Poultry Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⣿⠛⠳⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠙⠎⢀⣠⣋⣽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⢢⣿⢦⣩⣿⣱⣿⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠄⣠⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⣿⡟⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⢿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⡏⠁⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣧⣵⠂⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠕⠀⠀⠀⠀⢦⣿⣿⣿⣶⣦⡤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣽⣗⡛⢻⣧⠀⠽⠟⣻⡖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡰⠀⠀⢠⣤⣿⣿⡿⠯⣬⣟⣒⣿⡥⣿⣿⣎⢙⢿⣿⣿⣷⣌⣙⠛⣿⡋⠛⢫⣷⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⢀⣼⣿⣿⠟⠳⣶⣿⡯⣶⠞⠓⣶⠟⠉⡙⠃⠀⣈⡻⠝⢿⣭⡝⠒⢲⡿⠉⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣆⢠⣿⣿⠿⡙⠛⣷⠿⠷⣤⣴⠛⣿⠇⡾⢉⣡⣤⣤⠬⣴⣶⣾⣿⣿⣗⠋⢓⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⢸⣿⣏⣢⣈⣉⣿⣛⣶⣟⡿⣫⣕⣾⣟⣿⣟⠻⣦⣄⠆⠀⢿⡛⡙⠛⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡿⢛⠟⠟⢻⠿⣿⣯⣹⣷⣪⣷⣿⣷⣉⡈⠳⣄⣠⠾⠖⠒⠊⣁⡉⠀⠈⠃⠁⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠋⢩⠟⢻⠿⠶⠉⠀⠀⠀⠀⠂⠦⣀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣠⠀⢀⣀⣤⣂⠤⣄⣶⣤⡄⠀⠀⠀⢀⠀⢦⢀⣄⢸⣷⣶⣦⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣌⣾⣼⣝⣏⣑⡻⠻⣿⣿⡇⠀⠀⢻⡆⢀⡀⠙⢦⣽⣿⣧⣟⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⢙⣿⣿⣿⣿⣿⣷⣶⣿⣿⣷⡄⠀⠘⣿⣼⣧⠤⠀⢨⣿⣿⣿⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⡿⠿⢿⣿⣿⢿⣝⠻⣿⡄⢓⠜⣻⠿⣦⣀⣠⡞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠃⠀⠉⠙⠘⠋⠋⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡖⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⢋⡜⠁⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣷⣼⣧⣿⣧⣾⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⢤⣤⣀⠀⣅⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠀⣀⠀⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⢀⣀⣀⠉⠽⢿⣮⢶⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢨⠛⠀⠉⣠⣶⡤⠘⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⢻⣿⣷⣦⡙⣷⢳⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠈⠳⠤⣼⣿⠏⣧
⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠤⠴⠟⠒⠲⣿⣿⣮⢿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠕⡁⠀⠀⠀⠀⠀⠀⠸⢟⣠⠃
⠀⠀⠀⠀⠀⠀⣿⣡⡎⢁⣠⡶⠆⠀⠀⠀⠉⠛⠃⢹⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⢠⣯⣤⡴⠛⢱⣛⣯⣽⢿⣽⡦⡀⡄⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢫⠇⢀⢀⢀⠁⠀⠀⠀⠀⠠⡀⠀⢃
⠀⠀⠀⠀⢀⣿⣿⠿⣡⣦⣿⣿⣛⣟⣿⣷⣝⣦⣰⣦⢼⣄⣀⣀⣀⣀⣠⠤⠀⠘⠀⡁⠀⣳⠀⡈⣼⠘⢀⠀⢢⠀⠀⠁⠄⢸⡀
⠀⠀⠀⠀⢸⣿⡟⢠⣿⣿⣿⡵⠛⠋⠉⢀⣠⢯⣴⠾⠿⠶⠾⣿⣿⣁⣟⡗⠾⠀⡠⢫⡇⠿⠈⡁⠈⡄⠘⣇⠠⠣⠡⠀⠌⠠⡇
⠀⠀⠀⠀⢸⣿⡇⣾⣿⡿⠑⢋⣴⣿⢗⠔⣾⡿⢁⣠⣤⠀⠐⣾⠏⢁⠈⠷⠲⠚⠴⢿⡶⠃⡰⡇⣼⢃⠀⠸⡇⠇⠄⠀⡀⢰⠃
⠀⠀⠀⠀⠈⢿⣧⢿⣿⡇⢀⣿⢱⡏⡾⠃⣻⢃⡾⠋⠀⣠⣿⠏⢌⣲⣆⠐⠌⣻⡄⡰⣄⣠⠄⢐⣩⣬⢠⠆⡦⠰⠠⠁⢂⡞
