Biscuit Emojis & Symbols

⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⢀⡀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠢
⠀⠀⢰⠏⠀⠁⠀⠠⣄⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠐⠃⡀⠈⠁⠀⠀⠉⠁⣠⡠⠀⠀⡀⠋⣀⠀⠀⠀⠀⠠⠄
⠀⠀⡆⠀⠰⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠁⠠⠖⠀⣀⠀⣀⠔⠀⠀⣰⠄⠘⠉⠀⠀⠀⢲⠆
⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣤⣐⣈⡠⠤⠛⠉⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⡀
⠀⠀⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠉⣁⠀⠀⠛
⠀⠀⠈⠀⠀⠛⠂⠀⠀⠠⠆⠰⠏⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⢠⠞⡁⠀⠀⠀⣀⠀⠀⠈⠠⣴⡄⠀⠙⠂⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣡⡿⠀⠛⠉⢉⣛⢛⣿⢫⣡⣇⡤⡜⣧⡀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢿⠇⣼⠋⣆⣿⣸⣾⣿⡞⢹⡿⠯⠔⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡼⢿⡹⠉⠳⢋⣈⣤⡿⣶⣋⣙⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠻⠿⠛⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡄
⠀⠀⠀⣀⠀⠀⡀⠀⠀⠲⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠰⠦⠀⠀⠀⠀⠀⠀⣠⢣⣬⠤⠀⠠⠄⠐⠀⢠⠄
⠀⠀⢠⠁⠶⠀⠁⠀⠀⡤⠄⠀⠙⠛⠀⠀⠀⢀⠄⠚⠁⠐⠶⢀⠀⢠⡄⢀⡀⠀⠀⠀⢿⠋⣤⢀⣀⣂⣀⢠⠀⠀⠄
⠀⠀⣻⣦⣀⠰⠆⠲⠖⠀⠰⠖⠆⠀⡜⠀⠀⡞⠀⣭⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⣴⡶⠊⠁⢡⡌⠀⠈⠐⢠
⠀⠀⢿⡿⠛⠛⠛⣶⡿⠶⠶⢖⣶⢛⡄⠀⠐⠅⠀⠀⠀⠀⠀⠀⠸⠂⠀⠄⠀⠀⠀⠀⠻⠁⠰⠆⠀⠀⠀⠐⠂⠀⠠
⠀⠀⢰⡿⣿⡯⠤⣤⣤⣤⡶⠾⠟⠛⠃⡀⠀⠀⠀⢀⡁⠀⠀⠀⣀⠀⠠⠀⠀⣀⠀⠀⣇⠤⠀⠀⠀⠠⡄⠀⠀⡀
⠀⠀⠰⣶⠦⢤⠤⡤⢤⣾⣥⡤⠴⠶⠾⠁⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡀⠰⠆⠀⡀⠀⠀⠂⠉
⠀⠀⣼⡷⣤⢭⣄⣀⣨⣤⣥⡤⠖⠒⢊⡁⠀⠀⠀⠈⡀⠈⠀⢠⣄⡀⠀⠀⠟⢀⡄⠀⠀⠳⣄⠀⠀⠀⣤⠀⣀
⠀⠀⠘⠷⠿⣤⣔⠛⣀⢀⣀⣀⣸⡧⠖⠁⠀⠀⠀⢀⠉⣠⡀⠀⠉⠀⢠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⣩⡶⠀⠀⡤⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢀⠀⠀⠀⠈⠀⠀⠀⡠⠤⠀⠀⠰⡇⠀⣯⢁⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠋⠀⢠⣴⡆⠀⠀⠀⣠⣤⡄⠀⠀⠀⢠⣿⠀⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠛⠋⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⢾⠇⠀⠀⠀⢍⠀⣀⠀⠀⣀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡀⠀⠠⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⠇⠀⠀⠇⠈⣆
⠀⠀⠀⠀⠀⣠⠀⠀⠙⠁⠀⠀⠀⠀⣠⣤⠀⠀⠀⢀⠾⠁⠀⠀⣠⡾⠁⠀⠀⠀⠸⢿⣿⡆⠀⠀⠐⠄⠀⡰⠀⢠⣿
⠀⠀⠀⠀⡆⠋⡀⠀⠀⣠⠆⠀⠀⠀⣿⠿⠀⠀⠀⢈⠀⠀⣄⣴⡿⢾⡷⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢁⣴⡿⠃
⠀⠀⠀⠀⠇⢼⡷⡶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢀⣀⣴⣿⡷⣷⣄⡚⣀⣀⠠⣌⠀⠤⠃⠀⠀⢀⣤⣼⣿⠟⠁
⠀⠀⠀⠀⢢⣶⡈⠐⠀⠀⠀⠀⠒⠒⢋⣡⣤⣴⣶⣿⣿⣿⡟⠀⠐⣦⣤⣤⠀⠀⠀⠀⣀⣠⣴⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠓⠶⠶⣦⣤⣥⣼⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⡿⢿⠿⠶⠿⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⢀⣶⡿⢿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⣶⣶⣦⣠⣾⡿⠿⠿⣿⣾⡟⠻⣿⡟⠀⠀⠙⠟⠉⣻⣷
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠻⢿⣾⣿⠁⠈⢿⣿⠏⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⢠⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢰⣿⠟⠛⢿⣷⣶⡿⠿⣿⣿⡟⠀⠀⠀⠀⠙⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿
⠀⢀⣴⣶⣿⡿⠀⠀⠀⠙⠟⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟
⠀⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣁⣀⣽⠇⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⡀
⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⢠⣾⡿⠋⠀⠀⠀⠀⠀⢠⣾⠋⠙⣷⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣧
⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⣹⣿⠟⠀⠀⠀⠀⣴⠆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢹⡧⠀⠀⠀⠀⠐⠿⣿⣦⡀
⠀⠀⣿⣏⠀⠀⠀⢀⡾⠋⣴⠟⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠹⣿⣦⣀⠀⠀⠀⠘⠃⠰⠟⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡶⠟⠁⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣄
⠀⠀⠀⢈⣿⡿⠃⠀⠀⠀⠀⠀⠀⣾⡋⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⣤⣉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠈⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢾⣿⣧⡀
⠀⠀⠀⠀⠀⠉⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⣿⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⢠⡞⠛⢻⣆⠀⠀⠀⠀⠀⠀⠘⠷⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠘⢷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣤⡄⠀⠀⣸⣿⣧⣄⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣀⣀⣼⣿⣷⣶⣾⡿⠃⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣼⣿⣦⣤⣴⣿⠋⠛⠿⠟⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⣀⣠⣼⣷⡀⣀⣼⣿⢿⣷⣶⣶⣿⠟⠉⠙⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣉⣻⣿⣿⣛⣁⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠒⠒⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⣰⠏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢰⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⡄⠀⠀⠀⠀⠀⠘⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⡆⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠋⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⠀⠀⠀⠀⠹⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠀⠀⠀⣀⡀⠀⠀⢠⣤⡄⠈⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠘⠛⠃⠀⠀⠀⠈⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢸⣿⡇⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢰⣶⡆⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⡀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠘⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠠⣿⣷⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠁⠀⢀⣤⣄⠀⠀⠀⣤⣤⡀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠛⠛⠁⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠛⠛⢳⡦⠶⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡄⠀⣀⣤⠞⠉⠉⠉⠉⠉⠉⠙⠳⠶⠒⠲⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠏⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠀⠠⠆⢠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄⠀⠀⠀⠀⠀⠀⢀⡤⡝⣆⠀⠀⠷⠦⠴⠃⠀⠀⠀⠀⣠⠶⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡤⠟⠀⡼⠁
⠀⠀⠀⠀⠀⣰⠋⠈⠱⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⠀⠀⠀⣻⠦⠤⠔⠓⠂⠀⢀⣰⠋⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠰⣲⠄⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠁⠈⠳⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠤⣤⡴⠖⠺⣏
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⣼⣤⠤⠖⠋⠁⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⢹⡆⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠾⠁⠀⠀⠀⠀⢠⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡞⠀⠀⠀⢰⡋⠉⣦⠀⠀⠈⠑⠀⠀⣻
⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠰⡇⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⢠⠋⡠⠤⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠉⢧⣀⣀⠤⠄⠀⠀⠀⣩⣇⣀⣀⣤⣤⣤⡴⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣆⠀⠀⠀⠙⠒⠛⠲⠴⣇⠀⢀⣠⠜⠁⠀⠀⢀⣠⡴⠞⠁⠀⠉⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣠⣤⣄⣤⣤⣌⣉⣉⣡⠶⠲⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⣴⣏⣀⣻⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀⠀⡀⠀⡀⢀⡀⢀⠀⣀⡀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠈⠪⢰⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠒⠂⠤⠤⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠠⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠄⠀⠀⠀⠀⣕⠨⣐⡒⠀⠀⠀⠀⠁⠐⢂⠀⡪⠅
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢈⠐⠀⠀⠀⠀⠀⠁⠩⢒⠀⠀⠀⠄⠀⠄⠄⠀⡈⠔⠁
⠀⠀⠀⠀⠠⠀⠀⠠⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠠⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠔⠇⠮⠐⠍⠐⠅⠪⠕⠎⠰⠅⠪⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠃⠀⠁⠀⠀⠈⢀⠈⠈⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⡆
⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⢨⡀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⢼⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⢄⠀⠁⠂⠀⠐⠁⠀⠀⠀⡀⠀⠐⢈⠔⠁⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⣀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠁⠐⠀⠀⠂⠐⠁⡀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠀⠂⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⣠⠂⡠⢀⡤⢂⠖⡠⠐⢀
⠀⠀⠀⠀⢠⡀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⡄⢀⠴⠋⣠⠎⡠⠊⡀⢊⠐⠃⠈⡁⠠⡞
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⠀⠘⠁⡊⠄⠤⠒⠁⠈⠠⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠀⠨⠈⠉⠁⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⣉⢀⡀⠀⠀⠀⠀⠈⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⣶⣶⠂⠀⠀⣀⣠⡧⠀⠞⠓⠀⢥⣸⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⢋⠀⠛⠛⢩⣿⠁⠰⠾⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠁⠾⠃⠀⠀⠀⣀⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡄⠀⠁⠀⠀⠀⣾⣿⢠⣶⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⣾⣿⣿⠃⠀⠠⠀⠀⠘⠻⠏⠀⢀⣠⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠀⠀⣤⣦⡤⢠⣿⡿⡏⠛⠁⠀⠂⠀⠘⠾⡿⠿⠀⢠⣿⡏⠻⡿⠀⠀⠂⠀⠄⠀⠀⣠⡴⡿⣟⡰⠂
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠉⠉⠀⠀⣻⣿⡉⡆⠶⠆⠀⢀⡀⠀⣠⣤⠤⣾⣷⣅⣐⡀⠾⠁⠀⠀⠤⢤⣴⣟⢀⡽⠉⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⣿⡄⠀⡀⠀⢀⠲⠶⠦⠄⢸⣿⣿⣿⢺⣶⡄⠀⣁⠀⠙⢿⣿⣿⣿⣏⢡⠀⠀⠀⠀⢰⣶⣴⡿⠓⠟⢀⠀⢀⣴⠂⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠳⠀⠘⠷⠀⠀⢀⣼⣿⣿⣿⣷⡏⠁⠐⠊⡘⣳⣿⣿⣿⣿⣿⣶⡖⠀⠀⠀⠈⠁⢿⣧⡔⣰⠋⢱⠈⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣶⠦⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣯⠙⠀⠀⠀⠀⠀⣿⣄⡇⢿⠀⠊⠀⣤⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢠⠤⠈⠀⠀⠈⠉⢤⣾⣿⠋⢿⣿⣿⣿⣿⠛⠛⠉⠉⠉⠉⠹⣿⡆⠀⣀⣀⠀⣸⢥⠁⠀⠰⣢⡄⡉⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣀⣀⣀⣴⠀⠀⠯⠇⢂⣁⣩⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⠿⠀⢻⣿⠀⠀⢰⡿⢿⡷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠭⠼⠻⠖⠦⠈⠀⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⡀⠀⣀⢿⡧⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠒⢂⣤⣭⣝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢚⢛⣭⣶⣦⠀⠀⠀⠀⠘⠿⠿⢟⡶⠈⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠘⣘⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠘⠃⠀⠀⠀⣀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣄⣦⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠟⠀⠀⣠⢰⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠰⣿⣿⠃⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠛⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠮⠍⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡴⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣆⠀⠀⠀⠐⠇⢾⣿⣿⡿⠀⠰⢆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⣬⣭⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⡀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠰⠋⢀⣠⢤⡀⠀⠀⠾⠁⠀⠀⠈⠁⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠀⠀⠀⠀⠙⠛⠛⡁⠀⠀⢰⣶⡲⠶⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠀⠀⠀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⣇⠀⠀⠀⠀⠀⠙⠿⠷⢀⣀⣀⣀⣀⣠⠴⠖⠋⠉⠫⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠉⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠄⠀⢀⠀⠀⠉⠉⠀⠰⠾⣿⡶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠉⠀⠰⠿⠇⢠⠒⢶⠂⠀⠀⠀⠀⠀⠀⠈⠉⠰⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡍⠉⠙⠒⠶⠤⠤⣤⣤⢀⡀⠀⠈⠀⡀⠠⣶⠶⠆⠀⠀⣀⣴⡚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠠⠒⠂⠀⠀⠀⠐⠻⠿⠿⣀⠀⠒⠲⣐⡛⠂⠤⠒⡩⠍⠁⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⢠⠖⡋⠀⠀⠀⠀⠐⠛⠆⠀⠈⠉⣠⣶⣼⣷⠀⠀⠉⠀⠈⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠳⢤⠷⣀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠈⠒⡌⠙⠓⠶⣿⣿⠷⢄⡀⠀⣀⣴⣀⣠⠤⣄⣀⣄⣀⡤⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠸⠷⠿⠃⠀⠄⠀⢀⡐⠐⠾⠋⠉⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡶⠯⠀⠀⠀⠀⠀⡄⠀⠹⠀⠀⠀⢀⠀⠀⠈⠣⡄⠀⠀⠀⣾⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠉⠓⠶⠖⡾⠗⣶⣤⣀⣀⣠⡾⢓⣲⠦⠴⠷⠤⠒⠚⠁⠀⠩⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠀⠀⠾⠁⣀⠈⠁⢀⣙⠆⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢰⣿⡇⠀⠐⣿⣳⡀⠁⢀⣤⣤⡤⠀⠀⠀⠀⠤⡄⠈⠛⠛⠃⠀⠸⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠐⠲⠶⠤⣄⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣅⣀⠀⠰⠚⠋⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠉⠀⠛⠛⠓⠒⠖⢒⠀⠀⠀⠊⠉⠁⣀⠀⠀⠀⠀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡈⠁⠀⠀⠀⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⠒⠈⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠰⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⣰⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⢸⣿⠃⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣇⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠈⠉⠉⠁⠀⠀⠘⠟⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢰⣿⡏⠀⢀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠘⠟⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⢴⡆⠀⠀⣠⣶⣶⣄⠀⠙⣿⡗⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣼⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄⣁⣀⣋⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠯⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠛⠛⠲⢦⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠟⠁⢠⣶⣟⣷⣦⠀⠀⠀⠀⠀⠉⠛⠿⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢯⣝⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⢞⣋⣵⠏⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣿⣦⡀
