Buzzard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠁⠀⠀⠀⠀⡀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⣴⣶⣶⣶⣶⣶⣦⣤⣉⠳⢤⣀⢠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠳⡁⠀⠀⠲⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⡮⠀⡄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠟⢏⡉⣻⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⡀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢢⠞⠛⠛⢿⣤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠞⠁⠀⠀⠀⠀⠙⢿⣦⣈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣦⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣶⣀⣍⣉⡛⠛⠻⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠹⢾⣽⠻⣦⢹⡿⢿⣧⣾⣾⣿⣿⣌⡛⢯⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢋⠟⠶⠾⠇⢀⡀⠈⢻⣿⣿⣿⣿⣦⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢌⡴⠶⠚⠀⠚⠂⠒⡟⠀⠀⡀⠉⣉⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾⠾⣿⣤⣤⣀⡀⡈⠩⣯⣭⣭⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢀⣄⠛⡉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠐⠊⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠁⠀⠀⠡⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠀⣤⢀⠀⢀⠃⠀⠀⠀⢀⡇
⠀⠀⡔⠀⠀⠀⠀⠈⠀⠐⠐⠂⠀⠀⠀⠐⠀⠠⠂⠀⠁⠠⡀⠀⠀⡼⠀⡠⠀⠀⠐⢄
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⡠⠊⠀⢰⠀⠀⠀⡠⠀⠀⠈⠐⢄
⣎⠔⠈⢦⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠠⠈⠀⠀⠀⠄⠀⡇⠀⠀⠆⠀⠀⠐⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠁⠀⡘⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡠⠀⠀⠀⠀⠀⠔⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠎⠀⠀⠀⠀⢀⡊⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠓⠤⢎⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⡈⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠉⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢤⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠓⠠⠴⠀⠀⣎⠀⠀⠀⠀⠀⠀⠈⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠟⠀⠀⠀⠀⠈⠂⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⣀⠀⠀⠀⠀⠈⠂⠀⠀⢣⠀⣄⠀⠀⢠⠀⠀⣆⠹⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠐⣶⣶⠈⠀⠈⢣⡟⣄⠀⡏⠑⢄⡇⠁⠳⠀⢀⠀⠈⠙⠒⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠘⠀⠀⠀⠀⠀⠈⠢⡗⢤⡀⠉⠀⠀⠀⠀⠁⠂⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠊⠁⠊⡰⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢉⠖⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡠⠄⠐⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢻⣷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⣦⣀⢻⣿⣿⣯⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⣈⠻⣿⡿⣯⡿⢯⡷⣗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣜⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣝⠿⢭⣙⢮⡑⢌⠿⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢾⣩⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠰⣞⡟⠳⠦⣌⠳⠉⢾⢓⡿⣻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣏⡷⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣛⠣⠆⠄⣄⢁⣒⡛⠎⡵⠊⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⡻⣇⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣘⠂⠶⠠⠌⡡⠀⢀⠀⠘⡇⡇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⡽⡎⣹⣋⣽⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⡁⣓⢚⣑⣷⣀⠐⠈⠃⢠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⢍⡀⣶⣶⣿⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣩⢈⡤⠽⢛⠣⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢝⢄⣾⣿⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⠰⡼⣫⡉⡄⢀⠃⢀⠛⠄⠘⡀⠀⠀⠀⢠⣾⢿⣍⣫⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⣧⣥⡽⣗⡌⠀⠒⠊⡀⣀⠄⠹⡄⠀⣰⡿⢩⣽⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⠳⣟⣥⠄⡌⢤⢘⡹⣾⣦⣙⣶⣿⠿⡿⠿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣽⡶⠻⣥⣦⣻⣷⣿⣿⡟⠻⠁⠈⠻⣦⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⣽⣿⣿⢟⡛⡇⠉⠐⠀⣀⠄⠀⣀⣼⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⠗⠴⠌⠀⠂⠈⠀⠠⢐⣦⡬⠹⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠏⠁⠂⠠⠀⣠⣴⡄⢰⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠀⠀⠀⡄⠀⠀⠚⠀⢰⣇⠀⠀⣰⣿⠓⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉⠊⡀⡈⢀⠀⠀⠄⠀⠀⢈⣥⠀⣰⠿⣇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠁⢁⢀⡀⠄⠀⠀⠀⠘⠋⢀⠉⡈⢯⢀⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⡤⠠⠈⠀⣀⠄⠂⠀⠀⠀⣀⢴⠾⠟⢃⠹⠞⢉⠘⠳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢢⡇⠉⠀⣀⠒⠁⠈⠀⠉⠀⢀⠀⠘⣆⠀⠉⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⠀⣫⠐⠠⡌⡀⢀⡀⠀⠀⠀⣠⡌⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠖⠲⠶⠴⠃⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⢶⠶⢖⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣷⡂⠈⠋⠃⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⣷⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠹⠆⠀⠀⢸⠃⠤⠤⣤⣀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠨⠢⠀⠀⢁⠂⠩⠯⢛⣓⣦⡧⠀⠀⠀⠈⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣸⡿⣦⣄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠆⠀⡀⢀⠁⢶⡆⠀⢠⠀⠠⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣿⣷⡀⢍⡻⢷⣿⣷⠞⡁⠀⠠⠀⠀⠀⠀⠀⠠⢶⣿⠀⢀⠑⢀⠍⠆⢀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠈⠀⠻⠁⡠⠂⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⡀⡄⡁⠂⠘⡌⢸⠸⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⡀⠀⠀⢀⢺⡀⠀⠀⠀⠀⠄⢀⢀⠀⠒⠀⡽⣧⠀⣧⢰⡘⣗⢬⢧⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⣿⣿⣷⡀⠀⠀⠉⠨⠔⠀⠀⠀⠀⠀⠄⠁⠀⠀⠂⠀⠉⠻⣌⡇⠻⡈⢦⣅⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⣿⡅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡃⠀⢈⣿⣄⠙⢢⠙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⣿⣿⣿⣧⠠⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⣿⣿⣿⣄⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠤⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⠀⠀⠛⣢⠈⠀⠠⠀⠀⠁⠌⠐⣄⡀⠀⢠⡁⠸⢷⡝⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠻⢦⡃⠀⠀⡀⠂⠀⡀⠀⠁⠀⠀⠐⠀⣸⣷⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠄⠀⠤⢅⢄⠉⠈⠀⣈⣧⠀⠀⠀⠀⠀⠀⣷⢿⡂⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡅⠀⠀⠀⠈⣯⠓⣠⠂⣼⣿⡀⠀⠀⠀⠀⡀⣾⡃⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠀⠀⢿⡓⣜⣵⣿⣿⣿⡀⣀⠀⢠⢸⡏⠁⠀⠀⣿⣟⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣄⢀⡀⠀⠀⠈⢿⣀⣠⣀⡾⣻⣿⣿⣿⣿⣿⣿⣌⡇⠰⠟⠁⣠⡄⣸⣿⠿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠛⠛⠃⠘⠃⠀⠀⠀⠀⢙⠟⢅⠑⢿⣿⣿⣿⣷⣏⡿⠁⡙⠲⣶⣾⡿⠷⠤⠥⠦⠄⠘⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⡀⠱⡏⠀⠈⢹⡖⠉⢩⣍⣯⠁⠈⣀⡙⠈⠐⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠⠰⣤⡄⠀⠀⢁⣀⣀⠘⠁⣀⣀⣙⣇⣴⣷⣤⣀⣤⣤⣤⣤⣤⠴⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠘⠋⠉⠉⠉⠉⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠙⠁⠛⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢤⣱⣬⡳⠼⣦
⠀⠀⠀⠀⠀⠘⠶⠉⠃⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⠂⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣔⢢⢳⡀⠀⠀⢻⡷⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣱⣱⡀⠂⠀⠁⠽⠿⠂⡴⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣄⠀⠀⠀⡂⠀⠀⢠⣴⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣖⢪⠺⠇⣀⣾⡿⠃⠿⠿⠻⠿⠖⣆⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢧⣿⣯⣽⡟⢒⣰⡆⣒⠀⠒⠻⠾⣿⠿⠁⠐⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠥⣙⣿⠿⠛⢿⣾⣷⡾⣭⣕⣒⣰⡁⠀⡀⠀⠀⠀⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡿⠷⢿⠇⠀⠀⠈⠉⠛⠿⠾⣿⡿⢁⣭⣅⢹⣷⣾⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠑⠒⠂⠦⠀⠐⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠆⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⡅⠄⣀⠀⠀⠀⠀⠈⠈⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠠⠀⢐⡩⠄⠂⢀⠀⢀⡀⠀⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢨⣙⡲⠍⣀⢐⣠⣉⡤⠓⠖⠤⠀⠢⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣣⠩⣿⣲⡎⠣⠭⠿⢬⣜⢰⡈⠌⢰⡀⢞⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣛⢹⣿⢶⡶⣾⠟⣈⣃⣄⠏⡩⢂⠼⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣄⢿⡿⣟⠻⣊⣻⣿⣿⣶⣴⣲⣭⣭⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠚⣾⠩⣶⢲⣽⡵⣼⣛⡿⠗⠿⠻⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠩⠻⡶⢟⡹⠛⣎⣯⣷⡻⢮⡻⡡⡈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡏⣉⠖⣻⢞⣷⣽⣿⡼⣇⡱⡟⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠏⣽⣿⠖⠁⠈⠛⠛⢴⣮⣧⢿⣄⠄⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢫⠙⠉⢘⢌⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠿⢖⣠⡶⡵⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠭⠴⠖⠉⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠂⠆⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⣻⡛⠙⢬⡄⠀⣿⣾⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣀⣌⠁⠲⠄⠨⠿⠯⣿⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⣶⡆⠀⠀⠀⠀⠡⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣟⣿⣿⣿⣿⡓⠁⠄⠐⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡏⠈⠐⢁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⣟⣿⣉⠿⡆⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⡥⡶⢞⣍⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣻⣿⣿⢿⢿⣏⣚⡋⣹⣈⣔⢳⠈⡉⢉⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠛⠓⡋⣾⣯⣈⡟⣻⠓⣙⣫⣿⣷⠾⠾⢋⡜⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠅⢡⣿⡒⠝⠟⠹⢨⠽⠥⠀⠟⣤⠟⣦⣬⡵⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⢁⣾⣛⣭⣤⢶⣀⡀⠄⣀⣀⢡⣾⣽⣿⣿⠋⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠹⡃⢶⡿⠙⢋⡤⠋⠡⠈⠁⠀⣀⣼⠾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⡴⠀⢀⣽⣯⡗⠉⢀⣤⣦⠴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠐⣀⣴⣿⠟⠉⠙⠑⠈⠁
⠀⠀⠀⠀⠀⢀⣠⡟⣟⣶⡾⠛⠱
⠀⠀⠀⢀⠴⣻⣿⡿⠋⣥⠠⢀⣤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠁
⠀⠀⢠⣮⠾⢋⡕⠀⠀⢩⡀⢨⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰
⠀⠀⠀⠀⢀⡾⢃⡟⠀⠀⢁⣿⣷⢆⣀⢠⣆⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡜⠤⠐⠂⠀⠉
⠀⠀⠀⠀⠈⠤⠀⠸⡯⠃⣾⣿⣧⣶⣶⣿⣿⣿⣿⣷⣦⣴⡦⠆⠀⠀⠀⠀⠂⠂⠀⠀⠤⣀⠁
⠀⠀⢠⠃⠀⡁⠀⠙⠑⣼⡿⠦⣌⠁⠉⠻⣿⣶⡾⣭⣟⣿⣏⣦⣶⣆⡈⠀⠈⢁⠀⠀⠀⠀⠁
⠀⠀⣠⠀⠰⣻⣤⠀⣸⣿⣿⣽⣿⣛⠒⠄⡀⠀⠉⠙⠛⠿⣿⣿⢂⡉⠁⠁⠘⠋⠓
⠀⠀⠙⠆⢠⣾⡽⠞⢩⣿⣿⣿⣿⢿⣽⣖⠓⢻⡃⠀⠀⠀⠀⠉⠛⠶⡄⠀⠀⠒⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠙⠿⣿⣿⠟⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣤⣴⣶⠾⢶⣄⠸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠻⠀⠤⢾⠋⠁⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣄⠁⡘⢀⠛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠆⠐⠐⠀⠲⣞⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠠⣈⣤⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢠⡼⣿⣿⣿⢿⣿⣿⣻⡿⣿⣿⣯⣯⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃⠀⠀⣼⢿⣿⣾⣿⣯⣟⣷⣻⡟⠿⢿⣿⣧⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡈⠳⠀⠐⢿⣿⣿⣿⣯⣿⣿⣷⣿⣎⣿⣟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⠀⢤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡮⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⣙⣻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢤⡀⠐⢿⣿⣿⣿⣿⣎⣿⣿⢿⣻⣿⡿⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣺⣿⣿⣿⣿⣿⣿⣿⣯⣻⣦⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⢿⣿⣿⣿⣿⣮⣿⣿⢿⣻⣿⡿⣟⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⡠⠿⠋⡻⢿⣿⣿⣿⣿⣾⣿⣝⣯⡻⣟⢿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢑⣀⡀⠁⠄⢹⠛⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⠐⠀⢀⠅⠘⡟⠿⣿⣿⣿⣿⣿⣿⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣻⠋⣷⣾⠛⢦⠀⢻⠂⣈⠛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢿⣴⣿⣷⡖⠆⢐⣼⣇⠀⠠⠈⠻⣿⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠓⠛⢟⢁⡀⠙⠻⣦⠈⣈⣹⡄⠀⠀⠈⠙⠳⡍⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⡀⠀⠈⠛⡇⢠⣄⣵⠾⠟⠛⢳⠈⠂⠦⢰⣦⠀⡀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠐⣪⡿⠶⠚⠉⠉⠀⠀⠀⠀⠀⢷⣦⣄⣀⡀⠀⣷⡀⠸⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⣿⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣼⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠳⣄⠀⣀⣾⣿⠿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣉⠂⠈⢷⡋⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡋⢠⠎⣽⣆⠀⠈⣧⣧⣶⡆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⡠⠔⠋⠛⠃⣙⠛⣦⡾⡟⢉⡟⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⡼⠉⠀⠀⠀⣰⡮⠶⡞⠁⠟⢁⡞⠀⠀⠀⠤⣴⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢷⠀⠀⢀⡜⠁⠀⠀⢳⣀⣴⠛⠶⣶⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢧⡀⡼⠁⠀⠀⠀⠀⠈⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣟⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢸⡃⡏⢉⣿⣾⠋⡿⢹⠟⣻⣿⣿⣿⠿⠿⠿⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢾⡧⣷⣼⠁⢻⡾⣇⣿⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠛⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⢀⣿⣤⡿⠛⠀⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠱⠉⠉⣹⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠁⠀⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣭⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠙⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⡈⢣⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣫⡶⢄⠳⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣥⣶⣶⣆⣬⣛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⣿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢆⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠰⠃⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣙⢲⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙
⢃⠀⠀⠀⠀⠀⠂⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠈⣱⠀⠀⠀⠀⠈⠢⢄⡀⠀⠉⠉⠢⠄⠀⠀⠀⢠⠄⠁⠀⡠⠊
⠀⠱⡀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⠐⠁⠀⡆
⠀⠀⠈⠀⡀⣠⡀⠀⠀⣀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡢⠆⠀⠀⠀⡤⠂
⠀⠀⠀⡆⠉⠀⠁⡀⣀⠉⢆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡍⠄⠀⠠⡄⠀⠈
⠀⠀⠀⠈⠒⠤⠀⠠⠜⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇



⠀⠀⠀⡀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠊⡢
⠀⠀⠀⠃⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⢀⠑⠤⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡀⢀⠄⠊⠓⢊⠢⡀
⠀⠀⠀⠀⠀⠘⠘⠿⠖⠻⠁⠐⠄⡀⠀⠡⠀⠀⠀⠀⠀⠀⠰⡅⠀⠚⠁⠈⠀⠀⠀⠀⠞⠨⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣆⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣹⣋⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠉⠀⠀⠀⠀⠂⠀⣠⠀⠀⡠⠊⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡶⠶⢤⡀⠀⠀⡀⠀⡠⠊⠁⠈⠉⠢⡀⠈⣁⠈⠉⠉⠓⠄⠀⢠⡐
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢤⡑⠤⠀⠀⠄⠓⡀⢃⡀⢀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⢆⠁⠀⡄⠀⠀⠀⠀⡂⠐⡆⠉⠉⠀⡠⠀⠄⠀⠀⠋⢀⡀⠡
⠀⠀⠀⠀⠀⢸⠁⠀⢢⠀⠇⠀⠨⠀⠀⢃⢰⠀⠀⠀⠁⠑⣄⠀⠠⠠⠁⠊⠀⠀⠀⠀⠈⠉⠀⠙⠠⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠈⠙⠀⠀⣠⠄⠀⠸⡀⡀⠀⠀⠐⠀⠰⢄⠀⠀⠀⠖⠂⠀⠒⠀⠄⢀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡰⠀⠀⠀⢇⠄⠀⠀⠀⠄⠀⠀⠀⡑⡤⠀⠀⠈⢄⢹⠐⢄⠈⠂⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠂⠀⠀⠃⡀⠀⠀⠈⠰⠀⠀⠀⠘⠀⠀⠀⠈⢹⠀⠀⢀⡈⠾⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠋⠃⡠⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡕⠆⠀⠀⠀⠈⠒⠆⠀⡜
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⠘⢣⠰⠃⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡻⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢶⠡⠇⠀⠘⢆⠀⠀⠀⠀⢀⠀⠀⢠⡈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⡞⠀⠀⠀⠀⠚⠒⡠⠀⠀⠙⠶⠶⠖⠋⠀⠀⠘⠡⡄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡇⠀⠀⠠⠀⠈⡀⢁⡴⢀⡄⢀⠀⡀⣠⠼⠇⠀⠂⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡇⠀⠀⠀⡁⠀⠀⠈⠀⠏⠀⠉⠋⠀⠁⡀⠀⠑⠀⡼⠁
⠀⠀⠀⣀⣀⣀⡀⣀⣀⡜⠀⠸⡄⠀⢄⡡⠤⠍⠀⣀⡀⠠⠌⢥⠐⠂⣃⠀⢀⡾⠁⣀⠠⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣒⡦⠤⡄
⠀⠀⠂⠁⠀⠀⣉⠂⠀⠉⣐⠠⠻⢶⡁⠀⠀⢤⠐⠢⠈⡆⠐⠘⠐⠄⠀⣠⠋⠉⠂⠈⢂⠈⡂⠀⠀⣋⠉⠡⠤⠤⢤⣄⣉⡑
⠘⠉⠉⠉⠉⠀⠀⠈⠁⠀⠈⠙⣓⣚⣛⢶⡄⠀⣧⠀⡇⡰⢤⣤⡶⢖⣋⠅⠓⣄⠘⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡃⣼⠁⠛⡇⠀⠚⠙⠉⠉⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠇⠀⠀⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣟⣿⣻⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣟⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⣾⣫⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣶⣾⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠴⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠠⠴⠶⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠠⡴⠾⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠈⠀⢸⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠾⠟⠹⠿⢿⡿⢿⣿⠟⠁⢿⡿⠿⠿⠻⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢀⡀⠐⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡈⠛⠀⠲⠄⠀⡠⠁⠀⠀⠀⢸
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠃⠀⠁⠢⡀⠀⣠⠃⢠⠂⠀⠀⠑⠄⠀⡀
⢰⠃⣀⠠⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⢠⠃⡆⠀⠀⡇⠀⠀⠠⠀⠀⠀⠈⢢⡀
⠷⠈⠀⠣⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠁⠀⠀⠠⠀⢀⠇⠀⠸⠀⠀⢀⠃⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠠⠃⠀⠀⠌⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣠⠊⠀⠀⠀⠀⡐⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢰⠁⠀⠀⢀⠄⠘⠂⠐⠒⠀⢀⡄⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⢄⡀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠉⠁⠀⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠓⠒⠴⠂⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈⠲⠦⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⠀⠀⠀⠀⠑⠀⠀⠘⡄⢀⠀⠀⢀⡀⠀⣄⠸⡀⠀⠀⠀⠉⠁⠐⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠖⢠⣤⡤⠒⠀⠱⣼⢣⠀⢸⠑⢄⣇⠑⠷⠀⠀⠀⠁⠒⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠄⠀⢀⡄⠁⠀⠀⠀⠈⠀⠱⣼⢦⡀⠉⠀⠀⠀⠈⠑⠂⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠉⡚⣑⠆⠀⣀⠀⠠⠀⠀⠄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠈⠓⠋⠲⠁⠦⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⢀⠰⠄⠒

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⣴⣿⣿⣯⣍⡉⠉⠙⠛⠻⠿⠿⢻⣿⡄
⠀⠀⠀⠀⣼⣿⡿⠛⠛⢿⠿⠿⣷⡖⢚⠛⢻⣿⣹⡇
⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⢠⣦⣍⡀⠈⠁⢹⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣦⣤⣶⠚⠋⠉⠉⠁⠈⠀⢿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⣿⣷⣀
⠀⠀⠀⣿⣟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣹⢋⣤⠀⣰⠀⠶⠀⣄⠀⣠⠄⠈⠋⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠹⣇⠉⡾⢁⣴⠆⠀⢨⡀⠿⠀⠿⢠⣼⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢰⢿⡌⢀⡌⠛⣤⠻⡏⣥⠀⣼⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⢠⡔⢁⠈⢣⡴⠇⣀⠓⣈⠸⣅⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠘⢧⣸⠃⡾⠃⡔⠻⢠⡍⢰⣌⡰⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢰⣇⠘⠇⢸⠻⠆⡘⠁⢁⡈⢱⣆⣽⣿⣿⡟⣿⡉⣿⠿⣿⣧⣿⣿⣿⣿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠈⠁⢦⣀⢻⠆⠀⠿⠆⠸⠃⠆⠉⢸⣿⣿⣿⣿⡟⣻⣦⣿⡍⢻⡛⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡀⠁⠛⠀⣶⡀⠀⣶⠀⣠⣄⠻⢿⣿⣿⡿⢷⣼⣿⣿⣟⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠸⡇⢰⣆⠙⠜⣀⠙⠀⠬⠉⣷⢈⣿⣿⣿⣇⠙⡟⢿⣿⠈⣷⣹⣷⣹⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠰⣤⣆⠙⠂⡄⢹⡇⠐⣦⠠⠉⠸⢾⣿⣿⣿⣿⣯⡀⢿⠙⣿⠻⡟⢻⣿⣿⡟⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠩⡉⠁⠂⠀⢀⠁⢰⣄⠰⡁⣶⡄⣿⣿⣽⡏⢻⡇⢘⣧⠹⡆⢰⣌⢻⣞⢿⣞⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣠⠀⠈⠐⠀⠉⠀⠀⠸⠁⡭⢹⣿⡟⠻⣷⣿⣏⣹⣟⢣⡽⢶⣽⠷⡽⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⣿⣧⡙⡦⢻⣍⢿⣿⣿⣦⣼⣶⣿⡿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣛⠛⠣⠽⢧⢻⡄⢿⣍⣿⣌⢻⣌⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢚⢠⣄⡘⠈⠹⡛⢿⣷⣽⡧⣽⡷⣾⣩⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠟⣿⣯⠿⠻⢿⢿⣤⣷⣤⣵⣬⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣿⣿⣿⣦⣶⣙⣈⢋⡉⢻⡉⢿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⢡⡾⠋⣸⡟⣿⣾⣍⢻⠟⢾⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣹⣿⣷⣀⠀⠀⠀⠀⠈⠐⠛⢃⣬⢉⣴⣿⢿⣿⣷⣿⣎⣀⠙⣦⠹⣦⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡿⣿⣿⣇⡦⠀⠀⠀⢀⣌⠛⣁⣾⡇⣶⣿⠿⣿⣿⣿⣿⣷⣿⣦⣿⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠉⢛⡑⠀⢀⠀⠘⠋⣸⡏⢩⣥⣿⠏⢶⡟⣿⠙⢻⣿⣿⣭⣿⣛⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠋⠙⠃⣺⠛⢁⣶⠉⣡⣼⡿⣼⣟⠀⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠾⠋⣛⠛⣙⠛⢋⣸⡿⢋⠠⣿⡿⢠⡀⠸⣯⡛⢿⣿⣿⣿⡿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢀⣿⠟⠟⣉⠛⣛⣹⣿⣶⠿⠁⢀⡀⠀⣹⣧⣼⣿⣿⣿⣘⣯⣵⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣽⠃⡺⠏⣠⣿⡛⣹⡏⠀⠀⠀⠁⢀⣹⣿⣟⣹⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠷⣷⣾⣿⣽⡿⠋⠀⠀⠀⠀⢰⠿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⡿⢿⣿⣷⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣿⣿⣟⢿⣮⢻⣿⠀⠀⠘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣀⢀⡀⢰⡶⠋⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⣿⣿⣿⣿⠸⣿⡇⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠒⣉⠀⣑⣾⡥⠀⠀⠁⠘⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠉⠀⣿⡎⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏

⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣤⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣤⠀⢠⣐⠄⠈⢑⠀⢀⠀⣀⣀⡀⡄⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠈⠂⠀⠀⠈⠈⠉⠉⠁⠁⠀⠀⠀⢹⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠿⠟⡷
⠀⠀⠀⠀⠀⠀⢶⣿⣽⣏⢀⠀⠀⠑⠀⠀⠀⠀⣲⣤⡀⠄⠀⠄⠐⠀⠔⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣻⣿⣶⣷⢟⣼⣾⣟⣣⣞⣡⣤⠔
⠀⠀⠀⢀⢼⣿⣾⣏⣝⡻⢟⣵⣫⣿⣿⣿⡿⠟⢋⡏
⠀⢀⣴⣿⡿⠃⠉⠉⠈⠉⠉⠉⠉⠉⠋⠉⣧⠀⠈⠑⢄⡀
⠀⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠩⠄⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠙⠛⠛⠋⠀⠉⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⡈⠛⠑⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢂⠀⠀⠀⢀⡴⣛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢸⣋⣿⣏⠙⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡆⠹⠿⣿⣴⡀⠀⢸⣧⣭⠿⠗⠒⠺⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠋⠁⠐⠀⠂⠞⠩⢿⢷⣶⣦⣹⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⢤⠖⠀⠀⠐⠆⠠⠀⠄⣠⣿⡭⠉⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢓⡷⠋⠀⠔⠀⢀⡴⠀⡤⠎⢀⠾⠉⠀⢀⠀⢀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣫⣤⠎⠁⠂⠊⠄⢀⠎⠀⣠⠀⡀⠀⠀⠐⠇⠀⠃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⡾⠇⠀⢀⡌⠀⠃⡉⢠⠞⠁⠰⠁⢀⠴⠀⠀⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⡠⠊⠀⢀⠀⠸⡀⣠⡆⠀⠁⣠⠂⠀⠙⠁⣀⠀⠀⠀⠀⠰⢠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⡶⠃⠀⠐⠉⠁⡀⠔⡻⠛⠀⡚⠀⡈⠟⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⠀⠀⣀⣤⣾⣿⠁⠀⠀⣠⣴⣾⠀⠀⢀⡔⠀⠉⡐⠀⠈⠉⠀⡀⣰⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡠⢶⣞⠝⢡⢿⣿⣷⣿⣿⣿⣿⣿⣦⣶⣿⣿⣴⣾⣇⢀⣴⠀⠋⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⠟⠀⣰⠟⡣⢂⠔⣠⡽⣋⣵⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⡴⣾⠫⡪⢂⡠⠚⣥⢾⣽⢏⡉⠊⣉⣥⢶⠇⣉⣼⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠟⠇⣠⡼⣿⡬⠖⣩⠔⠋⣨⣼⣦⣵⣒⡿⠋⣔⣃⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⣠⠛⣻⡿⢀⡴⠋⡸⢋⡄⠊⣀⣤⠾⣛⣯⣾⠟⠡⣆⣬⣵⡛⢿⡉
⠀⠀⠀⠀⢀⣾⣽⣾⢇⣴⠋⢀⣏⡴⠿⣚⣻⢭⣷⣾⣿⡿⠿⠛⠛⠉⠈⣿⠃⠘⣇
⠀⠀⢀⡴⠟⠉⣰⣧⠞⢁⣴⡿⣭⢴⠞⠑⠒⠋⠉⠉⠀⠀⠀⣀⠠⢀⣠⣿⣄⠀⣹⡀⣀⣐⡒
⠀⠀⠀⠄⠀⣴⠞⠁⢀⠄⠔⠐⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠁⣠⠤⠉⠱⢛⣇⡉⠀⣒⣒⠊⠁⢀⡀
⠀⠀⠀⢀⠀⠀⠀⣠⢊⣘⡾⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⠛⠋⠉⠀⠉⠛⠒⠒⠻⠉⠉⠁
⢀⠃⠀⠁⠀⠀⣾⡥⠟⠋
⣞⣴⡇⢀⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⣿⣿⣿⢀⡀⢴⣶⣾⣿⡿⠋⠉⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⡏⠁⣤⣿⣿⡇⢠⠶⢤⣄⣶⣸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠉⠛⢷⣷⡈⠙⠻⣷⡄⠀⢙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⠿⠡⠀⠀⠀⡀⣿⣿⣿⣦⣿⣿⠠⠊⠄⢠⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⢠⣀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣆⢀⣴⣟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⡯⣤⣟⠀⠀⠀⠀⢀⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣉⣷⣿⠁⣠⣶⠀⠀⠈⠀⠀⢋⣿⣿⣿⣿⣿⡏⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣱⣿⣿⡿⣠⣿⠌⠀⠀⠱⣆⣴⣾⣿⣿⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣽⣿⣿⣇⣿⣿⣦⠀⠀⠀⠹⣿⡿⠿⣿⣿⣿⣿⢣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣿⣿⣿⣿⣏⠈⣃⢠⣦⢉⣠⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⣿⣿⣿⣿⡿⠰⡿⣿⣿⣿⣿⣿⠏⢻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣿⡿⢏⣥⣾⢿⣛⢈⠉⠛⢾⣟⠉⠘⢷⣿⢧⣠⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣛⡿⠋⣤⣿⡇⠀⠀⠀⢻⡆⠀⢀⣽⣘⣿⣿⣯⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣻⡿⢿⣷⣿⣶⣹⣿⣷⡖⠦⠀⢸⡇⢀⠼⢿⣯⣶⣖⠦⢬⣭⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡛⡿⠁⠀⢼⣿⡿⠛⢯⠙⠻⣿⣷⣶⣾⣿⣿⠸⠛⣫⣿⣿⣿⠸⣿⣷⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣿⡇⠠⠀⠀⠀⠀⠀⠀⢸⣷⠀⠉⠁⣼⣿⣞⣟⢳⡀⣿⣿⣧⡌⣠⣿⡛⠖⣸⢧⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠺⢽⣧⠋⣽⠃⠄⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣿⡿⢥⣙⣦⡙⣿⣿⣿⣷⢟⣿⣿⣼⣿⣽⡿⣷⣀
