⣧⣭⣷⡾⡟⢏⢣⠸⡈⠄⣔⣈⡑⣖⠟⢉⡬⣛⣿⢙⠀⢸⢻⣿⡈⣅⡛⢖⡖⡉⣐⡌⠎⡱⢣⣏⡟⡾⣷⣷⣥⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡍⡇⡇⠷⢛⢬⠤⢙⡙⣼⠀⠽⣿⢉⠘⠉⠕⠻⣴⡛⠀⠸⢻⡸⠗⠌⠙⠃⣩⣿⠥⣸⣞⣕⡠⢮⡑⡃⠇⠌⡬⣸⠇
⠀⠀⠀⠀⠀⠀⠰⠷⠂⠲⡉⠩⡚⢔⢹⢹⡏⣙⠁⠦⣠⢄⡀⣸⣶⢾⠇⠀⠀⢮⢷⣾⡇⣀⢤⣤⢖⠸⡇⢿⣼⢩⣘⢪⢅⡣⠂⠒⠾
⠀⠀⠀⠀⠀⠀⠀⠨⡬⡵⢚⣖⡟⣹⢏⣏⣷⣾⣑⣸⣈⣾⠿⢊⡥⠅⠀⠀⠀⠀⠠⣌⡸⠟⣿⣕⣎⢙⣷⣬⣏⢿⣩⡛⣚⠲⠽⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢲⡋⠯⡥⢳⣾⣟⣿⡿⣷⢳⡿⣠⣜⣾⠃⠀⠀⠀⠀⠀⠀⠻⣎⢭⡜⣷⢻⡟⣿⣟⣿⣿⡩⡽⠿⣋⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⢌⢙⠅⣺⣵⣿⣽⣏⢼⣿⣶⣹⡣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣽⣾⣿⣏⣿⣽⣿⣿⡇⢕⢑⠵⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢧⣟⣹⣛⣳⡶⠿⢜⡿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡟⡟⡫⠷⢶⣻⣛⣉⣳⠳⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣐⣂⡒⢶⣿⠅⣚⣚⣘⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣛⣁⣛⠰⣿⡏⣒⣀⣃⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⡓⠶⠖⢒⡹⢿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⠿⣙⡲⠶⡆⣹⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⣿⠞⣿⣛⣛⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⣳⡟⢚⣿⣷⣤⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠉⠉⠁

⠀⠀⢻⣿⣷⣄⠶⢾⣿⣆⢸⣿⣿⣿⣿⣿⠻⢤⠻⣿⢟⣻⣽⣿⣦⣯⣠⣤⣠⣿⣿⡇⠀⠀⠐⣾⣿⡇
⠀⠀⠀⢻⣿⠟⣣⣹⣿⠉⠱⢿⣏⣽⣿⡏⣡⠝⠂⠀⠽⠿⢯⠿⣿⣿⡥⣼⣿⠃⠛⠀⠰⡀⣠⣿⠟
⠀⠀⣤⣀⠳⣤⣾⣿⣿⡟⢿⠿⠛⠋⠃⠌⣀⡀⠀⣠⣤⣔⣀⢻⡛⣿⣟⡫⠠⡀⠀⠜⣭⠐⠛⠁⠀⡀
⠀⠂⣿⣿⣂⠈⠿⢟⣿⢧⣿⣿⣷⣾⣆⢠⣷⣧⡼⠄⠀⢪⡿⣿⣿⣿⣿⠖⠺⠇⣶⣠⣉⡆
⠀⠀⣿⠙⣯⢇⡀⠾⣿⣾⣬⣿⡿⠛⠻⠿⠻⠓⢄⢲⣦⣿⠶⠍⠉⢿⣟⣦⣄⣴⣿⡧⡎⢿⡃
⠀⠀⠙⠀⠉⣉⡥⢀⣘⡿⣿⣿⣇⣷⣆⢠⣀⡑⣰⣘⡙⠿⢸⡷⢷⣾⣾⣿⣿⣿⢿⣷⡾⠍
⠀⠀⠀⠀⠀⡟⠀⠀⣼⣷⣿⡿⢟⣃⣶⠃⣤⣿⡈⡝⢲⣅⣿⡷⣿⣿⣿⣿⣟⣯⢻⡉⣠⡤⣤⣶⠈⢆
⠀⠀⠀⠀⠠⣧⢺⣿⣒⠿⠋⠟⣼⡿⣧⠢⣹⣿⡇⢱⠂⡈⣻⣾⣷⣿⣯⡏⢿⡞⣹⣛⠉⣑⣯⣽⠀⠘
⠀⠀⠀⠀⣰⣿⣣⡉⣌⡀⢀⣤⣌⣀⣼⣿⣏⣭⢯⣾⠟⣵⠿⠿⣿⣌⣛⣧⡤⡤⠉⠈⢷⡌⠽⡿
⠀⠀⠈⠳⢠⡌⢿⣧⡠⣆⡚⢣⣾⣽⣺⣿⣿⣿⢟⠁⠈⠃⠂⢀⠨⣿⡇⡀⠐⢅⡠⠤⣊⣁⠘⣶⡄
⠀⠀⠀⢰⡾⠓⠀⠉⠻⠷⣭⠏⠻⣿⣡⣧⡼⢇⡀⢀⣅⠀⠀⠀⢠⠿⣋⣴⠀⠈⢁⣝⠉⢧⣆⡙⡧⡀
⠀⠀⠀⣿⠘⡄⠂⠀⠁⠂⢀⣠⣄⣿⣏⢁⢠⡾⢿⡋⢫⠄⠊⠠⢤⠽⠺⠇⠀⡼⠋⢀⠀⢀⣿⠇⠹⣿
⠀⠀⠀⠃⡿⠀⠀⡀⠀⠂⢛⠆⠰⠌⠉⠀⠋⡤⣶⣓⠃⠀⠀⠐⠈⠀⠀⠀⢼⠣⠻⡤⣠⣥⣦⣀⠀⢣
⠀⠀⠀⠘⠃⠀⠙⢤⡀⠒⠀⠀⠀⠀⡀⠀⠛⠖⠋⠁⠡⠀⠰⠠⠆⠀⠀⠀⠉⣠⣴⡎⣥⠄⠈⠁⠀⠈

⠀⠀⢄⠀⠀⠀⠀⠀⠀⠂⠀⠀⢢⡀⠀⠀⢠⡄⣤⠀⠀⠀⠀⡄⠀⠀⣻⣷⡀⠀⠀⡄⠀⣦
⠈⣂⠨⡇⠀⠈⠈⣄⢒⠀⡄⠈⠀⠁⠀⠀⠈⠁⠁⡀⢠⣄⠀⢿⠛⡲⠏⣿⣧⠀⠀⠻⣶⠃⠀⠀⣠⡀⠀⠀⠀⣞⡤⢹⠀⠀⠀⢠⡔
⠠⠀⠙⠁⢀⠆⠀⠈⠙⠀⠁⠄⢦⣘⡆⢠⡀⣄⠀⠇⠺⢪⡇⠈⠻⢷⣼⣿⣯⡀⠀⠀⠂⠀⠀⠀⠾⠓⢀⢀⣄⠙⠿⡿
⠀⠀⢀⠠⠈⣤⠀⢰⣄⠀⣠⣆⠠⡌⠀⢹⣰⠇⢰⠐⠟⣫⠀⡀⠀⠀⠠⠁⡞⠁⢀⡔⣦⠰⠀⡂⡠⠄⠘⠀⠠⠀⠀⣡⠘⠀⠀⡄
⠀⢂⢺⣧⠀⢻⡶⠨⢿⡀⠈⠛⢤⢟⠀⠀⠙⠠⠀⢰⡀⠀⠀⠁⠂⠀⢢⠔⣃⣀⡎⢰⣿⠁⣾⣿⡀⠀⠀⠀⠠⠄⣀⠰⡇⠀⠘
⠀⠈⠉⠁⠤⠈⠙⠷⡿⣷⠀⠸⣄⣈⠤⢌⣳⢴⡄⣺⣄⣰⣀⠀⣷⡠⢶⡞⣵⡿⢠⣿⢿⣄⢯⣹⠛⠃⢠⠀⣷⣾⣜⠃⢂⡀
⠀⠀⠄⢢⠀⠙⠀⠀⠐⠋⣄⠀⠾⠯⠶⣘⢿⡘⢏⢉⠐⣦⣸⠷⣿⢳⣾⣿⣿⣧⣿⡿⢀⣿⢧⣇⣤⡼⢋⣴⣄⣧⠻⣿⣠⣇