⠀⠀⠀⠀⠀⣰⠟⣱⠟⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦
⠀⠀⠀⠀⣠⢋⣼⠃⠀⠀⠀⠀⣼⣿⣿⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⢀⡴⡡⠚⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣿⡆⠀⠀⠀⠀⠀⠀⠸⡿⠻⢿⡿⠇⠀⠀⢠⣶⣿⣶⣦⡀⠀⠀⠘⡆⣿⡆
⠀⢠⡾⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣼⡷⠀⠀⠀⡿⣾⡇
⠀⣿⠃⠀⡀⣀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⠟⠁⠀⠀⠸⣧⢻⣇
⠀⡿⠀⠀⢻⣿⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠊⣿
⢠⠃⠀⠀⠀⠻⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⡄⠀⠀⠀⣠⣶⡄⡆⢸
⣾⣼⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⡿⠟⠚⠀⠀⢠⡞⣿⣿⡇⣷⢸
⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣝⣷⠀⠀⠀⠀⠹⢿⣿⡿⢟⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⢣⣿⡟
⠘⣯⢷⣄⠀⣤⡄⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣿
⠀⠘⢷⣿⠿⢿⣿⠿⢷⣾⣿⣥⣄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣿⡟
⠀⠀⠀⠀⡳⡌⠁⠠⠶⠁⠀⠿⠿⣷⡦⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢳⡄⠀⠀⢀⠴⣻⡟⣸⡏
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⢿⣿⣿⢟⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⢀⣴⡿⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣄⠙⠯⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⠗⠀⢠⣾⢫⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⣿⠂⠀⠀⠀⠀⣠⣶⠶⣤⡀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⢣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡤⠄⠀⠔⣿⣿⣾⣿⡇⢠⡶⡶⠿⢟⣋⣉⣭⣭⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣟⣛⣛⣓⣙⡻⠟⢋⣐⣭⣼⣷⣾⠿⠟⠛⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣽⣏⣙⣻⣿⣿⣛⣛⣻⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠚⠁⠀⠈⠀⠀⠀⣀⡀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⠁⠀⣠⡄⠀⠀⠈⣁⡀⠀⠈⠁⢸⡇⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠁⠀⠸⠋⠀⠀⠀⠸⠟⠁⠀⠀⠀⣾⠁⠀⠈⢉⠀⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⣰⡏⠀⠀⠀⣀⠀⠀⠀⠹⠗⠀⠀⠇⠙⡀
⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⢠⣶⠀⠀⠀⡜⠁⠀⠀⣴⢋⡀⠀⠀⠘⠻⠿⠄⠀⠀⠂⠀⡔⠀⣼⠇
⠀⠀⠀⢸⠀⣄⣀⠀⠞⠀⠀⠀⠸⠋⠀⠀⠀⠐⠀⣶⣞⡛⠛⡉⠀⠀⠠⠄⢀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⢈⣘⠛⢏⠀⠀⠀⠀⢀⡤⠆⠀⣠⣠⣶⣾⣿⠛⢛⠂⠀⠒⠀⠐⠀⠉⠀⣀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠙⠧⣤⣄⣀⣀⣀⣤⣶⣾⣿⡿⠿⠿⠿⠿⠦⣶⣿⣿⣶⣤⣤⣴⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⢐⠒⠫⠇⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⠃⠀⢷⠀⠈⠀⠀⡀⠂⢀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⡁⠀⢀⠀⠀⠀⠀⠀⠀⠣⠀⣀⠄⠜⠊⠋⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⡨
⠀⠀⢠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠂⠒⠂⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⠀⢠⠀⠸⠤⠒⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠄⠀⠙⢀
⠀⠀⠀⠐⠤⠤⠔⠀⠀⢀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⢁⣀⡠⠂⠐⠊
⠀⠀⠀⠀⠀⣀⠤⠀⠉⠀⢀⣱⠄⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⡤⠴⠒⠋⠁
⠀⠀⣄⠂⠉⠀⠀⣠⣴⣾⠟⠋⠀⠀⠢⢄⡀⠀⣠⣀⠀⠀⠀⠀⠈⠉⠉⡉
⠀⠀⢿⣧⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠠⠂⠀⠀⠘⠀⠀⠀⠴⠁⠀⡸⠁⠀⠀⠀⠀⠈⣆⠀⠀⠰⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⣟⢶⣤⣄⣀⣤⠶⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣬⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠻⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢹⡇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣠⡾⠃⠀⠀⠀⣰⡟⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠘⣷⡀⠀⠀⠀⠻⣦⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⣿⠁⠠⣿⣉⣻⡇⠀⠀⣿⣋⣹⡧⠀⢹⡇⠀⠀⠀⠀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣦⠀⠀⣿⡄⠀⠙⠻⠛⠁⠀⠀⠙⠻⠛⠁⠀⣼⡇⠀⢀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠸⣷⡀⠀⠀⣤⡀⠀⠀⣠⡄⠀⠀⣰⡿⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠈⠛⠛⠛⠋⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣿⠓⠀⠉⠛⠿⠶⠶⠶⠾⠟⠋⠀⠐⢻⡿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠿⠀⠀⠀⣤⣀⠀⠀⠀⢀⣤⡄⠀⠀⠸⠇⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣤⣤⣤⣤⣤⣬⣽⣷⣦⣶⣿⣭⣤⣤⣤⣤⣤⣄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⣿⣦⣼⠿⣧⣤⡿⢿⣤⣼⠿⣧⣤⡿⢿⣤⣾⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡿⠷⠶⠾⠿⠶⠶⠿⠷⠶⠾⠿⠶⠶⠿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⡾⢻⣷⠟⠻⣶⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⣠⡾⠟⠁⠀⢘⣿⣀⠀⠀⠙⠿⣦⡀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣿⠿⠋⠀⠀⣀⣴⠟⠋⠛⢷⣄⡀⠀⠈⠻⢿⣇⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⠛⠁⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠙⠻⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣾⣷⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣶⣶⣶⠆





⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣟⣿⣟⣛⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⢀⠼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⢸⠈⣆⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⣆⠀⠀⠀⡰⡃⡇⠀⠀⠀⠀⣼⢰⡿⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠞⣹⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡉⢣⡀⢈⢱⢁⠀⠀⠀⠀⠀⠒⢸⡇⢘⡃⠀⠀⠀⠀⠀⠀⢀⠞⣴⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⢿⡄⠳⡏⣿⢨⠀⠀⠀⠀⠀⡇⣸⡇⢨⠀⠀⠀⠀⠀⠀⢀⡞⢰⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⡄⢖⣿⢸⠀⠀⠀⠀⠀⠆⣿⡇⠾⠀⠀⠀⠀⠀⠀⡌⢡⣿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⣿⣸⡟⡶⢶⠶⣛⣶⠾⣿⣿⣷⠛⠶⠦⣤⣀⠀⡰⢣⣼⢇⣮⠠⠔⠖⠒⠺⢟⣻⠖⠛⠃
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠿⠋⠐⣃⠨⣭⠉⠃⢹⠍⢯⣉⣠⠀⠀⠈⠙⢷⣽⣿⣾⡶⠶⢿⣛⢫⠔⠋⠁
⠀⠀⠀⠀⠀⣵⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠶⠋⠀⠈⠉⠁⠀⠀⠀⠀⢀⡙⢿⣯⣴⢶⣎⠉
⠀⠀⠀⠀⠀⠘⣇⡝⢦⠀⠀⠀⢠⣾⠇⠙⠀⠀⢀⣤⣶⢤⣄⠀⠀⠀⠀⠀⠀⣀⡴⢾⣿⣟⣳⡄⠀⠀⠉⢸⠋⢨⡑⢍⠳⣄
⠀⠀⠀⠀⠀⠀⢡⡹⣦⠁⡄⠀⣾⠡⣤⡇⠀⢠⡿⠛⠛⠳⣤⡇⠀⠀⠀⠀⠀⣧⡞⠋⠁⠈⠙⣿⡀⠀⠀⢹⡄⣄⠀⣂⢧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠱⣛⣿⡹⡾⢏⣠⡟⠀⠀⠛⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⠀⠈⠀⠀⠀⠈⣿⢹⠰⡇⢸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⣀⠀⠀⠀⠙⣿⣿⠃⠈⠛⠁⠀⠀⢀⠖⠋⠛⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠳⣄⠀⠀⠀⠀⠙⣜⣃⡴⠋⢻⡆⢻⡄⠀⠀⢀⣠⠤⠖⠛⣋⠙
⠀⠀⠀⠰⣄⣤⣀⢠⣴⠏⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠈⠉⠐⠋⠀⠹⣆⢻⣦⠚⣫⣦⠞⢀⡴⠃
⠀⠀⠀⠀⠈⠺⢝⠻⣿⠀⠀⠰⢓⡆⠀⢀⠏⠀⣀⣵⣥⠀⢱⠀⠀⠀⠀⠀⠸⡇⠀⠀⢈⣬⣧⠀⢻⠀⠀⠀⠀⠀⣠⢶⡖⠄⠀⢻⣯⣴⡿⠟⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣺⠰⣧⠀⠸⠀⠰⣏⣠⣿⠆⡌⠀⠀⠀⠀⠀⠸⠄⠀⢀⣿⣩⣿⠃⠘⡆⠀⠀⠀⠐⣏⢈⡀⣴⠀⢸⣿⣭⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⢠⡟⠀⠀⠀⠀⣿⢻⡇⠀⠇⠀⠀⠀⠀⠀⠀⡄⠀⠈⠸⡟⣿⡀⠀⡇⠀⠀⠀⠀⠀⢯⠞⠁⠀⢸⡏⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⢠⠀⣦⣿⣿⢃⡻⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣄⣿⣿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡝⠳⠀⠀⠈⣄⠘⠿⢟⡾⠁⠀⠀⠴⠒⠤⠀⠀⠈⢦⠀⠹⢿⡿⢁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠾⠆⠀⠀⠀⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⠴⠋⠀⠀⠀⣤⠤⣄⠀⠀⠀⠀⠀⣸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⠋⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⣄⠀⣸⠛⠀⢸⠂⠀⠀⠸⢀⡿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡀⠀⠀⠈⠀⠸⣿⡟⠓⠤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢴⣶⣶⠀⠀⣿⠀⣷⢠⢃⣾⠃⠀⠀⢀⡾⢁⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠁⢸⠐⠀⠀⠀⠹⣿⣦⣤⣀⣉⣙⠛⠛⠉⢁⣀⣀⣼⣿⠟⠀⠀⠁⠀⠈⠛⠋⠁⠀⠀⠀⣾⣛⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡘⢷⣦⠀⠀⠀⠹⣿⣿⡿⠿⠻⣿⠿⠿⠛⠛⢛⠛⠋⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⣼⣿⠹⠿⠁⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⠀⠀⠀⠙⢧⡀⠀⠈⠀⠀⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠊⣀⣤⢴⣶⣦⡀⢀⣴⣿⡙⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡿⣷⠄⠀⠀⠀⠀⠙⠲⢤⣀⡀⢹⣿⣿⣿⠀⢀⣤⣤⣄⠀⣶⣾⣿⣿⠆⣠⣾⣿⡃⢸⣿⣿⣿⠸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠱⠏⠜⠛⢦⡄⠀⣀⣤⣤⣤⣤⡀⠀⢸⣿⣿⣿⣠⣿⡿⠟⠃⠀⢻⣿⣿⡟⠀⣿⣿⣿⠷⣿⡿⠟⠁⢀⣭⣿⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣤⡀⠀⢠⣾⣿⠇⠈⣿⣿⣿⡄⢸⣿⣿⣿⢻⣿⣿⣆⠀⠀⢸⣿⡿⠿⠀⣛⠻⣿⣆⠀⠀⣴⣶⡆⠻⠿⠷⠶⠛⠁
⠀⢀⣴⡶⠻⣿⣿⣦⠀⣴⣿⣿⠁⠀⣿⣿⣿⡄⣿⣿⣿⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣸⣿⣿⣿⣷⡄⣿⣿⡅⣾⣿⣿⠇⠙⠛⠛⠃⢙⣋⣁⠀⣠⡶⣶⣦
⣰⣿⣿⠇⢀⣿⣿⡿⢰⣿⣿⣿⠀⠀⣿⣿⣿⠃⣿⣿⣿⡀⢀⣿⣿⠟⠀⣿⣿⡿⠿⠃⠙⠛⠛⢉⡀⠀⠀⠀⢸⣿⣿⡴⣶⣶⣦⠸⣿⣿⡿⢸⣿⣷⣾⣯⡀
⣿⣿⣿⡀⠀⠛⠋⡁⠈⢿⣿⣿⡀⢠⣿⡿⠋⠀⠈⠛⠿⠿⠟⠛⠁⠀⣀⢀⣠⣤⡀⠀⠀⣴⣿⠋⢿⣿⣆⠀⢸⣿⣿⠀⢸⣿⣿⠀⣿⣿⣇⢀⣽⡿⢿⣿⡇
⠹⣿⣿⣧⣄⣠⡼⠃⠀⠀⠉⠛⠛⠋⠁⠀⣀⣠⣤⣠⣶⣶⣤⠀⢾⣿⣿⠏⠙⣿⣿⡆⢀⣽⣷⠞⢿⣿⣿⠀⣸⣿⣿⣀⣼⣿⠟⠰⣿⣿⠿⠈⠛⠛⠚⠉
⠀⠀⠉⠉⣉⣥⣄⣀⠀⣠⣶⡟⢻⣿⣦⠀⢻⣿⣿⠁⠘⣿⣿⡆⢸⣿⣿⡇⢀⣿⣿⣇⢸⣿⣿⣠⣾⣿⠿⠃⠻⠛⠋⠙⠋⠁
⠀⠀⣠⣾⡏⢨⣿⣿⠂⣹⣿⡶⢾⣿⣿⡀⢸⣿⣿⡄⢸⣿⣿⣧⢼⣿⣿⠇⠸⠟⠛⠋⠀⠉⠉⠁
⠀⠀⣿⣿⡇⠘⠛⢉⢰⣿⣿⣄⣸⣿⣿⠇⠺⠿⠟⠃⠈⠋⠉⠁
⣀⣀⣙⣻⣷⣄⣐⣋⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⢻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣬⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⠔⠊⢹⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⢀⣀⢀⡜⠁⡰⠃⠀⠀⢸⠀⣠⠀⡰⠊⠉⠲⣄
⣴⡖⠤⠋⠈⠋⠀⠰⡁⠀⠀⠀⣸⡤⠄⠠⢇⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠓⠊⠉⡆⢀⡠⠤⣈⢧⠀⢼⡡⠤⠄⠒⢶⣸⢶⡖⣒⡇⠀⣀⣀⠤⠤⠤⠔⠲⠤⢄⡀⠀⠀⠀⠀⢾⡵⠒⣷⣄⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠁⠀⣀⣈⣯⣷⠀⠉⠓⠒⠚⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠸⡅
⠀⠀⠀⠀⠀⢀⣸⡁⠠⠶⠮⠁⠀⠀⢀⡀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡔⠛⢤⠀⠀⡟⠲⡁
⠀⠀⠀⠀⠀⠈⢧⢈⣲⠶⠒⢢⠀⠈⠙⡧⣄⠀⠀⠀⠀⠉⠂⠄⢀⡀⠀⠄⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢱⠸⠊⠀⠀⠈⡯⣉⣱
⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⢀⡜⢣⠀⠀⠹⡜⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⠤⢤⣤⣀⠠⠄⣶⠤⠒⠁⣀⠷⠖⠒⠚⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠑⢤⣣⣀⡀⠈⠃⠀⠙⠂⠀⠀⣀⣀⣀⣀⡤⢄⣈⢻⠉⠓⠤⢀⠀⡇⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⡝⠀⠙⠢⠤⣄⣈⣉⠉⠉⠉⠁⠀⠀⡆⠀⠀⠉⠳⠝⠂⠀⠃⠉⠛⣢⣄⡀⠀⢀⣀⢠⣤⠤⠒⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂⠃⠀⠀⠀⠀⠀⠀⢀⣠⠒⠉⠀⠤⠋⠁⠀⡠⠊⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠒⠌⡒⢄⠄⠀⢀⡞⠁⠀⣀⠄⣠⣞⠾⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⡛⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⠞⠛⠂⠤⢾⣴⠊⢉⣼⡶⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢦⣕⡢⢄⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠑⠻⣁⠀⣀⡠⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠉⠓⠮⣕⣢⢄⣠⢎⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠃⠀⠀⢀⣀⠜⣹⠙⠓⠦⠭⣕⣢⣤⣄⣀⣀⣠⡯⠛⠉⡷⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠤⠖⣊⠭⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢩⢹⠀⠀⠀⡰⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠒⣹⠁⠀⠀⣘⠤⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠻⠔⣒⡺⠅⢴⣉⠦⢶⡇

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠤⠀⠐⣤⠄⠗⠲⠄⠀⠀⠦⣴⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣶⠖⠄⠁⠀⠀⠐⠂⠀⣀⠘⢀⣄⡖⢦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣰⣶⣮⣤⡄⢴⣿⡓⠻⠿⠁⢩⡿⣿⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣻⠋⡅⡇⠟⠀⠀⠀⠰⢄⠀⠱⠖⣳⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⡾⠃⠀⣠⠞⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠛⠁
⠀⠀⠀⠀⠄⠀⠀⠘⠻⣶⣿⣷⣿⠞⠉⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠓
⠀⠀⠀⠠⠀⠀⠀⣇⣼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣄⣀
⠀⠀⠀⠀⠠⠶⢿⣿⠓⢦⣀⠀⠀⠀⠀⠀⠠⠖⠶⠀⢀⢀⡌⠹⠿⠙⠄
⠀⠀⠀⠀⠀⠀⢾⢡⣽⣾⣼⢁⢄⠀⠀⠀⠀⠀⠉⢄⡈⠉⡁⠀⠐⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠘⠈⣿⢎⠈⢠⣆⠓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⣿⡠⡀
⠀⠀⠀⠀⠀⠀⡘⠴⠴⣦⠁⣸⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⡀⢠⣄
⠀⠀⠀⠀⠀⠀⣄⢂⣨⣿⣿⣾⣿⣿⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⡉⠓
⠀⠀⠀⠀⠀⠀⢹⢘⠙⠻⡟⠿⠿⣿⣟⠟⣆⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⣿⣿⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⡿⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣶⣶⣶⣶⠀⠀⠲⢦⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠖⠻⠷⣦⡀⠀⠀⠙⡿⠿⠿⠿⠦⠤⠯⠝⠀⠐⢂⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠛⠛⠀⠐⠻⢶⣶⣶⣇⡀⠀⠀⣤⠤⡄⠀⠀⠀⠐⠛⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢔⢶⣶⣿⠻⠙⠮⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⢿⣿⣶⣂⣀⣚⢛⠿⢷⡶⣖⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠈⠶⢦⣤⣤⣠⣤⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠃⠀⠀⠀⡌⠁⠉⠀⠈⠁⠈⡆⠀⠀⢠⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡠⠈⠀⠁⢄
⠀⠀⠀⠀⢸⠀⠉⢀⡀⠉⠀⡄⠀⠀⠀⠇⠀⠉⡀⡈⠁⠀⡆⠀⠀⠸⠀⠈⢁⢀⠉⠀⢸⠀⠀⠀⢠⠡⠄⠀⠠⠌⠄
⠀⠀⢀⠀⠀⠀⠀⡠⢄⠀⠀⠃⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢀⠂⠐⠂⠀⠸⠀⠀⠒⠀⠀⠄
⠀⠀⠀⠀⠀⡄⢠⠁⠀⠆⠀⠀⠀⠀⠀⢸⠀⢸⠀⠸⠀⢠⠀⢀⡀⠀⠀⢃⠀⠀⢀⠃⠀⠀⠀⠀⠀⠱⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠡⠜⠀⠀⠠⠜⠀⠀⠀⠀⠀⠦⠆⠢⠇⠃⠌⠀⠀⠀⠀⠀⠀⠢⠠⠊⠀⠀⠀⠀⠀⠀⠀⠣⠀⠜


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀
⠀⠀⠀⣀⣴⣬⡽⠿⠲⣷⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂⠀⠀⡠⣄⠀⠀⣀⢀⢀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⢨⣏⠙⠒⠂⠉⢁⢹⡆⠀⢀⠞⠁⠀⠀⠀⠀⠀⠢⡑⠄⠀⠀⠀⡠⢤⠤⡀⠀⠀⢨⠀⣯⠀⡎⠀⠹⡇⠀⢟⣏⡮⠁⠀⠀⢸⠁⢩
⠀⠀⠀⣾⠐⢒⣈⡉⠤⠄⣿⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠇⡘⠀⢀⠎⠀⠀⠱⣑⡀⠀⠈⠄⢸⠀⠘⠢⡔⠁⠀⢣⢠⠚⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢹⡁⠤⠤⠐⢒⡀⢸⡀⠀⠘⠢⠤⠤⢤⠤⡤⠤⠊⠀⢠⠃⠀⠱⠓⠀⠈⢷⡀⠀⠇⣨⡇⠀⠆⡇⠀⠀⡆⡈⠀⠀⠀⡠⠃⠀⠀⣷⡄
⠀⠀⠀⠀⡷⠖⠬⠵⢂⣒⠭⠇⠀⠀⠀⣇⣂⣈⣀⣁⣀⠀⠀⠧⢤⣦⣤⣤⣤⡤⠬⠇⠀⠀⢐⣠⠀⢁⡇⠀⠀⢇⠇⠀⠀⠀⠧⠤⠤⠤⠬⠬
⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⢸⠀⠠⡤⡂⠀⠈
⠀⠀⠀⠀⠀⣀⠤⠄⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⡉⣿⣻⣉⢉⣉⢉⣏⣟⣉⡉⠉⠉⠀⠀⢸⠄⢈⣀⣀⡀⢤
⠀⠀⠀⡠⢊⡤⠊⠉⠉⠁⠢⡑⢄⠀⠀⠀⠀⣠⣖⠉⡄⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠉⢁⣴⠞⠛⢶⡌⠉⢳⡇⠀⠀⠀⠀⠈⣄⣀⠀⠀⣀⡄
⠀⠀⢰⢁⢋⠔⢲⡄⣀⢀⢀⠘⡘⡄⠀⠀⡐⠕⠁⣼⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⣵⣶⢠⣤⣯⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⠘⣇⠀⢸⠅⣋⣿⣸⢀⢃⠃⢀⠤⠥⠤⠤⡤⠤⠶⠶⠶⠶⠶⠼⠤⡀⠀⠀⠀⠀⢻⡧⠤⣿⠃⠀⢸⡇⠀⡆⡶⡆⣖⢆⡶⡠⡆⢒⡞⡄
⠀⠀⠀⠣⣩⢲⠉⠀⠳⡄⡵⢋⠜⠀⠈⠋⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠁⠀⠂⠓⠒⠀⠁⠀⠈⠀⠒⠃⠇⠀⠥⠥⠥⠭⠭⠤⠥⠥⠭⠗⠃
⠀⠀⠀⠀⡄⡸⠭⠉⠘⠃⡆⠁⣠⡎⣙⢢⠀⠀⠀⢀⣀⣀⣀⠠⠤⢒⠒⡄⠀⠀⢠⠒⠒⠂⢐⣒⡦⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠳⠁⠀⠀⠀⣄⠃⠀⢟⢲⠲⢽⠂⠀⠐⠁⡜⡐⢠⢰⠀⠓⢁⡿⠀⠀⢨⣠⡤⠤⣤⣤⡄⠀⠀⣠⣴⣡⣤⣤⣤⣤⣤⣤⣤⣼⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠈⠀⡌⠀⠀⢻⣄⣀⠃⠘⣀⡡⢴⡩⠎⠀⠀⠘⡇⢀⣰⣿⢷⡇⠀⠀⠇⠊⣠⠤⣄⡤⣤⡤⢠⣄⡌⢱⠈
