Stuffy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠻⣦⣀⣤⣤⠶⠶⠟⠛⠛⠛⠛⠛⠛⠳⠶⣦⣤⣠⣴⠟⠁⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠰⢶⣤⣀⡀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣀⣠⣴⠶
⠀⠀⠀⠀⠉⠛⣿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⠛⠋⠁
⠈⠛⠛⠷⠶⢶⣿⣄⣀⣉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⢋⣁⣠⣿⣤⠶⠶⠞⠛⠋
⣀⣀⣀⣀⣀⣀⣇⢈⣉⣉⡛⠛⠶⢶⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⡷⠶⠞⠛⣋⣉⣉⣿⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⢻⡏⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⣹⡟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠟⠋⣿⠃⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠛⠛⠛⠻⢷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠟⠛⠛⠛⠻⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣤⣄⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⣠⡏⠀⠀⢰⡟⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢿⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠸⣿⠷⠶⠟⠋⠀⠀⣠⡿⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠰⣿⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠐⠿⠶⠶⠟⠋⠁⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣦⣄⡀⠀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⡀⠀⢀⣀⣴⠏⠀⠀⠀⠀⠈⢷⣄⣀⠀⢀⢀⣴⡟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣦⣤⣴⣶⠶⠾⠿⠶⠶⣶⣦⣤⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠁⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠘⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⢀⣀⡀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⢀⣾⠃⠀⠀⠰⣿⣿⣿⣿⣷⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠹⣿⠋⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠘⣧⠀⠹⣦⡀⠀⠸⣿⠀⠀⣠⡾⠁⢀⡿⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠛⢿⣷⣄⡘⢧⣄⠀⠙⠓⠾⠿⠶⠛⠉⠀⣠⡾⢁⣠⣾⣿⡉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣠⣿⠋⠙⠿⢿⣿⣷⣦⣤⣀⣀⣀⣠⣴⣾⣿⡾⠟⠋⠁⠹⣿⣄⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣴⣾⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣷⣦⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⣉⣤⣴⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣤⣀⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠉⠀⠀⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⠿⠿⣿⣿⡿⠿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣶⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣶⣶⠿⠛⠉


⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠇⠈⠁⠀⠀⠀⠀⠀⢀⣤⣴⣶⠶⠶⢠⢸⡇⠀⢀⣀⣀⡀⠀⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢉⣵⡶⠟⠛⢻⡘⡿⠿⠿⠛⠻⠿⢿⣶⡄⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣾⠁⠀⠀⣴⢸⡇⡇⣲⠶⠶⣦⡀⠀⣻⡟⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣿⠀⠀⠀⢻⣌⠃⢠⡇⠀⠀⠈⣷⠀⢹⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⢀⣴⠏⠀⠀⢀⣼⣿⠀⠸⣷⣀⠀⠀⠘⠀⠀⠙⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⣠⡴⠛⠁⠀⣀⣴⡿⠟⠁⠀⠀⠈⠉⠛⠛⠻⠷⢶⣤⡀⠈⠙⠶⣄⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢠⠞⠁⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠸⠆⠀⠙⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⠀⠀⣠⠞⢁⡄⠀⢠⣤⠶⠾⠟⠻⣦⠀⠀⠀⠐⠛⠛⠓⠦⠝⣷⡀⣄⡀⠀⠀⠀⠈⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⠏⢠⠎⠀⠀⢀⡤⠶⣒⣲⠄⠘⢆⣠⣤⣤⣝⡛⢻⡶⠄⠘⢿⣮⣙⠳⠶⣶⣤⡄⠈⠃⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠶⠒⠛⣋⣀⣤⡴⠟⢁⠀⠀⠀⠈⠒⠤⣄⡀⠀⣭⣛⠿⠶⣶⣶⠶⠄⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⢀⠀⢸⡆⣿⡄⠀⠀⢀⣴⠟⠋⠉⡡⣶⠞⠋⠁⠀⣾⠱⡄⣶⣆⠙⣦⣿⠈⠛⠶⠶⠂⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡍⠁⠀⠈⢇⠀⠓⠸⣿⡄⢀⡾⢋⡾⣦⡞⢁⠗⡀⠀⠀⡴⢿⡇⢻⡇⢹⡞⡿⣧⣠⡴⠂⢀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠳⣤⡀⠘⢿⡟⠀⡼⢡⡟⠀⡾⡰⡇⠀⢰⣷⠸⣧⠈⣷⠸⣧⡇⠘⣯⡀⠺⠿⠛⢁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⣤⣌⡻⣦⡀⠀⠀⢀⣴⠟⡾⣿⢃⡞⠀⡼⢡⡇⡏⠀⠸⡟⣧⢹⡀⢻⡆⢿⡇⠀⠈⢷⡄⠰⠟⠛⠋⡉⠛⠛⠿⣶⣄
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠉⠻⠦⠀⣤⡶⠛⠁⡼⢁⡏⣸⠃⣼⢡⡟⣼⠃⠀⠀⢻⡼⡄⠙⠀⠀⠘⣷⠀⠀⢀⡿⠀⠊⢀⢀⠈⢢⠀⠀⠀⠙⣷⡄
⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⠢⠀⠙⣧⠀⢸⡇⣼⢠⡏⢰⠃⡾⢰⡏⠀⠀⠀⢸⡇⠉⠀⠀⠀⠀⣹⡆⢀⡞⠁⠀⠀⠀⠐⢁⡼⠀⠀⠀⠀⠈⢿⡆
⢠⣿⠃⠀⠠⠰⠲⢤⠀⠀⠸⠀⠀⠀⠐⠁⠀⢹⡆⢸⡇⠀⠘⠀⠈⠘⠃⢸⣇⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⣤⠟⠀⠀⠀⠀⠀⠉⠁⢀⠠⠄⡀⠀⠀⠘⣿
⣼⡇⠀⠀⠀⠈⠑⠀⡁⠀⠀⠀⠀⠀⠀⠀⡀⠸⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢃⠔⠀⠀⠑⢆⠀⠀⢰⠁⡠⠀⡄⢡⠀⠀⣿
⣿⡇⠀⠀⡀⠢⠤⠊⠀⡠⠒⠀⠐⠄⠀⢰⢡⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇⠘⢠⠊⠡⠀⢠⠀⠀⠈⠀⠣⢈⡠⠎⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡠⠀⠈⢦⣀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠸⣷⠀⠀⠀⠀⢸⡇⠀⠀⠓⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣇⠀⠀⠀⣀⠠⠄⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⠞⢉⣀⠁⢢⠀⠀⠀⡠⠒⠀⣿
⠀⢻⡄⠀⡞⠀⡀⠀⠈⡆⠀⠀⡔⠂⡉⠐⡄⢀⡟⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⠀⠀⢻⡆⢀⢄⠀⠘⡀⢄⠀⢀⡼⠀⠀⢠⠁⡄⢠⡿
⠀⠀⠻⡄⠡⡀⠑⠀⠔⠁⠀⠸⡀⠨⢐⠠⠃⡼⠁⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣀⣤⣴⣦⢼⡇⠀⠀⠀⠀⢿⡈⠀⡇⠀⠐⠀⠈⠁⠀⠀⠀⠀⠂⠀⣼⠃
⠀⠀⠀⠙⣄⠈⠀⠀⠀⠀⠀⠀⠑⠤⠠⠀⡼⠁⠀⠀⠀⠀⠀⣼⠀⠷⣦⣄⣤⣾⠟⠋⠀⠀⠸⣇⠀⠀⠀⠀⠘⡇⠈⠀⠀⠀⠔⠊⡠⢌⠢⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢄⢀⡴⠀⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣹⡄⣀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢮⣿⣿⣿⣻⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠋⢿⡛⢿⣽⣿⣭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡘⢸⣾⣷⢷⣦⣙⠛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢳⣤⠀⣟⣿⢣⣿⢱⣟⡏⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣦⡟⣧⣽⣧⡿⣿⣿⣿⡾⣺⡿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢸⡟⣰⡟⣹⣿⡡⢋⣞⡿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣾⣾⣯⣴⣿⣾⣿⣿⣽⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠖⣏⣿⣷⣿⣽⣿⣿⣟⣉⣉⣙⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⣀⢸⣿⡻⣿⣿⠷⠾⠿⠻⣽⣿⣟⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣽⣉⠛⣿⣿⣦⡙⡏⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡘⣿⣦⣽⣿⣷⣿⡿⣿⣿⣿⣿⡆⣌⣿⣿⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⢿⣿⣿⣿⡿⣟⣻⣦⣝⠻⣿⢿⣿⣾⣿⣟⢛⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢶⣿⣿⣿⣝⣿⣿⣿⡾⣿⣎⠻⢷⣿⢸⣿⣿⡿⣿⣷⣷⣰⡖⣶⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡀⣿⡿⣿⣿⣿⣿⣿⡿⡏⠻⣿⢷⣻⣼⣿⣿⣶⣿⠃⣿⡟⠛⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⣿⣯⣽⣧⣘⣷⣿⠿⠟⠋⢁⣻⡼⡆⢹⡘⣿⣿⣿⣟⣒⣿⣧⣸⡇⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⢠⣼⡘⢿⡇⣧⢀⡿⠁⠀⣿⢿⣭⣭⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⢰⣿⣿⣿⣿⣇⣿⠈⠀⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢰⣾⣿⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣀⣩⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣤⠤⠤⠶⠶⠿⠿⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠚⠁⠀⡴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⡀
⠀⠀⣾⠏⠁
⠀⢰⡟⠀⢠⣶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠃
⠀⢸⣇⣠⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃
⠀⣼⣿⣿⣹⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣰⣆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣴⣿⡇⠀⠀⠀⠀⠀⠰⢻⡿
⠀⣿⠈⢿⣿⡿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠀⢿⠀⠀⣿⡇⠈⠙⠻⠿⠿⠿⠋⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠋
⢸⣿⣷⣄⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠉⠛⠛⠟⠛
⣀⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣡⣤⣤⣤⣤⠀⣀⣠⣤⣤⡀
⠸⣿⣿⣿⣿⠾⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣶⣶⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣦⡀⠀⠀⠀⠀⠀⠰⠶⡀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⡆⢻⣿⣦⡀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⣿⡀⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⣘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⢁⣉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣠⡂⡈⠻⠋⣼⣿⠃⠀⠀⠀⠀⠀⣀⣼⣟⣻⠇
⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⢸⣿⣮⣛⡁⠉⣠⣾⣿⣿⣿⣦⣤⣴⣶⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⣿⠏⠉⠛⠻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠘⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷
⠀⠀⠀⠀⠀⣀⡀⢀⣀⣤⣴⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⣀⠀⠲⣿⠄
⠀⠀⠀⠀⣿⣿⢻⣿⡿⠿⠛⠋⠁⠀⠀⠈⣿⣄⣀⣀⣤⣾⣿⣿⣿⣶⣶⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⠙⢿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⢿⣿⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠙⠏
⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋⠠⠆⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⡬⣷⣃⠀⠅⣈⠀⠀⠀⠀⠀⠸⣦⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡬⠀⣿⡅⣾⠈⠉⠀⠀⠀⠀⡀⠀⢿⡏⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡞⠀⢀⣪⠉⠳⣶⠂⠀⠀⠀⠀⠳⠀⡺⣏⠃⠢⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⠃⠠⢸⣇⡀⠁⠀⣀⠦⡠⢀⡀⠀⡀⣙⣿⣯⠂⢶⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⠣⠀⡦⠺⡍⣄⡂⠎⢅⡼⢤⠞⠉⣸⠝⡌⠫⡛⠑⠄⣄⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣁⣿⠀⣰⠳⠃⢰⣟⡿⡛⢱⡠⠾⠷⠍⢙⡆⣄⠳⡂⢻⡶⠆⠠⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⠈⣀⢁⠀⠀⡏⣋⠁⠇⠊⢠⡀⠀⠐⢋⢧⣒⠄⠐⢇⡹⠐⡉⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠯⠀⡹⡄⠀⠐⣣⠛⡠⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠁⠈⢽⣳⡀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⣸⢡⠀⠀⠀⡷⠆⣃⠀⠀⠀⡀⠀⠀⠠⢂⡷⠀⡄⠀⢸⠈⠃⢈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠁⡸⣿⡂⢩⠀⠀⠸⡷⠜⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⠀⢸⡀⠄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢰⡸⠁⣯⢟⡆⢧⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⡞⠀⠄⠀⠀⣠⠉⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢁⠤⠻⠔⣵⣀⠀⠀⠀⠈⢃⡂⢐⠀⠀⠀⢰⡁⠖⢠⠀⠀⣧⠤⢤⡤⢰⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡠⢀⠨⠀⠙⡌⠐⠀⠀⢀⠄⢸⠀⠀⢠⠷⠆⠺⠈⠠⠆⣠⠸⠡⡂⡄⡄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢦⣣⠀⢁⠐⠸⢦⣰⠀⠀⡇⢸⠀⠀⠀⣦⢊⡞⢴⡝⠺⠉⠀⠀⠙⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠈⢂⡙⢢⡇⠁⢸⡙⣤⠼⠃⣤⠀⠀⠀⣼⢿⠑⢘⡿⠃⠀⠄⠀⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡝⠡⣜⠁⡤⠑⠉⠓⠀⠀⠒⠉⠀⠈⠉⠛⠒⠳⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⠈⠁

⠀⠀⠀⡐⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢰
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⢠⡄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡇⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠁⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣇⢀⣁⣀⣀⣈⣉⡀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⢠⠴⠂⢙
⠀⠀⠀⡇⣤⠀⠠⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡂⠚⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⢸
⠀⠀⠀⡅⢸⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠒⠀⠀⠂⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀⣀⣀⢁⡀⢀⡉⡀⠀⠀⠀⠸
⠀⠀⠀⡟⠓⠒⠒⠚⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠁⠀⠈⠉⢈⠉⠉⠉⠁⠀⠀⠀⢨
