Water boy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠀⠐⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⡀
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⣤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣠⠊⣀⣸⠥⢀⠔⠉⠉⢢⠀⠀⠀⢸
⠀⠀⠀⢠⠎⠁⠁⠀⢶⡎⠀⠹⠀⢰⠠⠌⠀⢀⣀⠎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡔⠁⡀⠀⠀⠀⠈⠫⠀⠐⠐⠁⠀⠀⠀⠉⢁⠖⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈
⠀⠀⣥⠋⠀⠀⢠⠊⠭⠍⠀⢴⠂⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠸⡀⠀⠀⠀⠢⠤⠔⠱⡘⡧⠤⢒⡉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠈⠹⢇⠀⠀⠀⠀⢀⢥⢿⡏
⠀⠀⠀⠀⠀⠈⠓⠀⠐⠊⠁⢸⢸⠗⡖⡁⡠⢤⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡠⢃⡬⠎⢋⣒⣚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⣁⡤⠟⠘⠡⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣉⠤⠐⠒⢲⢺⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡼⠀⠀⠀⢣⣀⡀⠀⠁
⠆⠈⡁⠀⠎⠉⠉⡰⠑⠢⠱⠤⠤⠤⣼⣿⠐⠲⠄⠀⠒⢠⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⢲⡀⢠⠞⠛⠉⠉⠉⠙⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⣠⠮⢖⡶⠂⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢰⠾⠀⠉⠲⠤⣄⡀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⠓⢦⡀
⠀⠀⠀⢀⣀⠀⠈⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠉⠙⠚⠊⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣖⢽⣦⡀
⢀⡴⠚⢉⣉⡉⢢⡇⢀⡴⠁⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⢠⡀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠈⠳⣌⠁
⠸⣄⠀⠀⠈⠙⠀⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⡏⠀⠀⠙⢦⠧⠀⠀⠀⠀⠀⠉⢳⡦⣄⡀⢶⡤⣀⠘⢧⡀
⠀⠈⠉⠉⠉⢹⡀⠀⢠⣶⣿⡏⢹⡄⠀⠀⠀⣇⠀⠀⠀⠈⣧⠀⠀⠀⠠⠔⠒⠀⢳⠀⠉⠓⠳⠈⠙⠦⣽
⠀⠀⠀⠀⠀⠈⢧⠀⠋⢻⡋⣧⡎⠱⡄⠀⠀⢹⡀⠀⠀⠀⢸⣦⠔⠒⠋⠙⠛⢲⡈⣇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠹⣅⣱⣎⢹⢦⡀⠀⢧⣀⠀⢀⣸⠓⠀⠀⠀⢀⣠⠞⠀⠸⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄⠀⠈⠳⣼⠓⢄⡽⢦⡀⠈⠙⢾⠉⠑⠒⠒⠚⠉⠀⠀⠠⠴⠛⠉⠀⡳
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⡳⣄⠀⠀⠑⠮⡙⢢⣏⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠴⠚⠁
⠀⠀⠀⠀⡰⠋⠀⡴⠀⠀⢳⠈⠳⣄⠀⠀⠈⠙⠦⣽⡓⠤⣽⣛⡒⣲⠄⢤⠔⠒⠀⠀⢸
⠀⠀⢀⡞⠁⠀⢶⠇⠀⠀⠀⢣⡀⠀⠙⠢⢄⣀⠀⠀⠉⠙⠓⠦⠼⣏⣹⡋⠀⠀⠀⠀⡼
⠀⢠⠏⠀⠀⠀⡾⠀⣠⣤⣶⡲⠗⠀⠀⠀⢀⡟⢏⠓⠲⠤⠤⣄⣀⣀⣀⣁⣀⣠⠴⠞⠁
⢠⠏⠀⠀⠀⢠⣷⣋⠽⣿⠋⠑⠦⣤⢎⣵⡚⠛⠉⠛⢲⣩⣽⡀
⢯⡀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠲⢤⠀⠀⠤⣉⢳⡄⠀⠘⢧⠭⣛⡷
⠀⠻⡑⠲⠤⣄⣀⡀⠀⠀⠀⠀⡞⢧⡈⢢⣈⣿⠁⠀⠀⢀⡠⠼⡀
⠀⠀⠘⢦⡀⠀⠈⠉⠉⢉⡓⢺⡇⡇⠉⠉⠈⠀⠀⠀⢰⠚⠦⡤⠃
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠻⢇⢳⡀⠀⠀⠀⠀⢀⣾⡙⠛⢧
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢈⠧⣝⠢⠀⠀⠀⠀⢸⠉⠒⠋
⠀⠀⠀⠸⣛⠯⣷⣒⠲⠦⡤⠖⠛⠶⣬⢿⠶⠦⢤⠖⠋⣤
⠀⠀⠀⠀⠈⠙⠒⠭⣉⣛⣓⣲⣾⠿⠽⢾⣶⡶⠟⠀⢸⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠛⠛⠛⠛⠻⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠴⠶⠶⠲⠀⠀⠀⠀⠛⢛⣋⣍⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣠⡴⠺⠿⠿⠆⠀⠀⠀⠋⠃⠀⠉⠉⠛⠂⠀⠀⠀⢸⣿⣿⣿⠉⢳⡌⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⠈⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠶⠈⠐⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⣿⣿⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣆⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠐⠒⠛⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠐⠾⠿⠶⠒⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢫⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣏⣵⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢱⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠉⠻⢴⡀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⢸⣿⡞⢳⡆⠀⢀⣀⡤⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢰⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⢀⣠⣾⣿⣿⣿⣿⠿⡿⢿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢠⣾⣿⢿⠿⢿⢿⠃⠀⣀⡈⠉⠉⠀⣿⣿⣿⠿⠿⣿⣿⣿⣿
⢀⣿⣿⠏⠀⠀⠠⢦⣴⡾⠻⠟⢷⣄⣠⣿⣿⠇⠀⠀⠀⠀⠀⡿
⢸⠋⠀⢀⣀⣠⣄⣴⣦⣴⣶⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⢱
⣿⣀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⣇
⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⢀⣁⠀⠀⠀⢠⣾⠴⠚⠓⢄⡀
⠀⢻⣿⣿⣿⡿⠿⠿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠚⠙⠛⠀⠀⢀⠜⠁⠀⠀⢊⡦⠙⢦⡀
⠀⠀⢻⣿⣿⠠⠖⠒⢦⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠊⢀⡊⠀⠀⠒⣬⠲⣤⢤⣀⡀
⠀⠀⠀⠻⣿⡄⠀⠐⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠂⣰⠋⠀⡤⠂⠉⣀⠾⡈⠑⢄⡈⠳⣄
⠀⠀⠀⠀⠈⠻⣦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣼⠀⠀⠀⠈⠀⠀⠊⢁⡴⠁⠀⠀⠙⠦⡘⢧
⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⢀⣼⣮⣣
⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠉⠙⠻⠷⣤⣄⣀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠛⠉⠉⠙⠲⢦⣄⡀⠀⠈⢉⣻⣿⣶⡶⠞⠁⠀⠀⠀⠀⠀⣴⣿⣦⣄⠀⠀⠀⣸⠋⠀⠀⠀⢀⠇⠘⢆
⠀⠀⠀⢰⣿⠴⠚⠛⠲⠀⠀⠀⠈⠛⢦⣤⡿⠟⠛⠁⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⢿⣿⣷⣦⣀⠃⠀⠀⢀⡴⠃⠀⠀⠈⢇
⠀⠀⢠⡟⢁⣠⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠙⠟⠀⠘⣿⣿⣿⣿⣷⣶⣶⣿⠀⠀⠀⠀⠀⠘
⠀⠀⣼⡶⠋⢀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⢿⣧
⠀⣰⠟⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢻⡅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠙⢷⣄⠀⠀⢀
⣰⠏⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⣛⣻⣿⣿⣿⡿⠿⠿⠿⠿⠛
⢿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠁
⠈⢷⣄⣿⣷⣦⣀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠹⣿⣿⣿⣿⣿⣷⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡿⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⣸⠇⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣰⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠⢴⣶⠾⢿⠏⣩⣿⣿⡉⠉⠉⣡⣄⠀⢲⣦⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢿⣿⠄⢻⣟⣟⣠⣘⠛⢛⡋⠀⠉⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⢋⢹⠆⠷⣿⣿⣿⣿⣷⢆⢰⣷⣴⡿⢻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⢬⢡⣾⣿⣿⣿⣿⣿⣿⣭⣫⣾⣯⠛⣻⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣿⣿⣿⡙⣿⡿⢻⣿⣿⣷⣿⣿⣿⣽⢦⣙⣿⣵⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢟⣽⣿⣿⣿⣿⣴⣿⣿⣋⣽⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠉⢙⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⢿⣿
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢁⣬⣷⡿⠟⠛⠉⠉⠙⠻⣿⣭⣷⣶⣿
⠀⠀⣼⠣⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠶⣴⣤⣶⣤⠠⢀⣤⡔⣤⡀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣤⠙
⠀⣸⣭⣷⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⣬⠉⠁⣴⣿⣿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣽⣦⣀⣴⣤⣿⣀
⠀⠋⠉⢈⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣄⠀⡀⣆⠀⠀⠀⣉⠈⠀⢿⠟⠛⢛⡻⠧⠀⢀⠀⠀⡀⠀⢀⡀⠀⠂⡀⠀⠑⠀⠉⢁⢈⡀⠀⢸⡟⠇⢬
⠈⠙⠩⠙⠁⠐⠉⠀⠀⠠⢿⣿⣿⢿⠿⡰⢢⡀⢂⣠⠀⠨⣦⣤⡀⠀⠳⢦⠀⠀⠀⢈⡆⠀⢌⠀⠛⠗⠀⠀⠠⠄⠀⢠⣾⣿⠶⣿⠆⠚⢻⣦⣤
⠀⠀⠀⠈⠀⠀⠀⠀⠐⠃⠀⠙⡇⠀⠄⢡⠞⠉⠛⠉⠀⠀⠘⠂⠀⠀⠀⠀⠑⣄⡀⠀⣳⣤⠀⠂⡠⠰⠆⢀⡀⠐⠀⠀⢿⠛⢄⡩⠆⠀⠀⣧⠌
⠀⠀⠀⠀⢀⣤⣤⡄⢐⠀⠀⡘⠈⢤⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⢿⣿⣦⠀⠁⠀⠀⠀⠀⠀⠉⠀⡵⠀⠀⠀⠠⠐⠟⠿⢶
⠀⠀⠀⠀⢸⣿⣿⡍⢰⡷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⢈⣿⠟⠀⠀⠀⠐⠾⠆⠀⠀⠈⢀⣍⣀⣠⡀⠀⣵⡀
⠀⠀⠀⠀⠀⠹⣿⡆⢈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣇⠀⠈⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠐⠏⠱⠶⠟⢿⣹
⠀⠀⠀⠀⠀⠀⠀⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⡀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣵⣶⠶⠤⣴⣖⣚⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣦⣢⣾⣿
⠀⠀⠀⠀⠀⢶⣶⣧⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣎⠻⢷⣯⣿⣏⣉⠡⣄⣠⣤⣀⣀⣠⣀⣀⠛⠙⢲⡟⠗⠒⠋⠉⡉⠙⢛
⠀⠀⠀⠀⠀⠀⠉⠋⠉⠻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣶⣦⣴⣦⣈
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠙⠿⠛⠛⠛
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠻⡿⠿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣻⠃⠀⠀⠀⢠
⠀⠀⠀⠂⠀⠚⠛⣶⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⠀⠠⣤⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⡀⠀⣀⣀⣀⣀⣀⣠⣾⠟⠛⢹⣿⢈⡽⣽⣿⣿⣾⣿⣿⣿⣿⠉⠉⢰⡦⠴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲⣶⣮⣿⣿⣿⣿⣯⣵⣾⣿⣼⣿⣿⣿⣿⣿⡟⣽⠀⠉⠀⠀⢠⣿⢿⣿⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⣄⣀⣋⣭⠿⠯⣐⣒⣸⣿⣿⣿⣿⠟⢿⡏⠥⠘⠟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠭⢭⣽⣿⣿⣿⣿⣿⣿⠉⣄⢹⡗⠀⢀⠀⠀⠀⣀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣔⣒⣷⠖⣂⣤⣦⣀⣴⣤⣽⣾⠟⠟⢛⢿⠿⠻⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠯⠟⡓⠎⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠈⠋⠉⠉⠉⠉⠙⠛⠛⠫⠴⠒⢤⣄⡀⠘⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣿⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣀⣤⠶⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⢤⣄⣀⡀⠀⠀⠙⣇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀⣠⣇⡀⠄⠬⠥⠚⠶⢦⣬⣎⡆
⠀⠀⠀⠀⣀⣤⠤⢴⠿⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⣫⣭⠅⠶⠖⠒⠂⠉⠉⠉⠤⠒⠲⣬⡙⢦⡀
⠀⠀⠀⢰⠁⢀⢴⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⡩⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣦⢱
⠀⠀⠀⠘⠖⠁⢰⠀⠀⠀⠀⠀⠀⠀⣠⢋⡴⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⡸⠠
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⡋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⠂⣤⣀⠈⠉⠀⠀⢸⠇⣼
⠀⠀⠀⠀⠀⠀⢘⡄⠀⢀⡴⠋⠀⡷⣶⠸⡀⠀⠀⠀⠀⠀⠰⠒⣢⠞⡡⠊⠁⢀⡸⠦⣭⡐⢒⣊⡥⣾⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢣⡠⢞⡴⠚⠚⢛⠸⣧⡙⢄⡀⠀⠀⠀⣠⠞⣡⠞⠁⠀⠀⠈⢁⡀⠀⠀⠈⠀⠀⣾⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⡇⠀⠀⠀⠀⠈⠉⠲⢭⡭⢩⡭⠔⠊⠁⠀⢶⠉⠁⠀⠀⢙⠀⠀⠀⠀⢠⠇⢾⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⡀⣨⠀⠀⠀⢠⠇⠀⣸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠈⠀⣠⣔⣡⡤⣴⣃⡜⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠑⢏⣀⣀⠜⢋⣀⡠⠴⠊⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠙⠢⠤⢤⣍⣑⣒⣒⠒⠒⠒⠒⢲⡖⠂⠉⢫⡀⣀⢀⣤⠤⠖⠒⠉⠉⠁⠀⠀⣀⣤⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣤⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠉⠻⣶⡶⠷⠶⠶⣶⠦⠦⣴⣤⣤⣀⣨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⣆⠀⢀⣠⣀⠀⠀⠀⠀⢰⡏⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⠺⣇⣀⠈⠀⣀⠀⠀⠀⠀⣀⣈⠠⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⡼⠀⠀⢰⢧⣠⠋⠉⣀⣐⣃⣐⣀⠀⠘⠁⠀⠀⠒⠁⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⢀⡏⠀⠉⠉⠉⠉⠉⠉⠉⠉⢧⠀⠀⠀⠀⠀⠀⠄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣤⠜⠀⠀⠀⠀⠀⠀⢰⡆⠀⣀⣀⡳⣶⢤⣤⣤⣥⡤⠴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡞⠁⠈⠁⠛⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠢⠀⢸⠀⡰⠓⠶⢤⣤⣠⡴⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⡀⠀⠀⠃⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⢀⠍⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠂⠀⠀⣀⢃⡀⠀⠀⠈⠲⣄⡇⠀⠀⠀⣤⠀⠀⣀⣤⣀⠤⠤⣤⡄⠒⠉⠀⢀⠠⠦⠤⠤⠄⠀⢀
⠀⠀⠀⠀⠀⠐⠶⠀⣀⡀⠀⠀⠀⠀⢹⣉⠉⠉⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⢡⠄⠂⠀⠀⣰⠆⠂⠀⠀⠀⠠⣔⣁⢀⡤⠄⢤⡄⠶⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⣈⠉⣉⡀⠐⠒⠒⠒⠒⠠⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠖⠶⠄⠉⠁⠀⠀⠀⣀⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠠⠤⠤⠀⠀⠀⠀⠐⠒⠒⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢀⡡⠔⠈⠉⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡠⠚⠁⠀⠀⠀⣀⡀⠤⠔⠒⠂⠀⠀⠛⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠑⢤⣇⠞⠀⠀⠤⠐⠂⠉⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠀⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠉⠐⠀⠀⠠⡴⠦⠤⠤⠄⠒⠊⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⢄⡠⢤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⡌⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⠈⠑⠒⠐⣤⣤⣴⡧⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢀⣀⠀⠀⡇⠀⢸⣿⡿⠦⠒⠀⠀⠀⠀⠀⢀⠾⢂⣀⣀⣄⡠⢿⡿⠿⢧⣴⣇⣰⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢒⠈⠉⠉⣈⠥⠒⠉⠀⣀⡤⠀⠚⠻⣷⣶⠀⠀⠁⠉⣹⠁⠀⠐⣇⠀⠀⠤⢻⠛⢷⣞⣡⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠔⠋⠀⠀⢀⠔⠊⠙⠣⣀⣀⣤⣿⠏⠀⠀⣠⠞⠁⠀⠀⠀⠈⢑⡆⠀⠊⣀⠀⣸⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⡠⠚⠧⣤⣀⣀⡀⠀⠈⠉⠀⠀⠐⡫⠥⢄⡀⠀⠀⠀⠀⡼⠀⠀⢠⠋⠉⠁⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⡰⠋⠀⢀⡠⠤⢄⣀⠀⠑⠀⠀⠀⠀⠋⠀⠀⠀⠈⠑⠒⠉⠉⠱⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡠⠊⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⡧⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠞⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣰⣦⣀⠀⣸⣄⣀⣀⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣤⡿⠛⢿⣶⣾⠟⠉⠻⢿⡟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠋⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⠲⢍⡑⠚⠀⠀⠀⢸⠒⠒⠢⠤⠤⠤⠤⠤⠄⠒⡢⠟⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣯⠒⠀⠀⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠤⠤⠴⠒⡀⠀⠀⠀⠀⠀⠀⠨⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡴⢄⣄⡀⠀⠀⠀⠀⣀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣸⡦⢄⣀⡀⠀⠀⠀⠀⣀⣀⣀⢴⣛⣒⡶⠶⢖⠂⡞⠉⠉⠁⠈⠀⠈⣱⣴⣶⣿⣿⣿⠿⠛⠁
⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠉⣉⠝⠁⢀⡠⠃⠀⠀⠀⠀⠀⣀⣯⠀⡀⠤⠤⠄⢒⣨⡽⠿⠟⠋⠉
⠹⣿⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣉⣛⠛⠒⠒⠒⠃⠁⠀⠀⠒⠒⢂⣋⠩⠔⠓⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠑⠶⠊⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⡼⢇⡀⢀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⡸⢓⣶⣖⡉⠀⠈⠉⠁⠀⠀⠉⠙⡇⠀⢀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣴⠓⡾⠑⠋⡱⠁⠀⣀⣈⣆⠀⠀⠀⢀⠔⠒⠒⢌⠐⣮⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣇⠀⠀⠱⡀⣼⣿⣿⡟⠀⠦⡄⢍⠀⣠⣶⣶⣷⡧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⡛⣲⡄⠀⠈⢙⣏⣩⡴⣤⡴⠇⣌⠲⠿⠿⠟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢻⣭⣍⣉⣉⡒⠾⢤⣍⣉⣟⣁⣡⠟⠀⠀⢀⠞⠀⢀⣀⣀⣀⣀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⡿⠤⠤⢸⠟⠬⣍⣑⡒⠬⠟⠀⠀⣀⡠⠖⠋⡏⠉⠉⢁⣀⣀⣼⢁⡠⣝⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⠁⠀⢠⠏⠀⠀⠀⠙⠭⠭⠛⠉⢩⠁⠀⠀⠀⢸⠒⠋⠉⠁⠉⠛⢺⣿⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠤⠒⠊⠀⢀⡔⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣟⣒⣒⣚⡇⠀⠀⠀⠀⠈⠱⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⠉⡏⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠒⠶⠤⢄⣀⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⠋⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡎⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⠎⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠎⠀⠀⢸⠀⠀⠀⢰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⣤⣤⡎⠀⠀⠀⢸⣀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⡘⠓⢦⡀⠀⢸⣁⠈⢉⣙⡓⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠈⠉⠙⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⢰⣿⣦⠀⣰⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠚⠛⣿⣛⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠤⠤⠂⠀⠲⣟⠻⣿⡿⠿⣉⠐⠒⢦⣝⢿⣿⣿⣿⢸⣿⠇⠳⡄
