Piolet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠿⠛⠛⠛⠿⠶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣤⢤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⡤⡴⠖⢫⡴⠖⢡⡿⠃
⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣡⠴⢦⣀⠜⢀⣠⢾⡁
⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⢛⣩⡴⠋⠙⠳⢦⣄⡀
⠀⠀⠘⢿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⢛⣋⣥⡴⠞⠋⠁⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⠶⢶⡞⢛⣋⣩⣥⣶⡶⢛⣋⣉⣀⣀⣀⣠⣄⣤⣤⣤⡶⠶⠾⢿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⡟⠉⠉⠉⠙⣿⠋⢉⣭⡉⠉⠉⠘⣿⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣆⠻⡾⠇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠨⢿⣷⣤⣤⣤⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢿⣿⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠙⠃⠀⠀⢀⣴⡾⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⢀⣠⠀⠀⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠋⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⠿⣷⣶⣶⣶⣶⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢤⡀⠙⠿⣶⣦⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣠⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠉⠉⠛⠛⠻⡟⠛⢿⡟⠛⣿⡟⠛⠛⠛⣿⡉⠉⣹⢃⡏⣿⡶⢤⣀
⠀⠀⠀⢀⣠⠴⠛⠁⣼⡇⠀⠀⠀⠀⠀⠙⣆⠀⠻⣾⠋⢻⡄⠀⣰⠏⣧⡼⠃⡼⠁⢸⣷⣄⡈⠙⠲⢤⣀
⢀⣠⠖⠋⣀⣤⠴⢺⣿⠁⠀⠀⠀⠀⠀⠀⢈⣧⠆⠈⢷⡄⢹⣶⡏⢀⡿⠁⢈⡷⠀⠀⣿⡍⠙⠳⢦⣄⠉⠳⣤
⢾⡁⠈⠙⡿⣤⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠙⣿⡀⢙⡿⢁⡴⠋⢀⠀⠀⢹⣷⣴⡞⠛⡅⣀⡤⠟
⠀⠉⠓⢶⣄⣰⢿⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣾⠟⠋⠀⠐⠤⠑⠀⠘⣿⣾⠳⠾⠛⠉
⠀⠀⠀⢠⣨⣁⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣷⣴⣶⣆
⠀⠀⠀⢾⣿⠿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⢀⣀⣀⣠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢿⡿⠿⠿⢿⣿⡿⠿⡿⠿⠿⠿⣿⡿⠿⠿⠟⢿⡟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣻⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⡀⠀⢀⣽⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣠⡿⢳⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⢡⠈⠉⢋⡉⠹⣆⠀⠀⠀⠀⠀⣰⠏⢠⣭⣀⣀⣸⣅⣀⣠⡴⠟⠁

⠀⢀⢴⠘⡉⠂⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠘⢄⠵⠒⠜⠨⠀⢀⣼⡠⡄⣠⠉⢆
⠀⠀⠸⡀⠀⢀⡀⠀⠁⣠⣍⣉⢀⡀⠈⠀⠁⠢⡀
⠀⠀⠀⢀⠏⠀⠹⣆⠾⣿⢛⡛⠳⠦⣅⡂⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡏⢑
⠀⠀⠀⠈⣯⠴⠊⢉⠗⠋⢁⡻⠀⠰⡄⠜⣍⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡇⢸
⠀⠀⠀⠀⠈⢀⠀⠈⣤⣀⠤⠁⠀⠀⠈⠠⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣠⣄⣉⡠⣀⠓⠒⠒⠒⠁⠀⠀⠳⠚⠐⠠⠤⠤⠄⠀⠠⠀⠀⠠⠃⠀⠀⠀⣆⣉⣙⠀⠀⠤
⠀⣠⣾⣿⣿⣿⡿⢈⣧⡁⠂⠀⠀⣀⠴⠚⠲⣄⠀⣀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠁
⡐⠑⡝⠟⠛⠿⠃⣸⡿⠿⢦⠖⠛⠀⠀⠀⠀⡘⠈⠛⠁⠘⠟⠀⠛⠁⢀⣀⠠⠤⣒⣯⠏
⠁⠀⠘⠀⡀⠀⠈⠣⣖⡠⠴⢦⡬⠤⣤⠶⢋⣀⡒⠀⠀⠀⠀⠀⠉⠉⠀⢂⡨⠕⠊⠁
⠐⢄⢠⢠⠧⠤⢄⣀⣈⠀⠀⢒⣀⠀⠄⠊⢁⣠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⢠⣀⣀
⠀⠀⠉⠛⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣊⣉⠉⣁⣀⠀⠀⠀⠀⢀⣀⣀⣠⠖⠭⠐⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠀⠐⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠶⠛⠛⠛⠛⠶⢶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠛⠋⠉⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢛⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⡛⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠉⠿⠿⢷⣤⣤⡾⠿⠿⠉⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠛⠻⠶⠶⠶⠶⠟⠛⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⣾⠛⠛⠙⢿⡄⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⡿⠋⠛⠛⣷⠶⣦⣤⣄⣀
⠀⠀⢀⣠⣴⣶⠛⠛⠉⠉⠀⠀⠀⢸⡇⠀⠀⠈⢿⡄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⡇⠀⠀⠀⠉⠉⠛⠛⣶⣦⣄⡀
⠀⣴⠟⠁⠀⢿⣄⣠⣤⡴⠶⠞⠛⠋⠁⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⠻⣆⠀⠀⣰⠟⠁⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠈⠙⠛⠳⠶⢦⣤⣄⣀⡿⠀⠈⠻⣦
⣼⠏⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠙⣷⣾⠋⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣠⡾⠋⠙⢷⣄⠀⠀⢀⣼⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡾⢿⡀⠀⠀⠀⣿⢷⣄⣾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⣧⣀⣀⣸⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡛⠛⡟⠛⣻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⣠⣤⣤⡀⠀⢀⣤⣤⣄⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⢿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣄⢀⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠙⠋⠉⠀⠀⠈⠉⠛⠉⢸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣰⢿⣿⣿⣿⣿⣿⢷⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣻⢿⣫⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣦⣾⣧⣀⣠⣦⣀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣦⣤⣧⣤⣾⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠈⠻⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢶⣾⣿⡏⠀⠀⠀⠀⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⣷⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⣋⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠚⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠉⠉⠁⠀⠀⠀⠀⢻⡿⠋⣤⣤⣴⣶⡾⠿⠛⢿⠉⠁
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠟⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢹⠛⠿⢷⣶⣤⣤⣤⣆⣀⡬⠀⠁⠀⠀⢤⡀⠀⣿⣿⠇⠀⠀⠐⠚⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠛⠛⠿⠷⣶⣤⡦⠉⢰⣿⣿⢿⣿⣤⣤⣤⣦⣴⣶⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⡙⠿⠶⣦⣌⣙⣛⡛⠿⣿⣿⣿⣉⣁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣤⣀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⢶⡷⠿⣷⣾⡿⠏⠉⠉⠛⠛⠻⠿⠿⠷⢶⣤⣔⣀⠉⢀⢀⠠⠐⠁⠀⣀⣴⣿⣿⣆⢀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣠⣶⣤⡶⣀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⡏⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⡃⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⢸⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠉⠉⣉⡉⠉⠉⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠐⠶⣿⣷⠖⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢿
⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⡄⠀⠀⢹
⠀⠚⡴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠢⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢠⣄⠀⡇⠀⢀⠐
⡇⡃⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⡀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⡆⠁⢀⠀⠀⠀⢡
⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠁⠈⠿⠃⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⡘
⠡⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⢠⣦⣤⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⢠⣤⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢤⠀⠀⠀⢠⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⣀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠤⠀⠐⠂⠀⠀⠀⠀⡀⠈⢂⠀⠀⠀⠀⠀⠠⠀⢀⡏⠑⢄⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡆⠀⠑⣤⡔⠁⠀⡜⠀⠀⢨⠀⠀⠀⠈
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣄⢾⣿⣿⠆⡰⠀⠀⢀⠂⠀⠀⠀⠀⠀⢂
⠀⠉⠀⠀⠀⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠠⢠⣿⣧⢀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⢄⠀⢻⣿⣿⠂⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠄⣿⣇⠔⠁⢠⣦⡀⠀⠍⠢⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠛⠛⠛⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⠀⠀⣠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⢀⠔⠁⠀⠀⠀⢀⠌
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣀⣠⣥⣤⣤⣀⣀⣀⣁⣀
⣿⣿⢋⢛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣤⣤⣤⠀⢀⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣿⣇⣼⡿⢱⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⣿⣿⣿⣿⠿⠛⠆⠄
⠀⣿⣿⣿⣿⣿⣿⢡⣿⣷⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⢻⣿⣿⣿⣯⡀⣴⡄⢀
⠀⣿⣿⣿⣿⡿⣣⣿⣿⣿⡀⡇⠈⠃⠀⠀⣀⣀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⡟⣿⡿⠀⠀⠃
⠀⣿⣿⣿⣿⡵⣛⣉⣉⣹⣿⣿⡐⣿⡿⠗⠋⠁⢠⠞⠛⠉⠉⠉⠛⠿⣿⠙⣧⠀⡤⠔⠀⠀⠨⠭⣍⣃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢧⢻⣋⡄⣠
⠀⣿⣿⠋⣿⣨⣭⣭⣭⣿⣿⣿⠃⣬⣁⢄⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠘⢧⣸⠀⠙⢶⣶⣿⣿⣿⣿⣿⡗⠦⡑⠄⠀⠀⠸⠇⠀⠀⠘⡆⠈⠚⠉
⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⣘⣿⣿⠀⣟⡟⠀⠀⠀⢷⡄⠀⠀⢀⠄⡘⡟⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⢠⢹⣷⡀⠀⠀⠀⠀⠀⠀⢱
⠀⣿⡿⢰⣿⣽⣿⣿⣿⡿⠋⠋⡝⠛⠛⠂⠀⠀⠀⠀⠀⠀⠹⢦⡀⠄⢾⡘⠀⠀⠀⠐⠚⢙⣍⠻⣿⣿⣿⠈⣃⢻⣧⠀⠀⠀⢠⡄⠀⠀⡇
⠀⣿⣿⣟⢿⣼⣧⣿⣿⣷⣤⣠⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠋⠛⢈⡂⠉⠀⡀⠀⣠⣾⣿⣿⣟⣿⣿⣿⠀⠿⠸⣿⠀⠀⠀⠘⠁⠀⠀⡇⠀⠀⠀⢀
⠀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠰⠁⢠⣟⡿⠈⠿⢿⣿⣿⣉⣶⡄⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸
⠀⣟⠛⠿⣿⡿⢿⣿⣾⣿⣿⣿⣿⢀⣶⣶⣄⢠⣀⣀⣀⡀⠀⢀⣼⣇⡀⠔⠀⢀⢏⣿⣷⡈⢐⣀⡟⢻⣿⣿⢀⣿⡟⠀⠀⠀⠀⠠⠇⠀⢠⠀⠀⠄⠂
⠀⣿⣷⣦⣤⣬⣥⣉⠺⠽⡿⣿⣿⣿⣿⡿⠿⢻⣄⣀⣀⠉⠉⠹⣿⡾⣄⣀⣴⣿⣿⠿⠃⠂⢬⠿⢷⡀⠉⠁⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣯⣿⣟⡄⢻⣯⠩⣛⠿⣿⣿⣿⣿⣿⣭⣟⣯⣾⣿⣶⣶⣶⠖⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢼⣿⣧⠹⡳⠘⠻⣿⣿⣿⣟⣯⠙⣿⣿⣿⣿⣿⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⣿⣿⡿⣯⣷⣿⣿⡿⠿⠛⢩⣻⣽⣿⣿⣿⣿⣷⣿⣿⣧⣙⠄⢳⠸⣿⣿⣿⣿⣷⣍⡽⠿⠿⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣿⣿⣿⣿⣿⣿⣿⣵⡀⠍⠉⡉⡐⣿⣿⣿⣿⣿⣿⣿⣿⣟⠳⡈⢷⢹⣿⣿⣿⣿⡉⣾⣧⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢀⠀⡊⢁⣿⣿⣿⣧⢿⡟⢻⣿⣿⡀⢰⡌⢹⣿⠟⠉⠉⠻⣿⣿⡫⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⡸⢠⡀
⠀⣿⣿⣿⣿⣾⣿⣿⣷⣿⠙⠈⢀⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⡇⡼⠀⡘⠁⠀⠀⠀⠀⠀⢻⣖⣈⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣾⠓⣿⣿⣇⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⣗⣒⣚⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡼⠀⠀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⢹⣿⣿⣿⣿⣿⣿⣿⠍⡐⡦⠄⠀⠀⠀⠀⠀⠀⢸⡗⠤⡬⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⢰⠃⠀⠀⡄⠀⣀
⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠙⢿⣿⣿⣿⣿⣏⢠⡄⢫⡆⠀⠀⠀⠀⠀⢀⣾⠬⣑⠂⠀⠀⠀⣠⣴⣶⣖⠿⣷⢦⠿⣖⣈⠄⠚⡿⠤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠏⠉⣿⣿⣾⣧⣨⡃⠀⠀⣀⣤⡶⢏⠢⡑⠂⠀⠀⠤⢸⣿⣽⣁⣀⠈⡀⣼⣾⣿⣻⣟⣓⢀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠸⣿⣿⣿⣯⣍⠒⠺⠿⣅⡐⣤⠓⠤⠔⠒⠚⠛⠛⠛⠛⠛⠛⠛⢺⡛⢿⣿⡴⠚⠿⢙⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠻⣿⣿⣿⣦⣠⠂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⡀⢣⡀⠹⣿⣯⣔⣿⠅
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠚⠁

⠀⠀⠀⠀⠀⢀⣔⢀⡤⠀⣀⣠⡤⠀⠒⠒⡒⢲⠖
⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⠿⠿⠶⣯⣿⣂⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣿⢉⠽⠟⠁⠀⣐⣛⠢⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⠀⠀⠀⠀⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠃⠀⠀⠀⡊⠀⠀⠠⠁
⠀⠀⠀⢀⡠⠄⠀⠠⠀⠈⠠⣄⠀⠀⠀⠄⠁
⠀⠈⢁⠠⠖⠃⠀⠀⠀⠠⡈⣿⣷⡶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠉⠛⠦⡤⠤⠄⠤⡄⠀⠀⠀⠀⠀⠀⠀⡂⣰⠢⢀⣀⣢⣥⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⡇⠀⠀⠀⠈⠀⠘⢖⠂⠀⠀⢀⢄⣲⣤⣭⣷⣶⣿⣿⣿⣿⣿⠿⠛⣿⣷⠶⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢳⣤⣰⣯⡙⣻⣿⣿⣿⣿⡿⠿⢟⣿⣶⠛⠋⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢰⠈⣯⣭⡍⠁⠩⣛⠛⠋⠉⠙⠦⠀⠁⢬⣡⠀⠆⢀⡘⠀⢀⠠
⠀⠀⠐⠀⠀⠀⠀⣀⡄⠀⣿⣿⡇⠀⠀⠒⠀⠒⣺⠀⢠⠀⠀⢀⠼⠀⠰⠄⠠⠤⠤⠤⠤⠼⠛⠖⠷⠾⠳⠐⠓⠂⠂
⠀⠀⠀⠀⠀⠀⠚⠁⡇⢰⣿⣿⣇⠀⠀⠀⠀⠀⣿⠀⠈⡄⠐
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢨⣭⣭⣽⠀⠀⠀⠀⠀⡿⠠⠂⠘⡀
⠀⠀⠈⠙⠂⠀⠐⠈⠀⢸⣿⣿⣿⡂⠀⠀⠀⣤⠆⠰⠚⠉⢷
⠀⠀⠀⠀⠀⣀⡤⠚⠀⠘⢿⣿⠟⠀⠀⠀⠀⣧⣀⣀⣠⣄⠈⠉⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⣀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⣿⣿⣿⠿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠋⠀⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠘⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡈⠉⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠈⠉⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠒⢻⣲⣤⡔⠋⠉⢰⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⢉⣡⠼⢆⠈⣯⣿⡁⢿⠀⠟⠓⠢⣬⣉⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠤⠛⠉⠁⠀⠀⠡⢸⣿⣿⣎⠂⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⢠⣤⣤⣤⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠈⠁⠈⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⣠⡏⢻⡀⠀⠀⠀⠀⡠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠙⢶⠤⠤⠴⠞⠁⠀⠀⠘⠛⠛⠛⠛⣧⠀⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠆⣸⠃⢸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡷⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣠⣄⠁⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣿⡖⠁⠰⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢠⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⡼⢷⣤⣤⣤⣴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠈⠛⢿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠏⠀⠀⠀⠀⠈⠛⠛⠛⠋
⣤⣤⣤⣤⣀⣤⣄⣠⣤⣀⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⢾⣳⣽⣯⣽⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⠁⠉⠁⠒⠠⢄⡀⠀⠀⠀⠀⢀⣀⡀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠐⠢⠄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠠⠄⣀⡉⠈⠉⠁⠀⠀⠀⠀⠘⠉⣉⣩⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠉⠁⠀⣀⠀⠈⠉⠉⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣤⣄⣀⣁⠀⠀⠀⠀⠀⢀⣛⣀⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠙⡀⠀⠀⡴⠊⠡⡄⠀⠼⡀⠀⠑⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⡀⠘⠃⠒⠛⠀⠤⠀⠐⠋⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⠀⠀⠀⠀⠀⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡚⢖⢫⠉⠈⠹⠀⠀⡼⠉⡏⠐⠂⢲⠛⠇⠀⠀⠀⠀⡠⢁⠜⢉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⠓⠓⠰⢲⠀⠀⠃⣠⠻⣶⣶⢎⠀⡜⠀⠀⠀⠐⠀⠑⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠈⠦⠄⣈⠳⠀⣿⣧⡰⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠇⠀⠐⢅⠀⠹⣿⣿⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠑⠂⠛⠃⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠐⠁⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⠐⠗⢸⠀⠉⠉⠉⠉⠑⠒⠀⠀⠀⢖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡄⠀⢀⠄⠐⠃⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠐⠀⠘⠀⠀⠀⠀⢰⠒⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⡴⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠀⠧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠄⠰⠀⠀⠀⠀⠀⠀⠰⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠷⢤⡀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢡⠀⣀⠀⡀⠈⠑⠊⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣀⠓⠀⠀⠀⠀⠀⠀⠀⡆⠸⠄⠜⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⢸⠁⠀⠀⠉⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⣠⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⢀⡠⠔⠒⠒⠉⠀⠀⠀⠀⠀⠀⠁⠒⠉⠉⠀⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⢅⣀⣀⡠⠤⢤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠠⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠿⢻⡟⢻⣶⣆⢢⠒⣦⢴⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣞⠟⠉⠀⠀⣠⡿⠱⠟⠛⢃⠆⠀⣿⣄⠐⠈⠉⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦⣿⡄⠸⢿⣶⣤⣴⣏⣀⣀⣤⠶⢋⡴⠻⢮⡛⢿⣤⣄⠁⠚⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡦⣿⣿⡟⠭⠋⠀⠙⠶⣤⣀⡉⣉⠉⠉⠁⣀⣴⠏⠀⠀⠈⠻⠄⣉⣙⢛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⠛⠉⠀⠀⠀⠀⠀⠀⣸⡿⠁⠈⠉⢉⣛⣉⣉⡀⠀⠀⠀⠀⠀⣀⣈⣉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⢀⠀⣉⠙⠓⠀⠀⢠⡟⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠲⣾⣿⡛⠻⣆⠀⠀⠀⣿⣿⡛⢳⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡄⠀⣿⠎⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢠⡸⠇⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣬⣟⡀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡈⢿⣷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠂⠁⡀⢻⠿⣿⣶⣤⣀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⢺⡄⠂⠘⠿⣿⣿⢿⣿⣤⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⠆⠀⠀⠈⠛⢠⣿⣿⠿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢸⡿⣃⠈⠙⠁
