Pe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣦⣄⢻⡟⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⣭⣽⣇⠀⠀⣀⣠⡿⢉⣴⡶⠶⣦⡙⠛⢁⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠉⠛⠻⣥⣬⣽⢾⡿⠶⠒⠾⢇⢸⡿⢿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠖⠒⠒⠒⡻⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⣷⣌⠲⢾⡿⠶⣄⢻⣧⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢗⣶⣶⣿⠿⢃⣀⠀⠀⠀⢀⣀⡠⢥⡟⠸⣿⠸⣷⠶⠶⠞⣿⠀⢿⠏⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢼⡉⠁⠀⠀⠀⠀⠉⠹⡭⣤⢴⣒⡿⣧⡀⠻⣦⡙⠷⠶⠞⣣⣶⡆⢀⣈⣛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠷⢿⣙⠳⢦⣄⡀⠀⠀⢠⣱⣃⡞⡡⠒⢮⢿⣤⡾⢿⡿⣾⣿⡋⣿⣀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠐⠀⠀⠈⠙⠲⢤⣍⣿⠞⣩⣵⡾⠸⠁⢀⡼⢋⠁⠀⠎⠀⠀⠈⢻⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠹⣷⣦⣄⡀⠀⠀⠀⡠⢉⣿⢇⣿⢇⠇⢠⣿⠁⠘⠀⠰⠀⠀⠀⡐⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠉⠛⠿⢷⣶⣤⣠⣾⡟⣼⣿⢸⠀⠀⢿⠈⡐⠂⢠⠂⠀⠀⠀⠐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⣈⣠⡟⣿⠧⣿⣿⡘⢤⢔⣽⣧⡀⠀⠨⡀⠀⠀⠆⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠶⠟⠈⠛⠶⠛⠁⠈⠻⢦⣤⣵⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠃⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠘⠁
⠀⡤⡄⡄⣠⢠⡠⢠⢤⢠⢀⡄⢀⢤⡀⡄⠀⠀⡤⠀⣤⡀⣤⢠⢀⡄⢀⢤⡀⡤⢀⡄⡤⡄⣤⣀⠀⠀⡤⡀⡤⡄⢠⢤⢠⢄⢠⠀⢠
⠀⡷⠃⡗⣿⠸⡇⠸⣈⢸⢸⠀⢸⢸⡇⡇⠀⠀⡗⠀⣇⡞⣿⢸⢸⠀⢸⢺⡇⡏⠸⡇⡇⡇⡿⣧⠀⠀⣇⡇⡇⡗⢾⢸⢸⣸⢸⠀⢸
⠀⠁⠀⠁⠉⠀⠁⠀⠉⠈⠈⠉⠈⠈⠉⠁⠀⠀⠑⠀⠉⠀⠈⠁⠈⠉⠈⠈⠁⠁⠀⠁⠉⠁⠁⠉⠀⠀⠋⠀⠉⠁⠈⠁⠘⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⣋⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣭⣤⣴⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣦⠀⣀⣀⣀⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⣈⠻⠿⢿⣿⡿⠿⠟⣁⣼⣿⣿⣿⣿⣿⣦⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⠀⣠⣤⣤⣤⡀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⠟⣉⣴⣾⠏⣰⣿⣿⣿⣿⣷⡈⢿⣾⣆⡀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣤⣭⣭⣥⠘⣿⣿⣿⣿⣿⣿⡟⢠⣭⣭⣥⣤⡀
⠀⠛⠛⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠲⠶⠶
⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠹⣿⣿⣿⣿⣿⣿⠃⣴⣤⣤⣤⣄⢻⣿⣿⣿⣿⣿⡿⢁
⠀⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣧⠙⠿⠿⠿⠿⢃⣾⣿⣿⣿⣿⣿⣆⠻⠿⠿⠿⠿⢡⣿
⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡟⢠⣶⣶⣶⣶⡌⢿⣿⣿⣿⣿⣿⠏⣰⣶⣶⣶⣦⡘⣿
⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣿⣿⣿⡄⠻⠿⠿⠿⠃⣼⣿⣿⣿⣿⣿⣷⡈
⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠻⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⠃
⠀⠛⠛⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠛⠛⠛⠛⢡⣿⣿⣿⣿⣿⣿⣧⠘⠛⠛⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⠻⢿⣆⠻⣿⣿⣿⣿⡿⢃⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠛⠛⠁⠈

⣠⣤⡀⣤⠀⣤⢠⡀⣠⢀⣤⠄⣤⢀⣤⣤⠀⣠⡄⢀⡄⠀⠀⢠⣤⠄⣤⢤⡀⢠⡀⢠⡀⣠⣤⠄⢠⣄⢠⣤⣤⢠⡀⣠⢤⡀⢠⣄⢠
⣿⠶⠃⣿⠒⣿⠀⣿⠃⠘⢳⡄⣿⢸⡅⠀⢰⡯⣷⢸⡇⠀⠀⢸⡗⠂⣿⢀⡿⢸⡇⣸⠰⣇⠀⠀⣾⢽⡄⢸⠀⢸⠗⣇⠀⡿⢸⠻⣾
⣾⣶⣶⢿⣶⣾⣶⣾⣶⣾⣿⣶⣿⣶⣿⣿⣾⣶⣾⣶⣿⣿⣶⣾⣿⣷⣿⣿⣷⣶⣿⣷⣶⣾⣿⣶⣷⣶⣷⣾⣶⣾⣶⣾⣿⣶⣾⣶⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⢠⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢛⠅⠌⡄⠈⠕⢵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⣿⢋⣿⠂⣰⣾⠵⠞⢀⠱⠀⠀⠠⠸⣮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣀⠀⠸⢸⡇⢀⡿⠀⣳⡇⢀⠊⠀⠀⢂⠁⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢰⡇⠙⡇⠰⡎⡇⢸⣧⠒⣣⠀⣀⢀⠀⠀⡆⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠹⣆⠘⡄⡇⣷⠸⣟⢸⠊⠠⠁⠀⠀⠈⠀⢚⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠘⢆⠹⣧⢹⠀⢻⣆⣇⢄⣂⠀⢀⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⢦⢹⣿⡄⠘⡿⡔⡈⠍⢜⣂⣥⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⣦⠛⢧⠀⠈⢝⠻⠥⢶⡦⠟⠁
⠀⠀⠀⠀⠀⠀⡏⢿⠀⠳⡀⠀⡇⢰⠋
⠀⠀⠀⠀⠀⠀⡆⠈⣧⠀⠙⣶⠁⢸
⠀⠀⠀⠀⠀⠀⠃⠀⠈⡇⠀⠈⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⢠⡄⠀⠀⢹⡄⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠖⠣⠀⠀⠀⠀⠈⠀⠀⠀⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢸⡄⠀⠀⠀⢣⠀⠀⠀⠀⡰⠁⠀⠀⠐⣤⠄⠀⠀⠀⠀⠀⠀⠀⠨⢦
⠀⠀⠀⠀⠀⠀⡄⠀⠈⢸⢸⣿⡀⠀⠀⠀⢳⡀⠀⠰⠁⠀⢀⠠⠊⠀⠀⠀⠀⡠⠂⠉⠀⠀⣀⡤⠷⠒⠒⠒⣤⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣸⣿⣷⡄⠀⠀⠀⠱⡶⠋⠲⣀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⡠⠊⠙⣀⠀⠀⠀⠀⠙⠢⡉⢢⡅
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣿⡉⠉⠻⣆⠀⠀⠀⠙⡄⠀⠘⣦⠄⠀⠀⠀⠀⠀⠀⠠⠜⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⢠⠚⠛⠸⢦⡀⠘⢦⠀⠀⠀⠘⣄⠀⠘⡄⠀⢀⠂⠀⠀⠀⢸⣆⠠⠐⠙⢿⡿⠟⠀⡀⠀⢀⠀⠛⠿
⠀⠀⡰⠊⠉⠀⢹⡀⠼⠆⡀⠀⠀⠙⠢⠀⠱⡀⠀⠀⠈⢆⡐⠋⠇⠈⠀⠂⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⡃⠀⠀⠀⠘⣇⠀⠀⠀⠁⠒⠤⠄⠀⠀⢳⡄⠀⠀⠈⠣⣄⡘⢦⠀⠀⣀⠀⠘⠧⣀⣀⣀⣴⡀⠀⠀⠀⣀
⠀⠀⠣⡀⠀⠀⠀⡋⢻⡡⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠘⡙⣄⠀⠉⠹⠤⠒⢣⠀⠹⣿⣿⣿⡶⠀⠀⠀⠀⢠⣤⡴
⠀⠀⠀⠈⠂⠒⠋⠑⠤⢹⢶⠋⠀⠀⠀⣀⡀⠀⠀⠠⢷⡀⢠⣀⡣⠽⠄⠚⠀⠀⠀⠀⠓⢄⠛⠛⠻⡀⠀⠀⠀⠀⡸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡰⣿⣿⣿⣿⢆⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣭⣭⣶⣿⣿⣿⣿⣷⠄⡀⣠⣤⣄⢀⡀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⣑⣛⡱⣿⣿⣿⢎⣛⣂⡀
⠀⠀⠀⠀⠀⠀⢨⣭⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⡷⣙⣛⣋⣾⣿⣿⣿⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢞⣛⣛⢱⣿⣿⣿⡎⣛⣛⣱⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣾⣿⣿⣷⢙⣛⡋⣾⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠝⢛⣛⢱⣿⣿⣿⡞⣛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠻⠿⠟⠘⠉
⠀⠀⠀⠀⠀⠀⣀⢀⠀⣀⢀⢀⠀⠀⡀⢀⠀⡀⠀⣀⢀⡀⢀⢀⢀⡀⢀⢀⣀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠟⠹⠛⠏⠏⠹⠾⠻⠄⠾⠇⠧⠀⠷⠸⠼⠻⠼⠫⠄⠟⠆⠇⠟⠧⠟⠻⠇


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣭⣽⣭⣭⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣝⡻⠿⠿⢟⣡⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣣⡴⣰⣿⣿⣆⢦⣄⠀⠀⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⣶⣦⢻⣿⣿⡿⣰⣶⣶⣆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⢻⣿⣿⣿⢣⣭⣭⡜⢿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢇⣭⣭⣅⢿⣿⣿⡿⣨⣭⣭⡸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⢎⣭⣭⡱⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠎⠛⣛⡃⣾⣿⣿⣿⢘⣛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠿⠿⠃⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⡀⣀⢀⢀⡀⢀⠀⡀⠀⣀⡀⣀⠀⠀⡀⣀⠀⡀⣀⡀⡀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠸⠋⠟⠺⠸⠁⠷⠸⠻⠄⠾⠇⠧⠀⠿⠸⠥⠟⠧⠟⠥⠰⠿⠸⠀⠟⠥⠗⠿⠇


