Yard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠔
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠧⠀⠂
⠀⠀⠀⠀⠀⢠⣾⣟⢁⣠⠿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⡛⠛⠿⡗
⠀⠀⠀⠀⠀⠈⢿⡿⢿⠁⡄⠀⠀⠀⠀⠀⠆⠀⠀⠄⠠⠀⠀⠀⠀⠀⣴⣦⠀⣀⠀⠀⠀⠀⠀⠀⣨⣶⣾⢀⠀⡄⣰⣄
⠀⠀⠀⠀⣀⠀⠀⠀⢉⡐⠀⠀⢀⢠⠆⣀⠛⠒⡀⣱⣳⠀⠀⠀⠀⠀⠛⣇⣸⡇⠀⠀⠀⣤⠀⢸⣿⣿⠁⠀⠀⢸⣷⡋⠂
⠀⠀⠀⢸⣿⡆⠀⠀⢿⣶⣾⡇⠀⢠⣿⣴⣠⣿⡇⠁⢐⣶⣶⢦⣶⣄⣤⣭⣽⣷⣦⣤⣤⠶⠶⠀⣿⣿⠀⠀⠀⣀⣃⣀⠀⣀⣤⡄⠀⠀⠄
⠀⠀⠀⢺⣿⣇⠀⠀⢸⠏⢙⠁⣸⣞⣿⡿⠟⢿⣷⠂⠈⠠⣿⣾⠛⠛⠿⣼⣿⣿⣿⣿⣿⣤⠤⡄⣧⡘⢿⡀⠔⠉⠁⠃⠀⣿⠈
⠀⠀⠀⠘⡟⢿⡄⠠⠁⠠⡁⠀⢿⣿⣿⣶⣦⣀⣀⡘⢀⡶⠋⠁⠀⠀⠈⣿⡿⣿⣿⣿⣿⠻⠇⢰⣿⢿⡤⠍⠀⠀⠀⠀⣤⡿⠀⢀
⠀⠀⠀⢾⣿⡶⠀⡀⢦⠀⠀⠀⠘⣿⠟⠋⣿⣿⠁⠀⢸⣷⣶⣄⣤⣦⣤⡿⠟⢹⣟⠛⢷⡴⠆⣾⣿⢻⠀⠀⠀⠀⠀⠀⢻⠁⣠⣿⠂
⠀⠀⢴⡟⠛⠍⣎⠷⠄⢣⠀⠀⠀⠈⣀⡀⢻⡉⠁⠀⠀⣹⠛⣛⣽⡟⣿⣄⢀⣨⣿⣧⡄⠈⠀⣾⡿⠀⠀⠀⡇⠀⠀⠀⢸⣼⣿⡏⢸⣧
⠀⠀⢸⣿⣶⣶⠋⠐⢀⡀⠀⠀⠀⠀⣹⡭⡫⠡⠤⠀⠀⠛⠶⣛⣿⢷⣼⡏⣾⣯⣿⠻⠣⠴⠷⠀⠻⢤⣀⢠⣤⠀⢤⠀⡞⠻⠿⠀⢾⡿⠆
⠀⠀⣿⣿⣿⣿⣀⣀⣼⣧⢈⣠⠶⢸⣿⣶⣶⣶⣾⣆⠳⣄⠀⠘⢦⣐⠋⣁⣌⣛⠉⠚⣷⡾⣿⣿⣷⣾⢿⡅⣠⣹⡇⡀
⠀⢻⣧⣬⣿⠿⣿⣿⡷⣹⡦⠀⣾⣿⣙⠛⠫⠽⢿⣿⣧⡈⠻⣦⣶⣭⡀⢀⣠⠉⠠⠆⢀⠀⠀⠉⠉⠛⠋⠁⠉⠙⠻⡾⠈
⠀⠀⠘⠻⢿⣷⣿⣅⣀⠹⡇⠐⠻⠿⠿⠿⠿⠿⠿⣯⣿⣷⡆⢻⠟⠛⢛⣙⡉⠃⠰⠾⠇⠀⠀⠀⡠⠀⣤⡐⠒⢂⣤⠤⠀⡀
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠷⡿⢄⣤⠀⠀⠀⠀⠀⠁⠉⠉⠛⠃⠈⠁⠀⠈⠁⠀⠀⠀⠀⢀⣀⣴⣿⣿⣷⣻⣷⣾⣿⣇⠀⠟⢻⡒⢶⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠙





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⠀⠒⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⣿⣿⡏⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢋⡿⣿⠏⠈
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡄⠀⠀⣾⢿⡅⠈⠇⡀⠀⢠⡀⠀⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⠃⡄⠀⠀⠀⠭⠀⠀⠀⠀⢠⡑⠆⣰⣦⡀⢰⠀⠀⡟
⠀⠀⣤⣴⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣲⣿⣿⡇⠘⠀⠘⠁
⠀⠀⢿⣿⠟⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⣞⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠈⢹
⠀⠀⠈⠁⠀⠀⠀⣀⠛⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡀⠙⢳⡄⠀⠀⠀⢀⠀⠈⡇⡀⠨⠀⣆⠀⠀⠀⠀⣠⠠⣶⡀
⠀⣆⠀⣀⣶⡆⠀⠙⢛⠦⠀⠀⠀⠀⣀⣠⣧⣶⡆⣧⠀⣼⠁⡄⢤⠀⠀⠀⠀⢷⠰⡰⠀⡿⠲⡄⠀⠀⠀⠘⠉⠁
⠀⠹⣼⠿⣿⣧⣤⢦⡞⠀⠀⠀⠐⢾⡿⠟⢿⠉⢱⣸⣆⠀⣸⠀⠀⠀⠰⡎⠀⢸⡄⢳⡄⡧⢀⠘⣆⠀⠀⠀⠀⢀⠄
⠀⠀⡟⡀⣿⣋⣛⣿⣇⣀⣠⡆⠀⠀⢹⣄⠘⡇⠀⣯⣏⠀⡏⠀⠀⠀⠀⡀⠀⣸⣇⣘⣷⡇⢸⡄⠹⡄⠀⠀⣠⡎
⠀⢸⡇⢧⠿⣿⠻⣿⣿⣍⣸⣿⣿⣷⠘⣿⣷⣇⠀⢸⣿⢰⡇⠀⠸⠀⢰⠁⡀⢈⣯⠉⣻⣷⢸⡇⠀⢳⡀⠘⣿
⠀⠘⡇⣸⡀⡏⣇⠡⡈⣯⠉⣿⣿⣿⡆⣿⣿⣿⢀⠈⣿⣼⠃⠰⣤⢸⡷⣟⠁⡼⡏⢧⡿⣿⣼⡇⢃⠸⡇⣠⡏⢀
⠀⡏⢻⣿⡗⣿⢹⣤⣷⣸⣤⣿⡽⣿⠀⣿⡏⢻⢸⠀⣿⣿⣴⣾⣿⣿⢷⡿⣸⠀⡇⢸⡇⣏⣿⡗⢾⡀⣿⣿⡇⠘
⠀⡇⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⠀⣺⡇⢸⢈⠀⣽⣿⡿⢹⢸⢇⡎⢀⡇⢀⣧⢼⣧⣿⣇⡇⠀⡿⣿⣿⣁⠀⡇
⠀⡇⣸⡿⣿⢹⣿⣿⣿⣿⣿⣿⢿⢸⠀⣿⡇⢸⠈⡅⡟⢻⣷⣾⣼⣾⣀⣸⡶⠋⡇⢸⣿⣿⣿⣧⣀⡷⣿⣿⢻⠀⢡
⠀⣧⣿⣇⣿⢸⡿⣿⣿⠿⣿⣿⣴⣾⣶⣿⣿⣼⣿⣧⡇⢸⢻⣿⣿⣿⣿⣿⣄⢀⠇⠘⣿⣿⣿⣿⠁⡇⣿⣿⠘⡆⢸
⠀⣿⢹⣿⣽⢸⣷⣿⣿⣴⣧⣿⠈⣿⣿⢹⠉⡙⠛⢉⡇⣼⢸⣿⣿⣿⣿⡇⠀⢸⠀⡆⣿⣿⣿⠸⠆⡇⣿⣿⠀⡇⢸
⠀⡇⢠⣿⣿⠻⣿⢹⢸⠉⡇⣿⠀⣿⡇⢸⠖⣿⠀⠀⣿⣿⠀⡿⡟⣹⣽⣧⠦⣾⢀⣿⢹⣿⠏⠀⠀⢸⣿⣿⡀⡇⢸
⠀⡇⢸⣿⠁⢰⡏⢹⣼⠀⡇⣿⠀⣿⡇⢸⡀⣿⠀⢀⣿⡏⠀⡇⡇⡿⢋⠙⠂⣿⢸⡇⢸⣿⠀⠀⠀⢸⢹⡏⡇⡇⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠈⡄⠀⠂⣠⣀⣀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⠁⠀⠀⠀⠀⠀⡀⠀⢰⣷⣄⣴⣆
⠀⠀⡄⠀⠀⠀⠂⡴⢻⣿⣿⣿⣷⣶⣦⣤⡧⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠈⠀⡈⠁⣠
⠀⠀⢆⠀⣦⡘⣸⣧⣽⣼⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⢰⢩⡇⠀⠀⠀⣄⢠⣿⣿⣇⣀⠀⠃⠐⠠
⠀⠀⠘⣆⢹⣿⣩⣬⣿⣿⣉⣽⣿⣶⣤⣿⣿⣿⣷⡀⠀⡀⢐⠂⠀⢸⠃⠀⡀⠰⡎⣼⡃⠀⢀⣾⡿⠻⡉⢂⠉⠻⡄⠠⡦⠐
⠀⠀⠩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⡛⠛⢓⣿⣥⣤⣅⡀⠀⠘⢺⣆⠾⠉⡴⢳⣟⣤⣦⣾⣟⠲⣧⣽⣿⣤⣧⣆⢠
⠀⠀⠀⢹⣻⣿⣿⣿⣿⣿⡿⣹⣿⣿⣷⣶⠘⣿⣿⣿⣿⣽⣿⣷⣶⣾⣿⣷⡖⡏⣿⣿⣟⣴⣬⣇⠀⣈⣏⢿⣿⣿⢿⣀⡀
⠀⠀⢀⣸⣿⡿⣏⣹⣿⠟⣿⠿⢷⣗⣽⠿⠶⢼⣿⣟⣻⣏⣿⣿⣻⣿⣟⣿⣯⠛⣿⣽⣿⣟⡿⣿⣾⣿⣧⣟⣳⢽⣾⣽⡇⠀⠄
⢀⡀⠀⠙⣯⢻⣿⠿⣿⣇⣹⣶⣾⣿⣹⢧⣾⣾⢈⣽⣿⣿⡿⢯⣿⣿⣿⣗⣻⢷⣿⣵⡌⣏⡾⣟⠐⠟⣯⡿⢟⣿⣿⣷⡇⢠⡄
⠘⢀⣀⠐⣰⡿⡟⢋⣿⣛⣿⣏⣙⢊⣹⢊⣉⣏⣿⣘⠿⣿⣀⣈⣻⣿⣿⣿⣿⡿⡿⣿⣛⢿⡞⣿⡀⢀⣿⣿⣟⣻⣿⣾⣷
⠀⠙⠏⠸⠀⢠⣴⢿⡴⠽⣿⣡⡿⢧⣼⢠⠟⢠⢧⡾⢸⣿⣽⣿⣻⣯⣿⣿⣿⡿⣿⣿⣯⣿⣿⣏⢿⣿⣿⡛⣯⣿⣻⢿⣿
⠀⠀⠘⡴⢣⠟⣽⣯⠗⣉⢛⣻⡟⢮⣜⣹⣠⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⡛⠈⠻⢟⣿⣿⣿⣭⢷⡾⣿⣷⣶⣿⠂
⠀⠀⠀⠨⣼⣻⣿⡟⠛⠻⣿⣷⣾⣾⣿⣿⣿⣿⣿⠋⣡⣥⣬⣿⣟⡿⠻⠛⠛⢻⣿⣆⣽⣿⡼⢻⣿⣻⣿⣾⠗⣿⣫⡟⠛⡇⠄
⢀⢖⣐⠂⣿⣃⣻⣿⣿⣷⣽⡿⣿⣿⣿⣿⣿⣿⢻⡼⢿⣿⣾⣿⣤⡅⢠⠞⢁⣄⣄⣶⠀⣩⣷⣾⣿⡿⣷⡁⠚⣈⣿⡗⣶⠤⠒
⠈⠁⠀⠙⢿⣮⣹⣿⣿⡿⠟⢧⣼⢿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⢿⠯⢀⣿⣿⣿⣿⣿⠟⣿⡇⠀⣩⣆⡉⢹⢷⣷⡄
⠀⠀⠀⠀⢸⣷⣿⣿⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡵⣷⣿⣿⣷⣾⣿⡘⣿⢟⣻⡏⠃⡨⢸⣧⣤⣿⡼⠿⣿⡿⠛⢯⣠⣯⡀
⠀⠀⠀⠀⠘⣷⠣⣿⠾⣰⠟⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣦⣾⣿⣿⣿⣷⣀⣡⣶⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠙⠀⠻⠘⠃⠀⠀⠈⠀⠉⠁⠀⠈⠉⠉⠀⠀⠁⠁⠙⠛⠋⠛⠋⠛⠉⠛⠙⠉⠉⠉⠙⠘⠙⠛⠛⠋⠉⠁

⠀⠒⠅⠀⠐⡄⠀⠀⠀⠈⣜⢪⡏⠀⠀⠀⠀⢛⢃⠀⠀⠀⣈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆⠠⣏⡀⣠⣤⡄⠀⠀⢚
⠸⢃⢰⣄⠀⠀⠀⢸⣇⠀⠀⢻⠃⠀⠀⠼⣿⣏⠀⠹⠃⠀⠘⠁⠀⠀⠀⠀⡄⣸⣤⢀⠄⠀⠀⠀⠄⠀⠀⠀⠀⡼⢋⣽⣷⡦⠀⠀⠀⠰⣜⠿⢦⠙⠛⠁⠀⣴⡿
⠐⣢⡰⣷⠀⠘⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⣦⣿⡇⣿⣿⣖⠁⠀⠀⠀⣀⣄⡀⠀⠠⡾⠹⠉⠂⠀⠀⠀⢆⢈⣆⠖⣿⠶⠂⠀⠈⠁
⠀⡾⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣀⡀⠙⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⣻⣿⣿⣿⣯⣽⡥⢀⠀⠀⠛⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡼⠷⠻⠄⠀⠀⠀⣀
⣭⠑⠢⠊⠀⠀⠘⠄⠀⠀⠀⢀⣿⣼⣯⣧⠀⠀⠀⠉⠉⣿⣟⠀⠀⠀⢀⣈⣟⡀⣘⣻⠾⡇⠀⠀⠀⠤⡘⠁⠀⠀⢀⠀⠀⡠⠀⠀⠀⠘⠛⠛⠀⠂⠀⠀⠀⠀⠷
⠨⢴⣶⠶⠀⠀⠀⡄⠀⠇⠀⣿⣏⣿⣻⡧⠀⠀⠀⢀⣤⣅⡀⠁⠀⠀⠀⠙⠉⢹⡟⠉⠀⠀⠀⠀⢀⣠⠐⠛⠂⠀⠀⢑⢠⠀⠀⠀⠀⠀⡀⠀⠀⢀⣰⣦⣤⣀
⠀⣈⡁⠄⠀⠀⠀⠙⠂⢤⣀⠀⢸⠁⠈⠁⢠⠀⠀⠸⣷⣿⠁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢐⠉⡙⢣⣶⢀⢄⡴⠋⠀⠀⠀⣾⠿⣿⣿⠆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠰⢀⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⣿⠟⠋⠀⣤⣶⣦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠀⣀⡀⠀⠈⢷⡐⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢸⣿⣿⣿⡧⠀⠀⠀⢣⡀
⠀⠀⡀⠀⢀⣤⠄⠀⢉⠉⡁⠀⠘⣧⡍⠀⠀⠀⢀⠀⢠⢀⣄⢠⣀⣀⠀⣄⡀⢠⡀⣀⢀⣀⣀⣠⠀⡄⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠋⠀⠀⠀⠀⠀⠁
⠀⠸⡇⠀⠀⠁⠀⢾⡿⠀⠙⠀⠀⢹⡇⠀⠀⠀⠘⠒⠈⠋⠛⠘⠚⠘⠀⠉⠃⠘⠁⠚⠈⠒⠁⠛⠃⠃⠀⠛⠀⠃⠀⠀⠰⠦⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆
⠀⡐⠇⠀⢴⣶⠀⠰⠀⠀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠇⠀⠀⠀⢢⡀⢱
⠀⡺⠃⠀⠈⠉⠀⡞⢧⠊⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⡀⠄⠀⡀⠀⠀⠀⠀⣹⣾⡄⡄
⢰⡇⠀⠀⠀⠁⠼⢱⠞⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠹⠀⠀⠶⠀⡇⠀⠀⣆⠀⠈⠙⣟⣿
⣏⠀⠀⠀⠀⠂⠐⣴⠋⠀⠀⠙⠀⠀⠆⠀⠀⠀⢰⣦⠀⣤⣤⠀⠀⠀⠀⠀⠃⢤⠀⠀⠀⠀⠀⠀⠸⠏⣀⡨⠷⢀⠀⠀⠀⠠⠀⠉⠀⠀⠁⠀⠀⣽⠀⠀⠁⢸⡎
⠀⠀⠀⠀⠀⠀⠠⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠈⡅⠀⠀⠀⠀⢘⣁⣠⠰⢆⡀⠀⠀⠠⠐⢾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢉⣻⡇⠀⠀⢸⡇
⢀⣀⠟⢰⡖⠀⠀⠈⠀⠀⠀⠀⠀⢌⣁⠀⠀⢰⣦⣹⡘⣓⡒⡛⠀⠀⠰⣿⣿⣤⣾⣿⠿⠀⠀⠀⠐⠉⠙⠛⢫⡀⠀⠀⣠⣶⣶⣶⣦⠀⢸⠀⢲⣾⡦⠀⠀⠀⠙
⡄⢠⠶⢎⠁⠀⠀⠀⠉⠆⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠃⠈⠙⠁⠀⠀⠀⠉⠉⠁⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⡇⠈⠀⣈⣉⣁⠀⢠⣾⣦
⣿⠀⠀⢀⠄⠀⠀⢠⣤⣤⡀⠀⠀⠀⣰⣧⣀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⢀⣤⡄⠀⠀⠀⢂⡐⡀⠀⠀⠀⢀⣀⡀⠈⠋⠉⡿⢡⣤⠀⠀⠀⠴⠦⠄⠀⢸⣿⣿
⢹⣤⡀⠞⠀⣠⣄⠀⠀⠋⠀⠀⠀⣾⣿⣿⣿⡷⠂⠀⢛⡀⠰⣦⡴⠆⠀⢀⣼⡿⢋⣀⠀⠾⣽⢿⣿⠿⠆⠀⢾⣿⠇⠀⠀⠤⢤⣌⠀⠀⠀⠀⣟⣋⣁⠀⠈⡈⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠂⡀⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⡀⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⣶⣶⡟⠁⠀⠀⠀⠀⠀⠀⢸⡏⢹⡏⠉⠉⠉⠉⢹⡏⢹⡇⠀⠀⠀⠀⠀⠀⠈⠲⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢜⣃⣘⣃⣀⣀⣀⣀⣘⣃⣘⡣⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⡵⠤⠶⠴⠶⠶⠷⠶⠴⢮⣽⡴⠀⠀⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠁⠀⣾⡏⠉⡍⠉⢹⡇⠀⠁⠁⠀⠀⡿⢻⡇⠛⠋⢻⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣇⣀⣼⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⣿⡇⠀⠀⠀⠀⠀⠿⠼⠧⠤⠤⠼⠧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⡏⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⢿⣗⣒⣓⣒⡺⢇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡴⠓⣌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⠎⠀⠀⠈⠳⡘⢯⣉⣉⣉⣉⣉⣉⣉⣉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⠞⠁⠀⠀⠀⠀⠀⠘⢦⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣴⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣌⣻⣉⣉⣉⣉⣉⣉⣉⣉⣉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡞⡉⠉⠉⠉⠉⢳⠀⠀⢸⠀⠀⡏⠉⢙⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡧⠀⠴⠄⠠⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⣀⠀⢸⠀⠀⢸⠀⠀⣇⣀⣨⣀⣀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣇⡇⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣀⡴⠋⠓⢦⡀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⡴⠞⠁⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠻⢿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠙⠻⣄⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠙⠦⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⡴⠚⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⠛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⣤⣤⣤⣼⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠙⠻⣿⣿⣿⣿
⣿⣿⡿⠛⠁⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠈⠙⠿⣿
⣿⣏⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠤⠼⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⣸
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⣧⣤⣤⣤⣼⣤⣤⣤⣤⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣤⣤⣤⣼⣧⣤⣤⣤⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⢧⡤⣤⢤⠼⢤⣤⣤⢤⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠤⠤⠤⠼⠧⠤⠤⠤⠿⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠶⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢆⣠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣿⣿⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠟⠽⠋⠿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠿⠚⠿⣤⠀⢐⣺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠰⠃⡤⠵⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⢀⣮⡁⢠⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⢠⠦⣥⢦⣟⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⣾⣿⣏⠱⣬⡳⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡐⠁⠀⣿⡃⡌⢷⣌⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠎⣀⣠⡞⠁⢀⣼⣿⡿⣮⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⠥⠤⣤⣀⣄⣹⣤⣨⣹⣟⣰⣿⣿⡿⣝⡾⠉⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢜⣿⡥⠂⠉⠉⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠻⣷⣾⡿⢾⡦⠆⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁⠀⠀⠄⠀⠀⣀⣥⣬⣙⠚⠆⠀⠤⠄⠈⠳⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠡⠀⠂⢁⣀⣀⢤⢶⣿⣾⣿⡿⠟⠋⠈⢳⣾⣴⡎⢘⣷⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠰⣿⡀⠀⣀⣄⣤⢾⣿⣿⣦⣬⣿⠿⠋⠀⠀⠀⠀⠀⢻⡝⠻⠿⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣧⣂⢰⢿⢉⢻⢿⣿⣾⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠋⠁
⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣉⣧⣷⣾⣶⡿⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄⠀⢀⣤⣾⠿⠋⠀⠀⠀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⣿⣇⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⡿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⣿⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⡀⢀⠀⠀⠄⢀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠀⠈⠁⠁⠀⠀⠂⠀⠁⠈⠁⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠛⠛⢻⡟⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀
⠀⠀⠀⣰⡶⠶⠶⠶⠶⢾⡇⠀⠀⠸⠷⠶⠶⢶⡾⠋⣀⡤⣄⠀⠙⢷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄
⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠖⠉⣠⠞⠁⠀⠈⠑⢤⡀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠘⠃⠀⢀⡤⠚⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣀⠴⠋⣠⠞⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⢤⡀⠙⠦⣀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠳⣄⠀⠀⠀⠀⠘⣧
⠀⢀⡟⠀⠀⢀⡴⠋⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠙⢦⡀⠀⠀⢿
⠀⢸⡇⣠⠔⢉⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠉⠢⣄⢸⡇
⣠⡾⠛⣡⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⡈⠛⢷
⠙⠛⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠣⠬⠤⠤⠥⠤⠤⠤⠬⠤⠤⠤⠤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⢀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⣿
⣶⣶⡶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠸⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⣋⣙⢿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢛⣵⣾⣿⣿⣷⣮⡛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⢟⣥⣾⣿⣿⣁⣀⣀⣈⣿⣿⣷⣬⡻⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢟⣩⣾⣿⣿⣿⣿⣿⣄⣸⣇⣠⣿⣿⣿⣿⣿⣷⣍⡻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠦⠤⠤⢼⠦⠤⠤⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢤⡀⠀⢸⡇⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢀⣀⣤⡤⠷⠖⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣀⣤⣤⡶⣾⡟⠉⠁
⠀⠠⠤⠤⠤⠤⢤⡼⢿⣦⣤⡤⣤⣄⣀⣀⢀⣀⠲⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠻⠋⠋⠙⢍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⡩⠔⣊⢭⡥⢛⡿⢻⢶⠖⡖⣦⢤⣠⣀⡀⡀⠀⠉
⠈⡠⠄⠀⡉⠀⠐⠂⢁⠠⠀⠀⠀⠐⠉⠔⠉⡠⠊⡴⠃⡜⠀⡇⠸⠄⠹⣀⠙⢌⠙⠃⠐⠒⠤⠄⢀⣀
⠀⠀⠀⠈⠀⡠⠔⠊⠀⠠⠀⢁⠠⠂⠀⠠⠊⠀⠐⠁⠀⠀⠀⡃⠀⢣⠀⠘⢆⠀⠐⠄⠀⠀⠀⠀⠀⠀⠁⠀⠒⠢⠤⣀
⠀⢀⡀⠚⠁⠀⡠⠒⠀⢀⡀⠀⠀⢀⡀⠀⢀⡈⠀⠀⡈⠀⠀⠯⠀⠀⢂⠀⠀⢢⠀⠀⠑⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⡰⠂⠀⢀⠄⠀⠀⠠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣴⡿⠟⠻⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢫⣿⣿⡾⡿⠁⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠛⠉⢻⡷⢺⣷⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠃⠀⠀⣠⡆⣠⣇⠈⠉⠀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡴⢶⡿⠁⣿⢿⡿⢳⣿⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⡁⠘⠁⠘⠁⠀⢸⡇⢠⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣌⠻⣿⠿⠗⠶⠄⢀⣸⡇⠈⣿⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡿⠿⣿⣿⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠣⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢼⠟⣁⢾⣟⣿⣿⠀⢰⣧⠀⠀⠀⣴⠀⠀⠰⠋⠈⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢻⣄⠀⣶⡄⣿⡿⢻⣾⣿⡇⠈⠁⠀⠀⢸⣿⠆⣴⠇⠀⢿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠉⠀⠛⣤⣬⡆⢾⠿⣉⠀⠀⠀⠀⠀⠸⠋⠠⢀⣶⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⢉⡤⢶⡿⠋⠀⢀⡄⠀⠀⠀⠀⠰⡟⢿⡀⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣠⡶⠀⣠⣶⢟⣧⣞⠃⢀⣠⣤⠅⠀⠀⠀⠠⡴⢱⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣠⡦⠀⠿⠁⢰⠟⣱⡿⣿⣿⠀⢸⠉⡈⠀⢴⠄⠀⠀⢠⣾⠇⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⣰⡟⢸⠇⢀⣴⣿⣿⣿⡿⠀⠿⠛⠁⠀⢠⣤⠀⠀⠈⠁⠀⠻⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⢠⣾⣤⣿⣿⣿⣿⣿⠇⢰⡦⠤⠀⠀⠀⠂⠀⢰⡆⣀⠀⠀⣤⡿⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⢈⣹⣿⣿⣿⣿⣿⡟⢰⣷⣶⠆⠀⣀⠀⠖⢲⡀⠀⠋⢠⡀⠚⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠐⡇⢸⠃⢸⣿⣟⣼⣿⣿⣿⣡⣿⣿⠷⠀⠚⠇⠀⠀⠘⠃⠈⠀⠈⣽⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⠌⣻⣿⣿⣿⣿⣿⣧⣧⡀⠀⠀⠀⠀⠀⢠⠿⠇⠀⣶⠟⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣼⡇⣀⠀⣸⣟⣹⠟⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⣧⠐⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⢻⡇⣿⢰⣿⣿⣧⣾⣿⣿⡿⠋⠀⠙⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣷⠚⠀⠟⣿⣿⣿⣿⣿⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⣾⡟⢋⣭⡀⢀⢈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⢀⠊
⠀⠀⠀⠀⠀⠀⢠⡿⣇⠈⣿⣆⣿⢸⣿⣿⣿⣿⣿⡿⠹⠆⠀⠁⢸⠀⡀⠀⢠⠏⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠸⣿⠉⣡⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠀⠈⠁⠀⠃⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⢠⣄⠶⠀⢸⣶⠁⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⢠⣾⣿⡶⠀⢸⠟⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣽⣿⡅⠀⢀⡀⣠⠈
⠀⠀⠀⠀⣿⣿⡟⠁⠀⢾⣿⣿⣿⣿⣿⣿⣿⣞⢁⣤⣀⣰⣿⣿⣏⠀⣠⣾⡇⠁
⠀⠀⠀⠀⠟⠋⠀⠁⣤⢹⣿⣿⣿⣿⣿⣿⣧⣽⠛⢋⣿⡏⠀⣿⣷⣾⠿⠛⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⢉⣾⣿⣿⣿⣿⣿⢳⠟⠀⠀⣼⡿⠇⠀⡀⠛⠛⢲⣾⣷⣄⡀⠀⠁⠐⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⡛⢄⡻⠂⣯⣀⣴⡟⠛⠛⠛⠛⠙⢻⠉⠉⣿⠉⠛⢛⡛⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⡸⠀⠃⠘⠋⣸⣿⡇⠀⠀⡇⠀⣷⢸⠀⠃⠃⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣧⣶⢣⣾⡀⠀⠀⠹⣿⡇⢠⠀⡇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠡⣿⣿⣿⣿⠁⠀⡀⠀⣿⡇⠸⣦⣧⠀⢿⡇⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⣴⣿⣿⣿⣿⠀⠀⣀⠀⢹⡇⠀⢸⡇⠀⢸⣇⡘⠀⠀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣸⣿⡟⢿⣿⡇⠀⠀⡀⠀⠘⣧⠀⢸⠃⠀⣼⣿⡇⠀⠀⠀⠃⠀⡏⠈
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣿⠋⢰⣾⡟⠀⠀⢠⡿⠀⢰⣿⠀⠘⠀⢀⡟⢿⠁⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣿⡿⠀⣼⠃⠁⠀⢰⣿⠁⢀⡿⣿⠀⠀⠀⠘⠇⠸⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⠁⣶⠛⠀⠀⠀⠺⣿⣧⣼⣇⣿⠀⠰⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⣼⠁⢀⣿⠇⠸⠉⠃⠀⠀⠀⠀⢻⡿⢹⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡟⠀⣸⡟⢀⠃⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⢿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠛⠄⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠄⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠱⣀⠉⠐⠃⡀
⠀⠃⢠⠀⠁⣸⠁⠀⠐⠁⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠐⣿⡀⠈⡛⠓⠌⠠⢔⠴⠁⡠
⠈⠀⠀⠀⠀⠀⠀⠀⠤⡤⠂⣰⡇⢰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠂⠀⠁⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⣻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢀⣰⡾⠇
⠀⠈⠁⠀⠀⡄⠀⠎⡠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⡅⠀⠀⠀⠐⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⡠⠀⢀
⠀⠀⠀⠀⠰⠇⠄⠀⠀⣃⡀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠠⠤⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡀⠀⠠⠄⠀⠀⡀⡀⡀⢀⣄⡀⠀⣲⣷⣴⠖⠂⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⣦⣄⣀⣀⣼⣤
⣤⣤⣤⣤⡤⠴⠿⠟⠛⠋⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠛⠉⠉⠉⠉⠉
⠀⠉⠉⠁


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡋⠈⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣀⣤⣀⣴⡿⣫⣾⡇⠀⣿⣮⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠋⣾⣿⣿⡇⠀⣿⣿⣿⡎⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣫⣷⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠐⣮⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣮⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⡂⢨⡻⣷⣄
⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⡿⣫⣴⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣷⣶⣾⣿⣿⣶⣶⣿⣿⣿⣶⣶⣷⢬⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣷⡾⣫⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣮⣾⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢹⠀⢸⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣼⣀⣸⣿⣿⡯⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢹⠀⢸⣿⣿⣶⣶⣶⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣾⣤⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣟⣿⣯⣽⣿⣿⣿⣯⣿⣟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣿⣿⣿⣿⡏⣾⣿⣿⣷⣶⣾⣿⣿⣿⣷⡮⣿⣿⣿⣿⡇⣿⣿⣿⣿





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣇⢀⠀⢤⣦⡤⠞⠉
⠀⠀⠀⠀⠀⢀⡿⡿⣳⣳⢫⢖⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣞⣽⡿⣣⡆⠚⠁
⠀⠀⠀⠀⠀⢸⢛⣿⣯⣿⣯⣣⠞⢻⢗⡶⣄⢀⣴⣿⡿⠿⠃⠰⣜⣴⠟⢡
⠀⠀⠀⠀⠀⣬⢮⣿⣿⡿⣵⡟⣵⣱⣋⡼⣾⣿⣿⡛⢏⢀⣀⠾⠛⢡⠖⠃
⠀⠀⠀⠀⠀⣯⣷⣽⣷⣿⣟⣿⠿⡿⣭⣽⣭⠋⠉⠁⠑⢸⡮⣡⠾⣇⢄⡀
⠀⠀⠀⠀⠀⠻⢿⠿⣿⡿⣫⡜⠋⠋⠁⠀⠀⠀⠀⢀⣼⣻⣿⣷⣾⣧⣾⣽⠐⠀⣠⠆
⠀⠀⠀⠀⠀⣀⣠⣾⣯⡜⠋⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡽⣿⣿⢻⣿⣿⣿⣷⠺⠏⠌
⠀⠀⠀⣠⣾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣻⣿⣾⣿⡿⢎⠿⣫⠘⠁⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⢠⠿⠿⣴⠄⠒⠂⠀⠀⠀⠀⢠⣴⡿⢿⣿⢯⣿⣿⣴⣯⢾⡯⣷⢄⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣤⣴⠆
⠀⠴⠿⠷⠂⠀⣄⣤⣤⣶⣶⣤⣴⣾⣟⣷⣾⣟⣿⣿⣿⣿⣿⣿⣏⣵⣯⣪⣽⣫⣟⠼⣋⣤⣟⣭⣟⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣮⣷⣮⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡛⣛⡛⡛⠛⠛⠛⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣻⡻⠼⠴⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠦⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⡾⠿⠿⢿⡟⠛⠛⠛⣟⠙⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⠿⠿⠿⠛⠛⢿⠛⠉⠉⠉⠉⢳⡀⠀⠀⠙⣄⠀⠀⠹⣄⠀⠀⠘⢦⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⣿⡄⠀⠈⢆⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢳⡀⢀⣀⣈⣦⠤⣤⠼⠶⠶⠒⣞⠛⠉⠉⢯⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⣸⣿⡄⠀⠈⢧⣀⣀⣀⣤⠤⠷⠴⠒⢲⠊⠉⠉⠁⠹⡄⠀⠀⠈⢣⡀⠀⠀⠈⢧⡀⠀⠀⠱⡝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢀⡼⠃⠸⣿⡖⠋⠉⠉⠀⠈⢧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠙⣄⠀⠀⠀⠱⣄⣀⣀⣤⠵⠶⠖⢖⠛⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢠⡾⠁⠀⢀⣾⣿⡄⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣈⣧⣠⣠⡤⠤⠞⠒⠛⣏⠉⠉⠀⠱⣄⠀⠀⠀⠈⠣⡀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣰⠏⠀⠀⢠⣾⣉⣙⣿⣄⣀⣀⣤⡤⠖⠛⠚⠙⢯⠉⠀⠀⠀⠱⡄⠀⠀⠀⠈⢦⡀⠀⠀⠈⢦⠀⠀⢀⣀⣹⠦⠜⢻⣧
⠀⠀⠀⠀⢀⣾⠟⠀⠀⣼⠋⠀⠀⢠⡿⠲⠤⠬⢽⣿⡅⠀⠀⠹⡀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠘⢆⠀⣀⣀⡤⠵⠴⠒⢶⡚⠉⠉⠉⠳⡄⠀⠀⠀⠻⣷⡀
⠀⠀⠀⣠⣿⠋⠀⢀⡼⠃⠀⠀⠀⠈⠉⠙⠛⠶⠶⠾⣿⡆⠀⠀⠹⡄⠀⣀⣀⣤⠷⠴⠖⢲⡚⠉⠉⠀⠈⢣⡀⠀⠀⠀⠳⡄⠀⠀⠀⣘⣦⣤⣴⣶⡿⠇
⠀⠀⢰⣿⣥⣤⣤⡾⠷⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠘⣿⡶⠖⠒⠉⠉⠀⠱⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠱⣄⣀⣤⣤⣾⡶⠿⠟⠛⣿⡏⠁
⠀⠀⠈⠉⠉⢹⣿⠛⠛⠓⠶⠶⠦⣬⣍⣉⣛⠳⠶⠶⢤⣼⣿⡄⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣀⣨⣧⣤⣶⡶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⠶⢤⣼⣿⣆⠀⣀⣠⣤⣴⣾⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡴⠶⠺⡟⠋⢹⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡏⠉⠛⠛⠿⠷⣶⣤⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⡴⢶⠞⠛⠛⣷⠀⠀⢸⡇⠀⠀⡇⠀⢸⡇⠀⢠⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⢈⣇⣤⣼⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⢀⣠⡟⠀⠀⢸⣿⣿⡿⠿⠛⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⣠⣤⠾⣟⣫⡽⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⠃⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢿⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣚⣯⣿⡧
⠀⠀⠀⠀⠀⠀⣿⡗⠲⢤⣨⣿⣄⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⣚⣭⣵⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⢿⣟⡾⢯⣟⡳⢦⣤⣀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣋⣭⣴⡾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠶⣬⣙⡳⠮⣭⣛⠶⣭⣿⢿⣿⣀⡀⠀⠀⣿⠀⢀⣠⠤⠖⣋⣩⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣬⣙⠳⢶⣽⡷⠿⢀⣸⣿⣤⣭⣉⡓⣿⣊⣩⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣶⣷⣶⡾⠟⠋⠉⠈⠉⠛⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⣠⠀⠀⠸⠀⠀⠀⠀⠀⠀⢽⣿⡤⠀⠀⣠⣾⣤⠀⠀⡀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⣀⠀⠀⣸⡿⠛⠀⠀⠀⠁⠀⢶⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠺⠷⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⣿⣶⣦⡀⠀⣘⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣄⡀⠀⠀⡈⢻⣻⣿⣿⣿⣿⣿⢶⠀⠀⠀⠀⠀⠀⠀⠐⠒⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠑⣷⣿⣿⣟⣿⡯⠁⢄⠀⠹⠞⢿⣿⣿⣾⠀⠀⢠⣤⣄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠁⠙⠛⣿⣷⣿⡿⠓⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠿⢿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣿⠃⠂⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⣠⡾⢿⣿⣿⠐⠀⠙⢿⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⣿⣷⣜⣿⣧⡄⠈⠀⡀⢁⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⣀⡴⠀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠀⢀⣀⣴⣿⣿⡇⣀⣾⣿⣿⣿⣟⣿⣿⣄⠀⠀⠀⠈⠀⠀⠈⣿⣿⠃⠀⠔⠐⣻⡽⠀⠀⠄
