Extension Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⢸⡇⢀⢠⣼⢀⡇⢸⢠⡆⣶⡀⢠⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣆⢷⡈⣇⣿⣼⣼⣼⣿⣷⣸⣿⣷⣿⣿⣾⣿⣶⣼⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣶⡞⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⠿⠿⢻⠟⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡟⠀⢠⣄⣠⣿⣿⣿⣿⣿⣇⠙⢿⣿⣿⣮⣴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣄⠀⠀⠀⠀⠀⣿⠀⢸⣾⣿⣿⣿⣿⣿⣿⡇⠀⣾⠉⠻⣿⣯⣥⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣋⡽⢷⣄⡀⠀⠀⠘⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⣰⠇⠀⠀⠈⠻⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠖⣋⡽⢟⣦⣄⡀⠀⠢⣄⣀⠀⠉⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡴⠋⡴⢋⡽⢳⠶⣤⣬⣉⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠁⠏⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠠⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠀⡀⣠⢐⢄⡮⢀⡤⠰⠀⠀⡆⠀⡞⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠐⠣⣸⠐⠏⠁⠸⠅⢾⠀⠯⠰⠇⠿⠸⠇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠄⠰⠀⠀⠀⠀⠀⠠⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣒⣢⣶⣶⣶⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣄⣒⡀
⠀⠀⠀⠀⠀⠔⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠫⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠟⠉⠚⠉⠉⠉⠉⠉⠉⠉⠉⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢄⣀⠀⢄⣄⡀⢄⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⡠⢀⣠⡠⠂⣀⡠
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⠛⢟⢹⣿⣿⠟⡷⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⠻⣿⣿⡏⡻⠛⢿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⡬⠦⡭⣥⠤⠤⠐⠒⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒⠂⠤⠤⣬⢭⠴⢥⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⡤⠶⠚⠋⠉⠉⢸⢠⠃⠀⠀⠀⠀⠈⠉⣹⢛⡗⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡋⠁⢀⠀⠀⢰⡇⣤⢸⢸⡀⢰⠂⡄⢀⡆⢠⡏⢸⠃⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢸⣬⣇⢀⢸⡀⣇⢸⣇⣽⢸⣼⡇⣸⢠⠇⣼⢇⡾⣄⡟⢀⠀⢀⠄⠀⠀⠀⠉⠲⣄⣰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⣼⣆⢰⣻⣿⣬⣷⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣾⣟⡿⡹⣿⡵⢃⣴⠋⠀⣠⡾⠁⠀⢀⣼⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣴⣾⠏⢀⣠⢞⡿⠃⣠⠈⠳⡄
⠀⠀⠀⠀⠀⣠⡴⠒⠀⠀⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⠟⣋⣹⣿⣿⣭⣭⣵⣾⣭⣤⢞⡥⠞⢁⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠳⠶⠶⠦⠤⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢿⣿⣿⣿⣿⣦⠖⣡⠄⠀⠀⠀⠀⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠻⣿⣿⣿⣶⣿⣥⣔⣁⣀⣴⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣡⠴⣛⡥⠖⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⣛⣛⣋⣀⣠⣤⣤⠤⠦⠒⠀⠀⠀⠈⠻⣿⣿⣵⣖⣿⣯⢶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠂⢀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⠢⠀⠈⠉⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠆⠀⠀⠐⠀⡄⢄⢀⠀⢆⠰⠘⢆⠢⠙⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠃
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⣶⡶⠶⢶⣶⣶⣶⠶⠶⣶⣶⣶⡶⠶⢶⣶⣤
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣿⡏⣀⠀⢀⠘⣿⢁⠀⠀⡈⢿⠏⡀⠀⢀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⡀⠀⢀⣰⣿⣄⠀⠀⣠⣿⣧⡀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⠛⢛⠛⠛⢛⡛⠛⡛⠛⠛⡛⠛⢛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣸⡀⠀⣘⣃⣀⣇⠀⢀⣟⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⣾⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠛⠁⠀⠈⢻⡟⠃⠀⠀⠛⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠛⠻⢦⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠉⠉⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠒⣿⣿⣿⣶⣤⣤⣀⣀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⢿⡿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⢿⠿⣿⣿⣿⢿⡿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡤⠀⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠙⠘⢉⠁⠈⡉⠀⢀⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⢠⢸⡀⣼⣶⣰⣷⣷⣾⣷⣿⣤⣵⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣜⢦⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠛⠛⢉⡏⠁⠈⢻⣿⣿⣿⣿⢿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⢸⡀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠻⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣺⠿⣦⣀⠀⠀⠃⠀⠻⢿⣿⣿⠿⠃⣰⠃⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⣻⢷⣦⣄⡐⠢⢤⣤⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠊⡴⢹⢫⠟⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡣⠈⠀⠀⠀⡀⢀⡎⢀⢀⠀⣀⠀⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠇⡞⠼⠈⠸⠰⡃⠆⠴⠸⠸⠇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⡀⠀⣸⢣⣄⣀⢀⠆⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣸⡄⣶⢠⡟⣼⢃⣴⢇⣸⣿⣷⣶⣿⣿⣽⣿⣿⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢷⣽⣷⣿⣿⣿⣿⣿⣿⣷⣟⣌⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣼⡿⠋⢻⠛⢧⣼⡿⠻⡟⠙⠻⣿⣶⣮⣭⣽⣿⣛⠥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠏⢀⣀⣈⠳⣄⣀⣠⢞⣡⣴⡾⢏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠋⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡄⢠⠄⣠⠄⢸⠇⢀⡄⡤⢠⡆⢀⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⠇⣿⠀⠯⠀⢹⠀⠙⠈⠇⠀⣷⠸⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠋⠘⠀⠁⠑⠘
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⠿⠿⢷⣄
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠈⠉⠙⠛⠿⣷⠄⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⢙⣛⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣋⠝⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠐⠀⠐⠀⠄⢰⠄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⡠⠾⠛⠛⠛⠿⠿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠿⠟⠛⠛⠛⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠠⣼⣿⠟⢻⣿⣿⢿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢿⣿⢿⠛⢻⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠌⠢⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠵⠶⠥⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡄⢀⣄⣨⣤⠤⢤⠤⠤⣴⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀⣸⡄⢳⡀⢻⡍⣧⢠⢷⢠⣸⠀⠀⣿⠀⢰⠀⠈⡙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡴⠚⣿⡘⢧⡘⣷⣈⣿⣿⣷⣾⣾⣿⡇⣴⣿⣴⣿⢀⣰⡇⣀⡠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⢾⣉⢳⣷⣝⣿⣌⢿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣽⣿⣿⣷⣿⣣⡾⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣠⢾⡳⣤⡛⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣾⠻⣦⣍⡺⢿⣶⣽⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠙⠓⠮⠅
⠀⠀⠀⠀⠀⠀⠠⣔⢤⣀⡿⣦⣝⣾⣿⣿⣷⣾⣿⣿⣿⠟⠁⣼⣿⣿⣿⡿⠛⠋⠉⣉⣽⣿⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣯⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣀⣠⠤⠖⢋⡉
⠀⠀⠀⠀⠀⠲⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⢢⠒⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠽⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡿⠿⠟⠛⠉⠀⠀⡀⢤⡓⠆⠉
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠛⠁⠀⠐⠒⠒⠒⠒⠛⠛⠉⠉⠉⠀⣀⠀⠀⢄⠀⠦⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡴⢂⠖⠄⠄⠀⠀⢠⠆⡐⠆⠀⠎⠉⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡠⠴⠒⠛⠓⠒⠛⠦⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠴⠟⠒⠚⠛⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⢀⣤⣴⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠐⢛⠉⠀⠚⠻⠯⠙⠋⠙⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠙⠛⠿⠍⠓⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠀⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⡀⠀⠀⠀⠀⣇⡇⡆⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⠀⠀⠀⠀⣷⣆⣴⢠⡤⢼⡧⣧⣤⣤⣰⣰⣿⢰⣡⢃⢀⡇⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢘⣼⡷⣖⡂⠉⣿⣿⣤⣾⣇⣾⣇⣿⣿⣏⣾⣼⣯⣿⢿⣦⣼⣣⣮⡆⡠⠀⠀⠀⠀⡔⡀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡴⠚⢧⡳⣿⣻⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⢟⣵⢿⣿⣷⣡⡆⡔⡔⣸⠟⡼⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⢑⣦⠚⠁⡙⢎⣦⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣫⣾⣿⣯⣾⡾⣯⢞⡜⡱⣡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣙⢦⣾⣳⣕⢶⣬⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡿⣪⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣺⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣵⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡫⣽⣿⣻⣿⣿⡿⠛⠁⣿⣿⣿⣿⡇⢸⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⠀⠀⠀⠈⣹⠟⠿⢿⣿⠿⠿⠟⠚⠉
⠀⠀⠀⠀⠀⠀⠠⣼⣉⣛⣛⣻⣿⣿⠛⠀⠀⠀⢿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣿⣿⢿⣧⣾⡟⠀⠀⠠⠚⠁⠀⣠⠤⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠧⢭⣭⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⣿⡯⠖⠋⠁⢀⣤⣄⡒⠢⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠚⠀⣀⣀⣀⣀⣤⡤⠄⠙⠻⠿⠿⠿⠿⠷⠛⠛⠋⠉⠀⣀⡀⠠⢔⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠰⠒⠋⠉⡠⢠⢀⠄⡠⢀⠄⠀⠀⠀⣀⠀⠀⢀⣀⠀⠐⠫⡑⠍⠦
⠀⠀⠀⠀⠀⠀⣠⣮⠖⠉⠀⠀⠀⠀⠀⠁⠀⠸⠱⠃⠸⠀⡤⢳⠙⠀⠘⡄⠈⠻⠣
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢲⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⢰⠁⠀⡀⣠⣄⠀⠀⢧
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠀⠀⠀⠷⠾⠿⠤⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠭⠀⠀⡇⠀⣄
⠀⠠⡠⣠⡱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⢄⠀⠀⠀⠀⠀⠀⡠⠄⠊⣀⢴⠏
⠀⠀⡔⠯⢤⠀⠀⠀⠀⠀⡲⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠆⠀⠀⠀⡌⠀⣰⠟⠉⠁
⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠄⠂⠁⠀⠀⠀⠀⠀⠧⠲⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⠀⠀⠀⠀⠀⠀⢀⠼⠜⢿⠉⠀⠀⠲⢀⠈⡘⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢲⠂
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⠴⠺
⢒⠆⠀⠀⠈⢠⠒⠄⢳⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⠒⡉⠀⠀⠀⠀⠀⢨⠂
⠈⠀⠀⠀⠈⠉⠔⢁⡼⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣽⣳
⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠉⠐⠢⠤⠐⠂⠻⡍⠋⠉⠈⠋⠉⠁⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣨⢯
⠀⠀⠀⠀⠀⢠⢊⡆⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠀⠀⠈⠁⠰⠶⡦⣤⡀⣀⡀⠄⡅⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⡱⠀⠀⢠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡓⡾⢿⠄⡀⠀⠈⠀⠀⠀⠊⣠⢄⠙⣆⠀⠀⠀⠀⡁⠀⠀⢿⡺⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡰⡽⠙⣓⣒⢫⠦⡈⠢⡀⠀⠀⠀⠑⠋⡠⢁⡟⠀⠀⠀⠀⠁⠀⠀⠈⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⡇⠀⡎⠀⢀⣀⠀⢈⠴⠇⠀⠀⠀⠀⡎⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⢠⡶⣿⣗⣢⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠇⠀⡇⠘⢊⡡⡄⡄⠀⠀⠀⠀⡌⠨⡀
⠀⠀⠀⠘⠛⢁⣼⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣁⣡⠀⣾⡫⠊⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣗⣷⠈⠀⠀⠀⣀⠄⠂⠱⠀⠀⠈⠉⠉⠉⠀⠈⠙⠟⠛⠂⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⢀⣥⣥⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠤⠤⠄⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠹⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⢀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢸⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣴⣎⡔
⠀⠀⠠⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠐⣾⣿⣿⣿⣿⡟⠉⠙⢻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣿⣿⠟⠉⠉⠀⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⠏
⠀⠒⠶⠿⠿⠟⠋⣽⣷⠀⠀⢸⣿⡟⠿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡏⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠻⣿⣿⡾⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢾⣷⣤⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣴⣶⡾⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢸⠈⠛⠻⠿⠿⠛⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢀⣤⠾⠿⠿⠿⠿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠿⠿⠿⢦⣄
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠈⠙⠛⠟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠉⢀⣀⣀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠈⠙⢿⡁⢿⣙⣋⡿⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⣯⣛⣹⢇⣨⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢀⣤⠾⠿⠿⠿⠿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠿⠿⠿⢦⣄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢶⣶⣦⣄⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣤⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠿⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⣆⠉⠉⢹⡏⠉⠉⢰⠖⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡔⠊⠁⠘⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠀⠋⠈⠑⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠈⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢮⡀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢁⡵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠀⠹⣷⣄⠀⠀⠀⠀⠀⠸⢓⣀⠠⠤⢰⡇⣿⡇⠤⠤⣀⡚⠇⠀⠀⠀⠀⠀⣠⣶⠟⠀⠜⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠈⢻⡻⣦⡀⢀⠔⠊⠁⣠⡀⠀⢸⠁⣿⡇⠀⠀⣠⠈⠙⠢⣄⢀⡴⣊⡿⠋⠀⠀⠀⠀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⠀⠀⠙⣎⠻⣷⣄⡀⠀⠉⠁⠀⣼⠀⣿⣿⠀⠀⠉⠀⠀⣠⠞⣩⣾⠏⠀⠀⠀⠀⠀⠀⠉⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡌⠻⣿⣦⣀⠀⠀⡏⠀⣿⣿⠀⠀⣀⡴⠋⣡⣾⡟⠉⢣⡀⠀⠀⠀⠀⠀⠀⢀⡬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠆⡜⠀⣀⠀⠙⢦⡈⠻⣿⣷⣤⡇⠀⣿⣿⣦⠞⠁⣠⣾⣿⠋⠀⢀⡀⠱⠐⡶⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠧⠄⠀⠀⠀⠀⠀⠀⡘⠀⠘⠛⠀⠀⠈⠳⣄⠈⠻⣿⠁⠀⣿⣿⡇⣠⣾⣿⠟⠁⠀⠀⠙⠃⠀⢣⠀⠀⠀⠀⠀⠀⠠⠴⠆
⠀⡄⠄⢠⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣤⣤⣤⣤⣶⣶⣾⣶⣾⣿⣷⣄⣿⣿⠟⠛⠛⠛⠶⠶⠦⠤⣤⣤⣄⣸⣀⣀⠀⠀⠀⠀⠀⠀⡆⠀⢠⢀⡄
⠀⢠⡔⣋⠀⠀⠀⢺⠒⠲⠿⠿⠿⢿⣿⣉⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣉⣿⣿⣥⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⡿⠿⠿⠿⠖⠒⡷⠀⡠⢈⡃
⠀⠉⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⠉⠉⠛⠛⠓⣲⣶⣶⢾⣿⣿⠈⠻⣿⠿⠟⢿⠛⠛⠛⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡖⠁⠀⠀⠀⠀⠀⠀⢸⠀⢰⣶⠀⠀⢀⣾⣿⠟⠁⢸⣿⣿⠀⠀⣿⣷⣄⠈⠳⣄⠀⠀⢴⡆⠀⡎⠀⠀⠀⠀⠀⠀⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠞⠇⢡⡀⠀⠀⣰⣿⠟⠁⣠⠖⠋⣿⣿⠀⢸⠏⠻⣿⣷⣄⠙⢦⡀⠀⠀⡜⠰⠷⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⠟⢁⡴⠚⠁⠀⠀⣿⣿⠀⢸⠀⠀⠀⠙⠿⣷⣄⠳⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⠀⣴⠟⣡⣞⠉⠀⠀⣤⡀⠀⢹⣿⠀⡿⠀⠀⣤⠀⠀⠈⣻⢷⣌⢧⡀⠀⠀⠀⠀⠀⢤⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⣠⣾⡷⠋⠁⠀⠑⠢⢄⣉⠀⠀⢸⣿⠀⡇⠀⠀⢉⣠⠴⠊⠁⠀⠉⠷⣵⣄⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⢀⡴⠟⠁⠀⠀⠀⠀⠀⢸⡶⠈⠉⠉⠘⣿⢸⡇⠉⠉⠈⠲⡇⠀⠀⠀⠀⠀⠈⠻⢷⡀⢳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠄⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠄⣰⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⣦⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠃⠤⠤⠼⠧⠤⠤⠘⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⢀⡴⠟⠋⠉⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⣠⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣄⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣧⣆
⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⠉⣿⠛⢾⠏⣿⠏⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠹⣿⠱⠿⠋⣿⠈⢙⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣌⣳⣶⣾⣋⣠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣝⣿⣶⣟⣥⠴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⣿⡟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⠁⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠹⣿⠿⠄⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣛⣛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⣤⣤⣬⣙⠻⢷⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⣿⣿⣿⣿⡇⠲⣶⣶⣷⣤⠙⠻⠿⠟⠓⠀⠉⠉⠉⠉⠁⠀⠈⠙⠋
⠀⠀⠀⠀⢀⣠⣦⠀⢠⣤⣤⣤⡀⢰⣶⣶⣷⡄⠙⠿⠿⠟⠃⠈⠙⣩⣭⣭⣤⣄
⠀⠀⠀⠺⠿⢿⣿⡆⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣋⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠛⠛⠂⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣟⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⣀⣀⣠⣤⣶⣾⣿⣶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠍⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣸⣶⣄⢰⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠻⣿⣿⣿⣿⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣟⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣴⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⣿⣿⣿⣿⡇⠀⢠⣄⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣿⣿⣿⣿⡇⠀⣼⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣃⢀⣀⡀⣸⣿⣿⣿
⣿⣿⣿⣿⠀⢀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡀⢸⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠹⠈⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢠⠁⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣏⠀⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠇⠀⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⠉⠉
⣿⣿⣿⣿⢆⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⣧⠀⠈⣿⣿⣿⣿⣿⣶⣶⣶⡄
⣿⣿⣿⣿⣿⣿⣿⡟⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠰⠀⠀⠀⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⡷⣧⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠄⠀⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠃⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣍⡛⠿⣿⣿⣿⠿⠿⠛⠁⠀⠄⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣷⣶⠤⠄⠀⠆⠶⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣄⣀⡀
⠀⢀⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⢤⣄
⠀⠘⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠛⠉⢁⣠⣴⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀⡀
⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠟
⠀⠤⣿⡿⠟⣻⡅⠀⢿⡿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡏⠀⠀⠸⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⢰⣿⣷⣄⡈⢻⣿⣟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠷⣦⣤⣤⣤⣾⡿⠿⠛⠁
⠀⠀⠀⠀⢸⠛⠿⠿⠿⠋⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠑⣔⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠆⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⢳⡀⠀⡤⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢧
⠀⠀⠲⡀⠊⢙⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠠⠘⡄⠀⠀⡰⢹⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢅⠠⠀⠀⠀⡼⠉⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠗⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⣫⠃⠀⠁⠈⠉⠡⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⡋⢀⡤⠖⠁
⠀⠀⠀⠠⠊⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⡮⠚⠉⠀⠀⢄⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⢻⠃⠀⠀⠀⠀⠀⠀⠀⠁⢉⠎⢀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⣀⢀⠎⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡄⠊⠙⠛⠼⠗⠃⠉⠒⢀⣤
⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢀⣀⡤⠔⠈⠁⢸⠀⠀⠀⠐⠶⣤⣄⣀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡃⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⢉⣋⣀⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠄⠈⠀⠀⠀⠀⠘⠨⢀⠀⠀⠀⠰⠀⢀⡠⠔⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠼⠚⠃⠀⠀⠀⠀⠀⠑⢠⠀⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣨⢩⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠔⠛⠉⠉⠁⠀⠀⠉⠙⠻⠿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠉⠉⠁⠈⠉⠉⠛⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠉⠻⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⠀⠀⠈⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠁⠈⠉⠙⠲⠄
⠀⠀⠀⢀⡀⢀⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁⠀⢀
⠀⠀⠀⠢⢭⣽⣿⣿⣿⣶⣤⣤⣀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣭⡥⠄
⠀⠀⠀⠀⠀⠛⠿⠟⣿⣿⣿⣿⣿⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣿⣿⡟⠿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁⠉

⣶⡶⢶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢄⣠⣤⣄⣄⣀
⠀⠀⠀⠀⠤⠒⠒⠛⠋⠉⠙⠛⠿⠛⠿⢿⡿⠷⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠾⢟⡿⠿⠛⠿⠛⠋⠉⠙⠛⠒⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠏

⠀⠀⠀⠀⠀⠀⠳⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⠞
