Asteraceae Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⣤⡁⣀⣶⡀⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠃⠤⠘⢿⡅⠿⣿⣇⣥⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢀⣀⡙⢛⢿⣷⣿⢿⣛⠿⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠷⡴⢙⠈⠙⠓⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠎⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⢠⣿⢺⣤⠆⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⣶⡡⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣞⣧⣼⣿⣏⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⡴⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣻⣯⡿⡟⠋⠙⢶⣄⠀⠀⠀⠀⠄⠀⢀⠆⢀⠀⠀⠀⠀⠀⠁⠐⠀⡩⢡⣄⡈⣉⡛⠞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠿⠇⠙⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢬⣰⢻⣄⣾⣤⢀⡀⠀⠀⠀⣰⠤⢌⠡⠬⢓⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠘⠻⡄⢿⡾⢿⣿⣧⡘⠁⠀⠀⡸⠃⠀⠈⢠⡀⠠
⠀⠀⠀⠀⠀⠁⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⡁⢨⡿⢀⣿⣿⠯⠜⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁
⠀⣀⠰⠀⠀⢠⣾⣿⡭⠄⠀⠀⠀⠀⠀⠰⠀⠀⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣻⡼⣠⡞⢋⠓⠄⠉⡌⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⡘⣿⠗⠀⠀⠀⠀⠀⠀⢠⡇⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⠹⠎⠿⡁⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⡄⢀⠄
⠀⠀⠘⠣⡾⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢿⠂⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⣇⠀⠀⠀⠀⢫⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠂⠨⢶⠖⠀⢉
⠀⠀⡀⠃⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠘⣀⣹⠇⠀⠀⠀⢀⣤⣀⡀⠀⢸⠀⣿⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢀⠀⠤⢤⠀⢤⠈⠀⢚⡄⠀⠀⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢠⣼⣿⣯⣿⣧⡄⠀⡿⠀⢠⣤⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⢻⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⡇⠀⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠖⠀⡤⠀⡐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠆⠀⠀⠈⣿⣿⣿⣿⣿⠿⠷⠇⠈⠻⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⣦⠀⠀⢃⡁⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣤⣀⣀⡈⢀⡀⠀⠁⠀⠀⠀⠙⣿⣿⣿⡁⣿⠇⠀⢸⡷⠊⢀⡀⣄⢀⡀⠀⣀⣤⣤⢸⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠓⣭⣽⣿⣯⡰⣶⣦⣤⣀⠀⠀⠸⣿⣿⡇⣿⠀⠀⠘⢁⠄⡾⢡⡿⡏⣠⣾⣿⣿⡟⢸⣿⠀⠀⠀⢀⡠⠖⠉⠁⠀⠈⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠳⣾⣯⣿⡍⣿⡿⢸⣷⣄⠀⠀⣿⡇⢹⠀⣴⠃⣼⣷⣿⠰⠖⢰⣿⣿⣿⣿⢁⣯⡟⠀⣠⠔⠁⠀⠀⡀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠋⠑⠁⠀⠈⣿⣿⣇⠀⢿⡇⣇⣾⣿⠇⠙⠍⠉⠀⠀⢸⣿⣿⣿⡟⣾⠟⢠⡞⠁⠀⢀⣀⣀⣀⣤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢣⠈⣿⣆⢹⡇⣿⣿⡟⠘⢁⣤⣤⣴⣦⣼⣿⣿⡿⠉⠁⡴⠋⠀⠀⣴⣿⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⢻⣶⣶⣶⣶⣶⣤⣀⣀⠀⠈⢆⣀⣠⣈⣿⣿⡇⣿⣏⢀⡐⣿⣿⡿⣿⣿⣿⣿⢋⣝⢠⠞⣠⣴⣴⣾⣿⣿⣾⠹⢟⠙⣆
⢀⣤⣤⣤⣄⣀⣀⡀⠀⠈⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⡇⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣣⣾⡷⢁⣾⣿⣿⣿⣿⣿⣿⡿⠻⠚⢀⠋
⠸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⣤⣿⣵⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣍⣉⣤⣤⡄⠀⠚
⠀⠘⢿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣄⣻⠼⠻⣿⣿⣿⣿⠞⢋⠉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣶⣿⠿⣏⣁⡀
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠛⠻⢿⣶⣯⣿⣿⣿⡀⢡⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢀⠉⠈⡙⢳⣀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠀⠤⠀⠓⢒⣤⣝⣻⣿⣿⣳⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠛⣁⣤⣾⣿⣿⣿⣿⣿⡶⠶⣤⡀
⠀⠀⠀⠀⢀⡠⠀⠁⠀⠀⡄⠲⠶⠛⠛⢛⠘⢋⣩⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡃⠹⣿⣿⣯⣤⣾⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠸⣦⣀⠀⠀⠀⠀⢁⣀⣠⣾⣿⣿⣿⠟⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁⣀⡠⠚⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣽⡛⢦⡙⣿⣿⣯⡭⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢉⣯⡿⢿⣿⣯⠀⠀⠁⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣯⣿⣟⠻⢿⣿⣿⣿⣏⡙⠉⠙⠀⠉⠉⠛⠃⠀⠙⠿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⣉⣻⣤⣶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⡿⠻⣯⡛⠃⠘⣿⣿⣿⣿⣿⣿⠷⠶⡶⣶⠖⠚⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠬⠾⣋⠉⠻⠿⢮⠅⢹⣛⠙⠛⠻⠟⠻⣿⠻⢿⣦⡀⠀⠃⠙⠧⠀⢸⣿⣿⣿⠟⠻⠓⠄⠠⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠂⠈⠏⢲⣤⣤⡤⢄⡀⠙⣿⠁⠰⠄⠤⠄⠴⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣄⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣚⣂⣀⣈⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⢤⠀⠀⠀⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠐⠤⣸⠈⡑⢦⡀⡼⠁⠀⣧⡠⠊⢁⢄⡷⠐⠒⢂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠯⣉⠓⠲⢦⡀⠀⠘⠆⠠⠀⣷⡆⠀⢀⡏⠁⠀⣼⠎⠀⠀⣠⡯⠔⠒⠓⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⠉⠐⠈⢷⡀⠀⡙⡄⠈⢂⡀⡄⢸⠀⠀⢸⢀⡄⢸⠃⠀⢠⡾⠋⠀⠀⣀⠤⣰⣶⡥⠤⠤⠠⠤⣀
⠀⠀⠀⠀⠠⣤⡤⠶⠦⠤⣄⣙⡒⠤⠀⠂⢑⠀⠀⣾⠄⢀⠃⣀⣸⣀⣀⡘⢘⠀⣸⡀⠀⣿⠁⠀⡠⢛⡠⠊⠉⠀⠀⢀⣤⢶⣒⡋
⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀⠀⠉⠙⠢⡀⠒⢀⠀⢂⢸⡶⣞⡋⠇⠈⡉⢎⡑⢟⡈⠃⡿⠛⣧⣀⡰⠿⠋⠀⠀⠀⢤⡶⠿⠒⠒⠚⠻⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠘⠷⠿⠷⢤⣀⠀⠈⠰⣌⢀⣿⣬⠇⢤⢳⣐⣧⡙⠁⠬⠳⢲⠀⡁⠲⡁⣼⠑⢧⣀⡔⡫⠈⠉⠀⠠⠄⠀⠀⢀⣀⣀⣭⣶
⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠸⣯⡟⢚⠳⢎⡰⡯⠸⢧⠀⡤⢤⡏⠀⣈⠳⣉⣌⣱⣿⣿⠮⣠⡴⣶⡶⠛⠉⢉⠉⣉⣙⠃⠤⣀
⠀⠀⠐⠛⠒⠒⠒⢒⠶⠀⠠⠤⠤⠤⣤⣄⣱⣿⡉⢾⠡⢬⣹⣦⡽⣤⣾⡣⣾⠄⣁⢠⣼⡆⠹⣴⢿⣗⣿⣥⠴⠖⠰⢠⠿⢴⣆⣀⣀⣉⣒⣌⣳
⠀⠀⠀⠀⠀⣀⡸⠶⢒⠀⠀⠀⠀⣂⣀⣤⣄⣹⣷⠎⢶⣤⠿⢶⠿⣷⣦⢢⣭⣥⣲⠿⡷⢞⣻⢴⣯⣽⣿⣿⠋⠑⠲⠿⣄⡐⠠⠌⡉⠙⠢⡀⠈
⠀⠀⠀⣠⠚⠁⣀⣈⣀⢤⡴⠚⠉⣤⠴⢛⠓⢟⣿⣿⢶⣯⣽⢏⢙⣿⣻⡿⢭⠷⣯⣩⠳⣮⣝⣿⣿⡯⠭⡽⢿⣷⣦⣄⠀⢍⠛⠶⣌⣁⠀⠙⢆
⠀⠀⢼⣥⠤⠖⠚⠉⣡⠊⠈⣠⠛⠁⠔⢡⣶⡟⠉⢀⡠⠟⢛⣶⠿⣷⣏⣳⣾⣿⢥⣛⣿⣾⣿⣟⠈⡙⢶⣌⠂⢀⠹⢿⣿⣦⡙⠦⠈⠱⣍⠒⠾⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡋⢊⡾⠁⠀⠀⣰⢋⠏⠀⡠⠋⢀⣴⠟⢁⣾⡿⠉⣿⠉⢳⡄⠙⢿⡆⠁⠹⣳⡀⠀⠹⢷⣄⠱⡄⠻⣿⣿⡢⣌⡂⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠛⠟⢱⠃⠀⢀⡴⢣⡏⠀⡰⠁⢠⡿⠁⠀⣾⡟⠀⢀⡇⠀⢸⡇⠀⠀⢷⠀⠀⠁⢣⠀⠀⠘⠏⢢⡈⢄⠹⣿⣿⠈⠙⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⡴⠋⡷⠛⠀⢰⠁⢀⣿⠁⠀⣸⡿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡄⠀⠀⠌⡇⠈⠀⠀⢠⡗⣄⠀⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⡇⢀⠃⠀⣾⡏⠀⣠⠣⡇⠀⠀⣼⠁⠀⠀⢿⠀⠀⠸⢇⠀⠀⠀⠰⡀⠀⠸⠀⠀⠈⠣⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣾⢀⡞⠉⠀⢠⡟⠃⡇⠀⢠⡟⠁⠀⠀⠸⣇⠀⠀⠈⡄⠀⡇⠀⠙⢦⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⣆⣼⠃⠀⢧⣀⣼⡇⢀⢠⠀⠀⠙⢦⠇⠀⠈⢦⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡏⣼⡇⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⡴⠚⢻⣦⡀⠀⠄⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠄⠀⠀⢠⠾⠀⢴⣴⣤⠀⠐⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⡰⢀⡇⠀⠀⠀⠘⢷⡀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⢿⡷⡅⠀⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⣸⠁⢸⠃⠀⠀⠀⠀⠀⠻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣧⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢞⠀⠀⠀⠀⠀⠀⠀⠈⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢻⡆⠀⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣶⠎⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢿⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠘⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⣤⠀⡰⢲⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠛⣇⣸⠀⠀⢿⣧⡀⠀⠀⢠⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⣽⠁⠀⠀⠈⢿⣷⣆⡀⠀⠴⠟⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⠀⠀⢈⡷⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠯⡒⢠⠀⢞⡭⠤⠭⡷⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⡤⢒⣒⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡄⠀⠀⠀⢀⣄⠘⣿⡀⠀⠈⡗⢛⡾⠁⠀⣀⣷⠇⠀⢠⢎⠩⠊⠢⡙⠦⡀⢠⡮⡼⠋⠉⠈⠙⢃⢧⠀⠈⡇
⠀⠀⢀⠲⣶⠀⠘⣷⠛⣷⣽⠀⠀⠀⠈⢹⡆⠈⠛⠁⠀⢹⠸⡇⠀⠈⠉⠁⠀⠀⡳⠀⠀⢰⡄⠈⠀⢳⡌⢱⠁⠀⢾⡀⠀⣸⡯⠀⠀⠁
⠀⠀⠈⠀⢺⠀⠀⠁⠀⢸⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠳⣤⣄⡾⠃⠀⠀⠈⡇⣽⠀⠀⠈⠛⠺⠛⠀⠀⠀⠣
⠀⠀⠀⠇⢘⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿
⠀⠀⠀⢰⠀⡃⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⡀⠀⢠⣤⠀⠀⠀⠸⠀⣿⠀⠸
⠀⠀⠀⢸⠂⣷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡉⠑⠀⣰⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⢸⡖⣟⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣡⢄⡄⣚⠍⠀⠰⠆⠀⠀⠇⢸⡇
⠀⠀⠀⡛⠛⡿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⠠⡌⠀⠁⣄⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣴⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠘⠻⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠇

⠀⠀⠀⢀⡀⣀⡀⠀⡔⢀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠣⡀⠀⣺⢀⡇⠀⣨⠄⠀⠀⡒⠀⠀⠀⠀⡆⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⢀⠀⠠⠐⠊⠛
⠀⣀⡤⠤⠀⠼⣴⡏⠀⢱⡄⠀⠀⣀⠎⠀⣴⣄⠀⠀⡷⢄⣀⠀⠆⡠⢀⠄⠀⠀⠀⡀⠔⠠⢤⣤⠁⠀⠀⠆⠀⢀⠠⠀⠻
⠈⠙⠓⠤⣀⡆⠉⢹⡖⠀⠹⠖⠊⠀⠀⠀⢀⣡⡵⠄⣈⣤⡼⢀⠮⡤⠂⠀⠖⠀⠀⠁⠐⠦⡄⠙⠇⠀⠀⢀⢀⡎⠉⠒⠙
⠀⠀⡤⠊⠉⠳⣦⠈⠑⠠⢷⣄⡤⠉⠐⢏⠁⠀⠑⠤⡴⣿⠛⣀⡔⠺⣛⡁⠠⢔⡃⠴⢖⢶⠞⣒⡆⠐⢆⢐⡾⠁
⠀⠸⠔⠒⠈⠍⠙⡿⠲⠇⡀⠉⠦⢤⣤⣄⡵⠄⠀⡠⠦⠘⣳⣊⣀⡄⠣⡀⠀⠀⠀⠼⠆⠹⠗⠃⢹⠀⠀⡞⣁⠸⠃
⠀⠀⠀⠀⠈⠃⢸⠀⢀⣰⣷⣄⠀⢠⠀⠀⠀⠀⣨⠴⠒⢈⣿⠉⠸⡉⠂⠬⡄⠀⠀⠀⠀⠀⢀⠀⠈⢧⡀⠁⢇
⠀⠀⠀⠀⢦⡀⠘⣀⠎⠘⢿⡅⠉⠚⠀⠀⠀⠈⠁⠀⠀⡜⠈⠂⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⡀⠘⡗⡀⡺⡌
⠀⠀⠀⠀⠘⢧⡁⢤⡀⠀⢸⣇⠀⠀⠀⠀⠀⠈⡄⠀⠰⠃⣶⠖⠎⠑⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠙⡀⢿⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢿⣦⣸⡶⠀⠀⢀⣤⠖⡒⠁⠀⣶⡻⠃⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣈⠣⡀⠀⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠗⠀⠀⣞⠁⠐⠁⠀⢰⠏⠀⢠⢾⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠃⠀⡘⣴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⡘⠋⠀⠀⢠⠀⣎⠁⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠆⠄⠃⢹⠀⠀⢀⠴⣺⠂
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢿⠀⣿⢳⡄⢠⠘⠀⠆⢠⣿⡟⠀⠀⠀⠀⠀⠀⠤⠤⢤⠤⠐⠂⠀⢠⠾⠁⠰⢁⠘⡏
⠀⢴⣾⣿⣭⡿⢟⣶⣤⣀⠀⢠⢰⡏⣸⠛⡄⠀⠷⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠃⠀⠀⠀⠀⠃⠀⠀⣁⠞
⠀⠀⠈⠉⠉⠉⠉⢙⠛⠻⢷⣼⢘⠃⠷⠀⠑⡸⠆⣼⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢇⠀⢀⠀⡄⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠠⡈⡂⠀⣄⠀⢸⠁⠘⠍⢀⠄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠹⣅⣤⣌⡐⣰⠊⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⢷⣌⠸⣄⡆⣠⣥⢠⣤⠂⠁⠀⠀⠀⠀⠀⠘⡏⠣⠄⣔⣂⠘⣿⡿⠁⠁⠀⠈⠉⠐⣒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣿⣷⠧⣦⡾⠏⢀⠰⡶⠓⠀⠀⠀⠀⢸⣦⠀⠘⢯⡇⣿⣡⠯⠐⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠉⢉⣻⠤⣴⣠⡦⢁⢀⠄⣒⣘⡚⢋⢂⠰⢾⢛⣀⣴⣂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⠤⠶⠖⣿⠔⢺⡏⣰⣿⢿⣛⡋⣯⠻⢯⢹⣿⣶⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⣠⠴⣻⢼⣿⣫⣷⡋⡀⢘⠃⠿⣶⣴⢉⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠾⣼⣿⣷⡑⠈⡱⠈⠘⠀⣄⠁⠘⠂⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⢓⠛⢟⢹⢛⠀⠀⢨⡈⠄⠁⠦⠂⠀⣀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣾⠀⢐⡤⠏⡘⢠⡇⢀⠇⢀⠀⠊⠐⣄⠀⡐⠂⠐⠓⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠁⣯⠐⠻⢡⢰⠄⠘⠁⠙⠀⣄⠐⠆⠈⠛⡆⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡏⠀⠂⢸⡀⡂⠀⠈⠀⠀⠘⢢⠰⡀⠀⢆⠓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠄⣶⠀⠘⠃⠹⠘⠓⠄⠀⠀⠄⢆⠘⠂⠀⢦⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢎⡄⠸⢀⡇⠀⠆⠀⠀⠀⢠⠈⠤⠆⠐⠘⣄⠀⠀⠀⠧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠁⠈⠑⠀⢰⡀⢂⠀⠘⠀⢀⢡⠈⠀⠀⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡜⢠⠀⠀⠀⠀⠀⠇⠘⡀⠀⠠⠀⡎⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡆⠘⠀⡀⢸⠀⡆⠃⠀⡅⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠇⠀⢰⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⢝⣀⠂⠀⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣠⠗⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠀⠐⣄⠀⠈⢂⠀⠀⠀⠁⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠂⣄⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢣⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠇⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠂⠰⢠⡠⠀⠄⣠⠃⠀⢀⠄
⠘⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠑⠦⣄⠐⠄⠀⠀⠀⠀⠀⣀⣠⣤⣴⣦⣄⢶⣧⣀⢼⣃⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⣤⡀⢘⠇⡙⠭⠌⠻⣓⡤⠮⣽⣿⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠊
⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣟⡿⣃⢚⠨⣂⠇⣇⢻⣩⠻⢷⡿⣧⠤⣀⣀⠀⣀⡀⠤⠐⠒⠋⠁
⠀⠀⠁⠒⠤⣀⡀⠈⠁⠛⠒⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣘⣿⡝⢉⠫⡆⢉⠸⢸⢷⢏⢿⣿⣿⠟⠍⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠤⠤⠤⢬⣀⡉⠀⠀⣀⣀⠀⣀⡼⠿⡈⣆⢷⢸⢰⢸⢸⢸⠂⢸⠏⣮⠞⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠐⠐⡝⠞⠘⠉⠙⡟⠺⣄⡾⢰⠁⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡀⠀⠀⠻⣦⣄⠫⠀⠀⠀⠀⠀⠀⠁⢃⡆⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⠀⠄⠈⠙⠃⣠⠦⡀⠀⡖⠢⣄⡩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢁⠀⢣
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢼⠁⡏⢣⠋⡶⠻⢀⡄⠀⠀⠀⠀⠀⢸⠣⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢾⣀⣤⣿⣿⣿⣿⣧⣞⡈⠁⣦⣄⢀⡀⣷⣾⡀⠀⡌⢸⠀⠀⢰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡬⠓⢬⣿⣿⣿⣿⣿⣿⣿⣬⣧⡌⠣⡙⢏⠻⣄⢻⣶⣤⣬⡄⡠⠖⢋⠝
⠀⠀⠀⠀⠀⠀⠈⠉⣠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢲⣷⣿⣿⣿⣿⣿⣿⣠⠴⠁⠐⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡘⡟⣻⣿⢿⢿⣩⣶⣤⠬⠴⢶⣿⣯⢻⣿⣿⣿⣿⣿⣷⠂⠒⠊⣤⣄⡀
⠀⠀⠀⠀⠀⢠⣶⣠⡿⡏⢠⣿⢱⣷⡞⢾⠙⣿⠷⠤⢤⣾⣿⣿⣿⡿⣿⣻⣿⣿⣿⠭⢭⣿⠿⡜⠃⠰⡖
⠀⠀⣸⡿⠲⠾⡁⢸⠁⢸⠉⡹⢻⠏⡀⠰⠿⢷⡤⠒⠉⢻⣿⣿⣭⣿⣿⣿⣿⣟⠻⣦⢶⣻⣿⣿⣦⡄⢶⣦⡀
⠀⠀⠻⡇⢠⣴⣿⣿⣷⣾⣶⡶⠋⢉⢿⡘⣳⢿⠶⡖⡷⢁⠼⢿⢿⣿⣿⡿⣿⠛⡾⢟⣿⣿⣿⣿⣿⣟⠼⣯⠄
⠀⢠⣶⠋⣻⣿⣿⣷⣿⣿⣿⡗⠒⣷⡆⣾⣿⣿⣿⣿⡿⠁⣠⠊⠘⡀⡘⢀⢸⣷⣿⣿⢿⣿⣿⣿⣿⡟⠩⣀
⠀⠀⣽⣂⣻⣿⣿⣿⣿⣿⣿⠛⠿⠙⡩⣿⣿⣿⣿⣿⡿⠞⠁⠀⠀⠻⠃⠈⠛⠐⠫⠟⣪⠀⢉⣷⠁⣤⠀⠉
⠀⠀⠀⠉⠃⠉⠛⠙⠟⠓⠚⠀⠀⠘⠃⣩⡽⣽⠯⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⣿⣳⡿⠖⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⢿⡴⠋⣁⠔⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡝⠃⣈⠜⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠷⠲⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢸⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⢄⠀⠀⢂⠁⢸⠀⡀⢀⡌⠀⢀⠴⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⢀⣹⡶⠃⠟⣀⣴⣷⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⣲⢶⣶⣿⣿⣛⣁⣨⡟⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣙⣽⣿⠟⠉⠛⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁
⠀⠀⠀⢸⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠓
⠀⠀⠀⠈⠛⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⢸⠀⠀⠀⠀⠀⢠⡐⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⡣⡄⠀⠀⠀⠀⣴⠶⠷⠀⠀⢀⣾⠈⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠙⢦⢠⡞⠁⠀⠀⢰⠁⠀⡏⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⢤⣧⡄⠀⠀⡔⠁⠀⠀⣯⠀⠀⠀⡾⠀⠈⢶⡀⠀⠀⢠⢼⣿⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢷⡿⢿⣀⡄⠁⠀⠀⠀⢸⡀⢾⣶⣄⠀⠀⠀⢳⡄⠀⣾⢸⣿⡟⠀⠀⢀⡀⢀⠄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠨⠇⣶⠿⣿⣿⡆⠀⠀⣧⠀⠈⣿⣿⠁⠀⡀⣿⣡⣶⠦⠿⢿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡤⣘⡄⠀⠀⠀⢰⣾⠇⠀⠀⠙⡇⣀⡀⢸⠀⣰⣯⣭⠴⠖⠛⠉⣁⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠟⠛⠋⠁⠀⠀⠀⠀⠱⡀⢨⠁⣸⣏⠔⠁⠀⢰⣳⣷⣥⡿⠞⠛⠀⠀⠀⠀⠒⠢⢽⡉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠴⠶⠖⠒⠤⣷⣶⣦⣯⢁⡀⠀⢰⣞⡿⠿⠻⠿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣿⣿⣿⡶⠶⠻⠉⠉⠉⢻⣿⢻⡷⢿⣭⣥⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣧⣰⣿⣿⣷⣦⣤⣀⣀⠀⢻⣿⣾⣿⠟⢁⣀⣠⡾⢴⡶⠆⠘⢻⠇⠀⠹⠿⠏⠙⠉
⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⠿⠻⠟⠋⠁⠀⠀⠉⠉⠛⠲⣿⣿⣥⣜⠑⠿⠟⠻⠦⠀⠒⠾⣀⣠⡴⢦⣠⡀
⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⢀⣦⣀⣐⣦⣤⣌⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠻⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠉⠋⠉⠉⣀⣬⣿⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⢻⡟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣼⠏⢸⡏⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠟⣿⣿⣿⣷⢾⠃⠀⠀⠘⠢⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣩⣿⡟⠠⣶⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣻⣿⣿⡟⠀⢻⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠗⢽⠁⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠦⠀⠤⢈⠀⡀
⠀⠀⠀⠀⠀⠰⢄⠀⠀⣠⡤⠀⢃⠐⠚⠾⠀⢈⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡿⠀⢴⡷⠺⡇⢀⠀⣿⣤⣄⡀⡰⢷⣆⡰⠖⠊⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⢌⣠⣀⠀⠀⠘⠁⢀⡤
⠀⠀⠀⠈⣵⠒⠺⢦⡀⠈⢢⡞⠡⠀⣾⡷⠃⣠⣯⠴⠤⠴⢾⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣠⣿⣿⣿⣿⣾⣿⣷⣇⡀⠈
⠀⢀⠀⠘⠧⢠⡄⠒⠓⠂⢾⣧⣥⣼⣿⢳⠿⠳⣤⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠀⢠⣄⣀⣀⣀⣼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣤⡀⠀⣠
⠀⠈⠀⠆⠀⠴⣿⣤⣴⠚⠙⣿⢻⢻⡙⣿⢷⠦⢼⡗⠆⠀⠀⠀⠁⠀⠀⠀⠐⠒⠉⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠤⠀⢨⠟⠶⡞⠃⠊⣶⣶⣿⠗⠀⢀⣵⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠸⣿⣿⣯⣿⣿⣿⣿⣾⣿⣏⡈⠀⠀⠉
⠀⠀⠀⠀⠀⠰⢓⡟⠀⡀⣠⢶⣴⠏⠈⣿⡀⠈⢻⠁⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⠁⡿⢻⣿⡿⠋⠻⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢈⣉⣀⠀⢴⠏⠷⠶⠟⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣷⠀⠀⠀⢻⡄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣿⣟⣤⣿⣿⣷⣿⣿⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢤⡀⠘⢣⡽⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠦⡈⠰⠟⡁⣠⢿⡟⢸⣿⣿⢿⠏⠀⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡡⠈⠈⢋⡟⢸⣿⠇⡇⠀⣼⠃⠀⠆⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢹⡇⠘
⠀⠀⠀⠀⣴⠶⢤⣿⢤⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⡇⣿⠁⣰⣇⡾⠁⣠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠆
⠀⠀⠀⠀⠙⠇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡈⢀⣧⣴⢋⠟⠀⣼⠋⢀⡴⠋⣀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⡀⠀⢀⠘⠀⠀⠀⠳⠀⣾⢡⡟⢀⣼⢑⡾⢋⣠⠞⣁⠔⠂⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⣁⣉⠉⠘⢳⡆⠀⡅⠀⢐⡁⣾⣷⣾⣷⣿⣷⣿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡀⡇⣴⠀⢰⠈⢇⡿⣡⣿⣿⣿⣿⣯⣤⣶⡶⠾⠟⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠛⢀⡜⠛⠛⣛⣄⢘⣤⡾⠟⢻⡿⠛⢿⣿⣭⣭⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢿⣿⠟⢸⣿⣿⣦⡀⢿⣿⣶⣅⣄⡀⠈⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⣸⡾⠁⢀⡼⠁⣤⢻⡯⢿⣿⣿⡆⢹⡛⠿⠿⠛⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⡼⠀⣰⠃⠌⠀⣾⡇⠙⢿⣄⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡼⠁⠀⠁⠰⢀⣸⣿⡍⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢠⣐⠯⠷⠛⠿⠶⣦⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠄⠀⠀⢧⠁⠀⠀⠀⠀⠀⠀⠀⣠⣞⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⡞⣧⡄
⠀⠀⠀⠀⢠⠀⠀⠘⠄⠀⠀⠀⠛⢒⠅⠄⠀⠀⠀⠀⠀⠀⢶⡿⣿⣿⠃⠀⠀⠀⠀⠈⠑⠦⠀⠀⠘⣿⣌⣷⡀
⠀⠀⠀⠀⠀⠁⠰⢤⣏⠀⠀⣁⡇⢙⠂⠀⠧⠄⠀⠀⠀⠀⣾⣓⣿⡧⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⣿⡻⣅⣏
⠀⠀⠀⠀⠀⠀⢁⣀⠁⣄⣤⢈⣵⠈⢋⠄⠀⠀⠀⠀⠀⠠⣾⡝⣮⢿⣴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣰⢋⣏
⠀⠀⠀⣄⠀⢆⣤⠙⠶⠈⣄⢉⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠲⡷⢾⣦⣭⣻⡳⢿⢦⡖⢶⠲⣴⣃⡿⠛⣉⢿⠅
⠀⠀⠀⠈⠁⠆⢰⣉⠹⣇⠹⠉⣦⣩⠐⠦⠀⣦⠀⠀⠀⠀⠀⠱⡾⠟⣿⣻⣿⡷⢷⡷⠾⣟⠛⣥⣸⠟⠋
⠀⠀⠀⠀⢠⠃⡸⠃⢀⠆⢛⣠⢐⡂⢲⡄⠀⠘⠀⣄⠀⠀⠀⠀⠈⠙⠻⡭⠿⣽⣏⠛⣾⣉⣼⣳⡀
⠀⠀⠀⠀⠐⣅⡇⢰⢀⣧⣿⠁⢿⢁⣤⡄⡠⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⡇⠂⠀⠶⣇⢠⠞⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠙⢿⣏⡇
⠀⠀⠀⠀⠀⠠⢶⡁⠑⢤⣤⡼⡎⣀⡘⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⢸⣧⣃⣴⣾⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⢗⠠⢸⡀⣦⠃⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣷⣄⢀⣀⣸⣿⣫⣿⠟⣱⣾⣿⠁
⠀⠀⠀⠀⠀⠠⡀⢸⡔⢇⠱⢙⠔⠁⠹⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣷⣽⣯⣴⣋⡥⠞⢫⣿⡇
⠀⠀⠀⠀⠀⠀⡸⡾⡿⣾⡑⣬⡀⣣⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠸⢁⡐⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠹⡄⡈⢳⢸⡟⡁⠠⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⣴⣿⣥⠞⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢌⢸⡄⠅⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣴⡿⠏⢀⡿⠗⣋⣉⣤⠞
⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠘⣷⡁⢸⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⣿⣁⢔⣵⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠈⠈⣧⡾⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠟⣫⣵⠿⢫⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣄⣸⣧⣇⡀⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⣾⣿⠏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠁⣀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⠂⠀⠀⢠⣿⣶⣿⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣤⣿⣶⣦⡀⠀⢸⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡎⠀⠁⠦⠀⠀⡀
⠲⣶⠀⠘⢛⣿⣿⣿⣶⣦⣀⣻⡓⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠸⣄⠁⠀⠀⠀⣠⣴⣷⣆⣀⡀
⠀⠛⠍⠤⡏⠻⠉⠁⠛⢿⣿⠙⣓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠤⡈⢲⡄⢠⠀⡀⠈⣹⡿⠿⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣽⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣙⢶⣿⣿⣼⡵⠞⠉⠀⢐⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣾⣿⣿⣿⣯⣵⣮⣤⣴⣾⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠂⠛⣿⢿⣿⢻⣿⣿⡁⠀⠉⠻⢿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⢠⣿⡿⠃⣨⡿⠘⡇⠘⣷⠄⢤
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⠀⣠⠞⠋⠀⡘⠀⠀⢿
⠀⠀⠀⣿⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠉⠠⠀⠀⢧⠰⡄⢺⠀⠉⠂⠀⠀⠀⠀⢠⠤⠄
⠀⠀⠀⠘⢧⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠱⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⡁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢠⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢸⠀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇


⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⡈⠠⠀⠨⡄⠈⠢
⠀⠀⠀⠶⠀⣴⠆⡎⠁⢰⣇⣀⡀⡴⣄⡠⠄⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠘⠀⣀
⠀⠀⠀⡸⠀⠳⢄⠀⢢⠏⠀⣼⡟⣠⠮⠔⠒⠒⣥⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⣤⣼⣿⢿⣿⣿⣷⣤
⠀⠄⠀⠉⢰⡆⠁⡠⠽⣶⣾⢿⣍⣉⢲⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠂⠤⢴⣿⣿⣿⣊⣿⣿⣿⣿⡆⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⡉⠋⣻⣄⡴⠃⢠⣠⣷⠍⠈⣃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⡏⠃⠁⠒
⠀⠀⠀⠀⠰⢲⠃⡀⣠⣤⡜⠉⣧⠀⠺⠁⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡽⠛⣿⠟⠻⡀
⠀⠀⠀⠀⠀⠉⠀⠀⢈⣄⠀⠰⠃⠒⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠨⣫⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠠⠟⢀⣷⠟⢻⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⣿⡟⡀⢀⡞⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⢹⠀⠃
⠀⠀⠀⣴⡒⢼⠓⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢱⡏⢠⣧⠎⢠⠖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠁⢺⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡿⢣⠃⣰⢋⣤⠊⡠⠀⠀⠀⠂⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠂⠐⢤⠀⠀⠀⢰⠃⣿⣴⣣⣾⣵⣯⠶⠊⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣘⣄⠀⢀⠘⣰⢏⣾⣿⣿⣿⣥⠤⠶⠖⠚⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⠉⣩⣧⣀⣶⣛⠙⣟⣛⠻⠾⠦⠦⣤⣤⣄⡀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀⠔⠈⡹⠋⢸⣿⣿⣧⡘⢿⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⡴⠁⡜⠀⠉⣾⠛⢧⡈⠢⣌⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠰⠁⠈⠀⢄⣼⡧⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⡦⡘⢧⠀⣀⢲⡅⢰⠖⠆⡀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⡱⢰⡾⠂⢳⠟⣿⢳⡼⣧⣴⡿⢠⣿⣿⣻⠇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠁⢸⣿⠀⣸⣿⣿⣾⡤⣿⣿⢁⣽⣏⣿⡏⣴⣿⢏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣐⠒⠒⢒⣂⡠⠤⠤⣤⠐⣿⣏⢿⣴⡿⣿⣷⣿⣧⠿⣿⠉⣤⣿⣽⡾⣫⡴⣻⡿⢁⣴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣶⣾⢿⣟⡛⠿⠖⠚⣉⣀⡱⢦⠤⢤⣀⣸⠻⣮⣽⣾⣿⣿⢉⠏⣸⢇⡾⠫⣉⡿⣼⢯⡴⣿⣴⣿⣇⣼⣵
⠀⠀⠀⠀⠀⣠⡤⠞⠛⠁⠉⠀⠀⠀⠀⠒⠖⠙⠈⠀⠻⣄⠑⠀⢀⠹⣧⢸⡿⣿⢹⣿⣟⣴⣯⣾⣣⣾⣿⡶⢫⡀⠞⣵⣯⣴⣿⢟⣥⠀⣠⡄
⠀⠀⠠⠊⠉⠀⠀⠀⢀⡀⠠⠤⠔⠒⠒⠛⠖⠒⠘⠃⠈⠡⣠⣤⣈⣀⡞⠘⢧⣿⢿⡿⣼⣿⣿⢮⣿⣿⣿⣷⢿⣽⠞⡁⠉⣾⡿⠋⣡⡾⠋
⡰⠇⠀⠀⠐⠠⠞⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢤⣬⡁⠈⠙⢻⡀⠈⢿⣿⠱⣯⡟⢁⣼⣿⣿⡾⠇⣪⡷⣨⣾⠋⣿⣿⣾⣫⣤⣴⡦⡀
⢳⡏⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⣠⣶⠶⠖⠒⠐⠋⠙⠃⢲⣿⣤⣿⣶⣽⣶⣞⣡⡴⣿⣿⣿⣷⣏⣸⣿⣿⣿⣿⣿⣿⡶⠋
⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢁⠄⠀⢉⣀⣡⠤⠖⠒⠿⣷⠀⢀⡘⠋⠀⣈⣿⠋⣻⠿⠿⢿⣿⣷⣿⣿⣿⣽⣟⣛⡻⠿⣫⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠴⠃⢠⡴⠛⠉⠉⠀⣀⣴⠖⣋⣥⣄⣀⣤⣾⢻⣿⣷⣾⢏⣸⣶⣿⣽⠛⣛⣩⠾⠛⠛⠛⠛⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢠⢦⠄⡔⢋⡤⢒⣠⡴⢚⣡⠖⠀⠉⠉⠉⣹⠟⢡⣿⡏⠈⠻⣟⢿⣄⡈⠽⢺⣿⣓⠾⢿⣭⣭⣟⣿⡿⢿⣷⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠭⠀⠀⣮⠃⣾⢔⣭⠖⣋⣵⠾⠋⠁⠀⠀⠀⠀⡾⢃⣾⠟⢉⣶⠀⢰⡍⣠⣍⠛⠛⠛⠉⠙⠿⣢⣽⣤⣨⣅⢶⡌⣷⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣾⠈⢘⣥⣾⣷⠟⠋⠀⠀⠀⠀⠀⠀⢀⡜⢠⣾⠋⠀⣼⠏⣦⡀⣡⣿⣿⠷⣤⣠⣄⠀⢹⠇⠀⠙⢷⣝⣦⠀⣿⡇⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣠⠿⠝⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⣿⠃⡐⣼⠃⣾⣿⡿⠛⠘⣉⡶⣷⣯⡝⣷⣯⠀⠀⠀⠈⣿⣿⣆⠹⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢷⡄⠀⠀⠀⠀⠀⢀⡤⠊⠀⣰⣾⠇⠋⣰⠏⢘⣿⢹⠀⣠⡼⣫⣾⢋⣼⠇⢋⣿⣆⠀⠀⣸⣿⣿⠿⠆⢸⢟⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢀⣴⣿⠂⠀⣴⡟⠀⢠⠟⡇⢠⡇⣾⣿⣷⠿⠋⠀⠸⢫⡿⣄⣰⣿⣿⣿⣃⢰⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣒⣥⣤⡻⠟⣫⢟⣡⣾⣿⣿⠃⢐⡃⢠⠁⣼⠟⠋⠉⠁⠀⠀⠀⠀⡿⣶⠈⣿⠿⠋⠑⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣛⠛⠋⠁⠀⢈⣫⠴⠋⢹⣹⣿⡿⣠⡾⠁⡞⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡞⣧⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢸⢸⡟⣵⠟⠀⣴⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢻⣇⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡼⣡⡶⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡄⠙⣦⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠛⢡⡾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢻⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣺⣽⠶⠟⠋⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠋⠁⠀⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⠟⠉⠉⠙⠻⣿⣿⡎⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠈⢿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⢶⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⠇⢀⣴⣶⠿⢛⣩⣥⣴⣶⣶⣿⡿⠟⠋⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣰⡿⢋⣥⣾⡿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⣷⣿⣷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠁⠉⠀⠀⠀⠀⠀⠐⠃⠈⠃⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠄⠀⠄⠀⠁⠀⠀⠙⠀⠂⠀⠄⠀⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣀⡀⠀⠀⠀⢀⣠⣴⣶⠾⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠉⠙⢦⣴⣯⠩⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡰⠻⢤⠦⠀⠀⠀⠀⠀⠀⡸⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠑⠂⠄⠴⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⡰⠊⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠐⠰⠉⠀⠀⠀⠀⢀⠀⠀⢁⠀⠀⠀⠀⠐⠀⣼⣦⢀⡞⠀⢀⡴⠂⠀⠀⠀⣀⣠⠤⠤⠦⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢇⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⡀⠀⠀⠀⢉⠀⠀⠀⠀⢀⢰⠉⢡⡏⠀⣰⠟⠀⠀⠘⣀⠤⠅⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣼⣷⣄⠀⠀⠀⠡⠀⡄⠀⠀⠁⠀⠤⠦⠀⠀⠀⠀⠀⢀⣀⣠⠈⢀⡾⠃⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢀⠂⠨⠁
⠀⠀⢀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠛⠻⢿⣶⣶⡄⡀⢣⠀⠀⡾⠳⣶⢧⣤⣷⠶⠶⠦⡤⢩⣤⡀⢠⣼⣤⠄⢴⡻⣦⡀⠀⠀⠀⠀⠀⣀⢃⠀⠐⠀⢦⡀
⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠻⣿⣦⣧⡈⠰⢸⡿⣦⡀⠙⠠⢦⡀⢀⣀⣉⣯⡞⣈⠋⠲⢿⠇⣸⡇⠀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⢸
⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡲⠀⠙⢿⣯⣦⣘⣿⡟⣷⣴⣾⣦⣝⣿⡿⠿⣿⣿⠋⣁⣤⣭⣞⠋⠐⢣⡴⠊⠀⠀⠀⠤⠀⢀⣤⣴⠟
⠀⠀⠈⠀⡀⠀⣀⠶⠖⠒⠒⠂⠘⠛⠛⠙⠙⣿⣓⣶⣶⠟⠛⠁⠀⡾⠿⠷⠿⣇⡼⠃⠀⠀⢸⡧⠘⢿⠖⣽⣿⣿⢇⣀⣤⣤⡦⢺⡤⡴⣒⣿⠟⠁
⠀⢀⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣶⡤⠐⡀⠀⠀⢀⣈⣁⠀⢈⠀⢸⣧⣤⠚⠚⢿⣿⣿⣾⣿⣿⣿⠿⠤⠿⠛⠉
⣰⠃⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⣀⣄⣠⣤⣾⣛⣿⣿⣿⣯⣥⠠⢿⡤⠀⢛⣾⣿⣿⣿⡷⠀⠈⠉⠻⣆⠀⣹⣬⣷⢌⡉⡩⠐⠶⠦⣄⡀
⠈⠳⣤⡀⣀⡀⣬⡷⣶⣿⡿⢷⣿⣿⡿⢻⢿⣿⢿⡋⣽⠟⢹⠃⢂⠋⠀⠀⠛⠻⠻⠛⠋⢀⡀⠀⠁⢀⣼⣿⣯⣇⠐⠀⠂⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠙⠁⠀⢈⣉⡁⠀⠰⢤⣴⣶⠞⠋⣠⣞⡛⠣⢺⣷⣾⣦⣬⣀⣀⣠⣴⣄⠈⠄⠀⣸⡟⣿⡛⢻⢻⡿⠏⠙⠁⠲⢦⣄⡀⠀⠀⠀⠀⠀⠙
⠀⠀⠠⠀⠀⣀⣈⣶⡎⢂⣹⣼⠟⠁⠀⠞⠋⠈⢁⣼⠏⠐⢉⡿⢻⣷⠉⢁⡿⣿⢦⣤⣶⣿⣷⢦⣄⡸⠉⠀⠀⠀⠀⠁⠀⠈⠙⣤⡀⠀⠀⠐
⠀⠀⠘⠆⣠⣠⡉⠽⠛⠋⡡⠃⢀⠐⠀⠀⢀⡴⠿⠃⠀⢰⣾⣷⠿⣥⣾⠆⠀⣿⠆⣿⡀⠛⠗⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⢀⡴⠟⠑⠀⠀⠀⡼⠹⠁⠀⠀⢀⠄⠀⣿⡀⢸⡇⠩⡀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠀⠀⣀⣀⣩⣷⡿⠀⠄⠂⠀⠀⠀⠀⠀⢀⡾⠀⠀⣟⠁⣼⣷⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣿⣷⠊⠀⠀⠀⠀⠀⠀⠀⡀⢾⠁⠀⠀⠃⠀⣿⣸⣧⡀⠀⢀⠀⠀⠀⠀⠁⠀⣀⣀⡀⠤⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠠⠊⠁⠸⠀⠀⠐⠀⢰⣿⣿⣿⣷⡄⠀⠐⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣀⠔⠒⡄⠀⠀⠀⠀⢀⣾⡟⣿⣿⣾⣿⣷⣄⡀⢀⣠⠞⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠢⣀⠀⣠⡟⢸⠍⣿⢁⠈⠛⢇⣈⠙⣯⣳⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠧⣿⠀⠀⠀⢨⣿⣾⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠙⠁⠑⡀⢸⠟⠁⠘⣿⡏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢇⠲⣬⠄⠐⣾⡄⠠⡴⢸⣿⡻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡛⠂⣄⣼⠇⠀⣴⣄⢻⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⠃⣈⠛⡀⠈⢉⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡋⣈⢷⡄⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⣀⠙⠛⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣣⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⠻⢣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠉⠤⠁⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⢠⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠀⡀⢀⠀⠀⠀⠀⠠⠂
⠀⠀⠱⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣺⣠⡄⢌⣻⡃⢀⡴⠀⣠⠖⠁
⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⣀⡀⠀⢦⠲⣛⣙⢿⠛⠩⣟⣿⣿⣿⣿⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⢳⢇⠴⡄⠒⠏⡏⢾⡹⢯⣿⢿⡠⠀⠀⠀⠀⠀⠀⠤⠐⠒⠉
⠀⠀⠀⠈⠐⠂⢄⡀⠈⠉⠓⠒⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⢈⣽⣟⢊⠃⡆⠙⠙⢹⡿⣣⢶⣿⡿⠛⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠤⢌⣀⣁⠀⣀⣀⡀⠀⣤⠿⠄⣧⢻⢸⢠⡆⡅⣼⠃⡿⢱⠔⠋⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡈⠀⣎⠛⠘⠉⠉⠟⢳⢴⠇⠂⠀⣀⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠀⠀⠈⠳⣦⡘⠀⠀⠀⠀⠀⠈⠈⡴⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠈⣉⠴⠓⡀⠀⠏⠓⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⡀⠀⡸⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⠀⠀⠀⡰⠃⠁⠀⠄⠄⣤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡼⣡⡖⠁⠀⠀⠂⠀⠀⠀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣠⠞⢛⡷⠛⠀⠀⠀⠀⡜⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⡞⠁⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⠑⢄⡀⠀⠀⠀⢻⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢛⡻⠻⡓⠢⠀⡀⠢⡑
⠀⠀⢠⡄⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⠶⠚⠛⠙⠁⢠⠊⠒⡀⠈⡦⠀⠈⠀⡉⢳⠣⠀⢀
⠀⠀⠘⠈⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢁⠀⢠⠊⣀⠀⠐⡌⠀⠇⢠
⠀⠀⡟⢃⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠍⠑⠂⠐⠋⠀⠀⢐⠄⠀⠀⢮⢀⢀⠌⠁⠀⠉⢻⡀⠀⠺⡆
⠀⠀⢻⠈⡸⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠁⠀⠠⡀⠉⠀⠀⠀⠘⡼⠃⠀⠀⠀⢡⠈⢸⠏⣸⠃
⠀⠀⢸⣦⡇⠆⢀⠠⠀⠤⠀⠀⠀⠀⡀⠈⢠⡝⣀⣀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠅⠀⠀⠄⠀⠈⠀⠈⠀⣼
⠀⠀⠀⣿⡇⠀⡸⢆⣿⠦⠀⠃⠀⠀⠀⠞⡿⠋⠋⠈⠓⣾⡤⠀⠄⠀⠈⠐⢀⣤⡄⠀⠀⠀⢠⣧⠀⣦⡏
⠀⠀⠀⠘⣷⡿⠁⠂⣇⡀⢈⠀⠁⣀⢜⠅⠀⠀⠀⠀⠀⠸⡴⠀⠀⠘⡀⢐⠼⣯⣿⣲⠄⡴⢉⠘⣇⡟
⠀⠀⠀⠀⠙⡇⠀⡀⠀⠀⠈⠀⠔⠁⡚⢀⠆⡀⠂⡐⠀⠀⠀⠀⠀⠉⡛⡪⠳⠛⢿⡿⠚⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⡄⠀⢄⠀⠀⠀⠀⠀⡀⠉⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠐⡂⡏⠓⣴⠏
⠀⠀⠀⠀⠀⠀⠙⡀⠀⠉⠉⠀⠴⠀⡁⣤⡀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣇⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⣴⡀⠴⢃⣱⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠉⠸⠥⢤⣁⡀⠀⠀⠀⠀⣀⣀⠠⠤⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⡔⣂⠀⠀⠀⠀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠐⠀⠀⠀⣰⠛⠤⠀⠀⠀⠀⠉⢢⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⠼⠑⠄⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠀⠀⠀⠀⠀⠀⠙⠙⠂⠀⠀⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⡎⡖⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⣤⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⡀⠀⠃⠀⡀⠀⠀⡠⠀⢓⠴⠊⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡠⠀⠀⠑⢤⠠⣡⠜⠉⠀⠀⠀⠀⠀⠀⠀⣠⡄⡠⠒⠰⠈⠁⠛⠁
⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡠⠀⠴⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠲⠀⠀⠉
⠀⠀⠀⠀⠀⠂⠂⠀⠐⠁⢀⠀⠊⠀⠈⠠⠄⠈
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐⠁⠀⠀⠂⠀⠀⠀⠈⠁
⠀⠀⠀⡰⠁⠈⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⢰⣧⣡⠥⠒
⠀⠀⠛⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠃⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠤⡀⠀⠄⠀⠀⢀⠀⠠⡀⠀⣿
⣿⣿⣿⣿⣿⣁⠙⣿⣿⣿⣿⣿⡿⠀⢻⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢷⣦⠀⠀⠀⠀⠀⢰⣿⣰⣿⠆
⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣧⣀⣤⣄⣬⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠐⠀⠺⠿⠛⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠔⠈⠁⠉⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢀⠀⡏⠀⣄⢀⣤⡀
⠀⠄⠀⢀⠀⠀⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠨⠀⢳⠗⠛⠛⢹⡷⠀⠀⡄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⢀⣸⠏⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠠⠄⠸⠤⠄⠂
⠀⠀⠀⠈⠒⠀⠀⠀⠂⠀⠀⠀⠈⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄⠀⠀⠀⠠⡂⠁⠄⢀⠈⠄⠄⠀⢀⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡀⠘⢛⠿⢿⣿⣿⣿⣿⡆⠀⠄⠀⠂⠐⠐⠂⠀⠐⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠈⠀⠄
⠀⠁⠀⠠⡀⠙⢿⣿⣿⣿⠇⠀⠀⡐⠠⠐⠐⠀⠀⠈⠀⠀⠸⣯⠡⠀⠍⠠⢈⡿⠃⠀⠀⢀⠠⠀⠀⡄
⠀⠀⠀⠀⠙⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠂⢰⠃⠀⢀⠀⠀⠀⢀⣄⣄⡰⢂⢂⠼⡦⠘⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠄⡀⠠⠁
⠀⠀⠀⠀⠀⠓⠰⡜⣀⡏⡡⡅⣜⠄⠈⡆⠀⢁⣀⡼⠁⢀⠀⠁⢲⡀⡈⠃⢀⣀⠀⡀⠤⢤⠄⠤⠤
⠀⠀⠀⠀⠸⠱⠀⠐⠃⠈⠀⠀⠒⠀⢀⢁⢀⡄⠀⠀⢄⣀⡤⢀⡈⠀⢸⡸⢷⣁⡁⢟⢸⣸⠸⠅
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠒⠄⢸⢤⠦⡻⣘⣨⢋⡹⠁⠈⠈⠃⡙⠁⠀⠉⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣤⡀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠐⠂⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠈⠀⠃⣀⠈⣡⣄⡠⠤⢦⡀⠀⠀⠀⢴⠂⡄⠀⠀⣀⠀⠀⠸⡀⠀⢀⠀⡄⢲⠢⠄⣀⣤⢤⣀⠰⠞⡆
⠀⠀⣵⢋⣡⣴⣶⠿⡁⡄⢠⣰⠋⢦⡀⠁⢠⣾⡆⣀⠀⠀⠀⣬⣤⣌⣦⠀⢀⣀⠀⢹⣠⣧⡦⣦⣾⠰⠒⡇
⠀⠀⢰⣯⠛⢨⢸⡆⠘⢠⠃⡗⠱⠛⢧⡀⣘⠰⠙⠋⢣⢄⣾⣏⡋⢁⣿⢓⡯⠈⠀⡟⡼⡏⠡⢁⣿⣇⢠⠇
⠀⠀⠁⠈⠳⠒⠋⢠⠚⡁⠁⣧⡀⢀⣿⡞⣻⣰⠛⡀⡎⣀⡭⡿⣭⡋⠘⠁⠙⠶⣿⢮⡐⣛⣿⣿⠿⣯⣿⢻
⠀⠀⠀⢅⣀⡴⡊⢉⣠⣽⣦⠀⠙⡛⠹⡋⠈⠀⠱⡡⢻⡁⠆⠀⢸⡅⠀⢸⣭⣤⡥⢀⡥⡭⠭⢩⢾⡇⠸⠹
⠀⠀⠀⣰⡍⠉⢻⠋⢽⠄⣹⠀⢸⠀⣧⡋⠛⢻⠳⡏⠀⠳⡤⡖⢉⠛⠄⣾⡿⠛⣿⠉⠘⣇⠞⢀⣼⡁⠀⠇
⠀⠀⠀⢡⣴⠄⢬⣷⡴⡾⠃⢐⣩⣤⣼⣇⣐⣃⡞⡑⣠⡾⠁⠘⢻⣆⢰⠉⡟⠝⣺⡠⠂⠈⠛⠋⠀⠀⡰
⠀⠀⡈⠉⢂⢀⠴⠟⢳⡄⠀⡷⠟⠹⣿⠽⠙⣡⣄⠁⠫⡿⠊⠘⢛⡆⢩⡛⠖⠾⠉⠠⠄⢀⡆⣷⠀⡅
⠀⠰⡏⢣⠈⢻⣀⣀⡹⣧⠄⢻⣄⣠⡯⡀⢲⢧⠚⠀⠀⢹⠤⠤⠼⡀⡃⠌⠀⠀⣀⡠⠤⠊⠀⠁⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣤⣶⣦⣀⣠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢾⣿⠀⠙⢿⣦⠀⠀⠀⢀⣾⡟⢉⣿⣿⠟⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣾⡿⢿⣷⡀⢻⣯⠀⠀⠈⢿⣧⠀⠀⣾⡟⠀⣸⣿⠋⠀⠀⣿⡇⠀⢀⣴⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡉⠙⠿⣦⣿⣿⣿⣿⠀⠀⠀⠀⣿⣇⣰⣿⠁⢰⡿⠁⠀⠀⢠⣿⣧⣶⡿⠋⠁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠻⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⡇⠀⣿⠃⠀⠀⢀⣾⣿⡿⠋⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣄⡀⠀⠀⠀⢻⣧⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⣿⣿⠀⣼⠏⠀⠀⢠⣾⡿⠋⠀⠀⠀⢀⣿⠏⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠈⠙⠻⢷⣤⠀⠀⠻⣦⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⢸⡟⢀⡿⠀⠀⢠⣾⠏⠀⠀⠀⠀⣠⣿⣿⣴⣶⡿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠙⠻⣦⡀⠘⢧⠀⠀⠀⠈⢿⡆⠀⠀⠀⢸⡇⢸⡇⠀⢠⡿⠁⠀⠀⠀⢀⣴⣿⡿⠛⠉⠁⠀⠀⣠⣾⠟
⠀⠀⠀⢀⣤⣤⣤⣤⣤⣽⣿⣦⣄⠀⠀⠀⠈⠳⣤⠀⠓⠀⠐⠀⠘⣷⠀⠀⠀⢸⡇⡆⠀⣰⡟⠁⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⣠⣾⡟⠁⣀⣀⣀
⠀⠀⠀⠸⣿⣍⠉⠉⠉⠙⠻⠿⣿⣷⣦⡀⠀⠀⠀⠐⣄⠀⢀⠃⣀⣿⣶⣿⣶⣿⣷⣧⣠⡟⢀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣀⣴⣿⣿⡿⠿⠿⠟⠛⠻⣿
⠀⠀⠀⠀⠈⣻⣿⣶⣤⣤⣤⣤⣤⣉⣿⣿⣷⣄⠑⠤⡈⢳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡔⠉⠀⠀⠀⣀⣤⣶⠿⠟⠛⠉⠀⠀⠀⠀⢀⣠⣾⠿
⠀⢀⣤⣾⡿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⡾⠿⠛⠋⠁⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠁
⠀⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⠶⢶⣶⣶⣶⣿⣿⣿⡟⠋⠁
⠀⠙⠻⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣤⣀
⠀⠀⠀⣀⣤⣾⡿⡿⠶⣶⣶⣦⣤⣤⣶⣾⠿⠟⠛⢉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿
⢀⣴⡿⠟⠋⠀⠀⠀⠀⢀⣠⣼⡿⠛⠁⠀⠀⠄⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠙⠻⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿
⣿⣿⣤⣤⣤⣤⣤⣶⣾⡿⠟⠁⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠛⠛⣿⣿⣿⣿⣿⣿⣿⡉⠃⠙⢷⣄⠀⠀⠀⠈⠙⠿⣿⣍⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠈⠉⠉⠉⠉⢁⣴⡿⠋⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠋⠀⠀⠀⢀⣾⠋⠀⠀⣿⡇⠀⠘⢷⡄⠀⠀⠙⢷⣤⡀⠀⠂⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⢠⣿⡟⠀⠀⢀⣠⣴⣶⠿⠟⢉⣿⠟⠁⠀⠀⠀⢀⣾⠇⠀⠀⢰⣿⣷⠀⠀⠈⢿⣆⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠸⣿⣷⡾⠿⠟⠛⠉⠀⠀⣰⣿⠃⠀⠀⠀⡀⢠⣿⡿⠀⠀⠀⣿⣿⣿⡆⠀⠀⠘⣿⣆⠀⠀⠀⠙⣿⡿⢷⣤⣀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⡴⣡⣿⣿⠁⠀⠀⢰⣿⠙⣿⣿⡀⠀⠀⠘⣿⣧⡀⠀⠀⠹⣿⡄⠙⠻⢿⣷⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢀⣴⠟⣠⣿⣿⡟⠀⠀⠀⣾⡏⠀⠘⣿⣧⡀⠀⠀⢹⣿⣷⡀⠀⠀⢹⣷⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣴⡿⢁⣼⡿⠋⣿⡇⠀⠀⣸⣿⠁⠀⠀⢻⣿⣷⡀⠀⢸⣿⠛⢿⣦⣄⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⢠⣾⠟⠁⠀⣿⡇⠀⢠⣿⠃⠀⠀⠀⠘⣿⡿⣿⣄⣼⡿⠀⠀⠙⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⠀⠀⣿⣇⣠⣿⠏⠀⠀⠀⠀⠀⣿⡇⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣦⣄⣀⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣷⣦⣽⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣠⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⡄⠀⠀⠀⠀⢸⣿⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⣸⣿⠀⠀⢸⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⣿⡇⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⣿⡇⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢹⠆⠀⠀⢀⣤⣠⡏⠉⠃⠀⠈⡗⠒⠀⠐
⠀⠀⠀⠒⠶⡶⢼⡶⣶⣦⡀⠀⠀⠀⠂⠀⠀⠀⠂⢤⠤⣘⣤⡟⠲⡁⠠⠒⠱⠋⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡌⢿⠄⠈⠀⠀⠀⠀⠀⠀⡸⢐⠛⠀⠀⢀⣄⠀⠀⣤⠐⠉⠁⠀⢠⣾⢻⣻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣣⠀⡇⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠏⠈⠲⠒⠊⠉⠙⠋⠁⠀⠀⠀⠈
⠈⠺⡄⠀⠀⠀⠜⡿⠾⡟⠦⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠄⠀⠀⠉
⠀⠀⠈⢩⠖⠀⠀⡅⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠀⠀⠀⠋⠀⠀⣠
⠀⠀⠀⠈⠀⠀⠤⢰⡤⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶⠀⠃⠆⠀⠇⠶⠘⠰⠆⠖⠒⠓⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⢠⡴⠶⣶⠉⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠄
⢀⠔⠚⠂⠀⠈⠀⠀⠜⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣤⣾⡛⠛⠯⣃⢄
⢸⠄⠀⠀⠀⢀⣤⣤⡄⠀⠐⠋⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⣴⡜⠛⠦⠀⠙⠦⡈⢳⣃
⢀⠀⠀⢀⣨⣿⣭⣷⠁⠀⠀⠠⢄⠀⠀⠂⠐⢀⡀⠉⠀⠀⠀⠀⠆⠠⢤⣿⣤⣔⠉⠀⠇⠀⠀⠑⠄⠀⠱⠄⣿
⠈⠟⠉⠁⠀⠈⠉⠀⠀⠀⢀⣀⠈⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢡⠈⢇⠀⠀⠀⠀⠀⠈⠆⠀⣧⡟
⠀⠀⡀⢀⠄⠀⠀⢠⠀⠀⠀⠁⣠⠎⠀⠀⠀⠠⠀⠀⠀⠀⠢⣀⡀⢈⠀⠀⡰⢀⣿⠄⠀⠐⠀⠀⠀⢸⡴⠋
⣤⣤⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣥⣤⣽⣶⣶⣿⣬⣥⣤⣵⣤⣤⣵⣦⣥⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡠⠀⡀⠤
⠀⠀⠀⠀⠀⠐⠀⠘⠂⠀⢠⠅⢉⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠊⣡⠉⢁⠐⡄⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠄⠘⠁⡄⠀⢆⡹⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢃⠀⠀⠀⠀⢀⣀⠀⠀⠋⠀⡧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢁⠀⠀⣀
⠀⠀⠀⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⡀⠈⠣⠈⠘⢁⣠⠀⣀⠀⠀⠀⠀⠠⠤⡀⠘⣀⠣⠐⠈⠁
⠀⠀⠀⠂⠢⡤⡀⢆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠒⢀⠀⡓⠈⠤⢤⡘⠀⠀⠀⠀⠀⠴⠂⠲⠊⢀⠁⠉⠱
⠀⠀⠀⠀⠄⣨⡼⠋⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡅⡼⠋⢠⢌⠐⠦⠀⠨⠀⠀⢀⠀⠀⠰⠋⡄⠀⡄⢧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠤⠴⠋⠉⠘⠑⠋⠢⡀⠀⠀⠀⠇⠀⠀⡀⢁⠏⠐⠄⢛⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠣⠈⠀⠐⠁⠁⠆⠀⠀⠀⠀⠀⠀⠄⠈⡖⠉⡆⣀
⠀⠀⠀⠀⢈⡼⠀⠀⠀⠀⠱⡀⠀⠀⢰⠀⡀⢁⠘⠃⠀⠀⠘⠔⠈⠤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠁⠲⡄⠀⠰⠃⢀⠂
⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠘⡄⠀⢸⠀⠃⡌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢄⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠠⢄⠐⠉⠈⠁
⠀⠀⠀⠀⠀⢤⠤⢠⡀⠀⠀⠀⠹⡀⠘⡆⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⡀⢹⣦⣿⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⠠⣴⠊⢀⠲⠀⡀⠉
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⠢⣀⠀⢣⢸⣇⠃⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠔⠋⠻⠿⠿⡀⠀⠀⠀⠀⠇⠀⠀⢀⣾⡗⠃⠀⠞⢐⡀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠖⡀⠀⠉⣀⡈⡄⡏⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣤⠀⠀⠀⠈⢦⠀⠀⠀⡆⠀⢸⣿⣿⡄⠀⠀⠀⡨⠁⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⡉⠀⣿⣇⠐⠁⠀⠀⠀⠀⠀⠀⣀⠀⠰⠀⠀⠀⠘⠃⠠⠀⠀⠀⠀⢣⠀⠀⠇⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⢠⠌⢁⡀⢾⡏⠀⠄⠀⠐⠀⠐⠀⠈⢀⡀⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢃⠀⢠⣼⣿⢯⡟
⠀⠀⠀⠀⠀⠖⠶⠀⠀⡄⠀⡀⢈⡀⢸⣧⠀⢠⠄⣠⡄⠀⠴⠂⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠈⠆⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⠀⠸⢸⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣶⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠙⠿⢿⣿⡟⢁⣀⣠⣤⣀⣤⣠⣴⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠛⠛⠋⠉⡟⠛⠛⠉⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⢰⡄⠁⠀⠉⠤⡀⠀⡀⠀⣴⣾⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⣄
⠀⠀⣦⠀⣰⠀⠋⡀⠀⠀⠀⣤⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠃⠀⠀⣤⠀⠋⣤⠀⠀⠐⡟⠀⡀⠀⠛⠀⠀⠀⢠⡀⠀⠀⠘⢦⠀⠠
⡳⣄⠀⠀⠀⠀⠈⠁⡀⠀⠀⡿⣀⠀⣴⡀⠀⢘⠋⠀⢣⠀⠀⢱⡄⠀⠾⠀⠀⠀⠋⠀⠈⠛⢻⡄⠀⠀⠀⠿⣦⡀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠀⡄
⠀⢈⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⣄⠉⢳⡀⠈⠇⠀⠺⠀⠀⠘⢀⣄⠀⠀⠀⢀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣿⠀⠀⠀⠀⢀⣧
⣀⣸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣸⣀⣀⣻⣀⣀⣢⣀⣂⣀⣁⣈⣿⣄⣀⣀⣸⣀⣀⣀⣟⣀⣴⣀⣀⣀⣀⣆⣀⣀⣉⣀⣀⣉⣆⣀⣀⣀⣘⣃⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⣤⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⢸⣧⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⢀⣼⠗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣶⡶⠟⠁
⠀⠀⠀⠀⢠⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⡿⠋⠁⠀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣄⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⡉⠉⢹⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠁⠀⣱⠀⠀⠀⢰⠁⠀⠱
⠀⠀⠀⠀⠀⢀⡠⠔⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡠⠔⠚⠋⠉⠉⠳⠖⠄⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠐⠦⣄⡀⠀⠀⣰⠏⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢂⣀
⠀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⣥⣤⣤⡶⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⣷⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣼⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⡟⠋⠉⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣦⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟⠛⢿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣧⣤⣼⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠖⠉⡒⠒⣒⣿⣿⣦⣍⡉⠛⡦⣝⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣫⡢⠞⠋⠉⢙⣾⣿⣿⣶⣼⣿⣷⣬⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣱⡟⡇⢰⣏⣔⡬⠶⠚⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣪⢞⣼⣷⢸⣏⡁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡣⡀⠣⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡻⣿⣿⣿⡶⣤⡀⠀⠈⠉⢉⡙⠢⠀⠹⣿⣷⡄⠱⣌⠣⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡸⡱⣿⣿⡏⠀⠈⠀⠀⠀⠀⠛⠙⠀⠀⠀⡿⣷⠝⢦⡘⢷⣌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡰⡑⣹⢹⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⢇⣿⠣⡐⢌⣓⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠇⠈⢸⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣧⣿⣦⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⢀⢢⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡸⠺⣸⣿⣷⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⢹⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣷⣶⣽⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢇⢆⢻⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡕⢉⢜⢽⡿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠗⠋⠀⣨⣿⡇⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⣠⠖⣠⡆⢀⡾⣡⠃⡞⡼⢹⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡘⠹⡅⢸⡱⢃⢎⢜⣍⣾⡇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡻⣶⣿⠘⢠⠂⢊⣾⢞⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢹⣧⣸⢸⠀⡞⡱⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠈⠻⣎⢀⠐⡀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢐⡽⢾⡣⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⠻⠛⡛⡛⢉⣋⣈⣙⣛⢛⠛⡛⢛⣙⡛⡛⠛⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠋⠙⡄⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠃⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢃⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⠀⠀⣠⠞⠁⣠⣶⣶⣲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣦⣾⡔⢀⠀⠀⠀⢆⠉⠀⣶⡇⢸⣷⣯⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⣿⣷⢮⣤⢄⣄⢻⡿⠘⢿⣼⣿⡏⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠠⠛⢿⠾⢀⣉⣺⡇⣸⢃⡨⠷⡴⢊⠡⢒⡨⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠶⢮⡓⢿⡺⢃⡔⣻⣿⢁⡿⢛⣴⣨⠌⠱⡾⠷⠦
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡾⢺⣥⣤⣘⣯⣑⣅⣌⠲⣿⣿⣼⡗⣫⠞⠚⠋⠉⢀⣀⣀⣀
⠀⠀⠀⠀⡀⣦⡙⣿⣷⠃⡀⠈⠱⣿⣿⠟⢿⣼⡎⣹⡻⢯⡿⢋⣠⢴⣲⡾⠛⠀⠀⠉⠁
⠀⠀⠀⣀⣼⠻⣧⣽⣿⣿⣯⢉⡉⠩⣶⢍⠢⢿⣷⣿⣽⣦⣳⣾⢗⢏⠍
⠀⠀⣴⡿⢿⣟⣿⡿⣿⣿⣝⡉⣀⣀⢹⡃⠕⣬⢻⢿⣿⡏⣤⠝⣲⣍⠤
⠀⠀⠁⠀⣙⣿⣿⣿⣿⡋⣀⡈⠅⠛⠉⠻⣾⣿⡟⢶⡛⢿⣿⡼⠞⣙⣳⡴⠊
⠀⠀⠀⠀⡉⣿⣙⣻⢿⡟⢡⠀⠀⠀⠀⢤⡟⡍⢻⣾⣿⡜⡬⣴⣾⡏⡈
⠀⠀⠀⢠⣸⠛⣿⣳⣿⣿⣏⡤⠀⠀⠀⢼⠐⢿⣼⡿⠻⣷⣷⣞⣿⢿⠴⣯⠒
⠀⠀⠀⢈⣿⣶⣿⣿⣿⣏⣧⠄⠀⠀⠀⣀⣽⣼⣿⣷⡀⣹⣟⣿⣷⣿⡄⠃
⠀⠀⢀⣾⣟⣿⣻⣾⣿⣿⣿⣆⠶⠦⣾⣃⣱⣟⢻⣿⣎⣿⣿⢏⣇⣾⡟⠂
⠀⠴⠟⠛⠿⠿⠿⢿⣿⠿⠿⠫⠍⠀⠄⢈⣭⣿⣾⣿⣿⡸⢿⠚⠋⣼⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡠⢾⣼⣧⣿⣿⠿⣿⣇⠈⣸⣅⢹⣿⢰⢲⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣞⣿⢱⣿⠇⣟⣶⢸⢸⡇⢃⣿⣌⣿⡆⣠⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣻⢻⡿⡀⢻⢿⢔⢿⣇⣜⡹⣿⣿⣷⣿⣧⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢽⣾⢿⣾⣿⣿⣩⢀⣿⣾⡏⡟⡟⣿⣿⡿⣯⢹⡟⣿
⠀⠀⠀⠀⣄⣀⣀⠠⠤⢤⣤⣰⢾⣿⣷⣿⡟⣰⢹⣧⣸⣿⣷⣿⣧⣻⣿⢹⣿⣾⣷⣿⠆
⠀⠀⠀⠘⠼⡗⡡⣠⠖⣃⠿⠾⣿⡟⢿⢿⣷⡿⢸⣿⣿⢸⡿⣿⢿⣿⣟⣿⣿⣿⣿⣿⠾⠖⢒⠶⢶⡴⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⣿⣷⠁⡾⠁⠀⢀⡇⢹⠀⡇⣻⡏⠻⢿⣿⣴⣾⣽⡸⣿⣻⡻⣿⣿⡟⠁⣠⠊⢠⠂⠁⢹⡟⣏⠹⡙⣦⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠁
⠀⠀⠀⢠⠄⠉⡻⢺⡇⠀⣤⣾⠧⠿⠾⣷⠋⢭⣄⣸⡏⡿⣿⣿⣿⣿⣿⣽⣿⣿⣇⢀⡇⢰⡇⠀⠀⣼⠁⣼⢰⣇⣿⠄⣼⣅⣤⣶⡉⢱⣌⡢
⠀⠀⠀⠐⡆⣐⣬⣿⣷⣼⣫⡼⠶⢅⠐⠠⡀⣠⣿⠿⣿⡿⢿⢌⠻⣿⣿⡟⡿⢙⣻⠾⠟⣙⣧⣶⡞⢻⣾⣿⣾⣿⡟⠺⠯⠿⣍⡱⣶⢀⡈
⠀⠀⠀⠀⠈⠿⣹⡞⢡⢠⡏⢀⠀⠈⢳⡄⣻⣶⣺⡦⣌⣥⡀⣄⢠⣙⠛⠡⣶⣯⠈⠑⠚⠪⣿⢑⢔⢿⣛⡿⢻⣻⣾⣿⡿⠤⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣇⢸⣀⣧⡘⣧⣄⣸⣿⣉⢻⣷⣅⣤⣴⣯⣥⡿⠏⠛⠒⠋⠁⠀⠀⠀⠀⠈⠉⠉⠈⢉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢽⣿⠿⠶⠞⠄⠼⣟⣽⠋⠹⠁⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣦⠀⡴⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⣸⣧⡀⠀⠀⢻⣷⡄⠀⢀⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⢨⣿⠇⠀⡿⠉⠻⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢰⠿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⣴⡆⠐⢂⣼⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠙⠓⢿⣦⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢄⣘⢿⡀⠀⣽⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈⠁⠀⠘⣿⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠀⠙⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣯⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠓⡦
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠿⣿⣿⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⢀⡠⠴⠚⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠈
⣤⣤⣤⣨⣿⣿⣿⣿⣿⣿⣿⣡⣤⣠⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣠⣤⣠⣠⣤⣠⣤⣄⣤⣀⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣿⣽⣭⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣭⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠏⠸⠀⠀⠀⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⠲⢲⡆⠈⠀⠀⠀⣤⢤⡀⠀⠀⠀⠔⠛⢻⣿⠀⡀⠄⠐⣶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠏⠉⠉⠐⠄⠀⡿⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⢀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⢠⢁⡟⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠁⠂⠈⠣⠀⠀⠀⠀⠘⡄⠠⠀⠀⠀⠀⡌⣼⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⢀⣀⠀⣴⡖⠒⠢⠤⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣰⣦⣤⣤⣼⣠⣧⣄⠀⢀⡜⠁⠀⠀⣀⠴⠊⠀⠒⢒⡺⠿⠛⠛⠒⢙⣷
⠀⠀⠀⠀⠀⠻⠛⠶⠟⠁⠂⠄⠀⠀⠀⠈⠛⠶⣄⠀⠀⠀⢀⠰⠚⠉⠈⠀⠀⠀⠁⠍⠙⠻⣷⣿⣆⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠖⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠔⡸⣿⡗⢁⣄⣠⠴⠟⠒⠛⠋⠉⠉⠉⠉⠉⠭⢭⣖⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢛⣀⣀⣀⡁⢰⡟⠀⢅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠒⣿⣿⡟⠋⠀⠀⠀⠀⠀⣀⣀⢐⣂⡀⠚⠉⢭⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⣀⣀⣀⣛⣻⡇⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⡐⠀⠀⣈⠀⢀⠄⣴⣿⣿⠷⠖⢒⠒⠛⠓⠒
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠴⣿⣿⡀⠀⠀⠪⣅⢾⠀⠁⠡⢞⠁⠸⡚⠁⠀⣼⣿⣿⣿⡛⠓⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⣄
⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡨⣽⣿⣿⣿⣧⣄⣛⣤⠈⠳⠠⣂⣀⣏⣵⣿⣿⣿⢿⣦⠄⠀⠀⠀⠀⠀⠀⠈⠹⢶⣄⡀⠀⠀⠀⢀⠀⣸⣷⡄
⠀⠀⠀⠀⠐⠒⠀⠀⠀⣀⡠⠤⠀⠀⠀⠂⠁⠈⠙⠟⢛⣿⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⣿⡄⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡖⠶⠤⠾⠿⠛⠁
⠀⠀⠀⣀⡀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⢿⠋⠟⠛⢻⡟⠙⠙⣧⠑⠈⢫⡀⠀⠀⠄⠙⢶⡀⠄⠤⠀⡀⠀⣀⣤⡽⠏⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡠⠜⠁⠀⠀⡰⢣⠃⠀⠀⠀⠈⣷⠀⠀⠘⡆⠀⠀⠹⣆⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠉⠛⠶⠶⠋
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠜⠁⠈⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢀⡅⠀⠀⠀⢸⠀⠀⠀⠀⢿⠷⣤⣀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢐⣀⣠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡜⢃⠀⠀⠀⠘⢇⠀⠀⠀⠸⠢⣤⣽⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⡠⢃⠀⠀⠀⠀⠇⠀⠱⢄⠀⠘⢀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢆⣀⣀⣼⠀⠀⠀⠀⠉⠀⠀⠉⠲⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠢⢀⣀⣠⠃



⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⢀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠘⢿⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠀⠹⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠐⢚⣡⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣺⣿⢇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠐⠀⢸⣿⣿⡜⠀⠀⠀⠀⠆
⠀⠀⠐⠚⠄⠠⠀⢀⠄⠀⠀⢤⠄⠀⠀⠤⠄⠀⠀⠈⠙⣀⡇⠀⠀⠀⣀⠿⢿⣦⡀
⠀⠉⠭⣍⣈⠀⠉⠉⠀⢀⡉⠁⠀⠀⠀⠉⠀⠂⠀⠀⠀⠨⣽⠖⣦⠊⢠⠴⠈⠁
⠀⠀⠀⠀⠀⠀⠁⢤⠒⠉⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⡢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠰⠄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡧⡄⠰⠸⠘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠦⡤⠄⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⠀⠲⠊⠦⠀⠀⠰⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠘⠀⠀⠁⠀⠀⠘⠢⠀⢃⠄⠀⠂⠈⠁⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⠀⠀⠆⠀⠀⠀⠀⢠⣠⠀⠀⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⠀⠀⠀⠀⠐⢂⠁⠀⠀⠀⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠜⢁⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⠄⠀⠀⢠⠁⠀⠀⠁⠀⠠⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢁⠎
⠀⠀⠀⠀⠀⠀⠳⡀⠀⡜⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣨⠏⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠸⡴⡀⠀⠀⠀⠀⠀⠂⠀⢀⠐⠢⢊⠀⠇
⠀⠀⠀⠀⠀⠀⠙⢣⣀⠀⠀⢠⠃⠀⠀⠀⠂⠀⠀⠀⠀⠑⢀⡠⣀⣀⡀⠀⠀⢊⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⠋⠍⠁⠀⢀⡄⠀⠈⠘⠿⠛⣲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣃⠤⠒⠂⠀⠀⠈⠉⠀⠀⠀⠐⠨⠾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⢫⣿⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡆⣦⣦⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢇⡾⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣄⣷⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠊⢀⡀⠈⠀⠀⠀⠀⠀⠈⠀⠀⣧⢸⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇⣼⣄⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡵⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡤⢶⣿⡿⠿⠛⡉⣩⣯⣤⣾⣿⣿⣿⡍⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⢀⣼⡃⣽⢋⠃⠀⢸⢃⣿⡿⢻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣸⢿⣿⡏⠈⠀⠀⠀⣸⣿⣷⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⠟⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⡟⣿⣿⡧⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠝⣿⣿⣿⣷⣹⢿⣿⣿⣭⣽⣷⣮⣿⡿⠿⠃⠀⠀⠀⣀⡛⣳⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠋⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡷⠡⡤⠈⡙⢃⡖⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠄⠨⠭⠀⠓⣆⡉⢙⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣦⣤⡀⠀⠰⣿⣿⣿⣿⣿⢿⣿⣿⢻⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠿⠻⠛⠛⠿⢿⣿⣿⣭⠭⠉⠉⠉⠁⣈⣈⣄⣿⣿⣿⣿⣽⣿⣷⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣷⠄⠀⠁⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢣⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣿⣿⣿⣆⣠⣾⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣷⣦⣠⡿
⠀⠀⠀⠈⠹⠿⠛⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣠⠀⠀⠀⠀⢀⡾⠂⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣿⣧⢶⣀⣠⣴⠟⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠻⣆⡘⢿⣿⣿⣿⣿⣿⣯⣅⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠈⠛⠛⠛⠉⣻⣿⣿⣿⣿⣿⡷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣶⠄⣴⣶⣆⢈⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⣿⠁
⠀⠀⠀⠀⢠⣿⠀⣻⣿⣯⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠟⠀⠀⢤⣄⣀⠀⣀⣻⣿⣟⣀⢾⣿⣿⠟⢐⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⢻⣿⣿⡿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠟⠉⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡿⠀⠙⣿⣟
⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠀⠀⠙⠈

⠀⠀⡀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢦⠤⡄⠀⠀⠀⠀⠀⠀⠄⠰⢠⠄⠀⡀⣀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⡋⢰⡄⠀⠄⠀⠄⠀⠀⠀⣀⠰⢊⡋⠀⠀⠀⡀⠐⠄⢀⠀⢐⠒⠳⠀⠸⠾⠶⢧⠀⠠⡀⠀⢂⠐⡄⠀⠃⠘⢃⣀⠈⡃⣀⠀⢰⢽⣳⠀⠇⠀⠀⣎⣷⠄
⠀⠀⠀⠉⣡⠟⠀⠠⣤⠲⠐⠀⠈⠤⠜⠂⣡⠤⠒⠒⠓⠒⠢⠄⠀⠀⢢⠄⠀⠐⠀⡐⠀⠀⠀⣈⠉⠋⠉⠀⡰⠄⠉⠀⢈⠀⠀⠉⣤⠈⠋⠀⢀⠀⠂⠐⠁
⠀⠀⠀⠀⠈⣀⢤⣅⠈⠏⢀⡀⠀⠀⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⠧⠄⠀⠀⠘⠙⠃⢀⣠⠈⠢⠠⠥⡂⠀⠵⠋⣀⠀⠆⢄⡀⠐⠂⠐⠂⠁
⠀⠀⠈⠑⢤⣿⡬⠅⠀⠀⠨⠁⠀⣴⠤⣤⣄⣀⣠⣤⡀⣀⣀⡀⡀⢤⣴⠀⠀⢈⣍⠓⢀⣀⢰⣥⣤⡆⠈⡤⠀⠀⠀⢠⠀⠤⠀⠀⠈⠀⡄⠀⢃⠀⠀⠀⢀⠆
⠀⠀⢀⠀⠀⠁⠀⢀⡒⠆⠀⠀⠀⠀⢠⣦⢀⠉⠀⢨⡇⠈⠁⠀⡀⠀⠀⣄⠱⠦⠀⠀⠀⠄⠀⠉⢉⢀⡀⠀⣎⡁⠰⣿⠀⠀⠀⢀⠤⡈⠀⢀⠊⠀⠦⠆⠈⢦
⠀⠀⢸⣦⣴⠆⢀⠠⠤⣤⣄⠀⡠⠀⠀⠉⠆⡀⠀⢸⡇⠀⠰⠀⠓⠀⠀⡈⡄⠀⢰⢻⢢⠀⠀⡠⢄⢈⡀⠀⡯⠅⢩⣿⠀⠀⢠⠃⠀⠀⢢⠏⡀⠀⠀⠀⠀⠈⢃
⠀⠠⠽⠯⠇⠠⠁⠀⢰⠁⠀⠀⣤⠄⠀⠈⢈⡴⠀⠘⡇⠀⠃⣔⠒⠲⠀⠰⣧⠀⠘⠨⡃⠄⠀⢧⣀⠬⠁⠀⢳⢨⠼⠿⣝⡆⢀⠀⠀⠀⡌⠀⢁⡀⠀⢠⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢒⢚⡇⠙⠂⠐⠢⠂⠀⠀⡇⠀⠒⠃⠄⢀⠀⠐⡂⠣⠀⠡⣷⠃⠀⣀⠀⠀⡀⠀⠘⠂⠸⠴⠛⠁⠈⠀⠀⠀⠃⠀⠈⠀⠊⠃⠘
⠀⢀⣀⡠⢃⠔⠂⠐⠀⠀⠪⢹⡍⠑⠀⠀⠚⠁⠸⠥⠁⠀⡀⡄⠀⠈⠀⠀⠀⢄⢃⠀⢿⠀⠀⡏⠄⠀⣙⠄⠀⣀⠀⠈⠃⠀⠀⠈⠠⣀⢀⠀⠀⠀⢹⠀⡀
⠀⠈⠁⠰⠑⠖⠃⡀⠀⠀⢈⡐⠃⡄⠀⠀⢸⠂⠀⢠⡎⡃⢈⠤⠁⢰⠀⠀⠀⠀⠲⠀⠈⠀⠀⢃⠀⢊⠜⠀⠀⠙⠀⠀⢂⠀⡆⠀⣴⡀⠉⠃⠀⣁⡙⣨⠁⠈
⠀⠀⠑⠀⢤⠄⠈⠀⠀⣀⡀⠀⠀⠀⢀⠰⠋⠀⠀⠈⠓⠀⠀⡀⠐⠈⠀⠀⠀⢀⢶⡄⢀⡀⠀⠈⠉⠉⠀⠀⠀⠦⠁⠀⠾⢯⠃⢀⠆⠀⡀⠐⠆⠀⠀⠀⠀⣤⣠
⠀⠰⡀⠀⠀⠁⠀⢴⡀⠀⠧⠀⡄⠀⠉⠁⢢⠀⠀⠒⠉⠁⠈⠉⠎⠋⠀⠀⠀⡽⢀⡁⣀⠁⡔⠄⠀⢀⠔⠀⠢⠀⠐⠠⢦⠀⠀⠸⡀⢀⠄⢈⣀⣦⣀⡀⠀⢠⡀
⠀⠀⠷⠀⠖⠀⠀⠀⠉⠉⠁⠀⠐⠀⠀⠠⠎⠠⡂⣀⠊⠀⠈⠠⠿⡆⠀⣀⡌⠁⢮⡷⠶⠒⢂⣤⣞⢣⡄⢀⡄⠀⠀⡄⠀⢹⠀⠀⠀⡠⢖⠥⠎⡐⣁⣌⡴⡄
⠀⠀⣸⠀⢀⠀⡀⢠⠖⠀⠀⣤⣶⣴⡶⡆⠀⢰⡟⡄⠀⣀⡨⠤⠤⠤⠀⠀⠀⠈⠀⠠⠤⠐⠚⢼⠉⠹⡀⠀⠃⠀⠀⢀⠀⡀⠀⠀⣜⠽⡳⡴⢟⠴⡹⣾⣪⡵⡆
⠀⠀⠀⠀⣀⠀⣿⡶⠀⠀⠈⠉⠉⠀⠇⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣧⡄⡀⠀⠘⠑⠃⢼⡀⢠⠶⢮⠒⢺⡿⢿⣾⠖⢻⠟⢿
⠀⠀⢠⢀⠿⠶⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠻⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡽⡷⣜⡖⠸⡀⠀⠀⠀⠸⠀⠀⠀⠀⠃⠘⠸⠀⠀⠀⠘
⠐⣂⠸⠿⢤⠀⠄⠀⠀⠀⠀⡀⠀⠀⠉⠠⡭⠀⠀⢰⡴⢦⡔⠁⠁⢀⠈⠀⠀⠀⠀⠉⠳⣄⣤⣠⠀⠘⢓⠈⠉⠀⠈⡏⢁⠀⠀⣤⣤⣤
⠀⠙⡧⠤⠀⠄⣄⡰⢞⠀⢀⢉⡁⠉⠀⠈⠀⣀⣀⠐⠂⠑⢁⢀⠀⠦⡤⡤⠤⣤⠐⠁⠀⠛⠀⠊⠀⢀⡉⣄⠀⠀⣠⠴⠋⡀⠀⢳⠒⠻⠀⠀⠀⠀⠀⠷⣤⡤⣼
⠀⠀⠀⠀⢠⣀⠙⠓⠀⢰⡓⣭⠘⣦⠀⠀⠎⠀⠂⡀⠀⠤⡀⡀⠀⠀⠉⠙⠛⠉⠀⠤⡀⠀⠀⠐⠂⡀⠀⠀⠅⠀⡱⠚⠋⠀⠀⣈⠀⣰⣄⠀⣆⡄⠀⠀⠐⠂
⠀⠀⠰⢷⡐⠻⠂⠀⠀⠀⠛⠰⠼⠋⡀⣀⠐⠒⢎⢀⠀⠀⠠⣤⣠⣖⣆⠀⣠⠀⣈⡀⠅⠠⠈⠁⠀⡐⢀⠐⠀⠂⠈⠄⠀⣀⠀⢤⣀⣈⣹⠆⠀⡀⠀⠐⠦⠴⠃
⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⣟⠂⠀⣾⡟⠁⠀⠀⠟⣿⠾⢿⡇⠀⠀⠠⡄⠀⠃⠀⢠⠐⠠⠞⠀⢢⠂⠀⠂⠀⠨⠁⠀⠀⡀⠀⠀⣿⣿⡆⠀⣠⡀⡀
⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠉⡀⠀⠈⠷⠆⢀⢀⢀⣷⠀⢰⠃⠀⠀⢠⡄⠀⢡⣤⡈⠀⠀⠀⠀⠈⠀⠀⣈⠀⠀⢠⣀⣁⣀⢠⣀⠈⠉⠀⠘⡳⢮⠜
⠀⠀⠀⠈⠀⠁⠸⠀⠲⠁⠇⠐⠈⠄⠀⠁⠀⠀⠀⠀⡀⢳⣟⠛⠁⡀⢸⡄⠀⠀⠘⠃⠀⡟⢀⣷⡃⠀⠄⠀⠀⠀⢉⠁⠀⠄⠈⠀⠁⠈⠀⠁⠀⠀⢔⣠⢉⡠
⠀⠚⠀⠒⠀⠐⠂⠙⠒⠀⡐⠊⠰⠚⠀⠀⠁⣘⢳⡖⠱⣀⠈⠉⠉⠉⢋⢅⠀⠀⣀⠀⠀⣓⢬⡧⠀⠀⠀⠀⣀⡀⢀⣥⡄⠈⠃⠤⢄⣠⠤⠘⠁⠀⠀⠁⢠⡛
⠀⠀⢠⡆⢰⡄⠀⠸⠀⢠⢇⠀⠫⣞⠃⠘⠁⠼⢏⠀⣀⠣⠄⠀⠀⢮⠸⠐⠀⠀⢧⡠⠀⡅⠘⠟⢂⡄⠐⠀⠋⠇⡀⠀⠀⠀⠀⠀⣠⠔⢂⠲⠄⠀⣄⠀⡀
⠀⢴⣆⡬⢤⠆⠀⠘⠀⠈⠀⠀⠀⠓⠀⠀⠀⠛⠈⣶⠚⢀⠃⠄⠀⠀⠈⢲⠐⡳⠀⠙⢀⠀⠈⠈⠀⢠⣶⠀⠀⠸⠀⣈⡄⠀⠀⡰⡁⠁⠈⠂⠀⠄⠀⠀⠀⠈⠙
⠀⣀⣈⢡⡄⠀⠀⢸⠀⠀⠀⠀⠁⡁⡀⠀⠀⠘⠦⣀⠀⠂⠀⠀⡖⠀⠂⠈⠀⠀⠃⢀⡏⡁⠀⠀⠀⣾⡇⠀⢀⡀⠛⢾⡇⠀⢰⢀⠀⠀⠀⠇⠘⠀⠀⣀⣱⠂
⠀⢈⢡⡤⠇⠀⠇⡸⠀⠀⣄⣀⡀⠈⠀⢸⠂⠘⢦⣡⠄⠂⠀⠈⠁⠀⣰⠀⠀⠀⠾⣘⡗⠁⠀⠀⠀⢱⠂⠀⠣⣠⢺⣦⢵⠀⠸⡆⠀⠀⢀⢇⠸⠀⠀⡠⡝⡀
⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⢿⣿⣛⣻⢿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣯⣿⣷⣯⣿⣿⣿⣿⣿⣿⢭⣿⣏⣿⠿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣯⣿⣿⣟⣟⣻⣿⣿⣿⣟⣿⡙⢻⣿⢿⣿⣾⣷⣟⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣯⣿⣿⣿⣿⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣿⣾⡿⣿⣟⠩⣽⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⢿⢿⣿⡿⠙⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣃⣼⣯⣽⣝⣿⣄⠸⠃⠀⠈⠉⠪⠛⠟⠛⠛⠁⠚⠃⠀⠚⠿⣫⣿⡆⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣭⣘⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡙⠷⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣺⡿⠛⠛⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⢀⣴⣆⠀⠀⢈⣙⣟⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⡽⢿⡿⠃⠀⠙⣶⣤⡀⢴⣬⠿⠶⠀⠀⠀⠀⡤⠝⣯⠄⢀⠔⠺⣿⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣮⠀⠴⢶⠶⠀⠀⠉⠻⡄⠀⠀⠀⠀⠃⣀⡘⠁⢀⡤⠶⠟⠉⢁⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⣰⣶⣄⡀⡀⠈⠓⠾⢤⣀⣿⣩⣹⣦⠞⠁⠀⠀⢀⢮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣦⡁⠉⠙⢹⠿⣶⣆⣀⣀⡀⠉⠉⣉⣀⠀⢀⠀⣰⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠹⣶⣀⠀⠀⢿⠛⠛⠛⠛⠛⠛⠛⣿⠃⠈⣹⣿⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣆⣷⣨⡳⣦⣀⣀⣀⣴⢞⣅⣤⣰⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡄⠀⠀⣠⠀⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⠻⠋⡯⡿⣏⡿⢿⠽⠹⠿⡝⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠘⠃


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⠀⠀⠀⠀⢼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⡀⣀⡀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⣀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣤⣾⣿⣿⣿⣦⣤⣄⣄⣦⣄⣀⣤⣤⡴⢶⣾⣿⣿⣿⣷⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣙⡛⠛⠟⠁⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠛⠛⠛⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠻⠿⠿⠿⠛⠉⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠛⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⠒⢤⡀⠁⠀⠀⠀⠀⠀⠀⠦⠨⠀⡇⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠁
⠀⠀⠀⠁⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢰⣀⣀⠀⣁⣩⡄⠀⠀⠀⠀⠀⠘⠁⡠⠈⠀⠈
⠀⠀⠀⠱⠆⡇⢠⡛⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣸⢒⢒⢒⢰⢶⢆⡀⠀⠀⠀⠀⡠⠿⢀⡀
⠀⠀⠀⠃⠀⣀⢤⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢀⡠⣀⠈⣀⠤⠀⠀⠀⠀⠀⠀⠈⠁⠂⠉⠀⠀⠀⠀⠀⡔⠀⠀⢀⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⢰⠈⠁⠁⡆⠀⠀⠀⠀⠀⡂⠐⠀⠀⠄⡁⠀⠀⠀⠀⠏⠀⠤⠀⠕⠊⠀⠀⠀⠀⠀⠄⡀⢀⣀⠀⢞⢒⡀
⠀⠀⠀⠁⠰⠙⠂⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠀⠙⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠐⢂
⠀⠀⠀⢀⡀⠁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠎⠀⠀⠀⠀⠀⠀⠅⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠐⠰⡂⠂⡀
⠀⠀⠀⠀⠄⠀⠄⠇⠀⠀⠀⠀⣀⣄⣀⣀⣠⡨⠄⠀⠀⠀⠀⠀⠂⠘⢀⡀⠳⠟⠀⠀⠀⠀⠀⠄⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠲⠐⠠
⠀⠀⠈⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⣇⠀⣞⠄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠊⢐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⢷⣽⣷⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣦⣤⣄⡀⠀⠀⠆⠀⠀⠀⠈⢻⠀⠀⠸⢧⠀⠀⠀⢀⣴⣾⢿⠿⠟⠉⠉⠋⠛⠛⢻⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠒⠶⣤⣀⣈⠀⠀⠀⣈⡀⠀⠔⠙⠉⢀⣤⣄⣦⣴⣶⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢻⣿⣿⣿⣄⠀⠀⠀⣻⡿⣿⣦⣾⣿⣿⣷⢰⣶⣿⠾⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠙⠀⣸⣼⣿⣿⣿⣿⣿⣁⣀⠄
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⢿⠂⠀⠀⣰⣈⢠⣿⠿⠟⠛⠋⠁⠈⠁
⠀⣠⣴⣿⣿⣿⣿⣶⣦⠌⠿⢿⡆⠀⠀⠀⠀⣄⡴⠖⠓⠲⠋⠀⠀⠀⢄⣀⠀⠀⠢
⠠⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠉⠑⢢⣄⠀⠀⢳⠠⣠⣤⣤⣤⣶⣶⣦⣤⣄⣤⣤⣤⡤
⠀⠀⠀⠀⠀⣀⣀⣄⣠⣤⣴⣂⡀⠀⣰⠏⠋⠡⣀⣠⣽⣷⣶⣾⠟⠋⠉⠉⠛⠿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠖⠛⠋⠉⠁⠀⠀⠀⠀⠨⣡⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⡹⣏⠉⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣁⡀⠤⠚⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⠀⡀⠀⠀⠀⠀⠀⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⣤⣀⣠⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⣿⡟⠻⠟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⠀⠀⢠⣶⠖⢲⡦⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠘⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⣿⡇⠀⠀⠀⠀⢳⠛⠀⢀⠀⠪⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠠⣿⠃⠀⠀⠀⠀⠈⢣⣤⠞⠀⠀⠺⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⡀
⠀⠀⢠⣶⣿⣿⣿⣿⣶⣶⣤⠀⠀⠀⠀⢸⣿⣍⠁⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⡰⠆⠁⣠⠀⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠈⣆
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⡏⠀⠐⢼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠂⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠈⠛⠆⠉⣿⠿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣆⣰⠋⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣿⣿⣿⡟⠀⠀⠒⠀⡀⢰
⠀⠀⠀⠀⢀⣤⠶⣿⡿⠿⣿⣿⣿⡇⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣳⠂⠀⠀⠀⠀⠀⠀⠀⡀⠖⠀⠀⣘⣿⣿⣏⠁⡌⠀⠀⠀⠐⢸⠃
⠀⠀⠀⡴⠛⣁⣀⣄⣈⣉⠿⠋⠁⠰⠶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣠⣤⣿⣿⣿⣿⣿⡷⠿⠄⠀⠀⠀⢸
⠀⠀⠈⠙⠳⠯⠽⠟⠉⠁⠐⠒⠀⠀⠀⠀⠁⠚⢦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣻⣿⣿⡿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠄⡽⢹⠟⠲⠙⢷⠑⠀⠀⠀⠀⡀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠠⣄⠀⣤⠀⠀⠀⠁⠀⠀⢐⠀⠀⠐⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡲⠤⠀⠀⠀⠠⠄⠀⠐⠒⠁⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣤⣤⣀⣉⡉⠤⠶⠚⠉⠁
⠀⠀⠠⣤⣤⣚⡛⠋⠉⠛⠛⠻⠿⣦⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠙⠻⣷⣤⡀⠈⢉⣙⠛⠿⣟⠪⣿⣦⣴⢘⣉⡋⠀⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⣿⣿⣿⡷⡻⢯⣿⣿⣦⣤⣄⡀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠻⢿⣿⠟⠂⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⠀⠀⢠⡏⠀⢀⡀⠀⠀⠀⠀⣶⣤⣼⣿⣿⡅⡂⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢶⣿⣿⡿⠏⠋⢻⣿⣿⣧⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⢿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⣿⣧⣄⣀⣼⣿⣿⡗⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠇⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⠁⠻⢛⣿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣤⡀⣀⠀⠀⠀⠀⢀⣀⣀⢰⣶⣩⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢦⣄⣀⠀⠀⠀⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⡟⠀⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⢈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣽⣆⡀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⢺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣧⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢹⡾⣿⡿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣗⡿⣾⣿⣄⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣿⡿⣿⣆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠈⠛⣷⣞⣛⣿⣿⣿⣛⣶⣤⡀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠻⣷⣄⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠈⠀⠉⠀⠈⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡗⠄⠀⠀⠒⠀⠒⠀⠀⠀⠀⠀⢘⢙⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣧⠐⠂⠀⠀⠀⠀⠀⠀⠀⠃⠂⠂⣳⣿⠿⠛⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⢠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠘⢿⣿⣿⣿⣿⣟⡻⢯⡒⡲⢭⣙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⣿⣿⣿⣿⣝⠯⡳⢭⡪⢕⠪⢍⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣕⢬⡓⠭⡒⠬⣑⠢⢽⣢⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠳⡀⠀⠀⠀⢿⣿⣿⣿⣷⣝⡻⣗⢬⡑⠲⢤⣄⡈⠐⠩⡻⣿⣦⣶⠶⣒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠶⠤⠄⠀⠀⠀⠀⠀⠘⣿⣿⣷⣯⡛⢿⡲⢍⡪⢕⠠⠀⢭⡳⢦⡈⢿⣿⣿⡽⢖⣒⣒⠢⣜⣆
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣝⠷⣌⢦⡙⢦⡙⣆⠑⣝⢦⣽⣾⣿⣯⠞⢁⣒⣶⣦⣌⢻⡄
⠀⠀⠀⠀⠀⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣜⢆⢳⡈⠻⣌⠂⢌⢷⣿⣿⣿⡇⡴⡹⠟⠡⣿⣿⣾⠃
⠀⠀⠀⠀⠀⢀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠛⠛⠛⠛⠻⣎⢦⢣⠻⡄⠙⢷⣌⠳⣿⣿⣿⡎⠘⠀⡄⢸⢹⢸⡋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⣿⣧⡁⠱⡙⢮⡢⣝⢷⣾⣿⣿⣇⢸⢸⣠⣸⢸⢸⡇
⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⡷⡄⠹⣮⣙⢮⣿⣾⣿⣿⣿⣸⣸⢇⠇⢊⠞⠁
⠀⠀⠀⠀⠀⠈⠒⠀⠠⠤⣀⣀⡀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣳⡱⡝⢦⣘⠻⢿⣿⣿⢿⣿⣿⣿⣿⡯⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⣿⣿⣿⢷⡱⣜⠶⣭⣝⣫⣥⠶⣻⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⡻⣧⡻⣮⣓⡲⠖⣒⣤⠾⢣⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡱⣌⡻⠦⢭⡭⠭⢍⣠⡴⢋⡾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡦⣍⣓⣒⣒⣛⡩⢅⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣕⣒⠠⠤⢒⣺⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠩⠍⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠒⠊⠉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⠀⠘⡄
⢠⠿⠗⠈⠭⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⣀⣁⠰⡀⠀⠀⢀⠤⠷⢄
⠈⢻⠀⠀⡄⠀⡀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⠁⠀⠀⠑⠷⠀⠀⠁⠀⡀⠈⢳⠀⢀⠄⠀⠰
⠀⢸⠀⠀⠀⠂⡀⠀⢀⠼⠄⣲⢾⢗⠒⠒⠒⠒⠒⠚⠌⠋⡢⣀⢈⣿⡄⢡⠀⣀⠐⠀⠀⠀⠳⣆⡀⠐⢀
⠀⠀⢣⠀⠀⠀⢠⢞⣧⢖⠏⣯⢃⠾⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠄⡀⠂⠀⢢⠈⠀⠀⠀⠀⠐⢏⠁⠀⠁
⠀⠀⠀⠑⠤⠔⡫⠎⡇⢸⢠⣇⠎⣸⠂⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠉⠂⠄⠱⣄⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢠⡇⠇⡇⢸⡜⣡⢞⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⡠⠔⠀⠁⢄⠀⠀⡈⣆
⠀⠀⠀⠀⠀⢸⣇⠸⡰⡼⢋⢔⠥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡠⣴⢄⠀⠀⠀⠑⣠⠋⠈⢱⣀⣀
⠀⠀⢀⡠⢄⡀⢾⣷⣳⡯⢓⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣩⡤⡴⢸⠗⣟⢋⠀⠀⢫⠀⠀⠀⠀⠀⢸⡀
⠀⠠⠋⠀⢀⡨⠛⠳⡷⡬⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠤⣔⢫⠔⠁⠀⠑⠄⠫⠬⡤⠚⠠⠄⠠⢍
⢀⠃⠀⠆⠀⠀⡇⢀⠇⠀⠀⢠⠤⠤⠔⢢⡀⢀⡤⠳⢄
⠀⡆⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠠⡀⠀⠑⠋⠀⠂⠘⢄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠸⠤⠒⠂⠈⠁⠀⠀⠀⠀⠰⡄⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡀⢐⠁⠀⣠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⡼⢉⠉⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⢀⠄⠀⡼⠯⢭⠤⠤⠤⠤⠤⢀⣀⡀⠀⠀⢀⠇⢄⠡⠢⡆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⢏⣀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⡄⠄⣄⠡⣠⠈⢸
⠀⢀⣀⠀⢀⡤⠊⢉⡀⠀⠀⠀⠀⠈⠁⠀⡼⠹⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⡈⠉⣽⡖⠁
⠀⢻⠸⡶⠷⠒⣋⣉⡽⠀⠀⠀⠀⠀⠀⡰⢃⡄⢧⠀⠀⠀⠀⠀⢸⢹⠆⠀⠀⠀⠀⢀⣀⠈⠉⠈⠿⢦⡀
⢯⣉⡀⡶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⢧⢸⡀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠙⣄
⠞⠀⡇⢧⠀⠀⠀⣀⡤⠴⣄⠀⠀⢀⠟⡸⠁⢸⡀⡇⠀⠀⠀⢀⡇⣿⠀⠀⠀⠀⠀⠘⡇⣇⠀⠀⠀⠀⠀⠈⢆
⠀⠀⣻⠸⠶⢚⣉⣥⠴⠒⠃⢴⠒⠚⠐⠧⠤⣬⡇⢿⠀⠀⠀⢸⢡⡇⠀⠀⠀⠀⠀⠀⣇⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢳⡀⡞⠉⠀⠀⠀⠀⠀⠀⡹⢁⠟⠓⠒⠒⢦⠀⣽⠆⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⢾⢸
⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⢸⡆⡇⠀⠰⡇⡏⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆
⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠈⡇⢻⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇
⠀⠀⠀⢸⢸⡄⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⣤⣌⣉⣉⣉⣛⣛⡆⠀⠀⣧⠹⠤⡤⠴⠖⠚⣙
⠀⠀⠀⠘⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠻⠴⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡀⢀⠀⣄⣲⠐⣄⣦⢠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣢⢶⡄⢦⣴⢽⣷⣸⣆⠻⡙⡹⣻⣼⣼⣧⣧⣿⣧⣏⣶⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣣⣿⣿⣷⣮⣿⣻⣽⣿⣷⣽⣷⣷⣼⣿⠛⣿⢻⣿⣿⣾⡛⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⠲⣽⣿⣶⠟⢿⣟⣯⣽⣿⣯⠛⢮⣻⣷⣿⣿⣇⣿⣸⣿⢿⣧⢿⣯⣿⣧⡿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠈⣻⣟⢲⣮⣿⣧⣘⣯⣿⢟⣿⣿⡦⢾⣿⣿⣿⣷⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⠷⠂⡀
⠀⠀⠀⠀⠀⠀⡙⣻⣎⠻⣿⣭⡕⢻⣭⣻⣿⣿⣿⠛⣺⣿⠛⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡆⢰⡇
⠀⠀⠀⠀⠀⠀⢭⣾⡇⢅⣽⡾⣗⣪⡿⣝⣿⣿⢏⣹⣾⡿⣆⠈⠛⠀⠈⢹⣿⣿⣿⣿⣿⣿⠿⣿⠛⠁⠈⠋⠀⣼⡅
⠀⠀⠀⠀⠀⢀⣓⡎⢃⣘⣿⣟⣭⣛⣴⠾⢷⣶⣶⠊⠉⠑⠀⠀⠀⠀⠂⠈⠀⠀⠁⠸⠻⠁⠀⠀⠀⠀⠆⠀⠸⣿⡇
⠀⠀⠀⠀⠀⣨⣿⣿⣿⡶⢏⣭⣙⡭⣿⣻⣿⣿⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⡀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⣰⡿⢨⢞⣯⡿⡷⢛⡿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⡀⠐⠀⢀⣤⣶⡄⠁
⠀⠀⠀⠀⢈⣭⣴⣿⣿⣧⢾⣽⣾⣲⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠇⠘⠀⠀⠀⢸⠿⠿⠁
⠀⠀⠀⠀⢐⡏⣥⡬⣫⢟⣝⣯⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠛⢀⠆
⠀⠀⠀⠀⢈⣾⣿⣸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡀⠀⠰⠄⠀⠀⠠⠀⡐⡾⠀⠀⣐
⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠉⠙⠛⢦⣘⣧⣶⣶⣄⡰⢃⠆⠀⠃
⠀⠀⠀⠀⠀⠉⣿⡟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⡀⢠⡇⠎⠀⡶⢀⡉⠻⠿⠿⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣸⠅⠘⣷⠄⠚⠁⠀⠀⠀⠀⠈⠁⠙⢠⠈⠈⣴⠋⠃⠚⢻⢻⠿⠿⣤⣏⠞⢡⡆
⠀⠀⠀⠀⠀⠀⢛⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⡔⠂⠀⠐⣾⣾⠀⡃⠘⣏⣤⣾
⠀⠀⠀⠀⠀⠀⠀⣦⣶⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢴⣅⠀⠀⠘⣿⣶⣧⣰⡅⢉⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡾⣿⣀⠀⠀⠻⠶⠞⢻⣵⣷⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣰⣿⣿⣿⢟⣵⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣴⣿⣿⡿⣫⣿⣿⣿⠟⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾⣿⣿⣿⣿⣿⣿⡿⢡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⢟⣻⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠤⢒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠛⢋⣉⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⠿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢋⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⡀⣸⣿⣿⣸⣿⣿⡇⢻⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⣧⣹⣿⣿⣿⣿⣿⣧⣿⣤⣸⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⡿⣿⣿⣿⣿⢿⡟⢿⣿⡍⠳⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡋⠛⢿⡟⠉⠀⠑⠀⠈⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠁⠀⠈⠻⠟⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡀
⠀⠀⠀⠀⠙⣶⠟⡉⢹⣶⠚⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣱⣄⣀⠀⠀⠀⠀⠀⠀⣈⡛⢻⣿⣇⡀⢱⣄⠀⠀⠀⠀⠀⠈⣇⠀⢀⠣
⠀⠀⠠⠂⡉⠿⢦⣴⠞⡉⠉⠁⠀⠀⠀⠠⠠⢺⠿⡟⣿⠛⠟⠆⠑⠀⠀⠀⠀⠀⡠⠟⣋⡟⢙⣿⠻⣟⠛⠁⠀⠈⠦⡀⠙⠆⠀⠷
⠀⠀⠀⠈⠁⠒⢀⣠⢰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠾⠃⢷⠈⡁⠀⠀⠀⠀⠀⠑⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⡷⡆⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠇⠀⡜⠀⠀⠀⠀⠀⠀⠀⣶⣠⡖⣁⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⠶⢻⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠇⠀⠀⠀⠀⠒⣶⡖⠩⠉⣿⣉⣱⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠺⠊⢧⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⣷⣶⠾⢯⣠⡀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⠀⢃⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⢰⠀⠀⡴⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠇⠀⡇⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⢘⠀⠌⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢄⠀⠀⣀⣀⡄
⠀⠀⠀⠙⠤⡄⠠⡆⠀⠀⠀⠀⠀⡄⠀⠀⠈⠺⢀⠞⠓⠚⠀⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠾⣷⡶⠟⣛⠛⠓⢃
⠀⠀⠀⠀⠢⡖⢄⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⢰⠇⠀⠀⠀⠠⠄⠀⠀⢀⢠⣾⠏⢀⣀⡀⠀⠀⠀⠀⠀⠀⠛⠓⢾⣾⣧⣿⠴⢼⠶⠚⠉
⠀⠀⠀⠀⠀⠈⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⠋⡙⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠂⣩⣿⣦⣤⡟⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣄⢀⠔⠋⠀⡟⠋⢦⡀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⡀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⢠⣦⣅⠀⠀⠄⠀⡩⠒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠄⠀⠦
⠀⠀⠀⠐⠮⠂⠜⠷⠀⡀⠒⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⡄⠀⡁⢅⠀⠂⠄⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⡠⠄⠢⠀⠀⡂⡸
⢤⡾⡤⡄⠀⢀⡀⠰⠀⠐⠀⠔⠼⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠀⠴⠂⠀⠀⠂⠁⠀⠀⠀⠀⠀⠟
⠃⠂⠱⣭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⣠⠄⡄⠀⠣⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠄⠀⠀⠀⡄⠀⠤⠀⠀⠀⠃⠀⠀⠘⠃⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠠⠀⠸⠀⡀⠀⠄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡠⠀⠰⣄⡀⠠⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⠈⠁⣀⠀⠐⠀⠀⣁
⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢈
⠠⠠⠀⠘⠋⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⠀⠦⠠⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⢀⠀⠆
⠀⠦⡠⢀⢄⠂⠀⢀⠀⠑⠀⠀⠘⠁⠀⠀⠤⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠚⠁⠀⠀⡀
⠀⠀⠹⠀⠀⠑⠄⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⠀⠈⠈⢣⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢡⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡲⢿⠒⠐⠉⠠⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠪
⠀⠀⠀⠀⠀⢦⣬⡻⠂⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⠁⠈⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠓⠐⠀⠀⠀⠀⠲
⠀⠀⠺⢭⢻⡄⣗⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣈⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣤⡔⣤⢖⡆⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡖⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠂
⠀⠀⠀⠢⢩⠘⢤⡀⠀⠀⢿⣵⠥⠊⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⢄⡀⠀⠀⠈⠉⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠱⣄⠘⠆⠀⠁⠀⣠⢚⠁⠀⠀⠀⠀⠶⠭⣡⡄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢀⣤⡜⢠⣾⡿
⠀⠀⠀⠈⠃⠀⠀⣠⠞⡉⢺⡇⠀⠀⠀⠀⠰⠾⠡⠤⠤⠄⣈⡑⠀⣌⠉⠉⠐⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⢶⣞⢒⣻⠞⠀⠘⠛⠁⠀⠉
⠀⠀⠀⠀⠀⣠⠞⠁⢄⠘⣾⠃⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀⠀⠀⠢⣄⠀⠀⠰⠞⠉⠓⠦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠞⠡⡈⠲⡄⢳⣼⠀⠀⠐⢯⠇⠀⠀⢈⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠊⠀⢀⣈⠣⡀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠏⠈⠙⣷⠆⣉⡁
⠀⢠⡾⠥⠴⣄⣘⣦⠘⣆⣿⠀⠀⠀⡟⣧⠀⣰⢫⡇⠀⠀⢀⣴⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡇⠉⠳⠀⠀⢀⡀⠈⣄⠴⠛⢏⠀⠀⢀⠇
⠀⠈⠳⣄⡀⠀⠈⠙⠦⣼⡟⠀⠀⠀⡇⠘⠿⠁⢸⣇⣠⠔⣺⠟⠁⣠⠖⢛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣠⠔⢋⠀⠀⠉⠀⠀⠈⠉⡹⠉⢄⡀
⠀⠀⠀⠀⠉⠛⠲⠶⠦⠾⠃⠀⠀⢸⣗⣒⣒⠶⢬⣍⡀⢠⠏⠀⠀⠉⣿⠔⡹⠫⢛⡶⠦⣤⣤⣀⣀⣠⣤⣤⡴⠚⡟⣄⠀⣬⣆⠀⠀⠫⣭⣴⣌⠁⠂⠀⢈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣭⣉⠓⢮⣹⠏⠀⢀⡀⠀⠀⠀⠣⠤⠊⣿⠄⢸⠀⠃⡌⣧⠃⢸⠘⢦⡇⠈⠳⠇⠀⠀⠀⠀⠀⢀⠆⠐⠒⠀⢸⡀⡁
⠀⠀⠀⢀⣀⣤⡤⠤⠤⢤⣤⣄⠀⠈⠓⠭⣓⣛⣻⣦⠁⢠⠞⣉⣉⠳⣄⠀⠀⠀⠀⠉⠀⠸⠋⠑⠇⠘⠲⠞⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⢠⣊⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠘⢦⡀⠀⠀⢠⠾⠥⣬⣇⠀⠀⠀⠀⠀⠉⠀⠀⢾⢸⠃⠈⠳⡜⡄⠀⣠⣾⢯⡛⢦⠀⠀⠀⢀⡀⠀⠀⠀⠀⡦⢰⠔⠀⠀⢀⡠⠞⢁⠜⡽⠒⠦⠄
⠀⠀⠀⠀⠀⠳⣄⠀⣿⡊⠁⢀⣈⣙⣦⣄⠀⠀⠳⡶⢤⣘⡞⣧⠀⠀⢹⡇⣸⡟⡁⠀⠸⢸⠇⠀⡀⠈⠻⡄⠀⠀⠀⠃⣀⡀⠀⠀⠯⠡⠖⣡⠊⢀⠌
⠀⠀⡤⡤⣀⠀⠈⠳⢯⡙⣲⣍⠀⠀⠀⠈⠳⣄⣀⣬⠦⠈⠙⠮⣷⡸⡄⣿⣿⠎⠀⢐⣡⠟⠀⣈⠉⠉⢀⡤⣄⣰⠛⠛⠉⠉⠀⠀⠀⠀⠀⠧⠔⠃
⠀⠀⣦⣈⠠⠇⠀⡴⣄⠉⠁⠈⢳⡎⠉⠉⢉⣻⣇⠉⠙⠒⠶⢤⣌⣻⡟⠛⢧⣤⠶⠛⠛⠉⣿⡁⠀⠀⡞⠓⠤⠌⣽⠂⠀⠀⣆⠀⠀⠀⣠⠔⢊⠉⡁⠒⢄
⠀⠐⣁⣙⠓⠀⠀⠱⣮⠆⠀⠀⠀⢻⡤⠞⣉⣩⡿⠀⠀⡏⠉⠉⠉⡛⠉⠂⠚⠛⠛⠛⠻⠍⠂⠛⠂⢰⡗⠀⠀⢠⢦⠀⠀⠀⡟⠀⠀⡴⠋⠀⠀⢀⠀⠀⠈⢃
⠀⠀⡟⠢⠽⠀⠀⣴⠾⡦⠀⠀⠀⠈⠛⠛⢻⡅⠀⠀⠀⢳⠤⠤⠤⡇⠀⠀⡤⠤⠤⣀⠼⠀⠀⠀⠀⠸⡇⠀⣠⠃⠈⠳⢤⠼⠁⠀⢠⠆⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⡇⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣝⠋⠁⠀⠀⠀⠈⣾⠀⠀⠸⠀⡔⠁⠀⠀⠀⠀⢀⠆
⠀⠀⠀⢠⣞⠛⠙⢧⡼⠉⠁⠒⣤⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⡇⠀⡆⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⠀⢀⠔⠈⠀⣠⠊
⠀⠀⠀⢸⡈⣩⡛⢛⡛⠛⣛⠉⡅⠀⠀⢀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢠⠁⠀⠀⡸⠀⠀⠀⠀⠀⠶⠿⡄⠀⠀⠀⠀⠀⣴⠾⠓⠚⢀⣸⠦⠼⠤⠚⠉
⠀⠀⠀⠘⡷⠃⠙⠋⢣⣤⠁⢰⠁⠀⠀⣼⡛⠿⣿⠀⠀⠙⠚⠒⠚⠓⠚⠦⠤⢤⠇⠀⠀⠀⣀⣠⣤⣤⣀⣀⣤⢤⡤⢤⣀⡀⠀⠀⠙⠶
⠀⠀⣴⠚⢷⡀⠀⠀⠀⠀⢀⡟⠠⡀⠀⠘⣳⡶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠃⠀⣠⢴⣶⣬⣉⠀⢀⡦⡍⡆⢀⠂⠀⠀⠀⠀⢠⢤⣀⣀⡀
⠀⠀⠙⠲⢤⣭⣛⣒⣒⣚⡩⠤⠖⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡴⡇⠀⠀⢀⠀⠀⠀⠀⡇⢸⠀⢸⢱⡏⣴⣶⡌⢧⠸⠵⢯⠁⡌⠀⠀⠀⠀⣀⣎⡀⣉⡽⠛
⠀⠀⠀⠀⠀⠙⠶⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠠⡀⠀⢇⠘⢎⠢⡀⠑⢄⠀⠀⣧⢸⠀⠘⢾⣧⡻⠿⣣⠟⠀⢀⠆⠰⠀⠀⠀⣠⢎⠀⠈⠻⡍
⠀⠀⠀⠀⠀⣠⠶⢮⡙⢢⡀⠀⠀⠀⢀⣀⡀⠀⡇⠳⡔⢄⠈⠢⡀⠑⢄⠉⠓⢤⣬⡿⡷⠈⠻⢷⣄⣀⠀⠉⠉⠁⠀⢀⠜⠀⡘⠀⠀⣰⠧⡀⠙⠒⢴⠏
⠀⠀⠀⠀⡞⠁⡄⠀⢹⣄⡧⠤⠴⠒⠻⣆⡷⠀⠈⠢⡈⠂⠥⣀⣸⣶⣦⠑⢄⠛⠛⠀⠈⠄⠀⠀⠀⠈⠉⠉⠛⠓⠚⠉⠀⠀⡇⠀⠀⣯⡓⠬⢑⣰⠋
⠀⠀⠀⠀⢳⡀⠓⠶⠛⠋⠀⠀⣀⣠⠴⠋⠁⠀⠀⠀⠈⠢⡐⠾⣿⠉⠑⡀⠀⠑⠂⠀⠊⠀⠀⢀⠀⠠⡄⠀⡏⢣⠀⠀⠀⠀⠣⠠⣞⠡⢛⠖⠊⠁
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠔⠀⠀⠀⠀⠀⡠⢄⣀⣨⠧⣀⠟⢤⡈⡾⠒⠂⠀⠀⠀⠀⠉⠷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣗⣪⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⣤⠀⠀⢀⣠⣥⣀⡝⣎⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⢣⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢀⣿⣷⠐⠚⢿⡗⢉⣉⣭⠵⠒⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠣⠅⠀⠀⠈⠀⠀⠀⢢⣀⡀⢀
⠀⠀⠀⢸⣿⣿⡇⢦⢞⡃⢐⣿⠶⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢙⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⢸⡿⠿⣷⢿⣿⣯⡜⠿⡿⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⡀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠐⣂
⠀⠀⠀⠈⢻⡀⠛⢛⣿⣈⠻⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠐⠄⠀⠀⠀⠰⠸⠦⡄⠀⠴⢀
⠀⠀⠀⢀⣼⡷⣦⣀⣙⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⢤⣀⠀⠀⢀⣴⣾⣠⣿⢀⡌⣸
⠀⠀⢠⣾⣿⡝⠳⢽⣋⣏⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠒⢿⣿⣿⣏⢁⠼⠋⠥⠂
⠀⠀⢸⣿⣿⣿⣦⠸⢿⣿⢸⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠶⠒⢀⠀⠉⢀⣀⣬⠔
⠀⠀⢸⣿⣿⣿⣿⣦⣸⣿⣄⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⣍⡙⠛⢿⡿⠢⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠦⠦⠠⠙⠻⠿⠿⠛⠒⠂⠀⠀⠀⠹
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡃⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⠤⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⠹⠆⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣷⣶⣤⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⠇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⡼⢩⠟⣹⣿⣿⠛⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⢀⠏⢀⣿⣿⣿⡆⢠⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠀⠈⠁⡟⢸⡇⣼⡟⠃⠇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢨⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢿⠁⣿⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡁⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠠⠀⠀⠀⡐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡀⡠⠎⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠒⢠⠁⠰⠁⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⠃⡀⠀⠈⠀⠀⣠⠆⡠⠁⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡇⠈⠁⠊⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠐⠊⠀⠠⠂⠀⠔⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡜⠋⢀⡔⠀⢀⠠⠊⠁⢀⠀⠂⠀⠀⠀⢀⠔⠀⠀⢉⠀⠀⡴⢈⣁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠙⡁⠀⢐⣊⠁⢨⠂⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⡔⠁⣦⢃⠀⡾⠀⡁⠀⠐⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠤⠀⢄⠈⢃⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠈⠀⠜⠀⠀⠃⠈⢆⠑⠒⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠈⠁⠒⠒⠂⠁⢄⠣⠤⢀⠀⡄⠰⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⠁⠠⠔⠁⠀⠈⢢⠀⠀⠀⠀⠀⡏⠉⠁⠈⠁⠀⠀⠈⠀⢄⠁⠀⠀⠀⢠⠀⡄⠀⠀⠀⠉⠀⠀⠘⠄⠉⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⠎⠳⡀⢀⠒⢢⠉⠂⠈⠹⠀⠈⠠⠀⠀⠀⠂⠀⠀⠀⠈⠒⣀⠀⠤⠤⠤⠀⡀⠀⠈⠀⠤⠀⠠⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢄⠰⠀⠠⠀⠀⠸⠠⡀⠈⢎⠓⠃⠀⠀⠀⠀⢃⠀⠀⠡⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠂⠀⠀⠈⠀⠀⠀⠑⠠⠤⠜⠛⢄⡀⠰⣎⡆⢸⠀⠀⠀⠁⢄⣀⣀⡀⠄⠐⠀⠁⠀⠀⠀⠀⠱⡀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⠁⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⢈⢓⡤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠘⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⠡⠦⠐⠂⠁⠀⠆⢲⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣠⡅⠀⢠⣧⣀⠤⠤⠔⠒⠂⠉⠉⠀⠀⠀⠀⠀⠈⠡⠀⠀⠜⠀⠀⠀⠇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡸
⠀⠀⠀⠀⠘⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⢀⠀⠀⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠆
⠀⣍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⡨⠋⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢠⠂⠀⠀⢀⠃⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⢢⣠⠖⠩⢠⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠦⠤⠖⠒⠒⠢⡀⠰⡇⣀⡴⠤⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⣴⡟⠁⢥⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢆⠀⠀⢀⣀⣸⣿⣿⣤⣮⣀⡀⢀⡤⠤⠤⠤⠖⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠰⠒⠚⡶⠞⠋⠉⠙⣿⣿⡏⠉⠙⠛⠛⠒⠒⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣄⣄⣤⣤⣤⣀⣀⣀⣦⣠⣀⣀⣀
⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣾⣿⣾⣏⣿⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢠⣟⣿⡿⣾⣿⣿⣿⣿⣷⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢆⠀⠀⠀⠀⠀⠀⣠⣼⡿⠁
⠀⠀⠀⠀⣰⣿⣷⣽⣿⣃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠈⡄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠈⠀⢧⣿⣿⣿⣭⣿⣿⠉⠉⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⣠⢯⣀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠠⡀
⠀⠀⠀⠀⢿⢿⣿⡿⣛⡿⣻⣦⣶⣿⣷⣿⣯⣽⣟⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢀⠃⠀⠆⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⢀
⠀⠀⠀⠀⠘⢏⢻⣿⣾⣻⣿⡏⣿⣿⣯⣿⣿⣷⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠣⡼⣯⣿⣿⣷⣿⣿⣿⣿⣷⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠁⠀⠀⠀⢀⣠⣤⣶⡶⣀⣀⣸⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⣤⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡃⠀⠀⠙⠿⠟⠸⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠛⠶⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠡⠉⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠤⠒⣀⣠⣤⣤⣄⣤⣤⣈⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢵⠋⡝⣷⠊⠀⠀⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢶⣿⣿⣟⣵⣾⣿⣿⣿⣿⣿⡄⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠋⠁⢀⠀⠀⢒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢛⣯⣿⣿⣿⣿⣼⣿⣿⣿⡿⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⢚⡖⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣼⠟⠛⡛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠔⢀⣿⣯⣿⡋⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⣿⣥⣴⣾⣿⣿⣿⢿⣿⠟⢋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣟⣮⣼⡟⣻⣫⣛⠀⠀⠂⠀⠀⠀⠀⠀⢻⡍⠀⣀⣠⣾⣿⣿⣿⣿⠇⢹⣿⣿⠞⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠾⢾⠓⣟⣿⠏⡵⠏⠀⡀⠀⠀⠀⠀⠀⠈⢄⠙⠛⠻⢯⣭⣤⣤⡶⠿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠏⠘⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣠⣄⣀⣀⣀⣠⠐⡂⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⢿⡿⡦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠮⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠙⢷⣷⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⢠⣤⡄⠀⢙⣗⣾⣿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣶⣤⣀⠀⠀⠀⠠⠿⡿⠿⠯⣭⣭⣓⢾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣿⣿⣿⣾⣿⡷⠶⠶⢶⣶⠤⢒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣿⡷⠦⡄⠀⠐⠛⣫⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠷⡀⠀⠀⠻⠛⠛⢿⣿⣻⣷⡹⣿⣿⣿⣿⣿⣿⣿⡋⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡏⠀⣀⠀⡀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⣧⠀⢩⢀⣀⡀⠀⠀⠀⠙⠘⣿⣯⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡴⣾⣿⣿⣿⣧⠸⣃⡉⠉⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣵⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀⣤⣴⠾⢛⣿⣿⣿⣿⣝⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⣿⣿⣿⣷⣿⢿⣿⣿⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⠏⠃⢸⣿⣿⣿⣿⡇⠙⠂⠀⠀⢸⣿⣹⣿⣿⣟⠷⣭⣿⣿⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⢀⣴⢻⡿⢻⣿⣿⠃⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⠋⣠⣾⡵⢋⣴⢿⡟⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣿⠋⢠⣿⡟⠺⠀⠀⠀⠀⢰⠀⠀⠀⢿⣿⣻⣿⣿⣿⣿⡟⣿⢯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡿⢟⣯⣶⠟⣿⢃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⡝⣏⣿⣇⡟⣼⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣵⡾⢋⡴⣿⣼⣳⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⣿⣿⣾⡟⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⠏⣠⡞⣼⡿⣱⣿⣿⣿⣧⡠⠀⠀⣀⢠⣊⣶⣿⢻⡏⢰⡟⣿⣿⣿⣯⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⢸⣿⣼⡿⢱⣿⣿⣿⣿⣿⣿⣮⣭⣷⣿⣿⣿⡿⢼⣇⣾⣿⠸⡿⣿⣝⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡅⠙⠝⠛⠛⠟⠋⠛⠯⠟⠋⠩⠈⠙⠁⢺⣿⣿⣿⣇⣧⡈⢻⡎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡋⢚⠏⣿⡇⠀⠀⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠃⠀⠀⠀⠀⠀⠀⠈⢘⣿⢤⣄⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠂⢄⠀⢀⡇⠀⠇⠀⠀⠀⢀⠏⡳⠀⠀⠀⠀⠀⢘⠏⠐⠈⠀⠘⢩⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⠀⠁⢼⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠁⠀⠀⡰⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠀⠘⢆⠀⠀⠀⠈⢆⠈⠀⠀⠀⠀⠆⡼⠀⠀⠀⢀⡜⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⢀⠀⣠⠤⠄⠤⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⣀⣀⣴⣠⣅⡀⠀⣠⠋⠀⠀⣀⠤⠊⠐⠒⢒⡴⠶⠒⠐⢲⡆
⠀⠀⠀⠀⠘⠟⠲⠞⠁⠂⠄⠀⠀⠀⠙⠳⢄⠀⠀⠀⡀⠖⠋⠁⠁⠀⠀⠉⠉⠙⢻⣾⣗⣀⡤⠞⠉⠀⠀⠐⠀⠀⠀⠀⢀⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⠒⡀⠀⠀⠀⠀⠀⢀⣠⣶⣏⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⣀⠂⣤⠌⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠠⣿⣟⣠⣦⠴⠚⠓⠉⠉⠉⠀⠀⠀⢀⣐⣺⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢤⣠⡿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢿⣿⣏⠀⠀⣀⣀⡀⠠⠤⠀⠈⠉⠁⠒⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠐⠶⠶⠲⣾⣇⠀⠀⠀⠀⠀⠄⠀⠀⢠⡆⢀⠠⠆⠀⢤⣰⣿⣿⣍⣋⠉⠁⠀⠀⠀⠀⠀⠐⠠⠄
⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠒⠚⠛⠛⠭⢝⣿⣦⣤⣀⠠⡤⠁⢠⡀⠁⠈⡅⣠⣤⣴⣿⣏⠛⠃⠀⠈⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠉⣢⡀
⠀⠀⠀⠈⠄⠄⠀⠀⠀⠀⣀⠀⠀⠄⠚⠙⠻⡿⠿⣿⣿⣿⣥⣦⣤⣶⣶⣷⣿⣿⡏⠉⠻⢏⡀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⣬⣀⣒⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢠⡿⢻⢿⢿⣿⠻⠻⡍⢎⠻⡂⠀⠀⠉⠲⣄⠤⠀⣀⠀⠀⠀⢀⣸⠳⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢀⠠⠂⠁⠀⡰⢻⠁⠀⠀⠀⣿⠀⠀⢹⡀⠀⠘⣆⠀⠀⠀⠘⢧⠀⠀⠀⠉⠙⠻⠥⠤⠇
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠜⠁⡆⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠘⣧⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⢹⠻⢦⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢧⡤⠄⠀⠀⠀⠀⡄⠀⠀⠀⣐⠱⡀⠀⠀⢘⢆⠀⠀⠈⡗⠴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠌⢡⠀⠀⠀⠛⠀⠘⠢⡤⠉⢦⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠞⠀⠀⠀⠀⠆⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠆⠀⠠⠸⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠤⠀⠀⠀⠀⠰⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢞⢂⠐⠾⠁⠆⠀⠉⠀⠀⠀⠀⢰⡤⠈⣛⣀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠓⠉⠁⠑⠂⣀⠁⠁⠛⠀⠈⠉⠓⠈⠁⠘⡏⣩⡍⠁
⠀⠀⠀⠀⢀⠀⠀⠸⣿⡇⠀⠀⠰⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⢀⣀⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣀⠀⢠⠤
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠚⣩⣴⣯⣀⠀⠀⠀⠀⢀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣷⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⣿⣋⡉⠀⠀⠀⢀⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠋⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⢘⣰⠤⡜⣱⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⠀⠀⠸⢠⡎⣥⡀⠀⠀⣼⡉⢻⣾⣧⣹⠁⣯⠄⡴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣽⣿⡀⠠⠘⣿⣿⣿⡷⠀⠀⠘⣑⣚⠷⢟⠑⡍⠀⠚⢩⡍⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢻⡿⣷⡀⠹⠝⠮⢿⣷⠀⠀⣰⣿⢯⡦⣤⣈⠛⠂⠀⢴⣚⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠀⠀⠴⣻⣿⠲⣤⣀⠠⠄⠐⠋⠀⠀⠀⠈⠦⠿⡘⢆⢓⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⢶⠖⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠺⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠀⠄⢠⠀⠀⠀⠀⠀⠀⣀⠀⢸⡆⠀⠀⢄⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠇⠀⠘⠁⠀⠀⢖⠻⠭⠷⡙⠀⠀⠀⠀⠰⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠒⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⢀⡄⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⣶⠲⣶⠖
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠁⠀⠀⡞⢁⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢵⣦⢦⣈⠀⠟⠢⢀⠀⠀⠀⠀⠹⣶⣤⠤⢤⣤⣶⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⠈⡁⣟⠯⠄⣤⡀⠈⠻⡆⠠⣦⣤⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠞⢧⠙⢿⡤⠞⣡⠀⣀⠁⠀⠉⠉⠙⠻⠦⢄⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠃⠀⢄⠘⠁⠊⡐⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⢀⢎⣰⡀⣑⢈⠃⠀⠘⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⣈⡘⡏⢉⡁⠻⠅⠉⢀⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⣐⠇⢠⡀⠴⢂⠸⣏⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠜⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣄⡠⠄⠊⢀⣠⣾⣿⣦⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠈⠉⠈⠁⠀⠀⢀⣴⣾⣿⣿⠿⠛⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠘⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣾⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠉⠉⠛⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠖⠒⠊⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠈⣿⡟⠀⠀⢠⣶⣿⣿⡿⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢧⣀⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⡀⠈⠀⣀⣠⣶⡄⠀⠀⢀⡴⠊⠀⠀⠂⡀⢀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠛⣾⢿⣤⡄⠙⠐⠤⣿⣿⣿⣷⣄⡀⢠⠀⠀⠀⠀⠀⢹⡦⢑⢦⣶⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠢⠀⠀⠀⠈⣷⣾⣿⡇⠀⠀⠀⡄⠀⢹⠏⠈⠀⢸⢀⠀⠀⠀⠀⠀⠙⠚⠀⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⡈⠹⡅⠀⣠⠤⡿⠀⠀⠀⣠⡟⣸⣿⠽⠟⠛⢷⣦⣄⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢉⡷⢞⡻⠁⠀⠀⠀⢀⣼⡟⠁⠘⠋⠀⠀⠀⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⣀⣔⣁⠀⠈⢹⣷⣶⣤⣀⣾⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⡄
⠀⠀⠀⢀⡔⢎⠩⡯⠥⣴⣀⠀⠀⠀⠀⠒⠛⠛⠛⠉⠉⠁⡈⢻⡙⢾⣿⠑⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣯⠿⡇
⠀⠀⣄⡠⠶⠶⠛⠛⠛⠋⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢁⣸⡿⢦⠈⢳⣿⣾⡿⠷⠄⠀⠀⠀⠀⢀⣿⣿⣶⠇
⠠⠞⠁⠀⠀⠀⠀⠀⠠⡿⠻⠃⠀⠀⠀⠀⠀⠀⠀⠁⢰⢏⣠⠏⢸⣾⣧⠈⠻⣿⠀⠀⠀⠀⠀⣀⢾⣾⣿⣽⠋
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣰⡄⠀⠀⠀⠀⠀⠀⢠⢰⣫⡞⠁⠀⠀⢹⣼⠀⠀⠉⠠⢤⣈⣁⣉⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⢀⡫⠿⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣞⠨⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⣶⣶⣿⣿⡿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣩⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣦⣾⣿⣏⠉⢹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⣾⢃⣆⠀⠀⠀⣠⣤⣤⣴⡾⠻⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠃⢸⣿⠀⠀⠀⠈⠉⠙⠋⢻⣶⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢻⣧⣤⣀⣀⡀
⠀⣸⡿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⠁⠈⠙⢿⣿⣶⣤⣶⣶⣴⣶⣾⣿⣿⠋⠉
⣴⣿⡃⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀
⡿⠙⠻⠃⠀⠀⠀⠀⠀⣧⣿⣯⣀⠹⢧⠀⢹⣿⣿⣷⣴⢛⣿⣳⡿⣿⣿⣿⡿⠋⠙⣻⣏⢿⡿⠀⠀⠀⠀⢘⣟
⠣⠀⢰⠀⠀⠀⠀⠀⠀⠇⠈⣛⣿⣿⣿⣿⡷⣾⣵⣟⡍⠙⣿⣿⣵⣿⣿⢷⡻⣷⣿⢿⣾⣹⡏⠀⠀⠀⠀⠈⢿⣧
⡗⠀⢺⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡿⣬⣀⡀⢻⣿⣿⣶⣿⣿⣿⣏⣤⣾⡟⢫⣁⣾⣧⠼⣇⠀⠀⠀⠀⠀⠀⢻⣷
⠁⢰⣾⠀⠀⠀⠀⠀⠀⠀⢿⡓⣱⣸⣥⣬⣽⡿⠿⢻⣿⣿⣏⣒⡻⠿⣿⣿⣖⣺⣟⣺⣗⡓⣿⠆⠀⠀⠀⠀⠀⠈⢿
⢀⣿⣿⠀⣤⠀⠀⠀⠺⣷⣿⠟⠛⣽⡿⠉⠴⡶⠃⠘⢛⣿⣿⣿⣥⣤⢀⡉⠙⣿⣿⣿⣿⣷⣿⣶
⢸⣿⡇⣾⣿⠀⠀⠀⠀⠙⠻⢿⡻⣿⣷⣶⡮⢱⣾⣿⣿⣿⣿⡿⠿⠟⠃⢹⡛⢉⣁⣙⣾⠿⠛⠉⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠀⠉⢻⣿⣯⣿⣿⣿⣿⣿⣟⠃⠀⣀⣶⢶⣶⣾⣻⣿⠏⢡⣶⡄⠀⠀⠀⠀⠀⠀⢸⡄
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠛⠃⡤⠀⢸⡏⠻⣿⣿⣿⡿⠛⣿⣿⢛⣿⣩⣽⠏⣾⣿⡏⠠⠤⢉⠀⠀⠀⠀⠀⠀⠘⣿⣧
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠰⠄⠀⢠⢸⡇⢀⣼⣷⣿⣷⣾⢸⠉⢨⣿⣿⣿⡿⠋⣿⡃⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠖⣈⡣⠿⢿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠃⠀⣿⠿⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⡏⢿⣿⣄⣿⡟⠁⠀⠀⠀⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡻
⣿⣿⣿⣧⣾⣿⡷⠀⠀⠀⠀⠀⠀⠈⣰⠀⢸⣿⣿⡇⢸⣿⡿⢛⣶⣶⡖⠀⠀⠟⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⢿⣿⡇⠸⡇⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⡰⠄⢀⡄⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⡓⠎⠙⡁⡗⠀⠘⠃⠐⠎⠷⠊⢹⠤⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⡗⠶⡈⠑⢄⠀⠃⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠂⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠇⠀⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠒⠩⠧⡤
⠀⠀⠀⠀⠀⠀⠀⢲⡒⠚⠀⠘⠀⠀⠑⡀⣀⡰⡜⢦⠞⡔⠧⡤⢺⣦⠊⠀⠈⢡⠀⣀⠨⠶⣎
⠀⠀⠀⠀⠀⠐⠒⠩⣉⠂⠀⠐⡀⠀⠀⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠐⠛⠠⡀
⠀⠀⠀⠀⠀⠀⣰⣉⠀⠀⠀⠐⣏⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⣧⡄⠀⠀⠒⠒⢗⡂
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⡨⠄⠆⢠⣶⣶⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠈⢢⠀⠀⠀⠀⠀⠠⢬⠦
⠀⠀⠀⠀⢉⡲⠀⠀⠀⠀⣀⠕⠀⠐⡘⠛⠿⠃⠀⠀⠀⠀⠀⠸⠟⠛⢀⠃⠀⠐⢫⡄⠀⠀⠀⠤⢠⡗
⠀⠀⡠⠔⡳⠁⠉⠁⠀⢠⣊⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢉⡆⠀⠀⠀⠀⠀⣈⢢
⠀⠈⠀⠚⠠⡄⠀⠀⠀⢀⠋⠈⢢⠀⠀⠈⠢⡀⠠⣶⣶⣶⠆⢀⡠⠃⠀⠀⠀⡔⠡⡘⠂⠀⠀⠀⠀⠐⡄
⠀⠀⡠⠖⠟⠀⠀⠀⠀⢨⠃⠀⠀⢃⠀⠀⠀⠈⠂⠬⠛⠥⠐⠉⠀⠀⠀⠀⡘⠀⠀⣫⡀⠀⣀⡀⠈⢦⡘
⠔⠉⢀⣮⡴⠀⠀⠀⠀⢓⡆⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢿⠁⠀⠀⠀⢀⠀⠙
⠀⠊⠁⢐⠖⠀⠀⠀⠀⠘⡓⣄⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠰⡞⠂⠀⠀⠀⠀⠘⡂
⠀⠀⠀⠮⣤⠄⠀⠀⡀⠀⠉⣁⠑⢄⡀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢎⠀⠀⣀⠔⠊⡼⠁⠀⠀⣀⠀⠐⣦⠴
⠀⢀⠔⠁⠧⠆⠐⠁⠀⠀⠀⠘⢯⠁⠉⠉⠓⡄⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠈⣹⠁⢀⠀⠀⠀⠑⡆⠧⠑
⠀⠀⠀⠀⠘⠚⡆⠀⡠⠀⠀⠀⠈⠑⢂⠀⠀⡴⠤⢤⢤⠤⡦⠼⡀⠀⢀⣀⠒⠁⠀⠀⠈⠂⠀⢬⢶
⠀⠀⠀⠀⠀⠀⠁⡖⣠⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢀⠀⠀⠱⡀⠀⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠧⠂⢿⢤⢀⡀⠀⡀⠀⠀⡀⡀⠀⡀⠀⠀⣤⠀⣀⡠⣡⡔⠠⠉⠓
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣬⣮⣬⣬⣾⣽⣭⣮⣵⣮⣵⣤⣭⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠺⣧⡈⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠇⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠡⡠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⣄⢳⡀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠂⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣦⠀⡄⣿⡆⠀⠀⢀⡀⣀⣀⠀⣀⡴⠞⠉
⠤⠤⢭⣷⣶⣶⣶⣤⣀⠀⠀⠀⠹⣿⣯⣷⣺⠄⠘⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇⢣⣿⡇⠀⠀⠈⣻⢏⡩⠟⢉⠴⠂
⠀⠀⠀⠈⠉⠙⠻⢿⣾⣿⣦⣄⠀⠘⢿⣿⣿⢣⠀⣇⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣿⣸⣿⡃⣠⣤⢟⡵⢟⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⡈⢿⣿⣿⣧⣸⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⣸⣿⣿⣷⡿⣻⢟⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⡸⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⡔⠁⠀⠀⠀⠀⠀⢸⣿⣿⣯⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠋⠀⠻⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠂⠙⠢⡀⠐⢄⠀⠀⠰⠀⠀⠀⢀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡈⠱⡀⣴⣆⣆⣼⣼⣿⣠⣠⣾⡧⠖⠀⠀⠉⠀⠀⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠘⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⣀⣀⣠⠤⠶⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠟⠻⣷⣶⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⢤⢤⣤⣾⣿⣿⣿⣿⣿⣆⡸⣴⣿⣿⣿⣿⣷⡦⠤⡀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠠⠖⠁⠔⠈⠀⠔⠻⠉⢹⠿⠿⣿⡿⣿⣿⡿⢿⡟⠟⠂⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⢠⡾⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣬⢀⠀⢀⣿⠇⠀⠀⢀⡎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠀⠸⣿⠀⢠⣠⠏⠀⠀⠀⠀⠐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠄⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠐⠦
⠀⠀⠀⣶⡤⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⣾⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣽⢷⡀
⠀⠀⢠⢽⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡟⠷⡄⠂⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⣸⡇⣸⡇
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠉⣠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⣽⣿⢿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠈⣿⣷⠀⠀⠀⠀⠀⠸⢿⣿⣷⣶⡀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣦
⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠾⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⠔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣾⣧⠀⣴⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣷⠉⣻⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠷⠀⠀⠐⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⠛⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⠛⠛⡉⠉⠁⠀⠀⣀⣀⣤⣴⣶⣿⣡⠀⠀⠀⣰⠀⡀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⠀⣠⠀⣠⣤⣤⣷⣴⣶⣾⣿⣿⣿⣿⣿⡟⠡⣿⡄⡂⠀⣿⡂⠠⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢀⣸⡄⡠⠲⢾⡇⡀⠀⢀⡷⢤⡴⠒⠛⡦⠌⣿⣿⡿⠻⠙⠟⠛⡿⢏⢁⠀⠘⠿⠇⣈⢰⣿⣇⠲⠇⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡿⠛⠋⠀⣠⠚⠉⠒⠣⠀⠂⠀⢠⣆⢁⠔⠀⠀⠀⠀⢄⠈⢰⡆⠀⠁⠋⠳⠴⢄⣆⡀⠁⠉⠑⣄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠾⠋⢀⠤⠀⠀⢊⠉⠁⣷⢠⠀⢢⠤⠀⠀⠀⠀⡀⠠⠒⠉⠀⢠⡆⣼⣇⠀⠐⠂⠀⠠⠐⠩⠙⡄⠀⠀⡇
⠀⠀⡀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠡⠿⠜⠂⢀⡀⠀⠀⠀⢰⡇⣀⣐⡀⠀⣼⣧⡿⠿⠖⠒⢂⠀⠀⠀⠠⠀⡇⠀⠀⠇
⠀⠀⠇⠠⠄⠀⠠⠄⠁⠒⠂⠀⠶⣿⣿⣽⣷⠀⠀⠉⠁⠀⠚⠃⠼⣷⡐⢉⡁⠨⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠀⠈⠠⠆⠀⠆⢴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠂⠀⠠⠤⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠣⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠔⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⠠⢀⣀⣀⡠⢞⢅⠀⠀⠀⠀⡎⠁⠀⣺⠋⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⡇⠸⠀⠀⠀⢠⢇⣀⣰⡗⠂⠀⠈⠛⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠃⠀⠀⣤⠀⢰⠉⠀⡐⠀⢀⡾⡠⢒⠟⠂⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡔⠤⡀⠹⣧⠸⢘⡐⠠⢠⣻⡉⠤⢊⡆⠀⠠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠓⣟⣇⠀⠡⢔⠊⣏⠡⢲⡞⠉⠐⠲⠀⠀⠇⠀⠤⠤⢤⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠂⢁⢻⡖⢃⠲⣄⣁⠴⣉⡏⠨⠀⠈⠀⠂⠀⠀⠀⢠⠞
⠀⠀⠀⢄⣀⣀⡤⠞⢛⢾⡻⢁⡆⠃⠼⠋⠽⣦⠖⢅⡈⢢⡏⢠⠄⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠈⢷⢯⣒⢌⢆⠱⣎⢩⣷⣭⣭⡭⡼⢿⡂⠨⣤⡆⠁⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠘⢤⠈⠛⣻⣦⣈⠢⣝⢦⡀⠀⠀⠀⠙⢶⣄⣥⠁⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠇⢈⠃⠀⠉⠛⠾⢿⣿⣶⣶⣶⢶⡶⣿⣵⡷⠖⠐⠐⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠙⡄⠻⠄⠐⠈⢉⡽⢫⠟⠉⠝⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⡐⠊⠀⠀⢠⠀⢘⠔⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⡤⠥⠄⠐⢢⡠⠔⠊⠁⠛⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃

⠑⢤⣀⠐⠀⠠⡀⠀⠀⠀⠠⠌⢀⠀⠀⠀⠀⣀⠐⠀⢀⠀⢠⠖⠀⡓⠀⠠⠀⠀⠂⡢⣰⠀⠀⠀⠁⢀⠄
⠀⠀⠉⠢⣄⠐⢅⠐⠀⠀⠀⠀⡄⠙⡀⠀⠀⠀⠀⠠⡜⠀⠆⠀⡌⠀⠀⠀⠄⠁⠀⡴⠩⡊⠁⠀⠂⠀⠰
⠀⠀⢀⠄⠈⠂⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠠⠈⠀⠀⢀⠀⠐⠄⠐⠐⠀⠀⠀⠀⠀⠀⠡
⠀⠀⡀⠀⠒⠁⠀⠀⠀⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⠀⠀⠄⠊⢠
⠤⢠⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢀
⠀⢀⠀⠈⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⠖⠀⠀⣰⢢⢠⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⡈⠀⠐⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠏⠀⠀⣿⡜⣸⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐
⢁⣐⠋⠄⠀⠀⠁⠀⠀⠀⠠⠔⠋⠀⢸⡷⣻⢠⡖⢀⡟⠀⣿⠁⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠑⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠿⠜⠓⠁⠷⠊⠻⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠠
⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠽⠃⠀⠀⣾⢱⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣽⠖⠀⠀⠀⣇⠏⣿⡜⠠⠴⠤⠄
⡀⠴⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣴⠃⡇⣠⡏⢀⣿⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠇⠙⠚⠋⠀⠛⠁⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢄⡊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠧⠈
⢀⠆⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠠
⠁⠠⠀⠐⡉⢀⡂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣌⢂⠬
⠀⠀⠀⠀⠀⠁⠸⢀⣘⠆⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡈⠠⢾⠈⠀⢀
⠀⠠⠀⢠⠀⠀⡀⠠⠀⠰⠀⠀⢃⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄⠈⣀⠈⠐⠀⠀⠀⠀⠀⠀⠈⡆⢀⠁
⠈⠀⠀⠈⠂⠐⠀⠂⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠦⠄⣀⣀⡀⠀⠀⠀⠀⡔⠉⠑⠀⠀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠰⢒⠖⠊⠀⡸⠁
⠀⠀⠀⠀⠀⠴⠠⠀⢺⠉⠉⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢄⠀⠀⠀⡄⠀⡇⠀⠄⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠈⠓⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢢⢂⠀⢱⢠⡇⣸⠀⢀⡼⠁⠀⣠⠎⠀⠀⠔⠋⠀⠀⠀⡀⠜
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⢤⡀⠀⠈⢦⣈⢯⣷⣿⣿⡿⠿⣶⣿⡁⣠⠔⢁⡤⠚⠀⢀⡠⠔⠀⠈⢙⠆
⠀⠀⠀⡤⠚⠋⠉⠁⠐⠀⠤⣈⠒⢤⣈⣻⠟⡁⠁⠀⠉⠀⠀⠙⢻⣿⣾⣥⡴⠖⠋⠁⠀⣀⠀⠒⠁
⠀⠀⠈⠓⢤⣀⢀⡀⠀⠀⠀⠀⠉⢲⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⡴⠶⠚⠋⠉⠀⠀⠀⠈⠁⠐
⠀⠀⣶⠉⠀⠀⠀⠈⠉⠒⠒⠲⢶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠶⠶⠦⠤⠤⠤⢄⣀⠤⠄⠒⠈
⠀⠀⠉⠓⠒⠲⠄⠀⠐⢒⣲⡶⠿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⡎⣀⠤⠂⠉⠀⠀⢀⣠⣴⠾⢷⡀⠀⢀⠀⠀⢀⡀⢠⣴⠿⣄⠀⠈⠉⠉⠙⠒⠖⠒⠂⠀⠈
⠀⠀⠀⢠⠏⠁⢀⣠⠤⠒⡈⠉⠉⠀⣠⠎⣿⢷⣾⢶⣶⣵⣾⣻⡉⢧⠈⠑⠂⠀⡀⠀⠀⠀⠈⠐⢤⡄
⠀⠀⠀⠈⢱⠛⢉⡠⡤⠊⠀⠀⡠⠞⠁⣰⠃⠀⡟⢸⡟⠈⢻⡹⡇⠈⠓⠀⠀⠀⠈⢄⠀⠠⠤⠤⠤⠇
⠀⠀⠀⠀⠈⠉⠁⢀⠀⢀⡴⠉⠀⠀⡰⠃⠀⢠⠀⢸⡇⠀⠀⢣⢡⠀⠀⠈⢦⣀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⡴⠁⠀⢠⠏⠀⠘⠻⠀⠀⠀⠄⠄⠀⠀⠀⠈⠓⢄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⡠⠪⣷⠀⢀⠞⠀⠀⠀⡇⠆⠀⠀⢰⠘⣄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠷⠊⠈⡀⠀⢀⡇⠘⢆⠀⢸⡗⠚⢦⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣸⠓⠂⠈⠢⡼⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠢⠀⢂⠀⠀⠀⢀⢀⣈
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠤⣀⠀⠀⠀⠀⡠⠀⠀⠀⣉⠁⠆⠀⠀⠀⠚⠤⠀⠀⢀⡚⠟⢩⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠈⠀⠀⠑⠉⢳⣦⠀⠗
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⡀⠀⠀⠠⡀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⡐⡰⢰⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡂⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠄⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠸⠃⠀⠀⠀⠘⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠄⠘⣿⣴⣥⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⡄⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠩⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⢄⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣀⣀⣀⡰⠄⠈⠀⠂⠀⠀⡖⠒⢆⠀⠀⠀⠀⠰⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠅⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⢂⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠐⠃⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⡦⠀⠀⠀⠂⠀⣀⠀⠈⠀⠀⠀⠀⣾⠃⠉⠀⠀⠀⢀⡀⠠
⠀⠀⣿⣷⡔⠆⠈⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡀⢀⣤⣭⠀⠀⢠⣤⠀⢀⠀⠀⠀⣠⣤⡀⠀⠸⠀⠀⠀⠀⠉⠀⠀⢠⣤⠀⠀⠀⠈⠀⢀⡀⠀⣻⠟⠠
⠀⠸⠿⠛⠁⠀⠀⢰⣶⡆⠀⢤⣤⠀⢰⣶⡀⠀⠘⠈⠁⠀⠀⠀⣠⡄⠈⠁⠀⠀⠙⠛⠀⠀⠐⠀⠐⠂⣶⠄⠀⠀⠀⠁⠀⣀⠀⠉⠀⠌⠁⠀⠀⠂
⠀⠀⠀⠀⡰⡶⠀⠀⠉⠓⠀⠄⠀⠀⠀⠉⠀⠀⣠⡄⠐⢺⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣶⡦
⠀⣠⣾⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁⠀⢠⣶⠀⠀⢀⣄
⠀⠈⠉⠙⡀⠀⠀⠀⠀⠀⠋⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉
⠀⠶⠀⣈⣁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀
⠀⠀⠸⣿⡿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀
⠀⡀⠀⢀⡀⠀⠀⠀⠀⣦⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⠀⠿⠷
⠐⠟⠃⠘⡿⠆⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠄⢠⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⡄⠀⠻⠛⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴
⠀⡁⠀⠀⢠⡆⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠩⠄⠈⣀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣄⠀⡉⠀⠀⠀⠀⠀⠉
⠀⡀⠰⣿⡿⠀⠀⠀⠀⠛⠀⠀⠰⠆⠀⠀⠀⠀⠀⣀⠀⢠⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⢰⣦⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠁⠀⠀⣤⡄
⠀⠆⠀⠀⠉⠀⠁⠀⠀⠀⠚⠀⠀⠀⢠⣦⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣤⡀⠀⠀⠀⣀⣀⠀⠀⠉⠀⠀⠀⠀⢰⠀⠈⠀⠛⠁
⠀⡆⢴⡶⠀⠀⢠⡶⠆⠀⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣄⡀⠀⠀⣶⡄⠈⠉⠀⠀⠾⠂⠙⠃⠀⢰⠆⠀⠐⠂⠛⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⢴⡆⠾⠂⢀⠀⠀⠀⠘⠉⠀⢀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⢿⣿⣿⣷
⢀⣿⡟⣹⣿⣿⣿⣿⣿⠿⠿⠛⠻⠿⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⣸⣿⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢹⣿⣷
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣄
⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠙⠂⠈⠋⢹⡇
⠘⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⣠⣄⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⠉⠈⣻⡄
⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠟⠀⠀⠀⢠⡄⠀⠀⢸⡇⢀⡏⠀⢀⣿⣀
⠀⠀⠀⣼⢋⣩⣉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢸⣿⠋⠉⠱⡟⠀⢹⠷⣦
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣤⣤⣤⠴⠶⢾⠋⠀⠀⠀⠀⠀⣸⣿⠀⠶⢾⣇⣀⡟⠀⣼⠁
⠀⠀⠀⠘⠷⢦⢾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⣈⡆⠀⠀⠀⠀⢠⡟⠹⣦⡀⠀⠉⠉⠙⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠈⠉⠀⠀⠀⢀⣴⠟⠀⠀⣸⠇⠀⠀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠛⣶⣦⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⢞⣻⡟⠁⠀⢄⡀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠀⢿⠉⠉⠉⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠈⠻⣷⣤⣄⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣤⣼⠷⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠀⠀⢠⡾⠁⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠻⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠙⠶⣆⠀⢀⠀⠀⣦⡾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠛⠉⠙⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠋⠀⠀⠀⠀⠸⣧⣠⣤⡴⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠒⣶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣴⠶⠶⠛⠛⠉⠀⠀⠀⢀⡟⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⡀⠀⠹⣧⡜⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠷⢦⣍⣑⠂⠤⢨⡟⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣏⣹⣌⡟⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣍⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠤⠤⠧⠤⠤⠤⠤⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠰⡀⠠⢀⡀⠀⠀⣠⠄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⣞⣰⠄⠀⠀⢡⠂⢾⣵⣀⣌⣑⠲⠄⠤⠆⢨⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡂⠲⣿⣴⢻⡅⡃⡸⢉⠷⡜⢀⣭⡷⡦⠌⠰⢿⡆⠻⣤⡤⣽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢈⣸⣏⡧⣷⣾⡿⢻⡇⢟⢼⣤⣁⣿⡘⢃⡎⡐⠆⠈⠻⠀⠚⣡⣮⣿⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⣶⡾⢼⣿⢻⣿⣾⣷⣿⣵⣧⣾⣾⢻⣏⢀⣿⣈⠤⣤⡔⢃⣾⣉⠽⠲⡿⠭⣟⣹⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠹⣿⡟⣾⢿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡟⣿⣶⠿⣷⠶⡀⠔⠁⣍⢥⣠⣟⡿⠗⢾⣷⣟⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢂⣴⣤⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⡶⣿⡷⠤⠐⠀⣩⠘⡛⢻⠿⣿⢿⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣿⣶⡬⢓⡿⣿⢻⣿⢿⢿⣿⣿⡿⡿⣿⢿⣿⣿⡿⠿⠿⣯⢭⣗⣉⣳⠎⠲⣿⠾⠿⢿⣞⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢻⣿⣿⡗⠜⠛⠊⠑⠁⡄⠏⠘⠋⠈⠁⠀⠋⠊⣴⣾⣿⡬⢿⡲⢬⢿⡹⠛⢷⠰⡆⢰⠟⣬⣬⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⢻⡿⣽⣿⣍⢦⢸⡇⣸⢿⣷⣀⣾⣃⣲⡜⠻⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⣤⣄⡈⢧⣼⣦⡔⢾⠅⢀⣀⣀⠀⠈⠂⠀⠿⠮⢿⣖⣤⣾⡡⣝⢿⡛⣿⣿⣶⣴⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⣿⣿⡟⠈⢿⣯⡵⠁⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⢭⡙⠿⢿⣿⣇⣽⡋⢭⡟⢳⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠉⠀⠀⣠⡀⣀⢀⣀⢑⠀⠀⠀⡀⠀⡙⠻⣿⣿⣴⣿⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡈⠠⠄⣼⣿⣷⡀⠀⠀⠀⠀⢀⠀⣿⣿⣇⣳⡄⠈⠃⠀⠀⠁⠁⠜⢯⡽⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡼⠟⠁⠪⠛⠿⠛⠘⠽⠛⢲⠶⠀⠁⠉⠀⠉⠈⠀⠁⠈⠒⠆⠁⠐⠆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠴⠤⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢐⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣽⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⡀⠀⠈⠙⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠻⠿⢿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⢠⡞⢭⣱⠂⠀⠀⠀⠀⠀⣿⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⢻⡿⠆⠀⠀⠀⠀⢀⡴⢯⣍⡢⠁⠀⠀⠀⠀⠐⠛⠻⣿⠟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠞⠣⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡈⠳⠃⠀⠀⢀⠤⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠳⠎⠀⠀⠀⠀⢠⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡋⠁⠀⠀⢎⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠂⠀⠀⠀⠀⠀⡔⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢙⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠢⢄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⠀⠀⠀⠀⠀⣤⠰⠃⠑⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠈⠢⣄⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣔⠲⠈⢃⠿⠟⠀⠀⠀⠀⠀⢙⣀⠤⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠓⢄⠀⠀⢈⠀⠀⢸⠀⠀⠀⢀⡤⢤⠀⠙⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠈⠢⡀⠀⠑⠚⠙⢄⣀⠤⡈⠻⢸⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡀⣴⣿⠙⠬⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⣠⣿⣦⡀⠀⠀⠀⠙⡆⡹⠶⠃⠀⠀⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⣴⠛⠫⣸⡿⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢀⣼⣿⣿⣿⣿⣦⡀⠰⠿⠟⠣⡀⠀⠀⠀⠀⢂⠀⢀⡄⠀⠀⠀⠀⠘⠺⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣴⣿⣿⣿⣿⡿⠋⠁⠈⠲⢄⠀⠀⠘⢄⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⣀⣤⣶⣿⣦⣤⣨
⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣼⣿⣿⡿⠋⠁⠀⢀⣠⣴⣾⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⢰⡟⠀⣴⣿⣿⠟⠁⠀⣠⣴⣾⣿⡿⠟⠋⠉
⠀⠀⠀⠀⢠⡟⣠⣾⡿⠋⣁⣤⣶⡿⠟⠛⠉
⠀⠀⠀⢀⣿⣼⢟⣩⡴⠟⠛⠉
⠀⠀⠀⣾⠿⠛⠉⠁


⣿⣿⣿⣶⣶⡿⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣶⡿⡶⠿⡶⠶⠾⣶⡶⠶⠾⠶⠶⠾⠿⡾⠷⠿⠶⠿⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⡄⠀⠀⠀⢿⣶⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡏⠈⠃⢀⠈⠀⠀⢀⣤⢼⣿⣏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣬⣇⣨⠀⠀⢾⠋⠛⣽⢿⣿⣿⣿⣾⣦⣴⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣧⣏⣙⣿⣷⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣨⢦⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⠎⢹⡏⠁⠹⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣾⣿⣿⣿⡿⠦⠈⡋⢻⠉⠙⡄⠐⠀⠀⠘⠃⠈⣿⣶⢠⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⡿⠋⠈⣿⣴⡿⣧⡤⠄⠀⠀⠀⠀⠀⠷⢤⣿⣿⣿⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⡾⠢⣿⣿⣄⣀⣁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⢿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠻⠛⢿⡿⠗⠈⠀⢀⣤⣄⠀⢘⣿⣄⣹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠘⣿⣷⣶⣾⣾⡿⢿⣷⣼⣿⣿⡇⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣯⣶⣶⣿⣿⣻⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣿⣿⠏⢨⣿⠿⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠛⣀⠙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⢿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⢀⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣤⣴⡖⠀⢀⣠⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⣿⣍⡁⢀⣾⠛⢛⡀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⢀⠠⣴⣶⣾⣿⣻⡿⢿⣿⣿⣿⣿⣶⣤⣄
⠀⠠⢶⣿⣶⣶⣲⡤⠌⣀⣀⣈⣁⡁⠈⠉⠉⠻⠿⢿⣷⣦⣀⡀⠀⠠⠀⠀⠠⣤⣤⣀⣀⣀
⠀⠀⠀⠈⠙⢿⣿⣷⣤⣀⣄⠀⠉⠍⡙⠛⠛⠿⣶⣶⣮⣝⡳⣌⢳⣷⣦⡁⢲⡑⠻⠿⠻⠿⠃⠘⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⣤⣠⠀⣶⣭⡶⢐⢋⣴⢦⣍⣙⠛⣮⣙⣿⣼⣇⣤⡴⠤⠖⠒⠃⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⢿⣿⣿⣴⣾⣿⣿⣿⣷⣿⣿⣿⠟⣛⣯⣽⡞⢿⣷⣦⣦⣤⣤⣀⠀⠀⠀⠀⠀⠈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠒⠀⠋⠛⠿⠿⠋⠱⠊⠁⠀⠉⠙⠯⣍⣽⣿⣿⣿⣿⣷⡆⣀⠀⠀⠀⠁⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⡿⢟⠉⠈⠀⠀⠀⠀⠈⢎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⡈⣈⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣙⣼⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣯⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⢜⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢨⣳⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢨⡾⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣰⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⢿⣷⣺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣵⣦⠠⡠⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣽⣷⡄⣀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⡀⢾⢖⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣀⣴⣄⠀⣀⡀⢀⠀⢀⠀⣴⡶⢢⣄⣀⣬⣼⡿⣳⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣐⣷⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢨⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠠⠐⠀⡈⠀⠀⠸⠔⠀⠀⠀⠰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠘⠁⢀⡠⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⠀⠀⡀⠀⠈⠀⠀⠂⠐⠁⠀⡁⢀⡴⢦⡄⠛⠓⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠁⢠⡄⠠⢀⠀⠄⠀⡀⠀⠁⡀⣈⣀⡐⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠉⠀⠁⠀⠂⠖⠠⠤⠀⠀⠉⠀⠀⠀⠀⠀⠀⢾⠇⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠤⠁⠀⡀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨
⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⢏⡀⠀⢀⣀⣀⠤⢤⠀⠰⣋⢫⡀⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠉⠉⠁⠉⠀⠀⠁⠒⠛⠛⠉⠉⠉