⠀⠀⠀⠀⠀⠈⠻⡌⢿⠃⢸⣧⣿⣷⣧⣞⣿⡟⣠⠀⣾⢫⡳⣌⡀⢒⠆⡛⢛⣭⡬⠡⠶⢖⡶⣪⣝⣧⠏⣴⠃⠍⠄⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢸⣿⣿⣿⡿⣸⣿⡴⢁⡼⠁⣨⣽⠀⠥⡈⠳⠶⠖⣲⣭⠜⢣⣬⣥⡿⣉⣥⡾⢋⣁⠸⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡏⣿⣿⣿⡇⣿⣿⣣⣾⣷⣿⡿⠿⢓⣶⣦⣝⣻⣴⣶⣶⣿⣶⣶⡶⣿⣿⣿⡿⡞⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠣⣿⣿⣿⡇⣿⣿⣿⣿⡿⠃⠀⢀⠀⢒⢀⡉⠉⡰⡝⣻⢿⣟⣿⣿⣿⣿⣿⣆⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠻⡝⢩⣿⣿⣿⣿⣧⣶⣦⢈⡀⠲⢤⢠⣴⣇⣿⣟⣿⣿⣿⢿⣻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡛⠁⠘⣿⡹⠃⠙⢷⣶⣰⢴⣦⣤⣶⣾⣿⣿⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⢥⡭⠟⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢧⣀⠿⣶⡤⠴⣴⣦⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡮⠼⢮⣓⠦⠬⣥⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢸⡏⣹⠅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠋⠓⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠃⠄
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡤⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠌⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⢀⡇⠀⠀⠀⠀⠠⣴⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⢁⡠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠊⠀⠀⢀⠀⣰⠔⠊⠀⠀⠈⠒⠚⠺⠶⠵⠀⠀⢀⠀⠀⠀⠂⢁⠄⠋
⠀⠀⠀⠀⠀⠀⠓⠛⠄⠒⠄⠄⠓⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⡀⠔⠉⠸⠀⠐⠊⢁⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠉⡤⠔⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠡⡀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢬⡄⠔⠂⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠤⢀⡀⠀⠂⠀⣄⠐⠒⠛⢩⢶⣖⠆
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠭⠠⢉⡡⡨⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠠⠀⡠⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣯⡟⠋⠑⢠⣤⠄⠒⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠞⠭⢭⢿⠯⢑⡿⣿⡗⠺⢙⣔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠒⠛⢋⠥⠖⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣯⣽⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⢀⣠⣀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣶⡬⠉⠉⠀⢀⡙⠄
⠀⠀⠀⠀⠀⠀⠿⠅⢁⡀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠻⣿⣶⣾⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠘⠧⣄⢀⡄⠀⠂⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢫
⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⡶⠆⠀⠶⠆⠲⠀⠀⠛⡻⠷⠶⠶⠢⠤⠤⢀⣀⣀⣀⣀⣀⣀⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠹⣄⡀⠀⠄⠀⠀⢠⠸⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⢦⣤⣈⣙⠂⠀⢤⣠⣼⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣽
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣉⡙⠓⠀⠐⣿⠋⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⣓⠀⢠⣾⠂⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣀⠼⠗⠀⢶⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⡖⠀⠈⠳⡦⠀⠀⡠⠔⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠴⠚⢷⣖⠊⢀⣀⡀⠀⠀⠂⠀⠀⢀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⢀⡄⠠⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣣⡀⠀⠀⠀⡉⠉⢉⠈⠉⠉⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⠉⣮⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡅⡠⠊⢈⠁⠀⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡤⠂⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣤⡶⠙⣾⠻⠀⠀⠀⢀⠀⠀⠀⠔⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣾⣧⣀⡶⢠⣶⢀⠈⠀⠈⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣷⣿⣄⣤⣤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠃⠀⠀⠀⢀⣠⣿⣿⣦⣜⣿⣷⡄⣀⣀⠀⠀⠴⠀⡀⠀⠀⢠⡜⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⣀⣑⣢⡀⠛⣿⣤⣤⣤⣈⣉⣉⣠⣠⠿⠿⠛⠻⠷⣿⣿⣷⠶⠶⠾⠷⠄⠰⢾⣧⡰⡥⠀⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⠀⠼⠛⠛⠣⠛⢿⣿⡷⠄⡀⠀⢿⡦⠀⠀⢍⠉⠀⣄⠀⢰⣰⣴⣾⡷⢳⣷⣶⡭⢿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠒⠻⠃⠀⠉⠙⠛⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⠤⠤⠤⠤⠖⠚⠉⠉⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⡆⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠈⠋⠿⢠
⠀⠀⠘⢶⣌⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⢦⣤⣀⣀⣠⡴⢞⡿⠓⢦⣀⠴⠊⠉⠉⠉⠙⠦⣄⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠈⠙⠛⣻⠃⠀⠀⠀⠀⢀⣼⡿⣡⣾⣋⠁⠁⠉⠙⠛⠉⢠⣯⣶⣶⣶⢿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠽⢦⡀
⠀⠀⠀⠀⠠⠄⠊⢁⠀⠀⠀⠀⠀⣼⣿⢁⣯⣶⣾⡅⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠀⠀⢀⠤⣶⣶⢶⡦⢤⡀⠀⠀⠈⡍⠙⣯⠐⠂⢄⡀
⠀⠈⠳⠶⠶⠶⠞⣿⠀⠀⠀⠀⢰⣿⡇⢀⠙⠯⠟⣡⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⢀⣀⡀⢷⣭⣪⠅⠀⠀⡀⢀⡠⠴⠚⠓⠒⠒⢤⣑⡄
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢀⣿⣿⡇⠸⡄⠐⠚⠁⠀⠀⠈⢿⣿⡯⠭⠬⣭⣿⣷⡀⠀⠈⠙⠁⠀⠀⠀⣌⣠⠀⡀⠳⣾⣯⡀⠀⠀⠈⠁
⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⣾⣿⣿⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣤⡬⢽⢿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠙⣿⣷
⣀⣠⣴⣿⣿⣤⣤⣤⣴⣾⣿⣿⣿⠛⠀⡌⠁⡀⠀⠀⠀⢀⣠⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢸⡏
⠈⠙⠛⠿⠿⠿⢿⣿⠿⠿⠟⠋⠁⡀⣰⢿⣆⠙⠿⢿⣿⣿⠇⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⢸⣿⡆⠀⠀⠈⠿⠀⠘⡆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠸⠀⠀⠈⢧⣀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡿⢹⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⠉⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡼⠁⣼⣿⣦⣤⣼⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⡇⠀⠀⢀⠈⢧⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⢰⡟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⠀⡇⠀⣄⠈⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡟⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢷⡀⠀⢠⠃⠀⠙⠀⠀⢦⠀⠘⢦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠊⢠⡿⢫⣿⠟⣿⣷⣤⣀⣀⣠⣾⣿⣿⣿⣿⣿⣏⠳⠤⠊⠀⠀⠀⠀⡄⠈⠁⠀⠀⠳⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢠⠟⠁⣸⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠠⣄⠀⠀⠀⠀⠹
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠰⠋⠀⣰⠋⠀⣸⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠻⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⡄⠀⠘⣆
