Hot dog Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⢀⣠⡤⠴⠶⠒⡶⠦⢽⣷⠶⠛⠋⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠉⠀⣀⡤⠖⠛⠉⠀⠀⠀⣀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⢀⣀⡤⢶⠫⠁⠒⠂⠤⢀⣤⠶⠛⢉⠌⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠀⣀⣠⠴⠚⠋⠡⠊⠀⠀⣀⣀⣀⡀⠀⠈⠐⠊⠁⣀⣠⣴⠷⠶⠶⢦⣤⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉⠁⠀⣀⣠⡴⠖⠋⠉⠀⠀⣀⣠⡇⠀⢠⠞⠉⠀⠈⠉⢓⣦⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠚⠋⠉⠁⠀⣀⣠⡤⠶⡺⠛⠉⠁⠀⠈⠉⠒⠶⡛⠉⢁⠇⠀⢠⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⣠⡴⠞⠉⠁⢀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠎⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠈⠀⣀⣤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣠⠞⠁⠀⣠⠴⠚⢉⠡⢀⡀⠀⢀⣀⣤⠶⡞⠀⠀⢀⡏⠀⠀⠀⠈⢙⣶⡶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⣿
⠀⣰⠋⢀⡴⠋⠁⠀⠀⡇⠀⠀⠈⠉⠛⠉⠂⠁⠀⠀⢀⣞⣠⡤⠶⠚⠋⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣸⠏
⠀⣿⢠⢏⣠⡤⠶⠞⠋⠙⠦⣄⣀⡀⢀⣠⡤⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⣹⡿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠟⠋⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⡾⠃
⣰⠋⢀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⣿⠐⡶⠄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⡴⠞⠋
⠙⢆⠁⠀⠀⠀⢸⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠁
⠀⠀⠙⠲⠶⠶⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠖⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣲⣶⣦⣤⣶⣶⣶⣚⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⣩⠟⠋⢅⠀⠉⠛⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⢀⣼⠉⣉⣉⡥⠇⣠⠞⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⢀⣶⠛⠒⠷⠈⡆⣠⠞⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⢀⡤⠤⠶⣏⣀⣱⠄⣿⣩⠟⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⣀⣤⠞⠓⢦⡈⢲⡤⠶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⣀⣴⣟⠛⣛⠦⠤⠤⠤⠟⣠⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢀⡤⠤⢴⣶⣛⡉⠀⠉⠳⡌⠻⡛⢛⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⢠⡾⠋⠀⠀⠀⣀⣤⠶⠚⠻⡄⠸⡖⠦⠤⣭⣉⣉⣉⣡⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⢀⡟⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⢙⣦⠙⡆⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⢸⡇⠀⢠⡏⠀⡀⡰⠀⠀⠉⠛⠻⣷⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠘⣧⠀⢸⡆⠀⠀⠁⠤⠔⢂⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠹⣦⠈⢷⡄⠀⠐⠢⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠘⢷⣄⡛⢶⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠉⠻⢶⣤⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢷⣦⣤⣀⣀⣀⣤⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢉⣩⣽⠿⢶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⣀⡤⠾⣍⣀⡒⠒⣿⡿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣋⠥⣂⣴⠈⢩⣉⣉⡘⣛⡿⠛⠁⠀⠀⠀⢜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣚⡭⠖⡫⠑⡚⠥⢉⡓⠒⡆⢠⡶⢋⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢚⣫⣭⡶⠻⠥⢶⡿⠷⢤⣍⡛⢶⢀⣬⠟⢉⠔⠁⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠖⢛⣋⠭⣵⣖⣋⣭⢴⠾⣄⡐⠲⢤⣈⣉⠐⢒⣣⠾⢋⣡⠞⠁⠀⠀⠀⠀⠀⠀⡄⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠁⢀⣴⠶⢭⣵⣒⣺⣍⠀⢤⣬⣉⡒⠄⢉⠻⢆⣰⣬⠿⢋⣡⠶⠋⠁⠀⠀⠀⠀⠀⠀⢠⣶⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⠴⠒⠾⣟⣙⢷⣄⠐⢮⣉⡲⠨⠍⠶⠈⢻⣍⣽⡷⠖⢛⣩⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢠⢰⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⡷⠛⢿⡀⠲⡀⠠⣈⣀⣠⣧⣤⡷⠼⠶⠶⠚⣛⡉⠥⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣼⣾⡿⠋
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢳⡀⢹⡶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⣦⣿⣿⠿⠋
⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⢸⠇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣸⣷⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⢻⣎⠁⠀⢀⣽⣿⣇⠄⠀⠀⡀⠀⠀⠀⠀⡀⡀⡀⢀⠀⠠⡀⡠⣔⣼⣼⣼⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢾⣿⣿⣿⣯⣞⡼⣱⣱⢸⢶⢰⣧⣇⣧⣧⣧⣷⣿⣷⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⡴⠟⢆
⠀⠀⠀⠀⠀⠀⠀⢣⠀⣏⠀⣀⣈⢯⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⣠⠔⠳⢬⡉⠉⢉⣿⠳⡄⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢰⢁⠀⡀⠀⠙⢆⢿⣥⡤⣼⣦⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠸⡄⡀⢀⠀⠀⠈⠳⡀⢠⡞⠀⠑⢄⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢳⢡⠀⠀⠀⠀⠀⠙⢎⠛⠚⠛⢻⡗⢄⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢳⣣⠀⠀⠀⠀⠀⠈⠳⣄⢰⡏⢀⣀⣱⣦⡀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡕⡀⠀⠀⠀⠀⠀⠈⠳⣍⠉⠉⣸⠇⠉⠲⣄⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡻⠶⠞⠋⣿⠙⠲⢄⡀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡸⣧⡴⠞⠛⡯⠓⠦⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡈⠷⣤⠴⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠠⠀⠀⢀⠀⢠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠛⠿⣶⣄⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣿⣿⣤⡀⠀⠀⠀⠈⠙⠿⣶⣤⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠈⠙⠿⣷⣦⣀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡉⠛⣉⣠⣤⡄⠹⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠟⠿⢿⣿⣿⣿⡀⢿⣿⣿⠿⠿⠶⣤⣀⡀⠀⠀⠉⠙⠛⠿⠿⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⡿⠁⠀⠀⠀⠀⠉⠻⢿⣷⣄⣉⣤⣴⣶⣆⠘⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡄⠻⡿⠿⠛⣉⡉⠻⣿⣿⣷⣶⣶⣦⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣦⣤⣶⣿⣿⣿⣧⡈⠻⣿⣿⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣦⣤⣤⣴⣾⣿⣷⡄⠹⠿⠿⠛⢉⣉⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠶⠦⣶⣾⣿⣿⣷⣈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠻⠿⠦⠀⠀⠀⢀⣤⠆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⣠⠾⠒⠉⠠⢒⡶⣤⣽⣦⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠊⢀⠀⢀⣠⣾⣿⣟⠉⣉⣉⣶⡬⠙⣿⠟⠁⠀⠈⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠎⠁⠀⠀⣰⣏⣁⣀⡾⠶⢶⣤⣤⣤⣶⠾⠁⡀⠀⠀⢠⣐⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⠀⠀⠀⠀⣀⣤⣾⡿⢏⠁⣨⠙⣳⠄⠈⣿⠟⠁⠀⠀⠀⠀⠰⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⠀⠀⠀⠀⣠⡞⠉⢴⠖⠒⣶⠶⢤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⣴⣷⣌⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⡀⢀⣠⣾⢿⡟⠋⠛⠶⢦⠀⢻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⠀⠁⣴⣀⠀⠶⠒⣦⣤⣉⣐⣋⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⡙⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠄⠊⠠⠴⠿⠟⠉⠉⠱⠶⣌⠑⢾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣴⣽⣿⣿⣅⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⣠⠤⠂⣀⣴⣿⣤⣀⡐⢒⡲⠦⣤⣌⣉⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣹⣿⣬⣿⣿⣷⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠄⢀⡀⠀⢀⡤⠞⠛⠟⠁⢀⣤⢹⠆⠀⢻⣿⠿⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⣦⣀⣽⣿⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⣀⠀⠔⠖⣁⣴⣾⠿⠦⠬⣍⠹⣶⣶⣶⣶⣶⡿⠛⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠐⢮⠙⣿⡟⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠒⠀⠊⠀⣀⣤⠛⢉⣭⣑⡀⠀⢀⡼⠂⣨⣿⠿⠏⠉⠀⠂⠀⠀⠄⠀⢀⠀⠀⠀⠀⠠⣌⣿⡖⡄⢫⣽⣿⣿⣿⣿⠋