⠀⠀⠀⠀⠀⢀⣠⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⢃⡀
⠀⣀⣠⣤⣶⣿⡿⠟⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢀⡈⢁⠂⠀⠀⠀⠀⣿⣿⣷⣴⣶⣦
⡞⠉⠉⠉⠻⣆⠀⣀⣠⣼⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠁⣄⠀⢀⣾⣷⠘⣷⠀⠀⠀⠠⣿⣶⣤⣄⣸⣿⣧
⣿⣿⣿⣷⣦⡈⠙⠻⣿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠀⠀⢹⠆⣼⣿⣿⣌⣿⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣷⣤⠈⢻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⢽⣿⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠟⢿⣿⣿⣿⣿⡿⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⣠⣿⣿⣿⣿⣿⡘⣿⣧⣴⣿⣿⣿⣿⠻⠿⢿⡿⣿
⠀⠋⣸⡿⢹⡟⢠⣿⡗⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⡇⣿⣿⣿⣿⡿⠛⠃
⠀⢠⡿⠁⠛⠁⣸⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠇⠻⠻⠛⠛⠻⠻⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣻⣽⣷⣿⣿⡿⢿⣿⣿⣿⣯⣟⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠻⠟⠛⠛⠋⠉⠉⠙⠛⠛⠻⠟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⠒⠀⠀⠂⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⣄⠈⠑⠠⢀⡀⠄⠊⠁⡠⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣏⡀⢀⠀⠑⢄⢀⢊⡑⡀⡠⠊⠀⠀⢀⣹⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⠀⠘⡀⠀⠀⠁⣾⣷⠈⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣰⠀⠀⢀⣿⣿⡀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⣤⣤⣤⣤⠤⠄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⢻⣾⠛⠛⠉⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣻⣭⡴⣊⡥⠖⣁⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⣠⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣵⣞⣭⣶⣫⡵⣂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡂
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣛⣛⣛⣋⣉⣉⣉⣉⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⣿⣿⠿⠧
⠀⠀⠉⠉⠉⠉⠁⠀⠀⠿⡿⡄⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⢁⣀⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠠⣤⣤⣤⣴⣦⣄⣘⠻⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣌⣉⡙⠻⠿⣿⣿⣷⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣍⣙⠻⠿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣯⣝⡛⠻⠿⠿⠷⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠉⠉⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⡤⠒⢊⣉⣁⣀⣄⣈⣉⡙⠒⠤⡀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⡇⠀⠀⠉⠓⠂⠤⠤⠐⠚⠉⠀⠀⢸⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠤⠔⠒⠒⠒⠒⠒⠒⠢⠤⠴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣀⠑⠒⠤⠄⠒⠉⣀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢾⡀⠀⠙⠢⢄⡠⠔⠋⠀⢀⡷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠁⠸⡤⢸⠈⠢⣄⢀⣾⣷⡀⣠⠔⠁⡇⢤⠇⠈⠉⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⢠⠞⠀⠀⡆⠀⠈⠻⡃⢘⠟⠁⠀⢰⠁⠀⠳⡄⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⢢⠀⠀⢃⠀⠀⠀⡇⢸⠀⠀⠀⡘⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⡀⠀⢰⡁⢈⡆⡀⢀⡇⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠈⠁⠁⠉⠈⠀⠁⠉⠉⠉⠈⠁⠈⠉⠉⠉⠉⠉⠁⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠀⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠘⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠞⠒⠀⠀⠉⠉⠉⠉⠀⠐⠒⠀⠤⢈⠐⠢⢄⡀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⡉⢁⠋⢀⣈⠉⠁⠐⠂⠠⠤⢀⣀⣀⢀⣈⠑⠆⣘⠢⡀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⢉⣉⡁⠀⠀⠀⠀⠀⢋⣉⡁⠀⠑⢄⠀⠀⠑⢌⠦⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢠⢫⣭⣿⠄⠀⠀⠀⠐⢩⣽⣿⡆⠀⠘⡄⠔⠂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠉⠉⢠⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠎⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠢⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⣀⠤⠃⠀⠀⠀⠀⢀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⢢⡡⠐⡴⢹⠂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠉⠀⠐⠁⠸⠀⠀⠀⠀⠀⣈⠁⡴⠳⣀⡩⠝⢩⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠑⢀⠀⣀⠄⠞⠛⠀⠈⠴⠋⠣⠔⠀⠀⠘⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⠛⠀⠀⠀⣀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈⢀⠀⢀⡇⠈⠀⠈⠀⢠⠀⠀⠀⠃⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠔⠂⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢸⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠰⠰⣀⣀⠦⠤⠆⠦⠀⡏⠀⠃⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢲⡄⠀⠀⠀⠀⢸⢧⠈⠂⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡧⠤⠤⠤⢤⠇⠸⡤⠤⠠⢤⢴⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠰⠞⠂⠀⠀⡄⢠⡁⠀⠐⠳⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠤⠤⠀⠤⠄⣒⠨⠐⠁⠈⠀⠥⣐⠠⠤⠀⠀⠤⢰
⣀⣀⣀⠀⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⠉⠉⠛⠓⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠢⡤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠊⠉⠉⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠁⠛⠂⠤⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠚⢩⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠒⠢⣤⢀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣤⡴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⢛⠚⠛⠛⠋⠉⠉⠁⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣄⣀⠀⠀⠰⠶⠐⠊⢉⡂⠀⠐⢉⠕⣠⣤⠀⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣮⣹⣿⣿⣿⡷⢶⣶⣤⣤⣤⣝⣶⣶⣡⣤⣤⣴⣶⢾⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠹⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡿⠟⠁⣸
⠀⠀⠀⠀⠀⠀⠀⡾⡄⠀⠀⠈⠻⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠣⢄⠀⠀⠀⠀⠈⡉⠿⡟⠛⠛⠛⢻⠋⠛⠡⢄⠀⠀⠀⣠⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⡂⠀⠀⢠⠟⠀⠀⢹⠖⠒⠈⠛⠤⣀⣀⣠⠟⠀⢠⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠙⣄⠀⠘⠓⠒⠚⠓⠤⠤⠤⠤⠔⠋⠁⠀⠀⢠⠏⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠛⠛⠉
⠀⠀⢀⣀⣀⢤⡤⠤⠤⠀⢤⡠⠤⠤⢍⡵⢦⣄⣀⣀⣀⣀⣠⡾⠭⠤⣠⡤⠤⠤⠀⣤⠄⢠⡤⢠⡄
⠀⠀⠈⢇⠹⣎⢳⡄⠀⠀⣾⠀⠀⢠⡟⠀⢸⠀⠀⠀⠀⣩⠀⠱⠀⠀⢰⡆⠀⢀⡼⢃⡴⠋⢀⡞
⠀⠀⠀⢈⠷⠼⠧⠿⠦⡤⢿⠀⠀⢸⡃⠀⡜⢿⣿⣷⡶⡇⠀⢰⠀⠀⣼⠓⠒⠛⠓⠛⠒⠒⠙⡇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⣟⠸⣆⣀⠀⠱⠾⡀⢈⣿⣯⠁⡸⣦⠂⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢸⣅
⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⡀⠈⠉⣁⠗⠀⢳⣸⣿⣿⣦⠃⠲⢂⠉⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⣧⠀⠀⠙⢦⡀⠀⢻⣿⣿⠇⠀⣠⠞⠁⠀⠀⣠⠻⡄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠉⠓⠤⢿⡯⠤⠊⠁⠀⠀⠀⣰⠃⠀⠙⢄⡀⠀⠀⢀⠴⠛⢉⡱⣄
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣼⢹⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⢺⡅⠀⠀⠀⠀⠙⢦⡴⠃⡠⠊⠀⡠⠜⠓⠒⢢⡀⢀⣀⡀
⠀⠀⠀⡜⠄⣀⠀⠀⠀⢰⠇⡘⠂⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠙⠾⣀⡴⠋⠀⢀⠀⠀⣀⢿⡬⢄⠙⣆
⠀⠀⢸⠓⠄⠀⠈⠉⢉⡟⣰⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣤⣀⣉⣱⠈⣦⡽⠈⣄⢹⡄
⠀⠀⣠⠓⠦⢄⣀⣠⠞⠰⣏⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠟⠦⣬⠆⠀⢸⡆⢻⡀
⠀⣾⠁⠀⠀⠀⢀⣀⡑⡄⣼⠓⠒⠤⠤⠤⠤⠾⠤⣄⣀⣀⣀⣀⣀⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢹⠀⠀⠀⠀⠀⠀⣻⡀⣧⣀⡀
⠀⠈⠦⣄⠀⡀⠹⣄⠙⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⡆⠀⢀⡴⠋⠀⠉⠁⠀⢈⡝⠳⡀
⠀⠀⠀⠀⠉⠉⠑⠚⠂⠀⣿⠀⠀⠀⠀⠀⠉⣿⠙⣏⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢹⡴⠋⠀⠀⠀⠀⠀⠐⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢰⠃⠀⢻⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡎⠀⠀⠸⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢱⠀⢸⡏⠉⠉⠉⢳⡀⠀⠀⠀⡀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠻⣇⠀⢷⠀⠀⠀⠀⢣⠀⢀⡄⠙⡅⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⢸⡄⠘⡗⢤⡀⠀⠘⡄⢸⣤⠜⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣷⠀⢱⠀⠈⠳⢤⡟⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⡆⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠘⢳⣤⡀⠀⠀⢀⡀⣀⣿⠀⠀⠀⠀⠀⠳⣤⣄⠀⢀⣀⠀⣀⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⢀⡞⠁⠀⠈⠉⠉⠉⠉⡟⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠉⠉⠉⠁⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡴⠃⣾⠃
⠀⠀⢀⠴⠋⠀⠀⠀⢀⣠⢤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠤⠛⠲⠦⠤⠤⠤⠤⠤⠗⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡀⢀⣠⡞⠋⠛⠁
⠀⢰⣅⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠒⠲⡎⠉⠉⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡤⠾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⡠⠀⢀⠀⡀⠀⠀⠀⠄⠀⠁⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢖⡀⠊⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⡆⠀⠀⢈⡻⠓⣦⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣥⣀⠀⠈⢳⡀⠀⠀⠀⠀⠒⠀⢟⠒⠒⠛⢦⣘⢛⣰⠛⠳⠆⠻⣷⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⣣⡠⠄⠷⠀⠀⠀⠀⢤⡀⠀⠀⢀⣀⢀⢈⣩⠞⣥⠘⢿⣦⣾⣶⣬⠙
⠀⠀⠀⠀⠀⠀⣸⠇⢀⡠⠒⢪⠀⠃⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠻⠼⠨⠇⢋⠁⠿⠷⠞⠋⠙⠻⠏⠀⠀⠀⡖⠒
⠀⠀⠀⠀⠀⠀⡟⠰⡁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⠀⠀⠈⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢀⠎⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠊⠀⠀⠀⠈⠁⢀⠠⡇⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠱⣀⠎⢀⠞⠀⠀⠉⠛⠛⠷⠶⠦⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢶⡭⠶⣏⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠷⠶⣤⣬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣤⣤⣤⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣴⣿⣿⡿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡀
⠀⠀⢰⡏⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠋⠁⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠉⠙⠋⠙⢿
⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣄⠀⠀⣠⣼⣿⡇
⠀⠀⠀⡼⢿⡄⢀⣿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢟⠋⠁⠀⠉⢛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠾
⠀⠀⠀⢧⠈⠛⠙⠣⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⡤⠴⠋
⠀⠀⠀⠈⠷⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠼⠀⠀⠀⠁⠀⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⡌⠄⠀⠀⠀⠀⠀⠌⡄⠀⠀⠀⠀⠀⠀⠄⠠⣧⣤⣤⣤⣤⣤⣴⠄
⠀⠀⠏⠀⠀⠀⠂⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⢠⠂⣀⠀⠀⠀⠀⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⣿⣿⣿
⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡀⣿⣿⣿
⠀⠀⢀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿
⠀⠀⣠⣿⣿⣿⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣿⣿⣿⣀
⠀⠘⠛⠛⠛⠛⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠛⠛⠛⠛

⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀
⠀⠰⣤⣤⣤⣤⣤⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠛⠉⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠛⠹⣿⣧
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⢀⣾⠟
⠀⠀⠀⣤⣄⠀⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢀⣄⡀
⠀⠀⢀⣿⣈⣈⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣶⡟⠓
⠀⠀⢸⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢤
⡀⠀⣿⣽⣿⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⡰⠀⡀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠘
⠁⡈⣿⣿⣿⠛⣿⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠂⠀⠀⠡⠉⠁⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠀⠀⠀⠏
⠈⠀⣿⣿⣿⠀⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⣿⣿⣿⢠⣿⣿⣿
⠀⠀⣿⣿⣿⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⡿⢉⣭⣭⣴⣶⣤⣭⢹⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⠀⠀⣗⢸⣿⣿⠿⢿⣿⣿⠘⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⠿⠿⠿⠿⢛⣩⣴⣌⠻⢿⡇⠀⠀⠀⠀⣿⣬⣭⣴⣶⣶⣬⣭⣴⡇⠀⠀⣿⠿⠿⠿⠿⢛⣩⣴⣜⠻⢿⡇
⠀⠀⠀⣠⡇⢶⣶⣿⣿⣿⣿⣿⣿⣿⡆⣹⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⡇⢶⣶⣿⣿⣿⣿⣿⣿⣿⡆⣱⡀
⠀⣴⣾⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣷⣄⠀⠀⠀⢉⣙⣛⣛⣉⠁⠀⠀⢠⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣷
⠘⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⠟⣡⣿⣿⣿⣿⣿⠀⣠⡄⢾⣿⣿⣿⣿⡆⣠⣀⠀⠙⠿⣿⡜⢿⣿⣿⣿⣿⠟⣱⣿⠿⠋
⠀⠹⣿⣿⣿⣿⣿⠆⣬⣭⣭⣅⢸⣿⣿⣿⣿⣿⢃⡜⢿⣿⡈⢿⣿⣿⡟⣰⣿⠟⣰⣶⣤⡀⠉⠀⣬⣭⣭⣄⠈⠉
⠀⠀⠙⠟⣩⡀⣶⡘⢿⣿⣿⡿⣠⣶⢨⡙⠿⢃⣿⣿⣶⢆⣵⡌⣿⡟⣰⣇⢲⣾⣿⠟⣩⡀⣶⡘⢿⣿⣿⡟⣠⣶⢠⡀
⠀⠀⠀⢸⣿⣧⣙⠻⣎⠻⠏⣴⠛⣋⣼⣿⡆⣿⡿⣿⣿⣌⠻⣿⡈⣰⡿⢋⣼⣿⡟⢸⣿⣧⣙⢻⣎⠻⢏⣴⠛⣃⣼⣿⡄
⠀⠀⠀⣾⣿⡏⣩⡘⢿⣷⣾⠟⣃⣏⢻⣿⡇⢹⡇⣿⣿⣿⣷⣌⠻⢋⣴⣿⣿⣿⡇⣿⣿⡏⣭⡘⢿⣷⣾⠟⣠⡍⢿⣿⡇
⠀⠀⠀⣿⣿⡀⣿⣿⣷⡌⣡⣾⣿⡿⢸⣿⣿⢸⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠃⣿⣿⡐⣿⣿⣷⡌⣡⣾⣿⡿⢸⣿⣷
⠀⠀⠀⣿⣿⡗⢨⣿⣿⡇⣿⣿⣯⡄⣿⣿⣿⠸⠇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢰⣿⣿⡇⢨⣿⣿⡇⣿⣿⣯⡄⣿⣿⣿