⠀⠀⠀⡇⠀⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⡅⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⢀⢨
⠀⠀⠀⡆⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡙⣈⣙⣏⣀⡋⣉⣿⢙⣸⡟⢻⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠂⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠀⢄⡀
⠀⠀⢀⠊⠠⠐⠂⠂⠒⠒⠰⠄⠀⠀⡄⠀⠀⠀⠊⡠⠠⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠸⠀⠀⡁⠀⠀⠀⡀⠀⠸⠀⠀⠀⠀⢰⠁⠐⠀⠄⠈⠉⠢⢀⠀⠀⠀⠀⠈⢂⠀⠈⠀⠄⠀⠊⠀⠈⠉⠐⢄⠀⠀⠐
⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠈⠀⢁⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠰⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠆
⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠉⡀⢶⡢⠴⢂⠖⠛⢄⠀⠀⠀⠘⢄⠀⣉⠖⠲⠴⠶⢊⠍⠉⠐⠀⠀⠀⠀⠀⠀⢠⢀⠀⢀⡠⡤⠒⠂⠠⢀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⠀⠱⠀⠀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⡠⠐⠨⠁⠀⠈⠁⠒⢄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠐⠀⠠⢄⠀⣀⣀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⡆⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠈⠆⠀⠀⢀⢊⠀⣀⠄⠀⠄⡀⠀⠀⠑⡤⠀⠈⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⠐⠓⠀⠀⠀⠛⠃⠀⡄⠀⠀⠀⠀⠀⡔⡡⠊⠀⠀⠀⠀⠈⢀⠀⠀⠐
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⡏⠀⠀⠄⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢀⠀⠈
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠉
⠀⠀⠀⠉⠂⣠⠄⣀⣠⠀⠒⠡⢀⠀⠀⠀⠀⠀⠑⠂⢀⣀⠀⠀⠀⠔⡁⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⢀⠠⢄⠐⠁
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⠁⠀⠀⠑⡄⠀⠀⠀⢀⠔⠋⣀⣀⣀⡀⠀⠌⠁⢂⠀⠀⠀⠀⢀⠔⠀⠉⠁⠁⠈⠀⠈⠐⡀
⠀⠀⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡠⠒⠀⠀⠀⠐⠪⡀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣤⣶⣶⣶⣦⣜⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢂⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠾⢂⢀⣀⠰⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⡿⣿⣿⣏⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠁⢶⠆⠑⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣾⡉⣻⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢄⠚⢒⠔⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡿⣭⡿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠜⠑⠀⠀⠀⠐⠘⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣧⡀⣀⡀
⠀⠀⠀⡠⠒⠈⠁⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠀⠢⡄⠀⠀⣠⣴⣶⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢈⠀⠀⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠑⠀⠀⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠐⠀⠐⠁⠀⠀⠈⠛⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠛⠉
⠀⠀⠀⠀⠀⢀⠂⠁⠀⠁⠢⡀⠀⠀⠀⠀⡠⠂⠁⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣟⣿⣿⣿⣿⣿⢿⣽⣿⣿⣶⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢠⡃⠀⠀⣡⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠿⣿⡿⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠁⠀⠀⠀⠀⠁⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⠀⠀⣀⣤⣴⣶⠾⠿⠿⠿⠿⠿⠿⢶⣶⣤⣄⡀⠀⣀⣤⣶⣶⣶⣶⣦⣄
⠀⣰⡿⠋⠁⠀⠀⠀⠈⣻⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⡉⠀⠀⠀⠀⠉⠻⣷⡄
⢰⣿⠁⠀⠀⠀⠀⣰⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⢹⣿
⢸⣿⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⢸⣿
⠈⢿⣧⡀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣠⣿⠏
⠀⠀⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⢀⣤⣴⣦⣤⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⣿⡿⠿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣿⣇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢻⡟⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠘⣿⡻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠻⣷⣄
⠀⠀⠀⢀⣴⡿⠋⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⠙⢷⣄
⠀⢀⣴⡿⠋⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠈⠹⣷⣄
⢠⡿⠋⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⢻⣷
⣿⡇⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣀⠀⠀⠀⠀⣿
⠙⠿⠶⠶⠿⠟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠿⠿⠶⠾⠟
⠀⠀⠀⠀⣀⡴⠾⠛⠛⠓⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠖⠛⠛⠻⠶⣄⡀
⠀⠀⠀⣼⢋⡴⠛⠉⢉⡓⢦⡈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣤⠞⢋⠉⣙⠳⣌⢻⡄
⠀⠀⢸⣧⡏⡔⠀⠀⠀⠀⢆⢻⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⢡⠎⠀⠀⠀⠁⡌⣧⣿
⠀⠀⢸⣿⢰⠀⠀⠀⠀⠀⢸⢸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢨⠀⠀⠀⠀⠀⣸⢸⣿
⠀⠀⠈⢻⣎⡳⣄⣀⣀⡤⢋⣾⡾⠿⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⠿⢷⣿⣜⢦⣀⣀⣀⡴⣣⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⢉⣁⡤⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠐⠁⠀⡄⠈⠓⠤⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣣⠜⠊⠁⠁⠁⢴⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠂⠀⢋⠀⢠⡄⠁⠀⠀⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠜⠁⠚⠁⠀⠀⠙⠀⠙⢄⠗
⠀⢀⡠⠤⣀⠀⢀⣸⢀⣀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢒⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠸⠀⠀⠀⠓⠀⠀⠀⢄
⡔⠁⠀⠀⠀⠛⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠔⠑⠖⠈⠂⠀⠀⠀⢈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠀⠤⠀⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠤⡞⠋⠁⠒⢄⠀⠁⡠⠤⠤⠤⣴⠋
⠘⢄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉⠲⡉⠀⠀⠀⠀⠀⠀⠀⡴⠁⠘⢄⠀⠀⠀⠀⠑⠋⠀⠀⠀⢀⡸⠑⢄
⠀⠀⠑⢄⡀⠀⢀⡠⢖⠟⠉⠁⠒⠀⠤⠤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣸
⠀⠀⠀⢠⠟⠞⢡⠤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡀⠀⠀⠀⠀⠀⠀⠑⠖⡾⡀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⠖⠁
⠀⠀⠀⡇⠀⠀⠈⠀⡇⠀⠀⢀⠒⠩⠓⠖⠁⠉⠂⢹⢹⠂⠀⠠⠊⠉⠉⠘⠘⠀⠈⠒⢄⠀⠀⢀⡠⠒⠉⠄⠐⠓⠒⠒⢦
⠀⠀⠀⢣⠀⠀⠀⠀⢱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠣⢄⡠⠤⠒⠳⢄⡈⣆⠀⠀⠀⠀⢀⣀⠜⠀⠀⠀⠰⡀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⢀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠉⠀⠀⠈⠙⢿⣿⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣾⣿⠟⠉⠀⠀⠈⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⣀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠾⠟⠿⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⣶⣶⣶⣶⣶⡆⠀⠈⠙⣷⡀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⢰⡏⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠈⣷⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⣿⣷⣄⠘⣷⡀⠀⠻⢶⣤⣀⣿⣇⣠⣤⠾⠃⠀⣰⡟⢀⣴⣿⡿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠉⠀⠀⠈⣻⣿⣷⣾⣷⣄⡀⠀⠈⠉⠉⠉⠉⠀⠀⣀⣴⣿⣶⣿⣿⡋⠀⠀⠈⠙⠿⣿⣷⣄⡀
⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⣠⣾⣿⠟⠉⠛⠿⣿⣿⣶⣦⣤⣤⣤⣤⣶⣾⣿⡿⠟⠋⠙⢿⣿⣦⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠈⢻⣿⣦
⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠹⣿⣧
⣾⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⢻⣿⣆⠀⠀⠀⠀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⠀⠀⢀⣾⣿
⠀⠻⣿⣷⣶⣶⣾⣿⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣿⣶⣶⣶⣿⡿⠃
⠀⠀⠀⠉⠉⠉⢉⣠⣴⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣷⣤⡀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠛⠉⠁⠀⠈⠉⠛⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠋⠉⠀⠀⠉⠙⠻⣿⣷⡀
⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠉⠉⠉⠉⠉⠉⠉⠁⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣀⣀⣀⣤⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⣤⣀⣀⣀⣤⣶⣿⡿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠁⠀⠀⠀⠈⠙⢯⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⢢⡑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣏⠀⠀⠀⠀⣨⣥⡀⠀⠀⠹⣌⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢰⣿⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠈⣧⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⣿⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣸⣟⡼⢦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡜⣹⠏⢠⠀⠙⢮⣦⣄⠀⠀⠀⠀⢀⣀⡼⣯⣿⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠌⡰⠋⠲⡌⠁⠀⠀⠈⢓⣯⢥⡖⠀⠬⠕⠚⣯⠙⠻⣙⣛⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⢀⠐⣡⠞⠁⠀⠀⢷⠀⠀⠀⠀⠀⡍⢨⣿⢶⡄⠀⠀⠈⠻⣟⠋⠛⠛⢛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡴⣞⣀⣀⣀⠠⠔⢊⣡⠖⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣇⠀⠙⢷⣽⠀⠀⠀⠀⠀⠙⠶⣴⡖⠾⠿⣛⣻⠆
⠀⠀⠀⠀⠀⠈⠷⢤⡀⣤⠤⠶⡾⠛⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡐⠀⠀⠀⠀⠀⠀⢈⠞⠁⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⣠⡾⠁⣠⠖⠋⠉⠉⠓⠢⢄⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢿⠭⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢰⡿⢹⣧⠾⠉⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⡧⠀⠀⠘⣆⣀⣀⠈⠻⣦⠀⠀⠀⠀⣠⢔⡲⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⢸⠁⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣆⠘⢧⡀⣠⠊⡕⢃⣤⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡡⡀⣀⡾⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢘⡇⠀⠀⣿⣿⣿⣿⣿⣷⣌⡟⢡⠆⢠⣾⣟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢨⣇⠀⢠⡇⠙⠛⣿⣿⡿⠈⠀⠀⣰⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡷⢾⠛⡗⠉⠁⠉⠉⠓⢾⢠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡴⠋⠔⠁⣤⡇⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠙⢲⢁⡞⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⢷⡄⠀⠀⠀⠀⠀⠰⠀⣼⡟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣸⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠤⠟⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢷⡄⣀⣤⣤⣤⣤⣤⣄⡀⣴⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣼⠿⠋⠉⠀⠀⠀⠈⠉⠻⢿⣄⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡿⠁⠀⢠⡄⠀⠀⠀⠀⣤⠀⠀⠹⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⣤⡶⠿⠿⠿⠷⣦⣄⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢰⡿⠁⠀⠰⣶⡶⠀⠀⢹⣷⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⣿⣄⠀⠀⣿⡇⠀⢀⣼⡟⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣧⣌⠙⠻⠷⠿⠷⠾⠛⢉⣤⡿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⣾⣿⠿⢶⣶⣶⣶⣶⠾⢿⣿⣆⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣼⣿⣿⣀⣀⣿⣄⣼⣇⣀⣸⣿⣿⡄⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢰⡿⠘⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⢸⣷⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠻⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⣀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣸⡟⠙⠛⠛⠛⠛⠙⣿⡀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣄⣠⣤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣄⣀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡝⠿⣿⣿⣿⣿⣿⣿⠟⣫⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢟⣱⣿⣶⣭⣭⣍⡱⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣫⣭⣭⣿⠿⠿⣿⣭⣭⣝⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⡶⣼⢧⡝⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⣸⢧⡟⣼⢣⡝⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢇⣿⣼⣣⡟⣼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡼⣿⣿⣿⣿⣿⣿⢸⢟⣵⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⣹⣿⣿⣿⡿⣣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠇⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⣀⣤⣄⠀⠀⣙⣧⣴⠶⠛⠛⠓⠒⠶⠶⠶⠤⣤⣠⡞⠉⢀⣀⣀⣈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢨⣿⣿⣟⠵⠋⠁⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠈⠙⠾⣿⣿⣦⠍⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠃⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠹⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠠⠴⠟⠛⠓⣤⣾⣿⣿⣻⣆⠿⠿⢥⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⢠⡾⠁⠀⠀⠉⠓⠶⣤⣙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢀⡇⠀⠀⢀⡄⠀⠀⠀⠀⢀⡼⢁⣠⡤⠶⠖⠒⠒⠲⡄
⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠰⠄⠉⠛⢷⣤⣀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠳⠶⠒⠋⠀⠀⠀⣀⣴⡿⠞⠋⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⢠⠟⠛⠿⣷⡶⢦⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡾⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠉⠙⠒⣶⡿⣿⢻⣏⣽⣏⠻⡻⡅⠀⠀⠈⢷⡀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡜⣽⣱⢃⣾⢻⡜⡿⣄⡇⢹⠀⠀⠀⠈⣷⡀⠩⢷⣢⠄⠀⣀⣤⠟
⠀⠀⠀⠈⠙⠻⠶⢤⣤⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠳⠬⠗⢫⠏⣸⠂⣷⠤⠴⠋⠀⠀⠀⠀⠸⣧⣤⣬⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣸⣄⣤⠴⠦⢤⣄⠀⠀⠀⠀⠀⣾⣀⣸⠑⠶⠚⠂⠀⠀⠀⠀⠀⢀⣠⣿⣇⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠈⠉⠲⣌⠛⢦⡀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡴⠋⣠⠋⠀⠙⢷⡄
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⢧⠈⢿⡄⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣼⠁⡼⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⣷⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣸⠃⡸⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢹⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⡏⢀⠇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢸⣿⡄⡆⠠⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⡼⠳⠤⣄⣀⡀⢀⣀⣸⣿⠀⠀⣼⠃⢸⠀⠀⠀⠀⠀⡀⠀⠀⣼
⠀⠀⠀⠈⣿⣧⡇⠀⢨⢰⢠⠀⠀⣸⢠⡿⢀⡴⠋⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⣷⡀⣿⠀⢸⠀⣸⠸⣸⢰⠟⠀⢠⡟
⠀⠀⠀⠀⠈⠻⣷⣾⣼⣸⣚⣀⣴⣧⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣧⠸⡀⠉⠃⠉⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣤⡴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠳⣶⣶⠟⠛⠛⠳⣶⣶⠖⠚⠉⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠟⠠⡇⠀⠀⠀⠀⢷⡀⠐⠀⠀⠀⢸⠇⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠒⢮⠔⠄⣈⠂⠤⠄⡀⠰⠒⣤⠆⠀⠀⢺⡟⡩⠷⠞⢻⣧⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠈⠌⠀⠀⠀⠀⠀⠀⢸⠃⠀⡭⠀⠀⠀⠔⠃⢤⠒⠂⠛⡶⢛⡉⣹⣧⣶⡄
⠀⠀⠀⠀⢀⡾⠁⠇⠀⠀⡰⡎⠐⢆⣀⡀⠠⠾⠁⢰⠁⠀⠀⠀⠀⠀⠀⣱⠒⠠⠘⠉⠀⠀⠢⣟⢛⢛⣶⣦⣀⡀
⠀⠀⠀⠀⢾⣧⣤⣶⣖⣿⣦⡁⠈⠁⠀⠠⠃⠀⣠⣺⣶⣦⡀⠀⠈⠛⣀⡁⠚⠐⠁⡈⠀⣀⣜⡓⠳⠨⣤⣞⡛⢻⣿⣧⣄⡀
⠀⠀⠀⠀⠀⢸⣿⡹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢠⡇⠀⠈⠀⠀⠹⢀⠀⢨⡁⠀⡴⠃⠚⢿⡯⣶⢟⣝⣷⢻⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠺⣯⠄⠀⣅⠀⠀⢈⡀⠀⠈⠻⠿⠯⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠠⠸⢄⡀⢹⡿⠋⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢃⠸⠉⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣌⠂⠀⠑⢰⣦⣀⣠⡿
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⣹⣄⣠⠞⠋⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⣢⣀⣀⣠⡀⠀⠀⢀⣴⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⠾⣼⢶⡿⠶⠒⢲⣿⣧⡔⣤⣰⠶⠞⠛⠛⠛⠛⠚⠛⠛⠉⠀⠉⠉⠙⠻⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⢁⡴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣷⠋⣼⣡⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⠁⡼⢫⠟
⠀⠀⠀⢀⠀⠀⠀⠀⣸⠃⠀⠁⢈⡴⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣰⠋⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢋⡤⠖⠒⠦⣄⠀⠀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠘⠤⢄⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡾⠁⠀⠀⠀⠙⣶⠻⠅⠀⠈⢦
⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠐⠳⠤⣀⠀⠀⠀⠙⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣄⣀⣀⡠⠋
⠀⠀⠀⠀⠀⢸⡇⠀⣠⢾⠁⣴⠀⠀⣠⣤⣟⣦⡀⠀⠀⠉⠳⢾⡇⠀⠀⠀⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠈⢷⣼⠃⣾⠀⡟⣆⡼⠁⠀⠉⠻⣿⣦⠀⠀⠀⠀⠻⣄⠀⠀⠶⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠀⠀⡴⠒⠛⠛⠿⢧⠀⠀⠀⠀⠈⠙⢛⡿⠋⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠴⠖⢲⢤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⣄⠀⠉⠙⠓⠶⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠉⠓⠦⣌⣀⡉⠂⠀⢂⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⣦⣄⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠄⠐⣯⠙⣧⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡰⠀⠀⠀⠀⠀⠀⠈⣇⠈⠙⠾⣦⡀⠹⣮⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⠀⠀⣹⣦⣙⠀⠀⠀⠈⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠛⠦⣄⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢺⣁⣀⡽⠟⠢⠤⣄⣀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣲⠶⢤⣀⣈⡏⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠳⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⠀⠀⠈⠉⠉⠒⠒⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⣷⢤⡿⠆⠀⢀⣀⣘⣦⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣄⡀⠈⠀⠂⠀⠀⠀⠀⠈⠉⢛⠶⣄⢠⠇⠀⠙⠚⠛⠉⠉⣿⣿⠱⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⢉⡙⠢⣤⣀⠀⠀⠀⠀⣰⠏⠀⠘⠻⢦⡀⠀⠈⢦⡀⠀⣿⣿⣿⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠛⠲⢦⣴⠏⠀⠀⠠⢴⣾⣷⠀⠀⠀⠙⢶⣿⡛⠛⠒⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣦⣿⡄⠀⢠⠞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠷⠛⡄⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣳⠞⠉⢳⡄⠀⣠⣿⣿⢿⣠⠏⠰⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣸⠁⢀⡖⠈⡇⣰⢻⡟⠉⠻⠏⠀⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠖⠒⡙⠶⠞⠀⠀⣿⠃⡾⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢶⠚⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠻⣄⣐⣦⠀⠀⠐⢇⡀⠀⠀⠀⠉⢻⠖⠲⠲⠶⠶⠶⠒⠒⠒⠛⠛⠉⠉⠁⠀⢸⠁⡼⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠈⠉⢈⡷⠒⠲⡬⠻⡄⠀⠀⢀⡟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡇⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⡴⠇⠀⠀⠀⢙⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣼⣀⣤⡤⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠃⠄⠀⠀⠀⠀⠀⠀⠐⠚⠛⣛⠾⢻⠿⠍⡁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣑⣀⢤⡐⠄⠀⠀⠀⠐⠂⠉⠀⠀⢀⡏⠀⠀⠀⠀⠀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣻⣿⠿⠛⠛⠲⠤⠴⠦⠤⠶⠶⠒⠚⠛⠓⠒⠒⠶⠞⣽⣩⠏⠉⠙⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠲⣿⣿⣿⡗⢲⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⢛⣛⣛⠛⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣼⣿⣏⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⣿⣿⣿⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣿⣿⣿⣭⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣾⣿⣛⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣻⡿⣭⣹⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠀⠀⠈⠻⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟





⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠁⠒⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠐⠢⢄
⠀⠀⠀⠀⢠⠊⢀⠤⠂⠉⠀⠈⠑⢠⡀⠕⠂⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⢄⣠⠒⠉⠀⠈⠁⠢⣄⠉⢦
⠀⠀⠀⢠⠃⠠⠃⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠈⢢⠀⢃
⠀⠀⠀⡆⠀⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡇⠈⡀
⠀⠀⠀⠇⠀⢆⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⠃⢀⠁
⠀⠀⠀⠘⢄⠈⠲⢤⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣦⠔⠁⢀⠎
⠀⠀⠀⠀⠈⠑⠤⣌⠀⠀⠀⠀⠀⠀⠐⢋⠩⢉⠀⠀⠀⠀⠀⠀⠀⢈⠩⡙⠓⠀⠀⠀⠀⠀⠀⢸⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠄⠜⠀⠀⡠⠄⠤⡀⠀⠰⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠹⡀⠀⠀⡸⠁⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡁⠀⠲⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐⠀⠀⡆⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠑⡄⠀⠑⠤⣀⠀⠀⠀⢀⠠⠒⠁⢀⠜⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢄⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⢀⠴⠥⢀
⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⠑⠢⣄⡀⠀⠈⠐⠂⠀⠐⠂⠉⠀⠀⣀⡤⠂⠁⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠉⠐⠂⠀⠀⠀⠀⠂⠈⠁⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠔⠊⠐⣦⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠁⠀⠀⠀⠈⢡⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠀⢀⠠⡚⠄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⢠⠁⠀⠀⠀⠀⠀⠀⠀⢠⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⣀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢡
⡈⠀⢀⠀⠀⠄⢀⡠⠂⠉⠀⠀⠀⠉⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠈⠑⠠⡀⠀⠤⠀⢀
⡇⠀⠀⠀⠀⢀⠊⠀⣠⠖⠋⠉⠉⠓⠦⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡤⠒⠋⠉⠉⠓⢦⡀⠀⢆
⢣⠀⠀⠀⠀⠂⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⠄⠀⠀⠀⢀⠌⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠄⠀⠀⠀⢠
⠀⠑⠠⠄⣨⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⠄⠀⠀⡊⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢃⠀⢠⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠦⢾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡈
⠀⠀⠀⠀⠀⢃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡄⠀⠈⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠁
⠀⠀⠀⠀⠀⠀⢂⠀⠳⣄⡀⠀⠀⠀⢀⣠⠖⠋⡠⠊⠀⠀⠀⠘⠄⠈⠳⢤⣀⠀⠀⠀⠀⣠⠜⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠛⠛⠋⡉⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⢈⡉⠛⠛⠉⠁⣀⠔⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⡀⣀⣤⣤⢄⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡿⠃⢿⣟⡽⢙⠏⠀⠦⢄⡉⠢⠙⠂⠠⠤⣄⡹⣿⣿⣿⣿⣿⣿⡷⠋⢀
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣤⣭⣀⡀⣑⡛⠓⠦⣍⢲⡋⠀⣀⣀⠀⢹⣈⣉⣻⣿⡟⠗⠒⠉⠥
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠤⣌⢸⡅⠉⠉⠉⠙⠀⣯⠅⠀⠒⠁⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠠⢤⣀⣹⣇⠀⠂⠉⠉⠁⠷⠶⠖⠒⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⠿⠿⠟⣗⠢⢄⣀⠀⠉⠀⠤⣤⣀⣈⠙⠒⠂⠉⠀⢀⡀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠉⢀⣀⢈⡉⠙⠛⠁⠀⠀⠉⠛⠦⣄⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠻⣄⣀⣤⣄
⠀⠀⠀⠀⠰⠀⠀⠀⠚⠛⠛⠻⣷⣄⠈⢧⠀⠀⠀⠀⣤⣄⠸⣘⡿⡠⢒⣦⠀⠀⣀⣤⣿⠆⢠⡌⠉⠉⡹
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⢹⣿⡄⢸⠀⠀⠀⠀⠢⠽⠷⠘⠁⣛⣛⣋⠀⢸⣿⡁⠀⢴⣾⣿⣟⠰⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠿⠃⠀⠀⣼⡿⠀⣸⠀⠀⠀⠀⢴⠾⣻⢣⣦⡹⣷⣮⠀⠀⠉⢻⣷⠂⠸⢋⣉⠀⣹