⠀⠀⠀⣀⠤⠤⠤⠠⣄⠀⠀⠈⠙⠋⠙⠁⠀⠀⠈⠑⠣⠷⠾⠐⠋⠉⠀⠀⠀⠀⠀⣷⠻⠿⠀⣸⡃⠀⠀⡏⢇⢙⠖⢋⢋⡘⠧⠘⠘⢃⡼⢋
⠀⠀⠀⡗⠒⢀⡀⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠋⠉⠙⠋⠀⠀⠀⠁⠂⠐⠀⠂⠀⠂⠀⠀⠒⠀⠒⠁
⠀⠀⢠⢸⠟⠂⢒⠘⡇⡀⠀⠀⡏⡇⠤⠤⠤⠤⠴⠥⠭⠽⠥⠵⠤⠤⠤⠤⢄⢈⢸⠀⠀⠠⠒⠊⣑⣒⣉⣓⠒⠤⢦⡠⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠈⢌⢆⡀⢀⠰⡱⠁⠀⠀⡇⡇⠉⠉⠉⠉⠉⠉⠛⠙⡏⠉⠉⠉⢉⠉⠁⡀⢸⠀⠀⡸⢊⠉⡇⢂⢠⢠⠈⡑⠤⣰⠗⡂⠀⠮⠄⠄⠽⡀
⠀⠀⠀⠀⠑⠲⡖⠈⠀⠀⠀⠀⡇⡤⠤⠤⠈⡆⠀⠀⠀⠀⡇⠀⠀⢸⢠⠀⠀⠄⢸⠀⠀⡣⣀⠀⡇⠸⠸⠸⠀⣃⡴⠳⡀⠅⠀⡟⢻⠙⠋⡅
⠀⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⢸⠀⠀⠀⢸⠀⠀⠉⠢⠽⢓⡒⣒⡲⠩⠔⠁⠀⠈⠁⠀⣆⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠈⠈⠁⠀⡴⢌⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠊
⠀⠀⢠⢰⠰⣠⠤⠤⢤⡴⠖⠚⠛⠉⠀⠀⢀⠤⣒⠒⠀⡰⠖⠉⢳⠁⠀⠀⠀⠀⠀⠀⠀⢴⣶⡖⢲⠮⠅⠒⠀⠀⠀⠀⠀⠀⣀⢴⣂
⠀⠀⡬⠤⠤⠤⠥⠬⠭⠤⢤⠀⠀⠀⠀⡴⢁⠞⡝⠀⠀⡁⠀⣁⠎⢡⠀⠀⠀⠀⠀⠀⠀⠨⣿⠗⠃⠀⠀⠀⠀⠀⠀⠀⢀⠏⠙⠒⢋⣉⢶
⠀⠀⢇⢆⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⡇⠘⡄⠙⢦⠀⠈⢉⠡⢂⠜⠀⠀⠀⠀⡠⣺⣟⣻⣻⣿⣛⢷⢦⠀⠀⠀⠀⠀⡼⠔⢚⣉⡡⠤⠚
⠀⠀⢠⡓⠕⠀⠀⠠⠤⢺⡄⠀⠀⠐⢄⣉⣀⣈⣉⣉⣀⣀⣁⣈⣉⡰⠂⠠⡆⠐⠓⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⡤⠀⠀⣇⣐⣁⠬⠤⠒⠋
⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⣀⣤⡒⠐⠲⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⠖⠒⠢⢄
⠀⠀⠀⠀⢀⡤⠚⠋⣶⠰⠎⠀⠀⢀⡀⠀⠒⠀⢀⣄⠀⠀⠀⢀⠴⠏⠈⠋⠁⠀⠀⠀⣠⠈⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⣩⡜⠷⣄
⠀⠀⠀⡐⠉⣀⠀⠀⠀⠻⡄⠡⠀⠙⠁⠈⠀⠀⠉⠀⠀⠀⡐⠫⠇⠀⠀⠀⠀⠰⠇⠁⠀⠀⠸⠟⠈⠂⠀⠀⠀⡇⢸⠀⣴⡄⠀⠄⠠⡄⠀⠀⡀⠐⢃
⠀⠀⢰⡇⠘⠿⠀⠀⠀⠀⠸⠶⣆⣁⣒⣢⠶⠊⢂⠀⠀⢘⠀⠀⠀⢴⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⠃⢰⠀⠋⠀⡀⠀⢤⠛⠀⠈⠛⠀⠘⠆
⠀⠀⣺⠁⡀⠀⠀⠀⣦⠀⠀⣀⠈⠀⠉⠁⠀⠄⢘⠀⠀⢸⠀⠀⢠⠈⠉⠁⠠⠄⠀⠀⠀⢀⠀⠀⡀⢀⡅⠀⠀⠀⢹⠀⠀⠐⠃⠀⠀⠂⠀⠀⠀⠀⠀⡇
⠀⠀⢧⠀⠀⠀⠀⠇⠉⠀⠀⠻⠃⠀⠀⠀⣶⡄⡘⠀⠀⠸⠀⠀⢰⡆⠀⠀⠀⠀⠀⠄⠀⠀⠁⠸⢃⡐⠃⠀⠀⠘⡀⠳⣄⠀⠀⣶⡄⠀⣀⠀⢾⡄⡤⠃
⠀⠀⠸⡅⠘⠿⠀⠀⠀⠀⠀⠀⠀⠒⢾⠃⢩⣰⠁⠀⠀⠀⠳⣄⡀⠀⠀⠀⠈⠲⣶⠖⠀⠀⢠⡤⢔⠎⠀⠀⠀⠀⠘⢦⡙⢧⣄⠀⠀⠀⠻⠀⠀⡘
⠀⠀⠀⠙⠦⣆⡀⠀⠀⠀⢶⠄⡀⡀⢀⣰⣿⠃⠀⠀⣀⡀⠀⠈⠙⠧⠄⣀⡀⠀⠀⢀⣐⠠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⢷⣿⡷⠾⠆⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⢶⣶⠿⠟⠋⠀⠀⢠⠞⠉⠉⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡌⠀⠀⠐⠂⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⢀⡘⠃⠀⢰⠀⠀⠂⠉⣀⡤⢀
⠀⠀⠀⢀⣁⡸⠇⠐⠀⠀⠀⡰⣊⠙⠛⢀⠉⠉⠳⠶⣄⠀⢀⡀⠀⠙⠂⠐⠀⠀⠀⠀⠀⠐⠃⣠⣦⠀⠀⠈⠀⡆⠀⠀⠁⠇⠀⠀⠀⠰⡄⢣
⠀⠀⠀⠀⠁⠈⠀⠀⢀⠖⠁⠀⠛⠀⠀⠛⠁⠀⠀⣀⠘⠄⠘⠓⠀⠈⠛⡄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⡗⠁⠀⠀⠀⠀⠺⠗⠀⢷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠰⠀⠀⠐⢒⠀⠀⠀⠀⠙⠃⠈⣆⠀⠀⠀⢀⣁⠈⠃⠀⠀⠘⠂⠀⠀⡀⠀⠿⠀⢀⡅⠂⠀⡆⢰⠟⠀⠀⡀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⠙⠶⠀⢸⡀⠀⣶⠄⠁⠰⠃⠀⠀⡇⠀⠉⠘⠃⠀⠀⢠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠟⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⠃⠀⠀⠀⠐⠀⠀⠠⢠⠎⡈⠠⠠⡀⠀⠂⠀⠀⠀⢣⣠⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⣀⢄⠀⠀⠀⠱⣄⠲⢀⣀⠀⢐⣀⣠⣠⣤⠖⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡠⠋⠀⠀⡤⠶⠉⢂⠀⣆⠲⡶⠀⠀⡠⠁
⠀⠀⠀⠃⠂⠑⠄⠀⠀⠈⠑⠦⣍⣈⣉⣉⡉⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡩⠁⣀⠀⡂⠀⠀⣤⣄⠀⠀⡀⣀⠀⠈⠀⣀
⠀⠀⠀⡾⠷⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠩⡼⠟⠀⣑⠠⠆⢈⡅⠔⠀⠀⠒⡀⠉⠄⢈⠁⠀⢤
⠀⠀⠀⠙⡄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠱⡄⠀⠀⠐⠀⠉⠀⠀⠒⠀⠉⠀⠂⠀⠢⠀⠀⠀⢰⣦⠸⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠁⠙⠒⠊⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣔⠀⠀⢠⣦⠀⢀⠀⠈⣮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡆⠀⠀⣙⡁⠀⣐⠉⠄⠂⠀⠠⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠶⠚⠉⠈⠉⣉⠉⠛⠒⠒⠠⣄
⠀⠀⣠⠄⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠻⠁⢁⠤⠐⠀⠀⠀⠠⠒⠀⠐⠃⠀⠀⠀⠀⠀⢠⠊⠁⠀⠘⠁⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠾⢁
⠀⢸⠀⠀⢠⡈⠉⠁⠒⠲⠄⠀⠀⠀⠀⠲⠀⠠⠄⠒⣈⠁⠀⠰⠂⢀⠀⠀⠀⠠⠇⠤⠦⠄⣠⠀⠀⢠⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡎
⠀⠘⢦⠀⠀⠀⠀⠀⠘⠃⠀⠘⠖⠠⣀⠀⠀⠀⠀⣠⠄⠀⠀⡥⠌⢩⣬⡁⠀⠀⠠⣤⡄⠀⣸⠀⠀⢘⡃⠲⠦⠤⡤⢀⡀⠠⠠⣤⡤⠔⠒⠒⠋⠙⠏⢀
⠀⠀⠀⠙⢄⠂⢀⠀⠀⠀⠀⠰⣄⠀⡈⠁⠀⠀⠸⡀⠀⢠⣶⡇⢠⠆⠁⠀⠐⣤⠀⠀⠀⣰⠏⠀⠀⢨⠷⢟⡳⣶⠤⠀⠀⠀⠀⠀⠀⠉⠉⣀⠤⠔⣚⡽
⠀⠀⠀⠀⠀⠑⢤⡑⠀⠀⣀⠀⠀⠀⠉⠀⢠⠀⠀⠙⢦⣈⠉⠀⠁⠀⠀⠀⠀⠙⠁⣠⠾⠋⠀⠀⠀⢦⠀⠄⠁⠈⠸⣿⠒⠀⢩⣭⡬⠤⠴⠖⠚⢋⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄⢈⡙⠓⠲⠒⠤⢬⠃⠀⠀⠀⠈⠙⠳⠶⠶⠦⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠑⠢⢤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣥⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠉⠀⠐⢁⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⠿⠿⠿⢿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠐⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣍⣁⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⣔
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠟⠋⠀⠁⠀⠂⠀⠀⠁⠀⠀⢀⠀⠉⠛⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⠟⠁⠀⣜⠐⢶⣶⡄⠀⠀⠰⠿⠛⠀⠇⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀⠀⠰⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⠀⠄⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⣷⣉⠛⣛⣿⣿⠿⠱⠿⣿⣍⣉⣤⣴⣶⣶⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⠠⠄⢻⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⠏⢻⣿⣿⣿⣿⡧⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠈⢿⠿⠷⠀⠀⠀⠀⠐⠛⠛⠿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠤⠤⠤⢤⣤⣤⣤⠤⠔⠒⠒⠚⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠖⠊⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠑⠺⢧⣄
⠀⠀⠀⠀⠐⠀⢠⠞⡝⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢣⠹⡄
⠀⠀⠂⠀⠀⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⢷⠀⡇⠀⠀⢈⠎⠀⠀⠀⠀⢨⢀⠀⡁⡁⠀⣁⠀⠁⠀⠡⠀⠑⠀⢸⠀⠏⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣠⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠁⠀⠃⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⣿⣶⣿⢰⣷⢰⣷⢰⣶⣀⣾⡆⣶⣦⣠⣾⡆⢿⣶⣾⠆⢸⣼
⠀⠀⠀⠀⠀⠀⠀⡏⡇⠰⠿⠛⠁⠈⠻⠾⠟⠸⠟⠛⠻⠇⠻⠟⠛⠿⠇⠿⠟⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠄⡀⢤⠀⠄⠀⠀⠠⠄⠀⡠⠄⠠⠄⠀⠤⠀⠄⠀⢤⠀⢸⠋⡆
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡀⠀⠁⠀⠀⠀⠀⠐⠰⠀⠆⠄⠀⠢⠰⠄⠀⡀⠠⠊⠀⢸⠀⢇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢢⣱⡀⠀⠘⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣎⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠈⠻⡢⠤⠤⠤⠤⠤⢤⣤⣤⣬⣥⠤⠤⠤⠤⠤⠤⠤⠤⢔⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠴⠲⠒⠚⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠋⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣶⣶⣄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢿⣿⣿⡿⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠉⠉⠁⠈⠻⠿⣾⢿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⣀⡐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒
⠀⠀⠸⠃⠀⠀⠀⠠⡄⠀⠀⠀⠸⠆⠀⠀⠀⠀⠘⢁⡀⠈⠀⠀⠈⠁⠀⣤⠔⠀⠀⠈⣠⡄⠀⠀⠀⠐⠆
⠀⠀⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⢐⠃⢀⣄⡀⠔⠂⠀⢠⡖⠀⠉⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠶⠬⠤⠐⠒⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⣄⠀⠀⣀⡀
⠀⠐⠧⠀⠀⣀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠛⠀⠋⠀⣠⡤⢦⡀⣠⣴⣦⣤⣄⣼⣡⠇⡄⠀⣠⢤⠀⠀⠀⠙⠷⠀⡀⠁⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢰⡿⢁⡬⡀⣴⢦⣸⡯⣲⣴⢶⡯⢉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⢃⣾⢇⡷⣟⡼⢻⠻⣇⣙⡮⣅⠹⠧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡠⠊⠙⠊⣤⣬⣴⡶⠟⠉⠓⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠖⠀⡘⠋⢀
⠀⠀⠀⠒⣄⠐⠂⠀⢀⣉⠀⠠⠶⠄⠀⠀⠀⠀⣠⠖⢉⣭⠁⠀⠉⠀⠀⠀⠀⠀⢰⡧⠛⠋⠀⠀⠉⠀⠀⠘⠁
⠀⠀⣾⡀⠈⣀⠀⣤⡌⠁⣤⣄⠀⢄⠄⠀⣴⠁⠖⠀⠀⠀⠒⠀⠙⠃⠘⠂⠀⠀⢨⣧⣾⠷⠖⣒⠒⠺⢧⣀⡱
⠀⠀⣿⣿⣷⠶⢦⣤⣤⣤⣤⣤⡴⢎⠀⠀⡖⠀⠉⠀⠀⠀⠀⢲⠄⠀⡀⠀⠀⠀⠨⡟⢁⣤⠈⠉⠀⠤⡄⠈⢳
⠀⠀⢹⣷⣶⣒⣈⣉⣀⣠⣤⡿⠷⠿⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢰⢀⡀⠀⠀⠀⣀
⠀⠀⣸⣭⣍⣀⣀⣉⣩⣟⣀⣀⣤⣴⠃⠀⠀⠀⠈⠁⠀⠀⠀⠁⡀⠀⠀⠁⠁⠀⠘⡇⠠⣦⠀⠀⠈⠀⠤⠙
⠀⠀⣴⣭⣤⣛⣀⣀⣋⣘⣀⡤⠤⠖⢂⠀⠀⠀⠐⠂⠈⠁⢀⣀⠀⠀⠀⠆⢀⠀⠀⠙⢄⡀⠀⠉⢀⡀⢀⡀
⠀⠀⠹⠵⣿⣈⡹⠏⠉⢀⣀⣀⣻⠤⠊⠀⠀⠀⢀⠙⢀⡀⠈⠉⠀⣀⡀⠀⠛⠀⠀⠀⠀⠀⠙⠒⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠛⠶⣦⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡾⠋⠀⣠⣶⠶⣶⣄⠀⠉⠉⠉⠉⠉⠛⠻⢷⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢯⣝⣛⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢟⣫⠂⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣹⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣴⢏⣵⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⡄
⠀⠀⠀⠀⠀⢠⡟⢡⡞⠁⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⢠⠎⣰⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣌⠓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣿⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡉⢷⣄
⠀⠀⢀⣴⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠿⠟⠃⠀⠀⢀⣾⣿⣿⣿⣶⡀⠀⠀⠀⢹⡌⣿⡆
⠀⢠⣾⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠫⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⡿⠀⠀⠀⢸⡇⣿⡇
⠀⣾⡇⠀⢀⠀⣠⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠟⠁⠀⠀⠀⢿⡄⣿⡇
⠀⣿⠀⠀⠘⣿⣿⣇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸⣿
⢀⡏⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢠⠀⠀⠀⠀⢀⣴⣄⢠⠀⢻
⣸⢡⠂⠀⠀⠀⠈⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⣿⣿⣿⠟⠞⠀⠀⠀⣠⣾⣿⣿⢸⡇⢸
⣿⣺⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⣷⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠸⢷⣾⣿⡿⢸⣇⣿
⢻⡟⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡷⠀⠀⠀⠀⠀⠉⠛⢛⣯⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⡿⣿⠃
⠀⢿⣜⣛⣦⣐⣶⣦⣤⣙⢿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣾⡇
⠀⠀⠻⣿⢟⠛⢿⡿⠛⠻⣿⡿⣷⣶⣤⡀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣾⡇⣾⡿⠃
⠀⠀⠀⠀⢘⠣⠄⠀⠰⠿⠀⠀⠘⠟⠻⣧⡖⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡙⣷⠀⠀⠀⢀⢜⣱⡿⢰⣿⠁
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡆⢿⣿⣿⣿⢣⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⢠⣶⡿⢋⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠐⢭⣛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⣭⠭⠂⠀⣠⣿⢋⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠈⣿⡆⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢣⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⣿⣀⡀⢀⠀⢀⢸⣿⣿⣤⣿⣷⠀⢀⣤⣤⣾⣴⠿⠛⠛⢛⣋⣥⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣷⣭⣵⣶⠮⣌⢿⣿⣿⣿⠏⠴⠛⣽⣋⣭⣶⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡿⢷⣶⣶⣶⣶⣶⣤⣤⣴⣶⣾⠿⠛⠛⠛⠉⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣿⣿⣿⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢠⠄⡀⢠⠄⢠⣤⡄⢤
⠀⠀⠈⠶⢆⠸⣇⠠⢨⡇⢸⠶⠀⠰⠶⠀⠀⡇
⠀⠀⠘⠂⠋⡀⠘⠋⠋⠀⠸⠀⢂⠘⠀⠀⠀⠃⡀
⠀⠀⠈⢸⠁⢱⡎⠀⢣⠘⠀⢱⠂⢸⠁⠈⣅⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠰⠄⠈⠣⠤⠊⠐⠤⠜⠀⠸⠆⠀⠧⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣼⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⢿⣟⠛⠋⠉⢹⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣠⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⢸⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢸⣿⣷⢠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣿⡀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣀⣴⣿⣿⣿⣿⣧⣤⡐⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣿⣿⣿⣿⣿⣿⡟⣡⣿⠃⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢄⣸
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⡛⢌⡛⢿⣿⣿⣯⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣽⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡸⣽⣿⣿⣿⡟⠟⢿⣼⣷⡷⢻⣾⡟⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠄⠛
⠀⠀⠀⠀⠀⢀⡶⠁⣸⣿⣿⣿⣿⣄⠀⢻⣿⣷⣿⣿⡇⢸⠀⡆⢠⠀⡀⠀⠀⠰⣄⣀⡀⠀⠄⠀⢠⣀⣴⠃⠀⠀⠀⠀⠀⣰⠏⢸⡀
⠀⠀⠀⠀⠀⣿⢊⡉⣩⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⡇⢸⡄⠀⢸⣿⣧⠀⠀⢠⣾⣿⣿⡟⢂⡀⢻⡟⠁⠀⠀⠀⠀⠀⣰⠃⠀⢸⡇
⠀⠀⠀⠀⠘⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⣸⣿⣧⣤⣀⠉⢿⣿⡿⠉⠀⠀⢛⡿⢀⡀⠀⠀⠀⢰⠃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⡇⢠⣿⡿⣿⣧⠝⠂⣿⣿⡟⢦⠄⠚⠳⠐⠀⢰⡄⠀⣰⠃⠀⠀⢚⠳⡄⠛
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⢸⣿⣷⣿⡟⠀⠀⣿⠿⠿⠰⡧⠶⠤⡾⢷⣾⣀⣰⠋⠀⠀⠀⠰⠆⠀⠳
⢆⣨⠆⠀⣸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⡇⢸⣿⣧⣿⠆⠀⢰⣿⠀⠀⠀⡆⠀⠀⣿⠸⡏⢩⠏⠀⠀⠀⠀⠀⣾⠂
⠈⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣼⣿⣨⣿⣹⣟⠀⠀⣾⣇⡤⢺⠄⡉⡀⢀⣿⠀⢻⠈⠲⠐⣷⠀⠀⠀⣿⢀⡤⢤
⠀⠧⠀⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡀⠀⢰⣿⣯⡍⢸⣶⣶⡏⢿⣿⠏⢀⠀⠀⠀⠉⠀⠀⠀⣿
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠶⢸⣿⣿⣿⣿⣦⣴⣦⣽⣿⣿⣿⣿⣿⡁⠈⢳⡞⣿⣟⣀⣸⠛⣻⣏⠘⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠆⢻⠇⣿⣯⣭⣤⣤⣿⣿⡄⠀⣸⣧⠀⠀⠀⠀⠀⣀⡀⡇
⠀⠀⠀⠀⠀⠈⠉⠛⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠹⣆⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⠀⣾⡇⠀⠰⠿⠟⠁⠇
⠀⠀⠀⠀⢿⠀⠀⠀⢸⣸⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿⠇⠀⡄⠉⠀⠀⠀⠀⠀⢀⣠⢤⣀⠀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⡟⣼⣿⣿⡏⣿⣿⣿⣿⠀⢸⡇⠀⠀⠉⠉⠉⠛⠛⢿⣿⣿⣿⡟⠁⢠⡆⠀⠀⠀⠀⢴⣦⣤⣉⡁⠿⢾⣿⣄
⠀⣠⢀⣤⣼⡧⠖⠌⠉⣿⣿⣳⣿⡏⠀⠀⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠸⠗⠒⠊⠉⠉⠋⠛⠙⠿⠛⠉⠀⠉⠛
⣿⣿⣷⢋⠍⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⢐⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⢸⡦⢘⡀
⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⡿⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣷⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⠀⣠⡟⢿⣿⣿⡇⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣴⡟⠀⢸⣿⣿⡇⢠⣿⣿⠀⢰⣶⣦⣤⣄⠀⠀⢀⣿⠀⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣃⣤⣴⣿⣿⣷⣾⣿⢿⡄⠸⠿⠿⢿⣿⣿⣿⣿⣿⣺⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⢿⢏⣹⣿⣍⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡙⠀⣿⣿⡃⢀
⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣀⣙⣿⡭⡆⠰⢿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠀⣠⣤⡄⠀⠀⠀⠉⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣥⣤⣤⣀⣀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠈⠉⠀⠀⠲⠦⣭⣙⠳⢤⣀⠀⠀⠀⣀⡀⠀⠀⠘⣦⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠉⠰⣌⠳⣄⠀⢿⡹⣄⡀⠀⠈⢷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣾⣡⡟⠀⠀⠀⠀⠀⢰⣟⣽⠇⠀⠀⠀⠀⠈⡀⠈⢧⠀⢻⣬⡿⠆⠀⠀⢠⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡟⣧⠘⣧⠘⠁⠀⠀⠀⠀⠘⠃⣷
⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⢿⡆⠀⠀⠀⢀⡾⣦⠀⡟
⠀⠀⠀⠀⠀⠀⢀⡏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠿⠴⠋⣰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠸⡇⠈⡁⠀⠀⠀⠉⠳⠛⠀⠀⠀⠀⠀⠀⠀⠀⢾⣃⣼⠃⠀⠀⠀⡼⣹⢣⣾⠆⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⢡⠏⠀⣀⡴⢞⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⣴⣻⡽⢂⡞⣡⡿⢖⣋⡵⠞⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣌⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡴⢋⣴⠿⠚⠋⠁⠀⠀⠀⠀⣀⡼⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣭⣓⠶⠦⣤⣤⣀⣀⣀⣤⣤⡤⠶⠚⣉⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠓⠚⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⣠⡰⠊⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠈
⠀⠀⠀⠀⠀⠀⡀⢀⡠⠠⠄⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠄
⠀⡀⠀⠀⠈⠉⠠⠤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠀⢁⠀⡀⠀⠀⡀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⠀⠈⠀⢠⡀⠤⠄⠄
⠀⠈⠀⠂⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⡀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⢀⢀⠀⠀⠀⢠⠠⠀⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠈⠁⠉
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡁
⠀⠀⠁⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠈⣠⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⢒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠴
⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⢁⠐⠒⣒⢂⠉⢱⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠠⠁⠐⢀
⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠱⠐⠐⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⢂⡐⠐⠊⢠
⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠑⠠⠤⠤⠤⠒⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⣩⡶⠀⠀⡤⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢀⠀⠀⠀⠈⠀⠀⠀⡠⠤⠀⠀⠰⡇⠀⣯⢁⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠋⠀⢠⣴⡆⠀⠀⠀⣠⣤⡄⠀⠀⠀⢠⣿⠀⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠛⠋⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⢾⠇⠀⠀⠀⢍⠀⣀⠀⠀⣀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡀⠀⠠⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⠇⠀⠀⠇⠈⣆
⠀⠀⠀⠀⠀⣠⠀⠀⠙⠁⠀⠀⠀⠀⣠⣤⠀⠀⠀⢀⠾⠁⠀⠀⣠⡾⠁⠀⠀⠀⠸⢿⣿⡆⠀⠀⠐⠄⠀⡰⠀⢠⣿
⠀⠀⠀⠀⡆⠋⡀⠀⠀⣠⠆⠀⠀⠀⣿⠿⠀⠀⠀⢈⠀⠀⣄⣴⡿⢾⡷⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢁⣴⡿⠃
⠀⠀⠀⠀⠇⢼⡷⡶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢀⣀⣴⣿⡷⣷⣄⡚⣀⣀⠠⣌⠀⠤⠃⠀⠀⢀⣤⣼⣿⠟⠁
⠀⠀⠀⠀⢢⣶⡈⠐⠀⠀⠀⠀⠒⠒⢋⣡⣤⣴⣶⣿⣿⣿⡟⠀⠐⣦⣤⣤⠀⠀⠀⠀⣀⣠⣴⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠓⠶⠶⣦⣤⣥⣼⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⡿⢿⠿⠶⠿⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣉⣈⣀⡀⠈⠛⡉⠀⢀⡹⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣯⣓⣒⠋⠀⠠⠁⠀⠀⢬⠶⠤⠉⠉⢉⢉⡁⠁⠠⣾⣿⠟⠁⠀⠲⣿⢿⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶⠋⠀⠀⠀⠀⠀⠀⣬⢛⡋⠀⠀⠀⠙⠋⠉⠈⠛⠂⢀⣀⣬⣽⠟⢛⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⡿⣆⣀⣒⣆⣀⣭⣿⡛⢠⣤⣤⣀⡐⢤⣁⡂⠠⣀⣀⣀⣀⣈⣉⣩⣿⣿⣿
⠀⠀⠘⠈⠀⠀⠉⠉⣉⠉⠁⢈⣀⠙⠛⠉⠀⠈⢿⡷⠋⠋⠻⡿⢿⠍⠁⠩⠯⠤⣭⣤⣤⠭⠀⢈⣉⣁⠐⠈⢹⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⢠⣁⣬⢌⣵⡶⣰⠀⣼⣿⣿⠟⣱⠉⠉⠉⣹⣿⣷⣲⣤⡆⠀⣶⣾⣧⣤⣤⣤⣭⣟⠋⢙
⠀⠀⠀⠀⠀⠀⠀⣇⡿⣣⣿⡟⣰⣿⢠⣿⡿⣡⣾⣿⠀⢰⢰⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿
⠀⠀⣠⣤⠤⠤⢤⠈⡵⢿⠟⠼⠿⠇⠸⠯⠾⠿⠿⠿⢀⣸⢀⢿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠚⠀⣤⠴⡾⠅⠂⠙⠫⣭⣭⣶⣶⡖⠂⠀⠀⢰⣿⣿⠸⠈⣿⣿⣿⠁⠀⠀⠉⣿⣿⡟⠉⢹⣿⠇⠀⠀⡯⠄⠀⠄
⠀⠀⠀⠰⠁⠀⡂⡄⡍⠛⢁⣀⡊⠉⣉⡀⠀⣠⢴⡄⠙⢻⠀⠀⢻⣿⡏⠀⠀⠀⠀⠉⠉⠁⠀⣠⣤⡄⢠⡆⡦⠀⠀⠂
⠀⠀⠀⠰⣯⠿⡅⡅⡃⣂⠙⠾⡃⠀⣡⣶⣦⣌⠛⢁⣤⣼⡀⠀⠀⡌⠀⠀⢠⣾⣿⡧⠀⢠⣤⣬⡙⠇⠈⢣⠀⠀⠀⣉
⠀⠀⠀⢰⠂⠠⠴⠓⠐⠖⢒⣚⣛⣺⡿⠛⠛⣿⣦⡾⣟⣛⣛⣒⣒⣋⣛⣛⣓⣛⣛⣙⣳⣿⠛⠻⣿⣆⠁⢄⠆
⠀⠀⠀⠈⢹⣿⣿⣶⣾⠿⠿⠿⠛⢿⣧⣈⣀⣿⡟⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢉⣛⣛⣛⣿⣄⣤⣿⡋
⠀⠀⠀⠀⣀⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷
⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣐⡋⢩⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⡉⠉⠀⠐⠀⠀⢀⡀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠀⠰⠀⠰⠛⢿⣷⡠⡶⠀⠲⡷⠀⠙⠂⣙⠀⠀⠀⠀⠀⣰⠆⣴⡄⠀⠀⠀⠚⠃⠀⠀⢁
⠀⠀⠀⠀⠠⣄⠀⠰⠗⠀⠻⣻⡶⣧⠀⠀⠆⢀⠀⢀⠀⠀⠀⠀⠀⣿⣶⠄⠀⢀⣠⡄⠀⠂⠤⠔⠻⡄
⠀⠀⠀⠀⢀⠈⢳⡠⠕⠦⠤⠈⠻⡻⣶⡆⡀⠈⡀⡾⠀⠀⠀⠀⠀⢻⣳⠐⠀⠄⢨⣥⠀⠀⠲⠄
⠀⠀⠀⠀⢹⣄⠊⣍⢷⣄⠀⣖⡀⠘⢾⣧⡙⣠⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠐⠢⠤⠂⣁⡀⢠⠃
⠀⠀⠀⠀⠀⠙⠿⣟⡙⡍⠓⢤⡀⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⢰⠃⠴⠄⠀⠀⠀⠋⢀⣴⠺⠌⠁⡙⠋
⠀⠀⠀⠀⢠⠂⠡⠀⠉⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠃⠀⣀⡀⣿⡶⠆⠀⠠⣴⠄⣸⠿⡇⠀⠀⠈⢁⣤⡞⠃
⠀⠀⠀⠀⠁⠁⢠⣤⠄⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⡇⠀⠀⢉⡀⣿⣷⣀⡀⠂⢰⣶⣿⠆⠈⠀⣀⣲⣯⠎⢀⡐⠀⡀
⠀⠀⠀⣇⠀⠀⠀⢤⣀⠀⠀⠿⠀⠀⠘⢀⠁⠀⠀⢧⡀⠈⠃⠀⢾⣿⣿⣥⣈⣽⣿⣿⣿⠆⠀⠈⣿⡦⡏⠊⡀⠀⡃
⠀⠀⠀⢿⡈⠀⠙⠀⠀⠀⠀⠀⠁⠐⠀⠸⠀⠀⠀⠈⢳⡆⠀⢀⠑⡿⠻⣿⡿⠋⠉⠉⠙⠇⠠⡄⢽⠁⢠⣦⡁⠠
⠀⠀⠀⠈⢻⣤⠄⠀⠀⠂⣴⡄⠀⠀⠼⡇⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⠈⠉
⠀⠀⠀⠀⠈⠻⢦⡤⣶⣀⡀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⡄⠀⠀⣀⠀⠀⣤⣀⡄⠀⠀⠀⢀⡂⠀⠀⢀⠐⠲⠂⠀⠀⠀⠀⠀⠀⣦⡦
⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣀⣀⣛⢦⣀⣬⠶⣴⠚⠛⢯⡁⣈⠀⠀⠀⡀⠘⣆⠀⣉⣻⣿⣿⣿⣿⣿⡖
⠈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠉⠉⠉⠙⢿⣍⣢⣤⠬⠷⢶⡛⠉⠃⠀⠀⠀⠀⠀⠙⠀⣒⣶⠿⢿⠿⠛⠉⠁
⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠘⢿⣁⣈⣳⣤⡴⠛⠞⠀⡀⠀⠉⠀⠀⠀⠀⣭⡀⠀⠀⣀⡀⣀⠤
⠀⠀⠀⠀⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢈⠁⠈⠁⠀⠀⠀⠀⢀⣿⠉⠉⠀⠀⠀⠑⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⣷⠀⢀⣿⢳⡟⢁⣰⣶⣧⣄⢀
⠀⠀⠀⠀⢸⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠈⠋⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡤⢿⠇⠀⠀⠀⠀⠀⢸⣿⡇⠘⣿⣸⠃⣿⣿⣿⣿⣿⡆⣷⡀
⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠹⣿⣧⡙⢿⣿⣿⡿⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢯⣻⣤⣤⠄⠀⠙⠛⠲⠟⠻⠇⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠤⠔⠀⣠⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠐⠂⠀⠂⠀⠀⣹⣷⣀⣠⣤⣶⡶⠶⠶⠶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠈⠁⠠⠤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢠⠀⠀⠀⠀⣴⡀⠀⢰⡄⠀⠀⣸⣇⣾⡇⠀⠀⠀⢠⣄⢀⣶⣿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⢸⡄⠀⠀⠀⣿⣇⢰⣾⣇⠀⣶⣿⣿⣿⣷⢰⡇⠀⢸⣿⣸⣿⣿⣿⣼⣷⠀⢠⣿⣸⣷⠀⠀⣾⡀⣠⠀⢀⣾⠃
⠀⠀⢸⣧⠀⠀⢠⣿⣿⣿⣿⡇⠀⣿⣿⠙⢻⣿⣾⣧⠀⣼⣿⣿⢿⠏⠘⣿⡿⠀⠘⣿⣿⣿⠀⠀⣿⣿⣇⠀⣼⣿
⠀⠀⢸⣿⡆⠀⢾⠙⣿⣿⡇⠁⠄⢸⡇⠀⠈⣿⡏⠹⠏⠸⣿⡟⠀⠀⠀⠹⠀⢀⠀⢹⣿⠁⠀⠸⣿⣿⣿⣆⣿⡇
⠀⠀⠘⣿⣧⠀⠘⠀⢿⣿⠀⠀⠀⠀⠁⠀⠀⠿⠀⠀⠂⠀⣿⠇⠀⢠⠀⠀⠀⠸⠀⠈⠃⠀⠀⡆⣿⡏⠸⣿⣿⠃
⠀⠀⠀⣿⡿⡇⠀⡅⠸⡿⠠⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠘⣿
⠀⠀⠀⢹⠇⠀⠀⡄⠀⠀⠠⠀⢀⣀⣤⣄⣙⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠶⠀⠀⠀⣀⠀⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⠋⠉⠀⠉⠛⠀⠀⡙⣷⡀⠈⢆⣀⣀⠂⢀⣴⣿⣿⣿⣷⣶⣄⠀⢉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠳⣿⠘⣿⣶⡸⣿⡏⠀⣸⣿⡟⠋⠉⣿⣿⡿⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢠⣄⠀⠠⢰⢿⣤⣿⠅⣿⣿⣶⣿⣿⣿⣏⣿⠿⠎⣰⣿⢟⣷⣿⣿⡗
⠀⠀⠀⠀⢀⠀⣤⣶⣧⠈⣤⣀⣠⣿⣄⠀⢨⡿⡏⠁⣸⣿⣿⡟⠁⠀⠙⣿⣿⣶⣦⣴⣶⣿⠟⠉⠉
⠀⠀⠀⠀⠘⣷⣮⣿⣮⣷⣌⠻⠛⠛⢨⡆⠘⠷⣀⣴⣿⠿⠟⠁⠀⠀⣤⠘⠙⠿⢿⢿⠹⣯
⠀⠀⠀⠀⠀⠘⠛⠻⢿⣼⣿⣿⣶⣦⣬⣤⣶⠞⢻⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢻⣿⣿⣿⡞⠋⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢈⠉⢋⠁⠀⠀⠀⠀⠈⠀⠀⣦⢸⣦⢺⠀⣶⣶⡀⠉⠛⢦⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⣶⡶⠀⠉⠀⠀⠀⠀⠀⠘⠃⢸⠆⢀⠙⣦⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠁⠀⠀⠀⠀⠰⡀⠛⠀⠀⠀⠠⡀⠒⢀⣀⠀⠀⠀⠀⠘⠂⠘⣧⠀⠀⢻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠈⠙⠀⠀⠁⣸⣿⢀⠀⠀⠀⠀⢰⡀⣽⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣷⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠐⠀⢸⡉⣽⠛⠛⠂⠀⡰⡏⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠱⣴⣿⣀⣀⣠⡀⠀⡀⢀⠀⠉⠀⠀⠀⠘⢷⡏⢠⣤⠰⣞⠃⠀⢠⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⠙⠘⣇⢧⡰⣤⣦⡀⠛⠋⣿⠃⢰⣄⠀⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⣿⡀⢰⠀⠀⠙⣿⣿⣿⣿⣿⣷⣤⡁⠘⠃⠁⠀⠘⠀⠚⠁⠀⢈⣡⡾⣿⣿⣿⣿⡅⠀⢰⡆⠀⢸⣿
⠀⠀⠀⠀⠀⢾⣿⡇⢸⣇⠀⠀⢨⠙⠻⢿⣿⣿⣿⣿⣷⡆⣤⣤⣤⣤⣤⣶⣾⣿⣿⡇⣿⣿⣯⣸⣿⣁⣾⣧⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⠀⠀⣼⡇⠀⢠⣹⡿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣙⣼⠟⠛⢻⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠈⣿⠁⢹⣿⠀⠰⣿⣷⠀⣼⣿⣿⠟⣼⣻⣿⡿⢦⡿⡿⣿⣿⡿⢩⣿⢸⣬⠿⣽⣿⠃⠂⠸⣿⠐⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠻⠀⠀⡿⠀⠀⢻⣿⠀⣿⡏⢻⣿⢶⣟⣿⣝⠿⣿⢷⣿⡙⢡⣶⣦⣬⡍⠀⠀⢻⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠈⠁⠀⠉⠙⠛⠇⠓⠈⠿⠟⡷⠿⠏⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡌⠲⠄
⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠰⠒⠀⠀⠀⠀⢰⠆⠆⠀⠀⠀⠀⡆⢤⡄⠀⠐⠀⢲⣠⢢
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠁⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⡀⠁⡄⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠂⠀⡇⠀⠺⠂⠀⠀⠀⠄⠀⠘
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠓⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⣠⠞⠬⠎⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⡠⠁⣸⠖
⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠠⠈⠀⠈⠀⡄⠊⠀⠀⠀⠀⠀⠀⢴⠂⢀⠄⠉⡰⢳⠴⠀⠀⠐⠄
⠀⠈⠄⢢⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠂⠀⣠⠴⢚⠀⠀⠀⠀⠀⠀⠀⠊⢁⠄⠀⢠⠊⠁⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠐⠄⠀⠠⠀⠁⠀⢻⡄⠀⠀⡤⣖⣊⠁⠀⠉⠙⠀⠀⣀⠠⠤⠒⠉⠀⡠⠔⠉⠀⠀⠀⠀⠀⠠⡾⠆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠂⠤⣉⠉⠻⢝⠛⠋⢇⣁⠥⠒⠚⠉⠶⠤⠀⠀⠀⣠⢰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣙⠉⢔⠦⠐⣒⠤⠠⡄⠐⠂⠒⢃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠉⠀⠘⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⢠⡀⡀⠀⡀⣠⠀⠀⠀⠀⡔⠡⡂⠀⡀⢦⠀⠀⠀⠀⠀⠀⠿⣽⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣦⣤⡀
⠀⠀⠀⠀⠊⠙⠛⠛⠉⠁⠀⠀⠀⠀⠟⣛⡭⡉⣛⠺⠀⠀⠀⣠⢃⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣽⡽⠁
⠀⠀⠀⠀⠀⣰⡇⠠⡇⢸⡆⠀⠀⢀⢉⡈⢤⠐⠕⠡⠀⠀⠀⠋⠚⢄⣆⣀⠀⠀⣴⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠠⣩⣾⢱
⠀⠀⠀⠀⠀⠟⠀⢜⢡⡊⠀⠀⠀⠀⠈⠀⠄⠶⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠠⠶⡆⠚⠋⠀⠀⠀⠀⠀⠀⠐⠷⣷⠇
⠀⠀⠀⠀⢠⡞⠁⣻⡘⢯⡅⠀⠀⠀⠀⡄⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⢽⠁
⠀⠀⠀⠀⠈⠳⠄⠉⠿⠀⠀⠀⠀⠀⠀⠠⢠⡀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⡀
⠀⠀⠀⠀⠀⠀⣠⡶⣺⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮
⠀⠀⠀⠀⠀⠠⠏⠜⠁⠀⠀⠰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠩⠥⠘⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠈⠉⠀⠀⠀⢺⣟⣿⣿⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣽⠈⠀⠀⠀⠀⠛⢻⡶⠶⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⢨⡏⣿⠹⠿⠋
⠀⠀⢀⣄⣀⣀⣀⢀⠀⠀⠀⠀⠀⢭⠃⠀⠀⠀⠀⢀⣾⢿⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡇⣌
⠀⠀⠈⢀⡿⠿⡏⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣿⢹⠀⠀⠀⠀⠀⠀⣾⡛⡋⠛⠧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣡⠴
⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⠌⠀⠀⠀⠀⠀⠈⢿⠛⣉⣼⠗⠀⠀⠚⡁⠠⠔⠀⠀⠀⠀⣴⡜
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠰⠀⣍⢊⣨⠀⠀⠀⠐⢛⡔⠀⠀⠀⠀⣦⣸⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣘⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⣾⣴⣿⣱⠀⠀⠀⠀⠀⠙⣛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠌⣉⠟⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣶⢢⣬⣧
⠀⠀⠠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠢⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠟⠛⠛⠋⠙⠛⠛⠻⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢠⣶⣶⡄⠀⠀⢿⣿⡿⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⠀⠸⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⣿⣿⣿⠀⠐⠂⠀⢠⣾⣿⣦⠀⠀⠀⢠⣿⣿⣧⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠉⠛⠁⠀⠀⠀⠀⠈⢻⣿⣿⣶⠿⠿⠟⠛⠻⠿⢷⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠘⣿⣄⠀⠀⠀⢠⢄⣼⡿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣾⠏⠀⢰⣿⣿⣦⠀⠀⠀⠙⠛⠁⠀⠀⠀⣠⣤⣄⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣾⣿⣷⠀⠀⠀⠙⢿⣦⣼⡟⠀⠀⠈⠻⠟⠋⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠛⠿⠟⠀⠀⠦⠀⣴⣿⣿⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⣀⠀⠀⠀⠻⢿⣿⡀⠀⠀⠀⠀⠛⠁⠀⠀⢸⣿⣿⣷⠀⠀⠛⠀⠀⣀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⢿⣇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⣿⣿⣿⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠙⠛⠁⠀⠀⠘⠀⠘⣿⡄⠀⠀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠘⠀⢠⣾⣷⣆⠘⢿⣆⠀⠈⠉⠁⠀⠀⠀⠀⠀⣴⣿⣶⡀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⠀⠀⠀⠘⠿⠿⠋⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣤⣀⣠⣤⣤⣶⡾⠟⠻⢷⣶⣤⣤⣄⣠⣤⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣩⠔⠚⠓⠢⣍⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⡥⠒⠉⢀⣠⣼⣦⣄⡀⠉⠲⢌⡙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⣈⠳⢤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠴⢋⡠⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠑⠢⣌⡙⠿⣿⣿⣿⡿⠟⢉⠤⠒⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠉⠒⠤⣉⣡⠔⠊⠁⠀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⣧⣠⣄⡀⠀⢸⠀⠀⠀⢠⠿⢄⡀⠙⠲⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⡄⢸⠀⠀⠀⠀⠳⢄⡈⠑⡶⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⣀⣘⠀⢁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⡿⣿⣶⣿⠿⣶⣄⡀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⠟⠁⠀⢸⠀⠀⠀⢠⠛⢅⣉⠀⢸⠀⣿
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣛⣻⣿⡿⢟⣿⣻⢿⣿⣟⣿⡿⣷⣤⣀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⢄⡉⠀⠸⠀⣿⣶⣤⣄⡀
⠀⠀⠀⢰⣿⡿⣷⣾⣯⣿⣿⣯⡺⠿⠿⠿⢿⣪⣿⣿⣾⣟⣿⣿⢿⡆⠀⠉⠓⠦⣄⡀⠀⠀⢸⠀⠀⠀⢀⣠⣴⣿⣷⣿⣾⣿⣿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣷⣭⣻⢿⣭⣿⢿⣷⣾⣶⢿⣿⣭⣿⠿⣻⣽⣾⣿⣷⠀⠀⠀⠀⠀⠙⠳⢤⣸⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢸⡏⠻⣿⣿⣿⣷⣯⣛⠿⣿⣟⣻⡿⢟⣫⣷⣿⣿⡿⠿⠋⣿⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⠟⠋⠁⠈⠁