⠀⠀⠀⠀⢀⣠⣼⣯⣿⣯⣽⣇⣻⣷⣈⣦⣠⣤⣤⣾⣿⣃⠀⣀⣠⣼⣕⣿⣯⣿⣤⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠈⠉⠉⠉⢹⠿⢿⣿⠿⠿⠛⠛⢻⣿⣿⡿⣿⣿⠿⠿⠟⠛⠛⠻⠿⠿⣻⣿⣿⠏⠉⠉⠛⠛⠿⠿⠟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠉⠉⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⡇
⠀⠀⠀⠀⠀⠀⢶⡄⢸⡸⡄⣇⡆
⠀⠀⠀⠀⢀⠀⠘⡝⣄⢇⢱⡸⣸
⠀⠀⠀⠀⠈⢷⣄⠈⢪⡢⣢⣥⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠙⠷⡴⠛⠈⢸⡇⠾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠱⡄⠀⡏⢧
⠀⠀⠀⠀⠠⣖⣒⣒⣻⣴⠒⠶⣯⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠠⠀⢃⠘⡄⢰⠈⡆
⠀⠀⠀⠀⠈⢦⠐⡀⠀⠀⠀⠀⠨⠿⠤⠄⣀⠈⠉⠓⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣧⠀⢹⡙⠀⠰⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠐⠒⠰⠂⠴⢻⡄⠀⠈⠐⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠷⠀⠁⠰⠶⠳⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡒⠂⠀⠍⠁⠐⡂⠠⠜⣷⡀⠀⠀⠀⠀⠉⠳⠤⣤⡤⠴⣖⣖⣒⡛⢻⣿⡛⠻⠴⠶⠤⠄⠼⠻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠤⠒⠀⠀⠀⠀⠀⠀⢷⡂⠀⠀⠀⠀⠀⠀⠉⠛⠦⣔⠋⠉⠁⠀⠀⠀⠀⢀⣀⠤⠤⠤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢻⡤⠄⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠏⠑⣢⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣂⡀⠀⠀⠀⣔⠀⠓⠀⠀⠀⣤⠘⠋⡵⣊⡥⠀⠀⠀⠀⠠⠼⣦⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣄⡤⡽⠖⢀⠄⠀⠀⠀⠀⠀⠰⡿⠥⠶⢿⠟⠈⠁⠀⠀⢀⣀⣉⠝⢋⠏⠀⠬⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠿⠟⠁⢠⠊⠁⠚⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠷⠖⠂⢀⠀⠀⡉⣈⠉⠀⣏⣀⣀⣠⣤⡤⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡳⠛⠀⢠⢿⣬⣤⣂⣔⡤⣂⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣌⡾⢀⠞⠀⠐⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣶⣶⠃⠾⢺⠯⣭⠶⢩⣋⠠⠐⠒⠤⠄⢠⣄⣀⣀⣀⣈⡭⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⢘⡦⠎⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣣⠆⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠊⠁

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠐⠐⠂⠒⠠⠤⡂⠘⠁⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡉⠈⠀⢤⡀⢀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢠⡀⡒⠁⠀⠀⠐⠆⠐⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠊⠁⠈⡁⠑⠆⠀⠀⠀⠣⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠳⠄⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡖⠸⠆⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠁⡀⠠⠠
⠀⠀⠀⠀⠀⠉⢀⠔⠅⡁⠀⡜⠉⠉⠉⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢐⠁⠐⠂⠈
⠀⠀⠀⠀⢀⣔⠣⣐⠔⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⢠
⠀⠀⠀⠀⠉⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠇⠀⠀⠈⠀⢂
⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⡀⠑⡀
⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠀⠀⠀⢠⠀⠠⠉⠀⠀⠱⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠒⠉⡀⠀⠈⡄⠀⠤⠁⢀⠀⠂⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠈⠊⠁⠀⠀⠀⣆⠀⠀⠈⡄⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⠨⡀⠀⠀⠤⠁⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠆⢰⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣦⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡭⠉⠉⠩⡍⡉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠈⠁⢀⢄⠂
⠀⠀⠀⠀⠀⠀⢀⠀⡅⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣆⣠
⠀⠀⠀⠀⠁⠈⠤⠀⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡴⢋⡵⢋⣀
⣀⠐⠁⠀⠀⢀⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡮⠀⠋⢀⡤⠛
⡗⠀⠀⠀⡀⠐⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣥⣒⣉⣀⠤⠾⠂
⡇⣀⣐⡀⠛⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣭⣍⣉⣒⣤⣄
⡆⡹⠛⠻⠷⢤⣄⣀⡉⠙⠒⠦⣂⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠇⠶⣶⣶⣄⣀⣀⡀⠀⠈⠀⠠⢬⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⢀⣀⣀⠀⠀⠀⠀⢈⣀⣀⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠻⠷
⠀⠈⠉⠉⠉⠉⠀⠀⠀⢈⣉⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣤⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣹⣿⣿⣿⣿⣿⢻⠏⡜⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⢡⣾⣰⣱⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⢿⣿⣿⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠻⣿⢿⡿⡿⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢋⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠻⠟⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⣭⠉⠉⢉⣉⢉⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢑⠉⠁⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⡀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⣔⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣞⡡⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⢿⣿⡿⠋⠙⠿
⠀⠀⠀⠀⠐⠡⢏⠀⠀⠀⢁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡒
⠀⢀⠐⠀⠁⠀⢈⠐⠢⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣽⣷⠌⡷⠄⡀
⡆⡠⠁⠀⠀⢐⣮⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⢻⣷⣄⠃
⡏⠀⠀⠐⡠⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⡇⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡤⠞⠻⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠋⣀⣤⣠⡖⠁⠀⠀⠀⠀⠀⡂⠀⠀⠂
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⣠⠾⢋⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣴⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠿⠟⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠞⠀⣴⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠑⢒⣫⣽⠁⠀⠚⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⣴⡾⠁⣩⣀⣀⣀⣾⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⠟⡿⠀⢀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣀⣴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠏⠀⠁⠀⡄⢠⣿⠁⡇⠘⠂⠀⠁⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⣀⡄⢠⡇⠘⠋⠘⠁⡀⣠⣀⣴⣾⣿⣿⣿⣿⢿⣿⣿⣯⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣠⢰⣿⢏⡏⣴⣄⣠⣶⠶⣖⣡⣿⢿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣾⣻⣿⢿⣽⡷⣫⣿⣿⣿⣿⣿⢿⣿⣯⣿⣿⣿⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⢣⣿⣯⣿⣿⣿⣿⣿⠯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣻⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣾⠛⠀⣰⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⢡⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣟⡑⠁⠿⠃⢀⣾⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⡷⢿⣿⣿⣷⣶⠶⠶⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⣽⣿⡿⡙⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⢋⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠏⣽⣿⡿⡵⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡶⠟⠛⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠉⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣻⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠧⠦⣤⡤⠀⠹⡆⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⣀⡤⠬⢭⠢⣠⠃⠀⠀⡼⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⢸⠋⡽⠋⢉⠉⠛⢓⡾⢄⣰⡆⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢰⠃⠀⢮⡴⠛⡆⠈⣳⣹⡆⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡟⢦⡄⠀⠀⡰⠃⢀⢹⠈⡇⠀⠀⠀⠀⢢⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣡⢔⠀⠀⠱⡆⠉⠛⠈⢀⢠⣘⠓⠀⣇⠀⠀⠀⠀⡀⠻⡢⡀
⠀⠀⠀⠀⠀⠀⢠⠴⠲⠦⢤⣀⡀⠀⠠⠶⢟⣶⢯⣶⣡⢎⣴⠜⠙⢷⠓⢷⠺⠚⠈⠀⠀⢸⡀⠀⠀⠀⠘⢦⡱⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⡒⠀⠈⠉⠙⠒⠦⣤⢴⠶⢴⣦⢤⣀⣀⡴⡏⠀⠀⢙⡶⢤⠀⠀⠀⣳⡀⠀⡀⠱⡈⢗⠧
⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣒⠀⠀⠀⠀⢻⡁⣧⢸⠁⢠⡦⢤⣁⣴⠛⠞⠻⠏⠁⠈⣦⣀⢠⣯⠻⣄⠘⣄⠹⣌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠺⣇⣻⡈⡄⠉⣻⡁⠀⠈⠉⠒⢢⡾⢉⡹⢛⡦⢿⠃⠀⠘⢦⠸⣆⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢚⡿⣭⠷⠒⠒⠦⣧⣬⣇⡚⠀⡶⠧⢄⣀⣈⢳⣟⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⡷⠛⠛⠿⠦⠿⠂⠤⢤⣈⠉⡟⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠒⠒⠂⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⠴⠒⠒⠒⠺⢭⡶⠖⠂⠒⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⣴⣾⡇⣆⠰⡆⠀⢀⣠⣴⡾⣿⣻⣿⠟⠻⣧⣤⣄⡄⠀⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⠈⢁⣤⠀⠹⣷⣤⣤⣿⣿⣿⣷⡙⢿⣭⡆⢀⣿⣿⣿⣏⡁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣘⣛⣵⡶⠶⢦⣽⣿⣿⣿⣿⣛⢿⣶⣍⣡⣿⣿⣿⠿⣟⠆⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠿⣿⣷⣿⠿⠿⠿⢶⣶⣶⣶⣼⣮⣛⣋⠙⣉⣭⣤⣤⣄⣀⣢⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⡾⠛⡛⠻⠗⢛⣉⢄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠶⠶⠷⠒⣛⣭⠗⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣓⣶⣿⣻⣛⣻⣭⣭⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢈⣋⣟⣿⣯⣿⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⣰⠃⠀⠈⢈⣉⡻⡟⢼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠰⠵⣤⣻⣴⡇⠀⠀⠀⠀⠈⠸⠥⣾⢁⣤⣤⡲⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣴⣤⣼⣿⣶⣾⣿⣿⠁⠀⠀⠨⠭⣽⠿⠒⣿⣾⣿⣷⡟⢋⣁⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣽⣿⣿⣿⣿⡆⠀⠀⣐⣈⣸⡐⡄⢻⣿⣿⣿⣿⣿⣯⣄⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⣛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣣⢰⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣥⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⢉⣠⣴⣿⣿⣿⣿⣿⣿⣧⣐⡀⣁⣿⡤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣹⣣⠞⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⣠⣿⠟⣻⡷⢟⣛⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣿⣿⣷⣬⡛⠿⣿⣿⣿⠿⣟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⣯⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⢾⡿⠟⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠻⣿⣝⠢⠀⠉⠉⠳⠙⢦⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢋⣴⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣴⡤⠂⠀⠰⡄
⠀⠀⠀⠀⢀⡤⢒⡍⠉⠞⡿⠏⢀⠁⠀⢛⡀
⠀⠀⠀⣌⣴⠿⠶⠿⣷⢆⣄⠐⢆⣼⡀⢻⠁
⠀⠀⠀⠈⠀⠀⠀⠈⠀⠢⣴⢸⣶⠞⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡿⣽⢇⣧⠉⡻⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠚⡒⢂⡈⠅⣼⢉⢚⣷⡤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣉⡄⠖⠀⠀⠊⠑⠀⢹⡿⣴⠿⢹⣯⣬⣿⣶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⢰⣶⣶⣄⢮⣽⣋⣿⡿⣿⣿⣿⢯⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⠀⠀⠀⠀⠀⣿⣬⣿⣋⣷⡹⣿⡏⢹⣯⢿⡗⢿⣯⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠫⡤⠀⠀⠀⠀⠀⢹⢦⣿⣻⠻⣧⡷⣶⣿⡿⢷⣭⣼⣿⣿⢯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⠀⠀⠀⠀⠀⣼⣿⣿⡿⢾⣋⣿⣟⢱⣿⣽⣻⣟⡾⢳⣟⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡖⠆⠀⠀⠀⢰⢹⡯⣿⡿⣝⠚⣿⡻⣯⠼⣿⣽⡛⣺⣿⡷⠄⠀⠈⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⡀⠀⠀⠀⠣⠳⠸⣧⢿⡿⣷⢶⡟⠷⢬⣥⣿⣭⣟⣹⣶⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡗⣀⠀⠀⠀⠀⠢⣿⡸⡇⢻⡄⢻⣳⣾⢧⣿⣭⣿⢯⣻⠀⢠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢷⣥⠀⠀⠀⠀⠀⠀⡱⣿⡜⣷⠸⣇⣹⡇⢿⣹⣷⡾⠿⠃⠀⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⢧⣂⡐⡎⠁⠑⠀⢆⡘⣩⣯⣶⠽⠪⢛⠉⠁⠀⠠⠀⠀⠐⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣷⣦⣄⢀⣀⡿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⣣⠶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠻⡛⣻⣻⢹⠞⡅⠐⠃⠀⡀⠠⠤⢤⠠⠄⣶⢶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡏⠫⣞⢻⣦⣿⡆⠀⡤⣶⣶⣶⣾⣏⢿⣏⢿⣽⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣽⢽⡈⠹⢻⣤⣻⣍⠚⢿⣟⣿⣿⣿⡿⡙⢧⡿⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡹⡇⣸⣔⢃⡘⠺⠽⠭⠒⠓⠺⣿⣿⣿⣯⣿⣿⣾⡳⣮⡻⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠱⣏⠨⣏⣋⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⡿⠚⢿⣆
⠀⠀⠀⠀⠀⠀⠐⠁⡄⠀⠀⠀⠀⢠⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣿⣿⡈⢻⣿⣆⠀⠙⢇
⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⡀⠂⢤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣼⣿⣇⠀⠙⢿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣿⣿⣷⣿⣷⠀⠀⠀⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢯⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⠒⠀⠐⠐⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢿⣟⣿⣷⣄⣂⠀⣿⢟⣠⣤⣟⡓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠨⣥⡘⢿⣿⣿⡿⠗⠀⠸⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⢠⣤⣽⡄⠢⠀⢉⡢⣭⣤⣔⣿⣷⣶⣾⣷⣶⣽⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠯⣅⣿⣿⣿⣿⣷⣽⣿⣶⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⣭⣿⣷⣿⣟⡛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢛⣌⣹⡿⣿⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠌⢤⣧⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⡟⠰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠀⠀⠀⠀⠘⠓⢀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⠀⢻⣿⣿⣿⣿⣿⣥⡶⠋⢀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⢿⣿⣿⣶⡆⠈⠩⡝⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⣠⣾⣛⣫⡾⣀⠀⢉⣙⣉⣩⣥⣶⣦⣴⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡳⣿⡆⢠
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⠟⢉⣴⣯⠀⠘⣿⡿⠿⢛⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⡃⣀
⠀⠀⠀⠀⠀⠀⠘⢻⡏⣚⣿⣿⠷⠋⣹⣿⣷⠀⢸⣷⡺⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣹⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠞⣳⠿⢿⣿⣶⣟⣙⣽⣿⡆⢺⣌⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⢿⡿⣿⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣾⠻⣿⡿⢿⢿⣿⡗⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⡆⢰⡇⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠖⡿⠀⣠⠊⠉⣰⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣼⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⡻⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠾⠿⢶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡝⣷⠦⠀⠀⠀⠀⠀⠀⣤⠤⠻⣶⢱⣶⣄⡤⢀⣀⣠⣴⣾⣿⢶⡾⠅
⠀⠀⢠⣄⠀⠀⠀⠀⢧⣿⠁⠀⠀⠀⠀⠀⠀⣤⣿⣿⡇⠀⢿⣿⣿⣏⣴⣿⣿⣿⡟⢿⠿⠁
⠀⠀⠀⠀⠘⢦⠀⠀⣼⣯⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣷⣄⣼⣿⡏⠙⠛⠉⠙⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣬⣿⣦⠲⠀⠀⠀⠉⢋⣉⡔⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⡛⠿⠟⠟⠠⠄⠀⠀⠀⠀⠀⠉⠉⠒⠒⠶⠠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⠀⠀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⡭⠉⠉⣩⣭⣉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⠉⠁⢀
⠀⠀⠀⠀⠀⠀⢀⢀⡅⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠄⠑⠘⠄⠀⠄⡡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⡀⠐⠀⠀⣀⣠⣤⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠉⠀⢠⣕⠤⠲⣤⣀
⡇⠁⢀⣀⣽⣿⠿⠋⢀⠐⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡁⠈⢉⡙⣡⣈⣽⣴⣾⣷⣿⣷⣦⡀
⡇⠀⢸⣿⣿⣿⣿⣟⣷⣖⣒⣶⡾⢒⣢⣤⣀⡀⠀⠀⠀⠀⢂⠀⠀⠀⠵⣆⠀⡀⢹⡉⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⡇⠠⠀⠐⢶⣶⣿⣯⣿⣿⣿⣻⣭⣷⡿⣽⣿⣿⣷⣦⡀⠀⠀⠳⡰⡄⢀⢨⣦⣈⢶⡍⢲⣽⡇⢹⣿⣿⣿⣿⣿⣿⣷⣤
⠏⠀⠀⠀⠀⠙⢿⣿⣾⣯⣽⣞⡻⣿⣯⣿⣿⣿⣿⣿⣷⣶⣄⣄⡈⢿⣼⣿⢯⣿⣮⣿⣦⣿⣷⢘⣿⣟⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣻⣿⣿⣒⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠓⢮⣿⣏⢁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢦⣀⡀⠿⠧⠶⠖⣟⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣟⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣷⠶⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⢙⢻⣽⣖⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣈⠳⠶⠦⣤⠴⠖⠛⠛⣻⣿⣭⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⠀⣂⢀⣠⡀⠀⣴⣄⡀⠸⣷⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠿⠿⡷
⠀⠀⢠⢮⣭⣭⢭⢃⡠⠤⠶⠛⠉⠭⡜⠻⣽⣿⣿⣿⣿⣿⣃⢳⣛⣽⣿⣿⡇⠀⢉⢋⣟⡢⣤⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠛⠛⠉
⠀⢀⣭⣬⣿⣬⣷⣿⠅⠀⠀⣔⡒⣺⣾⣓⣿⣿⣿⣿⠏⠻⡿⣿⣟⠿⣿⣿⣿⣆⣬⣿⣿⣿⣿⣭⣆⣙⣿⣿⠮⡵⠋
⠀⢸⣟⠀⣻⠖⠁⠀⣈⢭⣽⣿⣯⣽⣾⣿⠿⣿⠉⣿⣶⣤⣳⣾⣫⡗⢻⣻⣿⣿⡿⣭⣾⣻⣿⣾⣿⡿⠿⠛⠙⠃
⠀⠚⠀⠒⠃⠀⠀⣸⣿⣿⣿⣿⣾⣿⡏⠀⣸⠁⢨⣿⢭⣿⣿⣦⣀⣈⠿⢻⣿⣿⠛⠛⠛⠛⠋⠹⠏
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠃⠚⣿⣶⢿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⡟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⠿⠋⠀⠀⠀⠀⠸⠃⠨⠟⠗⠛⢻⣿⣿⣿⡛⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠋⠈⠽⣿⣿⣿⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣄⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⣼⣿⣿⣷⣦⣄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣠⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣍⣉⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⡿⢿⣿⠁⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⠃⠛⢿⣿⡿⣿⣷⣸⠏
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠢⡀⣼⣾⣿⣹⣿⣿⠇⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⢀⡀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢀⣸⣾⣷⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢀⣀⣾⣿⣿⣿⠉⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣠⣾⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣿⣻⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡟⢷⣮⣸⠇
⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣨⡿⣶⣶⣄
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⢀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣭⢮⠹⣿⣿⣷⣝⡻⢿⣿⣿⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣝⣣⣜⢿⣿⣿⣿⣿⣷⣶⣯⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠝⣨⣭⣑⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣭⣿⡿⢸⣿⣿⣿⢶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⣿⣿⣿⣷⣙⣛⢫⣶⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣯⣭⣽⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⡛⠛⡳⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠛⠋⠉⣱⣾⣿⣿⠿⣿⣿⣶⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡽⠟⠉⠀⠀⠀⠀⢀⠀⠀⠠⢳⣿⢣⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣏⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢐⣦⣺⣧⢨⡀⠞⠛⠁⠂⡞⠆
⠀⠀⠀⠀⠀⠴⠞⠃⠛⠉⠉⠀⠀⠉⠂⠀⠀⠀⣀⠾⠛⠋⠛⠋⠉⠁⠳⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⡣⣄⣀⡀⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇⠀⠀⠰⢾⡿⠟⠛⣽⣾⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠏⠁⠀⠂⠀⠀⢸⡇⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⣉⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⣤⣶⠾⠿⠛⠛⠒⠄⠌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⠁⣔⡦⣖⢰⣰⢰⡆⢀⣶⢰⢲⢢⣲⢤⣤⡄⢠⢴⡂⣔⡦⡖⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠄⣀
⠀⣠⡴⠞⠛⠳⢦⣿⣿⣿⠟⠁⠀⠀⠀⣠⠖⢋⣉⣉⡿⠁⢈⡽⠟⠀⠀⠀⠀⠀⡰⢉⠔⢚⡒⠠⡑⠄
⢰⠏⣴⠟⠛⡳⢟⠑⣆⠀⠀⠀⠀⠀⢰⠃⡼⢋⣤⣿⣵⡞⢻⠀⠀⠀⠀⠀⠠⠤⡧⡇⠌⢸⠌⡦⠥⢼⠤
⢿⡀⣿⡘⠯⠅⣼⠀⣿⠀⠀⠀⠀⠀⢸⡀⢧⠘⠾⠟⣠⠇⣸⠀⠀⠀⠀⠀⠀⠀⢣⠑⢌⣐⣂⡠⢃⠌
⠈⢷⣌⠛⠒⠛⣡⡾⠃⠀⠀⠀⠀⠀⠀⠳⣌⡙⠒⠚⣁⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢼⠄⠒⠁
⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣳
⢀⠞⣋⣍⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⣉⣉⡓⢦⡧⠞⠁⠀⠀⠀⠀⢀⠔⠋⢉⡉⠙⣲⡗⠚⠁
⡞⣼⢡⠶⡵⠇⠀⠀⣀⣀⡀⠀⠀⠀⢰⠃⡞⣡⠴⢦⡝⢧⢱⡀⠀⠀⠀⠀⢠⠇⢠⠞⠉⣩⠿⡅⠘⡄
⡇⣧⢻⣘⣟⣛⠛⠛⠿⠿⠇⠀⠀⠀⢸⠸⡄⢇⠘⢃⡇⢸⢀⡇⠀⠀⠀⠀⠸⡄⠸⡄⠘⠃⢀⠇⢀⡇
⠹⡜⢦⡭⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠦⣍⡩⠔⢃⠞⠀⠀⠀⠀⠀⠀⣱⣄⠈⠓⠚⠉⣠⠞
⣤⣽⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣮⣥⣤⣤⣤⣤⣤⣤⣼⣥⣤⣭⣽⣯⣭⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠓⢄⡀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣧⡀⠀⠀⠀⠀⠈⠁⠀⠀⢤⡄⠈⠀⢀⣉⡀⠀⠁⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⠿⠇⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⡕⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣷⣤⣀⡀⣤⣀⢤⣤⡄⢴⡖⣶⣤⣤⣈⠂⠀⢸⠀⠂⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡯⢩⣕⠰⣶⡉⠛⠢⣹⡁⣼⣄⣾⠵⠿⢿⣧⡀⠠⠆⠈⠀⡄⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⡗⢄⣩⠳⣤⣈⠿⠆⣟⡁⠈⠃⠀⠀⠀⠀⠙⡇⠀⠀⠸⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠊⣭⣭⣤⣤⣴⣾⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣷⠿⡛⠿⠿⢿⣿⣿⣷⣿⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠞⣫⣾⣷⣦⡭⠳⣦⣀⡲⠾⠿⢿⣿⣿⣶⣦⣤⣤⢤⣀⣴⣾⣿⣿⣿⣷⣶⣠⢄⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢊⡴⣫⡵⠟⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠓⠲⣆⡩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⠟⠛⠿⢝⣿⣺⢶⣄⡀⡀⠂
⠀⠀⠀⠀⠀⡠⣢⣶⡿⠟⠋⢀⣀⡀⠠⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠪⢍⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠨⠉⠋⠒⠯⣄⡀
⠀⠀⠀⢀⣴⡼⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⢄⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⢤⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠂⠄⡀⠀⠀⠀⡀⠀⠠⢿⠸⣿⣿⣿⣧⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠈⠓⠀⠀⠀⣦⠔⠘⣰⣿⣿⢹⡿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠈⠀⢀⣴⢿⣿⡇⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢴⠀⢀⣀⣠⠼⠛⠛⠏⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢨⣿⣟⠋⠁⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢤⣑⢦⡳⠴⣦
⠀⠀⠀⠀⠀⠘⠶⠍⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⠈⠂⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣴⢢⠱⡀⠀⠀⠸⡿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡱⡰⡀⠡⠀⠈⠩⢿⠆⣤⢾⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣅⠀⠀⠀⠀⠁⠀⠀⣦⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⢀⡘⡇⢀⣰⣿⠓⠘⠿⠟⠷⠶⢒⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⢧⣶⣏⣉⡿⢒⡀⢤⠐⠀⠐⠛⠧⣿⣿⠇⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣘⣼⡿⠟⢿⣯⣷⣾⢭⣙⡀⠠⠎⠀⠀⠀⠀⠀⢸⣯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣶⢿⣾⡖⠀⠀⠈⠙⠛⠿⢷⣿⣽⠯⣵⣿⣀⣶⣴⣿⡓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⢀⣤
⣠⢀⣀⠀⠀⡇⠀⠳⡀⡠⠖⢉⠂
⠈⠲⣌⠉⠓⠓⠀⠀⠉⠀⠀⣯⣤⣄⣀⣀
⠀⠀⠈⠢⡀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⢀⡀
⠀⢀⡠⠖⠉⠀⠀⠀⠸⡓⢄⠀⠺⢤⣀⠀⠀⠀⠀⠀⢀⣸⠹⣤⢤⡄
⠀⠉⠉⠛⠛⠛⣻⠀⠀⣱⡀⠙⢶⡉⠉⠀⠀⠀⠀⠻⠭⣅⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⣏⡴⠊⠁⢳⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⣼⠖⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⡀⠀⠀⡈⠳⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠢⡈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⢄⠙⢦⡀
⠀⠀⠀⠀⢀⣀⢀⡞⠸⡄⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠑⠀⠉⠳⢄
⠀⠀⠀⠀⠈⠻⣍⠀⠀⠉⣳⡦⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠢⣄⠙⢦⡀
⠀⠀⠀⠀⢀⣴⡧⠶⢤⣸⡁⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠈⠑⢄⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡰⠀⢀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⢀⡔⠁⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⡴⠛⢦⡀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣶⣦⣶
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⡀⠀⠘⠁⣀⣀⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠀⡔⠚⠀⡐⢻⠁⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦⡑⠀⠂⠁⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂⢳⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⣟⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠃⠀⠀⣤⣌⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡻⣛⠷⢾⡧⢿⣾⣼⣿⢿⠃⠀⠀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣈⣄⣸⣼⠻⣻⣬⣿⣟⣿⣿⠀⠀⣴⢿⣯⠇⠀⠀⢀⢤⠀⣀⣠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣶⡟⠻⢩⣉⣯⣇⠀⣴⣻⠞⠁⠀⠀⡠⠋⠚⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⢋⡷⣯⡟⠗⡿⢿⠿⠋⢀⣼⣿⣁⡠⠴⠒⣉⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢹⡥⢴⣿⡵⠻⣁⠆⠈⠏⣡⣂⣧⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⣇⢸⢞⣋⣽⣿⡤⠖⠻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⢋⣩⣶⡀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠶⠚⠉⠀⠀⠉⢿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠟⠙⢦⡀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⡇⠀⠸⣄⠀⠀⠀⠀⠀⢠⣾⠂⠀⠀⠉⠢⣄⠀⠀⣴⠚⠉⠉⠉⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⡴⠁⠀⠀⠈⠑⠲⠤⠤⠖⠋⠁⠀⢀⣶⣄⠀⠀⠙⢾⣿⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣠⠴⠋⢀⡤⠲⢤⡀⠀⠀⣠⣶⠒⠂⠀⠀⣾⡿⠁⠀⠀⠀⠀⠻⣿⣶⡦⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⢻⣿⠁⠀⠀⠀⣪⣤⣤⣀⣧⠴⢺⣿⠇⠀⠀⠀⢸⣿⠁⠀⣠⣶⡖⠀⠀⠈⠻⣿⣦
⠀⠀⠀⢠⠞⠉⠀⠀⠀⢀⡠⠬⣿⣄⠤⠒⠉⢸⣿⡏⢁⡇⠀⠸⣿⡆⠀⢀⡆⠈⠿⠤⠚⠁⣿⠁⢀⠄⠀⠀⠈⢿⡇
⠀⠀⠀⠻⣶⣶⡄⠀⠀⠘⠒⠋⠹⣟⢆⠀⠀⠘⣿⢁⡜⠀⠀⠀⠙⢿⠶⠞⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⢸⠇⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣶⠀⠀⠀⠀⠹⣌⡆⠀⠀⠈⠉⠀⠠⠤⠄⣾⡏⡇⣾⣷⡜⠠⠤⠀⠀⢸⡀⠙⠦⠤⠤⢖⡉⠀⢀⣾⡟⠁⢸
⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠁⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠈⡇⣼⣿⠃⠀⢸
⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣤⣄⢰⡉⠉⠳⡄⠀⠀⠀⣞⠉⠑⣄⠀⢠⡞⠁⠀⠀⠳⠀⣠⡤⣄⠀⣰⡟⠀⠀⡰⢁⣿⡿⠀⢀⠞
⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⡰⠻⣿⣦⠙⣿⠗⡟⢉⣿⡷⠘⣶⣄⣸⠀⢸⣷⡀⠀⠀⠀⣼⡟⠀⣸⠀⣿⡇⣠⠞⠁⢸⣿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠹⣧⣠⠔⠁⠀⢹⣿⣇⢿⣰⠃⣾⡟⠁⣠⣈⠛⣿⡆⠀⢻⣿⣦⣀⣀⣿⡷⠚⠁⣠⣿⡏⠀⢀⣀⣼⣿⡏⢀⣀⣠⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡈⠁⠀⣿⢀⠜⣿⠏⠀⣿⡇⠀⡴⠻⣿⣿⠀⠙⢿⣄⡴⠁⢻⣷⠀⠈⢉⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠉⠁⠰⣿⠀⡼⢻⠀⠘⢇⣀⣸⡿⠀⠀⠀⠀⣀⠀⠈⠻⣇⢀⡜⢹⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⢾⣿⠧⣀⣀⠀⠀⠀⠿⠜⠁⢸⡄⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠉⠙⠲⣌⠉⠀⠸⣿⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⣿⡟⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠻⢄⡀⣸⠁⠀⣧⣀⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢻⡾
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⠈⠁⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣀⣀⣀⣀⡤⠚⠁

⠀⠀⠀⠀⠀⢤⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⣤⡀⠀⠀⠀⠰⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡄
⠀⢀⠀⠀⠀⢰⠾⠛⠻⠰⠄⠀⠀⠀⠀⠘⢂⠉⠁⠀⠀⠀⠀⠀⠠⠦⣄⠀⠘⣿⠀⠀⠀⠴⣿⡽⣿⣤⠀⠀⠈⣾⣧⡀⠀⠀⣰⣿⣿⣿⠗⠀⠀⠀⢈⣿⣶⡄
⠀⠚⢃⠀⠀⠀⠀⠀⢀⣤⠄⡁⠦⠀⠀⠀⠈⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣷⣷⡀⠀⠀⠈⣿⠃⠀⠀⠿⣿⡟⠉⠀⠀⠀⠀⠩⠿⠞⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⡄⠀⠀⠀⠀⠈⠑⠆⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠠⠀⠘⢶⢶⣤
⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡽⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡦⠀⠀⢈⡄⠬⠒⠒⠀⠀⠀⠘⠛⣯⣥⡂
⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠐⣂⠐⢾⣾⠆⠀⠀⠀⠀⠀⠀⢦⣤⠀⠀⣠⠆⠒⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⣨⠀⠀⠀⡄
⠀⠀⠠⣶⠃⠀⠀⠀⠀⢠⠂⠆⠀⠀⣀⠀⠀⢐⣾⡽⣷⡄⠀⠐⢀⢑⣠⣤⡤⠀⢤⣄⡀⠀⠈⠿⢟⣶⢺⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⠀⢠⠄⣀⠜
⠀⠀⠀⣿⠀⠀⠀⠀⡰⢋⠃⠀⠀⠀⠔⡞⠂⠴⠽⢿⡟⡿⢀⡀⠀⠀⠘⢿⠗⠰⢺⣯⢿⣦⠀⣰⣮⣤⣬⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣟⡿⡈⢺⠀⣬
⠀⠀⠀⠃⠀⠀⣠⠞⢡⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣽⡁⠶⠄⡖⠠⣀⠰⣀⠀⠈⠛⠿⢿⡟⠋⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣬⣿⡎
⢀⣀⣀⣀⣴⣋⡠⠄⠉⠀⠀⠀⠄⠀⠀⠀⠀⠐⠙⠛⠁⢡⣶⡾⣾⡷⢤⡀⠙⢄⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⣽⡇
⠀⠉⢣⣠⠌⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡷⡿⠟⠉⠀⠙⢆⠘⣆⣰⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⣤
⠀⠀⠀⠙⢤⡀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠱⡄⠂⠀⠀⠈⠓⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠈⣗⠂⡀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠈⠀⠤⠠⡄⠀⠀⣀⠀⠀⠟⠀⠀⠀⠂⠀⠀⠀⠀⢰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢴⠂⢄⠀⠀⠐⠀⠀⠀⠀⡼⠋⠀⠀⠀⡀⠂⢀⡤⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠄⡂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠁⠣⡀⠀⢀⡞⠁⠀⠀⢀⣆⡱⠂⠁⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢣⣠⠞⠀⠀⢈⣠⠚⠁⠀⠀⠀⣨⠄⠀⠀⠀⠀⠀⢀⢀⠆⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠀⠀⠀⡀⠀⠀⠀⠀⠙⢾⠋⣆⣀⣜⠁⠎⠀⡄⠀⣄⢀⣸⣤⡀⠠⣤⡴⠂⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠁⠸⡉⠘⠇⣄⠀⠀⢸⡦⠞⢉⣤⠼⠐⢲⠆⠙⢿⣤⢿⡄⢀⡴⠁⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡌⢦⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣖⠛⢃⣽⣙⣿⣿⢷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠐⣤⡿⠿⣿⣻⣯⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣴⣿⣿⣵⣿⡭⠶⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣯⣵⣽⡯⣯⣿⣃⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⠿⣽⣿⠟⣿⣿⣺⣺⣿⠫⡾⣽⣋⡽⠟⠒⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣿⠁⣵⢘⡔⠚⣟⠛⣍⠲⣉⣾⡿⣩⡶⢿⣿⣿⣿⣾⣵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠒⡷⠂⢳⡉⣇⣿⡍⡙⢢⢰⣮⡟⣲⡏⡕⢨⡼⡇⠙⣿⣿⣿⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡓⢻⡁⠰⢷⣩⠩⠃⢉⣧⡜⡸⣑⡟⠰⣎⡰⠰⣻⣤⡀⠛⣿⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣟⣥⡂⠀⠙⢦⠻⠞⢮⣄⣰⣾⣉⢃⣨⠣⢤⣘⠆⠁⠃⠘⠛⡿⢇
⠀⠀⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡧⠛⠇⡀⠀⠀⠀⠀⢀⠈⠻⡇⠈⢦⠞⠳⣎⠈⠁⡃⣀⠈⠗⣿⠘
⠀⠀⣷⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡷⡻⠀⠙⢴⣆⣸⣤⠞⠀⠀⢳⡀⢈⡆⠀⠀⠑⡤⠁⢠⠦⡆⢸
⠀⠀⡟⠠⡀⠉⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⢿⡀⢿⣧⣄⠀⠀⠸⡏⢛⣠⠀⠀⠘⣧⢈⡂⠀⠁⢠⡇⠘⠃⢆⠴⠊
⠀⠀⣤⣎⢬⣔⠢⠤⠄⠀⣇⠀⠀⠀⠀⠀⠀⠹⠾⢯⢿⢢⡀⡀⢻⡌⣟⣤⣀⠀⢹⡄⠇⠀⠀⢼⡇
⠀⠀⣿⡈⣢⡙⢷⡈⡟⢀⡈⠑⠢⢄⡀⠀⠀⠈⠀⠀⠉⠛⢿⡋⣺⡇⠈⠋⠿⢳⠞⢻⡌⠒⠧⣸⡷⣄⣠⡀⣤⣤
⠀⠀⠻⣿⣿⣟⣦⡙⢃⣌⠹⠻⡒⡄⠉⣷⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣆⠀⠀⢾⡀⠈⢿⣄⠀⠙⡇⠈⠙⠹⠀⢹
⠀⠀⠀⠙⣟⠛⣉⣠⡞⢳⣦⡄⢀⠁⢰⠋⠢⣄⠀⡄⠀⠀⠀⢈⣿⣿⣿⣧⠀⢸⣿⣿⣞⣿⣷⢠⣷
⠀⠀⠀⠀⠈⠉⠈⠙⣷⣞⢦⠿⠃⢠⣿⣁⠶⣌⡳⣾⣶⢄⣦⣼⣿⡏⡿⠛⣳⡌⣿⣿⡏⠁⢻⣧⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⣀⣴⣷⡥⣹⣿⣭⣷⠟⠙⢿⣟⢿⣿⣽⠃⠀⠇⢹⣿⣿⡇⠀⣾⡟⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⣽⠮⢾⣫⡭⠁⠀⢏⣼⢫⣟⠛⠿⢦⣠⡤⠦⣽⣿⣿⣴⡏⢡⠈⢿⡻⡶⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡉⣁⢤⣾⣆⠚⣷⡾⢯⡟⠀⣼⣵⣶⣶⣿⡿⢟⢯⣧⠘⡀⠈⠻⣝⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⣯⣤⠈⣰⠔⢁⡼⢦⣀⠤⢤⡙⣇⣼⢻⡿⣄⢁⠀⠐⠈⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⣉⣠⣴⢿⣷⣄⡀⡐⢖⠅⢸⢻⡈⠻⣎⣻⢦⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠻⣿⠋⢳⣿⡮⠁⢾⢸⡿⠷⣽⣿⠻⠧⢤⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠚⠉⣀⠰⡟⣿⠣⢠⣌⠻⠀⠀⣜⠲⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠛⠋⠉⠉⠉⠛⠉⠀⠈⠉⠉⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡉⠳⣄⡴⠟⠛⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⠾⢃⢔⣆⠀⠘⣆⡄⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⡸⠆⠾⠸⠿⣄⣴⡿⢻⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢠⠞⠁⠀⠀⣠⡮⣴⠋⠿⢃⡞⠁⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⡄⠀⢀⡞⠁⠀⠸⣄⣠⠾⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⢟⠿⣿⣿⡿⣿⢯⣿⣽⣿⣿⣿⣿⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⡶⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠿⣼⣼⡄⡞⢻⣰⡇⣞⣰⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠃⠻⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⠗⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠛⠁⠘⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣄⢀⠀⣄⣤⣶⣿⣿⣶⣤⣀⠀⠀⠀⣰⣶⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡿⣿⣿⣇⣶⣾⣿⣿⣾⣿⣿⡿⠃⠉⠀⠀⢀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⢻⣿⣿⣿⣿⣯⣬⣛⠉⠛⠋⠑⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⣭⣯⣭⣿⣿⣿⣿⣟⣛⣁⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⡏⠉⠉⠙⠛⠛⠛⠉⠉⠉⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⣻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⣂⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣳⢻⢿⣿⣇⠀⠀⢀⡤⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⢄⠀⠀⠀⠸⣟⣿⣿⣶⡠⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡐⣀⠀⠀⡀⠸⣟⢿⣿⣿⣄⠂⠈⠁⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⡜⠂⠰⡤⠀⡺⠟⢿⡍⠛⠒⠂⠀⢀⣀⡨⠛⣶⣄⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣦⡀⠤⢁⠀⠀⢀⠙⢤⡀⠀⢲⣦⣭⣽⣿⣿⣿⡿⠉⠹⣶⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠋⠀⠈⠀⠀⠀⠀⠀⢉⢶⣜⢿⣿⣿⣿⣿⣿⣛⣷⣾⣿⣇⠐⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⣀⠀⠀⡀⢳⡜⣎⢦⡻⣿⣿⣿⣿⣿⣿⡿⠷⢻⣿⣯⣤⣾⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⡄⢀⠀⢿⢸⡆⢹⣼⣿⣎⣷⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⡾⡿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⠘⣇⢻⣧⣿⣸⣿⣿⣿⣿⣿⣟⣛⣿⣿⣭⣉⣑⣊⣽⣿⣿⣛⣀⠿⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢻⡌⣿⣿⣿⣿⣿⣝⠻⠿⠛⣿⣽⣛⠛⠋⣛⣿⣿⣅⡙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⣿⣷⣹⣿⣿⣟⠿⢃⣿⣶⣾⡛⠿⠿⠻⢿⣿⣷⡤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⡏⠽⢫⣿⣿⣿⣽⣛⣹⣷⣤⣴⣶⡙⠛⠋⠈⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣼⣏⠫⠁⠬⣿⣿⡇⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡶⠖⠹⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠈⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣶⣦⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⢠⣾⣿⣿⣿⣽⣏⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠓⠒⠄⠘⢦⡾⠋⠉⠁⠀⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⡴⣚⡄⠀⠈⣷⢰⢀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣻⠦⣼⡀⠿⢿⡀⣠⣿⡾⢿⡯⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⢀⡠⢒⣠⡟⣿⡞⢀⡾⠂⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠿⠛⠉⣇⠀⠉⢠⡞⢁⣠⣀⣤⣵⣶⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣆⠀⠀⣰⠃⠀⠀⠀⠘⠦⠶⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣦⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⡿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡁⣟⢸⠉⢹⣿⣿⠋⡿⢩⡟⢛⣿⣿⣿⣿⣿⡿⠿⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⡆
⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣿⣼⣇⣿⠉⢿⡾⡇⣾⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁⠹⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⣿
⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⠤⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⣀⣠⣴⣿⣽⣿⣿⡿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⢲⡄⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⠋⠨⡳⣝⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢠⠠⠌⠁⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⢎⣡⠶⢂⣤⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣶⢸⣶⡶⢒⡶⢮⣝⣛⠶⠆⠤⠶⣖⣒⡰⠾⠿⠶⠏⣙⣛⠻⣿⣷⣌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣰⣮⢰⢸⣿⣇⢛⣛⡿⣛⣛⣫⡭⢉⠶⣾⠿⡛⣴⣿⣿⡇⣶⣝⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣿⢰⡌⣿⣿⡎⣭⣝⣫⣭⠶⢟⣫⣵⡶⠮⠁⠈⣭⣏⢶⣜⢿⡆⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⠧⢻⡜⣿⡇⣯⡭⢩⡶⠟⠛⠉⠀⠀⠀⠀⠀⣿⣿⢇⡻⣆⢷⢹⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠱⢹⣧⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣾⠇⠈⠘⣠⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠛⢧⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠀⠀⠀⠀⠀⠀⠀⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣀⣠⣤⣀⣀⣀⣠⣀⣀⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣛⣿⣾⣮⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⣻⢼⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠿⠟⠛⣉⣉⣙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣞⣶⣿⣽⣿⣿⣿⢿⣿⡏⣾⣿⣿⡷⣟⣿⣽⣿⢵⠞⠩⠉⢷⣦⣄⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣜⡾⣛⡿⣏⢿⣽⣿⣿⣿⣿⣿⣿⡛⣿⣶⣿⢿⣟⠊⠠⣤⣀⠸⠦⣍⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡜⠁⡻⡖⣿⠳⣾⣿⣿⣿⣿⣿⣾⣿⣝⣿⣏⣿⣿⣾⡟⣿⣿⣿⡏⠙⠓⠆⠙⢦⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⡶⠏⢯⣱⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠠⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⣅⢶⣟⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣾⣿⣶⣽⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡽⡌⡀⣏⣫⣿⡯⣇⢹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡵⡱⣷⢻⣿⡷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⣿⢿⣸⣿⢟⡾⡏⡄⣾⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡿⣼⣽⡿⣿⢰⣇⣿⡿⣿⣿⣿⣯⣿⣥⣠⣭⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣿⣿⡿⣿⡆⡇⣿⣿⣿⢡⢯⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡾⢻⢰⣿⡏⡇⣿⣿⡏⡞⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣾⣾⣿⣧⠀⣻⣿⣰⣼⣿⣿⣿⣿⣿⡿⡿⣟⠁⠀⠙⢮⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡜⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠁⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⡿⣸⣿⡽⣿⣿⠈⢿⣿⣿⠏⡙⢀⣤⡀⠢⠤⠽⠃⠀⠀⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡟⣿⣿⢡⣿⣿⣿⣿⠏⣥⠈⠛⣃⣐⡤⠉⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣼⣿⠟⠛⢁⡤⠴⡄⠉⠀⠀⡀⣀⠤⢀⠰⠀⠴⠄⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣻⠟⠙⠂⠐⠁⠀⠀⠈⠄⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠐⣞⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁⠀⠀⠀⠀⠐⠀⣀⣴⡶⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢙⣑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠼⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠙⢆⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡉⠁⠀⠀⠀⡎⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⡰⢉⡜⣠⣀⣠⣤⢜⡪⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⣸⣷⣯⠿⢿⡍⣰⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⠿⠛⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠛⠉⠁⠀⠈⢋⠁⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⢀⣀⣿⠀⣹⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠒⠴⠄⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣌⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⡗⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣾⣶⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠛⣦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣚⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠋⠀⠀⠀⢠⣷⠃
⠀⠀⠀⠀⠀⠀⡠⢓⢾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠁⠀⠀⠀⠀⣠⡿⠓⠤⠁⢀⣀⣰⣿⠃
⠀⠀⠀⠀⠀⢠⠃⡡⠀⠾⠛⠛⣛⣛⣻⡟⠉⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⡿⢻⣿⠏
⠀⠀⠀⠀⢀⢗⣨⣠⣠⣴⣾⡿⠛⣿⣿⡇⠀⠀⠀⠀⠀⢠⡿⣿⣿⡿⠋⠁⠀⢹⠁⢿
⠀⠀⠀⠀⠘⠋⠹⠿⠛⠉⠀⠀⠤⠿⣾⣷⠶⠆⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⢠⠼⠤⢾⡦⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣤⣄⣴⣴⣷⣶⣴⣶⣆⣄⢠⣄⡠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⣹⣿⣿⣿⡿⠋⠁⠀⣀⣀⣠⣄⡀⠀⠉⠙⠙⣿⣿⣿⣦⠈⠉⠛⢿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢰⣿⣿⣿⡁⠀⠀⣴⣾⣿⠟⠛⣻⣿⡄⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠈⠙⠻⣷⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠸⣿⣿⣿⣤⣴⣿⣿⡏⠀⠀⠀⠀⣻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢹⡯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠀⠀⠂⠿⢿⣿⣿⣿⡿⠿⠀⠠⠐⠚⠐⢯⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⢀⣠⠜⠀⣟⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣀⣀⢀⣠⣤⣤⡝⠇⠤⣠⣶⢤⡄⡀⣠⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⠀⠀⢠⡏⠁⠀⠀⠀⣿⢨⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠻⠾⠾⠧⡬⡉⠴⠛⠛⠛⠋⢋⣻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠁
⠀⠀⠈⠓⠤⠤⠤⣄⡘⣖⡖⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⠃⠀⠀⢀⣤⣤⣽⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢯⡃⣀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣶⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣀⣤⣶⣿⠟⠋⣉⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⣴⣀⣀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠛⠋⠁⠀⣀⣤⣴⣾⠿⠟⠛⡉⠠⠶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣿⡛⠻⢷⢶⣦⣤⣤⣄⣀⣀⣀⣀⣤⣤⡤⠶⠾⠟⢛⣩⡥⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠋⠂⠒⠤⠤⠠⣌⠉⣍⢩⣥⠤⠤⠤⠒⠞⢛⠉⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⡷⢞⡁⣶⠶⡆⠀⣠⢦⣤⣴⡆⠀⢴⡖⢺⣽⢸⠍⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⣀⡙⣞⡹⣙⠇⠀⠘⠃⠛⠛⠘⠓⠀⠀⠀⠀⠄⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣏⡿⣟⡇⣿⣳

⠀⠀⠀⠀⠀⣀⣠⣀⠀⢄⡀
⠀⠀⠀⣶⣾⣥⢛⢉⠃⠀⠳⠀⠀⠀⠀⠸⢷⣶⣴⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣤⣤⣶⣦⡴
⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠐⠀⠀⠀⠠⢤⣴⣬⣭⣟⠙⠏⢝⡻⠟⡿⢻⣟⣿⡽⣿⣿⠃⠯⣿⠟⡻⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠃⠀⠀⠀⠀⠀⠭⢿⣿⣷⣿⣼⣿⣷⣾⣷⣼⣿⣿⣧⣾⣭⣾⣬⣭⣼⡿⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢺⣶⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠁⠬⠀⠀⡀⣤⣤⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⢢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠸⡿⣥⢀⣾⢧⣌⣀⣄⡁⢣⣮⣰⡔⣴⣄⣹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠐⢿⣼⣼⣿⣿⣯⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣌⠣⠭⠣⠹⠻⣿⢿⣿⣻⣿⣿⣯⣿⣷⣿⣿⣿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢻⣮⡻⢾⣮⡻⣿⣿⡻⣭⡛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣮⣿⣿⣮⡙⢿⣎⢿⣏⠻⣷⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠐⢾⣿⣿⡿⢧⡝⢿⣶⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⠿⠛⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠀⣠⡾⣻⣥⣬⣟⢷⣄⠀⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢀⣀⣟⢸⣏⣷⣾⣹⡇⣻⣀⡀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⢷⡼⣷⣽⣯⣾⢧⡿⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠻⢷⡇⢸⠾⠟⠁⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣽⣶⣿⣿⣿⣿⣿⣿⣶⣯⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠐⠠⠤⡀⠐⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠒⠀⢀⡀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢄⠈⠢⠀⠀⠉⠁⢤⠀⡄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠒⠀⠐⠘⠄⡄⠀⠀⠈⢄⠀⠀⠀⠀⠀⠠⠁⠈⠀⠀⠲⣄⢀⠄
⠀⠀⠀⠀⠀⠂⣠⠠⣄⠀⠀⠀⢁⠀⠨⠀⠀⠠⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⡀⠑⠘⠀⡀⢁
⠀⠀⠀⠀⠀⠒⠂⢀⠔⢁⠀⠀⠄⠀⠀⠐⠊⠁⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⠂⠤⠉⠅⠘⠠
⠀⠀⠀⠀⠀⢀⠴⢃⠄⣼⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢌⠀⠈⠁⠈⠀⠁
⠀⠀⠀⠀⠴⢋⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠐⠀⠑⠀⠘⡀
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠑⡀⠀⠀⠀⠀⠀⢦⠀⠂⠐⡅⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠄⠀⠀⠀⠀⠈⠢⡄⠀⠘⢄⠈⠆
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠄⠀⠀⠣⠀⠀⠰⠀⠀⠘⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠤⠐⡀⠀⢃⠀⠒⠀⢁⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠤⠀⠀⠀⢀⡀⠐⢂⣂⠀⠀⠀⠀⠀⠀⠀⠐⢄⠘⢆⠀⢀⠈⡄⠰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠠⠀⠀⠀⡘⠀⠀⡊⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⠜⢃⠰⡄⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣴⣶⠾⠻⣦⡘⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠛⠂⠰⠾⡟⠁⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⠉⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠀⠀⠑⡈⠀⢀⣙⣾⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠢⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢧⠀⠖⢻⣿⣿⣿⣿⣿⣽⣟⣻⣿⣷⣷⣾⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⠸⠛⣿⣿⣿⣿⣯⣿⣿⣇⣽⢻⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠢⠈⠁⠀⢹⡿⣿⣿⣿⣿⣿⡿⣿⣧⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡘⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⢻⣿⣿⣿⣿⡻⣿⢿⣿⣿⣻⣟⢿⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠂⠀⢸⣿⣿⣿⣿⣷⣿⣿⡿⣾⡻⣭⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣉⡛⣻⡿⣿⣿⣾⣿⣽⣿⣻⣟⢷⡿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠇⠩⠀⠊⠘⢿⣿⣿⣿⣿⣿⣾⣻⣽⢷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡓⠂⢀⠁⠈⠂⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢿⡷⣤⣤⢾⡄⠐⣧⠌⠻⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣡⣿⣏⣠⡌⠠⣹⡄⠀⡈⠻⢿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⡴⣾⠛⠙⠻⣧⣄⠒⠙⢷⠀⠈⠁⠘⠻⣌⣻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠀⠀⠀⠈⠺⡧⢀⡀⣫⣴⡿⠿⢇⠰⠠⡄⣤⡈⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠔⣲⣤⠵⠚⠛⠉⠀⠀⠀⠈⣦⣄⡀⢈⠁⠸⡄⠈⢿⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⢧⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⠿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠲⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡟⢹⡦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⠞⠋⢁⣀⣤⡇⢸⣧⣀⡈⠙⠳⣤⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⠟⠁⣠⠞⠋⠁⠀⣻⣞⠁⠈⠙⠳⣄⠈⠻⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁⠀⣤⠞⠋⠉⠉⠙⠳⣤⠀⠘⢷⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⠁⠀⡾⠁⠀⣠⠞⠳⣄⠀⠈⢷⠀⠈⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣛⣛⣻⢼⡇⠀⣀⣽⣦⣴⣏⣀⠀⢸⡇⣾⣛⣛⣛⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⠉⢻⡁⠀⢷⡞⠋⠀⠀⠀⠀⠙⢳⡾⠀⢈⡟⠉⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢷⡀⠈⠻⢦⣄⣀⣀⣠⡴⠟⠁⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⣦⡀⠙⢦⣄⡀⠀⣩⣭⠀⢀⣠⡴⠋⢀⣴⠋⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⢦⣄⡈⠉⠛⡇⢸⡟⠉⢁⣠⡴⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⣇⢸⡗⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠆⠱⠸⠊⠸⠀⠀⠃⠸⠸⠺⠀⠇⠇⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣄
⠀⠀⠀⠀⢠⣶⣏⡉⠉⠙⣿⣿⣛⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠿⠻⢿⣄⡀⠀⠈⢿⣿⣿⣿⡋⢻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣶⣿⣿⣿⣿⣿⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠿⢻⣶⣿⣯⡻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⣿⣿⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣯⣽⣿⣆⣻⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣻⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢛⣧⣤⡄⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠈⣡⣿⣿⣿⣿⣿⣿⣟⣣⡽⢿⣿⣇⠼⣿⣿⣿⡿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⢿⣿⣡⠾⣿⣟⠴⣄⠹⢿⡿⠏⠉⢀⠻⣧⡿⣷⣄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⣿⢿⣾⣿⣶⣿⣹⡿⢿⣶⣜⣷⣴⣷⠈⣷⣮⠛⣿⣿⢧⡀
⠀⠀⠀⠀⣾⠋⣿⣿⣿⣿⣿⣿⡾⣋⣽⣿⡿⢷⣼⣤⣿⡼⣯⣁⠘⣿⣿⡛⢷⡝⠿⣿⣷⣄
⠀⠀⠀⠀⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣽⣧⣿⣿⡷⣿⣿⡇⢠⣽⣦⣹⣿⣿⣄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⣴⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⡄⡿⣿⣧⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⠷⣏⡰⣿⣇⣴⣦⡹⣿⣿⣿⣿⣧⠻⣆⢹⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⣷⣿⢸⡽⠛⣷⣿⡟⠛⣹⣿⣿⣿⣿⣿⣻⣟⣸⣷⣼⣷⢿⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣏⢻⣿⣾⣧⣾⣿⣿⣽⣷⣹⣯⠛⣿⣿⣉⡟⣿⣿⣿⣿⣿⣞⡀⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠱⢿⣿⢁⣾⣿⣻⡿⢿⣿⠟⢰⣏⠘⣿⣿⣿⣿⣟⡟⢻⣿⣅⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⢿⣤⡈⠃⠀⠿⠟⢣⡿⣿⣿⣷⣟⣧⣼⣿⣿⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣏⣿⣿⣏⣿⡿⢹⣿⠀⣆⢼⠿⣿⡀⣾⠁⣿⣿⡇⣿⣿⠉⣧⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⢹⣾⢴⣿⣷⣼⢇⣿⣟⣷⡾⢿⣦⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡼⡿⣿⣿⡈⣿⣿⣿⣷⣾⣿⣏⢹⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣼⢿⣿⣿⣷⣿⣿⣿⣇⣿⣿⣟⣿⣿⣿⣿⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣛⢿⣿⣷⣿⣿⢻⣿⣿⣿⣼⣿⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣽⡿⢹⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢿⣿⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⢿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢋⣉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⡜⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠿⠉⠛⠉⠻⠿⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠰⠀⢀⣄⡀⢶⠀⠀⠛⠃⠀⠀⠙⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡴⢶⣶⡉⠈⠙⠛⠓⠀⠀⢀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡇⢸⡯⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠁⠲⣦⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣷⡈⠀⠀⠐⠂⠀⠙⠆⠀⠀⠈⣧⡀⠀⠈⠀⣤⣤⣽⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢩⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢏⣤⣀⣤⣴⣿⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⡿
⠀⠀⠀⠀⠈⢿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡄⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠐⣿⡿⣏⣻⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⢦⡀⠀⠀⣽⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠸⣷⣦⡝⠚⠽⢳⣄⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠙⣂⡀⠈⣿⣿⣯⣿⣿⠻⣿⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠆⣤⠀⠀⠀⠛⢷⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠙⡿⢳⣿⣾⣿⣷⣭⡞⢛⡗⣩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡉⠘⠁⠁⠀⠙⠋⠙⢆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠛⣿⣿⣶⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢘⣧⠀⠀⠀⠀⠀⠉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⡕⣠⣟⣀⠀⠀⠀⠀⠀⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡜⣯⣻⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⡑⠀⠩⣿⣦⣤⣴⣾⣿⠋⣴⣿⣿⣿⣿⡿⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠉⠸⠀⣴⣿⣿⣿⠿⢯⣤⡸⠌⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢾⣿⣿⣿⣷⣤⡀⠈⢟⠀⠀⣠⣾⣿⣿⣟⠻⢣⣤⠀⢰⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⣿⡿⣿⣿⣿⣷⢾⣿⣿⣿⣿⣿⣿⣶⣏⣉⣦⣉⡎⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⣿⣾⠛⠾⢴⣿⣷⣿⣿⣿⣿⣿⣿⣶⣿⣿⣉⠓⠛⠃⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⡓⢻⣿⣿⣯⣟⣦⡹⣿⣿⣿⣿⣿⣿⣿⢿⣠⣇⣀⣀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡏⠁⡊⡃⢳⣬⡗⠦⠌⠙⢿⣿⣿⣿⣿⣿⠈⢻⣿⣿⠀⢣⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠈⠛⠃⢹⡤⢽⡄⢀⡍⣹⣿⣿⡟⣻⣿⣿⣦⡙⠻⡄⣉⡅⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⣿⠟⠛⣴⣿⣶⡶⢾⠐⠟⣿⣿⣼⣷⣿⠻⣿⣶⣷⣍⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢹⠿⣾⣿⣷⣼⡀⠀⣓⣬⢹⣿⣿⣿⣿⣶⣿⠛⣿⣯⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠒⣀⣼⣿⣯⣴⡆⡈⠟⠿⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⡟⠳⡎⣿⣿⣆⣉⠉⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠿⣿⣿⣛⣿⣷⣤⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣿⣿⢿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⢿⣿⡟⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⣠⠛⣝⢛⣻⣋⣭⣭⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢆⡽⠦⠊⣾⣴⣛⣿⣿⣿⣛⣽⣿⣿⡿⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠂⠀⠀⢁⢐⣰⣿⣿⠋⠀⢱⣿⣿⣿⣿⣧⣵⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣸⠟⢋⣮⠁⠀⠀⠈⢿⠙⢙⣿⣿⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠈⠀⢀⣿⣶⣀⠀⠀⠈⠓⣶⣾⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠠⢤⡄⠀⠂⠉⢹⣿⠿⠃⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⠠⢀⣀⠠⣤⣤⡤⢶⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠂⠘⣿⠯⠉⠙⠛⢛⠿⠶⣛⣯⣿⣾⣿⣷⡿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⣹⣾⣿⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⣀⣾⣿⠿⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣉⠭⠼⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣭⣭⡟⣛⣿⣿⣯⣵⢯⠿⣯⣻⡿⠿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⣿⣿⡿⡿⠛⣁⣠⣤⣤⣀⣈⠙⠢⡧⠀⠙⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⢛⡿⢟⣠⠶⠛⣩⣿⢿⣿⣿⣿⡿⣦⠘⣦⡛⣿⣿⡻⠒⠀⠀⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣿⣷⣯⣭⣴⣞⣋⣥⣶⣿⣿⡇⠿⣿⣿⣿⣿⠉⣴⣿⣿⣿⡋⠀⠈⠷⢆⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣽⣿⣿⣿⠿⣿⣿⣯⢥⣍⣉⣙⡻⠛⠛⢿⣦⣀⣛⣫⣿⠏⣸⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣷⣿⣿⣿⣶⣾⠿⢟⣻⠿⣄⡀⠀⠉⠛⠛⠋⠀⠠⡿⠿⠛⠋⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⣼⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣽⣶⣯⣿⣿⣶⡀⣀⢀⣠⣤⣶⡿⠞⢓⠃⠀⠉⠀⠀⢈⡹⠟⠚⠛⠋⠉⠁⠀⠘
⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣟⣿⣋⡭⣖⣫⣿⣿⣿⣿⣥⣤⠴⠊⠀⠀⠀⢀⡔⠉
⠀⢀⣼⣿⣽⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠊
⢠⠿⢿⠶⠋⠘⠛⠁⠘⠛⠉⠉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠼⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢧⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣾⣿⣟⣿⣿⡿⣼⡄⠁⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠻⠇⠘⠃⠀⠀⠀⢸

⣿⣌⢿⣿⡌⠻⣯⢷⠀⠙⠌⠄⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠻⣄⠀⠀⠀⠙⣆⠻⣄⠀⡣⡀⠀⢄⢻⣧⠳⡈⢿⣧⡠⡹⡄⢌⢻⣷⡹⣷
⢿⣿⣎⠻⣧⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢢⡀⠀⠀⠀⠙⣦⡀⠀⠀⠈⢷⣽⣧⡙⣷⡄⠈⢢⡹⣷⡔⡄⢻⣷⡕⡜⢎⢢⠹⣿⣜
⡄⠻⠻⣆⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡱⡄⠀⠀⠀⠈⢷⣤⢀⠀⠈⢻⣾⣷⡜⢿⣆⠀⠱⡜⢿⣎⢆⠹⣿⣌⢮⢣⡱⡘⢿
⠙⣆⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠱⡵⡜⢆⠀⠀⡀⠀⢻⣧⣳⡄⠀⠻⣿⣿⣎⢿⣷⡀⠘⢎⠻⣧⡣⡘⢿⣦⠣⡱⡜⢌
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢆⠠⠀⠀⠀⢀⢢⡀⠙⣿⣎⢣⡀⠀⠄⢦⡹⣿⣿⣆⠀⠹⣿⣿⣧⡻⣿⣄⢈⢧⡙⣷⡕⠌⢻⣷⡑⡜⢎
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢷⡑⡀⠀⠀⠠⡻⣄⠘⢿⣷⡱⡄⠀⢪⢳⡜⢿⣿⣧⡀⠘⢿⣿⣷⡝⢿⣆⠀⠳⡌⢿⣎⢂⠹⣿⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡀⠀⠀⠀⠱⡄⠀⠀⠀⠈⢧⡀⡻⣌⢆⢢⠀⠱⡝⣧⡘⣿⣿⣜⢆⠀⠡⡻⣮⢻⣿⣷⡄⠈⢻⣿⣿⣎⠻⣷⡀⠙⢆⠻⣧⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠹⣜⢆⠀⠀⠀⠙⣆⠀⠀⡀⠀⢳⡔⢽⣦⡣⡱⡀⠘⢾⣷⡌⢿⣿⣮⣣⡠⡘⢽⣷⡹⣿⣿⣆⠀⠹⣿⣿⣧⡙⠟⣄⠈⢣⡘⢷⡄⠀⠹
⠀⢦⠀⠀⠀⠈⢄⠀⠱⡀⠘⢯⣣⡀⠠⠀⠈⢷⡠⠘⢄⠀⢻⣎⢿⣷⡕⡜⢆⠈⢻⣿⣎⢻⣿⣷⡵⡄⠈⢫⢿⡜⢿⣿⢆⠀⠈⢿⣟⢷⡄⠈⢢⠀⠱⡄⠻⣄
⠀⠀⠳⣤⠀⠀⠀⠣⡀⠙⣄⠈⢿⣷⡄⠑⡄⠈⢻⣶⡈⢣⡀⠹⣧⡻⣿⣎⢎⢣⡀⠹⣽⣧⡹⣿⣿⣞⢆⠀⠱⡹⣎⠻⣷⡀⠀⠀⠙⢦⠻⣆⠀⠡⡀⠘⢆⠈⠢
⠀⠀⠀⠹⣷⡄⢀⠀⠹⣄⠈⢧⡀⠻⣿⣆⢪⢆⠀⠹⣿⣆⠱⡄⠘⣷⡝⣿⣧⡣⡳⡄⠘⢮⢷⡘⢽⣿⣯⡣⠀⠘⠌⠆⠙⣷⡀⠀⠀⠈⠣⠙⣦⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠘⢿⣦⠢⠀⠘⢦⠀⠳⡄⠙⣿⣧⡹⣦⡀⠘⢿⣧⡘⢆⠈⢿⣎⢿⣷⡔⠘⣆⠈⢳⡹⡌⢪⢻⣷⡄⠀⠈⢂⠀⠈⢷⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠑
⡄⠀⠐⡀⠀⠈⢻⣷⡵⡈⢮⢳⡀⠙⣆⢈⢿⣷⡜⣷⡄⠈⢻⣷⡌⢣⡀⠹⣧⠻⣿⣎⠈⢧⡀⠹⣌⠀⠡⡹⣿⣄⠀⠀⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠹⡄
⢻⣆⠰⣄⢄⠀⠠⡻⣿⣞⣌⢳⡽⣄⠈⢧⡡⡻⣿⣎⢿⣆⠀⠹⣿⣆⠱⡄⠙⢷⡙⣿⣧⡀⠳⡄⠘⢦⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈
⠀⠻⣷⡘⣮⡢⠀⠘⣝⣿⣿⣦⠻⣾⣦⠀⠳⡕⡝⣿⣧⡻⣧⡀⠘⢿⣧⠘⢆⠈⢂⠈⢿⣷⡄⠙⢆⠀⠳⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠹⣿⣌⢷⣵⡀⢈⢮⢿⣿⣷⡙⣿⣷⡁⠙⣌⢎⢿⣷⡝⣷⡄⠈⢻⣇⡀⠂⠀⠡⠀⠻⡿⣄⠈⢣⡀⠑⠄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣄⠀⠐⣜⢿⣮⢻⣾⢄⠀⠳⡿⣿⣿⣌⢿⣿⣄⠈⢦⠳⡹⣿⢎⢷⡀⠀⠙⣦⠀⠀⠀⠐⠀⠙⠝⢦⠀⠑⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢌⢷
⢚⣦⠠⠈⢮⣻⣷⡹⣿⣢⠁⠙⣜⢿⣿⣦⠻⣾⢦⠀⠱⡐⡜⢷⡂⠳⡄⠀⠈⢧⡀⠀⠀⠀⠀⠈⠊⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⢦⠑⣽⣄⠈⢿
⣧⡹⣷⡐⠀⠳⣽⣿⣌⢷⡕⡀⠈⢯⢻⣿⣧⡙⢧⡣⠀⠀⠈⢊⢳⡄⠙⣆⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠈⢳⡀⠀⢳⡌⢿⣦⠈
⠘⢷⡜⢿⣄⠀⠙⣞⢿⣦⠻⡌⢄⠀⠳⡙⣿⣆⠈⢳⡐⠀⠀⠀⠡⠙⣆⠈⢧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢳⣄⢠⡀⠻⣶⡀⠹⣎⢻⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣒⠐⡲⢤⣴⠖⣢⣶⣤⣦⣴⣦⣤⢤⡶⣾⣛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢡⣁⣻⣾⣷⣴⣍⡙⣭⣟⣽⢾⣷⣼⣿⡯⢿⣿⣯⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⣤⠴⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣩⣷⢾⣿⣿⢿⣟⢛⡋⣁⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⢁⡼⢛⣾⣉⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⠿⣾⣯⠉⣲⣍⣿⣿⣿⡘⠷⠦⠦⢤⡀
⠀⠀⠀⠀⠀⢀⣾⡏⢺⡞⣩⣦⣿⣷⣿⣿⣿⣿⢿⣿⠇⣽⣿⡧⠶⠿⣋⣴⣿⣿⣿⠍⢁⣦⡴⠴⡴⠖⠠⠀⢠⡀
⠀⠀⠀⠀⠀⣜⡏⡰⢋⡿⣯⣷⣾⣹⣗⢿⡎⣛⣿⣧⢸⣿⣿⣶⣟⠛⢑⣦⣤⣿⣳⠶⡚⠁⠆⡀⢹⣿⣶⣄⡀
⠀⠀⠀⢀⣾⣾⡥⢛⣾⠷⠷⢾⡝⢧⣉⣷⡞⣿⣿⣿⣏⣿⡷⠛⠳⣗⣌⢫⣽⡟⡋⠊⠔⢄⣀⠀⠰⢦⣌⡉⠁
⠀⠀⠀⣸⠡⢆⡰⣿⢤⣟⢳⣄⣹⣶⣿⢸⣷⣿⣿⢧⢿⣟⡛⠻⡜⠛⣿⣥⡤⡖⡿⣶⣶⣦⣽⠟⠲⠦⣌⠉⠲⡀⠘⣂
⠀⠀⠀⡟⠑⣨⢶⠑⠆⢻⡦⣞⠇⠟⣿⣾⣼⣿⣿⠟⣿⣯⣿⢻⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⠟⠀⠀⠀⠀⠈⠀⠘⠂
⠀⢠⠋⡜⠋⠁⣈⡦⣼⣱⣒⢯⣕⣰⣷⣿⣿⡷⣻⢸⣿⣿⡽⠿⢿⡟⠻⣿⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤
⠀⠄⣧⠐⣀⢻⢈⡇⣿⡽⣻⢼⡏⣿⢸⡿⣿⣯⡿⣾⣿⢿⡶⣶⣷⣿⣶⠶⣾⣿⡿⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠃⠏⡜⠐⠀⠸⢁⣻⢞⢻⠸⠇⢻⠸⣧⣽⣷⡷⣿⣿⣿⣾⣿⣼⣛⣧⣤⣿⡾⢁⠂
⠀⠀⢠⢃⢃⣏⣶⣮⣻⢸⢸⡀⡇⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⠃⠈
⡇⠆⢀⣤⣾⢹⡇⠶⢛⢬⢸⢀⡟⡛⢸⢸⣿⣿⣹⣿⣿⣿⡿⣿⢷⣽⣿⣿⠿
⡟⢸⣏⡧⣿⢸⡇⡖⠿⣎⡜⢻⡁⡇⡾⣼⢸⡏⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠗
⣇⢸⠇⠷⣿⢸⡇⣷⡷⣿⠀⣼⢀⣿⣷⡏⡾⣸⣿⣿⢿⣿⣿⣮⣻⣿⣿⡟
⠙⢸⣧⢳⡏⣾⣱⠋⡄⣿⢸⣿⣺⣿⠃⢴⣇⣿⣿⣿⣿⣯⣿⣿⢿⣿⡏⡇
⢸⠸⠟⣼⣵⠋⢸⣶⣇⡇⢸⣯⣿⡿⢠⣾⣿⣿⣿⣿⣿⣟⣿⣎⣿⣿⡿
⠀⠀⡞⡏⢈⢸⣿⣿⠈⠃⣼⣿⣿⠃⡏⢾⣿⣿⣿⢟⣿⣿⣯⣿⣻⣿⠇⠀⢄⠈⢧
⠀⠘⠁⡁⣾⣇⣿⣿⠸⠀⢉⣿⡟⡼⡾⣿⣿⣿⣟⣿⡿⡟⡿⠿⡝⠎⠀⠀⠀⠙⡂
⠀⠀⠀⡿⣿⣿⣿⡇⡼⢰⣿⣿⠱⡿⣽⣿⣿⣟⣾⣧⣿⣿⣷⣙⡄⠀⠀⠀⠀⠀⠀⠰⡄⡇
⠀⠀⠀⡘⣿⢻⣿⡷⡏⢸⣿⡯⣲⣹⣿⡏⢹⣿⣿⣿⣿⠟⢳⠁⠀⢀⠠⠀⠀⠀⣦⡄⠀⠀⠠⠄⣀⣀
⠀⠀⠀⡕⣧⣿⣻⢧⡇⣿⡟⣽⢱⣿⣿⠇⠀⢻⣿⣿⠋⠀⢦⣠⠞⠲⡀⠘⠁⠘⠁
⠀⠀⠀⡵⣡⢛⢹⣿⢸⣿⡿⣧⣿⣿⠃⢨⡄⠀⢁⣁⣐⣨⠗⠀⠀⢀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⢹⢿⢸⢸⡟⣼⣿⡟⠛⠛⢁⣠⣤⣵⡀⠈⠀⠀⡀⠀⢀⣀⠁⢀⠀⠀⠀⠤⠰⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣿⡎⣾⢳⡿⠛⡴⠀⠔⠋⠀⠀⠀⠡⡀⠀⠀⠀⠀⠉⠀⠢⠶⠔⠁⠀⠀⠀⠀⠀⠀⠐⠀⡀⠶⠆
⠀⠀⠀⠀⢸⣗⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⡟⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⠩⠿⣟⢿⣿⣾⠃

⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠉⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⢀⣀⣀⣀⠻⣄
⠀⠀⠀⠀⣀⣼⢧⣀⠀⠀⠀⣴⢋⡁⠀⠉⢷⡘⣦
⠀⠀⣠⠞⠉⠀⠀⠙⢦⣀⠀⢿⠹⠟⠁⠀⣸⠃⠘⢧
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠉⠛⠾⣷⠶⠴⠞⠋⠀⠀⠈⢧
⡾⠀⠀⠀⠀⢀⣀⣀⣀⣤⠶⢀⣽⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣇⢀⣤⡞⠛⠉⠉⠉⠁⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⡾
⠘⡾⠀⠙⠦⣤⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠹⢦⡴⠚⠋⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠘⢦⡀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠙⣦⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠋⠉⠀⠀⠀⠀⠀⠉⠳⣤
⠀⠀⠀⠀⣼⠁⠀⠀⠈⠷⣤⡀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⣰⢷⡆⠀⠀⠀⠀⠈⣷⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⢰⠏⠐⣇⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠀⠀⢻⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⡏⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣧⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⣆⠀⠀⠀⠘⣧⠀⢿⡀⠀⠀⠀⠀⢠⣀⡴⠃⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⣆⠀⠀⠀⠹⣆⠀⠙⠚⠷⣄⣀⡾⠁⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢧⡀⠀⠀⣿⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠘⣏⠳⣄⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣈⠳⣝⡳⣄⠀⠈⢧⡀⠀⠀⠀⠀⠀⡿⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⣷⡈⠓⢦⣄⠻⢦⡀⠀⠀⢸⡃⠀⠀⠀⠈⠙⢳⡶⠶⠶⠞⠃⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢹⡙⠓⢿⡀⠀⠈⣧⣀⣠⣤⣴⠞⠉⢀⣀⣀⠀⠀⣦⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⡇⠀⠀⢳⡀⠀⠸⡇⠀⠀⠈⠙⠛⠛⠉⠉⠙⠛⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢉⡽⠁⠀⠀⠀⣧⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⠟⠀⠀⠀⠀⢰⠏⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠞⣡⣾⣁⠀⠀⠀⠀⣰⠏⣠⠟
⠀⠀⠀⠀⣠⣶⣋⣩⡴⠚⠛⣋⣤⢥⣤⣬⣽⣷⣦⠞⣡⣾⣅⣀
⠀⠀⠀⠀⠛⠉⠿⠟⠓⠚⠋⣩⣶⣏⣡⡴⠚⠛⣉⣤⠧⢤⣤⡽⠿⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⢁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⢀⣤⣶⣶⣾⣷⣶⣶⡄⢀⣴⣿⣿⠟⢉⣉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⢋⣴⣿⣿⠟⠁⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⢀⣿⣿⡟⠁⠀⢀⡤⠀⣰⣿⣿⠟⣡⣴⡄⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⣴⣿⠠⣾⡿⠟⠁⠀⢿⣿⣧⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠹⣿⣦⣬⣤⠞⠀⠀⣾⣿⡟⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠈⣿⣿⣧⡀⠀⠈⠙⠋⠁⠀⢀⣴⣿⣿⠃⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠈⠻⣿⣿⣶⣤⣤⣤⣤⣶⣿⣿⡟⠁⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠈⠛⠻⠿⢿⠿⠿⠿⠛⠁⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣯⡉⠱⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠃⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠘⠆⠀⢸⠠⢤⣄⡀⠀⠀⡎⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣦⡀⠀⠀⠀⠀⠀⡘⢀⠛⠉⠷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣿⠙⠳⣤⣶⠔⢈⠀⠀⠀⠀⠀⢬⡇⠀⠀⡘⠤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠈⢋⠀⠀⠀⠀⠀⠀⢀⠘⣷⠀⡁⠀⡄⢋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⡄⠀⠘⠃⠀⠀⠀⠀⠐⠀⠀⠉⠓⣧⣣⢿⠫⡋⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡒⠙⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣷⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⠀⠓⡄⠀⠀⠀⠁⠖⢀⡀⠀⡀⢿⡻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠛⠎⠀⠄⠁⢠⠀⠀⠀⠀⢨⣷⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠙⣷⢄⢀⣿⡆⠀⠐⠂⢸⣿⠆⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⠀⠀⣿⣣⣽⣿⣿⣤⣄⢩⡭⠍⠀⣭⣭⢭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠇⠦⠀⠀⠙⢾⠖⠻⣿⣿⣿⣿⡟⠳⢨⣤⣾⣖⡋⠁⠸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠐⠇⠈⢷⠈⠹⣿⠉⠠⡜⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⡷⠤⠤⠴⠶⠄⢰⣶⣷⣾⣷⣶⣶⣶⣶⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣶⣶⣿⣶⣾⣿⣿⣿⣿⣶⣦⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣼⣿⣿⠟⠛⠋⠛⠂⡏⠘⠁⠀⠉⣋⠉⢻⣿⣾⣤⣖⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⣄⣠⣿⡿⠛⠊⡷⢂⠀⠀⠀⣠⡇⠐⠰⢣⠄⠀⠀⠉⠈⠉⠛⢿⣯⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣿⣿⠟⠣⢄⠀⠀⠀⣀⠂⢀⣼⣿⣿⡅⣹⢠⠀⠀⠀⠀⠀⠀⠀⢚⢹⣧⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠛⠉⠁⠁⠀⠀⠌⠀⠀⠸⣇⣸⡿⠛⣉⠙⠻⣷⣴⣀⡶⠂⠀⢀⠀⠀⢀⠀⠃⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣠⣿⡏⣀⣾⣿⣷⠀⠙⣿⣿⡿⢇⣔⠁⠀⠀⠈⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠒⢆⠀⠈⣻⣿⣿⣀⣻⣏⣩⣿⣧⣼⠿⡻⣦⣺⠁⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣿⣿⣿⣿⣿⣯⣿⣿⠟⢣⠄⠓⠈⢻⠷⠂⠀⠀⠈⢀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡻⢿⣿⣿⣿⣿⢿⣿⣿⠃⠀⣴⣶⣄⠀⠈⠀⠀⠀⠀⠀⠀⠗⠀⣼⣶⣾⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣹⣿⣿⣿⣿⣿⣿⠻⣷⠀⢸⣿⣿⣿⠄⠠⣾⣤⣤⣴⠂⠚⢀⠀⠀⠗⠉⠙⡿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣺⣿⠉⣿⣿⣿⣿⣿⣇⣸⣿⣿⣾⣿⣅⡶⠀⠁⠀⣠⣾⡿⠛⠉⢡⠉⠉⠃⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠨⠥⠿⠛⠛⠛⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠉⠙⢿⣿⡿⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⢽⣧⠀⠈⣿⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡌⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣠⡄⢿⣟⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠱⠆⢀⣾⠇⡰⠰⠀⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⣠⣴⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⣤⣤⣨⣤⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣶⣝⣿⣿⣿⣿⣿⣿⣧⣴⣄⣀⣀⣤⡀⣀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠿⠿⣾⡿⠿⢷⣯⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣮⣿⣿⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⠉⠁⠨⠇⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣽⣖⣶⣶⣶⣶⣶⣶⣶⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⢿⣟⣛⣒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣾⣿⣿⣿⣿⣿⣻⣿⢹⣿⣿⣿⣿⣯⣿⣷⠟⠭⠹⣶⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣟⣿⡻⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣟⣿⣭⣔⣶⣤⣛⡲⣍⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⣺⣿⢯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠉⠀⠳⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⢫⢾