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

⠀⠀⠀⣀⣀⣤⡀⠀⠀⢀⠀⠀⣀⡀⣀⠀⢀⡀⣀⠀⠀⠀⠀⠀⣤⣄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣤
⠀⣌⣹⣿⣿⣿⠇⠠⠤⡟⠁⣨⡽⢛⠛⢳⢯⡅⢸⡇⠀⠀⠀⢘⠁⠸⠈⠋⠀⣰⠏⢿⣇⠀⠀⠀⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠏
⠀⠙⠿⣟⣻⢟⠀⠀⠀⡏⠛⠧⣄⣸⠁⠘⠦⠤⠞⠀⢀⡄⢠⠼⠃⢁⣀⠀⡀⠋⣶⢸⣿⠀⢀⡀⢠⣴⡞⣿⢀⠀⠀⠀⠀⢠⣶⡆
⠀⣠⡆⠈⠠⠜⠀⣰⢻⡏⠳⡄⠀⠉⠀⠀⠀⣀⣀⠀⠀⠀⣇⣀⠄⠀⠉⠉⠉⠀⠉⣸⡿⠉⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⡀⣤⡂
⠀⠉⠻⣦⣀⠀⠀⠻⣌⣃⣠⠇⠀⢀⣤⣶⣿⢿⣿⣧⠀⠲⠟⠋⠀⣦⡤⡤⠴⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠃⠦⠁
⠀⠀⠀⠈⠻⠗⠀⠀⠈⠉⢠⣤⡾⢻⣿⣽⣷⡿⠿⡛⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⢀⣠⡄⣠⣾⣿⣥⡄⠀⠀⠀⠀⠀⠀⠁⠀⢀⠴⠆⠄
⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⢻⣿⣾⠾⢛⣽⣿⠇⠀⠠⣿⣷⣦⠀⠀⠙⣀⢀⣴⣿⣟⣭⣿⣿⠟⠁⠀⠀⣠⠀⠀⠠⣶⡀⠀⠰⠃
⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡉⠀⠀⣼⣿⠃⠀⠀⠀⠸⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⡿⠛⠉⠀⡀⠀⠐⡙⠦⣄⡀⠈⠀⠀⢯⠀⠀⠀⠀⠄
⠈⢿⣿⣿⡿⠋⠀⠀⡄⣀⠠⡀⠈⠢⠀⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⠀⠈⢛⣿⡿⠟⠉⠀⡀⠀⢠⠃⠀⠘⢀⠆⢀⣤⠀⠀⢀⠈⠤⣀⣠⠞⡀
⠀⢀⣭⡍⠀⠀⣠⠃⠀⠸⣀⠈⠀⠄⠑⠘⣿⣆⠀⠀⠀⣸⣿⣿⢃⣠⡴⠞⢉⠀⠀⠁⠀⠁⠀⢈⡀⢤⢂⠁⠀⠿⠃⠀⢸⣿⡇⠀⠀⠈⠈
⠀⠸⠿⠀⠀⠈⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠈⠙⠳⠄⠴⠟⠋⠁⠽⠛⠁⠀⠈⠓⠶⢶⢖⡛⠋⠁⠀⠀⠀⠀⠀⠀⢤⡆⠀⠙⢁⣴⣶⡄
⠀⠠⠖⠀⠲⡀⢰⠛⠙⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢰⠖⠒⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⣀⣧⡀⠀⠀⠀⠀⠈⠻⡿⠇⢰⣆
⠀⠀⡀⠀⣠⠇⢸⣆⡴⠇⣹⠞⡦⡄⢰⢠⡆⢰⡄⡖⢂⣶⡆⢸⠀⠀⢸⡗⠂⠀⣠⣇⡀⢠⣀⡴⠆⢠⢶⠀⣿⢠⡅⡴⣆⣤⡴⡄⠀⠀⠀⠁
⠀⠀⣩⣉⡁⠀⠘⠇⠀⠀⠸⠀⠋⠙⢾⠀⠼⠃⠃⠳⠊⠛⠙⠸⠀⠀⠸⠦⠶⠂⠷⠟⠱⠞⠙⠦⠖⠘⠞⠃⠙⠘⠋⠧⠞⠹⠀⠃⠀⠈⠻⠂⢢
⠀⠠⠹⣿⡇⢢⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣤⡴⠶⢶⣦⣄⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠉
⠀⠀⢣⣿⠿⣾⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣰⡿⠁⠀⠀⡄⠀⠙⠿⣆⠀⠾⢿⣣⡾⠉⢷⠤⠤⢕⠒⠂⣴⣾⣿⣿⣽⡳⣄⢰⡶⠟
⠀⠀⢸⣿⣯⣽⡿⠿⠿⣶⣦⣤⣤⡶⠀⢸⣿⣄⠀⠀⢠⡀⠀⠀⠀⣿⡇⠀⠘⣿⣷⠶⠟⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠉⠛⠛⠛⠛⠓⠀⠈⠉⠛⠧⠀⢹⣿⡀⠀⠀⠀⠑⢀⠀⢀⣿⡇⣀⣤⡹⣿⡄⠀⢻⣶⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡶⢿⣦⡀⠘⣿⣷⣄⡀⣠⢀⣠⣴⣿⠟⠀⠉⢻⡄⠙⢿⠀⢰⣿⠀⡄⠀⡝⠿⢿⣿⡿⠟⠃⠀⠀⠀⠀⡀
⠀⠀⢀⢠⣤⣤⣤⣤⣤⣀⠳⣿⣪⠟⣿⣷⣌⠛⠿⢿⣿⠿⠟⠋⠁⣤⡶⠆⠘⣿⣆⠀⢀⣾⠇⢰⠃⠀⡇⢠⡖⠀⠀⠒⠂⠐⣆⠀⠀⣿⡀
⠀⣴⡏⣾⡿⠿⠿⠿⣿⢸⣷⡌⠻⣿⣿⣿⡻⣿⣦⠀⠀⠀⡐⠞⢉⡁⠔⠀⠀⢹⣿⣦⣾⡟⠀⠘⠀⠰⠃⢸⠁⣄⣐⣂⠀⢀⡟⡂⡀⠻⠃
⠀⠙⠧⣿⣿⣾⣿⣿⣿⢸⠟⠁⠀⠀⠙⢿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⡄⢈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⠆⠬⠈⢵⡶⠞⢇⠃⢸⢦⡀
⠀⠀⠀⠀⠉⠉⠉⡉⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣗⣸⣿⣿⣷⣤⡀⡀⠀⠀⠀⢀⣤⣶⡀⠀⠨⣁⠤⠻⡀⠀⢱⡄
⠀⠀⠀⠀⢀⣀⣴⣶⠏⣷⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡁⣿⢿⡿⠛⠁⠧⢖⣠⣾⡿⢿⣽⣿⡄⠀⢨⣄⠀⠘⣧⠀⠇
⠀⠀⢀⡐⣹⠁⠐⡟⣠⡟⠀⠠⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠈⠁⣿⢸⡇⠀⢠⣴⣿⣻⣷⣷⣿⠟⠋⢀⠔⠁⠈⢢⡰⠀⢠
⢀⣞⢡⣇⣵⣴⣾⠉⠁⠀⠀⢀⠠⠔⠋⠁⠀⠆⠀⢀⣀⣀⣀⣠⣊⣠⣤⣤⡀⣠⡿⠸⣧⠀⠘⣿⣿⡿⠟⠋⠀⠀⢠⣇⠐⠀⠉⣸⣁⡤⠘
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉
⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠰⠀⣀⡀⠀⠀⠠⠄⠀⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠰⠀⠀⠀⠀⠐⠒⠒⠂⠀⡠⢰⠂⠀⠀⠐⢎⠀⠰⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡠⠈⠀⠁
⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠄⠈⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠈⠐⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠰⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠁⠁⠉⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠒⡒⠒⢒⠒⠒⠒⠒⠒⠒⡒⠒⠀⠀⡡⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠒⠒⢄⡀⠀⡗⠛⡆⡤⠀⢾⠸⡄⢰⡇⠀⠀⣅⡆⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣴⣦⡄⠘⢷⠀⠃⠀⠁⠂⠀⠈⠐⠙⠋⠃⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⠃⠀⣀⠀⡇⢸⢰⡆⣶⡎⠀⠀⡀⡇⢰⡆⠑⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡀⠀⢠⡿⠃⠀⣁⡘⢁⡀⡁⢁⢀⡀⡁⡀⠈⠈⢁⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⢸⢿⠀⣼⢇⢀⡟⡇⣸⢾⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠡⠄⠨⠀⠁⠤⠬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠈⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣀⣀⣠⡄
⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⣄⣤⣤⣀⠀⠀⠀⠀⠂⢀⣤⣤⣤⣤⠂⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠂⠀⠀⠀⢠⠙⣿⡟⡋⠁⠤⡤⠤⡤⣤⢄
⠀⠀⢠⠚⠛⠛⠛⠛⠛⡆⠂⠀⣰⡟⠛⠛⠛⠛⢳⡄⠀⢀⣼⠛⠛⠛⠛⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⣿⠀⣁⡄⠀⠀⡀⢀⠀⣸⡧
⠀⠈⣿⠀⢀⠀⠀⠀⠀⣿⠦⠀⣿⡇⠀⠀⡀⠀⢸⡯⢀⠸⣿⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⢸⡇⠇⠀⠀⠀⡅⢸⡇⠃⠈⠀⢸⡇
⠀⠀⠸⢤⣿⣤⣤⣤⣤⠏⠀⠐⠺⣧⣤⣼⣧⣤⡼⠁⠀⠀⠻⣤⣷⣤⣤⣼⡿⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢸⡇⢲⢸⡎⣶⢹⣎⡗⢰⡗⣿⢸⡇
⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠈⠀⠈⠉⢛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠑⢋⠟⠓⠋⠙⠛⠛⠛⠓⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠙⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣧⠀⢀⣠⣤⣤⣤⣤⣤⣤⡤⠤⢶⠖⠋
⠀⠀⠀⠀⠀⠀⢹⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠉⠉⠉⠉⠉⠹⣄⡀
⠀⠀⣠⡴⠚⠛⠉⠉⢯⡀⠀⠀⠀⠀⠀⠀⣨⠏⠉⠙⠛⠲⢄
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠙⠲⠶⠴⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⣀⣀⣠⣤⣀⣀⡀
⣸⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⠟⠋⡽⠃⠀⠀⠀⠻⡍⠓⠀⣄⠀⠀⠘⠚⠛⠛⠛
⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠿⠶⠶⢾⡁⠀⠀⠀⠀⠀⣹⠶⠶⠾⠷⡄
⢿⣤⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠹⣆⠀⠰⠶⠶
⣸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⢣⡀⠀⠀⠀⠀⢠⢿⡀
⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⢧⣤⣤⣤⣴⠃⠀⠀⠀⠀⠀⢳⣤⣤⣤⣤⠏⠘⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⠏⠀⠀⠀⠘⢧⠀⠀⠀⠀⢠⠟⠀⠀⠀⠈⢷⣰⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢨⡷⠶⠲⠲⣯⠀⠀⠀⠀⠀⠀⡿
⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣀⡀⣠⠏⠀⠀⠀⠀⠈⢧⣀⡀⡀⣀⡼⠁
⠀⠀⠀⠘⠷⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠟⠻⣍⠉⠹⣆⠀⠀⠀⠀⢀⡾⠉⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠾⣦⣤⣤⣤⡾⠶⠚⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣽⠿⣍⡿⠛⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢛⣶⡟⢳⣶⠛⣳⣾⠋⣨⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⢿⣁⡿⢯⣨⡿⣏⣹⠿⣅⣴⠟⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢳⣴⠛⣧⡾⠛⣦⡞⢻⣴⠟⢳⣴⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣹⠿⣏⣹⢿⣉⣿⣿⣉⡿⣏⣩⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣉⣠⠟⢳⣴⠟⢧⣼⠛⣦⡾⠻⣤⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣴⠟⢉⣷⣏⣹⣾⡋⠙⣦⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣯⣤⣿⣿⣤⣽⣾⡿⢿⣷⣄⠀⠀⠀⠀⠀⢀⡼⠛⠒⠒⠒⠒⠒⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠙⢿⣷⣄⠀⠀⢀⡞⠓⠒⠒⠒⠒⠒⠒⠒⠒⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠶⠦⣄⡀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡾⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠹⣆⠀⠀⠀⠀⠀⣼⠋⠳⣄⣀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠰⣿⠶⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠘⣦⣤⣤⣤⣾⠃⠀⢸⠋⠉⢳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣿⡤⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡾⠁⠀⠀⠀⠘⣧⠀⣾⠓⠒⢻⡆⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⢾⡁⠀⠀⠀⠀⠀⢨⣷⣿⡿⡿⣿⣇⠀⠀⢀⣨⠷⠶⠶⣤⡀⠀⠛⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢳⡄⠀⠀⠀⣠⠏⣾⢹⠇⡇⢿⢹⡄⢠⡞⠁⣠⠤⢤⡀⠹⣆⣀⣀⣀⣀⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣰⠟⠉⠉⠉⠻⣼⠇⣾⠀⡇⢸⡆⢳⣸⠀⢸⡇⠀⠀⣷⠀⣿⣉⣉⣉⣉⣩⡿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⡋⠀⠀⠀⠀⢠⡏⢠⡏⢀⡇⠀⣇⠈⣿⢧⣸⡇⠀⠀⣿⣰⠏⠉⠉⢉⡽⢯⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠒⠛⠳⠋⠙⠟⠙⠚⠉⠛⠋⠀⠉⠛⠲⠚⠋⠉⠛⠲⠚⠋⠀⠀⠙⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡑⠊⠉⠉⢢
⠀⠀⠀⢀⡠⠔⠚⠉⠉⠁⠈⠁⠀⠀⠀⡤⠃
⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠋
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⢧⠀⠀⠀⠀⢰⢒⡆⠀⡔⠙⠁
⠈⣧⠀⠀⠀⠀⣿⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄
⠀⠘⣆⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠴⠒⠚⠂⠀⠣⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠒⠠⡀
⠀⠀⠈⣇⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢀⡖⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⣠⡄⣠⢤⠆⠀⣀⡀⠀⢠⢶⡀⠀⠞⠦⠤⣤⠀⠀⡠⢴⠀⠀⠀⠸⡀
⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⢡⠏⢀⡞⠉⠙⠚⠉⠀⠙⡆⠘⡄⢰⠃⢀⠎⠀⠀⠙⠒⠲⡄⢡
⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣘⠀⠀⠀⠀⠀⠀⠀⢹⡀⠰⣏⠀⠮⣄⠀⠀⠀⠀⠀⢻⣠⣕
⠀⠀⠀⠸⣄⠀⠀⠈⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠼⠂⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠉⠛⠉⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣴⠄⠀⠀⠀⠀⠹⣧
⠀⠀⠀⢰⡿⠀⠀⠀⢠⡄⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⣿⠃⠀⠀⠀⠈⢿⡄⠀⠀⢀⣿⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠈⣿⠀⠀⠀⠛⠷⢶⣿⣦⣤⣼⣧⣤⣤⣤⣿⣄⣀⣤⣤⣤⣤⣤⣴⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⢹⣇⠀⠀⠀⠀⣼⠏⠀⠀⠘⣿⠈⠉⠉⢻⣏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⢶⣤⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠈⠋⠀⠀⠀⠀⣿⡀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠙⠻⣦⠄⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠰⢶⣄⠀⣠⡾⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠆⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣀⡀⠀⠀⠀⣹⣿⣏⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⠻⢶⣤⡾⠋⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢀⣼⠟⣷⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⢶⣤⣤⣤⣤⣤⣤⡶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣫⣍⡉⠉⠉⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⢀⠴⣪⡈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠐⡶⠀⠀⠀⢀⠠⠐⠂⠉⠁⠜⠁⠙⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠐⢄⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⢢⠀⠃⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠈⣁⠀⠈⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⠀⢈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠠⠀⠀⠀⠀⠀⠱⡀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⡄⢀⠀⠐⠈⠀⠀⠀⢀⣠⣔⡱⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢚⠀⠀⠀⠀⠀⠀⢄⠏⠁⠀⠀⠀⠀⠀⠈⢢⠀⣷⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⡙⠐⡄⠀⢠⠅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇⣀⡤⢴⡶⠄⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣁⠤⠀⠀⠀⢀⠻⠶⠤⠶⠶⠒⢚⠉⠀⠀⡜⠓⠈⢀⠂⠀⠀⢢⡔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡆⢠⠂⠰⡁⠀⠀⠀⡔⠀⠀⠀⠁⠐⠴⠇⠐⡆⢸⣶⢋⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡏⠉⠀⠦⠂⡿⠲⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠔⠇⢞⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠷⠴⠃⠀⠰⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⢾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠒⠋⠀⠀⣀⠀⣴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠘⢯⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢠⠖⠻⠓⣲⠆⠀⠀⠀⠀⠀⢠⠊⠁⠈⠱⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⠞⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡴⢁⣠⠤⠞⠋⠀⠀⠀⠀⠀⠀⠸⡖⠀⠄⢀⡟⠀⠀⠀⠀⠀⢰⠉⠀⠈⠳⡀
⠀⠀⠀⢠⠞⠁⢀⡴⣻⠁⠀⠀⠀⠀⠀⣠⢶⡀⠀⠘⠖⠉⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠒⠠⠯⣀⠀⠀⠀⠀⠀⢸⠂⠐⠀⣠⠇
⠀⢀⡴⠁⢀⡴⢋⡜⠁⠀⠀⠀⠀⠀⢠⠃⠀⠳⡄⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⠼⠏⠀⠺⢅
⡖⠉⠀⠀⠛⠢⡾⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠁⠀⠀⠀⠀⠀⠱⡄
⠹⢄⣀⠀⢀⡴⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⢻⣸⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠙⡄⠈⢷⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠖⠩⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣠⠞⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⠀⢀⣄⣠⡄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣮⠃⠀⠀⠀⠀⠀⠸⡄⠀⣳
⠀⠀⠀⠀⠙⠢⠤⠤⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⣸⠓⢻⡀⠀⢠⡄⠀⢀⡼⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡆⠀⡞⠉⠁⠀⠀⠣⡤⠏⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠐⡇⠀⠀⠀⠀⠀⠀⢀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠙⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠔⠃⠀⠀⠀⠀⠀⢠⣊⣀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣤⣤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣀⣤⠤⣤⣤⣤⣤⡤⠴⠶⠶⠶⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡦⣤⣤⣠⣤⡤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠟⠁⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀
⠀⠀⠀⣠⠶⠛⠛⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠙⠛⠳⢦⡀
⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣀⣀
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⠶⢺⡿⠋⠉⠉⠙⢿⡗⠶⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⣾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡞⠋⠀⣠⡟⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⢷⣄
⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠉⠉⠻⣆⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠙⢷⡀
⢻⠦⢤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⡀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⠛⠛
⣸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⢳⡄⠀⠀⠀⠀⠀⢀⡼⣷
⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⢷⣀⣀⣀⣀⣠⠟⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣀⣀⡾⠁⢹⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⡾⠉⠉⠉⠉⠹⣆⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⠉⢷⡀⣸⠃
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠁⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠈⢷⡿
⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠉⢻⣄⠀⠀⠀⠀⠀⠀⣼⠃
⠿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠛⠲⠶⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠚⠛⢧⣉⠉⠙⣧⠀⠀⠀⠀⠀⠀⣼⠋⠉⣩⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣼⣧⣀⣀⣀⣀⣾⡧⠶⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣤⡀⠀⠀⢀⠀⠀⣀⡀⣀⠀⢀⡀⣀⠀⠀⠀⠀⠀⣤⣄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣤
⠀⣌⣹⣿⣿⣿⠇⠠⠤⡟⠁⣨⡽⢛⠛⢳⢯⡅⢸⡇⠀⠀⠀⢘⠁⠸⠈⠋⠀⣰⠏⢿⣇⠀⠀⠀⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠏
⠀⠙⠿⣟⣻⢟⠀⠀⠀⡏⠛⠧⣄⣸⠁⠘⠦⠤⠞⠀⢀⡄⢠⠼⠃⢁⣀⠀⡀⠋⣶⢸⣿⠀⢀⡀⢠⣴⡞⣿⢀⠀⠀⠀⠀⢠⣶⡆
⠀⣠⡆⠈⠠⠜⠀⣰⢻⡏⠳⡄⠀⠉⠀⠀⠀⣀⣀⠀⠀⠀⣇⣀⠄⠀⠉⠉⠉⠀⠉⣸⡿⠉⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⡀⣤⡂
⠀⠉⠻⣦⣀⠀⠀⠻⣌⣃⣠⠇⠀⢀⣤⣶⣿⢿⣿⣧⠀⠲⠟⠋⠀⣦⡤⡤⠴⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠃⠦⠁
⠀⠀⠀⠈⠻⠗⠀⠀⠈⠉⢠⣤⡾⢻⣿⣽⣷⡿⠿⡛⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⢀⣠⡄⣠⣾⣿⣥⡄⠀⠀⠀⠀⠀⠀⠁⠀⢀⠴⠆⠄
⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⢻⣿⣾⠾⢛⣽⣿⠇⠀⠠⣿⣷⣦⠀⠀⠙⣀⢀⣴⣿⣟⣭⣿⣿⠟⠁⠀⠀⣠⠀⠀⠠⣶⡀⠀⠰⠃
⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡉⠀⠀⣼⣿⠃⠀⠀⠀⠸⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⡿⠛⠉⠀⡀⠀⠐⡙⠦⣄⡀⠈⠀⠀⢯⠀⠀⠀⠀⠄
⠈⢿⣿⣿⡿⠋⠀⠀⡄⣀⠠⡀⠈⠢⠀⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⠀⠈⢛⣿⡿⠟⠉⠀⡀⠀⢠⠃⠀⠘⢀⠆⢀⣤⠀⠀⢀⠈⠤⣀⣠⠞⡀
⠀⢀⣭⡍⠀⠀⣠⠃⠀⠸⣀⠈⠀⠄⠑⠘⣿⣆⠀⠀⠀⣸⣿⣿⢃⣠⡴⠞⢉⠀⠀⠁⠀⠁⠀⢈⡀⢤⢂⠁⠀⠿⠃⠀⢸⣿⡇⠀⠀⠈⠈
⠀⠸⠿⠀⠀⠈⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠈⠙⠳⠄⠴⠟⠋⠁⠽⠛⠁⠀⠈⠓⠶⢶⢖⡛⠋⠁⠀⠀⠀⠀⠀⠀⢤⡆⠀⠙⢁⣴⣶⡄
⠀⠠⠖⠀⠲⡀⢰⠛⠙⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢰⠖⠒⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⣀⣧⡀⠀⠀⠀⠀⠈⠻⡿⠇⢰⣆
⠀⠀⡀⠀⣠⠇⢸⣆⡴⠇⣹⠞⡦⡄⢰⢠⡆⢰⡄⡖⢂⣶⡆⢸⠀⠀⢸⡗⠂⠀⣠⣇⡀⢠⣀⡴⠆⢠⢶⠀⣿⢠⡅⡴⣆⣤⡴⡄⠀⠀⠀⠁
⠀⠀⣩⣉⡁⠀⠘⠇⠀⠀⠸⠀⠋⠙⢾⠀⠼⠃⠃⠳⠊⠛⠙⠸⠀⠀⠸⠦⠶⠂⠷⠟⠱⠞⠙⠦⠖⠘⠞⠃⠙⠘⠋⠧⠞⠹⠀⠃⠀⠈⠻⠂⢢
⠀⠠⠹⣿⡇⢢⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣤⡴⠶⢶⣦⣄⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠉
⠀⠀⢣⣿⠿⣾⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣰⡿⠁⠀⠀⡄⠀⠙⠿⣆⠀⠾⢿⣣⡾⠉⢷⠤⠤⢕⠒⠂⣴⣾⣿⣿⣽⡳⣄⢰⡶⠟
⠀⠀⢸⣿⣯⣽⡿⠿⠿⣶⣦⣤⣤⡶⠀⢸⣿⣄⠀⠀⢠⡀⠀⠀⠀⣿⡇⠀⠘⣿⣷⠶⠟⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠉⠛⠛⠛⠛⠓⠀⠈⠉⠛⠧⠀⢹⣿⡀⠀⠀⠀⠑⢀⠀⢀⣿⡇⣀⣤⡹⣿⡄⠀⢻⣶⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡶⢿⣦⡀⠘⣿⣷⣄⡀⣠⢀⣠⣴⣿⠟⠀⠉⢻⡄⠙⢿⠀⢰⣿⠀⡄⠀⡝⠿⢿⣿⡿⠟⠃⠀⠀⠀⠀⡀
⠀⠀⢀⢠⣤⣤⣤⣤⣤⣀⠳⣿⣪⠟⣿⣷⣌⠛⠿⢿⣿⠿⠟⠋⠁⣤⡶⠆⠘⣿⣆⠀⢀⣾⠇⢰⠃⠀⡇⢠⡖⠀⠀⠒⠂⠐⣆⠀⠀⣿⡀
⠀⣴⡏⣾⡿⠿⠿⠿⣿⢸⣷⡌⠻⣿⣿⣿⡻⣿⣦⠀⠀⠀⡐⠞⢉⡁⠔⠀⠀⢹⣿⣦⣾⡟⠀⠘⠀⠰⠃⢸⠁⣄⣐⣂⠀⢀⡟⡂⡀⠻⠃
⠀⠙⠧⣿⣿⣾⣿⣿⣿⢸⠟⠁⠀⠀⠙⢿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⡄⢈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⠆⠬⠈⢵⡶⠞⢇⠃⢸⢦⡀
⠀⠀⠀⠀⠉⠉⠉⡉⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣗⣸⣿⣿⣷⣤⡀⡀⠀⠀⠀⢀⣤⣶⡀⠀⠨⣁⠤⠻⡀⠀⢱⡄
⠀⠀⠀⠀⢀⣀⣴⣶⠏⣷⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡁⣿⢿⡿⠛⠁⠧⢖⣠⣾⡿⢿⣽⣿⡄⠀⢨⣄⠀⠘⣧⠀⠇
⠀⠀⢀⡐⣹⠁⠐⡟⣠⡟⠀⠠⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠈⠁⣿⢸⡇⠀⢠⣴⣿⣻⣷⣷⣿⠟⠋⢀⠔⠁⠈⢢⡰⠀⢠
⢀⣞⢡⣇⣵⣴⣾⠉⠁⠀⠀⢀⠠⠔⠋⠁⠀⠆⠀⢀⣀⣀⣀⣠⣊⣠⣤⣤⡀⣠⡿⠸⣧⠀⠘⣿⣿⡿⠟⠋⠀⠀⢠⣇⠐⠀⠉⣸⣁⡤⠘
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉
⠀⢀⣀