⠀⠀⠀⠘⠳⣄⡀⠉⠙⠻⢿⣿⣿⣾⡍⣽⣾⣿⡿⠛⠁⠀⠀⣀⣤⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠀⠙⠻⢿⣧⡿⠟⠁⠀⢀⣠⣴⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⢸⠀⣀⣤⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⣼⡇⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢤⣾⡟⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⢀⣼⠟⠀⠈⠙⠛⠓⠒⠒⠒⠂⠀⠀⠈⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⢀⣿⡏⠀⢀⣠⡶⢛⣁⠀⠀⣀⣀⡀⠀⠀⠀⠀⢠⣾⠇⣀⡀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣼⡿⠉⠋⠉⢁⣶⡟⠹⣣⡾⢋⡤⠃⣼⣿⠄⣰⣿⠃⣴⡿⢁⣶⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠃⠀⠀⠀⢾⡏⠀⠴⣿⠗⢿⣇⠐⢿⠏⢰⣿⠃⢰⣿⠇⢸⣏⢰⣿⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠛⠛⠓⠒⠲⠤⢤⣈⡀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠄⢒⣀⣀⢀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⢲⣿⣿⠛⠋⡉⠛⠻⠷⣭⣔⣶⣀⣠⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠙⢇⣤⣄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣤⡌⠀⠀⠀⠀⢙⣿⣿⣿⠏⣭⢉⣉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢫⣷⡀
⠀⠀⠀⠀⢙⣿⣿⣿⣾⣿⣷⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣗⣀⣠⣄⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡻⣷
⠀⠀⠀⠀⠈⠻⠛⡛⠿⣿⣿⣿⣿⡿⠟⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠈⠀⠰⠠⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡆
⠀⠀⠀⠀⠀⠀⠁⠈⠟⢿⠛⠲⣖⠛⠚⠈⠈⠁⠉⠨⢍⣉⠿⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢛⣁⠀⠁⠀⢤⣜⣷⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⢠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠈⠁⠂⠈⠀⡀⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⢀⠀⠀⠀⠠⢀⣀⣽⣟⠃⠀⠀⠀⠈⢣⢀⠀⠀⣠⣎⡼⡛⠁
⠀⠀⠀⠀⠀⠀⡘⠢⣙⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠖⠁⠀⠀⠀⠀⠀⢰⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠍⠊
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠆⠀⠀⠀⣀⣤⠄⠁⠀⠐⠀⠠⠀⡀⠀⠇⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⢀⡀⠀⠀⠑⠀⢤⠤⠀⠒⠉⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢂⠀⠀⠀⠌⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⡀⠀⠀⡀⠀⢀⣴⡝⠁⠀⠀⠀⠀⠀⣀⣄⠤⠆⠖⠟⠒⠒⢤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⠋⠀⠀⠀⠐⠛⢡⣶⡰⢿⠏⠀⠀⠀⢀⡴⠟⠁⠐⠀⢀⠀⠀⠀⠀⢁⡀
⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣀⠀⣄⠀⣴⠂⡠⠚⢏⣽⡄⠀⠀⠀⢠⡏⠂⠀⠀⠀⠀⠁⠀⠀⠗⠀⠉⡀⠀⣾⡶
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠁⠋⠀⠙⠀⠁⣰⣅⡔⠹⠻⠁⠀⠀⠀⣼⣇⣁⠈⠀⠀⡀⠀⠀⠀⡀⠀⠬⣗⣼⣷⠃
⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⣭⠀⠀⢀⡀⢡⣿⠽⡋⠂⠀⠀⠀⠀⠀⠹⣿⣧⣶⡄⢰⡇⡀⢀⣼⢡⣴⣶⣿⠿⠁
⠀⠀⠀⠀⠈⠳⣈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠉⠀⡶⣏⣿⡿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠟⠿⣯⠷⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠟⠁

⠀⢀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⣀
⠀⠑⠤⠃⠀⠀⡖⠐⡄⠀⠀⠀⠀⢔⠉⣣⠀⠀⠀⢀⣠⠤⠖⠻⣄⠘⠤⠃⠀⠘⠒⠊⠀⠀⠀⠀⠐⢍⠆⠠⡞⠣⠁⢸⠀⠀⢇⣈⠆
⠀⠀⠀⢠⠒⢢⢉⠨⡀⠀⠀⠀⠀⠈⠉⣁⡤⠖⠊⢉⢤⠀⠁⠀⠚⣄⠀⣸⠏⠓⡶⠚⠉⠉⠉⠓⢦⡀⠀⠀⠙⢦⣠⠼⠦⣄
⠀⠀⠀⠀⠑⠁⠰⠤⠜⠀⢀⣠⠴⠚⠉⠀⣤⢄⠻⠣⠉⠁⠀⠐⠀⠉⣇⢳⣆⣈⣇⠀⠀⠀⠀⠀⢀⡏⠩⣽⠃⣏⠤⠐⣂⡘⡆
⠀⢪⣉⠆⠀⢀⣠⠤⠖⠋⠁⡤⠀⣖⣢⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠉⠀⠈⠓⠒⠂⠐⠒⠛⠦⣴⠎⠀⢷⢊⡡⠄⠀⢻⡀
⠀⢀⣠⠶⠛⡩⠀⣀⣦⠫⡹⠑⠒⠀⢀⡤⠚⠉⣿⠤⠶⣲⠀⠀⠀⣠⠴⠋⠀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠘⣇⠴⠚⠉⠉⣧
⠰⢯⠐⠀⠈⠧⠤⠈⠀⠀⡐⠒⣢⢩⡇⠀⠀⣰⣧⠤⠞⣁⡤⠖⠋⠀⢀⣀⠀⢇⣨⠃⢠⢶⠎⠁⠀⠀⠀⠙⣆⠀⠹⣤⢒⣉⣥⡟
⠀⠙⣆⡀⠀⠀⠀⠀⢀⠀⠼⠦⠔⠘⠖⠒⠊⣀⡤⠖⠋⠁⢄⣱⠀⠀⠣⠤⠃⠀⠀⠀⠘⣎⢦⡀⠀⠀⠀⠀⢸⣄⠀⠉⣽⢀⠀⣙⡆
⠀⠀⠘⣧⠀⢂⠀⠀⠀⠈⠁⠈⢁⣠⠴⠚⠋⢡⠤⣄⡀⢀⠤⠤⢤⣀⣀⣀⠀⠀⢨⣈⠇⠹⡑⠪⠅⢒⣒⣒⣪⠞⠀⠀⠹⢴⣤⠿
⠀⠀⠀⠙⣧⠀⠁⢄⣀⡤⠴⠚⠉⠀⠀⠀⢸⣉⠀⠀⠉⡧⠒⠒⢊⠔⣒⣋⡩⢷⡤⠤⢤⠀⠉⡶⢦⠤⠖⠋⢀⣠⣀⡀⠀⠀⢀⠠⡄
⠀⠀⠀⠀⠙⠧⠚⠋⠁⠀⠰⣉⡇⠀⠀⠀⢀⡙⠒⠒⠛⣷⣒⠒⠩⠔⠁⢀⡠⣾⠀⢀⣛⣶⢸⠁⡎⠀⠀⢠⡏⡎⠉⠉⠉⠲⡄⠙⠁
⠀⠀⢎⣈⡆⠀⡀⠀⣀⣠⡤⠔⡲⢚⢷⠞⠁⠛⢲⠀⠠⠒⢌⠉⠙⠒⠓⠒⠶⠋⠙⠒⠾⢁⡏⣸⠀⠀⠀⡏⢠⢣⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠜⣌⣶⢋⠏⡔⠀⡰⢡⠊⡰⣧⣤⠤⠿⠀⠘⢒⣃⣀⣀⣀⡤⠶⡆⠀⢎⢹⠀⡸⢠⠇⠀⣀⠴⢳⡀⢢⢣⡀⠀⠀⡸
⠀⠀⢸⡏⠉⠉⠹⡌⡘⠀⢰⢁⣃⡴⠴⠋⠀⠀⢀⡴⠊⠉⠀⠀⠈⢣⠀⢙⡇⠀⠀⠀⢠⠃⡾⠒⣉⠤⠒⠉⠳⠤⣵⣭⣒⣺⠇
⠀⠀⠸⠶⣎⡴⠚⠙⠓⠒⠉⠉⠀⠀⠀⣤⠒⠒⣿⠀⠀⠀⠀⠀⢀⠜⠒⠮⠟⣀⡤⠖⡟⢸⠓⠉⠀⠀⡀⠀⡂⢓⡄⠀⠀⠀⢀⡀
⠀⠀⢀⠤⡀⠀⠀⠀⠀⡔⠒⡄⢎⢙⠄⢼⣡⡀⢈⡦⠤⠤⠴⢚⣁⠀⠀⡖⣋⠥⠒⢻⠃⡇⠀⠀⠀⡏⢈⡆⠉⢉⣀⠀⠀⠀⢏⣩⠆
⠀⠀⠱⢤⠟⠀⠀⠀⠀⠑⠚⠁⠀⠁⠀⠀⠘⠖⠋⠀⠀⠀⠀⠫⣠⠇⠀⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠉⠀⠀⢧⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠲⠄⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀
⠀⣌⢰⠤⡤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⠉⠒⠢⢄⡀
⠀⠱⣌⣱⡦⠻⠆⢈⠃⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠈⠑⠓⠚⠉⠓⠤⠦⣔⡺⠬⠐⠲⣤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠐⠠⢦⣅⠈⠑⠭⣶⣶⡀⠐⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠨⡑⠒⠢⠤⡿⠵⠊⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⣤⣀⣀⣀⣀⣀⡠⠤⠖⠋⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⠟⠉⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣾⣿⠻⡆⠘⢿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣼⣿⣿⣿⡿⠋⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⢱⣿⠿⠁⠀⠁⠀⠀⠻⣿
⣿⣿⣿⣿⣿⡟⣿⣿⣿⡟⢸⣿⣿⣿⣿⣇⣾⣿⣿⣿⠟⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⢠⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⠁
⣿⣿⣿⡿⣿⣎⣿⣿⣿⣾⣿⡟⣸⣿⣿⣿⣿⠃⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡟⠻⡟⠀⠀⠀⠀⠀⡀⣤⡀⠠
⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃⢀⣴⠏⠀⠀⠠⠂⠀⢀⣤⣠⣴⠞⣽⣿⣿⡏⠀⣴⡇⠀⠀⠀⠀⠀⡇⢹⡇
⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⢠⡿⠃⠀⠀⠀⣠⣴⣾⣿⣿⣿⣣⣾⣿⣿⣿⠀⣼⣿⣧⣤⠀⠀⠀⠀⠁⢸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⡆⡀⠀⠀⠀⢸⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⡿⣧⠃⠀⠀⠀⠸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣷⣼⣿⣿⣿⡿⢨⣿⢿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣧⠟⠀⠀⠀⡀⠀⠁
⣿⣿⣿⣿⣿⣿⣻⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣯⣾⣿⣿⣿⣿⡟⠁⢸⡞⣿⣿⣿⣿⣧⠀⠀⢠⣇
⣿⣿⣟⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⠿⣿⠟⠀⣿⣿⣿⡿⣿⠛⣿⣧⡄⠀⢇⠈⠻⢿⡇⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡟⠀⣠⢿⣿⣿⣿⣿⠀⠈⠻⣷⡀⠘⠆⠀⢸⣷⡀⠘⠀⠀⢻⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠇⠀⡇⠈⣿⣿⣿⢹⡆⠀⠀⠹⣧⠀⠀⠀⢸⣿⣷⡤⠀⠀⠈⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⣾⣿⣯⠇⠀⠀⠀⠙⡆⠀⠀⠀⢿⣿⣧⣇⠀⠀⢹⡆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢰⣿⠃⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⠀⠀⠀⠀⢰⡄⠀⠀⠈⢻⡿⣿⡀⠀⢼⠁
⣿⣿⣿⣿⣿⣿⣿⣟⣁⡀⠟⠁⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⠳⠀⠀⠀⠘⣇⣿⣧⠀⢸⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣾⣿⡟⢸⣷⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣷⣀⠀⣀⣤⣴⣄⣀⣤⣴⣿⣿⠀⠸⡇⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣿⡗⠀⠀⣿⠟⠿⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁⠀⢸⣿⡟⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⠸⠛⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣴⣟⠋⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⠻⢦⣀⣀⣀⣀⣴⣿⣿⡧⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡷⣰⡌⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠻⣿⡟⠛⠛⠉⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡟⣿⣿⣿⡿⣟⡿⠛
⣿⣿⣿⣿⣷⡜⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡚⢿⣿⣿⣶⣶⣾⣿⣶⣶⣶⣶⠂⣠⣤⣾⣿⣿⣿⣿⣿⠇⠿⣿⠿⠚⠉
⣿⣿⣿⣿⣿⣿⣞⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣾⡖
⣿⣿⣿⣿⣿⣿⣿⣯⠻⣆⠈⠛⠿⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣉⣉⣭⣿⣿⣿⣿⣿⠿⠟⠫⠐⠋⠁⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣷⣦⡀⠀⠉⠲⠬⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠁⠀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣶⣤⣬⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⡟⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠿⣿⣦⡀
⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⣿⡟⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⣿⣧⣄⣀⣀⣀⣤⣤⣶⣶⣿⠿⠿⠛⠛⢻⣿⡏⠉⠉⠉⠉⠛⠛⢻⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠇
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⣠⣿⡟⢀⣄⡀⠀⣀⣀⠀⠀⢿⣷⠈⠙⠻⠿⣿⣷⣶⣶⣶⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⣿⣿⣷⣼⣿⢿⣿⣶⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠋⠻⠟⠃⢸⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢴⣶⣿⣿⠿⠓⠀⠀⠀⠀⠈⠃⠀⠀⠀⠔⠚⢳⡶⣤⡄⠀⠀⠐⢦⠀⠀⠀⠀⠘⠁⠀⠚⠁⠀⠚⠁⠐⠂
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠇⠀⠀⠀⢸⠀⠀⢸⡷⣶⠿⡶⢾⡷⢶⡒⠿⠆⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⣤⡶⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠘⠿⣟⣦⣴⣶⡶⣾⡇⠈⠒⠂⠘
⠀⠀⠀⠠⠀⠀⠀⠀⠠⣠⠀⠀⠀⠀⣶⠟⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⡀⠀⠀⠀⠆
⠀⢠⡄⣀⠐⠀⠀⠀⠖⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠃⡈⠀⡄⣆
⠀⠈⠙⠛⠇⡵⠚⠁⠀⠀⢀⣴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⢄⡄⠀⠀⠒⠀⠀⠀⠈⠀⠀⡴⠀⢀⠓⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠐⡘⠁⠀⢀⣶⡟⢁⠀⠀⠀⠀⠀⡆⠀⢸⢰⠀⡇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠾⠿⢽⡷⢶⠀⠀⠀⠈⢢⡄⠈⠿⢧⠘⠁⠀⠀⠀⠀⠀⠀⢨⠘⢰⠁
⠀⠀⠀⠀⠖⡀⠠⡠⡄⡀⡠⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠐⢃⠔⠀⠱⡈⠅⠀⠀⠀⠀⠀⢀⠀⠤⣦⣄⠀⠀⠀⠀⠀⢠⡇⠀⡴⠶⠟⠿⠇⠐⠲⠀⠉⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⢠⣼⣿⣾⣞⣦⣠⣶⣄⠀⠀⢀⠜⠁⠀⠧⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⣐⠶⣿⣯⣳⢦⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠖⠉⠐⠀⠀⠘⠀⠀⠀⠀⠈⣹⠀⢀⡷⢶⣤⠿⡀
⠀⠀⠀⢀⣀⠀⠙⠻⣦⠙⠋⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢠⠁⣰⠏⠀⠀⠉⠈⠉⠒⠁
⠀⢰⠃⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡴⣶⠟⠀⠀⠀⠀⠁⢰⠏⠀⠀⠀⢠
⠀⢸⠀⠀⣤⠃⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⢈⢠⠡⠀⡀
⠀⢸⠀⠀⠚⡅⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠛⠁⠀⠀⠀⠀⠀⠀⠐⣶⡷⠀⠀⠀⠀⠂⠄⠸⣶⡀
⠀⠸⠀⠀⣷⠄⠀⠀⠀⠀⢀⢤⣄⣠⣀⢀⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣤⡀⡀⡀⠀⠐⠄⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠂⢠⣷⠦⠀⠠⢎⠉⠼⠿⠰⠶⠹⠿⠯⠾⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⡧⠤⠠⠡⠱⠡⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⠀⢆⠀⣶⡄⠙⠷⣄
⠀⠀⠀⠀⠀⣠⡶⠶⠀⠀⠀⠀⣀⠖⡶⣿⣹⠂⢸⣆⡿⣧⠞⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⢰⣿⠁⠀⠀⣀⠛⠖⠿⣤⠆⠻⠛⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠐⠳⣯⡎⠁⠀⠈⢁⠙⠀⠒
⠀⠀⠀⠀⠈⢿⣆⠀⢀⠿⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠊⠀⠀⠐⠛⠛⠁⠀⠀⠀⠛⠛⠳⠄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠁⢽⣿⠃⠀⠀⠀⠀⠠⣄⣀⡈⠁⠀⠐⠀⠀⣶⣶⡄⠉⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⠉⠀⠀⠃⠘⠃⣸⣿⡀⠀⠲⠀⠀⠰⡿⠛⠃⠀⣀⠀⠁⡀⠙⠿⢅⢀⡀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠉⠁⠚⢀⣂⠘⠰⠾⢿⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣽⡇⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⠀⠀⠸⠿⠇⠀⢰⣿⣿⣦⡀⠐⠁⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢈⠁⢠⣾⣿⡿⠠⡄⠀⢹⣿⣿⣦⡀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⢠⣌⠀⠀⠠⠀⠀⠀⠛⣿⣿⣷⣶⣦⣤⣤⣀⣠⡀⠀⠀⠀⠘⠋⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⡂⠹⢿⠗⠀⢰⣶⡀⠀⣼⠁⣼⡛⠿⣿⣿⣿⣿⣿⣿⡿⠶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠼⣎⠀⣄⣻⡇⠠⠜⠛⠋⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⢠⠰⡎⠙⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⡄⢿⡿⠿⠂⠒⠗⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠆⢴⣿⣷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡋⠀⠀⠈⠙⠉⠓⠈⣷⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢿⡿⠁⠀⢈⠀⠀⠀⠀⠀⠀⣾⣧⣄⠀⠰⠎⠀⠀⠀⠀⠀⠀⣠⠊⠀⠉⠀⠀⣀⡠⠀⠉⠀⠀⠀⠃⠸⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⢡⣴⣷⡶⡄⠀⢀⠀⠀⠀⠙⠛⠟⠂⠀⢀⡅⠀⠀⠀⠀⠐⠄⠀⠀⡀⠁⠀⠙⠀⠀⠀⣴⣶⣦⠀⣄⣄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢺⣤⠖⠀⠀⠻⠟⠵⠂⠀⠈⠃⠀⠀⠀⠀⠀⠀⠰⣿⠅⠀⠀⠀⢠⣿⠀⠀⣼⡇⠀⠀⠀⠀⠀⠛⣼⣿⠇⠀⠹⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⣶⡎⠀⠀⠀⠀⢀⠀⢀⠀⠈⠀⣀⠀⠀⠰⠄⠀⠀⠈⢀⡀⠀⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⠀⠀⠈⠑⠃⠀⠀⢠⣴⡶⠀⠀⠸⠏⠛⠀⠀⠀⠀⠀⠈⣀⣿⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡀⣤⠀⠈⠁⠈⠀⠀⣀⣠⡀⠊⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⣴⣦⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠄⢀⣀⠄⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠧⠀⠘⠛⠛⠛⠃⠀⠀⢀⠺⠋⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠴⠄⠀⢠⡄⠀⠐⠫⣌⠂⢀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢤⡐⡀⠦⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠋⠉⠀⢀⡶⠒⠶⢦⣄⡀⠀⠀⠀⠀⠀⠈⠙⠶⠦⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⢠⣿⣻⢿⣶⣦⣌⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣾⣿⠾⠿⠃⢀⣀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⢴⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⢸⠋⡙⠶⣄⠀⠀⠹⣆
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⣻⣷⣿⡆⠀⠀⠹⣆
⠀⠀⠀⣼⡟⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⣿⣽⣾⠟⠁⠀⠀⠀⣿⡄
⠀⠀⣼⡟⠀⠀⠀⢸⡏⡙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡴⢶⡀⠀⠀⢹⣧
⠀⣼⡟⠀⠀⠘⠁⠈⠻⢾⣯⣤⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣽⠇⠀⠀⠘⣿⡄
⠀⠘⣿⡄⠀⠀⠀⠀⠀⣎⣠⣤⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢈⣷
⠀⣰⡿⠁⠀⠀⠀⠀⠘⣯⣻⣟⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠐⠂⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢉⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⢿⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠈⣿⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⣧⠀⠀⢰⡟⠶⣤⡀⠘⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢮⡤⠀⠀⢀⣴⡾⠏
⠀⠀⠘⣿⡆⠀⢸⡇⣗⣦⣿⡄⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄⠀⠀⢀⣾⠋
⠀⠀⠀⠹⣷⣄⠸⣧⣽⢿⣼⡇⣸⡠⡎⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠂⢸⣇⣠⣾⠇⠀⢀⣾⠃
⠀⠀⠀⠀⠈⢿⡄⠈⠻⠿⠋⠀⠈⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣤⠞⠙⣦⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠘⠛⢛⣡⣴⣾⠿⠃
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡾⡷⣼⡇⠀⢀⠀⠀⠀⢀⡴⠋⢀⣉⣙⢳⡆⠀⢰⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠷⠿⠷⠛⠀⠐⢧⠴⠀⢀⡾⠀⣰⣟⣿⣷⡿⠃⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠉⠁⠀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠀⠙⠿⡾⠿⠛⠁⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣶⣤⣼⣯⣠⢌⡉⣿⡄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣥⣮⣿⠿⠛⠛⠿⣦⣄⣀⣀⣠⣤⣤⣶⡶⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⠾⠓⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⢱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠁⡑
⠀⠀⠀⡀⠀⠲⠀⠀⠐⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⢠⣤⣆⠟⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠄⠀⠀⠉
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠈⠀⠀⠔⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠐⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡁⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠰⠐⠒⠐⠖⠒⠒⠒⠆
⠀⠀⠀⠈⠢⢈⠉⣉⢀⣠⡤⠆⠒⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠒⠐⠀⡀⠀⠀⢀
⠀⠓⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠂⠉⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠃
⠀⠈⠆⠁⠆⠀⠠⠄⡔⠂⠉⠁⡬⣀⡀⠜⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⡇⠀⠀⡇⠀⠀⠒⠂⠛⠋⠈⠃⠁
⠀⠀⠈⠒⠓⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⢀⠔⠁⢐⠀⠇⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠅⠈⠀⠀⠢⠀⠠⢭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠤⠤⠈⠁⠀⠀⠀⡄⠀⢀⠀⠸⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠆⠀⠆⠀⠀⠀⠘⠀⠘⠋⠉⠀⠀⠠⠤⠤⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠊⠚⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⢀⡀⡇⡁⢇⡩
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠀⠙⠁⠙⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠛
⠀⠀⢠⠦⡤⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⡀
⠀⠠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢀⠀⡀⠤⠤⠤⠤⠀⠀⠀⡀
⠀⠐⠀⠀⠅⠀⡀⠀⠀⠐⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⠉⠀⠀⠀⠀⠐⠀⠀⠀⠉
⠀⢠⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⣆⠀⠾⠡⠄⠆⡰⠀⠀⠀⠀⠀⠀⣀⡀⠘⠫⠤⠤⠛⢀⣀⣤⠀⠀⠀⠀⠀⠀⠐⠛⠑⠘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠐⠂⠀⠐⠚⠁⠀⠀⠀⠀⠀⢠⠂⢈⡉⠉⠉⣉⢉⡉⠀⢸⠀⠀⠀⠀⠀⠀⡀⣰⠀⠐⢬⡄⠀⠀⢸⠀⠀⠀⠀⢀⠚⠉⠁⠀⠆⠔⠀⠉⠓⣄
⠀⠀⠀⠀⠥⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠧⠑⠚⠞⠃⠸⠀⠄⠃⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠰⠒⠂⠀⠂⠐⠔⠇⣸
⠀⢰⠀⠂⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠲⣅⡁⠘⠓⠁⢀⣀⠴⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⡀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢲⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⣾⣿⠃⠀⠈⠂⡀⠀⠀⠀⠉⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⢻⣿⣿⣿⣿⡷
⠀⠀⢸⣿⣿⣿⣿⠟⠀⣼⣿⣿⣶⣤⣤⣯⢄⠀⠀⢀⡀⢘⣃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠈⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⡏⡀⠘⠻⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⡷⠿⠟⠀⠀⠀⠀⠀⠀⢃⣤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣿⣏
⠀⠀⢸⣿⣿⣿⢇⠁⠀⠀⠈⣽⣿⡏⠁⠉⢻⣾⠀⠸⠂⠀⡀⠀⠀⠀⠀⠀⢀⣨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡎⣿⣿⣿⣿
⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⣀⣿⣿⡇⠀⠀⠀⠘⡇⠀⠀⠀⢿⡇⠀⠀⠀⠀⢠⣴⠟⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⣿⣿⡟
⠀⠀⢸⣿⡟⠁⡀⠀⠀⠀⣿⣿⣿⡁⠀⡀⠀⠀⣧⠀⠀⠘⡞⠒⠒⠀⠀⠀⣿⡏⢰⠀⡀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⢳⣿⣿⡷
⠀⠀⢸⣿⣴⣦⣿⣿⣦⣄⣸⣿⣿⡇⠀⠁⠀⠀⠃⠀⠀⡼⠁⠀⠀⠀⠀⣀⣽⣷⡆⠂⣷⣤⠂⢣⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⢹⣷
⠀⠀⢸⣿⣿⠛⠻⡿⠟⠙⢿⣿⡿⠟⣤⡀⠀⠰⣶⣶⡞⠀⠀⢴⠞⠀⠹⣿⠟⢿⣧⢤⣿⣿⡤⣼⣀⡄⠀⠀⠅⠀⠀⠀⠀⠀⡀⡄⠀⡀⢿⣿⡟
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠔⠀⢹⢿⠆⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣿⣿⡧⣼⣿⠁⠀⣤⠀⠀⣴⡂⣀⡀⡈⠀⢀⡀⢸⣿⣿
⠀⠀⢠⡀⠀⠀⠀⠀⠐⠀⠀⣸⢂⡠⠐⠾⠆⠰⢶⣿⣷⣾⣆⠀⠀⠀⠀⠀⠀⢸⣯⡴⠛⠛⠣⠤⠈⠀⠐⠂⠈⠉⠀⠈⠀⠉⢛⣭⣿⣤⣼⣿⡿
⠀⠀⢨⣭⣤⣤⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⢩⡍⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⡤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣩⠁
⠀⠀⢸⣿⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⢸⣿⣟⡉⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢸⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠁⠀⢀⡀⢠⡇⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢸
⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣬⢁⡰⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈
⠀⠀⢀⠀⠂⠀⠀⠀⢤⣤⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣿⡟⠛⡾⡇⣷⡄⠀⢀⣀⣴⣶⣶⣦⣤⡴⠃⣠⠚
⠀⠀⢸⡇⢠⠁⠀⠀⣼⣿⡍⠋⠉⠑⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣤⠆⢴⡇⡟⠈⠂⠐⣿⣿⣿⡿⣩⡶⠲⢯
⠀⠀⠘⣿⣄⡀⠙⠛⢿⡛⢻⡆⢰⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢀⠘⡇⠁⠀⠀⠲⢾⣶⣍⣿⣿⢅
⠀⠀⠀⠈⠀⠁⠀⢀⣸⣇⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠡⠾⠇⠀⠀⠀⠀⣩⣿⣿⣿⡋⠈
⠀⠀⠰⠀⢀⣤⣶⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢀⣠⣴⠾⢿⣿⠈⠋⠁
⠀⠀⢀⣤⣾⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡿⠀⠀⠀⠀⣿⣿⣿⣿⣄⡙
⠀⠀⢸⣿⣿⣿⡿⣿⣶⣿⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁⠀⠀⣴⠆⣿⣿⣿⣿⣟⣓⣠
⠀⠀⢸⣿⣿⣿⠃⠉⣿⢿⣿⠏⠀⠘⠚⠁⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⣴⣿⡆⣻⣿⡿⠁⢉⢿⣧⠚⠁⠀⠀⠲⠒⠒⠒⠤
⠀⠀⢠⣷⣶⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠙⣿⣿⣿⣿⣿⣿

⠀⠀⠸⢫⣶⡟⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⡇⣾⡇⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠃⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣻⣿⣿⣿
⠀⠀⢸⣿⡏⣼⡏⢀⠀⠀⡀⠀⠀⠀⣈⡀⠀⠀⠀⠠⠘⣤⣄⠀⢸⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢻⣿
⠀⠀⢸⣿⣾⣿⠀⡾⠀⣴⡇⠀⣴⡈⣙⣻⡆⠀⠀⠀⢀⠉⢟⠀⣸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡀⢻
⠀⠀⢸⣿⣿⣿⣶⣥⣾⣿⣷⣤⣸⣿⣿⡉⠀⠀⠀⠉⠀⢐⣿⣎⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋
⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠏⣇⠀⠀⠠⠀⠂⠻⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣿⣿⣷⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢧
⠀⠀⠀⠀⠀⢀⣤⣾⡟⠋⠄⠐⢛⣿⡿⣆⢉⡀⠀⠀⠀⣀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⡟⠃⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠐⠲⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⠇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠤⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⡀⠀⢀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡃⢻⡋⣿⣖⢒⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣩⠿⣧⡼⠻⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣞⢙⣶⡋⣳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣤⠿⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡉⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠤⠀⠰⠶⠴⢿⠛⠋⠙⠂⠋⠀⠀⠀⠀⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠣⠆⠀⠀⠀⠀⠠⢀⣰⠒⠛⠃⠀⠀⠀⠔⠒⠀⠚⠀⣠⣄⣬⠀⠀⢂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣹⠇⠀⠀⠚⣿⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠛⠆⢿⣷⠀⠀⢀⡀⠜⠳⢄⣀⡴⠖⠲⣄⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠠⠏⢠⣦⡄⠀⠀⠈⠀⠻⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣻⣍⣁⠀⠀⠀⠀⢀⣰⡆⠠⠆⠀⠀⠀⣤⣀⡀⠈⠉⠉⠠⣴⠐⠀⠠⣤⡤⠖⠒⠚⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠉⠈⢡⠉⠀⠀⠀⠀⠈⠁⣾⠤⠀⢠⡖⠛⠉⠛⠀⣁⣀⠀⠀⠁⠈⣷⣶⡏⠀⠀⠀⠀⠀⠙⡷
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢻⣶⣾⠋⠀⠀⠀⠠⣾⡥⠄⣤⠈⡴⢄⠀⠀⠀⠈⢃⡹⠂⢀⡸⠇⢀⠀⠚⢿⣿⠛⠀⢀⡸⠛⢀⣠⣴⣿⠐⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠋⣸⣆⠀⠀⠀⣤⠃⠘⣷⡘⠧⠀⠀⢦⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⣦⡟⠁⠀⠈⠁⣭⣤⣌⣀⠀⠀⠚⡂
⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⢀⣄⣈⣉⠇⠠⠾⡇⢠⡄⠘⠁⡈⠃⠀⠐⠊⠁⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠘⠛⠀⠀⠻⣄⠀⠀⠀⠀⠈⣹⣙⠉⠀⠐⡀⠉⠲⠶
⡀⣀⣰⣆⠀⠀⠀⠀⠁⠈⠿⢧⣄⠀⠀⠀⠀⠀⠀⣻⢾⡇⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠘⢻⠀⠀⠀⠀⠀⢳⣤⡄⣸⣷⠶⠖⠂⠈⠛⢀⣾⣿⣦⠀⠀⢀⡀
⠀⠀⠙⠟⠀⢀⠀⠀⠈⠀⠀⠀⣹⡄⠀⠀⠀⠀⢀⠣⠈⠁⠘⠲⣄⠙⣿⣶⠶⠖⠲⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⣾⠃⠀⠈⢙⠀⠀⠀⠀⢢⡈⠻⠛⠁⠀⠀⣼⡇
⠀⠀⠀⢦⣤⠾⣳⠴⢀⡀⠀⠀⠉⠁⠂⠐⠆⠀⠀⠐⠀⠀⣀⣼⣿⡄⠀⠀⠀⠀⠀⢰⡿⠰⢀⣠⣄⡄⠂⠐⠂⠛⢃⣉⠀⠈⠁⡟⠀⠀⣤⡄⠸⠻⠇⢠⡔⠉
⠀⠀⠀⠸⡇⠀⠀⠀⠘⠁⠀⡀⠀⠰⠓⠀⠀⣀⢠⠀⢤⣠⡈⢉⣿⣥⡀⠀⡀⠐⣦⠈⠀⠀⠀⢹⣿⣤⡀⠀⠀⢰⣦⠙⢶⡌⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣇⠀⠀⠂⠀⠄⣀⣽⠿⠇⠀⢰⣾⡟⢣⠀⠀⠉⠛⢀⣄⡙⠟⠈⢀⣸⣿⡇⠀⣠⣀⣠⣷⣄⢡⡀⣤⡈⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠛⢇⣸⠀⠀⠀⠀⠸⣿⣇⠀⠀⢿⣿⡄⠀⠀⠉⠹⠾⠿⠛⡛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⠣⢀⣠⣿⣄⡀⠘⠀⠀⠀⠈⡃⢀⠶⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠙⣻⡀⢀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠄⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠔⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢄⡁⠀⡉⠠⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣗⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣿⠀⠀⣀⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠊⠁⠀⠀⠀⢀⠄⠀⠀⠀⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣀⠤⠶⠐⠒⠐⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⠒⠀⠀⠀⠀⠀⠀⠶⠚⠉⠁⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠂⠐⠆⠀⠀⠀⠀⠀⠒⠐⣶⢦⠀⠀⠀⠀⠀⠈⡆⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠧⠜⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠦⠤⡀⠀⠀⠀⠘⣧⠀⠀⠀⡀⠀⠠⢀⠀⠀⠀⢸⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣆⠀⠈⠢⡀⠀⠀⠘⣧⠀⠄⠈⠑⠂⠀⠈⠀⠂⣸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⢿⠀⠀⠀⠸⡄⠀⠀⣿⠀⠀⠢⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠀⠀⢃⠀⢀⡟⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡄⠜⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠘⣠⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠘⣌⢂⡀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⡄⠁⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣈⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠤⠶⠒⠊⠉⠉⠀⠀⠀⠀⠈⠉⠉⠐⠒⠦⢤⡀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠔⠋⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠀⠀⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⠃⠐⠀⠀⠐⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠂⠀⠀⠀⠀⢀⠀⡀⢀⡈⠀⠀⢀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡰⠊⢁⡈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣤⣤⣤⣤⣴⠶⠦⠶⠶⠶⣶⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⢁⠔⡚⠛⡳⣦⡓⠺⠃⠁⣿⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠛⠛⠛⠛⠒⢦⣤⢤⣀⡤⠤⢤⣀⣠⠴⠋⢉⣩⠟⢿⣶⡄⠀⠀⡆⣠⠆⠃⡖⠛⠉⠄⠈⡇⢉⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⡀⠀⠀⠀⠀⠀⠀⢳⣤⡄⠛⣦⢀⡤⠤⠖⠚⠉⠀⢀⣼⡿⠛⠀⠀⠀⡸⠃⠆⠳⡤⢶⡎⡰⣧⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣄⠙⠿⣦⣄⠀⣞⠛⠛⠛⠁⠉⠙⠷⠛⢧⡀⠀⢀⣠⣴⠟⠋⡠⠀⣀⣀⣀⠀⠀⠀⠚⢀⡤⠛⢼⡌⣷⠠
⠀⠀⠀⠀⢸⣿⣏⠀⠀⠑⠀⡈⠙⠻⢿⣦⣤⣀⣀⣀⣀⣠⣤⣴⠾⠟⠛⠉⣀⢴⠀⡰⠋⠐⡔⢈⢳⡄⠀⣀⡨⠓⣤⠺⢃⠏
⠀⠀⠀⠀⠀⡘⠿⣷⣄⠀⠀⡤⠶⠦⠄⠀⠀⠉⠉⠉⠉⠁⠀⠀⢀⣴⣶⠟⢁⠌⠀⡃⢾⡇⠀⣿⠂⡷⠋⢁⡀⠀⢤⡀⠁
⠀⠀⠀⠀⠀⠈⠠⠈⠛⠷⣦⣷⣀⡀⠀⠀⠀⠀⠁⢀⣈⣷⣴⠾⠛⠋⡀⠔⠀⠀⠀⠰⢴⣤⡧⣬⡜⠁⢾⡁⠈⣿⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣹⠟⣧⢈⠙⠃⢩⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠒⠂⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠐⢻⣽⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁⠉⠠⠘⡡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⣿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠂⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⣀⣀⠦⢌⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠀⠄
⠀⠘⠅⠀⠨⠀⠀⠈⠁⠀⠒⠒⠒⠒⠀⠀⠉⠀⡉⡡⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠈⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠐⠀⠢⠀⠀⠀⢁⠀⢠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠂⠈⠉⠒⠒⠲⠤⠤⠤⠤⠆⠒⠒⠉⡀⠀⠀⠀⠀⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢠⠀⠀⠀⠀⠈⢢⣀⠐⠂⠠⠀⠀⠘⢄⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠦⠄⣠⣌⡿⢭⣤⣤⣤⡀⠀⠈⠣⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⠓⠒⣷⣒⣒⣈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠢⢄⡄
⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⣂⣐⣿⡒⣂⣀⠒⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠠⠤⣿⠤⠄⠀⡆⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠶⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⡀⠀⠀⠀⠬⠍⠭⠤⣽⠭⠭⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡃
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣤⢠⠀⠀⠠⢍⡄⠀⠂⠸⠀⠠⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈
⠀⠀⠀⠆⠀⠈⠀⠀⣔⠃⠠⠀⠀⠀⠀⠑⠀⠰⣦⠀⠉⠉⠰
⠀⠀⠀⠆⠀⢀⠀⠂⡍⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣩⠃⡆⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠂⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⣄⠀⠀⠀⠤⣤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠉⠀⠀⠐⠀⠀⠀⠙⢆
⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠈⠁⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠮⣫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠐⡀⠀⠀⠀⠀⢠⡪⠊⠲⠀⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⢀⠀⠀⠀⢀⢔⠏⠀⠀⣾⡀⠀⠑⢕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡖⠁⠀⠀⢐⢕⡇⠀⢀⣀⣉⣀⣀⠀⢸⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡠⣖⡴⢅⣦⣋⠉⣇⢤⡉⢠⠶⣆⠈⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢈⠸⠼⠀⡏⠚⠁⢸⠀⣿⠀⠓⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠦⠶⠾⠼⠦⠤⠴⠷⠤⠶⠿⠷⠿⠶⠶⠴⠦⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆⣠⡾⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠈⠪⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡻⠋⠀⢮⡇⠈⠻⣦⡀⠀⠀⠀⠀⠀⠨⠎⠀⠀⢭⡅⠀⢘⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡏⠀⠀⠀⠈⠀⠀⠀⠈⢫⡢⡀⠀⠀⠀⠀⠄⠀⠀⠚⠃⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠶⠴⠀⡏⢉⠃⠴⠆⠀⠈⠀⠀⠀⠀⡼⡇⠀⠀⢽⡇⠀⢈⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣉⠆⡇⡿⠚⢀⣧⣼⡀⠛⠷⢀⠪⡜⠀⠀⢠⣇⣇⢀⣀⣈⣁⣀⣸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠉⠯⠯⠭⠿⠿⠿⠋⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⣄⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣽⣾⣿⣿⣿⣿⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣞⢦⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⡟⣭⡿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢸⣿⡇⣿⡇⣿⢱⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢿⠇⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢞⡇⠀⠀⣿⠇
