Monday Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢩⢍⠢⡬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣁⠉⢪⠐⢧⢃⢤⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⣀⢀⡀⢠⠄⠠⣄⢀⣀⡀⢀⣀⡇⢐⣂⡅⡸⡐⡄⢩⠡⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢙⣮⠤⠤⠟⠀⠀⠘⠃⠀⠀⠘⠃⠀⠘⠃⠀⠘⠣⠤⢼⡬⢾⠔⣍⡑⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡳⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡉⠓⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣱⠤⠃⠀⠀⣀⡀⢀⣀⡀⠀⣀⡀⢀⠀⢀⣀⣀⣀⠀⠀⣀⡀⣀⡀⣀⣀⠀⠀⢱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠔⣘⡅⠀⠀⠀⠀⣇⡱⣏⠀⣶⢡⠤⡝⣮⢑⠏⡟⣷⢆⢱⡰⠿⢾⠣⡘⡠⠁⠀⠀⢸⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⡠⠮⢨⠶⡀⠀⠀⠿⠉⠹⠼⠉⠪⠩⠚⠰⠝⠲⠁⠏⠥⠺⠧⠉⠦⠇⠦⠃⠀⠀⠀⢸⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢐⡳⢄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢎
⠀⠀⠀⠀⠀⠀⠀⠠⠊⡝⠄⡵⢣⠎⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡤⠜
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠎⠂⠀⡩⡗⢤⠬⡗⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠄⠲⠅⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠒⠑⠀⠀⠊⠀⠓⠤⠤⠚⠢⠤⠜⠣⠠⠜⠤⠤⠣⣀⡜⠣⠄⠛⠢⠚⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⡹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠈⠗⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡠⢀⣀⣀⡠⠀⠀⠤⢀⣀⢀⢀⡤⠒⢀⡀⠀⢀⠤⢀⣀⣀⢀⠐⢀⡀⣄⢒⢤⢀⣀⠀⠀⠀⠂⡠⠤⠤⢄⠂⢤⠤⢤⡀⡠⢢⠒⠒⡆
⡄⠀⠠⠀⠀⠘⡔⡀⠸⠁⠀⠀⠘⡔⠂⠁⠀⠉⠢⡁⠀⠀⠀⢫⢤⢸⠀⢀⢸⠀⠀⠀⠑⢌⢀⠀⡇⢀⠀⠈⡄⠈⢣⠀⠱⣉⠎⠀⡘
⠄⠀⢸⠀⠀⠀⠰⣀⠃⠀⠀⠀⡄⠀⠀⡎⠑⡆⠀⢡⢸⠀⠀⠀⢣⢸⠀⢸⢸⢀⢨⠑⡄⡼⡜⢰⠀⡈⢡⠀⢡⠀⠀⠳⡀⠉⠀⡜
⠀⠀⢸⠀⢠⡀⠀⠙⠀⣠⠀⠀⡀⠆⠀⡄⠀⠃⠀⢸⢸⠀⢠⡄⠈⠻⠀⢸⢸⠀⢸⠀⡅⢨⡃⠘⠀⠱⠼⠀⠐⡀⠀⠀⢸⠀⢸
⠀⠀⢸⠀⢸⢣⠀⠀⠀⠛⡂⠀⡆⡇⠀⣇⠀⡄⠀⣭⢸⠀⢸⠙⡀⠀⠀⢸⠸⠀⢺⣠⠇⢨⠆⡃⠀⡀⣀⡀⠀⢡⠀⠀⢸⠂⢸
⠀⠀⢸⠀⠸⠀⣆⠀⢸⢀⡃⠀⡇⢢⡀⠀⠁⠀⣠⢃⢸⠀⣸⢠⢱⡀⠀⢸⠨⠄⠈⠄⠀⠼⢸⠀⢰⢡⠦⢃⠀⢘⡀⠀⢸⠀⢸
⠀⠀⣘⣒⣚⡋⣙⣖⡋⢙⡆⠶⢋⠁⢍⡒⠖⢚⡅⠀⣘⣒⣴⡂⠀⡳⠲⠞⡰⠤⠀⣚⡤⡞⠬⠤⠗⡎⠀⠸⣄⠤⠇⠀⢸⣀⣰
⠠⠮⠤⡤⠮⣤⣉⠄⣧⣀⣀⡰⢥⣀⠀⢀⡽⠋⣠⣐⣀⣠⠟⣀⣀⣈⣩⢉⠈⢉⣉⢵⡇⡉⠉⠉⠉⠀⡠⠋⠋⠉⡽⠓⠒⠀⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⢠⡀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⢀⣴⢶⡄⢸⣿⠟⠉⣿⠀⣼⠇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⣤⣄⣶⠀⣿⡟⠁⢨⡿⢸⡏⠀⢠⡿⠀⣿⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣾⡇⢠⡟⠁⣹⡿⢀⣿⡇⢀⣼⠃⢸⡷⠶⠟⠀⠀⠙⠛⢁⣿⣤⣤⣤⡶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠉⢸⡇⣾⠃⣰⢿⣧⣾⢿⡟⠛⠁⠀⣿⠇⠀⠀⢀⣤⡶⠟⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⠀⢸⡇⢻⣶⠏⠀⠛⠁⢸⡇⠀⠀⠀⣿⠀⠀⣴⠟⠁⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠼⠏⠀⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠋⠀⠀⠘⠻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠈⠀⠀⢢⠔⠀⠀⢄⠀⠀⠀⠀⠀⠀⢄⠀⠀⠘⣦⡀⠀⠀⣦⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⡄⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⣾⡆⠀⠀⢰⡄⠈⣷⡄⠀⣼⣿⡄⠈⣷⠀⠀⠘⣷⠀⢸⣿⠀⢸⣿⠆⠀⢳⣄⠀⢠⣶⡄⠘⣷⡀⠸⣿⠀⠀⣼⣿⠃
⠀⠀⠀⠀⢠⣿⡧⠀⠀⣿⣇⠀⢸⡇⠀⣿⡏⠀⠀⣿⡇⠀⡀⠈⠀⢸⣿⠀⢸⣿⠀⠀⢸⡇⠀⠛⠛⠃⠀⢸⣷⢀⠀⢠⣾⣿⠏
⠀⠀⢀⠀⢸⣿⠇⣀⣀⣿⣿⣀⣸⣷⡀⠘⠋⠀⣰⣿⡇⠀⣷⡀⠀⣿⣿⠀⠸⠛⠁⣠⣿⣃⣀⣶⣶⣶⣄⣀⣿⡄⠀⢸⣿⡋
⠀⠀⠀⠛⠿⠿⠀⠙⠿⠿⠏⠙⠿⠿⠋⠲⣶⣾⣿⠟⠙⢿⣿⡟⠻⠿⠿⣤⣤⣴⣾⣿⠿⠉⠻⠿⠏⠀⠈⠛⠛⠃⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⢰⠼⣹⣿⣭⣝⣻⣯⡯⡬⢧⡂⣍⡍⣂⠄⠺⠤⠉⠹⣮⠾⢩⡞⠀⢈⡴⠒⠦⣒⢄⠈⠃⠆⡓⠠⣆⣈⠉⢛
⠸⣥⣼⠷⢌⡉⢭⣳⡾⣓⢘⢬⢲⡥⢄⣩⢀⠴⢺⣄⠻⣬⣤⠾⡤⢖⣩⣎⡴⠊⣿⣸⢸⠂⠄⠀⠘⢁⠐⠃⠄
⠀⢀⠃⢌⢐⡵⠦⢢⠠⠥⠟⠡⠇⠓⢐⠀⣠⠦⣗⣙⣞⣿⣿⢶⠟⡯⠚⠟⢯⠆⣈⡁⣀⡀⠀⠘⠴⡦⠠⡒⠁
⠀⠀⠀⠁⠀⠁⠀⠈⠳⠤⠖⠐⠉⠃⠈⠂⠈⠋⠀⠈⠓⠒⠛⠉⠁⠁⠀⠙⠒⠊⠙⠀⠙⠁
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠤⣤⣄⣤⣤⡤⡄⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄







⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⢤⢤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠉⠡⡀⠆⣿⢄⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠤⢀⣀⡀⠙⠛⠀⠀⡇⠀⠀⠈⠁⠽⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣰⠁⠀⠀⠀⠀⠀⠈⠩⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣀⣀⣀⣀⣠⣤⠶⢋⠀⠑⢦⣀⡤⠤⠤⠤⢄⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢈⠉⡭⠋⡍⢀⡀⣰⢛⠃⢀⡼⠉⠳⢤⣀⣀⣀⡱⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠱⠆⠂⠀⠀⠊⡃⠉⠀⣯⠂⠀⣼⠃⠀⠀⠀⠀⠉⠉⠀⢸⡏⣻⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣧⢀⣿⡟⡇⢸⣷⡄⣠⣿⡀⠀⣴⡟⣹⠀⣠⣤⣌⠛⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡆⢿⡇⠀⡸⢯⣙⠟⠈⡇⡇⢸⡟⡇⠈⣿⠉⢁⣰⣿⠤⠟⣠⡾⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⣈⣛⣚⣁⣀⡀⠀⠀⢿⡟⠘⣇⡇⠀⠈⠁⣉⣀⡙⠒⠚⣁⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡍⠉⠍⣭⠉⢉⡉⣙⠛⠒⠶⠤⣽⣣⣤⣶⣿⣿⣏⡿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠃⠀⢹⠀⢸⡇⣸⡇⣰⠟⡄⢠⡄⢉⠙⠛⠻⠾⢦⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢸⠀⣸⢳⡋⠀⡟⠀⡀⠸⣇⢸⣰⠏⡂⣶⠢⣤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡇⢸⡀⣿⠀⡇⠒⢃⢀⠁⡇⢿⡘⣿⢀⣀⣏⡀⣿⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣥⣤⣤⣤⣤⣬⣀⣈⣉⠐⠃⠈⠃⢻⣀⠺⡇⠀⢻⠈⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣹⡿⢷⢧⢻⣿⣹⣷⢹⣿⡒⣿⣶⣶⣦⣥⣤⣍⣁⣈⣀⠈⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠛⠛⠓⠻⠾⠿⠿⠿⠾⠟⠛⠛⠛⢛⡛⠉⠉⠉⠁⣤⡄⣤⡄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⣿⡇⢀⣴⠿⣦⠀⣿⣆⠀⣿⢸⡟⢷⣄⠀⣸⣿⡇⢿⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣸⣿⡇⣸⣯⠀⣿⠀⣿⢻⣆⣿⢸⡇⠀⣿⢠⡿⢸⡇⠘⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⢸⡇⢹⡟⢷⡿⠛⣿⠀⢻⣿⢸⡇⢠⡿⣸⣇⣸⣷⢀⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠹⠃⠸⠇⠈⠻⠛⠁⠀⠻⠂⠀⠉⠸⣿⠟⠁⠻⠋⠉⡿⠘⠷⡾⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣦⣿⣞⣷⠹⣏⠀⢠⣿⣾⡭⠀⠀⠙⡏⢹⣾⣷⡯⠀⠀⣷⣾⢷⡯⢸⡯⢸⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠁⠁⠈⠉⠀⠀⠉⠈⠀⠀⠀⠀⠁⠈⠁⠉⠉⠀⠀⠀⠀⠈⠉⠈⠉⠈⠉⠁
⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠙⠛⠁
⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠤⠶⠴⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠉
⠀⠀⠀⠀⢀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣮⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡫⠔⠊⢉⣀⣀⠀⠀⠀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠒⠒⢢⠀⠀⡴⡦⠘⡡⠊⠀⡔⣊⢀⣠⣤⣄⠀⢸⠐⠷⠂⢱
⠀⠀⠀⠀⠀⠀⠀⢊⠁⠀⠀⡰⣊⣀⣀⠉⠁⣾⢀⣠⠘⠰⠃⣾⡟⢻⣿⣷⠘⢊⡐⠊⠓⠓
⠀⠀⠀⠀⠀⠀⠀⠠⠑⡊⠃⣸⣿⠿⣿⡇⠀⠻⠿⠉⠃⢀⠀⢀⣴⣾⣿⣿⢀⣿⡇⢀⣿⣿
⠀⠀⠀⠀⠰⠆⠶⢠⣾⣿⣦⡙⣿⣿⣿⢳⣿⣶⣶⣦⠀⢻⣧⢿⣿⣾⣿⣿⠸⣿⣷⣿⣿⡟
⠀⠀⠀⠀⠀⣴⣿⣦⡹⣿⣿⣿⡌⠉⠀⢸⣿⡏⢹⣿⡆⢘⣿⣇⠉⡉⠀⢠⣶⣀⣠⣾⡿⠁⣠⡀
⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣷⠀⠀⠈⠿⠃⠘⠛⣾⣿⣿⣿⠐⢫⠗⠸⢿⣿⡿⠟⠀⠤⠟
⠀⠀⠀⠀⠀⠈⢿⣷⣌⠛⠋⠘⠉⠀⠰⣃⣀⣄⠀⠀⢿⣧⣽⣿⡄⠀⠀⠀⢀⠤⡔
⠀⠀⠀⠀⠀⠀⢠⡟⠟⠀⠤⠤⠤⠒⠁⠁⢠⡞⠁⡸⠬⢍⠉⠉⠀⠀⠀⠐⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠈⠹⠇⠁⠒⠐⠒⠀⠒⠈⠁⠀⠀⢤⣅⡣⠃




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⢰⣿⠿⠿⣛⣋⣭⣵⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣶⡖⢿⡟⡟⢍⣑⣷⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣛⣲⡾⠛⠛⠉⠁⢀⣀⡀⠉⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡆⠀⠀⠀⠀⣏⣀⠀⠉⠉⠒⠢⢤⣿⠧⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⢇⣤⣠⡾⠳⣦⣤⡰⢻⡇⣠⠤⣼⡏⢰⡏⢱⣦⢰⣿⠀⣴⡄
⠀⠀⠀⠀⠀⣀⡀⢰⠇⣿⣸⣿⠈⣷⠴⠱⣿⠃⣾⡇⣿⠀⣿⡇⢸⡧⠾⣿⠜⠛⣚⣿⡇⠐⠓⡆
⠀⠀⠀⠀⠸⣅⣀⡼⠀⠈⠁⣿⢀⠇⠀⢰⣷⣶⢿⣡⢯⣽⣞⡧⡊⠀⠀⠀⡴⠉⢀⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣿⣿⣷⣷⣿⣿⣿⣿⠃⠀⠀⠀⠑⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠟⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠄⠀⠀⠀⠀⠀⢠⡾⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡠⣶⠀⠀⠀⠀⠀⢸⡏⢀⠊⣽⠇⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢲⡆⠰⢻⣤⠁⣿⠀⣠⠤⣀⠀⣸⣧⠂⣸⡟⢀⣾⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠇⢸⡇⢸⣿⢸⡏⢀⣿⠀⣼⣇⡼⠋⠀⢸⡏⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠈⠁⢸⡇⡼⣧⢾⡏⢰⢿⡟⣿⡀⠀⠼⣇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢸⡿⠀⠀⢀⠻⢇⠈⠁⠈⠳⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⣼⡇⠀⠀⠀⢤⡀⢀⠀⢸⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣿⡇⢀⠀⠀⡄⠁⠀⠁⢺⠇⠀⠈⠀⠀⢀⣴⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣾⠃⢰⢠⡾⢁⣿⠀⣰⡏⣵⡆⢸⡇⢠⣿⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⠀⠎⢸⡇⢠⣿⠀⣿⢀⣿⠃⠾⡧⢊⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠙⠂⠀⠘⠇⠋⣏⠀⠉⠁⠘⠋⠀⡰⠁⣿⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠠⣄⠀⠀⠀⢰⠃⢰⡏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣆⡀⢠⣀⠎⣿⡆⢹⡇⠀⣠⢄⠀⠀⠌⢀⡿⠀⠀⠠⠌⠂⠋⠠⠁⣿⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⢡⠉⣿⠀⣿⡇⣿⡇⢸⡏⢸⡿⢀⠠⠾⠡⣀⢀⣾⠀⡔⣿⠆⢠⣿⠀⣾⠁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠜⢿⡆⠀⠟⠀⠉⠀⣿⡇⠸⣧⠼⣇⠀⠀⢀⠄⣿⡄⣿⢐⠀⣿⠀⢸⣇⠘⣿⣰⣿⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠃⠀⠀⠀⢀⠔⠁⠀⣿⠇⠙⠁⠀⠁⠀⠀⠁⡠⠀⠁⣿⠁⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⡼⠏⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣿⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡷⠀⢠⣿⠟⠁⠀⠀⠀⠀⣼⡿⠃⠀⢀⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⢠⣿⣯⣴⣦⣶⡄⠀⣼⣯⣶⠀⣰⠟⠁⢀⡀⠀⣴⠀⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣴⣾⢋⣿⠿⣫⣾⡟⣼⡟⣡⡿⢺⡿⣫⡷⠚⢯⣼⣡⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠉⢸⠃⠘⠳⠛⠉⠀⢲⠋⣾⡟⠀⡞⢺⡟⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠀⠰⠇⢀⡴⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠘⠀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⠀⡰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠈⡴⢰⠇⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣀⣤⣾⡃⢀⣤⣶⣶⡆⠸⡇⠀⢰⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢀⡟⠀⣠⣶⡿⢛⡷⣸⣷⢟⡿⠃⣰⣿⢟⣋⢿⠀⣿⣻⡽⠛⠀⢀⣀⣴⡿⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⡼⠁⣾⠟⢋⡴⠋⢠⠟⣠⠟⣠⠞⠉⠛⠋⠁⠘⠀⠀⣁⣤⠶⠟⠛⣹⠏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠰⠇⠀⠉⠚⠉⠀⠀⠈⣰⡧⠚⠁⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠐⠊⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⢣⠆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⢈⠁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠢⡑⡬⠂⠉⠀⠀⠁⠀⣤⡄⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣿⢹⣰⡏⡗⢀⣀⠀⣀⠀⠨⠀⠀⠀⠀⠀⠀⠀⣿⡷⢿⣠⠞⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄
⣾⣷⡄⠀⠘⣿⣆⠀⠀⠀⠀⣿⣸⢸⣿⠇⣘⢻⡏⠁⠀⠀⢱⠄⠀⠐⠀⠀⠀⣿⡇⠙⢋⡾⠁⣠⣦⣠⣴⣤⡄⠀⣀⣰⡏
⠘⣿⣿⡀⠀⢹⣿⡄⢀⡶⡆⢸⡟⢸⡏⢠⣿⣼⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢿⡇⠀⠸⣧⠾⣇⡾⠀⣿⢸⣡⣾⣹⣿⢀⡴⣦⢀⢀
⠀⠹⣿⣷⡀⠀⣻⣧⢸⣧⠃⢰⢷⡾⠛⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠉⠙⠛⠟⠛⠯⣾
⠀⠀⠘⢿⣷⠚⠉⢿⣤⠛⠶⠋⠀⠀⠀⠀⢤⣄⣀⡀⠀⠀⡁⠀⠀⠀⢀⣀⣀⣀⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣠⡟
⠀⠀⠀⠈⠻⣧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢱⠀⠌⠑⠄⠀⠀⠀⠀⠀⠀⠀⠹⠦⠴⠞⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡄⢸⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡠⠃⠀⠀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⠊⠀⠀⠀⠀⠉⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⠀⠀⠠⠤⠒⠢⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⣤⣄⣀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠁⠀⠀⠀⠀⠘⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢺⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣀⣤⡤⠄⠀⠈⣛⡙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠀⠠⠞⣩⡤⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⣿⡀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⣤⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠀⠉⠉⠉⠉⠛⠛⠿⢿⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠇⡀⡀⣀⢀⢀⡀⡀⢀⢀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⣰⠇⡟⣾⢰⣣⣏⡾⣼⣡⢇⣟⡾⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⠀⠀⡼⠁⠀⢀⡿⠀⢸⠇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠴⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢄⠀⠀⣴⡇⢺⡄⣺⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⢀⡿⢁⡿⢡⡟⣠⠤⠀⣤⢠⠀⣠⢤⡟⢠⠤⣤⢀⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠃⣼⢱⡇⡰⣼⠃⣿⣸⠇⣼⢣⡟⣰⡏⣼⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⣰⠇⠀⠉⠀⠉⠀⠉⠈⠁⠈⠀⠉⢈⠁⠉⣾⠥⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠏⠀⠀⡰⠋⠂⠁⠀⠀⠀⠀⠀⠈⠁⠀⠰⠚⠀⠀⣼⠧⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠊⠁⠒⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡇
⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣼⠃
⠀⠀⠀⣿⣿⣿⡄⠀⠀⢰⣿⣿⠀⠀⢠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠤⣼⣿⣷⠞⠁
⠀⠀⢀⣿⣿⠏⣧⠀⢀⣿⣿⣿⠀⠀⠀⣀⣤⣤⡄⠀⠀⢀⣠⡄⠀⣀⣀⡀⠀⠀⠀⠀⠈⣿⣗⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠸⣿⣿⠀⠸⣆⣼⢿⣿⣿⠀⢠⣾⣿⡿⠁⣴⠳⣆⣻⣿⡷⠊⢻⣿⣿⠀⣀⣴⣾⣿⣿⣿⠀⠀⠀⣠⣾⣿⠟⠉⣡⡄⠀⢰⣿⣿⡄⢀⣴⣶⡄
⠀⠀⠸⣿⣿⡄⠀⢻⡿⢸⣿⣿⢠⣿⣿⣿⠁⠀⠙⠛⡏⣿⣿⠁⠀⢸⣿⡿⣶⣿⣿⠋⠀⣿⣿⠀⠀⢰⣿⣿⠏⠀⢰⣿⠇⠀⢸⣿⣿⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⠀⠀⠀⢀⡇⣿⣿⠀⠀⣾⣿⣿⣿⣿⣏⠀⠀⢸⣿⡆⠀⢸⣿⣿⠀⠀⢸⡏⠀⠀⣼⣿⣇⠀⣸⣿⣿
⠀⠀⠀⢿⣿⣇⠀⠀⠀⠸⣿⣿⡀⢿⣿⣿⣧⣀⣠⡾⠁⠹⡿⠀⠀⣿⣿⡏⢿⣿⣿⣄⣀⣠⣿⣷⣤⡼⠿⢿⣷⡤⠞⠳⠴⠊⠈⠛⠛⠛⢹⣿⡿
⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⢿⣿⣇⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⡄⠈⠙⠛⠛⠛⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢿⣿⡇⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⠤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⡶⠶⠛⠛⠉⠉⠁⠀⠈⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡄⢀⢠⡄⠀⠀⡀⠀⢀⣠⡏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⠶⢶⡂⣶⠀⣠⣼⢃⡤⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⡇⣿⣿⢱⠯⢶⣷⣧⣿⢿⣾⢳⣧⣯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡾⣰⢤⣆⡞⠿⣸⢿⡷⡯⣿⠺⢿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠇⡟⢿⡙⣂⣤⣼⠧⠤⠤⠤⠤⠈⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠿⠙⣋⡭⠾⠟⠒⠲⠶⠦⢠⠔⣾⠓⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠀⠀⢠⡟⡀⠀⢀⢠⡆⠀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⠛⠛⠃⢠⡤⠀⣀⡄⠀⢠⠟
⠀⠀⠀⠀⢠⡶⣿⣀⣸⡇⣴⣄⡞⣿⣼⢷⣶⠟⡷⣵⣾⣻⣧⡾⣥⣠⢤⡄⠀⠀⠀⠐⠚⠉⣸⢃⡟⡀⣠⢠⣄⡿⣾⣹⣥⣿⣟⣴⣧⢀⣄⡄
⠀⠀⠀⠀⠙⢀⣿⠹⣿⠸⡧⢉⣋⣉⣩⣼⣯⠯⠾⠧⣬⣄⣉⡁⠉⢻⣿⣀⠀⠀⠀⠀⠀⢀⡟⢸⣿⡇⠿⠻⠊⣋⣴⣏⣈⣉⣉⡋⠙⠻⢻⡃
⠀⠀⠀⠀⠀⠘⠏⠀⣠⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣭⠏⠉⠃⠀⠀⠀⠀⠈⠓⠀⠈⠁⡴⠚⠉⠉⠀⠀⠀⠈⠉⠛⢰⣭⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣿⢩⡍⠁⡴⢠⣤⡟⣡⠄⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣼⠇⢙⠓⣺⠛⡉⣤⢻⡆⡴⣾⢡⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣧⠜⡷⣸⡇⣿⢾⠷⠇⠿⠺⢿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⣉⣰⣿⣦⣿⢸⡗⠿⠻⠾⠗⠷⠿⠻⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⢋⣬⡵⠶⠖⠒⠚⠛⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠙⢃⣭⣴⠶⠖⠚⠛⠛⠛⠓⠲⢠⠼⣾⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣵⠀⠀⠀⢀⠀⠀⣀⡼⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠀⣚⠋⣴⣰⣧⣿⡾⣼⢽⣷⡿⢾⣼⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⡿⠛⣉⣡⡤⠷⠤⠤⣤⣤⡄⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠁⠚⠉⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡆⠀⠀⣼⣿⠛⣿⡇⠀⠀⢰⣿⠛⣿⣷⠀⠀⢰⣿⠟⢻⣷⠀⠀⢀⣿⡟⢻⣿⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇⠀⣿⣧⣤⣤⣿⣿⠀⣿⣷⣤⣤⣾⣿⠀⣿⣿⣤⣤⣼⣿⠀⢸⣿⣤⣤⣼⣿⡇⢸⣿⣦⣤⡀
⠀⠀⠀⠀⣴⣿⠟⠉⣿⡇⠀⣿⡏⠉⠉⣿⣿⠀⣿⡟⠉⠉⢹⣿⠀⣿⣿⠉⠉⢹⣿⡀⢸⣿⠉⠉⢹⣿⡇⢸⣿⡋⠛⢿⣷⡀
⠀⠀⠀⢸⣿⠃⠀⠀⠙⠿⠿⠟⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⠈⠿⠿⠿⠋⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠻⠿⠿⠟⠀⠀⠈⣿⡇
⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠃⠀⠀⠸⠿⠿⠟⠀⠀⠀⠿⠿⠿⠃⠀⠀⠺⠿⠿⠟⠀⠀⠐⠿⠿⠿⠂⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣶⣶⣦⣿⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⣶⣶⡄⠀⠀⢰⣶⣶⣦⠀⠀⠀⣶⣶⣶⡄⠀⠀⢴⣶⣶⣶⣾⠿⠛⠉⠁⠀⣴⣦⠀⠈⠉⠙⠿⣿⣦⣀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣽⣿⠟⠁⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠘⠛⠛⠋⠀⠀⠀⠛⠛⠛⠁⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠸⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠸⠿⠿⠷⠀⠀⠀⠿⠿⠿⠆⠸⣿⡇⠻⠿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣤⡀⠀⠀⠀⠀⠻⠿⠟⢀⣿⡇
⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡷⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣬⣥⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⡁⠰⣨⠃⢠⣄⡟⠀⠛⠀⡏⠀⡄⠘⠃⢠⡄⢹⠀⣾⣿⣿⣿⡦⠀⣶⠇⢸⠀⠘⠏⠀⡟⠀⠲⣾⣦⠀⣶⣿⠃⠀⢻⡇⠀⠂⣽⠀⣾⣿⠀⠐⢶⣿⣿
⣿⣿⣿⣯⣉⣀⣸⣆⣈⣩⣇⣀⣧⣀⣧⣀⣁⣼⣄⣈⣠⣾⣀⣉⣹⣿⣿⣧⣀⣿⣄⣸⣀⣶⣦⣀⣇⣀⣉⣹⣿⣀⣿⣇⣠⣄⣈⣇⣘⣀⣼⣀⣉⣹⣀⣈⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣯⣭⣭⣽⣭⣿⣿⣿⣿⣿⣯⣭⣯⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣭⣯⣭⣭⣯⣽⣿⣿⣿⣿⣭⣯⣭⣽⣿⣿⣿⣿
⣿⣟⣟⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿
⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣯⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣏⣿⣿⣧⣀⡀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⣴⣶⣄⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣰⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⣿⣿⣧⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠉⠉⠀⠀⠀⡀⣀⢀⠀⠀⠉⠉⠛⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⠀⠀⡀⢠⡰⠎⠁⠞⠛⠛⠘⠂⠃⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⣐⡞⠃⠈⠀⢀⣀⣤⣤⠤⠤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠓⠉⠀⢀⡴⠞⠋⣽⣷⣶⠀⠀⠀⠀⠈⠉⠛⢦⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠚⠻⠏⠁⠀⠀⠀⠀⢾⣆⣀⣬⣷⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣴⣿⡿⠀⠀⠀⠀⠀⢠⠟⠀⣀⡀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⢠⡟⣀⣘⣿⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⠻⣹⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⢟⠩⢻⢰⢙⡇⣠⣻⣷⣽⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢐⣛⣿⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⣿⡏⠙⢷⡜⣷⣙⣸⣴⣾⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢛⢻⠩⣹⣮⢸⣿⣟⣨⣧⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁⠀⠀⣿⡟⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡍⡃⣿⢀⣙⣦⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿⡏⠉⣽⠃⠀⠀⠀⠀⢠⣿⣿⣆
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠈⢁⡼⠃⠀⠀⠀⠀⠀⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠙⢿⣋⠉⢿⠆⠀⠀⠀⠀⣼⣷⣖⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⣀⣀⣀⣚⣻⣯⡴⠞⠋⠀⠀⡤⠀⠀⠀⢀⣾⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⢀⢀⣤⣆⠗⠈⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⣶⣄⡀⠀⠀⠀⠀⠰⢰⣶⢶⠶⡆⣢⠶⠉⠑⠉⠀⢀⣠⣶⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⢶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢿⣿⠿⠛⢿⣿⣿⡿⠛⠿⣿⣿⠏⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⡀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢀⡎⣵⡐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⢠⣚⢟⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠤⠃⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣘⡼⠤⠤⠤⠤⠤⠤⠤⠙⠚⠡⠤⠤⠀⢀⠠⠿⣁⣼⡇⢀⡠⢃⠀⢄⣀⣀⣀⣀⣀⣀⡀⠀⡠⢢⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡤⢋⣠⠤⠐⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠢⠤⠬⠥⠒⠉⠓⠒⠒⠒⠒⠀⠤⠤⣀⡉⠻⡉⠀⠀⣠⡤⡄⢀⡀
⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⡰⢣⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡱⡄⢸⢃⢈⣹
⠀⠀⢀⠀⢀⠀⠞⠀⠀⡜⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡌⠉⠚⠑⡄
⠀⠀⠈⣣⠏⠛⢘⡆⢰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠓⠁
⠀⠀⠀⠣⢤⢠⡌⠵⢀⢠⠀⠀⣀⠀⣤⣀⢤⣠⠤⡄⣄⡤⠤⣄⠀⣀⠀⡠⡄⡤⠤⢠⠀⡠⢤⣀⡠⠤⣀⠀⠀⡤⢤⢀⣠⢀⠤⢤⠀⠀⡇⠀⡶⢲⠄
⠀⠀⠀⡎⣀⠉⠀⠀⢸⠈⠀⠀⢇⠻⣀⠇⢸⠘⠐⣿⠋⡇⡶⡄⠁⡇⢸⠇⡇⠁⢸⡟⢈⢦⣸⡛⡅⢶⡄⢳⣰⠻⠂⡎⢦⠛⡴⠃⠀⠀⡇⢰⠉⠉
⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠆⠀⠸⠤⠻⠄⠆⠀⣆⡻⠇⢇⡉⠴⠃⢧⡜⠢⠇⠸⣈⡷⠺⢷⠧⠃⠷⠉⠴⠋⣧⠋⠣⠴⠨⣴⠀⠀⠀⠀⡇⢸⡠⢄
⠀⠀⠀⠀⠀⠀⠀⡤⣼⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠁⣣⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠢⡈⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠬⢉⡒⠢⠤⠤⢄⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠤⠤⠐⠊⠁⠖⣡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡊⠁⠠⠤⠄⠀⠀⠀⠀⠤⣤⡄⠤⣄⣠⠤⠤⢄⣠⠤⠤⠠⠄⠤⠔⠀⣏⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠘⠗⠀⠀⠀⠟⠀⠀⠀⠀⠀⣎⠀⠧⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⡀⢹⣀⡲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠙





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇
⣿⠛⠛⠛⠿⡿⠟⠟⠛⠛⠟⢿⡿⠻⠟⠛⠻⠻⣿⠟⠛⠛⠟⠟⠻⡿⠻⠿⠛⠛⠻⢿⡿⠛⠛⠛⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢀⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠛⠀⠀⠀⢸⡏⠀⠉⠀⢹⣿⠈⣿⠁⠉⠉⠈⣿⠀⣿⡏⠈⠉⠁⢹⡇⢸⡏⠀⠀⠈⢸⣿⠀⠀⠀⣿⡇
⣀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠸⠿⠿⠿⠿⠾⠟⠐⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⠠⣶⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⣶⡆⢰⣶⣶⡆⢀⣀⣤⣤⣴⣶⣿⣧⣤⣄⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⣇⣤⡶⠟⠋⠉⢀⣀⣀⣀⣀⠈⠉⠛⠷⣦⣀
⣿⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠘⠁⣼⡿⠋⢀⣤⡶⠟⠛⠋⠿⠋⠛⠻⠷⣦⣄⠈⠻⣷⡄
⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠃⢀⣾⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠈⢿⣆
⣿⠀⠀⠀⢰⡶⠶⠶⠶⢶⣶⠀⣶⠶⠶⠶⠶⣶⠀⣶⡶⠶⠶⠶⣾⠇⢀⣾⠏⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠈⢿⡆⠀⢿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⢸⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸
⣿⠀⠀⠀⢸⣇⣀⣀⣀⣸⣿⠀⣿⣀⣀⣀⣀⣿⠀⣿⣇⣀⣀⣸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠲⣿⡇
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢠⣿⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣾⠃⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣾⠏
⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣷⣄⠈⠙⠿⣶⣤⣤⣶⣦⣤⣴⡾⠟⠉⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⡈⠉⠉⠉⠁⣀⣠⣤⡾⠗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⢀⠴⠊⠀⠀⡸⠉⠑⠦⡀⠀⠰⠿
⠀⠀⠀⠀⡟⣀⠤⠖⠋⠀⠀⣀⣀⢵⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⣠⠶⠣⣄
⠀⠀⠀⢀⡟⠁⣀⢤⠴⠒⠉⠀⠀⠄⡇⣰⡿⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣀⡧⣀⣀⠞⡄
⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⣀⣀⠤⠲⡅⠻⣷⡀⠀⠀⠀⠀⠰⠂⠀⠘⠀⠀⠀⠀⢰⣋⣀⣀⣀⣀⠤⠛⠢⡀⠀⢀⣄
⠀⠀⠀⠀⠈⠓⢤⠒⠊⡍⠁⠐⠆⣀⣡⣀⣼⢧⣀⣀⠀⠀⢰⠀⠠⠀⠄⠀⢠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⣥⡀⡘⠉
⠀⠀⠀⠀⠀⡰⢁⣷⡀⠀⠰⡶⠟⢉⠤⠤⠤⠔⠲⣄⠉⠓⢤⡀⠁⠈⠀⢀⡞⠉⠉⠉⠉⠉⠑⠒⠒⠒⠊⠁⢯⣀⣉⠑⢦⡀
⠀⠀⠀⠀⢠⡇⣼⣾⢹⢦⠏⣀⠤⠊⠀⣤⠀⠀⡦⠀⠉⢱⠀⠙⣄⠀⠀⠘⡤⠠⠚⠓⠖⠒⠶⠒⠢⠖⢦⢤⡾⠿⣿⣷⡄⢱⡀
⠀⠀⠀⠀⠘⡇⢹⣿⣈⡇⢸⠁⠀⠂⠀⠠⣀⣀⠄⠀⠈⠀⢉⠆⢺⡄⠀⠘⡇⠀⠀⠀⡢⠀⠀⢠⡄⠀⢀⠼⡉⢢⠘⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠘⣄⠙⢽⠐⣏⠀⠒⠀⠀⣠⠒⢦⡀⠈⠀⢀⠈⢢⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⢺⡍⢡⠛⢉⡍⠙⠻⣼
⠀⠀⠀⠀⠀⣠⡌⠓⢼⠀⡜⠀⠀⠀⠀⠳⣄⣤⣃⣡⠤⠤⠤⠓⠒⠛⠛⠛⠳⠤⠤⢤⣌⡓⠒⠁⠀⠈⢲⣟⠀⢀⠁⢀⣄⠘⡄
⠀⠀⠀⠀⠈⠛⠃⠀⠀⢧⠈⢂⠈⠀⠀⠈⡔⠉⣤⣒⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⠈⢑⡤⢚⣉⡛⢮⡦⣈⠒⠂⠁⣠⠏⠸⡇
⠀⠀⠀⠀⠀⢠⠲⠔⠱⢞⠓⠺⣳⣀⣀⠀⢛⠒⠬⢝⣛⣛⣻⣿⣿⣿⣿⣿⣟⣛⣛⡫⠥⠒⠋⠗⠉⣸⠉⢦⢱⠈⠑⠒⠚⠳⢤⠖⠁
⠀⠀⠀⠀⢠⡚⡀⢁⠎⠉⢀⠀⠈⢯⣀⣑⣻⡀⠀⠀⠀⠀⣀⠉⠉⠉⠉⠉⢀⡀⠀⠀⠀⠀⢸⠒⠓⠁⢀⠞⡸⠀⢸⡷⠂
⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠈⠀⡀⠘⡆⠀⠀⢧⠀⠀⠀⠀⠷⠃⠀⠀⠀⠀⠸⠞⠀⠀⠀⢀⣇⣀⠤⠔⢋⠔⠁
⠀⠀⠀⠀⠀⠘⣄⠀⠈⠃⠀⠔⠃⠀⡇⠀⠀⠀⠳⡀⠀⠀⠀⠳⢄⣀⣀⡤⠃⠀⠀⠀⣠⠞⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⠴⠚⠀⠀⠀⠀⠀⠈⠲⢤⣄⣀⡀⠀⠀⠀⣀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⣤⠶⠒⢦⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⢤⡀
⢸⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣾⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⡟⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⡇⠀⠀⠀⠀⠀⠀⠀⠔⠒⠤⡀⠀⣋⠂⠀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻
⣇⠀⠀⡇⠀⠀⠀⠀⡄⢘⡆⠈⠈⡀⡅⡘⢡⡌⠀⡇⠈⢉⠔⠺⠐⠹⡄⠞⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣿⠀⠈⠇⠀⠀⠀⠀⠇⠈⢀⡴⠃⣇⡁⠡⠤⢤⡀⡇⢠⣸⠀⠀⠀⠣⠴⠀⠈⡀⢀⢄⠀⠀⠀⠀⠈⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠈⠋⠀⠀⠀⠀⠀⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠀⠀⠈⠉⠉⠑⠒⠒⠙⠁⢀⣀⠀⣠⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠍⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⢸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⡇⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⢠⠀⢠⠆⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠸⣆⠃⠰⠋⠉⠁⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠲⠶⣶
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠠⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⢉⣁⣀⠀⣟
⠀⠀⠀⢸⠀⣇⠀⠀⠠⠌⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠠⠄⠤⠤⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠈⢧⣌⣓⣒⠂⠀⠀⠀⠤⠤⠤⠤⠤⠤⠀⠈⠉⠁⠀⠒⠒⢒⣤⣤⣒⣒⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣿
⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣓⣒⣒⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣴
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣙⣉⣉⣉⣙⣳⠀⠀⠀⠀⢸⣙⣉⣉⣉⣛⡃⠀⠀⠀⠀⢘⣛⣉⣉⣉⣋⡇⠀⠀⠀⠀⣞⣋⣉⣉⣉⣋⡆
⠀⠀⠀⢸⠀⠴⡆⣲⠂⢸⠀⠀⠀⠀⢸⢰⣲⣖⣖⡆⠁⠀⠀⠀⠀⠈⠠⠄⠤⠠⠄⡇⠀⠀⠀⠀⡇⠀⠠⠤⠤⠀⡇
⠀⠀⠀⠸⣀⣀⣁⣉⣀⡼⠀⠀⠀⠀⠸⣘⣚⣓⣓⣃⡄⠀⠀⠀⠀⢠⣘⣁⣃⣘⣀⠇⠀⠀⠀⠀⢇⣀⣈⣉⣉⣀⠇


⠀⠀⠀⢠⠴⠤⠤⠤⠴⢤⠀⠀⠀⠀⢠⠴⠤⠤⠴⢤⠀⠀⠀⠀⠀⢠⠤⠤⠤⠴⢤⠀⠀⠀⠀⠀⡤⠦⠤⠤⠤⠦⡄
⠀⠀⠀⢸⠒⣒⣒⣒⠒⢺⠀⠀⠀⠀⢸⠒⠒⠒⣒⣺⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⣻⠀⠀⠀⠀⠀⡗⢒⠒⢒⠒⠒⡇
⠀⠀⠀⢸⠀⠣⢌⠴⠁⢸⠀⠀⠀⠀⢸⣀⣀⣸⠰⠆⡃⠀⠀⠀⠀⢂⣀⣀⣎⠀⠀⡇⠀⠀⠀⠀⡇⠐⠐⠐⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⢸⣉⣉⣉⣉⣉⣽⠀⠀⠀⠀⢸⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⢨⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⣯⠭⠭⠭⠭⣽
⠀⠀⠀⢸⠀⠀⠆⡖⢲⠞⠀⠀⠀⠀⢸⠀⠀⢠⠊⡙⠆⠀⠀⠀⠀⡀⠀⠀⡰⠊⡙⡄⠀⠀⠀⠀⡇⠀⠀⡴⢊⡛⡄
⠀⠀⠀⠸⠤⠥⠥⠗⠁⠀⠀⠀⠀⠀⠘⠒⠒⠚⠤⠥⠃⠀⠀⠀⠀⠘⠒⠒⠓⠬⠵⠃⠀⠀⠀⠀⠑⠒⠒⠳⠬⠴⠁


⠀⠀⠀⢀⣰⣒⣒⣒⣒⣢⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⡴⠶⠶⠶⠶⠦
⠀⠀⠀⢸⢺⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠘⠀⢀⢀⡀⠀⡇⠀⠀⠀⠀⡏⠉⠉⠉⡡⣶⠄
⠀⠀⠀⢸⠘⠤⠤⠤⢤⠜⠀⠀⠀⠀⢸⠀⠘⠲⠃⠀⡀⠀⠀⠀⠀⢀⠀⠰⠛⠄⠀⡇⠀⠀⠀⠀⢧⡀⢠⣪⠖⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠙⠛⠛⠛⠃⣿⡄
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠉⡍⠀⠈⠅⠐⠀⡼⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡞⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡆⢈⡉⠀⠀⣠⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢦⣤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⠀⠀⠀⠰⠒⠒⠤⠀⣹⡃⠀⣀⣀⡤⡀⢀⣀⣀⣀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⠀⠀⠀⠀⢰⠀⣱⠀⢀⠁⢠⠘⠠⡦⠀⠃⠀⢡⠖⠚⠨⠙⠤⠃⢡⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠸⣀⣀⡴⠃⠠⡼⠈⠲⠲⠄⠄⠠⠼⠀⠀⠈⠂⢶⠀⢸⠀⡴⡄⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠸⠀⢨⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠀⠀⠀⠀⠈⠉⠉⠀⢠⠤⣀⠜⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⠃⡴⠊⠒⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒⠒⠒⠂⠤⠤⠤⠤⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠒⠒⠒⠒⠒⢒⣒⣒⣒⣒⡂⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⡀⠀⠀⢀⡐⠒⣀⠀⠀⢀⣀⣀⠀⠀⠀⠤⡤⠀⠀⠀⠀⠈⠉⠉⢉⣁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣑⣒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠉⠉⠀⠒⠒⠒⣀⣀⣀⣀⣒⣒⣂⣀⣀⣀⣀⠀⠀⠀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠤⠶⠶⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⣠⢠⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡇⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣴⣤⠈⣛⠠⣦
⠀⠀⣿⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⣀⣄⡀⢠⣟⣶⠀⠀⠀⢳⡜⢤⣍⠁⣠⣄⠀⠀⣆⣰⠀⣰⡄⢰⢒⠄⡖⡂⢀⣠⠀⠈⠀⣾⢿⡇⠟
⠀⠀⠁⢰⣿⡇⠀⠀⢀⣿⡇⠀⢰⡟⢻⣿⠀⠈⠀⠀⠀⠘⢮⡙⠆⠀⣼⡏⣿⡇⠐⠃⠃⠸⠋⠇⠘⠁⠰⠋⠀⢨⠃⠀⠀⢸⡟⢸⡇
⠀⠀⠀⢸⣿⠁⠀⠀⢸⣿⡇⠀⢸⣇⣸⡿⠀⠀⠀⣤⡀⠀⠀⠁⠀⢰⣿⢠⡿⠀⠀⠀⠓⠊⠒⠁⠀⠀⠢⠖⠢⠊⠀⠀⠀⣿⡇⣾⠁
⠀⠀⠀⢸⣿⠀⠀⢀⣿⣿⡇⠀⠈⢻⣿⣷⣶⣶⣶⡘⢃⣴⣶⣦⡀⢸⣿⢟⣴⣦⠀⠀⣠⣶⣶⡀⢁⣶⢀⣤⣦⠀⠀⣴⡿⣿⣿⠇
⠀⠀⠀⢸⣿⡄⠀⣼⡟⣿⡇⠀⢀⣾⠏⢸⣿⣠⡿⠁⣸⣿⣠⡿⠁⣿⣧⡟⣩⡿⠀⣸⣿⣠⡿⠃⣸⣿⡾⣻⡟⠀⣸⡿⠀⣿⡏⠀⣀
⠀⠀⠀⠀⢿⣷⣾⠟⠀⢹⣷⣶⡿⠋⠀⠸⣿⣭⣤⡾⠿⣿⣭⣤⡾⢿⡿⠸⣿⣤⣾⠿⣿⣯⣤⡾⢿⣿⠁⣿⣧⣾⢿⣷⣴⢿⣷⣾⠏
⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⣁⣀⡈⠁⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⠉⠀⠀⠉⠁⠀⠉⠁
⠀⠀⠀⠀⠶⣆⠛⠀⢃⠀⢠⠀⠀⠦⠀⠀⠀⠀⠀⠀⣀⠀⠰⣿⣿⣿⣿⡿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⣠⠀⠘⠃⠒⢠⡶
⠀⠀⠀⠉⠁⣈⡙⠶⠤⣦⠄⢤⡠⣤⠤⠖⠂⢀⣀⠀⠙⠆⠀⠀⠐⠀⠀⠓⠀⠀⠼⠁⣀⣄⠀⠐⠀⢤⡄⣠⠅⢤⠄⠰⢏⡉⠈⠃
⠀⠀⠀⠀⠀⠉⠀⠴⠈⠀⢠⠂⣤⠁⠀⠀⠀⠈⠛⠿⠶⠄⠀⠀⠀⠈⠀⠂⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⢡⡄⣤⠀⠀⠦⠀⠉
⠀⠀⠀⡤⣐⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠂⠠⡀
⠀⠀⠀⡜⠿⣿⣿⣿⣿⣿⡿⠿⠿⡿⣿⡿⡿⢿⠿⣿⢿⡿⡿⣿⡿⡿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠅⠔⠲⠰⠃⠒⢠⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠈⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡶⠒⠒⠒⠒⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠋⣉⠐⠄⢨⢄⠀⡠⠤⡔⡄⢠⣤⠄⢄⢠⡄⠀⣤⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠧⠃⢨⠠⠸⠀⠐⠓⠀⠃⢸⢰⠉⠉⢠⠘⠘⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⠠⠔⠃⠀⠶⠀⠉⠙⠦⠂⠘⠚⠀⠀⠀⠁⠃⠀⠃⠀⠶⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣀⠀⠤⠤⠤⡤⡄⢤⠄⠀⠀⠀⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑⠐⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡰⠁⡤⠤⠔⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠠⠤⠤⢀⣀⣀⣀⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⡘⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⡀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⠦⠀⠀⠀⠉⠉⢁⣀⠀⠉⢁⣐⣒⣒⠀⠤⢄⣀⣀⣉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⠤⠤⠤⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠃

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

⡏⠉⠉⠉⠛⠻⣿⡿⠁⠹⠟⠉⢹⣉⠏⠉⢉⡉⠋⣽⢿⠝⠋⠉⠉⠛⠻⣍⠩⠛⠉⡭⢝⣯⡉⣩⢟⢭⡿⢫⣍⣛⡿⣟⠿⡟⠙⡿⡍
⡇⠀⣀⣴⠁⠀⢉⣉⣉⣀⣀⣄⣀⣉⠀⠀⠐⣄⡞⠉⠉⢠⢉⣠⣈⣁⣄⣈⣅⠀⠈⠰⢴⡏⢉⣁⣍⣈⣁⣍⣉⣈⣅⣍⣌⠉⠇⠃⣀
⣷⠋⠁⠺⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢰⡅⡗⠀⠀⠀⠈⠀⠈⠀⠁⠁⠀⠀⠀⢼⢠⠇⠀⠁⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⣐⠫
⡏⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⢇
⡗⡡⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡙
⡇⠉⠉⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠙
⡏⠠⠆⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣄
⣇⠘⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠛⠛
⡏⠙⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⡐
⡇⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠟
⣧⠚⢚⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣛
⣧⡾⢫⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⣣⢿
⣧⡿⠛⣿⠆⠰⠆⠆⠀⢲⠖⠀⢰⠆⠀⢰⢌⣤⣻⣶⠶⢶⠆⢶⢲⡆⠖⢶⠆⢶⡦⣉⠟⡓⡶⠀⠰⣆⡄⠰⢆⠀⠠⠒⡆⠰⣿⣦⣼
⢩⣷⢞⣹⠀⠀⠃⠲⠲⠶⠲⠂⠗⠖⠒⠀⠀⢹⡇⠀⠀⠀⠖⠶⠐⠖⠶⠲⠀⠀⠀⢸⣀⡇⠀⠸⠲⠒⠒⠶⠖⠖⠶⠒⠀⠀⠙⢣⣼
⣿⡍⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣅⣀
⡦⡅⢸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃
⣷⠌⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀
⣧⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠑⣌
⡆⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⡇⠀⠀⢀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠋⡠⠇
⡷⣤⣀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢬⠇⠀⠀⠈⠈⠉⠀⠁⠉⠈⠀⠀⠠⡯⢀⡈
⣇⣀⣙⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠍
⡇⢹⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠉
⡇⠦⢥⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣨
⡏⠉⠛⣿⣀⡀⣀⣀⢀⡀⣀⢀⣀⣀⢀⣀⣠⠦⡇⣀⠀⢀⡀⠀⢀⠀⠀⣀⠀⣀⣀⣨⠻⠄⠀⡀⣀⣀⡀⣀⠀⣀⣀⣀⠀⣀⡖⠛⠉
⡇⠘⣯⣐⣸⣮⣽⣿⣌⡛⢮⣼⣷⣫⠞⠘⢻⣗⣗⣁⣀⣌⡙⣦⣄⣁⢀⣁⣠⣣⡾⣁⣔⡂⢀⢠⠊⣃⠀⢠⣤⠀⣼⣿⣦⣄⣀
⡿⣿⢿⢿⢿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡄⣿⠀⣤⣤⣤⣤⣤⣤⢸⣿⢠⣄⡀⠀⠀⠀⢸⣿⣿⣿⡿⠟⣿⠀⣿⠛⠛⠛⠛⠛⠛⡇⠀⡟⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣧⣭⣼⣿⣿⣿⣿⣿⣿⣦⣭⣼⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠛⠶⠃⠀⢸⣿⣿⣿
⠀⠀⠀⢸⡏⠉⠉⢩⣭⡍⣭⣭⢩⣭⡍⣭⣭⢹⡇⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⡟⠛⣻⠛⠛⡟⠛⢻⠛⠛⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣀⣀⢘⣛⡃⣛⣛⢘⣛⡃⣛⣛⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠿⢿⡷⠤⣿⠦⠴⡷⠤⢾⠦⠴⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⠿⠿⠸⠿⠇⠿⠿⠸⠿⠇⠿⠿⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⣀⣸⣇⣀⣽⣀⣠⣇⣀⣼⣀⣠⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⢸⡇⠀⣻⠀⢀⡇⠀⣸⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⢸⣿⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢞⢒⡂⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⢻⠦⢄
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣼⡇⣸⡇⣰⣿⣦⡀⣿⠀⣠⣰⡶⠒⠂⠀⠀⠀⠀⢃
⠀⠀⠀⠠⠐⠀⠁⠀⠀⠀⠀⠂⠀⢠⣿⠉⣿⣇⣿⣯⣿⣧⣿⣾⡿⢸⣟⠂⠀⠀⠀⠀⠀⠈⠀⠐⢄
⠀⠀⠀⠁⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠛⠀⡛⡋⣛⣁⣙⡋⠙⠛⠀⠘⠛⠒⠀⠀⠁⠀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠂⠀⣀⡀⠀⣮⣭⣾⣾⣿⣿⣶⣾⣭⣭⣐⣒⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠄⠀⢀⣩⠅⣼⡿⠿⠿⠿⠿⠾⢿⣿⣿⣿⣏⢐⣊⡁⠀⠈⢀⠤
⠀⠀⠀⡀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⡠⠉⡙⠻⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⣼⣯⠙⣷⡄⣼⣿⠓⠂⠀⣴⣷⣄⠀⠀⠘⢛⣿⠃
⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿⠁⣀⣀⡀⣿⣿⡴⠟⢡⣿⣿⣤⡄⢰⣿⣿⢹⡄⠀⠀⢸⣿⠀⣀⣀⠀⠄
⠀⠀⠐⠀⠀⠀⠀⢸⣿⣿⣠⣤⣾⣧⣿⣿⡿⣦⣐⣿⣿⡀⠀⢸⣿⡿⠂⢿⡀⠀⣾⣿⡄⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠉⠙⠋⠉⠘⢋⠙⠛⢡⠀⠁⠉⠉⠉⠉⠈⠛⡁⠀⠈⠁⡀⠙⠛⠁⠀⠀⡀⠈
⠀⠀⠀⠀⠎⠀⠀⢀⠀⠀⢀⡄⠀⠀⢀⡀⢀⣀⠀⡀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⢘⠀⡠⡄⠀⠀⠀⢸⣷⣠⢆⣼⠇⣾⠅⢸⡇⢰⡷⠂⣾⠁⠀⣷⣠⡇⣿⢹⡆⠀⠀⡤
⠀⠀⠀⠀⠲⠒⠀⠀⠀⠀⠘⠿⠃⠘⠋⠀⠛⠂⠻⠧⠸⠟⠆⠿⠤⠸⡿⠻⠇⢿⠼⠃⠀⠀⠑⠗⠁
⠀⠀⠀⠀⠀⠢⣠⠖⠀⠀⠀⠀⠀⠀⢔⠀⢀⡀⠀⡀⠀⡀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠐⣆⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠁⠩⠄⠊⠐⠗⠈⠂⠜⠉⠀⠀⠀⠀⠈⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣁⡀⠀⠠⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢀⣀⡀⣀⠀⢠⣄⢀⡀
⠀⠀⣠⣴⣦⣤⣤⣤⠒⢲⣾⣿⡿⣽⡟⣧⣎⡏⢹⣞⠉⡏⠉⠉⠉⠉⠉⠉⢹⠀⡜⠙⠁⢀⣼⠁⢀⢹⠁⣾⣧
⠀⢠⣿⡿⣿⡯⢿⣔⣶⣞⣏⣟⣿⠿⡟⡯⠥⣭⠟⢿⠒⣇⣀⠀⢠⠨⠅⡀⠀⠙⠁⠀⡄⠏⠘⣦⣉⣼⠀⣿⡟⡆
⠀⢸⣿⡗⢻⡻⣿⣿⣿⡇⡤⠄⣀⣶⣬⣴⠶⣒⢲⢸⣷⣯⣿⠄⢸⠤⢤⠐⠀⠊⠉⠓⡓⠔⣀⣿⣿⣿⠀⣿⣿⣽⡀
⠀⢸⣿⠀⡏⠀⠛⠛⠋⠀⢧⢸⢹⣯⠥⠒⢲⢸⠰⢻⣿⣿⢿⣠⡎⠄⠈⠀⢘⠀⠈⠀⠁⣰⠓⢿⣿⣿⠀⣿⣿⣿⣿⡄
⠀⢸⣿⡇⢇⠀⢰⣒⡓⣲⢲⠀⠸⣼⠥⠴⠖⢚⣔⣞⣿⢿⡞⠑⢺⣿⡟⣃⣤⡅⢨⠀⡍⠉⢁⣾⡿⢻⣶⣾⢿⠟⠁
⠀⠸⣿⣇⣤⢲⠉⠉⡹⠛⡟⠛⠒⠰⠲⣶⢸⣿⣼⣿⡟⠋⣷⠰⡄⣜⣿⣿⣿⣿⠋⢸⣿⡄⡿⠉⣦⠈⠉⢻⢸
⠀⠀⠀⣿⣽⣾⢀⡴⠁⢰⠁⣿⡇⠀⣠⡟⣼⣿⣛⡋⢹⡤⠋⡎⢿⣿⣿⣿⣿⣿⣤⣿⣿⣿⠷⠶⢾⣤⠀⢸⣶
⠀⠀⣶⣿⣷⢿⣍⡠⣤⣼⡴⠟⠧⢬⡥⢿⣚⣁⠭⣅⣰⣤⡾⠇⠄⠦⢼⡿⢿⣿⣿⣿⣿⣯⠀⠀⢸⡟⠀⢸⠛
⠀⠀⣰⣿⣿⣿⣿⣿⡿⢿⣇⣩⢢⠀⠁⣠⢿⣿⣿⣿⣿⣗⢠⡄⣀⣀⠀⠄⠚⠛⡿⣿⢹⣶⣾⡅⣾⡧⠤⢸⡀
⠀⠀⣿⣿⣿⡈⣿⣿⣇⡼⠉⠉⣿⢇⡼⠛⢉⣍⣽⣿⣿⠁⣸⠀⠀⣧⣭⣭⣿⣿⣿⣿⣿⣿⡿⠜⠿⡆⡔⠀⣇
⠀⠀⣿⣿⡇⡇⣿⣿⣏⠇⠀⢸⣿⣟⣂⣛⣲⣿⣟⣻⠟⢠⡿⡀⠰⠿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⣅⣧⡷⣷⣸⡀
⠀⠀⣿⡏⢠⣇⣜⣛⣽⠀⠀⣼⢻⣯⣽⣿⡶⢻⣯⣿⠶⠟⠀⡃⣌⣠⣭⠿⣿⡿⣿⣿⣿⣿⣿⣶⣿⠍⣷⣿⣧⣇
⠀⠀⡿⢐⡿⣚⡿⢿⡇⠀⢠⡟⠛⣳⢾⣷⠶⣻⣧⡅⠒⠂⢠⣇⣽⠁⠀⠀⣿⠃⣀⠈⢻⣿⣿⣿⣿⣇⣿⣿⣿⣿⠄
⠀⠀⡿⣿⣽⣛⣛⣽⣀⣀⣸⠀⠀⡇⣿⡿⡏⠙⣄⣿⡀⠾⠛⠉⠁⠀⠀⠀⠛⠁⢶⠀⣎⡛⠛⠛⠟⢁⡤⢺⢯⡀
⠀⠀⠐⠓⠲⠿⠯⠟⠘⠛⠛⠒⠺⠞⢏⠉⢻⡄⢸⣏⢁⣶⠀⠄⠀⠀⠀⠀⠀⠀⢸⠹⡄⢄⠀⠀⠀⠀⠒⢬⣭⣁⡄
⠀⠀⠀⠀⠀⠐⠒⠦⠤⠼⠂⠀⠈⠀⠀⠀⠁⠐⠒⠚⠀⡜⣀⠂⠀⠀⠀⠀⠀⠀⠈⠆⠸⠀⠀⠀⠀⠀⠀⠸⠾⠻⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋⣁⣠⣤⣤⣄⣈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡇
⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⠀⣸⣿⣿⣿⣿⣿⠟⢀⣴⠟⢉⣠⣤⣤⣈⡉⠻⣦⡀⠻⣿⣿⣿⣿⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠏⢠⠟⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠻⡄⠹⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠻⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠀⢸⣿⣿⣿⣿⣶⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣶⣿⣿⣿⣿⡇⠀⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢸⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⣿⣿⣿⣿⣿⡇⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⠿⠿⠿⠿⠛⢁⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢁⣤⠶⠛⠛⠛⠳⠶⣤⡈⠻⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡟⢁⡴⠋⣠⣴⣾⣿⣿⣷⣦⣌⠙⣦⠈⢿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠁⣼⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⣧⠈⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣀⡁⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⣀⣿⣿⣿⣿⡇⠀⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⢸⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⡇⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⠟⢁⣼⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣈⡉⢉⣉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⢉⣀⣤⣤⣤⣤⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡇
⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢹⣿⣿⣿⣿⣿⠏⢠⡶⠛⢉⣠⣤⣤⣄⡉⠛⢦⡄⠹⣿⣿⣿⣿⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠃⢰⠏⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠹⡄⠸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⢸⣿⣿⣿⣿⣶⣆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢰⣶⣿⣿⣿⣿⡇⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢸⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⣿⣿⣿⣿⣿⣿⡇⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⠻⠿⠿⠟⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⠉⠉⡉⢉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⠆⡾⣴⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢰⡟⣸⣃⡄⢀⣄⡄⣀⣸⣀⢀⣀⣀⡄⠀⠀⠀⠀⠀⠀⢤⡀⢀⣴⠂⠀⢀⡞⠀⢀⠀⠀⠀⠀⢰⡆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⡠⢀⡟⠁⠛⠃⠀⠚⠙⠙⢱⠟⠃⠋⠘⡾⠁⠀⠀⠀⠀⠀⣴⠋⠃⣾⠃⠀⠀⣼⣁⢀⣈⢢⣼⡇⣀⣽⣀⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡾⠁⠀⢠⠤⠶⣖⠊⠉⡟⠰⣏⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣴⠀⠀⠃⠃⠛⠛⠘⠘⡇⠃⠋⠃⠃⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇⣠⠔⠂⠀⠀⠀⠃⠀⣿⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠂⠀⠀⠀⠀⠀⣠⠏⠀⢀⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣿⡅⠀⠀⢀⡷⠀⠐⢇⠀⠀⢰⠃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣇⡆⣾⢼⢁⡶⡶⢠⣷⡞⡿⠖⡼⢰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢇⡾⠁⢡⡟⠀⠀⠀⠀⠀⠠⡀⠇⢀⠄⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡏⠉⠀⠀⠀⠀⣄⠀⢹⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⡆⢀⡠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠑⠀⠀⠀⠀⣀⠾⢀⡾⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠈⣽⡇⢏⢀⡀⡀⣼⢋⣀⡀⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠠⠆⢰⡗⠀⠀⠠⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢢⡿⠀⠞⠼⠹⠣⠏⠾⠸⠹⢣⡇
⠀⠀⠀⠀⠀⠀⠀⣦⢘⣿⣼⣸⢢⢇⣶⡞⣼⣱⢢⢧⡇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠞⠁⠀⠀⠀⠀⠀⠀⡶⠦⣀⡜
⠀⠀⠀⠀⠀⠀⠐⠋⣸⠏⠉⠉⡈⠀⠀⠁⠉⠉⡈⢈⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⠉⠑⠒
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⡀⢘⡧⣀⠀⠀⠀⠀⠼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡡⠟⠀⠈⠑⠦⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⣦⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠁⡼⠁⠀⢠⠏⠀⠔⠀⡴⠀⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⠇⠀⠀⡷⢠⠆⡿⡼⠰⣶⠀⣿⡼⢸⠵⢰⠺⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠓⢠⡞⠀⠀⠀⠀⠀⠄⢠⡆⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠜⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⡄⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣠⠀⠀⣠⢠⢤⡆⢸⣇⡤⣾⣡⢢⢄⣆⡄⠀⠀⠀⠀⠀⡴⢃⠀⠀⠀⠀⠈⠁⠀⠁⠀⣼⠀⢁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠗⠀⠀⠉⠉⠈⠁⠈⣿⠁⠁⠉⠈⡀⣹⠃⠀⠀⠀⠀⠘⢧⡘⠂⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡎⠀⢠⠀⢈⡅⠀⢙⣿⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡈⠁⢈⣀⣠⠟⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣫⣝⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣽⣭⣭⣽⣿

⠀⠀⠀⠀⠀⠄⠄⠠⠄⠀⣀⢀⣀⣀⣀⣀⣀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠳⢿⣿⣸⣿⣿⣷⣶⣾⣷⢤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢻⣷⡆⢛⠳⢾⣙⣓⠾⣽⡆⣤⣦⣄⣸⠲⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢭⣿⣤⡽⢍⢽⣿⣖⡄⠀⢀⣤⣤⠝⠿⢿⣤⣍⣙⠛⠒⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣩⣿⣇⣈⣛⠹⠿⢾⠒⢄⡀⠤⠿⣛⢖⣈⣉⣻⣻⡆⡋⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠀⠈⢆⡙⠿⠿⠯⠑⠀⠐⠂⠶⣶⠄⠄⠀⠚⢛⣛⣢⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠄⠀⠰⠄⠄⠲⠗⠳⣤⠤⠤⠌⠘⠛⠛⢻⠿⡿⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣛⣓⠒⠒⠒⠒⠂⠛⡛⣛⣠⣤⡻⠿⠿⠿⠧⠤⠤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠦⠤⠤⠤⠼⠎⠁⠉⠉⠉⠁⠶⠶⠶⠶⠦⠒⠶⠶
⣶⡶⢶⡶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣾⣿⣿⣿⣷⣶⣶⣶⣷

⠀⠀⠀⠀⠀⠐⠀⢠⣤⣤⣤⢤⣤⢤⠀⠀⡆⠐⠀⠠⢤⣤⣤⣤⣤⡤⡄⠀⠂⢰⢠⣤⡄⡤⡤⡤⣄⣤⣤⢤⠀⠀⠀⠠
⠀⠂⢠⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡄⠐
⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠁
⠀⠀⢠⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⡤⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠡⢤
⠀⠑⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠊
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐
⠀⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⠉
⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⢐⡆⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠌⢰⡂
⠀⠀⢀⡅⠀⢀⣀⣁⣒⡒⡓⠒⠒⠒⠛⢀⡁⢸⣀⣀⣚⡚⡐⠓⠓⠃⠀⠀⡁⢸⣀⣒⣚⡈⡒⠚⠚⠚⠚⠀⡀⠀⢨⡀
⠀⠀⠘⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁
⠀⡀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠓⠒⠛⠚⠚⠚⠂⠀⡀⠀⠸⠇
⠀⠃⢠⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈
⠀⡆⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠄
⠀⡄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠁⠐⠇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠉⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⠖⠲⡄⠀⠀⠀⠀⠙⢦⣀⠀⣀⡤⠴⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡼⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⣃⣀⣀⡀⠀⢀⣤⠶⢦⣄⠀⠀⣠⣤⣭⣍⠉⠉⢻⡗⠲⣄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⠈⠉⠉⢻⢰⡿⠀⠀⠀⢘⡇⣸⠋⠀⠀⠉⣷⠀⢸⡇⠀⠈⡆⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⢻⣆⡀⣀⡿⠈⢿⣄⣀⣠⣾⠇⠸⣧⣀⣀⣠⡿⠀⢸⡇⠀⢀⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠐⠒⠚⠛⠛⠉⡞
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⣶⣄⡀⠀⣠⣴⢿⠀⠀⠀⠀⣠⠔⠚⠛⠛⠒⢤⠀⠀⠀⢀⣠⣤⣤⣀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⠙⢿⣿⠟⠁⢸⡂⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⣴⠟⠁⠀⠀⠈⠓⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡀⠘⠛⠛⠛⠛⢲⡆⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠈⠃⠀⠀⣠⣤⣤⣬⡁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠻⣶⣤⣤⣤⣤⡾⠃⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣾⠋⠀⠀⠀⢻⡆⠀⠘⠦⣄⡀⠀⠀⣀⠜⠁⠀⢀⡀⢀⣬⣍⡉⠁⠀⠀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠻⣦⣀⣀⣠⡾⢃⣀⡀⠀⢀⣉⣉⣍⡁⠀⢙⠀⢸⡿⠋⠁⢸⡇⠀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⢉⣩⠞⠋⠙⣿⠀⢸⡏⠀⠈⣿⠀⣿⠀⢸⡇⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⡟⠁⠀⣀⣠⡿⠀⢸⠀⠀⠀⣿⠀⣿⠀⢸⡇⠀⠀⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⡟⠛⠛⠻⣯⠀⠀⣿⠀⠀⢀⣿⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠹⠦⣤⣤⡄⠀⠸⡇⠀⡟⠀⢸⡇⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢄⣀⠁⠀⠈⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠀⠀⡠⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠤⠆⠐⡈⠀⠸⠐⣁⠂⢐⡄⡈⠪⠀⠀⠀⠀⣴⠀⣀⡈⣕⡠⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣅⠇⢈⠆⡤⠄⢁⡅⠀⢨⡈⢀⠤⡀⠉⡂⢠⠚⣉⠑⣠⣚⣬⡘⣄⢉⠔⠢⡀⠞
⠀⠀⠀⠀⠀⠀⣀⠤⣘⠖⠋⠉⠉⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠁⠁⠈⠀⠀⠁⠀⠀⠀⠈⠁⠃⣔⡒⢄
⠀⠀⠀⠤⡀⢰⠤⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠜⠠⡄⣤
⠀⠀⠀⠈⠉⣈⡒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⢦⡀
⠀⠀⠀⠠⡚⢃⠼⡆⠀⠀⠰⡁⢟⢢⠯⢱⠩⡩⠝⡏⢇⠏⢟⠴⢑⠈⣬⠑⣄⠷⢱⢍⢊⠜⠀⠀⠀⣒⠄⠶⡀⣀
⠀⠀⠀⠀⠥⢌⡸⡇⠀⠀⠪⠺⠚⠦⠙⠴⠀⠖⠀⠘⠠⠊⠘⠶⠴⠐⠬⠔⠧⠋⠲⠐⠆⠀⠀⠀⠀⠤⠖⠒⠙⠊
⠀⠀⡠⢒⠀⢠⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⢆⠈⠵
⠀⠀⠉⠁⠨⡀⠨⡉⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣊⡴⠃
⠀⠀⠀⠀⠀⠣⠐⠤⠒⢉⠖⢢⡤⠤⠤⠤⢤⣤⠤⠤⢤⠤⡤⠤⠄⠤⠠⠤⣤⢠⠀⠤⠐⣖⠎⡱⠀⠠⣂⠆
⠀⠀⠀⠀⠲⠶⠂⠀⡖⠄⠩⡠⠤⠏⠀⠈⠲⣀⠂⣀⠘⡆⠈⢉⣀⠇⢀⡉⠒⠁⠀⠈⠣⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠶⠝⠀⠉⠑⠔⢡⠄⠡⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⢰⣶⣶⣶⠀⠀⣶⣀⣶⡄⠀⢰⡆⠀⢰⣦⣀⣶⠀⠀⣶⣴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢁⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠛⣿⡇⠀⢸⡇⠀⢸⡏⠻⣿⠀⠀⣿⡿⣶
⠀⠀⠘⢢⠂⠐⠒⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠁⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡶⠿⠆⢰⣄⠀⠀⠀⡔⠒⠦⣄
⠀⠀⠀⠱⠌⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡆⢀⡞⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠚⠀⠀⠀⠿⠇⠀⠠⠶⠋⠀⠈⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠁
⠀⠒⠀⣠⡀⠐⠀⠀⢠⠀⠐⠆⠀⠀⡇⠀⢸⠀⠀⡇⠀⠲⣤⠁⢸⠀⠀⡇⠀⢰⡇⠀⢸⠀⠐⠒⠂⡏⠀⠒⣆⠄⡏⠀⠀⣤⠄⠀⠀⠀⢰⠆
⡀⠀⠀⠈⠀⠀⠉⠀⢸⠀⢀⡐⠂⠀⡇⠀⢸⠀⠀⠙⠦⡀⠀⠀⢸⠀⠀⡇⠀⡄⠁⠀⢸⠀⢀⡀⠀⠈⠢⣄⠀⠀⠘⠢⣄⠀⠀⠀⢀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠤⠀⠀⢸⠆⠀⠃⡀⠀⡃⠀⠘⠀⠀⡆⠀⠘⠀⠀⢸⠀⠀⠃⠀⢳⠀⠀⢸⡄⠀⠀⠄⢆⠀⠘⠁⡀⢆⠀⠘⠁⡀⠀⠙⠃⠀⠀⠛
⠘⣇⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠛⠛⠋⠀⠈⠉⠋⠉⠀⠈⠒⠀⠈⠀⠀⠉⠉⠈⠛⠚⠚⠁⠈⠉⠉⠉⠀⠈⠉⠋⠉⠀⠀⣠⠶⢄⠀⣦
⠀⡟⠉⠀⠀⠙⠣⠄⠀⠄⠒⠛⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠄⠀⢠⠀⡀⠀⠀⠀⠀⠀⡴⠊⠀⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠄⢀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠆⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⡹⠖⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠞⠋⠰⠆⠀⠀⠀⠀⡇⠁⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡
⠀⠀⢀⢀⡀⢀⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⢁⠀⠀⠀⠒⠀⠀⠀⠀⡇⠸⠁⢸⠀⢀⠡⠀⠠⠤⠇⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⢈⣑⣒⣛⣆⡌⠀⠀⠀⠁⣤⡄⠀⠀⠀⣶⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠈⠁⡇⠀⡆⠀⠀⢸⠀⣀⣀⠀⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠨⠥⠭⠙⠛⣁⠀⠀⠀⠀⢻⡆⢠⠔⠀⣿⠀⣴⣲⣄⠤⢤⡆⠀⢠⠀⡀⢐⡇⠀⠇⢀⠀⢸⠀⡇⠸⠀⠀⠀⠻⠟⠁⠀⠀⠂⠂
⣀⣀⠒⢀⠀⣀⢀⣀⡀⣀⢀⣀⡘⠃⣈⣛⣁⣘⡀⡙⢋⣈⢒⢋⢛⢀⡐⠘⠃⢘⣃⢈⡀⡀⠀⢚⠐⡓⠘⣀⣀⡀⣀⡀⡀⢀⣀⣀⣀⡀⠀⡀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡴⠒⠒⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⣶⠀⠀⡀⣀⠀⣀⠀⠀⣀⡀⠀⡀⡀⠀⣀⡀⠐⠂⡀⣀⡀⠀⣀⡀⡀⢰
⠀⠀⠸⡇⠀⢲⠠⡏⠀⣷⢸⠁⢸⡆⡏⠀⣿⠀⠀⡏⠈⡏⠈⡇⣾⠁⢹⡆⡏⠀⡟⠀⣷⢸⡇⣿⠀⣿⢸⡁⢈⡇⠸
⠀⠀⠀⠉⠓⠚⠀⠙⠚⠁⠈⠓⠋⠀⠙⠊⠉⠀⠀⠁⠀⠃⠀⠃⠈⠓⠋⠀⠃⠀⠁⠀⠉⠈⠁⠉⠀⠉⠀⣙⣩⠇⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⣾⣶⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⢠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠿⠗⠈⠹⣿⣿⣧
⠀⣴⣿⣷⡄⣠⣿⣿⣿⣷⣄⣀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣾⣿⣿⣶⣄⡀⠀⠀⢠⣿⣿⣿⣤⣾
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣦⣄⣀⣼⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡛⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣭⡟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡘⣱⡟⣩⣮⡙⣿⢹⣿⢹⣿⢩⣽⣿⡿⢩⣭⣿⢋⣽⡍⢿⡄⣽⡅⣽⠏⣽⡍⣿⢫⣯⠙⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⡻⠿⣣⣿⡸⢿⣸⣿⢸⣿⣿⣧⡙⠿⣿⣘⠿⢃⣾⡇⣿⡇⣿⣄⠿⠿⣿⡘⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠂⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠠⠤⠶⠶⠴⠀⠀⢸⣀⣀⣄⣤⡄⠀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣤⢤⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠙⠶⠤⣀⣀⣠⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠐⠒⠒⠒⠒⠚
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⠂⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠄⠀⠀⠒⠐⠐⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⡇⠀⠀⠀⢀⡀
⠀⢠⠄⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠁⠀⠁⠀⠀⡀⠀⠀⡁⠀⠀⠀⠈⠃⠀⠀⠀⠘⠓
⠀⢘⡀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠇⣀⣀⣀⢘⣃⡀⠀⢀
⠀⠈⡇⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠙⠃⠀⠀⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉
⠀⠀⢧⠀⠠⠤⠼⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠈⠀⠀⠀⡆⡀⠀⠀⠐⡦⠀⠸⠄⠀⠀⣀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠙⢧⡀⠀⠀⠈
⠀⠀⣄⡀⠀⠀⠀⣠⣴⠝⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢻⣆⢀⣠⡴⠞
⠀⠀⠙⠃⠉⠲⠛⠈⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⡠⠤⠤⠤⠀⠀⠀⠘⠒⠂⠐⠒⠓⠚⠙⠛⠃⠀⠀⠀⠀⠉⠋⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠈⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢡⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠓⢒⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠎⢀⡎⠦⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸⠀⡜⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠇⣸⠁⢠⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡼⢠⠃⢀⡞⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⢃⠎⠀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣞⡞⠀⣸⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⡟⢀⢠⠃⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡟⠀⢀⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠃⡾⠁⠀⡌⢠⠃
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠴⠚⠋⠀⢸⠁⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠖⠒⠒⠒⣚⣉⣉⣉⣉⣉⠉⠥⠤⠤⠤⠐⣲⠆⠉⠀⡰⢡⠇
⠀⠀⠀⠀⣴⡵⠚⠑⠒⠒⠒⠀⠀⠰⠒⠖⠀⠉⠩⠅⠀⠀⠐⠂⠀⠈⠉⠉⠉⠉⠉⠍⡍⠀⠀⣰⢃⡟
⠀⠀⠀⠸⡌⡿⠈⠀⠀⠒⡶⣶⠒⠒⠶⠾⠤⠤⠄⠒⣒⣒⣒⣒⠒⠒⠒⠒⠒⠀⠊⡴⠀⠀⢠⠃⡼
⠀⠀⠀⠀⠙⠪⢤⣦⣠⣠⠁⣇⣀⣀⣉⡉⠉⠉⠡⠄⢀⣀⠐⠒⠒⠚⠉⠉⠛⢒⣚⠓⠀⢀⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠚⠒⠒⠦⠭⠭⣉⣉⠒⠒⠀⠤⠬⣅⣀⣀⠀⠀⠀⠀⠀⢀⠞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠶⠤⠭⠭⣍⣉⣑⣉⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠎⢀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠢⠉⠅⠀⠀⠀⠀⢠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡘⢂⡕⠀⠀⠀⠼⡀⢀⠷⠀⠀⣀⣀⣀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠒⠋⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⡠⠖⠋⢩⣿⣿⢧⡀⠀⠹⠿⠟⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⡼⠁⠀⣰⣿⣿⣿⠀⢹⣦⠄⡀⠀⠀⠀⠀⡜⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣼⣡⣤⣾⣿⣿⣿⡏⠀⠇⣸⡧⢒⡀⠀⠀⠘⠓⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡹⢧⡀⠈⠻⣿⣿⡇⢀⣿⠧⣒⠥⣊⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡩⢔⡡⢛⠶⣄⣈⣻⣧⣾⠅⡊⠔⡪⠔⠀⠀⠀⠀⠀⠀⢰⡏⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣶⣿⣔⡢⠄⡀⢀⠀⢀⡠⢒⠥⢒⠥⣂⠥⣊⠅⡋⠕⡫⠔⡪⠔⡩⢐⡡⠀⠀⠀⠀⠀⠀⠘⡇⢰⣄⣳⡔⠊
⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣌⢒⢡⡶⠥⠶⠥⣊⠵⠾⠵⣎⠔⣨⠴⠯⢶⣩⡶⠦⠶⠥⢆⠀⠀⠀⠀⠀⠀⢀⡹⡄⠻⠇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢿⣿⣿⣧⣿⣿⣿⣷⠎⣴⠛⠛⠲⢀⡶⠛⠳⡌⠛⣡⠞⠛⢦⡉⢰⡟⠛⠳⣆⢳⡀⠀⠀⠀⠒⠁⠀⠘⢆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡿⢰⡇⠀⠀⠀⣸⠁⠀⠀⢻⠀⡟⠰⠷⠀⣧⢸⡇⠀⠀⢹⡆⣷⢀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠃
⠀⠀⠀⠀⠀⢀⡴⠇⠀⢿⣿⣿⣿⠟⠿⣿⣿⠛⣿⡇⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⡇⠐⠒⠀⣿⢸⡇⠀⠀⢸⡇⣿⠔⣊⠔⡢⠄⡀
⠀⠀⠀⠀⠰⠛⠀⠀⠀⠸⣿⣿⣷⣀⣰⣿⣿⣿⣿⣧⠸⣇⠀⠶⣶⢿⡀⠀⠀⣸⠀⣧⠘⠛⢀⡟⢸⡇⠀⠀⢸⡇⣟⡨⢔⡩⢖⡡⢂⠡⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠲⠶⢋⣄⡓⠶⠞⣡⣶⣌⠳⠶⠛⣡⡘⠷⠶⠶⢋⣴⡯⢂⠡⢒⠥⢊⠅⡊⠔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢙⡿⢿⣿⣿⠿⣿⣿⡿⠿⣿⣿⣿⠿⠿⢿⣿⣿⠿⣿⣽⡿⢿⣿⢿⣿⢿⣿⣿⠿⣆⠥⣊⡥⠎⢴⣊⢔⡨⢔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣆⠹⢁⣶⠈⢡⡶⠲⣦⠙⢠⡖⠓⠲⣮⠁⣶⠸⣿⢰⡆⠀⡆⢠⣆⢹⡇⢰⠈⡿⢡⡖⡒⠢⠙⡯⢒⠥⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⣿⢸⡿⣦⡞⣿⠀⡟⠀⠀⠘⡇⢸⡇⠀⠀⢸⡆⣿⢧⠉⢸⡇⠀⡇⠰⡿⣆⠁⢸⠀⢁⡟⢰⠭⢳⠶⢋⠅⣊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢈⡀⣿⢠⡇⠀⡂⠀⣿⢸⣧⣤⣤⠟⠀⣿⠈⢧⣸⡇⠀⡇⠀⡇⠘⣆⣺⠀⢸⡇⣿⣊⣕⣊⠔⡨⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⢿⡇⣿⠀⣇⠀⠅⠀⡟⢸⡇⠈⢷⡀⠀⣿⢠⡌⢻⡇⠀⡇⠀⡇⢠⡘⣿⠀⠸⣇⠉⠤⣤⡌⡏⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠐⡦⠤⣄⡼⢦⣠⡄⢸⠸⠇⣸⡅⠿⢠⡙⠦⡤⠞⣁⠸⠇⣿⡈⠷⠀⠿⢀⣿⠸⠇⠀⠇⠀⠇⢸⡇⠿⢈⣧⡙⠦⡤⠞⣰⠃⠀⠀⣴⠞⠁
⠀⠀⠀⠀⠀⣱⡤⢶⠇⠀⠙⢧⠈⠑⠚⠉⢻⣶⣿⣿⣷⣶⣾⣿⣷⣾⣿⣿⣶⣿⣶⣾⣿⠷⠚⠓⠲⢛⠶⣿⣿⣶⣾⣿⣿⠓⠒⠊⠁⢀⣰⠟⠁
⠀⠀⠀⠀⠈⠙⢦⡼⢀⣠⣴⠚⠁⠀⠀⠀⢘⣿⣿⣿⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠣⠤⠄⣊⠴⣋⠜⠀⠉⠛⠛⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠰⢻⡉⡗⠞⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⠟⠃⠤⢭⠩⠭⠤⠚⠁⣠⡶⠀⠀⣠⠄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠻⢿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠞⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

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

⠘⣿⠠⠼⡦⣤⣬⣴⠃⡘⢧⠇⣠⡼⢿⣷⣄⡀⠁⢄⠞⡛⣗⢯⡤⣒⢈⠿⢬⠷⠞⢷⠘⣿⠷⠊⠠⠄
⠀⠤⢤⣬⠟⡄⣂⢘⠀⢀⢂⠀⢈⣦⡌⠻⠘⠻⠟⠊⢸⠿⠀⠈⢓⠤⠤⠔⠋⠞⠁⠀⡇⠈⠻⣧⡤
⠀⠀⠀⠂⠙⠰⢲⡕⠓⠈⠭⠑⠠⠜⠃⠈⠰⠒⠘⠀⣐⠳⠴⠙⠈⠳⠆⠀⠀⠸⣥⠔⡱
⠀⠀⠀⠀⠈⠈⠘⠋⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀
⠀⠀⠠⠀⠈⠉⠉⠉⠉⠉⠁⠀⠄⠸⠀⠀⠉⠉⠉⠉⠉⠉⠀⢠⠀⡇⠉⠉⠉⠉⠉⠉⠉⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠠⠄⠀⠉⠉⠉⠉⠉⠉⠀⠄⠸⠄⠀⠈⠉⠈⠉⠉⠀⠀⢠⠀⡧⠀⠉⠉⠈⠉⠉⠉⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⠒⠒⠒⠒⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⠄⣤⡄⡄⣤⣤⣄⡄⣤⠀⠀⣤⠀⢀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢿⣿⡿⣿⠿⢻⣿⣿⡿⢿⢿⣿
⠀⠀⠀⠀⠀⠈⠋⠉⠃⠓⠉⠙⠉⠉⠉⠀⠀⠈⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⣠⢤⡀⠀⢀⡤⣄⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠿⠤⢼⠦⠼⠧⠤⡷⠤⠿⠤⢼⡦⠴⠧⠤⣿⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠋⠀⠀⠀⠒⠁⠀⠀⠀⠚⠀⠀⠀⠐⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤⠀⠀⠄⠀⠠⠄⠀⠠⠀⠀⠤⠀⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠁⠀⠈⠁⠀⠈⠀⠀⢉⡤⠖⠚⢛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠂⠀⠐⠂⠀⠐⠀⢠⠎⠀⠀⠀⣈⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⡄⠀⢀⡀⠀⢠⠀⣾⢠⡀⠀⠀⣿⠀⠀⢀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠚⠁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⡀⠀⠴⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢰⣿⡆⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢠⣶⡆⠀⠀⠀⢠⣶⡄⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⠶⠶⠶⠶⡿⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡄⠀⠴⠀⠀⢀⢀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⠤⠶⣾⢹⠀⣻⡇⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣄⣀⢸⣯⠀⢡⠆⠠⣯⣿⣤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣦⣀⣀⣿⣸⠀⢸⡷⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⠀⠘⠀⠀⠘⠻⡏⡰⠛⠛⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⢸⡇⠀⠀⠱⠀⠔⢠⠞⠁⠉⢲⠀⠀⠀⠀⠀⠀⡴⣒⠒⠈⢿⣠⠀⣼⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠐⠖⠛⣿⠇⠁⢸⡇⠀⠀⠀⠐⢠⣿⠤⠔⠒⠊⠀⠀⠀⠀⠀⠀⠉⠛⠓⠄⣜⣭⠀⢻⣀⠤⠄⠈⠛⣷⠖⠚⠓
⠀⠀⠀⠀⠀⠀⠘⠷⢿⣦⣶⠤⣿⡄⡄⢸⡇⠀⢀⣀⣱⣿⡧⡤⠄⠤⠄⡀⠀⠀⠀⠀⠀⣠⣤⠰⠾⣻⡏⠀⢸⠉⡄⠂⢑⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡿⡁⢸⡷⣾⣿⠟⣿⣿⡇⠀⢀⠀⠠⠕⠀⠀⠀⠀⠀⠙⠛⠦⠔⢸⡇⡂⣼⠃⠇⠸⠀⠃⣦⡙⠏⠁⠢
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠉⠀⢹⡇⡇⣸⡇⣿⡟⣀⠸⣽⣷⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⣻⠒⠖⠒⠒⢃⣿⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣾⡇⡇⣻⡇⣿⠁⠘⠀⣿⣿⣆⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⣿⡇⢿⠾⢾⣶⣿⣿⠿⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠠⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡶⠶⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⠥⠄⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠛⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡗⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠛⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⣼⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠌⠴⠷⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⣦⢰⣠⢟⣤⣦⠈⢉⡏⠀⣀⠀⢶⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠟⠛⠁⡾⠳⡌⠛⠳⠛⢻⠇⢮⣤⠴⠾⠇⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⣄⣀⣀⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣄⣀⣀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⠁⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢮⠀⠀⣸⢤⡀⠀⠐⡄⢸⠀⡖⠀⢀⣠⢼⡀⠀⢨⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠉⠀⠀⠈⠓⢄⠀⠀⠀⢀⡔⠋⠀⠀⠉⠉⠁⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡴⢾⠉⡏⠉⡏⠉⠉⢩⠉⣉⣉⡉⣉⠉⢹⠲⡄⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⡇⠀⠀⢸⠀⠀⠒⠒⠂⠀⢸⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸⠀⠟⠙⠇⠀⠀⢸⠀⠠⠄⠤⠤⠄⢸⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⢸⠀⡀⢀⣀⣀⠀⢸⠀⠀⠶⠶⠶⠆⢸⢐⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣀⣀⡀⢀⠀⢸⠀⠒⠒⠒⠀⠂⢸⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠁⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠉⢳⡀⠤⠤⣄⣉⠑⠢⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡰⠓⠦⠤⠤⠤⡄⣧⠀⠀⠀⠈⢙⡆⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣠⠊⡝⠁⢹⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣄⣀⠴⣏⠀⠀⠀⠀⠀⠀⢀⣡⠴⠧⠤⠼⠦⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠘⢇⡀⠀⣀⠞⠓⠦⣄⠀⠀⠀⠙⡞⠉⠉⠉⠉⠉⠑⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⢦⠀⠀⢣⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢳⢄⡀⣠⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡤⡖⢻⡀⠉⠁⣸⠃⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢣⠀⠉⠛⠋⠁⢀⠇⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠣⣀⡴⢦⣠⠞⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠈⠦⣤⡇⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠸⣅⠀⠀⠀⠀⠀⠀⡧⠤⠤⢸⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⠤⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠦⣌⠙⠒⠤⠤⠤⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⠶⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠁⠀⠀⠈⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣻⣃⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡝⠛⠛⠛⠛⠛⠙⢋⡉⠛⠛⠛⠛⠛⠛⠋⡉⠛⠛⠛⠛⠋⠛⢩⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢇⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣻⣄⣀⣠⣞⣉⣳⣄⣀⣀⣴⣛⣶⣀⣀⣠⣖⣛⣲⣀⣀⣠⠖⠒⣆
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣏⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⠙⣯⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⢻⣛⠛⠛⠛⣷⣦
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣉⣀⣀⡀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡤⠒⠒⢢⠠⠒⠒⢢⠠⠒⠒⢢⢰⠒⠒⢢⡔⠒⠒⢢⠄⠀⠒⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⢣⣤⣤⡜⢰⢤⣤⡜⢰⣤⣤⡜⢸⣤⣤⣜⢣⣤⣤⡜⣢⣤⣤⡇⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⣇⡀⠀⡀⢀⠀⠀⣸⢀⠀⠀⢸⢸⠀⠀⢸⣇⠀⠀⣸⡁⠀⠀⡇⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠒⠋⠱⡔⠒⠒⠂⠐⠒⠚⠂⠐⠓⠒⢢⠐⠛⠒⢢⡔⠒⠒⢢⠒⠒⠚⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⠴⠤⣔⡳⠤⢤⠆⡰⠤⢤⠄⡠⠤⢤⠎⡰⠤⠤⡜⡳⠤⢤⢚⠤⠤⡤⠃⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣀⣀⡿⣀⣀⢀⠀⡀⣀⣀⠇⣀⣀⢀⠇⡇⠀⣀⢮⣀⣀⣀⢇⡀⠀⡘⠀⣸⣿⣿⡇
⠀⠀⠀⠀⢀⣼⠃⠀⡎⠉⠉⢁⠆⠉⠉⢷⠂⠉⠉⣲⠂⠈⠉⡳⠂⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⠃
⠀⠀⠀⠠⠾⣥⣄⣀⣉⣉⣉⠁⠑⠒⠒⠈⠒⠒⠒⠉⠒⠒⠒⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠞⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣛⣛⣛⣟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠐⣆⣶⣆⡼⠀⠀⠀⠀⠀⠀⠀⡴⣤⡔⡆⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡄
⠀⠀⣴⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣼⡏
⠀⠀⣿⣿⣿⣿⣤⣿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⢸⡇⠀⠀⠈⡿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠇

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⠻⣿⢆⠀⢾⡷⡀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⠤⢤⡀
⠀⠀⠘⣿⣿⡟⢸⣿⣿⠻⠄⠀⠀⠀⠀⠀⠀⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢤⣇⣀⣼⣿⣤⣀⣣⠤
⠀⠀⠀⠈⠉⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣟⣛⣟⣉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣶⣷⣷⡆⠀⠀⠀⠀⠀⠸⠿⠋⠉⠉⠉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢸⠏⣿⣦⣤⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢣⠝⣻⢻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⠀⡟⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⢠⡀⣦⡾⠀⠀⠀⠀⠀⠀⠀⣀⣤⡐⣦⡎⠀⠀⠀⠀⣿⠀⢀⡄⢀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠷⠟⡇⢸⡇⠀⠀⠀⠀⢠⣄⠀⣯⣼⠃⢸⠀⠀⠀⠀⠀⡏⢀⢸⣧⣼⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼⠀⠀⠛⠈⠀⡤⢄⠀⡇⡾⠜⢦⡇⠀⠙⠈⠀⠀⢀⣤⠾⠛⠛⠻⢿⣿⣿⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠁⠀⠀⠀⠀⠀⡇⢨⡗⡷⠇⠀⠀⠀⠀⢀⣠⣴⠚⢻⡇⠀⠀⠀⠀⠀⠉⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠋⠀⠀⠀⠀⠀⢠⡼⠋⣥⣾⠀⣸⣧⣶⡿⠿⠷⣦⣄⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⢁⣠⣾⣿⠟⠉⢹⠁⠀⠀⠈⢻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠉⠀⠀⠀⣼⠀⠀⠀⠀⢸⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠀⠀⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞⢰⠄⡀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢼⠀⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⠇⣾⣶⢿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠋⠀⣠⣿⠇⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠁⠀⣠⣌⣡⢉⣴⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⠀⠀⠀⠀⢈⣩⠴⣋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠋⠁⠀⠀⠀⠀⠀⣀⡤⢞⣫⠵⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⢖⣫⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⢖⡋⠅⠀⠀⠀⠀⠀⣠⡤⢖⣫⡵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣠⠶⢛⡩⠒⠉⠀⠀⠀⣀⡤⠖⣋⡥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡞⣋⢴⡊⠁⢀⡤⢀⣤⠶⣋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⣤⣪⠵⣚⡿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⣖⡓⠲⠤⣄⣉⠙⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠉⠛⠲⠤⣍⣙⠒⢦⡄⠀⠈⠉⢓⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠀⠀⢀⡴⠋⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠰⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⢥⣀⣀⠉⠛⠒⠦⢤⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠓⠲⠦⢤⣄⣁⡀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣠⢖⡟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣳⠶⠦⢤⣀⣀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠦⣤⠚⢁⡤⠚⢡⠞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠉⠙⠒⠲⠤⢤⣄⣀⡀⢀⣧⠴⠋⠀⣠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡴⢛⣛⣲⣶⡖⠲⠶⠶⠤⠤⠤⠤⢤⣤⣄⣈⣉⣉⣁⣀⠀⣰⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠘⣗⣂⣺⠁⣸⠃⠁⠀⠄⠄⠀⠁⠀⠀⣐⣒⣂⠀⠀⠉⠉⢹⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⡟⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠒⠲⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠃

⠀⣴⠀⠀⠀⠀⠀⠀⣀⡄⢀⡢⢀⣠⠀⣰⠀⠀⠀⠀⠠⣤⠀⣐⣶⠖
⠀⣶⠀⠀⠀⠀⢀⡈⠁⠈⠁⠀⠁⠀⠉⠀⠉⢀⠀⠀⠠⣦⠀⠘⠁
⠀⣶⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣁⠚⠀⠠⣦
⠀⠶⠀⠀⠀⠐⡹⠁⠀⠀⠀⣰⠄⠈⠀⠀⠀⠈⠘⠀⠰⡶
⠀⠶⠀⠀⠀⠀⢲⠀⠀⠀⠐⣧⠄⠺⠂⠀⠀⠉⠔⠀⠐⠶⠀⠀⠇⢀⡀
⠀⠀⣀⠀⠀⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠠⠄⠂⠀⢠⣆⠀⡆
⠀⠁⠀⠀⠊⠀⠉⠈⡁⠆⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⢸⡏⠀⡅
⠀⠀⠀⠀⠀⠚⠀⠀⢀⡐⠁⠀⠀⠀⢒⡀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⡀⠀⠒⠂⢀⣀⠐⠂⠀⢀⣀⠂⠀⠀⢀⡀⠀⠀⣀⠀⠀⢀⡃⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⢇⠀⠀⠀⠀⠙⢬⡴⠀⢛⢟⠀⠀⠀⠀⠁⢂⣀⠈⠐⠀⠀⠁⡂⠀⠈⠐⠀⠀⠁⠂⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣿⣿⠆⠀⠀⠀⠀⢤⠈⠀⢴⣶⠄⠀⠐⡀⠀⠛⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀
⢀⡆⠀⠀⠀⠀⠀⠀⠙
⢀⡉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠐⠂⠧⠤⠚⠉⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠠⠔⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠶⠚⠈⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡄⠠⠔⠀⠀⢀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠄
⠀⠀⠀⠀⠀⠐⠋⢁⣀⡤⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣀⠀⢀⠀⢂
⠀⠘⠀⠑⠀⠠⠀⢒⠉⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠆⠃⠠⠀⠀⠀⢸⠀⠀⠓⠂⠉⠂⠉⢀
⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠊⠀⢸⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠀⡀⢀⠀⠀⠀⢈⡤⠤⠀⠀⠀⠀⢀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠤⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⢠⠡⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠐⠈⠘⠒⠘⠻
⠀⠀⠀⡀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄
⠀⠀⡰⠤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⣀
⠀⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⢀⡀⠈⠐⠀⠀⢰⠀⠀⠀⠀⠀⢀⡴⠒⢢⣄⠀⠀⠨⠀⠀⠀⠀⠀⣀⡀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠃⣈⠁⠉⣈⠌⠀⠸⠀⠀⠀⡐⠢⠄⠭⠭⠬⠥⠔⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⢀⠤⠐⠒⠀⡒⠤⢄⡀
⠀⠀⠀⠠⢀⠀⠄⠀⠀⠀⠀⠀⠀⢃⠐⡠⠰⠄⠒⠐⡠⢀⡇⠀⠀⠀⠀⠘⠢⢃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡈⢀⠀⣄⠄⢰
⠀⢀⠀⠀⠘⠑⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠴⠄⠀⠀⢀⠞
⠀⠀⡀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠑⠒⠒⠒⠉⠁⣠
⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠀⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠐⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡁⣄⠀⠀⢀⠾⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠂⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣷⠈⠑⢤⡀⠀⠀⠱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡤⠀⠐⠶⠀⢈⣷⠖⠊⠉⠠⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠄⠀⠀⠀⢀⠘⢃⠀⠈⠛⠂⠀⠀⣧⡀⢀⠀⡀⡀⠀⠀⠀⠀⢀⢦⡉⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠑⠄⠀⠀⠀⠘⢦⣌⡶⠒⠉⢋⢠⡐⠛⢷⡄⢾⡄⣿⢎⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⣴⣤⡖⠁⠀⠻⠆⠀⠀⠘⢧⣄⠀⠄⣼⡻⣧⠋⠀⢀⠀⡀⠀⠀⢰⡄⢀⡄⠀⠀⠀⣀⡤⠶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⣂⢘⡦⣀⡈⠋⠀⠀⠀⠀⢒⡀⣠⡀⠠⠈⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠤⠼⣷⠲⢤⣤⣄⠀⢀⠀⢀⠛⠀⠘⠆⠀⠀⠀⠀⠀⠰⠃⠀⠀⠳⠀⠀⠀⠐⣿⣿⣿⣅⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠉⠉⡿⡇⠁⠀⠀⡀
⠀⣠⠞⠁⠀⣀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠐⠒⡒⢏⣉⠂⠀⠀⠐⠀⢀⢂⠀⣾⣎⣠⡀⠀⠀⠀⠒⠉⠋⠀⠀⢀⡄⠀⠁⠀⠀⠀⠈⣢⠁
⠈⠁⠀⠀⢀⠛⡄⢠⠰⠶⡴⣂⡠⠔⠂⢀⡠⠿⠻⠁⠄⠀⠀⠀⠀⠈⠛⢷⣿⣿⠟⣈⣤⡀⠀⠀⠀⠰⠀⠀⠈⠀⠀⢸⡀⠀⡀⠄⠑⠂⠄⡀
⠀⠀⠀⠀⠈⡶⢷⡞⢣⠀⠀⠀⢀⣦⣀⣀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠰⠈⠙⠉⠉⠀⠀⠀⠀⢀⠀⠀⠈⠁⠁⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⣤⠀⠀⠀⠀⢸⠃⠉⠲⣍⠀⠀⠀⠀⠚⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣼⢟⣒⡖⠀⢀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⢠⠀⠀⠀⠩⠇⢤⣤⣄⡀⠄⠂⡄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠚⠁⡡⠈⠀⡀⢀⣾⣴⣤⡄⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣈⠱⢆⡨⠶⠃⠀⠀⠰⣿⣿⡇⠀⠀⢀⠁⠀⠀⡙⠿⡿⠁⠀⠀⠀⠾⠀⢀⣤⣽⣿⣯⠉⠁⠀⢀⣀⠀⠠⠄
⠀⠀⡠⢀⣿⣃⠈⡱⠊⠉⠀⢀⣬⡄⠀⠀⠀⠐⠁⠀⠙⢓⣠⠄⠡⠀⠀⢠⠄⠀⠀⠀⠀⠀⠄⠀⠰⠺⡿⣿⢯⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠉⠁⠀⠛⠂⠀⠀⠀⢀⣾⠋⣠⢳⠄⠀⠀⠀⡀⠀⠉⢻⡀⡄⠀⢠⡏⠀⢸⠀⠀⠀⢰⠀⡠⠀⠀⠏⠁⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣃⠜⠁⠀⠠⡄⣠⠞⠁⠀⠀⡸⣿⠃⡠⠀⠇⣱⡄⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣤⠈⠏⢀⠀⠀⠀⠊⠣⠀⡀⢀⠀⠀⣘⡇⠀⠀⡐⠋⠛⣽⠅⠆⠀⠆⠀⠄⠀⠀⠀⠀⢀⣣⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢰⡗⠁⠿⠷⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠉⠀⠀⠢⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⣸⡁⠈⠑⠂⠤⠤⠤⠔⢠⣏⠀⠀⣤⡄
⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠀⠀⠀⠀⠀⠀⠁⠀⢳⠙⢄⠀⠀⠀⠀⠀⠀⠀⢻⠅⠀⠊
⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡦⠄⠿⠉⠀⠐⠚⠹⣆⠈⢣
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠙⠀⠀⡧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡄
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠘⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠂⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠂⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠐
⢀⠄⡠⠀⠀⠠⠄⠁
⠈⣰⣴⠄⡄⠀⠀⠀⠀⢀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠇⠑⣀⠀⠀⠆⢀⡠⠯⡤⣌⠀⠄⢀⡀⠀⠀⠀⠀⠀⡄⠠⠀⢶⠀⡀⠦⠀⠀⠀⠔⠤⠀⠈⢁
⠚⠀⠀⢀⠅⠀⢀⠀⠂⡁⠇⠒⠠⠐⠂⠀⢱⠀⠀⠀⠀⠈⢠⡈⠀⠉⠹⠃⠈⡀⠀⠐⢀⠀⠀⢀⠀⠭⠄⠁⠆
⠀⠀⠀⠈⣦⢀⠈⠀⢉⠁⠀⠀⠀⠂⠀⢲⡀⠀⢠⠀⠀⠀⠘⡆⠈⠀⠈⠀⢠⠆⠀⠀⠄⠄⠀⢨⡆⠀⠁⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠂⠒⠶⠶⠶⠤⠤⠤⢄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠅⡪⠐⠡⢊⠔⡈⠐⡡⢂⠥⡪⢐⡡⢊⠔⡡⢊⠀⢂⠔⡠⢐⡡⢊⠔⣁⠔⡠⢂⠐⢁⠀⡿⡊⢝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠊⠔⠁⠐⠡⠊⠔⠡⢔⠥⢂⠔⡡⢊⠤⢊⠔⡡⢒⠅⢊⠔⡁⠔⡡⠆⡡⢊⠔⢁⠔⡡⠀⡯⢂⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠑⠲⠒⠖⠤⢤⢤⣄⣄⣀⣀⢀⠀⠁⠀⠁⠈⠐⠁⠖⠡⠊⡠⠊⡠⢊⠔⢁⠎⡠⠊⠔⢐⡏⠔⡀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠒⠀⠤⠤⠀⢀⣀⣀⠀⠀⠁⠉⠉⠉⠛⠚⠓⠒⠲⠦⢤⠤⣤⣀⣀⣀⣀⠀⠈⠀⠑⠁⢸⡃⠕⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⠀⠀⠀⠀⠂⠀⠀⠀⡀⠈⠉⠀⠀⠐⠒⠒⠀⠤⠠⠀⢠⠀⡀⣀⡀⠀⠈⠉⠉⠙⠒⠒⢢⢹⣶⣅⣾
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠁⠈⠀⠀⠐⠒⠀⠀⡜⡈⡆⣟⠃
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇⠈⠉⠀⢀⠐⠂⠲⠆⠁⠠⠄⣀⡀⠀⠀⠃⠀⠀⠀⠮⠀⠄⠀⣸⠀⢸⠀⣤⠀⠀⡇⢠⢡⢹⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠇⢰⠀⠸⠂⠀⠀⠀⣴⠀⠈⠁⠤⡄⠀⡄⠀⠉⠉⠁⠀⠒⠒⠀⠀⠠⠤⠄⠈⢀⡀⠀⢰⠀⡌⡌⡟⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠠⠤⠀⠃⣀⣈⡀⢰⠀⠀⠁⠀⠁⠠⠟⠀⢘⠀⢘⠆⠀⠀⣤⡤⠀⠀⡀⣀⠀⡆⢠⢥⢠⡏⢸
⠀⠀⠀⠀⠀⠀⡌⠠⠄⢰⠠⠄⢠⠀⡀⡀⠀⡄⠈⠁⠒⠀⠀⠲⠦⠄⠀⢠⣄⣀⠀⢀⠉⠁⠐⠀⠇⠯⢰⠀⢈⠀⡘⡔⡟
⠀⠀⠀⠀⠀⢀⠃⢰⠀⣈⠘⠂⠘⠠⠇⠅⠀⡇⢚⠺⠄⠗⢠⡄⡄⠀⡆⣀⢀⠀⠀⠀⠈⠉⠁⠒⠒⠀⡌⠀⠸⢀⢁⢇⠃
⠀⠀⠀⠀⠀⢸⠀⡈⢠⠀⣄⢀⠀⢙⠓⠒⢀⠀⠤⢀⡀⠀⠈⠁⠁⠰⠀⠓⠳⠀⡆⣛⢨⠃⠃⡤⣠⢀⠁⢀⡇⠈⠸⢸
⠀⠀⠀⠀⠀⡇⠀⡇⣘⡂⠊⢸⠀⠼⠳⠀⡌⢐⠄⡄⢀⠀⠈⠉⠁⠀⠐⠒⠐⠀⠀⣤⡀⠀⠀⠃⠋⠀⠀⢸⠀⠀⠇⡇
⠀⠀⠀⠀⢠⡇⢠⠀⢀⠈⠁⡄⠐⠂⠤⠄⠃⣈⡁⠁⠸⠀⠦⠬⠀⡇⢀⡀⡆⢠⠠⣄⡀⠀⠈⠉⠀⡍⠀⡇⠀⠰⢰⡋
⠀⠀⠀⠀⢘⠁⠘⠀⠞⠷⠀⡃⢀⠒⡆⠸⠀⢀⣈⠀⡀⠐⠒⠄⠀⠁⠤⠄⠀⠀⠘⠙⠁⡆⠌⠁⠀⠆⢠⡟⠀⠆⡬⠃
⠀⠀⠀⠀⠼⠀⠑⠒⠤⢤⣄⣁⣀⠀⠀⠆⠀⠋⠛⠀⠁⣸⠰⡆⣀⠀⣄⡄⢠⠀⠉⠐⠂⠠⠤⠀⠸⠀⢸⡇⠀⢠⠸
⠀⠀⠀⠀⢰⡖⠶⠤⣤⣀⣀⠀⠉⠉⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠙⠂⡄⠀⠀⢰⠀⠀⠀⠀⠇⠀⣿⡈⡀⠄⠇
⠀⠀⠀⠀⠀⠙⣷⣶⣤⣤⣭⣉⠝⣒⠲⠤⣤⣀⡀⠀⠉⠑⠒⠦⠤⣀⣀⡀⠀⠀⠀⠈⠀⠀⠀⡘⠀⣸⢷⠁⠁⠸
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢷⣟⣿⠿⣿⣶⣦⣭⣍⣓⡒⡤⢤⣄⣀⡀⠀⠉⠁⠒⠢⠤⢤⠐⠁⢠⡟⡌⡈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠛⠭⠿⣿⣿⣿⣶⣮⣭⣙⡿⢓⣲⡦⠤⠤⢤⢀⣀⠎⡨⢁⢃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣶⣶⣭⣥⣔⣂⣆⣊⡄⡘⡄⡍
⠀⠀⠀⠀⠀⠀⡀⢄⡠⠄⢀⠠⠄⣀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠷⣥⡕
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠂⠁⠐⠂⠉⠐⠀⠠⠀⢀⠠⠐⡀⠄⢂⡀⠀⠐⡀⠀⢀⠠⠀⡀⠄⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠊⠁⠒⠈⠄⠂⡡⠐⠈⠔⢂⡡⠐⡈⠀⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣮⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣽⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣛⣛⣛⣛⣛⣛⣛⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⡇⣿⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣓⣛⣛⣛⣛⣛⣛⣛⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣵⣷⠍⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣿⢟⣵⣿⢟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢟⡅⣫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢟⣵⣿⢟⣵⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣲⡿⣛⣵⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠀⣀⡠⠴⠺⡆⠀⢀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣩⣳⣦⠀⠀⠚⣻⣵⣶⣶⠆⠉⠀⠘⢁⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠈⠰⢻⣿⣿⣝⠛⠂⠀⢐⡀⠤⠞⠋⠀⠀⠀⠀⣦⡀⠀⠀⣸
⠀⠀⢀⠆⠄⠀⠊⢀⠀⠀⢿⣿⡇⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⢈⡄⠀⡀⠸⡄⡀
⠀⠀⠀⠀⠀⠴⡀⠛⠀⠀⠈⣿⣷⡀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠊⠀⠇⠀⠀⠀⠘⣿⣷⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢀⠔⠁⠀⢡⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣶⣄⠀⡊⠁⠀⠀⠀⠈⢇⠀⢀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢀⡤⠀⡀⠔⠉⣧⠁⠂⠀⠀⠀⠀⠀⠈⠦
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠐⢈⣄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⢶⣤⣴⠦⠀⠁⠂
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣨⣁⡐⠾⠿⠿⠆⠀⠀⠀⠼⠷⠆⠀⢐⠀⠙⠛⠃⠀⠁⣤⠔⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣛⠳⢦⣤⣄⣀⡀⠀⠠⠤⢤⠄⠘⠁⠀⠚⠉⠁⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠻⣿⣶⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣾
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢸⣿⠇
⠀⠀⠀⠂⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠸⣿⣷⣍⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟
⠀⠀⠀⠀⠀⠀⠢⠐⢌⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⡄⠉⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠾⢶⡤⠴⠿⢦⠤⠴⠷⣦⠤⠶⠶⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠲⠚⠀⠀⠲⠊⠀⠐⠖⠃⢀⣲⣞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⡭⠇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⢠⠤⡄⣤⢤⠀⡖⢦⢰⠒⡄⣖⢲⠀⡖⣆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⣒⡃⢘⣚⠀⣉⢉⢈⡭⡅⢨⠤⡀⡬⢤⠀⣇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠧⠼⠘⠤⠇⠙⢚⠂⢓⣛⢘⣒⡃⢙⣚⡀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣇⣀⣳⠀⢻⣹⠄⣏⣳⠸⠭⠇⠯⠼⠀⠧⠗⠘⠤⠇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⠀⢰⡒⡆⢴⢲⡀⣖⣲⢰⣚⡆⢯⣹⠀⣏⡧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣩⡥⠬⠭⠤⠬⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⣀⠉⢻⡅⠁⠀⢂⠀⣁⣾⠋⠀⠉⠑⡍⠈⢹⣦⡾⢛⣿⠛⠛⠋⢿⡀⡸⠏⠛⠛⢻⣟⠿⣖⣾⠉⠉⡕⠉⠁⠈⢹⣆⡁⠀⠐⠀⠁⣽⠋⢁⡀
⠏⠁⠐⠚⠓⠒⠁⠈⢫⠈⠙⠃⠘⠃⠀⢹⠀⠒⠓⠛⠈⠛⢀⡀⠀⠒⠛⠙⠒⠀⢠⠀⠈⠚⠓⠀⠀⣹⠁⠑⠋⠒⠛⠋⢨⠋⠀⠑⠒⠛⠂⠀⠉
⠠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠦
⠲⡆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡖
⠑⠃⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠓
⢐⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢒
⣿⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢼
⣾⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢹
⡶⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠴
⠲⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⢝⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡴
⠳⠄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠴
⢸⡅⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿
⠒⣳⣒⣶⢲⡶⣒⣶⣶⢆⠐⢲⠐⠒⠀⢛⣷⠂⠒⢂⠀⠀⢺⣅⠒⠒⠀⠐⠒⢀⣼⠂⠀⢀⠒⠂⢲⣞⠃⠐⠒⢰⠒⢀⢶⢶⣖⡲⡶⣲⣖⣲⡛
⠈⠉⠉⠉⠁⠉⠉⠉⠈⠉⠀⠁⠉⠀⠀⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠈⠉⠈⠉⠉⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁⠘⡄
⠀⠀⠀⢸⡟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠔⠒⠉⠁⣠⡤⣦⠻⠷⠈⢆
⠀⠀⠀⠸⡇⠀⠀⠉⠉⠀⠐⠒⠒⠈⠉⠉⣀⣀⣀⣠⠀⢠⣄⣺⣏⠻⣤⣴⣶⣦⡄⠈⢣
⠀⠀⠀⠀⢳⠀⠐⠒⠒⠀⢺⣶⣄⢿⣦⢻⣏⣁⣙⠈⢳⣄⣻⡿⢿⣆⢰⣦⣤⡼⠇⣀⠀⠳⣄
⠀⠀⠀⡖⠞⣧⠀⠉⠉⠁⡀⢻⣟⠷⣿⣧⡹⣿⣍⣀⣠⣍⠛⠗⠀⢉⣋⣤⡖⠬⠑⣂⣠⠤⡈⠢⡀
⠀⠀⢈⣿⡄⠈⣧⠀⠙⠦⠼⠆⠹⣷⡌⠙⠻⠊⣛⣋⣩⠴⠚⠁⠀⢀⣠⠤⢖⣚⡭⠵⠖⢛⠁⠀⠈⠓⢤⣀
⠀⠀⠈⢻⣿⣆⠈⢷⡀⠈⢩⣤⣶⡯⠤⠖⠙⠉⣩⡥⠴⠖⠚⠉⠳⡄⠓⠋⣭⣴⣶⡿⠟⣁⣠⣔⣢⣤⣢⠌⢉⣓⡒⠒⠲⢤⡀
⠀⠀⠀⠀⠻⣿⣧⠀⠹⣆⠀⠠⣴⣒⣺⣭⡓⠀⠙⢦⡀⠐⠀⠀⠀⠈⣢⣄⠁⠚⠷⠶⣛⣋⣭⡶⠾⠛⠀⠀⠀⠀⠀⡨⠓⡀⢹⡄
⠀⠀⠀⠀⠀⠘⢿⣷⡄⠈⢷⡀⠀⠈⠭⡑⣦⣶⠣⣄⠙⢦⡀⠁⠀⠀⠀⠈⠓⣦⣤⠙⠫⣅⠀⢀⠀⠀⠐⠀⠀⠀⠀⢀⣴⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠻⣆⠀⠀⠸⢛⠺⢋⣙⡥⣀⠑⢦⣀⣠⠤⠒⠋⢉⣠⡴⠆⠈⢳⡁⠀⢀⣀⡤⣰⣎⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠘⢷⣄⠀⠀⠙⣒⡾⣟⣻⠽⣚⣁⣤⣴⠶⢟⣛⣭⣖⣀⠀⣸⣿⣤⠾⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⢩⠓⢦⣄⡈⠩⣶⠯⠭⣒⣩⡥⢖⣻⣿⡻⣫⣿⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠈⠘⢰⢋⡻⣆⠀⠀⢜⣫⣵⣾⡽⠼⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠈⠁⢹⡆⣈⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⡶⠞⠛⠉


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠤⠤⠄⠀⠀⠤⠤⠤⠤⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣧⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⡿⢀⠀⠀⠀⠀⠐⠒⠂⣶⢋⡙⠃⢸⣀⣀⣀⠁⠁⠀⠀⠀⠀⠀⢀⠀⠘⡩⢆⠀⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠴⣾⣔⣡⣀⣠⣀⣠⣠⣀⣾⢧⠏⠠⣼⣿⣟⠀⣠⠀⠀⠀⠀⡀⠄⢊⠠⠐⢀⡠⢞⡧⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠠⠀⠀⠀⠉⢰⠋⣝⡍⠙⠉⠘⢋⡛⠛⣯⡝⠽⣻⢿⡿⠿⠞⠙⠀⡀⠐⠈⡠⠀⢀⡤⢚⠭⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡦⣌⢤⡜⢀⠏⠰⠿⠀⢰⡔⠀⣿⠃⠠⠿⣽⢶⣿⣿⠁⠠⠀⠖⠈⠠⠂⣁⠤⢊⠧⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠐⢰⡾⠠⢁⡞⢀⢠⠄⠀⠀⠀⠀⠀⢠⣅⢀⣀⣾⣾⡇⡠⢏⠀⠀⣡⠔⣚⠤⠚⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⡞⠀⠾⠏⠀⣼⡛⢹⠀⠀⠼⣷⡷⣻⣿⣏⡎⠀⠀⢺⡧⠖⠊⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠁⠁⠀⠦⠴⠇⠀⠀⣀⣰⣟⣀⣯⠀⠈⡀⠀⢰⣿⡿⠛⠯⣼⡡⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠒⠒⠀⠴⠄⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠉⠉⠁⠙⠛⠓⠐⠜⠀⠀⠠⠐⠤⠂⣄⣀⣸⠃
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠉⠉⠩⡹⠋⠒⠂⠄
⠀⠀⠀⠈⠙⠒⠢⠦⢤⠤⡄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠈⠙⠁⠂⠒⠒⠒⠖⠴⠠⠠⠤⢤⢤⣤⣤⠞⠨⠆⠁

⡯⡝⠋⠛⢉⠏⠛⠹⣻⣭⠉⡇⣘⣱⠟⠛⠻⢭⣟⣿⣹⣧⣭⣯⢫⣝⣻⢿⡛⣍⡟⢋⣯⢙⣿⣵⣿⣛⡯⢉
⡏⠧⢰⡁⠀⠰⠦⠤⠦⠦⠆⠆⠀⠈⡁⡇⠀⠐⠢⠴⠤⠴⠴⠶⠀⠀⣃⡍⠠⠤⠦⠦⠦⠦⠤⠴⠶⠄⢙⡷⣾
⡷⠯⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠥⠈
⣏⣒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⣴
⠏⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⡗
⣤⡖⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠈
⣇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠃
⡧⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡏
⡶⢂⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢔⣼
⣷⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣩⠿
⣛⠀⣾⠍⠹⡙⠉⠨⠿⠉⠹⠉⠉⠟⡛⡿⠧⠸⢿⠿⠷⠿⠻⠿⠾⠵⡡⡭⠙⠧⠿⠿⠶⠧⠠⠦⠀⠏⢅⡀⡞
⣟⠶⡉⠄⠀⠁⠉⠋⠉⠉⠉⠉⠁⠠⡴⠇⠀⠀⠉⠙⠉⠉⠉⠀⠀⠀⡅⡧⠀⠈⠉⠉⠙⠉⠉⠋⠁⠀⣤⠈⣍
⡟⠒⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄
⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢙⢷
⠗⢫⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣝
⡆⠤⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡏⠀⠀⡤⣤⢤⠤⡤⠤⠀⠀⢹⡐⢌
⡁⣄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⣼
⢧⣋⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢹⡃
⣶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂
⡛⠦⣰⣀⣀⣀⣀⢀⡀⠀⡀⠀⣀⢀⢯⢄⣀⡀⡀⠀⠀⣀⠀⣀⡀⣀⠆⣆⣀⣀⣀⠀⣀⣀⢀⣀⣀⣀⠴⠾⡿
⠧⠅⠀⠀⠤⠾⠟⠀⠌⠡⠤⠤⠀⠨⠥⠸⠧⠧⠑⠤⠄⠁⠠⠿⠀⠸⠄⠼⠿⠼⠷⠵⠎⠧⠤⠼⠟⠶⠾⠤⠦

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄
⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⠋⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⣾⡿⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢀
⠀⠀⠀⠀⣿⡇⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⢸⡿⠀⢀⣀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⢻⡆⠀⠀⠟⠁⡴⠚⣻⡇
⠀⠀⠀⠀⣿⡇⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⣾⡇⢰⣏⣸⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠺⡷⠒⢲⡾⢀⠇⣰⡟⢀⣴⠞⣶
⠀⠀⠀⠀⢸⣷⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣹⣿⡄⠀⠀⠀⣀⣀⠀⣿⡆⠀⠀⡿⢀⣿⠃⡜⣰⡿⢀⣾⠃⣰⠏
⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⢠⣿⢁⡾⣱⢟⡴⢶⡀⣰⠏⣍⢸⡿⣧⠀⢠⡇⢸⣟⡼⣱⡿⠁⢸⣿⣼⠏
⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⢀⣸⡿⠋⣼⢋⡾⠁⣿⡥⢿⠔⠷⣿⠁⢿⡀⣼⠁⠸⠟⣱⣿⠃⠀⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⡏⠙⠓⠶⠶⠶⠚⢋⣿⠃⢰⡏⠸⡷⢁⣿⠁⠀⠀⠀⠛⠀⠈⠛⠃⠀⠀⢰⣿⠃⣠⡾⢻⣿⡀
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠸⠇⠀⢠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⡾⠋⠀⠀⠻⢿⣶⠖
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢻⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠿⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋⠁⠀⠈⠙⠲⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⡠⠒⣺⠨⡴⣌⠉⠛⠳⠶⣤⣄⣀
⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⢚⡍⠳⣄⣩⠗⠈⠋⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣤⣀
⠀⣠⠞⠥⣀⠀⠀⠀⠀⠀⠀⣀⣀⡐⣕⠬⡓⠼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⣀⡀
⠀⠻⣄⠀⠀⠀⠐⠠⠀⠀⠈⠢⢖⣨⠏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠃
⠀⠀⠉⠛⠶⢤⣤⣀⡀⠈⠁⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠈⠁⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣁⡴⣺⠽⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠈⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⣴⠾⢛⣉⠤⢚⣡⠼⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⡀⠉⠓⠆⠠⣀⣠⠞⡡⠖⢛⣫⣑⣮⠽⠖⠋⢉⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⡘⢏⣙⣶⡿⠟⠋⣁⣠⡤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠖⠚⠋⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢔⠄⡀⠀⠆⢠⠄⠀⢀⠀⢈⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⡠⢄⠀⡀⠀⠀⠀⠀⠀⢿⣤⠀⠿⠲⠶⠄⢀⢀⡄⠒⣆⣶⠠⠀⠈⠀⠉⠈
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠣⡄⠂⠉⢁⠉⠉⠀⢀⢀⡀⢀⣇⣦⢰⠀⠈⠉⠘⡁⠈⠁
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⣿⣟⣧⣀⠀⠆⡇⠸⠃⠀⣀⠤⠐⠈⠁
⠀⠀⠀⠛⠿⠖⢀⠀⠀⠀⠀⣷⠀⠠⠸⠁⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋

⠀⡀⠴⠫⣿⣞⡭⡛⡿⢯⠧⠸⢧⡀⠌⡍⢂⢠⠢⠇⡀⠍⣵⣽⠾⣉⡞⠁⠀⣤⢖⣒⣒⡦⡀⠙⠲⠌⠶⠀⣆⣈⡙⠛⡋
⢸⣊⣡⡷⢄⣛⣛⣝⢣⡿⠢⠳⢬⠳⣏⣄⢉⠀⣠⡴⣇⡐⢥⣍⣠⣾⣤⠴⣞⡴⣡⡶⢻⡆⡴⣰⢀⡀⠁⠰⠃⡁⠖⢠⢄
⠀⠀⠪⢁⡂⢞⣂⡨⡍⢸⣩⠛⢊⡎⠷⠨⠉⢀⡥⣰⠍⢯⢞⣿⣷⣴⡶⣞⠿⡮⢾⡂⠈⠉⠑⠁⠀⠀⠰⣀⣄⢁⡰⠊⠁
⠀⠈⠁⠞⠀⠝⠀⠀⠳⣄⣁⣨⠠⠑⠆⢘⡢⠘⠲⠊⠙⠲⣎⡹⠧⠾⠂⠝⠀⠪⠮⠥⠚⠯⠐⠿⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠛⠙⠙⠋⠋⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠘⠘⠙⠉⠁⠀⠀⠀⠀⠘⠛⠃⠋⠋⠋⠋⠋⠋⠉








⠀⠀⠀⠀⠀⠘⠘⠚⠚⠒⠒⠓⠓⠃⠀⠀⠀⠀⠀⠀⠋⠓⠚⠒⠛⠃⠀⠀⠀⠀⠀⠀⠚⠚⠈⠒⠚⠚⠚⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠛⠙⠛⠙⠃




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

⠀⡰⠜⣫⡷⠚⠻⢲⡤⡤⠶⢿⣄⠀⠈⣷⢵⡒⠳⣙⣦⠴⠿⠦⠤⠴⠔⢃⢖⣿⠥⠤⠴⣮⡜⣝⠢⠬⢭⡣⢸⡀⣼⠃⠂⠀⣄⢾⣁
⠀⡗⣊⠳⣯⣴⡤⠿⡷⠿⠀⠀⠙⢷⠀⣸⠀⠀⠀⠘⣤⣇⣚⠛⠛⠚⠛⣩⠞⣁⡠⠤⣠⡀⠙⢿⡗⢀⡠⡑⣢⢻⡁⣀⠘⠶⢣⡔⠄
⠀⠓⠞⠠⠈⣼⣤⢶⡁⢘⢷⣶⡀⠀⢨⣿⣦⣀⡀⢤⣤⠽⡈⡉⠱⠾⡻⠟⡋⢥⣤⡦⠸⡇⠀⣸⠀⢌⣿⢛⠼⣊⡱⠌⠑⠣⢤⡙
⠀⠀⠀⠀⢠⣶⠉⠉⠉⠇⠀⠨⠬⠐⠛⢣⠁⠈⢃⣞⠽⣏⠛⠁⠁⠁⠔⠋⠀⠀⠹⣴⠗⠁⣠⠟⠛⠋⠁⢩⣦⠘⡦⣄⢰⠳⠊
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀
⠀⠀⠀⣇⠀⠀⠲⠶⠖⠶⠲⠶⠖⠆⠀⠀⢈⠀⣇⠀⠀⠒⠰⠲⠲⠖⠖⠖⠒⠀⠀⠈⠀⣇⠐⠶⠲⠲⠒⠖⠆⠶⠶⠶⠒⠂⠈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣇⠀⠈⠑⠚⠒⠛⠸⠺⠒⠛⠃⠀⣈⠀⣇⠀⠀⠀⠛⠛⠒⠗⠟⠚⠀⠀⠀⢈⠀⣇⠀⠸⠒⠋⠉⠒⠚⠸⠚⠛⠃⠀⢈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡗⠀⠀⢀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠷⠀⠀⠈⠁⠉⠉⠈⠉⠁⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠘⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠞⣇⠀⠀⠀⠀⢀⣤⡀⠀⠐⠉⢀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⣤⡴⡋⠀⢹⠆⣰⡆⢈⣥⡤⠂⠀⠀⠐⠉
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⣦⢄⣠⠀⣀⡟⠀⠀⢸⡇⢰⠃⠀⢸⡇⢘⣃⣠⣀⣀⣀⣀⣤⡤
⠀⠀⠀⢀⡼⠇⠀⠀⢀⠀⡀⣠⣀⡀⠈⠓⠐⣷⡘⣿⠀⢿⠀⢹⡇⣾⠀⢸⡇⢀⣄⠀⣼⡇⠈⠁⣀⠀⠀⡤⠚⠁
⠀⠀⡀⠉⠀⠀⠀⠀⡰⢶⠦⠜⠻⠿⠿⣶⣄⢹⣷⠈⢸⡌⠀⣾⢀⠶⣶⣨⣧⠌⠻⠶⠚⠛⠛⠉⠁⠀⢸⠁
⠀⠀⣿⣄⡀⣀⣴⡯⠊⣡⣴⠶⡶⣶⣶⣤⡙⠏⠹⠷⠚⠛⠋⠉⢁⠀⣠⠀⣀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⢈⣿⣿⢋⠎⡰⠛⠁⢷⠀⠀⠀⠉⢻⣿⡄⠀⢀⣀⣀⣿⠀⣽⡆⣿⠀⢙⡃⢼⣇⠀⠀⠀⣀⡤⠶⠒⠒⠒⠛⠲⠦
⠀⠀⣼⣿⣧⡎⠐⢀⠀⠀⠈⣧⠀⠀⠀⠀⠹⣿⠈⠉⠀⢨⢿⣆⣙⡃⢾⣆⡽⠒⠺⠧⣄⡀⠀⡇
⠀⠀⣿⣿⣿⠀⠀⡀⠀⠀⠀⣘⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠙⠲⢿⡀
⠀⠀⢹⣿⣿⠀⠀⠁⠀⠀⠀⡿⠀⠉⠁⠒⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠹⣿⡆⠀⠀⠃⠀⠼⠁⡀⠀⠀⢀⡞⡐⠀⠀⠀⠀⠀⠀⡴⠀⢠⡇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢀⣠⠔⡫⠺⠄⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠾⠟⠂⠤⠤⠤⠄⠐⠈
⠀⠀⠀⠀⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠃⠀⠲⠦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡠⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠂⡜⠁⠀⠀⠀⣸⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⢰⠋⢳⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡌⠀⠀⠀⠀⢠⠏⠀⣸⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡸⠀⠀⠀⠀⢠⢏⠄⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⠀⠀⢠⢏⡜⣰⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⠀⠀⢠⠏⡞⡰⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⠀⠀⢠⠏⠜⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡼⠀⠀⠀⠀⢠⠏⠌⡰⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡸⠁⠀⠀⠀⢀⡏⡜⣸⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡰⠁⠀⠀⠀⢀⡞⠘⢰⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡰⠀⠀⠀⠀⢀⡞⡐⢠⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠁⠀⠀⠀⢀⡞⡐⢀⣏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣰⠁⠀⠀⠀⢀⠎⠀⢠⢏⡼
⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠃⠀⠀⠀⢀⡞⠠⢠⠏⡼
⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢠⠁⠀⠀⠀⠀⡜⠀⢠⠏⡼⠁
⠀⠀⢸⡏⠀⠀⠈⠉⠑⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠃⠀⠀⠀⠀⡼⠀⣠⠏⡾⠁
⠀⠀⠘⡇⣇⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⣸⠁⢰⠏⡾⠁
⠀⠀⠀⠑⠬⢀⠀⠀⢀⠀⠀⠈⠑⡐⠒⠠⠄⣀⠈⠉⠓⠢⣤⣀⠀⠀⠀⠀⢠⠃⠀⠂⠀⠀⠀⠀⢰⠃⢠⢇⡾⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠲⠬⣥⣒⡒⡆⠈⣄⡀⠀⠈⠉⠒⠀⠀⠈⠉⠒⠦⣤⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠇⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢸⠶⠤⣭⣐⡒⠤⠠⣀⣀⠤⡈⠀⢰⠀⠀⡋⠓⠒⠒⠊⠁⢀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠦⣄⠀⠀⠉⠙⠒⠶⢤⣉⡓⠒⠴⠀⠀⣇⡀⠀⠀⠀⢠⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡙⢆⠀⠀⠀⠀⠀⠀⠉⠙⠒⢂⡀⣍⣉⠙⡒⢒⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠺⣦⠀⠀⢰⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣿⠀⠀⣾⡷⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢀⣠⣿⠀⣴⠏⠉⣿⡇⢰⡇⢸⣽⣰⡟⢹⡇⢀⠀⢠⡾⠉⢹⣿⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⡏⠉⠀⣿⡀⣿⡀⣠⣿⢀⣿⡇⢀⡝⣿⡒⠉⢠⠎⠀⢸⣇⢀⣼⡇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⠃⠀⠀⠻⠇⠙⠛⠁⠛⠋⠈⠛⠋⠀⠈⠛⠛⠁⠀⠀⠀⠙⠋⠈⠛⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠙⠶⠝⢧⠀⠀⠀⠀⠂⢀⡀⢤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡤⢀⠀⠊⠀⠛⣰⠟
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣀⣀⡀⢠⠉⠖⠒⠦⠤⠀⠀⠀⣶⣾⡅⠀⠤⠤⠴⡒⠒⠁⠤⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⢿⠆⢤⣤⣉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣈⣀⣀⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢠⣄⠉⠉⠉⠉⢉⣉⣉⡉⠉⠉⢉⣉⣉⣉⡉⠉⢩⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣴⣟⣻⣦⣸⠿⣤⠀⠀⣤⣿⠁⣸⠇⢀⣴⡟⠁⢙⣿⠁⢀⣼⣿⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠈⠉⣹⠏⠀⢰⣿⢀⡜⢸⣿⠋⠁⣠⠏⣿⡇⢀⣾⣿⢀⡞⢸⡟⢀⡞⢀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠈⠛⠿⠶⠶⠛⠁⠀⠀⠈⠻⠋⠀⠀⠛⠓⠚⠁⠀⠘⠛⠛⠘⠛⠋⠀⠘⠛⠋⠀⠘⠛
⠀⠀⠀⠀⠀⣩⡟⠀⢰⡄⢀⡀⠀⣿⠀⣶⠖⠀⣶⠶⠀⢰⡀⣰⠇⢰⡶⠆⢀⣆⠀⠀⣶⢠⡶⢦⣄⠀⢼⠶⠄
⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⢸⡇⠀⡿⠀⣿⠀⠀⣿⠀⠀⢸⣰⠏⠀⢸⡇⠀⢸⣿⡆⠀⡇⠀⡇⠀⢻⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣷⢸⡇⠀⣿⠄⠀⣿⠤⠀⢸⠿⡄⠀⢸⣧⠄⢸⡇⣷⠀⡇⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣾⠀⠀⡿⠀⠀⣿⠀⠀⣸⠀⢳⠀⢸⡇⠀⢸⡇⠸⣆⡇⠀⣿⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⣇⣀⠀⣿⣀⠀⣿⠀⠘⣇⢸⡇⡀⢸⡇⠀⢻⡇⠀⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣁⡀⠁⠀⠀⠉⠉⠀⠉⠉⠁⠉⢀⣀⣋⣈⣉⣉⣈⣁⣀⣈⣃⠈⠉⠁⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣇⣀⣀⣀⣿⣇⣀⣀⣀⣿⣿⣀⣀⣀⣸⣿⣀⣀⣀⣠⣿⣇⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛⠃⠀⠀⠀⠛⠃⠀⠀⠀⠙⠋⠀⠀⠀⠘⠛⠀⠀⠀⠀⠛⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⣿⡇⠀⢠⡤⠦⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⢤⡄⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠋⠉⠉⠉⠉⠉⠉⢻⡏⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠙⡟⠉⠉⠉⠉⠉⠉⠙⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠋⠉⠉⠉⠉⠋⠋⢻⡋⠉⠉⠉⠉⠉⠉⢹⡟⠉⠉⠉⠉⠉⠉⠙⡟⠉⠉⠉⠉⠉⠉⠙⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠘⠓⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠚⠃⠀⢸⣿
⠀⠀⠀⠀⢻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⡀⠐⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⡀⠀⠀⣈⠀⠀⠀⣀⠀⠠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠁⠒⠂⠁⠀⠀⠀⠀⠀⠐⠒⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡒⢂⡀⠀⣨⠡⠤⠈⢀⠂⠤⠄⠉⠐⡒⢀⣀⠈⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠤⠄⠀⡬⠭⠄⠤⠮⠉⠀⠒⣒⠀⠈⠉⠐⣒⡂⠀⠀⠒⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣨⣭⠁⠀⠐⢀⣭⢭⠈⠁⠀⠐⡶⡈⠃⠀⢐⣒⡀⠀⠀⡉⠉⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠈⠀⠨⠽⠶⠄⠀⡈⢱⣶⠀⠉⠀⠁⠒⠂⠀⠀⠆⠂⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⠰⠶⠆⠀⠀⠴⠆⣭⡍⠀⠈⠉⠀⣭⣯⡄⠀⠀⠐⢒⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠅⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠛⣃⣀⠀⠀⢀⣐⠒⠂⠀⠀⠀⡀⠭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠠⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣉⣁⠀⠀⠈⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠷⠀⠀⠀⠀⠉⠍⠀⠀⠀⠀⠠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⣤⡴⠠⠄⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣷⣶⣾⣿⢠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠲⠂⠂⠀⠀⠀⠀⠘⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠠⣶⣶⣶⠀⠀⠀⠀⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠟⠚⠓⢶⡞⠀⠀⠀⠰⠶⠶⠆⠉⠁⠀⠀⠀⠀⠠⠶⠒⠒⠆⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⢠⣦⣴⣤⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⢀⣀⡀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠠⠶⠶⠶⠄⠀⠀⠀⠀⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⡈⠛⠀⠀⠀⠀⠈⠛⠛⠉⠙⠀⠀⠀⠀⠀⣀⡠⠴⣀⡀⠀⠀⠀⠀⢠⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠉⠀⠁⠀⠀⠀⠀⠿⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣤⡖⠂⠀⠀⠀⠀⣤⣤⣤⣤⠈⠡⠁⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢘⠂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣧⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣧⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⡘⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡼⢁⣠⣞⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢫⡾⠟⠡⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣶⣉⠉⠦⠀⠻⡜⢠⠔⠚⠃⠈⢍⣙⡦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠉⠶⣾⣿⣿⣎⢿⣟⣯⣭⣾⣿⡲⠮⢹⣿⣿⣿⣿
⠀⠀⠀⠀⢀⢊⡽⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡵⣒⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢹⡸⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢇⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠲⢭⣿⣿⣧⣴⣥⣽⣷⣤⣴⣔⣻⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣝⣒⠀⠀⠙⣋⠋⢋⣉⡛⠋⠻⠇⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠴⠂⠙⢿⣯⣧⣶⣿⣷⣶⣼⡿⠋⠐⠦⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣫⣽⣦⡀⠒⠘⠋⠈⠘⠉⠃⣀⣴⣯⣛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠚⠿⠥⠤⠬⣭⣭⣭⣭⣭⠥⠤⠴⠟⠛⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣭⣄⣛⣛⣓⣒⣶⣶⣖⣒⣛⣛⣛⣤⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠈⠳⣄⠐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠈⢳⣽⣷⣤⣹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢿⣷⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡿⡟⠛⢁⣀⣉⠛⢶⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⣷⣾⣯⡇⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡀⠈⠉⠀⠉⠉⠉⠓⢶⣴⣶⣲⣻⡿⠯⠄⠀⠈⠉⠉⠁⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⠓⠲⢯⣀⡀⡀⠀⠀⠀⢀⠀⡀⠐⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⣴⣦⣴⣾⣍⡀⠈⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣈⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣮⡾⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⡞
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡗⡀⠁⠓⢦⡀⡲⠀⠀⠀⢀⠄⠀⠀⠀⠀⢠⣠⠀⠀⠀⡾⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⡀⠀⠀⠸⣿⣡⣤⠤⡶⠋⠀⠀⠀⠀⣠⣛⢫⡧⠀⣠⣴⡇
⠀⠀⠀⠀⠀⡀⡀⠄⠹⣿⣻⡿⣃⠀⢀⣿⣿⣿⣾⡇⠀⠀⠀⠀⣼⣿⣿⣾⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⠿⠵⠂⣘⣿⠞⢁⣤⡾⠿⢿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠟⢿⣿⣿⡿⡄
⠀⠀⠀⠀⢐⠀⠀⠄⢉⠉⠀⠀⠀⠀⠀⣄⢤⠿⣿⠿⠧⠀⠀⠰⡋⡅⠀⠀⠀⠙⠛⠁⠀⢀⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠒⠂⠀⠈⠀⠀⠀⠒⠂⠃⠀⠚⠃⣉⡀⠘⠀⠊⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣥⣬⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⡇⠀⠀⢰⡏⠙⡇⠀⠀⢰⠋⠙⡆⠀⠀⢸⠋⢹⡆⠀⠀⢸⠉⢹⡄
⠀⠀⠀⠀⢰⠶⠶⠶⢾⡇⠀⡷⠶⠶⢾⡇⠀⡷⠶⠶⢾⠀⠀⡷⠶⠶⢾⠀⢸⡷⠶⠶⢾⠀⢸⡷⠶⠶⠶⡆
⠀⠀⠀⠀⢸⠀⠀⠀⣸⡇⠀⣷⡀⠀⣼⡇⠀⣧⡀⠀⣼⠀⠀⣧⠀⢀⣼⠀⢸⣧⠀⢀⣾⠀⢸⣇⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢿⡳⠴⣻⠇⠀⢿⡧⠴⣻⠃⠘⣟⠧⠼⣻⠃⠸⣟⠦⢼⡿⠀⠸⣟⠦⢞⡿⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⡄⠀⣠⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣄⠀⢠⣤⣤⣤⣤⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⣧⣤⣤⣤⠇⠀⢿⣤⣤⣤⡼⠀⠸⣧⣤⣤⣼⠇⠀⢧⣤⣤⣤⡿⠀⠸⣤⣤⣤⣼⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡶⠶⠶⠶⡆⠀⣴⠶⠶⠶⢶⠀⢠⡶⠶⠶⢶⡄⠀⡶⠶⠶⠶⣦⠀⢰⠶⠶⠶⢶⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠷⠶⠶⠶⠇⠀⠻⠶⠶⠶⠾⠀⠘⠷⠶⠶⠾⠃⠀⠷⠶⠶⠶⠟⠀⠸⠶⠶⠶⠾⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡟⠛⠛⠛⡇⠀⣾⠛⠛⠛⢻⠀⢰⡟⠛⠛⢻⡆⠀⡟⠛⠛⠛⣷⠀⢸⠛⠛⠛⢻⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠛⠛⠛⠛⠃⠀⠙⠛⠛⠛⠛⠀⠈⠛⠛⠛⠛⠁⠀⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠀⠀⡇
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢶⠀⠀⠀⠸⡇⠀⠀⡾⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚
⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⢀⠀⠘⢦⠀⠀⠈⢧⠀⠀⠀⡇⡀⢸⠃⠀⠀⡴⠃⠀⠀⣀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠑⢄⠀⠳⡀⢢⡈⠆⢰⢰⠇⠃⠏⡜⢀⠞⠀⠀⣠⠞⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⡑⢄⠈⢆⠑⠈⠈⢸⡀⠘⠌⡠⢃⠜⡡⠊⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠈⡑⠌⡂⠀⠀⡁⠀⣀⡈⣉⡀⠘⢄⠁⢈⠠⠚⣁⠄⠀⣀⣠⠴⠒
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⠀⠈⠉⡓⠨⠅⠀⠰⠊⡔⠉⠀⠀⠀⠈⠐⡄⠂⠁⢀⠩⠄⠒⠉⠁⣀⡀⢀⣀⢀⡤⠤⣤⣄⣠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠉⠉⠉⠁⠒⠂⠤⠄⠥⠸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠒⠋⡩⠛⠚⠉⠉⠉
⠠⠶⠒⠒⠒⠚⠋⠁⠀⠀⠀⠉⠑⠢⣀⣀⠀⠀⠀⠠⠸⡀⠀⠀⠀⠀⠀⣀⡼⠄⠒⠖⠒⢂⡶⠊⠁⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢄⠒⠋⠉⠉⠉⠉⠉⠀⠀⠀⣀⣀⠴⠚⣠⡴⠉⠀⠀⠀⣀⣠⠔⠒⠂⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠳⠠⠤⠠⠀⠄⠑⠔⠀⠀⠮⠿⢯⣉⠀⠐⠒⠒⠋⣱⠋⠀⠠⠀⠉⠂⠀⠀⠫⡄⢞⠤⢄⡠⢄⡀
⠀⠀⠒⠂⠘⠓⠘⠂⠀⠒⠀⠉⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣰⠚⢡⣆⢠⣀⣊⡲⠀⠀⠐⠊⠀⠎⠅⠀⠒⠶⠎⠡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣈⣡⣤⠼⠵⠞⠉⠉⢉⣁⠀⠉⢓⣒⣒⠲⠚⠙⠓⠓⠶⠒⠚⠒⠓⠷⠄
⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⠤⣤⠖⠒⠛⠛⠉⠒⣒⣒⠫⠅⠠⠸⠂⠁⠉⠉⠁⠀⣉⣉⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠖⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠁⠀⡀⠂⠠⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠸⠀⠈⠁⠀⠀⠀⠤⠠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠉⠈⠈⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠶⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠮⠤⠴⠖⠒⠚⠛⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⣀⣀⣀⣘⣆⣀⣀
⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣡⠖⠊⠉⠉⠒⢤⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣁⣀⣀⣀⣀⣀⣀⣣⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⢹⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠼⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠼⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⢹⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠼⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠼⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⢹⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚





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

⣯⡟⠏⠉⠛⠩⠍⠷⢻⣛⣯⡉⢹⡍⡸⢩⡜⠛⠻⣿⣹⣟⣹⢻⣿⣈⣹⣝⢉⡽⠯⢿⡛⢯⣙⠿⠍⣍⠍⣿⣯⣼⣵⠫⡯⢹⡁
⣇⣧⠀⡘⠈⠉⢠⣄⣤⣬⢭⣡⣬⡅⠉⠉⡏⢰⠀⠀⢩⢍⣩⣌⣍⣍⣭⣭⠈⠉⠻⣹⠉⣄⣭⣤⣭⣭⣌⣍⣭⣥⢭⠁⠩⣼⣤
⡷⡔⣲⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠃⠈
⡯⣎⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⠄
⣗⡧⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣙
⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢧
⣷⠎⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠳⡄
⣇⠀⣸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠹
⣏⡞⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢹
⣥⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢃
⣯⢐⣲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣿
⣾⠎⢳⡤⠀⣀⣀⠀⠀⣄⠀⡀⣄⠀⡀⣀⣾⣧⡤⠀⢀⣀⣀⣀⢀⣀⣄⣄⣠⢀⡧⠻⠀⣄⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠜⠐⢏
⣧⡂⣺⠟⠀⠩⠌⣄⣀⣉⣀⣡⣩⣠⡀⠁⡆⢹⠉⠁⠈⣩⣡⢉⣌⣉⣌⠉⠁⠉⠢⢹⠀⠈⣅⣍⠭⣉⣩⣠⣩⣡⡀⠁⠲⢄⡨
⣯⠛⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⢳
⡏⠒⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⢧
⡇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠻
⡧⢖⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⣕
⡁⢸⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⢀⢀⠀⠀⡀⠀⠀⢖⣒⡨
⠧⠌⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⠀⠀⠀⠋⠋⠉⠙⠉⠉⠁⠀⠀⣴⣲⠌
⡄⡪⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡾
⠓⠬⠤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⡭
⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠜⠛
⡟⠦⣄⣢⣀⣀⣀⣀⢀⣀⠀⢀⡀⠀⡀⠀⢧⣤⣀⢀⡀⡀⠀⠀⢀⡀⢀⣀⡀⣀⠔⣸⣀⣀⣀⢀⢀⢀⣀⠀⡀⣀⣀⣀⠴⡾⠿
⣏⣁⣂⢀⣀⣀⣾⡟⠂⣈⣒⣀⣠⣄⣀⣀⣁⣀⣿⣸⣄⠑⢄⣀⢊⢀⣠⣏⣀⣸⣀⣩⣿⣇⣏⣾⣰⣌⣓⣠⣃⣿⣖⣒⣒⣂⣼