⠀⠀⢸⣿⣿⡇⣾⣿⣿⡇⣿⣿⣿⡇⢻⣿⣿⡀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡇⣾⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠘⠛⠛⠃⠛⠛⠛⠃⠛⠛⠛⠓⠘⠛⠛⠃⠀⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋⠘⠛⠛⠃⠛⠛⠛⠃⠛⠛⠛⠃⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢹⡦⡏⠉⣾⠉⢰⡇⢸⢸⣯⠇⣾⣧⢹⡏⢸⣷⡏⢹⡦⡿⣼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣿⣷⣼⣿⣦⣿⣿⣼⣧⣼⣧⣼⣾⣧⣼⣧⣿⣿⣴⣧⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⢀⣀⠤⠤⠄⠀⠤⠤⢀⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⢠⠊⠁⠰⢶⣒⣒⢑⣒⡶⠆⠈⠑⡄⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢺⡤⠤⠄⠐⠒⠚⠓⠒⠒⠠⠤⢤⡗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠤⠤⠤⠤⠤⠤⢀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⣈⠉⠒⠂⠉⣀⠔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡶⠙⡧⣄⠀⠉⢢⡖⠉⠀⣠⢜⠋⢶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠤⠒⠉⠁⠀⠀⡹⠂⠃⠈⠳⢤⡟⢻⡤⠞⠁⠘⠐⢏⠀⠀⠈⠉⠒⠤⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠺⡀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⢀⠗⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡆⠀⠀⡇⢸⠀⠀⢰⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠈⠁⠘⠃⠀⠙⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⢤⡤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠂⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠍⢁⡀⠀⢀⣀⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠉⠉⠉⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⣀⡀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⢀⠀⠰⢄⡀⡀⠴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⡀⢀⡴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠇⡀⣀⠸⠂⠄
⠀⠀⠀⠀⠀⠔⠁⠀⠘⠈⠁⠁⢒⠀⠀⠑⡀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠳⠞⢠⠤⠤⠈⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠂⠙⠋⠐⢀⣀⠐⠄
⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⠀⠀⠀⠀⠀⠃⠄⡀⠀⠀⠀⠀⠈⠀⢀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠘⠃⠀⠘⢉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠉⣂⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠠⢈⠈⠁⠈⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠈⠁⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⣖⠔⠀⠀⠀⠐⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠄⠀⠀⠀⡊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢀⠝⠀⠐⠈⠁⠐⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢈⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠊⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⢇⢠⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡀⠘⠃⢀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠀⠤⡰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⣀⣀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⡠⠀⢚⢂⡤⣄⢚⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⠃⡠⣤⠘⡒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡗⢠⡄⢺⠒⠢⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠈⠀⠓⠂⠉⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠅⠐⠂⠨⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠡⠌⠈⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠁⠀⠤⢀⣀⡀⣀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⣐⡄⢰⣀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⣀⣐⣂⣀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⢤⡤⠀⠁⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡠⠂⠉⠈⠁⠢⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠤⠤⠤⠤⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢉⡉⠉⠁⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠠⠀⠠⠐⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠂⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢙⢞⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⣀⣀⠔
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠪⠀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⠧⣀⣀⠸⠲⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠙⠋⠉⠂⠠⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡐⠁⠀⠕⠈⡕⢊⠁⠪⣀⣈⠂⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢴⡦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠑⠀⢀⠈⡄⠃⠸⢀⢃⡀⠄⠊⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠘⠃⠀⢀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠂
⠒⡀⠀⠂⢄⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀
⠀⠈⠂⡀⠀⠀⠀⠀⠀⣘⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠖⠫⡀⢠⠀⡄⠀⢁⣢⡄
⠀⠀⠀⠀⠦⣄⣀⡀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢉⠋⠉⠉⠑⢒⡤⢄⠫⢠⠀⠀⠠⠊⣀⣀⠀⠈⠆⠰⠖⡨⠐⠀⠈⢢
⠀⠀⠀⠀⠐⢄⡀⣀⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠠⣀⢀⡎⠀⠀⠹⣄⡁⠀⢡⣾⣿⣿⣿⣿⣶⣶⣷⣶⣶⣤⡀⠀⢃
⠀⠀⠀⠐⠢⠁⠀⠤⠄⠐⢟⠠⠄⠂⠀⠀⠐⠒⠀⠤⢀⠀⠀⠀⠈⠑⠲⢶⠒⠊⠅⠒⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠖⠣⠈⢤⣀⣀⠂⢀⠘⡸⣿⣿⣿⣿⠛⠛⣿⣿⣿⠿⣿⣿⠃⠂
⠀⠀⠀⠀⠚⠠⠄⠀⠀⠤⣀⡀⣀⡠⠤⠔⠂⠛⠢⡀⠉⡃⠐⠇⢉⣙⠒⠤⢈⣤⣊⡰⢻⣿⠛⠋⢁⡖⠉⠙⠿⢿⣶⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠘⣶⣀⠀⠐⠊⠀⠀⠀⠀⠀⢡⣤⠀⡹⠀⠀⠀⠠⣴⢦⡼⠀⠛⡆⠀⣘⠏⡿⠾⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⢨⡇⠀⠀⠀⠀⠀⠀⠐⢼⡁⠞⠁⠈⠒⠀⠅⡣⠎⠁⣠⣬⡱⣦⡴⠿⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣀⣀⠎⠁⣰⠎⡩⡧⠺⠉⣶⠀⠠⠂⠂⢢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⡏⠀⡞⠀⣖⠲⢷⣧⢴⣏⠀⠁⠐⠒⠙⢁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢗⠐⠧⣐⣿⣤⣿⠁⠀⠘⠇⢀⠢⠄⠎⢹⠡⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢘⢃⠎⠁⠀⢈⡉⡂⠀⠀⠀⠩⡀⠄⢤⠠⠂⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣨⡄⠀⠀⠼⣤⣀⡀⠀⠀⡈⠀⠀⠐⠄⠒⠈⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⡀⠀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠸⡤⠐⠋⠹⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠑⢄⢀⠔⠁⠀⠀⠀⡞⠀⢀⠈⢢⠐⠈⠉⠉⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢄⡀⠈⠑⢌⢠⠄⠀⠀⠀⠈⠰⢄⠀⠀⠀⠀⠀⠣⡠⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⣠⠞⢢⡀⠌⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣴⠋⢀⠌⣔⡒⠒⠋⠀⣠⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣆⣶⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣛⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠚⣿⡿⢿⡿⢟⠛⠛⣻⢟⣿⣇⠀⣠⣿⣟⣿⠻⠿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠐⠻⣿⣿⠻⠂⠀⢠⣿⣶⣿⣿⣿⣿⣿⣯⣿⣥⡀⠀⠈⣻⣿⣿⡟⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⣿⣧⠀⠀⣴⣿⣿⢿⣯⣽⣉⣩⣭⢻⢷⣿⣿⣿⣷⣿⣿⣿⣧⣤⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠘⢿⡁⢊⣿⣿⣽⣿⣛⣛⣛⣉⣛⣿⣶⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⡇⠀⠀⠀⢳⠀⠀⠻⢿⣿⣋⡉⠉⢉⣉⣹⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠸⣇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⣄⡀⠀⠛⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣭⣽⣿⣿⣿⣿⣿⠋⣿⣯⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣧⠈⠻⣷⣬⣾⣷⣶⣼⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⠉⡂⣿⣿⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⢰⣌⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣾⣿⣿⣿⠿⠻⠿⠿⠿⠿⠛⣻⣿⠿⢿⣛⣿⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠿⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠻⠛

⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⢤⡀⡠⢘⣶⡃⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⢠⣤⣶⣴⣦⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠑⠊⠂⠃⣿⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣯⣇⠀⠀⢸⣿⣿⣿⣿⣷⣯⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⣀⣀⡀⣿⠀⠱⠇⢈⠆⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣷⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡍⠀⢏⣀⣀⡀⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⢸⣿⣾⣿⣿⣭⣽⡇
⠀⠀⠀⠀⠀⣠⣴⣾⡿⡿⣿⣶⣤⡀⠀⠀⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡻⡿⣷⣦⣄⡀⢠⣿⣿⣷⢿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣶⣾⣿⣷⣶⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣿⡇⢰⣿⣿⣟⣿⣿⣽⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⡇⣿⣯⣭⣭⣾⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣻⣿⣿⡇⠈⠀⠉⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠻⠛⠟⠛⠾⠻⠟⠻⠃⠀⠀⠀⠈⠻⠻⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢺⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣭⣭⣽⡆⠀⠀⢸⣟⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⣶⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣵⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣯⡇⠀⠀⠀⢿⣯⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⠀⠀⠠⣄⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡇⠀⠀⣤⣤⣽⣿⣯⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣧⣀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⡀⠀⠀⠀⣿⣬⢿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣧⣧⣿⣿⣿⡆⢰⡇⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣭⣿⣿⣷⡆⠀⣿⣷⣿⡶⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢶⣶⢾⡿⣿⣷⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣯⣻⣽⣟⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⡿⣿⡇⢿⣿⠏⠁⢿⣿⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣾⣿⣿⣿⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠐⠢⡄
⠀⠀⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠐⠄⡙⠂⠤⢀⠄⢐⢀⠰⣄⠀⠀⠀⣼⡇⠀⠀⠀⣼⡇⠀⠀⠀⣠⠆⠀⡂⠠⡀⠤⠐⢋⠠⠂
⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠬⠕⣒⣦⠀⠀⢿⣿⠲⠤⠂⢹⠃⠀⢀⣴⣒⠪⠤⠀⠐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠠⣀⠉⡀⠈⠂⠉⠀⣼⢹⠀⠀⠀⢸⡇⠀⠀⠈⠑⢀⡉⣀⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠥⠀⠠⠄⠀⠀⠀⣿⢸⡅⠀⠀⢸⠁⠀⠀⠀⠀⠄⠀⠬⣀

⠀⠀⠀⠀⠀⠀⢠⡄⠈⡀⠀⠀⠀⢀⠄⠒⠔⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢀⢶⣶⣶⣶⡖⢁⣾⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠉⠁⠀⠀⠀⠑⠒⢦⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠱⡤⠒⠄⠴⣱⠞⠁⠀⠀⠀⠀⠀⠀⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡞⠁⠀⠀⠀⢠⡀⠀⠂⢠⠀⡀⠈⠑⠒⠢⢄⠠⠤⢤⠔⠚⠃
⠀⠀⠀⢀⣤⣴⣎⠉⢳⡞⣦⡀⠀⠀⠀⠈⠀⣜⠀⠀⠀⠜⠁⠀⠀⠐⠀⠀⠠⠄
⠀⠀⠀⠈⢶⣾⣟⡛⢚⣿⣩⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠙⠚⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢖⡐⢤⢀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠁⠀⠀⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⠊⠐⣤⢏⣀⣞⡂⠀⠐⢿⣿⠿⠷⠂⠀⠀⠾⠟⠁⠀⠀⠐⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠈⢜⠷⠞⠚⠒⠓⠺⢭⣨⡅⠀⠀⠸⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠕⠐⠀⠆⡌⠀⠀⠀⠀⠀⠿⢿⣶⣿⣶⣶
⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⢦⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡂⠠⢀⡀⠠⠄⠀⠠⣔⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⡂⠠⠶⠂⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠉⠀⠦⠀⠀⠠⠌⠃
⠀⠀⠀⠈⠂⢀⠈⠁⠀⠀⠠⣁⠖⡀⡡⢀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠄⠙⢁⡦⠀⠀⠀⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⣤⣤⠷⢶⡶⣄⣨⡵
⠀⠀⠀⠀⢠⠄⢄⠀⠀⠀⠀⠀⠀⣴⣦⡴⣶⡶⡆⠀⠀⠀⠀⣠⠄⠒⠂⠉⠉⠻⣶⡀⠀⣼⡟⠀⠀⠀⠀⢻⡏
⠀⠠⠢⢀⡈⣆⠈⠖⠐⠀⠀⠀⣴⣾⣿⣿⣾⣿⣿⣤⠀⠀⠸⣯⡀⠀⠀⠀⢀⢸⡟⠀⠀⠙⣧⠀⠀⠀⣠⣾⠇
⠀⠀⠉⠄⠘⠀⢆⠓⠀⠀⠀⠀⠈⠁⠈⠓⠋⠋⠙⠛⠧⠀⠀⠈⠻⠶⠚⠑⠺⠛⠁⠀⠀⠀⠈⠛⠛⢷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⡄⠀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⣖⣚⠑⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⣠⠤⠐⠒⠒⠤⣄⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⢄⡀⣀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⠒⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡞⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠑⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡄⢀⡀
⠀⠀⠀⠀⠀⣰⠖⢆⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠊⠠⣐⡄⠠⠄⠀⠀⠀⠀⠀⠈⡀⠬⠖⠦⢄⡀⠀⠀⠀⠀⠰⠥⢤
⠀⡤⠒⠲⠎⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠑⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⠢⡄
⠘⢦⢀⡤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱
⠀⠀⠈⠀⠀⠉⠉⠋⠹⣁⣀⠉⠈⠉⠁⢀⣀⠉⠛⠀⠛⠉⢁⠘⠒⠐⠂⠒⠁⠉⠘⠛⠋⡛⠒⠻⠭⠭⠤⠔⠒⠙⠒⠂⠐⠒⠛⢯⣲⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⢽⣉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠃⠀⠀⢄⢀⡀⣀⠀⠀⠀⢀⡠⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠚⡩⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠚⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢼⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗
⠀⠒⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠊⠁⠀⠀⠈⠢⠤⠀⠀⡀⠉⠉⠋⠉⠉⠁⠈⠉⠁⠈⠉⠒⠒⠒⠉⠙⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡤⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠠⡔⠒⢄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⡤⢄⣀⣀⢄⣀⣀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣒⣢⣤⣔⣒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡾⣿⣿⡿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣄⠈⠁⠉⢉⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣛⣋⣉⣉⣉⣛⣛⡛⠛⠻⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⢟⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⠟⠛⢛⣛⣛⣛⡛⠛⠛⠛⠛⠛⣛⣛⡛⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠀⢠⣶⣾⣿⣷⣦⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡞⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⣿⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⠤⠄⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⡒⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣤⣤⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣶⣤⣤⣤⣤⣤⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠤⠶⢤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠳⢦⠶⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⡆⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠈⢷⣀⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣀⣴⠟⡉⠉⠙⣏⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠛⠉⢹⡜⠟⠀⢀⡿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣹⡶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡈⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠳⠄⠐⣿⣀⣀⣴⣿⡿⠿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⡀⠀⠈⠙⠛⢩⡏⠷⣄⣈⣉⠉⠉⣿⣿⠛⣛⠒⠛⠛⠛⠉⣠⠊⠁⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠘⣷⣄⣀⣹⠀⠀⠀⡟⠻⣤⡾⠀⠀⠀⠀⠀⢷⢫⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠷⠖⠚⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⢶⡇⡇
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣶⡶⠶⠶⠶⠤⣤⣶⠶⠶⢒⣛⡿⠛⣿⠶⣤⣄⣀⣀⣀⣼⣷⣿⣷⠆
⠶⣤⣬⣷⣶⣄⣤⣤⡤⠶⠟⠋⣿⠀⠀⠉⠛⠒⣶⢾⡉⣉⣷⡛⠉⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⢻⢻⠿⠇
⠲⠶⣚⡿⠛⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣁⣰⠏⢻⣄⣻⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣸
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠀⠤⢸⠈⠉⠀⠀⠀⠀⠀⢻⠀⠀⣏⢄⠤⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣤⣤⣄⣤⣤⣼⣤⣤⣤⣤⣤⡤⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣥⣉⡉⢙⡦⠀⢰⣋⣥⣄⣀⣀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠀⠀⠘⠒⢲⣀⣀⠀⠀⠀⠀⢰⠉⠉⠑⠂⠐⡄⠀⠀⠀⠀⣀⣀⣰⠋⠀⠀⠈⢀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡀⠀⠀⢫⠀⠀⠀⠀⠀⠀⣰⠀⠀⡸⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠒⠐⢄⡀