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⣿⣿⣿⣦⡀
⣿⣟⠀⣸⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣶⣷⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣤⠖⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣴⣦⣄⠁⠸⡆⠀⠈⠙⠒⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⡀⠘⠛⠋⠉⠀⠀⠐⡄⠀⠀⠀⠐⠄⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢿⡿⢻⣶⣏⡀⠀⢀⡀⠀⠁⠀⣀⠀⠘⠦⣄⠈⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⢞⠵⠓⠾⣷⣿⣇⢰⣿⡿⠀⠐⢿⣿⡄⠀⠀⠋⠒⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠹⣾⢻⢶⣤⣴⡆⠀⡀⠁⢠⠀⠀⠐⠂⠉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠻⣗⢄⠀⠈⢉⡉⠀⠀⠈⠳⣶⡿⢋⣿⡀⠐⠄⢈⠓⠘⣇⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠻⡦⢠⣍⡇⣶⣀⣀⠀⠈⢷⡾⢙⣿⣦⡴⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⠀⡄⢿⣮⣭⡀⠀⠀⠉⠀⣀⠤⠈⢣⡸⣹⣇⠰⣖⠀⠀⣗⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡟⢂⣴⡈⣷⣦⡄⠀⡾⢃⣠⠐⠀⠻⣿⣿⣦⠈⣠⣨⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣾⡿⠟⢘⠻⠿⢦⡄⠁⠀⢠⠄⠀⠀⠙⠿⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣷⣤⣈⣙⡆⠀⢈⡳⣲⠄⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠛⠲⠖⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡄⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⢀⠠⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⠀⢀⣦⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠠⠀⠂⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢿⣿⣿⡿⠋⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠠⠟⠋⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠐⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡆⠀⠀⠀⢠⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⢆⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⣰⣶⣌⣢⠣⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡏⠀⠀⠀⠀⠘⠿⠿⠋⠀⠘⢄⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢣⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠛⠁⢢⡀⠀⠀⠀⠀⠐⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢠⡞⢆⡇⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠖⢋⣀⠀⠤⠐⠂⠈⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⣷⣿⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡔⡂⢸
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠉⠉⠁⠀⠀⣀⠀⠀⠀⠀⠒⠂⠉⠉⠻⣅⣃⣻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠏⢀⠃⣿
⠀⠀⠀⢀⣤⣾⣉⠠⠤⠄⠴⠖⠒⠈⢉⡤⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⠀⠀⠀⠀⠀⠀⠘⢷⡞⡃⡿
⠀⠀⠀⢸⠯⣿⡇⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢡⠆⠀⠀⠀⠀⠀⠀⠈⡅⡇
⠀⠀⠀⠸⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⣦⡇
⠀⠀⠀⠀⢿⣤⡀⠀⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⣠⢔⣲⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠛⠿⢻⡟⣷⡄⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣹⣄⣴⡷⠦⠿⠾⠛⠛⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢿⣄⣀⣐⣀⣀⡤⠭⠭⠗⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢉⣣⡀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠊⠉⠉⠻⠿⢿⣿⣿⣏⠂⢀⡈⠉⠙⠲⠄⠻⣷⣌⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠮⠤⠀⠰⠶⠶⠄⠀⠀⢄⠀⠻⣿⣿⣿⣄⠀⠁⠀⠀⣤⣤⡀⠈⢋⠀⠘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⣶⣦⣄⠀⠀⠀⠹⠦⣉⡁⠀⠀⣿⣿⣇⠈⠁⢀⠀⠀⠉⠛⠻⠂⠈⠛⠆⠘⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣍⡁⠀⠀⣀⠴⠀⠀⠉⣁⣤⣤⣽⣿⣿⣄⢠⣼⡶⠋⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣮⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⣿⣍⠙⢿⡁⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣼⣟⠀⡀⢀⠀⠀⢴⣷⣦⠀⠀⠀⡈⠁⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠍⣿⣿⣿⣷⣽⣻⣿⣷⣆⠛⠀⡀⢳⣄⡀⠀⢀⣉⣿⣿⣿⣿⣿⣦⡑⠀⠀⠀⠀⠈⢉⠀⠀⡀⠸⢀⠀⠠⠙⡇
⠀⠀⠀⠀⠀⠀⢠⣰⡎⣙⢉⡭⠿⠛⠻⠿⣿⣿⣤⣦⣽⣤⣉⣳⣶⣤⣤⣵⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⢀⡼⠃⠈⠈⢣⡈⣥⣄⡐⢶
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣟⣶⣆⣀⠀⡾⠁⠀⠉⠉⠉⠉⠉⠈⣧⢈⣽⡿⣽⣿⣿⣿⣿⣮⢅⡀⠂⠉⢠⠐⠀⠀⠀⣱⠜⢿⣟⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣻⣿⣬⣿⣤⡭⠀⠳⠀⠁⠀⠀⠈⣈⡀⠀⣿⡯⣠⣿⣿⣿⣿⣿⠟⠁⠀⠈⠐⠺⣶⣅⣂⣙⣰⡷⠂⢀⣶⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⡭⢻⣧⠤⣴⣦⣤⣤⣶⣾⣿⣵⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⡃⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠉⠉⠛⠷⣦⣄⡀⢠⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⠛⠉⠉⠉⠀⠈⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣀⡀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣤⣀⣤⣶⣿⣯⣭⣤⣤⣶⠶⠶⠶⠶⠶⠶⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⣽⣿⣿⠿⠿⠿⢿⣿⣿⠿⠿⠿⣿⡿⢿⣿⣟⡻⠿⠶⠶⠶⠶⠶⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣟⣋⣉⣁⣤⣤⣴⠶⠞⠛⠛⠉⠀⠀⠀⠀⠈⠻⠶⣦⣽⣿⠷⣶⣿⡿⠶⣦⣤⣀
⠀⠀⠀⣠⡾⠋⢩⡿⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣌⠉
⠀⠀⣼⠏⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠔
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠐⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠈⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠉
⠀⠀⠀⠀⠀⢸⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠖⠁⠀⠀⠀⢄
⠀⠀⠀⠀⠀⢸⣷⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡇
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠤⠤⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠄⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠒⠒⠐⠂⠤⠄⠒⠀⠉

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⠁⢀⡞⢇⠣⡐⢺⣿⣿⣷⣿⡿⣿⠅⠀⠀⠀⠀⡤⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⡿⡿⠀⠀⠀⠀⠠⣴⡃⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠂
⣄⣠⠏⠀⣸⡀⠈⢂⠻⠛⠶⠈⠁⠀⠀⠀⠀⠀⠀⠠⠽⠶⠀⠀⣰⣶⡾⠿⢷⣤⡀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠠⠧⠦⠀⠀⠾⠧⠀⠀⠀⣀⡀
⣿⢿⡄⠦⠁⢘⣰⡄⡁⠄⠒⠀⠀⠀⠀⠀⠀⠵⠄⠀⠀⠠⡀⢠⣿⣿⣷⣶⣿⣿⣗⠂⠀⠀⠀⡀⠀⢠⡷⠀⠀⡠⠄⠀⠀⠀⠀⠀⠐⢾⢿⠁
⣿⣿⣏⡑⡤⢤⡉⠸⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠁⣼⣿⣿⣿⣻⣿⣿⡿⠖⢒⡎⠀⠾⠂⣉⠀⠀⠐⠲⢤⡀⠀⡀⠂⠀⠀⠂⠀⠀⢀⠀⠀⡀
⠟⠉⠈⠠⣷⡖⢃⢉⢄⣦⣶⣿⣾⣤⣦⠀⠀⠀⢀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⢸⠿⢿⣷⣴⣶⣤⣤⣀⡀⠀⠀⠁⠖⠀⠀⠀⠁⠀⠀⠀⡻⠃⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⢠⣀⡤⢠⡀⠀⢿⣿⣿⣿⣿⣧⣀⠃⠀⠀⠈⠉⠙⠻⢿⡿⣿⣤⡀⠀⠀⠀⠀⢈⠀⠀⢀⠄⠀⠀⢢
⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣼⠿⠀⣾⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠉⠙⠂⢰⠞⠀⢠⣶⣶⡎⠀⠀⠀⡾⣿⣿⣿⡇
⠃⢠⠀⢴⣾⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣧⠄⠀⢁⣿⡿⣿⣿⣏⡀⠉⠈⠙⠿⠿⢶⣶⣶⣤⣄⡀⠀⠀⡰⠃⠠⠘⠿⣛⣁⣀⠂⠸⣷⣿⣿⣿⡾
⢀⡈⠐⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠇⣤⣴⣿⣿⣶⡏⠉⠉⠙⠷⢿⠁⠀⠀⠀⠈⠛⠉⠻⣿⣿⣶⢻⣤⠄⢢⡘⠻⣿⠇⠀⣾⣦⡝⠛⠉⠱
⠀⢐⡀⠀⣠⣶⣾⣿⢿⣿⣿⠀⢙⣀⠀⣭⠉⠉⠉⣬⣡⣴⣿⣿⣿⣿⣿⡀⠀⠉⠀⠀⢤⣷⣦⣀⠀⠀⣶⣄⣀⣽⡿⠁⣤⢠⡄⠀⢽⣦⡘⣿⣵⡿⠟⠋⠁⠀⠂
⣀⠘⠃⠰⢿⡿⡋⠀⠚⠻⠛⠀⠈⠁⢀⣴⠟⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠉⠙⠁⣴⣿⡔⠈⣩⣾⣿⣿⣿⣷⣾⡛⠂⠄⠀⠦
⡟⠀⠶⠀⠀⠀⠤⠐⠀⠀⣠⣶⡶⠲⣾⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠁⠀⣶⣾⣟⢉⣴⣿⣿⡿⢋⢾⡿⢹⣿⣿⡍⠉⠁
⠀⠀⠈⠤⠀⢰⡄⢰⣦⣼⣿⠀⠈⢢⠉⣀⡟⠋⠙⢀⢉⠻⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣰⣿⣿⣇⠚⠛⠋⠁⠔⠋⣼⣍⡁⠹⣿⣿⡄⢸⣿
⣶⡈⢡⡄⠀⠀⠰⢶⣿⣿⣿⣷⣤⠜⠣⠉⢤⣄⠀⠀⠋⠀⠀⠛⣿⣿⢄⠀⠂⠀⠈⠉⠙⢿⠿⠋⣿⡟⠻⣿⣿⣿⣿⣿⡟⠻⡀⠰⠂⣸⣿⣿⣿⡗⠈⠻⠷⠈⣿
⠉⠁⠀⢀⣶⠘⠃⠘⠿⣿⠿⢁⣴⣶⠶⠶⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⢰⣿⣷⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⡽⢏⠻⡛⣯⢼⣖⠋⣡⣶⡝
⠀⠀⠀⠈⢿⡇⠀⠘⢃⠞⢳⣿⠿⠿⢃⠀⠐⣶⠂⠀⢰⣆⣀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠞⢑⢄⠣⡈⢿⣿⠟⠛⠜⠛⠃
⣀⣀⣀⣀⣸⣹⣀⣀⣀⣀⣘⣿⣧⣀⣰⣶⣆⣐⣀⣚⣁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣏⣀⣰⣲⣁⣈⣂⣉⣙⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠀⠀⠀⠀⣴⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣤⣼⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡴⡋⣔⢮⢳⡀⠀⠀⠀⢸⡇⠀⠀⣴⠋⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⣎⢯⡳⢳⠀⠀⠀⢸⡇⢀⡞⠁⠀⢠⡞⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⢮⣅⣠⣼⠁⠀⠀⠈⣷⠋⠀⠀⣰⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣦⠀⢀⡾⠁⠀⢀⡼⠁⢀⣀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠒⠒⠒⠚⠛⣉⣩⣭⢿⡆⠀⠈⣷⣏⣠⡤⠶⠿⢛⣋⠉⣥⠖⠆⠀⣷⣀⣤⠤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⣶⣛⣭⡥⠶⠾⢓⡛⢉⣭⡅⣰⢰⡏⣳⢸⡥⠀⢨⣉⡷⠀⣿⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢸⡏⠁⡶⠒⡦⢸⡏⠀⣾⠀⡀⣿⠸⠟⠁⠘⠛⢁⣀⣁⣤⣤⣿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣃⣀⣀⣸⡇⠀⡷⠾⣅⢸⣯⠤⠘⠛⢁⣀⣠⣤⠶⠚⢻⠋⣁⣤⡴⠞⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⡇⠀⠁⠀⢀⣀⣤⠴⣶⠟⠋⠉⡇⠘⣆⠀⢸⠟⠁⢀⣀⣠⡤⠶⠶⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠙⠒⠛⣻⠏⠁⢀⣼⠋⠀⠀⠀⡇⠀⠘⣧⡈⠛⠋⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡼⢥⡀⢀⡾⠁⢀⣀⣤⣀⡇⠀⠀⠸⡿⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣷⣽⠟⠀⣞⣉⣤⡤⠾⡇⠀⠀⠀⠙⣮⡷⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠁⠀⠀⠀⢻⡉⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⡠⠊⢀⡀⠑⢄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⠀⠑⢄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢠⠎⠀⠀⢣⡸⠀⠀⠑⢄⠀⢀⠔⠁⠀⠸⡿⠀⠀⠑⢄⠀⠀⠀⢀⠔⠁⠀⠘⣀⠅⠀⠀⡑⡄⠀⣴⣿⣿⣿⡄⣸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⠀⢰⠅⠀⡥⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⢳⠀⣿⠀⠀⠀⠀⠀⠉⡇⠀⡮⠀⢨⡆⠀⡏⠀⠀⠈⣿⣿⡇⣴⣦⢹⣿⡏⠁
⠀⠀⠀⠀⠀⢸⣀⣉⣁⣀⣉⣀⡀⠀⠀⠀⠀⣀⣈⣁⣀⣉⣀⡿⠀⠀⠀⠀⠀⠀⢆⠈⠉⠀⠈⠁⠠⠇⠀⠀⠀⠻⣿⣶⣿⣷⣶⣿⠇

⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⢸⣿⡿⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡖⢚⣓⠒⠒⠆⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠐⡟⣿⡟⠙⣿⠀⢒⣖⠛⢒⡒⠂⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣅⠀⠀⠀⢀⣤⣽⣥⣄
⠀⠀⠀⠁⠉⠉⠉⠉⠀⢨⣿⠀⢸⡏⠀⠀⠀⠀⠀⢀⣛⣿⣿⣥⣤⠼⠿⣧⣤⣆⠀⠀⠀⣀⠀⣶⣿⣿⣿⣿⣷⡄⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠭⠽⠿⠷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠀⠀⠀⢿⠃⢻⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⡏⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣬⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠐⠀⠒⣲⠄⠒⠠⠤⠄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋⠁⠉⠀⢶⣾⠷⠀⠀⢉⣀⢀⡀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⢀⡀⠀⠀⠀⢀⡆⠀⠀⠈⠉⣉⡡⠂⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠖⠂⢁⠠⠄⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⡀⠀⣆⠀⠀⠀⠂⠘⠿⡷⠉⢁⣀⠀⣀⣠⡤⠖⠚⢫⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⠛⠉⠉⠒⠒⠒⠂⠉⠉⠉⠉⠉⠀⠀⠀⠐⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠐⢀⡀⠀⡈⠉⠀⠠⠾⢿⠶⠀⠀⠀⠀⣄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡁⠀⠀⠁⠀⠚⠛⢀⠒⢺⠁⠀⠈⠀⠀⠀⠀⠉⠙⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⣀⠈⠉⠓⠒⣲⣶⡒⠤⡀⠀⠀⣄⠰⠟⢛⡀⣀⠤⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉⠁⠀⢀⡠⠈⠉⠉⠑⣤⡈⠉⠯⠭⠁⣋⣵⡊⠀⠠⠔⠲⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⣐⠁⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀⠘⣏⡓⠬⢤⣤⣴⣦⡀⠀⠀⠀⣠⠀⢀⣀⠀⠀⠂⠀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠰⣶⣷⠌⠀⡈⠉⢉⠈⢱⡖⠚⣋⠟⠋⠉⠉⠉⠊⠁⠀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢴⡁⠀⠀⠀⢀⠁⠐⡏⠁⠠⢀⠀⠀⠐⢆⡀⠀⠀⢰⣖⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁⠀⠉⠓⠶⠶⡾⠶⣦⣄⣀⣀⡼⠷⣦⠤⠼⠧⠴⠖⠋⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃⠀⠸⠁⣠⠈⠂⢠⡑⠀⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠰⢿⠇⠀⠸⣯⣦⠀⡀⠤⠦⠖⠀⠀⠀⠒⢦⠀⠉⠉⠁⣀⠇⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠀⠀⠈⠉⠛⠒⠦⠄⣀⣀⣀⣀⠀⠀⢀⣀⠀⠀⣠⡥⠴⠤⠀⠐⠉⠁⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠉⠥⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠶⠀⠓⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⡢⠀⠀⠀⠺⠶⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠒⠒⠒⠒⠒⠛⠛⠒⠒⠢⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣤⣞⠉⠲⠉⠉⠀⠀⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠶⡀
⠀⠀⠀⠀⠀⣿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⢠⣶⣦⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠹⡄⢄⡀⠀⠀⢠⣶⡦⠀⠀⠀⠹⠿⣿⠟⠀⠀⠀⠉⠁⠀⢀⡠⠄⡠⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠲⣬⣥⣒⣀⠬⠤⢄⣀⣀⣀⣀⡀⠤⠄⠀⠀⢀⣤⣤⡠⠴⠾⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠟⠒⠶⠶⠶⣤⣦⠤⢴⡶⠶⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⣦⣤⡀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠰⣶⠀⠀⠀⢀⣠⣄⠀⢐⣡⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠉⠓⠮⢍⣒⡒⠢⠤⣤⣤⠀⠀⠤⠤⠤⠄⢀⣈⡭⠵⠒⠉⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⠀⣀⠀⠈⠉⠉⠛⠛⠓⠒⠛⠯⠍⠉⣁⣄⡀⠀⠠⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠭⢭⣐⣒⣶⣦⣐⣒⣂⣀⣀⣀⣙⣿⠥⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠏
⣀⡴⣶⣿⣶⣵⣦⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠹⠉⣃⣤⠶⢛⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡀⠀⠀⢀⣤⢞⣫⡴⠞⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣤⣶⣮⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣴⣫⣗⣛⣭⣨⣔⡀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣗⠒⠤⢀⠀⠀⠀⠈⠻⣿⣟⠛⠋⠉⠉⠙⢿⣿⣿⣿⡟⠀⠀⠀⢀
⠭⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣾⣿⣿⣾⣿⡍⠑⢄⠀⠀⠀⠀⠹⢿⣿⣥⣤⣴⣶⡿
⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⠀⠑⣄⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠘⢿⣿⣿⣿⡿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠀⢸⡀