⠀⠀⠀⢀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢦⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢐⡂⡖⢂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⡇⠠⠰⠀⠈⡍⢩⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⣦⠝⢽⡦⠀⣤⣞⣱⣶⠀⡒⣤⠀⢅⣀⡨⡷⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢻⠘⣿⠁⢰⣿⠉⢦⡶⢀⢁⠆⠈⠈⢄⠐⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡴⠛⣶⠀⢈⡴⣿⠀⣾⢃⣰⡌⠁⠐⢂⠀⠡⠀⡐⡀⣬⣿
⠀⠘⠷⡄⠈⢆⠘⡇⣹⠀⣿⡯⢀⣺⠀⠥⠄⢂⠀⢀⠥⢐⣼⡷
⠀⠀⠀⠻⡄⠸⣆⣧⢹⡄⢻⣇⠹⡐⡐⠀⠁⠀⠀⡀⠀⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠙⣄⠹⣷⣨⡇⠘⣿⣑⡧⠢⣈⡄⡀⢀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠘⣆⢉⣿⣇⠀⢈⣷⡴⡀⢭⢠⠒⣈⣬⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣇⢻⢿⡀⠀⠑⡽⢷⣚⣤⡥⣈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡟⢧⠂⠹⣄⠀⠈⡆⠀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠈⣧⠐⠘⢦⡀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⣇⠀⠈⢻⣇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⢒⠖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢿⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⡄⣷⡀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡠⠊⠀⠀⠀⠁⠀⠀⠀⠀⣈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⡇⣿⣷⠀⠀⠀⠀⠘⣆⠀⠀⠀⡞⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⢀⡤⠖⠂⠀⠀⠀⢀⣷⣤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠘⢦⣀⡰⢅⡠⠐⠈⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⢀⡴⢺⠉⠀⠀⠀⠀⠀⠿⣌⠒⢄⠠
⠀⠀⠀⠀⠀⠀⠀⣧⠸⠀⠀⢻⣿⡿⠟⣷⡄⠀⠀⠀⠈⢿⠀⠀⠘⢦⠀⠀⠔⠀⠀⢀⠎⠀⠑⡄⠀⡴⠋⠀⢀⣷⣤⣄⡠⠂⠀⠀⠈⠑⢤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⢄⠀⠈⠻⣄⠀⠀⠀⠀⠳⡀⠀⠈⣷⠂⠀⠀⠀⠌⠀⠀⠀⠀⡛⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⡀⠀⣀⡀⣸⡄⠀⡖⠉⠁⠚⠷⣄⠀⠙⢆⠀⠀⠀⠀⠱⡀⠀⢉⡆⠀⠀⣘⠀⠀⠀⠀⢠⣷⡤⠔⠉⠉⠻⡿⠟⠓⠄⡀⠀⢀⠀⠈⠙⠻
⠀⠀⢠⠟⠉⠉⠀⢹⣇⠤⠷⢂⡀⠀⠀⠉⠓⠤⠀⢳⡀⠀⠀⠀⠑⢄⡘⠋⢿⠀⠀⠁⠂⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⡟⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠈⢷⡀⠀⠀⠀⠘⢧⣀⠘⢦⡀⠀⠀⠐⠀⠀⠿⣄⠀⠀⢀⣠⡄
⠀⠀⢻⠀⠀⠀⠀⠀⡜⠳⡶⢀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠙⢆⠀⠀⠀⠀⠹⡓⣄⠁⠂⠒⢻⣀⡠⠴⡀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠉⠀⢀⠀⠀⣸
⠀⠀⠀⠙⢄⡀⣠⡤⢶⡀⠙⢤⢱⡦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣵⢸⡆⠀⡤⠁⠀⠀⢀⢱⡄⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠛⠛⠠⠀⢀⡀⠀⠀⠀⠀⠀⣐⣋⠟⠤⣖⣤⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠱⣄⠀⠀⢀⠠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣟⡿⣟⣻⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢙⣶⣛⣷⣞⣳⣾⢁⡽⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡽⢻⣤⠟⣧⡞⢧⣼⠻⣤⠞⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⢿⣩⠿⣏⡽⢯⣹⠿⣉⡿⣏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠞⢙⣶⡋⣳⣞⢹⣶⣋⣷⡏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⣠⠟⢧⡼⢻⣴⠛⢦⣹⣿⠏⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣋⣽⣿⣉⣷⣾⣿⣿⣄⠀⠀⠀⠀⠀⣰⠿⠶⠶⠶⠶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠙⢿⣷⣄⠀⠀⡼⠓⠒⠒⠒⠒⠒⠒⠒⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠲⢦⣄⠀⠀⠀⠀⠙⢿⣷⣼⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠙⣆⠀⠀⠀⢠⡞⠈⣳⠤⣄⠀⠀⠀⠙⡏⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠛⠛⠛⢻⡄⢀⣧⣀⣸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠚⡟⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣼⠁⠀⠀⠀⠀⢹⣼⣧⣤⣴⡇⠀⠀⠀⣷⣠⣄⣀⠀⠚⡟⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⣦⠀⠀⠀⢠⠞⡾⣿⢹⢹⢹⡄⢠⠞⢁⣠⣄⠉⢳⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⡼⠛⠛⠛⠻⣼⢁⡇⣼⠘⡇⢷⣾⠀⣿⠀⠈⣧⠀⣏⣉⣉⣉⣩⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣾⣁⠀⠀⠀⣰⡇⣸⡁⣿⡀⣷⠘⡟⢦⣿⠀⠀⣯⣼⡉⠉⣉⡽⢣⣀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠙⠁⠙⠉⠉⠉⠉⠁⠀⠉⠉⠉⠁⠀⠉⠋⠁⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡑⠉⠉⠑⡄
⠀⠀⢀⡠⠖⠊⠉⠀⠀⠀⠀⠀⠀⡠⠂
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⡇⠀⠀⠀⠀⣀⣀⠀⢀⣤⠎
⢳⡀⠀⠀⠀⢣⡏⢠⠋
⠀⢳⡀⠀⠀⠘⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠎⠉⢆⢀⡀⠤⠀⠀⠐⠂⠀⠤⠄⡀
⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⢀⡀⣀⣠⡄⠀⡀⠀⢀⣤⡀⠀⡶⣄⣀⡀⠀⢀⣤⡀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢘⠀⠀⠀⠀⠀⠀⠙⠊⠓⢋⠎⢀⠎⠙⠒⠚⠀⠙⣆⠘⡀⡰⠁⡰⠃⠀⠙⠒⢲⡀⠆
⠀⠀⠀⡗⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⠀⠀⠀⠀⠀⠀⢸⡄⡱⣇⡰⢥⡀⠀⠀⠀⠀⢷⡬⣶
⠀⠀⠀⢷⡄⠀⠀⠾⢄⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠤⣤⡀⠀⠀⠀⠀⢠⣶⠟⠋⣉⣉⠛⢷⣮⣿⠄⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠒⠲⠾⣧⣄⣀⣤⠤⣿⡃⣰⡿⢛⣛⢿⣆⠀⣀⣤⡀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⠙⢧⣶⣴⠟⢿⠟⠛⠉⠛⠻⣀⣿⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⠛⠛⣻⠃⠀⠀⠀⠀⠀⠀⠀⠠⣹⡇⣻⡷⡈⠲⠿⠿⠟⣦⠈⣿⣤⡶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣶⣾⣿⡿⠏⣀⠀⠀⠀⠀⢀⣀⣠⣥⣿⠀⢿⡇⢸⣿⠶⠶⠟⣿⠇⠻⣿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣞⠋⠉⠀⠀⠀⠀⠉⠑⢾⣉⣡⣤⢔⣾⣯⣄⠸⣷⣌⠻⢷⣶⠾⠋⣰⣦⠀⣀⡙⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣵⡟⠳⢦⣄⡀⠀⠀⠀⠀⠸⢰⠁⡼⢋⠴⢮⢹⣆⣨⣿⣷⣶⣶⣶⠟⣿⠏⢠⡟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠉⠛⠶⣬⣙⠛⢶⣤⠴⢛⣋⣽⢱⠃⠀⢀⡷⠛⠉⠀⠌⠁⠉⠙⢷⣟⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⣤⣀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣶⢋⣿⡇⡎⠀⣴⡟⠀⡆⠀⡘⠀⠀⠀⢀⠔⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣁⠀⠛⠿⢿⣶⣦⣄⡀⠀⠞⢠⣾⡏⣼⣿⢰⢁⠌⣿⠣⢄⣃⠀⠁⠀⠀⠀⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⠈⠉⠛⠿⠟⣶⡿⣿⢰⣿⣿⢸⠀⠀⣸⡇⡰⠀⠀⡎⠁⠉⠁⡈⠈⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣴⡟⢸⡟⢚⣿⣿⡈⠲⣊⡽⢿⣆⠀⠀⠰⡀⠀⠀⠑⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠈⠙⠛⠋⠀⠀⠙⠻⢶⣦⣵⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠁⠐⢂⠀⠀⠁⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⢀⢠⠄⠀⠀⠄⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⠄⠀⠃
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠐⠒⠒⡐⠒⠒⠒⠒⡒⠒⠀⢈⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠀⢠⡀⢰⠉⡆⡆⠐⡇⢆⣰⠁⠀⢰⣀⠀⠈⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⣿⡆⠈⠃⠠⢠⢀⠁⣀⠌⠁⠀⡄⣀⠀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣴⠂⠨⠸⠜⠐⠿⠀⠀⠆⠃⠛⠐⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠁⠀⢸⢰⣾⠉⣶⠁⣷⡇⣲⡆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠁⠁⠁⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣠⡀
⠀⠀⠀⣄⣤⣤⣄⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⢠⣤⣤⣤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢿⠟⡉⠡⠤⠤⠤⡤⣄
⠀⣀⡎⠉⠉⠉⠉⢱⡀⠀⣾⠉⠉⠉⠉⣳⡀⠀⣼⠉⠉⠉⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠾⠠⠴⠂⡄⠀⠀⠀⣿⠂
⠀⠀⡇⢠⠀⠀⠀⢸⠉⢀⣿⠀⠀⡄⠀⣸⠁⠀⢻⠀⡄⠀⠀⣿⠇⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⢺⢠⣠⡤⡀⣆⢧⢀⣄⡄⣿
⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠈⠙⠻⠿⠟⠁⠀⠀⠀⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠘⠼⡿⠧⠗⠿⠿⠼⠧⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⢾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠒⠋⠀⠀⣀⠀⣴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠘⢯⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢠⠖⠻⠓⣲⠆⠀⠀⠀⠀⠀⢠⠊⠁⠈⠱⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⠞⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡴⢁⣠⠤⠞⠋⠀⠀⠀⠀⠀⠀⠸⡖⠀⠄⢀⡟⠀⠀⠀⠀⠀⢰⠉⠀⠈⠳⡀
⠀⠀⠀⢠⠞⠁⢀⡴⣻⠁⠀⠀⠀⠀⠀⣠⢶⡀⠀⠘⠖⠉⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠒⠠⠯⣀⠀⠀⠀⠀⠀⢸⠂⠐⠀⣠⠇
⠀⢀⡴⠁⢀⡴⢋⡜⠁⠀⠀⠀⠀⠀⢠⠃⠀⠳⡄⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⠼⠏⠀⠺⢅
⡖⠉⠀⠀⠛⠢⡾⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠁⠀⠀⠀⠀⠀⠱⡄
⠹⢄⣀⠀⢀⡴⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⢻⣸⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠙⡄⠈⢷⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠖⠩⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣠⠞⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⠀⢀⣄⣠⡄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣮⠃⠀⠀⠀⠀⠀⠸⡄⠀⣳
⠀⠀⠀⠀⠙⠢⠤⠤⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⣸⠓⢻⡀⠀⢠⡄⠀⢀⡼⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡆⠀⡞⠉⠁⠀⠀⠣⡤⠏⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠐⡇⠀⠀⠀⠀⠀⠀⢀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠙⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠔⠃⠀⠀⠀⠀⠀⢠⣊⣀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣤⣤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⡊⠁⠈⡰⠌⠠⠀⠀⡀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠅⠈⠉⠉⢉⡄⠈⠄⠈⢀⠃⠀⠰⠀⠀⡀⠀⣀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠘⢉⠈⠁⠀⢸⠀⢰⠀⠌⢠⡐⠀⠈⠉⣠⢳⡝⣊⢗⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣺⡶⠠⠘⠀⠀⠈⠁⠄⠀⢠⠀⠘⠁⢜⠢⡐⢀⡀⠀⣯⢷⡹⢞⢯⢲⢥⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣴⠿⠚⡀⡠⠄⠁⠀⠀⠀⠀⣀⢀⠀⠔⠀⠀⠀⠐⠏⡁⠀⢹⠱⡛⠔⢫⠵⣹⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢵⣈⢆⠹⡟⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡀⡀⣀⢀⣄⢀⠀⣀⡄⢀⡀⢀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⣧⠟⣿⢼⡇⢳⡇⠻⡄⢸⢰⡇⠀⣸⢇⢸⡇⠀⠀⠀⠀⠀⠁⠐⣀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠋⠀⠀⠀⠀⠀⡇⠀⠿⠸⠇⢸⠄⢠⡽⢸⠈⢧⡄⡟⢺⠼⣧⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠈⠀⢀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠀⣠⠀⠤⠄⠀⢹⡉⢸⠙⡆⣿⢸⢠⡏⠁⢸⡇⢹⡇⡏⢸⠉⡇⣿⣸⠀⠀⠄⠀⠡⡐⠄⣈⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠦⠀⡠⠀⢸⣁⢸⣠⠇⣿⣸⠘⣇⡀⡷⢿⢸⡇⡷⢸⣀⡇⣟⢿⠀⠀⢀⡴⢱⠌⢀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡞⠂⡈⠚⢉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⣴⠂⠀⠀⢶⣄⠀⠀⠀⡅⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠄⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠙⢄⣀⡔⠉⠉⠁⠄⠀⠀⠀⠂⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠐⡉⡴⠀⠀⠀⠀⠀⠀⠴⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠾⣿⠟⠠⢀⣴⣦⠀⠴⠂⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⣃⠀⠀⠀⠤⠀⡈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡸⠿⡏⠑⠠⠡⠔⠀⠀⠁⠀⠀⠀⠠⠀⠀⠔⠀⠀⠀⠄⠀⠀⠐⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇⢀⡄⠀⢀⠄⠀⠓⠀⠀⠀⠆⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠂⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⡄⠀⠀⠠⠚⠓⠂⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⠀⠤⠄⠁⠀⠠⠀⠌⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣩⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⣤⣶⣶⣶⣦⡘⢿⣿⣿⣿⣿⣿⣿⡿⢃⣶⣶⣶⣦⣤⡀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣭⣉⣛⣋⣉⣥⣴⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⣃⣤⡖⣠⣿⣿⣿⣿⡄⢶⣤⣄⠀⠀⠀⠐⠒⠒⠒⠒⠒
⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠁⠚⠛⠛⢰⣿⣿⣿⣿⣿⣿⡄⠛⠛⠓
⠀⢿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣷⡀⠀⠀⢤⣤⣤
⠀⣶⣶⣶⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⠆⣉⣉⣉⣉⠰⣿⣿⣿⣿⣿⣿⠟
⠀⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣆⠻⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⠏⣴⡆
⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⢀⣤⣤⣤⣤⠸⣿⣿⣿⣿⣿⡿⠃⣤⣤⣤⣤⡘⣿⡇
⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢡⣿⣿⣿⣿⣿⣷⡘⠿⠿⠿⠿⢁⣾⣿⣿⣿⣿⣷⡌⠁
⠀⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⡿⢡⣶⣶⣶⣶⡈⢿⣿⣿⣿⣿⣿⠟
⠀⠛⠛⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠛⠛⠛⠛⢠⣿⣿⣿⣿⣿⣿⡌⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⠻⣿⡌⢻⣿⣿⣿⣿⡟⣠⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠛⠋⠀⠁