⠀⠀⠀⠀⠀⠀⢀⣼⣷⠶⢶⠟⠛⠁⠐⠁⠀⠀⠀⠀⠈⠒⠁⠻⠇⠓⠀⠀⠀⠀⠀⠘⣿⣆⠴⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⡀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⠻⣿⣿⣟⣡⣶⣿⣿⣿⣿⣷⣦⣌⣻⣿⣿⡟⣩⣍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣿⣿⣿⣿⣏⣨⣿⣿⣿⣇⣹⣿⣿⣿⣿⡄⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⣀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣭⣍⡉⠉⠉⣉⣥⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣛⠻⢿⣿⣿⣿⣦⣾⣿⣿⣿⠿⢛⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠆⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣬⣍⡙⠛⣩⣼⣿⣿⣿⣿⣿⣷⣌⡛⠿⣋⣩⣥⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⡿⢣⣬⣭⣭⣭⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⣩⣍⡛⢿⣿⣿⡌⢿⣿⣿⣿⣿⠟⢡⣿⣾⡶⢒⣭⣤⡲⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣾⣿⣿⣷⣸⣿⣿⡇⢠⣍⠻⢋⣡⡆⢸⣿⣿⣇⣾⣿⣿⣷⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠋⠉⠉⠉⠉⠙⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣀⡀⠀⡀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢠⣶⣤⣀⣠⣤⣶⡿⠟⠈⠙⠻⠿⠿⣿⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⣿⡆⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⢰⣿⠇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⣷⣾⠿⠿⠿⢿⣿⠿⠿⠿⣷⣾⡟⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⡟⠁⠀⠀⠀⢸⣿⠃⠀⠀⠈⢻⣷⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣠⣼⣿⡁⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⢈⣿⣧⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⠋⢿⣿⣷⣦⣴⣿⠟⠻⣿⣦⣴⣾⣿⡿⠙⣿⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠉⣻⣿⢿⣷⣄⣠⣾⡿⣿⣟⠉⠀⠀⠈⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠛⠁⠀⠘⢿⣷⡀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣠⣼⡿⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⢿⣧⣄⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠖⣀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⣠⣤⡶⠖⢉⣤⡞⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⢻⡆⠀⠀⠀⠄⠘⢿⣄⠐⢛⣉⡀⠀⠸⣿⣷⡆
⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠞⠀⠀⠀⠀⠀⠀⠉⢉⡙⠋⠙⠓⠀⠀⠉⠙⠃
⠀⠀⢀⠀⢠⣶⠏⠀⠘⠁⠀⠀⠀⠐⠀⠀⠀⠀⢴⣿⡄⠀⠀⠀⠀⢠⣴⡶⠟⢡⡞⠉⣠⡴⠋⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⣴⠀⣄⣰⣦⣶⡄
⠀⠀⠻⠀⠀⣱⠖⠁⢀⣠⡶⢲⣦⠀⡀⠀⢀⠀⣨⡖⠀⠀⠀⠀⣀⣤⠖⠀⠀⠋⠀⠀⠁⠀⠀⢠⣤⡤⢾⣯⠁⠀⢰⣷⣴⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠴⠞⠛⠉⠁⣼⡿⣿⣿⣦⣿⡇⠁⠀⣀⣤⣴⣿⣿⣥⡀⠀⠀⠀⠀⠠⠆⠘⠓⠒⠂⠀⠁⠀⠀⠰⠿⠿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠿⠟⠛⠛⠙⠛⠛⠙⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢤⠥⢶⡄⠊⢽⠍⢙⠓⠂
⠀⠀⠀⣀⠠⢘⡋⠁⣰⣿⣧⠀⢀⣀⡤⠤⢄⣀⣀⣀⡀
⠀⠀⠠⠦⠛⠃⠀⠸⣿⣿⣿⣶⣾⣅⡀⢀⠌⢉⣉⣉⣉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⣠⣿⣷⣄⠀⠤⢐⠲⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿
⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡌⢛⡛⠿⢿⣥⣤⠴⠶⠟⠻⠷⣷⢶⣶⣤⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣿⣿⣿
⠀⠀⠐⠒⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠂⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠤⠄⠲⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠓⠀⠀⠐⠚⠀⠀⠀⠀⠂⠀⠐⠐⠂⠐⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⠳⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⡄⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⢀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⣀⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡶⠿⠿⠟⠋⠉⠉⠈⠁⠉⠉⠙⠛⠻⢿⣶⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀
⠀⣠⣾⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⢀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠈⠳⣤⣄⣀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣀⣠⡶⠋
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠤⢄⡀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⠀⠀⠂⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠘⢇⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠙⠷⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠈⠉⠉⠃⠲⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣦⣴⣤⣀⣀⣀⣀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⢻⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣿⡄⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣄⣀⣠⡾⠃⠀⠀⠀⠹⠓⢦⣤⣤⣤⡾
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣮⣤⣤⣤⣤⣤⣤⣤⣭⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠉⠉⠉⠉⠒⠲⣄⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⢄
⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⠞⠉⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠸⡀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⡻⠥⠤⠤⠄⣀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣰⠋⠀⠀⢠⠤⠤⢤⡈⠑⠦⡘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢧⠀⠀⠀⠘⢄⣀⣀⡨⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠵⢤⡘⣆⠀⢢⣀⣠⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠛⠳⢄⡀⠀⠙⠚⠁⣀⡤⠞⠀⠀⣀⣠⢴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢠⠀⠀⠀⠀⠉⠉⠉⠋⠉⠙⠓⠛⡉⠉⠉⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡠⠖⠒⠢⣷⠤⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣀⣀⡼⠷⠲⠤⣄
⠀⣼⠁⠀⠀⠀⠈⠣⠈⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠞⡱⠋⠀⠀⠀⠀⠈⣇
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠐⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠤⣀⣀⣀⣀⣀⣀⣠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⣴⢒⣿⣿⣿⣿⣶⣴⣶⣶⢶⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠞⠿⠿⡿⣤⣶⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⢹⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣾⠖⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⣡⣼⣿⣿⣿⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⢹⣧⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠞⣻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⣸⡿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⠷⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢰⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣤⡤⠀⠀⠀⢀⣼⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⣟⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣿⣶⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣼⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠉⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⣼⣟⣡⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠆
⠀⠀⠀⠀⠀⢀⡀⠤⠂⢚⠻⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡶⠆
⠀⠀⠀⠀⠀⠉⠀⣢⣤⣤⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠈⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣛⡛⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠷⢣⡖⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠏⠉⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡍⠿⣶⣅⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠈⢢⡀⠁⠈⠉⠻⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢢⡀⠉⢲⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠈⣇⠙⢦⡀⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢱⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠁⣿⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⢯⠸⡀
⠀⠀⠀⠀⠀⢠⠃⢸⣿⠀⠀⠀⠀⠀⣸⣠⣀⣤⠖⡀⠈⡇⡇
⠀⠀⠀⠀⠀⡜⢠⣿⣿⠀⠀⠀⠀⢠⣭⣽⡧⠔⣹⠇⣠⡇⢇
⠀⠀⠀⠀⢀⡇⠘⢻⣿⠃⠀⠀⠀⠀⡿⠛⣞⣿⣵⡿⠁⢿⣾
⠀⠀⠀⠀⠘⡇⣦⢸⣿⡀⠀⠀⠀⢠⣇⠀⠀⣼⡯⠃⠀⢸⢻
⠀⠀⠀⠀⠀⣷⣿⡄⣿⠗⣉⡑⠛⢦⣹⠀⢠⣿⣷⡀⢀⣼⠆
⠀⠀⠀⠀⠀⡘⢻⣿⣿⣾⣯⠍⠙⢦⡻⠀⢸⡈⠓⠞⠒
⠀⠀⠀⠀⢀⡇⠀⠹⣿⣿⢹⡆⠀⠈⠁⠀⠈⡇
⠀⠀⠀⠀⠈⢧⣳⣤⣹⣿⣆⣇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⣶⣾⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢂⠈⠻⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⠆⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡻⢿⣿⣿⣧⣤⡙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡳⣍⠀⠉⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⣄⣀⣠⠴⠒⠛⠋⠉⠉⠙⠓⠒⠢⢤⣀⡠⠖⠚⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⢀⣀⣤⣄⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⢠⠞⠛⠛⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⢹⡆⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠸⣆⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⠟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⠴⠤⡀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢻⣼⣷⡗⠀⠀⠀⠀⠀⣇⣶⣮⠇⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢀⠞⢳⣄⠀⠀⠀⠀⠀⠉⠉⠀⣀⣠⣤⣤⣄⡈⠉⠁⠀⠀⠀⠀⢠⡟⠢⣄⣀
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠙⢦⣀⠀⠀⠀⡖⠚⠉⠀⢀⣠⣄⣀⠉⠓⢶⡄⠀⠀⣰⠏⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠉⡷⢦⣄⣹⠀⠀⠀⠘⢿⣿⠟⠀⢀⡞⣀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⠷⣤⣀⣀⣠⣇⣀⡤⠟⠉⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⢰⡏⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣦⣄⣀⣀⠀⠀⠀⢀⣼
⢿⣄⠀⠀⠀⣀⣤⢶⡿⠛⠉⠉⠁⠀⠈⠉⠙⠲⣄⡀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠉⠉⠉⠛⠷⣌⠉⠉⠉⠛⠋⠁
⠀⠉⠛⠛⠛⠉⠁⡼⠁⠀⠀⠀⣤⠤⣄⢠⡶⣄⠈⢳⡀⠀⢠⡟⠁⣠⣤⡀⢀⠖⠲⠆⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⢰⠇⢰⠞⠒⢄⢹⡀⢸⡜⣇⠈⣧⠀⢳⣤⣾⠃⢰⡏⠀⡇⢾⠀⣸⢀⠔⠚⢦⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠸⣤⣀⣸⠈⢓⣚⣀⣈⠙⠁⠀⠘⡆⣾⠀⠈⠓⠚⠀⠈⠉⠁⢻⣀⣠⠞⢸⠇
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠚⠁⠀⠀⠀⠉⠓⢤⠀⣇⢹⡀⠀⡴⠚⠉⠉⠉⠓⢤⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⢦⣀⠀⠀⠀⠀⢀⡼⠀⡏⠈⣧⡀⠳⣄⠀⠀⠀⠀⢀⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠈⠉⠛⠒⠚⠉⢀⡴⠃⠀⠈⢷⣄⠈⠉⠛⠛⠋⠉⣀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠘⣿⡹⣧⠹⣿⡻⣿⣷⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣾⣷⠹⣧⠘⢷⡙⢷⡝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢹⡆⠈⣷⡈⢿⡄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⡀⠸⣧⠈⢿⡄⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⣿⠀⠸⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⢂⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⢂⠠⠐⠀⠉⠀⠀⠀⠈⠁⠂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢰⣷⣤⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠙⠃⠀⠀⠠⠐⠄⠀⠀⠉⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡀⠰⠆⣀⣤⠤⣾⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⡧⢼⣛⠄⠉⠀⠐⢀⡏⢰⡏⠡⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣣⠘⣿⡆⠀⠀⠀⠸⢃⠏⣱⠀⠈⢄⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣿⠟⠋⠀⠀⠀⠀⢶⠸⠀⢻⡆⠀⠀⠠⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡥⣾⠃⠀⠀⠀⠀⠀⠰⠆⠁⡆⢰⢉⠧⣂⣀⡷⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠐⠈⢁⣤⠃⠀⠀⠀⠀⠀⢀⡟⡆⢸⢃⠈⠘⠠⠀⠃⢾⡆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⢀⡴⠀⠁⡠⠂⠁⡎⠀⠀⠀⠀⠀⢀⡞⢠⠁⡀⣸⠀⠀⠆⠄⣸⢸⡇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄⠌⠀⢀⠰⠄⢀⠀⡁⠀⢠⡄⢀⣰⡷⣧⡜⢠⠀⡇⣤⠰⠘⠉⢁⠂⠡⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠀⣐⠋⢀⠊⢀⣀⣀⠀⠘⡇⠀⢸⡯⠬⣿⣟⣾⣣⠶⠚⠁⠀⠀⠀⠀⠂⠀⠀⠑⠄⠀⢀⠔