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢠⠟⢀⣿⠻⣿⠿⣿⡏⢻⡿⠸⡟⠘⣇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣸⠃⢰⠏⠀⣿⠃⢸⡇⠀⡇⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠌⠀⠀⡟⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣶⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢩⣤⠊⠉⠟⠯⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡶⣤⠉⠛⢖⢺⠿⣶⡌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠙⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠁⣺⣆⡸⠀⢈⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠛⠳⠏⠁⠀⠁⠉⡔⠀⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠡⠠⠘⠂⢀⠀⡄⠀⡠⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠂⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢚⡲⢦⣤⠁⠂⢠⡁⠀⠀⠿⡄⢠⠀⢤⠀⠀⠀⠀⠀⢠⣀⢀⠀⡃⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠠⢉⡤⠋⣀⠔⠀⠂⠐⢀⠔⡹⡊⠠⠔⠀⠀⠀⠆⠀⠙⠛⠓⢿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠠⠤⢄⠹⠄⣘⣳⠀⡄⢁⠈⡠⢁⠚⡠⠈⡁⠀⠀⠠⠀⠀⠀⠀⠘⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⢈⡀⣉⣈⡃⠊⠄⠹⠐⡡⠀⠂⡖⢠⠃⠚⡀⠁⡠⡆⣤⠌⣢⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⢋⢯⠔⣊⣡⠐⢣⠄⠼⢡⠀⠨⠐⠀⠐⠋⣛⣖⣀⠀⣲⣯⣥⠴⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢙⣛⣃⣩⣍⡠⠠⠄⢀⠀⠀⠂⠀⠀⠐⠉⢿⣿⣣⣿⢱⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡗⠐⢂⣿⣥⡈⠤⢶⢧⣈⠐⠁⠀⠀⠀⠀⢀⠀⠙⢂⠛⢉⡍⢀⢼⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡓⠺⠭⠭⣢⠁⣀⠄⠐⠀⠀⡀⠀⠀⢀⠀⡅⠀⢚⠖⣩⢃⡴⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣴⡟⠃⣈⣠⢄⠀⠀⠀⠀⠀⠀⠌⠀⠰⠀⠡⠋⠡⣁⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣧⣴⣭⣴⣄⡀⣠⣛⣨⣌⣂⣥⠋⣠⣦⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⢿⡭⠟⢉⣱⣲⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠛⠛⣧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⣀⡤⡆⣰⢟⠩⠀⠖⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠈⠀⠁⠀⢴⣩⠴⠶⡀⠀⣸⣧⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠐⠣⠤⠖⢛⠛⡛⢠⠏⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣿⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⡉⢉⣛⣻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠰⡅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢾⣗⣬⣧⣤⣢⣝⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠠⡐⠛⣎⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⠀⠉⢿⣥⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠱⠳⢹⡄⠜⢆⠀⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣢⠀⢀⢣⡡⠈⢆⢁⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠠⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡈⢧⠈⢆⢳⠠⣸⠦⠷⠾⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⢆⢈⠁⢀⠀⠀⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣵⢄⠳⣿⣤⡂⣑⣒⣠⢖⠏⣡⢵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⠤⡀⡛⣚⠉⣙⠟⢡⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⢴⠀⣯⠐⠒⢚⣧⠬⢽⣚⢂⣰⣿⠀⠂⣤⣤⠠⣤⡄⠰⣧⣀⡿⡈⢄⣭⠹⢻⠉⠺⠦⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡟⠛⢲⣤⢞⡒⣚⡿⢳⣀⡗⠶⡧⠬⣙⣢⡚⠠⢾⡥⠘⡯⣤⣉⣼⣊⣷⢈⠖⠚⣋⠶⣤⠎⢰⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⢾⡏⣤⠺⠶⢮⠀⠴⠷⠶⢛⡖⣀⠿⣜⣁⣰⣉⣿⠄⠐⣾⢧⣎⣩⢴⠰⠾⡴⣄⣀⣱⣀⢳⠖⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣁⡍⣷⠐⣇⣠⡶⣶⢒⡉⠛⡇⠚⠫⠀⡴⣾⠛⣲⣾⣇⡼⢯⠘⢿⣶⣤⡄⡈⢋⢁⣄⣭⡴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆⣴⣿⡀⠘⡁⠾⣩⠙⡙⢋⠿⣾⣧⣴⣟⣩⠴⣯⢦⡀⡻⠲⣵⣏⣠⠃⡔⢻⡧⡻⣧⢧⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⣡⣾⢻⣇⡹⣎⣱⡹⠾⣤⢮⣄⡉⢳⠞⢛⣁⡤⢶⡆⣄⢴⡅⠹⢣⣆⡜⢱⡠⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣻⣎⣻⣤⣷⣼⣌⣿⣱⢿⣿⡿⢛⣻⣿⢋⢻⣤⣾⢈⡿⢐⣯⣀⡁⢛⣴⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⡯⣽⣉⣭⠍⢛⠿⡿⣿⠺⢉⣆⣭⣩⣤⣩⣍⣨⣷⣶⣿⣏⣤⣿⣯⡬⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣻⣺⣿⣥⡶⢖⣯⡤⢾⣿⠌⠿⣍⣛⠧⠴⠾⠷⣻⡿⢿⣿⡿⣯⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣷⣶⣿⣭⣒⣫⡽⣉⣿⠟⣿⣿⣿⣾⣛⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣦⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡟⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡽⣳⣿⣋⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠉⣻⣿⣽⣿⣓⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣦⣼⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⣤⠦⠉⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⢀⠎⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡮⢭⣤⡄⠀⠸⣨⠇⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢀⠊⢠⢀⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣧⠏⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡏⣰⠃⡜⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠠⠀⠜⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠀⠰⠦⠂⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⡴⣃⣀⠀⠠⠴⠖⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⢠⣀⠀⠀⠀⠀⠀⠙⠢⢤⣤⣴⠖⠋⠁⠉⠉⠁⠀⠀⠀⡀⠀⢄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⢠⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⠠⢀⠀⠀⠢⣀⠀⠈⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡀⠀⠁⠐⠆⠀⠀⠀⠀⠐⠀⢰⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠒⠀⠀⢀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠀⣀⡁⠀⠈⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠉⠈⠀⠀⠀⠀⠀⠈⠈⠁⠉⠀⠉⠂⠉⠊⠉⠀⠁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣭⠀⠲⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⡠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠓⠂⣤⣀⠀⠀⠀⠐⠀⠀⠀⣄⣤⡦⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠈⢑⣵⡾⠣⣤⡀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢛⡟⠀⠀⠀⠀⠉⠓⠲⠤⡴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢐⣠⠳⠙⣇⡤⠄⠄⠀⠒⠊⠥⠤⠍⣉⢠⣿⡄⠨⡷⠆⠰⠭⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠾⠟⢁⣟⠉⠛⠂⠉⠁⠀⠀⠀⠀⡀⠩⢄⣈⠸⠥⠒⠶⣈⠉⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢒⣩⡤⠄⣀⠔⠀⢀⣀⣀⠀⠀⠲⠖⠒⠊⠑⠂⠈⠀⠀⢂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⣀⡀
⠀⠀⠀⢀⡖⢚⣭⠽⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡏⠀⣴⠀⣿⢹⡤⣄
⠀⠀⠀⢸⡔⠁⡤⢰⡄⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠈⠀⢁⡏⠀⡾⢲⡄
⠀⠀⠠⠯⣼⠠⡀⢸⠇⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠤⠀⠀⠀⡀⠼⠀⠘⠀⢊⡹⡀
⠀⠀⠀⠀⢳⣀⣽⠀⠀⠀⠀⠀⠀⢲⣤⣌⡳⡄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠘⢆⠀⢠⡶⠛⠀⠷⡄
⠀⠀⠀⠀⠀⠈⡟⠀⢀⠀⡀⣀⠀⢠⠀⠋⠛⠈⢦⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⣤⠖⣸⠁
⠀⠀⠀⠀⠀⢰⡇⠀⠞⡄⠧⠙⠂⠀⢀⡠⠀⠀⠉⠉⣁⠀⠘⠣⠤⣄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣨⠇
⠀⠀⠀⠀⠀⢸⠀⠀⣤⠇⠰⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠤⣀⠊⠄⠀⢀⡆⠀⠀⠀⠛