⠀⠀⠀⠀⠉⠉⠉⢡⡿⡘⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡏⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠻⣇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢹⡄⢁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠁⠸⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣤⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⠿⣷⣶⣤⣤⣶⠆⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣦⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⣤⣤⣼⣿⣿⣿⣿⣿⡁⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠛⠉⠁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣇⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⡏⢠⣠⣠⣤⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢧⣿⣿⡿⠛⢻⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣤⣄⣀⣠⣤⣶⣾⡿⢃⣾⠟⠏⠀⠀⠀⣼⠏⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢟⣿⠟⠀⢘⠇⠀⠀⠀⣠⢞⣵⡾⠟⠋
⣶⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠘⢾⣟⢿⣦⣤⣤⣤⣤⣶⣿⣿⢿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⡁⠁⡰⠁⠀⠀⠀⠀⠀⢘⠐⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠙⢦⣌⠻⢿⢿⣿⣻⠿⠓⠠⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡧⢈⠿⠊⠀⠀⠀⠀⠀⡀⠖⠁⠰⠶⠾⡉⡀⣁⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⠁⢀⡲⠈⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠐⠞⠃⣁⡜⡖⡆⡄⡆⢄⣤⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣤⣈⣠⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠁⠄⣰⣤⣠⠀⢀⣠⣷⣿⡻⡽⠙⣹⣼⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢶⣤⣄⣀⣀⣀⣠⣴⣾⢏⠊⢊⠊⠜⠈⣅⣶⣿⣿⢃⢁⢡⣷⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⡿⠟⠑⠁⠁⠀⠀⣴⡼⣹⣿⣏⣅⣆⣧⣾⣿⣿⣿⣿⡿⣿⣿⢹
⣦⣤⣤⣤⣶⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⣠⣾⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⡟⣯⣷⣷⡿⠛⢼
⣽⠋⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠕⠁⠀⣀⣴⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⡟⠃⠃⠇⢿⢻⡀⣀⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀⢀⡄⣀⢠⣀⠀⠁⠖⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⡠⣾⣴⠟⡹⠋⠈⠛⣇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢿⡀⠈⠙⠓⣿⠀⢁⡀⠈⢿⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡿⠯⠀⠁⢁⢀⠜⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠷⢤⡤⠒⠈⡏⣰⠁⡦⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣧⡴⡂⢠⡼⠂⠀⠀⠀⠀⠤⣽⠀⠀⠴⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢉⣹⣿⣧⣙⠉⠁⠀⠾⡋⠀⠀⠀⣸⠀⠃⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⠃⠈⡻⢿⣼⠉⠉⠀⡀⠀⠄⠀⠀⠀⠀⠀⡧⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡶⣮⠺⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⠋⠀⠀⠘⠠⡐⢀⡀⠂⠀⠀⡀⣴⣧⣠⡏⠒⢠⠀⠀⠰⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⢎⠹⣿⣷⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⠀⠀⠀⠀⠀⠀⠇⣠⢴⡯⠁⡇⢿⣿⢿⡲⠯⣐⠀⣀⠀⡟⠃⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣘⣷⣿⡿⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⣿⣦⠀⠀⡀⡀⠀⠈⠓⣏⠀⠀⡶⠀⡉⡿⣃⣿⣷⣤⠄⠊⢹⣦⡀⠰⠀⠀⣀⠀⠀⠀⠀⠀⠀⢶⠉⢻⢟⣿⠇
⠀⣠⠞⣿⢟⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⢻⣹⢷⡤⣭⣀⣀⣀⢴⠞⠁⢤⠦⣹⣿⡘⠋⠻⣿⠷⢦⣄⠈⡋⢃⠞⠉⢁⣀⣀⡀⠀⠀⠀⢠⠸⣴⣏⣼⠏
⢀⢻⣞⢣⣼⠁⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⣿⣧⣵⠆⠉⠉⢙⣀⣠⡠⠞⠿⠋⠉⠙⢿⠀⠄⠀⠀⠈⠀⠟⠁⠴⠾⠋⢵⡂⣀⣠⣤⣶⠾⠳⠾⠍⠋
⠘⣿⣷⠐⠃⣴⡧⢄⡀⠀⠀⠀⢀⠀⠀⠀⣠⠟⣿⣷⣟⠀⠀⠀⠀⣿⣭⣿⢀⡀⠀⠠⠀⢉⠛⠛⠓⠒⠶⠶⣧⣤⠧⠤⠤⠔⠒⠋⠉⠁
⠀⠈⢿⣶⡾⣿⠉⢺⡁⠐⠀⠙⠟⠁⠀⠒⢾⣦⣾⡋⠉⠉⠉⠉⠉⡩⠁⠀⠀⠻⠇⢀⡀⠉
⠀⠀⠀⠉⠳⣼⣧⣾⣇⠀⠀⠀⠀⠀⣄⠀⠀⢻⣿⣷⣄⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠳⢬⣭⣛⣶⡶⠃⠀⠀⣸⣿⠍⠙⠳⠄⣀⠙⠧⡀⠀⡀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠠⠄⣺⣿⡇⠀⠁⠠⠀⠀⠉⢞⢷⣶⣥⣴⡇⣸⣐⢸⡷⠒⠀⡠⠶⠁⠀⠀⠀⠀⠀⠀⡉⠀⠀⠈⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠋⠁⠀⠀⢰⢦⣦⡉⠢⣍⠉⠘⠏⠓⠻⠟⠛⢳⡶⠛⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⢀⣠⣄⠶⠮⠉⠓⠀⡁⢨⠀⠀⠘⠀⣐⠀⠄⠢⠀⠀⠀⠀⡊⠄⠄⡀⠀⠐⠰⢱⠀⠀⢀⠉⢈⠂
⠀⠀⠀⡁⠀⣀⣀⠀⣤⠤⠌⠂⠀⠀⠠⢳⠘⠐⡁⢄⢴⢂⠀⢀⠀⠡⠐⢂⣀⡀⠐⠠⠙⠀⠀⢨⣦⠆
⠀⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣠⡳⠑⠉⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣠⢒⠂⡉⢙⣢
⠀⠀⠀⠀⠘⠐⢲⠀⡠⢢⠄⠀⠀⠰⠍⠃⠋⠁⠀⠀⣆⣀⣀⣀⣐⡇⠀⠀⣔⡒⢄⠀⢧⠀⠀⠀⠀⠨
⠀⠀⠀⠀⡘⠀⡼⢋⡴⠋⠀⠀⢀⠄⠀⠠⡀⠀⠀⠀⣟⢋⣉⣒⠋⡃⠀⡔⠈⠀⠀⠀⠀⠗⠒⠒⠲⠁
⠀⠀⠀⢸⣧⢊⣔⣗⠀⠀⠀⠀⠂⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡢⠀⠀⠌⠀⠀⣀
⠀⠀⠀⠀⠈⠉⠐⠚⠀⢖⢳⡀⠴⠀⠀⠀⠂⠀⠀⣠⠁⠐⡀⠀⠀⣀⣬⠀⠀⠊⠀⡴⠻⡉⠉⠢
⠀⠀⢀⡄⢀⠀⡀⣀⣀⠀⠁⠀⠀⠀⠀⠀⡠⠐⠁⠀⣁⠈⠀⣑⡀⠈⠉⠁⠀⢀⠞⢑⠌⠀⠀⠀⠑⡀
⠀⠀⢺⠀⢱⠀⡁⠘⠻⠠⠀⢂⠈⠀⢄⠀⣅⣈⣀⣀⣭⣥⣤⣤⠁⠀⠀⠀⠀⣾⠪⠂⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠈⠲⠊⠀⠈⠐⠁⠄⠰⢁⣠⠃⡠⠀⠩⠭⠉⠉⠭⠉⠀⠐⠀⠀⢀⠦⡀⠀⠉⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠠⢀⢀⠄⠀⠀⠀⡄⠀⠂⠠⢀⠀⠀⠀⢀⡃⠉⠨⠄⠀⠀⠀⡄⠡⠐⡄
⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠈⣢⠀⠸⠀⠐⠀⠲⠀⠀⣚⠀⠀⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠄⠐⠀⠁⠀⠀⠀⢈⠄⠀⡀⠀⠀⠀⠀⠀⠮⠁⠀⠈⣳⡴⡻⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠠⠔⠄⠐⠣⠀⠀⠀⠀⠄⠐⠈⠀⠀⠀⠡⠆⢄⢀⠀⡀⠂⠀⠀⠀⠘⣵⡞⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣄⡐⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠻⠻⠶⠷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣶⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⣉⠷⠿⣟⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⡘⠁⢀⠖⠉⢉⠿⠷⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣧⣀⡆⠀⠀⠁⠀⡰⠋⠀⡹⠿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣦⣀⣈⠀⢀⡞⠀⢀⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣦⡀⠀⠀⠀⣼⠿⣿⣿⣿⣿⠙⠷⣾⡀⠀⠈⣷⠦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣴⣌⣿⠀⠀⢸⣷⣿⣿⣿⣿⠁⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣄⠀⠈⢿⡅⠀⠀⠀⠻⢿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⣠⡀⣀⡀⠀⠀⢸⡏⠀⢠⣥⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣻⣾⣧⡾⠛⠀⠀⠉⣿⣴⣦⣤⣤⡀⣀⠀⠀⠀⢀⠀⣠⡶⢦⣤⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣿⣟⣽⣇⠀⠀⠻⣶⣿⡿⣿⣿⣻⣿⣿⣿⢠⡞⠉⠙⢻⡇⠀⢙⠀⢀⠟⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠈⠛⠁⠙⣿⠃⠀⠀⢶⣾⣛⣿⠶⢿⠉⠹⣟⣱⠟⠛⠳⣇⠀⠀⢅⠀⣸⣥⣬⣀⣴⠏
⠀⠀⠀⠀⠀⠀⡴⠋⠉⡽⢷⣄⠀⠀⠀⢸⡋⠑⢦⣄⣠⠛⠉⢣⡀⠈⢧⠀⣹⣿⣤⣤⠀⠈⣦⣤⠾⢛⣋⣩⣿⣿⣧⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠰⠖⠊⢈⣇⠀⣀⣸⡷⠒⠺⣅⠈⢳⡀⠈⣧⣀⣼⣶⡏⠀⠀⠈⢣⣤⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣦⡄⡀⠠⠚⠉⣿⠋⠉⠳⡀⠀⢹⠆⠈⣷⣶⠟⠋⢡⡾⠋⠙⠻⢆⣠⡿⣿⣿⣿⣿⣿⣻⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⡤⠞⢹⠐⢶⠀⢡⣀⡼⢃⡶⠜⡿⠶⣤⣼⣧⠀⠲⣄⠀⣿⠿⠿⢿⡟⠋⢉⣹⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠠⠎⢱⣏⣠⡾⠻⣦⡏⢀⡞⠀⣠⠋⠀⣀⠏⠀⠀⠙⣯⠤⢄⣈⣧⡔⠁⠀⣀⠼⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⢀⠤⠞⢈⣯⡀⠀⠈⠷⣤⣄⠈⣛⠀⠘⠈⠀⠺⢿⣶⣿⠠⢀⣈⣿⠙⢶⡏⠀⡠⠔⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡦⠤⠔⠊⠉⣿⡀⠀⠀⠀⠉⠛⠙⠶⣤⠾⢦⡴⠟⠹⣇⠀⣀⡈⣿⠐⣿⣳⣾⣡⠒⠢⠄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠠⠤⠚⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⢹⣿⠀⢼⣿⣟⣻⣦⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⡄⢋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⡀⠈⣿⠀⢰⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⢀⡴⠶⢶⡶⠟⠛⠳⣆⣀⣀⠀⠀⠙⢿⣭⣴⠟⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⠀⢀⣀⠔⠋⠉⠙⢷⡀⠀⠀⠀⣀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣿⡇⠀⣴⡶⠐⠋⠛⣶⠒⠿⡉⠻⣦⠀⠀⠈⠉
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣀⣹⡿⠿⣁⢀⣤⣀⠿⠄⣀⠈⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠛⠋⡉⠁⢀⠈⢻⠋⢀⣀⡀⠈⣾⠏
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⢩⡇⠀⠘⠀⢸⣀⣴⢿⡿⠫⠑⠚⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠀⣀⣼⡶⠟⠛⠉⠘⢧⣤⣤⠀⠀⣿⠆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⡑⠂⣰⠟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠹⣧
⢠⡿⠀⠀⠀⠺⡿⠀⢠⣶⡄⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣧⣀⣀
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⠀⣠⣴⠶⠛⠋⠉⠉⠉⠉⠉⠉⠛⠳⢶⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⣿⠀⠀⢰⣿⠆⠀⠀⠀⣾⡷⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⢹⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⢰⣶⡆⠀⠀⠀⠈⢷⡀
⠈⢿⡀⠀⠘⠿⠇⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣷
⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⣶⠄⣾⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢹
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣀⡀⠀⠀⠘⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢾⣇⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢦⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢶⣾⠃⠀⠀⠺⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⣾⠃⠀⠀⠈⠻⠗⠀⠀⢰⣆⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠄⠉⠌⠉⠉⢨⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⠀⠈⠀⠁⠀⠀⠀⣈⠁⢀⡤⠀⠀⡌⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠂⠠⡀⠀⠀⠊⠀⠈⠁⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡀⠉⠈⠀⠠⠅⠀⠀⠰⠮⠶⢂⠄⠀⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠱⠀⢀⠉⣑⠢⠤⠀⢀⠀⣀⠁⡠⠤⠀⡔⣺⡅
⠀⢀⠀⠄⠢⡄⠀⠀⠉⡤⡄⠀⠭⠑⢆⠀⠄⠀⣀⡀⠀⠀⠤⠈⠈⠁⠖⠦⠀⠀⠉⠁⡇
⠠⠋⠤⠀⠈⠀⠐⠋⠀⡀⠀⠀⠀⠀⠈⠳⣄⠈⠒⠉⠙⠠⠠⠀⠠⠠⠒⠐⠒⠈⢀⡔⠢⠀⠀⠀⢀⣀
⠃⠢⡞⠀⠀⠀⠘⠀⠀⠐⠶⠂⠁⠀⡉⠈⠂⠉⠛⠒⢠⠀⢤⡤⠄⢠⣤⠤⠀⠈⡁⠀⠠⠓⠚⠛⠁⠤⠈⠋⠈⠒⠢⢄
⢇⠸⡇⠲⠶⠀⣀⠀⢀⡠⠀⠠⠤⠄⡀⠸⡀⠀⣀⠁⠀⠀⢀⡀⣂⡈⠁⠈⠁⠀⡔⠁⠄⠀⠁⠀⠀⠀⠀⠾⠃⠀⠂⠈⠱⣄
⠀⠑⠤⣀⠖⠀⠉⠀⠀⠤⠀⠀⡠⠤⠝⢆⠘⠳⢤⣀⠁⠀⠀⠀⠇⠀⠞⠀⢀⡛⠁⠄⠀⠀⠄⠘⠋⠀⢠⠀⠠⠀⠂⠀⠀⠀⢳
⠀⠀⠀⠀⠁⠒⠄⠠⠀⠀⠀⠈⠡⠤⡆⠀⠁⠠⢌⡀⠉⠉⢒⠒⠒⠂⠘⠉⢉⡇⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⠺⢥⠀⠀⠤⠀⠀⠋⠁⠁⠩⠆⠀⠀⢡⡀⠛⠀⠀⠀⠀⠏⠉⠀⠘⠀⠉⣀⠂⠀⠈⣀
⠀⠀⠀⠀⣀⡀⠐⠒⢁⠀⠠⠀⠀⠀⠉⠉⢓⣊⢭⡒⠂⠠⠤⠤⠚⠶⠒⠂⠚⠉⢣⡄⠀⠀⢀⠀⠀⠠⠀⠀⠀⡘⠆⠃⠀⠸⡤
⠀⠀⣰⠉⠄⠈⢀⡄⠀⠀⠀⠀⣀⠀⠄⠀⠈⠉⠀⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣈⠁⠧⠀⠀⠀⠀⠈⠊⠈⠉⠴⢊⡜
⠀⢀⠁⠠⠄⠠⢀⡤⣨⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠰⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠤⢈⣙⣛⣈⣉⣀⡤⠖⠃
⠀⠘⠀⠐⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠘⠯⠂⣠⠆
⠀⠀⢧⡰⠗⠀⢀⠀⠈⠉⠀⡤⠤⢤⡄⠈⠀⠐⠂⠀⠐⣽⡼
⠀⠀⠀⠈⠲⢤⣈⠀⠀⠀⠀⠈⠙⠛⣀⠀⠈⠉⠀⢈⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣦⣀
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡿⠋⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠹⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣤⣤⣴⣶⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣷⣄⡀⠀⠀⠀⠀⢠⣴⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠄⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣀⣀⣀
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣇⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣰⣾⠿⠟⠛⠻⣿⣦⣤⣄
⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣶⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠋⠁⠀⠀⠀⠀⠀⠉⠉⢻⣷⣤⡀
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠟⠋⠉⠉⠉⠻⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣷⡀⠀⠀⣀⣤⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡿⠟⠋⠁⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡄⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠀⠀⠘⣿⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠃⠀⠀⠀⠹⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠹⣷⣄⣴⣶⡀⠀⢀⣀⠀⠀⣴⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣠⣄⠀⠀⠀⠀⠀⢀⣀⣠⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣶⣾⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣦⣤⣾⣶⣴⣿⠟⠛⠋⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣷⣄⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠿⣷⣦⣤⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣀⣀⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠉⠉⠉⠉⠉⠉⢹⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠔⠉⣤⠆⠀⠁⠠⢄⠀⡠⠠⠤⠠⡄⢄⡀⣀⣀⡀⣀⠀⠀⠀⠀⠇⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠔⠁⠉⠈⠉⠀⠈⠈⡇⠃⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⠀⠠⠀⡼⠄⠚⢧⠀⠀⡀⢀⣀⣀⣀⣀⣀⠸
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠄⠀⠠⠀⠒⠀⠇⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢴⠀⠀⠀⠀⣀
⠀⠀⢰⠒⠶⠶⠖⠒⠒⠒⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠋⠉⢳⡄⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⣠⠒⠉⠉⠑⢦⡀⢠⠊⠉⠑⡀⡞⠉⠙⡆⡇⠀⠀⠤⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⠀⢠⠞⠃⠀⢀⠀⠀⠀⠀⡀
⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⢣⠈⠦⡠⠔⠁⠳⢤⠴⠃⡇⠀⠀⠀⠁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠖⠲⣆⠁
⠀⠀⢸⠸⡀⠀⠀⠀⠀⠀⢸⢠⠊⠈⠳⡀⠎⠉⠑⡄⡇⠀⠀⠀⠀⠀⠀⡞⠀⠘⡀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⢀⠀⠀⠀⠀⠧⡀⠀⠐⠅
⠀⠀⢸⠀⠱⣄⠀⠀⠀⡠⠃⠘⠦⠠⠔⠀⠣⠤⠔⠁⡇⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠊⠉⠁⠑⠀⢠⠁⠀⠘⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠸⠤⠤⠤⠭⠭⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⢰⠀⢰⣷
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⢀⠄⠂⠐⠒⠂⠀⠤⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⢸⠀⠈⢻
⠀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⡇⢹⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠐⠄⢀⡀⠀⠀⠀⡀⠜⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠈
⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠺⣠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣴⡮⡭⡫⢽⡗⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢼⠀⡀⠤⢀
⠀⠀⠀⢤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⢄⡀⠀⠀⢸⢣⡖⠘⢑⡕⢀⣘⡇⠀⢀⠔⠛⠚⠲⣄⠀⠀⠀⠀⠀⡇⠀⢰⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⠸⣮⢲⢀⢡⢇⢕⣔⡇⠀⡇⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠤⠀⠤⠂⠀⠀⡠⡈⠲⠽⠖⠯⠾⠊⠀⠀⠘⢦⣀⣀⣰⠜⠀⠀⠀⠀⠀⡇⠀⢘⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣧⠀⡀⠀⠐⠀⠀⠂⠀⠀⢠⠊⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⡇⠀⠘
⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠃⠀⢀⠀⠀⢐⠀⠀⡇⠀⠀⠀⠀⢸⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢢⠀⠀⡇⠀⠠⠀⡇
⠀⠀⠀⠀⠆⠀⠀⠇⠀⠀⠀⠀⢠⠀⠀⢨⠀⠀⢨⠀⠀⢃⠀⠀⠀⢀⠜⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠠⠀⡰⠀⠀⠀⢃⠀⠀⠀⠀⠨⠀⠀⠄⠀⠀⠈⠢⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠴⠀⠃⠀⠠