Bonfire Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠘⣆⠀⠀⣿⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡄⠀⠀⢀⠜⠀⠀⠀⣄⠀⠀⠀⠘⣆⢀⡇⠱⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠢⣄⠎⠀⠀⠀⢀⠛⡄⠀⠀⠀⠸⠞⠀⠀⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⣴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣸⠀⠀⣄⠀⠀⠀⠀⢰⠃⠀⠀⢱⠀⠀⠀⢰⡆⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⠀⠀⡏⠱⣄⠀⠀⡎⠀⠀⠀⠈⡇⠀⡰⠃⢱⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣄⡇⠀⠈⢣⡀⡃⠀⠀⠀⠀⣧⠊⠀⠀⢸⠀⣠⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡏⠃⠀⠀⠀⠙⠇⠀⣷⠀⠀⠁⢀⣶⠀⠘⠋⢸⠟⡈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠹⡄⠀⠀⢶⣀⠀⣰⠁⢣⣠⠞⠁⢸⠀⠀⡰⠧⣪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⡄⠀⢸⠈⠳⠇⠀⠘⠁⠀⢀⡼⢀⡴⢥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠻⠤⠬⠦⣤⠔⠒⡒⠒⠲⢾⠒⢭⡰⢄⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡴⠊⠀⠀⣠⠔⢑⣦⡈⠳⠄⠀⠀⠰⢍⠂⠉⠲⠌⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣷⡦⡜⢁⠄⣠⠔⠋⠀⣠⠵⠚⠉⠲⢄⡐⢦⡀⢄⠙⠆⡴⢊⠵⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣟⠷⡜⠀⡠⢖⡩⠖⠉⠀⠀⠀⠀⠀⠀⠉⠲⣌⠳⠁⡞⡰⣟⡼⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣒⣱⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣧⣠⠭⠒⠁


⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⡇⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣇⠀⠈⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⠻⣶⣴⣿⠃⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⡄⠀⠀⠁⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⢻⡇⢠⡀⠀⠀⣼⡇⢀⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣇⠀⠀⠀⣸⡏⢸⡟⠶⠞⢹⠇⣸⣿⣿⠏⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠘⣿⡄⠀⢀⣿⠇⣸⠁⠀⢠⠏⣰⣿⢹⡿⢰⣆⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⠿⡶⠿⠋⣰⠏⠀⠀⣾⠀⢿⣷⡿⠃⡾⢹⡄⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡿⢦⣤⠴⠞⠁⠀⠀⠀⢻⡄⠀⠀⢀⡼⠃⠈⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢻⡄⠀⠀⠀⠀⠀⢀⣀⠀⠉⠓⠚⠋⠀⠀⢸⠇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠀⢁⣀⡤⠶⢒⣾⠉⠉⣿⣄⡀⠀⠀⠀⢠⡟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡩⠥⢤⡾⢫⢻⡄⡸⣸⣏⢏⢻⠲⠦⣤⣀⡉⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠯⠭⢤⠤⠀⢸⣠⠯⣔⠥⢛⣧⣽⠁⣿⠻⢷⡯⢶⢤⣬⣍⣹⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠛⠲⣄⠀⢰⣖⣰⣿⡳⠞⠢⣀⡞⣿⣷⠋⢹⡆⠠⠛⢶⣖⣒⠒⠢⡿⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠸⠧⢤⣤⣽⡴⠟⣹⡟⠥⠀⠤⣠⡟⠀⢹⣿⣀⣈⣷⣇⢰⠉⠉⠻⣝⠓⠷⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⠦⣄⣠⠏⠀⠀⢸⡿⠉⠉⢻⠙⢶⣠⡟⢉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡤⠾⠋⠀⠀⠀⠀⠓⠒⠛⠛⠀⠈⠻⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⡄⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⢶⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠊⡈⠀⠀⠀⢀⡆⠀⢀⡀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⣀⣼⠀⢰⡁⠉⠀⠎⠰⣋⡈⢪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⢳⢸⢸⣿⠀⡌⠳⢦⡀⢄⠸⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠂⠣⡱⡌⢆⢣⠻⣷⡙⡄⠀⢻⡌⣆⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠈⢀⢩⠑⣌⢻⣾⣷⣇⠹⣷⢹⣴⣼⡿⢸⢀⠀⠈⡇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢑⡏⣝⡳⣿⣿⣿⣶⡿⢸⣿⣟⡵⠋⡈⠀⣸⠇⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠠⠓⠂⢀⡇⠘⣿⣿⣿⣧⣿⣿⡋⣰⠾⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣀⣀⡀⢐⡄⠸⣧⠰⡘⣿⣿⣿⣿⣿⣿⣷⣷⢄⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠀⠟⠹⢭⠏⠛⢿⣿⠿⡿⠋⠉⠁⠀⠀⠀⠃⠈⠈⢉⡶⡲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣠⠇⠀⠀⣀⠴⠊⠉⠀⠀⠀⢀⡀⠀⠀⠉⢀⣦⡄⣾⢰⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠙⠛⠂⠙⡟⠲⢖⢦⡀⠀⢠⣰⣯⣦⢀⠏⠙⠋⢹⣿⠾⢦⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⣿⡅⠀⠀⣧⠀⡌⣿⣇⣤⠞⠛⠈⢿⣾⣄⣉⣀⣿⠻⡇⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣷⣀⣨⡥⢴⣿⠟⠛⢲⢿⣿⢿⣧⣈⣠⡼⠏⡉⠫⠁⠀⠀⠂⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠷⠀⠐⠈⠁⠀⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⣻⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠠⠀⠐⠄⠀⢉⡃⠀⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⣺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⢼⣷⣀⠀⠀⠀⣸⡇⠀⠀⡀⠈⣽⣏⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢀⡾⠁⠐⠀⠀⣴⡿⠁⠀⠀⢸⡄⠁⠙⠦⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁⠉⣰⠄⠀⡆⢸⡿⠁⠀⠀⠀⣾⡇⣾⠀⠀⢨⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠦⠚⠁⠀⢰⠇⢸⠇⠀⠀⠘⠀⣿⠁⢸⡄⠠⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⡐⠀⠀⣿⠀⠈⠇⠀⡆⠀⠀⢹⡇⠸⣿⠀⡀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣏⠀⠀⠟⣿⠀⠀⠀⠀⠱⣦⠀⠋⠀⢸⡿⠀⠀⣿⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⣿⡇⢀⠀⠀⢻⡀⠀⠄⠀⠀⢹⣇⡠⠀⢸⠁⠀⡀⢸⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⣿⣟⢸⡆⠀⠀⠁⠀⠀⠰⠀⢸⠋⠀⠀⠀⠀⢠⡇⣽⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⠿⠾⢿⣄⠸⣄⢰⡇⠀⣴⠀⠀⣸⠀⠀⣠⣾⠷⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣬⢻⣷⣿⣶⣿⣶⣿⣷⣶⣿⣾⣿⠟⠁⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣹⢟⢁⡽⠋⢛⠉⢻⡋⠙⢯⡈⣻⣿⣥⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⣿⣧⡶⠉⢂⡠⠊⢷⠀⠑⢌⡀⠈⢶⣼⣇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢈⣿⢷⣤⠮⣤⣄⡼⢦⣠⣤⣵⣦⣾⠏⠉⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣷⣤⡿⠃⠀⠉⢳⣾⠿⢦⣶⡟⢉⠉⠙⣇⠀⠀⣸⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⠈⠛⢯⣤⣤⡔⠛⠡⠀⢀⠉⣣⣀⣁⣠⣾⣶⡶⠋⠀⢴⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠙⠶⠶⠶⠾⠛⠋⠉⠙⠉⠉⠁⡀⠴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠖⠄⢠⡄⡄⡄⡀⠀⡄⠀⠤⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡟⠀⣼⣿⣿⠀⣸⡇⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⡿⢠⠿⠏⢸⠟⠘⣷⠀⢰⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠛⠋⠀⣠⠋⠀⠀⠈⠀⠀⣹⡇⠾⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡴⠏⠀⠄⠀⠋⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢼⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⣾⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⡟⣧⠀⡄⠀⠀⠀⠀⠰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡴⣊⢃⣿⣿⡸⣿⠁⠀⠀⡀⠀⠀⡀⣤⣰⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣶⠿⠁⢡⠎⠘⣿⡇⣹⡀⠀⢠⠃⠀⠀⠀⣿⣿⢦⡖⠐⢤⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⢴⣿⡏⠁⣠⠋⢁⡆⠀⡄⠀⣠⠏⣷⠋⣷⡀⢸⡆⠀⠀⠀⣧⡈⠣⡙⢷⣿⣇⠀⣼⣥⠽⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⣁⣀⠀⠀⠀⠈⣿⣿⠞⢁⣾⣿⣿⠎⠀⢰⣿⠀⣿⠀⠘⣿⣾⣇⠀⢀⣼⣷⢍⠀⠈⠐⡙⢿⣴⡟⠁⠀⢀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣥⣱⡄⣀⣤⡿⢃⣴⣟⣿⢣⠏⠀⢠⣿⡇⠀⡇⠀⢸⣿⣈⡎⣷⠻⢿⣿⡎⢳⣄⠀⠈⢲⣹⣁⡀⠀⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⠀⠀⠈⠙⠛⠻⣿⣵⣿⣋⣾⢣⡟⠀⢀⣿⢙⡇⣸⠁⠀⠀⣿⣏⣿⢸⡀⠈⢞⣿⡈⣿⣷⡶⠖⠿⣧⣈⠰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠀⠀⠀⠀⠌⣱⣾⠛⣙⣙⠛⢳⠖⠚⠛⠛⠓⣿⢾⣁⡥⠿⠿⠿⡧⠿⠄⠺⠿⣿⡋⠁⢀⣀⣠⠄⠘⣶⡄⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡌⢀⣻⡇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡟⠀⢀⣀⡀⠀⠀⠀⠀⠀⠹⡇⠀⠏⠁⠀⠀⠀⢹⣷⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢠⠁⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡇⠸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⣾⣤⣤⣏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣽⡇⠈⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡼⠋⢱⣦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠓⠦⠤⠤⠤⠄⠤⠤⠤⠧⠤⠛⢇⣄⣀⣀⣤⣤⣀⡤⣴⠁⠀⠀⣼⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠒⣲⡉⠒⠚⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠥⠤⠭⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⡜⠀⡀⠀⠑⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠄⠁⠠⡈⠐⢄⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⢠⠀⡀⡆⠀⠀⠀⠀⠰⢄⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠀⡠⠁⠀⠓⠀⠀⠀⠁⠀⠡⡐⠀⠀⠀⠘⠄⠠⠊⠀⠀⠀⠀⡐⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦⠀⠒⠈⠂⠘⡡⠀⠀⠀⠀⠌⠢⡀⠀⠀⢈⠢⡁⠀⠁⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡐⠈⠃⠢⡀⠀⠀⠀⠀⣀⠀⠀⠀⡞⠀⠀⢻⠀⡄⠠⠀⡀⠀⠐⠀⢀⠀⡠⠌⠢⡀⠚⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠈⠢⢀⠁⢦⠏⢦⡀⡙⠀⠀⠀⢸⠀⢰⠁⠨⡝⡆⠀⠀⠀⢨⡆⠊⠁⠀⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⠙⡀⠶⠀⠀⠑⡇⠀⠀⠀⢸⡀⠀⠀⡜⠀⢸⠀⠞⣀⡜⠀⣄⠀⠀⠀⠁⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⡤⡁⠀⢸⠀⠀⠀⠙⠀⠀⠀⠘⡄⠀⠼⠁⠀⢸⢀⠀⠌⠠⠀⡸⠀⠀⡀⠀⠁⢰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⡇⠈⠓⢼⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠁⠀⠀⢸⠁⠂⠀⠀⡰⠃⠀⠀⠐⠒⠒⠂⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢱⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠘⢦⠀⠀⢰⠃⡞⠔⡮⢻⠅⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠂⡠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢠⠖⠋⢀⠞⢀⠈⠐⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠉⢀⠀⠀⠀⠀⠰⠄⠀⠴⠀⠀⠀⠀⠀⠀⡠⠇⢤⠀⠀⠘⠃⠄⠀⠊⢀⠟⠐⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⣨⣅⡂⠀⠀⠐⠀⠀⠄⡙⠠⠀⠂⠖⡄⠀⢀⠀⠁⠠⠔⡂⠀⠀⢀⠤⠶⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠠⡦⢷⣒⣒⠒⠈⣉⣭⡤⠐⠒⠂⠀⠀⣠⣀⣀⢁⡀⣴⢡⢄⣱⡀⠤⢄⣀⡀⠀⠀⠴⣾⢾⣶⣶⠄⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢯⣧⣩⣯⣭⡤⢾⣁⡇⣀⣽⣤⡾⠿⡏⠑⣢⠈⣹⠇⡧⢅⡄⠀⢿⡆⠄⠽⠄⡷⠂⠀⢠⣤⡯⣖⣛⣐⣨⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠑⠒⠚⠛⠋⠉⠁⠚⢻⣧⠼⠫⠁⠻⣄⣡⣠⣞⡛⢦⣖⣸⣛⡛⠳⠾⠾⠤⠟⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢠⠇⡄⣰⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⣠⠏⠀⠀⢿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⡇⢿⠀⢠⡇⢸⡇⠰⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣼⠃⠀⠘⡀⣿⡇⠘⠀⠀⠉⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠃⡀⢻⡄⠀⠀⠁⣻⠏⣠⠆⠀⠀⡇⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣧⠈⠇⠀⠀⣰⡏⢰⡏⠀⠀⣼⠃⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡇⠹⢀⢠⠆⠀⢹⡁⢸⠀⠀⢸⡇⢠⠟⢀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⢀⠎⠋⠀⠀⢸⠃⠈⠀⠀⣸⣼⡏⠀⡞⠠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠘⣠⠀⢠⠎⢠⠇⠈⠀⠀⠀⠀⠀⠘⠀⡀⠇⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⡇⠀⡟⠀⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠱⠸⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣧⠀⢁⠀⢹⡄⠀⣷⠀⠀⠀⠀⢀⠀⠀⢠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠶⠤⠍⠢⠀⢧⡀⠃⢸⡇⠀⢀⠆⣠⠏⢠⢠⠯⠶⠶⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⠿⠟⠳⠟⠓⠊⠁⣸⣷⡄⣼⠧⣀⣾⡴⠿⠃⢬⣼⡘⡛⠛⠋⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠆⠀⢠⣄⣀⣤⠖⠋⠀⠉⣹⢁⠀⠘⣿⡀⡀⠀⠀⠈⣝⠻⠷⣶⣤⣶⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⣠⣾⠿⠛⠱⠂⡀⠀⢀⣼⡃⠙⠀⠁⠙⠻⣦⣶⠂⠀⠩⢀⢀⠄⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠂⢰⣾⢿⠋⠙⠂⣤⢀⠀⠀⢸⡇⠊⠀⠘⠋⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠋⠀⠈⠠⣴⠀⠘⠁⣈⡴⠉⠳⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⢸⣇⠀⠀⢠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠁⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⡄⠀⢰⠏⢧⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⡞⠀⠹⣤⠏⠀⠘⣇⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣼⠀⠀⠀⠈⠀⠀⠀⢹⡀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡘⢦⡀⠀⠀⠀⠀⣠⢟⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⣛⠻⠶⣦⣤⣀⣀⠀⠉⠛⠻⠿⣷⣶⣶⠿⠿⠛⠉⠀⢀⣀⣤⣴⠶⠟⢛⣛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠲⠦⣭⣉⡀⠀⠈⠉⠛⠻⠶⣶⣤⣄⣀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠁⠀⠀⣈⣩⡤⠶⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠰⠦⣬⣉⡛⠲⠦⣬⣙⣓⣶⣤⣽⠿⠟⣛⣩⡥⠶⢚⣋⣡⠤⠶⢚⣋⣩⠤⠖⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢶⣬⣭⣿⡶⠶⠟⠛⠉⠀⠐⠚⣋⠉⠀⠀⢈⣉⡅⠀⠀⣛⣩⣥⣶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⣿⣋⣩⠤⠖⢚⣋⣥⠴⠖⠛⠉⠁⠀⢐⣛⣩⣥⣶⡾⠻⠿⣭⣙⣿⡷⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢡⠖⠲⣌⢻⡆⠀⣈⣩⡥⠴⢒⣋⣤⣴⠶⣾⣿⣙⡛⠶⢤⣍⣉⠀⢠⣿⢡⠶⠲⣎⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⣤⣠⠟⣸⣇⣈⣭⣴⠶⠟⠛⠉⠁⠀⠀⠀⠈⠉⠛⠻⠶⣶⣭⣁⣘⣷⡸⢦⣠⠞⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢶⣶⠾⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⢀⣴⣿⡏⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⡀⠸⣿⣿⣿⣿⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⡇⠀⠀⣄⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡇⠀⢀⣿⣇⠈⣿⣿⡇⣷⠀⢻⣿⣿⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⣸⠇⠀⣼⣿⣿⠀⢿⣿⢡⡟⠀⢸⣿⣿⡏⠀⢸⡿⢀⣴⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠏⠀⣰⣿⣿⠏⠀⣾⠏⣼⠃⠀⢸⣿⣿⠀⣰⠟⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⡟⠀⢰⡟⣸⡏⠀⠀⢸⣿⡟⢰⡏⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⣰⡟⢀⣿⠀⠀⠀⣼⣿⠃⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡿⠿⢿⣿⡟⠉⠀⢸⣏⠀⠀⣰⡿⠃⠀⢿⡀⠀⣼⠋⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡟⠁⠀⠀⠈⠉⠀⠀⢸⣿⣤⡾⠋⠀⢀⠀⠈⠛⠛⠁⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣟⣼⠁⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⡈⢧⣀⣀⣿⠈⠀⠀⠀⣠⠀⠀⠀⡾⠁⠀⠀⠀⠀⢰⣆⢻⡆⠀⠀⠀⣰⣿⠿⠿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⣼⡿⢻⡏⠉⠛⠻⠃⠀⠙⠋⢹⣧⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠘⠃⠀⢀⣴⣿⠃⠀⠀⠀⠠⠶⠶⠖⠒⠒⠒⠂⠀⣀⣸⣇
⠀⠀⠀⠀⠀⢰⡟⠀⡀⣿⠀⠀⠀⠀⠀⠒⠂⠀⠙⢧⡀⠀⢿⡈⣙⣦⡀⠀⠀⠀⠀⣼⣷⡟⢀⣠⣴⣿⠟⠛⠋⠉⣉⣉⣤⣤⣤⠤⠤⠤⠤⠀⠀⠈⢩⡿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠐⠊⠹⣥⣤⣤⣆⣠⣤⣿⣷⣬⣿⣿⣿⣇⠀⠀⣠⡾⢿⣿⣴⣟⠙⠻⢿⣟⠳⢶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠤⠄⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢸⡇⠸⢆⣿⡆⠀⣀⣤⡭⠿⠿⠟⣛⣛⣻⣯⡭⠭⠉⠉⠛⣧⣰⣿⠀⠘⠋⠀⡀⠀⠀⠀⠀⠉⠐⠀⠉⠙⠳⢾⣿⣛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⠛⠛⠉⠀⠀⠒⠊⠉⠟⠛⠛⠉⠉⠀⠀⠀⠀⠘⣿⡟⢿⣦⣄⡲⣬⡓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣨⡽⠿⠶⢦⡀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⢷⣄⠀⠀⣀⣠⣷⠄⠖⠀⠀⢀⣀⡤⠴⠂⢀⡀⣿⣟⠲⠌⠛⠿⣿⡿⠷⣦⣀⠀⠀⠉⠲⡶⣠⡞⡉⠀⠀⠲⣆⢻⡄
⠀⠀⠀⠀⣸⢃⡄⡄⢠⡈⡆⢻⡄⠈⠉⠀⠀⠀⠀⣀⣩⣥⣶⣶⣿⠿⣋⣼⠟⠛⠻⢦⣷⠀⣀⠉⠓⠦⣍⠓⠂⠀⠸⣿⢋⡔⠀⣴⡆⠀⠙⣸⡇
⠀⠀⠀⠀⢻⡌⣧⢓⠀⠇⢹⠸⣇⠀⠀⠀⠀⣂⣭⠵⢒⣫⣽⠶⠟⠛⠉⠁⠀⠀⠀⠀⢹⡀⢣⡿⢶⣄⡲⢤⡀⠀⢰⡏⡾⢁⠀⡼⠃⠀⢠⡿
⠀⠀⠀⠀⠘⣧⢳⡘⠦⠄⡎⢸⡏⠀⠤⠖⣋⣭⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠈⠙⠷⣮⣀⣸⡏⣧⠺⡤⢞⡥⣠⡿⠁
⠀⠀⠀⠀⠀⠙⣧⡓⠄⠈⢀⣾⣥⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣌⣁⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⢁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠸⡉⠀⣘⢾⣆⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡼⠛⢃⣴⣿⠻⣮⣳⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣯⠁⣴⢋⡞⠁⠐⣿⠇⡋⢳⠹⠆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⠐⣬⣐⣿⠈⠀⢀⠀⢿⠞⣷⡾⢲⡐⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡸⠇⡿⠀⣿⡄⠙⡇⢀⣼⢷⣄⠀⠈⡗⠼⣷⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⣞⡞⢦⣿⡝⠦⢠⡞⠁⢰⡿⠀⢸⢵⡆⣿⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⣿⢱⣷⡄⠈⠃⣰⢿⣄⠀⢸⡟⢧⣄⠈⣿⢿⡀⢹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢿⡺⣧⠀⢰⢀⡇⠀⠙⠀⠀⠱⠀⡏⠘⠁⣾⣼⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢿⡈⠣⣿⡙⡅⢈⠧⠀⠀⠀⠚⠗⠀⣴⠿⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⣄⡉⠀⠀⠁⠀⠀⠀⠀⠀⣠⢴⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⡬⠝⠋⠛⠀⠀⠀⠀⢀⠀⠈⠁⠀⣠⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢾⣥⣴⠶⠋⠂⠀⠀⠀⠺⠿⠿⠷⠿⢻⡟⠙⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⣆⡉⢁⣀⣀⣶⣶⡿⢛⣻⣿⣿⣶⣟⢋⣉⢉⣙⣳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⠿⣿⣯⣤⡾⣇⣿⣨⣿⢯⡿⠛⠻⠿⠿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⣼⣿⣿⣿⡇⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⠀⢀⣿⣿⣿⣿⡇⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⢀⣿⣿⡿⠉⣿⣿⣿⠀⠀⢻⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⠃⠀⣿⣿⣿⠀⠀⠈⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⠀⠀⠀⢻⣿⡇⠈⣿⣿⣿⣿⢻⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⣿⣿⡿⣿⣿⣿⣿⠀⠀⠘⣿⣿⡇⠀⠀⢸⣿⡇⠀⣿⣿⣿⡏⢸⣿⡇⢸⣿⡇⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣿⣿⠃⣿⣿⣿⣿⣇⠀⠀⢹⣿⣿⠀⠀⢸⣿⠀⠀⣿⣿⡿⠁⢸⣿⣷⣿⣿⡇⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⢹⣿⡿⣿⣿⡄⠀⠀⢿⣿⡇⠀⣸⠏⠀⠀⣿⣿⠃⢀⣾⣿⣿⣿⣿⣇⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣷⣄⠀⣿⣿⢿⣿⣿⡀⠘⣿⣇⠘⣿⣿⡄⠀⠘⣿⡇⠀⠈⠀⠀⢰⣿⠃⠀⣼⣿⣿⠃⣿⣿⣿⡟⢡⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⡀⠹⣿⡇⠀⠹⣿⣆⠈⢻⣿⡀⠀⢿⡇⠀⠀⠀⠀⣿⠃⠀⢰⣿⣿⠃⢀⣿⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢻⣿⠀⠀⢹⣿⡆⠀⢿⡇⠀⣾⠃⠀⠀⠀⠼⠃⠀⠀⣿⣿⠃⠀⢸⣿⣿⠃⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠈⣿⠀⠀⠀⣿⣿⠀⠘⠃⠀⣁⣤⣤⣶⣦⣄⠀⠀⠀⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⣿⡆⠀⠛⠀⠀⠀⣿⡏⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠸⠇⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠂⠘⣿⣿⡄⠀⠀⠀⠀⠛⣠⣾⣿⣿⣿⣟⠻⣧⣙⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⢷⣌⢳⣌⢻⣷⣝⢿⡿⣿⣿⣦⠀⠀⠀⣽⠃⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢻⣿⠀⠀⠀⠘⣿⣿⣿⣧⡹⢦⡙⣧⡹⣦⣿⣍⢿⣿⣌⠻⣿⣷⣄⠀⠉⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⢸⡇⠀⠀⢀⣴⡿⣿⣿⣿⠻⣮⣻⣮⣿⣾⣶⡝⢿⣿⣍⠳⣌⢻⣿⣦⡀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⣴⡿⣫⡿⠋⣹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⡻⣷⡈⢿⣿⣿⣿⣄⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣠⣾⣯⣾⣿⣷⣿⣿⣿⣋⡥⠞⣋⣤⠾⣿⣿⣿⣿⣷⣜⣿⣬⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠟⣛⣩⣥⣶⠾⠛⠉⢐⣛⣩⡴⠟⣋⡿⣿⣿⣿⣛⣛⣛⠛⠿⢿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⢛⣡⣶⠟⠋⣩⠄⠀⢀⣤⠾⠛⣩⡵⠶⢛⣥⣾⣿⣿⣿⡋⣉⡉⠛⠿⣶⣤⣈⣙⡻⠿⣿⣛⠻⠿⣶⣤⣄⣀
⠀⠀⠀⣀⣤⡾⠟⣫⡟⢋⣥⡴⠟⣉⡤⠖⣫⣵⡶⢂⣈⣥⣶⣠⠿⣷⣾⠟⣋⣴⣿⣿⣿⣷⣬⣭⣛⠒⠶⢤⡍⠉⠛⠳⠾⠿⣿⣶⣿⣿⣿⣿
⢀⣴⣾⣿⣯⣴⣾⠿⠟⢋⣀⠀⣛⣭⡴⠟⢉⣠⣴⢟⣫⣿⣿⣷⣾⣯⣶⣿⣿⣿⣿⡿⠋⠀⢤⣭⣙⣻⣷⣶⣶⣶⡦⠄⠶⣤⣤⣹⣿⢋⡀⠈
⣾⡿⠛⠻⣿⣟⣁⣤⠶⠛⠁⣻⣉⣤⠀⣼⣿⣿⠿⠛⣋⡵⠞⣋⣥⣾⣿⣿⣿⡟⠉⠀⠛⠳⣶⣍⣉⠙⠛⢿⣿⣿⣷⣦⣠⣄⣩⣿⡏⣼⣿⢃
⢿⣇⢻⣴⡌⢿⣿⣁⣤⣤⣾⣟⣉⣴⣾⢿⣿⣷⣾⣿⣯⣴⣿⣿⣿⠛⣿⣦⣝⣿⣷⣦⣄⣀⠀⣬⣉⣃⡰⣦⣿⣏⢹⣿⣍⣙⣻⣿⣇⣹⠏⣼
⠈⣿⣎⢿⣟⢘⣿⣿⣯⣤⣾⠿⢟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣧⣬⣽⣿⣿⣶⣬⣉⠛⠶⣮⣭⣛⠀⢉⣙⣛⣿⠟⠹⣿⣯⣍⣹⣿⣇⠙⣋⣾⡟
⠀⠈⢿⣦⡙⠿⠏⣹⣿⣿⣷⣶⣿⣿⣿⠿⠟⠋⠙⠛⠿⢿⣶⣬⣙⡛⢿⣦⣬⣿⠻⠶⣾⣟⠻⢿⣿⣿⣿⠇⢸⣧⣿⠿⠿⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣭⣛⣷⣶⣿⠻⠷⣶⣤⣿⣿⠲⢞⣾⠏
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣷⣦⣬⣿⠿⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡉⠁⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠾⢧⣷⣠⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠋⠉⠁⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠷⣤⡀⡴⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠙⢶⠀⠀⠀⣰⠋⠹⡄⠀⠀⠀⠀⠈⠛⠛⠉⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢦⡀⣠⡾⠃⠀⣰⠃⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠛⠛⠉⠀⢠⣠⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡟⠻⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠨⣄⡀⠀⠀⠀⠀⢀⡬⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠁⠂⠴⢩⠁⠀⠀⢀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠛⠻⠿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠡⡀⢀⣀⣬⣿⣷⣶⣤⣀⡀⠀⠀⠀⠠⠎⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢹⣿⠿⠟⠛⠉⠀⠀⠈⠙⠛⠶⢤⣍⠠⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡄⠀⠀⠀⢀⠜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣇⣠⠤⠴⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⣄⠀⢠⡟⠀⠀⠀⠀⢀⡜⠀⠀⡀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠢⣼⡇⠀⠀⠀⢀⡎⠀⠀⣸⠇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⢀⠔⡙⠀⢰⠀⣀⠴⠂⠉⠉⠉⠉⠉⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡀⠀⠀⠘⢦⠏⢀⠇⠀⢸⠞⠁⠀⠀⠀⠀⢀⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠑⠲⢤⡀⠀⢣⠀⠀⢀⠀⢇⠀⠀⠀⢸⠀⠀⡇⢠⠏⠀⢠⢠⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠱⡄⠀⡇⠀⢸⡄⢸⠀⠀⠀⢸⠀⠀⠑⣼⠠⡀⢠⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⢀⠇⠀⢠⡟⢦⡀⠀⠀⠈⡄⠀⠀⠈⢇⠱⠘⡆⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡏⠀⠀⢀⡇⠀⠈⠳⡄⢠⠁⠀⡀⠀⠈⡄⡆⣇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⡁⢀⡄⠀⠀⡆⠀⣾⠀⢀⡄⠀⢸⠀⠀⢾⠇⡀⠀⣇⣠⠟⠏⢘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⢸⡇⡄⠀⣃⡞⢹⠀⡎⠀⠀⡎⠀⠀⢶⢀⡆⢠⠙⢇⠔⠉⡠⠊⠵⠒⠒⠂⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣆⠘⣾⡻⡀⠀⣿⢀⠘⡄⠃⢀⠈⡇⠐⢦⡸⡼⣧⣼⠀⡞⠀⢸⠀⠠⡀⠀⣠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢦⡀⠙⢷⡀⢿⠸⡄⢱⠸⠼⡆⢧⠀⠘⣷⡙⢹⣿⣮⡇⠀⢸⠀⢷⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡏⠒⠦⣝⠺⣷⠀⠘⠂⣸⠃⠀⣀⡼⢡⣇⠿⢟⣿⠀⠀⠀⡆⣸⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠹⣆⡘⣌⢳⡈⠦⠀⣴⣿⢠⠞⠁⠀⡞⠹⡄⠙⠁⠀⣤⡀⡃⢡⣤⣾⣄⣶⡤⢒⠶⣠⣄
⠀⠀⣀⣤⡄⢤⣖⣶⡶⢒⢺⡛⣋⣉⡙⢦⡈⠙⠪⠳⡇⠀⢸⠘⣿⢸⠀⠀⠀⢻⢃⠆⣴⣾⣼⠟⣠⣿⣿⣿⣿⣛⣯⠭⠼⢞⣽⣯⡃
⠀⢼⠁⢻⣿⣷⠲⠷⠿⠰⠚⠙⠁⠀⡩⠟⠉⢑⡒⣾⠁⠀⠀⠣⠹⠉⠖⠀⠀⣫⢵⣬⣯⣭⣴⢾⣯⣭⠥⣈⣭⡋⠀⠀⠀⠀⠙⣿⠇
⠀⠀⠙⠋⠦⠶⠶⠶⣶⢶⣶⡶⠖⠋⣀⣤⣶⡿⠟⠇⠀⣴⠋⠙⠦⣘⠉⠉⠳⠭⠽⠯⠭⢿⣟⣛⣓⠧⢺⡁⢸⣷⡶⣶⢾⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣵⣄⣤⣾⣿⣟⣥⠴⣮⣿⠷⣦⣿⡓⣤⡈⠑⠢⢄⣀⣀⠀⠀⠀⠀⢀⣈⡉⠉⢣⣘⠙⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⠒⣿⣿⡿⠟⠹⠶⢾⣿⠿⣥⠿⣿⣿⣝⣿⣦⡀⠀⠉⠻⢿⣿⣿⣿⡛⠂⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠻⣽⣿⣿⣿⣶⣴⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⢫⠟⠞⢳⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣏⢟⣦⡀⠙⠁⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⠶⣤⣄⡈⠻⢞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢵⣚⡽⢶⣄⠉⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⢹⡭⠖⢚⣯⠽⢧⡀⠀⠀⠀⣰⣯⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⡀⠀⣴⢟⣿⠀⠀⠀⣯⠅⣦⡄⠚⣩⣧⠀⠀⠀⣏⠐⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣻⠃⣸⣋⣩⡭⢷⡀⢠⣟⠀⡇⠹⡄⠴⢸⡆⠀⣶⣜⢦⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡟⠀⡿⢅⠐⣫⠭⣧⣾⠅⢠⠇⠀⣇⣨⢽⡇⠀⣿⣚⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣻⡀⣿⣸⡷⣄⠋⠿⣆⢀⡞⢴⠂⡏⣒⣿⠀⣰⣛⠅⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣟⡱⡌⠳⢦⢁⡞⠐⡯⢸⠃⢽⡇⣴⣫⠥⠐⠒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡘⢿⡷⢦⡀⠀⢸⣇⣹⡄⠈⢻⡇⢨⠆⣾⢐⣻⣿⡥⠖⠀⣘⣩⠏⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⡄⢹⡭⢿⡄⣸⢗⠂⡇⢃⡄⠀⠔⠁⣿⠨⢜⠿⠚⢁⣨⡽⠋⢰⡯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢸⣗⣩⠿⣛⡉⣸⠃⣫⠆⠀⠋⠀⠸⡄⠉⣰⡟⣠⡼⢁⣄⠘⢧⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢱⡔⢯⠵⣰⠃⣘⡵⠂⠀⠀⢠⣆⠹⣴⠃⣷⠴⡇⢸⣏⡷⡄⢳⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⣴⣛⢃⡞⢹⡈⡽⠃⠠⢟⠉⠀⠀⠀⠈⣾⡆⠈⠀⢹⣚⣿⣸⠷⣚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠥⢠⡞⠀⢸⡟⠁⠀⣚⡅⠀⠀⠀⠀⠀⠀⠳⣣⣄⠈⣷⢺⠿⣊⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣛⢡⠏⢀⠄⠘⠃⠀⠀⠘⠃⠀⠀⠀⢚⠀⠀⠀⠻⠋⠀⢘⢁⣿⣵⠟⣱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠅⣾⠀⣴⡃⠀⠀⠀⠀⢀⠄⠀⠀⠀⢈⠖⠀⠀⠀⠀⠀⣼⢴⡿⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⣀⣈⣷⡂⢹⡀⠞⠔⡀⠀⠀⠀⡴⡂⠀⠀⠀⠱⣢⠄⠀⠀⠀⢸⡯⣽⠃
⠀⠀⠀⠀⠀⢠⡞⡄⣀⠀⠀⠈⠉⠙⢶⡄⣧⠀⠈⠀⠀⠀⣰⡯⠁⠀⢀⠀⠘⡵⢋⠀⠀⢀⡘⣧⢾⡙⠛⠒⠛⠉⠉⠙⠛⢦
⠀⠀⠀⠀⢀⣸⣷⡇⡇⣿⣼⣤⠶⠖⠚⠓⠋⠙⣆⠀⠀⢚⣡⣴⠖⠚⠉⠳⠤⡔⣫⢴⡏⠉⠉⠉⠙⠳⢦⣴⣠⡄⣆⣄⢠⢸⡆
⠒⢻⠯⠭⠽⣾⣿⣷⣷⡯⣛⣉⢍⡳⡄⢀⢀⢀⠉⠙⣲⠟⣩⠭⠭⢛⠲⣴⣄⢻⣴⠋⡄⡤⡄⢀⠀⢠⣾⣷⣍⣧⣏⡟⣸⢸⣧⣄⣀⣀⡀
⠀⠀⠀⠉⠉⠶⠒⣚⡗⣇⢳⢬⡇⢱⣿⢸⢹⢸⣷⠀⡇⡴⣻⢮⣽⣊⢳⠘⣿⣾⠿⠾⣷⣷⣇⣿⠃⣿⣿⠙⣿⣿⣷⣷⡿⠯⠭⠭⠭⠭⠥⠤⠶
⠀⠀⠀⠀⠀⠀⡤⠭⢷⣼⣌⣿⠴⣫⢿⣞⣼⣾⣾⣞⣧⡻⣌⣛⣳⢊⡞⢠⣿⣿⠛⠉⠩⡽⠾⣿⡄⡿⡿⢻⣿⣗⣂⡐⠒⠛⠛⠛⠛⠛⠛⠙⠋
⠀⠀⠀⠀⠀⠐⣒⣋⡭⠽⠶⠛⠚⠿⣟⠱⠶⠖⠀⠰⠶⢛⣲⣬⣭⣭⣶⠛⠒⠂⠀⠀⠀⠀⠋⠩⠽⠿⠶⣾⣿⣿⣓⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠽⠽⠏⠉⠁⠀⠀⠀⠀⠀⠀⠭⠵⠶⠒⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠐⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠚⡦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⣄⡀⠷⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠉⠀⢺⡆⣴⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡄⢂⢠⣦⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠠⠂⢀⡿⠀⠀⠀⡘⠃⠈⠀⢻⡆⢀⡡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⣇⢺⡀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⣀⠈⠀⠴⠁⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠄⠀⠠⠻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠈⠀⢠⡟⠀⠀⠀⠀⠀⠀⣤⡴⣣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠓⠄⠀⠀⢂⡴⠉⣴⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠇⢀⣠⠏⢡⡎⢶⣄⠀⠀⢙⢦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣿⣿⣣⣷⣿⠕⣠⣻⣿⣄⢹⣄⠀⡼⢤⣙⠶⣤⣰⡟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣿⠉⣿⣿⣿⣯⣾⣻⠥⣎⠉⠹⠥⢈⡽⠿⣦⡀⣀⣽⡷⠶⢄⣨⣯⣡⡀⠑
⠀⠀⠀⠀⠀⠀⠘⠛⢻⣋⣡⣌⣉⣙⠛⠿⠤⠿⠿⣠⠤⠾⠷⠦⠼⠿⠻⣉⡐⣢⡏⠥⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠉⠓⠒⠀⠶⠶⠶⠖⠒⠒⠌⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⢀⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂⣸⠱⡄⠈⠈⢳⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⢦⢄⡀⠂⠄⠀⣤⣀⠀⣠⠗⠊⢺⠀⠀⠌⡢⡈⠀⠀⡰⡅⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⣇⠉⢦⠀⠀⢸⠈⣶⠇⠐⠃⢸⠀⠊⣠⠥⢹⠀⠀⠁⢃⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣒⠀⠀⣼⠀⠈⢇⠀⢸⠁⠆⠌⠀⠆⢸⠀⡰⠁⠀⢀⠁⣰⢀⠌⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠱⣌⢉⣳⣿⠂⢀⡈⠢⣜⠘⢄⠀⠀⠠⠈⢶⢃⠀⠀⣜⡴⡽⠀⢀⢦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢧⡀⠤⢀⡈⠂⢈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠐⠟⡞⢁⠔⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢱⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠈⡤⢰⡔⠁⠔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢵⣸⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠰⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢉⣝⠁⠀⢀⡤⡤⢄
⠀⠀⠀⠀⠀⢻⣡⣀⣠⣠⡤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠤⠶⠲⠶⠶⣿⣿
⠀⠀⠀⠀⢀⡏⠉⠉⠀⢀⠀⢀⠖⡎⠙⢦⡀⠠⣴⠅⠂⠺⡦⣷⡄⠀⠈⠉⠩⠉⠒⢦⡄⠀⠀⣸⢉⣸⡆⠀⠀⠀⡀
⠀⠀⠐⠒⠤⠆⢾⡆⠀⣉⣨⣿⡀⠁⠀⡂⡅⢀⡷⠀⠀⢵⠚⠉⢷⡐⠷⣤⣀⡂⠀⣾⡷⠛⠛⡿⢎⡼⠁⠁⠀⠁⠈
⠀⠀⠀⠀⢀⡘⠶⠓⠛⠉⠉⠉⢻⣀⣢⣴⡧⠛⢛⣉⠠⠛⣢⡒⠛⢦⣰⠀⠹⡁⠿⡿⠷⠤⣤⣶⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠹⠷⠶⠿⠁⠀⠀⠈⠙⠲⠆⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣵⡖⠄⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⢀⣾⣿⣿⢇⠀⠀⠀⣠⣾⣿⣧⡀⠀⠀⠀⠀⠀⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠈⠃⠀⠈⠋⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣧⠇⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣧⣾⣿⣾⡟⠀⠀⠀⢀⣸⢃⣡⣖⡀⠀⠀⠀⠀⣿⣿⡜⣦
⠀⠀⠀⣺⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⠻⡛⠻⠋⠀⠀⠀⢠⣿⣿⣿⣿⠈⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠂⠀⠀⠀⠀⠀⠀⠆⢸⢢⡀⠀⠉⠳⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⣀⡁⠱⠄⠀⠀⣀⡀⠀⠀⢠⡯⠔⢹⠀⠀⢠⢃⡀⠠⠀⠀⣠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣟⠓⢄⠀⠀⠈⡏⠳⣠⠏⢀⡄⢸⠀⠰⠂⡸⠙⡆⠀⠀⠃⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⡅⠀⠀⢹⠀⠈⣆⠀⠀⡿⠔⡛⠰⠁⡅⢸⠀⠀⡼⠋⠀⢹⠀⢠⠁⠱⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠘⡟⠲⢤⣀⣻⠂⠂⠘⢆⠀⡏⡀⢀⠀⠀⠁⠘⡄⡼⠁⠀⠀⡔⢠⣿⠐⠁⡀⣀⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠤⣠⠾⠛⠣⠀⠦⡈⠳⠇⠈⠂⠀⠀⠈⢄⠹⠃⠆⠀⢰⡴⣣⠃⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠐⡂⢤⠀⠑⠠⡀⠂⠀⠀⠀⠈⠄⠠⠀⡀⠙⢹⢃⡴⠋⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⡀⠸⡄⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠂⠀⢠⠆⢸⠊⠄⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢙⣄⡇⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠘⠘⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠌⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣣⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢰⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡵⠁⠀⣀⡀⣀⢀
⠀⠀⠀⠀⠀⠸⡟⣄⠈⠉⣁⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠮⣁⣤⢤⣤⣀⣀⣠⡆
⠀⠀⠀⠀⠀⢰⠛⠛⠋⠉⠉⠁⠀⡠⢴⠪⣦⡀⠀⢀⣰⡂⠄⢴⣄⣦⡀⠀⠀⠒⠒⡲⢦⣤⣀⡀⠀⠀⢀⡎⠎⣧
⠀⢀⣀⣀⡀⣉⢀⣤⡀⠠⠂⠒⣼⠀⠺⠐⠄⢑⠀⢹⡏⠄⠠⣂⠩⠳⣿⠠⣄⡀⠀⢠⡁⠀⣿⣧⣤⣦⣼⣴⢛⡟⠠⠀⠤⠊⠒
⠀⠀⠀⠀⠈⠉⣄⣻⣥⡤⠾⠛⠿⣦⠀⡀⠠⣸⢠⣾⣥⠆⠀⠜⠀⣀⣯⣇⠈⡙⠳⡶⢂⢼⣿⣃⠀⠀⡏⣕⣿
⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠘⠳⠷⠟⠛⠁⠀⠱⣾⣈⠀⣾⡦⠀⠉⠳⢧⣄⣑⠎⢹⠏⡉⠉⠛⠛⠛⠁⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⠁⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠿⣆⣠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡄⢋⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢖⣵⡇⡀
⠀⠀⠀⣀⠘⡇⣠⠈⡁⣤⢿⠀⠀⠀⠀⠀⠀⠀⡀⡏⢸⣇⣾⣆⣆⠀⠀⠀⠀⠀⠀⢀⠀⢈⡾⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⡟⡟⠜⣥
⠀⠀⠀⠈⣧⣸⢆⣰⢧⣠⢴⣰⠇⠀⠀⠀⠀⠀⣷⣶⠈⡛⠃⠟⡿⠀⠀⠀⠀⠀⠀⠈⢿⠛⣽⣾⡀⢰⠀⠀⠀⠀⠀⡀⣤⡇⢰⣀⣿⣦⡄
⠀⠀⠀⠀⠻⣍⢠⣿⠈⢷⠆⠎⠀⠀⠀⠀⠀⠀⠘⠉⢠⡿⣧⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠫⣟⡻⠀⠀⠀⠀⠀⣿⣦⡟⠰⡍⠁⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠘⠁⠀⠀⠀⠀⠀⠈⠣⠀⠴⠈⠂

⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⢠⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀
⠀⠀⠀⠀⠀⢰⣻⣿⠗⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣿⣷⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⣿⢈⠇
⠀⠀⠀⠀⠀⢀⡇⠸⣴⡃⠀⠀⠀⠀⠀⠀⠀⠦⢶⣀⣿⠘⣷⣿⡴⠄⠀⠀⠀⠀⠀⢠⡀⣾⡏⢿⣸⣧⠐⠀⠀⠀⠀⠀⢈⠃⡟⠟⠰⡞⢀⡄
⠀⠀⠀⠀⢠⠀⣧⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⣀⣯⠉⣧⡀⢠⣴⠀⠀⠀⠀⠀⠀⢠⣷⣼⡇⠸⠟⣿⡶⠀⠀⠀⠀⠀⠀⣄⢱⣰⡄⡇⢘⠃
⠀⠀⠀⠀⠸⣧⣼⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⢨⣯⢉⢀⣾⢻⣈⢹⠀⠀⠀⠀⠀⠀⠀⢿⠂⡛⣼⡀⢸⠃⠀⠀⠀⠀⠀⠰⣼⡟⠛⡇⠙⢻
⠀⠀⠀⠀⠀⠀⠐⢾⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠻⠍⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠹⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢶⠙⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣤⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣶⣦⣿⣧⢸⣿⣿⢸⣿⣿⡃⠀⣼⡟⠻⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣿⣷⡸⠿⠛⠛⠉⠉⠉⠈⠁⠈⠁⠀⠘⠻⠃⣴⣿⣿⠀⣠⣄
⠀⠀⠀⠀⠀⣠⡺⣷⣌⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢾⣿⣷⣶⠆
⠀⠀⠀⢠⣤⣈⠻⡾⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣣⣶⢷⡄
⠀⠀⠀⠈⠙⢿⠇⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⢻⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠈⢻⣾⣿⣿⠟⠃⣾⣿⣿⣿⡿⠘⣿⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣀⣴⡟⢉⢍⠛⣿⠏⠁⣀⠐⠻⣿⠟⠰⢢⣿⣿⣿⣿⣯⠀⠈⣿⠀⢻⣿⠙⠋⢻⣶⣄
⠀⠀⠀⠚⠛⠛⠃⠈⠉⠀⠛⠀⠘⠃⠀⠘⠃⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⣤⣤⡀⣤⢠⣄⣤⣤⣠⣤⢠⣄⣤⣤⣠⡄⣤⢠⣤⣄⢠⣤⣄
⠀⠀⠈⠉⠉⠁⢰⣿⣿⣼⣿⣼⡇⢻⣾⠇⣿⣿⣼⡿⢿⡇⢸⡇⢸⣧⣿⢸⣿⣿⢹⣿⣇⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠖⠀⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠞⠉⠀⣀⣤⠀⠀⢠⣄⡛⠐⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⣁⠴⠟⠁⠀⠀⠀⠀⠀⠶⣌⣉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠌⠥⠶⡖⠂⠀⠀⠀⠾⡶⠆⠠⣭⣄⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠙⡳⠶⣀⣀⡀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠙⠛⠧⣦⣤⠉⠁⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⠠⠤⡤⠤⠀⠀⠀⣠⡄⠰⠶⠿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣼⡘⠋⠀⠈⠉⠛⠳⠖⢢⣄⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠰⢦⣈⣉⡉⠉⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠉⠙⢋⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣛⣁⡀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢈⣭⡏⠀⣠⡶⠋⠀⢠⣴⢆⣠⠿⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠯⠹⣍⣀⠘⢥⣤⣤⡀⢀⣉⡉⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⢀⣠⣤⠝⠟⠋⠁⡾⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠚⣶⠆⠙⠿⢦⡰⠞⠃⠀⡅⣸⣷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢴⣛⡋⠀⠀⢼⣯⡄⣠⣤⣘⡇⣿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⣀⣀⣀⣀⡀⢈⡿⢽⣧⣨⡴⢉⣿⣧⣶⣸⣿⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣸⣿⣿⡟⣳⡛⠱⢿⣿⣟⠛⢿⢿⣿⣿⣿⣿⣿⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣡⣿⣿⣿⣷⣿⣭⢉⣻⡿⣿⡇⣡⣿⣿⣿⣿⣿⣿⣷⡄⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣟⠿⠋⣿⣿⣿⣿⣉⢔⣫⣵⣾⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⣣⣶⣿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡟⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣟⠏⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⠘⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣧⣿⣿⣟⢡⣿⣿⣿⣿⣿⣷⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣟⣻⣿⣿⣿⡿⢻⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣽⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⡿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠷⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⣿⣧⡀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⠀⠀⠀⠀⢻⣿⡇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣿⡆⠀⠀⠀⢸⣿⠇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣄⣀⣠⣿⡿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠀⠈⣿⣿⣿⣿⠃⠀⢀⣾⣿⠏⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢹⣿⣿⣿⣷⣶⣿⣿⡟⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⡻⣿⣿⣿⣿⡇⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢀⣼⣿⣿⡇⠸⣿⣿⣿⣿⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠘⠛⠁⢰⣿⣿⣿⣿⣿⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⡄⠀⢰⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⣿⣿⠃⠀⠈⠛⠀⢻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣦⡹⣿⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠂⡀⠀⠀⠀⢿⣿⣷⢹⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠐⢦⣄⠀⠀⠀⠸⠃⣼⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠋⠇⠀⠀⠀⢸⣿⣿⣦⣤⣴⣿⣿⡏⠀⠸⣷⣄⠀⠈⣿⡷⠈⢷⡄
⠀⠀⠀⠀⠀⢀⣆⠀⠀⢸⣿⡇⠀⠀⠀⠘⣦⣸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣿⣿⣇⠀⠸⡇⠀⣼⡗
⠀⠀⠀⠀⠀⢸⣿⣶⣴⣾⣿⠁⠸⣦⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡄⣼⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣾⣿⣿⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣰⣧⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠁⠈⢻⠀⠀⠀⣤⡀⠀⠀⢸⠋⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢎⣦⡴⠋⠀⠀⣰⣿⣷⡄⠀⠈⠳⣤⣎⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠁⠀⠀⢀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⢻⣿⣷⣿⡄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠍⠙⠚⠻⢦⣄⠀⢀⣠⠾⠟⢊⠭⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣅⡀⠀⠄⠐⠩⢝⠲⠥⣀⣤⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⡙⠓⣶⣤⣄⡀⠀⢀⠀⠈⣙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣷⣾⡿⠟⠛⠉⠙⠻⠶⣭⣵⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣷⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣉⣉⣉⣉⣉⠉⢉⣛⣛⣛⣛⡛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡟⠛⠛⣿⠀⢸⡟⠛⠛⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⣤⣸⡏⣤⠈⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣇⣄⣠⣿⠀⢸⣧⣀⣠⣿⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⢿⡇⣿⠀⣏⡍⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠃⠉⠉⠉⢉⣉⠀⠈⠉⠉⠉⣉⠁⠘⠛⠛⠛⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣼⣷⣤⣴⡏⠒⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣸⣿⡆⠀⢀⣠⣤⣄⣀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣭⣤⣀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⣿⠟⢀⣾⠟⠋⠉⠉⠻⣷⡄⠀⠀⠀⢀⣴⠟⠋⠉⠉⠛⢷⣄⠀⣿⡇⠀⠀⠀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋
⠀⠀⠀⢸⣿⣶⣾⡏⠀⣴⣿⣷⠀⠘⣿⣶⣶⣶⣾⡏⠀⣴⣿⣷⡀⠈⣿⣶⣿⡇⠰⠆⠘⠟⠀⠉⠉⠉⠙⠛⠻⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠘⠛⠋⠀⣰⡿⠀⠀⠀⠸⣷⡀⠘⠛⠋⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⣶⡾⠛⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠔⠁⠀⠀⠀⠘⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠣⡀⠀⠀⠀⠀⠀⠘⢆⠀⠣⡀⠀⠰⠀⠇⠀⠂⠀⡁⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⠜⢠⡀⠠⠀⠘⠀⠈⢣⠀⠈⠢⣄⠀⠀⠀⠀⠀⠣⡈⠢⡀⠀⠃⢠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣆⠈⠢⠀⠀⠀⠀⠀⢹⠀⢡⠀⠀⠀⠈⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣸⠻⣆⠀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠑⠄⠀⠀⠠⣤⣀⠀⠀⠀⢀⣴⠏⡪⢚⣇⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⡄⢠⢛⠅⠋⠈⠡⢺⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡰⢀⠎⠀⠊⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢠⠀⠀⠀⠻⡳⣄⠀⠀⢸⡗⠹⣗⠅⡠⣾⠀⡪⠼⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⠀⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢇⠘⣄⠀⠀⠀⢻⠝⠆⠀⢰⡫⠂⠁⡜⠁⠺⠈⠕⡃⠀⠀⣸⡄⠀⠀⢀⣾⡇⠀⢠⠀⠳⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⢢⠀⠀⡖⠐⣺⡄⢸⢗⠁⡼⠀⠀⠸⡀⠪⠅⠀⢠⠖⡹⡄⠀⣸⣻⠁⢀⢎⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡰⠁⢰⡫⢊⡀⢷⣼⡻⢀⠃⠀⡀⠀⢣⠘⣧⠀⡾⡊⣣⢾⠀⡿⣺⣴⢟⠅⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡛⠔⡬⠱⡄⢙⡞⢸⠀⠀⢀⠀⠀⢣⠘⣶⡋⢰⡀⠹⢠⡋⠞⢑⡡⢊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⠁⡆⠀⠈⠳⣌⢸⠀⠀⠀⡠⠀⠈⣇⠹⠃⣼⡀⣽⡿⢋⡄⡩⣪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡵⠀⡃⠀⠀⠀⠈⢾⡄⠀⢀⠔⠁⠀⢩⢈⠞⡸⢸⡿⣠⢺⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢸⡵⣿⡄⣇⠀⠀⠂⠀⠈⠃⠀⠀⡫⠂⠀⢸⡏⠀⣆⢀⡔⠁⢸⠨⣧⢔⡽⠛⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢦⡼⠆⠀⠐⡡⠀⠀⠀⠠⢖⠄⠀⠈⠁⡆⠈⠁⠠⠂⢸⠐⢱⠎⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⣀⣀⣀⣉⣲⣌⠦⠀⠈⠔⡡⠀⠀⠀⠁⡁⠀⠀⡬⠂⠀⠀⠚⠀⡼⢸⡇⣀⣀⡀⠀⠀⢀⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣔⣍⢈⡶⢳⡲⡶⠈⢣⠀⢀⢀⡠⢄⣶⠤⡀⠀⢀⢀⠈⠀⠐⣲⢖⡺⣅⣞⣙⠀⠀⢳⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⠒⣀⢹⢟⠕⠜⡰⣁⠄⡏⡐⠕⠇⣉⣘⣆⡤⠣⢪⢀⢀⡧⡜⣈⣸⡣⠡⠣⢢⢸⡤⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢋⡹⠷⣤⡷⢾⡿⠾⢛⠋⠝⢛⠳⢤⣲⠈⠡⢂⡿⠐⡛⠿⣚⠾⣕⣒⣹⡟⠿⠻⠟⠛⠿⠊⠕⠂⠝⠂⠭⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠒⠁⠠⠛⠉⠵⠊⠁⠀⠉⠔⠊⠽⠋⠡⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠸⣿⣿⣶⣄⠀⠀⠈⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠁⠀⢀⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡀⠀⠘⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣷⡀⠀⢹⠇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠉⠛⢷⣜⣥⣤⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⣧⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢹⣿⣿⣿⣿⣿⣇
⠀⣾⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄
⢰⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⣧
⢸⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⣿⣿⡇⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⣽⣯⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠁⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⢀⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⢀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⢀⣴⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢀⣠⣾⣿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣷⣶⣾⣾⣿⣿⣿⡇⠀⣀⣽⣿⣅⡀⢀⣿⣿⣿⣿⣿⣷⣶⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣭⡽⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⡿⣽⡽⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣹⣿⣿⣿⣿⣿⣿⣿⣿⢯⣴⣿⣿⣿⣿⣷⣽⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣮⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣛⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⣛⣻⣭⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢻⣿⣿⣿⣷⣌⢻⣿⣿⣻⣿⣿⣿⣵⣿⣾⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣷⣙⢿⣿⣾⣮⣻⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⣿⣿⣿⣻⣿⣿⣿⡧⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡧⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣻⣿⠗⣻⣿⣯⣝⡿⢿⣿⣿⣿⣿⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣭⣽⣿⡅⣻⣿⣻⣦⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣿⡿⣹⣿⡏⣿⣻⡿⣾⣗⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⣿⣿⣮⣛⣛⣩⣾⣿⣿⣿⢟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⠿⠿⠿⡿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇⠀⠀⠀⠀⢀⣾⡟⠁⠠⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⢀⠀⢀⣾⣿⠃⢰⠆⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⠀⡆⠀⣼⣿⣯⠀⣿⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⣠⣾⣿⠃⠀⠀⢀⣰⣿⣿⣿⣿⡟⠁⠀⠀⣴⠀⢸⣿⢿⡟⢸⡇⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠁⠀⢀⣼⣿⣿⠃⠀⠀⣠⣾⣿⣿⣿⣿⠟⢀⠀⠀⢀⣿⠀⢸⣏⣿⠇⡘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⣿⠇⠀⣠⣾⣿⣿⣿⣿⡿⠋⢀⠌⠀⠀⣸⡿⠀⢘⣿⠏⢰⡇⠐
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⣽⣿⣿⡇⢰⡟⣿⣿⠏⣠⣾⣿⣿⣿⣿⣿⠟⠁⣰⠏⡰⠀⣰⣿⠇⠀⡼⠋⣰⡟⣱⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⣿⡿⠉⠀⣿⢸⡿⢡⡟⢠⣿⡏⣴⣿⣿⠟⣸⣿⡿⢁⠀⣰⢋⡼⢃⣼⡿⠃⠀⠀⢀⣼⣟⣴⠏
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⡇⢸⡇⣾⠁⣸⣿⣿⣿⡿⠁⣴⣿⡿⢁⠆⣸⣧⣾⣵⣿⡟⠁⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠄
⠀⠀⠀⢠⡄⠀⢻⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⢸⡇⠀⡇⣿⠀⣿⣿⣿⠏⠁⣸⣿⣿⢁⣾⢠⣿⣿⣿⣿⡟⠀⢠⣾⣿⣿⠟⠁⠀⠀⣀⣴⡿
⠀⠀⠀⢸⡇⠀⢿⣸⣧⠀⠀⠀⣿⡏⠀⢰⡆⣸⠇⠀⠀⠉⠀⣿⣿⠋⠀⢠⣿⣿⡇⣼⡇⢸⣿⣿⣿⣿⠀⣴⣿⣿⡿⠁⠀⢀⣴⣾⣿⡿⠁⢀
⠀⠀⠀⢸⡇⠀⠈⣷⣿⡆⠀⠀⣿⡇⠀⣸⣷⣿⠀⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⣿⢡⣿⡇⢼⣿⣿⣿⣿⢰⣿⣿⡟⠀⢀⣾⣿⢿⡿⠋⠀⣠⠆⢀⡔
⠀⠀⠀⢸⣿⠀⠆⡸⢿⣿⠀⠀⣿⠀⠀⣿⣿⡟⠀⠀⠆⠀⢸⣿⠀⠀⢰⣿⡟⣷⢸⣿⡇⢸⣿⣿⣿⣽⣿⣿⡟⠀⢠⡶⡯⠚⠉⣠⣴⢟⣡⠖⠋
⠀⠀⠀⢸⣿⣇⠈⣷⣞⣿⡆⠀⠛⠀⣼⣿⣿⠇⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⢸⡇⣿⣿⣧⢸⣿⣿⢸⣿⣿⣿⠀⢠⣾⠋⢀⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠘⣿⣿⣆⢸⣿⣿⡇⠀⡆⢠⣿⣿⡟⠀⢀⠀⠀⠀⢸⠀⠀⠀⣾⠃⡿⢰⣿⣿⡟⢸⣿⣿⠀⣿⣿⣿⠀⢸⡇⢠⣿⣽⢟⡡
⠀⠀⠀⠀⠙⣿⣿⣾⣿⣿⡇⠀⠃⣾⣿⡟⠀⠀⢸⠀⠀⠀⠸⠀⠀⣇⣿⣸⢃⣿⣿⣿⢃⣿⣿⣿⢠⣿⣿⣿⠄⡌⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇⠀⣼⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣿⣿⠇⣼⠛⣿⢯⣾⣿⣿⢃⣾⣿⣿⡿⠀⣀⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⠀⣼⣿⠏⠀⡌⠀⣠⣿⠀⠀⠀⠀⢀⣿⣿⠟⡸⠁⣼⢫⣾⣿⣿⣷⣿⣿⣿⠟⠁⣠⡿⠟⠁⠀⠀⠀⠀⠀⢀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⣸⣿⠃⢀⡾⠀⣰⣿⡟⠀⠀⠀⠀⣼⣿⠏⠄⣱⠐⣤⡿⣫⣾⣿⣿⠏⠛⢁⠠⠚⠉⠀⠀⠀⠀⣀⣤⣴⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⡜⢠⡿⠃⢀⡾⠁⣰⣿⣿⠁⠀⠀⢀⣾⣿⠃⠈⢠⡿⣴⢯⣷⣿⡿⠋⠁⠀⣀⣤⠶⠂⠀⣠⣴⣶⠿⠛⠉⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡿⠁⢀⣼⢃⣴⣿⣿⠇⠀⠀⢀⣾⡟⠁⠀⢠⣿⠃⣏⣾⡿⠋⠀⢀⣤⡾⠋⠀⢀⣴⡾⠋⠁⠀⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣾⠁⠀⣼⣿⣿⣿⣿⠋⠀⠀⢠⣿⠏⠀⠀⢠⣿⣿⠃⢻⣿⠃⠀⣠⣿⣿⠁⠀⣴⠟⠋⠀⠀⣠⣾⣿⣿⣿⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢠⣿⣿⣿⡟⠁⠀⠀⢀⡿⠃⠀⠀⢀⣾⣿⡿⣿⢸⡟⠀⣼⣿⣿⠃⠀⠊⠁⠀⠀⣠⣾⣿⡟⣼⣿⠏⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠘⠃⠀⠀⢠⡿⢻⣿⡇⠘⢸⡇⣸⣿⣿⡏⠀⠀⠀⠀⠀⣴⣿⣿⣟⣼⣿⠏⠀⣠⠃⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣅⣾⣿⠙⣶⣿⣵⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⢿⣣⣾⣿⠏⠀⢠⠏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢀⡜⢁⣼⣿⣿⡹⢠⣷⣿⣿⣿⠋⠀⠀⠀⠀⣠⣾⣿⣿⣷⣿⠟⠁⠀⣠⠃⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠞⣰⣿⣿⢿⣿⣷⣿⡿⠟⠋⠀⠀⠀⠀⢀⣴⣿⣏⣽⡿⠟⠁⠀⢀⡼⠁⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡆⠀⠀⠀⢠⡾⢛⣽⣿⣾⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⡷⠟⠋⠀⠀⣠⣶⣿⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠁⠀⠀⣰⠋⣰⡿⠟⠉⠀⠀⠀⢀⣠⡤⢊⣠⣴⠛⠯⠙⠁⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣷⣦⠏⠀⠀⠐⢁⡾⠋⠀⠀⢀⣤⣶⣿⠟⠃⠛⠋⢁⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣼⣿⡟⠀⠀⠌⡠⠋⠀⠀⣠⣾⠟⠉⠁⠀⠀⢀⡴⠞⢛⣩⣭⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣿⠟⠀⠀⠀⠀⣠⠆⢀⡴⠋⣀⡴⠊⠀⠀⠊⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⢀⣠⠞⠁⠔⠉⣠⡾⠋⠀⠀⠀⣠⣾⡿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣀⣴⣾⣿⠋⠀⢀⣴⣿⠟⠀⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⡾⠋⢀⣴⣿⣿⡿⠋⠁⣠⣴⠿⠟⠁⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡦
⠀⠀⠀⠀⠐⠀⡴⠟⠋⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠚⠐⠃

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠒⠀⠀⠀⠂
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠂⠂⢀⣀⣿⣿⠛⠉⠉⠁⠉⠉⠙⠛⣿⣿⠀⠀⠀⠂⠘⠂⠐⠄
⠀⠀⠀⠉⠀⠀⠀⠛⠂⠀⠀⠂⡀⠀⠙⠋⠉⠀⠀⠀⣀⠀⠀⠀⠀⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠈⡀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠦⡤⠆⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡀⠀⡀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⡀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠐⠈⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠄⢈⠃⠆⠀⣀⠀⠀⠀⠀⠀⢀⢰⡀⢀⡄⡀⠀⠀⠨⠤⠀⠉⠁⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⠶⠃⠀⠠⠂⠀⠀⠀⢠⠠⢤⣤⡄⡤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠙⠋⠉⠉⠋⢀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠈⠃⢀⡃⠀⠘⠆⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⡀⢼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠈⣿⣿⣿⣄⢻⡇⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠊⣠⠀⠀⠀⠀⠘⣿⣿⣿⠈⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠲⣤⣄⠀⠀⠀⠀⠸⠇⣰⣿⡀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡄⠈⠟⢧⠀⠀⠀⠀⠀⣿⣿⣷⣄⣀⣀⣴⣿⣿⡟⠀⠈⢷⣦⡀⠀⠈⢻⣿⣦⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣿⣿⠀⠀⠈⠀⠀⢰⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣆⠀⠀⢿⡏⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⢀⣿⣿⠀⢀⡀⠀⠀⠈⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠀⠀⡘⠃⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣏⠀⠈⣿⣦⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣄⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⠋⠁⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣦⣀⣠⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⠤⠤⠤⠴⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣦⣿⣿⣧⣴⣶⣶⣾⣥⣾⣮⣿⣷⡄⠘⠓⣦⠀⠛⣒⣶
⠀⠀⢀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣧⡀⢸⡻⣿⣿⡇⠀⢀⢀⠐⣾⢈⢦⣠⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣴⣶⢟⡤⣿⡋⢐⡆
⠀⠀⣼⣿⣿⣷⣷⡀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣇⠀⠁⠘⠋⠀⢠⡌⣎⢠⢦⡟⣿⣯⣾⣦⡰⠀⠉⠉⠉⣉⢉⢉⣉⣉⣉⡭⣁⡀⢀⡻⣿⣬⣽⠟⢻⣟⣾⡋
⠀⣼⣿⠟⠁⣿⣿⣧⠀⠀⠘⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⢀⢰⣾⣧⣿⣿⣾⢻⣿⢿⣼⣟⣷⣷⣤⡀⣜⣿⣏⠻⠬⣟⠚⠋⠛⠃⠀⠻⣷⠾⣷⣴⣿⡿⠟⠃
⠀⢻⣯⠀⠀⢨⣿⡿⠀⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠺⠿⠿⠛⠿⠻⠿⠻⠟⠛⠛⠛⠟⠛⠿⠛⢸⣿⣿⣿⠿⠇⠛⠆⠀⠀⠀⠀⠀⠀⠟⠛⠃
⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠀⡄⠠⠀⠀⠀⠀⢀⠀⡀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡆⣱⣸⠸⣄⡏⢴⠀⣆⡄⠀⠀⠐⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠿⠟⠛⢁⠀⠅⠆⠀⠀⠘⢧⣀⡴⠀⠀⠀⠀⢀
⠀⠀⠀⠸⡄⠀⠀⣸⢸⣭⣏⣷⣿⣾⣧⣿⣼⣿⣥⣧⠀⠀⠀⠀⠀⠀⠀⣰⣆⣄⣀⡀⠀⠀⠀⠀⠀⠰⠄⠄⠸⢸⣿⠏⠃⣨⢿⡿⣿⠴⡿⠀⠀⢀⡈⣲⡄
⠀⢀⠄⣼⠇⠀⠀⠘⠿⠿⠿⠟⠟⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⢀⣰⣺⣷⣿⣿⣿⣿⡀⠀⠀⠀⢀⡵⣴⣠⡆⠀⠀⠀⠀⠙⣿⣿⣶⡞⠁⠀⣠⠬⡝⢿⡄
⠀⣾⠘⣯⢰⡇⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⢠⡠⣄⠠⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⠀⠀⢰⢿⡐⣤⢿⣄⡤⠀⠀⠶⠶⠿⠿⠿⠷⠴⠆⠙⠓⣿⣿⠏
⠀⠸⣷⢸⣿⣻⡆⠀⢀⡴⣫⣿⣿⢧⠀⠀⢠⣌⢿⣾⣡⡘⢶⡄⠐⠉⢻⣿⣿⣿⣿⣿⡟⠠⣏⣿⣼⣏⣼⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿
⠀⣼⣏⣸⣿⣿⠁⠀⣼⡿⠁⠿⠿⣾⡇⠀⢢⣹⣿⣸⣟⣹⣤⠇⡄⠀⣼⣿⡿⢿⣿⠗⠀⠀⢹⡿⣿⣿⣿⣿⣾⣷⠀⠀⠀⠀⠘⣄⠄⠀⠀⠀⠘⠿⣷⣤⡀
⠀⠻⣿⣿⣿⡿⠀⠀⠙⢷⡀⠀⣰⠟⠁⠀⠸⣿⡿⢿⢿⠿⢿⣾⠟⠀⠈⢻⠃⢹⡿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⠀⢠⣴⣾⣿⣿⣶⣦⣤⡄⢀⣤⣾⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⣠⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠱⠋⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⢀⡄⠀⢸⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⣴⡠⠚⢭⣆⣠⡏⠓⢄⡩⢳⣍⣂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⡤⠐⣉⣠⠐⠉⣽⠛⢻⡀⠀⠙⣀⡀⢈⣷⣌⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢟⣿⣿⡀⠀⣡⣠⣮⣿⣄⠀⠈⣻⡿⠿⠟⠛⠿⠈⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣁⠈⠉⠉⠙⠛⠛⠉⠛⢷⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⡀⢮⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢩⠆⣀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠴⡟⢦⠁⠀⠠⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠜⡷⠄⠀⠀⠂⡄⠀⠠
⠀⠀⠀⠀⠀⠺⢊⠄⠁⠀⢐⢄⠑⠀⠈⠢⠑⢦⡀⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⣠⠞⠡⢊⣄⠲⣇⣄⠀⠈⠀⡈⠺⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⢀⠀⣡⣌⠢⡙⠦⠄⠤⠠⠠⠤⠄⠀⠠⠜⡡⠊⠀⠉⠫⣷⣌⠣⠀⣄⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠙⢅⠀⠀⠄⣠⣾⡿⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢢⡀⠿⠁⠈⠛⠁⢀⠍⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠑⠙⠁⠘⠳⢇⡠⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠝⠢⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠒⠀⠤⠊⠓⢄⠁⠀⠤⠒⠂⢤⠄⠀⠀⠀⠁⠀⠁⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢠⠃⡄⠰⢠⠒⣒⠢⠁⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⡇⣀⠌⠃⢢⡘⠀⢀⠂⢳⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡡⡄⠀⠉⠤⡡⠂⡜⠀⠀⠀⠀⠀⢮⠗⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠷⣌⠀⠳⠁⢈⣡⠞⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⣀⠀⠀⠂⠀⠐⠀⠀⠁⠀⠀⠘⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠤⠤⠤⠤⠌⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠋⠀⠛⠁⠘⠃⠈⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢠⣾⣇
⠀⠀⢀⢠⣿⡿⢿⡿⠀⠀⠀⠀⡀⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⣼⣿⣿⡇⡀⠀⠀⠀⠀⢠⣶⣾⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄
⠀⠀⢸⣿⠟⢀⠀⣿⣿⠀⠀⢸⣿⣿⠋⢿⣿⣤⡆⠀⠀⢠⣼⣿⣿⣿⣿⣷⠀⠀⠀⢀⣹⣿⠏⢹⣿⡆⠀⠀⠀⣦⣿⡟⠈⢻⣿⡆
⠀⠀⠘⣿⠠⣿⡇⢸⠟⠀⠀⠈⣿⡋⠀⠀⢻⡿⠁⠀⠀⠈⢿⣿⣿⣿⣿⡏⠀⠀⠀⠸⣿⠃⣴⡈⢹⠇⠀⠀⠀⢿⡏⠀⠀⠘⣿⠃
⠀⠀⠀⠈⠑⠊⠂⠁⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠈⠈⠛⠁⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⡀⢸⣦⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⣴⣇
⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⣀⣿⣾⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⢀⢀⣾⣿⣿⡇⢀⠀⠀⠀⠀⣠⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⢰⡄⠀⠀⠀⣿⣿⡟⠿⣿⣿⡄⠀⠀⢀⣿⣿⡟⢻⣿⣧⠀⠀⠀⣿⣿⣿⠏⢻⣿⣿⠀⠀⠀⣄⣿⣿⣿⣿⣼⡄
⠀⠀⠰⣾⣿⣿⣿⣿⡇⠀⠀⠘⣿⡏⢀⣄⠹⣿⠃⠀⠀⠸⣿⡏⠀⢸⢿⡿⠀⠀⠀⢻⡟⠋⣠⡘⢻⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⠈⠓⠘⠿⠀⠁⠀⠀⠀⠀⠙⠁⠀⠀⠘⠁⠀⠀⠀⠀⠁⠐⠿⠃⠘⠁⠀⠀⠀⠈⠛⠿⠛⠟⠋

⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⣴⡇⡀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⢀⣼⣆
⠀⠀⠀⣠⣠⣿⣿⣦⠀⠀⠀⠀⣀⠀⣿⣷⣿⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⠀⠀⠀⠀⠀⣾⣿⣿⢀⡀
⠀⠀⢰⣿⣿⡟⢿⣿⣧⠀⠀⠀⣿⣦⣿⣿⠃⣠⠀⠀⠀⢠⣘⣿⠏⠻⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⠁⡀⠀⠀⠀⣄⣿⣿⣿⣿⣇⡀
⠀⠀⠸⣿⠃⠀⠀⢻⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⡏⣠⣆⢸⡿⠀⠀⠀⢰⣿⣿⡈⣻⣿⡗⠀⠀⠘⣿⣿⡟⠈⢿⣿⡇
⠀⠀⠀⠙⠣⠤⠤⠛⠁⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠙⠳⠿⠿⠞⠁⠀⠀⠀⠀⠙⠃⠀⠁⠟⠁⠀⠀⠀⠙⠏⠋⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠃⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⣿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠸⡇⠀⠀⠹⣧⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⣿⠀⠀⠀⢹⣇⠀⣾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⢹⡇⠀⠀⠀⢻⡄⢻⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⣿⠀⠘⣷⠀⢸⡇⠀⠀⠀⠘⣧⠘⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢻⣆⠀⠀⠀⢹⡀⠀⠙⠷⠾⠃⠀⠀⠀⠀⣿⠀⢸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡿⠁⠀⢸⡇⠀⣠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⢸⣷⣄⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⠋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠈⣿⠙⣷⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡏⠀⢠⣿⠀⢸⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⢸⡇⠀⠀⠈⣷⣤⣾⠃⠀⠸⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠁⠀⠀⣿⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠿⣯⣅⡀⠈⠻⣦⠀⢸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⣴⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣾⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⢺⣿⡇⠀⣸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⡀⢀⣷⡄⠀⢸⡇⠿⠾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣧⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠻⠾⠋⣿⠀⣼⠃⠀⠀⠀⣿⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⣧⣼⠏⠀⠀⠀⠘⣷⣼⢸⡇⠀⠀⠀⠀⡀⠀⠀⢻⣿⡟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⡇⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠹⣧⡀⠀⣀⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣤⡾⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡇⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⣤⣤⣤⣤⣤⣀⣀⣘⣿⣄⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣴⠂⠀⠀⠀⠀⠀⣠⣿⣃⣤⣤⣤⣤⣤⣤⣤
⣼⡯⠒⢿⣆⣀⣈⣉⠉⠉⠻⣷⣄⠀⠀⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣽⠏⠀⠀⠀⢀⣤⡾⠛⠉⠉⠉⠀⠀⠀⣿⠀⠘
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠙⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠟⠁⣀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⡇
⢻⣆⠀⣾⠁⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣷⣦⣤⣽⣿⣷⣦⣤⣤⣀⣠⣠⣤⣤⣶⣾⣿⣿⣥⡶⣾⡿⠿⠿⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⠐⣷⠢⣺
⠀⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⣀⣤⡶⠟⠋⠀⠀⠀⠈⢉⡽⢿⡿⠛⠉⠛⠙⣿⠉⠙⢧⡈⠀⡀⠀⠈⠛⠷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⣠⠔⠀⣠⡾⠋⠁⣼⡇⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⡈⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⣁⠄⠀⠀⠀⠔⠉⠀⣠⡾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⡀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⣾⢏⡥⠖⠠⠈⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡏⠀⠀⣠⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣄⢀⠀⠀⠈⢳⠞⣉⣉⡻⢷⣄
⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠈⢘⡇⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⣿⠃⡀⡀⠃⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣰⡏⠓⠀⠀⠙⣎⣿⡄
⠀⠀⠀⠀⠈⢿⣄⠁⢰⠀⠀⣸⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣟⣠⣧⣥⣤⣄⣀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡀⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠷⢾⡷⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣉⠢⠀⠚⠀⠙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣾⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣓⠀⠀⢈⣩⡾⠋

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣭⣰⣀⠀⠐⠐⠲⠦⣤⣀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡶⠛⠁⠹⡇⠀⠀⠀⠀⠀⠀⠈⠓⢦⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⡶⠋⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡏⠁⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⠙⣶⠖⠂⠀⠀⠀⠀⠀⠀⠠⡀⢴⡦⠄⠀⣸⢣⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⠈⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡿⡇⢠⣜⡀⢹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⠗⡂⢠⡟⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⡠⡀⠃⠀⠀⠀⠀⣀⣈⣀⡁⣤⣤⣄⡀
⠀⠀⠀⣀⢀⡀⡿⡀⠘⠆⠀⠈⣇⣀⣐⣖⠂⠚⠁⠀⠒⠈⢀⠁⠙⢏⠁⠉⠲⠶⢤⣤⣤⣀⡀⠀⡀⡀⡟⢡⠘⣾⣿⡀⢀⣠⡀⠀⠀⠀⢀⠀⠈⢫⣿⣿⡄
⠀⠀⠀⠩⣽⣿⣾⣿⡏⠀⢶⠎⠙⠿⡅⢠⠳⠆⠀⠀⠀⠀⠨⢧⠀⠸⠆⠤⠤⠀⠀⠀⠐⠉⠁⠙⠻⠶⠿⠸⠇⢛⡀⣗⠂⣈⠀⠀⠀⠀⡼⢣⠐⠉⣿⣿⣿
⠀⠀⠀⠦⢤⣝⢻⣳⣦⣼⣴⣄⠀⣺⣗⣺⡗⣶⣦⣤⣴⣶⣿⣿⣿⣿⣏⣉⣉⣩⣩⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠭⣽⣍⣹⣬⣶⣶⣷⡀⢻⣆⠐⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠾⡧⣤⡉⡟⠛⣛⠛⠛⠛⠛⠛⢿⣿⣿⣿⡇⠀⠀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⢰⣦⣾⣿⣨⣿⣿⡻⣯⣿⣥⢻⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡔⣈⣹⠁⠉⠵⡎⠤⠐⠒⣤⣄⣿⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⣿⣿⣟⣛⣿⠿⢣⣿⣿⣿⣧⠑⠃⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠖⠒⢺⠁⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣭⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣯⣀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣤⠞⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢠⣤⣠⣤⡀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⢀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⣾⣿⣦⣴⣦⣠⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿⣟⣛⣛⣻⣿⠢⣿⣧⣺⣿⣿⣿⣿⣧⣼⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⠋⠋⠉⢋⠉⠁⠁⠀⡷⠀⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⠿⣿⣿⣷⣏⣬⣛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠔⠒⠲⠶⢯⣽⣟⡉⠉⢉⡭⣷⠀⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢧⠀⠀⠀⠀⣾⠀⠀⠹⣦⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⠀⠀⠀⢸⡏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⢦⣀⡌⠀⠀⢂⠀⠀⢠⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢈⡄⠀⢸⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢰⡏⠀⣇⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⡄⠀⠀⢰⡄⠻⡄⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠙⠦⠤⠞⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣾⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡏⢠⠀⡄⠀⢀⡾⠁⢀⣽⡿
⠀⠀⠀⢠⡞⠻⣦⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠀⠆⡼⢁⣴⣿⣀⣴⣿⡥⠶⠞⠛⠏⠹⣦
⢀⣤⣦⣿⡀⠀⠀⠙⠛⠓⠚⠛⠛⣻⣿⣦⣄⡀⣀⣹⣦⡀⠀⠀⢀⡇⠀⢀⣼⣷⣷⣾⣿⣿⠛⠛⢋⡉⠀⠀⠀⠀⢀⡠⠀⢿⣧⣀⣤
⠻⣿⣿⣿⣷⣄⡠⢄⡀⠀⠀⠀⢉⣿⣿⡿⠻⠿⢿⣿⣿⣿⣷⣴⣿⣿⣾⣿⣿⣫⣝⣟⠻⣿⡀⠀⠀⠀⠀⠠⢤⣀⣠⣤⣴⣾⣿⣿⣿
⠀⠈⠙⠻⠿⣿⣿⣷⣶⣾⡿⠿⢿⣿⠋⠀⠠⠀⣼⠟⠻⣿⣿⣿⣿⠿⠋⠈⠙⢿⣛⠻⣦⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⢠⣾⡿⣆⠀⠘⣟⠻⢿⣦⡀⠀⠀⠀⠈⠉⠛⣛⣷⣶⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⢀⣴⡿⠟⠀⠸⣧⠀⢹⠀⠀⠙⢿⣦⡀⣀⠀⠀⣼⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣤⣤⣾⣿⡟⠁⠀⠀⠀⣿⣄⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠀⢸⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠘⢼⠄⡸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠁⠋⢸⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠀⠀⢀⠀⡄⠂⠀⠀⠈⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⠀⠀⢰⡋⠁⠀⢀⡦⠀⢸⣿⡟⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⠀⣴⣒⡶⣛⣷⡀⣰⣟⠷⣄⢈⠽⣯⣑⡦⣀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⣀⣀⡥⢛⡡⣠⠖⢛⣻⣿⣿⡀⠈⠻⡾⠉⠉⠙⣦⣙⠢⣄⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⢿⣧⣶⣿⣿⠃⠀⠸⠇⢠⣼⣷⣀⠀⣌⠻⣿⣿⣿⣿⣦⣶⣟⠓⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠦⠀⠀⠀⠉⠹⠿⠿⠷⢤⡴⣿⣾⣿⣿⣿⣧⣹⣴⣿⡏⠉⠁⠀⠀⢁⡀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⢀⣴⣿⠇⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢠⣿⣿⡏⠀⠀⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠃⢀⣿⣿⡿⠀⠀⢸⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠘⣿⠀⣿⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⢹⡄⠋⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣯⠙⣷⡀⢹⣿⣿⣿⣿⣧⡀⠀⠃⠀⠀⠀⢠⡿⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠉⠃⠘⡇⠘⣿⣿⣿⣿⣿⣧⠀⠀⢀⡀⠀⠸⠁⠀⠀⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢀⣀⠀⠀⢹⣿⣿⣿⣿⣿⡆⠀⣾⡇⠀⠀⠀⠀⣼⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠹⣷⣄⠀⢿⣿⣿⣿⣿⡇⠀⣿⣿⡄⠀⠀⠀⠏⣼⠟⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⢀⣤⠘⠃⠀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⣿⣿⣆⢸⣿⣿⡟⠻⣿⣿⣿⣿⡏⢀⣿⡇⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⡀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠘⢿⣿⣿⣧⣾⣿⣧⠰⠋⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡌⣷⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⢈⡵⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⣿⣿⣿⠻⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⡀⠈⠛⠧⠀⠀⠀⢰⣿⡿⠃⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⠀⠸⠟⠁⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣄⡀⠀⠀⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢀⣾⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⣾⣿⡄⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⣿⣿⣿⣄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⣿⣿⣿⣿⣧⡀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⢻⣿⣦⠀⣿⣿⣿⣿⣿⣷⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⣿⣿⣦⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠇⠀⣿⣿⣿⣿⣿⡟⠙⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢶⣦⣄⠀⣿⣿⠿⣿⣿⠁⠀⢻⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⠘⣿⣿⣿⣿⡏⠀⢹⣿⠀⢀⣸⣷⣤⣄⣀⣽⣿⠉⣿⣿⠿⣿⣿⣿⣶⡿⠋⢀⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⠀⢹⣿⣿⣿⡇⠀⣸⣿⣿⡿⠿⠟⠛⠛⠿⠿⣿⣷⣿⡇⠀⣿⣿⣿⡿⠀⠀⠹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⢸⣿⠈⠙⣷⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣿⡿⢿⣇⠀⠀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⠀⢸⣿⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠸⣿⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄
⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄
⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣇⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣟⣧⢼⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⣽⣶⣭⣌⠻⣿⣿⣿⣿⣝⣿⣿⡹⣟⡫⠀⣽⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⢿⣿⢂⢿⣿⣿⣫⡿⢿⠉⣌⣷⣶⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣵⡿⣾⣷⡟⠉⣭⣴⣾⢿⣿⣿⣼⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⡏⠽⠯⠝⢺⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⣿⣽⣿⣿⡞⣿⣷⣜⣽⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡕⣟⣿⣿⣿⡏⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢟⡯⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⡟⠿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣯⣃⡇⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠏⠀⣿⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⠀⢸⣿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣦⣿⠀⢸⣿⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⡏⠉⣿⡀⢸⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣿⡇⠀⣿⣷⠘⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⣿⡇⢸⡟⢻⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣿⣿⣿⣿⡇⣼⡇⢸⣧⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠉⢻⣿⣿⡇⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠘⣿⠀⠘⣿⣿⡄⣸⣿⣿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣿⡇⠀⢹⣿⣿⣿⢿⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣷⣾⣿⡟⣿⣿⣿⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⣿⣿⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⠻⣿⣿⣷⣤⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢹⡇⢹⣿⣿⣏⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⠈⣿⣾⣿⢹⣿⣿⠁⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣿⠀⣿⣿⣿⣼⣿⡟⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢘⣿⠀⣿⣿⣿⢹⣿⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣰⣿⣀⣸⣿⣀⣿⣿⡇⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣤
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣢⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠋⠉⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠋⠉⠙⠉⠉⠛⠛⠋⠉⠛⠋⠛⠛⠛⠛⠛⠛⠋⠛⠋⠛⠛⠛⠛⠛⠛⠋⠉⠋⠙⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠀⠀⠀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠻⣿⠋⠀⠀⠀⠀⠐⠄⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠒⡼⠃⠀⠀⠀⠀⠀⠀⠈⢠⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢅⣈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⢤⠏⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠹⣆⡴⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠘⢶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠰⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠈⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡏⠉⠉⠉⠹⢿⠭⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢤⡄⠤⠠⡄⡄⠠⢠⡤⣠⡠⣤⣄⡄⣤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣼⡿⡟⢄⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⣧⡀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢇⠀⣤⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⣶⣷⠧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢡⡘⠁⠀⠀⠄⠀⠀⢻⠧⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⢀⠀⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠂⠈⢣⠀⠀⠀⣠⠀⠀⢈⠷⢋⢴⣄⡀⢩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠄⠤⠤⣄⠀⠀⠉⠀⠀⠈⠚⠁⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠉⠃
⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠠⠀⠀⠰⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠆⡀⠀⠀⠀⠀⡈⠀⠐⠠⢀⠀⠀⡀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡶⣶⣿⣦⣄⣀⠀⡅
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢠⡖⠁⠀⣀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠑⠀⠐⠘⠁⠐⠃⠈⠀⠀⠀⠀⠀⠀⠘⠯⠉⠛⢻⠟⡻⠁⠀⠀⢀
⠀⠀⢴⣉⡠⠀⠚⣧⡀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⠀⠀⠁⠈⠁⠁⠈⠀⠀⠀⠀⠈⠀⠁⠀⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⡸⢃⠀⠀⠁⠘⠂
⠀⠀⣸⠾⣅⠦⠶⠇⠀⠀⡄⠀⢠⡟⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿⠇⠀⣼⠀⡔⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⡏⠀⠀⠀⠀⠀⢀⠀⠈⢦⣤⠀⠀⢀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣈⠉⠀⠈⠁
⠀⠀⠀⠀⣠⢺⠃⠀⠀⠀⠐⠘⠀⠀⣾⠇⠀⢀⠜⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠒⠂
⠀⠀⠀⢰⣇⡞⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⢠⡆⠀⠀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠾⠁⣀⣴⠿⠀⠀⠀⠀⠀⠀⢠⢴⠏
⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣾⡾⠋⠁⠀⠀⠐⠀⠀⠀⡴⢁⠎⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣤⣤⣰⣶⣶⡀⢹⣜⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠈⠁⠋⠀⢸⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠌⠜⠁⠀⠀⠀⠋⠉⠙⣦
⠀⠀⠀⠀⡰⡕⠿⠙⠏⠻⠝⠂⡌⠀⠇⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠄⠀⠀⠀⢀⡀⠀⠀⠀⣬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢰⠀⠰⠀⠀⠀⠀⠀⢃⠀⠃⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⡠⠴⣤⣠⡤⠤⠶⠂⠀⠸⣉⢂⠀⠊⠁⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠈⡀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡴⠉⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄
⠀⠀⠀⠀⣠⣾⣿⡿⠀⠀⠀⠀⢠⣀⣾⣿⣿⠀⡀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⢀⣾⣿⣿⣧⡀
⠀⠀⠀⠀⣿⣿⣿⣇⣦⠀⠀⠀⣿⣿⣿⠇⢻⣿⣿⠀⠀⠀⣼⣿⡿⠙⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣧⣦⠀⠀⢠⣸⣿⣿⣿⣿⡷
⠀⠀⠀⢷⣿⣿⣿⣿⡿⠀⠀⠀⠹⡏⢁⣾⡌⢹⠏⠀⠀⠀⢻⣿⠀⢠⠿⡿⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⢀⠀⢿⣧⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠀⠀⠀⠀⢠⣿⣿⡇⢀
⠀⠀⠀⣼⣦⣼⣿⠏⣠⠀⠀⠀⢠⣼⣿⣿⣿⣷⡀⠀⠀⠀⣄⣻⡿⠙⢿⣷⡀⠀⠀⠀⢸⣿⣿⣿⠁⡀⠀⠀⢠⣜⣿⣿⣿⣿⣀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇⠀⠀⠀⢿⣿⢁⣴⠈⣿⠃⠀⠀⠰⣾⡿⡈⣹⣿⡇⠀⠀⢸⣿⣿⡏⠘⣿⣿
⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⠈⠙⠾⠿⠛⠁⠀⠀⠀⠀⠈⠃⠀⠀⠛⠀⠀⠀⠀⠙⠋⠁⠀⠘⠁

⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀
⠀⠀⠀⢠⣄⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣼⣿⣧⣦⠀⠀⠀⠀⠀⣸⣿⣷⣄⠀⠀⠀⠀⣠⣠⣿⣿⣷⡄
⠀⠀⠀⣸⣿⡟⠙⣿⣾⡄⠀⠀⢰⣾⣿⡿⣿⣇⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⡏⢿⣿⡆⠀⠀⢰⣿⣿⡟⠻⣿⣿
⠀⠀⠀⣿⣟⠀⠰⠿⣿⠇⠀⠀⢸⣿⠏⠃⠹⣿⡇⠀⠀⠈⢿⣿⡟⣭⢿⣿⠇⠀⠀⢸⣿⠎⠁⠈⣿⡇⠀⠀⠘⣿⡇⠀⠀⢸⡿
⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠀⠙⠂⠀⠐⠋⠀⠀⠀⠀⠈⠙⠘⠛⠙⠉⠀⠀⠀⠀⠙⠀⠀⠀⠋⠀⠀⠀⠀⠈⠓⠤⠴⠚⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣦⣴⣦⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢳⣸⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠁⡇⠀⠈⣷⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⠀⢷⠀⠀⠘⣧⠀⡞⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⢸⡆⠀⠀⢹⡄⢧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠲⣄⠀⠀⠀⢿⠀⠹⣆⣸⠇⠀⠀⠀⣧⠘⣧⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣇⠀⠀⢸⡆⠀⠈⠁⠀⠀⠀⠀⣿⠀⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢻⣴⠏⠀⢸⠇⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡟⠀⣷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⡇⠀⣿⠈⣷⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢠⡿⠀⢸⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢴⣟⡉⠙⠳⣄⠀⣿⠀⠀⠀⠙⠋⠀⠀⠸⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⢹⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢻⣼⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⠀⢠⣄⠀⢸⠸⣦⡾⢡⡇⠀⠀⠀⠀⠀⠀⠈⣿⢿⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠷⠞⢻⠀⣸⠀⠀⠀⣼⢣⢶⠀⠀⠀⠀⠀⠀⢹⡼⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⡴⠋⠀⠀⠀⢿⡾⢸⠀⠀⠀⢠⠀⠀⠸⣿⠃⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⣆⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠹⢦⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠞⠋⢸⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣴⠶⡦⠤⠤⣀⣹⣧⡀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠀⠀⠀⠀⣠⣾⡧⠤⠤⠴⢶⠶⢦⡀
⠀⢸⡇⠀⢸⠀⠀⠀⠀⣈⠻⣦⣀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠞⠁⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⣬⠀⢸⡇
⠀⢸⡇⠀⡙⠀⠀⠀⠀⠀⠀⣀⣉⣛⣶⣤⣄⣙⡳⣦⣤⣀⠀⠀⠀⠀⣀⣀⣠⣤⣖⣋⣡⣤⣶⣟⣉⣀⣀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇
⠀⠈⠻⠾⠟⠛⠛⠛⠛⠛⠋⠉⢉⣩⠿⠛⠉⠉⠉⠙⣻⢿⡟⠛⠛⢻⡟⠻⣯⡉⠉⠉⠉⠳⢦⡈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠚⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⡠⠄⢀⡴⠟⠁⣾⠁⠀⠘⠘⣇⠀⠈⠻⣄⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠶⠟⠩⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡏⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣄⠀⠀⠀⢤⡀⣙⢳⣤⡀
⠀⠀⠀⠀⠀⣼⡁⠊⠀⠀⠜⠀⠀⢀⡴⠋⠀⠀⠀⠀⣼⠃⠀⡰⠀⠀⣿⡀⠀⠀⠀⠀⠙⢷⡀⠀⠀⣽⡉⠀⠊⡙⣦
⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⢀⣿⡾⠋⠀⠀⠀⠀⠀⠀⡟⢀⣀⣀⡀⠠⢹⡇⠀⠀⠀⠀⠀⠀⠙⢦⣼⠁⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠈⠻⠶⡶⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⢫⠍⠉⠉⠛⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣄⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢁⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣥⣤⣤⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⢸⠇⢈⡀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⡞⢱⡋⠀⠀⠎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡼⠀⢠⠞⠁⠀⠑⢀⠀⠀⡀⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⠦⡄⠀⣄⣦⠈⠀⠀⠀⠀⠸⣇⡶⡉⠁⠸⠀⢀⡆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠑⠄⠁⡴⠙⣿⡀⠀⢰⠃⡆⢰⠋⠀⠈⡄⢁⡼⠻⡄⠣⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠆⢠⠀⣼⠀⢀⠀⠀⠀⠁⠀⡇⠀⠀⠀⢸⠗⠋⠀⡸⠅⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⢀⠏⢶⣟⠀⠈⠧⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⢱⠀⠉⠀⡀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⢸⠀⠀⠉⠀⡜⠋⢠⠶⠀⠀⠀⠀⠀⢠⠞⠀⡻⠀⣸⢠⡆⣔⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⢠⠈⢄⠀⠠⣸⠀⢠⠃⠀⠀⠀⠂⠀⠀⠀⠐⠘⠀⣠⠇⠈⣠⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠞⠗⠮⡳⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⡔⢁⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡛⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠈⠀⡀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣵⠀⠀⠆⡀⠀⠁⠀⠀⠀⠀⠀⢰⠆⠀⠀⠄⠀⠀⠀⡜⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠃⠀⠀⢀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢐⠀⠄⣀⠀⠀⠀⢰⣄⡀⠀⠠⠤⠤⠤⠄⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣲⣬⣥⢤⠴⠶⢤⣂⣿⡶⣾⣿⣧⣾⣿⣖⠋⢻⣿⣿⣷⡾⠶⡤⢄⣀⣀⡀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢯⠀⠀⢀⣀⣠⡼⠟⠁⢀⡏⠈⠉⣿⠿⣯⠀⠀⠙⢿⡻⢿⡖⠲⣌⡁⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢿⢿⣿⡟⠁⠀⠀⢀⣼⠁⠀⢰⣿⡄⠻⣧⠘⠀⠀⠻⣷⣬⣄⣿⠛⠳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣠⡾⣿⡃⢀⡀⠸⣿⠗⣀⣻⣧⠀⠀⠀⢹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠈⠀⣷⡻⣦⡎⠸⠛⠉⠘⢶⣶⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠠⢠⠥⢠⡀⠀⠀⠀⢀⡀⢐⠀⠀⠀⠀⠀⢇⣠⠆⠀⠀⠀⠠⣀⣸⣇⠀⠀⠂⡀⠀⢂⠎⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⡼⠻⠹⠀⠄⠀⠀⠐⠚⠛⠃⠀⠀⠀⠈⠉⠉⠐⠀⠀⠀⠀⠀⠀⠌⠒
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢤⣄
⠀⠀⠀⢠⢲⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢉⡟⢂⣮⠖⣶⡖
⠀⠈⠸⠛⢲⠚⠑⠈⠄⠰⠀⠐⠁⠐⢄⠂⠀⠀⠐⣂⠁⠓⠥⠔⠀⣾⢉⠺⠁⠀⠀⢠⣴⣀⣀⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡇⠀⠀⠀⠀⢀⣟⡘⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡇⠀⠀⠀⠀⡼⢸⠀⠠⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⢷⠀⠀⢀⡼⢁⠎⡀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢠⠈⠓⠒⠋⠠⠊⡸⠁⠘⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⢤⡀⠀⠀⠘⣿⡸⠀⠘⡄⢰⠁⡴⢁⡀⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢳⡀⠀⡇⣇⠱⡀⠃⠈⡜⣰⣿⠁⠀⢸⠁⠀⠀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⢰⠇⡅⠸⠈⣧⣠⠇⡟⢀⡤⣶⠆⣿⠃⢿⠀⠀⢸⡀⠀⣸⢣⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡟⢠⠁⠘⠀⠀⠁⢀⢣⠏⣰⠃⠀⢻⠀⠘⣆⠘⡀⠓⠚⠃⠘⡀⠈⡇
⠀⠀⠀⠀⠀⣼⠀⡆⢰⠀⠑⢄⠀⠘⢸⡀⣏⠀⠀⣼⠀⠀⠘⡇⠁⠠⢰⣷⠀⡇⢸⠇
⠀⠀⠀⠀⠀⣿⠀⡇⠸⠀⢀⡀⠡⠀⠈⢧⠈⠳⠞⠁⠀⠀⢰⠇⠐⣱⠋⡏⢀⢃⡟
⠀⠀⠀⠀⠀⠸⡄⢣⠀⢇⢸⣧⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢯⠴⠞⠁⢸⠃⡜⠸⣇⣀⣤⡦
⠀⠀⠀⠀⠀⠀⠻⡄⢂⠘⣾⡌⠳⢤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠐⠀⠀⢈⢍⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣲⡿⡣⢔⡵⠋
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠙⠦⣅⡳⢤⣀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⣉⡤⠖⠋⢀⣀⣠⣄
⠀⠀⠀⢰⠏⢉⠉⠓⠶⢤⣀⡀⠉⠓⠿⠽⢿⣿⣯⣭⡭⠭⠤⢖⣚⣫⠥⠶⠒⠛⠉⠁⠀⠈⣧
⠀⠀⠀⢸⠀⢀⠉⠑⠂⠀⠠⠉⡛⠲⢤⣄⣀⡀⣀⣤⠴⠒⠛⠉⡁⠀⠀⠐⢀⡠⠄⠂⠀⣀⡾
⠀⠀⠀⠘⠷⣄⡀⠉⠐⠢⠄⠀⢀⣉⡤⠶⢛⡉⠁⠀⠐⠀⠀⠀⠀⠀⠀⠈⣀⣤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠙⣲⡶⠶⠚⠉⠀⠀⠀⠁⠀⡀⠠⠔⠒⢀⣀⡤⠴⠶⢯⣁⡀
⠀⠀⠀⣀⣠⠴⠖⠋⠁⠤⠒⣀⠠⠐⠀⠁⠀⣁⣠⠤⠖⠛⠉⠀⠀⠀⠒⠠⢀⢉⠙⠒⠦⢤⣀⡀
⠀⢠⡞⣁⠬⡙⣆⠀⠐⢈⡀⠤⣤⣤⠴⠒⠋⠉⠉⠓⠲⠤⣤⣀⡀⠀⠐⠀⠀⠀⠀⠀⢠⢞⠭⠍⠙⢦
⠀⢸⡹⣍⡣⢧⣏⣀⣨⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢬⣁⠂⠀⡟⢀⢶⠐⡄⣼
⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠿⠤⠥⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣾⣽⣯⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠿⢿⣿⣿⣿⡿⠟⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣷⣖⣠⣾⣿⣿⣿⣿⣿⡿⣦⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣟⣦⠀⠀⠀⠀⢨⣿⣷⣿⠿⢿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣷⠀⢈⣛⡿⢿⣶⣿⡿⠿⠛⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⣸⣿⣿⣿⣿⣷⣾⣖⡀⢠⣤⣸⣽⣿⣻⠿⠿⡿⠛⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⢀⣾⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣧⣾⣿⣿⣿⣿⣿⣿⣍⣰⣶⣦⡀⠀⠀⠉⢛⣿⣿⠋
⠀⠀⠀⢠⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣾⣿⣿⣿⡟⣻⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⡟⣰⣿⣿⣿⡇⠀⣴⣿⣿⣿⠏⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⡿⣿⣿⣧⣼⣿⣿⢽⣿⢴⣿⣿⣿⠀⠀⣴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢻⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣿⣿⣗⣿⣿⣷⣿⣿⣿⣾⣷⣿⣿⣿⣿⣶⡄⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣮⣭⣸⣿⣿⣿⡿⠛⣿⣿⢯⣿⠆⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⠀⠘⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠈⣿⣿⣿⡟⠀⢸⣿⡷⢽⡿⣿⣿⣿⡿⠁⢸⣿⣿⢠⣼⣰⣿⣿⣿⡾⠛⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⡏⠀⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠘⣿⡿⠪⠿⢸⣿⣿⡿⠁⢀⣿⣿⣿⡟⠀⣿⣿⣿⣿⠛⠙⠃⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢸⣿⣿⣿⠃⢀⣿⣿⣿⠇⠀⠀⠨⠅⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠋⠂⠀⠀⠀⡄⠀⠀⠀⠀⢀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⣸⠀⢰⠀⠃⠐⢡⣊⡁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠘⡆⡞⣿⡀⡈⠳⣄⠠⠘⢿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠡⢱⡘⣜⡙⣷⡼⡄⠈⣷⢱⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠀⠀⠠⢲⡼⣦⣻⣿⣷⣸⡿⢻⣷⡟⡸⠠⠀⢸⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⢳⠙⣿⣿⣿⢇⣿⣯⠚⣡⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⢀⠀⣾⠠⡙⣿⣿⣿⣿⣧⣾⣭⡁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⡉⠟⠺⡱⠘⠷⣿⡾⠿⠛⠉⠉⠉⠀⠊⠁⠑⢲⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣡⠇⠀⢀⠠⠚⠉⠀⠀⠀⣀⠀⠀⣉⢀⣢⡠⡏⡌⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢉⡉⠉⠹⠃⠛⡲⡆⠀⢠⣾⣷⡆⡏⢉⠉⣹⣿⠓⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠺⠇⢀⣨⣇⣐⣸⣧⣶⡉⠅⠀⣿⡷⣤⠤⠟⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⢚⠽⢛⡅⠈⡻⠟⠛⠷⠶⠞⠉⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