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⡿⠗⣺⣻⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣇⣾⣿⣿⣿⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⣷⣿⣿⢻⢰⣾⡯⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⣿⣾⣾⡿⣿⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣏⣽⣿⡇⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣣⣷⣿⣷⡘⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠪⡃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣼⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣾⣿⣻⣿⡆⢻⣿⠟⢨⣠⠦⡈⠒⠘⠉⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⢰⠀⠭⠀⠟⠀⠠⠈⡀⠠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡾⠿⠄⠔⠉⠉⠡⠀⠀⠑⠈⠹⠴⠃⠀⠀⠀⠁⠂⢤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠠⠂⣀⣤⡖⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⣄⡀⠀⣠⣤⣤⣤⡀⠀⢀⣀⣀⣴⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠡⢄⠀⢙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠏⠀⣠⠤⡄⠁⠀⠙⢿⡿⢻⠟⠁⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠥⣀⣀⡼⢹⣿⣾⠧⠀⢀⢈⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⠒⠛⠋⠉⠭⠈⠉⢀⡄⠀⠀⠀⣷⢀⡔⠢⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡴⡇⡼⠀⠀⣻⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⢀⣴⡋⠀⢷⠃⠀⠀⡇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠤⠔⠒⠊⣯⣿⠟⣁⠔⠉⢸⣧⠀⠀⠀⠀⡼⠀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡸⠁⠀⠀⠀⠀⢧⠴⠚⠁⠀⠀⠸⣿⣦⣀⣠⣾⣤⡀⡿⠋⡍⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣅⡻⢉⠻⠧⢦⢠⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠻⣛⠷⣿⣿⣷⠟⢃⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠙⢦⣌⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿⡹⢰⡪⣀⡲⢤⢈⡠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣶⠸⠗⠁⢸⢻⡴⢮⣏⠑⠖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠴⠒⠾⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡸⣠⠤⢤⠼⡇⢶⡀⠆⠾⠄⡡⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣦⣄⣀⠀⠀⠀⠀⠘⣧⢱⡏⣀⣃⡜⣀⢮⠛⠀⠀⠈⣾⢱⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠲⣤⣀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠛⠿⢿⣿⣿⣿⣶⣶⣤⣤⡼⢧⣕⣤⠼⣓⠀⠀⠰⢀⡄⢔⠽⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠳⢶⣶⡿⠋⠏⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⠋⡤⢀⡉⠛⣷⣬⣤⣬⣭⣴⡭⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⠀⠀⠐⠚⠛⠛⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣧⣸⠁⢼⣿⣿⠁⡆⠀⡆⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠋⠉⠈⢦⣵⣌⣯⣀⣽⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠻⣿⣿⣿⣿⣿⣿⡟⠙⠛⠻⠷⢿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡰⠶⠶
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⡟⣡⠖⠋⠉⠙⠲⡍⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⠿⠿⠠⠇⠀⠀⠀⠀⠀⠹⠈⠿⠿⠿⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡐⣆⠀⠀⠀⠀⠀⡼⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣌⡓⠦⠤⠴⢊⣵⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠃⡀⠀⢀⣜⣈⣾⣇⣀⣀⣤⣤⣤⣬⣶⣤⣤⣤⣠⣿⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣦⣤⣠⡤⣷⣤⣤⣾⣟⣛⣿⣿⣿⣿⠿⠟⠛⠟⠛⠊⠙⠛⠛⠛⠟⠛⠛⢻⣿⣽⣧⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣾⣿⣿⠿⠛⣿⢿⠋⠉⠉⠉⠉⠙⠆⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⠗
⠀⠀⠀⠀⣴⣶⣄⠔⠀⣴⣾⣿⣿⠟⠛⠁⠫⠑⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⢀⣠⣤⣾⠿⣿⣿⣿⡷⠔
⠀⠀⠀⠈⢻⣿⣿⣿⣾⢃⣾⠟⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣅⣄⣿⣶⣿⣽⠯⠄
⠀⠀⠀⠀⠈⡷⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢒⣿⣛⣹⣿⠿⣿⣿⣿⡁
⠀⠀⠀⠠⠈⣿⣿⣿⣿⣾⣁⡤⢶⣏⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⣷⠀⣀⣾⣿⣿⡿⢣⡞⠁⠀⠀⠁
⠤⢤⣀⣀⠾⢛⡿⣿⣿⣿⣿⣀⣼⣯⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠇⠀⠀⠤⢤⠆⢈⢛⡿⣿⣿⣿⡿⠟⠋⠛⢦⡀
⠀⣀⣀⣨⣿⣿⣽⢿⣿⠿⢿⣿⣿⣈⣧⠀⠀⠀⠀⠀⡇⠀⠀⣰⠇⡄⠀⠀⣀⣶⣖⠛⣛⠟⠻⠛⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠿⠟⠻⢿⣟⣿⣿⣿⣶⣤⣴⣾⣷⣄⣤⣿⣧⣿⣶⡿⠛⣹⣿⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⠛⢛⡿⠿⠿⠟⠛⠛⢯⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣾⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠄⢀⡵⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⢸⠀⠀⢀⣤⠘⢿⣿⣿⣷⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⢋⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡄⣦⣄⣾⠸⠀⣻⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡾⠟⣠⣯⣼⣷⣆⠀⠀⣤⠀⡠⠀⠀⠀⠀⠀⠀⠃⢰⣾⣏⠃⡈⢹⣯⣠⠘⠋⠉⠉⠉⠉⠙⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣾⣿⣿⣿⣿⡟⠿⠟⢹⣯⠁⡸⠁⠀⣠⣀⣀⣤⣼⣿⣿⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣴⣾⣿⣿⣿⠉⢁⢱⣿⢆⣾⢿⣶⣷⣾⣿⢹⢯⡿⣼⣿⣿⡿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠛⢿⣿⠃⠘⠈⣵⡆⢃⣿⣿⣿⢟⣿⣯⣿⣷⣿⣿⣿⠾⠏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡟⠀⠀⠨⣿⠀⣶⣾⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠰⠈⢰⡇⠀⠀⠀⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⠁
⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⡇⠀⠀⣤⠁⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣸⠀⠀⣾⡇⣠⠀⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣸⠀⠀⡿⢇⡉⠀⢘⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⡄⠛⣆⣀⣠⣾⣿⣿⣿⣿⣿⣿⢿⡆⠙⠀⠛⠉⠀⣿⣿
⠀⠀⢀⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⣼⢫⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⣿⢏⡀⠀⠀⠀⠀⠀⢼⡧⠒
⠀⠀⠁⠸⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⣿⣾⣿⣶⠦⢠⣾⣁⣿⣷⡼⠛⠛⠓⠐⠋⠓⠛⠒
⠀⠀⠀⠒⠿⠿⠛⠛⠋⠁⢀⡐⢜⣳⣿⣷⣿⣿⣧⣴⣿⣿⢯⣛⣉⣿⣯⣾⣋⢙⣶⣶⠟⠈⠀⠠⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠿⠿⠷⠿⠶⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢤⣤⣀
⠀⠀⠀⠀⠀⢀⣾⣟⣡⣤⣀⠀⠙⢷⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣷⡀
⠀⠀⢀⣾⣿⡿⠟⠛⠛⢻⣿⡇⠀⠀⣿⡿⡅
⠀⠀⠘⡿⠉⠀⠀⠀⣰⣿⣿⠃⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⣾⣿⣿⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣽⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠉⠻⣿⣏⣉⠛⠛⢾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣿⣿⣿⣿⡇⣾⣆⡀⡀⠀⠀⠀⠘⠻⠿⣾⣾⣦⣇⢌⢛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⣿⣿⣏⣿⣿⣷⣇⣾⣰⡄⡆⢰⠀⠀⠉⠉⠙⠛⠟⠛⠛⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣸⣧⢡⢣⢠⢀⢀⢨⢢⡠⣺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣯⣧⣯⣦⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣷⣷⡝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⢿⣿⣿⣿⣿⣝⢿⣿⣿⣿⣮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⢮⡻⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣟⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣮⢻⣜⢟⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣷⡻⣷⡹⣮⢮⠫⢻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠘⢿⣿⣿⣿⠈⠘⢹⢿⡽⣿⣜⢧⡳⡕⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠃⠀⠀⢻⣿⡏⠀⠀⠈⠀⠻⣿⣿⣮⢷⡽⣎⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣻⣼⣷⡹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢷⣻⣿⣿⣞⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠶⠀⠀⠀⢸⣿⣦⣄⠀⠀⠀⠀⠀⠀⠙⣿⡎⢿⣿⡝⢿⣿⣷⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⣦⣤⣶⣿⠿⠟⣛⣓⣀⣠⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣷⠀⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⠀⠀⠋⠉⠛⠉⡟⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⡀⠀⢄⡀
⠀⢀⣀⠀⠈⠓⠦⣭⣢⡄
⠀⠀⠉⠛⠿⠷⠶⠦⠬⠍⢽⣷⣶⠤⣀⣠⣀
⠀⠈⠙⠛⠿⠿⢓⡒⠒⡂⣛⣿⡅⢠⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠐⠺⢿⣟⣛⣋⡉⠥⠛⡿⢿⡿⣿⣿⣿⣿⣿⣿⠶⡀
⠀⠀⠀⠀⠀⠛⢻⣿⡵⠖⣃⣡⢴⣮⢶⣾⡿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⢻⣾⣿⣿⢿⣿⣻⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣀⢑⡶⠦⣄⣀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣒⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣚⢲⢦⡈⠉⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣍⠓⣶⣌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣯⡛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣼⣷⠹⠷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣬⣿⣿⣈⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠻⠛⠛⠁⠛⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠿⠖⣒⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠏⢋⡠⢾⣿⣿⠏⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢁⣴⣿⣏⣆⣦⡿⡛⢁⣿⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢻⣿⡿⢿⣿⣷⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⣶⣤⣇⠰⠋⠁⡀⠐⢦
⠀⠀⠀⠀⠀⣀⠤⠒⠒⣄⢹⣿⣿⠇⠀⠰⢤⠀⣠⣧
⠀⡀⠴⡾⡏⠰⢾⣃⠀⠈⠀⡻⠛⠂⠴⠀⡰⠀⢤⢴⣆
⡞⢁⣴⣾⣿⣿⣿⡿⣧⡾⢀⣠⠂⠰⣏⠀⣯⡀⠼⢊⡅
⠡⠾⠉⠁⠀⠀⠀⠶⠆⣺⠋⣙⠉⢀⣬⠋⢤⠁⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣛⣩⡝⢛⠈⣷⣿⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⠥⢭⣥⣘⣃⠨⠝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣐⣦⣉⣁⡘⠛⠶⠉⢷⡶⣆⣋⣧⡀
⠀⠀⠀⠀⠀⠀⢠⡶⠉⢡⡍⠡⡤⢀⡟⢡⡟⠲⢖⣉⣽⣿⡏⢲⣦⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠓⠀⠋⡆⠀⠁⠠⡤⠛⠠⠀⠉⢫⣿⣏⡤⣿⡏⢿⣿⣏⣀⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠞⢾⠁⠆⡄⠀⡀⢀⠀⠀⠀⠀⣀⡛⠃⠸⠛⠠⢤⣾⣷⢭⣟⣋⣽⣿⣵⡶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠋⠸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⡹⣦⣭⣁⣬⡿⣿⢿⣿⣽⣿⣿⠿⣶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠿⢤⠰⠀⠀⠀⠀⠀⢀⠀⠀⢸⣻⡏⢙⡿⠷⠜⣧⠻⣿⣿⣟⠿⣛⠿⢋⣿⣿⣮⣝⣿⡟⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⠀⡀⠀⠀⠂⠀⠀⠈⣿⡻⢿⣿⢶⣾⣿⣧⠸⣏⠻⠃⣸⣿⣯⡙⢿⣇⠙⣿⣿⣟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⠦⠀⠀⠀⠀⠀⠁⠀⠀⢸⢷⣮⣿⢍⣝⠻⣿⣴⡟⣷⣿⣟⢿⡟⢿⣶⣥⣤⣷⣬⣏⡽⣷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣤⠀⠀⠀⠀⠀⠀⠀⠘⢠⣿⣿⣿⣿⣷⣌⠿⠛⢸⡿⣿⠻⣦⣾⡛⢿⡾⢿⢷⣍⣷⣮⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣍⡀⢀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣷⣿⡓⣸⣿⣶⡾⣿⣧⣄⣽⣿⡿⢷⣶⢿⣦⣟⠀⣹⣿⣷⡷⠀⠀⠀⠀⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡱⠞⠶⠀⠂⠀⠀⠀⠸⡄⢸⣿⢨⣿⣿⣧⣉⠻⠚⣿⣏⢻⣬⣅⣼⣿⣦⣝⠟⠙⣻⣿⣿⣷⡄⠀⠀⠀⢿⠈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠄⠀⠀⠀⠀⠀⢻⡌⢿⡆⢿⣿⠘⣿⡿⢾⣧⣝⣠⣿⣷⣄⡙⠿⠈⣻⣿⠛⣿⡏⣧⣨⡀⠀⠀⠀⠄⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡌⠲⡀⠀⠀⠀⠀⠀⠙⠒⢷⠘⣿⡷⢻⣿⡻⣿⡘⣿⣦⡙⠏⢻⡷⣶⣿⣿⢳⣎⢴⣿⡿⣇⠀⠀⠀⠂⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⢿⣐⣇⠐⡀⠀⠀⠀⠀⠀⠘⢧⣹⣷⠸⣿⠀⢿⣇⣬⢿⡟⣳⣾⠻⣦⣙⡿⣠⣿⣷⣍⣛⡟⠀⠀⣠⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣿⢁⡜⠃⠀⠀⠀⡀⠀⠀⠈⠉⣙⢧⡹⣧⠺⣿⡀⢸⣿⠙⢿⣏⠻⣿⡟⢿⣍⠓⣸⣿⣿⡄⠔⠁⠈⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣾⣿⠂⠀⡀⠀⣀⡀⠀⠀⠀⢀⡳⡽⣻⣏⣩⣇⠘⢿⣆⢻⣿⠶⣽⣧⣹⠿⠿⠛⠋⠉⠀⠊⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡜⣿⡰⠆⠀⠲⣉⠁⣉⠡⠄⢠⠄⠆⠔⢈⣻⣈⣷⣄⡻⠎⡿⠟⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢡⡘⣷⣾⣶⣈⡁⠆⠀⠠⠀⠙⢦⣷⢰⣟⢿⣭⠿⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⣀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⡽⣿⣿⣿⣿⣦⡀⢠⣎⣰⡿⢻⡷⠟⠋⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠒⢀⠑⠊⢁⠴⢟⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸⢿⣿⠿⢿⡿⢿⡁⣿⡴⡏⠀⠠⠶⠁⠀⠀⢀⠀⣀⡀⣠⡀⣄⣤⣀⣶⣿⣤⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣦⣾⠗⣌⣀⣿⠀⣷⣄⠀⠀⡀⠠⠐⢈⠙⠓⠈⢉⣷⢤⣴⣞⣿⣯⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢮⣟⣿⠃⢸⡆⣛⣩⢿⣷⣾⣿⡇⠀⣠⠴⢚⣿⣿⢿⣿⣿⣿⣌⠻⣿⡄⠹⣷⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣯⢾⢿⣿⠘⠸⣾⣿⡇⣨⢿⡝⠳⢬⣷⣶⡞⣿⣿⣿⣷⣿⣿⡝⠻⣦⠙⣿⣦⡙⢿⣷⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣻⣿⠴⣿⠉⢻⣆⢘⣅⢨⢿⡷⣯⡿⠧⣀⣘⣿⣷⢸⣿⣿⣿⣿⣏⢻⣄⠹⣷⡜⢿⣷⣮⡈⠹⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⢽⣷⠀⣨⣴⠙⡇⠘⠛⠻⠤⢷⠶⠙⠒⠛⠛⢿⣿⣎⣿⣿⣿⣤⣿⡿⣿⣷⣏⠹⣦⣍⡙⢿⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠙⣼⡹⢧⠙⣼⠷⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣳⣿⣿⣿⣰⣾⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠌⢻⡦⠙⠷⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣟⡿⣿⡿⡎⢿⣟⣿⡟⠁⠀⠻⢿⣆
⠀⠀⠀⠀⠀⢀⠃⢠⣤⠆⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣼⣿⡟⣷⠀⠹⣿⣿⣧⠀⠀⠈⠻⣇
⠀⠀⠀⠀⣠⠋⠀⢠⡀⠀⠀⠀⣠⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⡗⣻⣿⣿⡄⠀⠈⢻⣿⣷⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠔⠁⠀⠘⠗⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣿⣿⣿⣿⣼⣇⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⣤⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣿⢻⣿⣧⣿⣧⢿⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⢿⣿⣿⣿⢾⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣾⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⣻⡿⡿⠁