⠀⠀⠀⠀⠀⠑⠤⠤⠤⠶⣤⣤⣤⣴⠴⠦⠤⠴⠞⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠀⠀⠉⠛⠒⠒⠲⠶⠶⠶⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠚⠉⠁⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⡀⠀⠀⠀⠀⠀⢀⡄⠒⠁⠀⠀⠀⠀⠀⠀⠉⣠⠵
⠀⠀⠀⠀⠀⠀⡀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢄⠀⠀⠀⠀⠙⠒⠒⠠⠔⠒⠢⠤⠚⠉⠉
⠀⠀⠀⠀⠀⠈⠒⠀⠠⠤⠤⢤⣀⣀⣀⣀⣀⡤⠤⠤⠤⠟⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢤⠋⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣦⣀
⠀⠀⠀⠀⠀⠀⢠⠴⠚⠀⠀⠀⠀⠹⠒⢦⠖⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⢀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠘⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠐⠁⠀⠀⠀⠈⠒⠂⠀⠤⠤⠤⢄⠀⠀⠀⢀⣤⡤⣄⠀⠀⠀⣠⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠔⠒⠚⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡍⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢸⣿⣿⣿⣿⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣃⡈⢸⣿⡿⣸⣿⣿⣿⣿⡸⣿⣿⢈⣀⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣷⣙⣃⣯⢉⡇⣿⣿⣿⣿⣿⡇⣏⢩⣇⣛⣱⣿⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣇⢿⢇⣿⣿⠿⠿⠿⠟⠘⠃⣿⣿⣿⣿⣿⡇⠛⠘⠿⠿⠿⣿⣿⣇⢿⢇⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⢿⣿⣿⢇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⡸⣿⡿⣸⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠻⠃⠈⠉⠉⠉⠁⠉⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣆⠀⠀⠀⠀⢀⣀⣤⣶⣶⡷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣿⠿⠿⣿⣷⡶⢶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⣰⡿⠋⣀⡀⠀⠀⠙⣿⡄⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⣄⡀
⠀⣀⣀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⡇⢸⣭⣿⡆⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣤⣀
⣾⣿⠋⠀⢀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣧⡀⠙⠉⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄
⢻⣿⣦⣴⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠙⠛⠛⠛⠛⢁⣿⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⣠⣾⣧⣀⠀⠀⠀⠀⠀⣼⡟⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⢠⣾⡿⠋⠉⠛⠷⠀⠀⠀⢰⣿⠀⣸⠻⣿⣄⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣰⠇⠀⢈⠿⢷⣄⡀⠀⠀⠀⠈⠉⠙⠛⠛⠻⣿⣿⢶⣶⣶⣶⣦⣤⣤⣴⣶⣶⣶⠾⠟⠋
⠀⠀⠀⠘⢿⣷⣤⣠⣴⣶⠀⠀⠀⣿⣧⠏⠀⢀⡞⠀⠀⣹⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠙⣿⡿⠛⠁⠀⠀⠀⣿⡟⠀⢀⡞⠀⠀⣰⠏⠀⢠⡟⠙⠛⠃⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠸⣿⣀⡾⠁⠀⢠⡏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣄⠀⠀⠙⣿⣇⡀⢠⡟⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡄⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⣦⡀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣜⠛⠛⢹⣷⣟⣱⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡟⡀⣠⣿⣿⣯⣭⡵⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⣠⠖⠉⢳⣼⣿⣤⡀⠀⠀⠀⠀⠀⢀⣼⡿⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡏⠀⠉⢳⡏⠀⠙⣧⣠⠄⢸⡇⠸⣿⠻⣷⣤⣀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⢀⣤⣾⡏⠀⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⡏⢱⠀⠉⢉⡷⢤⣴⠳⣄⠀⢸⡇⠀⣿⡆⠀⠉⠻⢿⣋⣀⣀⣀⣀⠀⠀⠀⣿⠉⠛⠒⠒⠚⠛⠙⡇
⠀⠀⠀⢀⣴⡿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⢈⣧⡴⠋⠀⠀⢸⡀⠈⠳⢾⡇⠀⢻⣿⠀⠀⢰⣏⣁⡀⠀⢀⣉⣿⡀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣴⡿⠋⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠟⠁⠀⠀⠰⠖⢸⡇⠀⠀⠀⣷⠀⢺⢹⡇⠀⡟⠀⠉⠉⠛⠉⠉⠈⣇⠈⣷⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠸⡿⡻⠋⠉⠉⠁⣢⣄⣀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⢸⠘⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠳⣦⣤⣤⣤⠾⠋
⠀⣰⢁⡇⠀⠀⠀⢠⡏⠉⠙⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠶⠀⣿⠀⠀⠀⢸⡇⢸⠀⣿⡄⢳⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠙⠈⠁
⠀⡇⢸⠁⠀⠀⠀⢸⣇⣠⣴⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⣷⣀⣀⣀⣤⣤⣤⣿⣤⣤⣤⣼⣧⣾⣦⣿⣧⣤⣽⣷⣦⣴⣶⣏⣁
⢸⠀⣼⠀⠀⠀⠀⢸⠨⣿⠿⠋⢀⣀⣤⡤⠶⠶⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉⠉⠙⠛⠛⠲⠶⢴⡄⣄⡀
⢸⡀⣿⠀⠀⠀⠀⢸⣆⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡥⣯⣉⡛⢶⣄⡀
⠀⠉⠳⠦⠤⢤⣤⢼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⢮⣁⣈⣉⣁⣀⣤⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢴⣗⡶⠄⠤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣌⠛⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⣤⣤⡤⠶⠋⠀⠀⠀⣀⣀⣤⣴⣶⠿⠟⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣌⡉⠛⠛⠿⠿⠿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠿⠛⠛⢉⣁⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⣶⣦⣤⣤⣄⣀⣈⣉⡉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣉⣉⣁⣀⣠⣤⣤⣴⣶⠶⠿⠟⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⢉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠤⢄⣰⠁⠀⣠⠞⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠲⢄⡀⠙⢦⡾⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠇⢴⣤⣍⣉⠙⠉⠓⠦⣝⡲⣄⡀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠷⠶⠚⠛⣿⠟⠛⠛⢿⡟⠻⠶⣯⣙⠛⠷⣶⡤⠞
⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⢸⡇⠀⠰⣿⡆⣿⠀⠀⠀⠙⢷⡄⠀⠘⠉⢆
⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣴⣿⠤⠄⠀⠀⠈⢿⡄⠀⠈⡃
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⡲⠁
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣦⣤⡀⠀⠀⠀⣿⠉⠁
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⡾⠹⡆⠀⠀⠠⠾⠛⢿⣦
⠀⠉⠛⠲⠶⠶⠶⠶⠶⠶⠾⣿⠛⠋⠉⠁⠀⠀⣀⡴⠋⠀⢀⣧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣠⠶⡆⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⢻⡋⠁⠀⠀⣠⣾⡇⠀⠀⠈⠻⣶⡟⠃⠀⠀⠀⠀⢸⠇⠀⡇⠀⢸⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⡶⠶⠛⠉⡼⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣄⠀⠙⠛⠋⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠲⠶⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢳⡆⢠⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠲⣦⣤⣴⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡿⠉⠙⣿⠛⢶⠶⣶⡶⠶⣶⡶⢶⠶⡶⠶⠟⢻⣇⠀⠀⠀⠀⠀⠀⠸⣶⡿⠀⡏
⠀⠀⠀⠀⣠⡴⠞⠋⠀⣼⠃⠀⢀⠏⠀⠘⡆⢠⠞⡟⠣⣄⡟⠀⢳⡀⠀⠈⣿⡶⣤⣄⣀⠀⠀⢠⣾⣿⣾⣿⡄
⠀⠀⠀⠀⠈⠙⠷⣤⣰⡟⠒⠚⠁⠀⠀⠀⠙⣇⡀⡇⣀⠞⠀⠀⠀⠳⢄⣀⣸⣇⠀⠉⠉⠛⠛⠻⢿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣀⣼⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣁⣀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⣀⣼⠁⣽
⠀⠀⠀⠀⠀⠛⠛⠁⣿⠉⠉⠉⠉⠉⠁⠀⡏⠔⠒⠒⢢⢹⠒⠒⠒⠒⠒⠒⠒⢻⡆⠀⠀⠀⣼⢋⡴⠒⢮⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡀⠀⠀⠀⠀⣇⢧⠤⠤⠼⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡘⢦⣤⠾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⡿⠛⠛⠛⠛⠛⢻⡟⠃⠀⠀⠀⠀⠙⠲⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⢲⣶⠒⠚⠀⠀⠀⠀⠀⠛⠛⡿⣞⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠛⠓⠚⠉⣇⠀⠀⠀⡴⠚⠛⠛⠛⠁⠸⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣛⣶⣶⣶⣖⣻⣦⣞⣀⣀⣀⣳⣶⣶⣖⣛⣹⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠁⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠾⣿⣿⣿⠛⢻⡿⣿⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢸⣿⡏⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⣿⣿⠀⡴⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠉⠈⠻⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⣷⠆⠀⠀⠠⠄⠉⠉⠉⠉⠛⠻⣿⠇⠀⠉⠁⠠⠐⠀⠈⢿⡘⣿⣷⡀
⠛⠛⢿⡿⠿⠯⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⠀⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡥⠀⠀⠀⠀⠀⢰⣶⣶⣤⣰⣶⠏⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠈⠙⢛⣿⣿⡿⠀⠀⠴⠀⣴⣿⠇⠀⠘⠀⠀⠀⠀⢠⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠧⠰⢠⣤⣾⣿⡟⡶⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡇⠀⠀⠀⠀⢀⠠⠀⣸⣿⣿⡿⠶⢀⣿⣿⣿⠟⡜⠃⠀⠀⠀⠀⢰⠀⠘⡇⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⣿⣿⣀⠀⠀⠀⢿⢿⣿⣿⣿⣿⠃⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠷⠀⠀⠀⠈⠋
⣿⡀⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢷⣿⣿⣿⡿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢰⠀⠀⠀⠀⣀⣀⠀⣀
⣟⠄⣤⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢻⡏⠉⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⢀⣿
⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣼⣿
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠴⢬⣶⣾⣿⣿
⣿⣿⣿⣷⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⢿⠀⢀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠋⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣾⣿⣿⣿⡆⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆
⠀⠂⠀⡆⠀⠐⠀⠰⠂⠀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠃⠤⢄⡑⠀⠀⠄
⠀⡆⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠐⠂⠠⠤⠄⠀⠀⠀⢰⡁⠠⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠁⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⢴⠦⠤⠐⠒⠂⢂⠀⠀⠀⠀⠀⠀⠈
⠀⠁⢰⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⡀⠀⠈⠁⠀⠘⠒⠊
⡀⠀⢸⠀⠀⠀⠸⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⡇⠢⠜⠐⢄⠔⠀⠴⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸
⢇⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀
⠀⠉⠐⠆⠀⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣀⣀⡀⠀⢣⠠⣄⢀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡇⠀⣿⠉⢆⣤⡴⣻⠀⣇⠀⠘⢦⡀
⠀⠀⠀⢧⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡇⠀⡏⠀⠁⠌⠁⠿⠀⢠⠠⡄⠈⠉⠒⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡷⡇⢠⠇⠀⠀⢐⠀⠘⡀⢸⠄⡛⢠⡀⠀⠀⡙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠐⠰⡊⣨⠃⢸⡃⠀⠀⣨⢀⠀⡇⢸⠂⡇⠀⠙⠖⠁⠀⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⠊⠉⠀⠀⣡⡀⠐⠁⣿⠀⠸⠂⠀⠀⠸⠐⠀⡇⢸⠀⣃⠀⠀⠢⣠⡜⠁
⠀⠀⠀⣀⡀⠚⡀⠡⠐⠊⠀⠀⠀⣀⠠⠚⠁⠀⠀⠰⠙⠀⠀⠀⠀⠀⠨⠈⠀⡇⢘⠤⢸⠀⠀⢠⠟⠒⢀
⠀⠀⡀⠑⠀⠶⠁⠀⠆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡤⡄⠀⡀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠈⡇⠠⠇⠀⢀⠂
⠀⠀⣍⠀⠀⣒⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠧⠀⢀⢠⡅⢀⢘⡃⠀⣇⢀⡧⠀⢀⠔⠁
⠀⠀⠢⠈⣉⣭⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠈⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⠀⢸⣀⣀⣀⣀⣀⢠⠅
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡸⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠡⠐⠛⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠂⠀⠀⠀⢀⡇⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠶⣠⡄⠀⠀⢤⣀⡀⠴⢤⡀⠉⢈⣡⠀⠀⠁⠘⠷⣀⠄⠀⠄⢐⠡
⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠋⠉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⠛⠛⠋⠉⠈⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠛⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⣾⣿⡇⢸⣿⣼⡟⣾⣿⠗⣴⣿⣿⡆⣿⡿⠷⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⣿⣿⢸⣿⣿⡇⣿⣿⠆⣿⣿⣿⡇⣿⡿⠶⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢻⣿⣼⣿⣿⣷⣿⣿⣶⠻⣿⣿⠇⣿⡇⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠻⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣭⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⣿⣿⣼⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠄⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡿⣿⣿⣿⣿⣿⣽⢿⠿⠛⠁⠀⠻⠿⠟⣣⣼⣾⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⢾⣿⣿⣾⣿⣧⣤⣤⣀⣠⣤⡤⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⡿⠛⠉⠁⠸⣯⣿⠧⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠉⠉⠁⠈⠹⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠈⠿⣮⣭⣵⠟⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⣀⣤⣄⢤⣥⠄⣤⣤⡀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣫⣽⠏⠀⣼⣿⣿⣿⣷⣶⣾⣿⣿⣿⡆⠈⢿⣵⣶⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣬⣥⢄⣤⣄⠻⣿⣿⠿⣿⢻⣿⢿⣿⠟⣣⣤⣄⢤⣥⠄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣿⣿⠿⢗⣛⣭⠗⣫⣭⡓⣯⣝⣚⠿⢿⣿⣷⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⣷⣾⣿⡿⠿⣿⢸⣿⣿⣿⢸⡿⠿⣿⣷⣶⡏⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡿⢂⣭⣐⠾⢿⣿⣷⣽⣻⣵⣿⣿⣿⣖⣨⣅⠺⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣤⣛⠿⢿⣿⣷⣶⣭⡛⠛⣫⣵⣾⣿⣿⠿⠟⣣⣼⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠁⢸⣷⣮⣭⣛⣫⣾⢸⣮⣛⣫⣿⣶⣿⡇⠉⠉⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠘⠟⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠋⠉⠉⠉⠉⠉⠛⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⣀⡀⠤⠤⠤⠤⠔⠒⠲⠶⠤⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠈⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠻⡄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡴⠇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣾⣿⣌⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏⠀⠈⠻⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣾⣿⠟⠉⠉⠚⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⢁⠀⠀⠠⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⣶⣿⡿⠁⠀⠀⠀⠀⠀⠈⠻⣿⡿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠋⡶⣶⡄⠀⠀⠀⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⢩⠁⠠⠊⠀⠀⠀⠀⠀⠀⠈⠙⠶⣮⣭⣉⣉⡉⠉⢠⡧⠤⠐⠛⢻⡇⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢶⡇⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⡀⠀⠈⠻⣦⡉⠉⠉⠙⢻⡇⠀⠀⢸⡇⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⢀⠀⠀⠀⠀⠀⡴⢋⠋⡁⠀⠀⠈⢿⡆⠀⢰⡾⠃⣀⣤⣾⣇⣤⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠊⠀⡀⠀⠀⠀⠀⢀⣇⠈⡀⠀⠀⠄⠀⢸⡇⠀⣈⣿⡯⠭⠭⣭⠉⣠⣞⣁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠉⠉⠁⠀⠀⠀⠀⢈⣻⣿⠋⠛⠛⠀⠀⢰⡗⠊⠁⢻⣧⣤⣦⣾⣾⣟⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠉⠒⢄⠀⠀⠀⠀⠘⠁⠁⢸⡆⠀⠀⠐⢤⠟⣵⣤⣴⣿⣿⣉⣉⡙⢫⡉⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠂⠱⡄⠄⠀⠀⣀⣠⡞⠀⠀⠀⠀⠀⣟⣿⣏⣠⣵⣿⠟⣿⢻⡏⠀⠀⠀⠀⠈⣧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣐⣄⠀⠀⠀⠀⠀⢡⣠⣶⣿⠿⠿⣿⣧⠤⠴⢖⣪⠽⣿⡿⢟⡿⠛⢠⡟⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣕⡠⠄⢀⣠⣾⡿⢉⡷⣄⡀⠈⣷⣟⡩⠭⠔⣊⣿⡖⠋⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠉⢁⡿⢄⣨⠁⠀⠉⠉⡽⣚⣗⣚⣫⠵⣚⣇⣀⠴⠃⡎⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠉⠛⠲⠦⠤⠤⠽⢷⠵⢖⣛⠵⣺⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⢿⡭⠶⢛⣩⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣯⡭⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠻⠿⣿⣿⣿⣿⣶⣶⣬⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠄⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠉⠉⠛⠃⠀⠀⠀⠘⠋⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⡆⠀⠰⠀⠀⠀⠀⣶⡶⣦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠸⢿⣿⠏⠀⠀⠀⠀⠀⠁⠿⣿⠿⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⢀⣴⠾⠛⠉⠙⣷⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⣰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠙⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠴⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⣷⣄⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⠋⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠈⣿⣿⣦⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣡⣾⣷⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⣀⣤⣿⣿⣿⣿⣇⣤
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠏⠀⠀⠀⠘⠛⠛⠿⣿⡿⠟⠀⣨⠁
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⡆⢹⣿⣷⡀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠘⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⢠⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣿⣷⣶⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠼⢟⣀⣤⠜⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠁⠀⠀⠘⠛⠁⠀⠀⠉⠱⠶⠦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡠⠴⠶⢀⣀⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠋⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⢦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡰⠟⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⢇⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡸⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⣀⠀⣀⡀⢀⣠⣤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠒⠛⢲⠿⣷⠒⢿⡇⠀⠀⠀⠀⢸⠟⠐⣿⠿⡖⠚⠓⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⡴⠏⠀⢻⣄⠈⢳⣄⠀⠀⣰⠇⠀⣰⡟⠀⠹⣦⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⢣⡄⠀⠈⢻⡀⠀⢉⡷⣾⠁⠀⢀⡟⠁⠀⢠⡌⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⢇⣤⡟⠀⠈⣷⣄⡸⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠹⣇⡀⠀⠸⠏⣹⠗⠛⡁⠸⠇⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⠀⢸⣏⠀⠀⣿⡇⠀⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠙⢦⡀⣿⠆⣸⡟⢠⡼⠋⠀⠀⠀⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡾⠋⣀⣠⣿⣆⣀⠀⠋⠛⢦⡀⠀⠀⠀⠀⠀⠈⠱⣆
⠀⣴⠏⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠉⠻⠿⠟⠋⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠘⣧
⣾⠁⠀⠀⠀⠀⢶⣴⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⠀⠀⢻⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠙⢧⡴⠀⠀⠀⠀⠈
⠙⣦⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠟⠃⠀⢻⠀⠀⠀⠻⠋⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⣰
⠀⠘⢧⡄⠀⠀⠀⠀⠈⠳⣆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢈⣷⣀⡇⠀⠀⠀⠀⢰⣶⠄⠀⣰⠀⠀⠀⢾⠆⠀⠀⠀⠀⢸⣃⣴⡋⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠙⢦⡀⢀⣴⡿⠋⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⠙⢿⣦⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⣠⣾⠟⢉⣽⣿⣷⡶⢶⣶⡆⠀⢺⠀⠀⢠⣶⣶⠲⣶⣷⣾⡉⠻⣷⣄⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡴⠻⣾⠟⣷⣟⣿⠛⠁⢀⣾⣄⡀⠈⠛⢿⣻⣬⠻⣷⠟⢦⣬⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣴⣦⣜⣤⣿⣁⠰⠶⠋⠀⠈⠑⠶⠦⢈⣿⣤⣇⣴⣆⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⡦⢴⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠈⠛⠓⠂⠚⠛⠻⡇⠀⠀⠀⠀⢸⡟⠛⠀⠐⠛⠛⠁⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⢠⡼⠃⠀⠀⠀⠀⠈⢳⡄⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠈⠙⠣⣄
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣠⡞⠃⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇
⣀⣤⣠⣄⣉⣵⣶⣶⣶⣦⣶⣬⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣀⣤⣤⣈⣩⣴⣶⣶⣶⣶⣶⣍⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣿⣶⣿⣶⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠦⣄⣀⣠⠴⠒⠶⠶⠒⠋⠳⠤⣀⣀⣀⣠⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠡⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠀⠀⣈⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠀⠈⡆⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢀⣤⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠁⠀⠀⡁⠐⠐⢦⡀⠀⠀⠁⢢
⠀⠀⠀⠀⠀⢀⢠⠖⠃⠀⠤⠀⢅⣄⠀⠀⠀⠐⠂⠀⠚⠲⣄⠀⢀⣤⠀⣀⣀⣄⣀⣀⣀⣰⠤⡀⠓⢀⡀
⠀⠀⠐⠒⠓⠒⠒⠲⠢⠴⠚⠒⠒⠒⠤⢤⣀⣤⣉⡤⠤⠤⠤⠭⠡⠑⣟⡠⠠⠀⠀⠀⠀⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠐⠚⠂⠠⢀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠓⡀⡀⠀⠀⠀⠂⠁⠈⠉⠉⠉⠉⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡤⠀⠐⠀⠀⠀⠀⠀⠀⣀⠠⠦⠀⠃⠀⠀⠑⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠀⠀⠀⡴⡄⠃⠰⠄⠀⠀⡧⠀⢀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢐⠀⠘⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠄⠀⠂⠀⢄⠀⠀⣀⣀⣀⠀⠀⠤⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠰⠊⢀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠀⠀⠀⠀⠀⠁⠎⠉⠂⠄⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣐⢪⠕⢚⣉⢩⡆
⠀⠀⠀⠀⢠⠛⠛⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢀⡆⠠⠄⠒⠀⢁⠡⠤⠒⠁⣀⡠⠃
⠀⠀⠀⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⡀⠀⠀⠀⠂⣁⢉⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂⢠⠇⠀⠈⢠⠂⠀⠈⠀⠀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⠀⠀⢐⡁⠀⠀⠀⠶⠒⡂⠀⠌⠀⠀⠀⠀⠐⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⡴⠠⢀⠀⠠⠐⠺⠀⠀⠀⢀⠀⠀⢀⠌
⠀⠀⠀⠈⠀⡠⠀⠠⡀⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠈⠀⠀⠀⣐
⠀⠀⠀⠀⠨⠀⢀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡔⠀⠀⠐⠂⠨⠁⠀⠀⠀⠀⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⢄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⢨⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣀
⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠐⢀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡠⠤⢀⠀⠅⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠓⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠀⡀⠀⡀⠀⠀⠁⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠂⠒⠚⠇⠁⢒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡠⠔⠒⠂⠀⠀⠀⠄
⠀⠠⡈⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠀⠉⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠄⢀
⠀⠀⣁⠀⠂⢀⠐⠐⠒⡅⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⠒⠀⠀⠄⠐⠈⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠈
⠀⠐⠀⠀⠉⠀⠐⠂⠩⠆⠀⠂⠀⠀⠀⢀⣖⠠⠀⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠉⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⠀⠀⡀⠀⢀⠶⢶⠦⣤⣤⣀⣀⠀⠀⠀⠀⠐⢁⠄⠠⡈⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠀⡀
⠀⠀⠀⢄⣀⠀⠀⢀⠀⢀⡀⢀⡀⡀⢀⠀⣀⠙⠦⡁⠈⠀⢀⡿⠀⠀⠀⠀⠀⠉⠉⢻⡗⠀⠀⠀⠘⡀⡠⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢰
⠀⠂⠕⠚⠃⠀⠀⠛⠂⠘⠃⠘⠁⠛⠛⠂⠁⠀⠀⠝⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢰⡉⣷⠀⠀⠀⠀⢸⢼⠆⠀⢈
⠀⠑⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⠏⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠀⠀⠈⠈⠀⠀⢠
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠉⠀⢶⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠝
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠂⠉⠉⠁⠉⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠛⠉⠉⠉⠉⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠒⠒⠒⠒⠒⠋⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣀⣰⡬⠥⠶⠚⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠈⢉⡳⠒⠒⢒⣛⣻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⢀⣴⠞⠻⣝⠿⠋⠉⡿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⡾⠽⠇⣠⠃⠉⠐⠉⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠙⠀⠖⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡶⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢷⡀⠀⢀⠀⠀⠀⠙⠛⠻⣿⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠈⠛⠞⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢡⠇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣸⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠖⠒⠒⡶⠒⢶⢲⢶⡖⢲⠒⠒⠒⣿⡄⠀⠀⠀⠀⢻⡛⠒⠒⠀⡏⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⠀⠀⣡⡀⢸⡏⢲⡇⣸⠀⠀⢠⢏⡷⣄⠀⠀⠀⠀⢻⡄⠀⠀⢧⣀⠀⠀⠸
⠀⠀⠀⠀⠀⢠⡾⠁⠀⣿⠀⠀⠈⢣⡀⠀⣇⡼⠀⢠⠏⡴⢋⣾⠀⠈⠳⢦⣄⡀⠸⣿⡿⠇⢠⠋⠓⢦⣄
⠀⠀⠀⠀⢠⡿⠁⠀⠀⣿⠀⠀⠀⠀⠙⣄⢹⠁⣠⡧⠚⡠⠋⣿⠀⠀⠀⠀⠉⠙⠛⢹⡇⢀⠏⠀⢀⣤⠾
⠀⠀⢀⣤⣾⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢨⣿⠖⢋⡴⠊⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⡦⠶⠛⠉
⢀⡆⢺⣿⣿⡟⠳⠿⠋⢻⡄⣀⣀⠤⠒⣉⣤⡞⡁⢀⣀⠀⠤⣿
⣼⠀⠸⠟⢿⠿⡆⠀⠀⣸⣿⣉⡤⠖⠛⠛⢲⠖⠓⠒⠂⠈⠉⣿
⡏⠑⠒⠒⢺⣀⡗⠒⠉⠁⢸⠷⠾⢿⡿⠿⠾⠷⠶⠶⡶⠶⠶⠿
⢧⣄⣀⣀⣀⣉⣀⣀⣠⣤⠼⠀⢀⢸⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠈⠉⠉⠉⠁⢀⡖⠋⠙⡉⠉⣙⢷⠀⠀⣰⣛⠉⠉⠙⠛⠲⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠆⡀⢠⣔⠀⠀⠑⣶⣿⣿⡿⠋⠁⠀⡀
⠀⠤⠀⠀⠀⠀⢉⣉⣉⣩⣈⣈⣉⣉⡉⣩⣉⣉⠉⢉⣽⡟⠁⣨⣧⣥⠀⠀⠀⣀⣴⣿⣿⠟⢓⣦⠂⠘⣅⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⣿⡟⠛⠛⠟⠻⣛⠿⠿⣻⣷⣷⣾⡿⠃⠀⢀⣤⣾⣿⣿⡿⣫⣿⡿⣳⣤⣾⣯⡍⠉⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⡠⠆⠁⡀⠐⠁⢰⡏⢉⣀⣴⣾⠿⠛⠉⠉⠉⠙⠁⠀⠈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⡊⠀⠀⠈⠀⣠⣴⣿⡿⠟⠋⠁
⠀⠀⠀⠈⠉⠉⠙⠛⠛⠒⣲⠒⠋⣀⣀⠠⠤⠜⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠛⠛⠛⠛⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠞⠛⠋⠉⢁⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⡈⠉⠙⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⣿⣽⣟⠁⠈⣿⣿⣯⣯⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠛⠛⠻⠶⠶⠟⠛⠛⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠛⠷⠶⠶⠾⠛⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠛⢿⡏⠉⠈⢷⡄⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⢠⡾⠁⠉⢙⡿⠛⠶⢶⣤⣤⣀⡀
⠀⢀⣤⠶⢻⡏⠉⠁⠀⢀⣀⣀⣼⡇⠀⠀⠈⢻⣄⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⡟⠁⠀⠀⢸⣧⣀⣀⠀⠀⠈⠉⢹⡟⠶⣤⡀
⢠⡿⠁⠀⠀⢿⡶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠘⢷⣄⣠⡾⠃⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⡿⠀⠀⠈⢷
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⣠⡿⢿⣄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⣠⣾⡋⠀⠀⠙⣷⣄⢀⣼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠸⣇⠀⠀⢸⡏⠙⠿⠃
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠾⠁⠀⠀⢳⡤⣤⠀⠀⢠⡞⠁⠀⠀⢛⡆⠀⠀⢀⣠⣤⣤⡾⠋⠀⠀⠀⠉⢳⣀⣀⡀
⠀⠀⠀⠀⠀⣰⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠉⠉⠛⠉⠉⠁⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢧
⠀⠀⠀⠀⠀⠹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⠃⠀⠀⠀⠀⠀⣠⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠞⢦⣤⡤⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠛⢦⡄⠀⢹⣆⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣀⡀⢀⠀⠀⢛⡆⠀⠙⠳⠶⠴⠞⠦⣄⣀⣀⣹⣄⠀⠀⠀⠀⢀⡰⠦⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣤⣀⡤⠤⣄⡈⠉⠓⠚⠛⠋⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠛⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⠀⠀⠛⠀⠀⠀⢷⣀⠀⠀⠀⠀⣴⠒⠏⠉⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣄
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢳⡄⠀⠰⣏⣠⠀⠀⠀⣁⣹⠇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⠀⠸⠦⣄
⠀⠀⠀⠀⠀⠘⢦⣄⣤⠀⠀⠀⢠⡄⠀⢀⣄⣠⡼⠁⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⣠⣤⣀⠀⠀⠀⠀⠹⢦⣄⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠞⠛⠒⠋⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠿⠀⠀⠀⠈⠛⣧⠀⠀⠀⠘⢧⣄⠀⣄⠀⣲⡼⠇⠀⠀⢀⣤⠶⢦⣤⠤⣤
⠀⠀⠀⠀⠀⣠⠟⠉⠁⠉⢳⠀⠀⠀⠀⠈⠷⣤⣤⣤⣤⣤⡤⠼⠃⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⡞⠉⠁⠀⠀⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠙⠚⠦⠶⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡀⠀⠀⣀⠀⣠⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠀⠇⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢴⡞⠉⠀⠈⠻⣤⠤⣤⡀⠀⢷⣀⡆⠀⠀⡀⠀⠀⠀⢀⣸⠇⠀⢀⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠇⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠈⠳⣤⣤⠷⣤⣤⡴⠋⢀⣀⣴⠋⠀⠀⠀⠀⠙⣧⠴⠶⣤⡀
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠻⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠹⣧⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣴⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣄⠀⠀⣀⣀⠀⣀⡼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣀⢀⣴⠖⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢄
⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡅⠀⠀⠀⠀⠀⠀⠳⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠜⠠⣶⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⢶⡟⠁⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠈⠢⢤⠔⠁
⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁

⠀⠀⠐⠀⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⠉⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠉⡆
⠀⠀⠐⠒⢂⢀⠄⠒⠒⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⣘⠓⡒⠷⣄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣠⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⣤
⠀⠀⠀⢸⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⠆⠀⠀⠀⢸⡇⠀⢀⣶⠇⢸
⠀⠀⠀⠘⣷⣤⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠓⠀⠀⠀⠀⠀⠘⠛⠛⣷⣶⣶⡦⠀⠀⠀⠀⢸⡇⣴⠋⠀⠀⢸
⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣭⡄⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⣦⡀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠰⣤⡆⠀⠀⠀⠀⠀⣰⣆
⠀⠠⠶⠀⠀⠀⠀⠀⠀⡶⠄⠀⠀⢀⣴⣿⣧⣤⣼⣿⣦⡀⠀⠀⠀⢸⠀⠀⢸⠆⠀⠀⣠⡟⠀⠀⠀⠀⠀⢠⣿⠘⡄
⠀⠀⠀⠀⠂⡀⢀⠐⠀⠀⠀⠀⠀⠀⣿⡋⢹⣿⠉⠙⣿⠀⠀⠀⠀⠈⣿⣶⡟⣀⣤⠾⠋⠀⠀⠀⠀⠀⢠⣿⣿⡀⠸⡀
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠿⠷⠾⠿⠀⠘⠿⠀⠀⠀⠀⠀⠘⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠈⠑⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣶⣴⣶⣦⣤⣤⣤⣤⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣦⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣽⣿⣿⡆⣻⣿⡿⣫⣽⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣧⣼⢿⣿⣽⡷⢿⣿⡚⣯⣽⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⣍⣤⣶⣿⣿⡇⢸⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢹⣿⡋⠀⠈⢿⣿⠋⠙⢿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢻⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣟⠻⠶⢾⣿⡷⠀⠀⣼⣿⣿⣿⣿⡿⠃
⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣴⣿⣿⠧⠤⢾⣿⣧⣤⣴⣿⣿⣿⣿⠿⠃
⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣁⣀⣨⡭⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒⠒⢲⣿⣿⣿⣿⣿⣿⡟
⠟⠋⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣄⣤⣄⡀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣤⣀⡀