⠀⠀⠀⠀⣶⣾⣏⠈⢁⣴⣾⡿⠀⠀⠉⢻⡿⣿⡇⠿⣿⡿⡲⠟⠫⢿⠿⣿⣷⣶⣄⠈⠀⣀⠈⡿⠇⣠⣰⠆⠀⠻⠃⢀⡄
⠀⠀⠀⠘⠿⠿⠳⢠⢼⡏⡟⣷⢰⣀⢰⡈⢱⡉⠿⢠⡉⢄⣧⠀⡀⠀⠀⠈⠉⠋⢿⡶⣦⡌⢑⢃⣦⢁⣿⡀⢸⡆⢰⡬⣿⣄⡀
⠀⠀⠀⠀⣴⣴⣿⣿⣿⣾⣷⣷⣿⣿⣾⣷⣾⣯⣾⣿⣧⢠⣿⡇⣶⠀⠀⠀⢀⣀⣀⣴⣝⣧⣤⣼⣿⣤⣿⣧⣾⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⣍⣙⠛⠛⠿⠿⣿⣿⣿⣏⣿⣧⣿⢿⣿⢛⣿⣷⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠀⠀⠀⣀⢹⢿⣿⡿⢹⠸⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢋⠉⠉⠉⠀⠀⢀⠀⣾⣆⣀⣧⣿⣯⣇⣈⠟⠎⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠻⡏⣿⣿⡴⢳⣿⣿⣿⣏⠀⣴⡆⠀⠀⣸⡇⠀⠀⣀⣠⣿⣶⣿⣿⣿⣿⣿⣾⣿⣴⣤⣀⣻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠡⢟⡘⢱⣮⣿⣿⣟⣏⡀⢹⡿⡀⢐⣾⣷⡂⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠟⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢶⣬⣿⣿⣿⣭⢄⣿⣿⢋⡠⣽⣿⡵⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣶⣧⠄⠀⠀⢘⣿⣵⢇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠀⣘⣻⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡍⢬⣻⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠛⠉⠉

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⣠⣶⣤⣀⡀
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣷⣦⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣷⣾⣿⣿⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⣿⣷⣶⣾⣿⡿⠿⣿⣿⣿⣿⣶⣤⣀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣌⡋⡻⢿⡿⠋⠁⠀⠘⠻⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣾⣷⣶⠿⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠈⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⣀⣠⠶⠚⠁
⠀⠀⠀⠀⠈⠳⡦⠀⢀⡎⢻⣶⣦⣄⠀⠀⠀⠈⠳⣄⣠⣽⣿⣿⣿⣿⣯⣉⣤⣴⣶⣦⡀⢻⡁
⠀⠀⠀⠀⠀⠀⠙⣮⠛⠀⠀⠙⠻⣿⣷⣤⣴⡿⣿⣿⣿⣯⣿⣿⣿⣿⡟⠉⠁⠀⠀⠻⠷⢷⣯⣖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡤⡾⠿⠉⠉⠁⣁⣠⣼⣿⣿⣿⠟⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⢀⣀⣤⣤⣶⡼⣿⣟⣉⣉⡉⠁⠀⣀⣀⣀⣀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠙⠋⣽⣿⠟⠉⠁⠀⠀⠀⠉⠉⣩⢟⡿⠙⢧⡉⣦⠀⠙⠍⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠸⣿⡷⢷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠄⠀⢀⡴⠋⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠖⠶⢶⡞⠃⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣛⣻⣧⡞⠁⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢀⡽⠋⠀⠀⠀⠀⡾⠉⣿⠉⢳⡄⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠏⠀⠀⠀⠀⠀⠸⡟⠛⣿⠛⢻⡇⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣽⡦
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣿⠀⠀⠀⠀⢠⡶⠶⢶⡶⠶⢶⡆⠀⠀⠀⠀⠀⠀⣶⠶⠶⡶⠶⢶⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣾⠉⠙⢷⣰⣏⠀⠙⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣏⠀⢀⣇⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⢰⠏⠉⢁⣀⣼⣯⣉⡙⠂⣿⠀⠀⠀⠀⢠⡟⠛⢻⡟⠛⢻⡇⠀⠀⠀⠀⠀⠀⣿⠛⠛⡟⠛⢻⡇⠀⠀⠀⠀⣿⠻⢦⣀
⠀⠀⠀⠘⢳⡞⠋⠉⠀⠀⠉⠉⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⢾⡀⠀⠀⢀⣤⠀⠀⠀⣿⠀⠀⠀⠀⠘⠷⠶⠾⠷⠶⠾⠃⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿⠶⠞⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠈⠛⠶⠚⢻⣇⣠⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⣩⠻⠦⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠛⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⣯⣀⢀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣉⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⢿⣯⣟⣽⣿⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣧⠐⠛⠕⢹⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠀⠀⢸⠋⡔⠀⢭⡴⠟⢻⣿⣷⣏⠁⠀⠈⢀⡀⠀⠈
⠀⠀⠀⠀⠀⢀⣿⠤⠓⠃⠉⠀⠀⣽⣿⣿⡌⠀⠀⠀⢀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⣸⣿⣷⠃⢀⢠⣶⡻⣿⣿⣿⣿⢶⣵⣾⣧⣤⡀
⠀⠀⠀⠀⠀⠸⣿⣿⡄⣧⡏⣸⣰⣿⣿⣿⣿⡄⠙⢿⣿⣿⢷⡀⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣤⡌⢿⠸⠁⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡾⠀⠀⢻⢻⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠁⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠗⣠⣾⠉⠁⠹⢿⣿⣿⣿⣿⣿⣧⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⣿⠀⣠⠺⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣺⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⠋⢣⣿⡟⠻⠿⠃⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡏⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠳⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠐⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⢕⢆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠣⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢔⠁⠀⠀⠀⣀⡀⠀⠀⠑⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠀⠠⡀⢀⠔⠁⠀⠀⣴⡿⠛⠛⢷⡄⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⡣⡀⠀⠀⠀⣿⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠨⡢⢀
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⠁⠀⠀⠀⠀⠀⠀⠈⠢⡠⠀⠀⠙⠿⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠪⠅

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⢽⠛⠛⠛⠛⠛⠛⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠒⠒⠒⠒⠒⠒⠒⢿⠀⠀⣿⣟⣟⣻⣳⢹⡇⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⡧⡧⢼⢼⣻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠄⠐⠘⠓⣦⣤⣤⣤⣤⣤⣤⣿⠀⠀⣿⣗⣗⣺⣺⣿⡇⠀⢀⣼⣦⣤⣤⣤⣤⣤⣤⣤⣴⣧⠄⠂⠤⠠⡀
⠀⠀⠀⠀⠀⢸⡀⢀⠀⢀⣸⠔⠂⠂⡀⠀⠀⠈⠀⠀⣗⠈⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠉⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⢰⢾⠭⠭⢽⠶⢶⠶⣶⠳⠶⠶⠴⠀⣀⣧⣀⣀⣀⣀⣿⣇⠀⠶⠶⠶⠶⠦⠶⣤⣤⡶⠶⠶⠶⢿⠭⠭⢽⠶⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠈⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠞⠉⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⢀⡄⣀⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠀⠀⠀⠠⣀⣤⡌⠄⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⡇⠶⣶⠀⣿⠀⠀⠀⠀⣿⠰⡶⡦⢸⡏⠀⠀⠀⢐⡂⢰⢀⠠⢠⣤⣐⡂⠀⠀⠀⡏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣮⠀⣿⠀⠀⠀⠀⣿⠀⡇⣥⢸⡇⠀⠀⠀⣈⣈⠸⠘⠙⠸⠙⣁⣂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠒⠒⠀⠿⠀⠀⠀⠀⠿⠐⠒⠒⠸⠇⠀⠀⠀⠨⠁⢰⠠⢆⢴⢤⠍⠅⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠘⠐⠘⠈⠛⠙⠉⠁⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠙⢯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⢠⠀⢹⡏⠈⠁⠀⠀⠀⣤⣤⣤⡄⠈⠉⠉⠀⣿⡇⠀⡄⡄⡄⠀⣿⡇⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠜⠘⢨⠃⢸⡇⠀⠀⠘⠛⠀⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇⠃⡂⡆⡃⠃⣿⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠌⢈⠩⠁⢸⠇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢿⡇⡁⠍⠌⡅⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠈⠈⠰⠁⢸⣦⣀⣀⡀⠀⠀⣿⣿⣿⡇⠀⣀⣀⣀⣼⡇⠁⠍⠏⠇⡁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠼⠿⠉⠉⠀⠀⠠⠿⠿⠿⠇⠈⠉⠉⠁⠿⠇⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠐








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣴⠏⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡄⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢿⣆⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣏⣀⣀⣠⣏⣀⣀⣀⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⣩⣭⣥⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣭⣭⣉⡒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠴⠶⠾⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠞⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢊⣁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉
⠀⠀⠀⡠⠔⠒⢊⣉⣭⣭⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣭⣭⣉⡑⠒⠢⢄
⠀⠀⠀⠙⠢⠤⣈⣙⡛⠿⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⠿⠿⢛⣋⣁⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢖⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⡲⠽⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠇
⠀⠀⠀⢜⡂⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢿⣿⣧⡀⠀⠀⡀⠀⠀⠎
⣦⠀⢄⠽⣂⠀⠄⠀⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⡈⠉⠻⠿⣷⡀⠈⠓⢄⡀⠴⣸⡇⠀⠠
⢨⠁⣨⠞⢧⡀⠀⡤⢋⠀⢀⡢⣤⣀⠀⡀⢈⣁⠦⠀⢀⠀⠀⠁⠀⠀⠘⢧⡀⢄⣸⣆⢀⡍⢀⠀⠀⢄
⡄⠀⢩⡞⠡⡉⢠⣴⣼⢤⣾⢇⣴⣠⢠⣷⣤⡌⣅⣸⡀⣄⣠⣴⢠⣤⣾⣠⣠⡆⣀⡄⣾⣷⢀⣰⡇⡀⡀
⡾⡗⣉⠟⢡⡈⠾⢿⣿⠘⠿⠿⠿⠿⢿⡯⣿⣿⠯⣯⡴⠿⠿⣿⡿⢿⢿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣽⠷
⠑⣴⢫⡖⢻⡖⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠿⠿⠿⠊
⠀⢈⡯⠖⢿⡯⠄⠀⠀⠀⠀⣤⣾⣿⠟⠛⠉⠉⠉⠀⠀⠈⠉⠉⠉⠋⠙⠛⠛⠿⢷⣶⣦⣤⡀⠀⠀⠀⠂
⠀⠚⢡⠆⣾⣮⡢⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠠⠀⠂
⡀⠐⣉⣠⣿⣽⢄⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠌⠀⠀⢤⡀
⠀⠀⣿⠁⠙⠿⠏⠢⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣬⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣷⣽⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣤⣶⣹⣿⣽⣿⣿⣿⣿⣽⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣛⢿⡙⠢⠙⣆
⠀⠀⠀⢀⣴⣿⣽⣿⣟⡿⣟⣯⣽⣿⣿⣿⣹⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⡲⣝⢧⠑⢈⡀⣟⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣟⣾⣦⣵⣶⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣟⣿⣿⣿⣾⠿⣫⣽⢿⣦⣄⣀⡀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⣾⠓⠗⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣯⣿⣿⢾⣗⢻⣿⣿⣿⣿⣿⣿⣟⡛⠃⠹⣿⢿⣿⡷⠆⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣮⣤⣆⣈⠁
⠀⠀⠀⠀⠀⠒⠲⠶⢶⣤⣤⣄⣀⣤⣄⣄⠟⠿⣿⡿⢟⡉⣙⣛⡛⡻⢿⣿⣿⣿⣟⣿⣿⣿⣿⡏⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⢧⣞⢿⡾⢿⣾⡻⣷⠌⠐⡝⢿⣿⣿⣿⡿⣿⡿⠉⠑⠻⠷⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣆⣨⣿⣿⠿⣶⡼⠀⠘⢿⠛⢟⠉⣀⡤⠤⠄⠐⠂⠛⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠨⢟⣊⢯⣰⠄⢀⣼⣄⣄⣹⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠳⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠄⠑⠤⡀⠀⢀⡤⠒⡝⠉⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⡠⠔⠉⣇⠀⠓⠀⠀⠀⠀⠙⠒⠦⡀⠀⠀⠀⠀⠡⠸⡄⡐⣀⠴⠋⠀⡰⠀⡀⠐⡀⠀⠈⠑⠢⢄⣀⣀⡀
⠀⠀⠀⣠⠔⠈⠀⠀⠀⣸⠛⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠠⢀⡀⣀⠵⠒⠋⠀⠀⠀⠈⠀⠀⡇⠀⠱⡀⠀⠀⠀⠀⠀⠀⠙
⠤⠖⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠱⠀⠀⠙⠦⡤⣀
⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠁⠂
⠀⠀⠀⢀⢺⡆⠡⠀⡰⢷⠄⠂⠀⠁⠀⠀⠰⡀⢃⠀⡀⠀⠀⠀⠄⠀⢀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆
⠤⠤⢀⡦⢼⠁⠄⢧⡍⢩⠃⢆⠀⠀⣀⣰⠀⢁⠀⢆⢤⠄⠠⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢈⢀⠀⠩
⣀⢀⣉⣹⣏⣡⣄⣼⡆⠜⠀⠐⠂⣀⣠⣇⣠⣌⣤⣜⣿⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣠⣀⣀⣀⣈⣀⠄
⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⠀⠀⠀⢠⡴⠖⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠲⢶⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢻
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼
⠀⠀⠀⣿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡴⠞⠁⠀⣼
⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣴⠞⠁
⠀⠀⠀⢹⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠁
⠀⠀⠀⠘⣧⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⢀⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣄
⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠉⠙⠛⠲⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠏
⠀⡀⣀⣰⠀⢀⡀⡀⣄⠀⢀⣰⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⢠⣷⣿⣿⣾⣾⣿⣷⣿⣷⣿⣿⣿⡿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠠⠄⠀⠀⠀⠀⠀⢬⡧
⠸⠟⠟⠿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠩⢍⡧⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠀⠳⠪⣽⣷⣂
⢀⣰⣷⣠⣼⣥⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⢠⣿⣀⠛⢭⣽⣯⠶
⠀⣿⣿⣿⣿⣿⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠰⢬⣿⠔⠂⠰⣬⣷⠞
⠀⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⣷⣄⠨⠽⣿⠭⠄⡼⣿⣿⣟

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⢮⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣠⢞⠜⠁⠿⠇⠙⢌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢄⠀⢀⠀⢸⡀⠀⠀⠀⠀⢀⠔⡵⠃⠀⢠⠤⢤⠀⠀⠱⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⡳⢋⡴⠀⡇⠀⠀⠀⡴⣫⠊⠀⠀⠀⢸⢺⣺⠀⠀⠀⠈⢣⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⢶⠊⠀⢀⡇⠀⠀⣪⠞⣹⠀⠀⢀⣀⣀⣉⣁⣀⣀⠀⠀⢸⢹⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢄⢸⡴⢃⡞⣠⢔⡫⠖⢹⢽⢀⣀⡛⠒⣒⣒⣒⡒⠚⣃⣀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣽⢲⠋⠋⡔⡯⠒⠢⢸⢽⡧⠤⡇⠀⣷⠉⢱⡇⠀⡧⠾⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⡇⡗⠒⠒⢸⢸⠈⠉⠁⠀⣿⠀⢸⡇⠀⠉⠉⢹⢸⢀⢄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣼⣼⣾⣭⣷⣧⣤⣤⣼⣾⣤⣤⣴⣵⣿⣷⣿⣿⣬⣦⣤⣼⣼⣤⣮⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠵⣝⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠰⡿⢿⠆⢀⡴⣫⠎⠙⢮⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁⠀⠈⠳⢝⠯⡏
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡴⣫⠞⠁⡤⠤⡀⠙⢌⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣊⠔⠁⠀⢠⢤⢤⠀⠀⠑⣌⠢
⠀⠀⠀⠀⠀⠀⢀⡷⣫⠞⠁⠀⠀⣟⣛⠇⠀⠀⠱⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠞⢹⠀⠀⠀⢸⠲⢾⠀⠀⠀⡏⠙
⠀⠀⠀⠀⠀⡠⣫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠛⠀⠀⠀⡇
⠀⠀⠀⠀⣪⡞⠁⡇⢠⢤⠤⡀⢰⡶⠶⢶⠄⢠⢤⢤⠀⡏⠣⠑⠀⠀⠀⠀⠀⣰⣼⠀⠀⠀⢠⠤⢤⠀⠀⠀⣇⣦
⠀⠀⠀⢈⣁⡇⠀⡇⢘⠚⠒⡇⢸⡾⢉⣾⠀⠟⢺⢚⠀⡇⣠⢄⠀⠀⠀⠀⢀⡇⢿⠀⠀⠀⢸⠚⢻⠀⠀⠀⣿⢘⡄
⠀⠀⢸⡁⣀⣙⡆⣇⢺⡍⢉⠓⢸⣯⣠⣿⡀⠋⠋⣽⢂⡇⢧⡴⠃⠀⠀⠀⣸⠁⣸⡀⠀⠀⠈⠋⠋⠀⠀⠀⣟⠀⣧
⠀⠀⠒⣿⣿⣶⣷⣿⣶⣶⣶⣶⣿⣷⣷⣿⣷⣶⣶⣶⣶⣷⣶⣷⡖⠀⠀⠐⢺⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠟⠒
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠄⠠⠄⠠⠤⠤⠤⠤⠄⠀⠈⠓⠊⠉⠉⠀⠀⠀⠀⠀⠙⠋⠛⠛⠋⠙⠋⠋⠉⠀⠉⠉⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢠⣾⠋⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠘⠻⣦⣄⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠠⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠙⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠿⠟⠛⠛⠋⢹⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣄⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⣄⠀⠀⣀⣀⠀⢀⣤⣄⡀⣤⣄⣀⣤⣤⣤⣴⠆⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢰⡟⠁⠀⠀⢼⡟⢻⡇⠈⣿⣭⣄⣿⣉⣸⣿⡤⢸⡷⠆⣼⣧⡾⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣇⠈⢿⣦⡄⠀⠘⠷⠟⠁⠀⡿⠉⢠⣿⠉⠹⣷⡄⢸⣷⢠⣿⣵⣶⣟⠋
⠀⠀⠀⢀⣤⣴⠶⠟⠛⠛⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠉⠙⠛⢶⣤⡀
⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣤⣤⣴⣶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠠⢶⣷⡿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⣄⣀⣀⢀⣿⠷⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⢠⡇⣽⣥⠄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣯⣔⣆⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⣠⣴⣿⣶⣞⣿⣹⣁
⠀⠀⠀⢰⣾⣦⡺⣦⡶⢾⡟⠙⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⣿⣿⣻⣉⠉
⠀⠀⠀⠀⠉⠛⠷⠾⠟⢾⣷⣿⣧⣦⣤⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠙⠛⠿⠦⢀⣆⠀⠀⠀⠀⢀⢸⣀⠉⢙⣿⠻⢿⣿⣿⣿⣄⣀
⠀⠀⠀⠠⣦⡀⠀⣄⢀⡼⣧⣿⣿⣿⡿⠋⠀⢀⠤⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠉⠛⠂⠀⠀⣈⣻⣑⣚⢛⣿⣞⢼⣿⣿⣿⡋⠀⠉
⠀⠀⠀⠀⠛⠿⣿⣾⡷⢋⣿⢿⢿⣟⢀⡀⠐⠈⢀⠀⠀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⡏⠩⠼⣿⣿⣿⣻⡿⣿⣷⣤⡄
⠀⢀⣠⣾⡷⠀⠀⠀⠰⣟⠻⢿⠿⣛⣿⣭⣤⠒⠁⠨⡁⠀⠀⢐⠀⣀⣀⣀⣥⣠⣴⣶⣤⣤⡴⣶⠿⣿⡟⠶⢻⡿⢿⢶⣿⣿⣿⣾⠄
⠀⠈⠛⠻⠻⣊⣗⡀⢢⣿⢸⣽⣿⣿⣿⢟⣿⠏⠀⠤⠬⠤⠀⠀⠀⠀⠀⠉⠀⢀⣀⣠⣭⣭⣧⣿⡿⢿⡿⣾⣥⣿⣿⣶⣶⠟⡟⠛⠳⠠⠄
⠀⠀⠀⠒⠾⠛⢛⣭⣶⣿⣌⠉⠉⠙⠛⠋⠉⠙⠻⣿⣶⣶⡿⢿⡶⠺⢿⣿⠻⢿⠟⠋⠀⠀⠀⠀⠀⣸⣆⡉⠉⠘⠀⢀⣘⡃⣅⡁
⠀⠀⠀⠀⣤⣤⣤⣤⣀⣿⠈⠗⠦⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣄⣀⣀⣀⠤⠀⠛⠚⠓⠈⠛⠋⠙⠃⠙⠈⠉⠉⠻⠛⠉⠉
⠀⠀⣤⣴⣢⣴⣮⡅⢀⣽⡄⠦⠀⢠⣿⣶⣦⣍⣠⣤⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉
⠀⠀⠈⠉⠛⠟⠋⠀⠀⣸⡇⠘⠀⢈⠛⠿⠿⠛⠿⠋⠁⠀⠀⠀⢀⠀⠠⢶⣶⡤⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢴⠶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣥⣤⣤⣄⣄⣠⡷⠶⠿⠿⠿⠇⠀⠸⠶⠿⠶⠾⠶⣶⣄
⠀⠀⠀⠀⣀⣤⣶⠿⠿⠟⠁⠰⠤⠀⣤⣤⣽⣄⣿⡆⣠⣀⣰⢀⣠⠀⣀⠀⠈⡹⠾⠛⠛⠁⣀⣤⣤⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠉⠛⠻⠿⢿⡿⢸⠛⠃⠻⠻⠻⠿⠒⠀⠀⡀⣀⣀⡀⣀⣀⣤⣤⣤⣤⣤⠶⠶⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⣄⡘⠛⠶⠤⠀⠀⠀⠀⠘⠙⠒⠀⠲⠖⠒⠲⠆⠀⠁⠉⠉⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠣⠤⢴⠃⠀⠀⠀⠀⠐⠃⠀⠂⢀⣑⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢢⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠗⣋⠔⠋⠀⠀⠑⢌⡐⢀⠈⠀⠀⠀⠀⠀⡀⢀⡀⠐⡆⠊⠀⡖⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡤⠊⠁⠀⠀⠀⠀⠀⠀⠉⠢⢍⡃⢄⠀⠀⠠⢀⠉⠀⠀⠊⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⡄⣂⡠⠒⠁⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⣀⣉⡒⠪⢔⡀⢤⡢⣤⣀⣉⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢸⡉⠈⠈⠸⠀⠀⠀⠸⠀⠀⠁⠀⠃⠀⠀⠁⠉⠀⠐⠳⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠮⠤⠦⠀⠀⠀⠸⠦⠦⠤⠤⠇⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⡀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⡀⠀⣀⢸⢀⠀⠀⡀
⠀⠀⠀⠀⠁⠠⠐⠀⠁⠄⠀⠀⠀⠘⠉⠋⠉⠙⠀⠀⢠⡀⢠⡀⢌⠀⠀⢠⡄⢠⢤⠈⣯⠇⢠⡄
⠀⠀⠀⠀⠀⠛⠀⠛⠀⠛⠀⠀⠀⢨⠄⠀⠐⠀⠀⠀⢸⡤⠊⣤⣼⠀⠀⠈⠁⠈⠀⠀⡏⠁⠈
⠀⠀⠀⠀⠀⠿⠀⡻⠀⠟⡄⠀⣠⢸⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠸⠍⠀⣯⠀⠰⠆
⠀⠀⠀⠐⠒⠓⠓⠚⠓⠛⠓⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠐⠀⠀⠂⠒⠒⠛⠖⠗⠲⠲⠓⠒⠒⠒⠖⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣍⡉⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⢒⣒⡒⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣤⣤⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦⠀⠀⣿⣿⣿⢀⣴⣿⠟⣁⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⠟⣡⣾⣿⣿⣦⡈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣷⣄
⠀⠀⠀⠀⢀⣄⠀⣀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⣠⣾⡇⢠⣤⢠⣤⡄⣿⣿⣿⣿⣿⡇⣤⣤⢠⣤⠈⣿⣷⡌⠻⣷
⠀⠀⢀⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢈⣉⢀⣉⡀⣿⣿⣿⣿⣿⡇⣉⣉⢈⣉⠀⣿⣿⡇⠀⠈
⠀⣴⣾⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⡇⠘⠛⠘⠛⠃⣿⣿⣿⣿⣿⡇⠛⠛⠘⠛⠀⣿⣿⡇
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⠿⠘⠿⠃⣿⣿⣿⣿⣿⠀⠿⠿⠸⠿⠀⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣇⣀⣿⣀⣸⣿⣀⣸⣿⠀⠀⠀⣀⣀⣀⣀⡀⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⡇⢠⣶⣄
⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡏⠉⣿⠉⢹⣿⠉⢹⣿⠀⠀⢸⣿⣿⣿⣿⡇⠈⠉⠉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⠉⠉⣻⣿⣾⣿⣯⢀⣀
⠀⠀⠀⢠⣿⣿⣿⠿⣿⣿⣿⣷⣦⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣏⠉⣿⠉⣿⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠻⡿⠷⠀⠠⣿⣿⣿⢿⣷⣴⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⢸⣏⢉⣿⠉⣿⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠃⠹⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠈⠁⡀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⢀⢀⠀⠀⠈⠦⣀⡀⢀⠀⠆⠀⠀⠀⠀⡀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠐⡀⠀⠐⠂⣅⠜⠀⠀⠀⣰⡿⠃⡈⠀⢀⠀⠀⠀⣿⣵⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠄⠀⠆⠈⠣⣄⡀⠐⠋⠀⠴⠑⠀⠀⠀⠀⠀⠈⡇⡄⠀⠀⠠⠀⠈⠁⣦⣤⡤⣴⠞⠚⡛⠶⠒⠖⠚⠉⢀⣀⣺
⠀⠘⠀⠀⢼⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡇⡤⢤⣿⣦⣠⡄⡀⠰⠀⠐⠀⠀⠀⠀⠀⠰⠀⠀⠀⠃⠀⣰⣿⣿⡾⠋⠉⠉⠙⠉⠋⠰⠘⠷⠤⢦⣤⠬⡁
⠀⠀⠀⠀⢨⡁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠿⠻⢶⣾⣤⣤⣀⢂⡀⠀⢀⠀⠀⠀⢀⣠⠴⠿⠿⠉⠁⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⡧⠀⠀⡀⣯⠀⠀⡀⢀⡀⠀⠘⡀⠀⠀⠀⠀⠠⠀⠀⣛⢲⠯⠽⣟⡿⠷⠘⠚⠛⠯⢄⣁⠈⠁⠀⠀⠀⢸⡄⠃⠀⡅⠀⠃⢄⣤⣀⡀⠄
⠀⠀⣀⠀⣸⡇⠀⠀⣧⢫⣉⠀⡇⡼⢷⡓⢺⠄⠈⠁⠀⠚⠃⠀⠒⠂⠍⠙⠉⠀⠀⡆⠀⠄⠀⠀⠀⠀⠉⠐⠠⡄⠀⠀⠁⠰⠀⠀⠀⠇⠘⢡⡟⠱
⠀⠨⡇⠈⢹⡟⠂⠀⣿⡈⠋⠱⡗⠚⠋⠀⢠⡇⡀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠀⠀⠁⠨⠅⡳
⠀⠀⢷⠰⠘⠛⢀⡐⠓⢀⠀⠈⡗⠀⠆⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠓⢀⡀⠀⠈⠈⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓
⣶⡶⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣥⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡀⣦⠀⠠⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠫⣅⣽⣿⣷⣴⣶⢀⢋⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⢠⡄⢾⣿⣿⣿⠟⣿⣥⣼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⡀⣸⣧⣶⣶⣾⡟⠁⠀⢀⠀⠀⠘⠟⢩⡷⢸⣿⢿⡁⠐⣷⣿⠾⢯⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣠⣤⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣾⣷⣤⣤⡄⢾⣇⣰⣿⣽⣷⣾⣿⠀⣠⣀⣤
⠀⠀⠀⠀⠀⠀⠤⢽⣾⣿⣛⠁⡭⢄⠀⠻⠛⠋⠉⠛⠛⡗⠰⡶⡄⣼⣿⣿⣿⡇⣸⣿⣿⣿⣿⣭⣙⢳⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣴⣿⡇⠐⣰⠀⠰⣶⠂⠀⠀⠑⠀⠀⣿⣸⣧⢇⣼⡿⠋⠉⢿⣅⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿
⠀⠂⠀⠀⠢⣶⣿⣿⣻⡣⠀⠀⠀⠒⠀⣿⡖⢒⣒⠒⠚⣿⡏⢫⢾⢿⣶⠖⠀⣿⣿⠙⣿⣿⡯⡌⡿⠁⠈⠀⠈⣿
⠀⠀⠀⠀⠈⠉⢹⡅⠘⢃⣠⣶⠀⠀⠀⣼⣷⣌⣉⠁⢀⣿⣿⣴⣾⣀⠀⠁⠀⣿⣿⣠⣿⣿⡇⠃⣴⠀⠀⠀⠀⣯
⠀⠁⣸⡄⠀⢒⣉⡋⣉⡉⠉⠙⠀⢰⣶⣿⡟⢻⣿⣿⡟⢿⢿⡿⠿⠿⠶⠶⠒⠚⣩⣉⣉⣉⣁⣤⣤⡔⠂⠀⠉⠁
⠀⠈⠉⣉⣙⠛⠛⠛⠻⠷⠬⠷⠤⠜⠟⠛⠛⠋⠉⠉⢁⡀⠀⢦⡄⠀⣤⣀⣴⣽⣿⣯
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣀⡉⠉⠉⠉⠙⠛⠚⠻⠻⠿⠿⠿⠿⠶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⠇⠀⠀⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⣠⡿⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡘⢷⣤⣤⣤⣾⠋⢀⣾⡿⠶⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⢳⣼⡏⠀⣿⣥⠶⠋⠁⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⡿⠏⢀⣀⣘⣏⣤⣤⣄⣀⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⡴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⢳⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠿⢧⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠟⠋⠉⠀⢸⡇⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠉⠉⠉⠉⠉⢉⡍⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠋⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⣠⣟⠁⠀⠀⠀⣠⡀⠀⠀⠀⣸⣯⣴⠶⠶⣄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⠀⠀⠈⣷⡶⠖⠛⠉⠙⠳⣦⣴⠞⠋⠀⠀⠀⠀⣿⡟⠁⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⢦⣤⡞⠁⠀⠀⠀⠀⠀⠀⣼⣧⣀⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⠀⠀⠀⠀⢀⣤⢴⡾⠋⠉⠋⠉⠻⣷⣄⢸⣿⠀⠀⠀⢀⡟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣴⣿⡄⠀⠀⢠⣿⣁⣀⣀⣀⣀⣀⣠⣤⡾⠟⣸⡟⠀⠀⢀⣼⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⢹⣦⣤⣀⣭⣿⣿⣏⣉⠉⠉⠉⠀⠀⠀⣿⣷⣤⠶⠛⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡼⠋⠀⢹⡏⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣦⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠚⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠶⠀⣦⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⢀⡀⠀⠀⢰⡬⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠃
⠀⠀⠀⠀⠈⢀⣔⠈⠁⠁⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠷⠷⣥⣿⠟⡆
⠀⠀⠀⠀⠀⢠⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢀⣅⠉⠉⠉⠉⠉⠀⡇⣠⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⣷⠉
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⡰⠁⢀⠀⠀⠀⠀⠀⠀⢿⠀⠁⣶⠀⠀⣤⠰⢷⡿⠃⠀⠀⠀⠀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⣥⣠⣷⣦⣤⣶⣿⣤⡤⠂⠀⠠⡄⠤⣤⠄⠿⡀⢸⡏⠀⠈⣷⠀⢸⡇⡀⠀⠀⠀⠀⣠⣼⣦⣀⣀⣀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣇⣀⣸⡧⠴⠞⠛⠛⠚⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢲⠆⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠎⠀⠀⠁
⠀⠀⠀⠀⢀⢀⡛⠛⠛⠛⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣁⠀⠀⠀⢉⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠶⠷⠾⠿⡆
⠀⠀⠀⢠⡅⠘⠉⠁⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣰⢀⠀⠉⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡟⠈
⠀⠀⠀⠀⢷⣄⠀⠀⠨⣶⡥⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⡔⢀⠀⠂⡂⠰⡆⢈⠄⠀⠀⠀⠀⠀⠀⡃⠑⢤⡀⠀⠀⠀⣧⣻⡀⠲
⠀⠀⠀⠀⠀⠉⣦⣀⠀⠁⠁⣀⡄⠀⠀⠀⠀⠀⠀⠠⠐⠂⠳⣄⠀⠀⢀⠼⠛⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠉⠉⠉⠉⢠⠃⢁⠖
⠀⠀⠀⠀⠀⠀⢼⣦⣭⡉⠉⠔⠁⠘⠃⠀⠀⠀⠀⠶⠶⣤⣄⣉⣛⠚⠉⠠⠤⠔⠂⠀⠀⠀⠀⠴⠷⣤⣑⡦⢤⡀⢀⣤⡴⠤⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⡀⠀⠀⣤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠆⠀⢀⡄
⠀⡾⡀⠀⣴⣯⣀⡀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠉⢷⠇⠀⠀⢀⡄⠀⠈⠑⢂⠀⠀⠀⢀⣞⣥⡀⠀⠀⠾⠧⠀⠘⢦⡀⠈⠉⢡⣶⣷⠄⠈⢠⠀⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠠⢾⣉⠀⠀⢰⡋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡪⠊⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠘⠃⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡤⠶⠷⠄⠴⣶⣴⣴⣠⣆⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠗⠄⣾⣾⠮⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢶⣤⣤⣀⣀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠤⠀⠀⠈
⡶⠒⢶⠶⡶⠶⠶⣶⢶⠶⠶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⡶⣶

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠁⡀⢀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⢀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣫⠵⠦⠄⠄⡀⠁⠀⠁⠠⣤⣀⣀⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢀⠠⠂⢀⡼⠋⡤⠀⠀⠀⠚⠿⠟⠻⠿⠿⠛⠛⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢉⠠⠊⣠⠀⠀⠀⠴⠷⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣲⠶⣂⠀⠀⠀⠚⡠⠔⠁⡰⠊⢠⣾⣡⠄⣠⠟⠀⠀⠂⠀⠈⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢊⣥⣦⡤⡖⠊⠛⠙⣠⢖⣀⠴⢚⡡⠈⡽⣷⡟⠁⠀⠀⠀⠐⠀⢠⠀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⣛⠭⣖⠯⠀⠀⢀⣠⣼⣏⣑⣚⠽⢚⡭⠖⠋⠀⠀⠄⠀⠀⢀⠀⢀⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠯⢍⠀⢈⣠⣤⣴⣾⣛⣻⣿⣿⣿⣖⣪⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⢳⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⠤⠶⠤⠭⣭⣍⣿⣷⣿⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⣪⣭⠟⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣇⣡⣤⣶⣶⣀⣶⣤⣤⢤⠴⠐⠒⠊⠉⠁⠀⠈⠀⠀⠀⠀⠀⢠⣤⡵⠗⠀⠀⠀⡀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠟⠛⠛⣫⠽⠿⡟⢛⣭⢁⡇⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⡥⢀⣀⣀⠜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⢀⡠⠔⠛⠁⠊⠉⠉⠑⠶⠙⠳⢆⡄⠐⠀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠔⠛⠉⠀⠈⠉⣹⣿⡖⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠝⠁⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⡶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⠕⠋⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⣠⢖⣭⠿⠋⠛⠛
⠀⠀⠀⠀⠀⠀⣀⠴⣊⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⡀⢐⣡⡶⠛⠁
⠀⠀⠀⣀⠴⣊⡵⣚⢅⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣽⣵⣾⣿⡅
⢀⣴⣿⡿⢞⡽⢊⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣲⣿⠿⣧⡭⠿⠛⠉
⠈⠱⣿⣾⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣭⠶⠛⠉
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠒⠒⠉

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

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸⣧⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣦⣬⡱⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠒⠀⠀⠀⠤⣀⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⡤⠈⠉⠉⠛⠳⠶⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⠎⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣳⡄
⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠄⠫⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢙⡇⠀⠀⠈⢎⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢛⠀⣠⣐⠝
⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢨⡇⠀⠀⠀⠈⢂⡑⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡍⠁
⣄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢃⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠑⠢⡈⠁⠢⣄⠀⠀⠀⠀⠀⠀⡠⢚⡴⠛
⢛⣷⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠉⠀⠄⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡑⢄⣀⣀⢠⢟⠐⠋
⠀⠀⠉⠛⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣿⣿⣶⣶⣤⣀⡀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⠿⡈⢛⠃
⠀⠀⠀⠀⠀⠈⠙⠋⢻⠯⣁⣶⠶⠶⠶⠶⠶⠻⠳⢶⠊⠀⢀⣉⢻⣿⣿⡯⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⡞
⠀⠀⠀⠀⠀⠀⡠⢾⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠐⢚⠿⠛⠃⠑⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡘⢯⠉⠛⠳⠶⠼⠶⣖⢦⡇⢠⡇
⢴⠋⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠂⣠⡀⣤⠄⠄⠀⠀⠉⠀⠈⢳⡀⠀⠀⠀⠀⠈⡁⢀⡟⠁
⠃⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠃⠀⠀⠀⠀⠠⠀⠐⡀⠀⠈⠀⠋⠀⠀⠀⠀⠰⣡⠞
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⠸⠂⠀⠀⢤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⣄⢠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠆⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢰⡟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠈⢃⣻⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢠⣀⠀⠀⠐⠖⠃⢠⣾⡅⠀⢸⠋⠤⠀⠒⠒⢲⢿⢙⠁⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠠⣄⣴⣦⣟⢻⣧⣿⣷⠸⠀⠀⠀⠀⠀⠀⠀⠸⣆⠠⢀⠸⣦⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣿⣿⣿⠻⣄⣘⢿⠠⡟⢹⣽⠉⠀⢀⣰⠂⠀⠀⣘⣃⣠⡿⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡶⢺⠃⠈⠛⣧⠸⡆⠈⠉⣈⡃⣠⣼⣿⡀⠀⢸⣗⣇⣸⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠉⠨⣿⠀⠛⠉⠁⡀⠀⡀⠀⣿⡆⠛⣤⢁⡜⠉⢏⠷⠲⠞⠉⠈⠉
⠀⣀⣀⢄⣀⠀⢀⣀⡠⠔⠒⡀⠒⠒⠂⢈⣍⠀⢰⣦⡶⡀⠀⠈⢀⣰⣷⠓⠲⣟⠶⣉⠃⣨⢋⡿⠁⠀⠈⡄
⠘⠿⠿⠿⠿⠀⣀⣈⣀⡾⠾⠇⠀⠤⠾⠛⠷⣀⡾⠛⠛⠛⠛⢀⠌⠁⠀⠀⠀⠈⠛⠛⣾⣯⠛⣿⡄⠀⠀⢿⣄⣀
⠐⠒⠉⠀⠂⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⡘⢀⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢸⣇⠀⠀⢸⡏⠙⠛⠷⠶⢦⣤⣀⡤⢄⣀⣀⡀⠠⠤⢤⣤