⠀⠀⠀⠀⠀⠀⡇⢠⠏⠀⠈⠐⣁⠒⠈⡵⣀⠇⢀⠊⠀⠀⠈⠉⠁⠀⠀⠠⠊⠉⠀⠒⢄⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⢣⠀⡌⠁⠰⠎⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡐⠊⠳⢄⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠈⣰⠃⠀⠀⠘⡐⠤⢤⠄⠛⣰⡇⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢣⡜⢂⠁⡴⠂⠁⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠐⢐⠦⢋⡴⢋⡴⠂⠁⠀⠑⠠⠄⠀⣀⠀⠦⠰⠀⠄⢀⠐⠇⢀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠂⡀⠰⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠥⠤⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣩⣍⣹⣩⣿⣥⣭⣹⣍⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣭⣭⣽⣿⣿⣧⣭⣭⣭⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⡾⠒⠵⠋⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡋⠀⠀⠀⠀⠀⡀⠀⡶⠒⠢⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠹⠀⠱⡀⠀⠜⡧⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢀⠀⠀⠀⠀⡠⣚⠀⠀⠙⠢⢤⠏⢉⠒⠒⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠒⠈⠃⠛⡀⠀⠀⠀⠀⠀⠀⠙⢔⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣕⠄⣶⡿⣄⠀⠀⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⠊⠀⠀⠈⠢⣀⠀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡜⠀⣇⣀⡀⡜⢰⠆⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠚⢤⣀⠀⠈⠸⡅⠀⠀⡕⠦⡠⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠶⠸⠟⠂⢠⠳⠴⠘⢄⠠⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⢀⠀⠤⡀⠃⠀⠀⠈⠀⠀⠘⠁⠀⠚⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠘⡦⢄⣈⠀⠂⠈⠀⠨⠀⡀
⠠⡐⠶⠶⡆⠀⠀⡀⢄⠀⡔⠁⠀⠌⠀⠀⠀⠀⠀⠉⠐⣶⠖⠒⠚⠃⠁
⠘⠃⠀⠐⠁⠸⠢⢀⠋⠆⠁⠁⠀⠀⠀⠀⠀⠀⢠⣔⣀⠖⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠒⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠞⠋⠉⠉⠉⠉⠉⠙⠳⢦⣟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⠋⠙⠛⠛⡿⠛⠛⠛⠛⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⣰⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣧⠈⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠋⠁⠀⢠⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⡆⠀⠀⠉⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄⣴⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣉⣭⡼⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⢧⣭⣉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡉⠙⠛⠉⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⠛⠋⢉⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠳⣦⣀⡀⠀⠀⡇⠀⠀⠀⣀⣤⠞⠃⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣉⣽⠟⠓⠛⠒⠻⣿⡉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⢀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⡀⣀⣠⣀⣆⡀⠀⠀⢰⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠿⠋⠉⠋⠻⣿⠿⣿⢿⣄⠉⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣁⠆⠀⠀⠀⠀⠈⠀⢰⡿⢿⣆⠀⠀⣹⠿⠻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⣃⣻⠗⠀⠀⠀⣠⠀⠀⠈⣿⣿⡏⠀⠈⣭⡟⠃⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⡿⣆⠐⠐⠛⠅⠀⠀⣰⣝⣿⣿⡈⠐⢻⢻⣿⣶⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢸⣿⣷⡹⡄⠃⠀⢀⢀⠘⡿⢘⣧⣙⣧⢰⡀⠠⣽⣿⣟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣼⣻⣓⣺⡟⠃⠀⢿⣿⣿⣿⣷⠁⠀⣿⠹⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⠹⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⠹⢿⣿⣷⡈⠁⠦⠿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣴⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⠹⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢿⠿⢿⠛⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⠶⠟⢿⣿⣿⣧⣠⣶⣷⣆⡀⠀⡠⢀⣼⠟⠁⠀⠘⠶⠆⠐⣨⠄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠉⠏⠀⠀⠤⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⢾⠀⢀⠁⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⣥⠰⡄⣺⣟⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⡀⠉⠙⠃⠛⠂⢘⡇⣦⣴⣿⡇
⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⠈⠂⢉⡙⣡⣟⣈⡙⠛⣿⣿⠙⣇⠀⠀⠀⠀⠀⠀⣠⠐⣝⠷⣺⣴⣿⣿⣿⡟
⠀⠀⠀⢀⠼⠟⠛⠿⣿⡿⠟⠋⠑⠒⠲⠛⢿⣿⣿⣿⣷⣿⣿⣶⣿⣷⣄⣠⣤⣤⣤⣤⣭⣷⣾⣿⣿⣿⢟⣿⡇
⠀⠀⠀⢢⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠿⠿⢿⣿⠿⠿⠋⢡⢉⠛⣽⣾⣾⣷⣿⣽⣯
⠀⠀⠀⢿⣿⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠊⣞⡙⡻⣿⣿⣿
⠀⠀⠀⠚⠙⣿⣿⣤⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣾⣤⣼⣿⣿
⠀⠀⠀⠀⣠⣼⣿⣧⡐⣛⠶⣤⣶⣗⡰⠀⠀⠁⠀⠀⣇⢲⣶⣀⡄⠀⡀⣀⣠⣀⡀⣠⣾⣶⣿⣿⣿⣿⣿⠿⣿⣄
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠶⠿⠿⠿⠷⠶⠞⠛⠛⠻⢿⣬⡉⡉⣽⣟⣭⣽⣿⣿⣿⣿⣿⣽⡿⠞⠋⠉⠁⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⠀⠐⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠠⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠋⠀⠀⠢⣦⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠉⠢⡈⠲⣌⣦⡖⠠⢥⡒⠷⠠⠀⠀⠈⣶⠏⠀⠀⠀⠀⢁⠀⠁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠀⣻⣋⠐⠓⢠⣐⠁⠈⠆⠀⠂⠀⡀⠻⠀⠀⠀⠀⠀⠘⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⡁⠑⢌⡀⣈⠬⠘⠂⢬⠀⢀⡼⠀⠐⣇⠀⠀⠀⢸⡔⡀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠌⢀⠄⡞⠀⠒⠀⠠⠤⠂⠀⠀⠊⠉⢀⠛⠀⠀⣤⠆⢀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠈⡔⢀⡀⠛⠀⠀⠀⠀⠀⠀⢀⠒⠂⠐⠁⠀⠡⡐⠈⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢷⠊⠉⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠐⠒⠒⣲⠚⣯⡀⠁⠱⠒⠁⠁⠺⠀⠂⠀⠀⠠⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠭⠧⡐⠀⠀⢃⣘⡀⢬⠈⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⢪⠁⠀⠔⠢⠤⠅⠀⠈⠙⡛⠁⢘⣀⠤⡐⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⠚⠦⠤⠄⠓⠶⠤⣲⡄⢠⠄⠀⠄⠠⡤⣀⡉⠀⣀⣡⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠄⣰⡃⠠⠀⠉⠐⠺⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡀⠀⠀⠈⠸⡁⠈⠈⠀⠤⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠈⢄⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⣀⣠⠤⠤⡤⠦⠤⢤⣠⠒⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢠⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣘⢿⠶⠶⢶⠼⢿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡜⠁⠘⠦⠤⠚⠀⠀⠱⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢇⢀⠀⢠⡀⠀⡠⠀⢠⠇⠀⠀⣠⢎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠙⠪⢷⣿⣥⣤⣒⣒⣒⣡⠤⠖⠋⠀⠀⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠑⣄⠀⠀⠀⢸⠁⠀⠀⠹⣄⣉⠔⠁⢠⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⡼⠶⠶⢤⣄⡀⠀⣸⠀⠀⠀⣾⠀⠀⠀⠀⠉⠁⠀⠀⢸⡀⠀⠀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠈⠙⠿⣏⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠉⠁⠉⠉⠛⢦⡀
⠀⠀⠀⢸⠃⢠⠞⠛⠓⢦⡀⠀⠀⠈⢷⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡴⠋⢀⡤⠖⠚⠶⣦⡄⠀⢳⡀
⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠙⢦⠀⠀⠈⣇⠀⠀⢳⠀⠀⠀⠀⠀⢸⠃⣰⠋⠀⠀⠀⠀⠈⢻⡀⢸⡇
⠀⠀⠀⠈⣇⣧⠀⠀⠀⠀⠀⢸⡇⠀⢀⡏⠀⠀⠀⢣⡀⠀⠀⠀⢻⡀⡇⠀⠀⠀⠀⠀⠀⣸⠃⣸⠃
⠀⠀⠀⠀⠘⢾⣦⡀⠀⠀⢀⡼⠁⢀⡼⣁⠤⠤⢄⣀⣱⣄⣠⣤⡼⢷⣧⣀⠀⠀⠀⣀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠈⠙⠿⣷⣞⣫⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣯⡭⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⣼⡇⠠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⡞⠀⠀⠀⠀⠀⠀⠀⣰⠿⠀⠀⠀⠀⠀⠴⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠄⠀⣠⡠⠆⠀⠀⠀⠉⠉⠀⠀⠀⢀⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣟⡁⠀⠀⠀⠀⠀⠀⠀⠰⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣰⣿⠁⠀⣠⣤⠀⠀⠀⠀⢸⣿⡷⢿⣿⡶⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠒⠚⠻⢿⣿⣿⡃⠀⠀⢿⣿⠁⠀⠀⠀⠀⠉⠁⠘⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠴⠶⠶⠶⢶⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠴⠚⠋⠁⠀⠀⠀⠀⠳⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢹⣿⣿⣿⡇⠀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠿⣿⣿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠉⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢾⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣀⣤⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠘⢧⠀⠉⠉⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣷⣀⡀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣷⣶⣴⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⣤⡄⠀⢀
⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣤⣄
⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧⣤⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠁⠉
⣿⣿⣿⣿⣿⣿⣿⣤⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡆⠀⡀⢀⣰⣾⠋⠉⠉
⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠀⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢣⡀⠘⠿⣿⡿⠟⠁⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⡀⠀⠀⠀⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⣤⣤⣀⣻⣆⣀⣀⣀⣀⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⣿⠉⠉⠉⠉


⠀⠄⠤⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢠⣿⣿⡟⠛⠛⠛⢿⣿⣿⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣿⣷⣾⡿⠿⠿⣿⣿⣯⠉⠀⠈
⠀⠀⠀⠾⠿⣿⣷⣦⣤⣤⣾⣿⣿⣶⣿⣏⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣼⣿⣿⣅⣀⣀⣠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢻⣿⡛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠶⠶⣾⣿⣿⣿⣿⣭⣉⣉⣉⣁⣤⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣦⣤⣤⣄⣀⣈⣉⣉⣥⣤⣤⣾⠃
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⣀⣀⣼⣟⣡⣾⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⡿⢿⣿⣆⠀⠀⠀⢀⣀⣴⣶⣿⣶⣦
⠀⠀⠀⠀⠈⢿⣿⣿⣏⠉⠉⠉⠙⠛⠿⠟⠁⠀⢸⣿⠁⠀⠀⠀⠀⠸⣿⣿⠿⠶⣬⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣀⡀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠈⠋⠉⠀⠀⠀⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣤⣤⣴⣶⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠦⣄⣀⣠⣤⠶⠶⠶⠶⣤⣄⣀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢰⠇⠀⠀⢀⣤⡄⠀⠀⠀⠀⢠⣤⡀⠀⠀⠸⡆⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣴⠛⠛⣦⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⢷⡞⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠛⠲⠾⠷⠖⠛⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⠶⣤⣀⣀⣀⣀⣤⠶⢿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⢧⡙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡾⠁⠀⠀⣠⠞⠋⠉⠉⠙⠳⣄⠀⠀⠈⢷⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⠟⠀⢸⡇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⡇⠀⠻⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⢻⡇⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⡇⠀⠀⢸⡿⠉⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠳⣦⠀⢻⣤⡤⠾⢷⠀⠀⠀⠹⣦⣀⠀⠀⣀⣴⠏⠀⠀⠀⡾⠷⣤⣤⡟⠀⣴⠞⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣶⣤⣀⡀⠀⠀⢀⣀⣤⣶⠋⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣿⠀⠀⣀⣠⣤⠶⠋⠁⠀⠉⠉⠙⠛⠉⠉⠀⠈⠛⠶⣤⣄⣀⠀⠀⣿⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠒⠚⠋






⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣆⠀⢠⡀⠀⢀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⢠⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣸⣿⣿⣷⣼⣿⣸⣺⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢀⣼⣧⠀⣸⠀⠀⣾⣿⣿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡺⣿⡇⠀⡏⠀⢰⣿⣿⣿⡟⣇⣇⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠼⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢆⣿⠅⢀⠀⠀⣸⣿⣷⣿⡇⣿⣿⠇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣼⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⢠⡿⠲⡄⣀⣀⠀⢀⣤⣀⣀⡀⠀⣀⠤⠓⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⣽⣧⣤⡇⣧⣀⣾⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⡀⣀⠀⠁⠀⠠⠄⣀⠀⠀⢘⣿⣿⣿⡇⠀⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⢀⣀⣤⡶⢿⣿⣿⣷⣿⣿⣿⡿⠛⠿⠿⣿⣧⣷⣤⡀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠉⠒⢸⣿⣿⣿⠇⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠈⢱⡤⢤⣥⣭⣭⣽⡿⠉⢻⣷⣟⣦⣴⣹⣿⣿⣮⣳⣶⣶⡀⢁⠈⠙⢿⡧⡀⠀⠀⠀⠈⢿⣿⣿⣿⣾⠛⠋⠉⠁⠈⠈⢀⠈⠛⠛⢿⣿⡇
⠀⣼⡇⠀⡖⠒⢺⣿⣧⣯⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⡋⠛⣛⣛⣙⣷⣾⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣟⠀⣠⣦⠼⠂⠙⢶⡷⢶⣒⣞⣉⣇
⢀⣿⡇⠀⣇⣀⣀⣉⣛⣉⣀⣀⣀⡀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣄⠀⠀⠀⠈⣿⣿⣿⣿⣾⠟⠁⠀⠀⢠⣿⣿⣿⡿⡿⠿⣿
⢘⣿⡇⠀⠆⠀⠀⠐⠆⠀⠰⠀⠀⠰⣷⠖⠲⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⡄⣾⣿⣿⣳⣿⣿⣿⣦
⠀⣿⡇⠀⠀⠀⠰⣆⠂⠀⠀⠀⠀⠀⣯⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⢿⠂⠀⡇⠀⣿⣿⣷⣿⣿⣿⣿⡿
⣠⣿⡇⠀⡀⣀⢀⣀⣀⠀⠀⠀⠀⣘⣟⡀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡄⠀⠀⢸⣿⣿⣿⣟⣸⡷⣾⡅⠃⣿⡟⣿⣿⣿⣿⣿⡇
⠟⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡇⠀⠀⢀⣿⣿⣿⣿⣹⡷⣏⣷⣴⣿⣽⣿⣿⣿⣿⣿⡇
⠈⣿⠃⠀⠃⠀⢀⡤⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡇⡷⠀⠀⠘⣿⣿⢻⣿⣿⣃⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅
⠰⣿⡆⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣦⣭⣽⢛⣛⣉⣽⣿⣷⡇
⠘⣿⡇⠀⠓⠐⠒⠒⠒⠒⠒⠂⠐⢲⣿⠒⠒⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢉⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡁
⢠⣿⡇⠀⣴⣶⣶⣦⣴⣤⣶⣶⣷⣿⣽⣷⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣼⣿⣿⡄
⠈⢿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣇⣿⣾⣿⡿⣿⣿
⠀⠀⠈⠛⠉⠫⠿⢟⠻⠿⠿⡿⠿⠦⣤⣤⣾⣿⣿⣿⣿⣿⣟⣋⣱⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⢿⠿⢿⣶⣷⡀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣉⣈⣁⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣉⣉⣛⣛⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠿⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠿⠶⣦⣄⡀
⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⢀⣾⠋⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⢻⣆⣀⣤⣴⡶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣦⣤⣀⣰⡿⠁⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠘⣿
⣼⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠸
⣿⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿
⢹⣇⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⣸
⠀⠻⣦⡀⠀⠘⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠃⠀⢀⣴⡟
⠀⠀⠈⠻⢷⣦⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⣠⣾⣯⣥⣬⣽⣷⣦⠀⠀⠘⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢀⣶⣿⣿⣿⣦⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⠀⠈⠙⠿⢿⡿⠿⠛⠁⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣰⣆⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⡿⢷⣦⣤⣄⣀⣀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣀⣀⣤⣴⡶⢿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⣀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⡀⢀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⣛⣭⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣬⣙⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠉⠙⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠁⠈⠉⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣠⣾⣿⣿⣿⣶⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠠⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣾⣿⣿⡿⠟⠋
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠛⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⡆⠀⠘⠉⠉⠉⠉⠓⠲⠞⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠁⠀⠀⡤⢤⡀⠀⠈⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢾⣀⣀⡷⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⢤⣳⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣞⡤⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠘⠛⠓⣦⣤⣀⣀⣀⣀⡤⢴⡚⠛⠃⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡇⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣅⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡜⠀⢀⣀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠈⠙⢦⡈⠳⢤⣀⣀⣇⣀⣠⠴⠋⣠⠖⠉⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠉⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢀⠧⣄⣀⣀⣀⣀⣠⠴⢧⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⡤⠔⠋⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠦⣤⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⣻⢦⡴⠭⠷⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡿⠉⠁⡞⠀⠀⠀⠀⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠦⠤⠀⠀⠀⠦⠒⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠋⠀⠀⠙⠒⠋⠉⠛⠚⠉⠉⠉⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠀⢩⠂⠐⠒⠚⠉⠉⠉⠉⠲⠰⡤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠁⠂⠧⣀⠤⠸⡠⠤⠀⠀⠀⢀⡗⠑⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠲⣄⣀⠈⠇⣀⡈⡢⠠⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢒⠞⢤⠢⠶⠦⢐⠖⠒⠋⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⣜⣁⡱⠭⠭⠭⠬⣄⣲⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢸⠀⠉⠉⠉⠉⠉⠉⠉⠀⡧⣤⡤⠠⢤⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠨⠗⠙⠄
⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⡄⢀⡤⠊⣀⣀⡤⠤⠶⢞⠉⠓⣄⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠈⡷⢤⣀⡈⠳⣄⠀⠀⠀⠀⠀⡧⣤⠤⢾
⠐⢰⠤⠼⠆⠒⡰⠿⠆⠒⢖⠛⠋⠉⠁⠀⠀⠒⠒⠊⢳⠦⠼⠀⠀⠀⠀⠀⠀⠀⣇⠜⡲⠖⠊⠐⠒⠊⠉⠓⠾⢷⡐⠒⠒⠂⡇⠀⠀⢸⠒⠒⠒
⠀⢸⣉⣹⠀⢸⣦⠤⠤⠤⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢾
⠀⢸⣀⣸⠀⢸⣋⠀⠀⠚⢩⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⡀⢤⠠⠀⢀⠀⠀⠀⠀⠀⠣⠤⠤⠊
⠀⠈⠉⠉⠀⠘⠲⠬⠭⠔⠀⠀⢀⣠⣤⣤⡀⠐⠛⠉⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡶⠶⠟⠛⠉⠁⠀⠀⠲⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠶⣤⣄⣀⣀⣀⣠⣤⠴⠞⠉⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠉⠉⠳⣤⣀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠉⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣸⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⠄⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣄⣼⠖⠋⠉⠉⢉⣀⢀⡤⣄⠀⠈⠉⠉⠉⠓⢾⣄⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢀⡟⠉⡿⠀⣼⠶⣄⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⠃⢸⠇⢠⡏⠀⣟⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⣦⠀⢀⡟⠀⡿⠀⣼⠁⢸⠏⣹⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆⣸⠁⢸⠇⢠⡇⠀⡞⠀⣿⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⡏⠀⠀⠀⠈⠁⠸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠟⠉⠉⠛⠓⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢈⡗⠀⠀⠀⠀⠀⠀⣀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧



⣇⢸⠆⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠂⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂⢾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⡇⠈⡆⡇⠈⠁⢠⢻⡀⡧⢼⢸⠀⠀⡇⠀⠀⠪⣄⠀⡇⠀⣅⡕⣸⠀⢱⢈⣖⠁⠬
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣷⣶⣥⣧⣤⣤⣦⣤⣥⣷⣮⣼⣶⣤⣷⣦⣤⣴⣮⣤⣥⣤⣤⣼⣬⣶⣮⣤⣤⣥⣴⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⡤⠤⠤⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⢒⡱⠖⠂⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⣠⠞⠋⠉⠀⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡁⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⠤⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣠⠾⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠋⠉⠑⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠉⠒⠤⢄⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠈⠛⢦⡀
⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣀⠉⠉⠁⠉⠉⠉⠉⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣄
⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠈⠙⠶⠶⠶⠶⢶⠏⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣿⠶⣄⣀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢀⣀⣈⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢠⡏⠀⠀⠉⠛⠶⠶⠶⠋⠁
⢀⡴⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⢀⡾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏
⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠉⠙⠳⠶⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢻⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠟⠛⠓⠲⠶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⢠⢀⣀⣄⣠⣾⣏⠀⣀⣀⠀⠺⣷⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠓⠒⢫⣾⣿⣿⣿⣿⣿⣦⣙⡛⠓⠀⠀⢀⣹⣿⣿⣿⡄⣴⣾⠆⠀⠈⢋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣼⣿⠶⠀⠀⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠉⠛⠙⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⢿⡗⢸⡄⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣴⣶⣾⣷⣤⣴⣶⣦⣤⣿⡿⢿⣿⣿⣿⡇⠀⣀⣄⠰⣾⣿⠄⠲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡿⠁⠀⠈⠙⣿⣷⡄⢸⣿⡷⣬⣉⣀⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠟⠉⠉⠉⢀⣀⠀⠘⠛⠉⠉⣀⠀⠀⢈⣿⣦⣸⣿⣿⣾⣿⠙⠂⠒⢶⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⣀⣠⣶⣤⣶⣿⣿⣿⣷⣷⣦⣴⣾⣿⣠⣶⣾⡟⠛⠻⣿⣷⣿⣧⠀⠀⠻⣿⣿⡟⠻
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⣿⣿⠏⠀⠀⠀⠘⣿⢿⣿⣷⣤⣴⡿⢯⡛⢂
⠀⠀⠀⠀⠀⠀⠿⠟⠋⠉⢩⠟⠉⢩⡾⠋⠀⠰⠟⠁⠀⠀⠙⠛⠉⠁⠀⠈⠁⠀⣤⡀⣀⣤⡄⠀⣼⣿⣿⣿⠧⠀⠉⠁
⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣤⣄⠀⠀⠀⣷⣤⣼⣿⣿⡿⢿⣇⠀⠘⢿⡿⠋
⠀⠘⠻⠿⣿⣤⣶⣶⣿⣃⣤⣶⣾⣴⣾⣧⣴⣿⣷⣤⣶⠟⠋⢿⣷⣤⣴⣿⣿⣿⠉⠻⠁⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⢻⣿⣥⣼⣿⣦⣤⣾⣿⣿⡟⠋⠉⠿⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⠟⠛⠉⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠈⠁⠐⠀⠤⣄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠒⠄⠈⠀⠠⠑⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠂⠤⠤⠀⣀⡂⠀⠄⠀⠀⢀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⣄⠫⣉⠁⠒⠀⠀⠁⠂⠀⠀⠠⠂⠀⠀⠈⠒⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢈⠳⠮⣙⡒⠢⠤⢄⣤⡀⠀⠀⠀⠀⠀⢠⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠂⡀⠀⠢⢀⠠⠊⢉⠵⠂⠤⠤⠀⠀⠀⠀⠀⠀⠀⠘⠿⡦
⠀⠀⠀⠀⠀⢀⡀⠌⠉⠀⠐⠒⢀⠤⠐⡐⠐⠖⢀⣰⣌⣀⠖⠁⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂
⠀⠀⠀⠀⡐⠃⠀⢒⠔⢒⠪⠭⠴⠂⡡⠴⠄⢁⠈⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⢀⠀⠎⡰⠁⠀⠀⠀⠀⠂⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠸⡐⠁⠀⠂⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠌⠐⡠⠁⠀⠀⢀⡤⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⠀⠀⢀⠀⠀⠀⠰