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠛⠛⢿⣿⠟⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠻⢿⣿⠛⡛⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠷⠬⠥⠴⠥⠄⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠠⠬⠦⠬⠥⠾⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⠋⠀⠀⠀⢉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⣷⠦⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠁⣠⠞⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢏⣴⣤⣾⣿⣷⡶⣌⢹⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣷⡀⠀⠀⣴⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟⠻⢦⡀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⢹⢿⣿⣿⠁⠀⠀⠀⡇⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣇⡙⢿⣷⣄⠀⠀⡷⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣙⢻⣿⣿⡄⢻⣿⣶⣤⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⠈⡎⢿⣿⣷⠘⣿⣿⠿⠿⣷⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⡀⠀⢣⠸⣿⣿⡆⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣌⣦⡙⢿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣤⣀⠉⠉⠉⠛⠛⠿⣷⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣻⡿⠿⠛⠻⣶⣥⡀⠀⠀⠉⠻⣷⣄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠉⠀⠉⠁⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣼⠿⠿⢿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣀⣧⡀⣀⢀⠈⠉⠛⠿⢿⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣦⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠿⡿⠿⠟⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⡄⣤⣠⣀⢀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠈⠻⢿⣿⣿⣿⠋⠀⠀⣿⣿⣻⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠻⣿⡿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠋⠉⡱⠎⠠⠌⠉⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⡛⠉⠉⣿⣿⣧⣾⣿⡟⠻⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠙⠶⠿⠿⠞⠁⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠔⢰⠀⡆⠘⡈⠿⠋⠻⠛⠛⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⠖⡪⠭⠤⢤⣾⣿⣭⣙⠛⠛⠶⣤⣙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡟⡑⡠⢊⢁⣶⠾⠟⣛⣛⣛⣿⣦⣌⣲⣽⣯⡆⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣾⡃⢁⣱⠕⢋⠠⢀⣀⣴⠖⠻⠟⣿⣿⣿⣿⣿⣿⣶⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡱⣻⢫⡇⢸⣯⣪⠵⠛⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⡻⠻⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣪⢞⣡⣿⣿⢰⣯⣄⠀⠀⠀⣀⣀⠀⠀⠀⠹⣿⣿⡮⡻⢬⡢⣄⠘⢜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢛⡽⣻⣿⣟⣼⢠⣄⠀⠀⠈⠉⠉⠉⠓⢦⠀⠹⣿⡻⣎⠀⠢⣄⠑⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⡏⣾⣿⣿⡟⠁⠀⠋⠀⠀⠀⠀⠸⠛⠓⠀⠀⠀⣿⣿⢎⠳⡑⠙⣕⢌⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠡⡱⡘⣟⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣷⡑⢌⠒⠬⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⢣⢰⢹⢸⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⢯⢸⣎⠛⢶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠏⠈⠄⣾⠀⠒⡀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⢨⣾⢹⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡄⡌⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠃⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡘⠜⣇⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢸⠋⡇⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣶⣭⣿⣿⣿⣿⣿⣶⠂⠉⠁⠀⠀⠀⠀⠀⠀⢸⢠⠹⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⢁⢿⡿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠁⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⢁⣔⣡⠋⠃⠱⣻⡇⣿⣿⣿⣷⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢷⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⡋⠀⠀⡠⢚⡿⡇⢹⡏⣿⢹⡄⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣴⢋⢴⡏⠀⣰⠏⣴⠃⡼⣱⡋⣸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⡘⡘⣇⠀⣟⡼⢁⠞⡴⣱⢡⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣝⢿⣾⣾⡇⠏⢠⠏⠌⣴⣵⢟⣽⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢹⣿⠀⡆⡎⡈⢸⢋⡕⣩⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⠟⢣⣧⠣⠡⢸⠘⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⣷⢱⡄⣦⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠥⠟⣻⠷⠋⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠠⠖⠚⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠉
⠀⢀⣄⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡛⠉⠁
⠀⠉⠛⠿⢿⡿⢻⡿⢿⣿⡟⡟⠛⠛⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡷⠶⠶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠹⣞⣿⣶⣯⠿⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠟⣸⡇⠀⢀⡝⠻⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⣋⣀⡤⣪⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⡿⠟⠋⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠂⠀⠒⠢⠀⠀⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠀⠀⠰⠆⠀⠀⠀⢻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠇⠁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠄⠠⠔⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠂⠠⢈⣄⡀⠀⠀⠀⠀⠀⠂⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠓⠒⠀⠀⠠⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⡠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀⠀⠀⠀⠀⢀⠔⠉⠐⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠊⠐⠀⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠐⠁⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢉⣙⠛⠳⢦⡀⠀⠀⠀⠀⠀⢀⡴⠾⠛⣉⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⠿⣿⣿⣷⣦⠈⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⣀⣁⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠛⠙⠂⠙⠂⠀⠀⠀⠀⠀⠐⢺⣿⣿⣿⣿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠉⠛⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⠳⠀⠀⠀⠀⠘⠛⠉⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⡿⠟⢟⢿⡷⠄⠀⠀⠀⠀⠀⡰⢿⡿⠿⠿⠿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣈⣉⡛⠓⠆⠀⠀⠀⠀⠠⠶⠛⠋⢉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢻⡟⣿⠆⠀⠀⠀⠀⠀⠀⠀⡶⠿⡿⣿⡿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠈⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠉⠉⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠂⠀⠀⠀⠀⠈⠉⠁⢠⣴⣴⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠻⠛⢉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⠈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⡖⠒⠶⠶⠒⠒⠓⠒⢚⠓⠒⠒⠒⠒⠒⠒⠒⠲⠖⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣬⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⡶⠶⠶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣲⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣧⣥⣤⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⣀⡀⢀⠀⠀⡀⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡿⠛⡿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠀⡭⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠄⣽⠀⣀⣠⠤⠖⡚⠍⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢄⣀⡤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠬⠥⠝⠒⡟⠙⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠸⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠇⣿⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠆⢸⠙⣇⠀⢐⡽⠊⣡⣇⣠⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢬⣢⠸⠙⢢⡇⡈⠉⠙⠟⠂⠄⠁⠘⠰⠊⠿⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⣿⣿⣿⡿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⡱⡑⢙⣯⣿⣿⣿⣿⣷⣦⣙⢟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠑⠑⠀⡟⣽⠟⠛⠛⠻⣟⢻⢿⠎⡞⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡄⠄⣆⢨⢸⡏⠀⠀⠀⠀⠘⠈⡟⡇⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡕⡜⡸⢸⡎⠀⡿⠀⠀⠀⠀⠀⠀⠀⠃⠃⠈⢸⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡜⡰⠁⡸⠁⢸⠧⣤⣀⠀⠀⢀⣠⡴⡆⠀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢈⢄⢎⠄⢠⠟⠶⠆⠉⠀⠀⠉⠀⠶⠚⢄⠀⠀⢿⣞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡴⡱⣣⠊⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡈⢪⡊⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⡜⣸⠝⣡⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡙⢦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⣿⢾⣿⣿⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⢸⡏⣿⡄⠻⣄⠀⠀⠁⠀⠀⠈⠀⠀⣠⡾⡹⡽⣿⣿⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⡎⢿⡜⣟⢄⢽⡄⠀⠀⠀⠀⠀⠀⣠⡿⠡⣱⢳⠇⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢰⠈⠌⢟⠜⢎⢎⢿⡄⠀⠀⠀⠀⡴⡿⠁⣰⢇⠟⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣧⡠⡀⢂⠈⣆⠀⢹⡀⠀⠀⢸⢸⢃⡾⡟⡌⡆⠇⣿⡱⣄
⠀⠀⠀⠀⠀⠀⠠⢒⡼⡹⢀⣿⣷⡑⡈⡆⢹⡀⢸⠃⠀⠀⠀⣏⢺⠇⢰⣇⡁⡀⣿⣯⡢⠑⠦⢄⡀
⠀⠀⠀⠀⠀⠀⢠⢊⠔⡡⢞⣿⢻⣷⣿⡇⣼⡇⠀⠀⠀⠀⠀⢪⡿⠀⢸⠸⣧⡀⠻⡝⢮⡲⣄⢠
⠀⠀⠀⠀⠀⠀⢰⡧⠊⢠⣿⠏⢈⡿⡟⢰⢿⠃⡀⠀⠀⠀⠀⣸⣟⢁⠸⡀⠈⠫⡢⡘⢆⠑⠌⢆
⠀⠀⠀⠀⠀⠀⠸⠀⣰⡿⢃⡰⠋⢚⠵⢋⡈⠊⠉⠒⠂⠒⠈⠀⠘⠃⠣⠑⢄⠀⠈⠈⡾⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠤⠶⠉⠀⠁⠀⠐⠁⠐⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣝⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡲⠄⠀⠀⢀⣤⡤⠤⠦⠤⠀⢿⣷⡀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠹⡷⠀⠀⠀⢰⠞⣿⡶⣦⡤⢸⣿⡇⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⡇⠀⠀⢸⡿⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣠⡿⢡⠃⣽⡗⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⣾⠟⠀⢀⡾⠋⠐⣡⣾⡿⠃⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⢠⣿⠋⢀⡴⢋⠔⣡⡾⠛⠁⢀⣠⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠁⠀⣼⡏⠀⡼⡡⢃⡾⠋⣀⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡧⢡⡟⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠘⣿⣿⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠉⠁⠀⠈⠙⠻⠶⣤⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⢁⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣭⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠸⠿⣿⣿⡟⢻⣿⣿⣿⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⣿⣿⣿⠛⣻⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢾⡿⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢿⠷⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣒⣒⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎⠁⠀⠉⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡀⡠⠚⠉⠙⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⣵⡆⠡⠶⠶⠷⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢀⣠⢤⠶⠰⠶⠲⠶⠦⡤⣄⣀⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠴⣋⡭⠴⠒⣟⠋⠙⠛⠛⠓⠧⢦⣎⣍⡖⠶⣤⣀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⣴⣻⠴⠋⠅⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣶⢤⡉⠲⢥⡀⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣰⡿⠋⠀⠂⠀⠀⠀⠀⠀⠐⠉⣳⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠓⢤⡙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢀⣀⠀⠙⢦⠙⣄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡰⠛⠳⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⢊⡽⠃⠀⠀⠀⠳⡘⣎⢄⠡
⠀⠀⠀⠀⠀⠀⠀⠏⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⣀⠤⠔⠒⠉⢁⢠⠠⣡⡞⠁⢰⠒⢲⠤⡤⠾⣺⠎⠄⠂
⠀⠀⠀⠀⠀⠀⡘⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢾⣩⣄⡆⢐⠊⡄⡰⢈⡾⠁⠀⠀⠀⢻⠏⠀⠀⠄⢹⣇⠆⡀
⠀⠀⠀⠀⠀⠀⡇⡆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠃⢎⣷⣹⡜⠀⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠒⠞⡛⢦⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⡔⡁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠧⠀⠀⡠⠿⡀⠀⠀⠀⢠⠊⠀⢀⠠⢐⠀⢀⡠⡔⠋⠁⠀⠀⠀⠀⠀⠉⠐⠠⠀⠀⠀⠀⢐⢀⡗⡰⡃
⠀⠀⠀⠀⠀⠀⠇⠸⡀⠸⡟⠀⠉⠒⠓⠀⠀⡰⠁⠀⠀⢘⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡘⡱⢠
⠀⠀⠀⠀⠀⠀⠘⡄⠱⡀⢫⠀⠀⠀⠀⢀⠎⢐⣡⡴⠞⠉⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⣟⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠣⡀⠙⢤⠀⠰⣷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠨⢫⣿⠎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⣄⠀⠙⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⠛⡽⠀⠀⠀⠀⠀⠀⠀⢠⠔⡵⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡐⠱⢦⣀⠈⠑⠲⣤⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⣀⡸⠖⡁⢌⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣆⡎⡕⡢⣤⣀⠈⠙⠒⠶⠤⠤⢠⢀⣄⣤⡤⠤⠽⠖⠖⠒⡉⠕⢐⡤⠖⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣸⢩⢓⢺⢦⠤⣀⣂⣂⡂⠂⠀⣂⣀⣘⣨⠤⠶⢊⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠺⠶⠤⣤⣴⣸⣉⣉⣁⣁⣄⣌⡬⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠄⡠⡂⠔⡠⢒⡠⠒⡠⢐⡠⢀⠤⢀⠀⡠⢀⡉⢉⠉⣉⠄⣀⠄⡠⢄⡠⢂⠤⡀⠄⡠⢀⠄⡀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠊⠐⠁⠔⠡⠊⠡⠊⠔⠉⠖⠡⠊⠡⠊⠡⠊⠐⠁⠔⠡⠊⠰⠊⠞⠰⠋⠔⠑⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡤⠴⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡀⠀⢢⣈⣷⡚⡅⢢⠀⣀⠀⡀⢆⢀⢠
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠹⣷⣌⢢⣔⠙⣌⣿⣞⢆⢳⡹⣧⣳⡘⣾⣾⣼⡄⡆⡆⡀⠀⡅⢠⠁
⠀⠀⠀⠀⢄⡀⠀⠈⠻⢦⣄⡀⢀⠑⠢⣝⣿⣷⣮⣓⣮⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣆⡇⡿⢠⢆⡔⡀
⠀⠀⠀⠀⠀⠙⢷⣦⣤⣀⣈⣙⠛⠿⣶⣦⣭⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⢉⣽⠿⢿⣿⣿⣿⣿⣿⣿⣇⣾⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣰⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣏⠛⠻⢿⣴
⠀⠀⠀⠀⠀⠙⠲⠶⢶⣤⣤⣤⣶⣶⡾⠿⠛⠋⣁⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡁⠀⠀⠀⠀⠀⠀⠘⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣧⡶⠟⠁⠀⠀⠀⠀⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠿⠻⠿⠿⠶⠦⠤⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⣀⠴⠚⠋⠉⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⡿⠿⠛⠉⠉⠀⠀⠈⠉⠙⠳⢤⣀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠻⢿⡄⠀⠀⠀⠀⢠⣿⡟⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡀⡀⢂⢄⢰⡀⠀⣷⢀⣄⡆⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⢠⢠⣀⡀⣶⠀⢀⡆⡀⡐⠀⢀
⠀⠀⠀⠢⡐⢜⣾⣦⡻⣿⣷⣶⣿⣾⣿⣿⣾⣣⢎⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡱⣼⣿⣿⣿⣷⣿⣤⣾⣿⢟⣵⣳⣣⢂⡄
⠀⠀⠀⠲⢭⣲⣭⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣯⣶⣫⠔
⠀⠀⠀⠉⠲⠿⢿⡿⠋⠁⢻⣿⣿⣿⣿⣿⣿⠃⠈⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠘⣿⣿⣿⣿⣿⣿⡿⠈⠙⢿⡿⠿⠖⠋
⠀⠀⠀⠀⠠⢚⡭⢛⣦⣄⡈⠻⢿⣿⣿⠟⣁⣠⡴⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⡘⠿⣿⣿⣿⠟⢁⣠⣴⡿⢍⠳⠄
⠀⠀⠀⠀⠀⠈⠰⢫⣾⢿⢻⢿⣷⣶⡿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣷⣶⣶⡿⡿⡿⣿⡙⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠑⠈⠈

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⣿⡎⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠳⣦⣀⠀⢠⣿⣿⣿⣇⠀⢀⣠⡶⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣀⢀⡀⣀⠀⢸⠇⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⣿⠀⠀⣀⣤⡄
⠀⠀⠀⠀⢸⣾⢷⡟⠀⠒⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠂⠀⣸⣒⡂
⠀⠀⠀⠀⠀⠁⠈⠁⠀⢸⡇⠀⠀⠀⠈⠉⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⡿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣴⣿⠿⢻⣿⣿⣿⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠼⠋⠁⠀⠘⣿⣿⣿⡏⠀⠀⠙⠳⠄⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣄⢿⢃⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⡛⠋⠉⠉⠉⠉⠛⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⠿⠛⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⡓⠦⣀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣍⠂⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣷⣦⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⠻⣿⡿⣿⣦⣀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠿⠿⠗⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣌⠻⣮⡻⣿⣦⡀⠀⠀⠀⢹⣟⣭⣵⣶⣾⠿⠿⣏⣙⢦⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣎⢻⣷⠈⣷⣿⣿⣷⡀⠀⠀⠀⢻⣿⡿⠿⣶⣖⡒⠾⣿⣷⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠀⢋⣿⣿⣿⣷⡀⠀⢤⣌⣿⡄⠀⠘⢢⣝⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⢿⣿⣿⣷⢸⡘⣿⣿⣿⢦⡀⠀⠀⣠⠍⡿⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⠿⢿⣏⢷⣿⣿⣿⣎⢿⡆⠘⠁⠀⢀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⣿⣿⣿⡿⣟⣾⣷⡄⠀⠐⠻⣤⣈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠻⣿⡃⠹⣿⡏⣰⣂⣀⣐⣿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⢿⣯⣉⣝⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⢷⡀⠹⣿⠇⢉⣉⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣦⣝⢿⣶⣶⣬⡻⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣷⡀⣹⣧⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣽⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣌⢿⡻⢿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣍⠳⣍⡻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⡻⣿⣿⣿⣷⣌⠻⣦⣝⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢝⠿⣿⣿⣷⣌⠻⣷⣬⡻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠙⠻⣿⣷⣌⢻⣿⣮⣽⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣝⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠥⠄⠐⠒⠈⡄⠀⠀⠀⢀⠰⠶⠤⢠⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡨⢃
⠀⠀⠀⠠⠄⡀⣀⠠⠂⡤⣄⠁⠀⠀⠀⢨⣠⣄⠀⠀⣠⡰⠄⠀⠀⠀⠀⠀⢀⢤⣝⡄⣎⣡⡞
⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠈⠐⠚⠁⠘⠃⠀⠀⠀⠀⠀⠢⣻⡿⠿⠻⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⢴⡅
⠀⠀⠀⠀⠠⠤⠄⠠⠄⠀⠄⠀⠤⠤⠵⠦⠤⠤⠤⠤⠤⠄⠀⠀⠀⠄⠳⠆⠀⠐⠒⠒⠒⠐⠆⠀⠀⠇
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣰⠿⠛⠛⠛⠿⢿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠿⠛⠛⠛⠿⣆
⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠛⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠛⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⢀⣤⣼⣿⣿⣿⣾⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣷⣿⣿⣿⣧⣤⡀
⠀⠀⠀⢾⣿⣿⣿⣿⠿⡿⢿⣋⣿⢿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⡿⢿⣉⡿⣿⠿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠉⠛⢿⣦⣤⣀⣀⣉⣁⣀⣀⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⣀⣀⣉⣀⣀⣤⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠙⠋⠉

⠀⠀⠀⠀⡠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀
⠀⠀⠐⢋⡤⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠱⣌⠑
⠀⠀⠀⠁⣠⠞⢡⠎⢙⠲⠦⢤⣀⣀⣀⣀⣀⡠⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⣄⣀⣀⣀⣀⣠⠤⡴⢚⡉⢦⠘⠦⡀⠁
⠀⠀⠀⠀⠀⠠⠋⢠⠏⢠⠇⢰⠃⣾⠀⡆⢰⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢰⠆⡆⠸⡆⢺⠀⢷⠈⢣⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠋⠀⠇⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢃⠀⠃⠀⠃⠀⠁







⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣦⡙⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣈⠳⣌⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠳⣌⠳⣌⠳⣌⡓⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⢲⣬⡓⢮⡛⠦⣝⣓⠶⠾⢽⣗⣲⣲⢿⣯⣙⣿⣿⣿⡟⢃⣿⠝⣳⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⡿⣿⢠⣝⡲⢭⣓⠦⠭⡭⠭⢽⣛⡶⢽⣛⠓⣶⢃⡴⢋⡴⢋⣥⠞⣋⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⢣⡏⡇⣍⣓⠮⣝⣓⣒⡯⢽⣲⡭⢡⢏⡞⣱⢏⡴⢋⡴⢋⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⡾⡼⡼⡽⡬⣝⣒⠶⢖⣪⡭⣰⢳⢏⡞⡼⢡⠟⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡇⡇⣇⢳⢳⢱⡹⡨⣭⣵⢰⢣⡇⡏⡾⢸⢣⡏⡾⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⣸⢃⡇⡾⣸⢸⢣⠇⢟⣛⡊⣎⢇⢧⢧⢹⢸⢸⢇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⣴⢣⡞⡼⣱⢣⠏⣚⡫⠵⠶⠭⣝⣘⣞⡞⡞⡼⡾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣡⠞⣡⠞⣱⢏⡼⣱⣃⣚⠯⣗⣺⠭⢭⣝⡲⢮⣙⢸⣱⢳⣇⡏⡇
⠀⠀⠀⠀⣤⠶⠛⣩⡴⢛⣡⠞⣡⠞⢡⠯⢤⣍⣓⠮⢭⣗⣒⣒⣒⠲⢭⣓⠮⣝⠃⡿⣾⠿⡇
⠀⠀⠀⠀⠈⠐⠻⢯⣔⣻⣡⣼⣿⣿⣿⣍⣻⣷⠼⠯⣽⣗⡲⠶⢭⣝⠲⣍⡳⢬⡛⠧⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⢮⡙⢦⡙⢦⡙⢦⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡙⢦⡉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠻⣌⠻⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡪⣰⢅⡢⢔⣨⣴⣶⣿⣭⣭⣥⠶⣒⣩⣤⣶⣤⠤⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣠⣢⣾⣿⡿⡫⢖⣥⣾⣿⢿⣿⣿⣿⣟⣿⠶⢛⠭⢟⣫⣥⠶⣶⠿⠿⣻⡻⠯⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣽⣿⢟⣿⣿⣯⣾⣿⢟⡫⣚⠽⠿⠛⠉⠉⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡇⣼⡟⣼⡿⣱⢫⡿⠻⠫⠋⠐⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣴⣼⣿⢡⢿⢠⠑⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡄⣿⡿⣿⡟⡏⠈
⠀⠀⠀⠀⠀⠀⢸⡇⡿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢒⠀⡍⡅⡀⢠⠀⡇⢠⠀⡠⢂⢀⠀⡠⢂⡀⣀⠔⡠⠂⠀⣀⣠⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢄⢢⣣⣧⣧⣷⣿⣾⣷⣷⣾⣿⣷⣷⣾⣶⣿⣾⣷⣾⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡀⣐⣠⣤⣷⣾⣿⣿⠿⢿⣿⡟⠻⣿⡛⠛⠛⡿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣐⣺⣿⣿⠿⠛⠙⡇⠀⠀⢻⣿⣷⣾⣿⠃⠀⠀⡇⠀⠀⠀⢀⣴⡿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⣥⣿⠟⠉⠀⠀⠀⠀⠱⣄⠀⠀⠈⠉⠉⠁⠀⢀⡜⠁⣀⣤⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣤⣤⣾⡭⢶⠛⠫⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⣀⣀⣀⣀⣤⢤⡤⠤⢶⠒⠺⡙⢋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠉⢹⠛⡁⠉⣀⡈⠀⠤⠘⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠚⠋⠩⠟⣛⠶⢒⡰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠛⠡⠚⠛⢿⣭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣷⣿⠅⠀⣠⠏⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢁⠜⠋⢀⣤⡞⠁⠀⠀⠀⢀⠈⠀⠀⠀⠒⠃⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡦⢖⠚⠿⠿⠒⢒⣡⣶⣿⣿⣿⢗⡤⣊⣭⡍⠀⠀⠀⠠⠐⠀⡀⠀⠀⠀⠲⣦⡀⠀⢸⡇⠐⣆