⠀⠀⠀⣠⠞⠁⡄⠠⠔⢊⠀⠀⠰⠾⠟⠉⠓⣄⠀⢠⣉⣷⣶⣶⡾⠏⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀⣳⣞⣾⡌⢳⣽⢧⣾⣷⣿⡿⠁
⠀⢀⡾⠁⠀⠦⠀⢀⣦⣀⠀⢶⠛⠛⠒⠒⢶⣤⣤⣴⣿⠿⠏⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⣠⣶⣻⣿⠟⢿⡫⣝⣮⣿⡿⠋
⠀⡾⠀⠀⢨⣀⣴⡿⠟⠛⣷⡄⢢⠀⣀⣴⡾⢿⠿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⡀⣄⢠⡍⠻⣿⣿⣶⣴⣑⣿⣿⠏
⣸⠀⠀⠀⣿⠿⠋⠀⠀⢈⣸⣿⣀⣿⠟⠛⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣼⡎⢷⢆⠹⣿⣿⣿⣿⠟⠁
⡟⠀⢠⣾⠋⠀⠀⡀⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⡀⡀⠀⠀⢠⣾⣿⣸⡿⢻⣿⣾⣮⣻⣿⣿⠟⠁
⢁⠀⢸⡧⣦⣠⣾⣷⠐⣼⡿⠁⠀⠀⢀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣠⢾⡇⢾⣿⣯⣿⡇⢿⣷⣿⣿⠟⠁
⠈⢆⠸⣷⣾⣿⣿⣿⠋⣽⠓⠛⠉⠛⠁⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⢠⡰⢲⡸⡿⣿⣷⠘⣷⣹⣿⣿⣽⣿⠟⠁
⠀⠈⢂⠹⣿⣿⣿⠿⣦⣯⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⡤⣶⡄⣞⣾⡇⠳⣃⣯⢙⡝⡶⣿⣯⣽⣿⠟⠁
⠀⠀⠀⠱⣌⠛⠻⣶⣿⣿⠀⠄⠤⡀⠀⠠⣶⣶⣾⣨⡏⣿⣿⣾⠼⡿⢆⣸⢿⣿⢻⣯⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠈⠙⠶⢮⣽⣿⡅⢀⣄⡅⡄⣚⢣⣸⣿⢩⣷⣿⣼⣿⣾⡇⡜⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣵⣿⣿⣺⣿⣾⣧⣿⣽⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠔⠒⠒⠿⠋⡉⠀⣀⡆⠀⣶⣼⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣴⣤⣤⣴⡖⠒⠒⠿⠟⠛⠋⠙⠦⢤⡉⠳⣦⣤⠬⠟⠃⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⢾⣽⡷⠿⠿⢿⠋⠉⡍⣉⠉⠉⠁⠀⠈⠆⡈⠈⠑⠢⢄⡀⠁⣂⡾⠗⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠴⠛⠋⠉⠀⡔⠒⠢⢤⡀⠀⠀⠀⠙⠆⠀⠀⠉⠢⢄⡐⠀⠀⠂⢀⣐⣠⠴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣷⣀⠤⠄⠨⠳⠦⠤⠀⠀⢤⠎⠤⠤⠠⠭⠷⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡸⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠘⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠲⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣠⡴⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠶⠦⠤⠤⣤⣤⡤⠤⠤⠤⠤⠶⠶⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣴⣶⣤⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣽⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⣁⡤⠤⠶⠦⠹⡦⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠴⠚⠉⠀⣀⡤⠶⠀⢀⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⣁⡠⠔⠊⠉⠁⣀⣤⣀⡈⠁⠀⣀⡤⠾⠶⠤⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⢉⣁⣤⠴⠒⠋⠁⣠⠤⠖⠀⠘⠁⢀⣠⠼⠛⠉⠁⠀⠀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣀⠴⠚⢉⣠⠤⠖⠒⠉⠁⡐⠀⢀⠤⠤⢄⡀⠀⣀⡠⠼⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⡜⢁⡴⠚⠡⠀⠀⠠⠴⠖⠛⠁⢀⣇⣀⡤⠴⠛⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢸⣧⣣⠴⠖⠊⠓⠤⣀⣠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⣰⠋⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⣇⠲⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁
⠀⠀⠀⠘⠢⢤⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⣤⡤⠤⠖⠒⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⢀⣤⣤⡀⣤⣶⣶⣶⡄⠀⠰⣶⣶⣤⡀⣠⣤⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣥⣽⡇⢾⡇⠈⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣷⡟⠀⣻⣶⡟⣩⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⢼⣿⠜⠷⠾⠋⠀⠾⠿⠂⠀⠀⠰⠿⠶⠟⠉⠻⠶⠟⠘⢷⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⣴⣶⣶⠶⢶⣶⣶⠶⢶⣶⡶⡤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⠉⠐⣖⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠿⣷⡄⠉⠳⢤⣌⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠁⠠⣄⣀⣙⡃⠀⠀⠉⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠈⠻⢿⡬⣵⡈⢷⣦⣀⣀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡢⡀⠀⠀⠀⠀⠀⠀⠙⢾⣧⠐⣄⠙⢳⣷⣄⠀⠈⠙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣂⣀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣥⣬⣥⠶⠶⣄⠀⣄⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡻⣦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢇⣦⣼⣧⠘⣟⡦⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢅⡢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢧⣈⣉⣉⣭⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠭⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢶⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣶⣢⣄⠀⠀⠀⠀⢀⠀⠐⢸⣿⡯⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠷⠿⢽⣶⣶⡮⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⢉⣹⣿⣿⣯⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⣤⠾⠿⠾⠷⠟⢻⣿⣿⣿⠿⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⣀⣼⣿⣄⠲⣶⢶⣾⣿⣿⠟⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⣠⣾⡉⠉⢛⠛⠃⢙⣿⣿⠟⠁⠀⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠛⠉⠀⠀⠀⠀⢀⣤⣤⣤⣾⣟⣿⣿⣶⡈⣿⣟⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣿⣧⣀⠐⢬⣉⣉⣉⣉⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⢿⢿⣿⣿⣿⣷⡄⠹⣿⣻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣯⡉⠩⣙⠛⠿⠿⠶⠶⠟⢁⣼⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣿⣿⣿⣷⣿⣿⣦⡈⠻⣿⡿⣶⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠁⢰⣤⣈⣉⡙⠛⠛⠿⠿⠿⠿⠃⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⢠⡾⠋⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣟⣿⣦⡀⠻⣯⣿⣻⣿⣷⣶⣶⣶⣶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⣿⠃⠀⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⢠⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠈⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠁
⠀⢻⣷⠀⢸⣿⣾⣿⣿⣿⣟⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⢻⣷⡀⢻⣿⣿⣿⣿⣿⣻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉
⠀⠀⠀⠙⢿⣦⡙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠙⢿⣷⣭⣝⣛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣦⣤⣤⣤⣶⣶⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⣠⣶⣾⣷⣿⡽⠍⢉⡉⢙⡌⠋⠑⠖⠤⣀⣀⡀⢀⡤⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠏⠁⡠⢀⡶⠯⠿⠯⠥⡄⢀⣶⡯⢁⣴⣚⣛⠍⢀⣠⠩⠀⣘⣋⠹⡒⣦⣀⡀
⠀⠀⠀⠸⡄⠰⠀⠀⠘⠁⠀⠠⠀⠀⡀⠈⠀⠀⠏⠉⠩⢁⡞⠛⠋⣠⠮⣿⡀⡰⠁⠋⣤⣎⣷⣄
⠀⠀⠀⠀⠧⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣀⠸⠀⠀⠀⠈⠀⠀⡞⠀⠀⠀⠀⢏⠙⢨⣌⠹⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠀⠀⠀⠀⠀⠐⠒⠉⠀⢣⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠢⠤⠬⢤⣀⣀⣀⣀⣀⣀⣈⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣀⣀⣄⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀
⠈⢧⣤⠞⠋⠉⠉⠉⠉⠑⠢⣄⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⢛⠉⢙⠦⡄
⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣤⠊⡰⢋⡆⠁
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠋⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠨⠙⢦⡾
⡇⠀⠀⠀⠀⢠⠶⠄⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣀⠀⠀⣀⡀⣠⠃⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠠⢄⠀⠀⣠⡀
⢻⡀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠲⢇⠀⠀⣦⣾⡆⠀⠀⠀⡼⢠⠉⠀⠑⣾⣄⠙⣧⠤⢾⡥⠂⠉⠉⠁⠢⣄⠀⢀⠴⠛⠂⠀⢀⠀⠀⠙⠶⣤⢈⡗
⠀⢳⡀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠳⡀⠉⠉⠀⠀⢀⣰⡫⠾⣆⠀⠀⠈⢺⠂⠀⠀⠜⢱⡒⠤⢄⡀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠑⣄⠀⠈⠪⢿⣀
⠀⠀⠱⣄⢰⡃⠀⠀⢰⣴⣷⠀⠀⠀⡟⣊⣵⠶⠒⠋⢈⠀⠀⠘⢗⢄⡀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠙⠢⠞⠀⠀⢠⡾⠋⢢⡀⠀⠈⠓⠒⠠⣄
⠀⠀⠀⢈⣾⣇⠀⠀⠈⠛⠉⠀⠀⢠⠷⠶⠋⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⡹⠀⠀⢀⣠⠔⡉⠢⢄⡀⠀⠀⢀⡠⠃⠀⠀⠀⠑⠢⣀⡀⠀⠀⡸⢲
⠀⢀⡴⠋⠀⠈⠳⣄⣀⣀⣀⣠⣴⣧⡤⠤⠤⠤⠤⠤⠤⠤⣬⣦⣤⣤⣞⣁⣠⣔⣁⣀⣀⣀⣀⣀⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡞
⠀⡾⠀⢀⡤⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠒⠒⠲⠤⢤⣀⡀⢀⡠⢾⡀
⠀⣷⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢄⠀⡇
⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠇
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣓⣒⣒⣲⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣖⣒⣚⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠖⠒⠋⣉⡠⠔⠊⠁⣢⣽⠂⣈⣉⣙⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢛⡉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣀⡀⢀⠠⣤⠠⠤⠒⠊⠉⣀⠤⠴⢶⣉⣁⡀⢸⠁⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⣏⣀⣀⣀⣀⠠⠤⠤⣀⣀⣀⣀⣀⣀⡠⠀⠐⠲⠤⣴⠒⠚⢉⠇⢠⠖⠒⠤⠤⠤⠞⢀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣉⣭⣍⣉⡒⠤⢀⠀⢀⠁⢠⡶⠦⢤⣉⡙⠓⠒⠋⢀⡴⠖⠒⠦⣤⣉⡉⠉⣀⠟⣀⣠⠤⠒⠚⠉⠉⠉⠙⢦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣴⢋⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⢤⣤⠾⠀⠀⠀⠀⠉⠙⠒⠚⠋⠀⠀⢀⣀⣀⠤⠭⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⣧⣼⡆⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢻⡈⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡿⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠋⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣂⣤⣾⣿⡿⠿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠿⠟⠛⠉⠀⢀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⢿⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⠀⠀⢠⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⡾⠀⠐⠢⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⡇⠀⠀⠀⡷⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⠷⠤⠤⠞⠁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣦⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣤⣤⣤⣴⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣰⣠⣀⢀⣀
⠀⠀⠀⣰⠊⠀⠐⠃⠀⢀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢹⣿⣿⡿⠿⠟⠛⣿⣿⠀⢠⣤⡄⣀⣀⣀⣀⡻⠿⠿⣿⣿⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⣤⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠀⣀⣤⠀⠘⣿⡀⢸⣿⠇⠻⠿⢿⣿⡗⠀⣸⣿⠃⢸⣿⠁⠘⣿⡇⠀⣿⣿⣧⠀⠀⣿⡇⢀⣴⣿⠟⢻⣿⡇
⠀⠀⠀⠈⣿⡇⠀⢻⣿⣶⣾⣿⡇⢸⣿⠀⠀⢀⣿⡟⠀⣰⣿⠃⠀⢸⣿⠀⠀⣿⡇⠀⣿⣿⣿⣆⢰⣿⡇⣼⣿⠃⠀⠀⠉⠁
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⢻⡇⣸⣿⠀⠀⣾⡟⠀⢠⣿⠃⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡏⢻⣿⣼⣿⢣⣿⡇⢰⣿⣿⣿⡟
⠀⠀⠀⠀⢸⡿⠗⢲⣿⣧⠀⢸⣿⣿⡏⠀⣼⡟⠀⠀⣿⣷⣶⣶⡄⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⢿⣿⣿⠸⣿⣇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠈⠋⠉⠀⣸⣿⣿⡇⢸⣿⣶⣶⣶⡎⠉⠉⠉⠀⢸⣿⣤⣤⣭⣄⠀⣿⡇⠀⠘⣿⣿⠀⢿⣿⣄⢀⣼⣿⠃
⠀⠀⠀⠀⠀⣿⣷⣦⣤⣶⣶⣿⡿⠃⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠉⠀⠀⠀⠈⠉⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣀⣤⣄⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⢿⣧⢰⣿⡏⢻⣷⡀⣿⡟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠘⣿⡆⢻⣿⠈⣿⡇⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⣸⣿⢣⣿⠏⢰⣿⢃⣾⡟⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠃⣿⡏⢰⣿⠏⣾⡟⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⣿⡆⢿⣧⠘⣿⣆⢻⣿⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡿⠃⠈⠿⣷⡿⠏⠈⠻⣷⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠋⣉⣠⣤⠴⠶⠶⠶⠒⠲⠶⠶⠶⢤⣤⣈⡉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠁⢀⣠⠶⠛⢉⣡⣤⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⢤⣌⠙⠳⢦⣀⠀⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⠀⠀⢀⡴⠋⠁⣠⡶⠋⠁⠀⠀⢶⣴⠂⠀⠀⠻⣦⡖⠀⠀⠈⢳⣄⠀⠉⠳⣄⠀⠀⠈⠙⠿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠀⢀⣀⣀⣴⠏⠀⣠⣾⣿⠁⠀⢦⣀⡴⠋⠙⢷⣄⣠⠞⠉⠻⣦⣠⡴⠈⡏⠻⣦⡀⠈⢷⣤⣄⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣰⠟⠁⣸⠃⠀⣴⢻⡏⢿⠀⢀⡴⠛⢦⡀⠀⣠⠟⢷⣄⠀⢀⡼⠻⣦⠀⡇⠀⠈⢳⡄⠀⢻⡀⠙⢷⡀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⡟⠀⢰⡇⠘⣇⠘⣧⡀⠀⠀⠀⣹⣾⡁⠀⠀⢙⣷⣏⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠘⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⣇⠀⢸⡇⠀⠻⣄⠈⠛⢦⣀⠀⠁⠀⠙⣦⡴⠋⠀⠙⠃⠀⠀⣼⠀⠀⠀⠀⣿⠀⢀⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⢻⡀⠀⢻⡀⠀⢷⡀⠀⠉⠳⣤⡀⠉⠳⣤⠀⠐⠋⢙⣤⣀⠀⠀⠀⢠⡏⠀⠀⠀⣰⠏⠀⣸⠃⠀⣼⠃⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠙⠳⠾⢷⡀⠈⠳⣄⠀⠀⠀⠙⠷⣄⠘⣇⠀⠠⣏⠀⢸⡇⠀⣠⡟⠀⠀⣀⡼⠋⠀⣰⠿⠶⠚⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠹⣦⡀⠈⠛⢦⣄⣀⠀⢻⡄⠙⢦⣀⠉⠛⢋⣠⣴⠟⣀⣤⠞⠋⠀⣠⡾⠁⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠙⠶⣄⡀⠈⠉⠛⠚⠷⣤⣀⠉⠙⣛⣿⠟⠛⠋⠉⠀⣀⡴⠞⠁⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⣈⠉⠛⠶⠦⣤⣄⣀⣉⣙⣋⣉⣀⣤⡤⠶⠞⠛⢉⣀⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣦⣤⣤⣬⣭⣉⣉⣥⣤⣤⣤⣶⣶⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁


⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠐⠀⠀⠁⠈⠀⠁⠉
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠻⣿⡟⢻⣿⡛⣻⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⣄⠀⢠⡄⠀⣴⠄⠠⠦
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⡿⢿⠈⡏⣰⣷⡄⢻⣿⠈⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⢀⣈⣿⢰⡏⠉⢻⡆⠀⣿⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣶⣾⠀⣧⡘⢿⢃⣼⣿⣄⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠉⠉⣿⠘⢧⣄⡼⠃⠀⠿⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⠟⢿⢛⡛⠟⢻⢛⣝⣿⢩⢽⣿⣿⣿⣿⠀⠀⠀⠀⠀⢘⠀⠀⠀⢈⣠⡀⣬⢤⣠⣄⡴⠶⠄⣖⣒⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣩⣼⠨⣅⢻⢸⣘⣟⣿⣘⣛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠶⠋⣷⠺⣝⡏⠧⠠⠆⠧⠤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⡿⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣠⣿⣧⣼⣿⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠄⠀⠀⠐⠟⠀⠘⠃⠀⠙⠁⠀⠉⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⢀⢀⠀⠄⠠⠠⠀⠀⠀⠐⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⠀⢼⣿⡏⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠈⣻⣿⡆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⣸⣿⡟⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠘⢿⣿⡄⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⢰⣿⡿⠁⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣄
⠀⠀⠀⢀⣾⣿⠏⠁⠀⢀⣠⣴⣶⣶⡆⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⢀⣿⣿⠃⠀⢀⣴⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⣸⣿⡇⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⣿⣿⠁⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣶⣶⣶⣿⡿⠛⠙⠻⣿⣧
⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⣿⣿
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠿⠿⠿⠿⠿⣿⣷⣄⠀⠀⠀⣻⣿⡆
⣠⣤⣤⣤⣤⣽⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⣤⣴⣿⣿⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⢹⣿⣇⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣰⣿
⠀⢻⣿⣦⡀⠙⠻⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⠿⠋⢀⣴⣿⡟
⠀⠀⠙⢿⣿⣦⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⢀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠄⠐⠦⣀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠰⠀⠰⢆⡀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠁⠀⠀⢀⠀⠀⠀⠘⠃⠈⠉⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠈⠙⠳⡄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠄⢠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠉⠦⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⠴⠃
⠀⠀⠀⢀⡔⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠈⠧⣤⣀⣱⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢦⣀⠀⠀⠀⠀⣠⠶⠒⣦
⣰⠒⠒⣦⡀⠀⠀⠉⠃⢀⣤⠶⠛⠉⠉⠁⠀⠀⠉⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⣠⠴⢊⠩⠷⣄⠀⠀⠷⠶⠖⠋
⠙⠶⠚⠉⠁⠀⠀⢀⠖⠋⢀⡠⠤⠒⠒⠒⠒⠒⠂⠤⢄⣀⡀⠈⠉⠓⠲⠶⠦⠴⠶⠒⠛⢉⣁⠠⠐⠊⠁⢰⠊⠁⡠⢖⡩⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠋⡠⠊⠁⢀⡠⠤⠄⠒⠒⠒⠒⠂⠤⠤⣈⠉⠓⠒⠒⠒⠒⠒⠒⡋⢉⣤⠤⠖⠊⠉⢀⣡⣴⡿⢊⣥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢧⣤⠞⠁⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠈⠳⠤⠚⠁⠉⠉⠀⠀⡀⠀⡠⠤⠆⠉⠉⠩⠟⠁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⡠⠗⠀⠀⠀⢁⣀⣀⢉⣉⣁⣒⣒⣒⣂⠀⢉⣁⡠⠴⠒⠒⠋⠁⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠊⠁⠀⣀⠤⢒⣩⠵⠶⠒⠒⠛⠓⠶⠦⢤⣤⣀⣀⣀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣊⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣀⠀⠉⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠀⣤⣶⣶⣿⣟⣶⡀⣰⣿⣿⣿⣶⣦⣤⡀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⢁⣤⣤⣤⡈⢿⣿⣯⣿⡿⣿⣧⣿⣿⣿⢿⣿⣯⣿⠇⣠⣤⣀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣠⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣷⣦⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⢰⣿⣿⣮⣿⣿⣿⣿⣧⠻⣿⣿⣿⣿⢸⣯⣿⣿⣿⢣⣾⣿⡿⣿⣿⣿⣿⣧⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠈⠿⣿⣿⣿⣟⣿⣿⣿⣧⡹⣿⣿⣿⢸⣿⣿⣿⢣⣿⣾⢿⣿⣿⣿⣽⡿⢋⣀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣶⣿⣿⣶⣤⣙⠻⢿⣿⣽⣿⣿⣷⡜⣿⣿⢸⣿⣿⣣⣿⣿⣿⡿⢟⣻⣭⣶⣿⣿⣿⡇⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣸⣿⣛⣿⣷⣿⡿⢿⣷⣮⣭⣟⣿⣿⣿⣎⠿⠟⠿⢳⣿⠿⣻⣵⣾⣿⣻⣿⣿⣿⣿⣿⣷⡀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠀⠀⠀⠈⢽⣿⣿⣿⣻⣿⣿⢿⣷⣿⣿⣿⣿⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢻⣿⣿⡿⢿⣿⣿⣷⣿⣿⡿⠾⢛⣩⣥⣶⢀⣀⢀⣀⢶⣿⣿⣯⣭⣟⣛⣿⣿⣿⣿⣿⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠛⠿⠿⠿⠿⣛⣋⣭⣤⣶⣿⣿⣿⣿⢣⣿⣿⡿⣿⣷⡻⣿⣿⣿⢿⣯⣿⣿⣿⣯⣭⡉⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣯⣾⠃⣾⣿⣿⣿⢿⣿⣿⣮⡻⣿⣿⣿⣿⣿⢿⣿⣿⣿⠄⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⡾⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⠏⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⢿⣿⣿⣻⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⡿⠋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⠻⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣾⣿⡾⣿⣿⣿⣿⣿⣿⣷⡈⠉⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠙⠛⠛⠁⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡿⣣⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⠀⠘⠿⣿⣿⣿⣾⣿⣿⣿⡿⠃⠈⠛⠛⠛⣉⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣌⣉⣉⣉⣉⣀⣀⣠⣤⣴⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠐⡐⢠⠀⣀⠀⠈⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠄⠀⡀⠐⠆⠀⠀⣄⠀⣠⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠁⢀⠀⠀⠀⠆⠀⠄⠈⠃⠈⠀⡄⠀⠈⡄⠀⢀⡄⠈⠀⠒⠰⠃⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠠⠐⠀⠀⣀⠐⠈⠀⡀⠐⠄⠀⠀⠈⠀⡀⠄⠃⠀⡀⠠⠆⢢⠄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠆⠀⠀⠘⠈⠀⠀⠀⠈⡀⠀⣀⠀⢈⠈⡇⠙⢁⠀⠁⠈⡡⡯⡻⢀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠰⠀⠂⠂⠄⢀⢌⠁⠀⠀⡀⣀⣰⢨⣄⠀⣅⠊⠀⠃⠀⣂⣀
⠀⠀⠀⡀⢀⠄⠠⠘⢀⠀⠀⢐⠂⡠⠌⢠⡶⠐⣐⡃⡄⠁⠠⠃⠄⠊⡄⠀⢍⡀⡄⡏⠿⡄⠤⣵⢌⢡⠆
⠀⠀⠀⠀⠀⠃⢠⠑⠈⠀⠰⢤⠀⠄⡀⠀⠁⠠⠈⣅⠀⣬⠠⢏⠌⠀⣻⡀⢘⢿⡍⠁⡌⠃⠃⡀⢠⡃
⠀⠀⠈⠋⠀⢠⠀⠄⠀⢠⠤⡐⠘⢠⡉⣀⣤⠳⠗⠂⣡⠉⠐⡄⣯⠈⡠⢶⢤⠈⣛⠛⡶⠀⢹⠿⠚⢁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⡈⠁⠀⠀⠀⠀⠀⠀⡞⠠⡊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢽⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡄⠀⠀⠀⠀⠀⢸⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⡆⠀⠀⢀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢃⠀⠀⠸⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⡆⠀⠀⠇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⡴⠀⠀⠀⠀⡇⠀⣼⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢤⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⡤⠀⠄⡀⠠⠀⠀⠀⠘⠀⢸⣆⠄⢼⠀⠀⣀⠄⠂⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢈⠁⡄⠀⣄⠀⠈⡢⣀⡀⠀⢀⠆⠀⠀⠀⢀⠱⡊⠠⠒⢁⠀⠱⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠
⠀⠀⠀⠀⠀⢠⡕⠀⠀⡇⠀⠘⢄⠀⡈⠪⣻⡦⠉⠀⠀⠀⢰⠀⡄⢇⡀⠈⠁⠔⠀⣱⠔⠂⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⠁
⠀⠀⠀⠀⠀⡞⠈⡓⠦⡘⡄⠀⠈⠳⡈⠂⠀⠙⣇⠀⠀⡀⠀⠀⡀⢸⠂⠉⠐⢂⡠⣤⠀⠈⠓⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⠬⠒⠤⠉⠒⢤⣀⡈⠢⣄⠀⠸⣆⠀⠡⡀⠀⠁⣼⠋⠩⠭⠅⠉⢀⡑⢦⣠⠔⠉⠀⠀⠈⢢⡀⠀⠀⠀⣀⠄⠀⢀⠜
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⢨⠃⠀⠱⠾⢋⡳⠢⢼⠶⢞⠡⠔⣒⣉⠩⠠⡨⢉⠓⠨⡀⢄⠀⠠⠖⠀⠧⣀⠤⠊⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⠃⠀⡞⠀⠀⠀⠀⠀⠀⢹⠃⠒⠴⠲⠅⢀⣀⢁⡀⡁⣁⡀⠁⠜⢦⡙⠄⣀⡈⢂⡡⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠰⡉⠉⠢⣄⡎⠀⢰⠁⠀⠀⠀⠀⠀⠀⡆⠀⢀⠇⠀⠀⢀⡀⠈⠉⠉⢨⣞⠁⡤⠂⠂⠒⠁⠰⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠈⠳⢬⠁⠀⡧⣀⡀⢀⣀⣀⣀⣸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⢠⠃⠈⠉⠉⠉⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠫⡑⢾⠀⢸⢦⣀⣀⣀⣀⣀⢀⡏⠀⢠⣧⡀⠀⠀⠀⠀⠀⣀⣠⠃⠀⡜⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⡞⠠⢀⣀⣀⣀⣀⢸⢁⠀⣼⣀⠈⠑⠒⠂⠒⠉⠀⣼⡀⢠⠃⠒⢄⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⡟⡌⢠⣇⠈⠙⠒⠶⠶⠖⠒⢐⢇⠁⡼⢢⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⡀⢰⠇⠁⢸⠀⠉⠓⠶⠶⠖⠒⠀⣸⣠⠴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠚⠶⠶⠶⠶⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠊⠀⠀⠀⠀⠀⠀⢰⢡⣄⠸⡇⠀⢇⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⢤⡀⠀⠀⠀⠀⠀⡇⢯⡔⢘⡇⠀⠀⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠆⠀⠀⠀⠈⢆⠱⠀⠀⠀⠀⠀⢸⢠⣬⠞⠀⢀⣴⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡡⠂⠀⣤⠀⡠⣣⠆⠘⠃⠀⠀⠀⡎⢼⠁⠀⠀⡇⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡑⡄⣤⠀⢸⢱⠁⠀⣤⠀⠙⠀⢸⢨⡽⠀⠀⠀⢱⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⡇⣼⣇⣀⣀⣀⣈⡋
⠀⠀⠀⠀⡠⠔⣚⡭⡵⢶⠶⢶⠳⡳⣚⡛⡚⡛⣚⠉⠉⢹⢱⡟⣒⣶⢒⠒⠒⠭⢭⡁⠒⢄
⠀⣀⠴⡺⣄⡀⠳⢼⣜⣺⠾⠷⠓⠛⠛⣻⠛⠻⠟⡻⣟⠻⠰⠟⠍⠛⢹⡏⠛⣓⣤⠼⠀⣸⠄
⡞⠁⡾⠛⣟⢝⡲⢴⣤⣨⣩⣉⣉⡉⣉⣉⡉⡉⡉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣭⣤⠦⠾⣚⣽⠀⠀⣰⢒⡆
⠹⣶⡙⢤⡟⠭⡂⠀⠙⠋⠊⠉⠉⠻⢿⣛⣟⣉⢉⣙⡋⠉⠛⠛⠛⠒⣒⣒⣒⣋⢭⣭⣷⡟⣳⢞⢡⡿⠃
⠀⠈⠛⠓⣿⡷⣢⠀⠀⠀⠀⠀⠀⠨⠭⠀⣐⣒⣒⡒⠂⠀⠀⠀⠀⠀⢐⣒⣒⡦⣭⣰⡾⡋⣅⠾⠋
⠀⠀⠀⠀⣗⠬⣒⠀⠀⠀⠀⠀⠀⢈⡻⠥⠶⠲⠄⠒⠂⠀⠀⠀⠀⠀⢀⣒⣾⠋⣬⣠⣴⡟⠁
⠀⠀⠀⠀⢳⡭⡒⠅⠀⠀⠀⠀⠀⠀⠨⠭⠄⣐⣒⣒⠂⠀⠀⠀⠀⠀⢐⣒⣛⣿⣯⣒⢽⡇
⠀⠀⠀⠀⢸⠭⣒⠅⠀⠀⠀⠀⠀⠠⠭⠍⢂⣒⣒⠶⠆⠀⠀⠀⠀⠀⢀⣛⡒⠭⠝⣊⢽⠁
⠀⠀⠀⠀⢸⡪⢐⠅⠀⠀⠀⠀⠀⠨⠭⣒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⢐⣒⡀⠭⢕⣊⢽
⠀⠀⠀⠀⢸⡫⢖⠭⠀⠀⠀⠀⠀⠀⡯⠴⢒⣂⣀⣐⠂⠀⠀⠀⠀⠀⢒⣒⣒⠩⠕⣒⣽
⠀⠀⠀⠀⠘⣟⢕⠭⠀⠀⠀⠀⠀⠀⣪⠭⠒⣒⣒⣒⠀⠀⠀⠀⠀⠀⠂⠒⣒⡂⠭⣂⣿⠶⢶⣀⣀⣀
⠀⠀⠀⢠⢾⣟⢵⡭⡀⠀⠀⠀⠀⠀⡨⢭⣐⣒⣂⣀⠂⠀⠀⠀⠀⠀⠭⠅⣐⠢⢭⠺⣗⣒⠒⢒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠸⡝⢭⠷⢯⣒⠀⠀⠀⠀⠀⣻⣭⣉⣉⣂⣈⠀⠀⠀⠀⢀⣀⣵⣶⣶⡿⢷⡿⠳⣏⢷⡀⠀⣀⣈⣉⣉⣉⣉⣉⣁⣐
⠀⠀⠀⠀⠈⠒⠭⣒⠤⣍⡓⠲⠦⠤⣤⣍⣉⣉⣉⠈⠀⠉⠉⣉⣟⣛⣫⡭⠷⠮⠭⣭⡭⢯⡽⠉⠩⠍⠉⠭⠉⠉⠭⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠭⣝⣒⡒⠲⠶⠶⠿⣿⣟⣻⣿⣟⣛⣛⡧⠶⠶⠚⢛⣯⠭⠿⠭⠭⠭⠥⠬⠭⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠞⠀⠶⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠈⢣⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠰⠀⠀⠇⠀⠠⠘⠒⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡔⠀⠀⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠀⢀⡄⢚⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠀⠀⣼⠊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡇⠈⡀⢉⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠐⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠦⠐⢀
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡁⠀⠁⠘⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⡀⠀⠆⡀⠆⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡴⢀⡀⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡤⢀⣀⡀⣀⣈⡄⠰⠆⠀⠀⠓⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⢁⠈⠁⢃⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠄⠠⠀⠘⠄⠘⡆⠀⠀⢀⡀⠀⠶⠀⠀⣑⡤⠀⠤⠄⡀⠀⠀⠀⢀⠠⠜⣶⣠⠀⠠⠆⠦⠠⡖⡠⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠀⠐⠀⠒⡀⡇⢀⠀⢀⡀⠀⢀⡤⠊⠀⡤⠀⠄⠀⡈⠑⢄⡰⡫⠊⢹⠛⠲⣎⠱⣞⢀⡊⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣀⠉⣄⢡⡽⢌⡁⣀⡠⠔⠉⡅⠚⡀⡠⠀⠆⢠⡄⠀⣼⠭⣀⠁⢀⡡⣄⠀⠓⣌⣏⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠄⠆⠺⡤⠧⠩⠄⠚⡀⠴⠈⡁⠠⠠⠄⣄⠀⠘⠉⡐⠊⢠⠌⠐⠮⠕⠢⣈⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠚⣇⣹⣆⣂⢙⡀⠴⠘⣃⠥⠔⠒⠰⢀⡈⠀⠄⢀⣀⢉⠜⢁⢂⣜⡵⠋⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⠱⣿⣶⠄⠲⠀⢃⢠⠴⠚⠓⠃⠦⢀⠚⠀⠖⡰⢊⢬⡥⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢯⣃⠩⡄⠆⢋⣽⣍⡚⠠⠂⣄⣉⠘⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠺⠼⠿⠾⠄⠡⠶⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡒⠓⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢸⡴⠒⢲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡜⢀⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡴⢹⠇⢀⡞⠀⢠⠇⣰⠉⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡜⠀⡘⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⢱⡇⡀⠺⠤⠤⠋⠀⠓⠚⣁⡤⠤⠤⢴⢀⡞⠋⠉⠉⡉⢰⠁⡸⠁⡌⢉⠇⡰⠋⠃⠒⠲⡄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠰⡇⠀⠻⣞⡆⠀⠀⡰⠉⡲⠀⠧⣤⠀⢀⠎⢀⡽⠋⡠⠞⢡⠃⢰⠃⡸⠀⡜⢰⠁⡰⢺⠃⣸⢁⠴⠋⣀⣀⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠙⠦⣤⡃⠀⠀⢰⠃⢠⠃⢠⠞⢁⠔⠁⡰⠋⠀⠾⢰⠀⡞⢠⠇⢠⠃⡸⢀⠃⣰⢃⡏⢠⠇⡏⠀⣞⡡⠦⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠹⡄⢠⠏⢠⠏⡰⠁⠀⢉⡹⠐⠧⠀⠒⠒⠋⠀⠉⠉⣀⣀⣉⣀⣀⣉⠁⠈⠙⠊⠀⠣⣤⣀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⠀⠀⡇⢼⣀⠞⠀⠗⠒⠋⠉⠀⠀⠀⠀⠀⠀⢠⣎⣉⣉⣭⣄⣤⣤⣤⣍⣉⣉⣙⠓⠒⠤⠤⠃⣰⠁
⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⢠⠟⢤⣸⠁⠀⣴⣶⣶⣶⣦⠀⣶⣶⣶⣶⣶⢰⣶⣶⣿⣶⠄⠀⢀⣶⣶⣦⠀⠀⢠⣾⣿⠙⣲⣶⡖⣡⣶⣶⣶⣦⡄
⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⣠⠏⠀⣸⡧⡄⢰⣿⣿⠋⣿⣿⠀⠛⣻⣿⡟⠋⢸⣿⣿⠛⠛⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣰⣿⡿⠀⣿⣿⡟⢻⣿⡇
⠀⠀⠀⠀⣟⠀⠈⠧⠤⠚⠁⢀⡴⠃⡷⠇⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⢸⣿⣿⣿⡿⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠘⠲⢤⣀⣀⡠⠔⠋⠀⡞⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⢻⣿⣇⠀⢸⣿⣿⣿⣇⠀⠀⠻⣿⣷⣄
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣽⣿⡿⣿⣤⣤⡿⡿⣧⣤⣼⠿⠿⠿⠿⣦⣴⡿⣿⣼⣿⣿⣤⣼⣿⢿⣿⣿⣤⣤⣤⣬⣿⣿⣿⣦
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⡇⠉⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⡏⠁⠘⠛⠈⠉⣿⣿⠀⢸⣇⠉⢻⣿⣿⣿⣿⡯⠛⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢱⣿⣿⣿⠛⢻⣿⡇⠀⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⣴⣶⡄⠀⢻⣿⠀⢸⣿⡀⠘⣿⡟⠛⣿⡇⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣮⢻⣿⣿⡀⠈⠉⠀⢀⣿⣿⡇⠀⣿⣿⣿⠀⠈⠉⠉⣿⠀⢰⣿⣿⣿⠀⢸⣿⠀⢸⣿⣇⠀⢻⡇⠀⠉⠁⠀⣼
⣿⣿⣿⡿⠿⠿⠛⡟⢻⣿⠿⠿⠿⠟⠘⢛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⣿⣿⣿⣿⣿
⣿⣿⣿⢁⠒⠗⢨⣥⣾⣶⣿⣿⡿⣇⣙⡋⢭⣿⣿⣿⣿⣶⣶⢦⣼⣕⡂⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⡿⠣⢿⡟⣸⣿⣿⣷⡿⢿⡏⠜⣽⣿⣿⣿⣿
⣻⣿⡧⣄⠱⠄⣹⢿⡏⣛⠛⣂⣭⣶⣶⢜⠻⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡧⡪⣳⣿⣿⣿⡿⢛⣉⡉⢀⣶⡏⠸⠉⡿⢱⠏⠼⣋⣴⣌⣵⣿⣿⣿⣿
⣼⡟⢘⡕⣪⣧⡼⢤⡑⢀⣈⣩⣭⣭⣭⣼⣿⣿⢿⣛⣙⠀⣽⣿⣿⠛⠛⠺⡴⠁⢻⣿⣿⣿⣿⡿⢡⣿⣿⣀⣾⣤⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⢿⣿
⣿⣷⡘⣻⠞⣱⢱⢇⡻⣿⣿⡟⣿⠏⠛⢛⣋⣙⣉⣢⣩⣤⣾⠿⠄⠄⠐⠊⠀⢀⣿⣿⣿⣿⡿⣷⣾⣿⢟⣭⠛⣿⣿⣿⣿⠿⡿⣫⡭⢋⡼⢁⣾⣿
⣿⣿⣿⣦⣙⠾⠈⠺⢇⣉⣉⠉⠉⠉⠉⠉⠉⠘⠛⢛⣻⣿⠷⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⠣⣶⣾⣿⣇⣼⡏⣰⡏⡼⠱⣋⣸⣤⣭⣾⠶⠂⣘⣻⣿
⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⢁⣾⠍⣽⣿⢟⡋⣰⣿⣷⣿⣿⣿⣿⣿⣟⣰⣫⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣵⣾⣿⡧⢞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠒⠛⠒⠶⠶⣤⣀
⣠⠴⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣤⣶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⣿⠦⣤⣀⣀⣀⣠⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⠀⢀⣺⣿⡦⢾⡇⣿
⢿⠀⣀⡈⠉⠉⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠋⠉⠁⠀⠀⣾⠶⠿⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠞
⠸⡇⠈⠉⠙⠓⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠛⠋⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠙⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠛⠛⠛⠓⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠛⢛⡛⣛⡛⠛⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⢛⣩⣭⣯⣿⠿⣿⣭⡉⠉⠉⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⡭⠥⢬⡀⠁⣸⠀⠐⠂⠀⠀⠒⡂⢹⡖⠀⠀⢠⠖⠒⠻⢿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⠞⠉⠀⠀⠀⣿⠀⢸⠀⠀⠀⠂⢀⠀⠀⢸⡇⣆⠀⣿⠀⠀⠀⠀⠉⠳⣌⠳⣄
⠀⠀⠀⠀⠀⣠⠏⣠⠞⠁⠀⠀⠀⠀⢀⡿⠀⡿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠀⣛⠀⠀⠀⠀⠀⠀⠈⠳⡌⢷⡀