⣠⢤⡄⣤⠀⡄⢤⢀⡄⣤⠤⢠⡄⣠⠤⠄⢠⣄⠀⡤⠀⠀⢠⡤⠄⣤⢤⡀⣤⠀⣤⢀⡤⠤⠀⣤⡀⢤⡤⢤⡄⣠⠤⣄⢠⣄⢠
⢿⠚⠁⡿⠛⡇⠈⡿⠀⣙⣷⢸⡇⣧⣀⢀⡿⢿⡄⣯⣀⠀⢸⣛⡀⣇⣠⡟⢺⣀⡿⠸⣄⣀⢸⠷⣷⠸⡇⠸⡇⢿⡀⡽⢸⡟⣾
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢀⣀⣀⠀⠀⠁⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣈⣉⠀⠀⢠⠈⡈⠉⠙⢄⡈⠀⠰⠐⠷⠠⠐⡔⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠠⠼⠿⠿⠷⢿⣿⣁⡤⠤⣴⡷⣀⠡⠄⠀⠈⠳⢼⡆⣱⠠⠄⠸⡑⠀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⡐⢠⣶⣴⣾⣿⡿⠦⠞⠛⡀⣄⡀⠤⣤⢞⣶⣌⢻⣦⣶⢞⠃⠀⢳⣄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣑⣦⣿⣷⣼⠧⣤⣷⠶⢾⣧⣬⣿⣿⣵⣞⣛⠻⣿⣿⡇⠘⣤⠠⠀⠀⢀⡷⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠍⢭⡡⣴⣍⣽⢉⢶⣋⣥⠵⠾⠛⠋⠛⣙⣛⣿⣿⣶⣿⣿⣿⣶⣶⠶⠟⠛⠋⡴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⢛⣿⣿⠛⢯⠥⠀⠺⠶⠿⠒⠛⠻⠿⣍⣙⠻⣿⡏⢿⣿⠗⠊
⠀⠀⠀⠀⠀⠀⠰⠍⠉⠣⠀⠀⠘⠿⠿⠸⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⡠⢰⡎⢑⡆⢄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠑⡌⢚⡗⢩⠊⠀⠀⠀⠀⠀⠀⠀⠳⡐⡀⠠⡡⠆⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⡃⠠⡇⢸⡅⠰
⠀⠀⢰⠈⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣟⡛⠀⠀⠀⠀⠀⠀⠀⡃⠀⡗⢸⡆⢰
⠀⠀⠘⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠃⠈⠓⠘⠃⠐


⠀⢠⣾⡭⢭⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠻⠉⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠰⣶⣶⠛⠳⣶⡶⠂⠀⠀⠀⠀⠀⡼⠁⠀⠄⠘⣿⠙⢦⠀⠀⠀⠀⠀⢀⣀⣶⣼⣷⣴⣀⡀
⠀⣿⣦⠐⢂⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⣾⣧⠀⠀⠀⠀⠀⠀⠀⢧⡀⠐⠀⢀⡿⠀⣸⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠛⠋⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠙⠓⠊⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠚⠋⠁


⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⡀
⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡬⠮⠆⠀⠀⠀⠀⠀⠀⠀⣰⡾⡆⢰⢿⣆
⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣇⠀⣀⠀⠀⠀⠀⢠⡆⢄⠒⠀⠐⠐⡄⠀⠀⠀⠀⠀⠀⣿⣿⠇⠘⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠚⠉⠛⠛⠉⠛⢻⠟⠋⠛⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⡟
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠡⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⡀
⠠⠶⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣷⣄
⠶⣏⣁⣉⣉⣉⣹⠶⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠀⢀⡤⠄⣠⣶⣦⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡿⠀⠀⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢸⡇⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⡄⠀⠀⠀⠀⢸⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣀⠃⠀⣿⠀⢠⡎⠁⣾⠄⠀⠀⣿⠀⠀⠀⡶⠀⠀⣤⠀⢠⡞⠑⣿⠈⠀⢾⡄⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣿⠐⢠⡏⠀⠸⣇⠀⣿⠀⠀⠀⣿⠀⠀⢀⡇⠀⢀⡏⠀⢸⣇⠀⠹⣆⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⠄⠞⠁⠀⠀⠀⠀⠘⠆⠀⠀⢹⠀⠀⠈⠙⠀⢸⡇⡀⠁⠈⠀⠈⢻⡀⠂⢹⡇⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⡸⠁⠀⣠⠇⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠁



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

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⢴⠁⠀⠈⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠏⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⢀⠁⠀⠢⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⠑⠒⠀⠘⠋⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣸⣿⣿⣿⠏⠀⠀⠀⠀⠁⠀⠐⠄⠈⠀⢀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⠼⢀⠀⠀⠀⠀⡐⠁⠀⢀⠀⠀⣀⣤⡀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⠐⠀⠐⠁⢀⡀⠑⢄⡀⠄⠙⠄⠀⠀⠀⢰⣿⣿⣿⢀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣸⣿⣿⣿⠟⠀⠀⠀⠀⠘⣿⣦⠀⢁⣴⣶⠃⠀⠀⣀⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠁⣰⣿⣿⣿⣷⣄⠈⠉⠈⠈⠛⠁⠀⠀⢸⣿⣿⡆⢈⣡⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠛⠉⠀⠢⠀⠀⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢠⣿⣿⡆⠻⣿⠇⠛⠿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣄⡑⠀⣿⣿⣿⣿⠟⠁⢀⠂⠀⣠⣶⣄⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢀⡀⢻⣿⡿⢀⠝⢷⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⠹⣷⣤⣤⣄⠉⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣥⣀⡀⠙⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡷⠀⠀⠠⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣤⣀⣤⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠿⢿⣿⣿⣿⣿⣿⣦⣠⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⣦⠈⣃⡀⢀⣀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣉⣀⣀⣀⡁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡤⠃⡞⠁⠉⠋⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣋⡐⢿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠀⠙⠒⢦⡀⣀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⠀⣤⡄⣰⠀⢳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠈⠒⢻⠉⠀⠉⢻⠒⠋⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣋⣼⣿⣿⣿⣧⣙⣿⠆
⠀⠀⠀⠀⠀⡜⠀⣠⠴⢒⡋⠀⠀⠀⠘⣒⠶⢤⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⣿⣿⣿⣟⠿⣿⣿⣿⣿⠇⣷⡀
⠀⠀⠀⠀⡼⠁⠀⠳⠤⠥⠭⠷⠶⠶⠭⠬⠤⠜⠁⠀⢱⡀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣶⣶⣶⣦⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠐⠘⡀⠢⡀⢁⠀⠀⢈⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠁⣈⠞⠳⡖⢠⣜⡋⣶⣀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠊⠁⢠⢄⠈⠙⠃⠈⢉⠀⠉⢒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⡀⠀⠘⠟⠀⠀⠀⠙⠟⠀⠀⡼⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢳⣄⠀⡖⢲⠒⠓⠒⡆⢀⡜⠁⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⠀⠉⠑⡇⠀⠀⠀⠀⡟⠋⠀⠀⡀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠒⠤⢄⡿⠀⢀⡤⡇⠀⠀⠀⠀⣇⡀⠀⠀⢻⠤⠤⠐⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣠⡧⠴⠋⠀⠉⣠⣦⣔⣒⠁⠉⠖⠒⠼⢄⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⣠⠤⠤⣽⠿⢿⡬⠴⠦⢄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⣤⠏⠀⠀⠀⠀⢳⠁⠀⠀⠠⠀⠹⣦⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢤⢿⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢠⡏⢶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠏⢸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠛⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡃⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⠒⠒⠒⢒⡟⣟⠒⠒⢒⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⡛⠛⢻⠳⣇⠷⣟⠛⢛⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠨⡾⠙⢮⠁⣿⠀⢶⠋⠳⡎⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠏⠉⠹⠀⢹⠀⢨⠁⠀⢅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠉⠁⠀⠈⠁⢸⡂⠉⠀⠀⠈⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡦⠤⠤⠤⢔⡿⣗⠤⣤⣤⠤⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⢿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠋⠁⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠘⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣋⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠
⠀⠀⠀⠀⠀⠀⠈⠙⠛⡀⢀⣆⠀⠀⢠⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿
⠀⠀⠀⣴⡿⠃⠀⠀⢸⣷⣾⣿⣆⠀⠀⠙⢿⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣿⣿⠏⠃⠀⠀⢰⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠟⠦⠀⠀⠀⠉⠻⣿⠀⠀⠀⠀⠀⠀⣠⡀⢀⡠⢽⣿⣿⣿⣶⠃⠶⠀⠀⠊⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣼⣿⡿⠀⠀⠀⠈⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠘⣿⡿⠆⢈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⡠⣴⣾⣿⣿⡆⠀⠀⢀⣴⣶⣾⣀
⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠻⣿⠋⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⠀⠀⢴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⠙⣻⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠟⢻⠿⣇⠀⠀⠀⢹⣿⣿⠿⠏
⠀⠀⠀⠀⠁⠛⠳⢿⡗⣍⠙⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⣾⠇⠉⢀⣀⣐⡤⠀⠀⠀⣶⣶⣧⡀⠀⠀⠀⠀⠈⠑⢦⣠⣴⡟⠤⠀⠤⠤
⠀⠀⠀⢤⡞⣤⣠⣿⡟⣿⠴⠆⠀⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⢙⣄⠀⣻⣷⣶⡃⠀⠀⠉⣹⠿⣿⡟⠳⠂⠀⠒⠒⠉⠁⠀⠈⢧⡀
⠀⠀⠀⡄⠁⣉⣁⣽⡗⣿⢠⡆⠀⢀⠀⠀⢀⠀⣜⡁⠀⠀⠀⣿⣿⠓⠄⣸⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⣀⣤⣤⠀⠉⠒⠦⠤
⠀⠀⠀⠀⠜⠉⢁⣼⡇⠙⠛⠛⠂⠀⠀⠀⢀⡉⣿⣇⠈⠀⠀⠉⠀⠀⠀⠉⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡘⠻⣿⣏
⠀⠀⠀⠀⣶⣀⠈⣿⣿⣷⠶⡇⠀⠀⠀⠀⠀⠀⣼⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡶⢤⠈⠉⠀⠤⢀⠚⠈
⠀⠀⠈⠀⣹⡿⠦⢀⣿⣷⣾⡷⠄⠀⠀⠀⠰⠶⣿⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⣤⢀⡰⣤⣽⣿⣿⡦⣤⡤⠄⠀⠀⢠⣦⣹⣿⣁
⠀⠀⠀⠈⠛⠃⠁⣨⣿⡷⢶⡦⠀⠀⠀⣀⠸⣿⣽⣿⣄⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⡀⡄⢷⣠⣄⠈⣀⡀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠊⠿⢡⠄⠙⠀⣻⡾⠏⠉⠀⠀⠀⠀⢺⠉⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⢀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠁⢠⠀⠠⠞⡁⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⣶⣄⠀⠙⠁⠀⢀⡄⣸⡏⠐⣆⡀⢀⡀⢀⢠⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣹⡧⠀⠀⡆⠀⣰⡇⠀⠀⢏⠳⠂⠀⠧⠀⠂
⢽⣿⣲⣶⣤⣠⣟⣍⣿⣇⣠⣼⣇⣾⣟⢠⢅⣿⠀⣠⡄⠀⢀⣐⣟⣀⠀⠒⠄⣈⣟⣷⣴⠆⣱⣾⣤⢽⢧⢖⣀⣾⣃⠀⢀⣰⡄⠀⣄⡀
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡎⢃⣷⡾⣿⣧⡿⢾⣿⣾⣿⣿⣾⣯⣾⣿⣿⣿⣷⣿⣾⣿⣿⡏⣴⣷⣿⣿⣤⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣬⣯⣭⣤⣿⣶⣬⣿⣥⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣷⣾⣿⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀⠘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠻⢿⣿⣿⣿⣿⣷⠌⠙⠛⠛⠛⠁⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠉⠉⠉⢁⣴⣶⣤⣀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠻⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⠈⠉⠙⠛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⣴⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣦⣴⣦⣰⠷⡴⠷⣴⠆⠀⠀⠀⠀⠀⠀⠀⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⠀⠀⠄⠀⠈⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣰⣫⣾⠿⢷⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⣤⣤⠀⢸⣿⠀⠀⠁⠀⠀⠀⠀⠀⢀⣷⣿⠇⠀⠀⠹
⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⠀⣼⣿⠀⢠⣄⠀⠀⢠⣶⡆⢸⣿⣿⡀⠀⠀⠀⠀⣤⣶⣤
⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⣿⣿⡿⠀⣾⣿⠉⠉⠍⠀⠀⣿⣿⠀⣿⣿⠀⣸⣿⡄⠀⢸⣿⡇⠀⣟⣿⣇⠀⠀⠀⢸⡏⠀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⠀⢠⣿⣿⡇⠀⡇⣷⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⢻⡀⢸⣿⠃⠀⠸⡜⣿⣆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣇⢸⣿⣿⡇⠀⣟⣿⣛⣛⣛⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⠀⢧⢸⣿⠀⣾⠂⠙⣞⢿⣦⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣼⣿⣿⡇⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⠀⠘⣿⣿⠀⣟⠃⠀⠘⣮⢿⣇⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⠙⠁⠀⠀⠘⡞⣿⡆⢿⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⠿⣿⣛⠀⠀⣿⣿⠀⣿⣷⠀⠀⣀⠐⠀⠐⠀⣀⡀⠀⠀⠀⠀⢳⢻⣇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢿⣿⣿⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣤⡄⠀⠀⠀⠉⠀⠁⠀⢀⣿⣳⡆⠀⠀⠀⢸⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⠀⠈⠛⡙⣇⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢤⠀⣠⠀⢀⣄⣀⣴⣀⠼⣧⠜⠧⠜⠀⠀⠀⠉⠉⠉⠀⠀⠀⢨⠿⡄⠀⠀⠀⠙⣏⣛⣧⣀⣠⣋⣙⡛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠉⠈⠁⠀⠁⠀⠀⣀⣀⠀⣀⣀⠀⢢⡞⠀⠀⠀⠀⠁⠀⠀⢀⣠⠀⠀⠈⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠁⠀⢠⠎⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡴⠀⢠⠞

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⣤⣤⡀
⠀⠀⠀⠀⠀⣠⣾⡿⣛⣛⣛⣛⣛⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⣫⣭⡖⣴⣶⣭⣭⢿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠶⢠⣒⣀⣿⣿⣦⡀
⠀⠀⠀⠀⣸⣿⣿⡇⣄⣿⣿⣿⣿⣫⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣷⣭⣯⣘⢟⣿⣿⣟⣣⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⠎⢁⣊⢤⣲⣶⣿⣾⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⠿⠿⢛⠅⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⠿⠿⠛⡇⠀⠀⠀⠀⣿⣿⡳⡞⣂⣋⣼⣿⣿⣿⡿⠿⠛⡇
⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⠏⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣾⡿⢿⣿⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⣿⣿⣏⠀⠀⠁⠀⠀⠀⠛⠁⠀⢳⠀⠀⠀⠀⠸⣿⣿⣯⠀⠀⠉⠀⠀⠀⠛⠃⠀⢱⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠉⠀⠀⠀⠓⠃⠀⢹
⠀⠀⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣰⠉
⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⢠⣀⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⢠⣀⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⢀⣀⣀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠀⠈⠉
⠀⠀⠀⢀⣤⣶⣿⣶⣦⣤⣴⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣦⣤⣴⣾⣷⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣦⣤⣴⣾⣷⣦⣄
⠀⠀⠀⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠛⠻⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⣤⡄⠀⠘⠛⠀⣤⡀⠀⠀⠀⢀⣄⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠙⠁
⠀⠀⣴⣿⣷⡾⠁⢀⣾⣿⣳⡄⠀⢼⣿⣿⡄⠀⠀⣼⣇⡀⠀⠀⢸⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷
⠀⠀⢻⣿⣿⠀⠀⠘⢻⣿⡿⠃⠀⢻⣿⡟⠀⠀⠀⠻⣿⣷⣤⠀⠸⣿⠟⠀⢀⡄⠀⠀⠀⠀⢀⣀⣚⣁⣀⣀⡀
⠀⠀⠀⣿⡟⠀⠀⠀⠸⣿⠇⠀⠀⡾⠁⢿⠀⠀⠀⠀⠀⠀⠿⠀⠀⢿⠀⢶⣾⡇⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⢛⠁
⠀⠀⢀⣿⣇⠀⠀⠀⣸⣿⡈⠛⠀⠀⣸⣿⡀⠀⠀⣀⣤⡾⠃⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠂⢹⠇
⠀⠀⢻⣿⣿⡇⠀⠘⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠙⣿⣿⡆⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⢛⣤⠀⠀⣤⣻⡄
⠀⠀⠸⣿⡿⠀⠀⠀⢿⣿⠇⠀⠀⠀⢿⣿⠇⠀⠀⠀⣸⡿⡇⠀⠀⢻⡟⣷⠁⠀⠀⠀⠀⠠⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠛⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠋⠀⠋⠀⠀⠘⠁⠃⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠈⠙⠁
⠀⠀⡀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⡀⣾⣿⡆⢀⣴⣿⣷⠀⠀⠀⢠⣾⣷⡿⠀⠀⠀⠀⡛⣷⣴⣦⣀
⠀⠀⣧⠀⠀⠀⢿⣿⣷⣤⠀⡀⠀⠀⡇⣩⣿⣷⢺⣿⣷⣤⢀⡀⠀⠀⣿⣿⠇⠀⠀⢤⣀⣈⣪⣻⣛⣭⣛⣂⡴
⠀⠀⡟⠛⠛⠛⠛⠻⡟⠿⠛⡇⠀⠀⡟⠟⢻⠛⠛⠛⡿⠻⠛⡇⠀⠀⢿⣿⠀⠀⠀⠀⠀⠴⠒⠋⠙⠓⠦

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⠊⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣤⣆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⠉⠉⠉⠹⣿⣬⣿⠀⣏⠂⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣯⡴⠛⠢⣴⡆⣤⡀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⠀⠀⠀⠀⣈⣿⣿⣿⡶⠀⢻⡗⠛⠀⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣧⣴⣏⠀⣦⣤⡼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠹⠿⠛⠿⣿⣿⣿⣿⣶⣿⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣃⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡏⢻⡿⠗⠂⠀⠀⢀⣾⠛⢿⣿⣿⣾⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣉⣀⢨⣿⣷⣶⠄⠀⠈⠁⠀⣸⣿⣿⡿⣿⣿⡇⢠⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⡙⣿⡟⢻⡍⠁⠀⢀⣠⣴⣿⠛⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⡆⢸⣿⠻⡯⠈⢻⣿⣿⣿⣿⣿⠟⠀⣿⣽⣿⣏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⣀⡹⠶⠀⠋⢹⣿⣿⡟⠛⠁⠀⠀⠹⣿⣿⣟⢿⣭⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⡄⢀⣿⣿⠏⠁⠀⠀⠀⠀⡀⠀⢿⣿⣶⠟⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⠟⠃⠀⠀⠀⡀⠀⣉⣴⣿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣝⣿⣿⡟⠉⠆⠀⠀⠀⠈⢀⣺⣿⠙⢿⣏⢰⠄⠀⠈⢷⠄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⡏⢷⣼⡌⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣀⢐⣄⣀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⡟⠀⠞⠻⡇⢁⡀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠂⠀⠸⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢻⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡈⡇⠀⠀⠀⠀⠀⢸⣸⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠚⠁

⢠⣶⣶⣦⣠⣶⣶⣤⠀⠀⠀⢠⣤⣤⡀⠀⢠⣤⣤⡄⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⢀⣹⣏⡀⣀⠀⠀⠀⠀⠀⣐⣷⣶⣂⡸⣋⡇
⢿⣿⣿⠻⡟⣿⣿⣿⠇⠀⠀⢸⣿⣿⣧⢠⣼⣿⣿⡇⠀⠀⢀⢤⣜⢇⠂⣂⢧⡤⣀⠀⠀⠀⠀⡿⠉⠀⣶⣝⢯⠀⠀⠀⠀⢸⣥⣯⣽⣭⣏⣥⡇
⠈⠛⢯⣶⣷⣸⠟⠉⠀⠀⠀⠸⣿⣿⠋⠀⠙⣿⣿⠃⠀⠀⢇⠘⢂⠞⢿⡁⢇⠑⣨⠂⠀⠀⠘⡇⠀⠀⢿⣿⣸⠀⠀⠀⠀⢸⡶⡷⢾⢶⣟⡶⡇
⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠘⠛⠛⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠦⣤⣤⠷⠃⠀⠀⠀⠀⠸⣷⡇⢸⣶⠿⠇⠇

⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⡀
⠀⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣛⣿⣋⠀⠀⠀⠀⠀⠀⣾⣿⡋⣻⣿⡄⠀⠀⠀⠀⣾⣯⣭⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣻⣽⡆⠈
⠀⣾⣿⣧⠀⣶⣶⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠈⠳⠿⠟⠁⠈⢄⠄
⠰⡇⠦⣹⠀⣿⣸⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⡀⡈⣉⡉⡐⠀⠀⠀⠀⠀⣿⣿⣿⣭⣿⣽⣿⣿⠀⠀⠀⡄⣾⡿⣷⣠⠀⢸
⠀⠿⠿⠿⠀⠿⠿⠇⠀⠀⠀⠀⠀⠈⠛⠙⠛⠁⠀⠀⠀⠀⠀⠈⠣⠄⠁⠣⠤⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁⠛⠛⠛⠈⠀⠘

⢀⣤⣤⠄⠢⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡄⠀⠀⠀⠀⣀⣠⣀⣀⣠⣀⡀
⢸⣿⣿⣦⣤⣼⣿⣿⠀⠀⠀⠀⢠⣄⣤⠀⣼⣟⣿⡆⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⡿⢿⡦⠀⠀⠀⢸⡿⠿⢿⢿⢿⠿⣿
⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡛⣿⣟⠈⠙⠋⠀⠀⠀⠀⠀⣚⠡⣹⠀⠀⠀⠀⠀⠀⠀⣸⢟⡻⣻⢟⣺⡇⠀⠀⠀⠀⢸⡇⠤⠼⠸⠼⠀⣿
⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢹⠿⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⠅⠀⠀⠀⠀⠸⠧⠭⠭⠤⠤⠤⠟
⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠉⠉⠁

⢠⣾⢀⣀⣀⣀⣸⣤⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢠⠚⠓⡄⠀⠀⠀⠀⠀⠀⠀⢿⡆⢀⡀
⠈⠻⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠐⢁⣤⠀⣿⡇⠀⠀⠀⠀⣴⣶⢷⡄⠀⡀⠈⠀⠀⠀⠀⠀⢸⡿⢿⠿⢿⡆⠀⠀⠀⠀⢰⣿⣶⠀⣿⡇⢀⡀
⢀⣶⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⢠⣄⢸⣿⠀⣿⡇⠀⠀⠀⠀⢻⣿⣿⣷⣄⣁⡴⠀⠀⠀⠀⠀⣿⣇⣘⡈⢻⣧⠀⠀⠀⠀⠈⢛⣉⠀⠿⠇⠛⠛
⠘⠿⠈⠉⠉⠁⠹⠛⠀⠀⠀⠀⢸⣿⠸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⢿⣶⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⣶⣶⢰⣶⡆⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢀⣘⡠⠊⠀⢀⣤⠀⠀⠀⠀⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⠀⠀⣴⠿⠿⣶⠀⣰⣶⠀⠀⠀⠀⣼⣯⣻⡆⢰⢟⡿⣆⠀⠀⠀⠀⢸⠿⠿⠿⠿⣷
⢨⣠⣤⣄⣴⢞⣥⣤⡀⠀⠀⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⠀⠀⣿⠀⡇⢸⠁⠙⡟⠀⠀⠀⠀⢹⣿⣿⠁⠰⣿⣿⠆⠀⠀⠀⠀⢸⠀⢁⠈⠁⣿
⠈⠙⣫⠟⠉⠛⠛⠛⠁⠀⠀⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⠀⠀⣿⡀⠀⣸⡀⠸⡷⠀⠀⠀⠀⢸⣿⣿⡆⢸⣿⣿⡇⠀⠀⠀⠀⢸⠀⡠⢈⡀⣿
⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣉⣉⣈⣉⣁⣀⣀⣀⣀⣙⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣉⣉⣀⣀⣀⣀⣀⣸⣶⣶⣶⣶⣟⣀
⣟⣻⣛⣛⣟⣙⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⢌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣤⣤⣄⣀⣄⡽⠆
⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⠀⠈⠈⠀⠀⣀⣤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⡠⠄
⠀⠀⠀⢈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣉⣉⠉⠉⠉⢉⣈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠋⠀⠀⠀⠈
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠈⠙⠛⠛⠛⠛⠉⠈⡂⠀⢀⠀⠇⠐⠒⣀⣀⡤⠀⠈⠁⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠐⠒⠒⠒⠒⠊⠉⠉⠁⠀⠠⡀⠀⠀⠁⠀⠁⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠄⡀⣀⠀⠀⠀⠀⢠⠀⠛⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⢀⢀⣄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣬⠠⠄⠀⠀⠀⠀⢃⠐⣄⠀⠸⠿⡋⡘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡷⠀⡎⠁⠠⠿⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠠⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠒⠃⠘⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢲⣀⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠤⣆⣀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠤⡀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⠈⠈⠂⡀⠀⡄⠠⠂⠀⠀⠢⣀⠀⠀⠃⢹⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⡜⠀⡀⠀⠀⠠⠀⠀⠀⠐⠁⠃⠀⠀⠆⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣦⠐⠠⠀⠀⠐⠊⠀⣀⠄⠀⠀⠄⠀⠈⠀⠰⠬⠂⠀⠀⠐⠈⠀⠀⣀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠁⠁⠈⠀⠀⠈⠁⠁⠉

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

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠰⡧⠀⠀⠀⠀⢀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠘⠿⠇⠀⢻⡇⣀⠤⣀⠋⣸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⢙⢧⡀⠠⣧⠀⠀⣀⠀⢠⣶⡄⠠⠄⠔⡂
⠀⠀⠀⠀⠀⠀⠀⠖⣄⡠⡴⠊⠲⣠⠁⠀⠈⣤⣿⠋⠉⠉⠑⡄⡔⠁⠀⠉⢢⣈⡪⠇⠃⣿⡇⠀⠉⠀⠀⠉⠀⡆⣿⢞⡀⡀
⠀⠀⠀⠀⠀⣤⠀⠀⡘⣻⠼⣄⡀⢸⣀⡔⠉⠀⢸⡀⠀⠁⠀⣿⡇⠀⠀⠀⣜⠀⠈⠑⡄⣿⡿⢂⡟⡠⠀⢄⡰⣳⠏⢀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠧⣄⡠⠬⣿⠟⡄⠀⠀⢃⠀⠀⢀⠜⣿⡟⢆⠀⠀⠠⠀⠀⢀⠗⢿⣵⡟⠹⠁⠀⢸⠿⠯⡀⠁
⠀⠀⠀⠀⠀⠰⢮⣥⣤⣴⡁⢀⠠⡇⠀⠊⠀⠠⠆⠀⠀⠁⢠⣿⣧⠀⠁⠀⠀⠃⠀⠈⠄⠀⠧⢅⠀⠀⠀⠀⠀⣀⠃⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡩⢛⡽⠁⠀⢈⠀⠀⠀⠀⡢⠀⠄⡀⠸⣿⡟⠀⡰⠄⠠⡁⠀⠀⠀⢀⠀⠀⠹⣷⠀⠀⠀⠁⢑⠂⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⢠⣪⡶⠟⢣⠀⠀⢈⠖⢤⠀⠘⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠜⠀⡠⠒⢇⠀⠀⣰⠋⠀⠀⠲⣴⣮⣴⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠁⠑⠤⡄⠀⠰⠄⠀⢠⠁⠂⠀⠘⣿⡟⠀⠀⠈⢡⠀⠀⢇⠀⢀⡤⠚⠁⡣⢸⣦⡤⠶⠖⠁⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⡀⢸⠀⠈⠆⠀⠘⡆⠀⠀⢠⣿⣷⣀⠄⠀⡇⠀⠀⠃⠀⠈⠀⠠⠒⢳⠊⠉⢪⡉⢲⠀⠰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⢀⡜⠁⠀⠀⢘⣿⡟⠁⠀⠀⠑⣄⠀⠀⠀⠀⠈⠀⠀⢸⣀⠛⠤⢂⣀⣀⣀⣠⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠃⡀⠈⣄⠀⢀⡎⠀⠐⠠⠤⢾⣿⣿⠦⠄⠒⠀⠘⣄⠀⣀⡏⠀⠈⠃⠋⠱⣾⣿⠿⢛⣽⡿⣁⣀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠚⠀⠉⠉⢄⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⣠⠋⠉⠀⠸⠈⢠⣭⣠⠀⠉⢈⠙⣌⠉⠉⠰⠶⠶⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⡴⠋⠉⠑⢦⠀⠠⠾⣿⡟⠢⠀⡠⠊⠉⠉⠳⡄⠀⣠⠾⠟⠢⢄⣀⡾⣦⣄⡱⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠀⠂⠀⠀⠀⡇⠀⠀⣿⠃⠀⢠⠅⠀⠀⠐⠁⢈⢩⠁⠀⢰⠀⠀⣥⡈⢢⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣀⠀⠈⠀⢀⠏⢇⠀⠀⠀⢀⣀⡀⠴⠊⠉⠒⢒⢚⣛⠹⣼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠒⠉⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣤⡄⡀⢠⡤⢄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠐⠀⢀⠀⠀⠀⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⠉⠀⠸⠀⠈⠉⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢢⣀⣀⣀⣤⠚⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠙⢦⠀⢠⠞⠁⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠲⢄⣳⣃⠴⠚⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⡇⠀⠀⡀⠀⠀⢸⠀⠀⠀⣀⠀⠀⣷⡀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣰⢿⢀⡤⠊⠁⠉⠉⠹⠉⠉⠁⠉⠢⣀⢸⢷⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢄⡀⠀⠀⠀⣰⣃⣾⣼⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣸⣇⣳⡀⠀⠀⠀⣀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⣀⠀⠙⣄⡠⠚⠉⠁⠀⠀⠈⠑⠒⠦⠤⠼⠤⠤⠖⠒⠉⠀⠀⠀⠉⠙⠦⣀⡜⠁⢀⡠⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠱⡔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡑⡴⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣜⢰⢁⡶⢠⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠢⣤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢧⠰⡄⢱⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠔⠂⠁⠲⢄⠀⠀⠀⠀⠀⠈⠒⢡⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡆⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠷⠤⠤⠴⠶⠶⠶⠦⠤⠴⠷⠶⠖⠒⠉



⣿⠿⠿⠿⡿⣟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣟⣻⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣴⠀⠀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡀⠀⢸⢹⣈⡃⣀⣷⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠷⣟⡬⣃⣼⣿⣿⣿⣻⣽⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡯⣭⢞⢽⢳⣷⣿⣿⣿⣿⣿⡿⣿⣽⡀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢬⣾⣿⣿⠿⠿⠛⢿⣿⣿⣦⣿⣽⡄⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠻⣿⣿⣿⠃⢀⠀⠀⠀⢈⠛⠿⣿⣿⣿⣦⣇⡐⣀⢡⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢴⣂⡈⠁⠠⣿⣿⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⣿⣿⣿⣻⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠉⠉⠁⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢉⣹⠈⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡧⠂⠾⠿⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⣨⡀⠘⠇⠀⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠹⢟⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⣧⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⢹⡏⠩⠙⢳⡄⡨⢁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠈⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⢌⠸⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠉⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠉⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⠋⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠸⣶⠞⠛⠛⠷⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠇⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⠶⠶⠶⠶⠶⣄⣾⢸⡇⢀⣤⠶⠏⠀⠀⠀⠀⠘⠷⣤⡶⠛⠻⢦⣤⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⢸⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠉⠐⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⡄⢻⣿⡿⠛⠋⠁⠀⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣿⣿⣿⣿⣿⣿⣇⠑⠟⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⠿⠛⠲⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠑⠢⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⠿⣷⣤⣤⣶⠃
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠉⠻⣦⣀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣤⣄⣀⣀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⢄⡀
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣀⢄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⠄
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠉
⢿⣿⣿⣿⡿⠟⠁
⠈⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠘⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⠶⠿⠛⠛⠛⠛⠛⠛⠻⠶⢶⣤⣤⣤⣤⣤⠴⠶⠒⠛⠉⠉⠉⠉⠙⠳⢦⣤⣀⣠⠆

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡏⠩⠿⠿⠿⠿⠄⠀⠀⠀⠠⠶⠒⠒⠶⠄⢸⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠘⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢇⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠍⠍⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⠈⢉⠛⠛⠛⠷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⡌⠱⡏⢣⠋⢣⠀⠀⠀⠈⠙⠻⠶⠟⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠉⠓⢄⠀⢨⠀⠸⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢸⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢓⡂⢀⣿⣻⣦⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⢦⣄⡈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⣉⣉⣁⠟⠫⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⢸⣈⠔⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠐⢍⡉⠓⠤⠬⠿⣄⣀
⠀⠀⠀⢖⠒⠢⠴⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠑⠒⢤⡎⠀⠀⠈⠉⠒⢦⡀
⠀⠀⠀⠀⠠⣔⡒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⣟⠒⠤⠤⠖⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢰⡓⠤⡤⠤⠖⠉⠉⠉⠑⠦⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣷⣝⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠦⣄⠀⠉⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢤⣀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠂⢄⠤⠀⠀⠐⠀⠀⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠘⠉⠋⠁⡡⢈⠏⠀⠈⠀⠱⠈⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⡠⠊⠀⠀⢀
⠀⠄⠀⠂⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⡠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⡀⠀⠄⠄⠄⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠈⠀⠀⠀⡁⢀⣀⣀⡀⣀⣀⢀⣀⣀⣀⣀⡀⠉⠐⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣂⣀⣀⠁⠀⠀⢄⣠⡀⠀⠀⢠⢠⠀⠀⡄⠀⡀⠄⡄⠀⠠⠇⠀⠀⠀⡴⠶⠦⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⣉⠀⠈⣦⠀⢿⠀⠇⢹⠀⠹⢈⢆⡽⡇⠀⠀⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⢺⣿⡷⠀⠘⠃⠠⢠⠀⡀⢠⡀⠄⠠⠀⡄⢀⠀⢄⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⡴⠀⠌⠸⠄⠣⠿⠇⠀⠀⠆⠃⠸⠡⠊⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠋⠀⠀⣶⠰⣤⡆⢒⣴⠀⣦⡆⢢⣤⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠉⠁⠚⡙⠐⠃⠃⣊⣹⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⢀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠄
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣄⡀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣀⣀⣀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⡙⢿⣿⠿⠛⣀⣀⣀⣀⣀⣀
⠀⠀⢀⠞⠻⠻⠿⠟⢦⡤⠀⠀⣼⠟⠿⠿⠟⠳⣄⠀⠀⢠⠾⠿⠟⠿⠾⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣄⣁⢸⡇⠈⣠⠀⠂⠀⠀⠈⢸⣇
⠀⠒⣻⠀⠀⠀⠀⠀⢸⡧⡄⢸⣿⠀⠀⠀⠀⠐⣿⡀⠀⣿⡀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣿⠩⠈⠁⠉⢸⠀⡇⠀⠐⠀⢸⡇
⠀⠀⠸⣀⣧⣠⣠⣀⡸⠃⠀⠲⢿⣀⣀⣿⣀⣠⠏⠀⠀⠹⣆⣼⣀⣠⣠⣿⠁⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⣿⠐⡆⡶⢰⡚⡆⣷⠀⡖⣾⠘⡇
⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠉⠑⠛⠛⠁⠀⠀⡀⠀⠘⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠈⠚⡳⠛⠚⠛⠛⠛⠚⠓⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠉

⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⡿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣏⠀⣸⢿⡀
⠀⠀⠠⢙⣉⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣦⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⢸⢸⢰⣅⣀⣵
⠀⠀⠀⢀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣳⡄⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠞⢸⡿⠿⢿
⠀⠀⠾⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢇⡀⠉⠁⠀⠀⠀⠀⠀⠸⠿⠻⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀
⠀⣀⣤⣄⡀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢾⣶⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⢰⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣿⣿⡟⣀
⠈⢯⣿⣯⡴⢩⣭⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄
⠀⠀⠉⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣶⣶⣶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡰⡦
⠀⠀⢀⣶⡀⣐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⡿⣧⡤⠄
⢀⣀⠞⣿⠳⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣂⣀⣀⣀⣀
⢾⡇⠀⣿⠀⢹⡏⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠁⠀⠉⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁


⠀⠐⡉⡉⢉⣉⠉⡇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣩⣉⣉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⢰⠀⠄⠀⠀⢀⣇⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠶⠶⢠⢳⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡶⣺⠛⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠞⢠⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⣭⣭⠁⡌⡜⠀⠀⠀⠀⠀⠀⠀⠄⢻⣿⣷⣌⣣⡾⠿⠿⠂⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⠛⠛⠋
⠀⠓⣒⣒⣒⣒⣋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠓⣒⣒⣒⣒⣩⠇⠀⠀⠀⠀⠀⠀⠀⠈⣨⡭⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⡿⠉⠉⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢠⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠃⠀⣽⣿⣿⠏
⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠋⠙⠳⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠢⣄⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣀⡄⠀⠀⠀⠀⠀⠀⠉⠳⣄⠁⠀⠀⠀⠀⠀⠀⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣤⣤⣤⣀⣀⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⣿⣿⡏⠀⠙⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣳⣿⣿⣿⣿⣿⣿⣷⣄⢀⣤⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠙⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣤⣤⡌⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢠⣀⣠⠀⡄⠀⣄⣀⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠢⠤⠴⠂⠀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣏⣀⣀⣀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠙⢦⣀⢀⣀⣀⣀⠀⢀⡼⠋⠀⠀⣠⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠙⢦⣀⠀⠀⢉⣙⠒⣄⠉⠛⠉⠀⠀⣀⡴⠃⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠓⠲⠦⣄⣈⠛⠒⠯⣌⣉⣁⣀⣠⡤⠶⢚⣁⣠⠤⠴⠖⠒⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠤⠤⠤⠶⠶⠖⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠰⢖⡶⠶⣶⠿⠒⠒⠲⠖⣠⣠⡖⢳⡞⠲⣲⠶⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠂⠀⠀⠘⣼⣶⣶⡶⡄⠿⠣⡴⡂⠠⠀⢠⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣤⢀⣀⣀⡿⣴⡏⢠⡄⠀⠔⣳⣿⣤⣼⢶⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⣱⣭⣽⣿⣦⣌⣵⡶⠀⠉⢀⣿⣿⣼⡅⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠘⠀⠀⢻⣤⣺⣟⣹⡷⣾⣶⣄⣟⠴⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⣀⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢛⡿⢿⣟⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣠⠀⢂⣼⣿⣿⣿⡟⣿⣿⣿⡿⣷⡿⠶⣾⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠞⠀⠰⢎⡹⣿⣿⣿⣿⣿⣿⣿⠏⠀⣀⢸⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢡⣧⣽⣿⣿⣟⣿⣿⣿⡻⠛⠻⢻⠄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⠀⠀⣼⡾⢻⡽⣯⠿⡧⣿⣿⣯⠌⣄⢰⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠸⠓⠚⣿⢿⣷⢱⣿⡆⣅⢹⣿⣿⡝⠘⠻⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡄⠀⢀⣿⣿⣿⡄⠙⠁⠋⣸⣿⣿⣄⣼⣤⡀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠓⢀⠈⠟⣻⠏⢁⣼⣿⣄⠹⡿⢧⡩⣿⣿⠇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠠⠖⠀⣿⠂⢼⡀⠈⠁⠉⠉⠀⡥⢸⠃⢉⣿⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠈⠁⠀⠀⢏⢰⡷⠂⠀⠀⠀⠲⡆⠀⠀⠈⢹⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠁⠔⡑⢦⠃⠀⠀⠀⠀⠀⠀⢹⠠⡄⠠⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡅⠂⡛⢻⠀⠀⠀⠀⠀⠀⠀⢸⣄⠃⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡀⠀⢰⡶⠄⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⣔⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣯⢹⣧⠀⠀⠰⠇⠀⠋⠀⠀⠉⠀⠉⡿⢘⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡃⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⣤⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠆⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣻⣿⣽⣷⡦⣤⣄⣀⡠⢤⡖⠚⠁⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⠛⠚⠿⣟⣿⡏⠛⠛⠿⠉⠟⠋⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⢀⠀⢀⡤⠔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠚⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣨⣉⡀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠋⠹⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⢀⣾⣿⣦⣶⣦⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣹⡝⠛⠿⠿⠇
⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣿⠀⠀⠄
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠠⠀⠙⠋⠁⠀⠈⡏⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠀⠀⠈⠒⠁⠀⠀⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⢂⠀⣀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣀⣀⣤⠐⠉⠘⣷⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⢾⣿⣿⠔⠋
⠀⠀⠀⠀⢀⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣰⣢⠀⠀⠉⠀⠀⠀⠀⠢
⠀⠀⠀⠸⠿⠼⠋⠓⠲⢴⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠿⠶⠀⠀⠀⠀⠠⠀⠷⠠⡤⠴⠲⠒⠀⡂⢨⠀⠀⠀⠀⠀⠀⠲⠆⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠐⠶⡆⣶⠀⠶⡆⢶⡆⠀⠀⠀⠰⣷⠀⠀⢠⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠰⠦⠄⠤⠤⠤⠤⠀⠀⠰⠀⠀⠠⣿⡿⠶⣾⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡆⠀⠀⠀⣀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢠⠁⠀⣠⣾⡀⡐⣾⣿⡇⠀⠀⠀⢐
⠀⠀⠘⠋⠉⠛⠷⡷⠶⠶⠦⣤⡾⠀⠀⣿⣿⠷⢾⣿⠟⠁⠀⠰⠄⠈⠒⠠⢤⡤⠤⠤⠤⠤⢤⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠈⠓⠒⠈⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⣼⠋⠀⠀⠀⣠⣾⠿⠉⠉⠉⠉⠛⢿⣶⡿⠟⠛⠛⠻⢷⣦⣀⣠⣤⣤⣤⣤⣀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⣦⡀
⠀⢀⣾⠁⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⠉⠙⢿⣿⠋⠉⠉⠙⠻⢷⣦⠀⠀⠀⠀⠈⢷⡀
⠀⣼⠃⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⣸⣿⠂⠀⠀⠀⠀⠀⣨⣿⠆⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠈⣇
⢠⡏⠀⠀⣴⡿⠛⠹⣿⣄⣀⠀⠀⠀⠾⠛⠋⠁⠀⠸⣷⣶⡾⠟⢻⡆⠀⠀⠀⠀⠛⠛⠁⠀⣠⣄⡀⠀⢀⣠⣿⣷⡄⠀⠀⠀⠸⡆
⣸⠃⠀⢸⣿⠁⠀⠀⠈⣹⣿⠿⠗⠀⠀⠰⣶⣤⡀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⢺⣗⠀⠀⠀⠉⢻⣿⡿⠛⠉⠉⠙⢿⣄⠀⠀⠀⣷
⡟⠀⠀⠈⢿⣦⣀⣀⣴⣿⣷⣶⣄⠀⠀⠀⢠⣿⡿⠿⠶⠟⠛⠉⠁⣀⡀⠀⠀⠘⢿⣧⡀⠀⠀⣿⠟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⡇⠀⠀⢀⣼⡿⠋⠉⠉⠀⠀⢈⣿⣄⠀⠀⠘⣿⡃⠀⠀⠀⢀⣴⡿⠛⠃⠀⠀⠀⠀⠉⠉⠀⠀⢿⡄⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⡟
⡟⠀⠀⢸⡟⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠈⠃⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⡶⠿⠿⠿⣶⣿⠿⠁⠀⠀⢀⠃
⢃⠀⠀⠘⢿⣄⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣧⣀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠚⠋⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢸⡇
⢠⠀⠀⠀⠀⠙⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢳⡄
⢸⡆⠀⠀⠀⠀⢿⣇⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠻⣆
⠈⣧⠀⠀⠀⠀⠘⢿⣄⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠶⡶⠶⠾⠛⢷⣄⣀⡀⠀⢀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⢻⡄⠀⠀⠀⠀⠈⠙⢷⣬⠛⠳⠶⠶⠶⣷⣶⣤⣄⣀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⣽⠇⠀⢀⣾⠟⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢹⡆⠀⠀⠀⠀⠀⢸⣧⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠋
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁







⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢼⣿⣿⣿⣿⣷⣆⣀⣀⣀⣀⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣤⣀
⣸⣿⣿⣿⣿⣿⣿⠟⠋⠁⠈⠉⠉⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⡗
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠃⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⢿⣿⣿⣿⣿⣦⡀⠀⠇⠀⣤⠄⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠈⠻⠿⣿⣿⣿⣿⣷⠦⠄⢀⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠹⣿⣿⡿⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣴⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣠⠀⠂⣈⢂⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠒⢀⠀⠒⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠠⠄⢀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡊⠀⠺⡀⢀⠗⠀⢑⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⢰⡦⠂⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡜⣏⠄⠂⠀⠘⠃⠀⠐⢄⢹⢧⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠢⡀⢀⠼⠔⠓⠢⠄⡀⠀⠀⢀⠠⠔⠚⠢⠧⡀⠀⠐⠘⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠑⠤⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⡇⠄⢆⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⡌⢁⢀⡀⡀⠀⠀⠀⠀⠐⣰⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠠⢀⠀⠀⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠒⠒⠚⠒⠚⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⣀⠀⠀⠀⠀⠀⣤⣦⣤⣶⡄⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⣼⣿⣿⣿⣇⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⠿⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡏


⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣋⣴⡄⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣼⣿⣿⣿⣿⣷⡄⠀⠀⠀⠿⠿⢿⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠙⠁⠙⠛⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁


⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢰⣶⡆⢶⣶⡆⠀⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⣶⣶⣶⡄⢦⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠽⠿⠿⠇⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠁⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠤⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠨⠤⠒⢂⠀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⣠⣤⣴⣯⣿⠿⠟⢡⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⣼⣿⣏⣈⠈⣍⣁⡠⠂⠑⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⠀⠐⡖⠘⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣤⣄⡙⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⠿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠙⣿⡄⢿⡏⠙⠉⠁⠀⠀⠀⡀⠀⠀⠉⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢈⣿⣾⣷⡔⠀⠀⠀⠀⠀⢹⣷⣦⣄⣠⣎⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠀⠉⠻⢿⣤⣀⣀⣰⣶⣶⠊⠀⠀⠈⠉⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⣼⣷⣶⡶⢂⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣴⡧⠼⢷⣆⠀⠆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⣸⣿⠆⠀⠀⠙⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢉⡽⣷⣿⣿⣦⣤⣄⣤⣴⣿⣿⣆⣄⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⢿⠋⣿⣿⣿⣷⣦⡜⣿⣻⣿⣿⣿⣟⣻⣥⡟⣾⣿⣿⣿⣿⣿⣿⣇⣀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠂⠨⠥⣀⣄⣈⣿⣿⣿⣿⣁⢀⣼⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⢶⣿⣿⣿⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣿⣛⢙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡿⣿⡿⠠⣤⡙⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠒⠁⣼⣿⣿⣿⣿⣁⣰⣾⠆⠀⠛⢧⡄⠈⠻⢏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣦⣌⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣄⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣿⡙⡡⠤⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⠉⠀⠀⠒⠛⠛⠉⠉⠙⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠤⢄⡀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠘⡄⡠⠚⠁⠀⠀⠀⠀⠈⠲⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⠔⠙⡄⠀⠀⠀⠀⢸⠁⠈⠒⠄⣀⣀⣀⣤⠀⠀⡘
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠉⠁⠀⠀⢣⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⠃⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢇⣀⠄⠒⢤⠇⠀⠀⠀⠀⠀⠀⠀⠆⢉⠀⠀⠀⠀⠀⠱⢄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠱⡩⠁
⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀⠼⡜⡀⠀⠀⠀⠀⠀⣠⠷⣤⠥⢄⡀⠀⠀⠀⢀⠔⠛⠁
⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠂⣀⡎⠀⠀⠀⠀⡀⠀⠘⢷⡄⠀⠀⢀⢔⠕⠀⠃⠀⠀⢱⣀⢀⡔⠉⣆
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣤⢤⡼⠀⠀⠀⠀⠑⢄⣀⣀⡈⠥⠟⠋⠁⠀⠀⠀⠀⠀⠈⣅⠜⠀⠀⢠⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⡇⢦⣌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⡔⠂⠀⠈⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⢸⠀⢀⣧⠀⠀⠀⣟⠒⣌⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⣠⠅⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⠤
⠀⠀⠀⠀⡌⠀⠘⢸⠀⠀⠀⠁⠈⡍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠊⠁⠘⠀⠀⠀⠀⠀⡠⡀⠀⠸⡀⠀⠀⠀⠀⡠⠔⠁⢀⠤⠊⠁
⠀⠀⠀⢀⠃⠀⠇⠸⠀⠀⠀⢰⠀⡇⢠⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⡆⠀⠀⠀⣰⠁⠃⠀⠀⠱⡀⣀⠴⠈⢀⠤⠊⠁
⠀⠀⠀⠸⠀⢀⠀⠀⡄⠀⠀⢸⢀⠀⣸⠀⠘⠸⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠀⠀⢀⡧⠤⠤⡠⣛⠀⠘⢄⣀⠴⠋⣀⠔⠊
⠀⠀⠀⠸⠀⡈⠀⠀⡇⠀⠀⠸⢻⠈⢹⠀⠀⡆⢇⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡆⢐⠙⣖⢂⠀⢳⠁⣀⡤⠚⡡⠔⢿
⠀⠀⠀⡂⢠⠁⠀⠀⠇⠀⠀⠀⣦⠇⠸⠀⠀⠰⠘⠄⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠃⠱⡀⠑⠳⣒⣂⡼⠚⣩⡔⠉⠀⠀⠀⠃
⠀⠀⢀⠃⡜⠀⠀⠀⠀⠀⠀⠀⡏⠀⡆⠀⠀⠀⠇⠈⢆⠀⠀⠈⠉⠒⠠⢄⡀⠀⢀⠀⠈⠉⠋⢩⡩⠔⠉⠀⠘⢆⠀⠀⠀⡆
⠀⠀⡘⢠⠁⠀⠀⠀⠘⠀⠀⠀⠇⢀⠁⠀⠀⠀⠘⠄⠈⠢⠀⠀⠀⠀⠀⠀⠁⠀⠈⡆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠙⠒⠊
⠀⠀⠃⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣴⣷⣦⠀⢰⡆⢰⣾⣶⠀⢲⠄⠀⢀⣶⣶⣄⠀⠀⠀⢀⣤⡀⣰⣶⣦⠀⠍⠀⠀⢠⣶⡆⠀⠀⠀⣴⣷⣦⠀⠀⠀⣠⠶⡄
⠀⠀⢀⣽⣿⣥⡀⠘⢧⣬⣿⣯⣤⠟⠀⠀⣨⣿⣿⣧⡀⠀⠀⢻⠯⢃⣽⣭⣿⣄⠀⠀⠀⢀⣿⣅⠀⠀⢀⣽⣿⣥⡀⠀⣄⣿⣶⣳⡆
⠀⢠⡿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⣼⣴⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⡄⢀⡿⣿⣿⣿⣿⠀⢸⣿⣿⡟⠷
⠀⢸⡇⣿⣿⣿⢸⠆⠀⣻⣿⣿⡃⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⢻⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⢸⡇⣿⣿⣿⢸⠅⠘⣿⣿⣇
⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⡏⣿⡇⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⢸⡏⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣻⡯⣿⡇
⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⡇⢻⡇⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⢸⡇⣿⠀⠀⠀⣿⣿⣿⠀⠀⢠⣿⠃⢸⡇
⠀⠀⠀⠉⠈⠉⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠉⠈⠉⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠄⠀⣴
⠀⠀⠀⡇⢸⣿⣻⠀⠀⠀⠀⠀⠸⣟⣻⠇⠀⠀⠀⡀⠄⠀⣾⡿⣧⠰⢀⡁⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣤⣬⡀⣿
⠀⠀⠀⠀⣶⣿⣷⣆⠀⠀⠀⣿⣶⣾⣷⣶⣿⠀⠀⣤⣠⡴⣾⣷⣷⠶⠟⠁⢠⡟⣾⣿⣿⣿⡟⣧⠀⠀⠀⠀⣤⠶⠶⣶⣾⣿⡿⠃⣿
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠘⢹⣿⣿⡏⠃⠀⠀⠀⠁⠀⣿⣿⣿⡀⠀⠀⠸⢣⣿⣿⣿⣿⣧⠻⠀⠀⠀⠀⠀⠠⣶⣿⣿⡿⠁⠀⣿
⠀⠀⠀⠸⢿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠘⠿⣿⢿⡿⠟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡧⠀⠀⣿
⠀⠀⠀⠀⣿⣷⣿⡇⠀⠀⠀⠀⣿⡇⢸⣷⠀⠀⠀⠀⠀⣰⣿⠁⣿⡇⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡿⠉⣽⡟⠁⠀⠀⣿
⠀⠀⠀⠀⠻⠟⢿⠇⠀⠀⠀⠀⢿⠇⠸⡿⠀⠀⠀⠀⠰⠿⠁⠰⡿⠁⠀⠀⠀⠀⠀⠿⠻⠇⠀⠀⠀⠀⠀⠈⠛⠁⠈⠛⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠄⢀⠄⠀⠀⠀⠀⠀⠀⣠⣶⣌⠘⠀⢠⣿⣿⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⢀⠖⠓⡂⠀⠀⠀⢈⣀⠀⠀⠠⠄⠀⣴⣿⣿⣿⣏⠀⢰⢿⣾⣷⣶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡄⠀⠀⢠⣄⠀⠀⢠⣄
⠀⠀⠄⣾⡿⣳⣴⡞⠋⠉⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠘⠀⣀⣀⠀⠀⢸⣿⣿⣿⣧⠀⢰⡟⠦⠠⢾⡿⠤
⠀⠀⠀⢈⡿⣿⣿⣿⣶⡿⣷⣄⡀⠀⠀⢸⣿⣿⡇⠟⠀⠀⢸⣿⣿⣿⠇⠀⠇⣠⣤⡼⣿⣿⠂⠀⠸⣿⣿⣿⠟⠠⠼⢿⡆⠠⠼⢿⡆
⠀⠀⢠⡿⢡⡞⢻⡿⠿⣿⠈⠙⠃⠀⠀⢸⣿⢹⣧⠀⠀⠀⣸⡟⢸⣿⠀⠀⢠⣿⣿⣷⣦⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠙⠉⠀⠀⢿⠇⠀⠀⠀⠀⠸⡿⠸⡿⠀⠀⠀⢿⠇⠸⡿⠀⠀⠸⠿⠿⠋⢿⠇⠀⠀⠀⢿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠀⠒⠂⠀⠉⠉⠉⠉⠉⠁⠐⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⢀⡀⢀⠀⢀⠀⢀⠠⢀⠠⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠀⠁⠋⠊⠁⠉⡰⠛⠒⠊⢀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠂⠀⠀⠀⢘⠀⠎⡗⢳⡢⠀⠉⠉⠉⠉⠒⠤⠀⠀⠀⠡⠐⠀⠄⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠠⠁⠀⠂⠐⠀⠐⡇⠀⠀⠰⣦⣀⣀⠠⠙⣤⠅⠀⠀⢀⠀⠀⠑⡀⠀⠈⠁⢂⠀⠀⢀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠁⠒⠁⡇⠀⠀⠀⠘⡆⠇⠀⠀⣰⣿⠀⠀⢨⠁⠀⠀⢁⠀⠐⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡈⠔⡈⠔⡀⠅⠀⠀⠀⢀⠽⢲⠀⠘⣯⡿⠀⠀⡬⠄⠀⠀⠸⠀⢘⡂⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠅⢂⠁⢀⠁⠀⠀⠐⠁⠀⠑⣃⠀⠘⠁⠀⡴⠁⠀⠀⠀⡈⠀⢸⠀⠀⠀⠀⡀⠔⠒⠁⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠄⠂⠁⢀⠀⠀⢰⠀⠐⠌⠈⠦⣀⣠⠎⣧⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠴⠮⣷⡛⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠔⠈⠄⢈⢸⠀⠀⠀⢆⠐⠀⡱⢄⠀⠀⡀⠈⠀⠁⢀⠂⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠛⠁⠀⠘⣄
⠀⠀⠀⠀⠀⢁⠀⠀⠈⠀⡈⠀⣘⠀⠀⠀⠈⢦⡀⠠⠈⠑⠬⣀⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠈⡀⠀⠀⡀⠐⠀⠐⡄⠀⠀⠀⠀⠑⠦⡀⠀⠀⠨⣏⠚⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠂⡡⢀⠄⡇⠀⠀⠀⠀⠀⢀⠆⠐⢠⠗⠉⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠊⡇⠀⠀⠀⠴⠦⠥⠤⠴⠛⠉⠉⠑⠈⠂⠐⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠲⢒⠔⠂⢀⠰⢂⠖⠰⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⢴⣶⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠈⠀⠠⠀⠡⠊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠞⢁⡝
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠡⠐⠈⠠⠀⢁⠠⠐⡉⠔⠊⠀⠐⠨⠀⠀⠀⣀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⠠⠀⡈⠀⢀⠡⠀⢀⡀⠠⠄⠒⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⡈⠀⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠓⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⣠⣶⡾⢀⠀⣀⣀⣼⣦⢹⣷⣤⡀⠀⢠⣶⣶⣶⣿⠁⠀⠀⠀⠀⠰⠾⠃⠠⠄
⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠃⠀⢀⣀⣼⠿⢿⣿⣷⣾⣿⣿⣽⡿⢉⣿⣿⣿⣼⠇⢀⠙⢿⣿⣧⡄⢠⣴⣤⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠐⠈⠙⠋⣀⠀⣤⣴⣿⠟⠁⢠⣾⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣒⣿⣷⣼⠛⢷⣥⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠉⠁⠀⠁⠀⢼⣿⠿⣿⣿⣧⣿⣿⣿⣿⣿⠿⠿⣿⠿⠿⣿⣟⡧⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠉⠋⢰⣿⡿⠟⠉⠛⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠗⠀⠑⢂⡀⢶⣄⠀⠀⠀⠀⢀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡆⠀⠶⠴⠄⠀⠀⠄
⣸⡟⣛⠛⠻⣾⡋⠀⠐⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡿⣴⡄⣴⣷⡐⠀⢀⣠⡀
⠋⠈⠁⠀⢸⣿⡄⠰⠾⠿⠿⠶⠒⠦⠤⠤⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⣷⡄⠐⠠⢈⣠⣈⠃⣱⣤⠘⣿⣿
⠀⠀⠀⠐⠚⢻⣶⠶⠶⠶⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⡾⠿⠿⠿⠿⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣧⣹⡷⣶⢟⣰⡿⣉⣏⠙⢀⣻⣿⠵
⠀⠀⠀⢀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠂⠙⠙⣿⣇⠤⣼⢟⡇⠈⠄⠀⠀⣿⣿⣜
⠀⠰⠋⡛⠛⠿⠷⠶⠶⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⡠⠉⠙⠛⠛⠿⢶⣄⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠈⠀⢹⣿⣿⡾⠟⠛⠒⠀⠠⠄⠿⠏⠁
⠀⠀⠞⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠉⠆⡀⠀⠀⠐⠋⣉⠀⠀⠀⢀⣤⣶⣶⣧⣤⣤⡀
⢠⡾⠛⠾⠶⠶⠄⢀⣠⠾⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣠⡞⠁⢀⣤⣶⣦⣤⣄⣀⣀⡀⠀⠈
⣼⣥⣀⠀⠀⢀⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣤⣤⡴⠖⠂⠀⠀⠐⠒⠒⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠉⠉⠙⣷⣦⣨⣤⣤⡴⠶⠀⢠⣴⠋⢠⣧⠂⠀⣠⣶⣿⡿⠟⠋⠁⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠋⢁⣀⡀
⠀⠀⣼⡗⠈⠉⠁⠀⢀⡄⠰⠋⠀⠀⠈⠁⠀⠀⡙⠉⣿⠃⢀⡜⠀⠈⠁⠀⠀⠀⠀⠀⢠⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⠛⠉⠉⠉
⠀⠀⠟⢀⣠⡴⠊⠀⣾⠀⠀⠀⢀⣁⠖⠁⠀⣾⣿⣆⠻⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠺⠁
⠠⢶⠾⠟⠉⠀⠀⠘⠃⣠⠄⠀⢹⠃⠀⢠⣿⡿⣿⣿⡀⠀⢀⡐⢃⠀⠀⠀⠀⠰⠿⢿⡿⠿⠿⠟⠛⠉⠁⢀⠀⠀⠀⠀⠀⢸⣥⣴⣴⠶⠛⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢠⠶⠟⠁⠀⠀⢀⣴⠎⢹⣿⠃⠸⣿⡇⠀⠀⠐⠉⠁⠀⠀⢶⣦⡀⣰⣇⡠⣤⣴⠶⣶⣶⠶⣶⣶⣶⡷⢶⣿⣅⣀⣀⣐⣄⣀⠀⠘⠁⠠⠀⢀
⠀⠀⣀⠀⣸⡄⣀⣴⣾⢤⠄⠀⠉⢹⡄⢸⣿⠀⠀⠙⣷⣼⠷⢀⣤⠀⠀⡀⠀⣈⡻⢿⣿⡇⠘⢿⣧⡈⠻⣧⡈⠛⠿⠷⢮⣉⠉⠉⣏⠹⣿⡏⠃⠀⠀⠀⠛⣾
⠀⠀⠀⡄⣹⡿⠋⠉⢩⣤⠀⠀⠀⢸⣿⣿⣿⣘⣀⠀⣟⠻⢶⣀⣿⡁⠀⠀⣸⣿⠇⣸⣿⠋⢦⠀⠻⢿⡄⣈⡿⢧⡴⢀⣤⣆⣿⣾⣧⣤⣬⣷⣤⣹⣆⠀⠀⢁
⣤⡆⣰⣏⣇⣄⠀⠀⣨⣽⣤⣤⡷⢺⢿⣷⣿⣿⣧⡀⢾⣶⣿⣹⣀⡂⢰⣾⣿⣿⣀⣽⣿⣠⠄⡀⣶⣄⡙⢠⣭⡴⣶⣼⣿⣿⣿⣿⣧⣬⣽⣿⣡⣥⣽⣠⣴⣿
⠛⣷⣿⣿⣷⣤⣶⡎⣹⣿⣿⣿⣷⣿⣤⣿⣿⣿⣽⣾⣬⣿⣷⣷⣿⣿⣾⣿⣿⣿⣯⣹⣿⡟⢴⣧⠛⢿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣿⣿⠏⠁⢿⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⣧⣿⠹⣿
⣿⣿⣿⣿⣷⣿⠟⠛⣿⠛⣿⣿⣿⣿⣿⠿⡟⢻⡿⠇⠘⠻⠛⢻⣿⣿⠃⠘⠃⣸⣿⣿⣤⣶⢸⣿⣿⣿⣿⠻⠛⣿⡇⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠁⣼⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣀⠠⠤⠒⠋⠀⠳⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⠚⠙⠓⠂⠀⠀⠐⠒⠂⠘⡶⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣽⠀⠈⠉⠁⣸⠀⠈⠉⠁⢠⣳⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢧⠀⠀⠀⠁⠀⠀⠀⠀⡜⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⠷⣄⠀⠉⠉⢀⣠⠾⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⠋⠀⠐⠾⡉⠑⠂⢉⠗⠂⠀⠙⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠚⠂⠀⢄⡀⠘⣆⣰⠃⢀⡀⠀⠐⠓⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠑⠾⠧⠊⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢳⡀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠒⢤⡀⠀⠀⣠⣮⠤⠶⠓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠶⠤⣵⣄⠀⠀⢀⡴⠒⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠿⠲⢤⡀⠀⢹⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠀⢀⡤⠖⠿⡄
⠀⠀⠀⠀⠀⣰⢃⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠃⠀⠀⠀⡙⣆
⠀⠀⠀⠀⠀⡏⡎⣸⢁⡶⢠⠇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⢤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⡆⢶⡈⣇⢱⢹
⠀⠀⠀⠀⠀⢳⡄⡇⢸⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠛⠓⡦⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠳⠎⡇⢸⢠⡞
⠀⠀⠀⠀⠀⠀⠉⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠑⠢⣄⣀⠀⠀⠀⠀⠀⠈⢊⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣵⣤⣤⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