⠀⠀⠀⠀⠀⠀⠒⢲⠠⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⡁⠉⠀⢨⣉⠉⠉⠙⠟⠿⠏⢛⢣⣤⣠⣿
⠀⠀⠀⠀⠀⣿⠀⠸⣄⡙⠟⠛⢿⣧⣄⢠⣄⣠⣼⠇⠈⠁⠀⠈⠳⢦⣤⢽⣔⡄⠈⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠶⠾⠻⠛⢋⠲⣌⠻⠿⢞⡉⠉⠀⠀⠀⠈⣿⠶⠖⠀⢸⣿⣷⠈⡽⣹⣿⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠑⠌⠛⠛⢋⣠⣤⡀⠀⠀⠀⠈⢑⣠⡄⣼⣿⣿⡆⢸⣿⡟⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠘⣤⠲⣦⠀⠀⠀⠀⠀⠘⠛⠁⠈⠉⠀⠀⠀⠉⢉⠉⡰⠃⣽⣿⠇⠸⣿⣧⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢯⣀⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠻⠞⠧⡴⢛⣿⠈⠳⣌⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢱⣿⠀⠀⠀⠀⠀⠁⣿⡿⠆⠀⠀⠀⠀⠀⡟⠁⣿⣾⠇⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡽⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠆⠐⠆⠑⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣋
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠃⣀⣤⣤⣄⣀⣀⡤⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⣧⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⠿⠿⠋⠀⠀⠀⢠⡔⠒⠉⠉⠉⠉⢉⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣹⠉⠙⠛⠛⠛⠋⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠠⠶⠶⣏⠘⠀⠀⠀⠀⠀⣀⠀⠀⠗⠚⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠈⢻⡅⣀⣙⠀⠏⣩⡟⣁⢀⡴⠋⠁⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡆⠀⣖⡚⠇⢹⣿⣇⠀⠁⣛⡟⢃⡴⢊⡴⢧⠀⢀⣠⠞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡾⠁⠀⠀⠙⣿⣾⣿⣿⡆⠘⣻⣶⣫⡴⢋⣤⡾⠶⣫⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄⢀⡾⠁⣿⠀⠀⠀⠀⠀⢨⣿⣿⣭⠀⣀⠀⣻⣟⠶⠋⣡⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⡾⠁⠾⣧⣀⡀⠀⠀⠀⠈⣟⣛⣋⠀⡈⢙⡛⡋⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠀⠀⠠⣬⣭⣿⡄⠀⠀⠈⣿⣿⠾⠿⠓⠾⢷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠰⢤⣈⣹⠀⠀⠀⠙⠿⠶⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣀⣀⣀⣈⠉⣳
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣰⡏⠀⢈⣥⠞⠋
⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣴⠟⠀⠀⠈
⠀⠀⣴⢶⣯⠀⠀⠀⢶⡟⠁
⠀⠀⢰⡞⣻⣷⣀⠀⠘⠃⣶⡄
⠀⠀⠀⠉⣅⣻⡿⣿⢋⣼⣏⠁
⠀⠀⠀⠀⠙⠉⢠⣿⡆⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠛⠛⢛⡿⠛⠛⠛⠛⠒⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢷⡀
⠀⡼⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠘⣧
⢰⠇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⢹
⣼⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈
⡟⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣠⠴⠖⠒⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⢀⣀⡤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢳⡀⢀⣀⣀⣠⠤⠖⠚⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⢻⡉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠉⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠖⠒⠒⠚⣹⠋
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⠂⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀⣴⣿⣿⣿⣿⣿⡀⠀⣹⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣆⠂⣼⣿⣿⣿⣿⣿⡯⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⡟⣡⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠋⣙⠻⣿⣿⣿⣿⠟⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠁⠉⠛⠋⠀⠀⠀⠀⠀⠈⡙⢂⠆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣈⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠈⠉⠉⠉⠀⠀⠀⣿⠿⠿⠋⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠂⢐⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣐⡞⢤⡤⠶⠀⠀⠀⠀⢸⢇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⢄⡀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢱⣦⣄⣀⣠⡴⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠋⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠀⠀⠒⠂⠀⠒⠀⠄⡀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡄⠀⢀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⣠⣿⡿⠀⠀⠸⣿⣿⡇⠀⠀⠀⢤⢾⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⣿⡿⠀⢀⣴⣦⡄⠀⣴⣶⡄⢠⣄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⢣⣤⣄⡀⠀⠀⠀⠈⠻⠿⠟⠀⠻⡿⠇⠘⢿⡇⠀⠐⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⠿⠿⠃⠒⠈⠀⠈⠋⢠⡄⠀⠀⢠⣶⣄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠋⣆⣰⡎⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠸⣿⠟⠀⣾⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⢠⣴⣤⠀⠙⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻⠿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢀⣤⡶⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⣰⡄⠉⠁⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠿⠇⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣦⣶⣴⣶⣦⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠀⠒⠒⠀⠀⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣩⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⢀⠀⠿⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠈⠐⠠⢄⣀⣀⣀⣴⣾⣤⡀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠚⠛⠿⠿⠟⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⠒⠒⠒⠒⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⢭⠀⢸⠀⣂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠭⠽⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠚⠲⣅⡀⠘⠀⢹⡀⠀⠀⣠⠖⠈⢁⠤⠀⢀⡀⠀⠫⠥⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠊⠀⡔⠁⢠⠂⢀⠜⢳⡴⠚⣓⠒⡚⠯⠁⠀⠠⠏⠤⠂⠁⠀⠀⠤⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠁⠀⡜⠀⢀⡡⠤⠓⠒⡏⠰⢇⣀⠤⠊⡠⠃⠈⣱⠖⠒⠦⢄⡐⠏⣁⣈⡄⠀⢴⠺⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠀⠀⠀⣠⠗⠋⠁⠀⠀⠀⠀⡸⡖⠒⠒⠒⠉⠀⠠⠊⠁⡠⠊⠀⠀⠉⠓⢤⡀⠀⠀⠀⠈⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢰⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⡠⠿⠞⠀⡀⢂⠀⠈⡀⠠⠐⠉⠀⣀⠔⠀⢀⠄⠀⠙⣦⡠⡀⠑⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⡴⠋⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢧⡣⣣⠂⢀⡀⡀⠤⠊⠀⢀⠔⠀⠀⢀⠞⠀⠱⣄⠀⠃⠀⢠⢌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⠡⢢⢀⠀⠀⠀⠀⠀⢸⠡⠀⢀⡠⠒⠁⠀⡠⠒⠁⠀⠠⠀⠘⣦⠀⠀⠈⠢⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠢⠷⠊⠀⢀⠄⠀⠀⠀⠙⢃⠀⠀⠀⠐⠉⠀⢀⠴⠊⠀⢀⠄⠈⢷⠀⢠⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠀⠀⠀⠀⠀⠀⢄⠠⠅⡄⠀⠀⠸⠆⠀⠀⡠⠔⠊⠁⠀⡠⠖⠁⠀⡠⠘⡇⠄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠩⡅⡀⠀⠀⠉⠉⠀⠀⡄⣒⣗⠈⠁⠀⢀⡠⠔⠉⠀⢀⠴⠊⠀⠀⣷⠀⢠⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠳⠭⠵⠁⠀⢀⠄⠄⠀⠀⢧⣓⣻⡄⠀⠀⠁⠀⢀⠤⠚⠁⠀⠀⣠⠖⠸⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀⠀⠀⠀⠀⠀⠀⡇⢈⠆⡆⠀⠀⠀⠀⢇⠀⠠⠐⠈⠀⠀⠀⡠⠔⠊⠀⠀⢠⡆⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘⠷⠃⠀⠠⢐⣂⠀⠀⠈⠑⠉⠀⠀⠐⠩⢅⢺⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠔⠁⡏⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⣔⣊⠔⠀⠀⡀⠄⠀⠀⠘⠬⠥⠊⡇⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⢀⢴⠓⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢈⡡⣹⢈⠀⠀⠀⠀⠀⣃⠀⠀⠤⠐⠈⠁⠀⠀⡠⠒⠁⡜⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢣⠀⡀⢒⡂⠀⠀⠀⠑⠒⠉⠀⠀⠀⢊⠭⣗⠀⠀⠀⢀⣀⠤⠒⠉⠀⡀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⣆⠣⣕⡢⠆⠀⠀⠀⢀⠠⠀⠀⠈⠢⠭⢾⠒⠒⠈⠉⠀⢀⡠⠐⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⢎⡄⡀⠀⠀⠀⡌⠀⣀⠤⠐⠊⠁⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠈⠳⣔⣀⠀⠀⠀⠈⠒⠊⠀⠀⢠⢊⠏⠁⠀⣀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢽⣶⣄⣀⣀⣠⣀⣀⣀⣈⣮⡥⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠉⠁⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⢀⣀⡀⠀⡠⠊⣁⣀⣀⠀⠐⠒⠤⠤⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⡥⠄⡶⠉⠀⠢⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢉⠄⠀⠀⠀⠈⠑⢍⠁⠒⠠⣀
⠀⠀⠀⠀⠰⢡⠀⠀⠀⠙⡆⠀⡗⠒⠠⠴⠤⠤⠤⠤⠽⠉⠐⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠖⠠⠃⠀⣴⠂⠀⠀⠀⠀⠑⡀⠀⠈⢳⣄⣀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣂⢀⡁⢀⡌⡇⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢯⠉⢆
⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠁⠀⢀⠖⠒⠒⠀⠓⠀⠉⡏⢑⡄⠀⠀⠀⠀⡍⠉⠀⠀⠀⠀⠈⠙⠃⠒⠀⢤⢀⣸⠀⠀⠀⠀⣼⠀⢨
⠀⠀⠀⠀⠈⠆⠀⠀⡜⠉⠱⠉⠀⠀⠀⠀⠀⠀⡀⢼⠀⠐⣆⠀⡤⠴⠾⣏⡁⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠰⢄⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠈⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⣆⠬⠁⣰⡇⠙⠢⠴⡖⠁⠀⠀⠀⠘⠈⠀⡄⠀⠀⠀⠓⠀⠀⠀⠀⠘⠤⠀⢄⠀⠀⠀⡌⢰
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢚⡩⠋⠉⡆⠀⠸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠜⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⡄⠀⠀⠲⡒⠉⡇⠀⢀⠜⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⢄⡠⠐⠈⠉⢘⠅⠀⠀⠀⠀⠀⠀⠀⢈⠦⠻⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠀⢤⣿⡦⣤⠴⠋⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠎⠀⡠⠶⣶⢻⠀⠀⠀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠒⠢⠼⠛⠉⠱⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠈⡎⠉⣿⠁⠀⠀⣸⠦⡀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢰⠁⠀⠀⠀⠀⠀⡸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⣇⠀⠘⠳⠶⠚⠋⠀⠈⢢⠀⠀⠀⠑⢄⡀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠄⠀⠀⠒⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣼⠀⠀⠀⠀⣀⣀⠀⠀⢡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⣂⡀⢰⡀⣀⠤⠚⡅⠀⠀⡠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣟⠀⠁⠀⠀⠀⠀⣀⠬⠦⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡨⠋⠉⠘⠃⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⠶⣐⣂⣙⢲⣖⣖⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠈⠀⠔⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠒⠀⢠⢠⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣉⣉⡉⠈⠉⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠸⠖⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢀⣀⢀⣀⣀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣈⢉⣉⣀⡀⠬⢥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⢆⡀⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠤⠴⠃⠀⠀⡗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠄⠤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠴⠶⠶⠶⠶⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳⡄
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⢳⡀
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣀⣀⠀⠀⡾⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠋⠉⠉⠉⠻⢧⡀⠀⠀⠀⠀⠀⢳⡀
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⣇
⠀⣸⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢹
⠀⣿⡤⠤⠖⠒⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠒⠶⢤⣸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢙
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⢸
⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⠉⠙⠉⠙⡆⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡟
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⠃
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢠⠏
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠏
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣟⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⡀⠀⢀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠈⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠐⠁⠀⠢⢀⠀⠀⠀⢀⠠⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⣿⡿⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⠇⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠟⣋⣹⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⠛⠻⠛⠛⠿⠿⠿⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣥⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⢀⣤⠶⠖⠂⠀⠈⠻⣿⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠛⣳⠀⠀⠰⠋⠙⠃⠀⠀⠘⠟⢡⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠳⠖⠂⠀⠠⢄⡀⠀⠀⠀⢰⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⡶⣶⡄⠖⠂⠀⠀⠀⠀⡼⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠶⠆⠺⣋⣵⠞⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢀⠀⠀⠒⢆⠙⢦⡴⠞⢁⣠⣼⠋⠙⡀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠟⣧
⠀⠀⠀⢂⡤⠤⠝⣳⡦⠌⣈⣤⣶⣿⣿⠟⠀⠀⠈⡿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⣠⢒⣥⣤⡀⢞⠁⣠⣾⣿⣿⣿⠁⠀⠀⠀⢀⠞⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠢⣄⡀
⠀⡇⣟⣽⣛⢿⡆⢿⣿⣿⣿⠟⠃⠀⠀⣀⡤⣾⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢠⠃⠀⠀⠉⠉⠓⠲⠆
⠀⠹⡝⠛⢫⣼⢇⣿⠟⠋⠀⠀⢀⠐⠏⠀⠀⠸⠀⠀⣡⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⠃
⠀⠀⠘⠻⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠙⢦⠀⠀⠁⢷⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠃⠀⠀⠈⠳⠤⠈⠁

⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣚⡩⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢀⡸⢟⣡⠶⠒⠛⠚⠛⠛⠒⠒⠒⠒⠒⠀⠀⠀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⠙⠓⠀⠀⠶⠶⠿⠷⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣀⡀⠀⠀⠀⠀⠀⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠉⢉⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠋⠀⠘⠁⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⠀⠀⣰⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡼⠀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡒⢠⡏⠀⢀⠀⡼⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⣳⡀⠉⠀⢠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⠛⠛⢉⡇⠀⡀⠀⠀⢀⠀⠉⠛⠓⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡑⢿⠁⠀⠀⠀⠀⠘⠁⢠⠇⠀⠸⠋⠀⠀⠀⠀⠀⢐⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠙⠓⠒⠶⠦⠤⠠⢤⣤⣤⣤⣤⣤⡤⠤⠴⠶⠚⠛⠉⠁⢸⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⢾⠿⠆⠀⢠⡄⠀⠀⢀⡀⠀⠀⠀⠀⣤⢶⣦⠀⢸⠁⢈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⣖⠛⠺⢇⣀⣀⡀⢳⡄⠁⠘⠳⡄⠀⢀⣠⣀⠈⠁⠀⢸⣀⡽⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⣃⣤⣀⠀⠉⠀⠟⢀⡇⢸⠃⣸⢃⡼⠉⠐⢹⠂⠀⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣞⣡⡄⠈⣧⠀⠀⠠⠞⠁⠀⠀⠟⣸⠇⣠⠞⠃⠀⣤⠀⢸⠇⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠈⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠤⠞⠁⠀⢸⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢷⠀⠀⠀⠀⠀⣰⡿⠁⠀⠈⠋⠉⠩⠽⠿⠍⠀⠀⠀⠀⠀⠀⣶⠀⢸⡷⠷⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⣠⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⡟⠳⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣼⠃⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢸⣷⣤⣤⡾⠃
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡟⠀⠀⣣
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠤⠴⠃
⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿
⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠁⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠉⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠁⠀⠉⠙⠒⠲⠤⣤⣬⣭⣭⣭⣤⣤⠤⠤⠔⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡁⠶⠂⠉⡑⡖⠒⠠⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠋⠈⢄⢣⡀⠇⡇⢀⡠⢊⠦⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣀⣀⣀⣈⣧⠙⠠⠖⣩⠖⠁⠀⢱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠐⠒⠀⢠⡄⢀⡀⢈⡉⣀⣐⣒⣚⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⢤⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣰⡀⣀⣴⢾⠶⠋⣽⡌⠣⡀⠉⠉⠹⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠍⠣⡉⠔⠀⠀⠀⣿⠈⠣⡹⢄⡰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠐⠀⠀⠄⣹⡀⡠⠜⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠶⢤⣤⣠⣤⠤⠶⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠦⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⠀⠀⠀⠀⠀⢀⡠⠔⠈⠉⠑⡉⠒⠢⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠈⣳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠐⠁⠀⣠⠴⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠃⠀⠈⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠐⣬⠒⠒⠒⠒⠒⠒⢦⡀⠀⠀⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠀⠈⢆⢄⡀⠀⢀⡠⠔⢸⠀⠀⢠⠀⠀⠀⠀⠘⢆⠀⠀⠀⣼⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠈⠢⠤⠚⠉⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠀⠀⠀⠀⠀⣠⠒⠢⢄⡘⢠⣀⠈⠀⠀⠀⠀⢀⢸⠀⠀⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠈⠙⠷⠄⠀⠀⠀⠀⠊⠜⠀⠀⢸⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢱⠀⠀⠀⠀⡐⠁⠀⡠⠉⠐⠢⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢃⠆⠀⠁⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢻⠀⠀⠀⠀⠀⠉⠂⠄⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠦⣈⠉⠙⠛⠛⠛⠛⠛⠋⢉⣁⠴⠖⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠋⠁


⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣬⣭⣭⣉⣍⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⣦⣤⣿⣿⣿⣭⣿⣿⣽⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠋⠉⠉⠉⠉⠉⠙⠛⠒⠒⠶⠤⠤⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣀⡀⠀⣠⠔⠒⠂⠉⠉⠉⠉⠑⣒⣤⣀⠀⠀⠀⠈⠒⠤⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠜⠁⠉⠉⣠⡤⠄⠀⠀⠀⠀⠀⠀⠈⠙⢧⠑⢦⣄⣀⠀⠰⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡌⠀⠀⠀⢈⠄⠐⠠⡀⠀⠀⠀⡠⠄⠀⢄⡀⠀⠀⠀⢫⢹⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⡇⢀⣴⡆⣘⠀⠀⢸⠀⣴⡖⡀⢱⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠳⢄⣙⠿⠋⣀⣀⡘⢦⣙⣛⣡⠊⠀⠀⠀⡼⡸
⠀⢀⡴⠚⠓⢲⠦⣀⡠⠞⠓⢦⡀⠀⣠⠤⠼⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠠⢶⠂⠀⠀⢰⣷⣁⣀
⢠⠯⠒⠒⠈⢁⠔⠉⢑⢦⠒⠐⢽⡄⡇⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢛⠁⠀⠈⡣
⡏⠀⠀⠀⢠⠊⠀⠀⠂⠀⠱⡀⠀⢳⠈⠓⠒⠢⠤⠪⣀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡠⠔⠁⠀⡠⢆⠀⢀⡴⠒⠃
⡇⠀⠀⠠⠁⠀⠀⠸⠀⠀⠀⢡⠀⣸⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⢀⡀⠀⠀⢰⣒⠤⠖⠋⠀⠀⠉⠉
⠸⡄⢠⠃⠀⠀⠀⢸⠀⠀⠀⠘⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸
⠀⠘⠦⣀⠀⠀⠀⡜⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠒⠚⠀⠈⡑⠒⢉⠓⣄
⠀⠀⠀⠈⠉⠓⠚⠓⠊⢽⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡜⢩⠆⠈⠂⠂⠁⠀⡆⠣⣈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠬⡖⢒⣳⠀⢀⡀⠠⠎⡘⠀⢸⠀⠀⠀⠀⠀⠀⢸⡠⢜⡁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⡙⣯⠈⢁⢀⡀⠔⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⢨⠘⠣⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣆⠂⢃⡝⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠤⠄⠄⢜⠢⠤⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢹⠈⠉⠉⠐⡂⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢼⠀⠀⠀⠀⡱⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞⠦⠴⡀⠀⠀⠀⢹⠒⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡉⠑⠂⠤⢔⡺⠃⠀⠀⢠⠛⠉⠉⠉⠉⢳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠃⠀⠈⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠉⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠂⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⡠⠐⠈⠳⡌⠂⠀⡀⢴⠧⠗⢄⡀⠀⠀⠀⠀⠀⣶⣶⣦⣨⣧⣤⡀⠀⠀⠀⠀⠀⠈⢡⠇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⢸⡄⠈⢆⠀⠀⠀⠀⢿⠟⠑⠀⠉⠉⠀⠀⠀⠀⠀⢀⠔⡝⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠃⠀⠀⠀⢀⣄⢸⡄⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠁⠀⡼⠀⠀⣤⣤⠀⠀⠀⠀⣠⢃⡴⠃⠀⠀⠀⠀⠀⠀⡈
⠀⠀⢀⠀⠀⠀⠀⣴⣿⡗⠈⣧⠀⠀⠀⠀⢹⡀⠘⡄⠀⡀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢾⡁⠀⠀⠀⢀⠀⠀⡰⠁
⠀⢀⠚⢀⠀⠀⠜⢿⣏⢈⠂⢸⡄⠀⠀⠀⠸⣷⡆⡷⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⢀⡼⠁⠀⠀⡇
⠀⠈⢒⣡⣀⡀⡇⢸⣿⠀⠀⠘⡇⠀⠀⠀⠀⣯⠎⠀⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⣵⣴⣇⣰⠟⠁⠀⠀⠀⠇
⠀⢻⡿⡅⠀⠈⠃⢸⠣⠀⠀⠀⡇⠀⠀⠀⠀⢿⣶⣋⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⣠
⠀⠀⠈⠘⠄⠀⠀⣸⣦⡀⠀⠀⠗⠀⢀⣀⣴⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣴⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⣼⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡏⢴⣾⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢃⣽⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠪⠻⣿⣿⡿⠛⠁⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠤⠀⠈⠉⠀⠀⠀⠀⢀⣀⡊⠁⠊
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠿⠿⠟⠻⠛⢿⣿⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠋⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⣀⣀⡋⠁⣀⠤⠀⠀⠀⢰⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠖⠚⠙⠉⠁⠀⠀⠀⠀⠘⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠐⠀⠀⠐⢀⣀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠷⠶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠰⡓⠐⢶⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⢀⣼⣿⠄⠀⢰⣿⣿⡆⠀⠀⠠⣠⣶⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⢀⣀⡀⠉⠉⠀⠀⠀⠀⠐⠌⠁⠐⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢿⣿⡷⠀⢀⣤⣤⡀⠀⣴⣤⠀⣠⡀⠠⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢟⣠⣄⡀⠀⠀⠀⠈⠻⢿⠗⠀⢿⡿⠀⢿⣷⠀⠈⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⠿⠃⠐⠀⠀⠐⠃⢠⠄⠀⠀⣠⣦⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⢧⣠⡞⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⢿⡿⠃⢰⣿⠆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣾⡇⠀⠀⠀⠀⠀⠀⣿⡿⠏⠀⣤⣶⡄⠈⠛⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠻⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣽⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⢠⣶⠂⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣶⡄⠀⢀⡀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠿⠁⢰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⠶⡶⢶⣶⡶⢶⠶⢶⣶

⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⢄⢀⠀⠀⠀⠀⠀⠀⡖⠈⣀⢀⢰⡆⠀⠀⠀⠈⠁⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⣟⣛⡟⠚⣿⡇⠀⠀⠀⠀⠀⡀⠰⠀⠌⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⡃⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠑⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠰⠆⠈⣴⠟⠀⠀⠀⠀⠀⠀⠳⠄⣀⣠⠾⠀⠀⠀⠀⠐⠦⠀⠐⠶⠃⠀⠀⠀⠀⠀⠠⠐⣠⣴⠇⠀⠀⠀⠀⠀⠸⠦⠀⣀⣀⣀⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢀⠀⣀⡀⠀⠀⠀⠀⢀⢀⣀⣀
⠀⠀⠀⠀⠰⣾⠀⠀⠨⢷⣩⡗⠀⠀⠀⠀⡋⠀⠀⠀⠁⠀⢹⠀⠀⠘⢆⠀⠀⠀⠀⣼⠃⠀⠀⢸⡛⠀⠒⠒⠒⠛⠻⠂
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠠⠋⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠀⠈⢢⠀⠀⡜⠁⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠰⠀⠀⢠⣷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⢃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⣿⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⣀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠐⠄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠘⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⡶⠊⠉⠉⣀⣨⣷⠀⠀⠀⠀⢠⣇⣤⣾⣧⣤⡄⠀⠀⠀⢠⠀⠀⠄⠀⣄⠀⠀⠀⠀⢰⠏⠉⠉⢭⣿⠀⠀⠀⠀⢠⣄⣒⣒⢒⢲⣦⡄
⠀⠀⠀⠀⠈⡇⠠⠆⠤⠖⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣁⠀⠀⠈⣿⠀⠀⠀⠀⢰⠂⣠⢿⡴⣣⢰⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠤⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣗⠀⠀⠀⡟⠀⠀⠀⠀⢸⠢⠔⠪⠃⡼⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⡀⠁⠀⢀⡅⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⢘⡀⠈⠉⠉⠁⣐⡀
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠚⠃⠀⠀⠀⠀⠀⠀⠘⠂⠐⠚⠃⠀⠀⠀⠀⠀⠂⠀⠐⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠚⠁⠀⠀⠀⠀⠈⠓⠂⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⢀⡠⠤⣄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢈⠆⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⣠⠎⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠈⠳⣜⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠁⠀⠀⠀⡤⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⣀⣀⣀⣈⣛⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⢢⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢀⠴⠋⠙⠿⣿⣿⣿⣟⣽⣿⡿⠋⠀⠀⠸⡀⠀⠀⣀⡤⠴⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡰⠉⠀⠀⠀⠀⠙⢿⣯⣾⡿⠋⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢨⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠰⢤⡀⠀⠀⠑⠤⣀
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣸⡀⠈⠉⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠛⠋⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠙⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤⣍⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠉⠳⣄⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⠛⠛⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣶⡖⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣼⠇⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢠⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢸⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣀⣀⣀⣠⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠐⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠒⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠁⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠦⠴⠁⠑
⠀⠀⠀⠐⠒⠒⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠
⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⡀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠠⣤⣤⠶⢦⣤⡶⢤⣀
⠀⠀⠈⠈⠅⠁⠙⢟⠄⠁⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠶⣌⡑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠈⠙⠢⡦⣈⠑⠢⢄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠡⠀⠀⠀⠀⢣⡠⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠈⠑⢤⢄⡈⠂⠠⢀⢠
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠱⡐⡀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠈⠾⢶⡤⢄⣹⣤⣧⣄⡀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⡌⠄⠀⠀⠈⠢⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢷⠤⠀⠀⠀⠤⠃⠀⠯⠙⠻⡿⣤⣇⠀⣠⠞⢳
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠐⢌⠂⠤⢌⠁⠀⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠂⡆⠀⠀⢰⠲⡖⠛⡛⢙⠽⢛⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢢⠡⡀⠀⠀⠀⠢⡀⢀⣁⣄⠀⠀⠀⠈⢢⠄⠀⠀⠀⠀⠀⡀⠤⠀⠃⠄⣀⡀⠌⠀⡠⠋⠀⠀⡌⡠⢢⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠡⡐⡀⠀⠀⠀⠀⢧⢀⠈⠧⣀⠂⠀⠺⠀⡀⠀⡀⠰⠊⠀⠀⠀⠁⠡⠀⠙⡀⠔⠀⠀⠀⠰⡻⢀⡼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠁⡈⢄⠀⠀⠀⠀⠁⡈⠳⡀⠀⠁⠐⢇⠠⡀⠠⠀⡇⠀⠀⠀⠦⢰⠀⠀⢸⣀⣀⡠⡪⠀⠁⡏⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠢⠀⠀⠀⠀⠈⢀⠘⠄⡀⠀⡇⠀⠐⣀⠁⠑⠀⠀⠀⡏⠀⠡⡀⠌⠀⢀⠌⠀⠀⢠⠠⠒⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢂⠡⡀⠀⠀⠀⠀⠑⡀⠀⠑⡆⠀⠀⠙⠀⠀⠠⠀⠀⡃⠀⠀⢙⡠⠂⠙⠀⠀⢀⡊⠒⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠢⠀⠐⠀⠀⢠⠀⠀⠀⠁⠀⠡⠀⠀⠀⠃⢀⠱⠤⠔⠋⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠈⠆⠀⠀⠀⠀⠀⠑⢄⠀⠀⠁⠠⡀⠠⠀⠀⠀⠁⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠉⠀⠙⢄⠀⠀⠀⠀⠈⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣠⠔⠉⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣀⠀⣀⣀⣀⣀⠀⠀⢀⡤⠒⠉⠉⠉⠙⠒⠾⡁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠐⠒⠒⠤⢄⡈⢧
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣀⠔⠁⠙⠢⣀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⡇⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠉⠉⢫⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⡇
⠀⠀⠀⠀⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇
⠀⠀⠀⠀⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠃
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠀⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⢃⡸⢶⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠈⣇⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠰⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⢠⠄⠀⣰⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠘⠀⠀⢠⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⡆⠀⠀⢠⣿⡿⠿⡟⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣃⣶⣿⣿⣽⢧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⡤⠤⠚⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠊⠘⢷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⣸⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⣾⣷⣿⣐⣂⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠦⣼⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢠⡌⠙⠛⠻⠿⠿⠛⢩⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣿⡇⠀⠀⠀⠀⣇⣀⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠲⡾
⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⣁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠋⠓⠒⠒⠛⠛⣛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠤⠶⠶⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠈⠷⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⣤⠄⢤⣤⣤⣤⣄⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⠀⠀⣀⡀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠠⠤⠤⠤⠀⠀⠐⢢
⠀⠀⠀⠀⠀⠀⡿⠀⠘⢻⣿⡿⠛⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⠁⢲⣤⣄⣉⡁⢀⣤⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠈⠁⠀⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢲⡇⠀⠀⠀⠀⠀⠀⣰⣃⠀⠐⠒⠒⠒⠶⠶⠦⠤⠴⡦⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⢠⣤⣶⡆⠀⠀⠀⠀⠀⠀⣿⣷⣶⣴⣶⣶⣦⣤⣶⣶⣶⣶⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣉⣻⣷⠊⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣔⣁⡈⠓⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⢀⣿⣥⠭⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡄⠀⢂⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣛⣆⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠐⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠶⠾⠿⣟⠁⠒⠂⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠿⠦⣷⣶⣒⣄⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣭⡿⣟⡻⠏⠉⠉⠉⠀⠒⠾⠚⢭⡽⢛⢷⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢲⡟⠁⠞⠋⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢞⠓⢮⡻⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⢃⠀⣸⠇⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⢻⣶⣾⣭⣟⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠈⠀⣼⠏⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⡇⣽⢿⣯⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡕⢃⢯⡹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠘⢘⢹⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⢀⡤⣤⣄⠀⠀⠀⠀⠀⠀⣾⢹⡏⠀⠈⠂⠞⣿⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠈⢠⡀⠈⣷⠀⠀⠀⠀⢀⡿⢸⡇⠀⠀⠀⠃⣿⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠇⣰⠏⠀⠀⠀⠀⣼⠃⣾⠃⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣠⠞⠷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠞⠃⠀⠀⠀⣠⡾⠃⢰⡟⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⡾⣋⢀⠀⠈⠳⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⣿⠁⠀⢀⡿⠁⠀⠀⠀⠀⢸⡏⣡⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣻⡝⠠⠄⣠⡾⠋⠙⠉⠋⠛⠋⢛⣿⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣵⣟⠛⢷⣏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣿⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢻⠀⣼⠃⠀⠀⠀⠀⠀⢠⣿⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣻⡌⠡⡙⣧⡀⠀⠀⣠⡞⠁⠀⠀⠀⠘⠓⠶⠖⠛⠉⠁⢷⠀⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⡏⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣽⣾⡃⠠⠈⣷⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⡏⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠃⡘⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡌⣼⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⡀⡾⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠹⣧⡉⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣧⢻⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⣏⣾⢏⢦⣦⣤⣤⣤⣴⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠛⠀⠠⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⡒⢒⢦⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠪⠄⠀⠀⡠⣈⡄⢙⠀⠀⢠⢄⣈⠿⠞⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡉⢹⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣈⣁⡠⣤⣞⠁⠀⠀⠀⣋⢙⡀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣈⠀⡎⢠⢷⢬⡄⠐⢋⠁⣀⠀⠀⣸⢁⣿⣿⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠉⠉⠠⠚⡋⠻⠃⢰⠁⣾⣿⡇⠀⠁⠱⡟⠁⠀⡇⠀⠀⠀⣠⣤⡖⠒⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠂⠐⡂⢛⡒⠁⠀⠀⠑⠛⠛⢁⣀⣒⠀⠋⠀⢠⠇⠀⢀⡞⠀⠈⢂⣀⣠⠊⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠉⠹⡧⠀⠀⠀⠀⠀⠚⠩⠙⠁⠀⣠⠎⠀⠀⢸⡷⠤⠤⣿⣿⣿⡤⠄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢡⡞⠦⠶⠦⡤⢀⣀⠀⠀⠀⢀⣀⠤⠞⠁⠀⠀⠀⠘⡅⠀⠀⠟⠛⢋⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡅⠀⠀⠀⠀⠀⠀⠈⠻⣬⠏⠙⡒⠀⡀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⣠⡶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠐⡀⠀⠀⣀⡄⠀⠀⢠⡴⠄⠀⠁⠀⠈⢠⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠁⡇⠀⠀⣿⡛⢻⡆⠀⠀⡠⠉⠁⠀⠀⠊⠀⡗⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀⠀⠠⢀⠃⠀⠀⠈⠙⠉⡀⠰⠉⣠⠀⠀⠀⠀⠠⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢆⠃⠊⢀⠂⡜⠀⠀⡀⠄⠊⠁⠀⠀⠀⢨⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⢱⠂⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣀⢀⠀⠀⠜⢢⡀⠀⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠈⠁⠀⠀⡄⠀⠀⠀⠱⡞⠋⠁⠈⢰⠶⣁⡞⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠐⡅⣴⠋⠁⢄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠰⢄⡓⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢀⡀⢀⣛⣛⣠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠚⠴⠇⠈⠉⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⠿⠛⠋⠉⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⢻⣿⡿⠿⠿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢰⠟⠁⠀⠀⠀⢀⣿⣴⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣤⣶⣿⣤⣀⣀⣠⡴⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣄⡀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢫⣭⣙⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⢁⣂⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⠶⣴⡏⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡶⠛⠿⠻⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠟⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢣⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡌⡈⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠰⢠⠅⠀⠀⠀⠐⣷⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢠⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡄⡘⠀⠀⠀⠀⠀⠫⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡⠁⠀⠀⠀⠀⠀⠸⢁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠘
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠦⠤⠐⠒⠀⠐⠂⠤⠤⠒⢒⡠⠃⠀⠀⡇⠈⠢⠤⠤⠤⠤⠔⠂⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠚⠚⠀⠀⠁⠉⠀⠀⠀⠈⠀⠀⠋⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣄⣠⣀⣤⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠛⠃
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡉⠹⠸
⠀⠀⣿⣻⠿⠿⠿⢿⣿⣿⡷⠀⠀⠀⠒⢶⠔
⠀⠀⢿⣿⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣀⠶⠤⠀⠀⢀⠀⠀⣄⢀⡀
⠀⠀⠈⢿⣿⣿⠀⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⢆⣀⠀⠀⢨⣁⠀⠀⠀⠀⠀⡄⠼⢦⣠⣷⠂
⠀⠀⠀⠈⠻⢿⣷⡎⠑⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠺⣋⣐⣶⣶⣶⡄⢠⠶⠾⠼⡿
⠀⠀⠀⠀⠀⠈⢿⠃⠀⠈⠳⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠪⡉⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠁
⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢸⡄⢀⣀⣀⣀⠤⠦⡀
⠀⠀⠀⠀⠀⡤⠊⠐⠀⠀⠀⠀⠀⠠⠏⡻⠀⠁
⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠣⡄⠀⠀⠀⠀⠀⠀⠘⠢⣀
⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⠀⠀⠀⠀⡌⢷⡀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠑⠢⣀⣀⢣⠘⣮⣳⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠱⠷⣌⡋

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣷⣌⡁⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠃⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣧⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠉⠙⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢆⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⢋⣀⣀⠠⠄⠂⠉


⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠙⠛⠻⠛⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⣠⣄⡀⠀⣀⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠊⠁⠀⠒⠚⠓⠀⡿⠓⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠴⢤⡴⠆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⠂⢒⡒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠈⠂⢌⠐⠀⢄⣀⠀⢠⣴⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠉⣉⠿⣿⣿⣿⡿⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠣⠹⣏⠉⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠈⣿⣄⠀⢹⡄⢠⣄⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡾⢠⡞⠀⠀⠀⠀⠀⠀⠰⡄⢀⠀⠀⢳⠀⠀⣿⣿⣆⠀⣧⠀⣿⣿⣷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣾⡇⠀⠀⠀⠀⠀⠀⠀⢹⣼⣦⣀⠈⣧⣀⣿⣿⣿⣾⣟⡀⣿⣿⣿⣿⣿⠐⢀⡀⠀⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢱⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⣿⣩⣿⡿⣿⣻⣯⢀⠿⠛⠛⣯⣄⠐⠸⠃⠘⠁⠙⠁⢨
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⠁⠀⠀⠀⠀⠀⠀⠀⣸⠿⠛⠟⠛⠚⠛⠛⠊⠉⠙⠆⠀⠀⠠⣄⠀⢼⢻⣷⣶⣶⣾⠟⡍
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠻⣷⡄⢠⡔⣡⢖⣀⣜⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⣞⠀⢻⣿⣿⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⣿⣬⣲⢏⣿⣿⡿⢇⠀⠀⠀⢀⠀⠀⢀⢀⠀⢰⣀⣶⣶⣶⣶⣆⠋⠿⢸⣿⣿⡷⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⢿⣷⣎⣃⣸⠛⣈⡃⣂⣀⣬⣥⣤⣬⣭⣇⣈⣛⣿⡿⠛⠁⠉⠐⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⣇⢠⡞⠀⠀⠠⣬⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⣉⠀⠸⠴⠟⢀⠄⡤⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⣾⡿⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠋⠶⠗⠉
⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⢉⣀⠀⣀⠀⠸⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠉⠉⠉⠉⠛⠛⠣⣿⣰⣾⣬⣿⣿⣿⣶⣦⣤⣤⣤⣄⣀⣠⣀⣤⣀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣋⣍⣭⣿⣿⣟⢻⠟⠉⡉⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠸⡟⠛⠉⠀⠀⠉⠛⠛⠿⠿⠿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⢰⡟⠁⣠⡾⠋⠁⢀⣀⣤⣴⣶⣿⡿⢷⣤⣀⣀⣉⣻⣿⣿⡑⢶⣿⣿⣿⠁
⠀⠀⠀⠀⢹⡇⠐⣿⡅⠠⢺⣿⣿⣿⣿⣿⠏⠀⠀⠙⠻⣿⣿⣿⣿⣭⡿⠋⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠙⢦⡙⢿⣦⡈⠻⣿⣿⣿⡏⠀⣀⡀⠀⠀⠀⠀⠉⢉⣁⡀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⠛⠷⣮⣿⣿⣿⣀⠀⠉⠳⠀⠀⠀⠘⠉⠁⢀⠀⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⠙⣿⣿⣿⠓⠖⠂⠀⠀⠀⠀⠒⠞⠋⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣾⣿⣿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⠺⢎⣧⡀⡶⣶⢈⣁⣁⡀⠀⢀⣴⡱⠖⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠋⡰⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢠⣾⣷⣄⣀⠀⠀⠀⠀⡄⣾⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠠⠤⠤⢤⣌⣉⣛⢶⢞⡼⠋⠗⢦⡤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⣿⡿⠁⠀⠀⠀⠐⠒⠲⢤⣙⣶⡛⠛⠋⠀⣸⣹⣿⣑⣿⣿⣿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⢠⡿⣿⣧⣀⣽⠁⠀⠀⢀⡴⢺⣯⠽⡛⠋⠉⠁⠀⢀⠔⣥⡿⢦⣿⣿⣿⣧⣠⣿⡿⢿⡀
⠀⠀⠀⠀⠀⢸⠃⠈⢿⣿⡟⠀⠀⢲⢿⣿⣎⣹⢷⣏⣔⢒⡔⢲⣉⡶⢏⣱⣾⣿⢿⣿⣿⣿⡿⠁⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⣿⠀⠀⢰⣠⣿⣿⣷⣶⣍⣟⢛⡟⢻⣉⣷⣾⣿⣿⣧⣠⣿⣿⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⠀⠀⠘⣿⣿⣯⣁⠀⢀⣀⣀⣀⣀⣀⣀⠀⢻⣯⣤⡉⠛⠧⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣴⣒⣒⣲⣶⠚⠛⠛⠛⠋⡉⠀⠀⠀⠱⣄⡿
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠀⢹⣿⡋⢙⣿⣿⠿⣿⣿⣿⠟⢿⣿⣯⣭⣤⡄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⢸⣿⣷⣿⣿⣷⣤⣾⣿⣿⣦⣾⣿⣿⣿⣦⣬⣽⣿⣿⠃⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇⠀⠀⠀⠀⢿⡁⣙⣿⣿⣿⣟⠁⣻⣿⣿⣿⡋⢙⣿⣿⣿⡏⣧⠀⠀⠈⡇
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣟⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠅⠀⢀⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⢀⠠⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠓⠀⣈⢀⠰⢄⠀⠐⠂⢀⠀⠀⠀⡀⣄⠐⠂⠀⠐⠊⠀⢤⠠⠄⠀⡀⠙⡱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣡⠖⠛⠢⡀⠈⠛⠀⠀⠘⠉⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⢔⠉⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣼⠁⠀⣀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠐⠂⠈⠁⠁
⠀⠀⡀⢀⠀⠈⠙⠀⠀⠂⠁⡀⠠⠄⠀⠙⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠉⠉⠈⠀⠀⠀⠈⠑⠉⠀⠠⠿⠀⠸⠻⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠁⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡲⠠⠤⡴⠀⠀⠀⠉⠉⠉⠲⣆⠀⠀⡀⠀⠠⠀⢤⠀⠠⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠐⠂⡐⠀⠀⠉⠀⢀⠁⠀⠀⢈⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁
⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠐⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢠⣵⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠂⠠⠀⠀⠀⠀⠰⡄⢻
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⡙⠢⡀⠀⠉⠻
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠙
⠀⠹⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞
⠀⠀⠘⡦⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠠⡶
⠀⠀⠀⡇⠀⠀⠐⠳⣦⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠈⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣷⣤⣄⣀⣀⣂⣀⡀⠤⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠘⠟⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠛⠛⠒⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣀⣠⠎⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠉⠐⠀⠠⠀⠀⠀⠀⠠⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠆⠀⢰⡀⠀⠀⣀⠀⠄⠀⠀⠀⠘⣦⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀
⠀⠀⠰⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠐⠁⣠⠟⠀⠀⠀⠹⡄⢰⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⡟
⠀⢠⡆⠀⠀⠀⠙⣦⠤⠄⠐⠀⢀⣠⡴⠟⠳⣄⠀⠀⠀⠀⢙⡄⠳⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⢠⠞
⠀⡈⠀⠀⠀⠀⠀⠈⠷⠲⠒⢿⡟⠁⠀⠀⠀⠀⠙⠒⠤⣤⠤⠀⠀⠀⠀⠃⠀⠀⠤⠤⠄⢀⡀⠠⠐⠊⠁⠀⠀⠀⢀⡴⠃
⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠓⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⣀⠀⠠⢈⡑⠒⠒⡲⠾⠃⠄⠒⠒⠂⠉⠁
⠀⠀⠀⢸⠁⠀⠀⠀⠐⠒⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⢲⠈⠙⠂⠤⠤⠜⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠁⠐⠂⠀⣤⣄⣀⣀⣘⣀⡀⠀⣤⡤⠾⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠇⠀⠀⠀⢸
⠀⠰⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠃⣀⠂⠀⠀⠀⠀⠈⠀⠀⣀⠠⠤⠀⠀⢤⡀
⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠤⠤⠀⠀⠀⢈⡆⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠢⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⣀⣄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠀⣀⣀⣀⡀⠤⢄⣀⡠⠞⠀⠀⠀⠈⠁⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣥⡄⠀⢰⢉⣉⣉⡉⣦⠀⢀⡀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⣿⡏⠋⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠛⡇⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡏⠉⠀⠈⠈⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⣻⣥⣬⣭⣭⣤⣤⡜
⠀⠀⠀⢠⠊⠉⠉⠙⠉⡼⣑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⠒⠤⢤⣀
⠀⠀⠀⣤⣤⣤⣤⡄⠀⡇⣿⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠑⢆
⠀⠀⠀⠘⣟⣛⣛⣣⣀⢳⡩⠭⠭⣉⢻⡀⠀⠀⢾⣿⡿⠿⣿⣷⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠷⠤⠤⠤⠜⠁⠈⠉⠲⡹⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠇⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⠦⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣠⣤⣤⠤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⣉⣭⣍⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡀⠀⣠⠄⠈⣀⢤⣬⣤⢤⡬⡀⠨⡈⡀⢀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⢄⠣⠃⠀⠑⠉⠛⠛⠛⠋⠒⢁⠠⠁⡃⠜⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⡁⠀⠈⢀⠀⠀⠀⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠦⡁⠀⣤⣉⡒⠐⠊⠁⣀⣀⣀⣀⡀⢃⣴⠖⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠃⠀⠀⠉⠙⠂⠀⠋⠉⠀⠀⠈⠁⣾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⣘⠀⠀⠀⣀⣀⡀⠀⠀⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⠑⠂⠀⠠⠀⡀⠤⠤⠤⠀⠈⢩⣾⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⢶⣶⣦⡄⠇⣤⣤⣤⢙⣂⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢰⣶⣆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠠⣄⠀⠀⠀⢸⣻⢶⡠⡀⢹⠄⠀⠘⠇⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠐⠀⠠⡟⠀⣤⠂⠈⡇⠀⠀⠘⣿⡗⠋⠡⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣵⣄⣀⡀⠀⠀⡀⣀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢨⣍⣀⣀⡀⠀⠀⠀⢈⣻⣍⣏⣐⠞⠁⠀⠀⠀⣿⣇⠶⢦⠠⠈⠳⠈⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⠖⠂⣤⠄⠀⢠⠋
⠀⠀⠀⠀⠀⠀⢸⣡⣍⠁⠿⢀⠀⠀⠈⠠⣿⠋⠁⠀⠀⠀⠀⠀⢺⣇⠳⣀⠶⠦⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣌⠁⠖⡔⠁
⠀⠀⠀⠀⠀⠀⠸⣿⢿⡷⠾⠽⠶⢶⠖⠈⣻⠀⠀⠀⠀⠀⠀⠀⣾⣄⢴⣟⡀⢠⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠈⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡟⠓⠂⠀⢶⠀⢠⠋⡏⠀⠀⠀⠀⠀⠀⠀⡿⣟⣟⠋⠀⠻⠄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⠳⢴⡀⡁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠸⡏⠀⠀⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣄⠀⠣⡖⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⠠⠃⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠄⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠓⠓⠛⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⢀⠠⠐⠀⠀⠡⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⠠⠠⠄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡟⠛⠉⠀⠀⠈⢦⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣤⠀⠀⠠⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡆⠒⢀⡬⠷⣄⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⣀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⡛⠳⠖⠉⠙⣣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⠛⠋⠉⠁⠿⠀⠀⠀⠀⠀⠀⣄⢦⢀⣠⠀⠀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠯⢅⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠎⣤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⣄⡰⠂⠁⠈⠣⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⡀⡀⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠹⡀⠶⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠀⠈⢀⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠶⠄⠸⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠀⠐⠁⠀⠀⠀⠀⠀⠀⠙⠚⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⣩⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠋⠙⠀⠀⠨⠭⢍⢠⠛⠛⠛⠿⣿⠿⠛⠉⠉⠙⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⢠⠎⠙⠶⠄⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣵
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⢃⠈⢣⠀⣆⠁⠀⠀⠀⠉⠉⠀⢷⣾⣿⣶⣶⣤⣶⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⠴⠛⡂⠔⢰⠀⠀⢙⠀⢳⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡠⠴⡏⠰⠈⢉⠃⠄⠓⡼⢳⠈⣡⢜⣁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠸⡄⢒⡩⢦⠒⠁⠀⠈⣈⠄⢸⢉⣴⣷⣄⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠁
⠀⠀⠱⡄⠱⡩⢦⡠⠖⠛⠁⡠⠊⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠼⠋
⠀⠀⠀⠈⠒⠛⠙⠤⣀⠀⢀⠗⢄⠀⠀⠀⠀⠀⠀⠈⡝⣇⣀⣈⣉⡹⢍⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡁⢀⡸⠑⣄⠀⠀⢀⠔⠒⣾⢀⠁⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠊⠀⠀⠘⢆⡰⠃⠀⡄⠀⠉⠉⠉⠉⠉⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀⣇⢀⡜⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⢉⡞⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡞

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣛⣛⣋⣉⣉⡁⠀⠀⠀⣀⣀⠈⠀⠠⣤⡤⠈⠁⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡿⠛⠋⠉⠹⠿⠿⣿⣿⣿⣿
⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣄⣀⣀⣤⣶⡾⣿⣿⣶⣶⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⡿⠛⠛⠈⠙⠋⠉⠛⡉⣁⠀⠉⠛⠛⠋⢩⣽⣉⣉⣋⣩⣿⣿⠟⣉⣭⣉⡉⢹⣽⣋⣉⣥⣼⣿⣿⣿⣿⣶⣶⣾⣿
⠿⢿⡿⠿⠶⠶⠛⠿⠿⣿⠶⠶⠶⠋⠻⠿⠳⠞⠛⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡉⢉⠙⠛⠋⠙⠛⠉⠙⠛⠛⠟⠫
⠓⠈⠻⠛⠓⠀⠘⠓⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⠤⣦⠤⠤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⢿⣿⣇
⣶⣶⣦⣤⣴⣾⣿⣿⣿⣤⣴⣶⣶⣶⣦⣤⣆⡤⣠⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣬⣿⣦⡀⠴⢶⣶⣢⣾⣿⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣶⠰⠛⠋⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣘⠃⠠⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠿⢿⣿⣟⣛⣯⣥⣶⣷⣿⣿⣦⣶⠀⠀⢻⣿⣿⣿⣟⣛⣙⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣶⣶⣿⣏⣉⣉⣉⣉⣿⣿⣫⡽⢷⡬⡿⠀⠀⢻⠿⠿⠿⠿⣯⣽⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠩⠦⢄⠉⠉⠛⠛⠛⠫⠭⠥⠤⠬⠙⠛⠂⠀⠀⠀⠀⠀⡏⠀⠀⠀⠐⠛⠚⠋⠻⢯⣟⣛⠛⠛⢻⣿⣿⣿⣿⣿
⣿⣿⣟⡂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠲⣿⣿⣿⣭⣿⣿⣿⣷⣾⣿⣿⣿⣿
⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⣄⣤⡤⠤⠄⠀⠀⠀⠛⠉⠉⠙⠉⠉⠉⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⢾⣋⡉⠉⠉⠉⠉⠉⠉⠙⣛⣲⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠈⠉⠳⢦⣀⣠⡴⠞⠋⠉⠀⠀⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠾⠳⠤⠴⠻⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣀⣤⠴⠚⠋⠉⠁⣴⠋⠀⠀⠀⠀⠀⠈⢳⡀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⢀⡟⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠸⡆
⠀⠀⠀⠀⠀⠀⣼⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣽⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⡿⢿⠿⠿⠿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡿⠀⠈⠳⣄⣠⣤⣤⠤⠶⠶⠶⠦⢤⣤⣴⣋⠀⠀⢹⣼⡆
⠀⠀⠀⠀⣠⠶⠚⣛⣿⡟⣁⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣹⠿⢿⠓⠦⣄
⠀⠀⣠⣞⣡⠤⣶⣿⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡟⢿⣒⠺⢷⡄
⠀⢠⡟⣿⠒⠛⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢉⡿⢻⡀
⠀⢸⡇⠘⢷⣄⢀⡏⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠟⠁⢸⡇
⠀⠘⣇⠀⠀⢉⡿⠀⠀⠀⠀⠀⢀⡏⠙⠦⣄⣀⣙⣷⣤⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠈⢷⣄⡀⣼⠁
⠀⠀⢿⡀⣴⠞⠁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⣠⡬⠁⠀⠠⣭⡉⠉⠉⠁⠉⠳⢦⣀⣀⠀⢀⣠⡼⢁⡟
⠀⠀⠘⣇⢹⣷⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠁⣴⠀⠀⢠⣦⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡍⢧⣼⠃
⠀⠀⠀⢻⣯⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡏
⠀⠀⠀⢸⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣤⣀⣤⠟⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠳⣤⣀
⠘⢯⣄⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⡿
⠀⠀⠉⠳⢤⣀⣀⣈⣳⣄⡀⠀⠀⠀⠀⢀⡴⠲⠤⢿⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⢠⡞⠃⠀⠀⠀⠀⠀⠛⣦⠀⠀⣀⣠⡴⠚⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣿⢿⣿⣶⣶⣶⣶⣶⣿⢻⣟⣻⣯⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡞⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⢹⡆⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢸⠃⢸⠀⢀⣀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠿⣤⠾⠛⠋⠹⣆⣀⣀⣀⣼⠋⠓⠶⣤⣽⠇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⣩⣭⣿⡁⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⢸⡀⠀⠀⠀⠀⠀⢠⣤⣤⠏⠀⠀⠀⠀⢸⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠾⠷⣤⣄⡀⠀⠀⢤⣤⡽⠂⠀⢀⣀⣤⡿⠶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠙⠛⣶⣴⣯⣤⠶⠛⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠴⠿⠭⠤⠭⠶⠆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠶⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⠞⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⢹⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣴⣶⣤⣤⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⣿⣿⣿⣿⣿⣿⣿⠇⣿⡁⣿⣿⣿⣿⣿⣿⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⣤⣀⣀⣀⣠⣤⢞⣿⣟⢦⣄⣀⣀⣀⣀⣤⢴⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠉⠉⠛⠛⠿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠉⠙⢿⠿⠟⠃⠀⠀⠀⠀⠐⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠰⠆⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡖⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⣀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠐⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⣰⣧⠀⠙⣶⣶⠠⠼⠀⠺⡉⠀⠀⢀⣀⡀⠤⠔⠁
⠀⠀⠀⠀⠀⠂⢀⠠⠐⠒⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠢⠄⣀⡎⠑⢄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠠⠄⢒⠦⠀⡰⠂⠀⠈⢿⡟⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠃⠨⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⡠⠄⠤⠄⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠈⠁⠀⠀⣀⠠⠤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠐⠂⠉⠲⠀⡁⠑⠀⠂⢀⡠⠔⠒⠒⡲⠦⢄
⠀⠀⠀⠀⠀⠀⢀⡠⠄⠚⢳⠀⠀⠀⠀⠎⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⠀⠀⠀⠑⢄
⠀⠀⠀⢀⡤⠚⠁⢀⡠⠤⢎⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⣄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢀⣴⠃⠀⠀⠀⢰⠀⠠⠀⢢⡸⣿⡛⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣔⠁⠀⠤⠄⠐⠈⠀⠈⢄⡔⠁⠙⠛⠋⠄⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⣀⡠⠔⠋⢸⡄⢀⣀⠴⢿⠃⠀⠱⡄⠀⠀⠀⠀⠀⢀⡀⠀⠠⡤⠘⠦⡀⢠⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢝⠄⠀⠃⠡⡀⡠⠂⠉⠄⠀⣤⠞⠁⠀⠀⠈⠉⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡐⠄⢈⠄⠂⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡠⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠐⠀⡲⢂⠈⢉⠉⠉⠉⠁⠐⠀⠀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⡼⡣⢔⠎⢀⠨⢐⡨⠔⢒⠫⠤⠒⡀⠠⠄⡉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣀⠄⡐⣸⠁⡊⠔⡨⠔⢊⠅⠂⠨⠐⡢⠄⢐⣀⢠⣭⡵⠶⢖⣠⣈⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠠⠦⢄⣀⠀⣠⣤⣶⡿⠋⢗⡿⣿⣖⡥⢔⡩⠀⠤⢂⣤⡴⠤⠄⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣛⠦⡀
⠀⠀⠀⠀⠀⠔⢀⠔⣒⡿⣿⢮⡩⠐⠒⠉⠀⠈⠉⠃⢾⣽⣿⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣈⡠⡄
⠀⠀⠀⢀⠎⠠⠂⢥⣓⣋⡝⠁⠀⢠⣠⡀⠄⡀⠀⠀⠈⠙⡅⢪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠪⢌⠢⡀
⠀⠀⠀⠂⣀⠁⡒⠿⢗⡎⠀⣠⡵⢋⠀⠙⢦⣪⣀⠀⠀⠈⢰⢀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⡂⠤⡑⠄
⠀⠀⡄⡀⢕⡡⡐⣿⣿⠁⢀⢼⡇⠁⢒⣃⣠⣿⠗⠀⢠⢂⠇⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡐⢌⠈⢢
⠀⡘⠠⢔⡵⠞⣙⣫⡉⡆⠀⢟⡿⣿⢿⠲⠛⠁⢀⠰⣡⠞⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠢⢈⠲⢅⠆⢣
⠀⣡⠔⠁⠠⠈⠃⢜⠧⡘⢄⡀⠈⠀⠀⠀⠀⣀⡡⠞⠃⠀⠈⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠄⡀⠑⢄⠲⡀⢃
⠸⠁⠀⠀⠁⠀⠀⠀⠈⠐⠀⢉⢟⡶⠒⠏⡉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⢦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⠑⢄⠀⢲⡅⠈⠄
⠈⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢣⠀⠏⢢⣈⠰⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⢄⠑⠄⠡⡈⢗⠀⡇
⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⠀⠠⢸⠈⣦⡀⠘⠠⠀⠀⠀⠀⠀⠀⠀⡀⠱⡈⠂⠈⢄⢱⡥⠂⡇
⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⢰⠀⠀⢸⠀⡇⡹⢄⠀⠑⠀⠀⠀⠀⠢⡀⠘⡄⠐⡄⢂⠈⡤⣃⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠁⡇⠸⠀⡆⢸⠀⡇⡇⠸⢢⡀⠀⠀⠀⠰⡀⢡⠀⠈⠀⠸⠈⡄⢹⠕⠀⡃
⢡⠰⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠁⡈⡈⢠⢀⠆⠀⠂⡄⢠⠀⠀⡆⡆⣷⡀⠀⠀⡄⢱⠀⢃⠀⠂⠀⡆⢰⠖⠂⢠⠁
⠨⠀⡀⡔⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⡀⡠⢠⠀⠃⡘⢠⢃⠆⢠⢀⠃⠌⡜⠀⡌⢠⠁⠈⢰⢰⢠⠀⡇⣷⠀⣂⠘⡄⣆⠈⡀⢰⠀⣠⣹⡍⢀⡆
⠀⢣⠠⠀⠀⡄⠀⠀⠀⠀⠐⠀⢙⡐⢡⢁⠃⡌⡐⢀⠃⡜⢀⠂⠆⡘⢰⠀⠰⠀⡌⠀⢄⠆⡏⠈⠘⠃⠋⡇⠘⡇⢿⢸⡄⣇⢸⡶⣾⡖⠀⡾
⠀⠀⢧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⣀⠇⠂⡌⡠⢁⠎⡐⠀⠎⡘⢠⢣⠃⢀⠃⡰⠁⡐⠘⡰⢱⠂⠚⠒⠐⢛⡂⠓⠘⠛⠃⠛⠈⠋⠙⠁⡼⠁
⠀⠀⠀⢣⡡⢀⠀⠀⠀⠀⠀⠀⠀⢯⠆⡜⠐⠁⠈⡠⠁⡌⡰⢁⠂⠆⢀⠆⢰⠁⠰⢠⠰⠁⠋⡸⢱⠃⠏⡜⢳⠘⡘⢇⡗⢺⠒⠃⠀⡞⠁
⠀⠀⠀⠀⠱⡌⢠⠀⢰⠀⠀⠀⠀⠘⡎⡠⢃⠌⠰⠁⠌⡰⠁⠄⡜⢀⠎⢠⠃⠠⢡⢃⠇⡌⡰⢁⠆⡸⢠⢃⠛⠀⡗⡧⣭⠝⠁⢠⠎
⠀⠀⠀⠀⠀⠈⠢⡉⠂⡄⠀⠀⠀⠀⢹⣁⠌⡰⠁⠜⡰⠁⠎⡘⢀⠎⢠⠃⡰⢁⢢⠊⡜⡰⠁⠌⢠⢁⠦⡮⣽⡇⣗⡗⠂⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁⠀⢁⠀⠀⢫⡴⢁⠎⡰⢁⠎⡔⢀⠊⣠⠃⡔⢡⢃⢂⠌⣐⠁⡬⣠⠣⡬⡽⣱⣻⠆⠇⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠑⠧⢢⢳⣦⡾⣁⢮⡜⢤⠂⣴⣁⠾⣴⣱⢧⡞⡶⢭⣟⣰⢃⡼⣵⢃⢏⢿⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠘⠓⢦⣄⣀⢹⡜⡡⢟⡰⢏⡾⣵⢫⠼⡴⢯⢏⡞⣡⠏⡘⢃⠌⡰⣡⡣⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠓⠊⠩⠵⢚⠴⢯⡴⣥⣯⣮⣴⣡⣎⣴⡥⢾⠔⡷⠞⠋⠑⠠⠅⣒⡤⢒⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠈⠀⠈⠀⠈⠁⠈⠁⠀⠁⠒⠉⠐⠂⠩⠄⠢⠅⢲⠭⢑⡂⠄⡒⠤⣀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠉⡚⠁⠀⠛⠖⠖⠰⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠋⠀⠀⠀⠀⠀⠈⠀⠀⠐⡄⠀⠀⠀⠀⠀⠙⠿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡀⠀⠂⠀⢠⣎⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⣇⠀⠰⠀⠘⣿⣼⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠈⠀⠙⣦⡀⠠⠀⠀⠈⠐⠶⣄⢀⣼⠃⠀⠀⠀⠀⢳⠀⠀⠀⠀⠉⠹⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣦⠄⠀⠳⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠳⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠱⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣼⠀⠀⠀⠸⠁⠀⠀⠀⢠⠀⠀⠀⠊
⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⢰⠀⠀⠀⠀⠉⠀⣠
⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⢦⡴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⡘⠀⢠⡾⠀⠶⠀⠀⠀⠰⣿⠘
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⠁⠀⠹⢧⠀⢰⣄⠀⠀⠀⣽⣄⠀⠁
⠞⠂⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⡉⠓⠀⠄⠈⣿⠃⠄
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⠃⢀⡀⠐⣿⠀⠀⠀⠀⢀⣤⠀⢻⣧⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡰⠊⠀⠃⠀⠠⡀⠀⠀⠀⠘⣯⢀⡞⣉⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠈⡝⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠉⠦⠀⠉⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢺⡄⠀⢿⡇⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⣰⡆⢌⠀⠀⠰⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣀⡄⠠⣄⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣾⣧⠎⠈⠛⢷⣷⣦⠶⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠀⠀⣧⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀⠀⢀⣀⠋⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡿⠓⠚⠉⡀⢀⡄⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢫⠏⠀⠉⠉⠛⠉⣸⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣦⣴⣤⣤⣦⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⠃⣸⣷⣦⣄⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄⠀⠀⠄⠰⣾⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⠈⠀⠀⠻⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠁⠀⡡⠀⠀⠀⠀⠀⠀⠀⠅⠀⠙⠇⠸⠟⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠆⣄⣠⠀⣶⣴⡟⠁⠀⠀⠀⠙⢻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠟⢻⠀⠀⠀⠀⠀⠀⢨⡉⠉⠛⠻⠿⣿⣿⣿⣿⣿
⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠕⠃⠀⠀⠀⠀⠀⠉⠛⢿⣿
⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠁⠄⠼⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡉⠉⠓⢦⠀⠀⠀⠀⠀⢀⣴⣿⡏⠉⠓⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⣿⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣷⡤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢠⡀⠀⠀⠀⠠⢄⠀⠈⠉⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠢⠁⠀⣀⠀⠴⠥⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⢠⠤⠀⣀⠀⠀⠀⠐⠐⠀⣉⠠⠊⠁⠀⠀⠀⠀⢰⣾⣤⡀⣰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢍⠀⠀⠋⡻⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⡆⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠰⠀⠀⠀⠀⠈⠓⠀⠚⠀⠂⠱⢶⣶⣤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⢠⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡀⠀⠀⠀⠑⠢⠤⠤⠤⠰⠒⠦⣶⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢄⣀⣀⡀⠀⠀⠐⢀⠃⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡦⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠁⠀⠀⠀⠁⠉⠐⠆⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠜⡂⠄⠀⠠⢄⣀⣀⣀⣀⠀⠀⡸⠀⠙⠿⣿⣷⣦⠀⠀⠀⠀⠀⣀⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢀⠞⠁⢠⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠐⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠄⠀⠠⠴⣊⠀⢶⡔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢘⠓⠀⡄⠀⠀⠀⠀⠀⠀⡸⠀⣸⠃⠀⡤⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂⠘⠢⢤⡀⠀⠀⣠⠇⠀⡡⠟⣁⡀⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢰⠋⠀⠀⠀⠁⠈⠁⠉⠀⠀⠁⠀⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡷⡀⠘⠓⠒⠀⠀⠀⠀⠀⠀⠐⠒⠚⠃⢠⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠀⡃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣷⣤⣄⣀⣠⣴⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢀⠁⠀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣮⡉⠙⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⣿⢿⣷⡄
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢠⣿⠃⠀⢻⣿⡄
⠀⢠⣿⡟⠀⢀⣀⣤⣤⣶⣶⠶⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⠶⣶⣶⣦⣤⣤⣀⣀⠀⠀⠀⣀⣿⣷⣤⣴⡿⠃⠀⠀⠀⢻⣿⡄
⠀⣿⣿⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠷⣾⡟⠉⠀⠈⠻⣷⠀⠀⠀⠀⠀⢿⣷
⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢠⣿⣶⣤⣀⠀⠀⠸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠷⣶⣾⡟⠁⠀⠉⠛⠿⣶⣄⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠙⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⣿
⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡿
⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣼⣿⠃
⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⣒⣯⣽⣵⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣷⣾⣿⣿⣿⣿⣿⣿⠿⠗⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠀⠈⠉⠉⠻⡶⣶⣦⣤⣤⣤⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠑⡎⠩⣿⣿⣿⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠰⠖⠂⠀⠀⠀⢞⣁⣘⣿⣿⣿⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠃⠀⠀⠈⠋⠀⠀⠀⢸⣽⡿⣛⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠈⣠⣶⣅⠀⠀⠀⠀⣠⠏⠀⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⠀⠂⠐⠂⡆⠀⠀⠀⠀⠀⡴⠚⠀⠐⠊⠁⠣⡰⠙⠃⡸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣬⠃⠀⣀⠀⠀⠃⠀⠀⣀⠴⠞⠁⠀⢀⠐⠢⠀⢠⠇⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⢀⣹⣄⣈⣀⣵⠒⠔⠉⠀⠀⠀⠀⠀⠎⠀⢠⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠄⠀⠀⠀⠀⢙⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡉⢻⡏⣾⣿⡇⠊⠀⢠⠎⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠊⠀⣀⣤⠴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠉⣻⣿⣼⣿⣿⠁⠀⠀⢸⡄⠀⠀⠀⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠁⢠⠎⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⢧⠀⠀⢸⠁⠙⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⢢⠀⠈⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⡴⠁⠀⠀⠀⠀⣰⠏⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢣⡀⠀⠀⢀⠀⢧⠀⠀⢺⡁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣇⣠⣖⣁⣤⡜⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⢀⣧⡴⠖⠒⠒⠒⠢⠤⢀⡀⠀⢸
⠠⣄⣀⡀⠀⠀⠀⢀⣸⣦⣤⣤⡔⠒⠙⠛⠉⠉⠁⢸⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸
⠀⠀⠀⠉⠙⣢⠞⠁⠀⠀⠀⠀⠉⠙⠓⠦⡀⢀⣀⣋⣀⣀⣘⠀⠀⠀⣀⡤⠤⠂⠀⠀⠑⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠀⠀⠀⠳⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠞
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠳⢦⣤⣀⠀⡠⠤⠀⢸⣅⡧⠠⣤⣾⢿⣛⠛⠋⢉⡉⣉⢉⡟⠛⠛⠋⠉⠁
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠉⠛⠦⡀⣀⣠⠼⠒⠋⠁⠀⢀⣿⢠⣆⣼⣷⠿⣾⠃⢀⠂
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⢀⣀⠤⠶⡞⣹⠏⣾⣿⠻⠏⠀⠁⠀⢸
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣠⠋⠀⢀⡤⠖⠋⠉⠀⢲⡄⣷⠏⠀⠀⠁⠀⠀⠀⠀⠀⣼

⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠔⠉⡡⠰⠄⠀⢀⡠⣀⠀⠠⡀⡠⠀⠠⢀
⠀⢀⠀⠤⢀⠀⣀⣠⠂⠉⠀⠀⠈⠘⠃⠁⠶⢦⢽⣜⣉⠑⠄⠀⠀⠀⢀⠌⠒⢇⣜⡴⠶⢎⠁⠃
⠴⠕⠢⠴⢤⢷⠥⢆⠈⢂⠀⠀⠀⠀⠠⣀⠘⠉⠊⠑⠁⠒⠆⠀⠀⠀⠡⠄⠉⠑⢚⠝⠡⣀⠆
⠀⢀⠁⡘⠉⢁⢉⠀⠈⠣⡄⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠰⠁⠀⠀⠘⠂⠀⠀⢠
⠀⠀⡈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠁⠀⠠
⠀⠀⠁⠀⠀⠀⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⢀⠀⠀⠄⠄⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠖⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⠲⡉⢄⣀⡀⠀⡇⠀⢀⠀⠠⠀⠐⠠⠄⣙⠙⡅⠁⠑⠆⠀⣃
⠀⠀⠀⠀⠀⠀⢀⡑⠊⠀⠰⠉⠀⢹⡄⡎⠀⠀⢨⠁⠀⢤⡹⠀⠃⠀⠠⠂⠀⠀⠁⠢⣀
⠀⠀⠀⠀⡠⠊⠁⠀⠂⢠⡀⠀⠦⡺⡌⠁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⠀⠄⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠘⢦⡀⠈⠲⣄⠀⠀⠀⠐⠒⡡⠀⠄⠀⠀⠀⠀⠀⠈⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉⠑⣄⠀⠈⠙⠒⠒⠒⠉⠀⠘⡩⠋⠉⠉⢶⠀⠀⠐
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⢀⠐⠙⠲⢤⠀⠀⠀⠀⢀
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⢨⠝⠋⠉⠉⠀⠀⠀⠀⢂
⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣀⣀⣀⢂⣀⣀⠠⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠤⠤⠤⠤⠴⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠛⠉⠉⠉⠉⠉⠙⠛⠒⠒⠦⣤⠀⠀⠀⢀⠔⢤
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠀⠀⠦⠴⠋
⠀⠺⢀⣑⡄⠀⠀⠀⠀⣀⡿⠁⠀⠀⠀⢀⣠⠤⠴⠖⠒⠛⠛⠋⠙⠛⠓⠒⠲⠤⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⣤⠞⠉⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠐⠲⡆⠈⠑⣿
⠀⠀⠀⠀⠀⠀⠘⣻⠗⠀⠀⢠⠞⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⠔⠒⢆⠀⠀⠀⠀⠀⠀⡔⠒⢆⠀⠀⣾⠀⠀⠀⡤⢀
⠀⠰⣁⠴⠃⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢀⠰⠆⠀⠀⠀⠀⢸⠀⠀⢈⠀⠀⣿⠀⠀⠀⠱⠤⠂
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⠢⠬⠛⠀⠀⠀⠀⠀⠀⠣⠬⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣀⣼⠇⠀⠀⠀⠀⠚⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⡶⠚⠉⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠈⢧⣀⡉⠁⣩⠇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⣀⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⢀⡼⠋⢀⣀⡀⠉⠉⠛⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣾⣅
⠀⠀⠀⠀⣰⠋⠀⣰⠟⠁⠉⠛⠶⣤⣀⠀⠀⠈⡿⢻⣿⣏⠉⠉⠁⠀⢀⣾⢿⣎⠛⢦⡀
⠀⢠⠶⠚⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⣼⡇⠻⢷⣤⣴⡶⠿⠋⠀⢿⣄⠀⠙⠷⣄⣀⡀
⠀⣏⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀⣀⣴⡟⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⣿⠳⣤⣀⡀⠉⠉⠛⠓⠶⠶⠶⠶⣤⣀⡀
⠀⠈⠳⢦⡴⠟⠁⠀⠀⠀⠀⠀⠀⣾⠉⠉⠛⠋⠉⠀⠀⠀⠀⠐⡁⠠⠀⠀⠀⠀⢻⡆⠀⠉⠉⠙⠛⠓⠒⢶⡀⠀⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠋⠉⠉⡟⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠶⣤⣄⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢦⡀⠀⠀⠀⢸⠀⠀⠀⠙⢦⡀
⠀⠀⢠⡾⢶⣄⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠁⠀⣼⠷⠶⢶⣤⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠙⢆⠀⠀⡇⠀⠀⠀⢀⣠⢷
⠀⣠⠟⡐⠁⢹⣆⠀⠀⠀⠀⣴⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⢠⡾⠳⠶⠦⣤⣄⡀⠀⢸⡗⠲⠤⣄⣀⠈⡇⢸⠀⡤⠖⠋⠉⠀⠈
⣴⣋⢌⣠⢶⡏⠘⢧⡀⠀⢀⡽⢷⣄⠀⠀⣠⡶⠶⠶⠦⠴⠶⠶⠞⠛⠻⠶⣦⣴⡿⠤⣤⣀⠀⠀⣨⠇⠀⢸⡇⠀⠀⠀⢈⡽⠃⡟⢾⠤⠤⣀⣀⣀⣀
⠈⠉⠉⠁⠀⠹⣆⠈⢳⡴⠋⠀⢀⡼⠷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠞⠁⣠⠞⠁⠀⠀⠀⢷⡴⠒⠊⠉⠀⢸⠃⠘⣄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡟⠀⠀⠈⠳⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠉⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢼⣥⣄⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣤⣀⣀⡈⣷
⣦⣶⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣦⣦⣤⣽⣿⣿⣴⣴⣤⣤⣤⣤⣤⣤⣤⣤⣴⣴⣤⣤⣤⣶⣤⣦⣴⣦⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⡿⣿⠟⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⠏⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠟⠟⠛⠉⠉⠈⠁⠀⢿⠿⣿⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⡀⡀⢀⡀⡀⣄⣴⣄⣠⣷⠀⢹⡿⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡄⠸⠁⠀⠀⣠⣤⣤⣤⢨⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠰⢛⣿⣽⣷⠋⠀⠀⠀⠀⠀⡠⠂⡷⠀⢦⡀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⣤⣍⠻⢿⣿⣿⣿⠟⠁⣤⡆⠀⠀⠀⠀⠀⠈⠛⠷⠀⠀⣰⢀⠞⠁⣼⣁⠀⠈⠛⠑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⢻⡌⢻⣦⠹⣿⣿⣇⡀⠞⢾⠂⠀⠀⠀⠀⠀⠸⣄⠾⠋⢡⠏⢀⣾⣿⣫⠖⠀⠀⠀⠘⡓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡜⢾⣯⡀⠙⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠺⡗⠖⠛⠛⠛⠛⠋⠀⢰⡀⠀⠙⠂⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠁⠈⠙⠛⠦⣄⠰⡛⠀⠉⢲⣾⡇⠀⢃⣄⡄⠀⠛⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣄⠀⠀⠉⠸⡄⠘⢢⣙⡇⢶⣄⡉⢼⠿⣶⣦⣭⡲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠱⢤⣀⠉⠙⠛⠻⢿⣷⡘⢿⣿⢏⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣻⡇⢹⣿⡁⠘⠛⠿⣶⣶⣶⡆⢀⡶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⣦⡀⠀⠀⠈⠳⡄⢳⣿⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠉⠁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣖⣀⠀⢀⠙⠋⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠻⣄⡄⠀⠸⣯⡅⠀⠀⠀⠀⠀⢰⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣭⣖⠡⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⡄⠀⠀⠙⠿⣦⣰⣼⣷⡀⠀⠀⠀⢠⠏⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣛⣾⡇⣀⣤⠞⣧
⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠙⣄⠀⠀⠀⠈⠙⠻⢮⣗⡀⠀⢀⡟⠙⢦⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣼⠿⠚⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠉⢳⡾⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠚⠛⠋⠉⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⡶⠈⢁⠸⠿⣶⣶⣀⣤⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⠟⣩⠀⡀⠖⠒⠚⣛⣉⣬⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠾⠃⢈⣡⡾⠟⠛⠓⠛⠻⠛⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⡠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢁⡽⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⡎⣁⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⡇⢠⡇⠀⠀⣠⣴⠶⠛⠋⠀⠀⠀⠿⠢⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣏⠹⠇⣸⡇⠀⠈⠁⢠⣤⡤⠀⠀⠀⢶⣶⠦⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⣧⣰⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠈⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⣢⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⢿⣦⡶⢧⠀⠀⠀⠀⠀⠀⠀⢀⣦⣠⣄⡀⠰⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⢀⡴⠿⡉⠀⣰⠟⣧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠹⣯⣽⠁⠀⢀⡴⠟⠧⠶⠀⠀⢀
⠀⠀⠀⢀⡴⠋⠀⠀⢠⡼⠃⠀⢿⣧⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⡐⠋⠀⠀⠀⠀⠀⢀⡠
⠀⠀⣠⠟⠀⠀⠀⠀⠈⠁⠀⠀⢸⡿⣆⠈⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠛⠙⠁
⠀⣴⠏⠀⠁⠀⠀⠀⠀⣤⠀⠀⠸⣇⠹⣧⡀⠈⠛⠿⣷⣶⣤⡀⠀⠀⠀⠀⠀⡉⠷⠶⠶⣤⡀⢄⣀⡀
⠈⠁⠀⠀⠀⣠⣴⠾⠋⠙⡆⠀⠀⢿⡄⠈⠻⣶⣤⡀⠀⠈⠙⣷⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠙⢦⣤⡄⢀⠔⠈⠉⠲
⠠⠤⢤⣤⡾⠋⠁⠀⠀⠀⢠⠀⠀⠈⢿⡀⠀⠈⠙⠿⣦⡀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢏⠙⣿⠁
⠀⢀⡀⠈⠻⣦⡀⠀⠀⠀⠘⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡴⠎⢸⠇⠈⠁
⠀⠻⣿⣦⠀⠘⣷⡄⠀⠀⠀⠙⡀⠀⠀⢻⣆⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠻⣷⠀⠸⣿⠀⠀⠀⠀⢣⡀⠀⠀⣿⣧⡀⠀⣠⣿⡇⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⢸⡇⠀⣽⡇⠀⠀⠀⢀⠈
⠀⠀⠀⠀⢹⣇⠀⣿⡇⠀⠀⠀⠘⡇⠀⠀⢸⡟⢿⣶⣿⣿⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡞⠀⠀⣿⡏⠒⠒⠂
⠀⠀⠀⠀⠀⣿⡀⢸⣿⠀⠀⠀⠀⠇⠀⠀⠈⣿⠈⣻⣿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠁⣠⠾⠈⠇
⠀⠀⠀⠀⠀⢸⣧⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢻⣾⡿⢉⢷⡄⠀⠀⠀⣠⠎⠑⠀⠀⠸⣇⠘⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠟⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠹⣦⠙⢆⠀⢰⡅⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢀⠀⠀⠀⢾⣿⠀⠀⡇⠀⠀⣤⣼⡟⠀⠀⠀⠹⣧⡈⠳⣹⣇⠀⠀⣴⡾⠟⣿⠀⠀⠀⠀⠀⠸⣧
⠀⠀⣀⣀⣀⣀⣀⣀⣸⣿⡆⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠘⢿⣤⣿⢿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠚⠛⠋⠉⠛⣉⣿⡿⢻⣇⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⣠⣾⣿⣏⣥⣶⠆⢻⡄⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⢰⣿⣿⡿⠛⣩⠋⠀⣸⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠄⢀⣾⡿⠁⠀⠀⣾⡇⠀⣀⣀⣀⣀⠀⠀⢻⡄
⠀⠀⣼⠿⡟⠁⡘⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⣶⢆⣾⡟⠀⠀⠀⢠⣿⣤⠾⠛⠛⠛⠛⠛⠆⢸⡇
⠀⠀⡟⠀⣤⡄⠃⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠲⢁⣾⡟⠀⠀⢀⣤⣾⠿⠷⠆⠀⠀⠀⠀⠀⠠⡀⢿
⠀⠐⠀⢰⣿⠛⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⢁⣾⡟⢀⣠⣴⠟⠉⠀⠠⣤⣀⣀⣀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⣾⡿⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢠⣿
⡀⠀⡀⣿⠃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠁⠀⠁⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠶⢤⣤⠀⠀⣠⣿⠇
⠀⠸⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢠⡀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⢀⠈⢦⠀⠀⠀⠀⢀⣀⣀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢸⠀⠀⢃⡀⣠⣾⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⢹⡇⢁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⢀⣿⣦⣴⣯⣿⠏⠁
⠀⠀⠀⠀⠈⢿⣄⠣⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⣀⣴⠟⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣴⣿⢛⣁⣤⣤⣴⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡮⠶⡀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⣶⣶⣋⠉⣫⢱⣇⠀⠘⢆⢿⠋⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠯⠤⢄⡈⠻⣿⡔⠣⡙⠧⣒⡬⠊⣀⡀⢄⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢻⣧⡀⠀⠈⠑⡎⡇⠀⠈⣁⡴⠚⠉⣠⡴⠖⢚⣙⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣌⠛⠷⠤⢮⡼⠁⣠⠚⠁⠀⠀⠀⠀⢀⡾⠋⠉⠉⠛⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⢀⡠⢉⠶⠒⠊⣠⠎⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⠳⣝⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣔⢉⠴⡇⡆⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠈⠁⢠⠁⣿⠁⠀⠀⠀⠀⠀⣴⠞⠀⠀⣸⠀⢠⠶⣤⡀⠀⠀⠀⠀⠀⣹⠀⠀⠈⢧⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠤⢤⣀⠀⠀⠳⣄⠀⠀⠀⢸⠀⠈⢱⡄⠀⠀⠀⠰⠃⢰⠞⠉⠉⠓⠲⣤⣾⠇⠀⠀⠀⠀⢀⠇⠀⢀⣀⣀⢳
⡟⠉⠉⡩⠝⣻⣁⣠⠀⣠⠀⣠⠤⠀⡄⠀⡰⠈⠓⢤⣀⠙⠒⢤⡀⡄⢀⠀⠀⠙⢦⠀⠀⠀⠀⣻⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⣠⠎⠀⡴⠋⠀⢨⠇
⡇⠀⠀⡧⣎⢱⠗⠂⣰⠉⣷⢧⣀⠀⡧⠤⡇⠀⠀⠀⣨⠇⠀⠈⠓⠣⠤⣤⡤⣄⠀⢷⠀⠀⠀⣈⣉⠳⢤⣀⡠⠮⣀⣀⣀⠤⠚⠁⠀⠀⠰⠦⠒⠉
⡧⠀⠀⠧⠞⠘⠋⠉⠉⠉⠙⠀⠀⠀⠁⠀⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⠞⠉⢉⣉⡉⠁⠀⠀⢻⡳⣄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰
⠰⠶⠶⠶⠖⠒⠒⠒⢲⡶⠶⡖⠒⠒⠚⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠚⢧⡤⠜⠛⠀⠀⠈⡟⢉⡱⠦⠤⣀⣀⣀⣀⣀⡠⠤⠖⠚⠉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⠳⠦⠤⠤⠶⠒⡏⠀⠀⢿⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣆⡔⢁⣀⣤⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣸⡒⠛⠀⠀⠀⠀⠀⣞⣀⣀⠀⡏⠉⠐⠢⣅⠂⠄⠀⡇⠀⠀⠀⠈⠉⠉⠀⠁⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠉⠋⡠⠖⠉⣀⠠⠤⠤⢤⠈⠛⠁⠀⠀⠀⠈⢳⡄⠈⠃⠀⠀⠀⢀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣠⠤⠔⠉⠀⢀⢀⡠⠔⠁⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠒⢮⣉⠀⠀⠀⠀⢰⠑⠢⣄⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⠘⣦⡀⠰⠂⡏⣼⢀⡀⠀⢀⠤⣠⠖⠁⠀⠀⠀⠀⠀⡀⠑⠢⠬⢍⣁⣒⠒⠚⠓⢤⡞⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⠀⠀⠀⠓⠁⠀⢷⣈⣏⠈⠉⠛⠘⢧⡀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠻⢍⣹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⡄⠀⠀⠉⠙⠲⠖⠒⠚⠖⢶⣶⠦⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣤⠴⠶⠶⠤⠤⢌⡀⠀⠁⢠⡞⡳⠒⠲⡄⠃⠀⣤⣤⣄⠀⠠⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣼⠞⠁⠀⡠⠋⠉⠉⠀⢀⣚⣳⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠄⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠒⠛⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣪⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠛⠉⠉⠀⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠙⠄⠀⠤⡀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠞⠀⠸⡤⠓⢄⢆⠀⣸⠏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢲⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠋⠁⠀⠈⣧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣼⠀⠀⠀⠀⠘⠛⠀⡤⠂⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠳⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⠀⠀⢠⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣴⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⡌⡸⠊⠔
⠀⠀⠀⠀⠀⢠⠖⢀⠢⣀⡖⡹⠂⠀⠀⠀⠈⠲⢄⣀⠀⠀⠈⠒⠊⠀⠀⢀⣀⣤⠴⠋⠁⠀⠀⠀⠀⠰⡄⠑
⠀⠀⠀⠀⠀⢸⢎⠢⠁⠈⢱⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡟⠚⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣤⣠⡤⠚
⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⣌⠲⢄⡀⠀⠀⠀⣀⣀⣀⣀⡤⠚⠃⠈⠓⠲⣄⣀⣠⠤⠤⢲⡀⡀⠠⠊⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠓⠦⣌⣉⠒⠲⡏⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢸⡧⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣤⣶⡖⠉⠙⠒⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢸
⢀⡔⠉⠟⠛⠃⠀⠀⠀⡸⠛⠳⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣸
⣜⣠⡀⠀⠀⢠⣿⣿⣿⡄⠀⠀⢸⡄⠀⠀⠀⠀⠳⠶⠶⡶⡶⠒⠒⠛⠋⡟⢹⠉⠉
⣿⣿⣿⠂⠀⠈⠻⣿⡿⠓⠀⢴⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡈
⢻⣿⣿⠀⠀⠀⠀⠄⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⣀⣀⣀⣇⡇⠀⠀⠀⢀⣧⣧⣤⣄⣀⡀
⠀⠳⣄⠀⢠⣶⣶⣷⣦⡄⢊⡴⠋⠀⣀⠴⢞⠋⠉⠁⠡⠨⠹⢦⠀⣴⠩⠀⠀⠀⠀⢀⠝⠓⣄
⠀⠀⠈⠓⠾⠿⠿⠿⠿⠟⠋⠀⠀⣼⠁⠀⠀⢡⣀⣀⣤⠴⠞⠉⠀⠙⠓⠶⢤⣤⣀⣘⡀⠀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⠦⢤⣤⣤⣤⡤⠴⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣀⢀⠀⠀⠀⡀⣠⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠾⠯⢭⣭⣭⠭⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡤⠖⠛⠉⠉⠀⠀⠀⠀⠈⠉⠓⠦⢄⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠦⣌⣉⠒⠲⠤⠤⠤⠤⠤⠴⠖⠒⣋⡥⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢉⠉⠛⠒⠒⠶⠶⠖⠒⠚⠋⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⡱⢰⣳⠆⡆⢰⣦⣠⠀⣷⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠘⠉⠀⠃⠘⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣧⢦⡖⣠⣆⠠⣶⠀⣶⠀⣾⣽⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢤⣀⡀⠀⠀⠉⠙⠀⠉⠀⠙⠁⠀⢀⣠⠴⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⠴⠶⠒⠛⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣝⣓⠲⠶⠶⠶⠶⣒⣛⣭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣛⣙⣻⣉⣟⣻⡟⣹⡿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠⠒⢆⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢟⣭⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣾⣤⣤⣤⣷⡄⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠛⠛⠛⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⢿⣿⣿⠟⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠉⠉⠁⠒⠢⢄⡀
⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠄⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⢼⡇⠙⠋⣿⢂⡥⡐⡄
⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⠖⣫⣽⠶⠖⠒⠈⣑⣳⣤⡇
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⣼⠋⣡⣴⣿⢧⠀⠀⠐⢮⣽⣿⣿⣿⡀
⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠐⡄⢰⠒⠒⠂⢙⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠋⠀⠀⢙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣄⠀⠀⢈⣙⠀⠀⠀⠊⠁⢀⠀⠀⠀⢶⣖⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠙⠀⠀⠀⠈⠙⠃⠀⠀⠈⠛⠉⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠤⠤⠤⣀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠶⠞⠁⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⠈⠀⠀⣴⣄⣠⣶⠟⠁⠀⠘⠆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠦⣥⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⠀⠀⢀⠀⠁⡀
⠀⠀⠀⠀⠀⠆⠂⠀⢀⣤⠀⠀⠀⠈⠁⣆⣀⡐⢏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣄⣀
⠀⠀⠀⠀⢠⠀⠀⠀⢸⡇⠀⠀⣠⣶⣿⣿⣿⢡⣿⣧⠁⠀⢀⣠⣴⣶⣦⠀⠀⢸⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣧⡀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠔⠂⢄⠀⠀⢂
⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣄⠀⠄⠆⠀⠀⠀⠀⠀⠴⠂⡀⠀⠀⢢⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠀⠀⠀⡤⠀⠁⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡤⠄⠀⠤⠤⠀⠂⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠾⠃⠀⠘⠛⠀⠀⠀⠀⠀⢀⡀⡛⠓⠀⠛⠶⠀⠀⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠣⠤⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⡖⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣉⠉⡀⠀⠀⡀⣀
⠀⠀⠀⠀⢀⠊⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠁⡉⢀⠀⠞⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⢄⡀
⠀⠀⡀⠔⠋⠀⠠⠂⠁⠀⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿
⠀⡞⡺⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢀⠀⡐⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⠀⢀⡄⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠐⡰⠀⠀⠀⠀⠀⠀⠁⠏⠀⠀⠘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠂⢀⡀⠀⠀⠀⠛⢿⠏⠂⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠞⠁⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⣀
⠀⠀⠀⠀⠀⠘⠀⠀⠄⠂⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠁⠐⠐
⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⢁
⠀⣤⣣⡝⣈⢢⠄⠀⠀⠀⠀⠃⠀⠒⠀⠄⠁⠆⠀⡔⠇⠂⠀⠀⢄⠀⠀⢀⡾⣀⠀⣣
⠀⠉⠑⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠲⡒⠋⠉⠉⠙⢲⠖⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⣠⣽⠖⠒⠒⠲⣧⣄⠀⠀⠀⠉⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣀⡤⠤⠒⠒⢻⠹⣑⠤⣄⣠⠤⣊⠏⡟⠓⠒⠢⠤⣀⡈⠳⣄
⠀⠀⠀⠀⠀⢠⢞⡥⠖⠉⠁⠀⠀⠀⠀⠀⠑⣮⠭⣭⣭⠭⢵⠊⠀⠀⠀⠀⠀⠀⠉⠓⠮⣳⡄
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠈⠙⠛⠉⠉⠀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⡌⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠐⠂⠀⠀⣠⠞⠀⠀⠀⢀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠴⣷⠀⣄⡀⢡⠀⠨⡐⠒⠀⠤⠄⠒⠎⠙⠀⠀⠀⠀⠂⠀⠀⠈⠢⠨⠁⠂
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠒⠓⡁⠀⢱
⠀⠀⢠⠁⠀⠀⠒⠒⢢⣤⠄⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⠔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠾⠯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣖⠀⠀⢀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠻⠉⠉⠑⠢⣄⠀⠠⡀
⠀⡀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠂⠀⠀⠀⢁⣀⡀⠠⠈⠳⡀
⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣷⡀⡀⠀⠠⣾⣿⣿⠀⠀⠀⢱
⠀⠀⠁⠀⠉⠂⠊⠀⠀⠀⠀⢀⠀⠙⢲⣤⠀⢀⠀⣀⢀⣠⠤⠒⠊⠁⠀⠞⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠻⢿⠃⠀⠀⠀⠈⠛⠋⠑⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⡇⠸⣿⠀⠀⠀⠘⠉⠀⠀⢀⣀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐⢄⠀⠀⡀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢧⠀⠻⣷⣤⣤⣶⠶⠶⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⣼⣿⣿⠆⠀⠀⢘⠟⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠐⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠤⠍⠩⠤⠤⠚⠁
⠀⠀⠀⠀⢀⠅⠀⠰⠀⠀⠀⠀⠃⠀⠀⠈⠉⠐⠒⠀⠈⠈⠉⠁⠀⠀⢤⠀⠀⠀⠐⠁⢀⡀⠔⠚⠉⠉⡷
⠀⠀⠀⡠⠁⠀⠀⢀⡆⠀⠀⠄⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠑⡀⡴⠊
⠀⠀⡰⠉⠂⠄⠔⠀⠀⠎⠀⠀⠀⠀⠈⠒⠠⠤⠤⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⡠⠒⠉
⠀⣰⠁⢀⡤⠊⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⡠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠉⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣤⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⡇⠀⠀⢠⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠖⠘⠛⠛⠋⠀⠀⠀⠀⣀⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠳⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣤⣶⠾⠿⠋⠉⠁⠀⠀⣀⣤⡔⠓⠂⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠷⠶⣤⣄⡀⠀⠠⣤⣄⠀⠀⠈⠉⠻⢷⣄
⠀⠀⢀⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣆⣀⣀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⠶⠾⠛⠀⠀⠀⠀⠀⢀⣠⣴⡿⢇
⠀⠀⠻⢿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣀⣤⣤⣤⣶⣾⡿⠿⠛⠁⠀⠘
⠀⠈⠶⠦⠞⣻⣟⠛⠻⠷⠶⠶⢶⢶⣶⣴⣦⠤⣤⢶⡶⠤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠖⠛⠋⠉⠉⠉⠁⠀⢠⠛⠉
⠀⠀⠀⠀⡆⠉⠉⠛⠛⠲⠶⠶⠆⣠⣴⣦⣤⣤⣤⡤⠤⠠⠄⣀⠴⠦⠤⠄⠀⠀⠀⠀⠀⣀⡴⠦⢀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⣀⣀⣠⡴⣶⠚⠋
⠀⠀⠀⠸⡧⠠⢤⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⡶⠾⠛⠛⠉⠉⠁⠀⣾
⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠛⠺⠷⠶⠶⠶⠶⣶⣶⣤⣤⣴⣦⣤⣤⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠚⠒⠂⠀⠋⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣧⢀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣼⠇
⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠁⠀⠀⠀⠀⢀⡰⠶⠛⠛⠛⠛⠲⠦⣀⡀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⣿⡏⠛⠢⠤⠄⠀⣰⡁⠀⠀⠂⠀⠀⠀⠀⠠⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠒⢦⣤⣾⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠒⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⠀⠉⠻⢈⠙⠉⠛⣻⠟⠛⠉⠉⠙⠛⠛⠛⠛⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠻⠇⡀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢠⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠂⣀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠠⠀⠀⠀⣠⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠻⠀⠀⠀⣻⠄⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠤⠤⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠈⠢⠀⠀⠈⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢀⣀⡄⠺⠶⠀⠈⠉⢣⣤⣀⣀⣀⣤⠶⠶⠀⠀⠀⠀⡄⢨⠀⠈⠉⠉⠛⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠤⠉⠁⠀⠀⢀⣴⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣀⡀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠋⢀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⠶⣤⣤⣀⣀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠿⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣶⣦⣤⣄⣀⣤⣀⣀⣀⣀⣀⣀⠤⡴⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠙⠻⢦⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣄⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠀⠈⠀⠀⢬⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⣀⣤⢤⠎⡇⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠤⠒⠉⠁⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠃⠀⣐⣲⣀⠀⠀⠀⢀⣐⣆⢄⠀⠈⠑⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⡀⢰⠁⢠⣖⣷⠀⡼⠁⣠⡬⢳⡀⠀⠀⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠢⠤⢽⠃⠀⢳⡀⠛⢟⡼⠁⠀⢠⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠘⠖⠀⠀⠈⠉⠉⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣀⣒⣁⣤⡴⠋⠣⡀⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠉⠫⠔⠊⠁⠀⠀⠘⢦⢀⣤⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠙⢦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⡀⠀⠀⢐⠗⢲⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠱⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠚⣢⢃⣐⣂⡥⠖⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠑⠈⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠈⠳⢖⣸⣮⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡅⠀⢄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡠⢾⣧⡎⠀⠀⠹⣗⣄⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢳⠆⠤⠤⠥⠤⠤⠤⠤⠬⠴⠒⠚⠁⠘⡄⠘⠢⠖⠚⠁
⠀⠀⠀⠀⠀⣜⣀⠀⠀⢀⣶⡒⠆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢠⠒⡄⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⣯⠢⠀⠘⢿⣿⠖⠀⢴⣹⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⡿⠇⠀⠀⢀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠋⠉⠉⠑⠃⠀⠙⠒⠲⠤⠤⡤⢤⠖⠒⠁⡀⢴⠓⡄
⠀⠀⠀⠀⠀⠀⠱⢄⣀⣿⣿⣮⣒⡤⠞⠀⠀⠀⠀⠀⠀⠀⣠⢎⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠔⣊⡤⠖⠛⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠘⢐⠂⠤⠤⣀⡴⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠘⡆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣒⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠙⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡀⠀⢸⡏⠉⠛⠛⣿⠛⠛⠛⢿⠟⠛⠛⠛⢻⡟⠛⠉⠙⣿⠁⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⢀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡿⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡉⠻⠿⣿⣷⣤⣀⣠⣧⣀⡀⠀⣠⡀⠀⣀⣀⣨⣥⣀⣠⣾⠿⠂⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⠿⢿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣷⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢀⣤⣤⡾⠿⠿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠿⣷⣤⣄⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⣿⡟⠀⠀⠀⣠⣴⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣤⣀⠀⠀⢸⣿⡆
⠀⢠⣿⡇⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⢿⣷
⠀⣾⣿⣷⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿
⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⢀⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣼⣿⠁⠹⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿
⣿⣿⣦⡀⠀⠉⠻⠿⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⠟⠋⢁⣴⣿
⣿⡏⠙⠿⣦⣄⠀⠀⠈⠉⠛⠻⠿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠿⠋⠉⠀⣀⣴⣿⠿⣿
⣿⡇⠀⠀⠈⠻⣿⣶⣄⣀⡀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠟⠛⠉⠉⠀⠀⢀⣀⣴⣾⡿⠛⠁⠀⢿
⣿⡇⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⡿⠟⠋⠁⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣷⣶⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣶⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠉⠉⢿⣷⠀⢀⣴⡿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣤⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣿⠀⢸⣿⠁⣀⣽⡿⠀⠀⣿⡇⠀⠀⣿⣿⣿⣆⠀⢸⣿⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⠿⠃⠀⢸⣿⣿⣿⡉⠀⠀⠀⣿⡇⠀⠀⣿⡇⠈⠻⣷⣼⣿⠀⠀⠀⣿⣿⠙⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠸⠿⠉⠙⢿⡶⠀⠀⣿⡷⠀⠀⢿⡿⠀⠀⠘⠻⠟⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣠⣄⠀⢀⣾⡇⠀⠀⢠⣤⡀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣤⣶⣶⡶⠀⠀⣾⡟⢉⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣷⣾⣿⠀⠀⣰⣿⣿⣿⣆⠀⠛⠻⣿⣿⠛⠃⠀⣿⣿⣁⣀⡀⠀⢠⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠈⢻⣿⠟⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⣿⣷⡄⠸⣿⡿⠛⠛⠛⠛⠀⠀⠀⠀⠀⢀⣴⣿
⣿⣿⠀⢰⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠈⠻⣿⠃⠀⠻⠿⠃⠀⠀⠙⠿⠿⠻⠟⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⣿
⣿⣿⠀⠀⠀⠿⢿⣾⣷⣦⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⠿⠛⠁⢠⣿
⣿⣿⠁⠀⠀⠀⠀⠉⠉⠉⠉⢻⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⡿⠿⠋⠉⠀⠀⠀⠀⢸⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⠿⠿⢿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⢀⣤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠀⠀⣸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠹⣿⣆⠈⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠈⢻⣷⣄⠈⠛⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠈⠛⠿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠛⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣄⠈⠉⠛⠿⠿⢿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⠿⠿⠟⠋⠉⢀⣤⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⣿⣦⣤⣤⣀⡀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⣀⣠⣤⣤⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⢿⣿⣿⣶⣶⣦⣤⣴⣤⣴⣶⣶⣶⣿⣿⣿⡿⠿⠿⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣩⠝⠂⠀⢸⠿⢿⡿⠿⠟⣻⣿⡿⠋⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠉⠉⠂⠀⠁⠀⠀⠀⠀⠙⣿⢻⣿⣿⣿⣿⠀⠀⣀⡀
⠀⠰⠊⠉⠁⠒⠠⠤⢤⠒⡖⢢⠔⢠⠤⡀⠀⠸⠀⡠⢦⠀⠀⠀⠀⠀⠀⠈⠙⠦⠹⠼⣿⣿⣿⡟⠐⠛⠦⣨⠆
⠀⠇⡎⠉⠒⣲⠤⢄⡀⠉⣱⣾⡦⢾⡤⢧⣤⣇⠀⠀⠈⠀⡤⠀⠀⠀⢀⢤⡀⠀⠀⢠⣿⣿⡿⠁⠀⠻⡍⢲⡀
⢸⠰⠬⠦⢐⡘⠭⠤⠃⢸⣿⣿⠇⣸⡆⣼⣿⣿⠣⡀⠀⠘⠀⠀⠀⠀⠀⠀⠁⡴⡄⠈⠉⠻⠁⠀⠀⠀⠈⠊
⠀⠓⠠⠤⣄⣀⡉⠉⠀⣾⣿⣿⠀⣿⢤⣿⣿⡟⢲⣼⣽⣧⠀⠀⠀⠀⠀⠀⠸⣥⠇⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠈⠏⠉⠙⠻⡿⠿⣭⣼⣿⣿⠧⠲⡞⠭⠏⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠏⠀⠀⠀⠉⠀⠀⠀⣈⡑⠦⠤⠄⠠⠤⠄⡴⠭⠀⣀
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡘⠀⣀⣀⣤⣤⣴⡶⠈⠀⢀⣾⣄⠀⠀⢀⡄⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠑⠤⢀⣀⣀⣸⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠿⠿⠿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠈⠻⢿⣿⠟⠁⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣦⣤⣾⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⣿⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡠⠔⠉⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢠⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢓⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡭⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⣉⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠻⠥⣄⡀⣠⠔⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠠⠖⢒⣠⣄⣀⡀⢸⠁⣀⡀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣴⠯⡤⠀⠀⠀⣰⠃⠀⠀⠀⠉⠛⢮⣽⣿⠀⠀⠀⢠⠇⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⣲⣾⡏⠁⢀⣠⠴⢃⡠⠴⠒⠒⢦⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⢉⡒⢄⠀⠈⢧⡀⠀⠀⠀⠉⠁⡿⠚⠉⠀⠀⠈⠀⠀⢀⣠⠞⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠀⠀⠀⢙⣦⠤⠶⠒⠾⠁⠀⠀⠀⠀⠐⠒⠋⠉⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠴⢻⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠟⢦⡀⢀⡠⠖⠛⣉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢻⡀⠐⠙⠓⠒⠉⢀⣠⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⡀⢻⡌⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠋⢩⠇⠀⠀⠀⠀⠈⠉⠉⠉⠉⡿⣍⠉⠉⠉⠁⠀⠉⠉⠙⣆⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⡇⠈⠳⣄⡴⣄⠂⠤⢄⡴⠃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠠⠷⡄⠀⠀⠀⢀⣜⡗⣧⠴⢾⡋⡟⠢⢝⡲⢼⣧⡞⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⢦⢀⢼⣅⡀⠉⠙⠓⠲⠛⠧⠤⣀⠈⠙⠚⠻⠿⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢤⣤⣀⣀⣀⣈⠛⠁⠀⠉⢙⣶⠦⢤⠀⠀⠒⠾⡁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⢹⢹⠉⠉⢯⠀⠀⢈⡦⠤⠴⠊⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠐⠲⠚⠾⠖⠂⣸⠀⠀⠀⠙⠲⠤⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣤⢤⡞⠢⣤⢿⠒⢦⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣗⡽⢮⣄⣀⣙⣉⣼⣴⣻⣴⣷⣺⣿⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠖⢄⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢘⠀⠀⠀⢠⡅⠀⠀⡠⠊⡘⠉⠉⠫⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⢁⠔⠊⠉⠑⠢⡀⠓⠦⠄⠊⠀⠀⡘⣶⣿⣿⡄⠤⣴⣿⣷⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡞⠲⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢂⠛⠛⠏⠀⠀⠈⠛⠛⢀
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⣦⣤⣆⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠟⠉⠀⠁⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠈⢂⢀⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠉⠀⠰⠀⢄⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⢸⠀⠀⢇⠀⠤⢤⡀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢂⡵⠗⠊⠉⠉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠈⡄⠀⢈⠠⢄⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡁⣀⡤⠒⠁⢀⠄⠈⠢⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠑⢄⡀⠠⠖⠁⣠⣎⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠬⠐⡁⠀⠀⠀⠀⠀⡀⢈⠠⠚⠐⠚⠡⠤⡐⢂⡀⠀⠀⠀⠀⠉⠙⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⢧⣠⠤⠀⠀⠀⠱⠁⠀⠀⠈⠐⠀⠀⠈⠄⠀⠀⠀⠀⢠⠖⡆⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠑⠤⣀⡠⠐⠀⢸⠀⠀⠀⠀⠂⠀⠉⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⡀⠀⠠⠀⠡⡀⠀⠀⠀⠀⠀⠀⢀⣬⣀⠀⠀⠀⡠⠁⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠙⠢⢄⣀⣀⡠⠔⠋⡠⢤⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠏⠉⠒⠲⠤⠿⠶⠒⢉⠀⠀⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣄⡀⠀⠀⠀⠀⠀⠘⢤⡚⠉⠀⠀⢀⠄⢂⠀⣠⠴⠂⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⣀⣀⣠⣤⣽⣾⣿⣞⠁⠀⢈⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠲⢄⡀⡐⠀⢴⣶⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⣼⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠙⣇⠀⠀⣿⣿⣿⠿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⡇⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⣆⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠑⠒⠾⡁⠈⠙⠒⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢰⣖⣬⠖⠒⠐⢪⠃⠀⠈⠉⠉⢲⠆⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡲⢤⠀⠀⢿⠆⣀⣀⣠⠋⠀⠀⠀⣀⡀⠘⣄⡀⠤⣀⡀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠂⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⣀⡴⡿⠋⠀⠠⣴⡶⠤⢹⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡠⠔⠚⡭⠟⣁⣀⡠⠀⠤⠤⣖⠋⠀⣠⡆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠈⠉⠉⠉⠉⠉⠀⢀⣀⣠⠤⠎⠀⠀⠉⠢⣄⠀⠀⠉⠉⠉⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⡍⢝⣷⣒⠛⡿⡩⠍⢠⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⣠⣶⠞⠉⠙⢷⣦⣀⠀⠀⠀⠀⠀⠨⣒⣳⣀⡉⣹⡟⡉⡠⢤⠑⠀⠀⠀⠐⣄⣠⣾⠿⡀
⠠⠊⠄⢼⠋⠙⣄⣀⣀⠜⠙⠻⣆⠀⢀⡠⢄⠀⢉⣒⣉⣠⡋⠀⠢⠈⠡⠊⠀⠀⠀⠀⡇⡠⠃⠀⠓⡄
⠁⠈⢰⣇⠀⣠⣿⣿⣿⡄⠀⠀⣸⠀⢸⡓⣎⢧⣼⡄⠀⠀⠈⠓⠦⢤⣀⣀⠀⠀⠀⢀⣧⠙⢤⢖⣊
⠀⠀⠈⡿⠁⠈⠛⡟⠋⠉⠑⣾⡟⠀⠀⣣⠘⢏⣱⠿⢆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢲⣀⡧⡈⠻⡆⠀⡐⢄⠀⠀⡄
⠀⠀⠀⠈⠢⣀⣠⣤⣀⣀⡤⠟⠀⠀⠀⠉⠳⠄⠀⠀⠀⠑⢄⡠⠟⠲⣄⠀⠀⠀⠀⠀⣠⡿⠉⣆⢷⡄⢳⡄⠐⠀⠂⢠⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⡤⠞⠛⠓⠉⠉⢷⠓⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠄⠄⠠⠠⠄⠄⠠⠀⠠⠀⠄⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠋⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⠿⠈⠙⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣠⠿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢀⠰⣆⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠘
⠀⠀⠀⠀⠀⠑⠢⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⢀⠤⠘⠣⡀⠀⠃⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⢱⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠀⠀⠀⢀⠤⠤⠀⢐⠞⠁⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡠⡀⠀⢀⠊⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠁⠀⠀⡥⢀⡌⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠑⢄⡀⠄⠑⣨⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠑⢄⢀⡐⠣⠀⠀⢻⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠠⡈⠑⢼⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡈⠢⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⢄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠐⠂⠤⠀⢀⣀⣀⣀⣀⠀⠤⠜⠀⠀⠀⠀⠉⠳⠄⣀⠀⠀⢀⡀⠔⠊⠀⠀⠀⣀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠁⠄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠠⠘⠂⡀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡠⣜⣃⡠⠔⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡁⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡂⠁⢀⡀⠄⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠉⠉⢹⡟⠛⠿⢻⠻⠟⠻⢟⠛⠿⠛⠿⢻⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠟⢿⡟⠻⢿⣿⡿⠿⠿⠿⠿⣿