Hmmm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⠀⡤⣤⣀⡄⠆⠀⠀⠊⠀⠀⠀⣶⣴⠤⡆⠄⠀⠀⠱⠀⠸⠃⠀⠀⠀⠀⢁⠀⠀⠈⠋⢩⣉⣍⣭⣉⢉
⠀⠀⠀⠀⠘⠁⠁⠀⡀⡀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⠁⠀⠀⠘⠚⠻⠛⠚⠉⠀⠀⠀⠀⠙⠛⠛⠋⠘⠊
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢀⣀⡀⡀⡀
⠀⠀⠀⡘⠘⣉⣡⣈⡁⠱⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠠⠐⠊⠀⠀⠄
⠀⠀⠀⡼⠀⠏⠙⠛⠋⠀⠈⠀⠀⠀⠀⠀⢿⢿⡁⠀⠆⠀⠀⠀⠈⠰⣤⡞⣿⣿⡜⠀⠈⠀⠀⠀⠀⢸⣿⡂⢠⣤⠠
⠀⠀⠀⠀⠕⠤⡀⣤⠤⠒⠁⠀⠀⠀⠠⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⢶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠉⣀⣀⣄⡀⡆⠀⡀⠀⠈⠀⣴⣤⣤⣤⣠⡄⠀⠀⠀⠀⢠⡄⢠⠀⣄⣀⢰
⠀⠀⠀⠀⠁⠀⠀⠈⠙⠃⡠⠁⠀⠀⠀⠈⠉⡉⢀⠤⡒⠉⠀⠀⠀⠐⠉⠈⠉⠉⢉⠤⣦⣤⣤⣤⡼⠙⠎⠙⠉⠀⢸⣤⣤
⠀⠀⠀⢀⡀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⢰⠒⢒⡀⠒⡚⠛⢻⣿⣿⣶⣶⣦⣶⣶⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⡀⠀⠀⠀⠀⢀⢠⠀⠖⢠⡔⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠈⠠⡀⢒⠀⢢⡾⠐⠀⠀⠀⠀⠠⠈⢰⣤⢢⠀⠁⠀⠀⠀⢀⡈⠀⢀⣤⣤⠈⠆⠀⠀⠉⠂⠀⢀⡀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⣂⣁⣀⣊⡁⣀⠄⠀⠀⠀⠀⠁⠀⣀⠁⠄⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⢂⠀⠀⠀⠀⠀⠂⠈⣥⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠀⠀⠀⠀⠀⠀⠈⠈
⠄

⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⣶⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠳⢦⣌⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠙⣧⣠⡴⠞⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣹⣧⡄⣀⢀⡀⣠⠀⣀⢀⡄⢀⡀⣀⡀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣽⣋⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢠⡞⠉⣼⣵⣇⣿⣷⣿⣸⣿⣾⣇⣿⣷⣿⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⢀⡿⠀⠀⠿⠸⠿⠻⠏⠿⠟⠿⠹⠿⠿⠏⠿⠶⠶⠶⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠃
⠀⠀⠀⠻⣄⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣌⠁⠉
⠀⠀⠀⠀⠈⠉⠛⠋⢹⡇⠀⠀⠀⠀⠀⠀⢀⡼⠙⠢⣄⠈⠑⠦⠤⠄⠀⢀⡀⠀⠀⢀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡻⣄⣀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠉⠓⠒⠒⠒⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣍⣉⣉⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⠀⠀⡀⠀⣴⢢⡀⢀⡠⢤⣀⡠⣤⡀⠀⡤⠀⢠⠖⠦⣤⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠇⠰⠋⢀⡾⠁⠀⢹⡍⠽⠓⠛⠙⠉⠉⠑⠃⠈⠁⠀⢿⠀⠀⠚⠃⠀⠚⠛⠲⡄
⢀⡀⠠⣤⡀⠀⠀⠘⢧⣳⣄⣿⡦⢰⡄⣠⢀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⡀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⢯⡀
⠀⣻⣦⠟⢷⣬⠷⣴⠃⠙⠊⠋⠀⠀⢀⡀⡈⢀⡆⢀⠀⠘⣯⣉⡷⠏⢷⠸⠉⠛⣿⠰⠃⠀⠈⢦⣀⠀⠠⣌⡳⣄⠐⡞⠃
⠀⠀⠙⠓⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣺⣯⠿⠛⠥⠤⠀⠀⠀⠀⠀⠀⠀⠶⢖⡀⡿⠒⠀⠀⠀⠀⠈⢧⣤⡼⠛⣞⢶⠃
⠀⠀⠀⠀⠀⠀⢀⣤⣄⠈⠆⠀⠀⠀⠨⢓⡠⣠⠍⠙⠀⠀⠀⠄⠀⡀⣀⢀⡄⠈⠁⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⣶⡾⣷⣿⣄⠠⠄⠀⠀⠀⠘⠁⠀⠀⠀⠰⣦⢶⡼⣽⠿⢿⡟⠃⢠⠤⠞⠉⠘⠓⠶⠄⠀⠀⠏⣆⠀⢹⡆
⠀⠀⣶⠲⡏⢻⣿⠟⠀⠈⣷⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠙⠃⠀⠀⠀⠀⣀⠈⠁⠀⠀⢤⡀⠢⠄⠀⠀⠠⠶⢿⣀⠘⠁
⠀⠐⢦⣤⠟⠋⠹⠇⠀⠉⠉⡀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⢉⡤⠂⢈⡉⢉⢀⡀⢀⣀⠀⢠⡄⡄⡄
⠀⠀⠀⠀⠀⠀⣀⣴⠾⠉⠈⢀⡀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠃⠀⠀⠀⠐⠀⠊⠀⠘⣄⠙⡞⢿⠟⢾⠬⣥⠘⠃⠃⠛
⠀⠀⠀⣀⢠⡖⠻⠄⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀
⠀⠀⢤⣼⡉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠷⠰⠛⠲⢤⠀⠀⢰⠰⣄⢀⠀⡀⣀⡸⣇⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⣠⠶⠂⠀⠀⣀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠴⢼⡇⠀⠘⣆⠈⢻⣇⢻⣿⣁⣿⡙⣆⠘
⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠐⣇⠀⠀⢰⠋⠉⠀⠀⢀⠈⠒⠚⠀⠀⠀⠀⠀⠀⢙⣆⠀⠈⠓⠾⠋⠉⠁⠉⠉⠓⠙⠐
⠀⠀⠀⠀⣼⢦⣠⣤⡀⠀⠀⠀⠈⢻⡃⠀⠀⠀⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⠀⠀⠀⣿⡛⠠⠤⠀⠈⠓⠞⠃⠀⠀⠀⠀⠑⢄⠀⢀⣷⠠⣀⣀⣠⠟⠲⢍⡁⣿⡄⣘⣆⣤⡠⣖⠀⣧
⠀⠀⠀⠠⡿⣤⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡉⠉⠁⠀⠀⠀⠀⠀⠀⠒⠚⠚⣷⠛⠙⠙⠛⠲⠓⠹⠘⠆
⠀⠀⠀⠐⠀⠌⠛⠉⠉⠁⠶⠂⠠⠀⠀⠀⠀⠀⠀⢀⡐⣦⠬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⣀⡀⠀⠀⠀⠀⠀⣶⡀⠀⠀⣴⢆⣤⠀⣀⠀⠀⠙⡿⠀⠀⠀⠐⡆⠀⠀⠀⢰⣤⡀⠀⠀⠀⢀⠀⠀⠀⠴⡶⠆⠈⣹
⠀⢠⢿⠝⢦⡯⣻⡿⠷⡜⣗⢦⠈⠁⠚⢡⡞⠋⠀⠀⣿⠁⠰⣆⣸⣔⡇⣤⣦⣴⣆⠈⠀⠱⣴⢷⡇⢾⢿⣼⣆⢿⠀⣠
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠋⠀⠀⠹⠟⠻⠃⠙⠙⠉⠋⠘⠀⠀⠙⠈⠁⠀⠀⠈⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⢠⣽⠆
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠳⠛⠛⠛⠚⠛⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠶⠞⠁
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠖⠤⠤⠚⠣⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠀⠀⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⣡⣤⢦⣶⠶⡂⠆⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠇⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠷⠛⠊⠃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠛⠛⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢝⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠐⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠁⠢
⠀⠀⠀⠀⣠⣠⣤⣠⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⡀⣀⣀⣈⣉⡲⡀⠀⠀⠀⠀⠀⢸⠀⡤⣤⣶⢲⡆⡆
⠀⠀⠀⠘⠗⠛⠛⠳⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢯⠿⡭⠿⠿⡝⠀⠀⠈⠀⠀⠀⠀⠸⡀⠷⠟⠛⠚⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢤⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠉⠁⠀⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡀⢀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠘⠲⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢙⣯⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢶⣴⣶⠖⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢊⠋⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣀⢀⡀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡆⢰⡀⡖⢸⣄⡖⠂⣷⡸⠉⢸⡆⡔⠒⢲⣄⠖⠒⣦⢀⠤⠤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡇⢸⢰⠁⠀⣿⠇⠀⢸⣧⡇⣀⢸⡏⡀⠀⣇⡞⢈⢀⡿⠃⠀⣽
⠀⠀⠀⠀⠀⢀⠈⡇⠸⠗⠃⢸⣿⠀⣆⠛⣸⡇⢸⡟⠀⣿⠈⣸⡇⢸⡿⠀⣿⠈⣱⡏⠀⣿⠤⢤⡤⢄⣀⣄
⠀⠀⠀⠀⢀⣏⠀⡷⢠⡟⡇⢸⡇⢸⣿⠀⣿⡇⢸⠇⣸⢟⣠⡟⢧⠼⢧⠾⠛⠶⠏⠳⠶⠛⠶⠾⢤⣼⣧⣠⡷
⠀⠀⠀⠀⠘⣿⣄⢹⠸⡇⡇⢸⣇⣸⠿⣼⠇⠛⠛⠛⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠋
⠀⠀⠀⠀⠀⠘⠿⣿⣠⣧⡟⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣿⣄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠶⠖⠋⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⠻⠿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣶⡶⠶⠚⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣦⣤⣀⣀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⣀⣀⣤⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣉⣉⣉⣉⣉⣹⣿⣿⣏⣉⣉⣉⣉⣉⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣭⠶⣒⣻⣭⢴⣾⣿⣿⣶⣶⣿⣿⣷⡦⣭⣟⣒⠶⣭⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢶⡿⢛⣵⣾⡋⠉⠉⠉⠉⠉⠉⠉⠉⢙⣷⣮⡛⢿⡶⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⠋⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⡀
⠀⠀⡔⡀⢰⠀⢠⣾⣻⢖⡆⠀⠀⢠⡀⠀⠆⡁⠤⡀⢄⣠⡄⠰⠒⠶⡀⠀⣗⡇⠀⠀⠀⠀⡎⢈⠥⠠⡄⢣⢀⣀⡌
⠀⡼⠗⠓⠛⠚⠋⠇⠄⠸⡇⠀⠀⠈⠁⠀⠂⠄⠀⠘⡙⠉⠀⣂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠲⠶⠷⠶⠿⠿⠋
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠒⠀⠈
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⠐⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⣀⢠⣦⡆⠀⠀⠀⢸⡏⢹⡇⣤⢠⡄⣤⠄⣿⢶⣦⠀⣿⢠⡿⡀
⠀⢀⣦⣠⣆⣀⡀⣠⣄⣤⢤⣠⢰⠾⡽⠹⠹⠃⠀⠀⠀⠈⠷⠾⠃⠿⠾⢃⠿⠆⢿⠾⡿⢀⣿⣘⣧⠆⠀⡀
⠀⣸⠙⢿⠻⠼⠃⠷⠋⠛⠊⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠓⠢⠤⣄⣀⣀⠈⠈⢐⣂⣁⣈⠠⠥⠐⠈
⠀⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣽⣶⠀⠀⠀⠀⠀⠁⡐⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠠⢀⡁⢠⠄⠒⢇⠚⠃⢘⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠃⢈⠈⣆⣉⡼⡒⣵⠋⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠫⣿⣷⣿⣾⡿⢛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠒⢤⣰⢂⢰⣧⣤⡼
⠀⠀⠀⠀⠀⠀⡀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⠉⠙⠛⠛⠉⡀⠤⠤⠌
⠀⠀⠀⠀⢠⡀⣇⡈⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠐⠊⠁
⠀⠀⣀⠀⠈⠍⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⠠⠀⠀⠀⠀⠀⠄⢀⠐⠄⣸⠀⢀⠈⠀⠀⢀⡀⡀
⠀⠀⠀⢀⣄⢀⣀⣤⣴⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠰⠾⣷⠾⢿⠷⠛⠂⠀⠀⢸⡷⣷⣷⣶⣶⣶⣶⣶⣶⣴⣀⣀
⠀⠀⠀⠸⠟⠛⠋⠁⠀⠀⠄⠀⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠟⢁⣘⡿⠋⡹⠋⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣷⡼⠋⡠⠊⠀⢠⠁⠖⠟⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠘⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠠
⣏⡠⠊⡀⣤⠫⡇⠇⠀⡆⠀⠀⠀⠈⡁⣤⣄⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡀⣄⠀⠀⠀⠀⠀⠂⠄
⠋⠠⡆⠑⠌⠀⠀⢐⠀⢧⡀⠀⠙⠂⠃⠁⢉⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠉⠛⠻⠾⠠⢱⠆⠀⠀⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢲⡆⠠⠒⠋⠉⠀⠀⠀⠈⠸⠀⠻⠧⠨⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⠇⠈⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⣐⣴⣾⣶⣯⣆⠀⠀⠀⠀⠀⠀⣠⣞⢯⣿⡆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣍⢁⡁⣈⡀⣉⣿⣿⣿⠀⠀⠀⠀⠤⣿⠛⡛⠍⣁⣿⠯⠀⠐⠀⢀⣼⣿⣻⡿⠟
⠀⠀⠀⠀⠈⠃⠀⠀⠁⠃⠃⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⢠⠿⣷⣿⣿⣿⡿⠄⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠈⠘⠉⠑⠉
⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠺⠛⠟⠻⠛
⠀⢠⣞⢻⡛⣻⡄⠀⢰⡶⡄⡆⢰⣴⣲⢴⣖⡆⣶⢰⡖⠀⠀⠀⠘⠺⠲⡆⠀⠀⠄
⠀⣸⠛⢻⡛⢻⡇⠀⢸⣏⡿⢷⣸⢿⣭⠿⣯⡿⢿⡸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⢿⠛⢿⠟⢻⡇⠀⠀⠤⠤⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⠀⠀⠆
⠀⢸⣟⢻⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠁⠀⡄⣤⠤⠄
⠀⠀⠙⠛⠛⠛⠛⠒⠀⢀⡄⠀⡄⡄⡄⡰⠤⠀⠀⠐⠀⠀⠀⢀⣀⣠⡀⣄⣤⠀⠂⠀⢀⠄⠂⠀⠀⠀⠀⠃⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠐⠂⠘⣧⡿⣾⣇⡽⢿⠿⡟⠀⠐
⠀⠀⠐⢊⣶⣺⠀⠀⠀⢒⡀⠖⠒⠒⠀⠀⠈⠁⠀⠐⠀⠈⠁⠁⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠇
⠀⠀⠀⢠⢠⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠈⣁⠀⠀⠀⠀⠀⠀⠰⠉⠨⠠⠘⠃⠀⠀⠀⢀⣤⣶⣶⣶⣦⣤⡀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠛⠉⠃⠀⠀⠀⠀⠀⠀⡷⡸⢿⢷⠰⠀⠀⠀⠀⣿⣿⡝⢹⢼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣆⣴
⠀⠀⠢⢀⠤⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣼⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣥⣾⣿⡿⣿⣿⣽⣿⡿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠛⠛⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⢟⣿⣿⣿⣿⣿⣿⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⠛⣿⡿⣟⣿⢿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡷⣿⣿⢿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠉⠁⠀⠉⠀⠈

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢂⠀⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⡒⣒⣶⣶⢒⠒⠂⠂
⠀⠀⠀⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡠⣤⣤⢠⣤⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣷⣿⣷⢰⠀⠀⠀⠀⠀⢠
⠀⠀⣤⣄⣠⡶⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠾⠿⠇⠿⠿⡟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠙⠀⠀⠀⠀⡠⠂
⠀⡼⢸⣿⣿⣷⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠓⠢⠶⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠉
⠀⢈⣆⡙⠻⠛⣃⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠁⠀⠉
⠀⠁⠐⠙⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡄⠁⠀⠐⠀⠁⠀⠀⠁⠤⡀⠀⠀⠉⠀⢀⠀⠐⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⢀⡀⢠⣠⣤⣏⡋⠴⣋⣭⠀⠀⡌⠀⢀⠀⠀⠀⠀⣀⢀⠀⢠⠑⣆⠀⡀⠈⠀⢂⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⣄⠀⢠⡴⣏⣷⢿⡇⠈⡀⠀⠈⠁⠨⢕⡒⠀⠀⡇⠀⠸⢇⣰⣦⡾⠏⠾⣷⣿⣶⡿⠤⡇⠀⢸⡄⠂⠈
⠀⠀⠀⢰⠀⠀⡆⠀⠀⠀⠀⠀⠂⠀⢸⡇⡆⣿⣠⢰⣿⣷⣽⣼⣦⢠⠁⠀⠀⠀⠈⢑⡆⠰⠀⠘⢦⡀⠀⠙⢫⡤⣶⣵⣿⡿⠏⡀⠀⠀⣰⠻⠐⣊⠥
⠀⠀⣄⠈⠀⠀⠁⠀⠀⣠⠀⠀⠀⡀⠘⣾⢿⣷⣿⣿⠾⠟⠛⠋⣉⣼⡿⠀⠀⠐⠀⢄⡂⢀⠀⠀⢂⠩⠛⣦⡀⠀⠈⠁⠀⣠⠾⣭⠛⡛⡅⢂⡩⡔⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣅⠛⠋⠁⠁⠀⠀⢠⣤⣤⠇⢈⠀⠀⠀⠀⡈⢔⠆⠈⠀⠀⠄⢂⠡⠘⣃⠀⠀⠈⠿⢡⡠⢄⣢⠱⣘⣬⢷⣹⡼
⠀⢠⢄⡄⣀⢢⠀⣀⠀⢀⠀⠀⠀⠀⠈⠁⠼⠦⢦⣀⣨⠿⡫⣪⢪⡆⠀⠀⠀⣄⢌⢦⡁⠘⢀⠀⠀⡄⢂⢡⡼⢆⢣⠐⣈⢥⢒⡉⡄⢢⣹⡜⣏⣷⣺
⠐⠾⠫⠤⠯⣬⣷⣌⢁⣜⡀⠀⠐⢤⡀⠀⠀⠀⢘⣋⣁⡋⠁⠁⠁⢁⣈⡈⠀⣈⣈⣀⠁⠈⠀⠀⢁⡀⠈⢡⡭⢉⡀⡀⣉⣀⣀⢁⢀⣠⠀⡠⠄⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠧⣀⣤⡞⢠⠀⠀⠀⢼⣿⣿⣿⣷⣶⡖⠀⠀⠀⠀⠀⠈⠁⣀⠀⢸⡦⣤⠀⣰⣿⣠⣾⣯⡪⠠⡂⣆⡌⡢⡐⢣⡰⡄⣇⢦
⠈⠀⠈⠀⠀⡀⠀⠤⠀⠀⠠⡄⠉⠉⠢⠰⠀⣠⣴⣾⣶⡈⠀⢈⡀⠀⢠⡶⣶⡀⢀⡚⢻⡄⢿⣟⡿⢠⠏⢿⡟⠹⠷⣍⡳⣼⣿⣯⣖⣟⠢⣗⡟⠿⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣼⣿⣁⠀⣯⡝⣧⢿⣿⣸⠃⠀⠿⠛⠀⠈⢩⣀⣠⣀⠁⢀⣀⣠⣋⠍⣿⣿⣿⣿⡟⢢⠬⣄⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢹⣿⡇⢿⣤⠏⠀⠈⢀⣀⣄⠀⢀⣀⣛⢛⣯⡭⣿⣿⣯⣽⡿⡿⡷⠛⠙⣿⣿⣷⣴⡬⣲⣬⣍
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠙⠟⠃⠀⠀⠀⠀⠀⠀⣁⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣄⣀⣠⣼⣿⣯⣿⣷⣼⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣾⠿⠛⠋⢹⣿⣿⣿⣿⠿⠿⠿⠟⠀⠈⠻⠛⠛⠛⠛⠛⠛⠛⠙⠋⠉⠉⠉
⠀⠀⠀⢀⡀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⢖⠂⠈⠁⢀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢖⠀⣰⠺⣀⣖⢲⡀⠞⠆⠔⠸⠴⠤⠀⠀⠀⠀⠀⢀⠀⢸⣿⣰⡄⣤⣠⡄⢡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠀⠀⠀⠀⠀⠂⢘⠋⠙⠁⠋⢙⢃⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠈⠡⣀⣠⣀⠀⠀⢀⣀⠄⠈⢀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠾⠷⠶⠷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢀⣥⠤⠤⣍⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣶⣶⣦⡀
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠹⣤⠖⠛⠙⠲⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣸⣦⢠⣠⣠⣄⣄⣤⣠⣠⡀⠀⢰⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⡼⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠋⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠙⠦⠤⠤⡆⠀⠀⠀⠀⠀⡤⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⡳⣀⣀⣀⡠⠞⠀⠈⠑⠂⠤⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠙⠲⠭⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠻⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⢠⠄⠀⠀⠀⢀⡴⠤⡀⠀⠀⠠⣀⢀⡠⣀⣀⠀⢀⡤⠀⢀⡴⠶⢤⣀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿⠀⢠⠏⠀⣠⠟⠀⠀⢿⠋⣛⡶⢽⠖⠞⠘⠑⠋⡿⠀⠚⠁⠀⣿⠀⠀⠠⠿⠁⠀⠸⠶⠦⣄
⠀⠀⠀⢀⡀⠀⠀⠀⠐⣦⡹⣆⢀⣤⣄⢀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠰⣼⡆
⠐⣶⡀⢹⣿⣄⠠⡤⠀⣬⡳⣝⢦⡻⠟⠀⠛⠐⠋⠰⠀⢀⠀⠀⠀⢻⡭⠿⣄⣠⠤⡀⢤⠤⣄⡤⡄⣸⠇⠀⠸⡇⠀⠀⠀⠀⠠⣄⠀⠀⣈⣷⡄
⠀⠼⢿⣯⡉⠛⠷⠝⠛⠍⠀⠈⠀⠀⠀⠀⠐⣦⡄⣄⣠⡷⠠⠆⠀⠈⠗⠲⠞⠛⠒⠇⠙⠀⠁⣹⡃⠋⠀⠀⠀⠙⠦⣤⠀⠙⢶⣌⢳⡄⢸⡏
⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢭⣻⡖⠛⠑⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠯⣅⣹⠋⡉⠂⠀⠀⠀⠀⠘⠶⣴⠾⠉⢳⡹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠘⠀⠀⠀⠀⠀⠜⢃⡠⣰⠏⠉⠁⠀⠀⠀⡄⠀⢀⠀⡀⢀⡤⠀⠀⠀⠀⡀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠐⣦⡼⣷⣿⣷⠀⢠⠄⠀⠀⠀⠀⠛⠁⠈⠀⠀⠀⢤⣄⣶⣰⣟⡿⡿⣿⡿⠗⢀⡄⡤⠚⠋⠐⠣⢶⣆⠀⠀⢠⠞⣆⠀⠈⣿
⠀⠀⣠⡤⣰⠷⣿⣷⣿⠏⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠿⠌⠃⠉⠀⠀⠈⠂⠘⠂⠁⠀⢀⡀⠠⢤⡀⡐⠀⠀⢀⣤⣿⠀⠀⠟
⠀⠀⡜⠂⢸⣦⡼⢻⡇⠀⠐⠶⠟⠁⠀⠀⢄⡀⠀⠀⠉⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠘⣋⠷⢀⢀⠀⠶⠽⠖⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁
⠀⠀⠈⠓⠛⠁⠀⠀⠋⣀⣤⣄⠠⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠃⣠⠏⠀⡀⠶⣄⣤⣤⣦⣠⡴⠦⠄⢸⡇⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠟⠃⠀⠀⢠⡆⠀⠀⠀⠠⠤⠄⠀⠀⣤⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⣄⣘⡏⠻⠛⠙⠗⠲⠶⠀⠡⠈⠈⣁
⠀⠀⠀⢠⣀⢴⡏⠙⠂⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢁
⠀⠀⢠⣤⣟⡉⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠆⠼⠛⠳⢦⡀⠀⠀⣦⠰⣤⠀⡀⢀⠀⣀⠸⣧⣀⠀⠘⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠰⣿⡇⠀⠀⠀⢀⣤⠶⠂⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣷⠀⠀⢻⡀⠈⢳⣷⠸⣿⣏⠁⣹⡟⣦⠀⠻
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣠⠶⠟⠂⠀⠀⠈⠲⢤⡤⠆⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠙⢦⣤⡿⠳⠟⠋⠻⠛⢷⠸⡇⢠
⠀⠀⠀⠀⠀⢤⡀⠀⠀⠉⠀⠀⠀⠀⠳⢦⣤⠀⠙⠀⠀⢦⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡿⠃
⠀⠀⠀⠀⣠⡿⠓⠶⠛⢦⣄⣀⠀⠀⠀⠀⢧⡀⢀⡄⠀⠀⠠⠀⠹⡄⠀⠀⠀⢤⠀⠀⠀⠀⠀⢹⡄⣀⣀⠀⣄⠀⠀⣄⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢼⡿⠀⠀⠀⢸⣿⠁⠐⠛⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣼⠇⠲⠤⢤⠴⠿⠁⠛⠦⣌⢿⡖⢠⣹⡤⣶⡜⠯⡁⢸⣦⡀
⠀⠀⠀⠀⠴⠿⣦⣀⣤⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⢻⡟⠁⠋⠙⠛⠒⠛⠈⠧⠙⠆
⠀⠀⠀⠀⠂⢀⠌⠛⠋⠉⠛⠰⠶⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣤⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⢀⡀⠀⠀⠀⠀⠙⢻⡟⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠘⢲
⠀⠀⣹⣓⢦⡰⣆⢶⣰⣦⡌⢿⣤⣀⠀⠾⢡⡾⢁⣴⠶⠀⠀⢰⡞⠃⢀⡀⢠⡆⢸⣅⡀⣀⢀⣀⠙⠟⠀⢦⠀⣦⢰⢃⣆⡆⣠⡀⢹⠀⠀⠐⠋
⠀⠘⠋⠿⠉⠛⠎⠛⠛⠉⠉⠊⠃⠙⠂⠀⢀⢀⡞⠁⠀⠀⠀⣸⠇⠀⠈⢿⣾⣷⣾⠸⣿⢿⡿⣽⠀⣄⠀⠈⢿⠙⣿⠀⠏⠋⠻⠻⠘⠇⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⣀⣀⣀⡀⣀⠀⢀⣀⣀⣀⣀⢀⣀⢀⣀⣀⢀⢠⣀⣀⠀⢀⣀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣷⡗⠀⠶⣟
⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠘⠛⠛⠛⠛⠞⠛⠛⠛⠘⠋⠛⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠛⠀⠲⠶⠋
⡿⢿⡿⠿⡿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣶⣖⠂
⠀⢀⡉⣁⣀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣷⣼⣷⣾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠆⠀⠀⠀⠈
⠠⠼⣿⣿⣾⢆⠀⠀⠀⠀⠀⠀⠉⠂⡒⠂⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠐⠬⠍⠅⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣶⣤⡆⠛⠋⠳⠨⡖⠇⠀⠀⠀⡆⣠⣠⣶⣤⣤⣼⣆⡆⠠⡀
⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⢰⣠⣳⣤⣿⣯⣯⣇⢈⠀⠀⠀⠈⠁⠀⠱⣄⡈⠛⠥⢦⣾⠿⢃⡀⢀⡖⡑⠈
⠀⠐⠀⠀⠁⠀⠰⠀⠀⢠⠜⠿⠿⠟⠛⠉⠉⡰⠞⠁⠀⠀⢐⠀⠀⠀⠀⠍⢷⠀⠀⢠⠔⠛⠊⢝⡌⣔⡬
⠀⣄⡀⡀⡄⡀⠀⠀⠀⠀⠑⠠⠄⣄⣠⣾⣿⢅⡀⠀⠠⢐⢄⠀⠀⠀⢀⠀⢂⢄⣀⢌⡊⢌⡎⣔⣮⣞⣮
⠐⠛⠒⠳⠵⠬⣬⡅⠀⢢⠀⠀⠀⣩⣥⣉⣉⡉⠁⠀⠀⠥⠌⠀⢀⡈⠄⢈⡼⢩⡄⠤⠤⠤⠤⢄⠠⠀⠠
⠀⠀⠀⠀⠀⡀⠀⢁⠲⠋⠀⡀⢀⣹⣟⠛⠛⠃⠀⣠⣀⠀⠰⣦⢴⣿⣇⣼⣷⢿⣷⠠⣠⣷⣊⡦⣪⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⣿⣅⣸⠽⡾⣷⡼⠀⠷⠃⢈⣩⣠⣄⣁⣠⣜⡋⣿⣿⣿⡗⠬⢅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⢨⣿⠙⠚⠁⠀⠰⠖⢀⣴⣯⣽⣗⣿⣿⣿⣿⡟⠉⢘⣿⣿⣿⣹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⣤⣴⡶⠞⢻⣿⣿⣿⣿⡿⠏⠙⠿⠿⠿⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢀⠀⠘⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⣠⠄⡤⡀⡤⠀⢤⣀⠈⠀⠀⠀⠀⢠⣤⣀⢀⢀⠀⣀
⠀⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⣋⠋⣛⣛⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠂⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠟⠛⠳⠶⡼⠋⠉⠉⠉⠻⣦⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠞⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁⠀⠀⠀⠀⠀⠠⠤⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⠀⣀⡀⠀⡀⠀⠀⣀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠸⠿⠆⢿⡯⣦⣠⣼⣤⣻⡗⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⢀⣠⣿⡟⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⡀⠀⠀⠀⠀⣀⣤⠞⠁⠉⠳⢦⣤⣤⣤⣤⣴⠶⠟⠃⠉⠛⠛⠛⠛⠁⢰⡏⠁⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣠⠿⢠⡖⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡿

⠀⠀⠀⠀⣤⣄⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⣶⠒⣶⣦⡶
⠀⠀⠀⠘⠧⠿⠿⠻⠻⠾⠀⠀⠀⠀⠀⠀⢸⣷⢻⠿⣝⢿⡛⠀⠀⠀⠀⠀⠀⠀⠰⠏⠼⠡⠾⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠂⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⣾⡷⣞⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢹⣼⣿⡏⠀⠀⠀⣀⠀⢀⣀⢀⡀⢀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣤⣤⣤⡤⣤⡄⠀⠀⠀⠀⠀⠀⢠⡵⣬⣠⣬⢥⠀⠀⠀⠀⠀⠸⠏⠯⠿⠿⠿⠿⠹⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠚⠀⠀⠀⠀⠀⠀⠀⠸⠇⠛⠟⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢆
⠀⠀⠀⠀⣀⣄⣠⣀⡀⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠟⠋⢧⠟⠿⠿⠫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠀⢀⡀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡴⣤⡆⢰⠄⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠾⠸⠋⠘⠋⠘⠋⠙⣹⣿⣟⣿⡏⣨⣟⣿⣿⣿⣿⣿⡟⡿⢠⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠳⠛⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠠⠒⠙⠛⠛⠙⠀⠈⠁⠈⠁⠁⠉⠉⡁⣱⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠒⠒⠛⠛⠘⠛⠓⠚⠓⠒⠒⠒⠒⠒⠚⠋⢛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣰⢳⣶⣰⢾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣰⢶⣾⣻⡆⠀⠀⣾⣹⣿⡞⠀⠀⠀⠀⢸⡴⣶⡋⣴⣷⣶⡶⡾⠀⠀⠀⠀⠀⢰⣤⢶⡖⣴⣦⡆
⠀⠀⠀⠀⠀⠈⠋⠉⠉⠈⠀⠀⠀⠀⣴⠶⣿⣿⡿⠟⠞⠉⠁⠀⠀⠀⠈⠁⠁⠁⠀⠀⠀⠀⠈⠁⠙⠉⠉⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠃⠛⠃⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠁
⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠴⠖⠊⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣒⣒⣲⣶⣶⣾⣿⣿⣽⣓⡶⣖⡂⠈⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣄⡈⠓⠥⠶⠶⠶⠾⠦⣤⣿⠾⠙⠛⢓⣂⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣛⣒⣒⣲⣶⣶⣶⣟⡁⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣺⣿⣿⣛⣻⣟⣯⣩⣞⣿⣿⣽⡻⡄⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣭⣛⣺⡿⠷⠶⠾⠿⠽⠾⣯⣖⣣⠃⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣤⣀⠀⢀⡀⣀⣮⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠉⠉⠉⠛⠋⠉⣩⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⡛⢯⠉⠿⠛⣿⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⣀⣀⣀⣀⣠⣴⣿⢿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿
⣀⢀⣀⣀⣀⣀⣀⣀⣸⣿⣧⣤⣽⣾⣿⠿⠶⣶⣿⣿⣤⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣿⠿⠋⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣶⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠻⣿⣦⣀⣀⠀⣀⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⡀⡀
⠀⠀⠀⣞⣿⣷⣿⣿⠀⠀⠀⠀⠀⣠⣴⣤⣦⣴⠄⠀⠀⠀⠀⠀⢨⢟⡟⣽⠃⠀⠀⠀⠀⠀⠀⠸⠋⠇⠿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⢠⣤⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠹⠞⠃⠀⠀⠀⠀⠀⣀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⣘⣘⢛⣛⡀⠀⠀⠂⢰⡶⣖⣦⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠿⠻⠻⠛⠀⠀⠀⠀⠀⠀⠇⠟⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⣴⣦⣶⣦⣤⣶⠆
⠀⠀⠀⠁⠀⠉⠈⠉⠈⠀⠀⠀⠀⢀⡀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠸⣶⡞⣹⢻⠟⠿⠲⠖⠖⣶⡶⣶⠖⣶⣶⣶⣶⣶⡴⡖⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠷⠻⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠚⠛⠛⠛⠈⠋⠙⠉⠉⠉⠉⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠉⠁⠉⠁⠉⠒⠀⠒⠐⠒⠉⢙⠋⠉⠉
⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⣠⣠⣠⠀⠀⠀⢀⢀⣀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⢀⡀⡀⡀
⠀⠀⠀⠀⠻⠟⠛⠛⠀⠀⠀⢀⣠⣤⣾⣾⡽⠗⠃⠀⠀⠓⠛⠃⠀⠀⠀⠘⠋⠿⠞⠻⠛⠻⠀⠀⠀⠀⠈⠟⠿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠋⠁
⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣄⡀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡴⠃⠉⠉⠉⠁⠀⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⡀⣀⣀⢀⡄⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⢀⣤⣄⡴⣶⣾⡶⣿⡃⡏⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⢸⠇⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⢸⣧⠟⠧⠏⠏⠐⠞⠓⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠿⠿⠲⠂⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⡀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠉⠒⠤⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢴⠋⠀⠀⠀⠀⠙⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⢀⡑⠄
⠀⠀⠀⠀⠀⣠⢶⣰⣦⣦⡖⠓⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⡀⣠⡄⣤⣠⢀⡌⢣⡀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⣾⣿⡞⢹⡍⣼⠀⢈
⠀⠀⠀⠀⠀⢯⠼⠿⠟⠟⠧⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⣿⣩⣿⣡⣿⣿⣿⣘⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⢸⡀⢸⣧⠿⠟⠿⠿⠖⠖⠀⠐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣉⣡⡈⠁⠈⠁⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⡤⠤⠖⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢤⡤⠤⠐⢋⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠀⠀⠣⢤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣶⡴⣶⢢⡶⠦⣴⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢀⣈⡸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⢱⣏⣡⣽⣆⡃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣶⣷⢛⣿⣿⣿⠟⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠛⠛⠙⠋⣘⠉⠉⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢀⠀⠀⠈⠀⠀⠠⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢨⠴⠂⠴⠋⠙

⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠄⣤⣠⣀⣀⠀⠀⠀⠐⠉⠀⠀⢠⣆⡆⣤⠒⠀⠀⠈⠂⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⡀⣀⣀⣀⣀⠐
⠀⠀⠀⠀⠘⠃⠉⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠈⢁⠈⠈⠀⠀⠐⠁⠀⢀⠰⠶⠿⠳⠶⠛⠀⠀⠀⠀⠘⠟⠿⠋⠩⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠰⠒⠓⡉⠓⠋⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⢀⡠⠀⠀⠠⢀
⠀⠀⠀⣹⠀⡾⠺⠿⡟⠆⢈⠃⠀⠀⠀⠘⢠⣶⡖⠠⢀⠁⠀⠀⠀⠔⢠⣀⡦⣶⣶⠰⠄⠑⠀⠀⠀⠀⢀⣦⠆⠀⣀⡁⡀
⠀⠀⠀⠉⣢⣀⠀⠀⢀⣠⠌⠀⠀⠀⠀⠆⠀⠉⠉⠀⠠⠀⠀⠀⠀⠀⠈⠋⠃⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⠙⠂⠔
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠐⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠊⠀⠀⢀⠀⠀⠀⠀⠖⠀⣀⢁⢀⣀⠀⡀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⢀⠉⠙⠛⠳⡷⠀⡀⠀⠀⠀⠀⠶⠾⠞⠓⢈⡤⠔⠀⠀⠀⠀⠿⠻⠻⠟⠚⠃⡀⠀⠀⠀⢠⠷⣸⢧⡿⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠠⠐⠢⠚⠑⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠐⣉⣻⣿⣿⣿⣯⣄⣁⣀⢀⣀⣴⣿⣿⣿
⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠤⠴⠤⡥⠤⠤⠼⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⢀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠰⠋⠀⠀⢠⡀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠐⢥⢸⠃⢀⠾⠁⠀⠀⠀⠀⠀⠢⠀⠷⠧⠝⠀⠀⠀⠀⠀⠤⠄⠀⠒⠲⠒⡐⠉⠀⠀⠐⠁⠀⠀⡄⢰⠰⠀⠡
⠀⠀⠀⠀⠂⠁⠀⠀⠀⠠⠠⠐⠀⠀⠀⠀⠀⠀⣤⡆⠐⠁⠀⠀⠀⠀⠀⠘⢉⡄⢀⠂⠠⠀⠀⠀⠀⠠⡠⡀⠶⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⣤⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀
⠀⢰⣿⣟⣿⢋⣟⣯⡟⣻⣿⠆⠀⠀⠀⣿⣿⣻⡝⣻⣿⣿⠀⠀⠀⢰⣿⡟⣿⣿⣟⣿⣿⡇⠀⠀⠀⣿⣿⣿⢿⣿⠿⣿⣿⡄⠀⠀⠘⠿⠃
⠀⠀⠻⣿⣿⣿⣿⣿⣣⡿⠋⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠾⣿⣏⠉⠉⠉⠁⠀⣀⣠⣤⣄⣀⠀⠀⠙⠂⢀⣤⣤⣶⣶⣤⣤⣀⠀⠙⠯⠉⠁⠀⢀⣀⣀⣀⠀⠈⠙⠛⡿⠟⠁⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠞⠁⠀⠀⠀⢀⣾⡿⣿⢿⣿⣿⢿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣷⡄⠀⠀⠀⣴⣿⣿⡿⣿⣷⣆⠀⠀⠀⠀⠀⣾⣿⢸⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣷⣿⣧⣽⣿⠇⠀⠀⢻⣷⣿⣾⣷⣿⣿⣾⡿⠁⠀⠀⠀⣿⣷⣿⣿⣼⣿⡿⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣿⡿⠃
⠀⠀⢀⣤⣶⣶⣤⣀⠀⠈⠛⢿⡿⠿⠛⠋⠀⠀⠐⢿⣿⡙⠛⠛⠛⠋⠁⡀⠀⠀⠀⠀⠈⠙⠛⢛⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⢠⣿⣿⣿⡟⣿⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⣴⣾⣿⣿⣿⣶⡀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠈⠿⣿⣿⣿⣿⡿⠃⠀⢠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣻⣿⣿⡇⠀⢠⣾⣿⣿⢽⣹⣿⣷⡄⠀⠀⣠⣾⣟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠜⠋⠀⠀⢰⣿⣿⣛⢿⣿⡿⣴⡄⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⠟⠛⠁⠀⠘⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⣿⣿⣿⠆⠀⠀⠀⠀⡀⠀⣀⣠⣤⣤⣍⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠏⠁⠀⠀⠀⠈⠻⠿⠿⠿⣿⡿⠛⠁
⠀⠀⣠⣤⣴⣶⣤⣤⡀⠀⠀⠀⠈⠋⢁⣀⣀⣀⠀⠀⠹⣾⣿⣿⡿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣦⡀⠀⠀⠀⠉
⠀⢰⣿⡿⣿⣯⣿⣿⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣷⣿⣿⣿⡿⠀⢀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠻⢿⣿⣿⡿⠿⠛⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⡏⠀⠀⠈⠙⠛⠛⠋⠁⢀⣾⣿⣿⣿⣿⣻⣷⡄⠈⠻⢿⣿⣿⡿⠟⣥⣄⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠉⢀⣠⣤⣴⣶⣤⣤⡈⠻⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣀⣼⣽⣻⡇⠀⠀⠀⠀⠉⠀⢸⣿⣿⣿⣟⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡉⠋⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠈⠛⢿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣾⡿⠿⠃⠀⠀⠀⢠⡖⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣷⡄⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠉⠁
⠀⠀⠀⣶⣶⣶⣶⣄⠀⠈⠁⠀⠀⠀⢀⣀⣀⣹⣿⡗⠀⠀⠘⢿⣷⣾⣿⣿⣿⣿⠃⠀⠀⣀⣤⣤⣬⣿⣦⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⢠⣿⣿⣿⣿⣿⡿⢁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡷⡄⠀⠀⠀⠉⠉⠉⠹⠋⠀⠀⠀⢰⣿⡿⣟⣿⣿⣟⣿⡇⠀⣼⣿⡿⣽⣿⣿⣿⣿⣦⡀
⠀⠈⠿⡿⣿⣿⣿⡇⠛⠣⠀⣼⣻⡏⣟⣿⣿⣥⣴⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⠃⠀⢻⣯⣷⣯⣿⣼⣥⣾⣿⠇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⠷⢿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⡄⠀⠈⠛⠻⠿⠛⠋⠁⠀⠠⣾⣿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⢠⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢿⣿⣯⣿⣽⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠁⢀⣀⣀⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣦⠀⠈⠙⠛⢿⠟⠛⠉⠀⢰⣤⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⣠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⢹⣿⣿⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣮⣛⣿⣮⣿⣿⠆⠀⠀⣿⣷⣿⣿⣭⣟⢵⡆
⠀⠀⠀⠀⠀⠀⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠀⠀⠀⠈⠛⠻⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⢿⣿⣷⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣟⣻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣻⣿⣿⣿⣿⠹⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢫⣾⣿⣿⣿⡻⠻⠀⡈⢿⣿⣏⢐⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣰⣿⣿⣿⢯⣿⣧⣰⣦⣷⡀⠹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣄⣹⣿⣿⢿⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢭⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⠟⠋⠉⠉⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠙⠢⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠚⠁⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠂⠈⠳⣗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡂⠄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⢤⠀⠀⠐⠂⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠦⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡆⠈⡙⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⣉⣉⣉⣉⣹⣿⣿⣏⣉⣉⣉⣉⣹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡵⣚⣫⡭⢷⣾⣿⣿⣷⣾⣿⣿⣷⣾⢯⣝⣓⠮⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⢟⣵⣾⣯⣀⣀⣀⣀⣀⣀⣀⣀⣽⣷⣮⡻⡟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉

⠀⠀⣠⣴⣶⣿⣿⣷⣶⣦⡀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣄⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀⠀⠀⠀⣀⣀
⠀⢸⣿⣻⣟⣼⣽⣽⢸⣿⠇⠀⠀⠀⣿⣿⣽⣭⣿⣿⠇⠀⠀⠰⣿⣏⣻⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣻⣟⣟⣿⣿⠀⠀⠀⠛⠋
⠀⠀⢹⣿⣿⣿⣿⡯⠟⠉⠀⠀⠀⠀⠙⢻⡿⠿⠛⠋⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠛⣿⠆⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠈⢀⣴⣾⣿⣿⣿⣷⣦⣄⠈⠑⠀⠀⣠⣤⣴⣤⣄⠀⠀⠈⠛⠁⠀⣠⡶⣶⣶⣶⣶⡄
⠀⠀⠈⠁⠀⠀⠀⠀⣿⡟⣿⣽⡿⢿⣿⡆⠀⢸⣿⣻⣟⣿⣿⣿⣻⣿⠆⠀⠀⢰⣿⣿⣿⢻⣿⣷⠀⠀⠀⠀⠀⣿⣧⣿⣿⣻⣹⡿
⠀⠀⠀⢀⣀⡀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⣠⣿⡿⢿⣿⣿⡿⠟⠋⠀⠀⠀⠘⠿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠈⠙⠛⡿⠛⠋⠁
⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠉⠛⠁⠀⠀⠀⠀⢀⠿⠃⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠐⠋
⠀⠸⣿⣿⣯⣾⣿⡿⠀⠀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⢽⣿⣿⡆⠀⢠⣶⣿⣿⢿⣿⣦⡀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠈⠉⠉⠿⠋⠀⠀⣴⣿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠿⠟⠁⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣟⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⠄⠉⠀⠀⠀⠀⠀⣀⣀⣀⣙⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⠟⠉⠀⠀⠀⠛⠿⣿⣿⣿⣿⠿⠃
⠀⠀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠘⠛⠉⣀⣀⡀⠀⠈⣷⣿⣿⡿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣄⠀⠀⠀⠘⠉
⠀⢸⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣾⣿⣿⠟⠀⣀⣀⣀⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠈⠻⠿⣿⡿⠿⠛⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠉⠙⠋⠉⠁⢠⣾⣻⣿⣿⣿⣿⣦⠀⠛⠿⠿⣿⠿⢋⣤⣤⣶⣶⣄
⠀⠀⠀⠀⠈⠁⣠⣶⣶⣶⣶⣦⡄⠙⠻⠿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣦⣠⣯⣿⡟⠀⠀⠀⠈⠁⠀⣼⣿⣿⣿⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣉⣽⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⡈⠙⢿⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⡿⠿⠃
⠀⠀⠀⢀⡀⠀⠉⠛⠛⡿⠛⠛⠁⠀⠀⠀⣴⣅⡀⠀⠀⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⣰⣿⣟⣿⣿⡆⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⡁⠀⠀⠙⠿⢿⣿⣿⡿⠟⠀⠀⣠⣶⣾⣿⣿⣿⣶⡄⠀⣀⣤⣶⣶⣶⣦⣤⣀
⠀⢸⣿⣿⣿⣿⣿⢡⡄⠀⢠⣾⡿⡿⣿⣿⡟⣻⣻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣯⣿⣿⣿⣿⡇⠀⣿⣿⣯⣿⡟⡟⢿⣿⣧
⠀⠀⠉⠉⠿⠿⠋⠀⠀⠀⠸⣯⣿⣷⣿⣿⣷⡿⠟⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⠀⠘⠿⣿⣿⣿⠿⠏⠀⠀⣹⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⠀⠉⠉⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⠀⠘⠻⠿⣿⠿⠟⠋⠀⣄⣀⣤⣶⣶⣦⣤⡀⠀⠁⠀⢠⣾⣷⣾⣶⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⠇⠀⠀⠀⠈⠀⠀⠀⠀⢹⣿⣯⣻⢻⣟⣿⣿⡄⠀⢰⣿⣽⣿⣟⣿⢣⣄
⠀⠀⠀⠈⠉⢉⡿⠋⠉⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⠿⠋⠀⠀⠀⠙⠛⢿⣿⠿⠀⠁

⠟⢡⣹⠟⠁⠋⢀⠖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⣷⠞⠁⠂⠀⡀⡄⠐⠚⠀⠀⠀⢄⠶⠀⠀⠀⠀⠀⠀⠀⠘⠃⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⡧⠈⠠⠐⡮⠺⠘⠀⡆⠀⠀⡀⠠⡰⠶⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣠⣴⡄⡀⠀⠀⠀⠀⡀
⠀⡼⠄⠁⠀⠀⠀⠀⠱⣄⣀⠉⠀⣀⣠⠴⠊⠀⠀⢀⠀⣀⡄⢤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠀⠋⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢹⠃⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠁⢀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣄⣰⣤⣴⡀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠟⠛⢻⣿⣷⠀⠀⠀⠀⣸⣿⠿⠿⠛⣷⡄⠀⡀⠀⢀⣼⡿⣮⣿⠃
⠀⠀⠀⠀⠲⠀⠀⠒⠠⠆⠀⠀⢿⣶⣿⣾⣷⣾⣿⣿⣿⣀⠀⠀⠀⣹⣧⣤⣶⣿⣿⡁⠀⠀⠒⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠹⠛⠻⠑
⠀⠀⠀⣐⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⠚⠗
⠀⢠⣟⣿⣛⣧⠀⠀⣶⡆⡆⢰⣶⣦⣴⣲⢰⠀⣖⠀⠀⠀⠘⠚⠶⠄⠀⠄
⠀⣿⣉⣿⣙⣿⠀⠀⣯⡿⠷⠼⠿⠽⠻⠼⠻⠤⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⣉⣿⣙⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣘⣯
⠀⠸⣏⣿⣹⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⢰⢰⡖⠠
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠂⠠⠇⠃⠃⠋⠁⠀⡀⠀⡄⣤⣤⡶⣶⣰⣶⡆⠀⠀⠐⠈⠀⠀⠀⠀⠈⠈
⠀⠀⢀⠀⡀⡄⠀⠀⢀⠀⣀⠀⠀⠀⠠⠤⠀⢀⠀⠻⠏⠟⠓⠋⠛⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⢀⠀⡁⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⠲⠀⠂⠆⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠘⠞⠸⠅⠀⠀⠀⠀⠀⣆⢰⣰⡀⠄⠀⠀⠀⣾⣿⠻⣿⡝⣿⣿⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣠
⠀⠀⠄⡀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣷⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⡄⢠⠄⣦⢠⠤⠄⢢⢀⡖⠂⣱⠀⡤⠦⠦⡄⣠⠤⠤⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡇⢸⠀⣿⢸⠀⠀⢸⡟⠀⠀⣿⢀⠇⠀⠐⣿⠃⠀⢸⡇⡜⠠⠀⣽⡴⠋⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⣇⣸⠀⣿⡜⠐⡀⢸⡇⣤⠂⣿⡸⠀⣶⠸⠇⣶⠀⢸⣷⠃⣰⡆⠟⣡⠆⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡴⠀⣷⠀⢉⣤⠀⣿⡏⢀⣿⠈⢸⡇⠀⣿⡇⢠⣿⠀⣸⡿⠀⢸⠇⢠⣿⠁⣰⣿⢀⣸⣟⠉⢱⠊⠙⣶⠒⠲⡀⠀⢀
⠀⠀⠀⠀⠀⢸⣇⠀⢹⠀⣿⢹⠀⣿⠇⢸⣿⠀⣾⢿⣀⣿⣠⣼⠻⣴⠟⠙⠟⠛⠛⠛⠁⠛⠋⠈⠙⠉⠉⠛⠛⠳⠾⠟⢶⣾⠏
⠀⠀⠀⠀⠀⠸⣿⣄⣸⠀⣿⢸⢀⣿⣀⣼⠛⠶⠏⠘⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣠⣿⣾⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⢿⣧⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣤⡴⠖⠉⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⣶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣤⣤⣤⣤⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠴⠖⠊⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣒⣒⣲⣶⣶⣾⣿⣿⣽⣓⡶⣖⡂⠈⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣄⡈⠓⠥⠶⠶⠶⠾⠦⣤⣿⠾⠙⠛⢓⣂⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣛⣒⣒⣲⣶⣶⣶⣟⡁⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣺⣿⣿⣛⣻⣟⣯⣩⣞⣿⣿⣽⡻⡄⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣭⣛⣺⡿⠷⠶⠾⠿⠽⠾⣯⣖⣣⠃⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣤⣀⠀⢀⡀⣀⣮⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠉⠉⠉⠛⠋⠉⣩⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⡛⢯⠉⠿⠛⣿⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⣀⣀⣀⣀⣠⣴⣿⢿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿
⣀⢀⣀⣀⣀⣀⣀⣀⣸⣿⣧⣤⣽⣾⣿⠿⠶⣶⣿⣿⣤⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣿⠿⠋⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣶⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠻⣿⣦⣀⣀⠀⣀⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣰⠋⠀⠀⠉⠁⠀⠀⠈⠳⡤⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⡀
⠀⠀⠀⠀⠀⣠⠖⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⡀
⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⣨⡶⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠻⠦⠤⠦⠽⠆⢀⣿
⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠘⠷⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⢀⡔⠢⣀⡀⠀⢀⣀⣠⠖⠤⣤⡴⠟⢡⠄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠚⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠦⢀⠼⢠⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⢀
⠀⢀⡆⡄⣆⢀⣴⡏⡗⣲⠀⠀⠀⣄⠀⠒⠅⠐⢄⣂⣰⠆⢐⠒⠒⠀⢘⣻⠀⠀⠀⠀⢸⠀⠵⠂⢤⢩⠤⣠⠔
⠀⠜⠋⠛⠉⠋⠉⠃⠀⢺⠀⠀⠀⠀⠀⠋⢐⠀⠘⠉⠋⠀⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠓⠛⠒⠻⠛⠋
⠀⠀⡠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠰⣶⠀⠀⢠⡌⢠⡄
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⡤⡄⡟⡟⠀⠀⠀⢸⡇⢸⡇⣷⣶⢰⡞⠀⣿⣿⡆⢸⡇⣿⠁
⠀⢸⢷⣼⣧⢢⢠⠶⣴⣎⡶⣇⠟⠓⠁⠃⠁⠀⠀⠀⠀⢙⡋⠀⠛⠋⠈⠛⡈⢟⣛⡣⠼⠯⠽⠂⢀
⠀⠞⠈⠛⠙⠊⠈⠉⠀⠁⠀⠀⠀⠀⢀⠀⠠⠀⠀⠄⠀⢤⡈⠉⠒⠒⠢⠦⠀⠀⠤⠤⠐⠒⠂⠉
⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠠⠠⠀⡀⡄⢀⡖⢻⣷⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⡈⢰⠲⠀⡇⠤⢹⣀⢄⡴⣺⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⣰⢤⣝⣲⣻⣮⣿⡄⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⡀⢈⢨⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠻⠿⠿⠟⡋⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⠧⢶⣿⠿⠋⠀⠀⠠⠂
⠀⠀⠀⠀⡀⠀⡄⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠢⣀⣀⣀⡠⠂⠈⠉⠁
⠀⠀⠀⠀⠚⠒⠓⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⡀⠀⠠⡀⣰⠘⠀⠉
⠀⠀⢀⠀⡀⠀⢀⣀⣤⣴⡶⠿⠀⠀⠀⠀⠀⠀⠠⣼⣦⣶⣿⡶⠷⠀⠀⠀⢸⣼⣶⣴⣤⣤⣤⣤⣤⣠⡀
⠀⠀⠀⢸⠟⠻⠛⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠘⠘⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣶⣤⣄⡉⠻⢿⣿⣿⣿⣏⣠⣤⣤⣤⣤⣈⣹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⣡⣄⡉⠻⣷⣾⣿⣿⣿⣿⠟⢉⣤⣄⡉⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧⠈⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠿⣷⣄⠙⠛⣁⣴⣿⣿⣿⣿⣿⣿⣦⣈⡉⣉⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣡⣶⠿⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢀⣿⠂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣠⣾⣋⣀⣉⣉⣁⣉⣉⣉⣉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆⢸⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⢠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠟⠻⠟⠿⠿⢿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⠇⠀⠈⠉⠉⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⢶⣶⣶⣶⣶⡶⣶⠾⠃




⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⡀⣀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⣿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠸⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡐⠐⣠⣆
⠀⠀⠀⠀⠀⠀⠁⠀⣀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠚⠘⠋⠓⠈
⠀⠀⠀⠀⠀⢸⠖⠇⠿⠎⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⢺⢻⢹⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⢺⡄⠳⡈⠈⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠄⠀⠀⢡⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢦⣦⠥⠶⠒⠛⠉
⠀⠀⠀⠀⢀⢀⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢀⠐⠂⠈⠁⠈⡈⢑⠑
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⣉⠈⠀⠀⠀⢀⡀⢠⠀⡀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠀⠘⠯⠷⡄⣶⠙⡶⠣⠂⠕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠊⠀⠀⠀⠚⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠴⠀⠰⣄⠸⠀⠂⠀⠀⢤⠄⠈⠀⠀⠀⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢒⠀⢀⡀⡆⢰⣄⡒⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓
⠀⠀⠀⠀⡤⢠⣤⢀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦
⠀⠀⠀⠀⠃⠈⠁⠚⠃⠺⠚⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢆⢀⠤⢄⠀⡤⠤⣀⠌⣇⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠄
⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢸⢠⠶⢠⠇⡜⠀⡏⠰⡃⢸⢀⢨⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠚⠙⠒⠑⠠⠆⠀⢄⠰⠆⡳⠀⡰⢰⡇⣇⠃⡟⢸⠀⣼⢰⠀⠀⠀⠀⡀⠠⠠⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠑⠒⠁⠈⠒⠊⡑⠂⡘⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠁⠀⠑⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠰⣶⣄⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠂⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠹⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣰⣿⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⢀⣾⡟⠁⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠙⠻⢷⣶⣤⣄⣀⣀⣀⣠⣤⣶⡾⠟⠋⠉⠉⠙⠻⢷⣶⣤⣤⣀⣀⣀⣠⣤⣶⡾⠟⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠻⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠶⠾⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡴⠖⠛⣋⣽⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠯⢭⡉⠙⠳⠤⠞⠁⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⢀⡞⣁⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠧⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠞⢀⠀⠀⠐⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⢷⠀⠀⣷⠈⠛⠦⣽⣟⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢀⣀⠌⠉⠛⠲⠦⣤⣀⡀⠹⣏⠙⠛⠲⣶⠤⣤⣈⣛⣦⣄⠉⠉⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⣩⠖⠚⠓⠶⣤⡀⠀⠀⠈⠉⠉⠁⣀⠀⠹⡄⠀⠀⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠙⣦⠀⢀⣤⠖⠒⠛⢧⠀⠹⡄
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⡾⣠⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣧⠞⠁⠀⠀⠀⠘⡇⠀⣹⠦⠤⡄
⠀⠀⠀⠀⢠⡟⢁⣀⣀⣀⠙⠻⣵⠋⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢉⠟⠒⠳⡄
⠀⠀⠀⠀⠈⣇⠀⠀⠙⠮⠿⡄⠙⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⡀⢠⡇⢠⣤⣄⠀⠀⢸⡇⠀⠈⢷⣉⠛⣧
⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣯⣿⡿⠣⣷⣾⡿⢀⣠⠟⢀⣠⠤⣄⠸⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠰⣄⣈⢙⣓⣶⠞⠁⠀⠀⠉⠉⠉⠀⠠⠋⠀⠀⣸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡄⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⢀⡄⢀⡴⠒⠛⣚⣧⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡼⠀⣼⠀⠀⠀⠀⢠⡏⠉⠙⢲⣒⣒⣾⣤⣬⠵⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⣇⣠⡇⠀⠀⠀⠀⢸⡗⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠚⢧⡞⠀⢸⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⡑⠀⠁⠀⡞⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠤⠤⠤⠞⠁⠀⠀⠀⠀⢀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣦⠤⠶⠶⠶⠒⣾⢻⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣿⣸⡤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠀⠉⣹⣉⣁⣼⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠢⠤⠤⢴⢶⢖⠟⣻⢀⣈⣷⡤⣾⠇⠀⠀⠀⠀⠀⠀⠲⠶⠶⠤⢤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢝⣒⣒⣒⣛⣛⣚⣉⡩⠥⠖⠋⠉⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⡶⠒⠒⠀⠠⠤⠬⠭⠉⠉⠻⠿⠿⠷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠶⠶⠯⠭⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣆⠀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠛⠉⠁⢀⣤⣤⣤⣉⡻⢿⣿⣿⣷⣌⠻⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⡄⠀⠁⠀⠀⠉⣙⣿⣾⣿⣿⣿⡏⠁⠘⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⣸⠀⠀⠀⢠⣾⡟⠛⣿⣿⣿⣇⣾⡇⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⡀⠀⠸⣄⠛⠁⢸⣿⣿⡿⢙⣿⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠙⢦⣤⡙⠿⡲⣿⣿⣿⣷⠞⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⣠⡄⣤⢒⢀⠀⠠⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣉⡟⣷⣧⡞⣫⠤⣰⣼⡟⣾⣛⣿⣷⡢
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠿⣾⣿⣿⠻⣴⡶⠀⡸⠟⣯⢽⢩⣟⢿⣿⣏⠈⣀⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⣀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣿⡷⠶⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢠⡄
⠀⠀⠀⠀⢀⣼⠟⠳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠸⣧
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⡿⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣾⢃⣴⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡄⠀⠀⠀⣠⣦⠀⢀⣼⡟⠀⢠⣾⠇⠀⣠⡾⠋⣽⠃⣴⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣤
⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣿⣇⠀⠀⢰⡿⣧⠀⣾⢻⡇⢠⡿⢻⣀⣾⠋⠀⠀⢻⣿⠃⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿
⠀⠛⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⠻⣿⡟⠻⢶⣾⡇⢻⣤⣿⠈⢷⣾⠃⢸⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢿⡄⡀⡀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀
⠀⠀⣾⠃⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⡏⠀⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣿⣿⣿⠀⠀⠀⠀⠠⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⣤⣤⠶⠞⠛⠛⠻⠆⠀⠀⠀⠀⢹⡿⠀⢀⣴⠶⣤⣰⡟
⠀⠈⠋⢸⡇⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠰⣶⣤⣤⣤⣤⡶⠾⠛⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⣅⡀⠈⢻⡇
⠀⠀⠀⠘⣿⠀⣀⣤⣄⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⢻⡆⠘⣿
⠀⠀⠀⠀⢻⣶⠋⠁⢹⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣼⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡿
⠀⠀⠀⠀⢸⡏⠀⣴⠋⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣠⡿⢁⣾⠁
⠀⠀⠀⠀⠘⣿⡀⢿⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠹⣿⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠙⣷⡈⢻⣄⡀⣿⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⢻⣿⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠉⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠛⣷⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⡶⢶⣿⣿⡟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠹⣯⡉⠉⠉⠉⠉⠉⠀⠀⢀⣾⠋⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⡀⠀⠀⠈⠛⠶⣦⣤⣤⣤⡴⠾⠛⠁⠀⠀⢀⡿⠛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⣴⣿⣤⣿⣦⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣠⣾⠁⠀⠀⠀⠙⣷⣄⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢶⣄⠀⠀⠀⢠⡄⠀⠀⢿⣀⣼⠋⢹⣇⠀⠀⠀⣸⠏⠹⣦⣀⡿⠀⠀⠀⠀⣠⡄⠀⣠⡾⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⢷⣄⠀⠈⣷⡀⠀⠘⠟⠁⠀⠀⢿⣤⣤⣴⡟⠀⠀⠈⠻⠃⠀⠀⠀⣰⡟⢀⣼⠏⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠹⣧⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠃⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠹⣧⠀⣿⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣸⡇⣸⡏⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⣀⠀⠀⠀⣿⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⡞⢋⣩⣽⠟⠛⠛⠿⢿⣄⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⠟⣿⠏⣠⡾⢃⣤⠀⠀⠙⣷⣄⣀⣤⣽⣧⣴⣾⠛⠷⣦⡜⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⣴⣟⣡⡾⠿⠟⠋⠉⠻⢷⣄⠀⠀⠉⢻⣆⠈⢻⣿⢁⣠⣤⣤⣤⣤⣤⣤⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣠⡶⠟⣹⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢹⡆⣼⡟⣋⣉⣀⣀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣠⡾⠋⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⠋⠛⠉⠉⠉⠉⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠋⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠛⠻⠶⢶⣤⣤⣤⣤⣤⣤⣤⣴⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⢠⡾⠛⠉⠀⠀⠉⠙⠷⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠉⠉⠛⠃⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢠⣤⣶⣦⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣼⣿⣿⣿⣷⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢦⣄⠀⣴⣾⣿⣿⣿⣿⣿⣆⣤⣴⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⡿⠛⠛⣟⠋⢉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣹⣆⡳⣨⣕⢾⣿⣿⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢩⣿⣿⣿⠏⢻⣿⠀⠉⢁⣀⣟⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣮⣿⣿⠏⠀⠀⠙⢢⣀⣀⣼⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣷⣶⣿⣿⣿⣿⠋⠀⠂⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢰⣿⣼⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢰⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⢸⡀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠘⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡼
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⣷⣧⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣆⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣶⣤⣌⣿⣿⣿⣿⣿⣿⣶⠌⠁⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢅⠆⠆⠺⠏⠳
⠀⠀⠀⠀⠀⠀⣦⣴⢰⣧⠸⠀⠀⠚⠊⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⣠⣤⢴⢶⢲
⠀⠀⠀⠀⠀⠀⠋⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⣀⣰⠺⡏⠻⠸⣸⣼⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠈⠀⠀⠀⠀⠀⠁⢏⠂⠹⡄⣙⣀⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠓⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠠⡄⡄⡀⢀
⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠰⠀⠁⠀⠀⠁⣤⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠄⢀⢀⠀⠀⠠⢄⠰⡀⠤⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢳⡏⣷⠊⠶⠳⠎⠘⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠂⠀⠀⠀⠘⠀⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠀⡴⠀⠰⢄⠘⠃⠐⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⡀⠀⢀⠀⡄⢰⢀⡓⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡿⠾⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠳⠄
⠀⠀⠀⠀⢀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉
⠀⠀⠀⠀⠳⠤⠿⠃⠾⠀⡯⡴⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣉⡉⢦⠊⣍⢁⡔⢡⡍⢶⠁⠻⡤⠃⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣀⣄⣠⡄⠀⠀⠀⢸⠀⡋⢥⡇⠘⠘⢸⠃⠆⡀⣾⢠⠸⢃⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠰⣜⣋⡜⢣⡰⢀⠜⣄⠴⢠⢇⡘⢠⡾⡰⠇⠀⠀⠀⠀⡆⠙⠐⠂⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠹⣄⠀⠀⠀⡤⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢶⠒⠤⠤⣀⠀⠀⠈⣇⠀⠀⢰⠋⠀⠀⢀⡼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠽⢳⡄
⢸⠀⠀⠀⠸⣀⠀⠀⢸⠀⠀⠈⡇⣠⠖⠉⠀⢀⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣶⡚⠛⠛⠒⠒⠃
⠸⣆⢤⣀⠀⠀⠑⠢⣼⠀⠀⠀⠙⠁⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠉⠒⠤⡀
⠀⠁⠀⠈⠑⠦⣄⠀⠀⡖⢦⡄⣖⢴⡆⣰⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠆⠀⢠⠎⠀⠈⠲⡀⠀⢑⠦
⠀⡠⠛⠦⠤⠤⠬⠿⠦⢦⣤⢰⣇⢲⠂⠑⠒⠒⠦⠤⠞⠳⣄⠀⢠⠖⠒⠲⠤⣄⡀⠀⠀⠀⠀⣠⠖⠒⠒⠲⣤⡠⠴⠒⠛⠓⣆⡴⠋⠀⢀⣀⠤⠼⠀⣌⢳⠆
⠾⣅⡀⢀⣀⣀⣄⠤⠤⢤⡄⠀⠀⡨⠤⠤⣄⣀⣀⡀⠀⠀⠈⢢⡘⢦⣂⡀⠀⠀⠈⠉⠓⠲⠤⣽⣄⡀⠈⡷⣯⠄⠀⠀⢀⡼⠃⠀⠀⣠⣿⡶⠚⠓⠦⠉⠃
⠀⠀⠉⠚⠀⠀⠀⣀⣀⡞⠀⠀⡰⠁⠀⠀⠀⠀⠀⢹⣀⠴⠚⠁⠀⠀⠈⠙⠓⠲⡦⠤⠤⠤⠤⠷⠂⠉⠉⢳⣄⣀⣠⠶⠋⠀⠀⣠⠾⠿⣏
⠀⠀⠀⠀⠀⠀⠀⢳⠉⠁⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠄⠀⠀⢀⡞⡵⠶⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣴⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠚⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡴⠚⣻⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣰⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡴⠋⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠒⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⠥⠾⣇⠀⠲⢄⡀⠀⠀⠀⣠⠴⠋⢀⡴⠋⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢒⣼⣆⠀⠀⠀⢀⡤⠚⠁⢀⡴⠋⠀⠀⢀⠀⠀⠙⠖⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⢯⣮⣬⣧⣠⢖⡉⠀⠀⡰⠋⠀⠀⣄⠀⠀⠣⡀⠀⠀⠀⠈⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡼⡏⠉⠁⣎⣿⣧⠀⠙⠢⠞⠁⠀⠀⠀⠈⣆⠀⣠⠟⠀⢀⡤⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢚⣿⣿⢀⣴⡏⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⡿⣧⠴⠋⢀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠿⢫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡜⢆⢠⢫⠷⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⠤⢤⣤⣤⣤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠘⢦⡛⠉⣈⣉⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠳⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡘⣌⣢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠉⠙⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠋⠽⣛⣙⣿⣿⣛⣷⠶⠶⠖⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⣀⡀⢸⣿⣿⡧
⠀⠀⠀⣤⣄⠀⠀⠀⠀⣶⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡆⠀⠀⣿⣿⣷⢸⣿⣿⣷⣤
⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⢸⣿⠀⠀⠀⣰⣿⣿⣷⠀⠀⣿⡿⣿⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⣿⣿⡇⠀⠀⠀⢹⣿⠀⠀⠀⣾⣿⣇⠀⠀⢿⣿⣿⣿⡄⢸⣿⡇⢿⣿⣿⣿⠻⣿⣿⠇⠀⢠⠀⠀⣾⠀⢸⣿
⠀⠀⠸⣿⣿⣿⠀⢀⣀⣼⣿⡇⠀⢠⣿⠁⣿⡆⢠⣿⣿⢻⣿⣧⢸⣿⠇⠘⣿⣿⣯⠀⠙⠿⠀⠀⣺⡆⢠⣿⡇⠘⠛
⣠⣤⣠⣿⣿⣿⡿⠛⠛⠉⣿⣿⠀⢸⣿⠀⠸⣿⣾⣿⡏⠘⣿⣿⣼⣿⡇⠀⠈⠛⠃⠀⠀⠀⠀⠀⠉⠃⠈⠉⠁
⠉⠉⠉⢻⣿⣿⡇⠀⠀⠀⢹⣿⣇⣿⣿⣄⠀⠙⠛⠟⠀⠀⢽⣿⣿⢿⠃
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠋⠋
⠀⠀⠀⠀⣿⡿⠿⠇⠀⠀⠀⠋⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠇⠀⢸⣿⣿⣿⣿⣿⢟⡭⢇⡴⠔⣤⠢⡵⠙⢿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣻⠇⣤⠚⣉⠀⠖⠠⡍⢳⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠑⡉⢸⣿⣿⣿⡗⣃⡸⠃⢸⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣥⢾⣷⠀⣯⣿⣶⣾⢰⣪⣾⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⣿⣿⣿⣙⣷⣻⣠⣈⡹⣿⣷⠿⣿⡿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣝⠳⣠⣴⣟⣫⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠶⡒⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠉⠂⠄⣀⣀⣀⣤⡀⠀⠀⠤⠤⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⠤⠄⠀⠐⠒⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠀⠐⠂⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣤⡀⠀⠀⠀⢀⡐⠠⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣄⠉⠠⠒⠀⠐⠂⠀⠐⠋⢀⣀⣀⡈⠁⠀⠀⠀⠉⠀⠀⢀⣀⠁⠀⠈⠒⠉⠈⡙⠀⠀⡀⠀⠛⠤⡀
⠀⠀⠀⠀⠀⠄⠤⣀⠀⠀⠀⠓⠠⠛⠿⢟⠃⠀⠀⢞⠉⠽⡿⠉⠂⠀⠀⠀⠰⠉⠨⠏⠏⠑⠀⡆⠐⠋⠽⢟⠃⠀⠇⠀⠠⠤⢻
⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⢠⠀⠈⠙⠃⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠀⠐⠐⠀⠀⡃⠀⠾⠀⠜
⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⡄⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠅⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠈⠢⠀⠀⠐⡀⢀⠀⠀⠀⠳⡀⠀⠉⣏⠁⠀⠀⠀⠀⣤⣀⣀⠀⠀⢀⣀⠀⠀⠄⠀⠀⢈⢱⡀⠀⠀⠰⠇⠀⠀⠀⠂⠔⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠀⠀⠀⠘⢂⠀⣙⠉⠙⠂⠀⠸⠃⠀⠀⠀⠀⠀⠀⠢⠆⠀⠀⠐⠃⠛⡄⠀⢔⠀⠀⠀⠀⠁⠈⠐⠁⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠽⣤⡮⣰⣠⠆⠀⢀⣤⣤⡶⠀⡠⣄⠔
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠔⠈⠉⠐⠁⠀⠰⠛⠛⠊⢀⣾⠊
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⠀⠀⢀⡄⠛⠛⠉⠁⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⠠⠄⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⢼⣿⣤⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⣾⣧⣼⣷⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⠀⠐⠛⠛⠛⠛⠿⠷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡟⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡇⠀⠀⠀⣿⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⢀⣴⣯⡴⠞⠛⠉⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠛⠉⠀⠀⠀⠀⢀⣀⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢿⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡄⠀⢀⣤⢠⡄⠀⢠⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⢀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⣠⣤⣄⢠⠏⠉⠙⣆⢸⢻⡀⣼⣿⢸⡇⠀⠈⣿⢳⡟⢹⣤⡄⣿⢳⡟⡆⢰⡏⠀⠉⠁⣿⣿⡗⢸⣿⣇⠺⣭⡃⣿⣼⠀⠇
⠀⠀⣶⢲⢴⡇⠀⢸⣾⡀⠀⠀⡿⢸⠀⢿⠇⣿⢘⡃⠀⠰⠟⠏⠿⠞⠀⠀⠋⠛⠛⠁⠘⣧⣀⣠⠄⡿⣷⡀⠼⠏⠿⠐⠶⠃⠛⠘⠀⠁
⠀⠀⡿⠙⣾⠳⣤⠞⠀⠙⠲⠚⠁⠀⠀⢠⢀⡄⢀⣄⠀⠀⠀⣤⣄⢀⣤⡀⣠⡀⣄⠀⢠⡀⠉⠁⣀⠀⠀⢀⡀⢀⣀⠀⣀⠀⠀⠀⡀⣤⡄
⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⣼⢿⡀⠶⠀⣿⣿⣸⣿⣿⣿⣿⣿⣆⣿⡇⠀⠀⢿⣤⣆⣾⢳⣿⣏⣷⢹⣇⣦⣸⡇⢻⠃
⠀⠀⠀⣶⠒⢰⣶⣶⣴⣶⣶⣶⣶⣦⠀⠘⠈⠃⠃⠈⠃⠀⠀⠛⠋⠈⠛⠁⠙⠁⠋⠙⠘⠁⠀⠀⠸⠟⠹⠟⠈⠻⠟⠃⠈⠿⠙⠿⠀⠰
⠀⠀⠀⣿⣁⢸⣿⡉⢸⢯⡁⣿⢯⠁⡀⣀⡀⠀⠀⠀⠀⣤⣶⣦⣄⠀⣤⣤⠀⢀⣤⣤⢀⣤⣶⣦⣄⠀⢠⡆⠀⠀⡀⣀⠀⣀⢀⢀⢀⣄
⠀⠀⠀⠈⠉⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢹⣧⣿⣿⣆⣾⣿⣿⢸⣿⣟⠉⣉⣀⢸⡇⠀⠀⡇⣿⣼⣏⢸⢺⣿⢨⠆
⠀⠀⢠⠶⠤⣴⡀⢠⡆⢠⣦⠀⣴⠶⠄⣆⡴⢰⠀⠀⠘⠿⣿⣷⠟⠁⠿⠏⠿⠏⠿⠿⠘⠿⣿⣾⠟⠉⠨⠇⠀⠀⠁⠁⠁⠉⠈⠈⠈⠉
⠀⠀⢈⣛⣶⣿⢷⡟⡇⣾⠻⡆⣧⣀⡀⡿⣄⢸⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⢠⣤⡄⢠⡄⠀⠀⠀⢠⡀⠀⡄⢠⢀⡄⢠⠤⠄⡄
⠀⠀⠀⠉⠁⠀⠈⠀⠁⠁⠀⠁⠀⠉⠀⠁⠈⠈⠀⠀⠀⣶⠛⠆⠉⡏⢱⠋⠀⠈⣧⢸⣿⠏⢸⠃⠀⠀⠀⢸⡇⠀⡧⢸⣎⠀⣽⠖⠀⡇
⠀⠀⠀⣴⠞⠶⢠⣶⣶⡄⣴⣾⣦⢰⡆⠀⠀⣶⠀⠀⠀⠈⠓⣦⠀⡇⠸⡄⠀⢀⡿⢸⠀⠀⠠⠀⠀⠀⠀⠘⠓⠂⠃⠘⠈⠃⠙⠒⠂⠂
⠀⠀⠀⣿⠀⢀⣸⣧⣼⡇⣿⣤⣿⢸⡇⠀⠀⠻⠀⠀⠀⠙⠶⠋⠀⠁⠀⠉⠛⢋⣤⡀⢀⣠⣀⢠⣤⣄⡀⢀⣀⡀⣠⠀⢰⡄⢂⡎⢢⢰
⠀⠀⠀⠙⠓⠛⠀⠛⠛⠀⠈⠛⠋⠘⠛⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣼⣿⢹⣾⣿⣿⠇⢿⣍⠃⣿⠀⠸⠈⠎⠧⠜⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⢀⣿⢸⡇⣿⣷⢸⡇⢸⡇⠀⠻⣷⡞⠹⣿⡾⢹⡇⠀⠀⣦⣽⠇⣩
⠀⠀⢰⡿⡇⠀⢠⣤⠀⠀⣀⠀⠀⠀⠀⠘⣿⡿⣿⡇⢸⡇⣿⠹⣿⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⠀⢀⡀⣀⠀⣀⡀⢀
⠀⠀⣾⠗⢿⠀⣾⢼⡆⢸⣿⡄⣆⣰⢰⠀⠙⠀⠙⠁⠈⠃⠙⠀⠙⠁⠈⠁⢀⣿⣷⣿⡞⠀⡇⠀⠀⢸⣿⣏⢰⣿⡆⢼⢷⣿⢸⠁⣁⣸
⠀⠀⠏⠀⠘⠗⠇⠀⠓⠛⠀⠃⠃⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿⠳⠄⠅⠀⠀⠘⠛⠛⠚⠀⠛⠘⠈⠛⠈⠓⠋⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢺⣶⣦⣤⣤⣀⣠⣴⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⣀⡀⠀⠹⣦⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣶⡆
⠀⠀⠀⠀⣰⡟⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⡀
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⣰⠏⢀⣤⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⢀⣤⠀⠀⣠⡦⠀⠀⣠⡾⢿⡏⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⢠⣿⣴⡿⠁⠀⠀⠀⠀⠀⠀⢹⣇⠀⢰⣿⠀⠀⠀⣴⣿⠀⢠⣾⣿⠀⣰⣿⡇⢠⡾⠋⠀⣿⣤⡟⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠈
⠀⣿⠋⣿⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⢷⣼⣿⢷⣄⣀⣯⢻⡄⣿⠁⢿⣠⡟⠸⣷⡟⠀⠀⠀⠘⠟⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠋⣸⡇⠀⠀⠐
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠙⠀⠉⠉⠉⠀⠛⠿⠆⠈⠛⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣶⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣴⡶⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣀⣀⣀⣤⡴⠶⣦⡄⠀⠀⠀⠀⢿⣷⠀⢀⣠⣄⡀⢰⡏
⠀⠐⠟⢹⡇⠀⠀⠀⠀⢈⣿⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⡴⠆⠀⠀⠘⢷⠈⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇⠀⣾⡋⠉⠻⣿⠁
⠀⠀⠀⢸⡇⠀⢀⣀⡀⠈⢿⡅⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠟⠛⣷⡀⢸⡇
⠀⠀⠀⠀⢿⣴⠟⠉⢻⣆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡏⠀⣴⠟⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⡿⢠⡿⠁
⠀⠀⠀⠀⠸⣧⠀⣿⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠋⣠⡟⠁