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⣴⠂⠀⡘⠀⢒⣈⣉⢉⡽⠀⠐⢀⠆⢀⠞
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⠀⢀⢖⡅⠀⠐⢁⣠⡭⠶⠒⠈⠰⢦⣄⣀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⢠⣈⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠠⠠⠤⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠇⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢀⣀⣀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣽⣛⣫⠽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⢀⡴⣯⣩⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⡹⠋⠉⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡼⠁⠀⠀⠀⠀⠸⣳⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⣤⡴⠃⢇⣀⠀⢠⡶⠒⣒⣻⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠽⠓⠂⠉⠉⠉⠉⠀⠈⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⣭⡽⢿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣶⣿⣿⠿⣥⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠶⠚⢓⣾⣿⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⣿⣿⣿⣾⡃⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣴⡚⣛⣳⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣤⣃⢠⡐⣓⣠⣤⣤⣄⣀⣀⣀⣤⠆⢠⢤⣄⡈⡀⠴⠚⠁⣸⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⠕⠚⠁⣐⠆⣨⠉⢙⡿⢿⡉⠀⠀⠀⡙⠰⢏⣱⡄⠛⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⡿⡟⢫⣰⢠⣌⡇⡁⡈⢶⣌⡀⠈⠧⣅⣀⣰⡟⣙⣯⠀⠀⠀⠀⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⡧⣞⣻⢼⣻⡼⢧⢿⡀⣺⣿⣷⣶⣞⡣⢬⡏⠠⠾⠤⢤⣀⣀⣲⣾⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣳⣯⣷⢺⢬⣗⣻⣚⣣⠼⠛⠻⣿⣯⣥⣀⣀⣘⣺⣴⣶⣶⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣷⠟⣿⠖⠼⡭⢏⣉⡶⢾⡉⣁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣮⣿⣳⡞⣯⣽⠛⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⢻⣤⣿⡻⣿⢿⠷⣽⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⢿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣠⣾⣦⣤⣀⣄⣀⣠⣄⣠⢂⡜⣸⣎⣴⣮⣤⣄⠀⠀⢀⣠⣀⢀⡀⢀
⠀⠀⠀⠀⠀⡰⣰⣶⣢⣶⡴⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣛⣻⣿⠟⢡⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⣖⣩⣿⣶⣤⣶⣦⠆
⠀⠀⠀⠀⠀⠃⠃⠃⠋⠛⠙⠛⠋⠙⠽⠿⠫⠾⠿⡿⢿⠿⢯⣿⢗⣢⣿⣿⠟⠉⠘⡿⠫⠽⠻⠾⠛⠉⠉⠉⠈⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣿⣿⣷⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠁⢀⣬⡙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⢟⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣯⠛⣾⣇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⢰⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠻⣿⣧⠈⢿⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠸⢿⡯⠟⠁
⠀⠀⠀⠀⠀⠀⢰⡶⢼⣿⣧⡙⢿⣧⡘⣿⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⡀⠀⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣦⡈⢿⣿⣆⠹⣷⣜⣿⡾⠦⠀⠀⠀⠀⠄⠀⠐⠀⠠⠄⠀⢈⣀⠀⡹⣷⣶⣿⣿
⠀⠀⠀⠀⠀⢠⣤⣬⣿⣿⣦⡙⠻⣷⣮⣿⡉⠙⠀⡄⠀⠀⠠⠀⠀⣑⠀⣀⣿⡶⠋⡿⣿⣵⣧⣏⣿⡿