⠀⠀⠀⢻⡀⠀⠀⠃⠀⠀⢌⣠⣶⣦⣄⣝⠺⣷⢼⠘⢿⠂⢘⡟⣿⢻⣿⡿⣿⣿⣿⣇⣸⣿⣿⢾⢽⣇⠘⠻⡿⣸⣧⣿⣏⢿⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠙⣶⣤⣿⣿⣍⡹⢽⠧⣧⢸⡆⣼⣶⡎⠻⣿⣿⠟⣰⡿⣻⣿⣼⡇⢹⣿⣿⣾⡿⣰⣿⡇⠻⣾⣿⡟⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⢣⠉⢧⢹⡈⠁⢻⠉⣇⣁⣙⣋⣈⣙⣁⣍⣹⣏⠀⠈⠻⣿⣿⢃⡿⢻⡿⣣⡙⠋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠜⠳⠶⠿⢈⣿⡹⣯⣿⣿⣿⣽⣥⣴⣿⣿⣿⠇⠀⠀⠘⣿⠞⠠⡃⣈⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠐⣄⠀⣦⣾⣧⢿⣽⣿⡿⠻⠿⠟⣿⣿⡿⠇⠉⠀⠀⠀⠐⣿⠂⡰⠁⠛⠣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣻⣏⢫⣽⣤⣄⣿⣍⣤⣙⣧⢈⠿⠉⠀⠀⠀⠀⠀⠀⣠⢀⡄⣤⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢦⣙⢷⣬⣿⣿⣿⣿⢿⣿⣿⡟⢤⠄⠀⠀⢠⣴⣶⣶⣿⣾⣟⣿⣿⣿⣯⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣮⣽⣿⣿⣾⣿⣿⡟⢾⣿⣿⣿⣿⣧⢾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⢿⣿⣿⡿⣿⣿⣿⣿⣿⣧⣵⣿⣿⣭⣤⣙⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣀⣀⣽⣿⣽⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡵⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢹⣟⣻⣿⣿⣿⣿⣿⣻⠿⣿⣿⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⢻⡿⠋⠛⣹⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣌⠉⠁⠉⠀⠀⠀⠙⣿⣦⣽⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠧⠿⣿⣇⣤⡀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣿⡶⠶⣔⣿⣿⣿⣿⣿⣿⢺⣼⣧⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠿⣋⣾⣭⣿⣿⡿⣟⣛⣶⠉⣼⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠾⡇⢿⡀⢿⣿⣟⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⢿⣳⣾⣿⠸⣿⣿⣿⣟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣿⣯⣿⠴⣿⣿⣿⣧⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠛⣶⠶⢿⣿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠟⠚⢫⣹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢾⣿⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⢯⣶⠟⠉⠹⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠲⣾⡶⠟⠛⠟⠀⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃⠀⣠⠄⠀⢠⡿⠃⡀⠐⢀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠁⢸⣿⡟⠀⠀⠃⠀⠀⡁⠀⠀⠀⡴⠋⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠏⠱⣾⠐⣿⣿⣧⢄⣸⣷⣶⣶⠀⢀⢠⠾⠁⠀⢸⡇⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠙⠀⣾⣳⡟⣉⠉⠻⠞⠛⠋⠀⠀⠀⢠⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡾⢳⣾⣤⡿⠷⢯⡠⠀⠀⠀⠀⠀⡄⠁⢈⠓⠦⠀⠄⠀⠀⢀⣠⡤⠚⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⢸⡍⠈⠀⠟⠁⠀⠂⠀⠀⠀⠀⠀⠀⠛⠲⢶⣴⠶⢾⣿⣝⣓⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢻⣿⣀⣸⣄⣈⠳⣦⠠