⠀⠀⠀⠀⠀⡀⠀⠀⠠⠋⠀⡮⠊⠅⠈⠀⡰⢞⢫⣵⢿⠋⣻⣯⡟⡿⢟⢿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠰⠃⠀⠀⠉⠀⣸⡇⠀⣿⣦⢄⡀⡀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠡⣚⣵⠟⢡⡇⣰⠜⢋⡈⠀⣠⠂⠀⠀⠀⠀⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⣟⠏⠀⣻⣏⢳⡍⠻⢯⣽⣒⠤
⠀⠀⠀⠀⣇⠣⢄⣀⡤⠖⡁⡀⠀⢀⠥⣪⡿⠃⣰⢻⣷⡕⡵⢋⡴⣾⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡙⠀⣸⠿⡜⢧⡙⠦⡀⠁⠪⡙⠲⢌⠢⡀
⠀⠀⠀⠀⠈⠻⢿⣟⣛⣩⡥⢴⣺⣷⢟⡿⢁⡼⣡⢿⣿⢉⡔⡇⣼⠇⡇⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣸⣿⠀⠹⡌⢻⣷⣌⠳⣄⠈⠣⡀⠀⠘
⠀⠀⠀⠀⢀⠠⣰⠴⣋⠥⣊⠽⠋⣠⢞⡴⢋⠜⣡⠋⡟⣸⢐⣿⠏⠦⡇⢸⠻⢷⣶⣦⣤⣤⡀⠀⠀⠀⠀⢀⣴⣿⡀⠙⠯⡏⠘⡤⢳⠈⢻⣏⢳⣌⢣⡀⠘⡄
⠀⠀⡠⣪⢕⣫⠶⢋⡥⠚⠁⣠⣾⢗⠉⢀⡴⢺⠇⢰⣸⣶⡏⣿⣿⠀⢣⣹⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⣫⡿⠿⠛⠀⠶⣖⣫⠃⣼⠂⡆⣟⢄⠱⡀⣧⠀⢰
⢀⢎⡼⢓⠟⣁⠔⠁⢀⣤⣾⠿⣂⣥⡾⠋⣰⢻⠀⣿⣿⡟⣼⣿⠸⠣⠀⠻⡀⠀⠀⠀⠀⠀⠐⠁⢀⣴⣿⡿⠀⠀⠀⠀⠐⠀⠑⣆⣿⣜⣵⡻⢸⡄⣷⢸⡆⢰⡀
⣈⡞⢠⢋⢜⠔⣠⣶⣿⣿⡶⡻⡫⠡⠁⢰⡇⢄⢢⣿⣿⡌⣾⡍⡄⢡⠡⡀⢱⠀⠀⠀⠀⠀⠀⣠⢻⡟⡜⡅⠀⠀⠀⠀⠀⠐⢄⠘⡿⡫⡻⢠⣿⢃⣯⣞⣥⡗⡘
⣿⢀⡯⠂⣡⣾⠿⢋⠕⢁⡾⢱⠃⡇⢀⣺⠿⠛⠉⠉⠩⢇⠋⢳⣷⠀⠳⣱⣬⠀⠀⣀⢴⠁⢰⣇⠻⢯⣱⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡴⢃⣾⡷⢟⠿⢛⡏⠃⠑
⠏⢸⡟⡜⣱⢳⡱⠁⢠⢺⢁⣿⠀⡇⣾⠁⠀⡀⠔⣸⠓⢹⡆⠀⣿⡣⠀⢻⡿⠑⠋⡰⢾⣠⣾⡷⣓⡣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣯⢴⣚⠁⣠⢎⡏⡀
⠀⢸⡟⢰⢿⣼⢀⠀⡟⠈⢸⢸⡆⣿⡇⢠⠎⠀⢀⣧⡀⠌⠣⢠⢸⡡⢀⢸⠃⢠⢊⠔⢉⢷⠹⣄⡕⠁⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠾⠾⠵⠾⠛⠁⠐
⠀⠘⣇⢸⠈⣧⣾⢸⡇⠀⢸⠘⢱⣸⡇⠈⠀⢔⢤⣶⢱⡄⢢⠐⣼⠄⢸⢸⡆⣴⠡⢊⡴⢫⠗⢬⠁⠀⠈⠂
⠀⠀⠘⣎⡀⠘⢿⡘⣷⠀⠈⡇⢇⢿⡇⠀⠀⠸⡇⢿⡉⣿⡀⣳⡿⡀⣨⡿⣇⢁⠀⣿⠏⡠⡢⠃⠀⠀⠀⡇
⠀⠀⠀⠈⠳⠄⠈⠳⣿⣆⢣⠹⡼⣼⡇⠀⠀⠀⠘⠀⠳⡌⢉⣽⠞⣰⢿⠀⠈⢿⣦⡑⠚⠟⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣷⡹⣿⡇⠀⠀⠀⠀⠀⠀⣸⣛⠳⠊⠛⠘⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇⠀⠀⡀⠀⠀⢀⠝⠀⠀⠀⠀⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣠⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠋⠉⠙⠛⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢿⣿⣿⣿⡿⠃⠀⠀⡀⠉⠉⠛⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣧⣤⣀⠘⢿⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢟⣭⣭⡙⢻⣷⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣯⣿⠀⠀⠀⠀⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡿⠋⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠙⣿⣿⣧⣀⢀⣠⠤⡀⠀⢀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣿⠇⡀⠙⢿⣿⣿⣿⣧⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡾⢳⣌⢢⠀⣿⣿⣿⣿⠿⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠛⠛⢟⣡⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣄⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠄⠀⠤⠄⣿⣶⡝⢿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠁⠀⠈⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠻⣷⣬⡻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠙⠛⠀⠉⠛⠃

⠀⠀⠀⠀⣀⣠⣤⡶⢶⣿⣶⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣷⣶⣿⠶⢶⣤⣀⣀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠳⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠻⠿⣿⣟⠛⡉⢻⡿⠻⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠛⠻⡿⠉⠛⣻⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠚⠓⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠐⠚⠓⠒⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⡳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⡀⡀⠙⢿⠻⣿⡶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢴⣿⠟⠿⠋⢀⠀⡀⣀
⢠⣄⡐⣆⣤⣰⣴⣶⣿⣿⣿⣾⣷⣿⣷⣧⡈⠙⠻⣍⢍⡛⠻⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣾⠟⢛⡩⡩⠟⠋⢁⣼⣾⣿⣾⣷⣿⣿⣿⣶⣦⣀⣤⣰⢂⣠
⠀⠹⣿⣿⣿⣿⣿⡿⠿⣿⣿⡛⣻⣿⡿⢿⣿⣆⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⢀⣰⣿⣿⠿⣻⣿⣿⠛⠿⠿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠩⢿⣿⣧⡀⠀⠈⢟⣧⣈⣹⡿⠀⠋⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠙⠻⣯⣈⡉⣛⡡⠀⠀⢀⣼⣿⡿⠍
⠀⠀⠀⠀⠀⠀⠿⢿⣦⣄⣀⠉⠉⠋⠁⣀⣀⣀⣤⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣀⣀⣀⠈⠉⠉⠉⣀⣠⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠻⠿⠿⠟⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠿⠿⠟⠛⠛⠟⠃








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣦⣤⣤⠤⣄⣀⣀⣀⠀⠀⠀⢀⢀⣀⣀⣠⣤⣤⣤⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⣿⣶⣶⣤⣭⣙⠛⣿⡿⠛⣛⣭⣭⣷⣾⣿⣿⣯⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡝⢿⡿⠿⠿⢿⣿⠿⠿⢿⡿⢟⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣷⣤⣀⣀⣈⢁⣀⣀⣀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣿⣿⣿⣿⣿⣿⢿⣟⣻⣿⣽⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣻⣿⣿⣿⣟⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣦⣀⠀⠀⠀⠀⠈⢿⣷⠀⢀⣤⠀⣿⣧⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠙⢿⣷⡀⠀⣠⡆⢸⣿⡆⣼⣿⠀⣿⡿⢀⣾⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠉⢻⡆⢀⣄⠀⢻⣿⡀⣿⡇⠸⣿⠁⢹⣿⠀⣿⡇⢸⣿⠁⣴⡿⢃⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢤⡼⠃⠘⣿⣆⠀⢻⣧⢹⣿⠀⢿⡆⠈⢿⡄⠹⣇⠘⣿⠀⡿⠁⣼⡟⠁⠈⢻⡇
⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⣤⣀⠘⢿⣦⡈⠻⣦⠙⢷⣌⠳⢤⡀⠙⠂⢈⣃⣈⣀⡙⠀⠸⡇⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠻⣿⣦⡀⠀⠙⢿⣷⣤⣌⣙⣂⠈⠛⠢⠈⣁⣤⣶⣾⣿⣿⣿⠿⠿⠿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠘⢦⡈⠙⠿⣷⣦⣤⣤⣭⣭⠭⠒⣀⣤⣶⣿⣿⠿⢟⣫⣭⣤⣄⡀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣈⣩⣤⣴⣶⣿⣿⠿⠟⠋⢁⣴⡿⠋⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⢸⡏⠀⠀⡞⠁⠀⠀⢿⣧⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢻⣤⣀⣤⣿⠇⣠⣴⠟⠁⠰⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣀⠀⠀⠀⠀⠀⠈⠓⠦⠀⢈⣉⣉⡷⠟⠛⠁⠲⣶⣤⡈⠻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠟⠋⣀⣀⡉⠛⠻⠿⢷⣶⣶⣶⣶⣿⠿⠿⠛⠋⠁⠠⣤⣄⡀⠈⢹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⣉⣤⡶⠒⠀⣀⡄⠀⣠⠀⠀⢰⡄⠀⠹⣦⡀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⣠⣾⠟⠀⣼⡇⠀⢀⣿⡇⠀⠀⣿⣧⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠃⠀⠠⣿⡇⠀⢸⣿⠀⠀⠀⠙⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠛⠁⠀⠀⠻⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣶⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠿⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠉⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠬⠓⠇⢿⡁⣇⠸⠸⠳⠇⣟⠀⠸⠺⣉⠸⣩⠾⠳⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠁⢀⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢤⣿⣿⣧⣄⡷⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⣿⠑⠈⠉⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠚⠀⢾⣿⣟⣷⡀⢀⣴⣿⣿⣿⠀⠑⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⣉⢀⠰⢚⣩⣿⣿⣿⣿⣿⣿⣍⡗⠂⡀⣅⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡾⠶⠌⠈⢯⣿⣍⣹⣯⡼⠋⠡⠶⢷⣿⣧⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣷⣌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣡⣾⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠅⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠚⠛⡿⡿⣿⣿⣿⣦⣶⠛⠓⠒⠟⢳⣴⡿⣿⣿⢿⡿⠋⠃⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠀⠀⢹⡇⠈⣼⣿⣿⣿⠆⢠⡤⠰⣿⣿⣿⣧⠁⢸⡟⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⡀⠈⣅⣸⣿⣮⡙⠛⢷⠦⠤⡾⠛⢋⣱⣿⣇⣸⠁⢀⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣮⣠⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⣄⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣍⣤⢿⡻⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣤⣉⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣿⣆⠁⠹⢿⡅⠂⠐⢨⣿⠏⠉⣠⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠙⠁⠀⠀⠙⠄⠀⠞⠁⠀⠀⠋⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣻⠟⠻⣿⣿⣛⠿⢟⣻⠿⠟⠿⢿⣿⢿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠞⠁⠀⠀⠈⠙⠿⣿⣒⠢⠭⢽⣒⡶⢬⣳⣝⣟⣿⣿⣿⣿⣿⣷⣮⡢
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢲⣤⣭⢿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⣠⣴⡶⠟⠓⠂⠀⠀⠀⠉⠻⣿⣿⣿⣯⡂⠈⢻⣿⣿⣿⣿⣜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢀⡀⠀⠘⢛⡁⣀⣠⣤⣀⣀⠤⠒⠈⠁⠈⢿⡿⣿⣿⣄⢸⣿⣿⣿⣿⣧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣇⠸⠛⠉⠀⡜⠁⠀⠀⠘⠏⠁⠀⠀⠀⠀⠀⠀⠘⠁⠸⢷⢻⣿⣿⣿⣿⣿⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢿⠉⣳⢮⡷⢇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡞⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣺⠉⠈⠀⡇⠀⠲⢄⣀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⣀⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠇⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⠀⠀⠑⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⣿⣿⣿⣿⡿⡝
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢃⢳⡀⠀⠀⠀⠢⡀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠰⡇⠁⠿⣿⣿⣿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠻⣄⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀⢀⡞⠀⠂⠄⠀⠀⠀⠀⠀⠓⢸⢃⠘⣿⠁⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠏⡜⣀⢣⢣⠰⣿⣟⣞⣷⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢸⡇⠠⢸⡄⠀⠀⠂⠀⠐⣡⢎⠈⠐⢿⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⡘⣡⢫⡌⣎⢦⠹⡜⣞⢦⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠀⠐⢨⠀⠄⠃⠜⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠡⠙⢯⡜⡌⢆⠳⡝⢎⠳⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⡈⢞⣜⢦⡑⣜⠦⣑⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⢈⣽⣿⣶⣶⣦⣤⡕⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢆⠻⢷⣝⣮⡓⢬⡓
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰⠐⠂⠀⢉⠀⡀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⢄⡙⢿⣟⣲⢬
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⠀⠀⠀⡀⠀⠢⠀⠈⠄⢀⢈⣅⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣦⣍⠢⢍⡑⠻
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⢀⠣⣄⠀⠈⠀⢀⠀⠀⢀⣴⡾⠋⢀⡀⢴⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡎⣿⢻⡟⣶⣬⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠈⠑⠂⠀⠀⠀⣠⣿⣿⠃⠀⠁⠀⣼⡀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⢸⠃⡇⣽⡈⡇⣏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣿⣿⣿⡁⣀⠲⣈⠰⣿⠀⠠⣤⣔⠶⣯⠀⠀⠀⠀⡀⠐⣒⣤⢌⡓⠤⣙⠳⣄⡀⠀⠀⠀⢸⠀⡅⣿⣧⡄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⣶⠏⠸⢻⣝⢚⣤⣍⠀⢸⡟⠒⠂⠒⠉⠄⣐⠀⠀⠀⠀⠀⠀⠉⡛⠿⣟⣷⣄⡑⠆⡙⢦⠀⠀⢸⡄⡇⢻⣿⡇⡏⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠍⠀⡀⠡⠈⠠⠉⠉⠀⡀⢾⠇⠀⠠⠤⠀⣀⠉⠀⠀⠀⠀⢀⡐⠂⠤⣓⠤⡙⠿⡻⣦⡙⢆⠳⣄⠈⣇⢣⠘⣿⣧⢱⢳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠘⢿⣥⠀⠀⠀⠀⠀⠀⠐⠮⠕⡂⢽⣆⠀⠤⠈⠁⣂⠨⠭⠥⠬⠭⢔⣒⠩⣐⢄⡑⢎⡲⡙⢿⣫⡤⠱⠘⢆⠸⡌⣆⠹⣷⢇⠳⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠜⢳⡚⠐⠀⠀⠀⠀⢤⣒⣮⣽⣿⠁⣐⣓⣂⡀⠤⠐⣛⣛⣛⣓⡢⠭⣛⠮⡓⢝⠢⡑⡜⢆⠙⡵⢃⠀⠈⢆⠹⡌⢆⠻⣯⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠹⣢⠔⣀⣴⣞⣿⣟⠓⣽⣿⣧⠶⠶⠶⢶⣞⡛⣯⣭⣭⣿⣻⢷⣮⡳⣌⢦⠓⡱⡘⡆⢀⢱⢫⡆⠠⠸⡀⠙⣌⠳⡜⠷⣮
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠄⠉⠀⠀⣙⡳⠮⣛⣿⣟⠻⠿⠿⠿⠷⠾⣙⣻⠿⢿⣛⢿⣿⣟⢷⡜⣆⢡⢱⢣⠀⠄⡄⡞⣱⠀⡆⢇⠀⠈⠳⢌⠲⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡶⣼⡻⢯⣿⣇⣬⠉⠉⠉⠉⢭⣄⠒⢾⡖⢬⠳⣹⣿⡼⣷⠸⡇⡇⡎⠀⢸⢠⠀⡟⡀⢠⢸⠀⠀⠀⣼⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⠤⢬⡛⠦⠝⠛⠻⡛⠶⢛⣛⣛⣛⠶⠮⣹⣿⢿⡇⡆⣿⢸⡃⣿⠀⠇⡇⡇⡄⢸⢸⢰⡇⡇⢸⢸⠀⠀⢸⠃⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡠⠬⠛⡳⠒⠒⠚⠛⠋⣅⠪⢒⠕⣠⣻⣼⢇⢿⢀⡆⡇⡇⠁⢨⢸⠸⡇⡇⢸⢸⠃⠀⢸⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡩⢔⠲⢬⣭⣉⡉⠉⠛⠓⠮⢑⣡⢞⣵⡿⣫⣾⢃⡞⡴⠱⠸⠀⡌⡆⣦⢿⠀⡄⡼⠀⠀⢸⠘⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢲⣸⡕⢊⡥⣒⢲⣶⣦⣤⣭⣽⣻⡾⡿⢫⣾⠟⣡⢚⠞⡡⣣⢃⡆⡔⢰⡝⡎⣰⢡⠇⠀⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⣒⡲⠶⣶⣶⣦⣤⣿⣿⣃⡔⣡⣶⢿⣛⣿⣿⣯⣽⣭⣭⡷⠶⢛⣡⢞⡵⡡⢊⠔⣡⢎⠌⣰⡷⡝⣠⢀⡎⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠔⠉⡳⠭⢛⣒⣶⣿⣿⣿⣿⣿⣷⣿⣎⣿⠿⢋⣭⣔⡲⠶⠶⠶⠶⢯⡝⠒⠥⠂⠄⣡⠊⠕⣣⣾⣿⠟⡱⢣⡞⠀⠀⠀⠀⠀⠀⢣⡘
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⢀⣡⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⡽⠓⠈⢉⡉⠉⠐⠒⠒⠂⠉⢂⡠⢀⠁⠀⢄⢙⠿⣩⠞⣵⠋⠀⠀⣀⣀⣠⢴⣓⡈⠿
⠀⠀⠀⠀⠀⢀⠀⠈⡄⠀⢀⢮⢾⠟⣿⣿⢟⣻⣭⣭⣭⣿⣿⣿⣿⣧⣿⡐⡴⢉⠄⠒⠤⠤⠤⢤⣬⠍⠐⠂⣀⠠⢀⠼⣡⠈⡥⠛⠀⣠⠔⣭⡔⠂⠲⣝⢶⣭⣙
⠀⠀⠀⠀⠀⠈⠀⡇⢘⠀⡞⣇⢣⡸⢏⡴⡫⠛⠂⢉⠉⠉⠛⣿⢿⣿⣷⣿⡐⡑⢊⢤⡐⠶⠶⡶⢶⢶⣬⡙⠢⢘⠠⢀⡠⠊⢀⡠⢖⣵⣮⢖⡮⣝⡛⣿⣛⡢⠬
⠀⠀⠀⠀⠀⠀⢠⢁⡄⠀⠇⣽⢈⡇⢸⢱⠁⡀⠊⡴⢮⢙⣻⣿⣿⣿⣿⣿⣿⡀⣾⡷⣪⠍⢒⣒⣒⣒⣀⣀⣭⡥⠖⠉⠀⡴⢫⢞⢡⠾⣿⡗⣩⠖⠒⣿⣗⣈⣭
⠀⠀⠀⠀⢀⠴⡻⣿⣿⡧⣼⣧⣾⣧⡾⣸⢀⡇⢸⢸⠰⣾⣟⢿⣿⣿⣿⣿⣿⣿⣿⠐⣡⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⣕⣇⢯⣻⢝⣻⣿⡕⣨⣤⣾⣿⣟⣛
⠀⠀⠀⢀⢇⢺⡁⡔⣰⣿⣿⢿⣿⣿⣿⣷⣮⣷⣄⣿⣸⣿⣥⡆⡞⠋⠛⠿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣱⣋⠖⣩⣿⣟⣽⣲⣿⣷⠺
⠀⠀⠀⢸⢸⡇⡇⠀⢻⣿⡟⣿⢣⡿⡟⠙⡁⣰⢟⢿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⢋⠼⣻⣿⣿⣧⣾⡟⣹⣿⣟⣭⣿⣟⣛
⠀⠀⠀⠀⢯⣾⣵⣆⢿⣼⣧⣿⡸⣿⢇⢷⡁⣵⢀⠜⣷⢻⠉⡽⠛⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡤⢺⡝⢾⣿⣿⣮⣵⣮⣿⡝⡿⣿⣿⣿⣟⣽⣿⣯⣾⣷⣶
⠀⠀⠀⠀⠀⠉⠛⢿⢿⣿⣿⣿⣿⣿⣿⣞⣷⣜⢧⣜⣿⣾⣷⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⢫⣾⢿⠿⣿⣿⣿⣿⡧⢞⠝⣹⣿⣿⣿⣿⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣸⡄⣅⢸⣿⡟⡻⢿⣿⡟⠳⢻⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣻⡜⣧⢣⡞⡱⢫⣽⣿⣿⣶⣾⣽⣿⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣼⣥⣹⡙⢁⣾⢯⣿⣽⡟⣼⣄⠂⠸⢸⠁⠁⡿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡴⡻⢿⣿⣿⣿⣼⣏⣠⣷⢫⣿⡿⣿⣿⣿⣟⣫⣾⢟⣿⣿⣿⣿⡏
⠀⠀⠀⡰⣺⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣼⣽⣅⣀⢀⣞⠄⣠⣳⡜⡾⢩⠇⠀⠀⠀⠀⠀⢸⡀⣇⢻⡹⢏⡟⡿⢿⣿⣿⣿⣏⣿⡟⡽⣻⣿⣿⣱⣫⣿⡟⣼⣿⣧
⠀⠀⣼⢱⡹⡝⢁⣾⢿⣿⣿⢟⣿⠿⠛⡩⣹⡿⢿⣿⣿⣿⣿⣿⣽⣷⠏⠀⠀⠀⠀⠀⠀⢸⣧⡏⣼⣇⡾⣸⢰⣿⣯⣿⢿⣿⣿⣿⣾⣿⣫⢿⣿⣿⣯⣼⣟⣿⣻
⠀⠀⣷⢸⠇⡇⢸⣼⡏⡟⣿⢸⢌⠀⡄⠰⢹⠀⣞⣷⢳⠁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣽⣿⢻⣿⣿⣧⡝⠻⠿⣯⠞⠁
⠀⠀⠹⣎⡾⣵⣄⢯⡾⣷⣿⣎⢾⡶⡷⢦⢻⢶⠼⡿⣿⣷⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧⣻⣿⣿⣯⡟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⢸⢠⡇⣿⢸⣷⣇⣿⣿⢸⡇⡆⢹⢘⠰⠀⡇⣿⠰⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢯⣿⣿⣿⣟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣛⠁
⠀⠀⠀⣷⡇⣿⡿⢸⡇⣿⢿⡏⣸⡇⢉⢸⢸⣶⠀⣷⢸⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡏⣿⢻⢻⡏⡟⣿⣿⢻⡟⣿⣿⠛⢻⡟⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠿⠿⢿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠿⠿⠿⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠈⠙⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠉⠉⠀⠈⠉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⡖⢻⣻⣿⣶⣦⣧⣼⣴⢦⣦⣢
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⡶⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣐⠲⠶⣞⣽⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢯⡀⠿⠿⢃⡇⠀⣹⣿⣿⡛⠁
⠀⠀⠀⠀⠙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠶⡟⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣝⣒⣚⣭⡴⠾⠟⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣩⠿⣿⣿⣿⢿⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣦⣷⣼⣦⣶⣶⣶⣤⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣮⣶⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⣤⣞⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⣶⣿⣿⣿⣿⡿⠛⠉⢹⣿⣿⠋⠰⣶⣿⣿⠉⠉⣿⣷⠉⠻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢿⣿⡀⠈⠙⠛⠋⠀⢐⣿⡇⠀⠀⠙⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠽⡷⣄⡀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣠⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠉⠗⠦⣤⣤⣀⣀⣀⡈⢙⠛⠿⠟⢋⣥⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠉⠉⠙⠙⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣙⣿⣭⣫⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣧⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⣿⣿⣷⣶⣾⣷⣷⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣴⣶⣶⣷⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠿⠿⠿⠿⠿⢭⣽⣓⡦⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣵⣶⣛⣛⣛⣛⣻⠷⢶⣦⣭⡙⠳⣤⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣭⣟⣛⡛⠿⠿⣿⣿⣶⣍⡻⢷⣌⠙⢦⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠛⠿⣿⣶⣦⣍⡛⢿⣷⡄⣍⠳⣌⢳⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⢉⡻⢿⣿⣞⣿⣿⡜⣧⠠⡀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⠀⠀⠀⠀⠈⣿⣦⠙⢿⣿⣿⣷⢹⣧⠹⡔⡔⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣧⠈⣿⣿⣿⢸⣿⡇⢻⣹⡌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡽⡇⢹⣿⡏⢸⣿⣗⢸⣿⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⢡⣿⣿⡇⣾⣿⠇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⣿⣿⣟⣼⣿⠋⣸⡿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣫⣾⠟⢁⡾⠋⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣴⡿⣳⠀⣩⢆⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣾⡿⣽⡇⢰⡿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⡇⣿⣇⣿⡇⠈⠀⢘⠁⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢹⣿⡀⣯⡓⠸⣿⣿⣿⡀⠀⠬⣯⡵⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡹⣷⡘⢿⣷⣌⠛⢿⣿⣦⣽⣳⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⢷⣬⡛⠄⢍⡻⢿⣦⡙⣯⡙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠆⠠⣻⣿⣿⣦⡀⢼⡶⡍⢷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠚⣛⣵⣿⣿⠇⠈⠵⣣⣧⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⡾⠛⠛⠋⠁⠀⠀⠨⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡽⡀⢁⣤⢀⣠⡀⣀⣀⢠⡴⣠⣀⠀⠀⣀⣤⢀⣀⡀⣀⣤⠇⠀⠀⡼⢁⣤⠀⣠⢀⣴⡁⣠⢂⣤⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠋⠹⠖⠛⠻⠘⠛⠗⠻⠘⢻⠟⠀⠀⠃⠛⠚⠙⠋⠛⠓⠀⠀⣟⠞⠓⠛⠚⠞⠚⠙⠙⠗⠳⠛⠛⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢰⣦⣤⣖⠀⠀⠀⢐⣦⣤⣴⠀⠀⠀⢀⣄⡀⢀⠀⠀⠀⠀⡀⠀⣠⡄⠀⠀⠀⠀⣤⣀⣀⡤⠀⠀⠠⣄⣀⣠⡄
⠀⠀⠀⠃⠈⠀⠀⠀⠈⠀⠃⠀⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠤⡤⠴⠀⠀⠀⠠⠤⡤⠄⠀⠀⠀⠀⠶⣤⣤⠆⠀⠀⠰⣤⣤⡴⠀⠀⠀⠀⠠⣴⣤⣤⡶⠀⠀⠰⣦⣤⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉
⠀⠐⠦⣤⠔⠂⠀⠀⠐⢤⣤⠖⠀⠀⠀⠐⠦⣤⡤⠂⠀⠀⠐⢦⣤⡴⠂⠀⠀⠀⠐⠷⢦⠴⠂⠀⠀⠀⠲⢤⡶⠗
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠉⠓⠲⠚⠁⠀⠀⠙⠲⠖⠛⠁⠀⠀⠘⠒⠶⠚⠁⠀⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠻⠶⠶⠋⠀⠀⠀⠳⠶⠾⠃
⠀⢀⡀⠀⣀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀
⠀⠈⠛⠛⠋⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠀⠙⠓⠞⠃⠀⠀⠀⠙⠖⠛⠁
⠀⢀⣄⠀⢀⡄⠀⠀⣀⠀⢀⣄⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⣤⣀⣀⡤⠀⠀⠠⣄⣀⣠⡄
⠀⠀⠙⠛⠋⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠈⠋⠉
⠀⠀⣄⡀⣀⠆⠀⠀⢦⡀⣀⡄⠀⠀⠀⠀⢄⡀⢀⡄⠀⠀⠠⡀⠀⣠⠀⠀⠀⠀⠀⣤⣄⣀⡤⠀⠀⠠⣄⣀⣤⡄
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁
⠀⢀⣀⠀⣠⠄⠀⠀⢤⡀⢀⣠⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣄⣀⣀⣤⠀⠀⢠⣄⣀⣀⣄
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣼⣾⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⡀⠀⠀⠀⠀⢰⠀⣰⠀⠀⠀⡄⠀⡀⠀⠀⢰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⡿⠿⣿⠿⡿⣿⣷⣾⣧⣿⢀⣿⢀⡆⣸⠃⣼⢡⡇⢠⡟⣠⠏⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⡧⢠⠇⣰⢏⡼⣱⣧⣿⡟⣽⢿⣿⣷⣾⣵⡟⣼⢏⡾⣠⣿⣵⠏⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⢿⣿⣿⣿⣿⠟⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣿⠋⠀⠠⣼⣿⣿⣿⣿⠂⢈⣿⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⡿⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⢿⣄⠀⠀⠿⡿⡿⠟⠃⠀⣼⠇⠀⠀⠀⣠⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠈⠻⣷⣤⣆⣀⢀⣄⣤⠿⠋⠀⢀⣤⣾⣿⣿⠿⣿⡛⠶⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠒⠒⠒⠶⠤⢤⣄⣀⣉⣙⣛⣛⣉⣁⣤⣴⣾⣿⢿⡟⢯⠉⠓⠀⠙⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⡇⠘⠀⠇⠈





⣿⣏⢹⡟⠻⣿⣛⠟⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠴⠶⠶⠶⠦⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡤⠴⢖⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⣤⣾⣿⣿⣿⣿⡿⠿⠟⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⢖⣕⣴⣾⡿⠿⠻⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡄⠀⠀⡴⢀⡄⣾⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣸⡇⠀⣼⡇⡼⣴⡏⢠⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣤⢼⣷⣴⣴⣿⣧⣾⣿⣼⣿⣿⣷⣧⣴⣿⠟⡠⣣⠎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡠⠘⣛⣥⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣾⡷⢀⣰⠏⡄⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣉⣵⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⠛⠛⣿⡿⢿⣿⣿⣿⣿⣟⣫⣴⣿⣥⣪⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⠟⢻⣿⡿⠀⠀⢿⣿⣿⣿⡏⠀⠀⢸⡿⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣻⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠋⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠃⣀⣴⣾⠿⣿⣛⠫⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠐⠒⢒⡶⢾⣿⣷⣦⣤⣠⣀⣤⣤⣴⣾⣿⠿⣿⡛⠷⣄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠟⢡⠏⢉⡟⠉⢿⢹⢻⠙⠳⡉⠑⠎⠳⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠛⣻⠦⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⡘⠀⠀⠀⠆⢠⠄⠀⠀⠀⠺⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⣰⠀⡠⢀⣬⠞⠀⢠⣁⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⣷⡇⠐⠁⡸⠃⣰⢆⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡗⠃⢰⡆⠀⢀⣤⣾⣿⣿⠿⠉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢁⣄⣼⣧⡆⡿⣸⠻⣽⠇⠀⠈⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢏⣿⣿⣿⣿⣿⡇⣛⣩⣿⠟⠀⠀⣿⡧⠁⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠆⠀⠀⠀⠀⢀⣿⠋⢺⣿⣿⣿⣿⣿⣧⣉⣿⠇⢠⣤⣼⣿⡇⠐⣃⠀⢧⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡆⢸⣿⣿⣿⣿⣿⣿⠟⡡⠀⣼⣿⣿⣿⡇⠀⠋⢰⠏⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢣⣼⣿⣿⣿⣿⣿⣿⠀⠀⣆⡼⣿⣯⣿⠁⢾⡇⠋⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡿⣯⣿⡟⠀⠀⠀⠈⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠐⢻⠁⣼⣿⠁⣴⠄⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡆⣰⣿⣿⣿⣿⣿⣿⣱⠀⣿⡿⠐⠟⢣⣼⡿⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡴⠊⢸⣿⣿⣿⣿⣿⣿⣄⠀⢠⡀⠀⢀⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⢀⣼⣿⣿⠇⢽⣿⣿⣿⣿⣿⣿⡀⢠⠛⠁⠀⡀⢾⣿⠃⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠁⠀⠀⡀⠋⠿⣹⢥⠀⣾⣿⣿⣿⣿⣿⣿⡇⠈⠀⠀⢀⣤⣸⠇⡞⣻⣿⣿⣿⠟⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣀⣸⣿⣿⣿⣿⣿⣇⠀⠀⣾⣦⡀⠀⠘⠀⠀⣿⣏⣼⣿⣿⣿⣿⠗⠃⠔⠀⣼⣵⡿⠃⣠⣿⣿⡟⠁⠀⣰