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠸⠦⣀⣣⠈⠀⠀⠀⠀⠀⠀⢀⣴⡏⠁⢇⠀⠘⣿⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠂⢍⡙⢧⡄⠒⠂⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠄⣼⡏⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠘⡆⠀⠙⡇⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣦
⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⢶⢹⠀⠀⠀⠀⠀⠀⡔⠿⠁⠀⠀⠀⠀⠘⢆⡀⢹⠇⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡣⡀
⠀⠀⠀⡄⣰⠀⠀⠀⠀⣀⣀⣀⣼⣇⣰⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⣠⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⣷⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣮
⠀⠀⠀⡿⣿⠀⠈⠉⠉⢀⠠⢊⡙⠀⢠⠀⠀⠀⢠⡟⠀⢀⠠⠔⣩⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⢠⠀⡟⠀⠀⠀⢀⠎⠀⡌⠀⠀⠐⠀⠀⢀⣾⡡⠚⠁⢀⣜⠏⠠⠏⡳⢤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⢽⣿⣟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣭⣹⣿⣯⣤⠴⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡈⢿⣿⣿⣿⠾⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⠴⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⣤⣐⣛⣿⣿⣿⣿⣿⡿⣞⡃
⠀⠀⠀⠀⠀⠀⠀⢈⠓⢾⣿⣿⡷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠰⢄⡀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⢋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⣿⣿⣿⡿⢃⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡄⠀⡸⢶⣭⣿⣿⣿⣿⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠬⣽⣿⣿⣿⡿⠟⠃⣠⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡼⢳⣤⣹⣿⣿⣿⣿⣿⣿⣷⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠙⠷⣿⣿⣿⣿⣿⡛⠋⠁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓⢾⣿⣞⣥⣙⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⢃⣀⣀⣀⣀⡀⢀⡀⢀⣀⣄⠴⠂⠀⠀⠀⣉⣻⣿⣛⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⡿⡿⠛⠋⠁⠁⠉⠁⠈⠀⠁⠈⠀⠁⡄⡀⢠⠦⣴⣍⣿⣿⣯⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡖
⠀⠀⠀⠀⢀⣀⠈⢿⣿⣿⣿⣿⣿⣷⡶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣷⢶⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠠⢤⠶⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠤⠒⠀⠀⣇⠀⣀⠀⠀⡀⠀⡄⢰⣇⣉⣛⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣍⣁⡀⠀⠀⢰⡿⠀⣿⠀⢰⡇⢸⡇⢸⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣭⣸⡇⠀⡷⣶⣿⣞⣿⣴⣾⣧⣸⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢃⠀⢳⡄
⠀⠀⠀⣨⠷⢤⣌⣙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⣰⠋⠳⣤⠿⠀⠀⠂⠀⠻⣄⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⢷⣀⠀⠀⠈⠉⠓⠲⠭⣝⣒⠦⢤⣤⣼⠁⠁⠄⠈⠳⢦⡤⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⡄
⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣈⣉⣿⠀⡜⠀⠀⠀⠐⢄⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠾⠄⠘⣯⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠀⣀⣀⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠹⡇⠀⠈⢣⡀⠀⠈⠀⠀⢠⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠲⢄⣀⠀⠐⠀⠀⠀⠀⣠⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠙⠒⠳⠶⠖⠚⠁⠀⠀⢀⠈⠀⢀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⡀⠰⡟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣳⣦⣬⣉⡛⣻⡦⢤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⢿⣴⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠷⢷⣄⡀⠀⣀⡤⠞⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡯⣤⠶⠲⢄⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠳⡌⢿⡀⢀⣸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢳⠖⠛⡏⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⣦⡤⢿⡏⠃⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢰⠇⠀⠀⢷⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠙⠲⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣉⣉⣭⣭⣹⡗⠀⠀⣠⡾⠋⣠⡦⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⣿⢁⣤⡾⠋⣠⡾⠋⣀⣴⠄⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠟⢁⣤⡾⠋⣠⡾⠋⣁⣴⠖⠀⠘⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣴⠟⠁⣰⠿⠁⣠⠾⠋⠀⠾⠋⠁⣠⡶⠇⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⠟⠁⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠟⠛⠛⣿⠛⠛⠛⢻⡇⠀⠀⠀⠀⣴⠾⠛⠛⢛⠛⠷⣦⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣾⠃⠀⠠⠞⣫⡶⢃⡈⢿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡷⠶⠶⣿⠟⠛⠛⢻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠁⠞⢋⣀⢸⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠛⠁⠸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣷⠶⠶⠿⠶⠶⠶⠿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⣤⣤⣤⣤⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⢶⣶⣶⣤⣤⣤⣤⣤⣿⣄⣀⣀⣸⣇
⠀⣀⣠⣴⠶⠟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠷⠶⣦⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡿⠀⠀⡰⠋⠀⢀⠞⠀⣠⠖⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣇⠀⢰⠃⠀⠀⡏⠀⢰⠁⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡆⢸⡀⠀⠀⠳⡀⢻⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠃⠀⠳⣄⣀⣠⠃⠈⠢⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠉⠲⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⡀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣈⣉⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢨⣏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡩⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣾⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣟⠀⠀⠀⠀⣤⠖⣛⡩⠶⢩⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣸⣿⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠟⠀⠀⠀⠀⠽⠶⠤⠶⠄⠀⠿⠟⣃⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧
⠀⠈⠁⣤⠞⣽⠃⡀⠀⠀⠀⠀⢠⡴⠂⠘⠁⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⢠⣤⠶⢢⠞⢁⡤⠂⠀⠀⠀⣀⡤⠎⠀⣠⢀⣠⣦⣆⣴⣠⣠⡄⡀⣾⠀⣤⢰⣿⣿
⠀⠢⠶⣣⣼⣿⣇⠀⣀⠐⠟⠃⠀⠔⢀⣠⡤⠂⣷⣠⡄⢀⡀⠴⠀⠀⣀⣠⡶⠂⠀⠁⠀⠀⢀⡀⠴⠤⠿⠁⠀⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠋⢿⠿⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣼⣯⣿⣿⣿⡧⠤⠶⠿⠿⠿⠷⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⢀⣰⠶⣿⣿⠿⣶⡜⠒⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠀⠰⠀⠀⠀⡇
⢀⣈⣹⣿⣿⣿⣤⡌⠙⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣧
⢸⣿⣿⢟⣰⣿⣿⣿⣉⣉⡝⠂⠉⠀⠐⢂⣤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⡿⢛⣈⡇
⢸⣹⣿⣿⣿⣿⣦⣹⣿⡦⢥⠨⠐⠀⠀⣿⣿⣧⣴⣋⠀⠈⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣰⣿⡿⢿⣭⠟⢉⣩⣧
⢠⢿⣿⣿⣷⣿⣿⣿⣗⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣼⣷⣦⠠⠀⢒⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⠿⠿⠿⢯⣸⣿⣿⣿
⠐⠸⡿⠟⣾⣿⣏⠧⠙⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠈⠉⠙⠚⠛⠉⠉⠙⠛⠻⢿⣦⡴⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⠛⢁⣾⡿⠀⠀⠙⣿⣿⣿
⠰⠶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠐⠂⠀⠀⠀⠒⠺⠟⠛⠛⠛⠛⠿⢿⣶⡾⠿⠛⠁⣀⡇
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠴⠲⠶⡤⣀⡀⠀⢀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣭⣭⣭⣩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡝⠂⠉⠐⣀⠈⠁⠀⠁⠀⠀⠀⠂⠀⠀⠈⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