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⣿⣟⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣽⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣾⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⣿⣿⣿⣟⣿⣿⣟⣟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣇⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡴⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣼⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⡟⢿⣿⣿⣿⣿⣿⣿⣷⢻⡽⣿⣻⣿⣿⣿⣿⣿⣿⡿⢁⣾⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣇⢻⣜⣿⣿⣿⣿⣿⣿⠟⣰⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢻⣮⣿⣿⣿⢉⢗⣴⣿⡿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣄⠀⠀⠀⠀⠾⣿⠱⠿⠿⠿⠿⠿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣻⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠿⠛⢿⣿⣿⠿⠛⢻⣿⠛⠋⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠛⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⡀⠄
⠀⠀⣀⠀⠀⠂⢠⡔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠀⡀⢤
⠀⠈⢹⠀⠸⡇⠘⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢲⡘⠀⠞
⠀⠐⠀⠀⠀⠤⠄⣠⡁⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⡄⠈⠋⠠⠖⠐⢀⠖
⠀⠀⢣⠀⠀⠀⠀⠀⠁⡰⠶⠾⠀⠀⠀⠀⠀⠰⣶⠀⣀⠀⠐⣀⠀⠶⠄⠠⢆⠀⠀⠀⠀⠈⠓⢦⠀⢥⠀⠀⠀⢀⠻⢦⡀
⠀⠀⠀⠓⣤⡀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⠀⠁⠀⠀⠞⠁⠈⣃⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⡜⠁⠒⠛
⠀⠀⠀⠀⠀⠹⠂⠸⡄⠀⠀⠀⠀⠀⠀⢀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣐⠆⠀⠀⠀⢱⠄⢀⡠⠴⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⠀⣘⡷⠄⠚⠓⠂⠘⠛⠒⠤⣀⠀⠀⠀⠀⢞⠁⠀⠀⠀⡰⠃⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⡄⠘⣧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⢄⠀⠀⠈⢓⠀⢀⡞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢃⡀⢘⣄⣠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠆⠀⠀⣸⣄⠎⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡞⠉⠉⢹⠇⠀⠀⠀⠀⠀⠊⠁⠙⠀⠀⣀⡤⠚⡁⠀⠀⠸⠀⠀⣿⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⣤⣤⠤⠤⠤⠔⠒⠋⠁⣠⣼⡇⠀⠀⠐⠓⢶⣟⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⠀⢻⣿⣄⣀⣤⣤⠤⠚⠛⢻⡟⠀⠀⠀⢦⡤⠎⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠀⠀⠀⠈⢿⣿⡿⠿⠇⠀⠀⣸⠋⠀⠀⠀⢠⠟⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⠀⠙⠓⠀⠴⠖⠊⠀⠀⠀⢀⣴⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠢⠤⠤⣄⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠰⠒⠒⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢣⡀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⠤⠒⣇⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄⠀⠀⠉⠉⢹⠉⠉⠚⠟⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠀⠀⠀⣧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣳⠀⠀⠀⠀⣠⠜⠁⠀⢠⡄
⠀⠀⠀⣀⠀⠀⢀⣤⠔⠊⠁⠀⠀⠀⢻⣀⠠⠔⠊⠁⠀⠀⠀⢸
⠀⠀⢘⠁⠄⢀⣈⣃⣀⣀⠀⢀⡠⠛⠊⠉⠣⠀⠀⠀⠀⡀⠁⢰
⠀⠀⠈⠁⠀⠤⠤⠤⠤⠤⠀⠿⣈⠀⠄⠀⠀⠰⠀⠤⠄⣛⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡐⢤⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢁⣀⣱⣮⡀⢸⡀⠀⠀⠀⠠⠝⠿⣄⠁⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡁⠀⠀⠈⢁⡨⠿⠯⠀⠀⠀⠀⠀⠀⠀⠛⣇⣀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠟⠃⠀⠀⠀⠻⠀⠀⠀⠈⢈⡧⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠐⣀⣠⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⣤⣦⡀⠀⢀⣄⠀⠀⣸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣶⣶⣤⣶⣾⣿⣶⣶⣦⣀⣠⣴⣿⣿⣷⣶⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠿⠛⠋⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⡀⠄⠀⠀⠘⠓⠲⣾⣿⣏⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠒⠇⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠀⢼⣿⣿⠟⠉⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢈⣿⣇⣀⣀⣠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⢶⣿⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⡀⣤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠒⠂⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠒⠂⠀⠐⠶⠂
⠀⠀⠀⠀⢀⣾⣿⣿⣼⣿⣷⣾⣿⣶⣶⣦⣤⣦⣤⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣦⣶⣤⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀
⢀⣀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠤⠤
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠆⠠⠄
⠀⠀⣸⢹⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄
⠀⢰⠇⠘⠛⢿⣿⣿⣿⣿⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⡌⠀⠀⠀⣸⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⢘⠀⠀⠀⢀⡿⠹⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠗
⡞⠀⡠⠶⣾⠷⠶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣏⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠙⠃
⠁⣠⣄⣸⣇⣁⣨⠽⠿⠾⠻⢿⠿⠛⠿⠿⠿⠿⠿⡿⠿⠿⠿⠽⠿⠿⠻⠿⠟⠿⢿⡿⠿⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠟⠂⠈⠁
⠊⠉⠀⠀⠠⠤⠤⠤⠤⠖⠒⠺⠤⣀⣀⣀⡀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠠⠤⠤⠶⠄⠀⡀⠀⠀⠀⠀⢈⣉⠍⠉⡉⠙⠓⠲⠶⠶⠶⠶⠦⠤⣄⢀⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢄⡀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣤⣤⣤⣤⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣀⣀⣀⡀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣧⣽⣿⣿⣿⣧⣼⣿⠀⣠⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⢠⣿⡿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⡟⣿⡿⣿⣿⡿⠸⣿⣿⣿⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣿⣹⡇⠀⠘⣧⣭⣯⠟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣽⣿⢿⣿⠶⣿⣿⢿⣯⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠱⣿⣿⠀⢿⣿⡄⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⢰⡟⣭⡝⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡽⣿⢏⣀⣀⠀⠘⢮⣉⣣⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢿⢻⣿⣷⠀⢶⣮⣼⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣷⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣻⣿⡗⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⣿⣳⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠛⠛⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋⠘⠧⣿⡯⠟






⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠰⢤⡀⠀⠀⠀⣤⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡆⠀⢹⡄⢀⣾⣻⣿⣿⣼⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠛⠀⠠⠆⠀⠀⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠁⠀⡸⠁⠙⢿⣿⣿⣿⣿⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⡀⣤⣴⣴⣆⠀⠖⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠒⠀⠋⠀⢀⠀⠈⠻⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⢿⠿⢿⡆⠀⠀⠀⠈⠏⠀⢐⣿⡆
⠀⠀⠀⠀⠀⠀⠠⣶⣘⣶⣶⣶⣖⣸⡶⢴⣶⣽⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡟⠀⠀⠀⠠⣶⡆⢨⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠀⠠⣶⡆⠀⣿⡇⠘⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢾⣿⡷⣿⣿⢾⣿⣿⣿⣻⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⡇⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣿⣿⣾⣿⣾⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⣿⡇⠀⣿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠿⠃⠘⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠛⠩⠶⠎⠛⠛⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⣤⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡇⠀⠉⢻⡛⠉⠀⣼⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡠⣪⣿⣿⣿⣿⣿⠀⠀⣿⡗⠀⠀⣿⣿⣿⣿⣮⣦⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⡇⠀⢰⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠛⠛⠛⠛⠓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⣿⠀⢹⡆⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⣿⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠛⠉⠛⠲⣤⡀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣷
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⠀⠀⣠⠏⠳⣄⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡧⣤⣤⠎⠀⠀⠈⠈⠉⠁⠀⠀⠈⠙⠓⢋⠀⠈⠳⠤⣤⣤⣤⠴⣞⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⠴⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⡤⠒⠋⠉⢶⡀⣿⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠛⠛⠉⠉⠁
⣿⠀⠀⠰⣄⡸⠂⠀⠀⠀⠐⢆⣠⠀⠀⢰⣄⣀⡴⠁⠀⢀⡇⠀⢀⣤⣀⠀⠱⠀⠀⠀⠀⠀⠀⢹⡼⠀⣠⣶⣦⠀⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠸⡇⠀⠺⣿⡿⠀⢀⡇⠀⠀⠀⠀⠀⠈⣧⠀⠙⠿⠛⠀⣸⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠲⣆⠀⢀⡇⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⢹⠳⢤⣀⡀⠴⠋⢹⡄⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⣀
⠸⣇⠀⠀⠢⠀⠀⢰⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠉⠉⠉⢹
⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠠⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⡾
⠀⠀⠹⣦⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣼⠁
⠀⠀⠀⠈⠳⣤⡀⠀⠈⠛⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠇
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣲⠶⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣀⣤⠶⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⡾⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢀⡾⢄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣸⠇⠀⠀⢠⡞⠁
⠀⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⢷⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⣯⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣴⠏⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠛⠶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠁⠀⠀⠀⠀⠀⠉⠛⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠙⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⡓⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⠐⠒⠂⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⣩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠓⠶⠦⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⢉⡉⠉⠉⠉⠁⠀⠀⠘⣓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣀⡀⠀⠘⠒⠒⠀⢌⠃⠚⢁⠤⠒⠶⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠆⢿⣿⣿⣿⠟⣷⣶⣶⣶⣶⣿⣶⣧⣶⣶⣶⣾⡿⣿⣿⣿⡧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠙⢢⠀⠀⠀⠀⡠⠚⠻⣉⠉⠉⣩⠃⠈⠁⠢⢄⡀⠀⡰⠒⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⡄⠀⠀⣜⠀⢀⣤⠞⠁⠁⠀⠉⢠⡶⠐⠊⠁⢠⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠘⣤⡀⠉⠉⠁⠈⠉⠁⠒⠂⠉⠁⠀⠀⠀⣴⣃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢄⣀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠉
⠀⠀⢰⠒⣶⠲⡎⠉⠉⠉⡓⠒⠒⢂⡶⠋⡟⠒⠂⠄⠰⡖⢻⠁⠐⠲⠎⠉⠉⢉⡽⢋⡽⠋⢩⠂
⠀⠀⠀⠣⡈⣧⡙⣆⠀⢀⡇⠀⠀⢸⠀⠀⣿⣀⣀⣀⣰⠁⠀⠃⠀⠀⣤⣀⣠⣟⣠⣏⣀⠴⡇
⠀⠀⠀⠰⠁⠈⠁⠁⢸⡇⢣⠀⠀⠸⣀⡜⠉⢻⣿⡿⠉⣆⡐⠃⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠈⢇⠘⠢⠤⠤⡌⢹⡄⣾⣿⣆⢰⢙⡡⠤⠘⠀⠀⢀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠻⡀⠀⠀⢎⠀⠀⠹⣿⣿⣿⠃⠀⢙⠆⠀⠀⠀⡼⡆⠀⠀⠀⠀⠈⠉⠳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠀⠈⠑⠤⢀⣹⣿⣇⡠⠔⠁⠀⠀⠀⡼⠁⠳⡀⠀⠀⠀⠀⢀⡤⠝⢦⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡜⡷⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠲⣄⢀⠔⢁⡠⠀⠈⣹⣄⣀⡀
⠀⠀⠀⣼⡀⠀⠀⠀⠀⣠⠃⡷⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠻⣤⠊⢀⠔⠋⠀⠀⠀⠨⣶⣒⡐⢦
⠀⠀⣠⢇⠀⠘⠀⠠⢴⡏⢰⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⣄⠀⠘⠂⠤⠰⡉⢣⠹⡀⢷
⠀⠀⢙⢦⣈⣁⠐⣢⡾⢡⠇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠛⢉⠿⣄⣹⠁⠀⢣⠘⣇
⠀⣴⠁⠀⠀⠉⠉⠙⢦⠈⡷⠦⢄⣀⣀⣀⣠⣇⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⣿⠀⠈⠁⠀⠀⠈⣦⢹⡆
⠀⠻⣤⡀⠀⠀⢏⡙⠚⠃⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠸⡆⠀⠀⡤⠖⠀⠉⠉⠉⢉⠶⣄
⠀⠀⠀⠉⠘⠓⠠⠵⠀⠀⡇⠀⠀⠀⠀⠀⢒⡖⢲⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢷⣠⠎⠀⠀⠀⠀⠀⡠⠂⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⠸⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⠇⠀⠀⣷⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⠀⢹⠁⠀⠈⠑⢦⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⡾⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠾⡆⠀⣇⠀⠀⠀⠘⡃⠀⢀⠘⢻⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢻⡀⠸⠦⢄⡀⠀⢻⠀⣡⡄⠊⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⣇⠀⠧⠀⠙⠲⣌⠟⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢸⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠘⣧⣄⣀⠀⠀⠀⣀⣀⠇⠀⠀⠀⠀⠘⣧⣄⣀⡀⠀⢀⣀⡠⢤⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠓⠒⠁⠀⠀⠀⢀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⡞⣡⡟
⠀⠀⡠⠖⠋⠀⠀⠀⣀⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⠞⠒⠒⠂⠤⠤⠤⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣀⣀⣠⣵⡏⠉⠉
⠀⠸⣥⣄⡠⠴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⢻⡋⠈⠁⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠖⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠻⡿⢋⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣻⣛⣛⣿⣿⣛⣛⠛⣻⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠈⠉⠉⠉⠛⢉⡙⠛⠉⠉⠉⠀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⠷⠯⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠒⠒⠒⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⠓⠒⢲⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⣠⣤⣤⠄⠒⠂⢤⣤⣤⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠛⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⣴⠛⠊⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠶⡄⠀⢸⠀⣠⣄⠀⠀⠀⠀⠀⢠⣤⡀⢿⠀⢀⡴⠿⢦⡀⠀⠀⠀⠀⠀⠀⢠⣤⠈⢹⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢹⡄⣠⠰⣿⣿⡄⠀⠀⠀⠀⣾⣿⡇⣤⠀⡜⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⡿⣤⣿⡅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠑⠋⠀⠻⠟⠀⠀⠀⠀⠀⠹⠿⠃⠈⠒⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡤⠞⠁⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣉⣉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠀⠀⠀⠀⣀⡴⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠒⢶⣶⠖⠶⣆⠀⠀⠀⠀⣿⠲⢶⣶⠖⠆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⢀⡼⠁⠹⣦⠀⠺⣀⠀⠀⣠⠋⢀⡜⠁⠳⡄⠀⠀⠀⠀⠙⠛⠛⠓⠀⠤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠀⠸⠇⠀⠀⢹⡄⠀⢉⣶⣶⠁⠀⡜⠃⠀⠠⡎
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢸⡆⠀⠈⢡⡴⢿⣿⣿⢧⣼⠃⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⢲⣄⡀⠈⣡⢀⣿⣯⡀⠛⢀⣀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠰⠶⠒⠂⠀⠉
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢇⡀⠘⣾⣿⣿⡟⠀⣸⠇⠀⠀⣀⡀⠤⠤⠤⠚⠛⠁
⠀⠀⢀⠾⠀⠀⠀⠀⠀⠀⢀⣰⡟⠁⠀⣤⡈⠙⢦⣙⣿⣿⣤⠞⠁⠀⠀⠀⢙⡃
⠀⣠⠞⠀⠀⠀⠀⠀⣀⡴⠋⠀⡟⠠⢶⣿⣷⠆⠀⠙⢻⡟⠀⠀⠀⠀⠀⠀⢸⡇
⡄⠃⠀⠀⠀⢀⣤⠖⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠛⡄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⡇⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠸⠇
⠀⠙⢦⡀⠀⠀⠀⠈⢳⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⣧⣀⠁⠀⠀⠀⠰⣿⠀⠀⢸⡇⠀⢾⡷⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠳⣄⣀⣴⡟⢉⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠛⢇⣴⡿⢋⡼⣿⡿⢏⣱⣶⠀⠀⢸⡇⠀⢶⡦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⠊⠘⢫⠝⣿⡏⠀⢀⣠⣼⣧⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣧⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⣠⡄⠀⠀⣸⠄⠀⠀⠀⠀⣧⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠛⠣⠤⠜⠃⠀⠀⠀⠀⠀⠀⠓⠤⠴⠛⠣⣄⠀⠀⠀⠀⠀⠀⠙⣶
⠀⡀⠀⠙⣢⣤⣤⣤⣤⣤⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⣤⣤⣤⣄⣠⣄⠛
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠈⠁⠀⠈⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠁⠀⠒⠿⢿⣯⡂
⠀⠀⠀⠀⢀⣠⣴⡶⢋⡥⢚⡭⠖⠀⠀⠀⠀⠩⣙⣿⣷⡦
⠀⠀⢠⣾⡿⠛⠁⠔⢋⠔⢋⡴⠚⠉⠀⠀⠀⠀⢒⠭⣿⣯⣶
⠀⠀⡞⠀⠀⠀⠀⠐⣡⠞⣁⠄⢀⣀⡀⠀⠀⠀⣀⣯⣻⣿⡮⣗⢂
⠀⠀⡇⠀⠀⠀⠀⠘⠡⢊⡥⠚⣉⣤⠶⠚⠛⢿⣿⡍⠉⠉⠛⠻⣮⣧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⣁⡴⠛⠉⠀⠀⠀⠀⠘⣿⡟⡄⠀⢀⢲⣻⣿⡄
⠀⠀⢸⠀⠀⠀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣘⣜⣎⣿⣿⣿⡇
⠀⠀⠀⠱⡞⠋⠁⠀⠀⠀⠀⠀⣀⡴⠒⠠⠀⠀⠀⠀⠙⠛⢻⣿⡟⣿
⠀⠀⠀⢠⢧⠖⠲⠆⠀⠀⠀⠀⠉⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⢹⣵⡏
⠀⠀⠀⢸⢸⠀⣀⣤⡄⢳⠀⠀⠀⢋⠐⠋⠀⠀⠀⠀⣄⠀⠀⣼⠉⢧⠀⣀
⠀⠀⠀⠸⣼⣷⡩⠸⠏⠘⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⡟⠀⠀⠃⠠⣤⣿⣿
⠀⠀⠀⠀⠙⠿⡇⠀⠀⠠⡄⠤⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣰⣶⣾⣟⡾⠃
⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⣀⣤⣴⡶⠶⣶⣮⡆⠀⠀⠀⣰⢊⡟⠛⣉⣤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⣿⣭⣤⣶⣿⠟⠁⠀⠀⢀⡴⠃⢸⡿⢉⢁⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠛⠛⠛⠁⠀⠀⣠⠴⠋⠀⣰⢾⠁⣷⠀⠙⢷⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⡠⠔⠋⣁⡤⠴⠛⠁⢾⠀⢹⠀⠀⠘⣧⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠒⠉⢁⣀⡴⠊⠀⣼⠀⠀⢣⡀⠀⣿⠉⠙⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣦⣴⡾⠟⠉⠀⢀⡰⠟⢳⡀⠀⠙⠂⣿⡴⠃⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⡛⠁⢀⣀⡤⠚⠋⠀⠀⢸⠋⠢⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⠉⠉⠋⠀⠀⠀⠀⢀⠎⠀⣠⠊⠢⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⠎⢀⠞⠁⣠⠀⢈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠎⡰⠋⠀⠀⣿⣄⡼⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⡰⢋⠞⠁⠀⠀⣠⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠐⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠁⠀⠀⠑⣄⠀⠀⠀⠀⢀⠔⠒⠢⠤⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠒⠠⡄
⠀⠀⢀⡴⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠼⠀⠀⠀⠀⠀⠀⠑⠒⠦⠒⠢⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠁⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠆⠀⠀⠘⠛⠛⠦⠰⢄⠀⠀⠀⠀⠀⣀⢀⡠⠬⠭⠭⠕
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⣁⠤⠤⡀
⠀⠀⢠⠊⠉⠉⢢⣀⣀⠤⠒⠤⡀⠀⠀⢀⠤⠤⠊⠀⠈⢳⠀⠀⠀⠀⣀⠤⠤⠤⠔⠉⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠁⠀⠀⠈⠦⢄⡀
⠀⢀⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠳⡄⠀⡇⠀⠀⠀⠀⠀⠸⡁⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⡀⠀⠀⠀⢀⡴⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢
⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠁⠀⠈⢇⠀⠀⡀⠀⠀⡇⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⠀⠀⠱⢄⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠜
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠢⡀⠀⠀⠀⠀⠀⠈⠒⠊⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠓⠠⠔⠊
⠀⠀⠀⡰⠉⠉⠋⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⢀⠔⠊⠉⠒⢄⠠⢄⡀⠀⠀⠀⢀⣀⡔⠁⠀⠈⢢⢀⡀⠀⠀⠀⠀⠀⣀⢀⠔⠉⠉⠑⢄
⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⡰⠉⠀⠉⠀⠀⠀⠀⢸⠒⠒⢯⣉⡉⢡
⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠴⠃⠀⣏⣁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠇⠸⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠱⣀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠈⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠒⠀⠒⠉⠒⠊⠁⠀⠀⠀⠀⠑⠒⢆⡀⢀⡠⢀⣀⣀⠜⠀⠀⠈⡀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⡤⠃
⠀⠀⠀⡠⠒⠠⣀⣀⡰⠊⣉⠵⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠉⠘⠣⠤⠒⠤⠴⠃
⠀⢀⠜⠀⠀⠀⢈⠔⠈⠒⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⢁⠔⠋⠑⠒⠉⠉⠐⢄⠀⠀⠀⠀⢠⠂⠉⠑⠴⠊⠉⠲⡄
⠀⠈⠢⠤⠤⡔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢴⠊⠀⠀⠀⢀⡔⠊⠋⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⡤⠶⠤⠴⠉⠉⠑⣄⣀⠱⠒⠢⠔⠒⢢
⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠉⠉⠉⢳⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⡀⣀⡀⡉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠂⠀⠀⠈⠉⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⡠⠴⠁⠀⠀⠙⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠂⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠒⠒⠢⡀
⠀⠀⢠⠔⠊⠀⠀⠀⠀⡠⠤⠎⠁⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡤⠒⠓⢤⡀⢀⡠⢄⡀⠀⠀⠀⠀⠀⣀⠤⠀⠤⡎⠀⠈⠀⠀⠀⠀⠥⠤⡀
⠀⠀⡇⠀⠀⠀⢀⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡀⠀⠀⠀⠈⢲⠀⡠⠜⠀⠀⠀⠀⠁⠉⡆⠀⠉⠙⡆⠀⠀⢀⠃⠀⠀⠀⠘⠒⠢⡔⠋⠉⠉⢢⠀⢈
⠀⠀⠘⠦⢀⠰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢠⡤⣤⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠾⣁⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡎
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠔⠢⢄⣀⠤⠊⠈⠁⠀⠀⠀⠀⠉⠉⠑⠈⠢⠄⠀⠤⣀⡠⠆⠔⠊⠉⠉⠉⠉⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⡀⠀⠈⠉⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣩⣉⡽⣷⣾⡿⣿⣾⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣀⣴⣾⡟⣴⢳⣿⢸⡜⣮⢻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣼⣿⣸⣧⣽⣶⡇⠈⠉⠉⠉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠁⠀⠈⠹⣿⣿⣿⡟⣤⡀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠠⠴⠖⠒⠶⣿⠟⠙⠁⠛⢸⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠀⠀⠀⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣄⣀⣰⣤⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣶⣶⣶⣖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠉⠀⣀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣄⡀⠈⠩⠤⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡙⠏⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠍⣷⡄⢀⠤⠀⢀⣿⢷⡽⣶⣤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣞⣛⢿⣾⢻⣤⣴⣿⣿⢀⠀⠈⠀⠙⠞⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⣼⣿⣝⡙⠙⡋⣿⠠⡦⠈⠰⣦⡶⢈⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⣾⣿⣿⣿⠃⠂⠳⣿⠈⠀⢀⣀⡈⠀⡄⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⣺⣿⣿⣿⣿⣾⣷⣶⣾⣿⣿⣿⡿⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠐⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠔⠂⠀⠒⢄⡀⣀⡀⣀⠔⠂⠀⠒⢬⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢄⠀⢀⡠⠗⣄⣀⣀⣠⠜⠉⣭⣉⠙⣤⣀⣀⣀⡴⠧⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⡍⢹⠀⡞⡇⢀⠃⣤⡀⠀⠑⠒⠶⠀⠉⠉⠀⣤⡀⠇⠀⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢃⠀⣄⠃⢸⠀⠙⠁⠀⠀⠀⢄⠀⠀⠀⠀⠙⠁⢸⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢣⣸⠀⣆⠀⠀⠰⡒⠒⠒⣒⠒⠒⠒⡲⠀⠀⠀⣇⢸⠁⠀⠀⣠⠔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠇⣼⡈⢢⡀⠀⠑⠢⢄⣀⣀⠠⠖⠁⠀⣠⠎⠘⡀⢇⢀⡜⠁⠀⠀⢱⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠎⡰⢁⠝⠒⠬⢷⣤⣤⣤⡤⢤⣤⣤⣐⣊⡀⠀⠀⠱⡈⢞⠀⠀⠀⠀⢸⢀⡔⠻⣌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠴⢧⡎⠀⠀⡠⢊⠜⠁⠑⠒⢢⠞⡱⠁⠀⠀⠉⠉⠉⡗⠬⠆⠀⠀⠉⠉⢉⣀⣉⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢤⠀⠓⣤⢞⣴⠁⠀⠀⠀⡰⢃⠎⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⣀⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣦⣞⡵⠋⡯⠀⠀⢀⠞⡰⠃⠀⠀⠀⠸⡋⠓⢒⣒⠻⠿⢥⠊⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠻⡄⠀⢸⣄⣀⣈⠉⠉⣐⣒⣧⠾⠤⠄⠒⠒⠈⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢔⡇⠀⠀⢧⠀⡆⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠒⠉⠹⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠒⠒⠊⠉⠁⠀⠀⠀⠈⠁⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠣⣀⠀⢀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠂⠀⠀⢄⣤⠄⠐⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡽⠉⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⠇⠀⢱⡀⠀⠀⠀⠇⠀⢀⣀⠠⠤⠒⠊⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡍⢉⣽⣚⡣⠟⠋⠁⠲⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⠤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⠉⠛⠛⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⢤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠲⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⢉⣉⣠⣤⣤⣤⣤⣤⣤⠤⠤⠄⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠰⣯⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣤⣤⣀⡀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢾⠏




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

⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢫⠄⠁⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠦⣄⠀⠀⠀⠀⠀⠀⠈⣇⠰⢇⣠⠀⢠⡍⠉⣗⠀⢀⣤⡀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⠦⣄⠀⠈⢧⠀⠀⠀⠀⠉⠀⠙⠈⠉⠀⠳⠬⠥⣾⠉⣹⡈⠹⡿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡨⢷⠒⠙⠉⠀⠀⠀⠐⢆⠀⠀⣢⢠⢤⡤⣾⠁⣾⠈⣻⣾⣿⣿⡆⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⢀⢴⠊⠈⡆⠈⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠈⠘⠁⠀⠈⠛⠋⠉⠀⠀⠙⠀⠀⠀⠨⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠃⠰⡼⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡞⠀⠀⢀⠄⠀⠻⢷⣦⣄⣀⡀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠂⡹⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⣹⢎⣠⠎⠀⠀⠀⠀⠀⠉⠍⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⡀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⡤⠔⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⡄⢀⡀⠀⠀⢀⣿⠃
⠀⠁⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢈⠇⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡄⠀⠀⣀⡀⠀⠀⢄⠈⠢⡀
⠀⠎⠀⠀⠀⢰⠓⢦⡀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣧⡀⠀⢀⣠⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣠⠞⢉⡿⠀⠀⠀⢳⠀⠹
⠀⠀⢠⡴⠤⣈⢳⡀⠳⡄⣾⠁⣠⠟⠀⠀⠀⠀⠀⠀⠀⣴⠟⣉⡀⠀⠀⠀⠀⠈⠉⠀⠉⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠒⠒⢦⣟⠈⠙⠃⣴⢛⣀⣀⠀⠀⠘⡇
⠀⠀⠀⠱⢦⡈⠳⠟⠁⣹⠿⢰⡃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣦⡙⣧⠀⠀⠀⠀⠘⠷⠒⠛⡦⠈⠻⣦⠀⠘⢋⣠⡼⠃⠀⠀⡇
⠀⠀⢀⡤⠴⠨⠗⠀⡾⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠁⠸⣇⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠘⠆⠀⠛⢤⡀⠀⠀⢸⠇
⠀⠀⠈⠑⠒⢲⠖⠀⠁⠀⠀⠰⢏⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇⠀⠙⠓⠶⠤⠤⠴⠖⠛⠁⢸⡀⠀⣿⡀⠀⠀⠀⠀⠀⢀⡽⠂⣤⣠⡴⠟⠓⠲⠃⠀⠀⠉
⠀⠀⠀⠀⠰⡥⠞⠛⠶⠶⠞⢧⡀⠳⣄⠀⠀⠀⢠⡞⡛⠂⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠍⢳⠀⠀⠀⣰⠏⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄⠀⠈⢳⡶⠀⠀⣿⣿⣷⣤⣄⣀⣀⣀⣀⣤⣴⣿⡇⠀⢲⣴⠟⠀⢠⡞⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⠳⣄⠈⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣗⢀⡴⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⠀⠀⠀⢹⡟⠿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣡⡿⠃⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣄⠀⠙⠳⠶⠶⠞⠋⠀⡀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠈⠳⣦⣄⣀⣀⣠⣴⠞⠁⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠳⠦⣄⣀⣉⠉⠉⠉⢀⣀⣀⣤⠴⠚⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣠⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡇⠘⣇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣧⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⣆⠀⠀⠀⠸⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⢀⣠⠶⠛⠙⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⣄⠀⠀⢠⠟⣻⠻⠶⠶⢶⣶⣶⣶⣶⡶⠶⠾⠟⠻⣏⠻⣄⢀⡴⠛⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠳⠶⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠋⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⢿⡿⠿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠉⠛⢷⡀
⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢀⣤⣤⣄⠀⣠⣤⣤⡀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠻⣿⣿⡿⠀⢿⣿⣿⡟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖



⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⢰⣿⣾⣽⣿⣿⣿⣿⣫⣷⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠀⠀⡀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣯⣭⣯⣭⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠙⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣏⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢀⠀⡀⢀⠌⠑⣀⠀⡠⡑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠇⠀⠠⠇⢈⠠⠔⠁⠀⠀⠀⠑⠄⡊⠿⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠛⡀⠢⠊⠅⠀⠄⠀⠀⠀⠀⠤⠤⠀⠉⠘⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠘⠀⠐⠺⠗⠂⠀⠀⠀⠐⢺⠗⠂⠀⡁⠀⡁⢂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣺⠀⠀⡀⡇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⡇⠀⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢹⠀⡐⢀⡆⠀⢐⠒⠒⠂⠒⠒⠒⠂⠀⠀⠀⡇⡔⣆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠹⡂⠀⠈⠢⢄⠤⠤⢀⠐⠀⠀⠀⠠⣋⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠢⣠⣄⡠⠤⠤⠒⡺⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠑⠦⣀⡀⠀⠀⢀⡠⠎⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣧⠀⠀⠀⠡⢂⠉⠈⠀⡁⡇⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢀⠠⣄⢀⢢⠀⠀⠈⠳⡀⠀⠀⠁⠈⠀⢀⠔⠀⠀⢸⠊⠀⠺⣥⡶⡖⡲⢦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⡟⢿⢹⣕⣗⣠⡇⠀⠀⢆⠀⠀⠀⠈⠢⢄⠀⠀⠔⠁⠀⠀⠀⡼⠀⠀⠀⢛⠯⣅⣾⣭⣏⣿⠞⠒
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠙⠯⠉⠙⠉⠃⠁⠀⠀⠈⠢⡐⠀⠄⠀⢀⠄⢢⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⠉⠉⠙⠃⠀⢂⠄⠈⡄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⣞⢁⠡⡹⣞⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣹⢘⠛⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠡⢊⠔⣆⠀⠀⠠⢤⠤⢐⠋⠂⣠⣠⠀⢀⠣⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⠀⣜⣣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡠⢂⣕⠀⠀⠀⠉⠙⠀⠂⠀⠘⠉⠄⠠⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⢼⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⠪⠊⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡊⢘⠅⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠿⠟⠛⠛⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⣉⣥⠞⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡞⠋⣉⣅⠀⢸⠇⠀⢠⡟
⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠙⣛⣭⠴⢚⣡⠿⢤⣀⡀
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⢛⣩⣤⠶⠋⠁⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠙⠻⠷⣶⣤⣄⣀⣀⡀⠀⣀⣀⣤⡤⠶⠞⠛⣉⣩⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⣯⣥⣤⣶⣶⣿⣿⣭⣥⣤⣤⣤⣤⣴⣶⣶⣶⠶⣿⡟⠛⠋⠛⠛⠿⢿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠁⠀⠀⠀⢸⣿⠁⢠⠶⣄⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣇⠙⠿⠟⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠰⠿⣿⣶⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠙⠇⠀⠀⠀⣠⣶⠾⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⢀⠀⠀⠀⠚⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⠟⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⢀⠀⠘⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠉⠉⠉⠉⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⣷⣄⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣥⣤⣤⣤⣶⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠉⠉⠙⠛⠛⢿⠛⠻⣿⠛⢛⣿⡟⠛⠛⠋⣿⣏⠉⢉⡟⣰⠛⣿⠲⢤⣀
⠀⠀⠀⠀⣀⣤⠖⠋⠁⣿⠃⠀⠀⠀⠀⠀⠈⢷⡀⠘⢷⣾⠙⢷⡀⠀⢠⡟⢿⣠⠟⢠⡏⠀⢻⣧⣀⠈⠙⠲⣤⣀
⠀⣀⡴⠞⠉⢀⣠⡴⢿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠄⠻⣦⡈⠻⣶⡟⠀⣸⠏⠀⣻⡆⠀⠘⣿⡉⠛⠶⣤⣀⡉⠳⢦⡀
⣾⠋⠀⠒⢾⣏⡁⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⢤⣀⠈⢳⣾⠁⠹⣾⠏⣠⠜⠃⠀⠀⡀⢻⣇⣠⡴⡾⢋⠁⢀⡼⠇
⠈⠓⢦⣴⣈⠏⣹⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣽⣧⣰⣯⠞⠃⠄⣌⡑⠆⠁⢸⣿⣽⣧⣡⡵⠞⠋
⠀⠀⠀⠈⠙⠛⢉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢻⠁⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣦⣁⣄⣄
⠀⠀⠀⣘⣷⣷⣾⠏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⢿⡿⡿⠂
⠀⠀⠀⠈⠛⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣇⣀⣤⣤⣤⣤⣶⣶⡾⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⢻⡟⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⢧⣄⣀⣀⣰⡇⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣤⠾⠛⣹⠉⠙⠒⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⠉⠁⠀⢰⠀⠉⢉⣉⠈⢷⡀⠀⠀⠀⠀⠀⣰⡏⢀⣶⣤⣄⣀⣼⣀⣤⣤⡶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⣄⣠⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠚⠁⠳⡄⣀⣀⣤⣤⠤⠴⠶⠶⠖⠒⠒⠛⠛⠛⠛⠒⠒⠲⠶⠶⠦⠤⣤⣤⣀⣀
⠀⠀⠀⢀⡴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣷
⠀⢰⡇⠀⠀⣠⣾⣿⣿⣿⣿⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣞⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⡆
⠀⢸⠁⠀⣼⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠿⠿⠿⣧⠀⠈⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⡀⠙⢿⡀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⢿
⠀⡇⢀⢀⠀⠀⣠⣞⣭⣶⣶⣶⣿⣿⣷⣦⠃⢀⠀⠀⠀⠀⠀⠊⣴⣾⣿⣿⣶⣶⣦⣭⣷⣄⠀⠀⡀⡀⢸
⠀⡇⠛⠣⠷⠿⠿⣬⣝⣛⣛⣛⣛⣿⡿⠟⠀⣼⠀⠀⠀⠀⡇⠀⠻⢿⣟⣛⣛⣛⣛⣫⣵⠿⠿⠿⠜⠛⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⢸
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡟⠀⠀⠀⠀⣿⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⠀⢰⣄⡀⠀⠀⢀⣀⣤⡶⠟⢳⠃⠀⢸⡇⠀⠀⠀⠀⢸⠆⠀⠸⠜⠻⢦⣤⣀⡀⠀⠀⣀⣠⠆⠀⡟
⠀⢸⡄⠀⢻⣏⢿⣿⡉⠉⠀⠀⠀⠀⠀⠘⠆⠛⠦⠤⠤⠶⠋⠲⠂⠀⠀⠀⠀⠀⠉⢩⣿⡟⣽⡟⠀⢠⡇
⠀⠈⣇⠀⠈⢿⣧⢻⣿⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⠷⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⣿⡟⣼⡿⠀⠀⣸⠁
⠀⠀⢻⡀⠀⠈⢿⣦⠻⣿⣶⣶⣶⣶⣾⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣷⣶⣶⣶⣾⣿⢏⣼⡟⠁⠀⢀⡟
⠀⠀⠈⣧⠀⠀⠀⠻⣷⠦⡉⠉⠉⠉⠉⠹⠿⠶⠶⠶⠶⠶⠶⠿⠏⠉⠉⠉⠉⢉⠴⣿⠟⠀⠀⠀⣼⠁
⠀⠀⠀⠈⢧⡀⠀⠀⠙⢧⡈⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠈⢁⡼⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠸⣿⣿⠃⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⠽⠧⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡟⠀⠀⠀⠀⢸⠃⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠶⣄⠀⠀⣀⡤⠶⠒⠚⠉⠉⠉⠉⠉⠙⠒⠶⢤⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡾⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠑⢾⠀⢸⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡴⠶⠦⣄⠀⠀⠀⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡠⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡠⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡞⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⢠⠙⠦⠀⠠⠞⠑⢄⣀⡰⠳⡄⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡰⠋⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠁⣠⠤⠤⡀⠀⠀⠀⠀⢠⡀⢈⡑⠒⢖⠹⡇⠀⠀⠀⠠⣞⣠⣤⠤⠶⠖⠋
⠀⠀⠀⠀⠀⠀⣾⠀⠠⠄⠴⠀⠀⠀⢄⡀⠀⢠⣄⡠⠃⠀⡜⠀⣠⣄⠈⡆⠀⠀⠀⠀⢷⠃⣠⣦⡈⢷⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡄⠀⠀⠀⢣⠀⠻⠟⠀⡆⠀⠀⠀⠀⢸⣄⠈⠋⢀⡾⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢄⡀⢀⠞⠁⠀⠈⠲⠔⠁⠀⠀⠀⠀⠑⠤⠴⠊⠀⠀⠀⠀⠀⠀⡆⠉⠀⠉⠀⢳⠀⠀⠀⠀⢾⡛⠓⠒⠒⠶⡦
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠰⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⣄⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠢⣄⣀⡀⠀⠀⠀⠀⢸⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠥⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⣆⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⢨⡇⠀⠀⠀⠀⠈⡄⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⡤⠋⠁⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠁⡞⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡎⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠲⢤⡾⢣⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢣⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠈⠢⡀⠀⠀⠀⣴⠋⠈⠓⠦⣄⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠲⣤⠎⠙⣇⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⣞⡁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣠⠞⠁⠀⠉⠓⠦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠇⢻⣿⣿⣿⣿⡇⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⣰⣾⣿⣿⣿⣿⣹⡄⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣿⣧⣿⣿⣿⡟⣿⡇⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⣿⠘⠛⠛⠹⡿⠁⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣷⣷⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣷⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣷⣦⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⡈⠉⠒⠂⠄⠀⠰⢾⡆⠀⠀⠀⠀⠀⠀⠀⠚⠋⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠠⠀⠀⠀⠀⢀⠮⢶⡶⠆
⠀⠁⠀⠉⠀⠀⠀⢀⣴⣿⣦⡀⠀⢠⠄⠀⠀⠀⠐⢳⣦⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠒⠢⢀⡀⠀⠀⡀
⠀⠀⠀⣀⣴⣦⡴⠟⣿⣿⣿⡇⠀⢰⣳⠦⠆⠀⠀⠈⣿⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣦
⠀⠀⠀⠁⠄⣀⢻⢀⣿⣿⣯⠁⠀⠘⣿⣷⠀⠀⠙⠺⠽⣤⠄⠀⠀⠞⣿⡟⠂⠀⠀⠀⠐⡀⠁⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠛⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢠⠰⠀⠀⠸⠋
⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠈⠀⢈⡀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠀⠹⠁⠀⠀⠆⠀⠀⠦⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠍⢉⠟
⠀⠀⢀⡴⠾⢛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⢷⣶⣿⡀
⠀⢠⣾⣿⣶⣿⣿⠛⠳⣾⠆⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠴⣿⣷⠀⠀⠉⣉⡀⠀⠀⠀⠀⠀⣰⡶⢿⣿⣤⣶⣿⣧
⢠⣿⠁⢸⣿⣿⣿⡀⠀⠀⠀⢀⣀⣠⠀⢹⣷⠀⠀⠀⠀⠂⢄⡀⣿⣿⠀⠀⢸⣿⣽⡶⠆⠄⠀⠀⠀⠀⢠⣝⣿⣿⠟⠁
⠀⠁⠀⣈⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⠀⠸⡿⠄⠐⣺⣤⠀⠀⠗⠟⠇⠀⠀⠀⣿⣿⠗⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⢈⡿⣿⠀⠀⠀⠀⡀⠀⢻⡯⠆⠀⣀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠉⠻⣷⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣷⡄
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⢷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⠛⠿⠛⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⠀⠀⠀⠀⠈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣿⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠈⠙⠛⠿⠛⠛⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠙⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠈⣿⡿⠋⠉⠉⠉⠉⠛⢿⣿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣿⣆⠀⠀⠀⠀⠀⢀⣸⣿⣦⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠛⠋⠹⣿⠏⠛⢷⣄⠀⠀⣤⡾⠋⠹⣿⠃⠀⠛⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢰⣷⣀⣤⣤⣌⠻⠿⠿⠋⣡⣤⣄⣀⣿⡆⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⡿⣿⡆⠀⠀⠀⠉⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣷⣤⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⢀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠁⠀⠀⠀⠀⠀⣰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠿⢿⣿⣿⠿⠟⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⢀⣴⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢳⣦⡀⠀⠀⠀⠀⠈⠹⣶⡄
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⣶⠋⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⢰⣟⠁⠀⠀⠀⠀⠀⠉⠿⣆⡀⠀⠀⠀⠀⠻⣧
⠀⠀⣸⠟⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡀⢠⣶⣤⣾⣷
⣀⣾⣿⣀⣤⣄⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠿⠃⠀⠀⠀⠈
⣿⠃⠀⠈⠉⠉⠰⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣀⣴⠾⠿⣧⡀⠀⠀⠀⠀⠀⣀⣶⠟⠉⠀⠀⠀⠀⠀⣠⡾
⠛⣶⡄⠀⠀⠀⠀⠀⠘⠻⣦⡄⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⣿⣿⣆⡀⠈⠿⣄⠀⢀⣴⡾⠛⠀⠀⠀⠀⠀⢀⣴⡞⠋
⠀⠀⠙⠿⣤⣄⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⣤⠿⠃⢀⣬⣿⣿⣄⠀⠀⠀⠸⠿⠃⠀⠀⠙⢿⣿⡿⢿⣄⠀⠻⣿⣿⡋⠀⠀⢀⣠⣤⠼⠿⠃
⠀⠀⠀⠀⠈⠉⠳⣶⣄⣀⡀⠀⠀⠘⣿⣾⠋⠀⢰⣿⣿⣿⠟⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠈⢻⣦⡀⣿⣿⣿⣶⡶⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢠⣴⣾⣿⣷⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣤⣀⣀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣀⣤⣶⣶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣼⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠛⠛⠿⢿⣿⣿⡿⠿⠟⠛⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⢻⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⣾⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⢻⠇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠘⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣀⣼⡇⠀⠀⠀⠀⣼⣯⡀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⣤⣾⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣤⡟⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠉⠁⠈⠛⠛⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄⡀⠀⠀⠀⠀⠀⠀⡤⢀⣸⠁⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠃⠀⢠⣤⣷⣶⡄⠀⠀⠀⠙⣷⣿⣇⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠂
⢀⠔⠈⠒⠒⠀⠤⣀⡀⣧⣠⡄⣸⠇⠀⠀⢸⠁⠀⠀⠀⠀⡟⠘⢿⣦⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠁⠄
⢸⠀⠀⠀⠀⡀⠀⠀⠀⢸⠟⠙⠆⠀⠀⠘⣏⠀⠀⠀⠀⠘⡄⠀⠼⠻⣿⣶⣤⣤⠎⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠐⣒⡲
⠀⠀⠀⠀⠂⠐⠀⢀⡴⡋⠨⠑⠢⡀⢰⠞⠉⠉⠒⡄⠀⠀⢇⣀⣀⣀⣙⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠮⠉⠓
⢀⠀⠀⠠⠂⠀⡰⠋⡰⠁⠀⠀⠀⠙⢍⡀⠀⠀⠀⠇⣀⡤⠖⠛⢹⠃⠉⠙⠳⢦⡀⠀⠀⠀⠀⢀⡀⠔⠂⠉⣁⣄⠀⡎⠱⡄⠂
⠸⠀⠀⠀⠀⡜⠁⠀⠁⠀⠀⠀⠀⡴⠊⠉⢢⡰⠚⠉⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠈⡙⠲⡄⠀⡇⡤⡄⡎⢱⢇⣸⠀⠣⠔⠀⠦
⠀⠀⠀⠀⡼⢆⠂⠂⠢⡀⠀⠀⡌⠀⣠⢄⠚⠃⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠎⠀⠀⢹⢈⢄⢑⠻⣌⠥⠀⠐⠢⠄⠀⠖⠈
⠀⠀⠀⡞⠀⠀⠈⢢⡀⠈⢢⡘⠀⡜⠁⠀⠑⢦⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣼⠘⠃⠀⢿
⢀⠀⢰⠀⠀⠀⠀⠀⠱⡀⠀⠱⡰⠀⠀⠀⠀⠀⠱⡀⠘⡆⠀⠀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢻⢆⡀⠠⠊
⠀⠁⠘⠀⠀⢠⢤⠀⠀⢣⠀⠀⢹⡄⠀⢀⢄⠀⠀⠱⡀⠘⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣌⠇
⠀⠀⠀⠇⠀⠘⠀⠇⠀⠀⡆⠀⠀⡇⠀⠆⠀⢣⠀⠀⢡⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⠈⠋⠉⠛⢹⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠛⠃⠀⢸⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⠀⠀⠒⠒⠒⠀⢸⣄⣀⣴⠃⠀⠀⠀⠀⠀⢀⣠⠔⠳⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣸⣄⣀⣀⣀⣀⣀⣸⣄⢰⠇⣰⠒⠒⠒⠚⠋⠉⠀⠀⠀⠈⠓⠲⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠹⣄⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⣠⠃⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⠀⣀⣀⡀⠀⣀⣸⠀⠀⠀⠀⠳⣄⠀⠀⠤⠄⠀⠀⣠⠏⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣤⡽⠀⢿⣤⡇⠨⣧⣼⠀⠀⠀⠀⠀⠈⡷⠦⠤⠤⠴⣞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠖⢶⠀⣶⠒⡆⢠⡖⢯⣀⣀⣠⣤⠤⢾⠁⠀⠀⠀⠀⠘⡦⠤⣤⣀⣀⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⣓⡛⠀⣻⣶⣧⣬⣷⡚⢡⠏⠀⢹⠀⠈⢧⠀⠀⠀⢀⡞⠀⢀⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣉⣽⠀⣿⣲⡤⣄⣠⠼⢿⠀⠀⠈⠓⢲⠈⢻⡙⢹⠏⠠⡖⠋⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠈⣇⠀⢠⠇⠻⣄⠀⢳⡏⢀⣤⠇⢰⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡀⢀⣀⣀⣀⣀⣸⠀⠀⠀⢹⠀⣿⠀⠀⠀⠙⢾⡇⠈⠀⠀⠀⡿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠶⠶⠶⠀⠸⡇⠀⠀⢸⡆⢸⣦⠀⠀⠀⢸⡇⠀⠀⠠⣴⡆⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⢸⣿⠀⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠃⡾⣻⠀⠀⠀⢸⠀⠀⠀⠀⡇⢷⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠽⠦⠤⠾⠥⠿⠤⠤⠤⠼⠦⠤⠤⠤⠧⠼⠤⠤⠤⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠋⠋⠙⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠰⢶⡒⠶⠶⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠯⠭⣉⣉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣤⠤⢄⠀⠀⢠⡬⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⠋⠀⠀⠀⠱⣄⠀⠉⠓⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡏⠀⠀⠀⠀⠀⢹⡀⠀⣀⡬⢍⠂⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⢀⠀⢘⡷⠋⠁⠀⠈⡇⠀⠀⠈⠁⠒⣲⠂⠀⠀⠀⠀⠀⠀⠀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣄⠈⣇⠀⠀⢰⣏⣡⠎⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠁⠹⣄⠀⠈⠋⣟⣤⠀⠀⢀⡀⠞⠀⠀⠀⡍⠁⢸⣇⣠⢤⣄⣀⠀⠀⠀⠀⢹⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠓⠠⢌⡓⠆⠚⢇⠉⣠⠖⠉⠀⠀⠀⠀⠼⠀⠀⠠⣤⡴⠦⣤⡈⠱⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠱⠀⠀⠈⠿⠁⠀⠀⠀⠀⢀⠎⣰⠀⠀⢀⠀⠉⠙⠋⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⡏⠀⠀⠌⠙⠓⠶⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⢠⠀⠑⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠐⣤⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠲⢤⣀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠘⠧⠀⠀⠀⠈⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡀⠐⠃⠄⠀⠤⣼⠑⠢⢄⡀⠀⡟⠉⠀⠀⠀⠸⣷⠁⠀⠀⠘⡄⢿⠲⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠞⠐⠉⠉⠉⠹⣄⡤⢴⠈⠒⠁⠀⠀⠐⡆⠚⠛⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠑⠚⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡍⠙⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠗⠓⠂⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢚⡤⠤⣼⣀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⣾⣤⢶⣹⣽⣭⠉⠑⠒⠒⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠛⠓⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠃