Bloating Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠘⣧⠀⣀⣤⣶⠶⠶⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠹⠟⠋⠁⠀⠀⠀⠀⠀⠈⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣴⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⢀⣼⣿⣿⣿⣅⣼⣿⣿⣿⣿⣿⣿⠃⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠟⠛⠋⣉⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠞⠳⠶⢦⣤⣴⠟⢉⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⠋⢠⣿⣿⣿⣿⣿⣿⣿⣯⣨⣿⣿⣿⣿⣿⡟⠙⣿⡿⠟⢁⣴⡿⠋
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣴⠶⢤⣤⣴⣦⡈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢉⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⡇⠀⠀⠀⠀⠙⢷⣤⣈⠉⠛⠛⠛⠛⠛⠛⠛⠋⣉⣁⣠⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠶⠶⠾⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢷

⠀⠀⠀⠀⠀⠀⢀⡖⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠦⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠎⢿⠀⠀⣤⠤⠤⠤⠤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠉⠉⠉⠛⠛⠲⠶⠦⣤⣄⣀⡀
⠀⠀⠀⠀⢠⠏⠀⢸⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠶⠦⣤⣀⣀⠀⠀⠀⠀⠉⠙⠛⠲⠦⣄⣀
⠀⠀⠀⠀⡎⠀⠀⠈⡇⠀⣿⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠉⠙⠓⠶⢤⣀⠀⠀⠀⠀⠀⠉⢙⣦
⠀⠀⠀⢸⠁⠀⠀⠀⣧⠀⢸⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢀⡤⠚⢧⠀⠳⡄⠀⠀⠀⠀⠀⠀⠉⣳⠂⠀⢀⡴⠋⢸
⠀⢀⠀⡿⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡠⠚⢲⣤⡀⢀⡔⠁⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡴⠋⠀⠀⢸⠁
⡆⢸⠀⡇⠀⠀⠀⠀⢹⠀⢸⠀⠀⣀⡴⠋⠀⠀⠈⠁⢱⡎⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⠞⠁⠀⠀⠀⢸
⠇⢸⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠘⣄⠀⠀⠀⠀⠀⣸⡅⣄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⠋⠀⣰⠋⠀⠀⠀⠀⠀⡿
⠀⠀⠁⢷⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠈⠓⠶⢦⡶⠚⠉⠿⡷⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇⢸⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⣰⠃⢀⡾⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢳⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠈⠲⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡞⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠈⣇⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⢀⡿⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰
⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⢸⠀⠀⠀⠀⠈⠁⠸⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡾⢠⠃⢀
⠀⠀⠀⠀⠀⠀⠹⡄⣼⠀⠸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡾⠀⣸⠁⠀⠀⠀⠀⠀⠀⢀⡾⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣏⣀⣀⠀⠀⠉⠁⠐⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⠇⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠭⣉⣛⠒⠶⠤⣄⣀⡀⠀⠈⢩⣇⢒⣤⡤⠴⠃⠀⡞⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⢀⡤⢞⣵⣪⠽⣱⣊⡭⣝⣒⠦⢬⣉⡛⠒⠮⢥⣜⣦⠽⠂⠀⢰⠁⠀⣀⣴⡊⠁
⠀⠀⠀⠀⠺⢭⣒⣛⠷⠯⣼⣥⣞⣽⢧⣞⣼⢭⣖⡯⢽⢖⡦⣤⣍⡙⠳⠦⣼⠴⠊⠉⠉⠛⠻⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠪⠭⣝⣛⡻⠿⣿⣷⣯⣿⣼⣺⡯⣗⡏⡟⢲⠀⠀⠀⠰⣎⠿⣛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠮⢭⣑⡛⠭⢟⡳⠿⣿⠸⡆⠀⠀⠀⠈⣩⠖⠛⢳⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠮⢍⣒⠢⣷⠀⠀⠀⠀⡧⠒⠒⠚⠤⢤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣿⢤⣀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠙⢫⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠉⠉⢩⢿⢿⠆⠀⠀⣴⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⠁⠀⠀⠀⢠⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠒⠢⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⡆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⡤⠤⠶⢋⠀⢠⣾⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣷⡄⠀⣀⣀⣅⠀⣈⣿⠃⢀
⠀⠀⠀⠀⠀⢷⠀⠀⠀⡇⠀⢸⠀⠀⢿⠀⢀⣀⡀⠀⡤⠀⢁⣠⠀⢨⡇⠀⠍⠉⣿⠀⢸⣿⡆⠀⠹⣦⣤⠄
⠀⠀⠀⠀⠀⠀⣀⣀⡠⣯⡀⠸⣄⢀⢧⠀⢄⣀⢀⢠⢧⣀⣈⣠⣄⣸⣇⣀⣵⢘⣿⣄⣀⣨⣿⣦⣀⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣼⣿⣧⣿⣀⣸⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣤⡴⠤⣤⡀⣀⣀⣀⣀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⡞⠉⠁⠉⠻⠶⠖⠒⠶⣟⡁⠀⠀⣼⠟⠉⠉⠉⠉⠛⢦⡀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⠀⠀⠀⠀⢀⣸⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⠛⠂⢀⡴⠋⠁⠀⠀⠈⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠚⠋⠈⣷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⠛⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠙⠳⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⡤⠞⠻⣦⣀⣀⣤⠞⠳⠦⢤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢠⡿⠒⠶⠶⠶⠶⠶⠦⠤⠤⢤⣤⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠶⠒⢻⡄
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⣼⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣧
⠀⠀⠀⢰⠏⠈⠉⠉⠉⠛⠓⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠉⠉⠉⠁⢹⡆
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠤⠤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇


⡄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⠠⡄⠀⠀⡄⠀⣤⠀⠠⠀⠀⡄⠀⡤⠀⠀⢀⠀⠀⢤⠀⣤⢠⠀⡄⠠⡀⡀⡠⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣇⢀⠟⠄⠀⡇⢰⠒⠆⢘⢹⠄⡇⠀⣏⠀⠀⠈⢱⠄⢸⠀⡟⢺⠸⡀⢀⠅⠛⢆⢸⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⣛⡻⣿⡿⣿⢟⣫⣝⠻⠟⣛⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠾⢿⣿⣶⣶⣶⣬⣿⣿⣾⣿⣿⣿⣦⡹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢼⡿⣩⣶⣿⣷⣾⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⡇⡯⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⢋⣼⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠻⣿⣿⣿⡿⢿⣿⣿⠏⣿⣿⣿⢟⣴⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣭⣴⣾⣦⣭⣥⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣫⣭⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣭⣝⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣫⣭⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠿⣿⣿⣿⣿⣿⣿⠟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣼⡇⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⢹⣇⣫⣥⣶⣭⣙⢿⣿⣧⠀⠀⠀⣤⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦⡛⢿⣿⣿⣿⣿⣧⢻⣿⣇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⡆⣿⣿⡀⠀⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣋⣙⡛⣣⣿⣿⣿⡿⢰⣿⣿⡇⠀⠠⠦⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡅⢿⣿⡿⣿⠿⠿⢛⣵⣿⣿⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡆⣿⡇⣶⣶⣾⣿⣿⣿⣿⡿⠀⠀⠈⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠛⢦⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠛⠛⠛⠛⠀⢀⣠⡼⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡤⢤⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠖⠋⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⢴⡃⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⠶⠦⢤⣄⡀⠀⠀⠀⠀⢠⠏⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⡟⠀⣶⠀⠀⠀⠀⠈⢳⡀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠿⡄⠀⠀⠀⠀⠀⠹⡄⠀⢹⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣀⣀⡤⠖⠋⠀⣠⠟⠀⠹⣄⠀⠀⠀⠀⠀⢻⠀⠈⠳⠀⠀⠀⠀⠙⠦⠤⡴⠚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠉⠉⠀⠀⢀⣠⠖⠁⠀⡀⠀⠈⠳⣄⡀⠀⠀⢸⡆⠀⠀⢰⠀⠀⢰⠀⠀⢠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠒⠲⣶⠶⠖⠚⠋⠁⠀⠀⠀⠉⠓⠒⠒⠻⡍⠀⠀⠘⡇⠀⠀⠘⠇⠀⢸⠀⠀⠸⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠃⢴⣦⣤⡤⠖⠀⠀⠀⠲⢤⣤⣴⠆⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢿⠀⠀⠀⠀⣠⠴⠚⠛⠛⠒⠦⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢿⢧⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⢠⢿⠃⠀⠀⢸⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠀⢸⠀⠘⣎⢳⡀⠀⠈⠻⢭⣭⠽⠃⠀⠀⣰⢋⡟⠀⠀⣠⢸⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡟⠀⠸⡄⠀⠸⡄⣿⢦⣄⠀⠀⠀⠀⢀⡤⢾⢁⡞⠀⠀⢠⡏⢸⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢧⠀⠀⢻⡏⠀⠈⠙⠓⠒⠛⠉⠀⠘⡿⠁⠀⢀⡞⠀⢀⡇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠼⠧⣤⣬⡧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣤⡤⠿⠶⠚⠉⠀⠀⠀⠈⠑⠦⣤⣀⣠⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⣉⣀⣉⡉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⣰⡿⠁⣠⣾⣿⣿⣿⣿⣿⣷⣦⡈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡄⢠⣿⠃⢰⠟⠋⣁⣠⣤⣤⣀⡉⠛⢷⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠇⢸⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣿⣿⣿⠟⠛⠋⣠⣾⠟⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡀⠙⠿⢿⣶⣶⠾⠟⠉⣠⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣷⣤⣤⣤⣤⣤⣴⣾⣿⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣴⣿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠁⠀⠈⠙⢿⣿⡁⠉⠛⠛⠛⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣄⠘⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣿⠇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣴⡿⠋⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⣈⠉⠉⠛⠛⠛⠋⠉⣁⣠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⡔⠖⠐⠚⠛⠲⢒⡖⠲⠶⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠐⠁⠀⠀⡐⠀⠠⡀⠀⢄⠀⠤⠀⠫⠭⠐⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠴⠂⠂⠀⠀⠀⠀⠠⠔⠀⠀⠘⠎⠠⢡⠀⠀⢀⠠⠀⡀⠩⠂⡵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⣦⠀⠀⠁⠂⠐⣄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⡂⠀⢰⣠⡀⠾⠂
⠀⠀⠀⢀⠠⢀⠀⠀⠀⢐⠄⠀⠨⠆⠀⠀⠀⠡⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠚
⠀⠀⠀⣡⣤⣄⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀
⠀⠀⠰⣸⣿⣿⠏⢰⣤⠒⣄⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣀⣸⣟⣸⠆
⠀⠀⠀⠈⣻⣷⣶⣀⣩⣀⠉⡷⠼⣷⡶⠒⣲⢦⣶⠀⣤⣤⠀⡤⢺⡇⢀⣭⡦⣴⠂⡶⠶⠏⢼⣟⡉⢛⣃⠀⠪⡀
⠀⠀⠀⠀⣙⠿⣿⣉⣀⠈⠁⠀⣒⣾⣿⠟⠁⠀⢹⡞⠓⢛⣿⣷⣞⣢⣾⣝⠁⠙⣿⣦⣤⣷⣦⣉⣠⣿⣿⢧⡄
⠀⠀⠀⠀⢰⣧⠀⠈⠉⠉⣉⣩⣭⣷⣾⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣌⡉⠒⠈⠉⠉⢀⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⢾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⠿⢛⣛⣛⣿⠿⣿⡯⢩
⠀⠀⠀⠸⡗⢹⣟⢫⡍⢻⣍⣉⣉⣉⣭⣴⣿⠮⢭⣋⡛⠛⠛⢛⣛⣿⣿⣿⠟⠓⣾⣿⠋⣿⣿⠷⠀⢿⠗⡻
⠀⠀⠀⠀⢸⣎⣁⠚⢿⡟⢠⣤⠁⠈⠿⠋⢹⣧⣬⡅⠸⣿⠿⡍⢹⡅⠛⢻⠅⣴⡮⡹⠀⠙⠫⢶⣦⣼⣿⡷⠂
⠀⠀⠀⠀⠈⡌⠉⠒⠛⠛⠋⠛⠻⣿⡇⠰⣶⣾⣿⠀⣶⣦⣴⣴⣬⡗⣶⣦⠶⠿⠧⠿⠛⠿⠷⠚⢋⢉⡿⠁
⠀⠀⠀⠀⠀⠈⠠⣀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⠛⠁⠉⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠄⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠠⠄⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣀⠋⠁⠀⠈⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢁⡀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠊⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣯⠁⠈⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠘⠤⠠⡘⠂⠹⠀⣿⡆⡄
⠀⠀⠀⠀⠀⠀⢠⣾⡰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠁⣠⣤⣶⢇⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⡀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠷⢛⣋⣥⢞⣿⡍⡥⠃
⠀⠀⠀⠀⠀⠀⢅⠋⢾⢿⣿⡿⠗⠶⣶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡞⠻⣿⠻⠌⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢆⣼⡿⡏⠀⠀⠈⣷⣶⠿⡟⠛⠻⣯⠹⣿⡿⠙⣿⣯⠉⣻⣿⣿⡦⢻⡟⠈⢧⣻⣿⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠳⢶⣾⠋⠃⠀⠀⠀⠂⢁⡄⠇⠁⢀⡿⣏⣼⠜⠀⠉⢁⣾⢷⣴⣾⢿⣿⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⠓⣦⣀⠈⠑⠒⠀⠀⠀⠀⠉⢣⡴⡶⢟⣩⠿⠿⠷⠦⠴⠛⠛⢉⣠⣴⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠈⠉⠉⠑⠒⠦⠤⢤⣤⣬⣽⠶⣿⡛⣻⣷⣶⡤⠤⢖⣶⣿⣿⣿⣭⣧⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣶⣾⣿⣿⣿⣿⣿⢶⣶⣭⣝⡒⠒⠀⠰⠒⠒⠺⠏⣼⣿⣿⣿⣿⣿⣿⣿⡋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢟⠛⠿⠻⠿⠿⠿⠿⠶⠦⣀⣀⢀⢰⠟⠛⠛⠛⠛⢑⣲⠲⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠉⠀⠀⠀⠂⢋⣁⢴⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢔⣫⡵⠟⠉⢉⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠠⠄⠀⠀⠤⠄⠒⠀⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠛⠃⠀⠀⣀⠀⠉⠉⠓⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⣀⠀⠀⠀⠀⣔⣲⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⣠⡄⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢡⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠴⠆⠀⠀⠀⠠⢊⠁⣶⠇
⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⠳⢥⣀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠔⠚⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠉⠙⠒⠦⢤⣀⣀⠀⠀⢤⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⢰⣲⠮⠭⠒⠊⠉⢁⣀⣀⣀⣀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠖⢚⣻⡆⠀⣀⣠⡄⠈⠉⣩⣹⠟⠓⠒⠒⠒⠒⠛⠉⢉⣉⡤⠤⠶⠒⢚⡋⠩⠉⠁⡤⠀⢻⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠛⠂⢩⣭⣡⡤⠳⠖⠊⠳⣄⠀⠀⠀⢀⣀⡤⠒⢫⡉⢠⣴⣆⠁⠠⡭⠝⠸⢆⣠⣠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠿⢿⣦⣤⣿⣅⡁⠈⠳⠟⠄⠸⣄⡤⠞⠉⠔⠻⣄⠺⣗⣀⣀⣉⣤⣤⣴⣶⡶⠿⣛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡂⠀⠉⠉⠛⠛⠛⠿⠿⠷⠶⠿⠶⠾⠿⠷⠿⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⢀⡉⠥⢞⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢐⣪⣽⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠷⢤⣄⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⣀⣀⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⠶⠶⠶⠿⠿⠿⠿⠟⠛⠛⠛⠿⠿⠷⠶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⢛⣋⣩⣭⣤⣤⠤⠶⠖⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣿⣭⣭⣝⣛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣭⡶⠾⠛⠉⠉⠀⠀⠀⠀⠀⣀⣤⣤⡴⠶⠶⠶⠶⠶⠶⢶⣦⣤⣤⡀⠀⠀⠉⠉⠛⠿⢶⣤⣍⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡠⣿⡁⢀⣠⣴⣶⣶⠀⠀⠀⠀⣿⠟⢁⣠⠶⠶⠶⠶⠶⠾⠿⠶⣦⣄⢹⣿⠀⠀⢠⣴⠶⣤⣄⣀⠙⠻⣦⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡙⢷⣟⣻⣿⣧⣤⣤⣿⣄⣀⣀⠀⣿⠀⢸⡿⣶⣶⣤⣶⣶⣶⣶⡿⢿⡿⢸⡏⠀⠀⣾⠁⣀⣤⣬⣿⣟⣛⣫⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⠉⠒⠾⠷⣾⣭⣭⣽⣿⡆⣿⡀⢸⣧⣄⣀⣄⣀⣰⣀⣠⣤⣼⣿⣾⠇⠀⠀⣿⣼⣿⣿⠟⠛⠋⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⡀⠀⠀⠀⠀⢹⣿⡇⠸⠿⠷⠟⠛⣛⠋⠛⣻⡿⠿⠟⣭⡿⢿⣦⣄⠀⠈⢻⣿⣿⠀⠀⢀⣠⡴⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⣿⡆⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⠀⣼⠋⠀⠀⢰⡏⢠⣾⠋⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠁⠀⠀⢹⡇⠀⠀⢸⣿⣇⣀⣤⣤⡶⠾⣿⡀⠹⢦⣄⣀⣼⡇⢸⣿⠀⣸⣧⣤⣼⣿⣿⠀⠀⣼⡏⠙⣿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣦⣤⣀⣀⣸⡇⠀⠀⢸⣿⡟⠉⠀⠀⠀⠀⠹⠷⠶⡶⠿⠋⠙⠷⠾⠿⠾⠋⠀⠀⢹⣿⣿⠀⠀⣿⣷⣾⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠛⠿⠿⠿⠇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠉⠀⠀⣠⣿⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠛⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣀⣀⣠⣤⣴⠿⠋⠁⠙⢷⣄
⠀⠀⠀⠀⣴⡾⠋⠀⠀⠀⠉⠙⠻⠶⢦⣤⣤⣤⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⡇
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠿⠛⠛⠻⠿⢿⣿⣶⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣾⣿⠃
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⣴⣿⣿⣁⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠿⠿⠿⠿⠟
⢸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠉⠛⠛⠿⣿⣦⣄
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠛⣿⣷⡀
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣷
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣀⣠⣾⣿⠏
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣾⣿⣷⣴⣶⣿⣿⠿⠟⠋⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠉⠛⢿⣿⡆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⢻⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣴⣾⣿⠟⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⣿⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠙⠻⢶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⠶⠛⠛⠉⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⢶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠷⠶⠶⢶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠾⠶⠿⠟⠛⠋⠉⠁

⠀⠀⠀⢠⠠⢠⠄⡀⡀
⠀⠀⠀⠸⠀⡆⠀⡀
⠀⠀⠀⢀⠉⠙⠚⠛⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⡀
⢠⡈⢹⠀⠑⠊⠐⠀⠈⡅⠋⠁⡇⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠋⠉⣠⠄⠀⠉⠙⠛⠿⣷⣦⣄
⠘⠥⠴⠤⠴⠤⠴⠤⠤⠦⠦⠤⠇⠀⠀⠀⠀⢠⣵⣿⣿⣿⣿⠏⠀⠀⠀⣠⡌⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡏
⠀⠀⡂⡦⠀⡤⠀⡤⠤⠀⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⠁⠀⠀⠴⣿⣿⣿⣶⠀⠁⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⡇⣀⠀⡂⠀⡀⣸⠀⢀⡇⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⠈⠀⠀⠀⠹⢿⣿⣭⡀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠈⡉⡩⢉⣉⢍⠉⣉⢍⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠴⣶⣄⠛⠛⢀⣤⣀⣀
⠀⠀⠀⡇⡄⠀⠉⢨⠀⢽⢸⠀⠀⠐⢴⣿⣿⣿⣿⣏⡄⠀⠀⢀⣴⢆⠀⠀⠀⢀⡀⢀⠀⠀⠀⣠⣿⡿⠓⠾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠳⠶⠶⠶⠶⠦⠾⠞⠀⠀⠀⣾⣿⣿⡯⣿⣿⣗⣦⡄⠐⢶⣿⣷⣤⣶⣿⣷⣴⡄⣠⣾⡿⠏⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠘⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⠻⡿⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠾⠁⠀⠈⠀⠁⠉⠉⠁

⠈⠻⠿⠿⠾⠓⠤⠀⢀⠀⠀⣠⡜⠋⠀⠈⠀⠉⠉⠙⠓⠲⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠿⡶⠾⠿⠟⠒⠲⣶⣧⣴⣶⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢸⠑⠈⠀⠀⠀⠀⠠⠀⠀⠀⠐⡀⠀⠈⠙⠓⠦⣄⣀⡀⠀⠀⠐⠂⠉⠁⠀⠀⠈⠑⠀⣀⣀⣺⣯⢱⡛⠽⣿⢻⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣤⡄⠀⠈⠉⠀⡀⠐⠀⠈⠈⠀⡀⠈⠐⠔⣩⣿⣄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣏⣊⣧⣖⢫⣿⣿⣿⠄
⢀⡀⣀⣤⣴⠎⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣛⠲⠶⠶⠶⣾⢤⠤⣤⣤⣶⣤⠤⡾⠾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠟⠛⠛⠉
⣿⠿⡁⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣴
⣄⣈⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⣠⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⠧⠒⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠞⣡
⡿⠟⠁⠀⠀⠀⢀⣀⠀⢀⠀⠂⢒⣂⣀⡾⢃⢴⡡⠞⠋⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣿⣃⣴⠏⠀⠀⠀⠀⠀⠀⢠⢟⣠⠾⠋⠄⢈⠟⢋
⠶⠂⠀⣠⣴⣿⣿⣿⢷⣾⡿⠋⠉⠉⠛⡻⠛⠛⠓⣠⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢸⡏⢀⠀⠀⠀⠀⠀⠀⣠⡿⠛⡙⠛⣀⢠⣥⣴⡿
⠀⢠⣾⣟⣧⢽⠛⡉⣏⢭⣿⣤⡤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠈⣧⢀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢈⡇⢸⠀⠀⠀⠀⣀⣼⢃⠴⣇⡀⠊⣠⡿⠛⠁
⠀⣿⣿⡿⣿⢫⣴⡌⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡁⣟⣿⠀⠀⠀⠀⠀⠀⠀⣼⣧⣦⣾⣇⠈⠀⠀⠀⠰⣟⠘⣁⠔⢠⣴⠾⠛⠨⠄
⠀⠈⠙⠛⠳⠻⠶⠞⠛⠁⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠛⢻⣆⠀⠀⠀⢀⣼⡿⢯⠁⣼⣿⣅⠀⠀⠀⠀⣀⠹⠯⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣻⡇⣤⣬⠠⣇⠸⠀⠀⢸⣏⣮⠇⣁⢧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁⡌⠩⠙⣦⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁⢘⠛⠄⣿⠀⠀⠀⠸⡿⢛⣼⣿⣻⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣹⠗⠀⠀⠀⡀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀⠀⡃⣼⡇⠂⠀⠈⣿⣿⣿⣶⡿⠋⠀⠀⠀⢀⣀⡀⠁⠄⠀⠀⠀⢀⣾⣿⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠲⣿⢻⠀⠀⠀⠀⠀⠀⠀⣽⡏⠸⠀⢷⢾⡇⣶⠁⠀⠘⠿⠿⠛⠀⠀⢀⣴⣞⠋⠉⠝⠺⡄⠀⠀⠀⢸⣿⡛⡆⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠅⠀⡂⠀⢀⡟⡘⠀⠀⠀⠀⠀⠘⠀⣾⡇⠐⠀⣺⢈⣷⠃⠀⠀⠀⠀⠀⠀⠀⡰⣻⠿⠇⠀⠀⠂⠀⢻⡀⠀⠀⠈⢧⣭⣸⣾⣿⠇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⡐⠘⠠⠁⣸⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⡓⢺⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⡞⠙⢻⡿
⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠁⢠⣰⡏⠀⠀⠀⢀⣶⣶⣦⣀⠀⢸⣧⣤⡆⢰⢸⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠠⠁⠂⠀⢀⡞⠁⠀⠀⠀⠈⠁⢀⡼⢁⠄
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⢟⣿⣿⢿⣧⡀⠙⢿⣳⣤⡿⠃⠀⠀⠀⠀⣠⠀⢠⣿⠆⠀⠀⢂⠀⢀⡾⠁⠀⠀⠀⠀⠀⣧⣾⡃⠈
⠀⠀⠀⠀⠄⢽⠀⠠⠁⠀⢠⣿⡀⠀⡇⠀⠀⣿⣮⡜⢿⡧⣿⣿⡄⠀⢩⡀⠀⠀⠀⠀⠀⠈⠀⢀⣾⠁⠀⢀⠀⢀⢀⡞⠀⠀⠀⠀⠀⡄⣸⣟⠈⢿
⠀⠀⠀⠊⠀⣿⠀⠌⡀⢠⡿⠙⡇⠀⠀⠀⠀⣿⣽⠗⢀⣘⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠁⢀⡁⢰⠄
⠁⠀⠀⠀⠀⣿⢠⠀⣤⡿⠁⠠⠀⠀⠀⠀⠀⠘⢿⣧⠆⣉⣏⣽⣃⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠀⠀⠈⠉
⠀⠀⠄⠀⠀⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠟⠻⣯⠀⡀⠀⠀⠎⠀⠀⠀⠀⣼⣦⠀⠈⣠⠎⠀⠀⠀⠀⠘⢶⡤⣤⣀⣦
⠀⠀⠆⠠⠀⠈⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⣽⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡾⠁⠀⠈⠀⠀⠀⠀⣈⣿⡆⠙⠻⣿⠿⢶⣈⡂
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠙⢻⣦⣑⡦⣀⣶⣌⠻⣦⣄
⠶⣜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠌⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⠂⠀⠀⠀⠀⠙⠻⢷⣌⠙⢅⡉⠢⢉
⠰⣄⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢎⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠻⢶⣶⣎⠒⠄
⠰⠞⠙⠢⡻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢷⠘⣇⠀⠀⠀⡄⠀⠀⠤⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣀
⠷⠄⠸⠦⠤⠾⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠦⠘⠄⠙⠷⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⣀⡠⢐⡲⠿⣇⠀⠀⢱⡀⠘⡦⣤⠄⠒⠀⠐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⣠⢴⡏⠀⠀⢀⡎⣰⠉⠈⢁⣹⢦⡀⠀⠁⠀⠟⠀⡠⠴⠛⠂⠀⣀⣴⠷⢦⣤⣀⡔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢶⠏⢩⠀⠽⢿⡓⡠⠼⢧⣀⡇⢠⠊⠀⡤⢹⣶⢤⠀⠀⠀⠀⢀⣠⣴⠞⠁⠉⠳⡔⠂⠹⢁⢂⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⡈⠀⠈⠀⡈⠸⢯⠀⠀⠀⠙⢢⡁⣠⠋⡠⠊⣌⠜⡑⢛⢟⠟⢻⡶⠋⠀⠀⠐⡄⠇⠈⠃⡞⠘⠃⣻⡳⣄⡠⠀⠄
⠀⠀⠀⠀⠀⢸⠉⠀⠀⠈⠳⣄⠑⣄⠀⢀⣴⣧⡀⠑⣤⠶⠒⢛⡿⣯⣩⡤⠊⡖⠚⠛⠛⢆⡠⠊⠀⠀⠀⢰⠁⠳⠤⢤⡴⠃⠰⣋⠆⠁⡸⠁⢠⠆
⠀⠀⠀⠀⠀⠘⣆⠀⠀⣄⠠⣎⠀⢈⢞⠏⠁⠀⢹⣦⡇⣀⣄⡛⠀⠴⢀⠘⠐⠉⢢⣄⣤⠞⠵⣂⣀⠁⠄⢉⠃⡀⠀⠀⠱⠄⢲⠟⠀⢈⠅⠴⢇
⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠘⠚⢽⡏⡜⠠⢠⡀⠈⣷⡾⠀⠠⠂⠠⠃⣸⡲⡀⠼⡱⠋⠀⠀⠀⠈⢻⣠⠞⠁⠀⠀⣀⡀⠀⣧⢀⡠⠖⠂⢀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⡦⢤⣼⢱⡇⠀⢀⡀⡆⢻⡁⠀⡀⠀⢠⠀⠀⢰⣷⡠⣷⣟⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⣌⠀⠀⣿⠋⠁⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠻⢶⣼⡄⢷⠰⡀⣀⠀⢌⢿⠋⠇⣀⣀⣠⠞⠃⠭⣉⠉⠛⣾⡁⠀⡎⠁⡅⢀⡀⠰⠄⢀⠀⢀⡟⠛⣩⡿⡋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠛⠾⣧⣱⣀⠠⠀⢻⢻⠉⠁⠑⣾⠀⠀⠀⠈⡗⠉⢩⣷⣦⣙⠦⢤⣤⣤⠤⠔⣉⣤⠿⠖⡋⠕⡊⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠉⠙⠒⠳⠶⠧⣤⣤⣿⣄⣀⣀⣸⣃⣠⣏⣈⣧⡭⠿⠶⠶⣒⣚⡭⠥⢒⡪⠅⠒⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣒⣒⣛⡭⠭⠕⠒⠺⢍⡓⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡓⠶⢬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢒⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢌⣉⡙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⣉⣉⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⣀⣶⣀⣀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⢴⡋⠉⣁⣿⣈⠉⠙⣦⣄⠀⠀⠀⢸⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡰⢃⡤⠛⠉⠉⠁⠉⠉⠓⢧⡈⢦⠀⠀⢸⡇⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣹⠛⠦⣄⡀⠀⠀⠁⠀⠀⠀⠘⣆⣆⣀⢸⡇⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠁⠀⣨⣽⡟⠁⢡⣤⣄⡀⠈⠉⠉⠉⢸⡇⠀⠀⠀⠀⣤⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠛⠉⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠋⠀⠀⠀⠀⠀⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣉⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠛⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢹⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢸⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣾⡇⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠁


⡟⠉⢙⡟⠟⠻⠿⠛⠿⡛⠿⠻⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠀⠒⠒⠒⠂⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⢁⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡾⡀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠒⠀⡈⠈⠂⣿⣦⢀
⠀⠀⠀⠀⠀⠺⣧⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⣳⡿⠿⡇
⠀⠀⠀⠀⠀⠠⢸⣿⣶⣤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⠵⢶⣿⡑⠛⠈
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣻⣼⠿⠛⠻⣏⢻⣿⠛⣿⡏⣻⣿⣿⠝⡿⠉⢆⣾⡆
⠀⠀⠀⠀⠀⠀⣰⣷⡈⠓⠾⣏⠁⠀⠀⠀⣠⠘⠀⣠⣟⣷⡟⠁⢉⣰⣟⠶⢟⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⢳⣿⠙⠶⠤⢄⣁⡀⠀⠀⢠⣖⣩⣽⣾⣛⣛⡉⣉⣡⣤⣶⠿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣴⣶⣭⣽⣯⣟⣛⠭⠄⠙⠻⠭⢥⠞⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠛⠿⠻⠿⠿⠷⠶⢄⣀⡀⡸⠛⠛⠛⠛⣳⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⣡⣴⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⢀⡀⢠⠽⠞⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⢀⣠⡶⠟⠛⠛⠛⠿⠿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠿⠛⠛⠛⠿⢶⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⠁⢀⣀⢀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠋⢿⡛⠾⢙⡟⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢹⡏⠾⢋⡿⠈⣻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠽⡿⡯⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠽⠿⠯⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣒⠭⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣫⠕⡊⠁⠀⠀⠀⣀⣤⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢀⡩⠖⠉⠀⢈⡠⣔⠿⠋⠁⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⡚⠉⢀⡤⠞⠉⠀⢀⣤⡾⠝⠊⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣀⠴⣊⠵⢊⠤⠚⡉⠔⣀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⢠⠖⠛⢲⡄⠘⢦
⠀⠀⢠⡊⠁⠈⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⡿⠀⠈⠳⣄
⠀⠀⠀⠱⣄⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⢀⠤⠤⠤⣤⠤⠤⣤⠖⣾⠻⣗⡋⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠈⠳⣖⠕⠉⠀⠀⠀⠀⠀⠀⡏⠐⠒⠲⢼⠆⠀⣯⠀⢿⢀⡿⢀⠵⡒⠒⣦⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀⡴⡟⠶⣶⠚⡿⢻⢉⡹⡗⠛⡿⢥⣯⣀⣬⠖⣻⣀⣀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢾⢧⠈⢦⢸⡇⢷⠘⣯⣀⡿⠀⡼⣋⣉⣙⣶⠊⣹⠉⠉⠙⡆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⢮⣛⣶⠟⣡⡎⠛⢞⢯⣠⠶⣷⣶⣖⣻⠟⠀⢿⣒⣖⣺⠇⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⢳⣙⠲⢤⣞⠻⡶⣄⠻⣍⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⡸⡋⠉⠀⣹⠈⠺⠶⠋⠀⢠⠤⠤⣄⠀⠀⠀⠀⢀⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠱⣌⣓⣲⠟⠀⠀⠀⠀⢠⣇⣀⣀⣼⠀⠀⠀⣪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡔⢻⠓⣆⠀⠛⠻⠛⠃⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢸⣄⣸⣠⣾⠀⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠛⠛⠛⠁⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡾⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⠈⠚⠉⠉⡷
⠀⠀⠀⠀⡼⠈⠓⢲⠤⠖⢚⣄⣥⣀⣱⣴⣶⠒⡷⠲⡔⠒⠈⢉⡀⠀⡷⠦⠒⠶⠚⢧⣶⡦⣤⠾⡄
⠀⠀⠀⠀⣣⠀⠀⠘⡿⢸⠥⠒⠤⠖⠓⡏⢈⠛⣄⢃⠷⠔⢊⣩⡿⢋⢧⡲⢤⣄⡮⢶⣎⡃⣠⠀⡇
⠀⢀⡠⠼⢇⠁⠌⠄⠉⡦⠒⠒⣒⡩⡖⢻⡌⠀⢘⠿⢲⠶⠟⢭⡜⠁⠄⠀⠠⢾⣃⠀⠀⡟⢀⡱⠓⣄
⠀⠸⣄⣈⠉⠓⠒⠒⠠⠿⠭⠥⠀⠹⠷⠾⠭⠤⠮⡴⠓⣈⡀⠠⠜⠈⠀⠉⠉⠉⣉⣉⣉⣤⠤⢀⡴⠃
⠀⠀⠀⠈⢣⡀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠲⠶⠂⢰⡡⠊⠁⠒⠒⠒⠒⠚⠋⠉⠉⠉⠀⢀⠀⢠⠏
⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠦⠤⠤⠤⣤⣀⣀⣀⣀⣀⣠⡤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠀⠀⠀⠀⠀⢀⠠⣂⡜⢣⠀⠣⡀⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⡀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠄⢣⠴⠀⡘⠈⠂⡠⢤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠐⡀⠀⠀⢠⠁⠠⠀⠀⠀⠡⠀⢡⠄⠂⣠⡉⠈⠂⠐⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠐⠁⢀⠔⠉⢡⡔⠈⠠⠄⠀⠈⢲⠈⡱⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠊⢀⠄⠒⢀⣰⠁⢈⠶⢀⠀⠈⡶⢁⡬⣀⡐⠀⠄⠀⠂⢈⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢁⠠⠁⠀⡀⡊⠀⡸⢅⠀⠀⢑⠒⢀⡈⠀⠘⠂⠀⠈⡀⠐⠁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠁⠀⢀⡊⠀⠀⠃⠀⠀⠑⠄⠂⣀⠈⠀⡃⠀⢀⠄⠊⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠁⢈⠖⠈⠒⢄⣀⠔⢄⠠⠃⢄⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠰⠁⠁⡤⢆⠀⠀⠀⡠⠀⠀⡜⠀⢀⠄⢀⠊⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢦⠂⠀⢀⠁⠀⠑⠁⡠⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⡯⠁⡀⠀⠀⡠⠁⠈⢲⠅⠀⠠⢀⠔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢢⠺⢀⠀⡠⠂⠀⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⠐⠉⡈⠌⠉⠁⠐⠒⠃⠀⠎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠐⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⢀⠨⠉⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢀⠐⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠊⠀⢀⠐⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠐⡀⠠⠋⠀⢀⠀⠂⠁⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⠄⠈⠐⠠⠀⠀⠠⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠒⠒⡒⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠆⠀⠀⠚⠂⠀⠀⠈⠁⠀⠀⠤⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠆⠀⠀⠀⡤⠀⠀⠀⠠⠆⠀⠀⠀⣤⠀⠀⠀⢤⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠁⠀⠘⠋⠀⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⠚⠁⠀⠀⠀⠐⠖⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⣻⣧⠀⠀⠠⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣸⡇⢄⡀
⠀⠀⢤⠤⢤⣉⠉⠙⢷⣄⠀⠀⠀⢰⠄⠀⠀⡀⠀⠀⢠⡄⠀⠘⠃⠀⠀⠀⠀⢰⡄⠀⠈⠁⣀⣴⠟⠀⠀⠘⡆
⠀⠀⠈⠑⠠⢌⡹⡄⠀⠉⣻⣶⣤⣀⡀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠘⠂⠀⠀⠀⠀⣀⣠⣴⠾⠟⠓⢠⡂⠘⠋⠁
⠀⠀⠀⡴⠂⡿⠀⠁⠀⠀⠀⠤⠾⠛⠛⠻⠷⢶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠟⠛⣋⠿⠂⠤⠴⠖⣋⠀⠉⢱⡆
⠀⠀⣠⠇⠀⠙⢄⠀⠀⠀⣀⣠⠴⠶⠶⢤⣄⣀⣀⠤⠚⠓⠦⠤⠤⠀⠤⠀⠒⠊⠁⠀⠀⠀⣠⠴⠟⠒⠂⠈⠙⣧
⠀⠈⠳⣄⡀⠉⠉⠙⢦⣀⣀⠀⠀⠀⢀⠠⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠃⠀⢀⢒⠲⠶⠚⠁
⠀⠀⠀⠀⠉⠳⡄⠀⠀⡴⠛⠳⢤⣐⣀⣀⣠⣤⣶⣶⣤⣤⣤⢴⡾⠟⠓⠚⣛⣉⡉⠑⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⣀⣤⠴⠶⢤⣀⣀⣠⠶⠥⣤⣀⣈⣀⣤⠖⠋⠉⠉⠉⠙⠛⠛⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠸⣯⡉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣶⣿⡿⢿⡛⠉⢩⠀⠀⠀⡇⠀⢀⠂⠀⣠⠈⠉⠛⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣤⣿⣿⠛⠉⢱⣀⣀⣧⣤⣼⣦⣶⣾⣷⣦⣼⣤⣠⣏⠀⠀⢀⡞⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⣠⣿⣿⣯⣤⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣅⡀⣠⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⠎⣿⣿⣷⡀
⠀⢤⣀⡀⠀⠀⠀⠙⠲⣤⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣻⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢰⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡆⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠠⠬⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣿⠏⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠤⠴⠶⠿⠿⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⢾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠋⣉⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠻⣿⣿⣽⡿⣿⣿⣿⣿⣿⣷⣽⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣁⣀⣀⣹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠋⣉⣽⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⣠⡶⠟⠛⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⢿⣿⠛⠛⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡾⠋⣹⠿⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣟⠻⢯⡌⠙⣆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠞⠁⡴⢋⡴⠛⢉⣿⠟⢻⣿⠿⠿⣿⡿⠿⣿⡿⣟⠻⣯⡉⠳⣌⠙⢷⡀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁⠀⢸⠃⠀⣼⠃⠀⠀⣿⠀⠀⢸⡇⠙⡇⠈⠳⠀⠈⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠛⠀⠀⠸⠁⠀⠈

⠀⠀⠀⠀⠀⣀⢀⣀⣠⣀⡀⣀⣀⣀⣤⣀⣀⢀⣀⣀⣀⣀⣀⢀⣤⣀⣀⣀⣀⣀⣄⢀⢀⡀
⠀⠀⠀⢀⠄⠀⡠⣲⣷⣿⣿⣿⣿⢲⣾⣦⣄⠀⠀⠀⠀⢀⢤⣶⣿⣿⣿⣿⡟⣲⣾⣦⣄⠈⠢
⠀⠀⠠⠀⠀⠘⢼⣇⢞⣿⣿⣿⣲⣽⣋⣦⣹⠇⠀⠀⠀⠘⢼⣠⢞⣯⣟⣟⣶⣿⣋⢆⡿⠀⠀⠑⡀
⠀⢰⠁⠀⠀⠀⠀⠈⠐⠒⠷⠿⠿⠟⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⣐⡺⠿⢟⣛⣛⣂⣉⣀⠀⢀⣀⣈⡄
⠀⢠⢰⠀⠀⠀⠀⠀⠀⣰⠔⣒⢶⡀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣄⠖⡒⢦⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢸⠘⡀⠀⠀⠀⠀⠀⠛⢯⣐⡩⠁⠀⠀⠓⠒⠒⠒⠒⠒⠂⠀⠀⠛⣯⣒⡹⠀⠀⠀⠀⠀⠀⢀⡇⠆
⠀⠈⠲⢒⠖⠂⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠒⠒⠚⠛⠒⠒⠒⠒⠒⠒⠲⠶⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠴⣾⣲⠴⡲⣆
⠀⠀⠀⠀⠀⠀⢠⠚⠳⠵⠲⣆⠀⠈⣥⣼⢏⣿⡇⢉⠌⠉⢡⣿⠐⠦⠀⠀⠀⣠⣤⣴⡤⠿⠿⢆⢀⡀
⠀⠀⠀⢀⡴⠚⠛⣤⣄⣠⡇⠉⢻⣿⡧⢠⡈⢸⣿⣿⣠⠄⠈⢉⣿⣿⣷⣶⡟⠁⠈⠋⡀⢰⣠⡾⢛⠟⣳⠄
⠀⠀⠀⢘⣧⠀⠰⣾⣿⠟⠿⠲⣿⣇⣐⢦⡹⣼⣿⣿⣵⡶⠂⠈⢿⡟⠁⠄⢀⠀⣳⢴⣥⡘⠛⠛⢃⣀⣻
⠀⢀⣴⣿⡏⠀⣶⡏⠉⣀⣦⡆⢸⢿⣿⣤⣿⢟⡿⠙⣿⣿⢁⣰⣿⣁⣦⡼⢿⣾⣅⣾⣿⣿⣷⠦⣼⣿⣯⡄
⢠⡟⣿⣿⣷⡶⠿⢷⣏⣿⣼⡇⣀⣘⣿⠋⡴⠋⣤⢯⣿⣿⡿⢟⣿⣟⣩⠤⣤⡀⠜⠻⠿⢍⣻⣿⣿⣿⣿⡿⣦
⣾⣿⣿⣧⠀⠠⣀⡆⠈⣿⡿⢿⣿⣿⣡⠞⡁⡾⣫⣾⣿⣫⠖⣛⣉⠉⠻⡛⠿⠹⣿⢿⣿⣿⣇⣈⠁⢈⣿⡇⢹
⣿⢿⣿⣿⣤⣍⡛⠳⣴⡇⠀⡄⣿⣿⣧⣬⣼⣾⢿⣿⣿⣶⣾⣿⡿⢷⣦⣶⣾⣶⠈⠣⠈⠛⠿⠏⣠⣾⣿⣇⣼
⣿⠈⠻⢿⣿⣿⠡⠤⢼⣧⣼⠀⣸⡿⠙⠿⣿⣶⡾⣿⣄⣰⣋⣀⣤⣾⣯⣽⡿⣿⣿⣶⣾⡟⠻⢿⣿⣿⣿⢟⣹
⢸⡈⣷⣤⡉⠻⢷⣦⣄⣹⣿⣿⠿⠞⠲⣾⠛⠻⣖⣻⡿⢷⣦⣇⠀⢣⠉⢿⣧⡘⢿⣻⣿⣿⣦⡽⢟⣫⣴⣿⡿
⠀⢧⠹⣿⣿⣷⡦⠤⠉⠛⠛⠷⠶⣶⣦⣤⣦⣴⣈⣁⣴⣀⣉⣿⣷⣮⣧⣼⣿⠿⢾⣛⣛⣭⣵⣾⣿⣿⡿⣽⠃
⠀⠈⢧⠹⣿⣿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃
⠀⠀⠈⢳⡘⢿⣿⣿⣽⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣟⡽⠃
⠀⠀⠀⠀⠙⢦⡙⢿⣿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣛⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣝⠻⣿⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣶⣿⣿⣿⣿⠿⣫⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣌⣛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⣿⣿⡿⠿⣛⣭⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣭⣭⣵⣶⣾⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⣄⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⣇⡀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⢴⠒⢣⡦⠔⢘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠖⠀⠀⠐⠠⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠙⠀⢀⠁⠠⡾⢸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢻⠊⠀⣠⠀⡠⢰⠀⠈⠀⠈⢲⡀⠀⡠⠀⠀⢉⣽⣷⣶⣴⢂⡴⠒⠲⡄
⠀⠀⠀⠀⢄⠀⢄⠀⢀⡀⡀⡂⠀⠀⠀⠀⠲⠀⡐⠁⠀⠀⠁⠀⡀⡀⠄⠀⠀⠀⠀⠀⣠⣿⣮⡵⢭⣿⡌⢠⠄⣀⡟⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠯⠅⠀⠾⠃⠈⣆⡀⠀⠀⠀⠀⠀⠈⠣⡤⠂⠸⡏⠉⠇⠀⠀⠀⠀⠀⠀⠉⠷⣟⣧⣾⡼⡗⠡⠂⠈⢳⣤⣤⢤⡄⠀⠀⠀⠸⢀⣴⣃⡀
⠀⠀⠀⠀⠀⡶⢀⡀⠶⠁⠀⠀⠁⠀⠀⠀⠂⠀⠀⢠⡽⠷⡆⠀⠁⠀⠀⠀⡰⠂⡤⠀⠀⢀⠀⡿⠤⠀⠑⠨⠀⢀⣋⠈⠂⢀⣾⣟⡭⢿⣯⣀
⠀⠀⠀⠀⡀⠀⠀⠀⢉⠵⠀⣠⣶⣾⣷⣤⡀⠀⠀⠀⠁⠐⠀⠀⣈⠂⠀⠀⡁⠄⠁⠄⠁⡄⡞⠀⠀⠀⠀⠀⠈⠁⢀⣤⠤⠾⢏⠣⣀⡔⣻⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣱⣿⡟⢘⢿⢽⡀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣠⣈⡀⢰⠃⢈⠸⠀⠀⠀⠄⠠⠊⠐⠐⠈⡓⢦⠘⣌⠮⠤⣴⣡⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⣿⢯⡉⣁⠞⡽⠉⠲⣄⠀⠀⠀⢀⣦⠈⠋⠉⠉⠑⠠⠍⠈⢠⡇⠐⠠⡺⠁⠔⠑⠶⠖⠾⢠⠀⠀⠉⠋⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠉⠓⠋⠀⠰⠞⠛⡟⠷⡒⣁⣀⣀⠰⠀⠀⠀⠊⠀⠀⡆⢁⠀⢀⡞⠒⠠⠌⠫⠄⠀⠘⠈
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁
⣀⣀⣀⣀⣘⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠉⠉⠉⠲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠛⠁⠉⠉⠉⠉⠋⠙⠃⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠤⢶⠦⠶⠶⠤⠦⠴⠶⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⣀⣀⣀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛⠋⠉⠛⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢻⡇⠀⠐⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⢸⡏⠛⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⣰⡇⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣽⣦⣤⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⣀⣧⢤⣤⣤⣤⡤⠠⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠤⣤⣤⣤⣤⣼⣄
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⢁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⢁⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢁⣴⣿⣿⡟⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⣾⣿⣿⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣾⣿⡿⠃⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣿⣿⣷⣄⣀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠱⢆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡰⠎
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⡤⠤⣤⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡧⠤⠤
⢷⡤⢤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⠤⠤⠤⠤⢤
⠸⣇⠀⢀⡀⠀⠀⡀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⣀⣸
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠁⠘⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡈⢦⢸⡰⢁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡾⡽⣷⢿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣳⢯⣿⣟⣿⣹⣥⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢤⣟⣿⢺⠧⢿⣤⣷⠶⢻⠀⠀⣴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⢠⠓⡾⢤⣏⣹⣗⣻⣗⣛⣖⣩⣧⠞⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⣦⠀⣼⣩⡗⢺⡟⢾⢦⣼⣯⣩⣯⣠⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⢳⡗⢼⣍⣹⡇⠈⣧⠾⡦⠴⣷⠤⠾⡆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⢳⡀⢸⣝⣻⠤⢼⠇⠀⢸⣞⣟⠒⢻⡖⢺⣷⡷⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢻⡇⢸⠦⣿⣉⡿⠀⠀⢸⣯⣿⣒⣺⣏⣩⣿⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⣾⣓⣳⣼⠃⠀⠀⢸⠷⣿⢬⣽⡧⢤⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡬⣿⡏⠀⠀⠀⣾⣉⣹⠒⠚⣟⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢷⢻⠇⠀⠀⠀⣿⢭⢿⡉⢉⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣿⠀⠀⠀⠀⣿⠲⢾⡽⠋⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡆⠀⠀⠀⢹⣽⠏⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢳⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡀⠀⠀⢸⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣭⣷⠤⣾⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣤⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠿⢿⣦⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣷⣾⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣷⣿⣷⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠟⢻⡛⢛⣿⡟⢻⡟⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣿⠋⢻⣿⠋⢻⠛⠿⣿⣿⣿⣶⡥⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡀⠘⢧⣄⣁⣠⡾⠁⡀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣀⠹⣦⣤⣀⣴⠟⢀⣀⠴⠋⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣧⣼⠇⠀⠀⠀⢀⡴⠋⠉⠙⢷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢛⡿⣿⡿⣿⡀⠀⠀⠀⠀⠀⣀⡤⠚⠃⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣿⢇⣿⢿⠀⠀⠀⠀⣸⠃⣼⠛⢦⣀⣉⣉⡙⢷
⠀⠀⠀⠀⠀⢀⡶⢋⡵⠒⢉⣡⢾⣿⣧⠘⣷⠀⠀⠀⡰⠊⠀⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡾⡼⣸⡿⢸⡤⠀⠀⣰⠟⡴⠃⠀⠀⠉⠀⠈⡷⢸⡇
⠀⠀⠀⠀⠀⣿⠁⡼⠀⡔⢉⠔⣿⡏⡏⣇⠸⡇⠀⢰⣅⠀⠀⠀⠀⠀⠀⠀⠤⠴⣧⠀⠀⠀⢰⠇⡇⠹⠃⢿⡇⠀⢰⢣⡞⠀⠀⠀⠀⠀⢀⡼⢃⡞
⠀⠀⠀⠀⠀⣿⠀⢿⠘⡇⣿⢰⣳⢃⡇⡿⢀⡇⠀⢸⡀⠀⠐⠤⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⣼⢸⡀⠀⠀⣸⠁⠀⢸⡌⠳⠤⠶⣄⠀⠀⣸⢣⡏
⠀⠀⠀⠀⠀⠹⣧⡈⢧⢱⣿⣾⣫⢞⡴⢃⣼⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢈⡽⠃⠀⠀⠀⡏⠸⠇⠀⢠⠇⠀⠀⠀⠙⠛⠲⡆⠈⠓⠚⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣿⣟⣛⣉⣤⠞⠁⠀⠀⠀⠀⠈⠳⡄⣰⠖⠚⠛⠉⠀⠀⠀⠀⠀⣧⣀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠈⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⡀⠀⠀⠀⠀⠀⠀⡾⠀⠘⠓⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⢰⠃⠀⣤⠀⠀⡼⠀⠀⠀⠀⢀⡴⡖⠉⠉⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⡴⠋⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⢀⣧⣀⠀⡀⠀⢀⠇⠀⠀⢀⡾⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠁⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢉⣩⣤⡞⠀⠀⠀⡟⠃⠀⠀⠀⠀⠀⠀⠀⢹⣄⡴⠃
⠀⠀⠀⠀⠀⡾⠀⠀⢀⠀⢠⣄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢰⢟⡆⠀⠀⠀⠀⢠⡷⢤⣀⠀⠀⡜⠀⠀⠀⠈⣇⠀⠀⠆⠀⣄⠀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⢷⠀⠀⣾⠆⠘⠿⠀⠀⠀⣿⣿⠃⠀⠀⠀⢰⡟⢠⠃⠀⠀⠀⠀⣼⠀⠀⠠⢤⡾⠁⠀⠀⠀⠀⠙⢦⣀⡀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⡿⢿⢸⠀⠀⠀⠀⢠⡇⠀⠠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠹⣄⡀⣀⣤⣤⣴⣾⡿⠟⠀⠀⠀⠀⠀⣿⢸⡏⠀⠀⠀⠀⢸⠋⠂⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢞⣩⡤⡤⣄⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣀⠀⠀⢸⡀⣠⠏⠀⠀⠀⢀⣤⠶⠒⠶⣄⠀⠀⠀⣼⣳⠏⡁⢸⡵⣀⡹⣆⢹⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡇⠀⠈⢣⡉⢧⡀⠈⠛⠁⠀⠀⠀⡴⢋⡴⠛⠛⢦⢹⡆⠀⠀⣿⣿⢺⡽⣠⣄⠳⠆⣿⢸⡇
⠀⠀⠀⠀⢠⡞⠉⢠⡇⠀⠀⢹⡿⢦⡀⠀⠀⢠⡏⠀⠸⠁⠀⠀⠀⠃⠈⡇⠀⠀⠀⠀⣀⡼⢡⡿⠀⠀⠀⣸⠀⣧⠀⠀⢻⡜⢦⡀⠳⠞⢀⣴⢃⡼⠁
⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⠀⠈⢻⡀⠀⢼⠀⢠⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡼⠋⡴⠋⠀⢀⡀⠀⠙⣦⠘⡆⠀⠀⠙⠶⣭⣙⣛⣩⡵⠞⠁
⠀⠀⠀⠀⠈⢻⣶⠀⢀⡴⠀⠀⢀⠀⣼⠁⠀⠘⣧⠘⣆⠀⠀⢸⣀⣠⡟⠁⠀⠀⢸⡃⠸⣇⣀⡴⠛⢷⡀⠀⣸⠀⡿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢧⣤⡤⠾⠞⠁⠀⠀⠀⠈⠳⠿⣦⣄⣼⣿⠉⠀⠀⠀⠀⠘⠷⣤⣬⣥⠴⠶⣌⠓⠚⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠈⠉⠉⠉⠁⠐⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠎
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⢒⡲⠒⠒⠒⠒⠒⠒⠛⢛⡉⠉⠉⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡉⠁⠒⠒⠀⠀⠀⠀⠀⠶⠒⠒⠈⠉⠀⠀⠈⠀⠀⠤⠤⢀⡀⢀⣀⠀⠀⠀⠤⠤⠒⠀⣇
⠀⠀⠀⠀⠁⠨⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠂⠠⠀⢀⣀⠀⡀⠀⠤⠀⠂⠁⠀⠐⡶⠢⠄⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠐⢀⣀⣀⠀⠠⠤⠀⠀⠀⢠⠀⠀⠀⠀⢈⡀⠀⠀⣀⠀⠀⣀⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠄⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠄⠐⠀⠀⢄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠄⡀⠀⠀⠢⠀⠀⠀⠀⠀⠂⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠂⣆⡀⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⡐⠉⠠⠀⠀⠀⠀⠀⠀⡠⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠓⠔⡡⠀⠐⡘⠃⢐⡀⣨⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠜⠀⠀⠓⡄⠀⠀⠀⠎⠒⠓⠢⡤⠀⢤⢀⠗⠈⢱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠤⡀⠂⠀⠀⠀⣂⡈⠁⠀⠀⢠⠖⠈⠉⠍⠣⠀⠾⡗⢀⠖⠚⢼⠄⢄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⡄⠄⠘⡀⠻⣤⠴⠆⡙⠛⠓⢄⠀⠀⡇⠀⠀⢀⡠⠄⣦⣈⣁⡉⠀⠀⣰⠀⠈⠁⠀⠥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠐⡘⡀⠡⡴⠅⡘⠀⠀⠀⠀⣸⠤⠒⣇⢀⡰⠉⠁⠀⠀⠀⠈⢅⡱⠜⠾⣷⡀⠀⠔⢢⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣄⠔⡪⠉⣛⡛⠣⡀⠈⣦⡤⡕⡼⠁⡀⡴⠛⡐⠅⠀⠀⠀⠀⠀⠀⣃⠔⠁⣠⠏⠱⠎⠀⢸⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⣢⣎⡔⠉⠄⠌⢷⠑⡤⠉⠀⡙⢀⣬⡊⢈⡀⢰⣌⠙⡳⠂⣶⠴⢦⠋⢀⡾⢁⣀⠀⠁⠀⠌⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⡀⠀⠱⣀⠜⠙⡬⠆⠗⠠⣀⣡⠊⠀⠈⠁⠉⠁⠘⠷⠉⠀⠈⠑⢦⠔⣡⡤⠤⡈⠻⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠀⡂⠀⠙⠲⠀⠀⢸⠁⠀⠄⢤⠵⠤⠤⡔⠀⠀⠀⠀⠀⢠⠹⠋⠀⠑⠌⢤⠋⢰⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡮⡣⢄⡇⠀⠀⠀⠄⠀⠸⠰⠁⣰⠁⠀⠀⡰⠇⡄⠀⠀⠄⠀⠈⢀⠃⠘⠐⢢⡸⢔⢵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠒⠤⢀⡺⠄⢠⣂⡇⠀⠙⡀⠈⠈⠁⢡⡀⢀⣀⣀⣴⢡⠮⠀⡒⠄⠒⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠉⠀⠒⠐⠈⠈⠩⠦⠤⠬⠥⠍⠁⠁⠂⠒⠀⠉⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠔⠊⠁

⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠹⡄
⠀⠀⠀⠀⢸⣏⠢⣄⠀⠀⠀⢀⢿⣆⡀⠀⢣
⠀⠀⠀⠀⢸⣿⠂⠘⠲⠖⠒⠺⠈⢿⣾⡅⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠂⠀⠀⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠟⠉⠘⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⡀
⠀⠀⠀⠀⡻⠁⠀⠀⢲⣾⣷⡀⠀⠐⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦
⣀⣄⣤⡊⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣿⠗⣿⠏⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠿⢶⣟⣛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠈⠙⠓⠒⢶⣀⠀⠀⢀⠄⠀⢀⢠⠴⡄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡀⠁⠀⠌⣠⣁⣌⡀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⣡⡀⠁⠀⠀⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣾⣧⣽⣿⣿⡘⠧⠀⠀⠀⠀⢰⡟⠈⣖⣡⣴⣾⣥⣤⡄⠀⠀⣠⣤⣿⣿⡄⠀⠀⠀⢀⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣿⠻⠂⠀⠀⠀⡰⠛⠛⠿⠿⠿⠟⠿⢿⣿⣴⡷⠟⠛⠉⢻⣿⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠿⢀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⣨⠀⢸⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⡼⡟⢀⠃⠀⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⡾⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠟⠁⢠⣾⣦⡾⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣿⣧⣠⠾⠇
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠒⠒⠒⠒⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⢀⡰⡀⡄⠠⣠⢀⡢⡠⡀⢀⡄⢈⡉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⢢⣄⣨⣷⣽⣾⣶⣷⣿⣿⣿⣷⣿⣶⣩⣶⣦⣢⠔⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢄⣀⣢⣽⣾⡿⢿⡿⠫⣿⣿⡿⢿⣯⡻⣿⠻⢿⣿⣿⣿⣶⣚⣥⣠⣌⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤⣿⠟⢻⡟⢀⡏⡀⢀⡼⣿⣷⣼⣿⣿⡜⡆⠈⣿⠉⠛⠿⣿⣿⣮⣔⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣲⢿⠉⠀⠀⢸⡇⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⣿⠀⠀⠀⠈⠙⠿⢿⣿⣥⣶⣊
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠉⠀⠀⠀⠀⠈⣷⠀⢣⡻⣿⣿⣿⣿⣿⣿⡿⣱⠃⢠⡟⠀⠀⠀⠀⠀⢀⡤⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣦⠀⠀⠀⠀⠀⠘⢷⣀⠙⠮⣝⣛⣛⣛⡭⠞⠁⣠⠟⠀⠀⠀⢀⡠⠶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣤⣴⠟⠡⣀⣠⠴⠊⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠒⠶⠛⠻⠟⠛⠟⠈⠙⠉⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣟⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣰⠭⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⡄⠀⢨⡯⠿⢹⡅⠀⢠⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⢿⣿⡗⣺⣒⡖⢺⣿⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⣚⣃⣛⣘⣃⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡈⣿⠥⠽⠤⡧⢼⡟⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⢽⠥⣭⢬⡧⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⡒⣛⠒⡓⢺⠃⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡓⣿⢚⢃⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠈⣇⣭⣉⡭⡼⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡩⣿⢍⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣭⣥⡀⠉⣤⢰⡶⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⢢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠀⠀⠀⠀⣀⣀⡀⣸⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⢤⣖⣒⣲⣬⣭⡭⠴⠶⠖⢊⠓
⠀⠀⠀⠀⠀⠀⣀⠠⠤⠶⠟⠛⠛⠃⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠠⠤⠄⠀⠀⣄⠀⡈⠀⢀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠉⠀⠁⠲⠶⠤⣀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠛⢻⣶⣶⣶⠶⠀⣶⣷⡄⠀⠒⠒⢒⠚⡖⠒⠐⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡘⠛⠁⠀⢠⠀⠀⠀⠀⣀⢀⣄⣰⢉⣛⡟⠉⣿⣷⣶⡀⣛⣉⡇⠀⡶⠀⣾⢠⡇
⠀⠀⠀⠀⡀⠄⠒⠛⠓⠘⠛⠻⢘⡷⡇⡴⠤⠛⠟⠿⠋⠈⠛⣉⣿⢃⢰⣿⣧⣬⣿⣿⣿⡇⣍⣉⣁⠀⣀⡀⢀⣬⡄⣠⣤
⠀⠀⠀⠈⢿⣶⣶⡶⣿⣷⠿⠿⠯⣶⣶⣟⣷⣤⣄⣀⣾⢻⣤⣭⣽⡿⠾⠍⠁⢀⣈⣁⡀⣍⡭⠭⠟⠛⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⣏⡛⢛⡛⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠲⠙⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠸⣿⣷⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠏⠁⢿⠿⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢒⣍⠀⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠀⠀⠈⢠⡇⠀⠀⢀⣃⠐⠀⡐⠂⠄⠤⠠⠄⠠⠤⠀⠀⣢⡄⢀⣲⠀⠈⣀⠐⠄⠀⠀⢰⡟⢯⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠀⠁⠀⠟⠉⣀⠀⢨⠞⠃⢀⠿⠀⠀⠐⠢⡀⠀⢠⣄⣠⠏⠙⠌⢶⠀⠀⠻⠂⠀⠀⠐⠊⢠⢦⢱⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠷⠶⠦⣄⣀⣀⠉⠀⠈⠉⠀⢉⣁⣠⣴⣿⣿⣿⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⣠⡾⠛⠿⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣤⣤⣭⣍⣉⡉⠛⠛⠟⠛⠉⠉⠉⠛⠢⢤⣤⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣻⣿⣯⢿⣟⣻⣿⣛⣿⣿⣿⣿⣶⣆⡈⠐⢄⡀⠀⠀⠀⠀⠽⣫⣿⣿⣿⣿⡿⠿⢻⣿⣿⡟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣷⡷⢹⣿⣷⣤⡙⢆⠀⠀⣠⣾⣿⣯⣸⣥⣏⡎⣸⣿⣿⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠻⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣀⣴⣿⡿⣿⣿⣿⢿⡿⣟⣼⣻⣿⡿⠿⠿⠿⢛⢽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⢻⠿⣿⣯⣿⠿⠿⢿⣿⣻⣦⣿⣟⠛⠟⢾⡛⠍⠛⠋⠉⠄⠔⠰⢒⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠧⠔⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣀⠀⠀⠀⢈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢉⠒⠶⢤⣄⠀⢠⣀⣤⡼⠟⠿⣄⡉⠇⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠿⢷⣶⡾⠀⠀⠀⠀⠁⠈⠂⠀⠀⠘⡄⢯⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠒⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⣿⢱⡖⠁⠀⠀⠁⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡏⠘⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⣠⠞⠑⠒⢻⡷⠆⠀⠀⠀⠀⡼⠀⠀⢀⣼⠇⠀⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⠋⠀⠃⠀⠁⡀⢀⠈⠀⢢⣤⡾⣧⠀⠈⠉⠁⠀⢀⠐⠀⠀⣾⠀⠁⠀⠀⢀⣤⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⠄⠀⠀⠈⠦⠀⠀⢀⠁⡀⠀⠀⠑⣔⡁⠀⣀⡔⣇⠀⠀⠀⠀⠈⠁⠀⡐
⠀⠀⠀⠐⠀⠤⠀⠀⣶⠀⠀⠀⠀⠀⣔⢀⠐⠛⢹⡗⢤⣤⣤⡅⣳⡆⠀⡀⠀⠀⠀⢸⡟⠛⠛⠉⠁⣤⠿⠦⡀⢇⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢽⠋⠀⠀⠀⢹⣀⣬⠋⠀⠀⠙⡄⢸⣿⣦⣀⢰⣅⣀⣀⣠⣴⡿⣌⠑⠂
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢦⣄⣡⣤⡾⠟⠛⠀⠀⠀⠀⠨⣿⠋⠉⠉⢿⣿⣿⣯⣁⠀⠘⡌⠑⢀⡀
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡙⢲⡖⠚⠛⢿⣿⣿⡇⠄⠀⠑⠼⡁⠀⠀⠢
⠀⠈⠀⠀⢀⢠⠭⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⢴⠀⠀⠀⣀⣾⣷⢚⡿⢶⣄⠀⠀⠈⠓⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠁⠀⠘⢿⠀⠲⠆⢀⣀⣀⣠⠑⠦⠤⠜⢰⠛⢙⠛⢻⣹⢀⣠⣤⣤⠛⠠⠄⠙⠦⠤⠐⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⡀⠘⠀⠀⠀⠀⠀⠈⢡⠀⠀⢠⡄⠀⠀⢸⠀⠰⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠀⠤⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⠞⡇⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢑⡤⠋⠁⠀⠀⡀⠐
⠀⠀⠀⣀⠜⠂⠀⠀⠀⠈⠿⠷⡀⠀⠀⠀⠀⠀⠀⢀⠜⠰⠀⢈⠀⠀⠀⠀⠀⢠⣤⠎⠀⠀⡀⠐⠊⠉⠀⠀⠀⠀⠐⠉⠀⠀⢦⡀
⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠛⠡⠖⠁⠀⢀⡀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣦⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣏⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠙⣫⣥⣶⠟⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠶⠚⠉⠁⠀⠀⣿⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣿⣛⣻⡿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣭⠈⣽⠀⢸⡥⠨⣮⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⡦⠠⡼⠀⠘⣗⠀⣻⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⡝⢹⠧⠠⣞⠃⠙⣧⠀⣽⡋⣧⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠲⠇⢰⠆⠠⠞⠁⠘⡧⠀⢴⠆⠲⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢺⠆⠲⣄⣀⡜⠃⠚⢇⡀⣴⠂⠸⠇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠻⠦⢄⠀⠀⠱⠆⠰⠄⠀⠀⠦⠴⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣠⣀⡈⠁⠈⢣⡀⢠⠎⠁⠙⣄⢠⡌⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠲⣇⠀⠘⠖⠒⢆⡀⢀⡶⠂⠲⡀⢀⡽⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⣿⣷⢤⡏⠉⢳⣄⣠⡌⠉⠉⣡⣄⣤⠋⠉⢳⡤⣿⡇⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⣛⣛⣛⠛⠿⣿⣿⣇⢀⠷⠲⢮⠀⠀⡸⠒⠒⢃⠀⢀⠗⠒⠿⣀⣼⣿⡿⠟⢛⣛⣛⣛⠛⠿⢿⣶⣦⣄
⠀⠀⠀⠀⣠⣾⠿⢋⣥⠶⠚⢩⣭⣭⣭⣍⡓⢦⣹⡏⢉⣄⣀⣜⠉⠉⢡⠤⢤⡎⠉⠙⣤⠄⣴⠉⢹⣯⠴⢚⣩⣭⣭⣭⡍⠓⠦⣌⡛⢿⣷⣄
⠀⠀⠀⣼⡿⠡⠖⢛⣋⣛⡛⠲⢬⣙⢿⣿⣿⣶⣽⠗⠚⡀⠀⠘⠒⠒⢆⣀⣀⠐⠂⠚⣀⠀⣈⠓⢻⣧⣾⣿⣿⠟⣫⡴⠖⣛⣋⣛⡛⠢⠙⢿⣷⡀
⠀⠀⣼⣿⣴⡾⠿⠟⣻⣿⣿⣿⣦⣍⢳⡝⢿⣿⣿⡆⢼⠉⠈⢹⡤⠤⡌⠀⠈⣠⠤⢴⠉⠀⠙⠤⢼⣿⣿⡟⣡⠞⣡⣶⣿⣿⣿⡛⠿⠿⣶⣤⣿⣧
⠀⠸⠟⠋⠀⠀⣰⣾⠟⣋⡭⠤⠤⣍⡀⠹⡎⢿⣿⣀⣰⠗⠒⢷⣀⣀⡘⠋⠙⢃⣀⣀⠛⠐⠻⣀⣠⣿⡟⣴⠃⣈⣭⠤⢬⣍⡙⠿⣷⣄⠀⠙⠻⣿
⠀⠀⠀⠀⢀⣾⡟⢡⠞⣡⠖⣛⣓⠶⣍⡳⣄⠘⣿⠁⢉⣤⣤⣜⠁⠈⠱⠦⠴⠋⠁⠉⡦⠤⣴⠁⢈⡿⢁⡴⢚⡥⠖⣛⣓⠶⣝⢷⡘⢿⣧⠀⠀⠈
⠀⠀⠀⠀⣼⡟⢠⢏⣾⣥⠤⢭⣍⡛⢮⡙⣮⠳⣌⠛⠛⣀⣀⡘⠓⠘⢃⣀⣀⡞⠓⠛⣄⣀⣀⠛⠛⣡⢏⡴⢋⡴⢛⣭⡭⠤⣼⣧⢳⡈⢿⣇
⠀⠀⠀⢰⣿⠁⡟⢈⡵⠖⣛⡓⢦⡙⣦⠻⣎⢳⡜⢆⠐⠉⠉⢱⠄⠠⠎⠁⠈⠠⠀⠴⠉⠀⠉⢀⡜⢡⢏⣴⢋⡴⣫⠔⣚⣛⠲⣬⠀⣧⠸⣿⡀
⠀⠀⠀⣼⡏⢸⣧⠏⣰⡿⠟⢿⣦⢹⡌⣦⢻⣧⡹⣎⢧⡀⠐⢣⣄⣀⡌⠀⠘⣠⣀⣠⠋⠀⢠⡞⣴⢣⣾⡏⡾⣱⠋⣾⡿⠻⣷⣌⠳⣼⡀⣿⡇
⠀⠀⠀⣿⡇⢸⠏⣼⡿⠁⠀⠘⣿⡇⢷⢸⡌⣿⣷⡘⣆⢳⡀⢮⠀⠀⡱⠆⠶⠁⠀⠈⠶⢠⠏⡼⢡⣿⣿⢸⢇⡏⣸⡿⠀⠀⠘⣿⣆⢻⡇⢻⡇
⠀⠀⠀⣿⡇⡟⣰⡿⠁⠀⠀⠀⢹⣿⢸⡎⣇⢻⣿⣷⡘⣆⢳⡈⠋⠉⢣⣄⣠⠞⠉⠙⢠⠏⡼⢡⣿⣿⡇⣿⢸⢃⣿⡇⠀⠀⠀⠘⣿⡆⠇⢸⡟
⠀⠀⠀⢹⣇⢠⣿⠃⠀⠀⠀⠀⢸⣿⡄⡇⣿⢸⣿⣿⣷⡘⣎⢣⠀⠰⢆⠀⢀⡰⠄⢀⡏⣼⢡⣿⣿⣿⢃⡇⣿⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⣿⡇
⠀⠀⠀⠘⣿⣾⡟⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⡘⣿⣿⣿⣷⠹⡌⣧⠀⡌⠉⠉⢡⠀⡾⣸⢃⣿⣿⣿⣿⢸⡇⡇⣸⣿⠀⠀⠀⠀⠀⠀⣿⣶⡿⠁
⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⣿⣇⢸⢸⡇⣿⣿⣿⣿⣇⢻⡘⡆⠐⠒⠒⠀⣸⢡⠇⣾⣿⣿⣿⡿⢸⢹⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⡎⡇⢿⣿⣿⣿⣿⡎⣧⢹⠀⠤⠤⢠⠇⡞⣸⣿⣿⣿⣿⡇⣿⢸⠁⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡇⣿⢸⣿⣿⣿⣿⣷⠸⡄⣇⠀⠀⡞⢸⢁⣿⣿⣿⣿⣿⠇⡟⣾⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢿⢹⡌⣿⣿⣿⣿⣿⡇⣧⢹⡀⢰⠇⡏⣼⣿⣿⣿⣿⣿⢸⢇⡏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠸⡎⣧⢻⣿⣿⣿⣿⣷⢸⡌⡇⣼⢸⢃⣿⣿⣿⣿⣿⠇⡾⣸⢃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢻⡜⣧⠻⣿⣿⣿⣿⡈⡇⣷⡇⣼⢸⣿⣿⣿⣿⢏⡼⣱⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⣎⠳⣌⡻⢿⣿⡇⠷⠾⠶⠇⣸⣿⡿⢛⡵⢟⡴⢃⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⡛⢮⣝⡳⠮⣭⡘⣿⡟⢣⡭⠶⢚⣩⠴⢋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣬⣉⢻⠠⣇⣙⣃⣼⢰⠋⣩⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡘⠷⠬⠭⠭⠥⠞⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣶⣶⣶⡾⠟⠋

⠀⠀⡠⠶⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧
⠀⣿⡏⣤⣶⣦⡄⣤⣶⣦⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣞⣽⣶⣲⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣀⣄⠀⣠⡀⢀⣄⡀⢰
⠀⣿⡇⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⢸⣇⡤⠶⠤⢤⠠⠤⡀⠀⠀⠀⠀⠀⣿⢹⠛⠛⢏⣻⠃⠀⠀⠀⠀⠀⢠⣾⣿⣧⣯⣭⣭⣭⣭⣭⣭⣧⣼⣤
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠙⠙⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠛⠟⠿⠻⠻⠟⠟⠻⠻⠛⠟⠟⠛

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠠⣤⡄⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠈⠙⠓⠂
⠀⠀⢰⣷⣖⡛⠻⡟⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⢀⢹⡄⠀⠀⠀⠀⠀⠀⢠⠶⣭⢛⠫⣵⢶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⢰⣽⣵⣶⣶⣶⢶
⠀⠀⢸⣿⣬⡍⢣⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣾⣿⣿⣧⡴⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢩⡷⣿⣿⣿⣿
⢠⣤⣼⡇⠨⣇⠈⠿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠼⣻⡻⠿⠿⠿⠞


⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⢀
⠀⢠⣶⣿⣶⡄⠀⠷⠿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣿⡇⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣠⠔⠊⢙⡏⠁⠒⢄⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⢿⣿⣿⣿⢰⡖⠒⢄
⠀⢿⣿⣿⣿⡇⢀⡍⠨⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢶⣄⠀⠀⠀⣿⣿⣿⣿⣯⣿⣽⣿⣿⣸⣷⣶⣶
⠀⠀⠃⠈⠘⠀⠘⠌⠱⠁⠀⠀⠀⠀⠀⠠⠬⠯⠽⠥⠄⠀⠀⠀⠀⠈⠙⠯⠟⠉⠉⠉⠉⠉⠉⠹⠭⠏⠉⠀⠀⠀⠙⠯⠗⠺⠽⠉⠉⠉⠉⠉⠹⠽⠋


⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⡀⣰⣆⣀⣴⡄
⠀⠀⠀⠀⠆⠀⠈⠀⠾⢠⡄⠀⠀⠀⠀⠀⣰⣿⣿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⢳⣿⡟⣿⣷⣿⣿
⠀⠀⠀⣿⣿⣷⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⠹⣿⣿⡜⡄⠀⠀⠀⠀⠀⢤⣶⣿⣾⣿⣿⣿⣾⣶⣦⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡇⣿⣿⣿⣿
⢴⡔⠋⠈⠭⠁⠙⢢⡆⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢿⡗⠀⠀⠀⠀⠀⠀⢾⣈⣏⢸⣏⢈⣿⢈⣇⣸⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡇⣿⣿⣿⡿
⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠫⠭⣭⣽⣿⣭⠭⠭⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠸⣿⠇⢿⡿⠋⠁

⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⢰⡸⠭⠇⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣀⠠⠤⢺
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠇⠠⡤⠔⠃
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡄⠠⠦⠠⢦⠁⣤⢟⢧⣷⢠⠴⠀⣤⢤⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀⠁⠞⠁⠐⡒⠄⠙⠸⠜⡧⠓⠯⠥⡈⠠⠭⠄⠀⠀⠠⠬⠐⠴⠤⠉⠂⠀⠾⠂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡸⣆⠀⠠⢸⠋⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠢⢠⣀⡀⠀⠊⠀⠀⠃⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠂⡴⠁⠸⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⠙⠀⠀⠀⢃⠀⠀⠉⠐⠒⠂⠶⠬⣀⢒⡞
⠀⠀⠀⠀⠀⠀⠀⡆⠈⣼⠀⠀⠀⠱⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡎⠳⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠀⠀⠱⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⣆⠀⠘⣄⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠒⠷⠞⠉⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣾⠛⣆⣀⣀⠀⠀⣠⣤⡇⢳⡀⠀⢧⡀⠀⣀⣀⣠⣄⣀⠀⠀⣀⡀⠤⠔⠚⢩⠹⡇⠀⢀⣀⠀⣀⡾⠃⣀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⢴⠛⠀⠈⠀⣸⢋⡋⢀⡤⠧⡴⠷⠴⠶⢿⣿⢿⡉⡡⠛⡍⢻⡁⠆⠂⣀⣤⣄⣀⣸⠉⠙⠋⠈⠋⠉⣙⠻⣿⣏⠳⠶⠉⢷
⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⠉⣿⠉⣻⠴⠚⢦⡴⠚⠉⢲⡇⢀⠉⠳⡌⢇⣹⠷⠒⠋⣉⣴⣾⠇⡹⢖⣚⡤⢤⢄⣮⣴⣶⣮⣹⠀⣤⠀⢸
⠀⠀⠀⢀⣀⣿⠉⠃⢈⣆⠀⠙⢶⣁⡤⠴⢤⠴⢚⣥⠾⣧⠐⢇⠀⢻⣿⣤⣤⣴⣮⣗⢋⡵⠋⠀⣈⠀⠀⣩⢽⠗⠀⠀⠈⣿⠊⠈⢧⡾⣤⡀
⠀⠀⡜⠉⠠⢤⣷⣄⣀⣈⠣⠀⢸⣧⠤⠴⠒⠒⢯⣠⡀⣹⢣⡀⢁⡼⢋⣤⣊⡩⠄⠈⠉⢧⣤⣬⣤⣤⠤⣭⠽⠶⠦⠤⠴⠧⠀⠒⠊⠀⢨⡇
⠀⠀⠳⠤⢤⣀⠀⠀⣉⡉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠛⠛⠓⠒⠛⠚⠓⡿⠋⣁⡤⠤⠒⠒⠋⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠶⠖⠋⣠⠔⠋
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠶⠶⠖⠂⣼⢠⠞⠁⠀⠐⠒⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⢚⠛⡦
⠀⠀⠀⠀⠀⠀⠎⠉⠉⠙⠢⢄⣀⣀⣀⣀⠀⠀⠀⠀⢀⠤⠚⠉⠐⠄⠠⠀⢀⡎
⠀⠀⠀⠀⢀⡴⠀⠀⠀⣠⡇⠀⠃⠱⠀⠈⢧⠀⡠⠚⠉⠐⠈⠀⡄⠁⠀⢀⡞
⠀⠀⠀⠀⠏⢀⣠⠞⠉⢩⠃⠀⠀⠀⠀⢀⠼⠋⠀⠀⡀⠀⠑⠊⠀⠀⣐⣏⡠⠤⠒⠊⠙⠦⠒⠒⢄
⠀⠀⠀⢰⡴⠁⠀⠀⢠⠃⠀⠀⠀⢀⡴⠋⡀⢀⠨⠀⠤⠈⠠⠀⢐⡰⠋⠀⠀⠀⡠⣤⠶⠒⠒⠒⠺⠒⣄
⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⢀⠄⡠⠎⠀⠀⠈⢈⡠⠁⠀⠂⠀⣠⠎⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣆⠜⠁⠀⠀⢀⡐⠂⠀⠀⢀⣠⣜⡡⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠏⠀⠀⠀⠠⠂⠀⢀⣠⠴⠚⠉⠁⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⢠⠋⠀⡏⠠⠀⠀⠀⣀⡤⠚⠋⣁⣀⣀⣠⣴⣞⣉⡤⠤⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡌⠀⢸⠃⠀⣠⢴⣊⠥⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠒⠒⢒⠦⠴⢄
⠀⠀⠀⢀⣀⣴⣡⣆⣧⡤⠞⢋⣁⣠⠤⠴⠖⠒⠚⠛⠋⠛⠙⠙⠓⠒⠒⠶⠤⠤⠤⠤⠤⢤⣀⣸⠀⣄⣈
⠀⠛⠛⠻⠲⠶⡦⠤⠶⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⣠⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⢊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠤⠒⠋⠁⢀⣀⠴⠚⠁
⠀⠀⠀⢈⣤⣤⡵⠊⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠋⠉⢁⣀⣠⠤⠖⠊⠉
⠀⠀⠀⠀⢀⡤⢭⡹⠋⣠⠤⠖⠛⠓⠒⠒⠒⠶⠶⠖⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠛⠀⠀⢀⡀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⢀⢀⠀⠀⠠⠢⠀⠀⠀⠃⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠄⠓⡀⣊⢂⡐⠂⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⢤⠤⠤⠤⠤⡀⠠⡄⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠰⠀⡂⡠⡌⠈⠂⢃⠀⠀⠠⠤⠈⠠⡀⠀⠀⢠⣐⡓⠊⠀⠁⢢⡄⢀⢉⣣⠀⣌⢐⡍⡥
⠀⠀⠀⠀⠀⡄⠅⠀⠗⡒⠂⡠⠃⠀⠀⠀⠤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠨⣙⠦⡀⢤⣠⢤⠈
⠀⠀⠀⠀⠀⣥⠄⠔⠀⠀⢀⠀⠈⢀⠀⠁⠀⠈⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⡘⠑⣄⡀⢈⠀⠄⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⠀⢂⣀⢀⠀⠀⣈⡀⢀⣀⡀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠉⠣⠀⢀⠀⢀⠀⠀⠀⢀⠀⠀⢀⠈⠁⠂⠀⠀⢀⠀⠀⠀⠀⢀⠔⠀⠀⡀⢀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠣⠤⡠⠈⠂⠠⠂⠀⠀⠀⠤⠉⠐⠀⠒⠠⠔⠁⠂⢂⠀⠀⢀⡄⠈⠈
⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠤⠤⠤⠍⠀⠀⠀⠀⠊




⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣻⣿⣿⣿⣿⣻⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠯⢭⣭⣭⣭⣍⣉⣉⣉⣉⣉⣭⣭⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⠙⠳⠦⢤⣤⣄⣀⣀⣀⣀⣉⣁⣀⣠⣤⡤⠾⠛⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⡶⢶⣶⣦⣤⣤⣭⣭⣭⣥⣤⣤⣴⣶⡶⢿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣼⣧⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣼⣧⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢾⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⠷⢶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣿⡇⠀⠀⠀⠀⢀⣴⠟⡽⠊⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣶⣦⣤⣄⡀⠀⠀⢸⣿⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⣿⣇⠀⠀⠀⣰⡿⠛⠳⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠒⠲⠍⣿⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠄⠀⣿⡀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣶⣦⣶⠟⠀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢿⡟⠛⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⠉⢹⡇⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⢀⣀⣐⣒⣒⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠀⠒⠋
⠀⠀⠀⠀⢀⠴⢶⠀⠀⠀⠉⠙⢫⣭⢈⣈⣀⣭⣭⠽⠛⠛⠋⠉⠀⣀⣤⠈⠀⠀⠀⠀⠐⠁⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠈⢢⣄⡤⠤⣊⡉⠁⠀⠉⠉⠁⠀⠀⢀⣠⣤⣖⣊⡁⡁⠁⠂⢐⠀⡊⣀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡎⣹⠀⠀⠢⠍⠉⠓⠒⠒⠒⠚⠋⣁⡠⠛⠉⠉⠩⠉⠙⠻⠍⠉⠡⠀⠀⢹⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠾⠀⠀⠀⣀⣤⢶⡚⠉⠉⠩⢀⠀⠂⠀⠀⠀⠀⣀⡀⠰⠀⢨⠀⠀⢀⣀⠈⠁⠀⣀⣀
⠀⠀⠀⠀⠀⢲⠀⠀⠀⢀⡴⠛⠙⠓⠒⠚⠓⠓⠒⠠⠄⠼⠁⠄⠀⠉⠤⠤⠄⠀⠐⠶⠖⠋⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠴⠻⠇⠀⠀⢠⡶⠶⣶⡆⠀⠀⠀⢀⡴⠶⠦⡀⠀⢀⡾⠷⡄⠀⠞⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢶⠀⠀⠀⠀⠤⠐⠁⠀⠀⠘⠃⠐⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠔⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡇⠀⠀⠉⠙⠳⢶⣤⡀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡄⠀⠀⠀⠀⠀⠉⠻⣇⡒⠉⡁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠐⡿
⣰⡂⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢨⡄⠿⣦⠀⠀⠀⠀⣀⠔⠀⠙⠦⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠠
⣏⠀⠀⠀⠈⢆⠀⠋⠉⠀⠁⠀⠰⠇⠀⠈⠳⡀⠀⠊⠀⠀⠀⠀⠀⠈⠣⡠⣀⠀⠀⡆⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠘⣆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡡⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⡹⢾⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⠖⠉⠁⠀⠀⠀⡀
⠀⠸⣧⣼⡖⢺⣿⣦⠀⠀⠀⠀⢰⣷⣿⢀⣰⢦⠟⢜⣳⣤⣤⣵⣦⣴⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⢛⡁⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠹⣿⣿⡄⢹⣿⣆⠀⢀⠠⠿⠁⠃⠀⠀⠀⠀⠀⠀⠈⡟⠛⠿⠿⠿⡟⣤⠀⠀⠀⠀⠀⠀⢰⠉⣋⣡⣴⣾⣿⣯⣀⡤⠶⠛⠉
⠀⠀⠀⠹⣷⣿⡄⢻⣿⡄⠀⠈⠐⢤⣀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⢡⡇⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠹⣿⣿⡏⠘⠁⡤⠄⠀⠀⠉⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⢀⠑⣼⣿⣷⣀⠀⠀⠒⠒⠂⠚⠉⠁⠀⠀⠠⢾⡿⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⣄⠙⠳⣶⣿⡆⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠛⠐⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠷⢤⣤⣤⠄⠠⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢢⠈⠛⢿⣿⣦⡈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⠻⠿⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡀
⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠘⣝⠶⣴⡧⠒⢲⢋⣽⠟

⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⡀
⠀⢀⣴⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣷⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⢷⣄
⠐⠋⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣦⣄⡉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⣁⣤⣶⣶⣶⣤⣤⣤⣄⣀⡀⠀⠀⠈⠓
⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⣶⣿⣿⣿⠿⠋⠁⡟⣿⣿⣿⣿⡟⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⣿⣿⣿⡟⡏⠉⠛⢿⣿⣿⣷⡄
⠀⠀⠈⠹⣯⣤⡀⢀⠀⠻⠑⢜⡋⠝⠁⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡀⠙⠍⢛⠔⠹⠃⠀⢀⣠⣬⠯⠉
⠀⠀⠀⠀⠀⠘⠛⠷⠶⢷⣶⡾⠿⠿⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠘⠻⠿⠿⣶⣶⠶⠶⠟⠛






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠶⠋⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣦⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⠞⠋⠹⣆
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠸⣆
⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣫⠞⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⢃⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠛⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣀⠄⢀⠀⠒⠂⠀⠀⠒⠂⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⡤⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠄⠀⠉⢠⣀⠀⠈⠁⠠⣄
⠀⠀⠀⠀⠄⠐⠒⠀⠀⢄⡤⠀⠀⠈⠀⡀⡀⠀⠀⠂⠀⠀⠀⠀⠠⠤⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⢌⠃
⠀⠀⠀⡾⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠈⠉
⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠤⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢙⡒⠂⢀⠀⠀⠤⢤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠠⣤⡀⠀⠀⠤⠄⡀⠐⠂⠀⢈⣡⠔⠂
⠀⠀⠀⢸⠗⠿⠉⠥⠬⠤⠶⣶⣂⣀⣠⣖⣀⣠⣀⣀⣀⣀⣀⣈⣀⣀⣀⣐⣘⣓⡦⠤⠤⠤⠤⠀⠠⠚⡅
⠀⠀⠀⣎⣀⠀⠀⠀⡤⠀⠀⠀⢂⣀⠀⢀⡀⡀⡀⠀⠀⠀⠀⡀⠀⡀⡀⢀⡀⠐⠀⠀⣄⠄⠀⠀⠀⢀⣉⡄
⠀⠀⠰⣎⠪⣟⢄⠒⣔⢖⠲⡤⢾⡁⡤⠴⠤⠭⡁⣄⠤⠤⠬⠥⠤⠤⠤⠬⠤⠤⠄⠆⠀⠂⠀⠂⠉⠁⠀⡙
⠀⠀⠀⢀⡑⠢⠣⠱⣌⢢⣕⣌⠣⡅⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣐⣀⠀⣀⣀⣤⠶⠂⣠
⠀⠀⠉⠉⠉⠁⠐⠀⠒⠧⠤⢈⣉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠀⠀⠀⢉⣁⡠⠄⣀⡀⣤⡤⠶⠋⠀⣀
⠀⠉⠙⠀⠀⣀⠠⠒⠠⢴⠀⠀⢀⡈⠀⠠⢀⣄⣀⠀⠀⢠⡤⠀⢀⣄⡈⠁⠛⠉⠁⠀⠀⢘⠂⠀⡠⢄⠀⠘⠉
⠀⠀⠀⠠⠞⠓⠉⠁⠄⢀⠀⣰⡡⠼⡀⣰⡥⠤⠚⠀⢀⡆⡼⠄⠘⠒⠪⢆⠀⠴⠤⣵⡀⠀⠀⢄⠩⠛⣳
⠀⠀⠀⠀⠑⠆⠈⠙⠀⠀⠚⠇⠀⢀⡉⠁⠀⠀⠀⠈⠈⠁⠀⠬⠁⠀⠀⠜⠃⠀⢠⠀⢀⡄⠈⠛⣠⠄⠉
⠀⠀⠀⠀⠐⢤⠀⠒⠆⠀⠀⠐⠂⠀⠀⠀⠀⠒⠀⢀⡀⠊⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠂⠀⠸⠇⢀⡀⠀⠀⠐⠆⠀⡀⠀⠀⠀⡀⠀⠄⠠⠦⠀⠀⠐⠀⠀⠀⣤⠄⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣭⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣧⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢹⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⢶⣾⠁⠀⠙⠷⣤⡀
⠀⠀⠀⠀⢰⣿⡿⠛⡛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠻⣄⠈⠛⠷⣦
⠀⠀⠀⠀⢸⣿⣧⡀⠈⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⣴⠿⣦⠀⠀⣿
⠀⠀⠀⠀⢸⣟⢿⣟⣛⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⢤⣄⣀⡀⠀⠀⢸⢇⡴⠆⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠉⠉⡹⣦⠘⠷⣴⠏
⠀⠀⠀⠀⠀⠹⣦⡌⠙⠋⠉⠙⠛⠛⠛⠛⠛⠛⠟⢛⡟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⢦⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣦⡿⠘⢷⣄
⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⡟⢠⠞⠁⠀⡴⠋⢷⠀⣾⠳⢄⠈⠳⡄⠀⠙⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠁⠀⣿⠀⠹⡄⠈⡇⠀⠹⡄⠀⠈⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠒⠈⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⣤⡴⠃⠀⠀⠉⠛⠁⠀⢠⠇⠀⠀⣾⠁
⠀⠀⠀⣿⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣄⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠊⠁⠀⠀⢀⡾⠋
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣦⣤⣤⣤⣤⣤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡤⠀⣀⣤⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⡏⠉⠓⣦⣤⣤⣀⣀⣠⡇⠀⠀⠀⣼⡗⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣦⣉⣽⡄⠀⠉⠉⠉⣧⠀⠀⢰⡿⣷⠄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢙⣧⠉⠉⠀⠀⠀⠀⠀⢻⣅⠀⢸⡇⠿⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠠⣸⣹⠇⠀⠀⠀⠀⠀⢀⣴⡧⠁⠸⣿⣎⠈⣿⡟⣧⠘⣷⡿⣶⣞⠀⠁⡚⣸⢻⣿⣿⣿⣿⡟
⠀⠀⠀⢀⠀⣠⣤⣤⠹⣿⣼⣿⣦⣰⣾⣶⣿⠂⠀⠀⠀⢻⣿⣷⠘⠏⣿⡎⣿⡇⠈⠛⢅⣿⡇⡿⣿⢿⣿⣿⡿⢦⢀⡀⠀⡀⣨
⠀⢀⣠⣾⢼⣿⣿⣿⠑⠜⢿⣿⣻⡙⢻⣿⠎⠂⠀⠀⠀⢸⠘⣿⣧⣤⢻⡷⣿⠂⠀⠀⢠⣿⢫⢷⣿⣿⣿⣿⠁⠀⢻⣷⡿⣫⠟
⣀⢀⢙⣽⣼⣿⣿⣿⡧⢀⣜⣿⡯⢿⣧⡈⠲⣤⣿⣆⠀⠘⣇⠘⢿⣿⢸⣟⣿⠀⠀⢀⠟⣿⣾⣺⡏⣿⣿⡏⠀⢼⠟⢉⠿⣷⣀
⢻⠟⣛⣩⣿⡿⣿⣿⢿⠀⢿⡾⣿⣌⣿⣷⣄⠈⠛⢿⣆⠀⢻⣤⡈⣿⣿⡇⣿⡇⠀⡼⠀⣿⣿⡟⣸⣿⠟⣒⢷⣇⡲⠁⠀⠀⠑
⣴⢺⠟⠉⠀⣿⢿⣸⡌⠃⠀⠀⠈⠙⣏⣿⣿⣷⡄⠀⠉⢳⡈⣿⣧⡈⣿⡇⣿⡇⣰⠃⢰⣿⡿⣱⣿⣿⣿⣾⣽⣿⣯⣐⣠⣀
⠈⠪⠀⢀⠀⣿⡘⣯⢧⠁⠋⣀⠠⡑⡜⣞⣿⣿⣿⡀⠀⠀⠁⠘⣿⣷⣿⣇⣿⣇⡟⠀⣸⡿⢀⣿⢷⣿⣿⣿⢋⣰⠄⣀⠉⠻⣆
⠀⠀⠀⡜⠀⢸⣷⣿⡏⣧⢰⠸⡌⢧⢱⠘⣼⣿⣿⡇⡄⠀⠀⠀⠘⢿⣿⡿⠹⣿⡇⠀⣽⢣⣿⡟⣹⣿⣿⢏⣾⢇⣾⣿⠀⠀⡼
⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⢘⡀⢫⠉⡌⢧⢹⣞⣿⣧⣿⡒⢠⣶⣶⣤⣿⣿⣿⢿⠇⠀⡟⡼⢏⡸⣿⣿⣏⣼⣏⠌⣿⣿⡀⡰⠁
⠀⠀⠀⠀⠀⠐⠋⣵⣿⣿⣇⢹⡈⡧⠼⡽⣆⢽⡞⣿⣿⣿⣿⡛⠯⠟⢛⣛⡛⠛⠃⠀⢬⣣⣼⣿⣿⣿⣿⡟⡜⠀⡿⠫⢠⠇
⠀⠀⢀⣴⠅⠀⡼⡿⣿⣿⣿⡂⠳⢘⡥⢪⠹⣊⣿⢻⣿⣻⣿⣧⠀⠀⠀⠨⣻⣿⠙⣳⡶⢿⣿⠿⢿⣿⢋⠎⠀⠀⣠⠀⡘
⠀⢰⣯⣿⢀⡾⠀⢻⡍⠟⣿⣷⡈⠄⢻⠄⠣⢰⡸⡿⣿⣟⢿⣿⡆⠀⠀⢈⣿⣿⡀⢺⡏⣾⣿⣷⣷⣵⡁⠀⣠⣾⢯⠀⡇
⠀⢸⣿⡏⣗⡇⠀⠈⣟⡜⠸⣿⣷⢩⠄⣧⠘⡭⢆⢿⢻⣿⣸⣿⣿⡀⠀⠀⣾⣿⡐⡸⣡⣿⣿⡿⠃⠈⢓⣻⣿⣿⣧⢻⠁
⠀⢸⣿⣷⣿⡇⠀⠀⠸⣷⠐⢹⣿⡎⠥⠘⡥⢻⠵⣊⢿⣿⣿⣿⣿⡇⠀⠀⢾⣿⣷⣃⣿⢿⠟⠁⠀⢠⣿⣿⣿⣏⢾⣿⠃
⠀⠐⣿⣿⣿⡇⠀⠀⠀⢹⣷⡀⢿⣷⠩⡏⢷⠈⢧⠵⡊⢿⣿⣿⣿⣿⠀⠀⡬⡿⢛⣫⠔⠁⠀⠀⢀⣾⣿⣿⣿⣟⣾⡏
⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⢻⣧⠸⣿⣇⠽⠄⢏⠉⡧⢒⡨⣿⣿⣿⣿⡇⠀⠈⣱⣿⠇⠀⠀⠀⠀⠜⡟⣱⡿⠭⡎⣼⣧⣴⠃
⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠈⢿⣵⡽⣿⣆⢯⠽⡇⢹⡄⢗⠹⣿⣿⣿⣷⠀⢀⣿⣿⠀⠀⠀⢀⠆⠀⢠⡏⠀⠀⣰⣿⣿⣿⣶⠞
⠀⠀⠀⣿⣿⢹⠀⠀⠀⠀⠀⠀⢻⣾⡝⣷⡌⢧⠸⡀⢧⡘⡏⣫⣿⣿⣿⣇⣰⣿⡇⠀⠀⠀⠂⠀⢔⠎⠀⠀⣴⡟⠁⢈⠞⠁
⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡭⢧⢻⠈⣇⠹⡺⣿⣿⣿⣿⣼⣿⡇⠀⠀⣰⣴⣿⣿⡣⢀⣼⣿⢇⡴⠃
⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣽⣂⢣⣸⡎⢣⣿⣿⣿⣿⣿⣿⠇⠀⣬⣿⣿⣿⣷⣷⣿⣿⣿⡿⠁
⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣿⣮⡇⢽⣸⢽⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⡿⠁
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣝⣧⢻⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⠋⡴⠁
⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⠙⢹⠁⡼

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡼⠒⠒⠠⢩⣶⠘⠟⠒⠈⠤⠒⠀⠖⠆⡶⠂⠧⠖⠗⠴⠐⠀⠂⠆⠃⢠⣮⠡⠐⠒⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠁⠈⠆⠀⠀⠀⠀⠀⠁⠶⠀⣀⡀⠀⠀⠀⢮⣧⠞⠀⢬⡷⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠠⠀⠠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠐⣿⠆⠀⠐⢷⠖
⠀⠀⠈⠀⣿⠦⠀⠀⠀⢀⠄⠒⠇⡈⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠚⢀⠀⠀⢀⣀⠀⠀⠀⣀⢠⣤⣤⡤
⠀⠀⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⠀⡀⠀⠀⡄⣠⣄⠀⠀⠙⠀⠀⠁⠠
⠀⠀⠀⠀⠰⠀⠀⠀⣀⡀⠀⠀⠀⠤⠉⠰⠄⢂⠀⠈⠀⠀⠀⠸⠀⠂⠀⢰⠇⠋⢸⠀⠀⠀⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠐⠀⠀⠀⠀⠘⠀⠈⠄⠀⠀⠀⠄⠁⠀⢸⠁⠀⠘⠀⠀⠀⠀⠘⠷
⠀⠐⡾⠚⠀⠀⠓⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠈⠂⠠⠤⠄⠀⠈⠀⣰⠓⠀⠀⠀⠀⠀⠀⠚⠶⢷⠚⠃⢐⡄⠀⠀⠀⡸⠀⠀⠁⠀⠀⠀⢰⡷⠝⣇
⠀⢠⣄⡀⠀⠀⠀⠐⠟⠛⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⡠⠄⠀⡀⠤⡁⠀⠀⠈⠁
⠀⠀⣠⡷⠻⠃⠀⠀⠀⠠⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⡴⠔⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠁⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠈⠃⠀⠀⠀⠂⢠⠀⠠⠀⠀⢀⣀⣴⣴⠀⠀⢰⣿⡀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⣿⣿⣿⠀⠀⠸⣿⠃⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠘⠀⡘
⠀⠀⠀⡡⠂⣢⢬⡳⡄⠀⠀⠸⠟⠀⠀⠀⠉⣷⣿⣿⡤⠀⠀⢻⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠉
⠀⠀⣄⠣⠀⠠⠄⠂⠇⡖⠀⣈⣡⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠛⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠠
⠀⠀⠈⠀⠀⠀⠀⣀⠜⠀⠀⠈⠃⠀⠀⠀⠀⣿⣿⣽⠃⠀⠢⣤⠄⠀⠀⠁⠤⠤⣤⣄⠀⠀⠀⠤⠖
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀

⢀⡴⠚⡿⢦⡀
⣞⣠⠞⠁⠀⠙⢦⣀
⠙⠷⣄⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠈⠳⣦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠘⣧⠀⠀⢀⣀⣤⣤⣶⣾⣿⡿⠿⠛⠛⠉⠉⠉⢛⣯⣙⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢹⣶⡿⠿⠛⠛⣹⠻⢭⣀⠀⣀⣠⡤⠶⠒⠛⣏⠁⠻⣄⡀⠈⠙⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡄⠀⠀⠀⢻⡀⠀⠀⣸⠃⠀⠀⢸⠋⠉⠀⠀⠀⠀⠀⠈⠛⠲⢿⣽⠶⠀⠀⠀⠉⠻⢿⣷
⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠉⠀⣿⡄⠀⠀⠈⣷⣶⣿⣛⢦⣀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿
⠀⠀⠀⠀⢸⣏⠁⠀⠀⠀⠀⣼⣿⣽⣓⣒⣛⣡⢾⣿⡉⠳⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣚⡯⠷⠛⣿
⠀⠀⠀⠀⢸⣿⡷⣄⡀⠀⠀⠹⣿⣶⣭⣭⣭⠴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢖⣻⡽⠿⠛⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡏⠻⢮⡛⢦⡀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⣚⣯⠵⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⣛⡭⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠳⣬⣛⠦⣤⣤⡤⠶⢛⣩⠴⠚⠋⠁⠀⠀⠀⠀⣼⠳⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡶⠖⠛⠉⠀⠀⠀⠀⠀⠀⣴⣄⢰⡇⣴⡌⠿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠁⡏⢷⣴⡆⢿⣤⡄⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠁⠀⠀⠀⠀⠀⠀⢰⡷⠿⠀⣦⣀⡇⠈⢹⣇⣬⢹⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣠⣤⣷⠀⠀⣧⠙⠃⠀⠘⢋⡏⠸⢧⡶⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣆⠀⢶⡶⠾⠄⠀⠀⠀⠘⢻⠇⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠙⢦⣄⡀⣀⠀⣴⢋⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣄⡀⠈⢉⣉⣉⣡⡾⠓⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠁⠀⣤⠶⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡼⠳⡄⠀⠓⠲⣄⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⢿⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢉⠀⠀⡷⠚⢻⠀⠀⠖⠋⠀⢀⣠⡴⢞⣫⠿
⠀⠀⠀⠀⠀⠀⠳⣜⢧⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠈⣹⠇⠀⠃⠀⠀⠀⢀⣠⡴⢞⣫⡵⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡹⣆⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣀⣤⠶⣛⣭⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢷⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⣋⣥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡙⢦⡀⠀⣿⠸⠀⠀⠀⠀⣀⣤⠴⠞⣋⡥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⠳⣧⣬⣤⡤⠶⣛⣩⡴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣦⣴⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣠⣤⣤⣀⣠
⠀⠀⠀⠀⠀⣴⣿⢿⡿⢿⣿⣿⣿⣿⠻⢿⣷⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⢀⡞⠋⠀⣾⠀⠀⣾⣿⣿⣿⡇⠀⠈⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠋⣿⠿⢿⣿⣏⠉⢿⠉⠙⠻⣿⣿⡮⠄
⠀⠀⠀⢘⣇⡀⠀⠘⢄⠈⠈⠃⢙⡿⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⠀⠀⠘⢦⣈⡉⢁⣴⠏⠀⠀⣠⣼⠻⠁
⠀⠀⠀⠀⠀⠉⠙⠒⠠⢭⣞⣛⣭⡤⠶⠛⠃⠀⠀⠀⠀⠀⠀⠀⣀⢠⣴⡄⡆⣤⢤⣠⡄⠀⠀⠀⠀⠀⠈⠑⠲⠦⣤⢉⡉⣭⣤⠴⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⠿⢻⣿⢿⣿⣿⣿⣿⡾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡟⠁⠀⢸⡇⢸⣿⣿⡿⢿⡇⠈⢻⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠈⠱⢄⣀⠘⢷⣄⡉⢉⣠⡿⠁⣀⣤⣿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⠛⠛⠋⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡴⣲⢄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣦⣤⣤⣴⣦⣀
⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⡥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣽⣿⡿⠛⣻⣿⣿⢛⣿⣷⣷⣄
⠀⠀⠀⠀⣶⠟⠉⠀⡿⣿⣿⣿⡟⣿⡆⠙⢿⣾⡭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⠀⢸⡏⠍⢠⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⡍⣀⠀⠀⠹⡉⠙⠛⠃⡹⠁⣀⣼⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠈⢿⡸⣌⢙⣿⣿⢓⣾⣿⡷⣄
⠀⠀⠀⠀⠉⠉⠉⠛⠓⠛⠓⠒⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⢬⡿⠾⠿⠿⠿⠿⠟⠿⣀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣾⣝⣿⣿⡇⡶⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⠋⡟⠻⣿⣿⠟⣿⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⢯⡀⠉⠋⣠⡟⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢿⣦⣬⣭⣼⠿⠿⠶⠒⠒⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠠⠤⢒⠖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⡀⣀⡀⣂⣄⣤⠆⠶⠒⠛⠁⠉⠉⠉⠉⠀⠉⠉⠁⠤⣄⡀
⠀⠀⠀⠀⣴⣴⣀⣔⣵⣢⣴⣶⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠧⣀
⠀⠀⠀⠀⠿⡿⠿⠿⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⣢⢤⣤⣴⡿⢿⠿⠿⢿⣿⣷⣖⣶⣒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣠⣶⡾⣟⣾⣧⣤⣶⣷⣾⣾⣾⣿⣿⣿⣿⡟⣶⣿⡵⣂
⠀⠀⠀⠀⠀⠄⢀⠄⢤⣜⣿⡿⣯⣷⣿⣿⣿⠿⠛⣻⣟⣽⣿⣿⣛⠿⣿⣿⣾⣿⣿⣿⣿⣝⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣵⢍⣿⡿⠛⣿⣿⣻⢠⣷⣾⣿⣿⣿⠁⠀⢸⣇⡉⢿⣿⣿⣿⣧⡟⠿⣆⡠
⠀⠀⠀⠀⠄⠠⠆⢄⣼⣉⣾⠁⠀⢀⣿⣿⢟⣿⣿⣿⣿⣿⣿⡦⢤⣾⣿⠄⢸⠈⠹⢿⣿⣿⣿⣯⠖
⠀⠀⠀⠀⠀⡄⠐⣻⣵⣿⣿⠀⢀⠀⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣶⣿⡟⢀⡼⠀⠀⣻⣿⣿⣿⣟⢋⠁
⠀⠀⠀⠀⠠⠀⠲⣿⣿⣿⣿⣷⣜⠄⠹⣿⣯⣜⣿⣿⣿⣽⣿⣿⣿⡿⢃⣼⠏⣀⣴⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠛⢿⣶⣥⣀⠈⠻⢿⣿⣿⣍⣿⣿⣿⣭⣾⣿⣵⣿⡿⠿⠛⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠿⢿⣿⣿⠿⠿⣿⣿⠻⠍⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡔⡻⢶⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠳⣌⣊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⢤⠤⠐⠒⣻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠫⠼⠀⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠆⠀⢀⠸⠸⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠀⢀⠐⠂⠀⠀⠀⠈⠀⠀⠎⠃⠂⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⣰⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡊⠀⣴⠀⠀⢐⢑⡶⠤⣄⣈⠀⠀⠀⠁⠀⡀⠀⢀⠀⢠⠀⠀⠀⢀⠀⡀⠅⠀⠀⠄⢀⠀⢠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡰⠉⣧⠀⠀⣼⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠒⣶⢆⠀⠀⠈⢰⠋⠙⠒⠤⠀⣀⡀⢀⡑⠄⢀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⡜⠁⠀⠸⣄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠁⠀⠀⠀⠘⡆⢀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠇⠈⢣⡠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢺⡀⠀⠀⠀⠀⠙⣌⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠘⣦⠀⠀⠳⡌⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣂⠈⡉⣱⠦⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⡤⠜⠉⠉⠀⠉⠉⠓⢦⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⢺⣋⡵
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⡾⠁⠀⠀⡴⠋⠉⠉⠁⠂⠀⣴⠏⠉
⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁⠀⠀⠸⡏⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⢐⣲⠆⣀⠀⣰⠶⠚⢹⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⠴⣦⠴⠦⣬⡑⠚⠋⠉⠒⠮⡙⠍⢂⡀⢸⠀⠀⠀⠀⠀⠀⠈⠟⢌⣀⣀⣠⡾⠁
⠠⠚⣅⡀⠁⠀⠀⠈⢻⡄⠀⠀⠀⠀⠘⢆⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣴⠓⢄
⡂⢦⠏⡋⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⢣⡀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡆⢱⡤⢤⡀
⣀⠚⠉⠃⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⡷⠊⠙⠺⢿⠛⠙⠲
⠘⣇⠀⠀⠒⠤⠄⣠⠴⠦⠽⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⣏⡐⢲⣦⣄⡀⠀⠀⠀⣼⠀⢰⠃⠀⠀⠀⠀⠇
⠀⠈⠳⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠵⢒⣿⣿⠴⠋⠀⠈⠹⣆⠀⠀⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢀⣴⠾⢯⣍⠉⢧⡉⠒⢦⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⢀⡴⠋⠂⠀⠁⠳⠈⠀⠙⣆⠐⠿⣦⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠋
⢠⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣇⠀⠀⠙⠢⠤⢄⣠⣤⠴⠞⠁⠀⠀⠀⠙⠲⠤⣤⣀⣠⣇⠴⠃
⢸⠀⠛⡆⠀⠀⠀⠀⠖⢠⡊⠰⠤⣤⣼⠎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢀
⣿⠀⠀⠳⠦⣀⣰⠧⠬⠀⠀⢀⡎⢀⠇⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠴⠃
⠹⣄⡇⠈⠲⢶⠶⣄⠀⠀⢀⢞⡔⠁⠀⡇⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠉⠀⠴⠛⠁⠚⠋⠉⠉
⠀⠘⢧⠆⠀⠨⣄⠈⠙⢯⣴⣂⠀⢀⡞⢁⣸⠃⠀⠀⣀⡤⠴⠒⠋⠁⠀⠀⠀⢀⠀⢀⣶⠀⢦⡉⠉⠙⠒⠤
⠀⠀⠘⢧⣀⣀⣈⠓⠙⢓⣺⡿⣻⠋⡰⠋⠀⢠⣶⣉⣀⣠⣀⣠⣤⣤⡶⠴⠶⠾⠛⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⢈⠓⡄⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⠀⢴⠿⣶⡀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢠⡏⠼⣿⣿⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⣿⡆⠁⡽⣤⣼⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠇⣿⣿⠃⢘⠃⢀⣐⣗⣲⠺⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⠀⠀⠘⣿⡏⣤⠈⡃⢸⠁⠭⠽⣯⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⠀⠀⠐⣿⠧⠬⢁⡍⢉⣁⣭⣉⣟⣛⡺⣿⡇⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠡⡀⠀⠛⠐⠓⠀⠃⠐⠖⠺⡴⠧⣽⢍⣻⡇⠀⠀⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣄⠀⢘⡃⣘⠁⢐⣋⣉⡿⠲⠷⠶⢾⠀⠀⠀⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⢨⡅⣨⠀⢈⣁⣀⣋⣈⣛⡛⢻⠀⠀⣰⣿⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⢠⡀⢴⠀⢠⡅⢨⣏⣉⣉⣉⣛⠀⣰⣿⡏⣼⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡴⠂⠶⠀⠰⡦⢤⣍⣤⣽⣩⣉⣳⣿⣿⠙⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠙⠀⠐⠂⠶⠦⠤⣼⠤⢯⣿⣿⣏⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠀⢚⠂⠒⠃⠐⠾⠖⠦⣿⣿⣿⣆⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣙⣉⣉⣏⣙⣺⣛⣟⣿⣿⣿⣿⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠄⢠⣯⣭⣽⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⢴⠀⠰⡶⠶⢶⠶⡿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⢸⡇⠈⣟⢚⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⢸⡇⠠⠧⢤⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢸⡇⢹⡏⣽⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⡇⢸⣃⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣷⢸⣷⡾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷⣿⡇⢿⠛⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠸⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠛⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⡆⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠘⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠀⠀⢀⣴⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣄⠀⠀⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣴⣶⣄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠈⠻⢦⣤⣤⣀⠀⢰⣿⣄⣿⣿⣿⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⢠⡙⣦⣤⣀⡀⣰⣿⣿⣿⣿⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠙⠿⡿⠟⠁⠙⠿⡿⠿⠛⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠄
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠶⢶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⠛⠋⠉⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠈⠉⠙⠻⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠦⢤⣀
⠀⠀⢸⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⠶⠶⠚⠋⠁⠀⢠⡶⠿
⠀⠀⡞⠁⠈⠉⠙⠛⠻⠶⠶⣦⣤⣤⣄⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠤⠤⠤⠶⢶⡶⠶⠖⠛⢛⣛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠶⣶⠾⢣⡀
⣠⡼⠿⠶⣶⠿⠛⣿⣦⣀⣴⢶⡄⠈⣨⣭⠉⠉⢉⣄⡀⠀⠀⢀⣤⣀⣀⣴⠞⠛⠻⣤⣀⣤⠶⠋⠉⠳⣤⣀⣀⣴⠛⠻⣦⣤⣤⣏⡴⠛⠀⠀⢻
⢿⣤⣤⡀⠀⠀⠀⠹⣦⡉⠁⠀⠙⠛⠁⠘⢧⡴⠟⢉⣷⣤⡴⠋⠻⢯⣝⡃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠉⠉⣉⠛⢛⣛⣿⣯⠴⠴⣶⠒⠛
⠀⠀⢸⡟⠓⠶⢬⣁⡊⠛⠓⠶⠤⠤⠤⠴⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠛⠋⠉⠠⢀⣂⣤⠴⠖⠛⠉⠉⠀⠀⠀⠀⣼
⠀⠀⠸⣇⠀⠀⠀⠉⠙⠓⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣂⣬⠾⠛⠳⣄
⠀⠀⠀⣼⠿⢶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠤⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠉⣡⣤⣤⣤⡾
⠀⠀⠀⢧⣄⣤⣀⠈⠙⠛⠷⠶⢤⣤⣤⣀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢦⣴⣴⠞⠉⢀⣀⣤⣠⣤⣤⠤⠖⠚⠋⠉⣹⠏⢀⣤⠶⠶⠶⠟⠀⢻
⠀⠀⠀⠀⠀⣿⠙⢧⡀⠀⠀⢀⣀⣀⠀⠈⠉⠉⠛⠛⠓⠲⠶⠒⠒⠚⠛⠛⠛⠛⣉⣩⣭⣀⠀⠀⣠⣾⣛⣛⣛⣫⠶⠋⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⣿⡀⠈⠙⠻⠛⠛⠉⠉⠻⢦⣀⠀⠀⣀⣴⠾⠛⠛⠷⠶⢤⣤⡴⠞⠋⠁⠀⠉⠛⠛⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠘⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⠤⠶⠶⠚⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣯⣫⣴⣭⣿⣧⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠢⢀⠌⠀⠨⢤⠄⠄⠀⡄⠚⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡂⠀⠸⡄⠁⠁⢠⡾⠴⠥⣀⠀⢀⠀⡧⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⡀⠀⠀⠀⢤⠐⠛⠀⠀⢀⠆⠂⠀⠚⡀⠸⡞⠒⡤⠦⣥⠠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠡⠈⠄⠳⡠⠴⢂⠚⠚⠆⠀⠀⡆⠀⠀⣀⠤⢼⣈⢃⡈⠀⠀⡄⠀⠁⠀⠡⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠰⠀⣄⣄⠀⡼⠅⡀⠈⠀⠀⣸⠄⠒⣇⢀⡎⠉⠀⠀⠀⠀⢥⠱⡚⠽⣆⠀⡀⠂⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠲⠔⡨⢂⡩⣍⡑⣄⢸⡶⢒⡘⠁⢔⢟⠉⠝⢄⣀⡀⠀⠀⠀⣡⠊⢀⡞⠉⠒⠀⢸⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠀⠘⠚⢴⠀⢠⣀⢹⠈⡆⠀⠠⢁⠞⠆⡠⢆⠜⢘⣨⡉⠸⢟⡳⢁⠴⠓⠠⢤⡀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⠀⠒⠆⠈⢿⣀⠁⠑⡶⠀⠀⠀⢇⠀⠀⡀⠁⠀⠀⠀⠙⢧⡶⠉⢑⠄⢡⢖⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠧⡀⠀⠀⠀⠘⠀⠀⡃⡐⠁⠜⠉⠉⢉⠅⠀⠀⠀⠀⠀⠈⠆⢠⢀⠈⣃⢘⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠂⠅⡠⠀⣰⠀⣠⢀⠀⠘⡀⠐⠒⡉⡀⠀⠀⠀⢀⠌⣐⠀⡀⠄⠫⠑⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠀⠒⠉⠉⠀⠠⠗⠤⠤⠲⠟⠦⠉⠉⠉⠈⠂⠈⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠤⠄⠀⠀⠀⠀⠠⠤⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠔⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⠁⠢⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠶⠲⠤⣀⠸⣄⣀⣀⣀⣀⣀⡀⠤⡏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠂⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠠⠀⣀⣑⣅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠻⡀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣠⠴⠒⠋⣉⠝⢓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣠⠤⠖⠁⠀⢠⠏⠀⢀⠄⢊⠤⠊⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠫⢍⡉⠈⠉⠒⢄⡀⠀⠀⠀⠀⢸⠤⠎⣀⣐⡁⠀⠄⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠁⠂⠌⡑⠢⢄⡀⠙⢆⠀⠀⠀⠀⠑⠒⠶⠶⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠐⠠⠀⢀⡀⢁⣢⡬⠑⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣦⡀⣦⣀⢀⡀⠀⠀⠀⠀⠀⢼⣿⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣴⣦⡀⠀⠀⠀⠀⣠⠸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⡗⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⣵⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡌⣿⢾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⢎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣭⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⣿⣿⡟⣿⣿⣷⠀⠀⢸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣼⣿⣿⢠⣿⣿⣿⠀⠀⢸⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠟⠿⠛⠛⠛⠛⠿⠅⠛⠿⠀⠒⠈⠛⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠖⠈⠋⠙⠉⠁⠉⠉⠙⠁⡒⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠎⠁⠐⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡀
⠀⠀⠀⠀⠀⠀⠈⠢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⡚⣑⠅⢳
⠀⠀⠀⠀⠀⠀⠀⣸⣠⡢⠤⠭⠛⢇⣒⣲⣶⣶⣶⠖⠒⠒⢲⠒⠺⠿⠿⠗⠚⠐⠒⢉⣁⡠⣴⠃
⠀⠀⠀⠀⠀⠀⢸⡥⠜⠛⠛⠯⣿⡍⡉⠉⠀⠑⠢⠭⠭⠭⠭⠝⠒⠒⢒⣒⡂⠉⠉⠁⢀⠀⢸
⠀⠀⠀⠀⢀⠓⠲⢇⠀⠀⠀⠀⠀⠉⠢⣕⢄⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⡯⡀
⠀⠀⠀⠀⢀⠗⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⢢⠑⠤⢊⠔⠉⠀⠀⠀⠂⣁⣈⡠⠒⠒⠢⠤⠜⠀⡣
⠀⠀⠀⠀⠘⠒⠒⡆⠀⠑⠙⠲⠖⠒⠤⠠⠤⠵⠮⢅⡠⠤⣒⠒⠤⠊⠁⠀⣀⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠏⠙⢄⣠⢄⣀⡠⠒⠢⠤⠤⠤⠤⠒⠊⠁⠉⠈⠒⠒⠊⠀⠈⠉⠉⠉⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠗⠒⡇⠉⣇⢸⢁⡌⢠⢉⡿⢳⠗⢠⡶⢀⢀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠯⠉⢀⠑⢤⣈⣷⣽⣦⣽⣾⣿⣴⣷⣿⣣⣿⡴⣯⣀⣠⢜⣫⡿⢴⣁⡤⠊⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⡘⢧⡈⡐⣦⣨⣶⣾⢿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣟⣽⣿⡦⠶⢰⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣶⠿⠋⠉⢡⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣯⣿⠻⢿⣿⣿⣿⣿⣓⣲⡽⠷⠖⣋
⠀⠀⠀⠀⠀⣒⣺⡿⠛⠁⠀⠀⠈⠀⠈⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⡟⠈⠀⠁⠈⠟⣿⣿⣿⣿⠶⠶⠮⠥⢀⠰⣂⣠⠴⠂
⠀⠀⠀⢴⡾⠿⢟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡌⠈⢻⣿⣿⣿⣿⣿⡿⠛⢳⣾⠇⠀⠀⠀⣰⡾⠏⣠⣭⣙⣛⠢⣾⠿⠭⠥⠄
⠀⠀⠰⣄⣀⣀⡀⢀⣂⡤⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠛⠛⠛⢋⣥⣴⡿⠃⣀⡠⠴⣚⣡⣔⣛⣛⠻⠯⡍⠁⠈⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠀⠗⡢⢤⡄⡠⣠⢰⢈⡁⠖⡶⠖⢛⣛⣓⣚⣩⡡⠶⢿⡺⣿⢯⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠈⡰⢋⠛⠃⣿⡮⢲⢿⡻⡙⠢⡹⠙⢆⠑⠌⠱⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠘⠀⠑⠄⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣚⡉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢃⠰⢤⡐⢠⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣉⠛⣖⣉⠽⢦⡔⠃⣞⢈⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⡆⡩⠔⢁⠿⢤⠚⣉⠴⣥⠔⠃⣈⠒⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠂⡎⢻⣤⠂⢹⠧⣤⠋⣩⢯⡤⠛⣅⢡⣜⣒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢻⡕⢃⡼⠈⣇⣤⠋⡹⢤⡎⣁⠼⢤⠔⠃⣄⠒⣀⡲⣔⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⢦⡤⠀⠨⠿⠭⠤⠞⣅⣜⣘⣀⡜⢉⣧⣤⢊⡩⠧⡤⠚⣠⠐⢊⣆⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⢩⣀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣉⣙⣉⣛⣓⠊⠽⠧⣤⠘⢂⡜⡆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣄⡙⠢⣠⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠯⢔⣀⠈⠁⠐⠻⢴⡀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⣹⢉⣓⡸⠀⡭⢴⢀⣆⠽⢓⡦⠥⠾⠒⠑⠂⠲⠤⣀⠀⠉⠒⠤⠀⠀⠉⠂⢤⣀⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣥⡎⢩⠗⢻⠔⣼⣡⠜⢀⠖⣠⢏⡉⢮⠔⠊⠍⠓⠲⢯⣶⣤⣀⡀⠀⠀⠀⠀⠈⠻⣷⡵⡀
⠀⠀⠀⣀⠠⠀⠀⠀⠠⢱⠠⠴⣀⠋⡻⠰⢧⣴⣁⡜⢉⡝⢺⠅⢺⣥⡞⣁⠖⣰⠯⣔⣋⠞⢙⢪⣟⣿⠶⣕⣤⡀⡠⡈⠻⣷
⠀⠀⠈⠉⠉⠉⠒⠢⢄⡀⠛⡔⡉⡕⠣⠴⡁⣼⠁⡙⢛⠖⢦⣴⣃⣰⢁⡜⢺⠧⣎⢤⡎⡩⠗⡤⠚⣷⡳⠄⠁⠻⣖⢌⢷⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡡⠔⡜⢄⠦⢨⡎⠳⠚⠤⣆⢠⠁⡹⠋⡗⢧⢴⣁⡜⢉⠗⢹⠗⢾⠤⣿⡄⠀⠀⠀⡙⢾⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣈⣢⠧⣰⠈⣲⢣⡼⣀⡅⢙⡊⣣⣆⣰⣁⣟⣙⡖⣧⢴⣇⣴⠉⣇⢀⠀⠀⠈⢧⣻⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢻⣿⣿⣷⠦⡒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⡒⠦⢄⣠⣸⣿⣧⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠐⣊⠝⡩⠂⣁⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣝⣿⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⣢⡶⠟⠒⠐⠒⠒⠓⠒⠂⠒⠠⢌⣑⠂⠄⠦⣄⡂⢕⡢⢀⠠⡀⠀⠀⠀⠀⠀⢹⠇⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡈⠛⠷⣮⣕⣯⣶⣭⣒⣀⣠⣴⣟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠬⣖⣿⣿⣿⣿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠲⠶⠶⠶⠶⠶⢤⡀
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢰⠇⠀⠀⠀⣠⢞⣫⠵⠶⠶⠶⢬⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣋⣭⡤⠬⢭⣍⣓⠦⣄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⡿⠀⠀⢀⡞⣱⠋⣠⢖⣒⣏⣒⡶⣌⠻⣆⠹⣆⠀⠀⠀⠀⢀⡞⣱⠟⢁⠴⠒⠒⡶⢤⡉⠙⢮⡻⡄⠀⠀⠀⢷
⠀⠀⠀⢰⡇⠀⠀⢸⡇⣯⠐⡃⢾⣁⣀⣨⠗⢸⠀⢸⡆⢹⠀⠀⠀⠀⢸⠃⡇⠀⣇⠀⢾⣉⣹⠀⢘⠆⢨⡇⢹⠀⠀⠀⠸⡇
⠀⠀⠀⣼⠀⠀⠀⠈⣧⠘⢦⡙⠢⠬⡭⠤⠴⠋⣀⡼⢁⡟⠀⠀⠀⠀⠘⣧⠹⣦⡈⠓⠤⠭⠤⠒⢉⣠⠞⣰⠏⠀⠀⠀⠀⢻⡀
⠀⠀⢀⡏⠀⠀⠀⠀⠈⠳⢦⣍⣛⣓⠓⠒⣛⣋⣥⠶⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⢭⣟⣛⣛⣛⣛⣩⡴⠞⠁⠀⠀⠀⠀⠀⠘⡇
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣼⣤⡤⠤⠴⠶⠶⠶⠖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣛⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⠄
⠀⠀⠸⣖⠈⠉⠉⠉⠉⠉⠀⠀⢸⢻⢹⢹⡆⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠉⡿⡈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣘⣻⣾⣛⣀⣀⣀⣀⣀⣀⣈⣀⣀⣁⣀⣀⣀⣀⣀⣈⣓⣳⣛⣃⣀⣀⣀⣀⣠⣤⣤⣤⣀⣼⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⠉⠉⠉⠉⠉⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣶⣦⣄⣀⠀⣿⣿⣷⣾⣿⣿⠿⠿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⠋⢀⣿⠉⠛⠻⢿⣿⣿⣿⡿⠛⠁⣀⣤⣤⣄⣈⠙⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢻⣿⣾⣏⡁⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠿⠟⠛⠻⢿⣷⣦⡀⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣦⡀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠈⢻⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⣠⣾⣿⡿⠛⠉⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀⠈⢿⣿⣷⡀
⠀⢀⣾⣿⡿⠉⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠙⢿⣿⣦⡀
⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠙⢿⣿⣦⡀
⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣿⣄
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠈⢿⣿⣧⡀
⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⢀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣄⣀⣀⣀⣀⣀⣠⣤⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⡀⡀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠂⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠳⢄⠀⡀⠀⠀⠑⠂⠉⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡆⢀⢜⠀⠑⣶⠒⣦⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠘⢇⢸⠁⡰⡀⠎⡇⠐⢂⠀⠀⠀⠀⠂⠱⢄⠀⠀⠁⠀⠸⠀⠰⢖⠢⢤⣀⣀
⠀⠀⠀⡠⣂⠵⡂⡉⢢⠄⠐⠒⠈⠉⠉⠉⡽⠄⡢⠰⢁⠄⠎⠢⣀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠖⡑⠠⡀⢱⣤⡀
⠀⢀⡾⡎⠁⠆⡈⡾⠁⠀⡠⠒⢐⢄⡠⢊⠜⠑⢦⠒⠂⢍⡀⢒⠨⠉⢡⠀⠀⠠⠤⠤⠤⡀⠀⢠⠀⠠⡀⠶⠀⠁⢸⠇⡅
⠀⠰⠀⠛⣈⣁⡜⠀⠀⢊⡀⢳⣼⣾⢗⡃⢅⠢⣡⡄⠧⠎⢥⣀⣠⢀⢠⠢⡀⠀⠀⠀⠀⠈⠢⡇⠀⠈⠠⡐⣤⢞⠍⡰
⠀⠀⢧⠀⠢⡀⠻⠀⠀⠢⣩⢛⡟⣁⠻⠂⠈⠁⢀⠄⣀⡀⠀⢡⠭⢡⠰⠑⢄⢹⢀⠁⢀⢱⢀⠐⢄⠀⢠⠀⡠⠊⡔⡃
⠀⠀⠰⠐⠠⡈⠉⠂⠠⠎⡸⠋⢌⣡⠶⣖⠒⠚⠍⠉⣉⣀⡠⠂⣭⡈⢰⡀⠱⡜⡇⡔⢀⡟⡎⣥⠂⡴⠇⠊⡠⠊⢀⠃
⠀⠀⠀⢀⠀⠈⠐⠢⠄⡀⠀⠁⠀⠐⠀⠈⠰⠖⠀⣀⠀⣅⠒⢛⣐⣲⣂⡱⠅⠤⠌⠙⠋⠐⠃⠉⢀⡠⠔⠊⠀⠀⡜
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠄⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠤⠄⠒⠉⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⡾⠿⠿⠛⠛⠛⠛⠛⠛⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⡞⠟⠛⠛⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣋⣡⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠉⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⠟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣤⣄⠄⠉⠙⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠉⠀⠀⣾⣿⣿⡟⠉⢹⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⢰⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢣⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⠁⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣈⣦⣀⣀⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠙⠻⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⠔⠒⠛⠿⠉⠉⠉⠉⢉⠉⢉⠙⠛⠓⡒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⠉⠀⠂⠀⠴⠀⠀⠁⠀⠀⠀⠈⠁⠴⠀⣁⠀⠀⠤⠀⠩⡗⠦⣤⡀
⠀⠀⠀⢀⡶⠞⠁⠀⠉⠁⠀⠆⠀⠀⠖⠀⠀⢀⡀⠀⠂⠀⠀⠀⠠⠀⠀⠄⠀⠀⢀⠀⠀⡀⠀⠐⠎⠙⢆
⠀⠀⢠⡞⠰⠂⠀⠔⠀⢠⠄⠀⠀⢀⠀⠐⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠊⢳⡀
⠀⠀⣟⠎⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠒⠀⠀⢳
⠀⢨⣇⣶⡄⣠⢤⠀⠀⠀⣤⣄⣴⣦⣤⣤⣤⣼⣦⣾⣶⣶⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣿⣿⣛⣓⣾⣿⣿⣾⣿⡿⠿⠟⣛⣉⣉⣉⢤⣠⠤⠴⢤⣀⣀⣀⠾⠿⠯⠍⣩⢿⣛⣓⡒⠶⠶⠶⠦⠴⢞⡀
⠀⠀⠘⣿⣿⠿⣿⣛⣋⣉⡀⢀⡴⠋⠁⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠂⠀⠙⠲⢶⠲⠲⢴⣶⡽⠦
⠀⠶⠚⣵⡾⠏⠉⣌⠉⢉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⡤⠀⢹
⠀⠀⠀⢿⣤⣤⡀⠀⡀⢈⣀⣠⣾⣭⡭⠽⠶⠶⠶⠶⠶⣄⣠⣄⠀⣠⣤⡴⠿⠿⠶⣶⡆⠀⠀⢀⣈⣀⣀⣀⣀⠞
⠀⠀⠀⠀⢹⣿⣿⢰⠛⠋⠉⠀⠀⠀⣀⣀⣀⣀⣀⣠⠤⠽⢿⣿⣤⡿⠟⠓⠒⠢⠤⠼⠿⣴⣛⣋⠉⠉⠉⢙⡇
⠀⠀⢀⣤⠾⣿⠟⠓⠒⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠳⣄
⠀⢰⣿⣿⡿⠿⡶⣦⣤⣀⠀⠀⠀⠀⠀⡄⠰⠒⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⢉⠉⠉⠉⠉⠙⠳⠴⠶⣄⠀⠀⠀⡬⣷
⠀⢺⣐⡀⣹⣋⠀⠀⠈⠙⠻⢶⣄⠀⠀⠳⠂⠀⠀⢠⠞⠋⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠒⠀⠹⡄⡞⠛⠀⢘
⠀⠈⢻⣿⣍⣁⣀⣉⣀⣀⡀⢀⡉⢻⣶⣤⡀⢠⡞⠋⠀⢀⣢⣀⣀⣀⣀⣀⣀⡀⠀⠘⠀⠀⠀⠀⠀⣤⡉⢁⣀⣡⠞
⠀⠀⣾⣿⣿⠉⣹⠿⡿⠟⠉⢻⡟⢻⣿⣿⣷⡾⠛⢿⠛⠋⣉⡉⢉⡟⠉⡏⠉⢻⠻⢯⣙⡽⠋⢿⠋⠛⢩⠿⣿⡇
⠀⠀⠈⢸⣿⡷⠃⠀⠀⠀⠀⠈⠳⠞⠀⠈⠉⠀⠀⠸⡄⢠⠉⠓⠋⠀⠀⠿⠖⠃⠀⠀⠀⠀⠀⠸⣆⣀⡟⠀⢸⡇
⠀⠀⠀⠸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠻⢿⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠀⠀⠀⠠⠤⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣾⡿⡫⣄
⠀⠀⠰⣌⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⡉⠀⠀⠘⠿⠞⠁
⠀⠀⠀⠈⠉⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⢩⡿⠁⠒⠲⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣦
⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡤⠀⠀⠀⠀⠀⡐⠺⠦⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⢠⣴⡾⢏⣀⣀⡀⠠⠤⠀⠐⠠⢀⠀⠀⢀⢁⣨⡋⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠿⣷⣶⣾⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣻⣿⣿⣿⣩⣿⣛⠀⠂⠀⠀⠄⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣾⣿⡿⠉⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠿⢇⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠁⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣸⣁⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣠⡀
⠀⠀⠀⠰⠒⠒⠲⠆⠐⠒⠎⠛⠉⠛⠚⠃⠈⠉⠉⠉⠙⠉⠙⠓⠂⠀⠀⠀⠉⠛⠉⠁⠉⠉⠉⠉⠉⠀⠈⠛⠒⠒⠒⠢⠤⠤


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠿⠛⣛⡛⢻⣛⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⠅⠤⠀⠀⢀⠈⠀⠀⠘⠈⠀⣨⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠍⢠⡀⠂⠐⠦⠀⠋⠀⠀⠀⠄⠀⠘⡾⠽⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡯⠀⠀⠈⠁⠀⡀⠀⠰⠄⠀⠀⠀⡀⢠⡄⣒⣤⢾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣄⣀⣁⡀⠐⠃⠀⠲⠀⢠⠆⠈⠉⣤⠈⢁⣸⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣷⣴⣶⣶⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠽⢿⣿⣿⠿⠛⣛⣻⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡿⠿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠋⠋⠉⢉⣉⣉⣬⠤⠾⠿⠃⠀⢐⣉⣭⡿⢛⡉⢁⡰⣬⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣶⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡑⣶⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⣿⣟⠉⢀⣁⣨⣄⣿⣋⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡉⠛⠻⢋⣿⣿⣿⣿⠿⠻⠿⢿⣿⣿⡿⠟⠛⢻⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣦⣈⠏⢉⣉⣉⣻⣃⡑⠢⠤⢿⣋⣭⣽⢋⣍⣉⣉⣩⣿⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣯⣘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡖⡈⣲⣿⣿⣿⣿⣿⡿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣦⣤⣤⣤⣤⣠⣤⣤⣤⣤⣥⣴⣶⣾⣿⣿⣿⣿⣿⣿⣾⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣄⣠⣀⣀
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧
⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇
⣿⣿⣿⣿⣿⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⠃
⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠉⠻⣿⡿⠟⠉⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠑⠖⠋⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠑⢦⠀⠀⠀⠀⠀⠀⢀⣀⡸⠀⠀⠀⠀⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⡠⠤⠔⠂⠁⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠤⠂⠀⠀⠀⣀⣀⡇⠀⠀⠀⠀⠀⢠⠖⠉⢙⡤⣄⢀⡴⣺⣕⢦
⠀⠀⠀⠀⢀⠎⠀⢀⠤⣤⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⡄⠀⠔⠁⠀⢀⣤⣀⣘⠅⠀⠀⠀⠀⠀⠀⠀⠈⡳⢒⡽⠊⠹⣾⠏⠀⢸⢸
⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀⡷⠀⢸⠁⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⣄⣀⣀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠀⠀⠙⣄⠀⣸⢸
⠀⠀⠀⠀⠈⢢⡀⠈⠛⠛⠁⢀⠏⠀⠀⠀⠀⠀⠀⠸⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⣈⠢⣽⢸
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⡁⠀⠑⠌⣙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡞⠁⠀⠈⣇⠤⢤⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⢱⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀⠀⡠⠖⠋⠉⠹⠶⣄⡤⠶⢼⡇⠘⡰⠊⡁⠀⡼⣀⣀⣀⠀⠀⠀⢸⠀⢄⠀⠀⢀⠠⢂⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠸⡟⠁⠀⠂⠀⠂⠀⢘⡏⣀⢉⡇⡞⠠⠊⡠⢾⠟⠁⠉⠈⠦⡄⠀⠘⢆⠀⠉⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⠁⠀⠀⡇⠀⠀⠹⡏⢉⣿⡫⢍⡲⡏⢉⣈⡉⠳⢧⣔⡋⣴⠛⠀⠀⠀⠀⢠⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢹⠣⢤⣀⡀⠀⠐⠀⣠⠟⠧⣴⠛⢖⠋⠍⢿⡴⣫⠟⠅⢈⡿⠀⠀⢀⣀⡠⠔⣻
⠀⠀⠀⠀⡼⠻⠷⠈⢳⡀⠀⠈⣇⠀⠀⠉⠉⠓⠒⠓⠲⠶⠷⠦⠤⠷⠤⠼⠶⠾⠾⠖⠚⠓⠚⠉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠸⡄⢀⢂⠀⢻⡄⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠹⡄⢂⠀⢀⠙⣆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⣠⢴⡲⡄
⠀⠀⠀⠀⠀⠀⠹⡔⠀⠐⣆⠘⣦⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⠴⠚⠉⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡔⠔⢌⢦⡈⢷⡀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠁⠀⠠⡖⣏⠀⠀⠀⠀⠀⡰⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠢⡁⢵⣌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠮⣑⠲⠤⠖⣊⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣀⡤⢄⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⢁⠄⣊⣭⡙⣯⡟⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⣀⡀⡀⢈⣩⣼⠻⣿⣿⣿⣷⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠟⢿⠔⠁⠈⠟⠁⢹⣿⣿⣿⣿⣱⢦
⠀⠀⢀⣠⠴⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡚⠂⠙⠖⠛⢀⣴⣿⣿⣿⡿⢀⣾
⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⣿⡗⠂⠀⠀⠀⠈⠻⢿⣿⣇⡸⠃
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⢎⣿⠗⠚⠀⠀⠀⠀⠘⢿⠟
⢨⡳⢤⣄⣀⣀⣀⣀⣀⡠⠄⢀⠀⠜⠀⠀⠀⢀⠀⠀⠈⠛⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠙⠷⢤⡄⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠀⠀⠁⣦⡀⠀⠀⠀⡅⠀⠀⠸⠂⠀⠀⠀⠀⠳⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢤⡂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠒⠒⠀⠀⠀⠀⠀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢾⣿⠏⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣝⣿⣻⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠻⣿⣿⣟⠛⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠐⢀⣦⣼⣿⠟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣇⣀⣤⣾⣿⣿⣷⣶⣦⣄⣰⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠉⢉⡉⣻⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣯⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣤⡾⢿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣀⣼⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟
⠀⠀⠀⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⣤⣤⣶⣿⣷⣦⣾⠆⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠻⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⡧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⡄⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠈⠉⢹⣿⣿⡏⠉⢹⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠘⠿⠿⠃⠀⠘⠛⠋
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣐⡊⠙⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠃⠀⣠⠀⣀⣀⠀⠀⠘⠀⠀⠨⠀⠊⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠠⣀⢀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⣀⠠⣤⣄⠀⠶⡄
⠀⠀⠀⠈⣇⣠⡉⡾⠛⢧⣦⡄⠀⠀⠀⠀⠀⣠⠀⠠⢀⢄⠀⠈⠛⡘⢷⣄⡀
⠀⠀⠀⠏⠛⠑⠀⠁⠀⠸⡏⢰⣤⣇⠁⢴⣦⢾⣾⠁⠀⠈⢁⣠⣀⡀⢀⠻⠿⡏⠑⠂⠀⠀⠓⠭⠒⠤⢀⡀
⠀⠀⢧⠀⡠⠀⠐⠀⠀⠀⢸⣛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣮⢶⣄⠈⠙⠢
⠀⠀⠐⢄⡀⣀⣀⣠⢴⣾⣿⣿⣿⣿⣯⣿⣿⠛⠑⢷⠄⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡆⠙⢎⠳⣽⠷⡆
⠀⠀⠀⠈⠛⠉⠸⠗⠛⠛⠻⠿⣿⣿⣿⡟⠁⠀⣸⣿⡖⠀⠀⢢⣠⠀⣞⠉⠁⢀⠀⢀⠚⠛⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡏⠀⠀⢀⠀⠀⢀⠀⠚⢻⡄⠙⠃⠴⢛⡅⠝⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠂⠀⠀⠀⣀⠁⠀⣀⠀⠐⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠿⠿⠿⠟⠻⠟⠛⠛⠟⠿⠿⢿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣦⣄
⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄
⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⡆⠠⢣⢸⠇⣿⠃⡰⠃⠂⡾⢃⡼⢣⢷⠇⡤⣠⡔⡀⠀⣠
⠘⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣦⣷⠇⢅⣰⣠⣬⠼⣶⣿⣶⣷⣿⣿⣿⣿⣿⣷⣶⣶⣤⣵⣿⣯⣿⣷⡧⣤⣀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⢀⡈⣷⡿⣿⡬⠚⠋⠉⠀⠀⣴⣿⣿⢿⣿⣿⣿⣿⠉⠉⠉⢹⣿⣷⡈⠙⠻⠿⣿⣾⣿⣿⢾⢦⣀⡄
⠀⠀⠀⠀⠀⡀⠐⠶⢝⣥⠞⠉⠀⠀⠀⠀⠀⢸⣿⠏⣴⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠉⠛⠶⣥⡞⠛⠁
⠀⠀⠀⠀⠢⢬⣽⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢉⡽⠟⢷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡅⣿⡿⠿⣿⣿⡿⣿⢿⣫⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠘⣿⣿⣷⣴⣷⣤⣷⣤⣴⣿⣿⣿⣿⠟⠀⠀⢀⣀⣠⣴⣶⠿⢿⡿⣼⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣄⣀⠈⠛⠿⢿⣿⣿⣿⠿⠿⠟⠛⢉⣠⣤⣶⡾⡟⠉⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠿⠶⢶⢶⠶⢶⠶⠶⠶⠾⠛⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⣀⠀⠀⠄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡀⠐⠰⣦⡀⠀⣁⠈⢧⣀⣉⡈⣴⠋⠘⠀⠀⠉⢰⡢⡄⢸⣁⠒⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠷⠽⠾⣆⡙⠿⣷⡤⠉⢀⡉⢉⠓⠔⠐⠒⠂⠀⣌⣴⡷⢎⠺⠷⢶⢸⡏⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠈⠄⣠⣤⡤⢠⣽⣦⡤⣤⠶⠔⠀⠍⠠⠀⠀⡂⠀⢀⠈⠙⠿⣤⡀⣁⣠⠞⠁⠀⠉⠀⡙⠄⠤⡀
⠀⠀⠀⠀⠀⡰⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⢘⣿⣷⡄⠟⠒⠓⢮⣀⠰⠶⢿⣞⠉⠁⠈⠒⠶⢤⣝⡁⠀⠀⠐⠀⠐⡂⣠⠆⠴
⠀⠀⠀⠀⠀⢆⣠⡈⠐⠒⠚⠓⣤⣀⣀⣤⣾⣿⠟⠁⢀⠀⠀⠈⣿⡄⠀⠈⠻⣷⡄⠀⠀⠐⠀⠙⣿⡆⠀⠑⠒⢖⡧⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠑⠙⠩⡛⠓⠋⠉⣿⠿⠿⢭⣴⣎⠀⢀⣽⡧⣿⠃⠀⠠⣸⣿⠀⡀⠄⠀⠀⠀⣸⢟⣄⣀⡗⢥⢶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠶⠄⠀⠀⡨⠛⡿⠿⠿⣿⣅⣀⣴⠟⣋⣏⣩⣲⡖⡒⠚⣉⡭⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⡜⢫⡞⠁⢗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠢⠄⣈⠧⠀⣘⠃⠯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠒⢤⠓⠤⠓⠀⣤⠶⢨⡂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠲⠄⣸⠁⠐⠛⠀⢠⣌⠀⠢⢈⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣼⠓⠤⠣⢤⠛⠀⢤⡤⠁⣙⡃⢦⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⣙⠉⡲⠓⠤⠚⠂⠀⡴⠆⢀⣈⠁⠓⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣑⠦⢀⡘⠓⠢⠞⠢⢄⣀⠓⠂⢴⣁⡙⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠓⡶⢀⡈⠉⠲⠾⠂⠀⢀⡜⠁⠲⡆⢀⡛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢍⠧⢀⡘⠓⠠⣴⠄⠀⠀⣈⠁⠲⠶⢤⣈⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⢄⡠⠆⢀⣠⠄⠀⠀⢨⡇⠁⡒⠃⢤⠌⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠒⠂⢤⠆⠀⢀⡀⠀⠀⢘⡆⠀⣨⠉⢐⡞⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠓⠲⡎⠁⠐⣎⠁⠀⠐⠧⠀⣀⣍⠁⠺⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⠚⠂⠄⠁⠐⠊⠁⠁⠐⡾⠀⢀⣭⠀⠈⠇⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠛⠃⢶⡁⠀⡒⠁⠀⠠⡴⠀⢀⣨⡀⠀⡛⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢈⠓⠀⠦⠄⣠⠋⠐⠀⣴⡆⠀⠀⠥⢀⣙⠓⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡊⠁⡲⢄⣀⡰⠄⠀⢀⠿⠀⣈⠿⠄⣤⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢤⠄⣨⠤⢀⣨⣀⠀⢀⡟⠀⠠⠜⠠⡄⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣈⡁⣀⡧⢀⣈⡁⠀⠈⣏⠀⠐⣖⠓⢠⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠴⠆⠸⠆⢠⡁⠀⠀⣫⠀⣀⠶⢀⣉⡋⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡒⠲⠖⠠⡌⠁⠀⠶⠣⣀⡛⠠⣤⠓⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡁⡘⠂⢰⠉⠁⣲⠃⠠⣬⡁⠲⠆⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⡉⣉⣀⡤⠀⣈⡛⠰⠶⢄⣬⠁⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡾⠴⣇⣰⠏⣁⠉⢠⠒⢤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣬⣗⡯⢶⣭⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⢉⠉⠉⠉⢉⡉⠓⠒⠒⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠰⠶⠀⠰⠟⠀⠀⠀⠛⣋⣀⠰⠶⠆⠀⠀⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠘⠛⠁⢀⡀⠀⣰⡆⠀⠀⠘⠻⠆⠀⠉⠉⠀⠀⠀⠀⠀⠾⠓⢀⣀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢀⡄⠀⠛⠁⠀⠈⠀⠶⠆⠀⠀⠶⠶⠀⠀⠰⠟⠀⣀⡀⠀⠀⣙⠉⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠲⣄⠀⠀⠀⠀⠀⠀⣠⣶⡗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣶⣿⣷⣶⡄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠲⢤⡀
⠀⠀⢖⠒⠂⠉⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⠃
⠀⠀⠀⠑⢄⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⢧⡀
⠀⠀⠀⠀⠀⠙⣦⡀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠙⣄
⠀⠀⠀⠀⢀⡜⠀⠙⠢⡈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠲⣄⠀⠉⠉⠒⠒⠤⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠴⠶⠒⠛⠛⠑⠊⠉⠉⠉⠉⢹⡖⠒⠤⠿
⠀⠀⠀⠀⠈⢳⠦⢄⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢺⡇
⠀⠀⠠⣤⠒⠚⡄⠀⠈⠉⠉⠉⠉⠉⠑⢄⠀⠀⠀⢀⣀⡤⠤⠖⠒⠋⠉⠀⡸⡏⠑⠦⣀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠐⠒⠒⠊⠉⠀⠀⢸⠧⣄
⠀⠀⠀⠈⠉⢲⠞⠑⡖⠒⠒⠒⠒⠒⠒⠚⠛⠒⠛⠛⠓⠒⠒⠒⠒⠒⠒⠉⠀⠈⠒⠒⠚⠛⠒⠒⠒⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⠃⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢪⡀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢻⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠈⢱⠦⢤⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⡭⠭⠭⠭⠭⠭⠭⣭⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣩⣥⣶⣥⠤⠤⠤⠤⠤⢤⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⢀⣠⠤⠔⠒⠒⠉⠉⠉⠉⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⣀⣀⠤⠤⠲⠒⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠠⠤⠤⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠔⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠈⠳⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠞⠀⢠⠀⠈⠀⠸⠓⠲⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠹⣤⠴⢦⣀⡤⠗⠀⠀⣀⣀⣗⣁⠔⠊⠉⠒⢋⠅⠀⠀⢸⣃⡤⡄
⠀⠀⠀⠀⠀⣠⠶⠾⠁⠀⠀⠀⠀⠀⠉⠑⠀⢀⢶⠻⡈⡇⡟⡆⡶⣪⠕⣏⠉⠉⠻⡀⠀⠀⠛⠖⠒⢲⡀
⠀⠀⠀⠀⢀⣸⠦⠀⠢⠄⣄⡠⠔⢀⡤⢚⠒⣚⢧⠻⠅⣻⠮⠧⣅⡟⢡⠛⠀⢒⠤⡀⠀⠀⠀⠀⠈⢿⠛
⠀⠀⠀⢠⡏⠀⠀⢄⠀⡠⡺⡒⣄⠈⠫⠆⣏⣂⢳⡗⠞⣵⠁⡨⢮⣧⢋⠄⠶⢜⢤⡌⢯⢤⣍⢢⠤⣾⡤⠖⢦⡀
⠀⠀⠀⠀⢙⡇⢠⢶⢻⣌⡓⠞⣌⠖⠊⢁⡄⣀⣉⢳⣇⠱⣝⡍⠡⡛⢋⣮⡢⣘⠀⠀⡰⠒⠹⠶⠒⢋⠄⠀⠀⣷
⠀⠀⢀⣠⣾⡟⠃⠈⣮⡑⠷⠊⡎⠀⢙⡅⣤⣖⢒⣬⡣⣑⠀⡝⡏⠘⣿⢵⣌⣲⣝⡫⠔⠉⠑⢂⠈⣉⠥⡤⡶⠃
⠀⠀⢺⡀⠀⠈⠛⠚⠛⠻⡛⡏⢳⡄⢀⡀⣡⡴⣪⢎⣿⡟⢿⡕⡿⣦⢤⢋⢆⡸⢨⢷⣦⠕⠂⢛⠀⡨⡆⢇⡇
⠀⠀⢰⠃⡠⠒⠉⠀⢳⢤⠸⣒⡭⠋⢠⠖⠁⠠⢸⠚⡷⠾⠭⢊⠗⠒⡽⡀⠑⠵⠁⣪⢆⣀⠀⠉⡳⣘⢝⣾⠃
⠀⠀⢻⣀⣠⡄⠀⣤⣤⡄⡙⠢⡇⠸⠁⠀⠈⠞⣁⠤⠘⠒⠺⡿⣛⢭⠫⠙⡔⢒⡈⣝⣁⡈⠛⠭⢔⡵⠋⣸
⠀⠀⠀⠀⠀⣧⡅⠀⠙⠺⢾⣢⠿⣐⡗⣤⣈⠋⠰⢦⠻⡤⠄⣼⡉⢔⡇⠁⠸⠇⣄⠈⠀⠙⡖⠀⢠⠶⣦⡏
⠀⠀⠀⠀⠀⢹⣄⠀⠫⠂⠀⣈⡉⠓⠺⠯⠽⣴⣒⣛⣀⣰⣊⣚⣛⣁⣲⡭⠬⠿⠓⡤⠀⠰⠇⣴⣆⠗⣹⠁
⠀⠀⠀⠀⠀⠀⢧⠉⠂⠤⣀⡘⠁⠀⠪⠆⠀⢠⡆⠀⢀⢤⠀⠀⣤⡀⠀⢔⡄⠀⢻⡁⠀⠐⠀⡨⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠑⠒⠂⠤⠤⠤⠤⠀⠉⠀⠠⠬⠠⠤⠄⠒⠒⠉⠙⣄⣀⠜⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⡆⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣠⡴⣫⢞⡽⠱⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣠⢿⣟⡽⣣⠞⣵⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⣮⡿⡻⢫⠊⣱⢏⡾⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⣰⡿⣛⠞⡳⢁⠞⡡⢗⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠘⠀⣀⣼⢯⣝⢫⠟⠩⡃⣜⣰⡟⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⠀⢀⠿⣱⢏⡶⢁⠬⣑⠮⣝⠏⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⡐⢣⡟⣉⠮⣑⠎⣱⢮⣛⠊⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⠀⣿⠯⡼⢇⡹⢉⡰⢍⡲⢥⡟⠁⠀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⢰⠁⠀⣏⠶⣉⠮⣁⠾⢁⡶⡽⣹⠃⣰⠋⣩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠂⠀⣟⣤⡉⠢⠍⣰⠮⡽⣵⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⢹⢦⣜⢓⣤⡭⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⣦⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠐⠾⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⡉⠛⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⣿⠞⣧⣬⣿⣿⣿⣿⣿⡟⣥⣾⣿⣿⣿⡏⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣌⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣟⠝⢿⣿⣿⣿⣿
⠀⠀⠀⠉⠹⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⣿⣾⣿⣿⣿⣇⠈⢿⣿⣿⣿
⠀⠀⠀⠀⣠⠞⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣈⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⢀⠀⡀⢤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢻⣿
⠀⠀⠀⢸⠋⡰⠋⠰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣈⡈⢁⠀⣀⢀⠀⣀⣀⣪⣿⣿⣷⣾⣿⣿⣿⣿⣿⣯⣟⣯⣿⣿⣫⣍⡛⢽⣿⡤⠄⢸⣿
⠀⠀⠀⠀⠀⠅⠀⠰⢣⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⣿⣿⠿⠿⠩⠷⠛⠋⠉⠁⠀⠐⠃⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⣀⢨⠁⠈⠀⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠿⣿⡎⠛⢹⡜⢯⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣧⢘⠃⢿⠛⣽⠯⣿⡽⡟⠿⣿⢿⣿⣿⡋⣿⠙⣿⡏⠈⠳⠰⠇⠀⠻⡀⠀⠃⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠋⠀⢹⡆⠇⠀⢻⡌⣿⢹⡆⢩⠀⢻⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⠄⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠄⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠄⠂⣠⣦⡀⠠⠀⠀⢀⠀⠀⠀⠀⠂⢠⣤⢴⠵⡀
⠀⠘⣄⠀⠄⠀⡿⠀⠀⠀⠀⠠⡀⠀⠀⠀⡊⠀⠀⠀⢄⠙⠮⣙⠒⠂⠀⠀⠀⠀⠈⠐⢺⠋⠑⠀⠑⠂⠀⠒⠉⠀⠀⠀⠀⣤⣲⡭⣆⣼⣽⣄
⠀⠀⠈⠓⠺⠀⠘⠆⠀⠀⣀⡴⠃⠀⠀⠀⣧⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⠻⠜⡰⢡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⡀
⠀⠀⠀⠘⠠⠟⠄⠐⢂⣠⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠤⠬⣳⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⠀⡜⢀⡴⠒⠀⣄⠀⠀⠀⢰⠀⠃⣀⣸⢼⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠩⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠈⢠⠀⠐⠄⢠⠈⠀⠀⠀⠈⠦⡴⠒⠁⠈⠣⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢰⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⡄⣀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠉⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠉⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⢀⢭⠘⠶
⠀⠻⠶⠄⠀⠁⠒⠦⢄⣀⠀⠠⠀⠀⠀⠀⢀⣀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⢀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠉⠓⠒⠂⠀⠀⠰⢄⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣤⡤⣦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠤⢺⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠐⠒⠛⠃⠈⠀⠀⠀⠀⠤⠀⠠⠜⠀⠃

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⡄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣥⣖⢦⡀⠠⠄⡂⣤⡇⠂⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⡠
⠀⠐⠄⠑⡀⠐⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⡿⣹⠃⠀⠀⢀⠴⠋⠀⠠⡔⠀⠀⠀⠈⠁⠠⠂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠒⡈⠆⠐⡀⠀⠛⢠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠁⠁⠃⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠⠴⠁⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠈⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠃⠀⠀⠀⡄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⢿⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⡄⠀⢀⠘⣇⣠⣤⣀⠀⠀⠀⠀⣇⢼⠆⡇⠀⢸⣇⣀⡾⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠹⡄⢡⠀⠛⠁⠀⢹⡄⢠⡴⠶⠿⠊⠁⡇⠀⠘⢩⡟⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣰⠖⠲⣄⠀⢀⣬⢷⡄⠹⠇⠹⡄⠀⠀⠀⠀⠸⠛⢻⣤⣄⡀⠀⠀⣴⠟⠳⣾⠁⠀⢀⡇⠀⠚⢻⡆⣀⣠⣀⠀⠀⠀⣀⣤⣤⡀
⠀⢠⡶⠲⢶⣿⢠⠄⢹⡾⠋⢳⣄⣀⠈⠉⠀⠉⣠⠾⠛⠉⠳⣴⠏⠀⠈⢷⡀⢸⡇⠀⠀⡀⣄⠀⡼⠀⠀⠀⣼⠟⠋⠀⠈⣧⠀⣾⠋⠀⠀⢹⣤⠶⠛⠳⡆
⠀⢸⡇⠀⠀⠉⠐⢂⠀⠀⠀⠘⢃⣹⡷⠶⠖⠲⣏⠀⢠⡀⠀⠙⠀⢠⣾⠋⠙⣛⠳⢦⣄⠉⠀⢰⠇⠀⡆⠀⠀⠀⠀⣠⠶⠿⠾⠇⠀⡆⠀⠘⠁⠀⠀⠀⣿
⣴⠶⠿⣦⠀⠀⠀⣁⣀⡀⢀⡴⠛⠁⢠⢒⡆⣠⠿⠆⣀⡙⠢⠈⠀⢸⡎⢣⡀⠣⡵⠀⠉⠳⣄⠈⠀⠈⠀⠀⠀⠀⢰⡇⠀⠀⢤⡐⢠⡇⣘⠀⠀⠀⠀⢠⡏
⣿⠀⠤⡤⠀⣰⠟⠩⡍⢻⡟⢁⣴⡦⠈⠉⣠⢿⣄⠀⠀⠋⠀⠀⢀⡀⠻⣆⠙⢦⡀⠀⢯⡷⠘⣧⣠⡴⠶⢒⡶⠛⠛⠛⢠⢠⡄⠁⢸⠀⢠⡠⠄⠀⠀⠙⠳
⠘⢷⣈⠓⠀⣿⠀⠀⢳⢸⣁⣀⣉⣀⠴⠚⠁⢀⡽⠗⠀⢀⡴⠚⠉⠉⠳⡌⠳⢦⣈⠓⠲⠤⠤⣿⠁⠀⠀⢸⡇⠀⠀⠲⣄⡈⠁⠀⠿⠀⠀⢁⣠⠖
⠀⠰⣟⣓⢀⣻⣄⣀⣘⣎⣿⣭⣭⣤⣤⣤⣾⣿⣤⣤⣤⣾⣤⣤⣤⡤⡤⠿⠤⠤⠭⠿⠷⠶⠶⠿⠤⠤⠴⣯⣿⣦⣤⣤⣤⣝⣂⣀⣀⣀⣐⣋⣀⣀⣀⣀⣠
⠀⠀⠘⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡿⠃
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠻⣟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⣷⡶⠶⠶⠶⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⣐⣲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠦⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣠⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢁⡞⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⢸⡇