⠀⠀⠀⠀⣰⠃⣴⠷⠶⠶⠶⠶⠶⠶⣏⠀⠀⡿⠒⠒⠒⠒⠒⠶⠟⠳⢾⠀⠐⣿⠀⢀⠀⠐⠄⠀⠀⠀⠙⣆⢳⡄
⠀⠀⠀⢰⠇⣸⠃⠀⠀⠀⠀⢀⠀⠀⠈⢷⠀⡏⠀⠀⠀⡀⠀⠁⠨⠅⠈⡇⠀⣿⠀⠁⠀⠀⠀⠄⣀⠠⠡⢹⡄⢷
⠀⠀⠀⡾⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢼⡇⠸⡆
⠀⠀⢀⡇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣒⣂⣼⠀⣇⣐⣠⣤⣤⣤⣀⣀⣀⣠⠇⠀⢹⣄⣀⣀⣀⣀⣀⣀⣀⣊⡼⢻⡄⢳⡀
⠀⠰⢯⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⡭⠭⠥⠤⠬⠭⠭⠭⠭⣭⣭⡭⠭⠥⣤⣤⣤⠭⢭⣭⢭⡭⡭⠭⢭⠍⠠⠤⠥⣤⠷⢀
⣎⣲⠼⡀⠀⠛⠛⠛⠛⠛⠛⠲⠶⣄⠀⠀⣶⡇⠀⠐⡟⠛⠋⠉⠛⠻⢦⠘⠋⠉⢹⠋⠉⠉⢉⠛⠀⠀⠀⠀⢀⣆⢸⠲⣉
⠈⠁⠀⣇⠘⣇⠀⠀⠀⠀⠀⢸⠀⣽⠀⢸⡟⡇⠀⠀⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣸⣇⡏
⠀⠀⠀⠘⡄⢻⡀⠀⠀⠀⠄⠀⠀⣿⠀⣼⠁⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢹⠀⠀⠀⠀⢠⡿⣼⠁
⠀⠀⠀⠀⢻⡀⢳⣀⣈⣀⡀⠀⣀⡟⢠⡏⠀⣻⠀⢀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⡇⠀⠠⠸⡄⠀⠀⢠⡿⣱⠇
⠀⠀⠀⠀⠀⢳⡄⢻⡉⠉⠉⠉⠉⠀⡼⠛⠋⢻⡀⢸⡷⢶⣖⠶⡶⠒⠋⢀⠀⠀⠀⡇⠀⠀⠀⠘⠶⢶⡞⣱⠏
⠀⠀⠀⠀⠀⠀⠻⣄⠱⣄⠈⠀⠀⢰⠇⠀⠀⠸⡇⢸⡇⠁⠙⢷⡁⠐⠂⠡⠀⡀⠀⡇⠀⠀⠀⢀⡴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣜⢦⠀⠀⡟⠀⠀⠀⠀⣷⢸⡇⠀⠃⠀⠹⡄⠀⠀⠀⠀⠀⡇⠀⣤⠔⢋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣘⠃⠀⠀⠀⠀⠹⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠁⣠⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠗⠲⢦⡤⠤⠌⣁⣀⣀⣠⣠⠤⠤⣤⠶⠒⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅⣀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣴⣿⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣷⣶⣿⣯⣽⣛⣛⣓⣖⣲⢤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⣤⠖⠈⠉⠉⠉⠉⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡞⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⠏⠀⠀⢸⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⢄⡀⠀⢀⣠⣀⣀⣀⠞⣱⠋⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⣤⠀⠀⣠⣾⡿⠟⠛⠛⢑⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⡡⠂⢀⣤⢨⣬⠀⣸⣿⣫⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢿⢷⠁⠀⢾⣿⠸⣷⣾⡿⠛⠉⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⡏⡇⠀⠀⠀⠈⠻⢿⡛⠳⠀⢀⠀⠀⣿⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢧⡇⠀⠀⠀⠀⠀⠘⣿⣄⣤⠛⠁⣼⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡰⠷⠀⠀⠀⠀⠀⠰⠞⠿⠃⠀⢀⣿⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠘⣥⠞⠀⠀⠀⣿⡄⠀⠀⠈⠉⢉⣉⡙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠲⠾⠟⠃⠰⣻⠁⠀⠀⢀⣐⡻⣧⢀⡤⣲⠽⠋⠉⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣥⣤⢀⡴⠀⢀⣀⣀⣀⣿⣤⣶⣿⠿⢿⠿⣿⠷⡟⣠⡄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⡀⠀⠀⢀⣹⣿⣿⣻⣿⡏⠉⠙⣿⠋⢹⠧⢂⣴⢿⣿⡴⣟⠼⣝⢦⡀⠀⠀⠀⢠⣆⢹⡄
⠀⠀⠀⠀⠀⢠⣲⢏⣴⡟⠀⠀⠀⠀⠙⡾⠟⠋⠽⠁⠀⠀⣿⠀⣞⣼⡿⢃⡾⠁⢱⣝⠀⠈⠙⠳⠀⠀⠀⡠⣿⢸⡇
⠀⠀⠀⠀⢀⣿⠃⢸⡿⠀⠀⠀⠀⠀⠀⠤⣤⠤⡞⠀⠈⠀⢹⡀⣿⣿⠀⣸⠁⠀⠈⡟⡄⠀⠀⠀⠀⠀⢀⣷⣯⣾⠇
⠀⠀⠀⠀⢸⡏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⣿⢸⡇⢻⣿⡘⠁⠀⠀⠀⢳⡇⠀⠀⠀⠀⢈⣾⡿⣹⡟
⠀⠀⠀⠀⢸⣷⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣧⠀⠛⠃⠀⠀⠀⠀⠀⠃⠀⣠⣶⢫⣾⢏⣴⡟
⠀⠀⠀⠀⠈⣿⣏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡆⠛⠉⠀⠀⠀⠈⠂⠠⠤⣴⡶⣿⣿⠷⣫⣴⡿⠏
⠀⠀⠀⠀⠀⠘⣽⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⣻⠷⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣎⠀⠀⠀⠀⠀⠀⣤⠾⢛⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣠⡀⣄⢠⡄⣠⣄⠰⢤⣄⢠⡄⣀⣤⣠⠆⣀⠀⡀
⠀⠀⠀⠀⢀⡀⣀⣘⣇⣼⡿⣯⣾⠋⢿⣾⠟⣷⡾⠋⠙⢷⠿⠛⠷⠿⠛⡿⠾⠛⣿⣤⣭⡦
⠀⠀⠀⣀⡈⢃⣼⠿⢿⡿⣤⣤⣤⣤⣠⣤⣤⣴⣶⣤⠄⢰⡲⠤⣶⣶⣢⣶⢖⣶⠿⠀⣾⢁⣀
⠀⠀⠀⠀⠺⠿⣷⣜⢛⣛⡛⢿⣝⠓⢿⣛⣛⠙⣛⣮⣁⠛⣫⣽⣭⣍⣉⣬⣭⣉⢠⠾⡛⠓⠈
⠀⠀⠀⠐⠂⠐⠛⡿⠉⠹⠟⣫⠛⠛⢫⠍⠙⠛⠉⠰⠙⠛⠉⢰⠂⠉⠉⠐⠮⠉⠛⠀⠉⠃
⠀⠀⠀⠀⠦⠲⠖⠤⠶⠶⠂⠓⠂⠘⠒⠒⠓⠒⠂⠀⠐⠒⠂⠂⠚⠀⠐⠓⠀⠀⠓⠀⠑⠋
⠀⠀⠀⠀⠔⠞⠚⠂⠘⠚⠒⠂⠚⠒⠛⠒⠒⠁⠘⠃⠘⠚⠚⠛⠓⣚⣀⠉⠓⠚⠓⠚⠐⠓
⠀⠀⠀⠀⢀⡴⠛⢉⢉⠑⠖⠋⠉⣧⣤⣤⠔⠒⠒⠖⢚⠳⣄⣤⣼⠋⢭⣙⠶⠒⠶⢤
⠀⠀⠀⠀⢿⡈⢂⣛⣭⡆⢟⣁⣤⡤⠜⣇⢰⠶⠶⠿⠈⢀⣉⠉⣦⠒⠄⠈⢙⠓⠶⢆⣟⡛⠲⢄
⠀⠀⠀⠠⣏⠘⢉⣉⠰⠞⠛⢉⣡⢶⠎⠰⠶⠖⠂⠘⠓⠙⣫⢡⡏⠘⡃⠤⣬⣉⡂⢶⣬⣙⡳⢚⠇
⠀⠀⠀⢰⣏⢘⣋⡡⠞⠉⠉⠉⠀⠈⠓⢲⠀⢰⠷⢷⠿⠧⣈⡰⢷⡄⠛⠆⣤⣈⠉⠐⠶⣉⣁⠏
⠀⠀⠀⠀⠉⠉⠁⠀⢻⣎⣉⡉⠉⡉⢉⣉⠉⠛⠙⠒⠒⠚⠉⠉⠉⠉⠛⠒⠒⢥⣘⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠙⠙⠀⠁⠈⠩⠭⠥⠌⠩⠭⠭⠄⠑⠘⠙⠊⠁⠀⠀⣇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⡇⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢠⣾⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢀⣿⣿⠃⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠇⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠈⢧⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠶⢀⣤⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⣠⣴⡿⠛⢉⣉⣉⣁⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⢀⣠⣴⣿⣿⡟⢀⣾⣿⣿⠿⠛⠉⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡾⠟⠛⠉⠀⢀⣠⣴⡾⠟⠛⠿⠿⠛⢁⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠉⠀⠀⣀⣤⣴⣾⣿⣿⠏⣠⣶⣿⣶⣶⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠏⠀⣀⣠⣤⣶⣾⠟⠛⠛⠛⠿⠟⠋⣰⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⣿⣿⣿⡿⠁⣴⣿⣿⣿⣶⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣦⣤⣉⣁⣤⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⠀⠀⣀⣀⣠⣤⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠋⠉⠁

⠑⠀⠈⢠⡀⣧⠛⠾⡓⠀⠀⠀⠀⠀⠀⢀⠸⡃⠀⢀⠀⡀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠐⣠⠄⠀⣠⠊
⢀⣶⣕⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠁⠀⠐⠁⠘⢆
⡉⠀⠁⢠⠶⠈⠀⠀⠀⡀⠆⠘⠖⠒⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠓⠀⠀⠀⢀⠐⠑⠆⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠐⠂⠁⢀⠀⣀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠁⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠋
⠀⢀⡄⠀⠂⢘⠀⠀⠀⠀⠏⠈⠀⠀⠢⠀⠀⠀⠁⠐⠀⠀⠀⠸⠂⢈⠄⠋⠈⠁⡆⠓⡀⠀⢄⠀⠀⠀⠐
⡈⠀⠀⠀⠀⠠⠛⠈⠁⢐⠀⠊⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠰
⠉⠁⠀⠀⢀⠤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠠⡄⠀⠀⢨⠥
⠓⠠⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰


⣉⣉⣀⠤
⣉⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣉⣤⡴⠖⡊
⠉⣥⠶⢃⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢿⣶
⣈⣴⣮⣅⣤⣴⣎⣠⣠⣄⣤⣄⣠⣇⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣀⣤⣤⣈⣤⣌⣵⣦⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢹⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣌⣉⣉⣉⣀⣡⣴⣾⣿⣿⣿⣿⡿⠋⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣄⡀⠀⢉⣿⣿⠋⠉⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⢀⣔⣩⡝⢳⡦⣥⣭⡴⠋⠀⠀⠀⣴⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⡀⠐⣛⣋⠩⠙⠆⣸⠟⠁⠀⠀⠀⣠⣼⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠠⢀⠠⠶⠛⠁⠲⣬⠙⣿⠟⠁⠀⠀⠀⠀⣠⣿⣿⣍⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠒⣠⣶⠟⠒⣢⡝⢩⣶⡶⠊⠁⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⢐⣨⣴⠟⠴⣞⢲⣶⣶⡶⠞⠁⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⡿⠃
⠀⠀⢀⠔⠁⡀⠀⠀⣠⡶⠧⡐⠐⢦⣤⣪⡴⠛⠁⠀⠀⠠⠎⠀⠀⠀⣠⣾⡿⣷⣽⣿⣿⡟⠁
⠀⡼⠁⠀⢀⣤⣄⠰⠓⠒⢲⣤⣤⡿⠟⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣿⡿⠋
⡸⠁⢰⣶⠟⠁⠘⣧⢡⡶⢟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣮⢻⣿⣿⡾⠛
⠇⢠⡞⠁⣠⡚⣿⡿⠏⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟
⠀⢸⣿⣾⣿⡇⡿⠤⠠⠀⠂⢀⠀⠀⠀⠀⢀⣀⣠⣾⣯⢻⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠶⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⡀⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠈⠀⠀⠀⠀⠠⠔⢁⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⣿
⠀⠀⢀⡴⠶⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿
⠀⠀⠀⡀⠂⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠜⠀⠀⠀⣬⡿
⠀⠀⠀⢇⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠘⣆⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⢠⣼⠟
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⠚⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣈⠛⠳⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣥⣽⣓⡲⠤⣤⣀⢀⣀⣀⢀⣀⠀⣀⣀⣤⣤⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁




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

⠬⠒⠃⠀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⢀⠀⠘⠦⠀⠀⡀⠀⠅⠀⠁⢧⡀⢹⠋⢻⣬⢺⣣⣼⣼⡿⠉⡽⠁⡨⢋⠁⠃⠼⢁⠞⠹⠃⢈⡼⠋
⣅⡀⠀⠀⠀⠀⠀⠈⠀⠈⢢⣠⣤⣄⣤⣀⡀⠀⠠⠀⠀⠀⠀⣠⠖⠾⣇⡀⢱⢋⡷⠻⠋⠀⠈⣷⡼⢁⣀⠀⡈⠀⠀⢀⡀⠀⣀⠐⠁⠀⠈⠀⠀⢀⡴
⢤⢉⠁⠤⣀⠀⠀⠀⠀⠀⣾⠉⠀⡀⢀⣉⣻⡖⡦⠥⡀⣀⣴⠧⠁⠀⠙⠉⢨⠮⣤⡀⠀⠀⠰⠟⠗⠛⠉⠙⠉⣹⠋⠁⠹⠏⢉⠷⡄⠀⡀⠔⠊⠉⠉
⣓⠀⠀⡀⠀⠐⠀⠀⠀⠀⢻⡍⠀⠈⠯⡉⠁⠀⠁⠀⠀⠉⠉⠗⠲⠄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠉⠀⠀⠸⠀⠀⣈⡗⠂⠁⠀⠀⠠⠐
⠦⣠⠀⠁⠉⠒⠀⠀⠀⠀⠻⣆⡄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⣲⢤⠤⢤⠀⠀⠀⠀⢲⡋⠀⠀⠀⠀⠠⣄⠼
⠂⠀⠅⠀⠂⠀⠀⡀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⢀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⡀⠀⠠⠆⠀⠈⠹⠀⢀⣸⠃⠀⠀⣀⣠⠇⠀⠀⠀⠀⠀⠀⢀
⠙⠃⠀⠀⠀⠀⢀⢀⣠⣔⡓⢺⡋⣷⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠠⠄⠀⠀⠱⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⠞⠁⣴⣃⣠⡀⠀⠀⠀⠀⠄
⠚⠛⠓⠲⠆⠀⢰⣏⡀⠉⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠹⠻⡟⢲⡄⣀
⣛⠖⠃⠀⠀⢀⣿⠉⠀⣀⠴⠚⠁⠀⠙⠂⠀⠀⠀⠒⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠀⠁⠀⠀⢀⡼⢋⣑
⠻⠍⣠⣶⠋⠙⠇⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶
⠉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠀⠀⠀⢠⣤⣄
⠤⠼⢿⣓⣤⣰⠯⠀⠸⣧⣀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠂⢠⠀⢀⡀⠀⠀⠀⠀⠁⠀⠉⠉
⠃⠉⠀⠀⠀⠉⢳⣀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠘⠆⠀⠀⠁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣸⠇⠀⣠⡼⠟⠂⠒⠠⠤
⣴⡖⠊⠀⠀⠀⠀⠈⠛⠷⠖⢡⡀⠀⠀⠀⠀⠺⢤⠴⠦⠄⠀⠀⠀⠀⠘⡄⠀⡀⠀⠀⠀⠠⠀⠀⠠⠨⠆⠀⢀⠀⠀⠀⣹⡝⠀⢙⠀⠀⢄⣀⠀⠘⠑
⡩⠀⠀⠀⠀⠀⣀⠄⠈⠁⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⢆⣀⡔⠀⠀⠰⣤⠤⡀⠙⠆⠀⠀⢀⣄⠀⠀⡀⠘⠒⠋⢹⣷⣛⠁⢠⣴⡏⠀⠀⠀⠀⠉⠀⠈
⠂⠀⢀⡀⠄⠊⠀⠀⣀⡤⡾⠊⠳⠴⠤⠴⠻⣄⣀⣰⣯⠉⣅⡀⡐⢀⣽⡆⠁⠘⠲⢤⢶⡞⠉⠓⠚⠻⣤⣤⡽⠛⠀⠉⠙⠋⠁⠀⠀⠀⠀⠑⠢⢈⡐
⣪⠕⠉⠀⠀⣀⠔⠛⡉⠈⠁⠀⠀⠀⠀⢀⠀⠀⠁⡰⠃⢠⠻⣤⠶⣞⠈⡻⣆⠀⡀⣀⡾⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⢂⡠⠋⣡⡴⡋⢀⣄⣠⠊⡰⢂⠔⣉⡴⠀⣰⠃⣰⣦⣢⠘⣶⡔⣾⣧⣨⣃⢹⡉⡀⠀⡀⠀⠪⠀⠰⣄⠀⠐⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠐⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠠⡄⠀⣀⠀⠀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢉⠉⣦⠠⣇⠀⠂⢸⡎⠀⢉⠆⠢⢼⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⡀⠱⡜⠀⠄⢈⠀⠄⢸⠀⠀⢸⡇⠈⢸⠀⠂⢸⡇⡴⠒⠳⠀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠠⠀⣾⡀⠤⣸⠦⢄⠘⣇⠤⣼⠀⠀⠸⡇⣆⠈⠀⣾⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡜⠁⠐⢄⡀⠀⠀⡦⠊⠘⣿⠀⠀⢸⢤⠀⢡⠁⠀⠀⡇⠀⢀⠋⠉⠙⡆⡌⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⢣⠀⠀⡇⠀⠀⡇⢀⠀⡄⠐⠀⣾⠀⡀⠀⠋⡉⠙⠀⡀⠐⢧⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠁⡀⢱⡄⠇⠀⢸⢡⠊⠉⢣⠀⡀⠽⢀⠀⡰⢄⠀⠀⠀⠁⠘⠈⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠠⠄⠽⠤⢤⡟⢸⠀⠁⢸⠸⠀⠀⢸⡏⢀⠀⠃⠀⡤⠔⠒⠖⠰⠲⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠤⠤⠀⠳⡈⠀⠀⠀⢰⠀⠀⢸⠁⠀⠀⠁⡴⠁⠐⠐⠒⠒⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⣗⠀⠀⢸⠀⣀⠴⠋⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠳⢄⡀⠉⠉⠒⠒⠋⠉⠀⣠⠞⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⠴⠚⠋⣁⣤⠤⠶⠦⠍⣷⠴⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠴⠚⠉⣀⠤⠒⠋⠁⢀⣠⠴⠆⠀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠤⠒⠋⣁⣠⠤⠚⠉⠉⢀⣀⡀⠘⠋⠀⠀⣀⣴⣧⣤⣄⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠚⠉⣁⣠⡤⠖⠋⠉⣀⣤⡄⠀⠜⠉⠉⢉⣳⠶⠞⠋⠉⠀⠀⠀⠈⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣀⡤⠖⠋⠉⣀⡠⠤⠒⠊⠉⠉⠀⢀⣀⠀⠈⠁⠀⠀⢀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⠞⢁⡤⠖⠊⠉⠀⢀⣀⣤⡶⠀⠀⡜⠉⠈⣉⣢⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⢃⠔⠉⣀⣠⢅⠀⠀⠈⠉⠀⣀⣠⠼⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠞⠉⠁⠀⠀⣉⡶⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠀⠀⢰⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣠⡤⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣤⠤⠤⠶⠒⠛⠉⠁


⣷⣶⣶⣷⣶⡾⣶⣶⣶⣾⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠾⣶⠷⠶⠾⣷⣶⣶⠶⠶⠶⣶⡾⠷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣰⢾⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢰⠟⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⠁⠈⠀⠀⠀⠀⣀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⢀⠀⣠⢤⣀⡴
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠻⢦⡀⠀⡠⠖⣷⠞⠙⠿⠛⠋⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠒⠉⠟⠁⠈⠉
⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⣀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⡀
⠀⠀⠀⣰⢿⠋⠉⣷⣠⡴⠞⠋⠉⠁⠀⣹⠀⠉⠻⠷⠄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠉
⠀⠀⠘⢷⡈⢷⣾⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢷⡞⢁⣠⠶⠛⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡿⢿⣿⠿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⢻⡆⠀⠀⢻⠀⠀⠸⠇⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠓⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡶⠿⠿⠿⠿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠀⠀⠀⢀⣀⣀⣀⣲⣄⣈⡉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠂⠉⠁⠁⠀⠀⠀⢣⠁⠀⠉⠑⠢⢙⠻⣷⣄
⠀⠀⠀⠀⢀⢀⠀⠀⠈⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⡅⠀⢣⠀⠈⡁⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢠⠇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⢀⠀⠀⠈⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣤⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢰⡏⠀⠉⠙⠛⠾⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⣸⠃⠀⠒⠀⠀⠀⠀⠀⠉⠉⠳⠴⠤⢄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⣿⡅⠀⠃⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⢸⣇⠰⠖⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠠⢈⠉⢹⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠠⣤⡄
⠀⠀⠸⣿⡀⢱⠉⢃⡴⠊⠀⠀⠠⠀⠀⠀⡀⠀⠀⠄⠈⠓⢤⠁⣰⠏⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢆⣰⡿⠋⠀⢀
⠀⠀⠀⢻⣧⠀⠱⡋⠀⠀⠄⠀⡆⠀⠀⠀⠸⡄⠀⢀⠞⠀⠀⡰⠃⠀⢠⣿⠀⠀⠀⠀⠀⠀⠰⣧⣼⡾⠋⠀⠀⠀⠈⠻⣶⣤
⠀⠀⠀⠈⢿⣆⠀⠙⣜⢮⠀⢠⠃⠀⣀⠀⠀⠹⡄⠀⠈⣠⠞⠁⠀⣰⢯⠇⠀⡀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠠⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠈⢿⣧⡀⠈⠳⣄⡚⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⣠⠞⢡⣿⠶⠷⠿⠛⠛⠉⠁⠀⠀⠀⠀⣄⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠉⠙⠓⠒⠚⠋⠉⠁⢀⣠⡴⢚⣡⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣧⡀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣦⣤⣤⣶⣶⣾⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⢠⡀⠀⢰⡀⠀⠘⣦⡀⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠛⠉⠉⢉⠙⠛⠒⠢⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠉⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠢⣀⠙⢀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢦⠘⢷⡄⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠆⠌⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠈⠀⠀⡁⠆⠀⠀⠀⠀⡐⠒⠀⠀⠀⠀⠀⠀⡐⠁⢀⠘⡇⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄⢹⠀⠀⠀⠀⠀⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠈⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠀⠈⠀⠀⠠⠴⣦⡄⠀⠀⠀⠳⠀⠀⠀⢀⡔⠀⠰⠁⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢐⠒⠊⠉⠴⡆⠀⢸⠃⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⢠⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡾⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⣀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢀⣉⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⠰⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠑⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⡀⠀⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣍⠀⠀⠈⠙⠶⠤⠴⠋⠐⠀⠤⠄⠀⠀⠀⣀⣠⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣬⣀⠐⠲⠤⠤⠴⠄⠐⠒⠲⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣶⣶⣷⣾⣷⣾⣿⠿⠟⠿⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⠀⠀⠀⢀⣀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠎⡀⠀⠈⢋⢲⠞⠋⠈⠉⠋⠉⠛⠁⠈⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡞⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣢⠖⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡉⠡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⠀⠀⣨⠀⠀⠀⠀⠀⠠⠀⢤⡠⠞⡀⠀⡀⢀⠖⠓⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣗⠀⠐⠠⠀⠀⠀⠀⢠⣾⡶⣿⣆⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⣦⣤⡄⠀⠀⠀⠀⠀⠩⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠃⠀⠠⠀⣀⡀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠠⠘⠋⠻⠟⠁⡀⣀⠀⣄⣀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢇⠆⢀⣤⠂⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡄⠀⢀⣀⡵⣅⣩⠴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠋⠛⠉⢷⠀⠀⠀⠀⠀⢸⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄⡴⠊⣿⡁⠀⠀⠀⠀⣿⠦⢤⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠐⡂⢹⡤⢄⡀⠤⣀⣿⢐⠤⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠆⢀⠀⣀⣄⠁⠐⠀⠀⣀⣀⣀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⡏⠈⠛⠦⢾⡶⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⢧⡀⠀⢀⣠⠊⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠐⠒⢴⡉⠉⠋⡇⠀⣀⢀⣿⡁⠈⠉⠡⠀⠀⠀⠳⢦⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⢀⠐⠋⠁⢀⠀⠊⠙⠞⢄⠁⠀⠁⠀⠁⠀⠀⠀⠈⠁⢀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡃⠆⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠰⢎
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠛⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠐⠀⠀⠉⠑⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠛⠃⠉⠀⠀⠀⠀⠁⠀⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠁⠈⠉⠙⠓⠊⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡦⠀⠰⠆⠀⠷⠀⠘⠂⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⣠⡤⠄⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣠⡬⡇⢾⠁⢉⡇⠀⣗⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠃⡀⠛⣈⡃⠚⣁⠀⣩⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⡏⢹⣮⡗⠂⢸⡿⣇⢹⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠋⠀⠉⠉⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠶⠀⠰⠆⠀⠻⠀⠘⠃⠀⠙⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣈⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⣆⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⡄⠀⠀⠀⠀⠀⠀⠙⣆⠈⢧⠀⣀⡤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠸⡄⢸⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠇⠀⠀⣠⠴⠖⠒⠶⢿⡎⠀⢀⠎⠀⣿⣠⣤⠤⠤⣤⣄⡀
⠀⠀⢠⡞⠛⠛⠶⣄⠀⠀⠀⠀⠈⠀⠀⠘⣷⡄⠀⠀⠀⠀⢣⡔⠁⣀⡜⠉⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠘⢧⣄⣀⣀⣸⡶⠀⠀⠀⠀⠀⠀⠀⣿⠘⡆⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢿⠛⠒⠒⠛⠁⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢶⡒⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠟⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢍⡁⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⣤⡴⣤⠶⠶⢤⣤⣤⣤⠴⠖⠋⠁


⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⠂⠉⠙⠆
⡇⠀⠀⠠⡂⠀⠀⣀⢀⡀
⠑⡄⠀⠀⠱⣤⠋⠀⠀⠈⠓⡄
⠀⠈⠐⠄⡀⠈⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⡇⠈⣇⡠⠐⠤⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠮⠤⠃⠀⠈⠂⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠑⢢⠄⠀⣁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠗⠤⠤⠤⣀⡀⢱⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠙⢆⢧⠀⢀⡴⠊⠉⠉⠉⠒⠤⣠⠴⠀⠀⠠⢀⡀⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠈⠶⠘⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡰⠂⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠁⠁⠂⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠐⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⠲⠦⣄⣀⣀⣤⡤⠤⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠢⡠⠒⠚⢱⣀⠔⠉⠉⣆⡠⠔⣦⠞⠁⠉⡆⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢺⠀⠀⣰⠃⠀⢀⡸⡇⠀⠀⠀⡏⠀⢰⣇⠀⠀⢸⠓⢊⣉⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡚⠁⠓⠚⠁⠀⠙⠤⣀⡠⠋⠁⠉⠀⠳⣤⠏⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⣀⡀⠀⢸⠉⠈⠢⡖⠢⢄⡞⠉⠁⠢⣰⠋⠒⠲⣄⠖⠒⢦
⠀⠀⠰⡅⠀⠀⠀⠀⠀⠈⣉⣉⠗⠺⡄⠀⢀⣽⠀⠀⣇⠀⠀⠀⡹⣄⡀⠀⢸⡄⠀⢸⠒⠒⠒
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠳⡤⠊⠀⠉⠉⠹⣄⡀⠴⠃⠀⠉⠓⠊⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀⠀⢀⠠⢄⡀⠀⢀⡀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢘⡎⠁⠀⢸⡔⠁⠀⠀⡷⠊⢁⡟⠁⠀⢠⣇⡠⠤⠤⠤⠤⠤⠴⠆⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⠾⣀⣀⠴⠚⢇⠀⠀⠀⣧⣀⡸⠳⡄⠀⡼⠉⠑⠠⠤⠤⠤⠤⠀⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⠒⠚⠀⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠢⣀⣀⡀⠀⡴⠒⠢⡀⢠⠤⢄⣀⢀⡤⢄⡀
⠀⠀⢠⠐⠒⠒⠒⠒⠒⠒⠒⠲⣄⣷⠀⠀⠘⣇⠈⠹⡇⠀⠀⠈⣮⠀⠀⠀⣏⠀⠀⢱
⠀⠀⠈⠒⠤⠤⠒⠒⠒⠒⠒⠊⠀⠸⡄⢀⠞⠙⠤⠴⣇⠀⠀⣠⠋⠑⠦⣄⠜⠦⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