⣶⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣆⠀⡹⣿⣿⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣏⠰⡄⠀⣶⠆⠞⣠⢦⣽⣿⡿⠁⣀⣶⣿
⣿⡿⢿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠛⢋⣹⣿⣿⠉⠀⢹⣿⣿⣧⠁⠛⠟⠁⠀⠋⣹⣿⣿⣿⣿⣿⣿⣿⣶⢁⣈⡿⠃⠰⠃⣺⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⢸⡀⣨⣾⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣾⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣽⡿⣤⡟⠀⠀⢰⣾⣿⣿⣿⡄⠀⣿⣻⣿
⡆⣤⣴⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⡘⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⡟⠙⣃⣿⠃⠀⡼⣿⣿⣿⣿⣿⢀⣾⣿⣿⣿
⠞⠋⠛⢿⣿⣿⣿⣀⣴⣿⠹⣿⣿⣿⠃⠟⠟⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣜⣿⣿⣿⣿⣿⣿⣷⣶⣿⢫⣾⣾⣿⣿⣿⣿⣿⡇⢘⣩⣽⣿⣿
⠀⠀⠀⣿⡿⠉⣿⣿⢿⣿⠀⠘⣿⡏⢠⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣼⢿⣿⣿
⢸⠉⠁⣿⣇⠀⣿⣿⣻⣟⠀⠀⢸⡇⣼⣤⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣴⣶⣿⣿⣿⣿⣿⢏⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣾⣿⣿
⣌⢣⣀⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⡇⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠁⠀⢸⣿⣿
⠹⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠀⠶⢻⣿⣿
⣀⣸⣿⣿⣿⣾⡿⢿⣿⣾⠿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⡿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀⣸⠳⣿⢿⣽⣿
⠙⠿⣿⣿⣿⣿⠛⣋⣉⣤⣶⣿⣿⣿⣿⣿⣿⠐⣿⣿⣶⣾⡿⡟⣿⣿⣿⠇⣸⣿⣿⣿⡂⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣥⡆⠀⣾⣿⣿
⢧⣰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⡿⢹⣿⣿⣷⣾⣿⣿⣃⣴⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣿⠀⣿⣾⣿⣿
⢈⣿⣿⣿⣿⣿⣿⠏⠉⠈⠀⣿⣿⣿⣿⣿⣿⠀⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⡿⢙⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣠⠟
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⣠⣄⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⣀⡴⣤⡀⠀⠀⢘⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⣿⣧⣤⠞⠉⠈⠻⣦⢴⠟⠁⠀⠀⠙⢷⢤⡾⠋⠀⠈⠙⡦⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠐⣛⣛⣛⣛⣿⡏⠉⠳⣄⣠⡴⠋⠙⢦⣄⠀⢀⣴⠞⠉⠳⣤⡀⣀⡴⠋⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⠀⠈⠙⠋⣠⢤⡀⢰⠟⢷⣄
⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⡿⠈⢷⡀⠈⣷
⠀⠈⠋⠙⠻⣿⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣴⣄⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⢹⣤⠾⠋⠀⠀⠀⠙⠚⠃
⠀⠀⠀⠀⠀⣸⠿⣄⣠⡴⠋⠙⢦⣄⣠⠞⠉⠀⠈⠳⣤⣀⡴⠋⠀⠙⢷⣄⣠⠞⠁
⠀⠀⠀⠀⠀⢻⣤⡾⠻⢦⡀⢀⡴⠓⠳⣄⡀⠀⠀⣠⠞⠚⢦⣀⠀⢀⣴⠗⠳⣄
⠀⠀⠀⠀⣀⣾⡟⠀⠀⠀⠙⠋⠀⠀⠀⠈⠛⠶⠞⠁⠀⠀⠀⠙⠷⠋⠀⠀⠀⣸⠷⣦⡀⠀⠀⠀⢀⣀
⠀⠸⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⣦⠀⣴⠟⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠈⠳⠶⠋⢸⡃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢶⣶⣦⣤⣿⣿⣦⠀⣠⡶⠳⣤⡀⢀⣠⠞⠉⠙⢦⣀⠀⣠⡴⠋⠳⣦⡀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⢿⣅⠀⠀⣈⡿⢽⡁⠀⠀⠀⢀⣹⠽⣏⠀⠀⠀⢈⡯⢿⡁
⠀⠀⠀⠀⠀⠀⠙⢋⣩⣽⡿⠛⠛⠀⠈⠳⠾⠋⠀⠀⠙⠷⣄⡴⠟⠁⠀⠈⠻⢦⡴⠋⠀⠀⢙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣄⠸⣅⠙⠷⣄
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣤⠟⠀⠙⢦⣠⡿
⠀⣠⣤⣿⣌⢻⡖⣆⠀⠀⣠⣄⠀⠀⠀⢀⣤⠶⣦⡀⠀⠀⠀⣠⡴⣄⠀⠀⠀⢹⡶⠟⠁
⢰⠃⠀⠀⠉⢸⡇⣿⣤⡞⠃⠈⠳⡤⡴⠛⠁⠀⠀⠙⢦⢤⠞⠉⠀⠈⠳⡦⡴⠋
⢸⣄⠀⠀⣠⣿⠁⣿⠉⠳⣄⣠⠾⠋⠛⢦⡀⠀⢀⡴⠛⠙⠳⣄⠀⣠⡾⠋⠛⢦⡀
⠀⠙⠿⠿⠟⠛⠷⠏⠀⠀⠈⠁⠀⠀⠀⠀⠙⠳⠋⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⣼⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣚⣃⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⣠⣶⣿⠿⢿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠿⢿⣷⣦⡀
⠀⠀⠋⠉⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠢
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣾⣿⢿⡿⣿⣿⣿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠿⣿⣿⣿⠛⠿⣿⣦
⠀⠀⠀⠀⠚⢿⣇⠸⣄⣛⣧⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⣵⠟⢀⡠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠺⠿⠷⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠚⠓⠁

⠀⠀⠀⠀⡀⠠⠄⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⢰⠏⠁
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠀⠀⡄⠀⠀⡀⠀⠀⢀⠘⠄⠀⠠⠂⣠⣀⡰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠈⠁⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⡤⠚⡭⠝⣿⡤⠿⠟⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⢈⠀⣀⡄
⠀⠀⠀⠀⠀⢀⡀⢸⡄⠀⣀⣀⡬⠤⠦⠤⠔⠒⡶⠂⠀⠈⣆⠬⣃⢄⠀⠀⠂⠀⠐⠶⠋⠁
⠀⠀⠀⠈⠀⠀⠣⣰⡠⠀⢈⡷⠀⠀⠀⠀⢠⠞⠁⠀⢠⡀⠀⠀⠈⢢⠑⡄⠀⠉⢠⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠉⡴⠚⠁⣀⡤⡄⢀⠜⠁⡠⠀⠀⢸⠀⠀⠉⠀⠀⠇⠁⠀⠀⠈⠁⠿⠆⠀⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡼⠁⠀⠹⠃⡀⠀⠈⠄⠀⠈⡆⠀⢀⣀⢎⠜⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣿⠸⠉⠂⠅⠀⠀⠄⠐⠄⣷⣤⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣠⠟⠀⠀⠀⡀⡀⢌⠀⠀⠀⡏⠀⠀⠀⠄⣀⠐⠙⠳⢶⣬⣅⡨⡍⠉⢷⢦
⠀⠀⠀⡁⠀⡀⠀⠀⠈⠚⠁⠀⠀⠀⢀⠄⠈⠀⠀⠂⢼⡄⠀⠀⢠⣀⠀⠄⠀⢰⠀⠈⠁⢛⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⠀⢠⡂⠀⢀⢔⠝⠁⠀⠀⠉⠙⢦⣄⣀⣀⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠇⠀⠑⠁⠀⠀⠰⠶⠀⠀⠀⠈⠁⠉⢹⠞⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣭⣭⣭⣭⣍⣛⡛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠉⢻⣿⣿⣿⣿⣶⣭⣝⡳⢶⡨⢙⠲⢬⣙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡶⣝⡳⣝⠲⣌⠲⣍⡒⠬⣅⡈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣮⣝⢮⡓⠮⣙⠲⢬⣑⠲⢬⣓⢌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠋⠻⣆⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣯⡻⢿⡷⣌⡓⠦⣬⣀⠀⠌⠐⠬⣛⢷⣿⣆⣠⣤⣤⡤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⣁⡀⠀⠀⠈⠃⠀⠀⠀⠀⢹⣿⣿⣯⣟⠿⣿⡦⣝⠲⢝⡲⢄⡉⠙⠳⢦⣤⡈⠳⣹⣿⣿⣯⢖⣂⣠⣄⣀⡙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⢷⣮⣙⠦⣝⢦⣍⠲⣬⡒⢌⠲⣌⠻⣦⡘⢿⣿⡿⣋⡥⠄⡀⠠⠍⠳⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣮⡙⢷⡘⢧⠙⢷⣌⠳⡄⠑⣌⢻⣮⣿⣾⣿⡾⠁⣈⣭⣽⣿⣷⣦⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠹⡌⢷⡀⠙⣧⡑⠐⣌⢷⣿⣿⣿⣿⢁⠞⣵⠟⠁⢪⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⠷⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠛⠻⠟⠛⠛⠻⣯⡳⡘⣎⢳⡀⢈⢿⣄⡘⢷⣽⣿⣿⣿⠈⠜⡁⡰⠀⡟⣿⢹⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠆⠀⠀⠀⣿⣷⡘⠌⢦⠻⣌⠢⣝⢿⣦⣝⣿⣿⣿⡇⢰⢡⢃⠀⡇⣿⢸⡇
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣄⠈⢷⣌⠷⣝⣷⣬⣟⣿⣿⣿⡇⢸⢸⣸⢰⠃⡇⣼⠇
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡹⡝⣦⠀⢻⣷⣬⣛⣿⣿⣿⣿⣿⣿⣾⣸⣧⠏⢈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⢦⠤⠤⢀⣀⣀⠠⠒⠁⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⣿⣧⢳⡹⣌⠳⣦⣉⡛⠿⠿⠿⣻⣿⣿⣿⣿⣿⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⡟⣧⠳⣌⠳⢦⣭⣭⣭⣵⠾⢫⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⡹⣿⡜⢷⣌⡛⠶⠤⠤⢐⣠⡾⠋⣼⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣧⠳⣌⠻⣦⣍⣛⡛⣛⣛⡋⠁⣠⡾⢣⡟⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣌⡳⠦⣭⣍⣉⣭⡥⠶⠛⠉⣠⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡺⣝⣒⠶⠤⠤⠴⢒⣊⣥⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⣭⣉⣉⣉⣭⣵⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣤⣶⣶⣶⣤⣴⣤⣄⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿
⠀⠀⢠⡞⢁⣿⠟⢿⣇⠙⣿⣿⣿⣿⠿⠊⠀⠀⠀⠀⣿⣿⣿⢫⣶⠃⢠⡄⠈⣷⡄⠀⠀⠀⣉⣭⣿⣿⣿
⠀⠀⠀⠑⢦⣿⣿⣿⣧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣝⡃⠀⠀⠰⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⠀⠀⠀⠀⢰⡇⢠⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣍⠛⠶⣄⡀⣾⠀⢸⡇⠀⠈⣧⠀⠀⣀⣤⠴⢺⡿⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢷⣄⠀⠙⢧⠀⢸⡇⠀⠀⣸⠖⠛⠉⣠⠞⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢧⡀⠈⢇⠸⡇⠀⣰⠃⠀⣠⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠧⣄⡀⠀⠙⣦⡘⣦⡇⢠⠇⣠⠞⠁⠀⢀⣰⣃
⠀⠀⢀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠉⠛⠶⢬⣻⣿⣧⣿⣞⣡⠴⠖⠋⠉⠈⠉⠙⠛⠲⠦⢤⣄⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠈⣿⣼⢷⣼⠋⠀⢠⣴⣶⣯⣭⣤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⣶⣾⣿⣿⣿⣿⣖⠲⠶⠒⠶⠶⠶⠶⠶⠲⠶⠶⠾⢿⣿⣶⠦⠀⠀⣿⠤⠤⠁
⠀⠀⠀⠸⠇⠈⠟⠀⠀⠀⠈⠉⠛⠓⠲⠦⢤⣤⣀⣀⣀⣠⡴⠖⠋⣩⠞⡟⣿⢻⠳⣌⠙⠓⠦⣄⡀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⠀⠀⠀⠻⠶⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠁⠀⣠⠞⠁⡸⠁⣿⠈⢧⠈⢳⣄⠀⠀⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠁⠀⢠⠇⠀⣿⠀⠈⢧⠀⠙⢷⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠁⠀⠀⣀⡞⠀⠀⣿⠀⠀⢘⣧⣄⠀⠙⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣁⣤⡴⠖⠛⠉⣧⠀⠀⣿⠀⠀⣸⠁⠈⠛⠶⣄⡙⢦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⢰⡗⢻⡞⣿⢠⣾⣻⡆⣿⠀⣷⢸⡟⢻⡄⣾⣹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠛⠘⠇⠸⠃⠻⠈⠻⠶⠂⠻⠀⠟⠸⠃⠸⠃⠛⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠭⠍⠩⢍⡩⣐⣤⣩⣉⣑⢶⣄⡠⣀⠂⢐⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠌⠁⠀⠀⠢⠌⠉⠿⢿⣻⣿⣿⣿⣿⣶⣝⡦⣽⠻⣿⣿⣶⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠙⠻⠟⢿⢿⡿⡷⣭⣿⡙⣿⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠊⠳⠛⠩⠻⣿⣿⡴⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣰⣠⢤⡸⣤⣰⣦⣤⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢦⢩⣴⡦⣶⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⣶⣿⣿⣷⣿⣿⣾⣿⣿⣿⠿⣿⣿⣿⣿⡿⢿⣿⠿⠿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⣿⣿⣿⣿⣿⣿⠿⣿⠉⠁⠐⢿⣽⣿⣿⠇⠐⣿⠀⠀⠀⠀⠉⠙⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠿⠿⠿⠿⠋⠁⠀⠹⡄⠀⠀⠀⠈⠉⠉⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠁⡰⠀⡀⢈⡉⠒⠤⠤⣀⣀⣀⣴⣋⣀⣀⡤⠤⠤⠤⠶⠖⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⣀⣽⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠁⠀⢀⣠⡶⠶⢟⡻⠶⢦⣄⡀⠀⠀⠀⠹⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡟⣩⡄⠀⠘⠋⠀⢠⣍⢻⣆⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟⠀⠈⠁⠀⢸⡇⠀⠈⠁⠀⢻⡆⢀⣴⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⣾⡇⠿⠂⠀⠀⠸⣧⡀⠀⠐⠿⢸⡿⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠏⠀⠘⣷⠀⣠⡀⠀⠀⠈⠁⢀⣄⠀⣼⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠙⢷⣍⡁⠀⢴⡦⠀⢈⣩⡾⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣦⠀⠀⠀⠀⠉⠛⠷⠶⠶⠾⠛⠉⠀⠀⢀⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠁⢹⡧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠄⠄⠀⠀⠀⠀⠀⠀⢠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢘⢹⠉⡇⢾⠘⢸⡇⡇⡏⢹⢸⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣠⠶⠛⠉⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠇⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⢻⡶⣤⡀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⡴⣾⠓⠶⣦⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⠻⣬⣻⡄⢀⣠⡼⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⣼⣫⡼⠋⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⢩⡏⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠙⣏⠉⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣛⣲⣶⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣳⣶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⡭⢽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣭⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⣤⡷⢤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣀⣤⠴⣷⣤⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⢠⡾⣩⡾⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⢠⡾⠉⠀⠀⠹⣮⡹⣦⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢺⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠈⠛⢻⠖⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⠴⠶⠋⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠠⢤⡴⠾⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⣉⣁⣀⣐⡒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⢀⠠⠒⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢖⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⣹⣆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢼⡀⠀⠀⠀⠀⠀⠟⠀⠀⣠⠔⠞⢥⠏⠀⠀⠀⠀⠀⠰⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣳⡀⠀⠀⠀⢐⣘⠄⢰⠁⢠⠚⠁⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠂⠉⠀⠀⠀⠸⡆⡇⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠸⡆⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢀⣧⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⠀⢀⡀⠈⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡬⠽⠶⣚⡫⠭⠝⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣩⢷⢈⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡪⡸⡄⠀⠀⠀⠀⠀⢸⠄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣈⣉⣉⣷⣦⡇⡀⣇⠀⠀⠀⠀⠀⠀⢻⡿⠿⠦⠾⠶⣀⣀⠉⠀⣸⡇
⠀⠀⡀⠀⠀⠀⢀⠤⠄⠈⢙⣒⣛⣟⣛⣛⣛⣋⣭⣍⣻⣿⣿⠯⣾⠀⠀⠀⢀⠀⢀⡈⠲⠾⣿⡿⠟⠛⠛⠻⠿⠟⡠⢄⡀
⣐⠓⠹⠯⢉⡉⢉⢙⣿⡙⠛⣒⠷⠿⠶⠶⠾⠟⠛⣛⣻⣭⣿⣿⡟⠀⣸⡇⠈⢿⣦⡙⠳⣶⣦⣬⣉⣐⣒⡒⠒⠚⠁⠀⠀⠁⣲
⠀⠀⠀⠐⠚⠓⠒⠀⠉⠻⣷⣤⣀⡿⠿⠿⠿⠿⠿⠿⠟⣋⠕⢁⡠⡾⢋⡟⠀⣸⠽⢿⣆⠘⢯⡉⠛⠛⠛⠛⠻⡇⣀⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⢀⡠⠔⠊⣠⣴⠿⠋⠀⣸⠃⢠⡟⠀⠀⠹⢷⣦⣈⠲⢄⡀⠀⠘⠛⠻⠑⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⠂⢁⣠⣴⠿⠚⠁⠀⠀⡰⠃⣰⡿⠁⠀⠀⠀⠀⠉⠛⢷⣦⣌⡑⠢⠤⣤⠎⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠈⠑⠊⢀⣤⣶⠽⠊⠁⠀⠀⠀⢀⡜⠁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡶⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⢀⣉⣲⣄⠀⠀⠀⠀⢠⠞⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣴⡶⠶⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠛⠿⠟⠛⠁⣰⠑⠂⠤⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⠀⠀⠉⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣢⣥⣴⡶⣟⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣾⣷⣾⣷⣾⣿⣷⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣠⣖⣤⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣾⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠈⣿⣿⠿⠋⠙⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⡀⣀⠄⠀⠀⢀⠀⠁⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠊⠀⠀⠀⠀⠀⣄⣸⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣾⣥⣀⣤⡞⠁⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠂⠀⠀⠀⠀⣠⣴⣿⣿⡿⢿⣿⣿⠟⢻⣿⣿⠉⠙⢿⠛⠻⢿⣿⣿⣿⣿⣷⣿⣴⣮⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⣾⡿⠟⠉⢹⡇⠐⠿⢿⣶⣾⣿⣿⣏⡀⢸⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⠟⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠟⠉⠀⠀⠀⠘⣇⠀⢔⡼⠛⠿⠛⠙⠀⠀⡼⠀⠀⠀⠀⣠⡾⠋⠀⠉⠉⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠞⢋⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠛⠛⠒⠒⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⢒⣚⣥⣴⣞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠶⠐⠶⠖⠋⠉⠛⠯⠉⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠚⠉⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡠⠀⠘⠋⠁⠀⠀⡘⠡⠀⠖⠆⠀⠒⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠋⠓⠀⠀⠀⠀⠀⠆⡄⠀⠀⠀⠰⠀⠀⠀⠀⠙⠂⢀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⢠⠄⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀⣨⠶⠈⠁
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡠⠀⠈⠀⠁⠸⠀⠖⣠⣠⠔⠋⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⢀⡀⠀⣡⡔⠉⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠉⠉⠁⠀⠈⠙⣎
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⣀⡀⢠⣫⠔⠋⠀⢀⠀⠄⡒⠠⠂⠀⠀⠀⠘⢆⡀⠂⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠀⢀⠀⣄⠐⠤⠖⠀⠈⠁⠈⠉⠆⣀⠔⠋⢀⡈⠀⠋⠁⠀⠀⠀⠀⠀⠈⣀⢴⣊⠉
⠀⠑⠂⠀⠒⠐⠒⠀⠂⠀⠦⠀⡠⢺⣥⠞⠋⠉⡉⢣⡀⠀⠀⠀⢀⣠⠔⡛⠁⢢⠌⠵
⢻⠀⠀⠀⠀⠠⣖⣀⡋⣁⣠⡾⠋⣱⠃⠤⣂⡈⠀⠀⣿⡄⢀⡴⠮⠄⠠⠒⠃⠀⠀⣤
⠀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠻⣄⣀⠃⠀⠀⠀⠉⠀⠁⠙⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⣾⡀
⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢉⠀⠈⠁⢸⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠚⠻⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇⢸⠀⣸⠀⡆⠹⣜⡄⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⢧⡇⢸⠀⣟⠀⡇⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠄⠀⠀⠀⠘⢳⢸⠀⢸⡀⡇⠀⢀⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠈⠂⣇⡌⠉⠀⠉⠉
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⠒⠦⣤⣀⠀⠀⠀⠀⠈⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠙⠓⠶⠶⠶⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠔⠂
⠀⠀⠀⠀⠐⠲⠶⠶⠾⠿⠛⣛⣽⣿⠟⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⡚
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠞⠛⢋⣩⣴⠿⢛⣵⡿⢋⣼⡟⣸⡿⢻⡿⣿⢿⣿⣿⣿⡿⣿⠿⣍⠛⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢉⣠⠶⠟⠉⣠⡾⠋⢠⡿⠁⣿⠃⣿⢸⡿⢸⡇⢿⠈⠳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠰⠋⠀⡼⠋⣸⠃⢸⠃⠘⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡇
⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢸⣿⣿⡇
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⠀⠸⣿⣿⡇
⡇⠀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢠⣿⣿⠆⠀⠀⣿⡇
⢵⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⣾⣿⣷⡄⠀⠀⠿⠃
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡆
⠸⢿⠟⠙⠁⠀⢀⣀⡄⡆⠀⠀⠨⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢠⣾⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡟⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⠙⢙⢹
⠀⠀⠀⠀⠘⠟⠁⠀⠀⢀⣾⣿⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⡀⠀⡇⠀⠀⠀⣶⣦
⢀⡀⢰⣄⠀⠀⠀⠀⠀⢸⣿⣿⡀⣀⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡀⣆⢀⣧⠀⠀⢰⣿⣿⡄
⢸⣿⣿⣿⣿⣧⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣟⣛⣀⣀⡀⠀⣀⣐⣛⣻⣻⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠟⠋⠁⠀⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠟⠛⠛⠛⠿⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⡐⠶⣷⡶⠲⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⣦⠀⠉⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠋⠙⣳⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣦⢴⡦⢠⠃⠈⢳⡼⠁⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⣿⣿⠁
⠀⠀⠀⠈⠁⠙⠓⢠⣉⣿⣷⣿⣧⡀⣠⠟⠀⠀⠀⠀⠀⢠⠞⠉⢀⣴⠄⠒⣒⣾⣿⣿⣷⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠘⠤⠿⣿⣿⡿⢙⢿⡁⠀⠀⠀⠀⠀⠀⢄⠀⣀⡾⠥⡶⢿⣯⠶⠾⢿⣷⠒⠻⣷⣶⡄⠤
⠀⠀⢀⡀⠀⠤⠂⠀⠐⢿⣿⡿⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⡀⠀⠀⠈⠀⠀⠀⣨⢹⡄⢠⡄⢸⣿⠂
⠀⠀⠀⠀⢀⢰⣤⣦⣀⠾⣿⣧⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠂⠀⠀⠀⠀⢠⢙⣸⣇⣀⡇⠘⠋
⠀⠀⣠⡾⣭⢻⣿⣿⠿⢙⣿⣿⠙⡄⠀⠀⠀⠀⠀⣄⠐⠀⠀⠀⡀⠀⡀⠤⠀⠴⠃⢠⣿⡅⠈⢙⢳⡄
⠀⢹⣿⣻⣿⣼⡿⠛⣶⣯⣿⣿⡄⠄⠈⠛⢀⣠⡤⠈⠀⠀⠀⠈⠿⠁⠀⠤⠴⠄⠀⢸⣿⠿⠤⢼⠈⠛
⠀⢸⣗⢼⣾⡿⠃⡸⣾⣿⣿⣿⣄⢸⣿⡓⠶⠄⠋⠈⠀⠀⠀⠀⣠⡄⠀⢀⣨⡤⠄⣼⣿⠄⡀⠘
⠀⠀⣠⠞⠛⣦⣤⣿⣿⢋⣿⣿⣴⢾⣿⣿⡁⠀⠀⠀⠀⠀⠈⠉⠹⡀⢀⣋⣵⣶⠆⣸⡟⠀⠀⠁⠀⢄
⠀⠰⠟⣀⡤⠤⠶⠋⣿⠙⣿⣿⡀⣿⠉⠉⠳⣤⠀⠀⠀⠀⠀⠀⣰⣮⣤⣽⣵⡇⠚⣿⡇⠀⠀⠀⠀⠈⠃
⠀⡀⠘⠉⠀⠀⠀⠀⣿⠀⣿⡧⠘⠛⠛⠳⣦⣤⣤⣤⣴⣄⣀⣍⣛⣟⠻⣿⣯⠁⠀⣿⡇
⠀⠙⢷⡞⢫⠄⠱⠦⣿⢰⣿⣧⠀⣠⣴⡿⡿⠃⠀⢠⠋⢤⣧⡉⠁⢙⢦⡟⣧⠀⠀⣿⡇⠀⠀⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣇⠞⠁⠀⠀⠀⠀⠀⢠⣾⠟⠻⣿⡆⠸⣿⣄⡟⠀⢰⣿⠀⣀⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⡀⢀⡀⠸⢀⣹⡿⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣿⠁⠀⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢮⣽⣾⣿⣿⠀⠀⠀⠀⣀⠵⠶⠚⠛⢛⠛⠒⢦⣄⠀⣸⣿⣾⡿⠛⡿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢱⣿⡏⠀⠀⠀⠀⣼⠟⠁⠀⠙⢶⠄⠀⠁⠀⢀⣼⣿⣿⣿⠋⠀⠀⠀⠈⡇⣸
⠀⠀⠀⠀⠀⣴⠏⠹⣏⠰⠾⣟⣾⠋⢰⡀⠀⢀⣀⠀⢀⣼⣿⣿⠟⠋⠀⠸⡆⠀⠀⠀⠀⣵⠋
⠀⠀⠀⠀⠐⠋⠀⠘⣿⠀⢀⣿⡏⠀⢨⣧⡀⠈⢛⣴⣿⠟⠋⠀⠀⢠⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠋⠹⢦⡈⣻⣥⣶⠿⠛⠛⠷⣤⠶⠶⠾⢿⣦⣀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠀⠀⣠⣿⣿⢟⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⠉⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣾⡿⠋⠁⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁

⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣷⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⢀⡴⠟⠋⠉⠀⠀⠉⠉⠉⠛⠛⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠈⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⠄

⠀⠀⠀⠀⠀⠠⢶⣾⣦⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⢀⣀⣠⣴⣾⣶⡆
⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉



⣿⣿⢟⠿⢛⠿⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠾⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠈⠛⠳⠶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠈⠒⠲⠶⠶⠶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤
⠀⠀⠀⠀⠐⠲⠶⠶⠟⢛⣻⣽⡿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡚
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⣉⣥⡶⠟⣫⡾⠟⣱⡿⢹⡿⢹⡿⣿⢻⡿⣿⢿⠻⡟⢯⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠴⠛⠉⢀⡼⠋⢠⡿⠁⣾⢃⡟⢸⡇⢹⠈⡇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠜⠁⠜⠁⠘⠀⠈