⠀⠀⠀⠀⠃⠀⠀⠀⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠄⢠⠁⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⡄⠀⠀⠀⠱⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠠⠂⠀⠀⠀⠑⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠈⠁⠒⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠊⠁⠀⠀⠀⠀⠈⢢
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⣀⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢰⠀⣀⣀⠤⠔⠊⠁⠀⠱
⡌⠢⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇
⡇⠀⠀⠁⠒⠠⠤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⡀⠀⠀⠀⠀⢀⠠⠊
⠐⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠈⠁⠀⠀⠈
⠀⠈⠑⠂⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠒⠒⠒⠒⠒⠋⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠤⠄⠠⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⡇⢰⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠱⠗⡠⠊
⠀⠀⠀⠀⠀⠀⣰⠳⢦⣨⠇⠀⠀⠀⢳⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⣀⠀⠀⠀⠘⡆⠀⠀⣇⢀⡞⠉⠉⠙⠓⠒⠶⢦⣀⡴⠊⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢦⠀⠀⢹⡀⠀⢹⡈⢳⠀⣠⢤⣤⣄⣀⡸⠊⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⡀⠈⡆⠀⣀⠷⠚⠺⣇⡏⢠⡟⠾⣄⡴⠋⢀⣴⡚⢧⡀
⠀⠀⠀⠀⠀⢸⡀⠀⡎⠀⠀⠈⢂⠹⡋⠁⢀⠴⠒⠋⠀⠸⣧⡾⢋⣀⣜⡧⠤⠽⠔⠛⠲⡄
⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠈⡄⢳⠀⡎⡤⠀⠦⠤⣀⠈⢱⠉⠀⢀⣀⣀⣀⣀⣀⠀⢻⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⡆⠈⢆⠀⠀⠀⡀⠀⢿⢰⠁⠀⠀⠀⢀⠀⢸⠀⡎⠁⠀⠀⠀⠀⠈⡆⢸⠋⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡔⠒⠻⡀⠈⠳⠤⠴⠃⠀⠈⠇⠀⠀⠀⠀⢸⠀⡽⠀⢰⠀⠀⠀⠀⠀⠀⢣⠈⡆⠀⢀⣄⠀⢠⠘⡄
⠀⠀⠀⠀⠀⢳⡄⠸⠥⣀⣀⣀⣀⣀⣀⠀⠈⣆⠀⠀⠀⡇⢰⢦⡆⢸⠀⠀⠀⠀⠀⠀⠘⡄⣧⠜⡅⠈⡇⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⠤⢀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠁⠀⡆⠀⢧
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢢⠀⠀⣇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡟⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠉

⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⣠⡶⣶⠴⠶⠚⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀⢀⡴⠶⠶⣤⡀
⠀⠀⠀⣰⠟⠉⠀⠈⠙⠷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢀⣀⠀⠀⢻
⠀⠀⠀⣿⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣠⡟
⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⡴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠑⣄⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢹⠀⠀⠀⠀⢸⡆⠀⠀⣴⠟⠉⠁⠈⠙⢳⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⡶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠃⠀⠀⢠⣄⠀⠀⠹
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠛⣾⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⠏⢀⡤⣄⠙⠞
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣤⠟⠶⠂⢠⡇⠀⠀⠀⠀⠀⠀⠀⡼⣱⠏⠀⠘⠦⠞⠀⣴⣲⠄
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⡤⠴⠖⠋⠀⠀⠀⠀⠀⠀⢀⡾⣱⠏⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⣠⡿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⡰⢦⡀⣤⠀⢀⡄⠀⣠⡀⠀⠀⠀⠀⢀⣠⡶⠋
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢤⡰⠋⠈⠙⠚⠀⠀⠉⠀⠳⠞⠙⠛⠁⢳⡀⠀⠀⠘⣯⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣇
⢿⡀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢻⡄
⠈⠳⣦⣤⣤⣤⠾⠋⣼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢤⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠳⠶⢶⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣠⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣶⣶⣶⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣶⣞⣛⣉⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠟⣋⣤⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢁⣶⣶⣦⣡⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⢃⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦⡙⠻⢿⣿⣿⣿⣿⣿⣿⡿⢡⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠈⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⢋⣩⣤⣤⣤⣍⣁⠀⠀⠀⠀⠀⠀⠈⢿⣟⠻⣿⣿⣦⡈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠈⢿⣧⣙⣿⡿⢿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⣿⣿⣿⡿⢋⣥⣤⣌⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⣾⣿⣿⣿⠆⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣼⣿⣿⣿⡟⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣸⣿⣿⣿⡟⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⠃⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣿⣿⣿⣿⣿⣿⣶⣤⣉⡛⢿⠀⠀⠘⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⡛⠻⢿⣿⣿⣿⣿⣿⣿⠄⠀⠀⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣉⡛⠿⠿⠋⡀⠀⢀⡉⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠿⠿⣿⣷⣤⢃⣔⣥⡶⠶⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢈⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⣿⣿⣿⡟⣩⡦⠀⠈⣿⣿⣿⡏⢛⣿⣿⣦⡙⠂
⠐⡀⠂⠠⠀⠂⠉⠀⠀⢸⢫⣾⣿⢃⠹⠠⢁⣀⠀⢸⡟⢸⠇⣀⠛⡜⢿⣿⡍⠒⠢⢄⡀
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠈⠀⠴⣟⠫⠥⡀⠀⠀⠘⠿⣷⠀⠈⣿⣿⡄⠀⠀⠈⠑⠢⢄
⠀⠀⠙⡦⠀⠀⠀⠀⢀⠀⣿⣿⡇⠀⠀⢰⠁⠠⠀⡀⠀⢀⡀⠘⠈⠀⢠⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⣸⠃⠀⠀⠀⢊⢊⠴⡿⢿⠃⠀⠀⠸⡀⠀⠈⠁⣀⠈⢀⡜⠀⠀⣼⣿⣏⠈⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢠⡏⠀⠀⠀⠀⢠⠏⠀⢢⣄⠀⠀⠀⠀⠒⠛⠁⢤⡠⠟⠛⠈⠀⠀⢻⣭⢌⢧⢙⠀⠀⠀⠀⣰⠀⠀⣷
⠀⣸⠁⠀⠀⠀⠀⠘⣆⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡼⠋⢊⡞⠀⠀⠀⠀⢠⡏⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⠶⠾⢧⣀⣀⠀⡴⠖⠒⠒⠚⢁⣀⡤⠖⠿⠤⠶⠋⠀⠀⠀⠀⢀⣿⠁⠀⢀⡇
⠀⣷⠀⠀⠀⠀⠀⠀⢀⣠⣐⠤⠑⣂⡥⠾⢶⣶⠂⢠⠞⠛⢿⣿⠀⠐⠢⢄⡀⠀⠀⠀⠀⢌⠆⠀⠀⢸⠃
⠀⣿⠀⠀⠀⠀⠀⠸⠶⡋⠄⠂⠁⡳⠂⣸⣦⡁⢠⣮⣶⠒⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡟
⠀⢻⡀⠀⠀⠀⠀⠀⣁⠀⠀⡠⢊⠝⣼⣿⠃⣿⡟⠸⣿⡆⠠⡀⠳⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡾⠁
⠀⠘⣇⠀⢀⠂⢁⡎⠀⠈⠈⣀⡏⣼⣿⡏⢠⣿⣧⠀⢻⣿⡀⢱⡀⠘⣦⠀⠀⠀⠀⢠⠁⠀⢀⡾⠁
⠀⣀⠿⠶⠺⡀⠈⣇⠀⣆⠀⡏⠉⠉⠉⠓⠺⠿⢟⠒⠉⠉⠉⠉⠉⠑⠚⠢⢤⣀⠀⠘⠀⢠⡟⠁
⠈⠀⠀⠀⠀⠉⠒⠉⠐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣴⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⣍⣩⣭⠤⠶⠶⠚⠛⢋⣉⡉⠥⠤⠤⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⢿⡿⡿⣿⣿⣿⣷⣒⣚⣿⣿⣯⣦⣒⣒⣂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠯⣍⡉⠑⢿⣏⡘⠷⠤⠋⣨⡿⠋⠉⢛⣛⣿⡓⠦⠤⣭⣽⣛⣶⣍⡒⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠒⠮⣍⡛⢛⣉⣭⣴⣶⡿⠛⠉⢹⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⣉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠁⠈⠉⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⡴⠾⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣾⣿⣷⣶⣤⣼⣧⠴⠞⠻⠽⠗⠀⠀⢸⣷⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣷⣾⣛⡿⠋⠠⠼⢋⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⣈⠉⠛⠓⢤⣤⣀⠀⠀⢀
⠀⠀⠀⠀⡀⣀⣴⣶⣿⠿⣿⠋⠀⠀⠈⣿⠁⠀⠃⠀⠀⢹⡷⠀⠀⣀⣠⡶⠟⢻⣿⣿⣿⣿⣶⣤⣄⡂⠉⠢⢘⣂⣀
⣴⣤⣾⡿⡟⠋⠉⠀⡀⢰⡏⠀⠀⠠⢠⡿⠀⠈⡟⠒⢾⣿⣱⡶⠚⠿⠟⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡈
⠉⢹⣾⣯⣧⠀⠀⡀⣼⣾⠇⠴⠦⢤⣁⣇⠀⠀⣧⠀⡟⣿⡗⠀⠀⠀⠀⠀⣇⣼⡿⣿⣿⡿⠟⢻⢽⣿⣿⣿⣿⣿⣿
⠀⢘⣛⣻⣽⣀⣤⣶⣿⢃⠆⠀⠐⠀⠈⣿⢦⣠⣿⢰⣇⠘⣿⡀⠀⣰⣴⣾⢿⣟⠗⠊⠁⣠⣵⣒⣲⡉⣽⠿⢿⣿⠿
⢀⣬⣽⠟⠛⠉⠛⠛⣿⢀⡂⠄⠀⠈⠠⠏⠸⠛⠛⢹⣿⡀⣿⣷⣾⠿⡿⠧⠂⠀⠀⢀⠔⣯⡻⠟⡹⣹⢃⡀⢚⠇⠙
⠙⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣠⣄⣀⡰⠀⠁⠀⠀⣸⣂⣛⣻⡏⠒⠀⠀⠀⠀⠀⠀⠠⡀⠉⣡⡞⢐⡇⠰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠞⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠓⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⠛⠛⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠠⡤⠄⢰⣀⡀⠠⠤⠤⠤⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⣠⠔⠋⣱⡀⠄⠀⠀⠒⠐⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠴⢦⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣏⣠⣒⠁⠈⢆⣀⠤⠒⢉⡠⠒⣽⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢤⡀
⠀⠀⠀⠀⡠⠛⠀⠀⠹⣤⡤⠀⠀⢿⣿⣿⣿⣿⣿⠟⡥⠞⢻⣿⠉⠉⠹⢤⠖⢁⡠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠳⣀⡀⠀⢀⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠊⠡⠽⡗⠒⠒⢞⠉⠉⠙⠟⠛⠛⠋⢩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢦⡀⠀⢸⠁⠀⠀⠘⣆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠓⠦⣾⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⠀⠈⢳⣿⣿⡿⢏⡀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⣼⠀⠀⣸⣷⠀⠀⠀⠀⠀⡏⠉⠉⠁⠈⢦⣀⡀⠀⣀⣀⣸⣄⣀⣀⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡤⠤⠴⠁⠀⠀⠀⠀⡿⠻⠴⠚⠁⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠣⠤⠤⠤⠴⠦⠤⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠤⠤⣤⣤⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⣄⠠⠤⠤⠤⠤⣀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠋⠀⡔⢡⠀⠀⠀⠀⠀⠀⠙⣄⠈⡫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠁⡎⠀⠀⠀⣀⠀⣀⠀⠀⠀⠘⡆⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠈⢛⣐⡛⠀⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠁⠀⠀⠀⠻⣿⠟⠀⠐⡄⠀⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⢣⡴⠤⡀⠂⠠⠀⠁⠀⣤⠞⠁⡼⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⡸⠁⠀⠀⠈⠝⠒⠒⠈⠁⠀⢀⡔⠁⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢰⠁⠀⠀⠀⢸⣀⣀⣀⠤⠄⠒⠉⣇⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠸⡄⠘⣄⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡼⠁⢧⠀⠀⠀⠀⠹⡄⠈⠳⡄⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡜⠁⠀⡸⠀⠀⠀⠀⠀⠹⠀⠀⢰⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠸⡄⠀⠀⠀⣠⠏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⡆⠀⢸⠄⠀⠀⠀⠀⠀⠘⡌⠢⣀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠑⠤⠴⠊⣏⠤⠔⠒⠒⠓⠶⣔⢦⣀⠀⡰⠒⠋⠁⣨⠔⠒⠒⠒⠤⣀⢡⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠉⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⠛⠋⠙⣄⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠛⠋⠉


⣿⡋⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⢀⣀⡀⠉⠻⢦⡀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⣤⣤⣀⣀⠀⠀⠀⠀⢀⡴⠛⠉⢀⣀⡀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡴⠛⠉⠉⠉⠛⢦⡈⣷⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡾⢁⡴⠛⠉⠉⠉⠛⢦⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⢰⠏⠀⠀⠀⠀⠀⢀⣤⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣄⡀⠀⠀⠀⠀⠀⠹⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣯⠀⢸⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⡇⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢧⡀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢀⡼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠉⢛⣆⣀⣀⣀⣤⣤⣤⣄⣀⣀⣀⣰⠟⠉⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠁⢀⡤⠶⠒⠲⢦⡀⠈⠉⠉⠛⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢘⣇⡀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡟⠀⢀⣀⣀⣀⣀⠀⠀⣾⠉⠙⣻⡋⠁⣹⠀⠀⢀⣀⣠⣄⣀⡀⠈⢷⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣨⠷⠛⠉⠉⠀⠉⠉⠙⠳⢬⡛⠛⠉⠙⠛⣡⡴⠛⠉⠁⠀⠀⠈⠙⠳⣼⠃⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠞⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠗⠛⠛⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀⣀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣀⡀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠉⠁⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠐⠓⠶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠶⠚⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⠟⣫⣉⢉⣭⡳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣭⡉⣉⣝⠳⣄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣾⡵⢾⣧⣼⣯⣤⡷⢦⣷⡀⠀⠀⠀⠀⢠⣯⡴⣿⣤⣿⣧⣼⠷⣼⣧⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣴⠟⠉⠉⠻⣧⠾⢹⡇⠀⠀⠀⠀⢾⠙⢦⡾⠋⠉⠉⢻⣶⠟⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡰⣇⠀⠀⠀⠀⣸⠇⣼⣛⣓⣶⣶⣚⣻⣆⢸⡁⠀⠀⠀⢀⡿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⣒⣒⣛⣥⠞⠉⠉⠉⠉⠉⠉⠉⠙⠳⣍⣓⣒⣚⣫⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⡀⣀⣀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⢳⡈⠳⢤⣤⠴⠚⠡⢄⡀⠀⠀⠀⠀⡇⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢧⠀⠈⢧⠀⠀⠀⠀⢳⡀⢦⠙⣄⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⡆⠀⠘⣆⠀⠀⠀⠀⠃⠘⠂⠘⡆⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡠⠋⠹⡄⠀⠉⠉⠉⠉⠉⠉⠉⢳⡄⠛⠀⠀⡇⠀⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠐⢦⡰⠊⠀⠀⠀⠙⠶⠤⠤⢤⡤⠤⠤⠤⠾⠤⠤⠤⠤⠇⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠓⠲⠶⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡖⠒⠒⠒⠒⠀⠀⠀⠒⠒⢺
⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠆⠈⠁⠧⠘⠀⠀⠈⠀⠐⠀⠉⠸⠀⠀⠀⠦⠸

⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⣀⣀⠀⡀⡀⠀⡀⠀⠀⠀⣀⣀⡠⠤⠤⠴⠖⠒⠒⠒⠒⠛⠛⠋⠉⠙⠛⠛⠒⠒⠒⢶⠤⣤⣀⣀⣤⠞⠉⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⣀⡠⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠔⡶⠀⠀⢀⣠⠤⠶⠒⠚⠛⠉⠉⠛⢒⣢⣤⠈⠉⠀⡀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡴⠚⠁⠀⣼⡡⠖⠛⠉⠀⠀⠀⠀⠀⢀⣤⣖⣛⣉⣁⣀⠀⠀⠀⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⡴⠋⡽⢀⡴⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⣴⠚⠋⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠋⣇⠀⠀⠀⠀⢸⠃⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⡠⠽⠆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⣿⣶⣿⠀⠀⡗⢄⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠒⠒⠢⠤⠞⠉⠉⢤⣀⠁⠙⣇⡠⠜⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠇⡏⣈⣉⣹⡆⠀⠹⣄⠁⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣤⡤⣍⡳⢄⠈⢂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣸⣿⠿⠿⢿⡇⢰⠖⢉⡀⠀⠀⠀⠉⣐⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠙⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣯⣤⣤⣤⣼⡇⢸⡴⢹⠃⠀⠀⠀⠀⠙⣦⠀⠀⠀⠠⠚⣡⢤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠛⡇⠀⠀⣿⣀⣀⣀⣤⡀⠀⢸⡇⠀⠀⠀⡞⠁⠀⠈⢻⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢼⣿⣿⣿⣿⣿⡇⠀⠀⢈⣩⠤⠤⠤⠷⠤⣼⠁⠀⠀⠰⡧⠖⠚⠿⠁⡏⠹⡿⠒⠶⠤⣄⣸⠃⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣸⡁⠀⠀⣀⢀⣧⠀⢠⠏⠀⡠⣖⣒⣦⠄⠀⠀⠀⠀⠀⠳⣄⠀⣀⣴⢷⣄⣀⣀⣀⣠⡼⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡿⠿⣿⠀⢸⣄⠀⠀⠈⠉⣀⣀⢤⡀⠀⠀⠀⠀⠘⠛⠛⠉⠀⢉⠙⠉⡉⠥⠤⠦⣄⠀⠀⡟⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣼⣇⣀⣀⣄⣀⣿⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢁⣝⠐⣠⡀⠀⣀⣠⠟⠀⢰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡏⡿⠟⠿⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢷⢿⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠐⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠾⣿⣿⡶⣶⣤⣤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡼⣏⣉⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠈⠓⢦⣀⣀⠤⢤⣴⠒⠚⠋⠉⠁⠀⠉⠉⠁⠀⠈⠈⠉⠛⠻⢿⣿⣟⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢻⡿⠿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠳⣤⠖⠁⢹⣤⠊⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢳⣾⣿⡇⠀⠀⣀⣀⣀⣠⠖⣶⢻⠗⣲⣇⣀⣀⣀⠈⠛⠃⢰⢸⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠉⠉⠉⢉⠽⡍⠉⠙⠒⠓⠼⠲⠋⠀⠀⠀⠉⠉⠉⠉⠛⠛⢶⣀⣇⣸⣉⣦⣀⣀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠰⣦⠒⡒⡤⠊⠀⣧⡤⢤⠤⢲⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣞
⠀⠀⠀⠐⠉⠀⠀⣀⠤⠖⠛⠉⠣⡙⡄⠀⠈⠖⠁⠂⠀⢀⡥⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⢃⠈⠀⠈⠀⠀⡠⠚⠁⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⡀⠀⠀⢀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⣀⠀⠉⠘⠒⠀⠤⠦⠚⠒⡠⠖⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣈⠒⠠⢽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠉⠉⠓⠒⠒⠢⡔⠋⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⣄⣀⡀⠀⠀⠃⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠐⠂⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠭⡑⠄⠀⣀⡠⠶⣆⣀⡀⢀⠔⡩⠭⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠜⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⡄⣀⣀⣀⠀⣤⡄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢉⠋⣴⣶⣶⠉⠫⡁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢨⠀⠉⢛⠉⠀⢀⠇⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠑⠤⣈⡢⠬⠉⠬⢔⣋⠤⠊⠁⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⠟⠉⠉⠢⡀⠀⠀⠀⠀⢠⠚⠉⠈⠳⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⠦⠤⠤⠤⠤⠧⣀⣀⡠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠄⢀⠈⠘⠧⠖⠒⠀⠠⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢡⣖⠄⠀⠀⠀⠐⠀⠤⠄⣀⣀⡀⠤⢄⣀⡀⠀⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⢠⡙⠀⠿⢀⠄⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢀⠄⠀⠘⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀⠀⠀⠀⠀⠢⠇⠀⠀⠀⠘⠜⠀⠀⠀⠀⠘⠌⠀⠀⠀⢀⣀⡠⠊⠀⠀⠙⠈⠀⠀⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠶⢀⣉⠀⠀⠀⠀⠀⢀⠀⠆⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠈⠐⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠨⠃⠀⠀⠀⠀⠀⠈⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠃⠘⠀⠁⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠻⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠽⠀⠀⠀⠀⣀⠱⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠤⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⢀⠀⠀⠀⡖⡄⠀⢀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⢺⠀⠀⠀⠀⠀⠀⣨⣷⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡷⣿⣿⣶⣿⣇⠠⠄⠀⠐⠂⠀⠤⠘⣿⣿⣿⣿⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠰⢙⣟⡻⠟⠁⠀⠀⣀⠀⢀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠄⠠⡙⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡔⣦⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠀⠠⠄⠀⠀⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⢀⡀⠄⢂⡨⢉⠆⣀⣀⣑⡀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢶⡄⠀⠀⠀⠀⠉⠁⠀⠒⠒⠒⠴⠖⠈⠉⣀⠔⢸⠃⠀⡸⠃⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠘⣄⠁⠠⠖⠛⠋⠭⣥⡒⠂⠁⠀⣀⠄⠊⠀⠐⠀⠀⡐⡠⢈⠄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠁⠄⠚⠀⠀⣀⠜⠀⠙⠀⠘⡀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⡀⠀⠀⠐⣀⣀⠐⠶⢀⠶⠤⡄⠉⠁⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠕⠂⠀⠉⠉⠀⠀⠀⠌⠀⠀⠀⣈⣵⠒⠂⢉⡀⠀⠀⠀⠀⠥⠃⠀⠀⠨⣥⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠔⠀⣠⠤⠠⠶⠀⠠⠤⠜⠃⠘⠂⠀⠀⠀⠰⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠂⢠⠞⠁⠠⣦⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡞⠁⡀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠘⠅⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠚⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠘⠢⠆⠀⠀⣀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⢠⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠤⠃⠀⢀⣀⠀⠀⠉⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢌⠆⠀⠀⠠⣌⠆⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢰⢲⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠓⣀⠒⠁⠀⠀⢀⡤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠠⠀⠀⢩⡀⠀⠠⣈⠆⠀⠀⡿⢀⠄⢈⠍⠁⠀⠀⠉⢨⠖⠢⠤⠿⣄⢀⣀⡀⢀⡴⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠉⠱⠶⣄⣀⠷⡤⠤⡤⠴⠤⠞⣷⠟⡓⠸⠀⠀⠀⠀⠀⢸⡇⢠⠀⡀⢸⢈⣥⢟⠘⡛⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣐⠁⣾⣠⣆⣴⣻⣷⡖⠷⠚⠢⡀⠀⠀⠀⢠⡿⠚⣿⡿⢿⣿⣷⣬⣥⣨⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠝⢻⣿⡟⣹⡟⢿⡋⣱⠀⠀⡰⠀⠠⠌⣠⠖⢦⣴⣿⢣⠗⣾⡛⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⣆⠈⣠⣿⣉⣥⡸⣷⠞⠓⡀⠀⠀⠀⠻⠋⠐⠉⠹⠀⠀⠈⠻⢧⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⣄⠉⠘⠀⠀⠑⠋⡈⠉⠙⠂⠀⠰⡇⠀⠀⠀⠈⠀⠀⠀⠀⠈⢳⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⣀⡀⠀⠸⠀⠀⠀⠈⡦⠀⣀⣀⣀⣀⡀⢤⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⠒⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣭⣭⣭⣭⣬⣭⣥⣬⣥⣽⣭⣤⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⣿⣶⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⢹⠟⣡⡆⢻⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢿⠿⠟⣛⣭⣶⣧⣶⣾⣿⣷⣶⣶⣶⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠀⠀⠀⢾⣿⣿⢫⣝⣿⡿⢿⣿⣋⡍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⢻⣿⣦⣿⣿⣿⡌⢿⣿⣵⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⣛⠻⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢦⠀⠀⠀⠈⢿⣿⢋⢸⢿⠘⡘⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡜⣎⢧⣅⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡐⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣦⠻⣿⣌⡻⣿⡿⢟⣠⣴⣆⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⠟⣤⣶⣿⣿⣿⣿⣌⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡁⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡇⣾⣇⣿⢸⠗⢯⣸⡯⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠈⠈⠉⠈⠉⠈⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⢦⣀⣀⣀⣀⣠⣄⣤⣀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣖⠋⠉⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⢿⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡴⠚⢻⣿⣿⡟⢒⡿⠷⣄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡞⠀⠀⠈⠻⣿⠟⠋⠀⠀⢸⡆⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢷⣷⡀⠠⣄⠀⠈⠀⢀⡀⢀⡼⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡼⠁⠈⠓⠮⣽⣓⣚⡭⠶⠿⠶⢿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣰⠁⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⢇⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣤⡀⠀⠀⣇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢞⠙⠉⠀⢷⠂⠀⢸⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠙⡄⠀⢸⡄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⢹⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⠾⠶⠛⠁⠀⢸⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣸⢢⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣠⠏⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⡼⣤⣀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣼⣀⠈⣇⣀⡀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⣄⡀⠉⢧⡀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠋⠀⠈⠳⡌⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⢻⠀⠈⣷⣄⣀⠀⢸⠀⣀⣠⡏⠀⡴⠛⢳⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠦⠞⠀⠀⡽⠋⠉⠉⠋⠉⠉⠻⣄⠈⢧⣀⠞⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠞⠁