Round Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⢀⡞⠳⣤⣀⡟⢻⣀⣤⠞⢳⡀⢀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠤⣄⢽⠀⢻⠋⡀⡆⣀⢘⠂⠐⣋⣀⢰⠀⠙⡟⠁⡯⣤⡤⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠦⣄⡺⠀⢆⢀⣈⡍⣥⢪⡿⠤⠝⢧⢹⡏⠼⠫⠬⢿⡕⣬⢩⣁⡀⡴⠀⢷⣠⠴⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣄⡨⢃⣸⣆⠽⠛⡂⠩⠄⠒⠂⡂⡀⣈⣁⢈⢀⠐⠒⠠⠍⢘⠛⠯⣰⣇⡘⢇⣠⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⠀⣤⢆⠳⢈⠣⠒⢉⣠⡴⠷⠒⠛⠋⠉⠉⠉⠉⠉⠛⠲⠾⢦⣄⡋⢒⠜⡡⡞⠰⣤⠄⡠⠀⣿
⠀⠀⠀⠀⠀⠀⠸⡔⠒⢿⠀⣤⣣⠞⣉⠆⢡⣢⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣔⡜⠠⣉⠳⣌⣴⠀⡿⠒⢢⠇
⠀⠀⠀⠀⠀⣀⣠⡹⠦⢊⢬⣑⠂⠜⢐⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡂⠣⠐⢊⡥⡑⠴⢯⣄⣀
⠀⠀⠀⠀⠈⣏⠠⢄⠲⣾⠦⡐⠡⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣖⠭⢊⠴⣿⠖⡠⠄⢹⠁
⠀⠀⢀⡠⣄⣝⡀⠀⠑⠃⣜⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⢣⠘⣶⡆⢀⣫⣀⣄⡀
⠀⠀⠈⠻⣤⣀⠇⣭⡍⡈⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢁⢡⡭⠸⣀⣤⠟⠁
⠀⠀⣠⠴⠋⠠⣮⡶⢠⠁⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠈⡅⢶⣵⠀⠙⠦⣄
⠀⠀⠙⣦⠘⢩⣅⠇⠈⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠁⠸⣠⡍⠁⣴⠋
⠀⢀⣤⠼⠳⠐⣍⣃⡁⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢈⣐⣩⠂⠞⠧⣤⡀
⠀⠈⠛⢲⡴⠀⣋⡍⡁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢈⢩⣙⠄⢦⡖⠛⠁
⠀⠀⣠⠟⢀⣘⠋⡆⢀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⡀⢰⠙⣃⡄⠻⣄
⠀⠀⠙⠲⣄⠀⢟⠶⠘⡀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢀⠃⠾⡻⠂⣠⠖⠋
⠀⠀⢀⣤⠟⠉⡆⣚⣃⢁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡈⡘⣓⢰⠉⠛⣤⡀
⠀⠀⠈⠙⠋⣫⠁⠸⡿⡄⢫⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡝⢠⢿⠇⠈⣽⠙⠉⠁
⠀⠀⠀⠀⢀⣏⠐⠋⠴⣿⠖⠡⣐⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢂⢌⠲⣿⠦⠑⠂⣸⡀
⠀⠀⠀⠀⠀⠉⠙⣳⠖⢌⢚⡉⠄⢦⠨⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠅⡴⠠⢈⡓⡡⠶⣞⠋⠉
⠀⠀⠀⠀⠀⠀⢰⠣⠤⣾⠀⠛⣙⡦⣉⠂⡘⠟⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⢣⠰⣋⣴⠋⠻⠀⣷⠤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠊⠀⠛⠎⡼⢊⡔⠤⣘⠙⠳⡶⠤⣤⣄⣀⣀⣀⣀⣀⣤⠤⢶⠞⠋⣇⠬⢢⡁⢧⠱⠛⠂⠑⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⠋⢐⡌⢻⠟⣲⣄⡅⣐⠂⠤⠄⠁⡁⢉⡉⢈⠨⠠⠤⠐⣂⠨⣠⣖⠽⡟⢡⡊⠙⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠖⠛⢷⠀⠎⠈⢉⣃⡋⢜⣶⡒⣢⡖⣸⣇⢢⣔⢒⣾⡣⢛⣘⡉⠁⠱⠀⡮⠙⠲⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠒⠋⣺⠀⣼⣄⠀⠇⠉⣨⠄⠠⣅⠉⠼⠈⣠⣧⠀⣗⠛⠒⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠈⢧⡴⠛⠉⣧⣼⠉⠓⢤⡼⠁⠈⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⢸⣄⣴⣉⠁⠈⣉⣦⣠⡇⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⡴⠋⣿⣸⢧⣄⢳⡞⣠⡼⣇⣿⠙⢦⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣸⡖⠛⠲⣧⡘⣳⢻⡘⠦⠎⠑⠴⢃⡟⣞⢃⣼⠖⠛⢲⣇⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢲⠾⡮⣻⡹⣡⠏⢉⢁⡠⣧⡖⠮⠵⢲⣼⢄⡈⡉⠹⣌⢏⣟⣥⠷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⢻⡀⠗⣪⠛⢄⡴⣒⠁⡜⢹⣌⠳⠞⣩⡏⢣⠈⣒⢦⡠⠛⣕⠺⢀⡟⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠸⡴⠋⣉⠻⡏⢁⢰⢫⡛⣬⣍⡃⢿⣿⣧⣼⣿⡿⢘⣩⣤⠟⡝⡆⡈⢹⠟⣉⠙⢦⠇
⠀⠀⠀⠀⠀⠀⠒⢶⣛⣻⣽⣚⠁⡰⣁⡤⠧⠹⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⠏⠼⠤⣈⢆⠈⣓⣯⣟⣛⡶⠒
⠀⠀⠀⠀⠀⠀⡔⢻⠉⢷⠲⡍⢩⡿⡗⢶⣿⣷⣾⣿⠟⣥⡷⢾⣬⠻⣟⣷⣾⣿⡶⢺⠿⡍⢩⠖⡾⠉⡟⢢
⠀⠀⠀⠀⠀⢾⡁⠀⣹⠶⢎⠀⢎⡇⣹⠆⣉⣿⣿⣿⣹⣏⠀⠀⣹⣏⣿⣿⣿⣉⠰⣏⢸⡱⠀⡵⠶⣏⠀⢈⠷
⠀⠀⠀⠀⠀⠀⠣⣼⣀⡾⠴⣃⣘⣶⡧⠾⣿⡿⢿⣿⣦⡛⡷⢾⢛⣴⣯⡿⢿⣿⠷⢼⣶⣃⣘⠦⢷⣀⣧⠜
⠀⠀⠀⠀⠀⠀⠤⠾⣭⣽⣻⢭⡀⠱⡍⠒⡖⣰⣿⣿⢿⣽⣿⣿⣯⡿⣿⣿⣆⢲⡒⢉⠎⢀⡭⣟⣯⣭⠷⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠳⣄⣉⣴⣇⡘⠸⣜⣴⠟⣋⡅⣾⣿⡟⢻⣿⣷⢨⣙⠛⣮⣣⠏⢁⣸⣦⣉⣠⠞⡆
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⣼⠁⡦⢝⣤⠊⠳⠭⡀⢣⣸⢋⡴⢦⡙⣇⡜⢀⠭⠞⠑⣤⡫⢴⠈⣧⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⢶⠟⣽⣱⡙⣆⣈⡈⠑⡟⠧⢖⡲⠼⢻⠊⢁⣉⣰⢋⣎⣯⡻⡶⠧⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢹⠧⣤⠴⡟⢡⡽⣼⢡⠖⢄⡤⠲⡜⣧⢯⡌⢻⠦⣤⠼⡏⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠳⣄⣿⢹⡞⠋⡼⢧⠙⢳⡏⣿⣠⠞⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⢸⠋⠻⣉⡀⢀⣩⠟⠙⡇⠈⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣁⠁⠞⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠶⢠⡆⣦⣀⠠⣴⣟⣁⡠⠤⠤⠄⠛⠙⠿⠏⠁⠲⣤⢦⡄⠐⡄⠀⠀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢠⣤⠶⠿⠀⠀⠀⠈⠉⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠲⣶⠔⢄⠀⠀⠈⠓⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠁⠂⠿⣆⡘⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⣐⠡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤
⠀⠀⠀⢀⡀⣤⠛⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢈⠁⡰
⠀⠀⠶⢀⡅⢀⣼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠌⠀⠀⠀⣠⢀⣀
⠀⠀⠀⠈⠁⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⡄⠩⣬⠁
⠀⠀⠀⠠⠂⢀⠆⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⡒⠆
⠀⠀⠀⠀⣴⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠃
⠀⠀⠀⠼⠅⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠎⠀⠔
⠀⠀⣘⢃⡀⠻⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⣀
⠀⠈⠉⠘⠁⠀⠀⢀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⢀⡋⠰⠄
⠀⠀⠀⠀⠀⠰⠁⣈⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡜⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠨⠋⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⠉⢷⡆⢄⡀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠢⣄⠀⠀⠈⠢⠺⠷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣀⣀⠀⠀⠀⢰⡶⠚⠛⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⠀⠣⠀⠛⠞⠳⠄⣀⣴⣦⣤⡄⠒⠒⠒⠍⣩⡿⠓⠊⠙⠇⠛⠰⠆⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢡⠆⡈⣁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⠻⠍⠉⠀⠀⠒⠒⠒⠒⠀⠀⠉⠩⢗⡲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣚⠍⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠩⣓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⡓⢄
⠀⠀⠀⠀⠀⠀⠀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡑⢆
⠀⠀⠀⠀⠀⢠⢞⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⡄
⠀⠀⠀⠀⣰⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆
⠀⠀⠀⣰⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆
⠀⠀⢰⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⡆
⠀⠀⡏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢹
⠀⢸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⣸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇
⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠈⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸
⠀⠀⢹⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⠏
⠀⠀⠀⢳⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⢳⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠞
⠀⠀⠀⠀⠀⠹⣜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣣⠋
⠀⠀⠀⠀⠀⠀⢰⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⠅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⡜⡘⢂⠢⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠲⠆⣸⠉⠀⠀⠀⢰⣀⡰⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠡⢀⠀⣀⡀⠀⠀⠀⠀⣇⣀⡆⠀⠀⠀⠉⣆⠰⠖
⠀⠀⠀⠀⠀⠐⠖⠉⢀⡧⢰⣤⡀⠉⠀⠈⣷⣦⡀⢿⣿⠂⠰⠀⠀⠀⠀⡆⠐⣿⡿⢀⣴⣾⠁⠈⠉⣀⣴⡆⣾⡀⠙⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⢁⣈⠣⠀⠀⠀⠀⠼⠿⠇⣀⠀⠈⢦⣀⢀⣀⣀⡴⢁⠀⣀⠾⠿⠧⠄⠀⠀⠀⠜⣀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢠⣶⣶⡈⠁⡠⣄⠀⢸⣿⣿⡇⠀⡠⠄⠈⢁⣶⣶⡄⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠸⢿⣿⡿⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠀⠠⣶⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠻⠆⠀⠃⠀⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⣰⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⠀⠀⠀⠀⠈⠛⠀⠀⠰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠀⠒⠀⠀⠀⠀⠀⠘⢁⢿⣿⣿⣿⣷⣶⣦⡀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⡿⠆
⠀⠀⠀⠀⢀⣴⠁⠀⣼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠝⢹⡿⠉⠙⠿⠻⠄⠀⠈⠙⣹⢿⣿⠿⠿⠂⣀⣤⣶⡾⠛
⠀⠀⠀⠀⠀⠻⠀⢠⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠠⠞⣻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⠀⢰⣟⠁⠀⠀⠀⠀⠀⠀⠁⠾⣿⣿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⡹⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⡀⠀⠀⠀⠀⣸⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠀⠀⢀⣸⡿⠏⠀⠀⠀⠀⠀⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⣾⣿⣅⠀⠀⠀⠀⠾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⡶⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠸⢿⣿⣿⠀⠀⠀⠠⠾⠟⣿⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠂⠀⠀⠀⢀⡀⠈⠉⠙⠉⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣦⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⢿⣿⣿⣿⣿⣷⣄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⢿⣿⣷⣤⠀⠀⠙⣿⣿⣆⠀⠉⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠙⣿⣿⣧⠀⠀⢹⣿⣿⠀⠀⠘⣿⣿⡆⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠙⠻⣿⣿⣧⠀⠀⢹⣿⡟⠀⠀⣾⣿⡟⠀⠀⣰⣿⣿⠇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⡀⠀⠘⣿⣿⠀⢀⣿⡟⠁⣤⣾⡟⠋⣀⣠⣴⣿⡿⠏⠀⠀⣰⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠛⠛⠻⢿⣿⣿⡄⠀⣿⡏⠤⠯⠥⢬⣍⡛⠶⣌⠛⣯⡻⣍⢉⣀⣀⣠⣶⣿⣿⠟⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡀⠀⠀⠙⣿⡇⣠⠽⠖⠒⠳⠶⣦⣌⠻⣦⡘⣷⡈⣷⡈⣧⢻⡛⠿⠟⠛⠁⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⢸⢋⡵⠞⠛⠛⠻⠶⣤⡙⣧⠘⣧⢸⡇⢸⡇⢸⡇⣷⢲⣤⣤⣤⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⣼⠟⠋⠉⠉⠉⠛⢿⣧⡼⣩⡴⠞⠛⠛⠛⠶⣬⡳⡜⡇⡿⢸⠇⣾⠃⣾⠃⣿⢸⡉⠛⠿⠿⠟⠋⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠋⠀⣀⣤⣤⣤⣀⠈⠿⡼⢋⣴⠾⠛⣋⣉⣛⣲⣕⣟⢥⡣⣫⠞⣡⡾⠃⣼⠏⣸⢹⣦⣄⡀⠀⣀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⣿⣷⣼⢡⡟⢁⣴⠟⣩⡴⢖⡾⣺⢿⣯⢬⣓⣚⣩⡴⠞⢋⣴⠏⡼⠙⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠉⠻⣼⢠⡿⢁⡾⢋⡴⢫⠞⡎⡷⡟⣝⠲⠦⠤⠶⠾⠛⣡⡾⣹⣇⠀⠈⠉⠉⠁⠀⢠⡆
⠀⠀⠀⠀⠀⠹⠀⠀⣠⣶⣿⣿⣷⣦⣼⢸⡇⣾⠃⣾⠃⣿⢸⡇⢷⡹⣮⡙⠳⠶⠶⠶⠛⣉⣼⠉⣿⣷⣦⣤⣤⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠉⠉⠉⠙⠚⣇⢿⡀⣿⠀⣿⠈⣷⡈⢷⣌⡙⠻⠶⠶⠶⢛⡡⣾⡀⠈⠻⢿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⢀⣴⣶⣶⣶⣾⣌⢧⡘⣧⠘⣧⡈⠻⣦⣉⠛⠳⠶⢖⣛⡍⠀⣿⣷⣄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⣿⣿⡿⠋⠀⠀⣀⣤⣽⣮⡳⠮⣙⡳⣤⠭⢉⣭⠏⢡⣿⡇⠀⠹⣿⣿⣿⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢠⣾⣿⡟⠉⠀⣠⣶⡿⠋⠁⣴⣿⠇⠀⢸⣿⣷⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⣿⣿⡟⠀⠀⢰⣿⣿⠃⠀⢰⣿⣿⡄⠀⠈⢿⣿⣿⣶⣤⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠹⣿⣧⠀⠀⢸⣿⣿⡀⠀⠈⢿⣿⣷⣄⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⡀⠀⠻⣿⣷⣄⠀⠈⠛⠿⠿⠿⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣣⣀⣄⣀⠀⣀⠀⠀⠰⣄⠀⠀⠀⠐⢦⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠃⠀⠀⣀⡀⠀⠛⠀⠀⡈⠁⠁⢉⠛⢣⣤⠀⠈⢉⣵⠆⢼⡄⠀⠑⠦⣉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣧⡀⠀⠀⠀⠀⢀⣉⡄⣜⡼⠯⠀⠀⢠⣼⠀⠀⠘⣾⣀⣀⠀⠀⠀⠈⠢⡈⢳⡀⠉⠒⠾⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡸⠏⢹⣁⠄⢚⣁⣦⡸⠮⢙⠀⠁⠀⢀⠘⠃⠀⠈⠉⠉⠻⠟⠀⠠⡤⣈⢘⠓⠸⢦⣻⡀⠀⡀⠀⠈⣷⣰
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠿⣤⣼⣯⡼⣾⡷⠟⠉⡠⠚⣘⡓⠂⠡⣤⡀⠀⠀⠀⠀⠀⢛⣤⠀⠀⠀⣹⠢⡀⠀⠈⡷⢄⡇⠀⠀⢻⣿⡆⠀⡀
⠀⠀⠀⠀⠀⢸⠀⣤⠋⠀⢹⣿⠟⣼⢋⡐⣈⠴⠀⠊⠉⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡇⢠⢸⣦⠀⠓⠀⢿⡄⢿⣦⠀⢦⢻⣇⣴⢿⡆
⠀⠀⠀⢸⡇⠘⡀⠀⠀⣾⡯⣿⣴⡷⢋⢁⠄⠄⡄⡠⢰⣦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠷⣽⢤⣦⣸⣀⠘⣏⠣⣸⠀⣿⣇⣸⡇
⠀⠀⢀⠀⣧⢀⢿⢀⢲⣧⢶⡏⡼⣧⠺⡒⡘⢎⠳⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣄⠹⣄⠉⢸⠟⣢⣿⣦⢹⡜⢹⣿⡟⠀⠄
⠀⢀⠀⡆⠸⣏⡆⣙⣿⢃⡟⣼⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠳⣼⡄⠀⠘⠻⣿⣧⣼⣿⡆⠶⠂
⠀⠘⣆⠘⡆⢹⣧⡾⢣⡿⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⣿⣷⢹⣿⠖⢶⢃
⠀⠀⢹⣤⠁⡌⢻⣿⡟⢀⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⠳⣿⣿⠀⣿⣇⢹⣁
⡀⠀⠀⣿⣶⡀⢸⣿⠗⢹⣿⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣼⣿⣄⢹⣿⢺⡅
⠹⣆⢸⣧⡝⣿⣿⣿⣶⣿⡧⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣌⣿⡏⢻⣿⣿⠂
⠀⠹⣿⢸⣸⢹⣿⣿⡿⠛⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⣿⣿⣿⡷⠍
⠀⠀⢻⣆⣧⠸⣿⣿⡇⣠⣦⡀⣰⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢾⡿⢸⣿⣿⣿⡇
⠀⠀⡄⢿⣾⢶⣿⣿⣇⡽⠋⠡⣹⣚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡖⡠⠀⡙⢾⣿⣿⣿⢻⡀
⠀⢦⣇⡼⣿⣢⣿⣿⣯⠀⠀⢹⣿⡗⡣⠀⠀⠀⡀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣿⡵⣾⡿⢾⣿⣿⣯⢸⣗
⠀⠠⣿⢖⠹⣷⣿⣿⣿⡄⠀⢭⣛⣟⣻⠅⢀⢦⣷⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠙⣾⣿⣿⡇⢸⣏⠀⣄
⠀⠀⠹⣧⢖⡙⢿⣿⣿⣷⡀⠐⣂⣿⡪⢄⢪⡿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣧⠤⢠⣾⣾⣷⡏⡿⠃⢰⠏
⠀⠀⠀⠲⣎⡖⡸⣿⣿⣿⣷⡄⢒⣾⡖⣺⣽⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⣖⣾⣳⢛⣿⣿⣿⣿⣷⠏⠀⠊
⠀⠀⠀⠀⠩⢾⡄⣿⣿⣿⣿⣿⣆⠿⣶⣿⣟⣒⣠⣤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⣺⣿⣤⡀⢰⠀⠁⢰⣶⣄⢺⣟⢅⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠉⢿⣹⣿⣿⣿⣿⣿⣷⣿⣟⢳⠛⣻⣟⣿⣷⣶⣤⠁⠀⠀⢀⠀⠀⠀⢫⣞⢯⡿⣸⣰⣀⡏⣿⢉⠾⣳⣾⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠿⠿⣿⣿⣶⣼⣧⣼⣽⣿⣿⣿⣁⠀⠈⣀⣀⡀⣀⣡⠷⠛⠓⣛⠛⠻⣿⣾⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡿⣦⣄⡀⠈⢻⣿⡟⠁⠀⠀⠀⡀⢀⠩⠛⠛⠳⢯⡔⠉⠀⣨⡐⠛⣟⢃⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢉⣿⠿⣿⣯⣉⣡⡾⢿⣥⠀⠀⣠⠔⠁⣙⣦⡤⣿⡷⠖⠛⠛⠛⠛⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠈⣿⣿⣧⣼⣾⣾⡿⠿⢤⣴⡛⠉⠿⣴⠃⠀⠀⠉⣀⣲⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠈⠻⠯⠀⠰⠋⠀⠈⠢⠴⠋⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⣴⢂⣴⠖⣀⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣸⠟⠑⣋⣁⠈⢁⡾⢉⠁⠀⠈⠁⠖⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡿⠃⠀⠰⠶⢶⣶⣍⠁⠀⢁⡠⡀⣔⣻⣷⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠈⠿⠀⢰⣿⡇⠀⠀⠙⠿⠇⠀⢿⡷⢃⢠⡉⠀⣰⣻⡌⠓⠟⢫⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⣴⣶⣰⣶⠘⢵⣥⣤⣄⠀⡷⢮⡛⠃⠀⠙⠿⠃⠀⠙⠛⢠⣏⡆⡛⢛⡁
⠀⠀⠀⠀⠀⠀⠈⢿⡷⠌⠋⠃⠓⠋⣀⠈⠚⠛⠃⠀⣠⡮⢁⣠⣀⡀⢠⡷⠃⣶⡆⠸⠿⠇⢸⡆⢻⣦
⠀⠀⠀⠀⠀⠰⠴⡄⠀⠀⢀⣢⣤⣌⢿⣿⠀⠀⠀⡄⠀⠰⢻⣿⣋⡀⠀⢀⠿⠀⠁⠀⢀⣴⢸⡷⣄⢩⡁
⠀⠀⠀⠀⢰⣿⡆⢀⠀⠀⢙⣲⣿⣋⠀⠀⠰⢧⠼⣣⢠⡀⠠⣽⣿⠋⠁⠞⢰⠀⠀⠴⠟⣡⣿⠇⣿⠸⣷⡀
⠀⠀⠀⠀⢀⣵⠿⡻⠗⢰⣫⠽⠿⣿⣷⠀⠀⠰⢾⡿⢿⣷⣿⠾⣿⣷⠀⠀⠘⢷⣶⣶⣾⠿⠋⣸⣿⠀⣿⡇
⠀⠀⠀⠀⠈⣉⠀⠀⠀⠚⠁⠀⠄⡘⠟⠀⠀⠤⠄⡔⣾⡿⠉⠰⢴⠒⠀⣠⡀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣿⠃
⠀⠀⠀⠀⠈⠻⢷⢆⠃⢀⣶⠆⠀⠈⠠⠀⠀⢀⣀⡙⠂⠀⡆⠀⠀⢶⡄⡟⡴⠗⠀⠀⠐⠛⠛⢁⡄⢠⡀
⠀⠀⠀⠀⠀⢴⣶⡄⠁⠈⢃⣴⡶⠀⠀⠀⡴⠋⠉⠙⠂⠘⢿⠀⣴⣷⡅⢀⣶⣷⠀⠀⠀⢠⣾⡿⣹⡘⡇
⠀⠀⠀⠀⠀⠀⠙⠫⠂⠀⠎⢋⡀⠀⠀⠰⢀⣤⣶⣶⣦⡀⠈⠃⠘⢿⠻⣼⢹⣯⣴⣦⠀⢀⣩⢠⡇⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⡇⢀⣴⡄⣾⠏⠉⠉⠉⠁⢠⣶⣿⣛⣻⡏⠛⠒⠛⠋⠸⠿⡳⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠘⠃⠘⣿⡇⢻⠀⣴⣿⠂⢀⠀⠉⠉⣰⣧⡇⠀⠠⠤⠤⠴⢊⣴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣄⡈⠻⡌⠃⢿⠁⠀⠘⣿⣶⣦⣘⠛⠁⣀⡰⢶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠶⠄⠀⠀⢀⣀⣀⣉⡭⢁⣴⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠟⠋⠀⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠊⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢄⡜⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢠⠋⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⣀⣄⣼
⠀⠀⠀⣠⠀⠤⠤⠖⠢⠄⠀⠲⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⠲⢦⣀⡀⠠⠞⠒⠀⠀⠈⠉⠓⠂⠀⠀⠀⢀⣤⡄⠞⠛⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣶⣤⣀⣀⣤⣶⣿⣶⣦⣤⣴⣶⣶⣿⣿⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢯⠁⠤⠭⢭⡭⣉⣀⣀⣀⣉⣉⣉⣉⣉⣙⡛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢿⠛⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠶⠿⠷⠖⠒⠒⠆⠠⣄⡀⠀⠉⠙⠻⡏
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠉⠛⠉⠛⠲⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⠠⣀⣀⠄⣠⡀⢄⣀⡠⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣐⠈⢂⡐⢿⠃⠆⠐⠊⢛⠋⠄⠠⡐⣿⡧⡘⠤⠤⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠃⢌⠻⠁⢉⡠⣴⡄⠰⢶⠒⡿⠤⠾⡖⢲⠇⣈⣤⣁⢰⠈⡻⠧⠰⢄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠻⢟⠨⣀⠴⣷⠐⣉⡃⡈⢦⣶⠆⣆⠰⣿⠆⢰⠐⣴⡤⢉⠸⠛⠤⣴⣌⠀⠄⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⠡⠀⣁⣦⠔⣉⡄⢄⠚⢿⡀⣶⡦⠙⠠⠿⠆⠋⠰⡿⠄⠻⣡⣾⢈⣿⠦⣈⠐⠿⣁⣁⠤⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡏⠋⠀⠺⠇⠈⡐⠻⣇⠸⠷⠊⠀⠄⢀⡈⢀⡠⣈⣀⣁⣀⠀⠀⠈⢉⠈⠡⣶⠇⢸⣷⠄⠛⢳⣬⢁⡿⠇
⠀⠀⠀⠀⠀⠐⠀⣨⡅⢀⢀⠜⠏⢰⣾⡄⢻⡶⠀⠀⢀⡠⣤⣾⡯⠿⠿⠗⠛⠿⠛⠻⡿⠿⣽⣦⣤⡀⠩⠉⠉⣤⡞⢠⣍⠡⣤⡐⠂⠈⠣⠀⠀⡀
⠀⠀⠀⠀⠄⢀⡆⢛⠛⠂⡻⢓⡐⢦⣤⠍⠀⠄⣡⣲⣿⡿⢉⡉⠄⠂⡒⣈⢍⣉⣨⡑⠒⠢⣘⢛⠳⣿⣦⡄⠀⠉⠐⡟⢛⣁⠉⢼⡤⢁⣶⡖⠀⠁
⠀⠀⠀⡀⠀⢒⠡⡀⢠⠛⠥⠻⠿⠄⠋⠐⡠⣶⡿⠏⡠⠒⣡⠆⢊⣁⣠⣤⣤⣤⣤⣬⣁⡐⠢⣌⠢⢍⠉⣾⣧⡌⠀⠹⠟⢡⣶⡄⣣⣄⠐⠀⢦⠀⠂
⠀⠀⠀⠀⡘⡿⠥⠈⠻⢃⣌⠲⣶⠂⠀⣤⣿⡇⠠⠊⡱⢃⣠⣶⣿⡿⠛⢋⣋⡙⠋⠛⠿⣿⣶⣄⠑⢄⠳⢌⠹⣷⡆⠠⠀⣩⣭⡥⠀⣼⣆⠂⣤⣁⠀⠄
⠀⠀⠀⠀⡇⠐⠀⠿⠇⠜⠛⢓⠁⠂⢰⣮⠉⡄⠃⡜⣡⣾⡟⠋⣠⣴⣿⣿⣿⡿⣿⣷⣦⣄⠘⢿⣿⣀⠻⡐⢄⠛⣟⣃⠈⡙⢏⣴⣷⠈⣄⡈⣈⠃⠄
⠀⠠⠀⢨⣴⢉⠲⠷⢂⡉⠻⡿⠀⢠⡿⡟⢡⢃⡔⣠⣿⡟⢠⣾⡿⣟⠮⢃⣃⣃⡳⢫⢿⣿⣧⡄⢹⣿⣆⢹⠀⡀⢿⣧⡄⠆⣨⣤⡬⠀⣉⡆⡠⠘⢇⠈
⠀⠀⠀⢄⢋⠁⢴⡇⠸⠛⢛⡀⡀⢨⣿⠆⠡⢸⠀⣿⡿⠀⣾⡿⢽⠂⠔⢉⣤⣤⡉⠢⡱⣺⣿⣷⠀⣿⣿⠀⡇⢰⢀⣷⠇⠀⡘⣥⣴⣧⢨⠉⠴⣾⡦⠀⠂
⠀⠈⠀⠝⡠⠄⣱⡤⢈⠚⠻⠇⠄⢬⣿⡐⡇⢪⠀⣿⡇⢰⣿⣿⠭⢸⠀⣿⣿⣿⣿⠀⡇⣒⣻⣿⡇⢸⣿⠀⡣⠰⠌⣿⣳⢈⢰⣦⡤⡁⠚⣏⠐⠊⣔⠀⡀
⠀⠠⠀⠾⡿⠖⣀⡃⢻⠟⢛⡌⠀⢰⢿⠃⠆⢸⠀⣿⣿⠀⢿⣟⡯⢌⠢⣈⠛⠛⢁⠔⠠⣓⣾⣿⠁⣾⣿⠀⠇⢂⠸⣿⡃⠐⠈⣤⣤⡆⢸⠷⢐⣡⡑
⠀⠀⡀⢑⡄⠂⠸⣍⡀⡒⠛⡋⠠⠘⣻⣷⠈⠀⣧⠹⣿⣧⡘⣻⣿⣷⣣⢮⢭⢭⣡⣲⣽⣾⡿⠃⣼⣿⠋⡜⢡⢊⣼⣿⠃⠠⣶⣦⣉⠣⢼⠦⢁⠛⠋⠀⠂
⠀⠀⠀⠐⢡⣅⡘⢙⡀⢿⠟⣱⣌⡀⢩⣽⣤⠑⠌⢦⠉⣿⣷⣄⠙⠻⢿⣿⣿⣿⣿⣿⠟⠋⣠⣼⡿⢋⡔⢀⠜⣠⡿⡇⠠⢀⠥⣤⡄⢰⣾⠀⡄⠘
⠀⠀⠐⠀⢙⠛⠠⠛⡟⠁⢊⣛⣛⠀⠂⢸⢟⣆⡑⢦⠑⢄⠙⠻⣿⣶⣦⣨⣈⣉⣁⣤⣴⣿⠿⠋⠡⠎⡠⠂⢸⣿⠋⠀⠐⠿⠦⡙⢁⣶⡀⢢⣶⡌
⠀⠀⠀⠀⠀⠣⠐⠆⠙⢫⠘⠿⠃⣴⣆⠀⠙⢻⡿⣀⣑⠢⡙⠢⢌⢉⡛⠛⠛⠛⠛⠋⢉⡡⠰⢊⠥⠊⣰⣿⠿⠊⠀⣠⠐⢶⣶⠒⣤⠃⠈⢂⠥⠀⠈
⠀⠀⠀⠀⢀⠀⠼⠿⢁⠺⣟⣀⠉⣥⣜⠄⣀⠠⠙⠻⣿⣦⣥⡉⠢⠤⢌⣃⣉⡑⣉⠌⠠⠐⣀⣁⣾⣿⠧⠋⠐⠀⣐⠛⠣⠌⢡⣮⠠⣬⣴⠸⠁⠐
⠀⠀⠀⠀⠈⠀⠐⢠⡄⠠⠌⠛⢂⣙⠃⡼⠛⣂⣘⡂⠍⠛⠻⢿⣶⣷⣦⣤⣴⣤⢴⣶⣶⢺⡿⠟⠚⡁⠀⠀⢾⣧⠘⡿⠃⣰⡜⠁⠢⠘⡋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⣦⢁⠛⠧⣤⠐⢿⡇⢰⠿⢂⡀⣁⡀⠀⠀⠉⢉⠉⡙⡈⠁⠀⢁⠐⠀⡠⢶⡆⢹⣦⠌⡀⢲⣮⠀⣀⣹⢰⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠐⡂⢉⠉⣶⠄⡉⠲⣿⠁⡿⢋⣦⠐⣾⠆⣤⠰⣶⠆⣄⠘⡿⠈⣷⣤⠑⠘⣈⠔⠻⢉⠐⢂⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠰⠀⡙⠟⠒⣤⡆⣁⠚⠟⠄⠏⠰⣿⠄⠹⠰⠿⠗⡈⢠⣉⠄⢾⠖⠋⡄⣡⣆⠀⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⠆⢲⣮⡀⠇⢉⠛⡉⢰⠧⠼⡶⠒⣾⠤⠷⠆⠘⠟⠊⣉⢀⣦⡑⠠⠄⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠒⠖⡌⢺⣿⠌⠂⠰⢡⣷⡐⠄⠠⢠⣷⠌⢡⠠⠍⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠑⠈⠋⠐⠙⠃⠂⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣶⣶⣦⣤⣍⡙⠲⢖⡶⣦⢄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠂⠉⠳⢍⡲⣝⡪⣕⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠉⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠙⣦⡙⢦⡙⢮⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⣀⣤⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡈⢻⣆⠙⣦⡙⢷⣌⠳⣄
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣠⣴⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣆⢻⣧⠈⢿⣄⠹⣧⡈⢷⣄
⠀⠀⠀⠀⣰⣿⠋⢀⣴⣾⡿⠛⠁⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠈⠙⢿⡄⢿⣇⠈⢿⣆⠘⣷⡄⠹⣦⡀
⠀⠀⠀⣼⡟⠁⣴⣿⠟⠁⢀⣤⣶⣿⣿⠿⠟⠛⠛⠋⠉⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀⠀⠀⠀⢸⣿⡄⠘⣿⡆⠘⣿⡄⠹⣷⡀
⠀⠀⣼⠏⢠⣾⠟⠁⣠⣾⣿⠟⠋⠁⠀⠀⣀⣀⣠⣤⣤⣤⣀⣀⣀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣷⣄⠀⠀⣿⣇⠀⢹⣿⡀⠸⣿⡀⠹⣷⡀
⠀⣸⡏⢠⡿⠃⣠⣾⡿⠋⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠈⠻⣿⣿⣷⠀⣿⣿⠀⢸⣿⡇⠀⢿⣧⠀⢻⣷
⢀⡟⢠⡿⠁⣴⡿⠋⢀⣴⣾⡿⠟⢛⣩⡅⠀⠀⢠⣶⣶⡦⠀⠈⠉⠛⠿⣿⣿⣿⣷⣄⠀⠀⠈⠻⡿⢸⣿⡿⠀⢸⣿⡇⠀⢸⣿⡄⠈⣿⡇
⣸⢃⡿⢁⣾⠏⢀⣴⡿⠟⠉⠀⣾⣿⡿⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄⠀⠀⠀⣾⣿⡇⠀⢸⣿⡇⠀⢸⣿⡇⠀⢿⣿⠀⠈
⡿⢸⠃⣾⠃⣠⣿⠋⡄⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⠀⣼⣿⡿⠀⠀⣼⣿⡇⠀⢸⣿⡇⠀⢸⣿⡄
⡇⡟⢸⠇⣰⡟⢡⣾⡇⠀⠀⣸⣿⣿⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢋⣼⣿⣿⠃⠀⢠⣿⣿⠁⠀⣸⣿⡇⠀⢸⣿⡇
⡇⡇⡟⢰⡟⢰⣿⣿⡇⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⢀⣿⣿⡏⠀⢀⣿⣿⠃⠀⣸⣿⡇
⢁⣳⡇⡿⠀⠘⣿⣿⡇⠀⠀⢸⣿⣿⡆⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠀⠀⢠⣿⣿⡟⠀⠀⣼⣿⡟⠀⠀⣿⣿⠁
⠘⣿⢱⡇⠀⠀⢿⣿⣿⠀⠀⠘⣿⣿⣇⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠟⠁⠀⢀⣴⣿⣿⠟⠀⠀⣰⣿⣿⠁⠀⢸⣿⡿
⠀⢹⢸⠀⠀⠀⠸⣿⣿⣇⠀⠀⢻⣿⣿⡄⠀⠈⢿⣿⣷⡐⣶⣶⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⣴⣿⣿⠃⠀⢠⣿⣿⠇
⠀⠈⣞⠀⣇⠀⠀⢹⣿⣿⡄⠀⠀⢿⣿⣷⡀⠀⠘⢿⣿⣷⡘⠛⠛⠛⠉⠉⠀⠀⠀⢀⣠⣶⣿⣿⡿⠋⠀⠀⢠⣾⣿⡿⠃⠀⢀⣾⣿⠟
⠀⠀⠘⠀⢿⣆⠀⠀⢻⣿⣿⡄⠀⠈⢻⣿⣿⡄⠀⠈⢻⣿⣷⣄⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⠟⠋⠀⠀⢀⣴⣿⣿⡟⠁⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠸⣿⣦⠀⠀⠹⣿⣿⣦⠀⠀⠹⣿⣿⣦⡀⠀⠙⢿⣿⣷⣌⠻⣿⣿⡿⠿⠛⠉⠀⠀⠀⣠⣶⣿⣿⡿⠋⠀⠀⣰⠟⢋⠄
⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠈⢿⣿⣷⣄⠀⠈⠻⣿⣿⣦⡀⠀⠙⠻⣿⣷⣦⣀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠋⠀⠀⠀⣈⣥⡶⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⠙⠿⣿⣷⣄⡀⠈⠙⢿⣿⣷⣤⣀⠀⠉⠛⠿⣿⣶⣶⣤⣭⣉⣉⣉⣁⣤⣤⠴⢖⡫⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣄⠀⠈⠻⢿⣿⣶⣄⡀⠈⠙⠻⠿⣿⣶⣦⣤⣄⣀⣈⣉⣉⣉⣉⣉⣭⡤⢖⣪⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⡀⠈⠙⠻⢿⣷⣶⣤⣄⣀⣈⠉⠉⠛⠛⠛⠛⣛⣉⣩⡭⠶⣚⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣤⣀⣀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⣛⣉⣭⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢌⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⡜⠁⠀⡠⠊⢈⣠⠋⠀⠀⢀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⣇⠀⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⡜⢀⠈⠔⠀⠘⢩⠟⠁⢀⡠⠄⢀⣔⡁⠀⠀⠖⣛⠖⠂⠁⠛⠻⣄⠀⢻⡀⠀⠀⢻⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⠊⠐⢠⠞⠀⠠⣀⠎⢠⢀⣴⣠⠖⠃⠀⠒⠒⠂⠁⠀⠀⢠⣿⣷⣾⠀⠑⢼⡏⠐⡄⢸⣇⢳⡆
⠀⠀⠀⠀⠀⠀⢀⡜⣱⠇⠀⡶⠁⢀⡴⢚⣡⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠠⠆⠙⢶⣤⣿⣿⡌⣇
⠀⠀⠀⠀⠀⢀⡎⠀⣿⠀⢸⣷⠞⠉⣴⡟⣁⠀⠀⠀⠀⠀⠀⠠⠄⢀⣒⠒⣢⣄⣀⣻⣿⣿⣿⣿⠀⠀⠀⠈⢻⣿⣿⣿⠈⢦
⠀⠀⠀⠀⠀⣼⠀⠀⣿⠀⣼⡏⠆⣸⡿⠁⠀⠀⠀⠀⠀⠀⠒⠲⠤⠤⠀⠛⠉⠁⠈⠉⠉⠛⣿⣿⣆⠀⠀⠈⢹⣿⣿⣿⠀⠈⣧⢰⠂
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣶⣿⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠳⢦⣀⣼⣿⣿⣿⠀⠀⣿⠋
⠀⠀⠀⠀⠀⢿⡄⠀⣿⣿⣿⣿⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡆⣼⠃⠀⢠⡀
⠀⠀⠀⠀⠀⠘⣧⠀⢸⣿⣿⣏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠹⡆⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢰⣿⣿⣿⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣑⣭⣽⣻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⢻⡄⠀⠀⠀⠀⠀⠀⠀⢐⣷⡀⠀⣠⠀⠀⠀⢠⡇⠀⣥⣠⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣀⢀⡀⠀⠀⠀⢠⣾⣗⢀⣿⡄⠀⢰⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⠋⠛⠛⢻⡟⠛⠛⠟⠷⠶⣴⣿⣿⣽⣿⡿⠿⠿⠿⢿⠛⠉⠀⢀⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢿⣶⣤⡴⣾⣷⢾⣷⠧⢄⡐⠶⣾⣟⣛⣛⣟⡥⢠⡄⣿⡀⠀⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠼⠿⠿⠿⠀⠀⠀⠒⠻⠿⠯⢭⠧⢀⠸⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⠒⠛⠛⠓⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣟⡻⣿⣦⣠⣿⢏⣵⣝⢿⣦⣤⣿⡿⣛⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣯⣄⣾⡟⡞⣿⡲⣽⣿⡳⣣⣾⣮⢲⣻⣻⠵⣺⡟⡘⣿⣄⣭⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⣲⣾⢽⢻⣻⢽⣺⣿⣿⣷⣕⣽⣿⣿⣿⣷⣓⣵⣿⣿⣿⣺⢽⣻⢫⢵⣶⢺⣿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣅⣼⣿⢘⣸⣿⣷⣮⣷⣿⣿⠻⡏⢻⡉⣛⣘⣋⣸⠋⠿⠛⣿⢿⣯⣯⣶⣿⣿⣺⢸⣿⣤⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢹⣽⢹⠽⢽⣺⣿⣿⢿⣏⢛⣤⣵⣶⣿⣿⣿⣿⠿⢿⣿⣿⣿⣶⣧⣼⣁⠿⢻⣿⣿⣺⢺⠽⠽⢽⣽⢹⣿⣷⡀
⠀⠀⠀⠀⠀⡴⣿⠟⣿⡞⠯⣿⣿⣿⣿⣾⠛⣦⣾⣿⣽⣿⠯⠼⡹⣛⡫⣕⠡⣻⡻⡹⠬⢿⣿⣭⣷⣮⣁⠿⢾⣿⣿⣿⡏⠿⣾⡟⢿⣿⡄
⠀⠀⠀⠀⣼⣿⣯⣶⣿⡷⣹⢽⣿⡿⠧⣨⣾⣿⣋⡝⡙⣵⢭⣟⣼⣯⣿⣶⣾⣯⣿⣜⣧⢵⡝⡙⣍⣿⣿⣦⡛⣻⣿⣿⢹⣽⣿⣷⣮⣿⣿⡄
⠀⠀⠀⣸⣿⣟⣯⣧⣢⣮⣭⣷⣟⢓⣼⣷⣿⡿⣒⣒⣡⣿⠿⠛⢩⣅⠀⣶⡆⢀⣭⠙⠻⢿⣇⣓⣒⡺⣿⣾⣿⣍⡼⢷⣷⣭⣦⣢⣯⢏⣿⣿⡀
⠀⠀⢠⣿⡿⣿⣎⢯⢻⣿⣿⣯⣉⣾⣿⣇⠗⡃⢿⣿⠟⠅⢻⣦⣘⣿⣬⣿⣯⣾⣟⣥⣾⠂⠙⢿⣿⠇⡑⢇⣿⣿⣦⠾⣿⣿⣿⣫⢏⣾⡿⣿⣇
⠀⠀⣸⣿⣇⣼⣿⠯⡳⡽⣿⡤⣼⣷⣿⡗⣭⢶⣿⠃⠻⣶⣮⣿⣿⣟⢿⢻⠟⡿⣿⣿⣯⣶⡾⠢⢻⣷⢮⡵⣿⣧⣿⡞⢻⡿⡽⡢⣿⣿⣄⣿⣿
⠀⠀⣿⣿⢟⣫⢳⣭⣷⣿⡷⠤⣿⡟⢱⢱⡹⣿⠇⢶⣷⣽⣿⣟⠓⣼⣾⣿⣿⣶⣍⠾⣿⣿⣶⣷⠆⢿⡿⣱⢱⠹⣿⡇⣙⣿⣷⣯⣵⣫⣛⢿⣿⡇
⠀⠀⣿⣿⣮⣛⢝⣿⣿⣿⠷⢺⡿⣿⣷⣩⣾⣿⢠⣤⣮⣿⣿⠦⣼⣿⣿⣿⣿⣿⣿⣎⣹⣿⣯⣤⣤⢸⡿⣮⣳⣿⡿⣿⣈⣽⣿⡿⣛⢝⣯⣾⣿⡇
⠀⠀⣿⣿⠙⣿⣾⢝⡩⣻⡟⢺⣿⡿⢝⢮⡻⣿⠀⢉⣩⣿⣿⣋⠹⣿⣿⣿⣿⣿⣿⠳⢾⣿⣯⣉⠁⢸⡿⣫⢞⠿⣿⣿⢤⣽⡻⣘⢽⣾⡟⢹⣿⡇
⠀⠀⢿⣿⣾⣟⡫⣻⣾⣿⣟⣉⣿⣷⣜⡜⣼⣻⣇⠛⠟⣹⣿⣿⣖⣩⢛⠿⡟⣏⢧⣽⣿⣿⡛⠛⢃⣿⢿⡜⣜⣼⣿⡧⢴⣿⣿⣾⡫⡙⣿⣾⣿⡃
⠀⠀⢸⣿⣏⡺⠟⡻⡿⣿⣿⣁⣹⣯⣿⣟⠭⡛⣿⣎⠺⠻⢫⣿⣿⣿⣾⣾⣷⣿⣿⣿⣯⠻⠻⢂⣾⢟⡫⠕⣿⣯⣿⠳⣶⣿⡿⡿⡛⠿⣊⣿⣿
⠀⠀⠀⣿⣿⠿⢿⣾⡽⣳⣿⣯⣤⠻⣿⣤⣗⡄⣿⣽⣷⣄⠺⠋⢸⣿⠋⣿⡏⢻⣷⢈⠻⢂⣴⣿⣽⡇⣖⣧⣿⣿⣉⢻⣿⣷⡳⣵⣾⠿⣿⣿⡏
⠀⠀⠀⠸⣿⣧⣸⡿⡵⣳⣿⣿⣷⡖⠹⣿⢿⣿⡒⡒⡎⣿⣿⣶⣬⣅⣀⣛⣀⣀⣭⣴⣾⣿⡇⡖⡒⣺⣿⢻⡿⣥⣹⣿⣿⣷⡱⠽⣿⣤⣿⡿
⠀⠀⠀⠀⠹⣿⣿⠣⢿⢛⢟⣻⣿⣻⣏⣨⠻⣿⣧⣽⣴⣝⠭⠲⡹⣯⡿⣋⡻⣯⡟⡵⠫⢭⣵⣾⣥⣿⡿⠋⢶⣾⣻⣻⣻⢟⢿⢧⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠹⣿⣿⡿⢾⣶⣞⡞⣾⣿⣿⣶⢋⡹⢿⣽⣿⣯⣹⣼⣶⣎⠔⢍⣶⣾⣼⣭⣿⣿⣭⡟⢫⣄⣿⣿⣿⡾⡾⣾⣾⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⣿⡇⣇⡿⠿⡿⣻⣻⣧⣶⣈⣽⢛⡿⠿⣿⣿⣿⣿⣿⠿⡿⠛⣋⢹⣤⣿⣾⣿⡻⡿⠿⣇⡇⣿⣇⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣗⣝⣯⣥⣿⡞⡽⣿⣿⣿⡿⢾⣧⣾⣤⣶⣰⣦⣼⣦⣿⣶⢻⣿⣿⣿⡿⡝⣶⣯⣭⣝⣇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣯⡉⢿⣇⢣⣟⣯⢗⣫⢷⢻⣿⣿⠿⡱⡽⣿⣿⣿⢓⢯⣒⢭⣻⣯⢓⣿⠏⣩⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣎⣽⣾⡿⢿⣯⣪⣿⡱⣫⣾⣮⡱⣹⣧⣢⣿⠿⢿⣾⣍⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣬⣿⣷⣹⣾⣿⣋⣛⣿⣮⣱⣿⣿⣶⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠟⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣶⣶⣄⠀⠛⠘⠃⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠋⠀⣼⡏⠈⢻⡆⠀⠀⢀⡾⠛⠁⣿⠄⠀⠛⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣄⠀⠀⠀⠛⠀⠀⠀⠈⠀⢀⠂⠀⠀⠀⠻⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢻⣇⠈⠙⠛⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠚⠛⠛⢉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠋⠀⠈⢿⠀⠀⠀⠀⡀⢀⡀⢰⠀⢇⢰⡄⡜⢀⠂⠀⠀⢈⠀⠀⠀⠀⣾⠋⠀⠸⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠐⠀⠠⢄⠑⢤⢹⣮⠀⡀⠛⠫⡀⠈⣴⡇⠤⠂⡀⠀⠀⠂⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⠛⠋⠁⠐⠀⠀⠠⣄⣃⠀⠈⠀⡄⠀⠱⠀⠀⣠⠈⠀⢀⣞⣠⠂⠀⠀⠂⠀⠲⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠐⠒⠚⠛⠀⣄⠀⢈⠆⡜⢢⠰⡁⢀⡌⠈⠛⠡⠖⠂⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠀⠐⠀⠈⢉⣑⣄⠀⡀⠀⠈⠤⡕⠉⠉⢪⠬⡁⠀⡀⠀⣠⣚⠈⠀⠀⠄⠘⠋⠁⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣀⣠⠄⠀⠀⠀⠈⠭⠋⠀⠊⠁⢁⠒⡢⡀⣀⣜⢒⡁⠀⠁⠀⠛⢍⣁⡀⠀⠀⢀⡀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⠁⠀⠀⠀⠀⠤⠖⢠⣤⠀⠖⠁⢈⠄⢃⠘⠠⠁⠈⠓⢀⣤⡤⠄⠄⠀⠀⠀⠀⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠤⠀⠠⠀⠀⠐⢉⠍⠑⢄⠀⠊⠀⠠⠄⠀⡙⠀⠀⠈⢍⠙⠀⠀⠀⠄⣀⣤⣤⣬⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⠐⠀⠐⠁⡠⠊⣼⢿⠀⠂⣤⡄⠀⢀⠿⡇⠢⡀⠑⠀⠀⠀⠀⠀⠈⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⣴⡏⠀⠀⠀⢀⠀⠀⠁⠘⠀⠎⠸⠁⠃⠘⡄⠈⠀⠀⠀⠀⠀⠀⣦⠀⠀⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣠⣤⣤⠄⠈⠀⠠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠄⠌⠠⣤⣀⠀⢿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⢠⡆⠀⠀⠀⡈⠀⠈⡀⠀⠀⢀⡆⠀⠀⠈⠛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣤⠀⢸⣿⢀⣴⡟⠀⠀⠀⢻⣦⠀⣾⡇⠀⣠⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⣤⣀⣤⠀⠹⢷⠟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡀⢀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⢠⢐⠋⡳⠗⠰⣻⠀⢁⡜⢃⠀⠓⡦⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣉⢱⡇⣸⠊⣻⡅⣀⢳⡽⢀⡄⠀⠑⠆⠀⠉⠑⠂⠠⢀⠈⢀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣤⣙⠎⠋⠙⢚⣩⣀⡯⢙⣿⢭⣤⣀⡀⢄⠀⠀⢀⠀⠩⡄⣭⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢸⣷⡤⣷⡞⠘⠂⢨⣷⠴⠒⠊⠉⠀⠀⠀⠀⠉⠁⠀⠮⢈⢑⣾⢅⣂⠐⠀⠀⢀
⠀⠀⠀⠀⠀⢤⣄⣀⣿⡉⠿⠼⠛⢹⣯⣷⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠣⣣⡀⠅⠀⠉⠙⠉⠆
⠀⠀⠀⠀⠀⠀⣹⡧⠄⠉⠀⠀⠀⠜⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⢀⠀⠠⠶⢦
⠀⠀⠀⠀⠘⠛⢯⡴⢀⠀⠐⠃⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⣠⡾⢤⡏⠀⢀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⣤⣶⠃
⠀⠀⠀⠀⠈⠀⠀⢠⣿⠖⣧⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠚⠭⡁⢸⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠒⠐⠊⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣾⡜⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠁⠀⠀⠈⠿⠓
⠀⠀⠀⠀⠀⠀⡔⠐⢙⡿⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⡑⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠠⠴⠶⠛⡿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢯⠀⠀⠀⢐⣧
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⣸⠏⠈⢻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⣴⣜⣢⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢛⣂⣀⣷⡟⡙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⠄⠁⠈⠪⠅⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢻⢤⣬⣧⡸⠿⣀⡩⣷⣦⠄⣀⣀⠀⠀⠀⠀⠀⣀⡠⠄⠘⠹⠟⠛⠀⠃⣦⠥⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡇⣴⠟⣱⢰⡏⠳⢤⣵⣊⣁⡭⢽⢟⠭⣉⠀⠀⠀⢠⠐⠃⠀⣱⣄⠀⠃⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠀⣩⣜⡛⢶⠍⣭⠉⠀⢀⠎⠘⠄⠀⢁⠠⡄⠤⢴⡣⠖⢽⡱⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠎⠈⠛⠀⠀⠀⢡⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⣶⣶⢶⣿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⣹⢻⣄⠀⢀⣀⠀⠀⠈⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⡟⣆⢈⢳⠈⣿⡷⠒⠳⠴⢦⣀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠀⢿⡻⠀⠀⠀⠀⢉⣧⣤⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⡖⠦⠴⡶⠤⠾⣿⠟⠀⠀⠈⢻⣦⢷⢿⣿⣿⡽⢋⡗⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣏⣠⣤⡴⣦⣀⡀⠀⠙⢶⣠⣿⠀⠀⢸⠶⢦⠀⠀⠀⣙⣦⣻⣿⣿⣽⣯⠗⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⡰⠁⠛⣿⣿⣿⣿⢿⣷⡄⠀⢸⣿⣧⣶⣤⠞⠀⢸⣇⢀⡞⠀⠻⣿⣿⣿⠟⠁⠀⣠⣤⡀⠀⣀⡀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣵⣴⢦⠶⡋⣸⡟⠀⠀⠀⣹⠹⡏⠀⠀⣸⠁⠀⠉⣿⣭⠿⡄⠀⠀⠘⡆
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⣿⣿⣽⣿⠏⠀⢀⡿⠟⠛⠻⠿⢿⣿⠶⣻⣷⣤⡤⠴⠋⠀⣧⣾⢛⢹⢀⣵⣰⣿⣿⠄⢹⡆⠀⠀⢹⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⠈⠻⡟⣀⡀⣠⠟⠁⠀⠀⠀⠀⠘⡇⡴⠟⠛⢦⡄⠀⠀⢀⡇⠉⢹⣿⣿⣷⣿⣛⣁⡤⢤⡗⠀⠀⠀⣇
⠀⠀⠀⢀⠇⠀⠀⠀⣼⠉⠈⢳⡟⠉⠛⠛⣦⢰⠛⠙⠳⣄⠀⣧⠞⣀⢤⣾⡷⠚⠓⢾⠁⠀⢸⣿⣿⣿⣿⣷⣾⡶⣷⡄⠀⠀⠀⢹
⠀⠀⠀⢸⠀⠀⠀⡔⠉⠀⠀⢠⣿⢀⠀⢀⣿⠇⣀⣴⣾⣿⣷⣧⣤⡈⣿⣿⣃⣤⠀⣸⠇⣠⠾⠛⠛⠻⣿⣿⣿⡿⡊⣷⠀⠀⠀⢸
⠀⠀⠀⠸⠀⠀⠀⣇⣀⣀⠀⢸⣿⢸⡴⠋⢿⠞⠋⠉⠛⠿⣿⣿⠃⢻⣿⣿⣿⣿⣿⠛⢻⡁⠀⠀⠀⠀⠘⠟⣭⠟⠛⠃⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⢠⡏⠉⠹⢷⣾⣿⡟⣧⠀⣿⠀⠀⠀⣲⠶⢾⣏⠀⠈⣿⡿⠿⢿⣟⣢⡼⠃⠀⢀⡠⠴⠖⠒⣄⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢰⠀⢾⡀⠔⣿⠋⡽⢻⠀⢹⠀⠻⢤⣀⡰⠏⠀⣿⠸⡆⢰⠏⠀⠀⣼⡥⠤⢴⣶⣿⣿⣿⣷⣦⣤⡜⣇⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢆⠀⣏⠠⢿⢰⠇⢸⠂⣸⠀⣀⣀⡜⠉⢲⡤⢿⠀⣷⠏⠀⢀⣼⠋⠀⠀⠀⠘⣿⣿⣿⣯⣍⣉⡉⢻⡆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠈⣄⠈⠛⠉⡿⠀⡟⠀⡟⠘⠁⠉⢻⣼⣿⣡⠀⣷⡿⠀⢠⠏⡿⠀⠀⠀⠀⠀⣿⢿⠿⣿⣯⣭⣄⣸⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⡇⢰⠇⠀⡟⢧⡴⠶⠿⣿⣿⣿⣹⢿⡇⢠⣯⡾⠓⠒⢦⡀⠀⠀⣿⣸⡇⣟⢸⣿⣩⠏⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣸⡇⢸⡀⠀⣇⠀⠀⠀⠀⡿⢳⠏⠀⣟⣴⣯⠁⠀⠀⠀⠀⠹⠀⣼⣯⡟⣷⠘⣈⣿⠛⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣘⡇⠀⢹⡄⠀⢀⠎⢀⡟⠀⢸⣿⢿⠟⠷⣄⠀⠀⠀⣸⠀⠉⣠⣝⣋⠛⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⡀⢻⡀⡟⠀⣸⠁⠀⢸⡇⢸⡀⠀⣿⠻⠤⠞⠁⢠⣾⣧⡧⢉⣷⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢧⣀⣿⠀⠀⠸⡇⠸⡇⠀⢹⡴⠒⠒⠚⢛⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠛⠶⠿⠶⠾⠓⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠛⠛⠉⠉⠉⠉⠙⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢍⠒⠐⠀⠀⠀⠀⣀⣞⣁⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⣶⣿⣿⣯⣹⠀⠀⠉⠳⡀⠙⢿⣄
⠀⠀⠀⠀⢀⣾⣃⣤⣴⡶⠶⠶⠶⠶⠶⠤⣤⣀⠁⠁⢀⡄⠀⠀⣸⣶⣮⠝⢿⡏⠀⠀⠀⠀⡗⠠⠈⣻⣷⡀
⠀⠀⠀⢀⣾⡿⠛⠉⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢙⣶⣻⡟⣿⠿⣯⠘⣖⠒⠟⡇⠀⠀⣠⠞⢁⣤⡾⣹⢣⣿⡄
⠀⠀⠀⡾⣡⠖⠛⠉⢉⡭⠭⠛⠛⠋⠿⡻⣷⣶⢾⠟⣿⣇⢟⣦⠈⢥⢹⠀⠀⣧⠔⢫⣴⡾⡿⣼⣳⠯⣟⣲⢷⡀
⠀⠀⣸⢿⠁⣀⢠⡞⠁⣤⣤⣤⣤⣄⣴⡿⠻⠇⠛⢺⣿⣿⡼⠋⠛⢾⣼⣿⣊⣷⣿⢳⣾⡳⢯⣷⢮⣽⢲⢏⡾⣇
⠀⠀⡟⠘⢤⣬⣼⡿⠖⢄⠀⠈⠭⠉⢿⣄⠤⠌⣈⢻⣦⣿⣧⣈⣠⡼⠟⣿⣿⣗⡧⢯⡞⣽⣓⢾⣿⣥⣟⠾⣽⣿
⠀⢠⡇⠀⠀⠈⠉⣦⠀⠚⠛⣻⡖⠋⠉⢹⣀⢾⣽⣏⣻⣿⣿⣿⠅⣂⠉⣟⢿⣿⣽⣻⠾⣵⢣⣟⡾⣱⠿⣽⣣⢿⡀
⠀⠈⡇⠰⠶⢤⠤⢽⣄⠀⣈⡿⣧⣈⣁⣿⣿⣠⡿⢏⢡⢻⣷⣻⣆⠏⠉⠄⢹⠈⣿⢟⣿⠖⢿⠞⠛⠷⠻⠓⡛⢻
⠀⠀⣷⠠⠜⠃⠀⠀⢨⡟⠁⠈⠻⠉⣩⣿⡯⠼⣷⡀⡀⠈⢻⣯⣮⡦⣄⡀⠀⠀⠸⣄⣿⠆⡼⠞⢩⡖⠀⢀⡄⣾
⠀⠀⢸⡄⠒⠓⢆⣰⠞⠰⡄⠀⣠⣾⣿⡛⠷⢡⡚⣧⠌⠀⠀⠉⠻⣿⣵⣯⣟⣖⣆⣻⡅⠠⡥⠀⡨⡷⠀⠸⢠⡇
⠀⠀⠀⢷⠀⠀⢺⢃⡀⠀⣡⣾⣿⣿⣿⡇⠀⠈⠳⣭⢳⢦⡀⢀⠀⡀⠊⠙⠙⠛⠻⠿⢿⡀⠀⠀⢠⠇⣠⢗⡿
⠀⠀⠀⠈⢧⠐⠋⠈⠛⣴⣏⡹⣿⢟⣥⣤⠀⠈⠆⣶⣵⡪⢛⡷⡶⣦⣂⠰⠠⠂⠀⠀⠀⠹⡄⠀⠸⡚⢁⡾⠁
⠀⠀⠀⠀⠈⢳⡀⠀⣾⣿⣿⡿⠨⡙⢿⣿⣆⠀⣸⣿⣿⣿⣷⣿⣼⣞⣽⡽⣟⡛⡟⣟⣟⣛⡟⣶⣀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⠃⠀⠠⠀⣿⣿⣾⠿⠟⠋⠙⠙⠙⠛⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⡀⢀⣴⡿⠋⢸⡷⢲⠶⠾⠶⢴⣤⣠⣀⠈⠛⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣛⣾⣷⡐⡘⣿⡝⠧⣀⠀⠀⠀⠉⠛⢷⣆⢈⣹⡯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣽⣷⣶⣿⣷⣤⣀⣤⠤⠴⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⠖⣀⠀⠀⣌⠓⠚⡡⠀⠀⢀⠲⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣃⡤⢀⠀⠀⠲⠀⠀⠓⣠⣤⣉⣀⣀⣁⣤⣄⠚⠀⠀⠖⠀⠀⡀⢤⣚⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠀⠙⢂⣤⣦⣴⡾⣿⣿⠹⣿⣯⣽⣿⠏⣿⣿⢶⣦⣶⣦⡀⠋⠀⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣢⠆⣤⠀⡀⣅⣤⡶⣿⣿⢈⡛⠟⣛⣡⣶⣦⣭⣭⣴⣶⣌⣛⠛⢛⡁⣿⣿⢷⣤⣨⢀⢀⣤⠰⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⣠⣿⡟⠻⠿⢟⣡⣾⣿⣿⣿⣿⣻⣿⣻⣟⣿⣟⣿⣿⣿⣿⣷⣌⡛⠿⠟⢻⣿⣄⠀⠀⣬⠁
⠀⠀⠀⠀⢀⢀⡀⡀⠀⠠⣤⣾⣯⣿⠇⣼⣿⣿⢿⣟⣶⣿⣿⣿⠿⠛⢻⡟⠛⠿⣿⣿⣿⣾⣻⡿⣿⣿⣧⠹⣿⣽⣷⣤⠄⠀⣀⢀⡀⡀
⠀⠀⠀⠀⠘⡽⠃⠉⠁⣽⣾⡙⣋⣵⣾⣿⣻⣿⣿⠿⡯⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⢽⠿⣿⣿⣝⣿⣷⣮⣙⢉⣿⣯⠈⠉⠘⢧⠋
⠀⠀⠀⠀⠀⠐⠓⢀⣾⣯⣿⢃⣿⣿⣯⣾⡿⣯⠁⠀⠀⣀⣀⣠⣷⣴⣼⣧⣦⣾⣆⣀⣀⠀⠀⠈⣽⢿⣷⣽⣿⣿⡸⣿⣽⣧⡀⠚⠂
⠀⠀⡀⣀⢠⡀⠰⣿⡛⢛⣡⣾⣿⢧⣿⡟⠁⠀⠀⣤⣰⣾⢯⣻⣽⣾⣿⣿⣷⣯⣟⡽⣿⣆⣤⠀⠀⠈⢻⣿⡾⢿⣷⣌⡛⢛⣿⠆⢀⡄⣀⢀
⠀⠐⢣⠟⠀⠀⣸⡿⣿⠈⣿⣿⣿⣿⠏⠋⠀⢠⣰⣾⣻⣿⣿⣏⣿⣼⣷⣾⣧⣿⣻⣿⣿⣿⡷⣆⡄⠀⠙⠹⣿⣧⣿⣿⠁⣿⢿⣆⠀⠀⠻⡜⠂
⠀⠀⠀⠈⠙⣠⣿⣿⠟⣸⣿⣿⣿⣿⡄⠀⢀⢸⣿⣳⡿⢿⣾⡻⢍⣣⣵⣮⣜⢩⢟⣷⡿⣿⣾⣽⣇⠀⠀⢠⣿⣿⣿⣿⣇⠻⣿⣿⣄⠋⠁
⠀⠀⡀⢄⡄⢺⣷⣦⡸⣿⣿⣾⣿⡃⠀⠀⢙⣿⣳⣿⣛⣿⠧⣩⣯⣿⣿⢿⣿⣽⣎⡼⢿⣛⣿⣞⣿⡋⠀⠀⢘⣿⣷⣿⣿⠇⣴⣾⡗⢠⡄⢀
⠀⠶⣹⠀⠀⢸⣏⣿⡇⣿⣿⢾⣿⠶⠀⠀⠶⣿⣿⣿⣹⣿⡵⣿⣿⣟⠀⠀⣻⣿⣿⢊⣿⣏⣿⣿⣿⠶⠀⠀⠶⣿⡷⣿⣿⢸⣿⣹⡇⠀⠀⣏⠶
⠀⠀⠁⠈⠃⢼⡿⠟⢰⣿⣿⢿⣿⡄⠀⠀⣨⣿⡽⣿⣯⣿⡖⣙⣟⣿⣷⣞⣿⣿⣋⢲⣿⣭⣿⢯⣿⣅⠀⠀⢠⣿⡿⣿⣿⡆⠻⣿⡧⠘⠓⠈
⠀⠀⠀⢀⣠⠙⣿⣿⣦⢹⣿⣿⣿⣿⠃⠀⠀⢹⣿⡿⣿⣾⢿⣴⣂⡝⡛⢛⢫⣛⣮⡿⣷⣾⣯⣻⡇⠀⠀⠘⣿⣿⣿⣿⡏⣴⣿⣿⠋⣄⡀
⠀⠠⡜⣆⠀⠀⠹⣷⣿⢁⣿⣿⢻⣿⣆⣄⠀⠘⠹⢿⣿⣿⣿⣯⣿⢻⡿⢿⡟⣿⣹⣿⣿⣿⡿⠏⠃⠀⣠⣰⣿⣿⣾⣿⡈⣿⣾⠇⠀⠀⣴⢣⠄
⠀⠀⠁⠉⠘⠁⠘⣿⣥⣬⡙⢿⣿⡾⣿⣧⡀⠀⠀⠛⠹⢿⣾⣽⣻⢿⣿⣿⡿⣟⣯⣷⡿⠏⠛⠀⠀⢀⣼⣿⢳⣿⡿⢋⣥⣬⣿⠆⠈⠃⠉⠈
⠀⠀⠀⠀⠀⠠⡤⠈⢿⣟⣿⡌⣿⣿⣟⢿⣷⣟⡀⠀⠀⠉⠉⠙⠿⠻⢻⡟⠟⢿⠋⠉⠉⠀⠀⢀⣻⣾⡿⣻⣿⣿⢡⣿⣻⡟⠁⢤⠄
⠀⠀⠀⠀⣠⢳⡄⣠⡀⣻⣿⣁⣍⡛⢿⣿⣝⣿⣿⣶⣗⡀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣺⣶⣿⣿⣫⣿⡿⢛⣉⣈⣿⣟⢀⣄⢀⣞⡄
⠀⠀⠀⠀⠈⠈⠁⠉⠀⠀⠛⢿⣟⣿⡆⢻⣿⣿⣶⣯⣿⣿⣿⣿⣶⣤⣼⣧⣤⣶⣿⣿⣿⠿⣽⣷⣿⣿⡟⢰⣿⣻⡿⠛⠀⠀⠈⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠙⣿⣧⣴⣶⣬⡙⢿⣿⣾⣿⣿⣽⣿⣽⣯⣿⣯⣿⣿⣿⣿⡿⢋⣥⣶⣦⡼⣿⠋⠀⠈⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⠆⠛⠀⠁⡋⠛⠷⣿⣿⢈⣥⣤⣭⡙⠿⠟⣛⣛⠻⠿⢋⣭⣴⣬⡁⣿⣿⠾⠛⢙⠈⠀⠛⠰⠩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⠀⣠⠈⠻⠟⠻⢷⣿⣿⣰⣿⣟⣻⣿⣆⣿⣿⠾⠟⠻⠟⠡⣄⠀⣘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡯⠓⠈⠀⠀⠴⠀⠀⡤⠙⠛⣉⠉⠉⣉⠋⠋⢤⠀⠀⠦⠀⠀⠁⠚⢭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⠦⠉⠀⠀⢋⡤⢤⡙⠀⠀⠈⠴⣚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣲⣂⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣠⠖⠋⠉⠉⠉⠉⠓⢦⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡯⠶⠚⠛⢓⣶⣼⡏⢷⡟⠉⠛⢦⡀⣠⠞⠋⠙⣷⠏⣿⣴⣖⠛⠛⠒⠬⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣴⠋⠀⡼⢧⢸⡟⣶⣤⠀⣷⡇⢠⣴⡞⣿⢠⠿⡄⠈⢳⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⣯⠖⠒⠲⢤⣿⠀⣾⠳⡌⢿⡇⢧⣀⣴⠋⢳⣄⣠⠇⣿⠏⡴⢻⡄⢸⣧⠴⠒⠒⢮⡇⠀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠛⢿⡤⢤⣤⡄⢻⠳⠤⠴⠃⠘⣧⠀⠀⠀⠤⠄⠀⠀⢠⡟⠀⠳⠤⠴⢻⠃⣤⣤⠤⣼⠟⢫⡟
⠀⠀⠀⠀⢀⢀⡴⠞⢫⠟⠛⡷⢬⣳⡜⢧⣤⠟⠀⣤⠀⣠⡤⢿⣞⡛⠛⠒⠚⠛⣛⣾⠧⣤⡀⠠⡄⠘⢧⣤⠟⣴⣣⠴⡟⠛⢯⠛⠶⣄⢀
⠀⠀⠀⠀⢻⡟⠀⠀⢸⡀⠀⡷⠶⡍⠻⣦⠀⢀⣤⠶⠚⠁⣴⠋⣯⡉⠳⣄⡴⠛⣩⡏⢳⡄⠙⠲⢦⣄⠀⣠⡾⠋⡵⠶⡇⠀⣸⠀⠀⠘⣿⠃
⠀⠀⠀⠀⣾⠀⢀⣠⣤⣷⣤⣉⣠⠏⠀⢈⡷⣯⡴⠶⠤⣤⡇⢸⣿⡿⣆⠙⢁⡾⣿⣿⠀⣧⡤⠴⠶⣬⡷⣏⠀⠈⢧⣈⣡⣴⣧⣤⣀⠀⢸⡆
⠀⠀⠀⠀⢹⣴⠋⠀⣀⣤⠙⡆⠀⡤⢠⡟⢰⡏⢳⣶⡦⣌⠁⢸⡞⣿⡸⡆⡾⣸⡟⣾⠀⢉⡤⣶⣶⠋⣷⠘⣧⢠⡄⠀⡞⢡⣄⡀⠈⢳⣼⠁
⠀⠐⠶⣶⡶⠿⠶⣟⡙⠦⠴⠃⠀⣠⠞⠀⠈⣧⠈⢯⠻⣮⣝⣮⣷⢧⡟⢿⠟⣧⢷⣯⣞⣭⡾⢫⠏⢠⡇⠀⠘⢦⡀⠀⠳⠤⠞⢙⡷⠾⠷⣶⡶⠖
⠀⠀⠀⣠⣽⠶⣶⠶⠿⠶⢦⣤⣠⣏⡴⠛⣉⣉⣁⣈⠳⣼⠾⣏⣹⠶⢗⠛⢓⠷⢾⣉⡿⢾⡴⢋⣀⣉⣉⡙⠲⣌⣧⣠⡤⠶⠾⠷⢆⡶⢾⣥⡀
⠀⠀⡼⠉⣧⠀⠙⣾⠛⢳⠀⠀⣾⠹⡏⠻⣿⠿⣯⣭⠽⣟⢳⡞⠱⣤⣼⢳⢻⣤⡴⠙⣶⢛⡿⢭⣭⡿⢿⡿⠋⡿⢹⡆⠀⢰⠛⢻⡞⠁⢠⡏⠹⡄
⢠⣸⠃⠀⠈⠳⢤⣤⡤⠟⢀⡄⢿⠀⠙⢦⣈⠙⠲⠯⣿⣍⣿⠑⢶⡛⠈⠀⠈⠘⣳⠖⢹⣏⣽⡯⠷⠚⢉⣠⠞⠁⢸⠇⣄⠘⠧⣤⣤⠴⠋⠀⠀⢻⣠
⠈⢹⡆⠀⣠⠞⠋⣉⠙⢦⠀⠁⣿⠀⣴⠋⣁⡴⢖⣻⠿⣭⢽⡔⠺⣶⣂⢀⢀⣰⡾⠒⣼⢭⡽⢿⣓⠶⣄⡉⢳⡄⢸⡇⠁⢠⠞⢉⡉⠛⢦⡀⠀⣾⠉
⠀⠀⠻⣄⣏⠀⣰⠿⠤⠞⠀⢀⣻⣼⡧⠾⣿⣟⣛⣻⡿⣷⡞⢳⣔⠋⢻⠞⢾⠋⢓⣴⠛⢶⡿⣿⣛⣛⣿⡿⠦⣿⣼⣃⠀⠘⠦⠼⢷⡀⢈⣇⡼⠃
⠀⠀⠀⢈⡽⠟⠛⣛⡻⠿⠛⠉⠙⣇⠙⠦⣤⣤⡄⢀⡞⣹⣻⣏⣼⢛⡷⢶⠶⣟⢻⣌⣿⣻⡙⣆⠀⣤⣤⡤⠞⢁⡟⠉⠙⠻⠿⣛⡛⠛⠿⣍
⠀⠐⠚⠛⠛⣶⠞⠧⣴⡋⠙⡆⠀⡘⢧⡀⢀⡏⢠⣯⡾⣋⡵⢋⡿⣏⡷⡿⡷⣏⡿⣏⠳⣝⡻⣮⣧⠈⣇⠀⣠⠟⡀⠀⡞⠉⣳⡤⠿⢶⣞⠛⠛⠒
⠀⠀⠀⠀⣼⠻⣄⣀⢈⣉⣴⣃⡀⠉⠈⢧⡘⣇⡼⠛⠛⣁⡄⢸⣧⡿⣹⠃⢻⡹⣧⣿⠀⣄⡙⠛⠻⣄⡟⣠⠏⠈⠁⣀⣳⣌⣉⢀⣀⡼⢻⡄
⠀⠀⠀⠀⢻⠀⠀⠉⢉⡿⠋⣭⠉⣧⠀⣀⡽⢯⡛⠛⠋⠙⡇⠸⣿⡷⠃⣠⡀⠳⣿⡿⢀⡟⠙⠛⠛⣫⠟⣅⠀⠀⡎⢩⡍⠻⣏⠉⠁⠀⢸⠃
⠀⠀⠀⠀⣮⢧⡀⠀⢸⠀⠀⡟⢛⣡⣾⠋⡀⠀⠙⡓⢦⣄⠙⢦⣏⣤⠞⠁⠙⢦⣌⣧⠞⢁⣤⠖⡋⠁⠀⠈⢻⣦⣙⠛⡇⠀⢸⠀⠀⣠⢯⡆
⠀⠀⠀⠀⠀⠀⠙⠶⢬⣷⣴⠟⢋⡞⢱⡏⡙⢧⠀⠉⠀⠈⠙⣿⠷⠦⠴⠶⠶⠤⠶⢿⡟⠉⠀⠈⠁⢰⠟⡉⣷⠙⣎⠛⢷⣴⣯⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⠶⣿⠛⠉⠙⢁⣾⠞⠋⠛⡆⢠⡏⢀⣀⡀⠙⠁⣀⣀⠈⣧⠀⡞⠋⠛⣾⣆⠉⠉⠉⢻⡶⢬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⣟⠳⠶⠞⠋⣿⠀⠻⡞⢡⢿⡇⣏⢉⠙⣦⡞⢁⢙⡇⣿⢧⠙⡾⠃⢸⡏⠛⠶⠶⢛⡇⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠹⣄⣀⣹⡏⢸⣧⠟⠉⢀⡿⣇⠈⠙⢦⣿⠈⣿⣁⣀⡾⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢯⠓⠶⠶⠶⠛⢹⣧⠞⣷⠤⠴⠋⠀⠈⠳⠤⣴⡟⢦⣿⠙⠻⠶⠶⠖⢫⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠳⢤⣄⣀⣀⣤⠴⠋⠀⠈⢿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠁⠀⠀⢀⠈⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⡠⠀⠀⣀⠤⠂⠒⠀⠉⠉⠁⠀⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢢
⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠓⠒⠒⠒⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢃
⠀⠀⠀⢻⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⢸⡠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⡰⠃
⠀⠀⠀⠀⢇⠄⠢⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠔⠈⢿⣿
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠐⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠢⣀⡀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠢⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠐⠒⠂⠤⠤⠤⠤⠤⠤⠄⠒⠂⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⠙⠓⠒⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⣀⡤⠴⠒⠖⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⣤⣀⡀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀⠈⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣴⠒⢦⡔⠛⣷⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠀⣀⢴⡄⠀⠀⠀⠀⠀⠙⠶⣤⣀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠳⣄⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⠟⠁⠀⠀⠀⢠⣞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠇⠀⠀⠀⠈⢳⣄⠈⢷⡀
⠀⠀⠀⠀⠀⣴⠏⠀⣰⠏⠀⢀⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⢾⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⣷
⠀⠀⠀⠀⢰⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠐⠯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⠀⠸⣧
⠀⠀⠀⠀⣿⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡇⠀⢹⡆
⠀⠀⠀⢰⡟⠀⢰⡇⠀⠀⠀⠀⠀⢠⡴⢤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⣾⠙⠶⠚⢳⠀⠀⠀⠈⣷⠀⠈⣧
⠀⠀⠀⢸⡇⠀⢸⠃⠀⢀⡀⠀⠀⢸⡄⠘⠋⣻⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⣄⢀⣴⠏⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡇⠀⣾⠇⠀⠻⠟⠀⠀⠀⠻⠶⠞⠉⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢰⣶⠀⣿⠀⠀⢹
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⢾⣿⠀⣿⠀⠀⣾
⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠟⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡆⢸⣷⠀⠀⠀⢀⣴⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠔⢀⡿⠀⠀⣿
⠀⠀⠀⢸⡇⠀⣿⡄⠀⢾⡽⠋⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠈⠓⠦⣤⣄⣀⣀⣀⣀⣠⡴⠛⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢰⡇
⠀⠀⠀⠀⢷⡀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣼
⠀⠀⠀⠀⠘⢧⡀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢠⡟⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠈⢷⣄⠈⠻⣆⡀⠀⠘⠟⠀⠀⠀⣠⣤⡴⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣤⠖⢄⣀⡀⠀⠀⠙⠚⠀⢠⠏⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠻⣦⡀⠀⠀⠀⠸⣧⣀⡁⣼⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⡽⠀⠀⠀⠀⢠⠏⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠙⢦⣄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠸⠯⣽⣓⣶⣤⣄⠀⠀⠀⠀⠈⠳⠶⠚⠁⠀⠀⢀⡴⠋⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⢶⣄⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠶⠀⠀⠀⠀⢀⣠⡴⠋⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠈⠻⢦⣄⡀⠀⠀⠻⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠉⠛⠳⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⡶⠾⠛⠉⠁⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⡀⢀⣴⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠙⣿⣿⡇⢸⣿⣿⠋⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣤⣀⣀⠀⠀⣀⣠⡾⠟⠁⠀⠀⠈⠻⢷⣄⣀⠀⠀⣀⣀⣤⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢀⡴⠺⣿⣟⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣻⣿⠗⢦⡀
⠀⢰⣿⠁⠀⠙⢿⣷⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⣾⡿⠋⠀⢈⣿⡄
⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⢠⣴⠆⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠰⣦
⣿⣿⡀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⢀⣿
⠈⠛⠻⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠟⠛
⣠⣾⡿⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢿⣷
⢿⣿⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿
⠀⠙⠁⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠈⠋
⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀
⠀⢸⣿⠏⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠹⣿⡇
⠀⠈⢿⡄⠀⣼⣿⠋⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠙⣿⣧⠀⢠⡿⠁
⠀⠀⠀⠙⢻⣿⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢾⣿⡟⠋
⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣤⣀⠀⠀⠀⢀⣀⣠⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⠀⠀⠀⠀⣀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣴⣿⣿⡇⢸⣿⣿⣦⣀⣸⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣋⣀⣀⣉⣻⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠶⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣄⠀⠀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢰⡆⠀⠀⣤⣤⣤⣤⣤⣶⡦⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⢹⡏⠉⠉⠉⠁⠀⢸⡇⠀⠀⠈⠉⠛⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡆⠀⣿⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠃⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣾⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣿⢀⣠⣴⡾⠟⠁⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣧⣤⣤⣴⣶⠆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠛⠋⠙⣷⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢷⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢻⣆⠀⠀⢸⣧⠀⠀⠀⣰⡟⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠻⢶⣴⡾⠋⠀⠀⠀⠀⠈⣿⣶⡶⠶⠿⠿⠿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⣴⣤⣤⣤⣴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠙⢿⣦⡀⢸⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠹⣷⠈⣿⡀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣧⠀⠀⠀⠀⢠⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡆⠀⠀⠀⣼⡟⠀⠀⠀⣿⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣷⠀⠀⢠⣿⠃⠀⠀⠀⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢻⣧⠀⣼⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠈⣿⣶⣿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢶⣶⣶⣶⣶⣶⣶⠀⠈⠻⢷⣶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⡆

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠋⠀⠈⡥⣄⠐⠄⠀⠂
⠀⠀⠀⠀⠐⠄⣰⡾⠃⢠⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠂⠀⠀⠀⠄⢠⣀⣀⡠⠊⢐⡄⠀⣀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣄⣷⢄⠀⠀⠀⠆⡯⠀⡰⠋⠠⢀⠀⠀⢃⢠⣶⣮
⠀⠀⡈⠀⠀⠀⢀⡇⡘⠀⠐⣿⡾⠖⣢⣾⡽⠿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⡄⢈⢁⠄⠀⠀⡏⠖⣀⡁⣽⣋⣵⠞⠋⢀⣀⡀⠄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠡
⠀⣨⠃⠀⡔⣼⢷⣯⠊⡸⢫⡟⠁⠀⣴⠋⠁⠙⡄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡸
⢰⡇⠀⠈⣰⠃⠀⢀⡼⢡⡏⠀⠀⠀⣇⠀⣾⠟⠁⡀⠂⠘⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠃⠠⣏
⡞⠀⠀⣰⡅⡀⠀⣼⡇⢹⠀⠀⠀⠀⠙⣄⠀⠉⠁⣀⡄⠀⠀⠀⠀⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡠⠐⠛⠓⢦⣔
⡇⠀⠀⠁⠔⠁⠀⡅⠑⠘⡇⠀⠀⠀⠀⠀⠀⠀⠋⠹⠙⠀⠐⢀⡀⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠤⡰⢀⣤⣀⡞⠃⠈⠉⠀⣠⣴⡤⠀⢉
⡃⢊⠄⢸⠁⠀⠠⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡚⠉⠉⡐⢄⠀⡨⠛⠉⠉⠁⠰⢄⡀⠀⢻⡥⠐⠁
⡿⠌⡆⡔⠀⠀⢰⡆⠀⠀⠀⠛⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢩⣢⡴⠟⠲⢄⡔⠂⠈⡷⠀⠀⠀⠀⠀⣀⠀⢱⡄⠄⠀⠀⠉⠣
⣇⠀⠘⠀⠀⡀⠀⢟⠠⡀⠠⠐⠤⠙⠦⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠒⠉⠀⠀⠀⠀⠈⢓⡈⣕⣷⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠂
⣧⠀⠀⠀⠀⠀⠀⡼⣇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠞⠉⠁⠀⠀⠀⠀⠀⢂⡨⠤⢀⡈⠻⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢻⡴⠁⠀⠀⡃⠀⠀⠘⣦⡐⠢⠄⠀⠀⠀⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠤⣄⠙⣄⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠁⠆⠀⠀⠆⢁⠀⠘⠀⠈⠝⢯⣍⢐⠸⡀⠤⠠⠄⢀⠝⠃⠀⠀⢀⠔⡪⠀⠀⠀⠀⠀⠸⣇⣾⠀⠀⠀⠑⣄⠀⠀⠀⠤⠐⣋⠉⠁⠀⠢⢀
⡀⠀⣄⠘⠀⠀⡆⢀⠀⠈⢄⠀⠌⠙⠷⣦⣤⣤⣈⣁⡠⣤⠴⠊⠡⠈⠀⠀⠀⠀⢀⡀⠀⠸⢿⡆⠀⠐⠀⠀⠙⠂⠀⠀⣀⣀⣈⢀⡀
⠀⠈⠸⣦⠀⠀⠀⠀⠃⠀⠀⠁⠀⠠⡀⠀⠀⠈⡀⠁⡸⢁⠔⠀⠒⠂⠀⠀⠀⠈⠁⢀⡀⠤⠷⢵⡀⢀⠈⠀⠀⠂⣲⣆⠠⠒⠒⠒⠤⣀
⠀⠀⠀⠙⣦⡐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠐⠡⠈⠀⠀⠀⠀⠀⠀⣀⠀⠂⠁⣀⠀⠀⠀⠀⠠⡀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠈⢻⣄⠀⠀⢡⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⣠⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⡖⠶⠀⠴⠴⠆⠀⠒⠐⢄
⠒⠒⠀⢄⠀⠀⠉⠳⣄⠀⠳⡔⠈⠂⠀⡀⠄⣀⠀⠀⠀⢀⡠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣂⠀⠀⠀⠠⢄⠀⠀⢸⠄
⠀⠀⠀⠀⠡⡐⢄⣀⠈⣙⣶⣿⣶⣤⣬⣭⡢⠶⠎⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠈⠰⢿⠃⠀⠊
⠀⣀⣀⠀⢀⣈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠊⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⡀⡀⡀⢀⣀⣀⡀⢀⡀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⣩⠝⠉⠀⠀⠀⠀⠉⠛⢭⡙⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣉⣤⠤⠔⠊⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⢭⣷⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⢠⠖⠚⢉⣁⡽⠀⠀⢰⠋⢳⠀⠀⠀⠀⢀⡠⠖⡄⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠈⠒⠋⠉⠀⠀⠀⠀⢸⠀⣾⠀⠀⢀⠔⠋⣠⠖⠁⠀⠀⠀⠀⢸⢧⡀
⠀⠀⠀⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⣀⠇⠀⠀⠘⠒⠋⠀⠀⠀⢸⡉⢣⠀⠸⡆⠱⡀
⠀⠀⠀⣰⠁⢰⠁⠀⠀⠀⣏⣉⣉⣉⣁⡵⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢣⠀⢣⡀⢱⡀
⠀⠀⢀⠇⡰⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠉⠑⠒⠉⠉⠛⢆⡀⠀⠀⠀⠀⠀⠀⠑⠜⠁⠀⠈⠢⣇
⠀⠀⢸⠞⠁⠀⣇⠙⣄⠀⠀⠀⠀⢀⡴⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡑⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠸⡄
⠀⠀⡏⠀⠀⠀⠈⢦⠘⢆⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⢀⡠⠚⢁⠼⠀⠀⠀⡇
⠀⠀⣇⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢏⡠⠞⠁⠀⠀⠀⣼⡇
⠀⠀⢹⢆⠀⠀⠀⠀⠀⡠⢄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⠃
⠀⠀⢸⠈⢧⠀⠀⢀⡜⢁⡜⠀⠀⢸⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠞⠀⠀⠀⠀⢠⠋⣳⢸⠀⢸
⠀⠀⠈⡆⠈⡇⠀⡞⢀⠞⠀⠀⠀⠀⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⢀⠏⢠⠃⠘⡄⡎
⠀⠀⠀⢹⠀⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢄⣀⡤⠤⣀⡠⠔⡿⠀⠐⣏⠙⠦⡀⠀⠸⠤⠏⠀⠀⢹⠁
⠀⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠀⠉⠉⣀⣀⠀⠀⠑⢦⡈⣣⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⢣⠀⣞⢉⣉⣉⣉⡱⠀⠀⠀⡠⠊⢁⠇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠉⠁⠀⡠⠖⣫⠋
⠀⠀⠀⠀⠀⠈⢣⡈⠉⠁⠀⠀⠀⠀⢠⡋⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⡜⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⡒⠢⠤⢤⣀⠉⠁⠀⠀⢰⡉⠲⢤⣀⠀⠈⠣⠞⠀⠀⠀⠀⠀⢰⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠙⠢⣄⠀⠀⠉⠓⠤⣈⡵⠀⣠⠴⠒⠦⢤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠽⠦⢤⣀⣀⣀⣠⡾⠥⠤⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡿⢻⣿⣿⣿⣿⢶⣾⡿⠍⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠺⣭⣙⣧⠘⠟⢟⠿⣙⡇⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣣⣾⣿⣞⣶⣿⣿⣦⣤⡈⠙⠛⠀⢠⣾⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠈⢹⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣻⠿⡿⠍⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⣿⠾⣿⣿⣿⣄⠀⠈⠙⠻⠿⣿⣿⣿⡏⢿⣧⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠉⠀⠀⢹⣿⣿⣷⣶⣶⣶⡄⣠⣶⣾⡇⢾⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠋⠈⢡⣾⠟⣿⣿⡿⣿⣿⣿⣟⣾⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣷⡟⢛⣿⣿⣷⣿⣿⡿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⢸⠏⠛⢻⣿⡟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢤⣼⡤⡰⠛⠁⠀⠀⠁⠠⣀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠰⠁⢸⣿⣿⢶⣶⣿⣷⠁⣄⣀⡀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⣿⣿⣿⣲⣬⣭⣾⣿⣶⣿⣿⡷⣄⠀⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣿⣧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠅⠀⣼⣿⣿⣿⣿⣿⣿⡿⡽⠋⠁⠰⠏⡌⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⡙⢛⣿⠋⠉⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣿⣷⠎⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣙⠃⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⡏⣀⣠⡤⢀⣈⣀⠀⠁⠀⢀⣀⡀⠬⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣋⡉⠉⠀⠈⠉⢙⡖⠋⠀⠀⣬⡿⣷⣿⣿⣆⠘⡄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢤⣵⣦⡀⠀⠀⠀⠀⠀⢸⠿⠿⢿⣿⣿⣷⣶⣤⣀⠉⠉⢈⣧⣄⣿⣿⣿⣿⣿⣿⣆⢰⣄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠙⠻⣧⣙⠂⠀⠀⠀⢀⣰⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠀⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⢀⣴⡆⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣶⣄⠀⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠘⠛⠁⣾⡿⠠⣿⡆⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣾⡗⠸⣿⡆⠙⠛⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢁⡀⠄⣀⣤⣿⠿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⢿⣧⣄⡀⠀⣀⠁⠈⠛⠁
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠴⠿⣿⠞⣽⡿⠟⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠿⣿⣝⢾⡿⠷⠄⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠈⠀⡨⠇⠀⠀⠀⣀⡴⠋⠀⠁⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠁⠈⠳⣄⡀⠀⠀⠀⠯⡀⢈
⠀⠀⠀⣠⣬⡲⠤⢤⠤⠖⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠦⢤⠤⠴⣪⣤⡀
⠀⠀⢘⣛⡋⢁⣴⣶⡤⠁⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡇⠱⣴⣶⣄⠉⣛⣛
⠀⠠⠾⠿⠃⠾⠿⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠿⠿⠆⠻⠿⠧
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠐⢶⣶⡄⢶⣶⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣶⣶⠆⣴⣶⠖
⠀⠀⢨⣭⣅⡈⠻⠿⠓⡀⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⡰⠻⠿⠋⣀⣭⣭
⠀⠀⠀⠙⢛⠵⠒⠚⠒⠦⣬⡂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣢⡤⠖⠒⠒⠲⠝⠛⠁
⠀⠀⠀⠀⢈⠀⢑⡆⠀⠀⠀⠉⠳⣄⠀⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣖⠁⢈
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠲⣶⣿⢦⣻⣷⣦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣶⣿⣫⢾⣷⡶⠂⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⡈⠁⠂⠉⠛⣿⣶⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣾⡟⠋⠁⠂⠉⡀⢀⣤⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢠⣤⡀⢿⣷⠀⣿⠇⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢿⡇⢰⣿⠇⣠⣤⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠂⠈⠻⠇⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠿⠋⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠄⠀⠀⠀⠀⠈⠣⠍⠌⠄⠋⠉⢠⠀⢌⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡺⠁⠈⠂⠀⠨⡹⠀⠀⠀⢀⠄⢄⠀⠀⠈⠒⠂⠀⠈⢫⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣊⠉⠁⠀⢀⠤⡀⠀⢀⣀⠀⠀⠀⠀⠑⠀⠀⠀⠤⠒⠀⠀⠀⠀⠁⢄⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠈⠤⠃⣀⠀⠀⠑⠁⠀⠡⠼⠀⠀⠀⠀⠀⡔⠒⡀⠑⠚⠀⠀⠀⠀⠀⢀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡄⡄⠀⢸⠀⠑⡰⠀⠀⠀⠀⠀⠀⢀⡤⠒⠢⡁⠊⠀⠀⠀⠔⢠⠀⠀⠀⠣⠔⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠑⠁⠀⠀⠁⠉⠀⠀⠀⡠⠤⠤⢠⡊⠠⠠⠄⢘⠠⢀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡈⠢⠀⠀⠀⢀⠔⠊⡂⠀⢠⡒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⢂⠀⠀⢘⣙⠀⠀⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠀⠀⠀⠀⠑⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠁⠁⠀⠀⠀⢀⠀⠀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠱⠀⠀⢠⠒⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⢀⠤⡀⠀⠀⠀⢇⠰⠀⠒⡔⡄⠀⠀⠀⠀⠀⠀⠀⠀⡔⡠⠀⠀⠈⠒⠁⠀⠀⠱⡠⠃⢐
⠀⠀⠀⠀⠀⠀⠀⠰⢠⠀⠈⠒⠁⠀⠀⣀⠀⠀⠀⠀⢃⣈⠐⠤⢀⣀⣀⣀⠠⢊⠸⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠣⡀⠀⠀⠀⠘⠴⠁⠀⠀⠀⡄⢄⠈⠆⣀⠄⠂⢐⠀⠊⠀⠀⠀⢆⡁⢸⠀⠀⠀⣎⢱⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡆⠀⠀⡎⢢⠀⠀⠀⡔⠱⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢔⢰⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⣀⠀⠀⠀⠈⠈⠁⠀⠰⣁⡑⠀⠐⡅⠠⠕⠀⠀⠀⢠⠂⠉⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣈⣚⠠⠀⢤⠀⠀⠀⡀⠀⠀⠀⡀⠀⢈⡉⠀⠀⠀⡰⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡇⠀⠈⠚⠀⡠⠊⠀⠀⠀⠈⠁⢀⡡⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠛⠒⠀⠀⠀⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⡠⠄⠒⠲⢤⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⢎⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠑⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠤⠶⠒⠛⠛⠛⠛⠛⠒⠲⠒⠚⠋⠁
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡾⠒⠋⠉⠉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣸⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣤⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣿⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣔⢜⣾⠏⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣶⣤⣴⣤⣤⣼⣶⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠂⠀⠠⠤⠤⠤⠤⠴⠷⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⠛⠳⠴⠴⢦⣠⢖⡥⢽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠚⠛⠲⡞⠋⠉⢋⠀⣐⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠸⡄⠰⣼⡇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠛⠋⢉⠇⠈⢿⡉⠙⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠃⠀⠀⢈⣈⣿⣿⠃⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣩⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠾⠾⠷⠷⠾⠿⠛⠋⠁


⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠀⠀⠀⠀⢂⠀⢀⠔⠀⠀⠀⠀⠀⠀⠠

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠦⠀⠀⠀⢀⠄⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡤⣄⠈⠢⡀⠃⠀⢀⣀⠀⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢹⣶⣿⣇⠀⠙⠀⠀⣿⣽⣇⠀⠈⠀⠄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡈⠉⠛⣠⠀⠇⠀⠙⢿⡗⠈⠀⢸⠀⣀⣴⣲⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⠀⠘⠀⡘⠂⠀⠀⠅⠀⠀⠒⠀⠸⣿⣿⠇⠀⠸⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⢀⡘⠀⢀⣂⠀⠀⠀⠀⠦⠀⠀⠘⠋⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠙⠀⣿⣹⡇⠀⠠⠀⠀⠀⢀⠀⠖⠁⠐⠀⠀⣴⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢄⠙⢿⠇⠀⠀⢀⣤⡀⠀⠀⠀⠀⠂⠀⢰⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠨⡀⠉⡠⠠⠂⢾⣷⡇⠀⡂⠀⠀⠀⠀⠈⣿⠟⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠄⠛⠀⠁⠘⠟⠀⢐⢇⣤⣄⠀⢀⢈⠅⠀⠀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠙⠀⠀⡁⢀⡌⢸⣿⠟⠀⠈⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢠⣶⣦⠀⠀⠀⠀⢸⡀⢈⠁⠀⣀⠀⠄⠈⠀⣰⣯⡽⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠐⠉⠀⢡⠔⠀⣠⡤⡄⠀⠢⠿⠛⠁⠠
⠀⠀⠀⠀⠀⠰⣧⣀⡓⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣴⠚⠒⠐⠁⠀⣰⣶⠀⢠⠀⢸⣿⡿⠃⠘⠁⠀⢀⠌⣀
⠀⠀⠀⠀⢾⣄⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠐⠈⠁⠀⣾⠆⢀⡠⠛⠉⠀⠈⠈⠀⠀⠀⡰⠃⢀⣠⣄⡀⠈⢁
⠀⠀⠀⣀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠸⠃⠀⠂⠀⠀⠒⠋⣂⠀⢈⠁⢠⣴⣦⠀⣤⢠⣿⣿⡾⠃⢀⠌
⠀⠀⠀⠾⢷⣦⣤⣤⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣝⠛⡧⣄⠁⠠⠶⣸⠁⠘⠟⠃⠹⡶⠿⠟⢛⣼⣿⣷⢵⣾⢛⠟⠳⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⡡⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⠾⣻⡟⢽⣫⣷⡠⠒⣻⡀⣀⠈⣙⣉⡉⣠⠈⢋⣉⣉⣨⣾⣿⣻⣯⡚⣸⡇
⠀⠀⠀⠂⠀⠐⠒⠊⢉⣠⡾⠷⠒⣒⠢⠄⡄⣀⣀⡠⠴⣞⠩⠔⣺⢇⡤⢃⠹⣍⡇⠀⣩⠔⠁⠈⠛⠋⣠⡁⣀⣤⡀⠁⠩⠙⠛⢫⢁⡀⠊
⠀⠀⠀⠀⣄⠄⠰⠒⠱⠣⠀⣀⠀⠀⠀⠂⡀⠀⠄⢀⡹⠅⣊⣤⡏⡁⠘⠘⠄⣝⢻⠀⠰⢀⠑⠂⠀⠒⠠⠀⠈⠉⠀⡐⠠⣤⣶⣶⣄⠀⠢
⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠀⠄⠠⠀⠀⠐⢀⡊⢥⠖⣪⠞⡋⣿⠌⡹⣋⢺⠝⡏⠄⣬⠙⠆⠀⠛⠛⡠⠈⠑⠋⠉⢐⠂⠀⠉⢍⠉⠀⡠
⠀⠀⠀⠀⠀⠀⠙⠦⣰⡄⠈⢀⣀⠠⠘⠠⠰⠴⣰⣯⠷⠋⣠⣡⠉⠖⠒⠙⠋⠙⠁⠀⠀⠉⠁⠁⠈⠁⠐⠒⠶⢶⡌⣧⠀⢠⣤⣤⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⢤⡀⢀⣀⠤⠴⠿⠋⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢉⠛⠃⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠁⠀⠀⠈⠀⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢿⡾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡃⡼⠃⠀⠀⠀⠀⣴⣶⣶⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢤⠀⠀⠀⠀⠈⢃⣀⣀⡸⢱⣀⠤⠙⠛⠛⠛⠿⣿⣿⡿⠃⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢠⣄⣀⣠⠊⠁⠀⠐⢁⠜⢣⠀⢸⣏⣶⣤⡀⠘⣿⣿⣷⡄⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠈⠒⢤⠤⠀⠠⠀⠘⠀⢀⠀⠆⢠⣉⡙⠻⠛⢹⡏⠉⠉⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⢏⠤⠒⠚⢄⠀⠢⠴⢶⡊⠁⠀⢰⠀⡏⢙⡀⠀⠠⡀⠀⠀⣴⠟⣻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠱⣁⣀⢶⣾⠂⢣⡀⠀⠸⣨⠗⠈⠙⠚⠀⠛⠳⡦⡁⠇⠀⠠⠵⣾⣿⣝⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠀⠐⡇⠛⠿⣤⣽⣿⣿⣇⣀⣰⠁⠀⣀⣀⠀⠠⢆⡀⠀⠙⣄⠀⠀⠀⠈⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⢠⣀⡇⠀⠒⢃⠜⠋⢡⣂⡄⠀⣉⠲⠄⠋⢀⣠⣴⠀⣀⠤⡀⠉⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡡⠨⢳⡀⠀⡏⠀⠀⢨⡽⢗⠻⠀⠀⡎⠁⡪⠽⠋⠰⠥⡀⡨⢰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠰⡅⡀⡇⠙⠂⠖⠂⠘⠁⢀⣎⠀⣀⠔⠋⠉⠑⠢⣾⠼⣀⣈⢣⠸⣌⠆⡀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠳⢌⣘⡦⡄⣠⡀⠀⣤⠈⠈⢶⠁⠠⡄⣠⣶⣴⠈⡞⠀⡏⡆⣠⠃⠊⠐⠂⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠕⠠⢊⠼⢿⠀⠉⣀⣀⣄⠀⠀⠸⠀⠀⢀⠁⢀⡀⠀⢰⠊⠀⠀⢹⠍⠉⡓⣎⠔⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠡⡐⠆⠆⠀⠻⠽⠟⠀⠀⠀⡆⠧⡸⠀⠿⠃⠀⠸⠀⠀⠇⢸⠈⢑⠈⢘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⠖⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⣶⣶⣶⣿⡖⠒⠒⠚⠧⠁⣂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠈⠛⠟⠛⠁⠀⠀⣠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣶⣤⣀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢀⣀⡀⠀⠀⠀⣰⣶⣶⣦⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀
⠀⠀⢀⠄⠘⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠊⠈⠁⠁⠑⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠉⢂
⠀⢐⠉⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠶⠦⡈⡅⠀⠀⠀⠀⠀⣀⠑⠐⠢
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⡀⠄⠠⡀
⠀⢀⠁⠀⠈⠒⠰⠈⠑⣆⠀⠠⠂⠊⠀⠀⠱⠀⠀⡠⠒⠀⠐⠁⠀⠑⠂⢄⠀⠀⠀⠀⠀⡠⠜⠉⠀⠀⠀⠘⠒⢄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠆⠀⠀⠀⢈⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣲⠀⠀⢎⡀⠀⠀⠀⠀⠀⢀⢀⡀⠅
⠀⠀⠀⢀⣀⠤⠒⠠⠀⠀⠀⠀⠈⠒⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡑⠂⠒⠁
⠀⢀⠴⠀⠀⠀⠀⠀⠓⠠⡀⠀⠀⡐⠁⠀⠑⠒⠂⢄⠀⠀⠄⠐⠁⠀⠈⠒⢄⠀⠀⢀⠄⠠⠊⠀⠀⢡⣀⣠⠤⠤
⠀⠠⡀⠀⠀⠀⠀⠀⠀⡠⠃⠐⠊⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠆⢂⠀⠀⠀⠀⠀⠸⡀⠀⠒⠂⡅
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠑⠀⠂⠀⠀⠁⠀⠀⠈⠒⠢⡀⡠⠄⣀⠠⠃⠀⠡⠀⠀⣀⠀⢀⠇⠀⠀⡔⠁
⠀⠀⡔⠈⠲⢄⠎⡠⠘⠓⢄⠀⠀⠀⠀⠀⠔⠈⠁⠢⠄⣤⢀⡀⠀⣀⠀⠀⠀⠀⢀⠀⡀⢈⠁⡀⠈⠁⠈⠑⠊
⠀⠨⣀⣀⠤⠃⠀⠀⠀⠀⠘⠒⢄⠀⡤⠊⠀⠀⠠⠤⠊⠀⠀⠀⠀⠀⢡⠀⢀⣠⣁⡠⠀⠢⡀⠈⠄⢄⡠⠄
⠀⠀⠀⠈⠄⣀⣀⣀⣀⣀⣀⣀⠜⠀⠉⠁⠐⡎⠁⠀⠀⠀⠀⠀⠀⢀⠌⠠⠉⠀⠀⠀⠀⠀⠀⠈⠆⢀⣀⣀⣁⣐⡄
⠀⠀⠀⠀⠀⡠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠉⡉⠁⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀
⠀⠀⡀⠔⠉⠀⠀⣀⡩⠓⠒⢄⠀⠀⠀⠀⠀⠀⠔⠐⠁⢀⠨⢄⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⣀⠔⠒⠊⠀⠈⢂
⠀⢰⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⢢⠀⠤⠁⠀⠀⠉⠙⡀⠉⠐⠀⠀⢐⠁⠀⠀⠑⠀⠄⠊⠁⠢⠈⡃
⠀⠀⠑⠀⠪⢄⢀⡀⡀⠀⠀⢀⠀⠤⠂⠀⠤⠴⢎⣉⠀⠀⠀⠀⠀⠀⣘⣒⡚⠄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢆⣽⣷⣦⡌⢽⠍⢴⣶⣿⣦⡙⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢻⡥⠚⠛⠿⡋⣰⣿⣿⣦⣙⡛⣉⣡⣝⠣⣭⣥⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⣿⠁⣾⣿⣿⠎⣫⡉⠻⡿⢁⣼⡉⢿⣿⣿⡆⢹⣿⣿⡄⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠇⡿⠟⠛⣤⠛⠛⠿⣀⣿⣿⡶⠖⣿⣿⣧⣜⣉⣡⡜⢣⣤⣤⣱⡞⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠟⠟⣴⣾⣿⡀⢿⣿⣶⠛⣍⠻⠃⣷⠸⢋⣤⢹⣿⣿⠇⣼⣿⣿⡇⢸⣦⣍⡄
⠀⠀⠀⠀⠀⢠⣠⣾⣇⠹⡿⠟⠓⣼⠛⡛⠇⢹⠗⣚⣉⣑⡻⢇⣈⣭⠔⣻⣤⣤⡩⡦⠻⣿⡿⢁
⠀⠀⠀⠀⠀⣉⢹⣿⠟⣺⠓⢾⣿⣿⣦⡙⠿⢡⡞⣍⣹⣉⡿⣦⢻⡁⠺⠿⣿⠋⣠⣾⣦⣌⣃⣾⡆
⠀⠀⠀⠀⠀⣿⣄⣅⣼⣿⣷⡤⢹⡍⠤⢶⡇⢺⡒⢸⣭⣽⠐⣿⢈⡟⣛⣡⣽⣍⠙⢿⡿⢃⣍⢻⡇
⠀⠀⠀⠀⠀⢻⠃⣥⡙⠻⠏⣠⣾⣿⡷⢂⣴⡌⠷⣥⣸⣤⡿⢏⡼⢷⣌⢻⡿⠿⠗⠾⢤⣾⣿⣄
⠀⠀⠀⠀⠀⠈⣸⣿⣿⠞⣫⣉⣉⡻⣆⣉⣥⢈⣷⠦⢭⣥⠒⢿⡄⣶⣶⣾⠓⣶⣿⣿⡀⢿⢏⡈
⠀⠀⠀⠀⠀⠀⠡⣝⠻⠀⣿⣿⣿⠁⣾⣿⣿⣘⣡⣾⡘⠇⣼⣦⣡⡼⠿⠿⢇⡹⠿⠛⠳⢧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡛⣍⣉⣙⡶⣉⣡⣤⠉⣿⣿⠗⠗⠻⣿⡇⢸⣶⣶⡎⢡⣶⣿⡇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢹⣿⣿⡄⢻⣿⣿⡧⣸⣁⣴⣿⣦⡈⣧⠺⠿⢿⣃⠼⠿⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣍⣁⣾⠍⣡⣤⣯⡈⠻⣿⠟⣠⣾⣶⣶⢆⣽⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠍⣻⡿⢛⣃⣿⣤⣙⡛⣿⠅⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⣉⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⡶⠶⠿⠿⠛⠛⠋⠛⠛⠀⠀⢤⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⢦⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢾⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠄⠀⢼⣿⡆⠀⠀⠀⠀⠀⣤⠔⠋⠁⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠀⠀⠴⢠⣷⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣯⢀⡄⠀⢀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⣿⣿⣿⣿⣿⡿⣋⣼⠷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠛⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⣿⠛⠏⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣁⣾⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠛⢀⠀⣹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣦⢠⣄⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠛⢷⣶⣶⣶⣶⣶⡶⠦⢴⣶⣶⣶⣶⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠄⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⠀⠀⠀⠀⣀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⣀⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢻
⠀⠀⠀⠀⠈⠀⠂⠈⠉⠒⠒⠒⡃⠀⠀⠀⠀⠀⠙⢻⡟⠀⠀⢀⡀⠤⠄⣴⠂⠃⠈⣹
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠈⠦⢤⣀⣀⠀⠀⠁⢀⡞⠃⠀⠀⠀⠉⡈⢻⡴⢛⡛⠁⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡒⠚⠦⣴⣿⣷⣍⡰⠟⠊⠀⠋⠁⢂⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣱⠦⠀⠙⠻⣽⣶⣶⡔⠂⠀⠐⠛⣠⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢘⣿⡃⣾⠙⣷⠄⠈⠻⣿⠧⠴⠛⠾⠉⣭⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠛⢛⠒⠾⠧⣾⣦⠀⠈⢿⣿⣆⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣡⠓⠦⣉⠃⠀⠘⣟⠁⠀⠀⠚⢛⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠀⠀⠀⡈⠾⣠⣬⢁⣄⠀⢹⠖⠶⠒⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣾⣿⢋⡝⣦⠻⡥⢒⠒⠴⣄⠀⠀⠉⠻⣦⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠉⠘⠷⡏⣵⣄⠊⡁⢆⣈⠳⡀⠀⠀⠈⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠀⠈⠐⢦⣤⣀⣉⠁⢀⡈⠁⠛⠛⠓⠆⠀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢶⠀⠀⠀⢑⣿⠉⣿⠉⠉⠉⢻⢐⣧⠈⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡂⠀⠀⠀⢄⢻⣷⡿⠁⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠁⠠⠀⢠⠆⠷⣿⡇⠂⠆⠄⠀⡀⢚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠁⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢹⣆⠀⠀⠀⠑⢂⢀⢹⡇⡄⡈⠄⠀⠀⠀⢰⣆⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢋⣿⣻⣦⡀⠀⠀⠈⠈⣿⡏⠥⠀⠀⠀⠀⣠⣿⣻⡆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣾⡿⠋⢉⠀⠁⠀⠀⠀⣹⣿⠀⠀⠀⡀⠀⠉⠙⠻⢿⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠋⠀⡄⠁⠀⣥⠶⠟⠛⢿⣿⡖⠛⠷⢤⡂⡀⠑⠄⠀⠘⣳⠗
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠠⠀⢀⣾⠃⠀⢀⠀⠾⠈⡧⠀⠀⠀⠹⣧⠀⠀⠀⠀⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⡀⠀⠀⠀⢸⡇⠀⠀⡀⢠⡁⠀⢀⠀⠑⡀⠀⢿⡆⠀⠈⠀⠈⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠸⣧⠀⠀⠀⠀⡘⠀⢇⠀⠈⠀⠀⣿⠇⠀⢀⠀⠠⠶⠷
⠀⠀⠀⠀⠀⠀⠀⠐⠞⠣⠀⠀⢀⠀⠙⢷⣄⠀⠀⠘⠒⠃⠀⢀⣠⣾⠏⠀⠀⠈⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠈⠻⣗⣤⡀⠀⠀⣴⣯⡿⠁⢀⠔⠠⠁⠀⠌⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⡀⠠⢀⡄⠀⠹⣧⣵⣄⡼⣧⡯⠀⠀⡀⠀⠁⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣦⣀⠀⡀⢀⣿⡁⠁⠀⢸⣇⠀⠀⠀⠀⠼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣢⠤⠛⢉⡿⠶⠶⠏⠙⣣⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠀⣿⠀⠀⢺⠀⠀⠹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣷⣦⣄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠂⠀⢀⣀⣤⡤⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⢀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣷⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠕⣡⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠧⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢃⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⣜⡿⢡⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⡀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡞⣼⣱⣿⣿⠁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣆⠀⠘⡆
⠀⠀⠀⠀⠀⡜⢰⣿⣿⢣⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⢻
⠀⠀⠀⠀⢰⠃⣿⣿⠇⣾⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⢸⡆
⠀⠀⠀⠀⡾⠀⣿⡟⠀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⡇⢸⣿⡇⠠⣇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇
⠀⠀⠀⠀⡇⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸⠁
⠀⠀⠀⠀⢧⢸⣿⡷⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡟
⠀⠀⠀⠀⢸⡈⣿⣇⠆⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⢧⢻⣿⡜⡀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠈⢞⣿⣿⣷⡄⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⢷⡕⢝⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡙⢆⢾⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠷⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣈⠛⠻⢿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣬⣭⣶⣶⣿⣿⣿⠿⣛⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣄⡀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠟⠛⢛⣋⠭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⠀⠀⠀⣀⡀⠤⠔⠒⠒⠒⠉⠉⠀⠀⠐⠒⠒⠤⠄⣀⠀⠀⠀⢀⠃⡀
⠀⢀⠀⠀⢀⡆⠀⠀⡀⠹⡔⢚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⡼⠁⡀⠀⠀⣦⠀⠀⢀
⠀⠀⠴⠆⠠⠏⠀⠁⠢⢄⠈⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡉⢀⠤⠂⠁⠈⠧⠀⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⡀⠀⠀⠀⠀⠈⣿⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⢲⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠧⡀⠀⠀⠀⠀⢹⢿⣻⣿⡇⡀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣠⠶⠋⠁
⠀⠀⢿⣿⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⣿⣿⡏⠴⠠⡄⠀⠀⠙⢿⣿⣿⣿⣛⠅
⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣉⣩⡽⢯⣶⣍⣲⠆⠐⠶⢄⣹⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣾⣏⡓⣶⣾⣿⣿⡍⣴⠄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣽⣿⠿⢙⣙⣿⣻⣿⣿⣽⣋⣽⣛⢻⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⣙⣉⣿⠉⠀⢻⡉⠻⣿⣿⣿⣿⡭⠉⠉⠀⢀⣄
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠸⡟⡿⢿⣯⣽⣿⣷⣄⠀⠙⣿⣿⣿⣿⣿⣷⣿⡆⢀⣾⣿⣿⣶⣤⡀
⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣭⣀⡸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠻⣮⡛⣿⣷⣄
⠀⠀⢀⣿⠟⠋⠀⠀⠀⣠⣾⣿⣿⣿⠟⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠉⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠐⣩⡿⠟⠋⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡖⠀⠀⠀⠀⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠆⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠂⠀⠀⠀⠒⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡱⣷⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠆⠀⠀⢀⣀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣸⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⣾⠯⠦⠴⠾⠷⢄⣀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣭⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣴⣶⣏⣥⡄⠀⠐⠂⠀⠀⡦⣴⣿⣷⠀⠀⠀⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣅⡈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠒⠀⠀⠀⠙⠿⣿⣏⣴⣶⠒⠒⢻⣷⡶⠟⠋⠁⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡺⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⡆⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⡀⠀⢸⣷⡀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⣤⠀⠀⣿⣿⡄⢾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⠀⣿⣿⣧⠀⢸⣷⡀⠀⠂⠀⢰⠀⠀⠀⣿⣧⠀⣿⣿⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⣇⣽⣿⡧⠀⠀⠀⠀⠀⠀⢠⣿⣿⣅⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⢠⡆⠀⠀⠀⢀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣠⡤⠆⠦⠤⠄⢘⣣⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠟⠋⠁⡀⠈⠛⠿⣿⠿⠛⠉⢀⠈⠙⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣠⣴⣿⣿⣷⣤⣀⠀⣀⣤⣾⣿⣷⣦⣄⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⣀⣠⣤⣄⠄⠀⠀⠀⠀⠀⠾⠓⠀⠀⠀⠀⠀⢠⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⠀⠀⠘⢙⣿⣭⢯⣠⠆⠀⠀⠀⣀⣄⣀⣤⣄⣠⠀⠀⣾⡅⠀⢀⣻⣿⡟⠛⢿⣿⠻⡟⢿⣿⡇⠀⣠⣄⣤⠀⣠⣾
⠀⠈⢁⡤⠀⠤⠠⠆⠀⠀⠀⠀⠀⣾⣿⠛⣿⣿⣿⣿⠀⠀⣿⡷⡤⢼⣿⣿⣿⣦⡀⠙⢛⢙⠛⢿⡇⣠⣿⣿⣿⣾⠟⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣿⡿⠗⠀⠀⣿⣧⢨⣿⣿⣿⣟⠛⠛⢦⡙⠉⣠⣿⡇⢿⣿⣷⣿⣧⣦⡶
⣿⡿⠛⣿⣿⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⡋⠉⢭⣼⣏⠻⣿⣿⡿⣿⡇⣿⣯⣵⣿⡇⠈⠿⠁⠀⠀⢀⠀⡀
⠋⣠⣼⢿⣿⣷⣦⡘⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⠀⢣⢠⠀⣾⢿⡿⠀⣿⣷⣇⠻⢿⡿⠋⣿⣿⡇⠀⢰⣄⣀⣀⣼⠟⠁
⠸⠤⠋⠸⣿⢿⡿⠷⢼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣐⣾⣌⠈⠥⠼⠷⣿⣿⣿⣿⢳⣦⣴⣾⠹⣿⡯⢀⣿⢹⠛⣻⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠀⢻⡇⣼⣿⣤⣉⣁⣀⣀
⣿⣿⣿⠛⢟⢛⠿⡻⢛⠛⡻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣠⣴⣦⠈⣷⣿⠿⠿⠿⣿⠟⠁
⣿⣿⣿⠻⠿⠿⢿⡿⢿⣿⠿⠶⠿⠟⣿⡿⠻⡛⠿⠟⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡷⠝⢿⣿⡶⣯⠉⠀⣠⣞⣥⣀⣀
⣿⣿⣿⠶⢷⣶⣶⡷⣾⣿⡶⣾⣷⣶⣿⣷⣷⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣶⣶⣵⣶⣷⣾⣦⣶⣷⣴⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⡇⣿⠇⠀⠀⠀⠀⣠⣤

⠀⠀⠀⠀⠀⠀⢀⢠⡀⡄⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣦⢀⣤
⠀⠀⠀⠀⢦⠳⡘⡆⠇⠇⡎⡜⡰⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣿⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⢿⡌⢿⠈⡟⢸⠇⣼⠟⣠⡄
⠀⠀⠠⣙⠢⡑⠐⠀⠀⠀⠁⠀⠑⡡⢞⡡⠄⠀⠀⠀⠠⣤⣀⠀⠙⣧⠀⣿⠀⣸⠏⠀⣀⣤⠄⠀⠀⠀⠠⣤⣈⠻⣌⢳⡘⡇⡇⡟⡸⢃⡼⢋⣠⡦
⠀⣈⣓⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢒⣋⡀⠀⠀⠀⠈⠛⠳⢦⡈⠆⠉⠰⢃⡤⠞⠛⠁⠀⠀⠀⠀⣦⣤⣉⡳⠮⡓⠑⠀⠁⠐⠕⡫⠖⣋⣡⣤⡆
⠀⠴⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠤⠄⠀⠠⠶⠶⠶⠶⠦⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠆⠀⢠⣤⣤⣬⠭⠥⠀⠀⠀⠀⠀⠀⠠⠭⣭⣤⣤⣤
⠀⢘⣩⠥⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢭⣙⠀⠀⠀⠀⠀⣠⣤⠖⢉⡄⡀⢠⡉⠲⣤⣄⡀⠀⠀⠀⠀⣤⡶⠶⢒⣡⢄⠀⠀⠀⠀⢀⢭⣑⠲⠶⣦⡄
⠀⠀⠰⣊⠔⡠⠂⠀⠀⠀⠀⠀⠢⡐⢍⡲⠄⠀⠀⠀⠰⠛⠋⠀⣠⡟⠀⣿⠀⢻⣄⠀⠙⠛⠆⠀⠀⠀⢠⣴⠞⣋⡴⢣⢎⡆⡆⡌⢮⠳⣌⠛⢶⣤
⠀⠀⠀⠁⠞⣱⢣⠇⡆⡆⢇⢦⢱⡙⠆⠁⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⣿⠀⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⣰⠏⣼⠁⡇⢹⡌⢷⡈⠻⠆
⠀⠀⠀⠀⠀⠀⠈⠘⠃⠃⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠰⡿⠰⣿⠘⡷⠈⠛

⠀⠀⠀⠀⠀⢤⡀⣷⢠⣧⢰⡆⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣄⡀
⠀⠀⢀⡈⢷⣜⢷⣹⡟⣿⣿⣵⣟⣴⠏⣀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢻⡀⣿⠀⣸⠃⣠⡾⠃⠀⠀⠀⠀⠀⠀⣴⢷⣽⣞⣿⣿⣿⣿⣿⡿⣽⣷⡄
⠀⢀⣈⣛⠷⣽⣿⣿⣿⣿⣿⣿⣿⣵⢟⣋⣀⠀⠀⠀⠚⠷⢤⣈⠳⠄⠃⠀⠐⢃⠔⢋⣠⡴⠶⠀⠀⠀⢀⣾⢯⣿⣿⣿⣿⠟⡿⣟⢿⣿⣿⣯⢿⣆
⠀⢠⣭⣽⣟⣿⣿⡏⠁⠀⠙⣿⣿⣟⣿⣭⣥⠀⠀⠰⢦⣤⣤⣈⡑⠀⠀⠀⠀⠀⢐⣉⣁⣠⣤⡄⠀⠀⢼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⠄
⠀⢠⣴⡶⢶⣿⣿⣧⡀⠀⣠⣾⣿⣿⠶⢶⣤⠀⠀⢀⣀⣤⡤⠤⢂⠀⠀⠀⠀⢀⠠⠤⣄⣀⡀⠀⠀⠀⣻⣿⣿⣿⣗⡢⠀⠀⠀⠀⠠⣓⣿⣿⣿⣻⡃
⠀⠀⠴⠿⣫⡿⣻⣿⣿⣷⣿⣿⡿⣯⡛⠷⠄⠀⠀⠈⠉⢀⣠⠖⢉⡔⢀⠀⡀⢆⠙⢦⣀⠉⠛⠁⠀⠀⠸⣿⣿⣿⣿⣿⣤⡀⡀⣠⣲⣿⣿⣿⣿⡿
⠀⠀⠀⠚⢋⣾⢫⡿⣿⢻⡟⣯⢻⣎⠛⠆⠀⠀⠀⠀⠀⠛⠁⣠⡞⠀⡾⠀⣇⠈⢧⡀⠙⠗⠀⠀⠀⠀⠀⠛⣿⢿⣿⣿⣿⣷⣷⣿⣿⡿⣟⢿⡝⠃
⠀⠀⠀⠀⠈⠁⠿⠁⡿⠸⡇⠹⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⡇⠀⣿⠄⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣾⣽⣿⣿⣿⣿⣿⣿⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠋⠉⠈
⠀⠀⠀⠀⠀⠀⢠⡀⡄⢤⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⣀⠲⣌⢢⠳⠸⠸⠸⢡⢣⠇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣸⡗⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠈⢿⠀⢸⠇⠀⣼⠆
⠀⠀⠐⠢⣝⠎⠄⠀⠀⠀⠀⠐⢁⠊⠴⣣⠄⠀⠀⠀⠀⠲⢶⡄⢸⡇⢰⠏⣠⠖⠛⠃⠀⢀⡄⠀⠀⠀⠀⠴⣤⡀⢹⡆⡞⠰⠋⡴⢛⣁⣤⡾⠃
⠀⠠⠭⠍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠵⢚⡀⠀⠀⠀⣀⠈⢿⡀⠳⠈⠀⠃⠔⠛⠛⠛⠋⠀⠀⠀⠀⢀⣀⣀⠹⡈⠃⠁⠀⠀⠀⠁⠴⠦⣤⡴⠗
⠀⢐⣃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠭⠥⠄⠀⠘⠛⠛⠻⣤⣈⠁⠀⠀⠀⠀⣉⠙⢶⣤⣤⠄⠀⠀⠉⠁⣉⠳⠌⠀⠀⠀⠀⠀⠀⠈⡳⢤⣤⡴⠆
⠀⠩⠖⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⣑⣚⠀⠀⠀⢀⣤⣶⣤⡤⠖⢀⠀⡄⢢⡈⢷⡀⠀⠀⠀⠀⠀⠸⠋⠉⣙⠒⢀⠀⠀⠀⠀⢀⢢⡙⣦⣀⣀
⠀⠀⠊⡴⢫⠔⡠⡀⠀⠀⠀⢀⢠⠢⣑⠲⠀⠀⠀⠀⠘⠁⠀⢠⣴⠶⠋⣰⠇⢸⡇⠈⠻⠶⠀⠀⠀⠀⠀⢰⠟⠉⣩⡥⠔⣠⢂⡆⣸⠀⣧⡀⠉⠉
⠀⠀⠀⠀⠃⠎⡼⢰⢨⠰⡐⡎⢦⡑⠌⠁⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⢸⡏⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⢠⡞⠁⣾⠀⢻⣄⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠁⠘⠈⠃⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⠛⠛⠛⠉⠉⠉⠛⠛⠛⠓⠒⠶⠶⠶⢦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠤⣀⠀⠀⠀⠀⠀⠘⠦⣤⣉⣉⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⣀⢴⡏⠀⠀⠀⠀⠀⠈⠢⣈⠑⣆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⣠⠞⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢃⡤⠞⠋⠉⠉⠓⠦⣤⣀⣀⡤⠎⠀⠀⠉⠉⠉⠀⠀⠀⠹⡆⠀⠘⣏⠛⢦⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⣼⠋⠀⡏⠀⠀⣠⠖⢋⡆⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢿⠀⠀⠈⠣⣄⠈⣧⠀⠀⠀⠀⠀⠘⣧
⠀⣼⠇⠀⠀⢹⠀⡞⣁⡴⡋⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀⠀⠨⣙⠲⣄⠈⠳⣄⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⢹⡆
⢸⡏⠀⠀⢀⡞⠀⠉⠁⠚⠁⠀⠀⠀⠀⠀⡇⠀⣠⡴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⣌⠳⣌⢳⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠏⠸
⣿⠀⢀⠀⠈⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⡼⠃⡽⢠⠏
⢻⡄⠸⡆⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⢠⡖⢉⠇⠀⠀⠀⢰⣃⡼⠁⠘⡆⡰⢀
⠘⣷⠀⢷⡀⠀⢀⡴⠋⠀⠀⠀⣠⠴⠒⠂⡄⠀⠀⠈⠙⠓⠶⠦⣤⣤⣀⣀⣀⣤⣤⠤⠶⠚⠋⠉⠀⠀⠀⢠⡏⣠⠊⠀⠀⠀⠀⠀⠀⣀⡤⠞⢱⠃⣸
⠀⠸⣧⠘⢷⡀⡞⠀⠀⠀⡞⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢠⠞⠁⠀⢠⠃⢠⡟
⠀⠀⠘⢧⡈⢳⡹⣄⠀⣰⠇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡤⠖⢋⡽⠃⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣰⠋⢠⡟⠁
⠀⠀⠀⠈⠻⣆⠙⢮⡉⠁⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠻⠴⠚⠁⠀⠀⠀⠀⠀⠘⠳⠦⠤⠌⠃⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣸⠇⢀⡾⠃⣴⠏
⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⡤⠖⠋⠁⠀⠉⠉⠒⠶⠞⢋⡴⢛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠦⣤⣤⠞⠉⠹⡄⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡼⠁⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠦⠤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⠤⠴⠶⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠤⢆⠀⣀⣀⣀⣤⠤⠤⣤⣀⣀⣀⠀⡠⠤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣲⣖⣢⣷⣄⣀⡀⠐⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠂⢀⣀⣠⣾⣔⣲⣶⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣼⠿⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠿⣧⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⠚⡾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⠑⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠶⠼⣶⣤⣠⡾⠁⢸⢄⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣠⡇⠈⢷⣄⣤⣶⠧⠶
⠀⠀⠀⠀⠀⢀⢴⣤⡶⢀⣞⡽⠁⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⢯⣳⡀⢶⣤⣦
⠀⠀⠀⠀⠀⠛⠛⠉⢠⣏⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣄⠉⠛⠛
⠀⠀⠀⡀⢀⠀⠀⣰⢋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡝⣆⠀⠀⣄⣀
⠀⠀⠀⠛⠭⠟⣳⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⣛⡓⣋⡀
⢀⣔⣼⠿⠋⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢆⠙⠟⠿⣣
⠀⠁⠀⠀⢰⣵⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢧
⠀⠀⢀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⡀
⠀⠀⠘⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠇
⠀⠀⠀⠘⣆⡇⠀⠀⠀⠀⠀⠀⠀⢠⡶⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠃
⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠆⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣄⡀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⠀⠀⢀⡀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢹⢳⡀⠀⠘⢶⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⠜⠀⠀⢀⡞⡏
⠀⠀⠀⠀⠀⠀⠀⢧⠱⡀⣼⡄⢠⠞⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⡄⠃⣴⡀⢀⠞⡼
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡹⡍⠑⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠉⢡⢎⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣶⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡉⢷⣰⡶⢠⡤⠀⠀⠀⠀⠀⠐⠁⠂⡀⠀⠀⢀⠴⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⣀⣤⢠⡦⣰⡇⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣻⣬⣃⣰⠀⢀⡠⠚⡣⢴⡀⢹⣀⡀⡃⠀⠈⢠⠶⢠⡖⠉⠉⢆⠀⣐⢠⢋⣼⠟⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠸⡋⢩⣿⠶⡋⠀⢺⠗⡂⢉⠇⡠⠯⢽⣦⢀⣯⡘⠳⢀⠆⠀⠈⣤⣾⣛⠙⢫⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠐⠠⡔⠛⠁⢸⣷⠲⠤⣃⡀⠀⠈⠋⣄⣀⠴⠊⠉⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠁⠶⣶⡤⠏⠉⢢⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠔⠃⠄⠠⠐

⣶⣶⣶⣾⣷⡿⣾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣾⣷⣾⡷⣶⠶⡶⠶⠿⣶⡶⢶⡿⠷⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡅⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⣴⢠⠀⠀⠀⡄⠀⢀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⢀⡘⠀⠀⢀⠋⠀⠀⠀⣀⠀⠉⠿⠀⠀⠀⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⡃⠀⠀⡧⣀⠀⠀⠀⠈⠐⠀⠀⢠⢿⡀⠀⠰⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢣⠁⠀⠀⡀⠘⠀⠀⠠⠀⠀⠀⢀⠳⠋⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⡄⠀⠀⠀⠁⠂⠀⠀⡄⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⡀⢹⡄
⠀⠀⡀⠄⠀⠀⢸⡀⠄⠀⠀⠐⣆⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⢠⠄⠁⣠⣣⣀⣀⡄⠀⠀⣸⠀⠀⠀⡠⠐⣗⠀⠀⠀⢀⡀
⠀⠀⠀⠸⡀⠀⢁⠀⢀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠁⢀⠀⠀⠀⠉⠀⠀⠀⠼⢈⢿⡇⠀⠀⢸⠛⠃
⠀⠀⠀⢀⢾⣿⠧⡤⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⢾⣷⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⠳⣌⣺⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⢸⠀⠀⢠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠀⠀⠠⡄⠀⠂⠀⡤⠄
⠀⠀⠀⠀⠀⠀⢴⡿⣿⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠐⢀
⠀⠀⠀⠀⠀⠀⢋⠶⠸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠛⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣆⠀⠂⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⠍⠁⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢋⠑⡄⠀⠀⠀⠀⠀⠀⢐⣞⣲⣧⡀⠋⠀⣐⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⡿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠈⠉⣏⠉⡁⠠⠆⠁⣇⠋⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⠠⠀⠀⠘⠀⠠⢿⡆⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⠀⠀⣐⠀⡄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠄⢠⡄⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠸⣷⠑⡀⢀⢀⡁⢀⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠀⠀⠀⠀⠀⢻⠆⢘⠨⠀⠀⠒⡧⡀⠰⡄⢧⠀⠀⠀⠀⠀⢴⣀⡀⢨⣎⠈⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⠀⢀⠀⠀⠈⠛⢦⣹⣄⣧⠀⠀⠀⠀⠀⠉⠀⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠀⢸⣷⣤⠀⠈⠙⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠐⠱⠂⠈⠿⠓⠆⠀⠐⠂⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠸⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢟⡿⠁
⠀⠀⠀⠙⢦⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣵⡏
⠀⠀⠀⠀⠀⠈⢳⣬⣙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⣋⡵⠞⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢹⡟⠶⣤⣍⣙⡛⠲⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠖⢚⣛⣉⣭⡤⠶⠚⠋⠹⡇⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡿⠈⠉⠉⠙⠛⠓⠲⠶⢶⡶⢶⡶⣦⠤⠤⣤⠶⢶⡶⠶⣶⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣧⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡏⠀⠀⣿⠀⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣸⣿⠇⠀⠀⢹⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⡄⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢁⢀⢶⡤⠓⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠘⠀⠀⠀⣀⠃⢆⠆⠀⠠⠀⠀⠤⢀⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡃⠀⠀⠀⠀⠀⠀⠲⠦⡆⢦⣤⠥⠄⣀⣐⠰⠀⢀⠏⠀⠀⡄
⠀⠀⠀⠀⠀⢀⢘⡀⠈⠀⠏⠗⠰⢤⠴⠠⡀⡤⠄⠀⢛⡇⠶⠂⡏⠀⠀⠀⠀⡾⠤⢛⠉⣁⡀
⠀⠀⠀⠀⢀⡀⠀⢹⡟⠞⠪⠀⠈⠉⡇⢨⡂⠀⠀⣠⡄⢹⣇⣽⣱⠀⠀⠈⠀⠓⣖⠞⠤⡼⠤⡀
⠀⠀⠀⠀⢶⠠⡍⠀⠀⠀⠀⠀⠠⣄⠈⡇⠴⠀⠀⣷⠾⠷⡇⢬⡀⣤⠛⠾⢷⡀⠆⠠⡤⠝⠣⠈⡀
⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⡥⡶⡚⠉⠈⠀⠀⠀⠀⠉⠰⡌⢫⡓⠀⠀⣀⣀⠀⠀⠀⠀⠀⢑
⠀⠀⠌⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣟⢐⠏⠀⡤⠄⠀⠸⠿⠀⠀⠀⡀⢿⡅⠀⠉⠘⢂⣤⡀⠁⠀⠀⠂
⠀⠀⠆⡀⠀⠀⠀⠀⣸⡅⢤⠤⠏⠀⠀⠓⠈⠥⠂⠄⢀⡀⢀⠀⠀⠀⡠⡓⡔⡄⠀⠈⠉⠓⢬⠀⠀⠄
⠀⠀⠀⠐⢄⠤⢐⠴⢗⡄⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⢼⡋⣭⠃⠲⢀⣍⣧⡼⠀⠀⠀⠘⡢⠜⠀⠐⡄
⠀⠀⠣⡴⠟⣰⠁⣀⢠⠀⢐⠀⡁⣶⠄⠀⠈⢀⣾⠁⢀⠗⠉⠀⠀⡄⡈⠀⠁⠀⠆⠀⡰⠠⡄⡀
⠀⠀⠰⣅⠀⠥⣢⠻⠾⣄⡤⢤⠟⠋⢉⡈⠁⣦⢡⠀⠀⠀⠀⠄⠀⢐⡓⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⡀⠛⢢⡀⡠⡜⠀⠀⠀⠀⠈⠛⢮⠩⣥⣀⣤⠀⠠⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠨⠇⠈⠖
⠀⠀⠀⠀⠃⠄⣘⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡶⠏⠈⠁⠀⠀⠰⠚⣋⠒⠀⠀⠀⠀⠀⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⢄⠀⠀⠀⠀⢀⡗⠀⠉⠀⠠⡄⠉⡀⠁⠀⢠⡄⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⡈⣌⢡⠂⠀⠚⠰⠀⠀⠀⠀⠀⢠⠀⠀⢉⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠒⣄⠀⠀⢀⡠⠁⠀⠀⠹⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠁⠈⠈⢱⠁⠀⡠⠠⠂

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⡄⠀⠀⢀⡇⠀⠀⢀⡎⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠹⡆⠀⢸⡇⠀⠀⣾⠃⠀⢀⡾⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣀⠀⠈⠷⠀⣀⡥⠶⠖⠒⠒⠲⠶⣤⣀⠛⠁⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙⠗⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠉⠀⠀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⠀⣼⠃⢠⡀⢠⠀⣠⣤⠀⣠⠀⠀⢠⠀⠀⣠⣤⣄⠈⢧⠀⠶⠖⠛⠉
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠶⣾⢀⡷⠆⠀⡿⠀⠀⣿⠀⠰⡇⠀⣹⠀⢸⡆⢠⣀⣀⣀⣀
⠀⠀⠀⢀⣾⠟⠋⢉⣿⠇⠐⠒⠚⠛⠛⠸⠇⠀⠙⠀⠋⠘⠛⠓⠀⠛⠛⠀⠛⠛⠀⠙⠛⠁⠀⢸⡇
⠀⠀⠀⠘⣿⡄⠀⠋⢁⣶⢂⣴⠄⣰⣆⣴⡶⣠⣶⠀⣰⣆⣴⡶⣠⣶⠀⣴⡶⣦⢰⣿⣴⡆⠀⣼⠃⠛⠶⠦⠤
⠀⠀⢀⡔⠈⢿⣦⢠⣿⣣⣾⣋⣼⣿⢫⣿⢟⣿⣃⣼⣿⢫⣿⠟⣿⣃⣼⡿⠾⣣⠎⣼⠏⢀⡼⠃⣤⡀
⠀⠀⠈⠿⣶⡾⠟⠀⠛⠋⠙⠋⠙⠁⠀⠉⠈⠛⠋⠙⠁⠀⠉⠀⠛⠋⠈⠛⠛⠁⠀⠛⠚⠉⠀⠀⠈⠉⠓⠦






⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢿⣿⣹⡿⡿⡿⡿⡿⣿⣻⣻⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⡿⢟⣏⣏⠏⠏⠇⠇⠇⠇⠇⢇⢯⢯⣟⣟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠝⠟⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠊⠿⢿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣫⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⡃
⠀⠀⠀⠀⠀⠀⠠⢿⡿⢿⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣷
⠀⠀⠀⠀⠀⠀⣤⡆⣀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣟⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣚⡆
⠀⠀⠀⠀⠀⢼⣿⣿⣓⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡇
⠀⠀⠀⠀⠀⠿⣿⣓⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠃
⠀⠀⠀⠀⠀⠀⣿⣛⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣖⣺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡝
⠀⠀⠀⠀⠀⠀⠸⣿⣶⡭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣖⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠛⠮⣞⣖⡖⡤⡄⡄⡄⡄⡄⡤⡤⡴⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠃⠓⠃


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⠞⠉⠉⠳⣄⣀⠀⠀⠀⠀⣠⠟⠛⠉⠹⣦⡀⠀⠀⠀⠀⢀⣀⣠⡶⠛⠉⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⢀⣤⢿⡀⠀⠈⠀⠀⠀⠀⠛⠉⢻⣀⠀⠈⠷⠤⠤⠤⠤⠾⠃⠀⢀⣴⠟⠋⠙⠋⠀⠀⠀⠀⠀⠀⠈⠟⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣀
⠀⠀⠀⠀⠀⠀⠻⠦⣤⣄⠀⢀⡀⠀⠀⠀⠀⠀⣠⠤⠞⠁⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠶⠶⠛⠛⠛⠁⠀⢀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠉⠳⢦⡀⠀⢻⣄⠀⠀⠀⢠⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⡀⢀⠀⠀⡋⣳⠀⠀⠙⠳⠶⠶⠞⠳⣤⣀⣀⣀⣷⣄⠀⠀⠀⠀⠀⣠⠾⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣤⣀⣠⡤⣤⣀⠉⠉⠛⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠓⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣸⠁⠀⠀⠀⠹⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⢠⡶⢾⠋⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢦⡀⠀⠀⣼⠀⡀⠀⠀⠀⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠛⠀⠀⠀⠀⣷⣄⡀
⠀⠀⠀⠀⠀⠈⢧⡀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⠛⠷⠶⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠁⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣆⣀⣀⣀⡿⣤⣤⠾⠛⠋⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠻⠦⣤⡀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣴⠚⠞⠛⠛⣦⣄⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡈⢳⡄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠴⠶⣤⣄⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠁⠘⣧⠀⠀⠀⠀⠙⠷⣄⣼⣦⣴⠟⠋⠀⠀⢀⣠⡞⠋⠙⠳⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⣞⢁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⠛⠶⠶⠶⠶⠖⠒⠒⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠻⣤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣄⠀⠀⣠⣀⣀⡶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠇⠀⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣞⠉⠀⠀⠉⢷⣤⢤⣤⡀⠀⠻⣄⣴⠀⠀⢀⠀⠀⠀⠀⢀⣠⡗⠀⠀⣠⡴⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠈⠹⣦⣀⣨⣦⣀⣀⣤⠟⠁⢀⣀⡼⠁⠀⠀⠀⠀⠙⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⢧⡀⠀⠀⠀⠉⠁⠀⠈⠉⠁⣠⠞⠋⠙⡇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠴⠞⠁⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⡶⠷⠦⡤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠞⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣆⣠⣤⣄⣀⣀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡤⠖⠚⠉⠉⠀⠀⠀⠈⢳⠀⠀⠀⠀⠉⠙⠓⠦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠇⢀⣠⠤⠴⠖⠒⠒⠛⠛⠛⠛⠛⠛⠒⠲⠦⣤⣀⡀⢻⠲⣄
⠀⠀⠀⠀⠀⣠⠞⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣉⣡⣤⣤⣀⠀⠀⢠⣈⡉⠻⢤⣈⠳⣄
⠀⠀⠀⠀⢰⢃⡴⠋⠀⢀⡴⢋⣠⠤⠶⠚⠉⠉⠛⠋⠁⠀⠀⠀⠈⠳⠦⠤⣤⡉⠂⠀⠈⢳⡹⡄
⠀⠀⠀⠀⣿⡞⠀⠀⠐⢁⡴⠋⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠲⡄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢯⣀⡀⠀⠀⣀⣠⠇⠀⠀⠀⠀⣀⡼⢁⡇⠀⠀⢰⡿
⠀⠀⠀⠀⢹⠙⢦⡀⠀⠘⢤⣖⣂⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠟⠩⠖⠉⠀⢀⡴⢫⡇
⠀⠀⠀⠀⠈⣧⠀⠙⠳⠦⣤⣀⣂⠙⠶⠤⠴⠿⠿⠧⢤⣤⣤⡤⠴⠚⠁⠀⢀⣠⠴⠚⠉⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠚⠋⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⡀⠈⣿⠀⠀⢸⡟⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣀⣄⣰⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠍⠒⠒⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠒⢲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠋⠉⠉⠓⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⡤⠴⠋

⠀⢀⠀⠀⠀⠀⠀⠀⢠⢀⣀⣀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⣀⣀⡀
⢀⣻⡠⠒⠀⠀⠀⠀⢂⢳⣄⡉⠉⠁⠀⣀⣇⡔⠂⠀⠀⠀⠀⢡⣑⠒⠖⠒⠀⠁⣀⣿⡰⠂⠀⡀⠀⠰⡹⣤⡉⠉⠁
⣿⣿⣿⣷⣆⢀⠀⡄⠘⠂⠭⠙⠀⢲⣾⣿⣿⣿⣷⡀⡀⢠⠀⠒⠛⢿⡉⠀⠰⣶⣿⣿⣿⣷⡄⢀⠀⠀⠳⠈⠉⠂⠰
⣿⣿⣿⣿⣿⠙⠃⢀⣮⣠⠆⠀⠲⣺⣿⣿⣿⣿⣿⡏⠙⠀⢠⣅⡴⠄⠐⢰⢒⣿⣿⣿⣿⣿⣟⠙⠀⢠⣇⡰⡄⠀⢲
⠿⠿⠿⠛⠉⢁⢦⣸⣿⣿⣿⠠⠄⡅⣹⠿⠿⠿⠏⠉⡐⢤⣾⣿⣿⡇⡤⠸⢤⢹⠿⢿⣿⢿⠉⣤⣦⣼⣿⣿⣧⣤⣨
⣠⣄⠠⠂⡄⣰⡔⢹⣾⣿⠏⠨⠕⠈⢁⣠⡀⠐⢠⠂⣤⠚⣷⣿⡿⠃⢽⠒⠒⢀⡐⣈⠸⢂⠓⣾⣞⣿⣿⣿⡏⢭⣿
⠀⠈⠁⢠⠈⡿⢁⠀⠉⠙⠠⠅⠀⠀⠀⠀⠉⠀⡄⣿⢁⡇⠀⠉⠁⠬⠈⠁⠈⠀⠀⠈⠡⡄⣿⠏⡀⠈⠉⠁⠬⠈⠉
⠀⠀⠀⠈⡷⠧⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⢛⠙⠉⠁⢁⡈⠉⠈⠋⠭⡷⡒⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡟⢀⣠⠤⠀⠀⠈⠱⠲⠀⠀⠀⠀⠬⣛⣹⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⣾⡿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢈⢈⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣾⣇⣿⣾⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠰⠋⠀⢀⡄⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⣿⣼⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⢋⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣄⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⢣⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠅⣾⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠿⠿⢛⣟⡋⣭⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢰⠏⢵⡿⣯⠉⠀⣴⠲⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⢀⠀⢰⣯⣤⣾⢡⠃⠀⠀⠋⢸⣿⣏⣼⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⢹⣿⠇⠈⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⠋⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣹⣻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡃⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⣿⣿⣞⣿⣿⣿⣿⣍⣙⣷⣮⣛⣿⡿⡯⠇⠀⠀⠀⠀⢘⠻⣵⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢐⠟⢿⣿⣿⣿⣧⣟⣿⣿⣿⣿⣿⣿⡍⣉⣀⠂⢶⠖⡤⠄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣤⣙⠁⢐⡒⠀⢱⣌⠛⠛⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⣩⣭⣤⡀⠘⠒⠀⠚⠛⣿⣿⣿⣿⣿⢿⣿⣭⣟⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣷⢀⣁⠉⠾⣽⣿⣿⣿⣚⣿⣿⠷⣼⡿⠿⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉⠉⠙⠛⠻⠻⠻⠚⠓⠀⢀⣀⣀⣂⡦⣤⣾⣷⣽⣿⠿⢷⢿⠧⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⢿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣾⣏⣀⡀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⡿⠀⠀⠀⠀⢻⡄⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⢻⣏⣹⠃⠀⠀⠀⢀⡾⠁⠀⠀⠀⣷⠀⠀⠀⣀⣺⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢀⡼⠛⠛⠛⠉⠛⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⠘⠷⠶⠟⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡟⣀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢾⡄⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣠⠟⢉⡽⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡠⠀⠀⠷⣤⡄⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣾⣧⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣠⣷⡀⢀⣀⡘⣷⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣤⣤⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢤⣤⣄⡀⠀⠀⠈⠻⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠞⢁⣤⡾⠃⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⡠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣤⣤⣤⣤⣤⣤⠶⠞⠛⠁⠀⠀⠀⠀⠀⢀⣀⣭⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣷⣄⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠾⢛⣛⣿⣿⡶⠶⠶⠶⢶⣝⡲⠶⣤⣂⣤⣤⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡼⠇⠀⣠⢴⡆⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣷⣦⢠⣶⣦⢠⣾⡿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⢹⣿⠀⠙⠁⢸⣿⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠄⢉⣤⠟⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢧⣌⠡⠄
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠉⠛⠒⠲⠤⠤⠤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⠤⠤⠶⠒⠚⠋⠁⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣀
⠀⠀⣠⠞⠉⠹⣿⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡿⠉⠙⢦
⠀⢰⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⣧
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⢈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢃
⠀⣼⣟⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷
⣤⡘⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠛
⠉⣰⣿⠿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢿⣶
⠀⠹⢷⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⠿
⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠈⣇⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢠⠏
⠀⠀⠈⠳⢤⣼⣿⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣤⠴⠋
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⣀⡤⠶⠒⠚⠋⠉⠉⠉⠛⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⠛⠉⠉⠉⠛⠒⠲⠦⣄⡀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢃⣤⠙⣷⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⡏⣡⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠀⣼⣿⢀⣴⣄⢸⣿⡀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣋⣈⣛⣋⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⢻⠝⡇⠀⠀⣾⣷⡀⢀⢀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⢹⡷⠀⠀⠀⠀⠀⣾⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠹⣿⣿⣦⡀⠲⠀⠀⠀⠀⠀⠀⠤⣬⣿⠃⠀⠀⠸⠇⠀⢀⡤⣴⠆⠀⠀⠀⠉⠛⠶⠻⠁⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣄
⠀⠀⠀⠀⠀⠀⠑⢕⠀⠀⠀⣴⣟⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠙⣆⠀⠀⠀⢼⡿⠾⢻⣿⠀⠀⠀⢀⡼⠝⠒⠋⠝⣆
⠀⠀⠁⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣢⡆⠀⠀⢀⠀⠀⠊⠀⠀⠄⠀⠀⢸⡦⠀⠀⠀⢴⡟⠀⠀⠈⣸⣇⣀⣠⣷⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠌⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠙⠛

⠀⠀⠀⡤⠢⠢⠂⠄⠀⠀⠀⢀⣴⣾⣿⣶⣦⠀⠀⠀⠀⡠⠒⠂⠲⢄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠶⠒⠲⢢⡀
⠀⠀⣐⠀⠀⠀⠀⠈⡄⠀⠀⠼⠿⠿⠿⠿⠿⡇⠀⠀⣜⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣉⣉⡄⠀⠀⠀⢰⠁⠀⠀⠀⠀⢧
⠀⠀⠐⣀⠀⠀⢀⡌⠀⠀⠀⠉⠍⠭⠭⠭⠍⠁⠀⠀⠘⣄⠀⠀⠀⣠⠃⠀⠀⠐⠿⠃⠀⠘⢻⠆⠀⠀⠀⠀⠛⠚⠚⠁⠀⠀⠀⠘⣧⡀⠀⠀⣤⠟
⠀⠀⠀⠀⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁
⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢀⠬⢠⡇⡱⢔⠀⠀⠀⠠⠊⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⢀⣾⣿⣼⣾⣷⡄⠀⠀⢠⣾⠛⠉⠙⢻⣆⠀⠀⠀⠀⠀⠀⠄⡱⣄
⠀⠀⠙⡪⢣⢱⢕⢐⠃⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠀⠸⣻⣿⣿⣿⣕⠗⠀⠀⢸⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠠⣢⣾⣿
⠀⠀⠀⠻⢿⣿⡿⠋⠀⠀⠀⠈⠢⢄⣀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢽⡳⠞⠀⠀⠀⠈⠻⣦⣤⣴⠿⠃⠀⠀⠀⠐⢪⣾⣿⠿⠃

⠀⠀⣠⠴⠚⢋⣉⣉⣓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠶⣦⣀
⢀⠞⠁⠀⣴⡿⠷⠒⠈⠑⢎⢳⡄⠀⠀⠀⠀⢀⣾⣿⣏⣠⣴⣶⣦⣄⠈⠙⣷⡄⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠚⠛⠓⠲⣄⠙⢧
⡞⠀⠀⠐⠿⠃⡄⠀⠠⣌⠈⣇⢻⡄⠀⠀⠀⣾⣿⣿⣿⠋⠀⣠⣿⣿⣷⠀⠸⣿⡀⠀⠀⠀⢠⣿⠀⣼⡏⠀⠀⣀⡀⠀⠈⢧⠈
⡇⠀⠄⠀⣀⡀⠀⠀⠀⢸⡇⡀⠈⡇⠀⠀⠀⣿⡿⢿⣧⠸⣧⣿⡇⠈⣿⠇⠀⣿⡇⠀⠀⠀⢸⣯⠀⣿⡀⠀⠀⠊⠃⠀⡆⢸
⢹⡀⠘⣦⠸⣿⠟⢀⡠⠛⠀⠁⡸⠀⠀⠀⠀⢹⣿⡈⠻⢷⣬⠁⠀⠀⠋⠀⣸⡿⠀⠀⠀⠀⠈⢿⣆⠈⠳⣦⣄⣀⣤⠞⢁⡎⢠
⠀⠑⢌⡁⠐⠁⠔⠉⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠹⢿⣦⣀⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣈⣁⣠⡴⣫⡢⠋
⠀⠀⠀⠉⠒⠢⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠁

⠀⠀⣀⡴⠒⠋⠉⠉⠓⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⢛⣻⣙⡓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣛⣭⣭⣬⣥⣀
⢀⡾⢃⢀⣤⠖⠂⣂⠀⢀⢀⠙⣄⠀⠀⠀⠀⢀⡞⡽⡁⢀⡴⠦⠑⠫⡑⠈⠳⡄⠀⠀⠀⠀⠀⣠⣟⣿⣿⣿⣿⡿⠙⠻⣿⣿⣦
⣾⠁⢁⣾⣿⣶⣾⣿⣿⣆⠀⠀⠸⡄⠀⠀⠀⡞⣸⠡⠁⡞⠀⠀⣠⡤⢭⡓⠀⢻⡀⠀⠀⠀⢠⣿⣿⣿⢹⡟⣽⣿⣿⠿⣿⣷⣹
⣇⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠠⠀⡇⠀⠀⠀⣧⡇⠀⢸⡇⠀⠀⠙⠓⢀⠇⡐⣸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣬⡿⢿⡇⡼⣿⣅
⢹⡀⠀⠈⠻⠿⠿⠟⠋⠀⡠⠃⡸⠀⠀⠀⠀⢻⣧⡀⣈⠙⠓⠶⠶⠒⣉⠔⠁⡼⠀⠀⠀⠀⠈⢿⡝⢿⣿⣿⣿⣿⣿⣽⡇⢋⣼
⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠹⣦⣄⠉⠀⢠⣭⡭⠔⣢⠞⠁⠀⠀⠀⠀⠀⠈⠻⣄⡹⣿⣿⣿⣿⣿⣷⠿⠋
⠀⠀⠀⠙⠒⠢⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠿⠭⠭⠟⠂

⠀⠀⢀⣤⢶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢖⣒⣀⡐⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠒⠒⠲⢦⢤⡀
⢀⣴⢿⢟⡭⣖⣛⣓⠲⣍⡻⣷⡀⠀⠀⠀⠀⠀⡔⡥⣪⣵⣶⣶⣶⣭⡢⡈⠳⡀⠀⠀⠀⠀⠀⢠⢺⢯⠋⠴⠖⢂⠀⠈⠙⢿⢦
⡼⢁⢃⣾⡞⣯⠝⢛⣷⠈⢻⡈⣿⠀⠀⠀⠀⡜⡌⣼⣿⣿⣿⣿⣮⢳⡹⡌⠂⠸⠀⠀⠀⠀⢀⢣⣼⣆⢿⣦⣴⡇⣾⡿⠷⣌⢇
⡇⠈⠘⡟⢇⣋⣷⢸⡟⢀⡀⡇⢸⡇⠀⠀⠀⡇⣷⣿⣿⣟⣿⣿⢿⢯⡇⡇⠠⠀⡆⠀⠀⠀⢸⢦⣻⡟⢮⣙⡋⠀⢁⣀⣠⣹⡞
⢻⡀⠐⢌⡳⠯⠕⣋⣴⣿⣿⠁⡾⠀⠀⠀⠀⢣⢻⣿⣿⣿⣿⣿⢞⡼⣰⠃⠀⢰⠀⠀⠀⠀⠈⣦⣙⠿⣲⠤⠒⠛⣛⠛⢿⣷⣇
⠀⠹⣄⠀⠈⠉⠉⠀⠾⠟⣡⠞⠁⠀⠀⠀⠀⠀⠣⡙⠿⣿⣿⣓⣫⠔⠁⢈⡴⠃⠀⠀⠀⠀⠀⠘⢮⣵⠎⣙⣖⣀⣠⠔⣇⢸⠟
⣀⣀⣀⣙⣲⣦⣤⣤⣶⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣲⣤⣤⣤⣄⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣶⣼⣯⣭⣔⣋⣁⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡾⢶⡄⠀⢴⡀⠀⠀⠀⠻⣟⣛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⡁⢈⡁⢀⠄⠀⠀⣀⣰⣶⣚⣿⣏⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⣮⣇⣠⡄⠀⠰⠟⠈⢹⣿⠿⣿⣿⣿⣿⣿⣦⣶⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣏⣙⣿⣷⣶⣶⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣟⣈⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⣴⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣣⠟⢻⡏⠂⠹⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣍⣿⣿⣿⣿⣿⣏⡉⠻⠛⡿⠛⠻⡂⠠⠈⠉⠉⡿⠀⠘⣷⣄⡀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⡿⠋⣤⡄⣰⣦⡅⠀⡀⠉⠀⠀⠀⠀⢠⣆⠀⣿⣾⣇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⣃⡠⢠⣾⡟⠛⠩⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⣾⣿⢷⣤⣤⣦⣀⢠⠄⠀⠀⠀⠀⠀⢸⣿⡁⢾⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢿⡉⠋⠉⣿⣟⡋⠀⠀⠀⣴⣶⣆⠀⢸⣿⣦⡼⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⢹⣿⣾⣿⣿⣿⣿⠟⢿⣷⣾⣿⣿⡇⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣹⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣘⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣿⡏⠈⣿⣿⠏⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣷⣿⣿⡦⠈⠻⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠟⣿⣿⣯⣄⣾⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⡻⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠿⡀⠒⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣶⣿⣿⡏⠀⢀⣴⣦⡀⢰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠦⠄⠀⡼⠋⠉⠥⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠠⠂⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣴⢾⣿⣿⣿⣿⣿⣿⣄⣸⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣇⠀⠀⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠴⠕⠂⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠸⠿⠿⢟⠛⠋⠉⠉⠀⣀⣀⣤⣶⢂⠀⠀⠀⡄⡀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⢀⡀⠀⡄⠀⠀⠀⣀⠀⣀⣠⣼⣤⣶⣶⣾⣿⣿⣿⣿⠃⣾⡅⠒⢸⡇⠀⠾⠿⢇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣧⡀⠒⢾⣃⡀⠀⠸⠦⠴⠒⡛⠦⢭⡿⠿⠛⢉⠛⠛⢟⡉⡀⠐⠻⠇⣈⣼⣿⡰⠎⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿⠠⠛⠉⠀⡠⡊⡈⠘⠃⠘⠀⠠⠧⠡⠒⠀⢀⠀⠰⠀⢸⡄⠀⠈⠘⠓⠢⣧⡀⠀⠈⢢⠔⠈
⠀⠀⠀⠀⠀⠀⠘⠁⠔⠀⠀⡀⠀⠀⣇⣰⠀⠒⠀⠀⠀⢠⡀⠊⠀⠄⢀⡇⣿⣇⠀⣁⡀⠀⠀⠀⢁⡄⠀⠰
⠀⠀⡄⠀⠀⠀⠀⠄⢀⡀⢀⣤⣤⣤⣝⠀⠀⠤⠀⣠⡄⣸⡇⠲⠥⠠⠚⠛⠛⠉⢁⠁⠂⠀⠀⠈
⠀⠀⠀⠉⠀⢀⡁⠀⡀⣀⣀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠻⠟⠀⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠺⠖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠸⠀⠃⠘⢃⠸⠃⠰⠂⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠆⠈⠀⣄⠀⠇⠈⠃⡘⠀⠆⢠⠆⢈⠐⠃⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠁⢤⠘⢀⡀⢰⠀⠻⠀⠇⠀⠇⣠⠄⠊⠰⠃⡀⠐⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠓⠀⣀⠙⡀⠰⠄⠃⡀⢰⠀⠇⠰⠃⢠⠄⠀⠚⢀⡬⠀⠴⠈⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠤⠀⠤⠉⣄⠘⠆⠉⠀⢠⠀⢰⠀⢀⠰⠋⢀⠉⠠⠄⠰⠒⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠒⠆⠠⡌⠀⠠⡄⠓⠀⢠⡄⢀⠋⣀⠈⢉⡀⠠⠄⠀⢚⣀⢚⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⣀⠐⠒⠀⠒⠀⠘⠒⠀⣀⡘⠀⠈⣉⠀⡀⠀⠠⡄⠠⠤⠄⠠⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢀⠐⠛⠀⠀⠆⠀⠚⢀⠀⠴⠂⠷⠈⢀⠉⠀⠤⡄⠐⠶⠄⠰⠖⢀⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠉⡀⠉⠁⠐⠚⢀⠈⠁⠰⠂⣴⠐⡆⠈⠃⡀⠦⠀⡐⠄⠀⠛⠂⠠⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠈⢁⠀⠁⣠⠀⠉⣰⠂⡀⢀⡀⡀⢠⡘⡆⠙⠀⢤⠙⠠⣌⠉⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠈⣠⠆⠁⠞⢁⡄⠘⠁⠘⠃⠿⡈⢁⠰⡄⠳⠀⠐⠦⠀⣀⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠘⠁⠤⠀⡀⠚⠀⠞⠰⠂⠹⠘⠃⠀⠰⡌⠙⢀⠲⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢀⡄⠈⠁⠰⠂⠄⠰⠄⠶⠸⠆⠙⠀⣄⠲⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠞⢠⠆⢠⢠⡄⡆⠰⡆⠲⠈⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠶⠶⠶⠤⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠙⠂⠀⠀⢦⡄⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⢦⡀⠀⠀⠛⠉⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⢤⣤⣄⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣅⡀⠀⠀⠈⠙⣦⣀⡴⠋⠀⢀⡀⠀⠀⠀⠈⠳⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣿⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⡇⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣷⠀⠘⢷⣄⣠⡴⠛⠻⡆⠀⠀⠀⠀⠲⣤⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠒⠶⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⣇⢀⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⡀⠀⠀⠀⣠⠤⠦⠤⠴⠞⠁⣾
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠈⠳⠶⠟⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠋⠛⢳⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡞⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣀⣤⠞⠀⠀⠀⣀⣄⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⣶⡀⠀⢀⣀⡀⠉⠛⢻⣿⣿⣧⡀⠀⠀⢺⣿⣿⡇⠀⠀⣾⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⡿⢸⡿⠁⢰⣿⣿⣽⠀⠀⢸⡟⠁⠉⠛⠂⠀⠉⠛⠉⠁⠀⠀⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿⣿⠏⠉⣷⡀⠙⢁⣠⡀⠀⠉⢀⣴⣶⢲⣤⠀⠀⠀⣴⠞⠂⠀⣿⣿⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣿⡏⠀⠀⢻⣷⢠⣾⣿⠇⠀⠀⣼⡿⢡⣿⡟⠀⠀⢸⡟⠀⠀⠀⣿⣿⠃⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣸⣿⠁⠀⠀⣼⡿⣾⡿⠋⠀⠀⢠⣿⣧⠏⣿⣷⠀⠀⣸⣷⣀⣰⠞⣿⡏⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠿⠃⢀⠀⢠⡿⠁⠻⣷⣀⣀⡴⠋⠈⠉⠀⠘⠿⡶⠞⣁⠈⠉⠀⠀⣿⠀⢀⣿⡇⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠀⠀⠀⠈⠉⠁⠀⠀⣀⠀⠀⡀⠀⢀⣾⣿⠀⠀⠀⠀⠁⠀⠸⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⡄⠀⣴⠟⣶⣄⣶⢟⣦⠀⢸⣿⢧⣾⡟⠠⣼⣿⣧⡶⠃⢀⣤⠀⠀⠀⠨⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠞⢰⢿⡇⣿⣿⣇⣼⣿⡀⠉⠀⣾⣿⠀⣠⢾⣿⡅⠀⢀⣾⣿⠀⣾⣇⠰⣤⡙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣡⣾⢸⢸⣷⢟⢡⣉⠁⠻⣿⣦⣤⡿⠿⠛⠁⠀⠻⣷⣤⠞⠁⠙⠟⠉⣿⡄⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢸⡌⣿⢸⣎⠛⠛⠠⣤⡉⣉⠀⣴⣿⣿⣦⡀⢀⣠⣤⡴⢂⣴⡖⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠘⣇⣿⠀⠻⠇⣠⢖⣿⡿⣏⠀⠿⠻⢿⡽⠡⣿⣿⢏⣴⣿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⢻⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⡀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠉



⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣖⣲⣶⣶⣶⣶⡄⢀⣶⣶⣾⣯⣴⣶⣶⣶⣶⣶⣶⣶⣶⠆⡼⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⡿⢿⣏⣭⣭⣿⠟⢻⡿⠿⠛⠛⠀⠈⠁⠚⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⢻⣿⢿⣿⡇⠰⣿⣿⣿⣿⠋⠁⢀⣠⠄⢀⣀⠀⠀⠀⣠⣶⣶⣄⣠⣄⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⣴⢶⢶⣄⢷⣄⠙⠿⠃⠉⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⠟⠀⣻⡷⠀⣸⣷⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⢀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢹⠏⠁⠀⠀⣛⣥⡶⢰⠂⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⢠⣶⣤⣾⣿⡿⠙⠘⢀⣼⣿⣿⣿⡟⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠈⠻⣯⣿⣿⠿⠛⠛⠻⣶⠳⠶⢾⣿⣧⣀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡼⠋⣠⠀⣹⡟⠁⢂⣀⡀⠈⠉⠀⠀⠠⠉⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⢻⣾⠟⠁⢆⠀⠈⠁⠁⠀⠀⠀⠀⠁⠸⠗⡄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣙⣡⡮⠨⠀⠈⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⣀⡐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⣦⣤⠀⠀⠑⠂⢀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠⢸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡇⠀⢿⣿⠓⢄⠀⠀⠊⠻⠷⣤⣤⡤⠠⣾⣷⣆⠀⠀⠙⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⢸⣿⡔⣆⢀⢀⣀⠀⠀⠀⠀⠀⣤⣼⣿⡘⠂⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣠⣼⣿⣥⣬⣿⣿⡛⠱⣤⣤⣠⣶⣿⢩⣿⣧⠄⠁⣬⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡟⣿⣽⢻⣿⣿⣿⣿⣿⣥⣌⣉⣁⡿⠿⣿⣿⣿⢻⡄⣢⠽⠺⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠙⠛⠿⠒⠛⠁⣿⡟⣿⣟⡉⠀⢈⣿⣿⣿⣿⣿⢻⡇⠀⠂⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣖⣲⡆⢀⣇⢀⡳⠄⠀⢀⣀⠈⠉⠉⠉⠋⠸⠇⡰⣤⣨⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠉⢡⠊⠉⠁⠉⠀⠀⠉⠉⣻⠀⠀⠈⠀⠀⢈⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠀⢠⡀⢽⣷⣶⣶⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠙⠛⠏⠉⠏⠱⠄

⣺⢠⠛⢀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠚
⠢⠣⠀⠁⠀⠀⠀⠀⠀⢀⣐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠤⡙⠃⢍⡀⠀⠀⠀⠀⢀⠈⠘⡀⠁⢑⢁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⣾⣷⡦⠄⠀⠀⠀⠀⡀⡘⠋⡡⣠⡠⢠⠡⠀⠀⠀⠀⣀⡈⠈⢈⢈⢈⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⠓⠀⠀⠀⠀⠀⢰⡆⠀⠁⢉⠃⣹⢼⠀⠀⠀⠀⢀⡁⢀⢀⢀⢀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⢌⡱⠂⡀⠔⠁⠀⠀⠀⠀⠀⠀⠤⠄⠄⠄⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⡠⠊⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠱⠂⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣭⣀⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠠⢽⣾⣿⣷⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠒⢂⠀⢂⢀⡒⠀⠀⠀⠀⠀⠀⠿⣿⠧⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉⠥⠄⠩⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠄⡄⡄⡂⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⡂⢕⠔⡢⡀
⠀⠀⠀⠸⣄⣄⡗⢕⡀⢀⠀⠀⠀⠀⡌⡀⢐⠀⠀⢀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡣⢀⠊⠑⠉⢊⠐⠄
⠀⠀⢄⡈⣽⣿⣿⡭⠍⠁⠀⠀⠀⣐⢒⠐⠂⠐⠒⠒⣒⠂⠀⠀⠀⠀⠀⠘⠛⠏⠀⠀⠀⠀⠀⠀⢉⠊⡑⠬⡊⢔⢕⠽⡂
⠀⠀⠠⠾⠋⡊⠇⡤⡑⠀⠀⠀⠀⠀⠄⠠⠀⠄⠀⠠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢉⠂⢔⢔⠒⣁⠞⠁
⠀⠀⠀⠀⠁⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡁⢀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣁⣈⣇⣃⡀⠀⠀⢀⡀⣀⡀⣀⠀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⠀⣀⡀⣀⢀⢀⠀⠀⠀⠀⢀⣂⣀⣋
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⣶⠶⠚⠋⠉⠉⠛⠲⢶⣤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠈⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⡁⣀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⣀⡈⠙⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠋⠉⠉⠉⠁⠀⣀⣠⣤⠴⠖⠒⢦⠀⠀⠀⠀⡤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠀⣀⣀⡤⠴⠋⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢳⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠀⠀⠀⠀⣠⠞⠋⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢿⢷⡀
⠀⠀⠀⠀⡴⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠸⣇⣠⡴⠋⠁⠀⠀⠀⣰⠞⢲⡄⠀⠀⢸⡄⠹⡄
⠀⠀⠀⡼⠁⢠⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⡄⠀⠸⡇⠀⠹⡆
⠀⠀⣼⠁⠀⣸⠁⠀⠀⠀⠀⢸⡋⠀⠀⢀⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⡄⠀⢻⡄⠀⢹⡄
⠀⢰⠇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣴⡿⠿⠛⠓⠦⠤⠴⠚⠛⠯⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡿⠀⠀⠙⠲⣄⢷
⠀⡟⢀⡞⠁⢀⡤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⢻⡆
⢸⣷⠋⠀⠀⠘⢇⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⣠⠖⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢻
⣼⠁⠀⠀⠀⠀⠘⢧⠀⠘⣆⠀⠀⠀⠀⠀⢰⡇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠃⠀⠀⠀⠀⠀⢀⡴⠚⠁⢨⡇⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⢘⡇⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣠⠖⠋⠀⣠⠞⠉⠀⠀⠀⠀⢰
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠻⣄⡴⠋⠁⠀⠀⠀⠀⠀⣰⢿
⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢸
⢸⡇⢳⡀⠀⠀⠀⠀⠀⢀⡞⠉⢹⠄⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠇⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣼⠁⠀⣿
⠀⡇⠀⠹⡄⠀⠀⠀⢠⠟⠀⣰⠏⠀⠀⠀⠀⣏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⠂⣿⠀⢀⡇
⠀⢻⠀⠀⢻⠀⠀⣠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠉⠳⠿⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⠏⠀⢹⡀⢸⠃
⠀⠸⡇⠀⢸⠀⠀⠘⠦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠀⣴⠖⢦⡀⠀⠀⠀⠀⣼⠁⢠⡟⠀⠀⠈⣧⡟
⠀⠀⢹⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣓⢶⣤⡴⠖⠓⠶⢤⣶⠚⢉⡼⠁⠀⠀⠻⢄⠀⠉⠢⣄⠀⠀⠙⠶⠛⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢻⡀⠀⢀⡤⠤⠶⠶⠶⠶⠶⣤⠀⠀⠀⠀⠀⢀⣤⠞⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣷⠀⠀⠀⠀⠀⠙⠲⠚⠁⠀⠀⢀⣠⠴⣾⠋
⠀⠀⠀⠀⠀⢧⠀⠸⢄⣤⣤⣤⣤⡤⠤⠞⠀⠀⠀⢀⡴⠋⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣤⣀⣀⣀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⣠⠶⢦⣄⡀⠀⠀⠀⠀⠸⣇⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣉⠉⠉⠛⠲⠦⣄⠀⠀⠀⠀⠀⠹⢤⣀⠈⠙⠓⠦⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠈⠙⠲⣤⣀⣸⠇⠀⢀⡴⠖⠒⠒⠶⢤⣤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣌⣙⣦⣄⡀⠀⠀⠀⠀⠉⠁⣠⠞⠉⣀⣀⣤⠴⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣓⣶⣶⣶⣶⣞⣛⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣰⠏⠀⠀⢀⡼⠃⠀⠀⢀⡾⠋⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠙⢧⡀⠀⠀⠘⣧⡀⠀⠀⠻
⡟⠀⠀⢀⡾⠁⠀⠀⣠⠟⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠶⣤⣀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠻⣄⠀⠀⠈⢷⡀
⠀⠀⠀⣼⠃⠀⠀⣰⠏⠀⠀⢀⡾⠃⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠘⢷⡀⠀⠀⠹⡆⠀⠀⠘⣧
⠀⠀⢰⡏⠀⠀⢠⡟⠀⠀⢀⡾⠁⠀⠀⣰⠟⠀⠀⠀⣀⡴⠶⠒⠒⠶⢦⣀⠀⠀⠀⠻⣆⠀⠀⠈⢷⠀⠀⠀⢻⡄⠀⠀⢹⡆
⠀⠀⣼⠁⠀⠀⣼⠁⠀⠀⣸⠃⠀⠀⢰⠏⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢹⡆⠀⠀⠸⣇⠀⠀⠘⣇⠀⠀⠈⣧
⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⣰⠖⠲⣆⠀⠀⠈⣷⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⣄⣠⡟⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⢿⠀⠀⠀⢻⠀⠀⠀⢻⡀⠀⠀⢹⡄⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢠⡏⠀⠀⢠⡟⠀⠀⢀⡟⠀⠀⠀⡿
⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠘⣧⠀⠀⠀⢻⣄⠀⠀⠈⠛⢦⣤⣀⣀⣤⡴⠛⠁⠀⠀⣠⡟⠀⠀⠀⣼⠁⠀⠀⣸⠇⠀⠀⢸⡇
⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠘⣧⡀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⣼⠃⠀⠀⢠⡏⠀⠀⢀⡿
⣆⠀⠀⠘⣧⠀⠀⠀⠻⣄⠀⠀⠈⠳⣄⠀⠀⠀⠉⠛⠶⣤⣤⣄⣠⣤⣤⠶⠛⠉⠀⠀⠀⣠⠞⠁⠀⠀⣠⠟⠀⠀⠀⣼⠁
⢻⡄⠀⠀⠘⢧⡀⠀⠀⠙⣦⡀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⢀⣴⠋⠀⠀⢀⡾⠃⠀⠀⢠

⣿⣿⣦⣀⣈⡧⢳⣟⠓⢧⠆⠉⣾⠻⣷⣿⣿⣿⣷⣽⣯⣯⠉⠀⠸⠏⠰⠀⠻⠈⠛⠈⠋⢿⣶⠼⠃⠀⡟⠀⢸
⣿⣿⣿⣿⣿⣶⠮⢍⡀⠈⠙⠆⠸⠛⠻⣚⣻⣿⣟⣝⡶⢿⣳⠉⠁⣯⣤⣀⠒⠀⠀⠀⠀⠈⡏⠀⠘⠠⡄⠔⢋
⣿⣿⠟⠛⣳⢾⣿⢸⢟⠖⠂⠀⠁⢸⡟⠯⠉⠛⠉⠉⠛⠃⠀⠰⠏⠛⡿⡻⡡⢀⠂⠄⠀⠢⠠⠀⢁⣠⡷⠖⠛
⣿⣿⣿⣿⣧⣼⡿⣛⣋⣴⠶⢀⣘⡓⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠋⠁⠰⠆⣠⠀⠑⢰⡄⠠⠀⠸⠁⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⠟⠘⠁⠀⠙⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡅⣿⢸⠃⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣁⣼⣴⡶⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⣶⠞⡿⣵⣿⢿⡇⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣴⣷⠀⠀⠀⠈⣅⡀⠀⠀⠀⠄⠈⠁⢄⠒⠋⠸⡆⠉⡟⠁⢀⠀⠀⢀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢀⣄⢤⠋⠁⠃⠀⠀⠀⠀⠀⡽⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡴⣢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠴⠠⠄⠀⠀⣀⠀⠀⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣾⠁⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠀⠈⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣰⣿⣅⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠏⣡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣬
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡶⡟⠃⣄⡀⠀⠀⠀⡶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠗⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⠿⠶⠀⣷⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⠏⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⢉⣶⣴⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠰⢤⣼⣿⣭⣿⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠀⠀⠀⠀⠀⠀⣨⠚⢻⣣⣾⠿⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⡠⠬⠍⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⣠⠆⠁⠀⢀⠔⠀⠀⠀⠐⠀⠀⠀⣀⡀⢄⠀⢀⡔⠀⠀⠀⠀⠀⠀⠠⠄⠉⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⠀⠐⠊⠏⠉⠉⠛⢶
⠸⣇⡀⠀⠀⢘⡤⠤⠴⠆⠐⠒⠀⠀⠀⠀⠊⠀⠘⠳⠄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⣸⠀⠀⠀⠀⠀⠀⠀⠠⠪⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡖⠘⠂⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⡠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠚⢉⠴⠂⠀⠀⠀⠀⠀⠀⢀⡬⠄⣁⠘⠀⣴⠒⠈⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠄⢰⠋⠀⣠⠄⠀⠀⠳⠒⠂⠄⠀⠀⢀⠉⠉⢱⠄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠔⠋⠁⠀⠀⠀⠈⠁⠘⠿⣤⣀⡀⠀⠀⠄⠀⠀⠀⠈⠑⠂⠀⠄⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⡖⠀⠀⠀⠀⠀⠀⠀⡴⠶⢦⡀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⣴⠁⠀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠈⠙⠀⠈
⠀⠙⠂⠀⠀⠀⠀⠀⠎⠀⠀⢠⠁⠀⠀⠀⢷⣄⣀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣄⣀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠄⠀⢤⣀⣁⠀⡀⠀⠀⠄⡀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠈⠉⠃⢀⣀⣀⡀⠈⢹⡆⢀⠤⠖⠒⠲
⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⢱⠰⣂⠀⠀⠋⠀⠐⠋⠀⠀⢀⡴⠊⢉⡅⠀⠐⠀⠀⠉⠓⠀⢀⣠⠔⠋⠉⠀⠀⠙⠒⠋⠀⠀⠀⠠⣄
⠀⠈⠙⠂⠀⠀⠀⠀⠀⠸⣄⢀⡂⠀⠀⠊⠀⠈⠉⠀⠰⠬⠤⠂⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⠀⠠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠄
⠀⠀⠀⠈⠀⠀⢠⠀⣠⠴⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄
⠀⠀⠀⠀⠀⢀⠀⠤⠀⠐⠀⠒⠀⠀⠄⢠⠄⠀⠀⠀⠀⠀⠒⠄⠀⣠⡆⠀⠀⣀⣤⠠⠞⠉⠀⠀⠀⠀⠈⡄⠀⠀⢀⣴⠁⠀⢱⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⢩⡏⠀⠀⠀⠀⡀⠀⠀⠀⡼⠋⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠇⠀⠀⢰⠃⠀⠀⣀⡀
⠀⠀⠀⢀⣤⣦⣤⡀⡀⠀⠀⠀⣿⠀⠀⠻⢦⣄⣀⡤⠔⠉⠀⠌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡰⠁⠀⣀⠀⠀⢀⠴⠋⠀⠀⡾⠏
⠀⠀⠀⠀⠀⠁⠀⠀⢣⡀⠀⢀⡝⠀⠀⠀⢀⣠⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠉⠁⠀⠀⠀⠒⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⡄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢠⢤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠱⡀⠀⢀⣀⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠈⠓⠒⠚⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠈⠓⢄⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠓⠈⠒⣄⣀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠐⠲⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢌⠀⠈⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⢀⠐⡠⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠡⠄⢀⠀⠀⠀⠀⢀⠂⠀⠀⠉⢦⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡬⠋⠈⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠢⡵⢀⠐⣀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠒⢤⡀⠀⠀⠀⠀⠙⠒⠒
⢀⡖⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠖⢦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣄⣀⡀
⠀⠳⢤⡤⠢⣀⣀⣀⣀⣀⡠⢄⡀⠀⢀⡀⠠⠦⣀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡠⣄⣀⣀⣠⠜⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣙
⠀⠀⠀⠀⠀⠀⠉⠉⠁⢫⣀⣄⠀⠉⠀⠀⢀⣀⡀⣉⠉⠉⠉⠀⠄⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⢍⠁⠉⠓⠠⠔⠒⠒⠉⠉⠑⠒⠒⠒⠉⠻⠭⣖⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠸⣉⣁⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡐⠈⠈⠁⠐⢄⢀⡀⢀⡀⠀⠀⠀⣠⠤⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠉⠑⠒⠒⠚⡩⠽⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠛⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠐⠒⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠤⣄⠔⠉⠀⠈⠈⠒⢄⠀⠀⢀⠀⠉⠒⠒⠋⠙⠒⠒⠒⠉⠉⠑⠒⠶⠶⠐⠊⠑⠒⠊⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠉⠒⠒⠒⠊⠉⠑⠒⡒⠚⠉⠠⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠐⠁⠈⠁⠀⠉⠈⠉⠑⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠓⠒⠊⠕⠒⠒⠗⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣈⠀⠀⠀⣀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣯⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⣴⠋⠁⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠱⠄
⠀⢠⠀⠀⠀⢀⡀⢀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢸⡄⠀⠐⠈⠉⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⠛⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠻⣄⡀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠈⠉⠓⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠂⠈⠀⠁⠂⠀⠠
⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⣦⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠀⠠⠀⠤⠂⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠄⣀
⠀⠀⠀⢠⠁⢠⣤⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠘⡀⠘⠛⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠘⠿⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠈⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⣀⣀⠴⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠿⣿⠟⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡟⠛⠛⠻⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣛⠻⣦⣈⠛⠶⣤⣤⣄⣨⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢋⡾⢻⡄⠀⠉⠛⠷⠶⣤⣬⣥⣤⡴⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡛⢷⣌⠙⠛⠃⠀⠀⣠⡶⣆⠀⠀⠀⢠⣄⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⠻⣦⡈⠻⣦⣀⠀⠐⢿⣤⡿⠀⠀⣰⡏⣿⡀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢤⣄⠀⠈⠙⠷⣤⣉⠛⠶⣤⣤⣀⣀⣀⣈⣹⣿⠶⠞⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⢶⡙⣧⡀⠀⣀⣈⡙⠻⠶⣤⣤⣉⣉⣉⣉⣥⣤⠶⠛⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢈⣿⣌⡛⢛⣉⣉⡛⢦⣀⠀⢀⣩⣭⣍⡉⠀⠀⣠⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣴⠏⣿⠈⠉⠉⠉⠉⠻⣆⠉⠛⢋⣥⡴⣬⡻⠶⠞⣋⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡝⠛⠛⠀⠀⢠⡞⢷⡀⠈⠛⠛⠋⠁⠀⠀⠙⣳⠛⠋⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⡀⠀⠀⠻⣦⡾⠃⠀⠀⣰⠿⣆⠀⠀⣾⠻⣇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠷⣦⣄⣀⡀⠀⠀⠀⠻⠶⠟⠀⠀⠙⣛⣫⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣌⡉⠙⠛⠛⠲⠶⠶⠶⠞⣛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣶⣶⣶⣶⣾⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠐⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⠀⢰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠄⠤⢤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢰⠃⠀⠀⠈⡇⠑⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣄⠐⠈⢀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⡄⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⢿⣿⡆⠤⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠘⢄⡐⣄⢀⢀⡠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠠⠄⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠈⠉⠹⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠠⠀⠠⠄⠂⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠄⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠐⠠⠀⠀⠀⢀⠀⣃⣤⠀⠀⠈⠀⠀⣿⠀⢸⡇⠀⠀⠘⣇⠰⡀⡀⠀⠀⣸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠚⢿⡟⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠨⡷⣬⡿⢶⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣠⠀⠸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣄⡴⠀⢱⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣚⣯⣍⣉⠉⠉⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣚⣉⣀⣠⠞⠁⠀⠀⠈⠙⠒⠋⠉⠉⠻⡟⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⠉⠈⠉⠀⢰⡆⠀⠑⠦⠀⠹⡄⠀⠀⠀⠻⢤⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣠⠞⠀⢤⡀⠀⠀⠘⠃⠀⠀⢀⣀⠀⠀⣠⠖⠀⠀⡄⠈⠳⡌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠉⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠰⠇⠀⠀⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣇⠀⢀⠤⠖⠂⢀⡇⠀⢀⣽⠶⠒⠒⠒⠶⣄⠀⠘⠓⠂⠀⡀⠀⢧⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣰⠃⠀⠀⠀⠀⠀⠘⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡀⠀⠙⠆⠀⠙⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⠲⠦⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠋⠀⡀⠀⠀⣹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠀⢲⡀⠀⠀⠀⢦⡀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡾⠀⠰⠋⠀⠀⣇⠀⢠⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡆⠀⠁⠀⣶⠀⠀⠁⠀⠘⠳⣄⣀⣀⣀⣤⠞⠁⡀⠀⢠⠞⠀⠈⠀⢸⡄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢧⣀⠀⢠⡇⠀⠀⠀⢲⠀⠀⣠⠍⠉⣉⣀⠀⠀⡇⠀⠀⠀⠲⡄⠀⠈⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢳⡀⠀⢠⣀⠀⠈⠃⠐⠃⠀⠀⠈⠉⠀⠀⠀⡤⠀⠀⠀⠀⣠⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢧⡀⠀⠈⠀⠀⠀⠰⣄⠀⠀⠲⡄⠀⠀⠸⠁⠀⠐⠲⢾⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢭⡛⠓⠦⣄⠀⠀⠘⢃⣀⡀⠙⠀⠀⠀⣠⠴⣲⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣬⣝⣓⣚⣉⣉⣉⣳⣶⣤⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠉⠉⢻⣿⣿⣿⣿⣿⡟⠛⠿⠿⠿⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⡿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⢀⠀⠀⠀⠀⣀⠹⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣷⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇⠀⠀⠀⠀⢹⡗⠘⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⠘⣿⡿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⡇⠀⠀⠀⠀⡸⣿⡆⢀⠉⠀⠀⠀⠀⠀⠀⠘⠋⠀⠛⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡏⠒⠒⠚⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠱⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⠛⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠀⠁⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠂⠐⠈⠀⠀⠀⢀⠜⢂⠀⠀⠀⠑⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⢀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠢⠀⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠐⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠠⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠔⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡐⠄⠀⠄⠀⠀⠀⠈⠂⠀⠠⠂⠐⠀⠀⠐⠁⠀⠀⠀⠢⠀⠠⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠂⠠⠀⡐⠀⠀⠁⠀⠀⢀⡀⠀⠀⠈⠀⠀⠂⡀⠀⠐⢈⠠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⡀⠠⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠀⡀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠑⣀⠀⠐⠂⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠢⡀⠀⡌⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡀⠀⠀⠀⠀⢈⠤⠇⠀⠀⠀⡠⠆⠌⠀⠀⠀⠀⢀⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡞⠉⠐⢀⡀⠀⠀⠈⢀⠈⠀⠀⢀⡄⠂⠉⠉⢃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠄⠂⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣄⠀⣤⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠢⣹⣤⣧⠛⠉⢛⠑⢄
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠐⢶⡤⣸⠀⢸⠁⠀⠀⠱⠀⠀⠀⢠⠖⠲⢤⢄⣠⠄⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠰⡀⢑⣦⢀⢵⣤⣷⣾⡆⢀⠤⡀⠀⠶⢶⣇⠀⢀⣥⣬⣿⣾⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⠋⠀⢸⠋⢁⡸⣿⠀⠈⠲⡶⣄⣿⣿⡁⢀⣀⣸⣇⠤⠞⠛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠎⠉⠛⡆⢀⡘⠀⠊⢀⡜⠓⣧⣯⣉⢃⠀⠀⠹⠋⠀⠙⢓⣶⣮⠉⡽⠋
⠀⠀⠀⠀⠰⠾⢩⡇⠀⢀⣀⡀⠠⠤⢄⡖⠉⢑⡴⣇⣀⡀⣇⣧⣠⣞⠀⠈⠀⠀⠀⣹⠊
⠀⠀⢀⣠⣤⣊⡹⠿⠛⢧⢈⠝⠉⠉⠁⠀⣠⠎⠀⠯⣤⠍⠛⠡⣭⡿⡗⠒⠉⠉⢻⣿⣉⠉⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠊⠉⠉⠁⠀⠀⠀⠘⠒⢶⣖⡀⣤⠗⠀⠤⡄⠀⢸⣿⣿⣁⣉⡙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢤⣏⣀⣼⣿⣗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢯⠒⠂⠬⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠚⠋⠉⠉⠉⠉⠉⠙⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢉⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⡉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢇⠪⠬
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢒⠰⡃⢠⢸
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠈⢱⣵⣦⠋⡁⠒
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠦⣀⠀⠣⣀⣩⢝⢿⠻⠢⡔⠊
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠑⢄⠀⡇⢀⢞⣄⡁⢸
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣀⡉⠤⠮⠤⠸⠇
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠤⠅⠀⢁⠀⡠⠊⠀⠀⣀⠀⠀⢠⠁
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⢀⣀⡈⠢⣈⠤⡅⡠⠒⠉⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⠀⠀⠈⢑⣟⣿⣾⠤⠄⠀⠠⠴⡋
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢆⠀⠀⠀⠉⡲⠒⠒⢊⠝⢹⠻⠫⣍⠐⠒⠠⢄⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢔⢇⠰⢈⠆⠒⠤⡜⠀⠀⡴⠃⠀⣸⠀⢡⠀⠱⡄⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢄⣀⣀⠀⣀⠤⠓⣂⣷⣿⣾⣒⡡⢴⠁⠀⠈⠀⠀⠠⠛⡀⠀⠇⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢀⡈⠉⣉⡒⠀⡶⢈⢿⡉⠂⢣⠸⡀⢀⡠⠄⠒⠁⠀⠱⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠢⠂⠀⡇⢈⠔⠉⠒⠐⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣸
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠋⠉⠋⠉⠙⣿⣿⠋⠋⣙⢛⠙⠉⢩⣩⠉⣛⡫⣿⠙⠋⠛⠛⣉⠉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣯⣽⠋⠉
⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠠⠄⠀⠠⠀⠫⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀

⠀⠀⠀⠀⣠⢤⣤⣤⣠⢤⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠙⠈⠙⠉⠙⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣌⢀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠒⠀⢀⠀⠉⠲⡒⠂⢦⠀⠀⠀⡀⠀⠀⡘⡀⠀⡆⠀⠈⡀
⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠄⠀⠀⠀⠀⠁⢀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣴⣴⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠲⡀⠀⠀⡇⢨⠁
⠀⠀⠀⢠⣄⣤⣩⡤⣤⣬⣥⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⡀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠈⠢⢀⣦⠃
⠀⠀⠀⢨⣅⣩⢬⢉⣍⣩⣉⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠘⠛⠙⠈⠉⠛⠙⠙⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡖⠀⠀⠀⢴⣄
⠀⠀⠀⠀⠐⠃⠓⠛⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠋⠱⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢠⣤⠀⢴⣆⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠐⠋⠀⠙⠂⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡁⠀⢈⣉⠀⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⣖⠘⠛⡠⠠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠤⠄⢀⠀⠀⠁
⠀⠀⠀⠀⠸⠿⠀⠘⠟⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠁
⠀⠀⠀⠀⢠⣤⡀⢰⣶⡄⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⣀⠂⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢻⠇⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢀⣀⡃⠀⠸⡀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⢀⡴⣊⠀⠈⣿⢆⣿⡇⠀⠀⣷⠀⠀⠀⠀⢀⠀⠀⢶⡀⠀⠀⡄
⠀⠀⠀⢀⡴⢀⠄⠁⠀⠀⠀⢀⣠⣠⣏⣾⣿⣿⣿⣵⣿⣿⣧⢀⠞⠛⠀⠀⠀⠀⠀⠰⢆⠘⣿⣄⣀⣀⠀⠀⠁
⠀⠀⢀⣿⣵⠏⠀⡄⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣤⣤⣄⠀⠀⠀⠀⠀⠠⠕⡌⠻⣿⣿⡆⠀⠒
⠀⠀⢸⣿⣄⡁⠀⠣⢀⡀⠤⠊⠁⠀⢀⠈⠀⠈⠀⡠⠚⠋⠀⢀⠜⠝⢿⣶⣄⠲⡀⠀⣳⣼⣧⠀⠈⠉⠁⠀⡄⠀⡀
⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⢸⣿⣿⣦⠀⣤⣔⣿⡿⠀⢀⡀
⠀⠀⠀⠀⠮⣻⢿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠟⣿⣿⡛⢾⣿⢷⡄
⠀⠀⠀⠱⣆⠀⠀⠫⠛⠻⠿⠿⠿⠶⠶⠶⠶⠒⠛⠛⠛⠉⠉⠉⠀⠀⠀⣠⡾⣫⣿⣿⡟⠀⠙⡄
⠀⠀⠀⠀⠈⠙⠒⠄⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠚⠉⠀⠙⠟⠉⠁⠀⠀⠈⠒⠂⠠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠶⠶⠶⠶⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠠⣀⠒⠈⠉⠈⠉⠁⠀⠈⠉⠀⠈⠁⠒⢠⠄⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠉⠈⡀⠂⡀⠀⠀⠄⣄⠀⠈⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠁⠀⣠⠠⠀⠀⢀⠐⠀⠁⠉⠁
⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠘⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⢀⣴⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⡀⠀⠀⠈⢻⣿⣿⣿⣆
⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⢠⣾⣿⣿⣿⠋⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣄
⠀⢠⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠋⠉⠉⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣷⣦⠀⠀⠀⠈⢿⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿⡄
⠀⣾⣿⣿⡿⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⢿⣿⣿⣷
⢰⣿⣿⣿⠇⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠈⢿⣿⣿⣿⡄⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿
⣼⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⡀⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿
⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿
⢻⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣷⡀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡿⠁⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿
⠸⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿
⠀⢿⣿⣿⣷⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⣾⣿⣿⡿
⠀⠘⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣦⣄⣀⣀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⠟⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠘⢿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠹⣿⣿⣿⣧⡀⠀⠀⠈⠻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟⠁⠀⠀⢀⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