⠀⠀⠀⠀⠀⠀⠀⡤⢠⣔⠿⠀⢠⣿⠯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠛⢿⡋⠉⠁
⠀⠀⠀⠀⠐⣆⡀⣩⣥⠨⠆⠈⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠙⢿⣷⠀⣀⡀⠤
⠀⠀⠀⠀⠀⠈⠛⣋⠀⠄⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⢸⣯⡀⠘⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢷⠀⢴⡦⡀⠀⣦⣼⡿⠉⣷⣄⠀⠀⠀⠀⢀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠘⢿⣅⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⢢⠛⡘⠛⣩⠀⢠⣦⡟⠉⠀⠀⠚⠁⠀⢀⣠⠍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢲⡤⠾⠆⠉⠻⠋⠁⣤⠀⠙⠃⠃⣤⣄⣃⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢙⣄
⠀⠀⠀⠀⠀⠘⢧⡀⠀⣤⡀⠠⠀⠀⢸⣟⠀⠀⠀⠀⠈⠉⠙⢷⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⣸⡅
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠻⠁⣠⣄⠀⢀⣿⢃⣴⠂⠀⠀⠀⠀⠈⡗⠀⠐⢷⢶⣤⣄⠀⠀⠀⠀⠀⠀⠘⠛⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣖⢈⡛⠛⠁⣼⢇⡼⡏⠀⠀⠀⠀⢄⠰⠃⠀⠀⢀⣄⠈⡿⢱⡄⠀⠀⠠⠀⠘⣶⣂⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣆⠙⢷⡎⢻⢠⣿⣦⣴⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⣦⣀⠀⠻⢦⣀⡀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⣇⠸⡏⣿⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠹⠟⠒⣦⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠹⣿⡄⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣤⣀⠀⠀⠉⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣜⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⠀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡈⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⠀⠀⡀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡦⡀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠀⠀⠀⠀⣴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣴⡆⢻⣦⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡛⣷⠈⠿⠃⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠉
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠛⣿⣶⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠶⠈⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⡄⠀⠀⠄