⠀⠀⠀⠀⠀⠹⣦⡙⢷⣄⠀⣿⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠘⠷⣤⡙⢻⡟⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠉⠉⠙⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⢋⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⢿⣿⢿⣟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠈⢿⣤⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠹⣧⠀⠀⠀⠈⠛⠻⢶⡶⣶⠾⠛⠉⠀⠀⠀⢰⡟⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⢀⣴⠿⠶⠾⢷⣄⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣠⢿⡇⠀⠀⠀⢀⣿⣦⡀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠉⠻⣦⡀⠀⠀⢷⡀⠀⠘⣧⡾⠋⠈⣿⡀⠀⠀⣼⠇⠈⢻⣴⡏⠀⠀⠀⢀⣴⠆⢀⡴⠟⠁⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣆⠀⠸⣧⠀⠀⠉⠀⠀⠀⢸⡿⠛⠻⣿⠀⠀⠀⠉⠀⠀⠀⠀⣼⠇⣠⡟⠁⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠹⣧⠀⣿⡀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢰⡟⢠⡿⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣆⢹⡇⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣼⠿⠟⣶⣶⣤⣤⣼⡃⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠸⣿⣴⣶⠟⢡⡴⠀⠀⠉⠻⣦⡀⠀⠀⢸⣇⢀⣤⣤⣤⣀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠋⣠⡾⣣⣶⣤⡾⠷⢾⣿⡛⠛⠛⠻⣦⡀⠙⢷⣿⠀⠀⢀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣼⠟⣿⡿⠞⠋⠁⠀⠀⠀⠀⠈⠻⣦⠀⠀⠈⢿⡄⢸⡿⠶⠛⠛⠉⠉⠉⠉⠃⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢠⡼⠋⢁⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢸⣷⠿⠳⠾⠟⠛⠗⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠰⠛⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠛⠛⠷⠶⣦⣤⣀⣀⣀⣀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣴⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣿⣭⣿⣭⣿⣿⣯⣿⡗⢿⣿⣿⣹⢿⣽⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣛⣿⣿⠿⢿⣿⣿⠿⣿⠿⠈⠀⠉⠙⠛⠿⠷⣿⢿⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣺⣷⣶⣶⣵⣚⣟⢿⣷⣽⡦⣄⠀⠀⠀⠀⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠯⢿⣾⣿⣿⣿⣿⣛⣯⠭⠭⣭⣿⣳⡆⠀⠀⠀⠀⢲⣦⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣛⣷⣽⣷⣲⣯⠷⢿⣟⣿⣿⣿⡓⠀⠀⠀⣠⣤⣀⣀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⢿⣿⣿⣿⡽⣒⣝⣻⣿⡛⠿⠟⠃⠀⠀⠞⠉⣉⣻⠁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣷⠙⡟⣇⢻⣯⣿⡟⠀⠀⠀⠀⠚⣿⠃⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⣻⣿⣞⣿⣯⠛⢼⠏⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣩⡿⣷⠋⠙⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡽⠛⠁⠀⠀⠈⠳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣄⡀⠀⠀⠀⠀⠘⢿⣲⢤⣀⠀⠀⠀⠀⢠⣤⣠⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦⡙⠲⣄⡀⠀⠀⠀⠉⠓⢮⣄⣀⠀⠀⠈⢧⣈⠙⠲⢍⡳⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⠻⣿⣿⣿⣦⡀⠙⠲⣄⡀⠀⠀⠀⣨⠿⠛⠛⢳⡾⣟⠛⢶⡀⠹⡜⢦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣦⡀⠀⠙⠳⢦⣴⠿⣿⣿⣶⣾⠁⡿⠋⠛⠻⢦⣘⣎⣳
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣆⠀⠀⠀⣿⢳⡹⣿⣿⣿⣟⡴⠋⠓⠶⢤⠈⢹⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣷⡄⠀⢿⣆⠻⠜⢿⣿⡿⣔⢾⡗⠲⠀⢠⡾⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⠿⠿⠛⢻⣷⣤⢠⠄⢤⡧⡌⣿⢁⡽⠟⠁⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠃⣾⣿⣿⣶⣄⢸⡇⠰⣶⣾⠀⠀⠀⢀⣶⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣾⡇⠀⠘⣿⡀⠀⠀⡞⠹⣯⡉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣄⠀⠸⣷⡄⠀⠀⠀⢹⠿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⣦⡀⡇⣹⣆⠀⢀⠞⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⢿⣿⣿⣿⣧⠿⡽⠗⣁⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣿⣿⣧⣿⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣷⣤⠙⢿⣿⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠶⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⣀⣀⡠⠤⠄⠀⢀⣀⡀⠀⠀⠀⠉⠁⠶⠆⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢾⠏⠉⠉⠉⠳⣄⠀⠀⠈⠉⠉⠶⢦⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠶⠶⠀⣀⣴
⢀⣠⣤⣀⣀⠀⡀⠐⠒⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡈⠿⣦⠀⠀⠀⠀⠘⠻⣶⠶⠶⠄⣀⡀⠀⠀⠀⠀⣀⠴⠋⠉⠰⠆⣀
⠈⠻⠃⠈⠁⣀⡀⠐⠲⠶⠖⠀⣼⣆⡀⠀⠀⠀⠈⠀⠀⠈⠶⡀⠀⠀⠀⠀⠘⠧⠀⠀⠈⢩⣶⣄⠰⠂⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠿⢧⣀⠀⠀⠀⠀⠀⢹⣤⣀⣠⣀⠀⠸⡿⠒⠂⠀⠈⢹⡆
⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠈⠀⠀⠉⢳⣤⡀⠚⠓⠀⠉⠻⣄⠀⠀⠀⢰⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⣦⡀⠀⢀⡄⠀⢻⡇⠀⡄⣼⠗⠚⠀⠺⠋⠙⣿⠟⠢⣤⡀⠀⠈⢳⣄⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠂⢀⡌⠀⠀⢸⠀⠀⠀⠈⣶⡄⠘⠿⠃⠛⠋⠙⠻⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣷⠀⠀⠀⠀⢸⡄⠀⠀⠘⠋⠀⠀⠀⠆⠀⠀⠀⠁⠀⠘⠛⣦⣤⠤⠄⠀⠀⡀⠀⠤⠤⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠈⠻⡆⠀⠀⠀⠀⠙⣷⣄⣀⠀⣴⡄⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⢤⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢋⡄⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠉⠙⢿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⢶
⠀⣴⣇⡀⠉⠀⠀⠀⠈⢣⡀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠁⠁⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠈⠁⠀⠀⠀⠶⠶⢾⣉⠉⢉⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⣀⣀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⢀⠀⠀⠂⠈⠉⠁⠈⠉⠉⠁⠲⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⢾⡷⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢀⣀⡀⠀⠀⠀⣀⡀⠀⠶⠦⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡆⠀⠉⠉⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠒⢀⣀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢂⣀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣄⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠘⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣤⣤⣦⣤⣤⣄⣀⣀
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠉⠁⠀⠀⠀⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣴⣦⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣯⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⡟⠀⠀⣿⡟⠁⣤⡄⠘⣿⣿⡿⠁⠀⣿⣿⡇
⠀⠀⣿⣿⠏⣠⡇⠀⣿⡇⠀⣿⣿⠀⣿⡟⢁⣾⠀⣿⣿⡇⠀⠀⠀⠀⣴⣶⣦⣀⣀⣤⣄
⠀⠀⣿⣧⣀⣀⡀⠀⣸⣇⠀⢿⠏⢀⣿⣀⣀⣀⠀⣨⣿⡇⠀⠀⠀⠀⠻⡿⠛⢿⣿⣿⣿⠃
⠀⠀⣾⣿⣿⣿⣧⣴⣿⣿⣶⣤⣴⣾⣿⣿⣿⣿⣤⣿⣿⡇⠀⠀⠀⠀⠀⠔⠐⠌⢸⣿⡀
⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣤⣾⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣭⣓⢶⣦⠀⠀⠀⠀⠀⠀⠐⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⠻⣿⡷⣾⣧⡀⠀⠀⠀⠀⠀⢰⣯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠹⠗⣛⠛⢄⠀⠀⠀⣠⡾⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀⠀⠀⠐⢋⣡⡴⠾⠳⣤⣰⣿⣷⡳⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣦⣀⠟⠛⠉⠀⠀⠀⠘⠁⠞⠃⢱⡌⣙⠻⣷⠟⢁⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢈⣡⣬⣦⣸⣦⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠉⠁⢈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣆⠀⠂⢦⠈⠢⣠⢴⣼⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠝⠉⠛⠂⠀⠀⠀⠙⣿⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀⣶⡆⠀⢀⣀⠛⠶⠀⠀⠀⠀⠀⠙⡦⣦⣄⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⣤⣶⣿⣹⡛⠂⠀⠀⠀⠀⠀⠀⠠⡞⠉⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡿⣿⣿⣿⣷⣶⣶⣶⣤⠀⢠⡏⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡹⠏⠀⠀⠻⣿⠿⣜⡓⢶⡅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠙⠷⢦⣤⣀⣀⣀⣀⣀⣤⡴⠞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠶⣤⣀⣀⠀⢀⠀⣀⣀⣀⣀⢤⡴⠚⡀⣀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠴⠟⠿⠍⠀⠴⢿⣄⠀⣸⡿⠹⡟⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠁⢹⣇⡼⣛⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⣦⣄⠀⣰⣶⣦⠠⢶⡿⠗⣸⣋⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣶⣿⣷⣿⣟⣿⣻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⣠⡆⠈⠉⣹⣿⣹⣿⡈⠃⢹⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣶⣿⡇⢘⣿⣿⣿⣿⡇⠀⣸⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⣀⣀⣴⡀⢈⣿⣧⣾⣤⣀⣽⣿⣟⣻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⢤⣿⣿⣟⣻⣿⣿⣿⣿⣾⣿⠛⢃⠉⡏⢫⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢶⡾⠃⢴⡿⠁⠀⠀⠀⠀⠈⠋⠟⢺⣿⣿⣿⣿⣿⡀⠈⠁⡐⡀⢵⣿
⠀⠀⠀⠀⠀⠔⠈⢀⣴⡆⣀⣴⠍⠀⢠⣼⠡⣤⡆⢀⡀⠀⠀⣀⠠⣤⣾⣿⡿⠿⢿⡷⠿⠶⠿⠿⠛⠋
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠶⠿⠗⠒⠚⠚⠛⠛⠛⠃

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣤⡄
⣿⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⣄⣀⣀⣶⣶⣿⣿⣿⣿⣿⣿⢣⠤⡤
⡅⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣦⣄⠀⣀⣠⣶⡄⣀⡀⣠⣶⣿⠃⣹⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠉⠀⠀⣀
⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣧⣿⣿⠀⠀⠀⠀⠀⣠⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⡿⠀⠀⢀⣤⣤⣿⣷⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀
⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿
⡿⠟⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣤
⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠛⠉⠁⠐
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉⢻⡇
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⣁⡀⠀⢀⣀⣠⣤⠔⠚⠛⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣋⣿⣛⣿⣿⣿⣿⣿⠁⠐⠛⢻⡇⠀⠀⠀⠀⢤⣤⣀⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢷⣿⣿⠏⠁⠤⠾⠋⣿⣷⡇⠀⡄⣾⣿⣦⣄⡀⠀⣾⣿⣿⣿⣿⡇⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡁⠀⠀⠀⣲⠏⠀⢻⠛⢋⠀⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣨⣿⣿⣿⢿⣿⣿⣿⣿⠋⠀⣾⣀⣤⣦⣤⣴⡿⠛⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⣼⣿⣿⣿⣿⣄⠀⣟⣹⣿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣭⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⡿⠇⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠻⣿⣿⣿⣿⡟⣿⣿⠏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣴⡄⢻⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡄⢀⣿⡇⠀⢺⣷⣿⠏⠀⣸⣿
⠀⠀⠀⠀⣰⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⠇⠀⠘⠿⠏⠀⠀⠛⠋
⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠉⠉⣿⣿⣿⣷⣴⣿⣿⣿⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠉⠉
⠀⠀⢰⣿⣿⣿⠙⠁⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠿⠟⠀⠙⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣏⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣶⣿
⠄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⢿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣴⣾⣿⣿⢿⡟⢻⡿⠿⣿⣿⢀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠸⣿⣿
⠀⣿⣿⣷⠂⠀⠀⠀⠀⢠⣴⠉⠙⣸⣿⣿⣿⣿⣿⣶⣾⣿⢿⣿⣿⣴⣇⣿⣡⣿⣃⣴⣿⣿⣽⡁⠀⢸⠁⠀⠀⣰⣦⡀⢀⣶⠂⠀⠀⠀⠈
⠀⣿⣿⣿⣿⣙⠋⣿⠃⠙⡋⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣰⣿⠀⣿⣿⣿⢸⣿⣦⡀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡠⠏⣀⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⡤⢄⣀⣀⣀⣀⣤⠤⠖⠒⠳⠬⠙⠃⠉⠀⠛⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠓⠺⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠿⢝⡂
⠀⠀⠀⠀⠀⠀⠻⣖⣒⣒⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡴⠃⣀⣤⠴⠶⠶⠦⣄⡀⠙⢆⡀
⠀⠀⠀⣀⡴⠚⠉⢉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠤⣄⣹⣞⡴⠚⠉⠀⠀⠀⠀⠀⠈⠹⣄⠀⠙⢦⡀
⠀⠀⠀⠈⠉⢉⠟⠁⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠹⠏⠀⠠⠶⢤⣄⡀⠀⠀⠀⠀⢹⡄⠀⠀⣳
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢬⣙⠳⡄⠀⠀⠘⡇⠀⠀⣧
⠀⠀⠀⣼⠖⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠁⠀⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠶⠒⢸
⠀⠀⠀⠀⠀⢸⠠⠤⢼⡄⠀⣀⣀⣀⣀⣤⣤⣴⠾⠛⠋⣴⡄⠀⢠⣆⠀⠀⠀⠀⠀⢀⣿⠀⢠⡏⠑⠂⢸
⠀⠀⠀⠀⠀⢸⠀⠈⠉⡇⠀⠈⢻⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣼⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢧⠀⠀⢳⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⠀⢠⡇⠀⠀⣰⣧⣄⡀
⠀⠀⠀⣠⠶⠒⠚⠢⠤⠼⠂⠀⠀⠀⠈⠉⠛⠛⣟⡿⠋⠀⠀⠠⠖⣦⠀⠀⠉⠉⠉⠀⠀⠀⠳⠶⠊⣡⡴⠒⠙⣆
⠀⠀⠀⡏⢀⡴⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠫⣿⠀⠀⢹
⠀⠀⠀⢷⡀⠀⠼⠈⡧⢸⡀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⠂⠁⠐⠒⠒⠤⣄⡀⠀⠀⠀⠀⢠⣆⠙⠀⠋⠀⢀⡿
⠀⠀⠀⠀⠛⢦⣄⣀⣠⠼⢧⠀⠀⠀⢠⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣠⣤⣾⣿⣿⡆⠀⠀⠀⣼⠉⠛⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢧⡀⠀⠘⢿⢿⣿⢿⣟⣋⣛⣛⣛⣿⣿⠿⠿⣿⣿⡇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠉⠲⠤⢄⣀⣀⣉⣁⣀⣀⣠⠴⠞⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠚⠛⠋⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣭⣿⣿⡿⣿⣿⠿⣿⣿⣿⢿⠿⠿⠿⠷⠦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣮⣿⣿⣾⣿⣿⠩⠯⠬⢍⣉⣂⠈⠛⠢⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⢿⣷⣶⣤⣄⡀⠙⠙⢦⡀⠘⢦⡘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠙⢿⣿⠀⠀⠈⠳⣿⡙⢿⠿⠟⢮⣀⠈⠉⠛⢿⣷⡀⠀⠑⢄⠀⠙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⠙⢿⣆⠀⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣆⠀⠀⠙⠃⠀⠀⠀⠁⢣⢻⡄
⠀⠀⠀⠀⠀⠀⣿⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣾⣆⠀⠀⠀⠀⠀⠀⠀⠈⠏⣷
⠀⠀⠀⠀⠀⠀⢹⢤⣌⡻⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢰⠏⣄⡘⠿⢳⡆⠀⢀⠸⠿⣶⣶⣶⣶⣦⣀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⣼⠁⠀⠋⠀⠻⠝⢻⣿⣭⣉⠛⠷⠀⠀⠹⡿⡽⢛⣧⡀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠃⠀⠀⠀⠀⠓⠶⠀⠉⠀⠈⠐⠒⠀⠀⠀⠀⢰⡏⢤⡻⣦⡀⠀⠀⠀⠀⢀⡼⢷⡀
⠀⠀⠀⠀⠀⣿⡆⠀⣿⠁⠀⠓⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⣠⢅⠌⠻⣦⡀⠀⢠⡞⠁⣼⢷⡀
⠀⠀⠀⠀⠀⢸⡇⢰⡈⠃⠀⠛⠂⠉⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡭⠔⣡⠋⠀⠀⠘⢷⣴⠏⢀⡼⠃⠈⢷⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢹⡶⢤⣀⣀⡀⠀⠀⠙⢦⡄⣶⠀⠀⠀⠀⣰⣿⣤⡤⠞⠁⠀⠀⠀⣠⠞⠁⣠⠟⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠶⢤⣭⡭⠽⠟⠀⠞⢠⠏⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⠴⠶⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⡀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠠⠄⠠⠂⠀⠀⠀⠀⠀⠰⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠣⡈⠀⠀⠀⠀⠄⠄⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠂⠑⢡⠄⢂⢀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⡄⢐⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠈⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠄⠂⠐⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⢀⠀⠀⡂⠀⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠀⠂⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠄⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠠⠁⡀⠄⠂⠀⡀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠆⠅⠌⡂⠀⠌⠀⠀⠄⠀⠀⠀⢀⠂⠀⠈⠀⠈⠀⠀⡘⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠠⠀⠌⠀⠀⢀⠈⠁⠠⠁⠨⠐⡕⠁⢀⣠⠸⣤⢤⠀⠀⠀⢀⣤⣀⣀⣨⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠇⢨⠀⠀⡈⠡⠀⢐⠈⢀⠨⢀⡉⡇⢠⡨⡁⣨⣛⣓⠀⣠⠭⠗⠚⠿⢿⣓⣻
⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢸⣦⠈⠊⠀⠀⠐⣀⠬⢐⠠⠀⠂⠒⠊⠀⠀⠒⡰⢿⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠑⠀⠀⠀⠀⠀⠀⠙⢷⡤⡐⠠⠤⢥⠂⢁⠁⠀⠀⠀⠀⢀⠤⢚⡵⠁⠰⠦⡀⠀⡀⠄
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠐⣐⣠⣤⣤⣄⣔⣶⡭⠕⠚⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠤⡀⢀⠁⠄⠀⠀⠀⠀⠈⠉⠉⠩⠅⠀⠀⠦⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠠⠁⠙⠊⢐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠂⠠⠤⠤⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⠻⢿⣶⣄⠀⠀⠀⢀⡴⢞⣡⡤
⠀⠀⠀⢀⣠⣤⣶⠶⢶⣤⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣠⣿⠟⠋⠁
⠀⢀⣾⡟⠉⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠿⠿⠶⠶⠶⢶⣦⣤⣄⣀
⣰⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠋⠉⢁⣠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀
⣿⣿⠀⠀⠀⢤⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠘⢿⣷⡀⠀⠀⠀⠈⠛⠶⠶⣶⡶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠉⠛⠷⠶⣤⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠠⣄⠈⢲⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢀⠀⠀⢠⣿
⠀⠀⠀⠀⢙⣿⣏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⠞⠻⣷⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡞⠀⠀⡾⢃
⠀⠀⠀⠀⢸⣿⠈⠳⢬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠈⠙⠻⠷⠶⠶⠚⠉⠀⣠⢴⣿⠋⠀⢀⣾⣵⠏
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠙⢮⡳⠄⠀⠀⠀⠀⢠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠖⣫⡴⠋⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠖⠋⠁⠀⠀⢠⣾⣟⣁⣀
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⢻⣷
⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠐⠉⠘⠃⣠⡶⢷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡦⠀⣿⣷⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣣⣴⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣄⣀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⡿⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢻⡏⢻⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠀⠀⠀⠘⠋⠉⣀⣀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟
⠀⠀⠀⠀⠀⣿⡄⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠷⣦⣤⣤⣤⣴⣶⠶⢶⣦⣀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⠶⠀⠀⢀⣾⠏
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⣀⠀⣄⠀⣀⠀⣠⣄⡀⣠⣤⣄⢀⣤⣄⡀⣠⣤⣄⠀⣠⡀⠀⣠⠀⣄⠀⣀⢀⣤⣤⡀⣄⠀⣠⠀⠀⢨⡇
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⡟⢧⣿⢸⡇⠈⠀⣿⠶⠀⢸⡧⣼⠃⠀⣿⠀⢰⣏⣷⠀⣿⠀⡟⢧⣿⠀⢸⡇⠀⠘⣷⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠙⠛⠋⠀⠃⠈⠛⠀⠙⠛⠁⠛⠛⠓⠘⠃⠘⠃⠀⠛⠀⠛⠀⠘⠃⠙⠀⠃⠈⠛⠀⠘⠁⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⠀⠀⠀⠀⡴⣃⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⢳⣤⡀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠹⡄⠀⠀⢀⣀⣤⠤⠶⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣆⣀⣽⢶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣩⣄⡸⣰⡿⠭⢗⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⣇⣿⣵⠿⠁⠀⢀⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣼⣡⠾⠿⠒⠛⠁⠀⠀⠀⢸⡞⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣁⡈⢻⠏⠀⠀⠀⠀⢀⣀⣠⣤⠤⣤⣶⡶⠶⠾⢯⣾⣿⣿⣿⠿⣟⡳⠦⣍⡁
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⠀⢀⣤⠖⠛⠉⠁⠀⠀⠘⠁⣈⣻⡇⠀⠀⠴⠖⢺⣭⠙⠛⠿⣷⣦⣈⠁
⠀⠀⠀⣴⣿⣿⣿⣿⣿⠃⣠⡾⠋⠀⠀⠀⠀⠀⢾⣉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠉⠻⢿⣦⡀
⠀⢠⣾⣿⣿⣿⣿⣿⠃⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠈⠛⠛⢿⣿⡿⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠛⠁⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠟⠛⠛⠻⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢸⣿⣅⣠⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⢀⣾⣏⣈⣷⡄⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⠀⠈⠋⠉⠉⠉⠛⠛⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠏⠀⠀⠀⣰⡟⣀⣤⣴⠾⠟⠛⠉⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠀⢠⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⡰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠉⠉⣩⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠁⣀⣀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⣿⡛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⢶⣤⣤⣤⣤⡶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠽⠼⢿⠿⠿⠿⠿⠿⠇
⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⡄
⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣟⢷⡛⠛⢛⡿⣻⢿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⡷⣳⡛⡿⣿⣇
⠀⠀⣿⣿⣿⡇⢿⣿⡿⢿⣿⣛⣿⣮⡟⠻⠛⠕⣳⣾⣿⣿⣿⣿⣿⣿⡿⠇⠅⡿⣯⡿⠙⢠⣤⡀⣿⣧
⠀⠀⣿⣿⣿⣿⣼⣿⣫⣽⣿⡴⠖⠿⣧⡈⣡⣿⣿⣿⣿⣿⣿⡷⢯⣭⣥⣮⠆⠰⣸⣀⢁⠘⠚⣡⣿⣿
⠀⠀⣿⣿⠟⠛⠓⣛⣠⣼⣿⣴⠿⡿⢛⣿⣿⣿⣿⣿⣿⡟⢿⣋⡤⡇⠽⠋⠴⠘⠷⠟⣀⣤⣄⣴⣿⣟
⠀⠀⣿⣿⠶⠟⡛⣋⢩⡥⠐⣼⢧⣼⣿⣿⣿⣿⣿⣿⣿⠇⣌⡥⡛⢫⢀⡶⣶⠐⠂⢿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣏⠜⠓⣀⣩⣴⣤⢶⣾⣿⣿⣿⣿⣿⣿⠉⣼⣳⠨⢟⡱⠀⠄⠉⠉⡀⣠⣼⣏⣿⣿⣿⡿⣿
⠀⠀⣿⣿⣷⠾⣛⡯⠽⣗⣲⣿⣿⣿⣿⣿⣿⡟⣾⣍⠬⣡⢖⣷⣿⣦⠀⠀⠐⢦⣙⣿⣿⠟⠛⢽⣷⣿
⠀⠀⣿⣿⣳⣽⡽⣳⠿⣻⣿⡿⣿⣿⣿⣿⣯⠰⠛⠏⡨⣿⣿⣿⣿⣿⡗⡀⢤⣽⣿⣿⢇⢭⣖⣿⣿⣿
⠀⠀⣿⣿⠃⠁⢀⣈⣿⣿⣿⣿⣿⣿⣿⡟⣉⠘⠁⠌⢛⣿⣿⣿⡿⠟⠉⣶⣿⡭⣶⣿⣿⣭⣽⡽⣿⣿
⠀⠀⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣗⣠⣤⣤⣤⢤⣤⣬⣽⣾⣾⣿⣶⣿⣿⣟⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⣿⣿⢡⣠⣼⣿⣿⣿⡛⠿⡙⢙⢻⣿⣟⣟⣫⣿⣿⣿⣾⣿⣿⣿⣯⣻⣿⣿⣷⡿⢿⣿⣿⣿⣿⣿
⠀⠀⣿⣹⢻⣛⣿⣿⣿⣿⠟⠣⣵⣤⣿⣿⣟⣻⣿⣿⣿⣯⣽⣿⣯⣽⣿⣿⣻⣿⣻⣿⣿⣿⢿⣻⣏⣿
⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣶⣴⢯⣿⣿⣭⣵⣾⣿⣿⣿⣿⣿⣿⣭⣽⣿⣶⣯⣫⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿⠿⡿⢿⠿⠿⠿⣷⠿⠿⢿⠿⢿⠿⠿⠿⠟⡿⢟⣿⣿⣿⣿⣿⣽⣯
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠓⠚⠓⠒⠒⠒⠀⠐⠀⠀⠙⠛⠛⠀⠑⠈⠐⠊⠀⠚⠁⠈⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣶⣶⣖⣷⣶⣶⣾⣶⡀

⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣥⣦⣴⣀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣼⣇
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠉⠈⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣿⣿⣿⣄⠀⠴⠖⣂⣂⡀⠀⣿⣿⣿⣿⣯⠿⠁
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠻⠛⠋⠀⢀⡟⠹⠀⠻⠇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡇⣼⡈⣿⠄⠀⠀⣠⣾⣿⣿⠇
⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢷⣾⣧⣤⠀⠀⠀⠀⣠⣿⡇⣧
⣿⣿⣿⣿⣿⣦⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⠸⠙⢿⠖⠀⢀⣠⣶⣿⣿⡇⢸⡆
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⣋⣶⡀⠌⠛⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡮⠟⣡⠔⠧⠌⢠⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣾⡟⠀⡀⢠⣾⡿⣻⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠉⠀⠀⢠⣿⡟⠀⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⢠⣿⣿⡁⢐⣿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⡿⠋⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⡄⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣷⣤⣄⣀⣈⡄⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⣼⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣧⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣽⠟⠁⠀⢿⣿⡇⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⢸⡀⠀⣿⡿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣶⡄⢰⢻⢶⢶⠀⡖⢳⠀⡿⠀⢰⠏⡇⢰⣶⡀⠀⡶⡆⠀⡀⠀⣀
⠀⠀⠀⠀⠀⢸⣿⠁⢸⢸⢸⠘⣦⣣⢸⢀⡇⡇⡞⡄⡇⠈⣷⣧⢠⡇⡇⢠⣿⢀⢿⢠⢦⠀⡞⡆
⠀⠀⠀⠀⠀⠘⣧⠟⡏⢸⢨⣶⣙⣿⣸⠸⢥⢿⢱⢇⡇⠀⡇⡹⣟⢓⣗⠘⣼⢻⣿⠀⣾⣾⢷⡇
⠀⠀⠀⠀⠀⠀⣿⡄⣷⢸⢸⡌⢷⠀⡇⣿⡼⣇⡷⠻⡇⢀⡇⡇⠙⢸⣸⢘⡎⡏⣽⠀⣿⠙⠘⣧
⠀⠀⠀⠀⠀⠀⣷⡇⠻⠞⣼⡇⠈⠠⠿⠯⠑⠈⠙⠂⠈⠺⢻⡁⠀⠀⠀⠘⠃⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢴⠀⠀⠀⠉⠀⠀⠀⠀⢱⠂⠀⠀⠀⢀⣤⡤⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣻⠀⡟⢀⡆⠀⠀⠀⢸⠀⢠⢀⠀⠀⠀⠀⠐⡧⢤⡀⠀⠀⢿⠀⠁⠀⢱⠊⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣭⠈⠁⠀⠀⠀⢸⡀⢎⡀⠀⢀⡔⢠⠒⢧⠀⢱⠀⠀⠀⠑⢄⠀⠈⢆⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⠢⢤⡌⠁⢄⢉⡝⢂⡀⠈⣁⡽⠀⢀⣾⠦⣄⣈⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣶⠆⠀⣄⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⣈⣤⡛⣯⣝⢛⢁⣀⣠⣾⣭⣙⣧⠾⣤⣑⣄⣠⠃
⠀⠀⠀⠀⢀⡴⠧⢄⠀⠀⠘⢀⡀⣀⣰⡠⠢⠞⠖⠛⠛⠉⠉⠀⠀⠀⢀⡄⢦⣁⠜⢙⣏⠳⣤⠋⡠⣤⣀
⠀⠀⠀⠀⣼⣿⣆⠀⠣⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⣾⠋⠀⠀⠀⠓⠈⣷⣼⣞⣷⣯⣼
⠀⠀⠀⠀⢧⣿⡿⠀⠞⠀⠀⠇⣀⣦⠁⢨⡼⣷⠞⢻⡘⢯⢉⣧⣸⡤⢽⡝⣦⣄⠐⢍⣯⢨⣿⠜⠁⣼⣼
⠀⠀⠀⠀⠘⢏⡤⠔⠀⠀⠻⣿⢧⡘⣆⢘⣧⣬⣷⣾⣿⣯⣽⣿⣾⡷⠾⠷⠧⠼⠿⠥⠟⠛⠛⠀⠀⡷⠾
⠀⠀⠀⠀⠀⠀⣷⠖⠄⠘⠉⠈⠙⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣒⣶⢲⡖⣾⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢉⡉⢉⡉⠉⣉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⣁⢈⣉⣀⣉⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⢷⣄⠀⠀⠀⠀⣠⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⢷⣄⣠⡾⠋⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⣠⣾⠟⠻⣷⣄⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⠟⠁⠀⠀⠈⠻⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠋⠉⠙⠛⠛⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠒⠛⠛⠉⠉⠉⠉⠉⠉⠛⠓⠒⠲⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢠⠏⠀⠀⣾⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⡾⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣤⣤⣠⣤⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⣿⠶⠾⠛⠛⢻⡇⠀⣿⣴⠟⣷⠾⠻⣧⠀⢸⡟⠁⢹⡏⠀⢿⡄⢸⣿⡾⢻⣷⠟⢿⡆
⢷⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⣿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⢸⣿⠀⢸⣇⠀⢸⣧⠀⢀⡀⠀⣀⠀⢀⡀⠀⠀⢰
⠘⣇⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠿⠀⠀⣿⠄⠀⢿⠄⠈⠁⠀⠘⠃⠀⠈⠁⠸⠿⠀⠀⣿⠀⠀⣿⠀⠙⠁⠀⠋⠀⠈⠋⠀⢀⡟
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠙⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⠤⠤⠴⣇⠀⠀⠀⠀⠀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣰⡏⠁⠀⠀⢈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⢯⡀⢈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⣴⣼⣡⣱⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⢿⣿⣾⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⡳⣫⣿⡗⢿⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠢⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⡀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣾⠙⣀⠁⠀⠀⠀⠀⠀⠀⠀⣖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡑⢋⠗⠀⠀⠀⠀⢀⡞⠤⣨⣍⣋⡭⢤⣫⢶⠖⣠⣀⣀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢥⡀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠀⠈⠁⠀⢃⠘⡥⢧⣶⠼⣎⡑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⡿⢻⡀⠀⠀⠀⡇⠀⡶⡤⠒⣶⣶⡄⠂⢸⠘⡴⠉⠀⠉⠁⠙⢇⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⢇⡜⢲⠒⣊⡇⠀⣷⣴⣻⠏⠿⡇⠀⢸⢠⡇⠚⠁⠀⣻⢤⠞⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠉⠉⢈⣉⡇⠀⠻⠦⣽⢿⡼⠇⠀⠸⠸⡇⠤⡤⠤⡬⣞⣦⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠶⠼⠂⠀⠀⠤⠴⢤⠀⠳⠟⡇⠁⠀⠀⠀⠀⠀⠋⠀⠰⠀⣇⡀⠀⠀⠈⠙⡽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠭⠞⡤⠤⢤⡁⠂⠤⠄⢀⣀⣀⠀⠠⠼⠚⠁⢰⣱⠀⠀⠀⠀⡞⣛⣵
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⡠⠔⠈⠑⠒⠚⡷⠀⡆⡶⠀⢀⡔⠒⣠⠆⠀⢀⣴⢰⣆⣆⢀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠘⠄⣀⣀⣠⣤⡮⡕⠲⡒⡄⠀⢎⡠⠼⣁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⡍⠉⢣⢀⣸⠤⠗⠋⠀⠈⠢⠎⣡⣻⢋⠞⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠖⠒⠉⠁⠀⠀⠦⠯⠭⠀⠀⠀⠀⠇⡎⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠧⠚⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣤⣿⣆⠀⢀⡀
⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠹⣿⣿⣟⡋⠁⠈⠉⠢⣄
⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢈⠻⢿⠁⠀⠀⠀⣰⣬⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠇⠀⠀⠀⠀⠀⠀⠠⠀⢀⣼⣿⣿⣷⡀⠀⠈⢤⣾⢿⣦⣬⣽⣿⣷⣙⢶
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠞⢀⡤⣆⠄⠠⠀⡀⢀⣿⣿⣿⡿⠛⠀⠀⠀⠈⠻⣿⣾⣿⣟⠿⠿⠀⠀⠑⢄
⢸⣆⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠻⠦⠚⢀⣐⠋⢰⡈⣁⢸⠃⠁⠀⠀⠀⣤⡄⠀⢀⣿⣿⡿⠿⣿⣷⣶⣴⠈⢦⡀
⠘⠛⠛⠣⣄⠉⠁⠀⠀⠀⠀⣁⢀⣦⣀⡉⣻⣷⢯⡽⠋⠔⣻⠟⣬⠁⠀⠀⠀⠀⠈⣿⠃⠀⣾⣿⠛⠉⠀⠀⡈⢻⣿⣷⣤⠈⠂
⠀⠀⠀⠀⠈⠛⣦⡀⣄⠀⠚⠉⠉⢀⡀⠠⢋⡥⢞⣠⣋⣠⣎⣷⣄⣶⡄⠀⠀⠠⠔⠁⣴⣾⡿⢋⠁⢠⣶⣿⣷⠀⢹⣬⣥⣤⡇
⢸⣦⣤⡀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈⠡⠉⠐⠞⠛⠻⠿⢛⡿⠿⠯⢄⠁⠀⠀⠀⠀⠀⠙⢷⣍⠺⢀⣤⣍⡻⢋⣶⠰⣿⣿⣿⡷⡀
⠀⢻⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⠄⠀⠐⠀⠀⠀⠀⡄⢀⡐⢠⡿⠀⠀⠀⢉⣿⣿⡟⠋⠉⠙⢫⠃⠈⣿⡟⠙⠘⡄
⠀⠈⠻⠿⠿⠷⠆⢠⡀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣴⣾⠟⠛⠂⠀⢿⣼⣷⣿⡇⠀⢀⡴⠛⡑⢸⣇⡧⠀⢀⣘⠀⠀⣇⠐⣴⣶⡙⡄
⠀⠀⠀⠙⣃⡤⠀⠘⠀⠁⠀⠀⠤⠤⠀⠀⢘⣧⠀⢸⣿⣇⠀⠀⠀⠀⠀⣹⣿⣯⢀⢰⣿⡟⣷⢷⡙⣻⣿⠀⠈⠙⠦⡀⠹⣿⣿⡿⢣⡆
⠀⠀⠀⠀⠈⢻⡁⠀⠀⠀⣠⣤⠄⣠⣤⠅⡞⣿⣷⣈⡛⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠤⠁⠠⠀⣾⣼⣿⡿⣴⡧⠄⢀⠀⠠⢈⢻⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡉⠀⠀⠐⠋⢀⡈⠀⡐⠀⠆⠃⢻⡟⣡⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡈⢛⠛⠿⢾⡐⠠⠐⢶⡈⠁⢻⡏⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠶⢏⣴⣾⣗⡆⢀⣴⠋⠀⠀⠀⠀⠀⢀⣴⠆⣠⠄⢠⠀⠀⠀⠀⠀⢸⠄⠻⠦⣌⠀⠐⢰⠃⠀⣸⣷⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠸⣀⠀⠀⠀⠀⡿⠘⠁⠀⠀⠀⠀⠀⠀⠘⡇⣰⣿⣇⣈⡄⠀⠀⠀⠘⠀⠀⠀⠀⠈⢱⣄⡜⠈⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡙⠢⢀⢀⣁⠀⡀⠀⢠⡗⡦⠀⣾⠞⡇⠉⢹⣮⣄⠁⠀⠀⠀⣀⠀⠀⢠⣀⠀⡔⠧⠀⢨⣦⣹⣙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣴⣾⣿⢸⣷⠀⢸⡇⣰⣄⣽⠄⠁⠀⠘⣿⣿⣀⣦⣴⡤⡉⣿⣦⠈⢿⣷⡈⠀⠀⢼⣿⣿⣿⣷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣿⣿⠸⣿⡆⣽⡇⢽⣿⣿⡇⢸⡆⠀⣿⡟⣿⣿⣿⡇⣷⠸⣿⣷⡈⢿⣿⣧⠸⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠶⠀⠀⠈⠀⠙⠛⠻⠟⠛⠁⠹⠛⠗⠪⠭⠿⠀⠳⣤⣞⠃⠹⣿⣿⡇⢸⣧⠙⠻⠇⠈⠿⠿⠦⠙⠿⠿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢡⣤⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣠⣤⣍⠛⠆⠀⠀⠉⣩⣤⣬⡉⠁⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠁⠀⠘⡇⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⣄⣀⣰⠇⠀⠀⠀⠘⣧⣀⣠⡿⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣤⣤⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠾⠷⠶⠶⠶⠶⠶⠶⣄⠓⠶⠶⠶⠄⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠟⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⣀⣀⣈⡷⠘⠛⠓⠒⠒⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⠞⠉⠉⠉⠛⠁⠀⠀⠈⠻⣄⣠⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠳⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠧⠿⠶⠤⠴⡬⠿⠀⢀⣿⠆
⠀⠀⠀⠀⠀⠀⠉⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣀⣠⡴⠃⠈⠲⠤⣤⣤⡤⠶⠞⠉⠉⠛⠛⠋⠁⣾⠉⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠚⠡⡎⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚
⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⢶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣻⡿⠃⠀⠀⠀⠀⠀⠀⢸⡟⠁⠉⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠻⣷⣦
⠀⠀⠀⠀⣼⡏⠉⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠹⠷⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⡗⠦⣀⠀⠀⠀⠙⣟⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠘⢋⣤⡄⠀⠀⠀⣀⡀⠀⢀⣹⠀⠀⠙⠲⠤⣤⠼⠀⠈⠳⣄⠀⡷⢦⡀⠀⠀⠸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠈⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠃⠀⠀⠀⠀⠀⠀⣈⡃⠀⠀⠀⠀⠄⠠⠀⠸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡴⠋⠉⠉⠓⣆⠀⠀⣠⣤⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⠀⠀⠘⡆⢸⠱⠀⠈⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠦⣀⡞⠀⠀⠀⠸⡄⠀⠀⠀⠀⢿⡇⠘⣆⠀⢀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⠛⠓⣌⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⡼⠀⠀⠈⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⢆⠀⠀⠀⠀⠀⠈⠦⠒⠚⠁⠀⠀⣈⠀⣸⠂⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠓⠦⣄⡀⠀⠀⠀⠀⡄⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⢀⡔⠀⠀⠀⠉⡷⠒⠶⠴⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⢲⠞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣍⡛⠃⠀⡞⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣖⣂⣀⡴⠧⣄⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠉⠁⠀⠠⢤⠬⢍⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣁⠉⢻⣙⣒⣶⣶⢟⡒⣻⠋⠉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠭⠭⠟⠚⠉⠈⠑⠛⠽⠿⠯⠝⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠢⣀⠤⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢷⠀⢀⣀⠀⠀⠉⠢⠀⠀⠀⠀⠀⡀⡀⣀⡀⠀⡴⠋⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠈⣳⠀⠀⣤⠴⠺⣏⠉⠀⠀⠀⠀⠜⠀⠀⠀⡴⠁⢀⡈⢠⣤⡼⠇⠀⠀⣴⣷⠄⠀⠀⣷⠀⣀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⠂⠀⡠⠐⠊⠉⠉⠉⠀⠉⠓⢿⣳⠊⠁⠀⠀⠀⢠⣿⠃⠀⠀⢀⣾⠘⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠑⢀⠀⢀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠻⣿⣷⣦⣤⣶⡖⢼⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⢒⡦⠖⠛⠉⠀⠀⠀⣀⠀⠀⢰⠁⠈⠀⠈⠉⠲⡄⠀⣠⡄⠀⠀⠀⠀⠀⣏⠀⠉⣰⠞⠉⠀⠙⠤⠖⠂
⠀⠀⠀⠀⠀⠀⢧⡀⠹⣺⠋⠀⠀⠀⠀⢀⠰⣏⠁⠀⠀⠈⠢⠤⢤⡀⠀⠀⢸⠈⠋⠀⠀⠀⣀⣠⣼⠟⠉⠀⠃⠀⠀⠀⠀⣀⡀⠀⣀⡀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢧⣀⡇⠀⠀⢤⣄⡀⢸⠀⠸⣦⠤⠀⠀⠀⠀⠀⢸⠀⢀⣾⢦⣤⣴⣶⣾⣿⣿⠁⠀⠀⢀⡀⠠⣄⢠⠂⣇⣜⢠⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠙⠿⡀⠀⠹⡄⢀⣠⠎⠳⠖⠃⠀⣿⢹⣫⠘⠛⠂⠼⡟⠀⢸⠲⢾⢸⣳⣄⠈⢇⠀⠧⠔⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠹⡄⠀⠙⠉⠀⠀⠀⠀⠀⢰⡏⠈⠀⠒⠲⢒⠦⠓⠀⠈⡆⠈⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⠁⠀⠀⠀⣀⡧⠀⠀⠀⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⡰⢏
⠀⠀⠀⠀⠀⢀⡤⠚⢀⡤⠶⡌⠑⠒⠒⢾⣧⣀⠀⠀⠀⠀⣀⠴⠋⠉⠀⠀⠀⢀⡞⠁⠀⡐⠇⢰⠠⢉⠈⠉⢻⠛⣲⢶⠒⠋⠀⣸⠂
⠀⠀⠀⠀⠀⡎⠀⠀⠾⠀⢠⡇⠀⠀⠀⢨⡇⠈⣛⣉⡽⠉⠁⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠘⠌⣀⠜⠀⢀⢾⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠘⠃⣀⠀⢰⠟⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⣄⡀⠀⣀⠔⠙⠁⠀⠑⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⢉⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⣤⣤⣤⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⣤⣤⣤⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⣾⣿⣿⣿⣿⣿⣆⣻⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣏⢀⣾⣿⣿⣿⣿⣿⣧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸⠿⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠄⠅⠀⢀⣴⠾⠿⠿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⠷⣦⡀⠀⠨⠥⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠐⠋⠀⠀⠀⢀⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⡀⠀⠀⠀⠙⠀⠀⣠⣴⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡤⠚⢉⣁⣤⣉⣑⠢⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⣉⣄⣈⡉⠒⢄⠀⠀⠀⠙⢩⢿⡽⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢠⠊⢀⣶⣿⣿⣿⣿⣿⡷⢌⢣⠀⠀⠀⠀⢀⡞⣱⣾⣿⣿⣿⣿⡿⠦⡀⠱⡀⠀⠀⣬⣏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⣾⣿⣿⣿⣿⣿⣿⣆⣸⡌⡇⠀⠀⠀⢸⢰⣿⣿⣿⣿⣿⣿⣷⣄⣥⠀⢧⠀⠀⠓⠋⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡜⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⡄⠘⢿⣿⣿⣿⣿⣿⣿⠏⡼⠁⣶⣴⡦⠈⢆⠻⣿⣿⣿⣿⣿⣿⡿⠁⣰⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠢⣀⠉⠛⠛⠛⢋⡡⠊⠀⠀⢈⣉⠀⠀⠈⠳⢌⡙⠛⠛⠛⢉⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡜⠉⠳⡀⠀⠀⠉⠉⠉⠉⠁⠀⢠⠶⠛⠛⠛⠛⠻⢦⡀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠀⠀⣹⡤⠖⠚⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⠞⠁⠈⠀⠀⢀⣀⠤⢴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⠀⠀⠀⠀⠀⠀⣀⡠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢚⣇⠀⠀⠀⠀⠀⣀⡠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠤⠳⣄⡀⠀⣀⣠⠴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢈⠐⠀⠀⠀⠀⠀⠀⠤⠤⠑⠂⠠⢄⡀
⠀⠀⢰⣄⢠⡄⢠⡤⣠⣀⢀⢄⢸⡂⡀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠈⠙⠘⠂⠈⠁⠉⠉⠘⠊⠙⡇⠛⠀⢸⠀⢠⠰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⡤⠀⣤⣠⣄⠀⣀⠀⣀⣁⠀⠀⢸⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠧⠀⠛⠈⠫⠸⠤⠃⠻⠝⡄⠀⢸⠀⠀⠀⠀⠀⠀⠢⠑⠄⠀⠀⡠⠁
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢸⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠚⠀⠉⠉⠉⠀⠀⣤⠀⠠⠉⠉⠉⠉⠁⠱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠤⠀⠸⠉⠁⠾⠿⠿⠷⠿⠿⠶⠿⠶⠿⠾⠭⠤⠄⠄⠀⠀⠄⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠒⠀⠤⠴⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠀⣀⡉
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠿⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠤⣀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⣩⣷⣄⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⣷⡀⠀⠀⠀⢰⢃⣶⣿⣻⣦⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⡇⠀⠀⠀⡏⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡇⠈⠛⠻⠛⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠰⠒⠚⠛⠉⠛⠛⠒⠀⠀⠉⠉⠙⠒⠦⠚⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⡤⠞⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠈⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣼⠵⠚⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠤⠤⠾⠗⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠶⠶⢷⣶⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠋⠉⢉⣉⣡⣤⣤⣀⣀⣀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⡾⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠟⠻⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⢰⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⣀⣤⣤⡉⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⠸⡇⡧⢸⣿⡏⠈⣿⡁⠀⠀⠀⠀⠙⠉⢆⠀⢠⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⡆⠳⠹⠘⣿⣿⡄⠈⢀⡀⠀⠀⠀⠀⠐⠂⢀⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⠘⣿⣦⣀⡀⠀⠀⡄⣸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡆⢰⣿⣿⡿⠉⠉⠉⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⠁⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣭⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⣭⣿⣏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