⠀⠀⠀⠀⠀⠘⢷⣦⣈⠛⠻⣿⣶⣝⣧⡀⠙⢖⠀⠈⠁⠀⢺⣀⣀⣨⣟⣁⡽⠳⣿⣠⡎⡹⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⣦⣄⡙⠻⣏⠉⠳⣖⠛⠲⠶⠤⢽⡋⠁⣚⣿⣯⣤⠞⢙⡿⣿⣷⣾⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣝⡛⠿⢿⣷⣶⣽⣶⣽⠷⠦⠤⣬⣭⣽⣏⣁⣀⡽⣿⣶⣏⣤⠋⡸⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣬⣭⣝⣛⣻⣷⣶⣤⣄⣐⣒⣿⣿⣁⣠⠔⠉⣻⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡿⢼⣧⣭⣇⣨⡿⣶⡏⢉⡽⣶⣿⣤⣞⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣾⣨⣿⣤⣾⠛⣻⣧⣾⣉⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣿⣴⣧⣴⣿⢚⣿⣤⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⠿⠾⢿⣫⣿⠷⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝⠀⠐⠒⣶⣄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠒⠒⠒⠂⠚⠓⣀⡨⠋⠀⢐⢉⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠒⠒⠂⠐⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠂⠉⠉⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠐⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⡀⠤⠄⠒⢒⡦⠀⠀⠈⠂⠤⢀⣀⡀⢀⣀⡀⠠⠔⠊⠁⠀⠀⠐⢍⡁⠒⠢⠤⢈⠐⠠⣀
⠀⠀⠀⢀⠔⠋⠀⠀⠀⢎⣁⣀⠠⠔⠊⠁⠀⣀⠔⡹⠀⠀⠀⠀⢀⠒⡄⠀⠀⠀⠐⡑⠢⡀⠀⠀⠈⠑⠢⠄⣀⣨⠂⠀⠑⢄
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡔⠁⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠈⢆⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⢄⡈⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠈⢗⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠤⠻
⠀⠀⠀⠀⠈⢆⠀⠀⠈⠉⠙⠢⠤⠴⢄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⢀⣀⠠⠂⠀⠈⠁⡀⠀⢠⠃
⠀⠀⠀⠀⠀⠈⢆⠘⡀⠀⢠⠀⠀⠀⡀⠀⠁⠀⠉⠒⠒⠉⠉⠐⠒⠊⠉⠑⠒⠊⠁⠈⠉⠉⠀⠀⠀⠀⢰⠀⠀⠀⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⣀⣆⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠒⠠⠤⠦⠀⠀⠀⣄⠀⣀⢀⣸⠀⠀⠀⠠⠥⠤⠐⠚⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⢋⡉⠙⠛⠦⠤⠦⠤⠤⠞⠙⠛⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢀⡴⠚⡽⠁⠀⠀⢠⠿⡄⠀⠀⠀⠻⡓⢦⡀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⢀⣀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠉⠚⠀⠀⠀⠀⠯⠤⠽⠀⠀⠀⠀⠳⠶⠛⠀⠀⠀⠀⠀⠀⠈⠳⣤⠴⠶⠤⣄
⠀⠀⣠⠖⠋⠉⠉⠙⠂⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⢳⡀
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠦⣀⣠⠤⠤⠤⠤⠖⠉⠉⠉⠉⠑⠢⣄⠀⣀⠤⠖⠚⠛⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠈⠓⢦⡤⢤⡄⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⢤⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⠈⠑⠂⠤⠖⡖⠉⠑⠢⠴⠚⡏⠑⠂⠁⠀⠉⡟⠋⠁⠈⠑⢢⠤⠤⠤⠤⠒⡟⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠇⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⠦⠤⠼⠦⠤⠤⠤⠤⠷⠤⠤⠤⠤⠴⠷⠴⠶⠦⠤⠴⠧⠤⠤⠤⠤⠾⠤⠤⠤⠤⠾⠴⠦⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣷⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣯⡙⠻⣿⣿⣦
⢀⣀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠉⠉⠿⠿⠙⡿⢀
⠈⠻⠥⣼⣿⣿⣿⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠠⣶⣦⣰⣿⡋⠉
⢺⣛⣺⣿⣿⣿⡻⣿⣷⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷
⢠⠟⢿⣿⣿⣿⣿⠦⣽⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡄⠀⠀⠀⠀⢠⠀⢿⣿⣿⠟
⣨⠽⣿⣿⣿⣿⠶⠾⠛⠦⠁⠹⣦⡀⠀⣀⣤⣤⣤⣤⣤⠤⠶⠶⠟⣿⣿⣤⡔⠀⠀⣠⢿⡄⠈⢿
⣙⡾⠟⢛⣿⠷⠚⠉⣀⣀⠈⠛⢿⡿⠟⠛⠙⠋⠀⠁⠀⠀⠀⠀⠀⠉⠛⠘⣷⣷⣴⡟⠸⣧⠀⢸
⠙⢶⣵⣿⣿⡶⣞⣿⠏⠉⢠⠖⠀⢀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠏⠁⠀⣿⣷⡿
⣟⠿⣿⣿⣿⣿⣿⣵⣦⣠⡏⠀⢰⠃⠀⢠⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠈⢙⣛⣻⣿⣻⣽⣿⡟⣼⠀⡴⣿⠀⠀⢸⠁⠀⡏⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠈⠿⣽⣿⣿⣿⣿⡼⡟⣼⡇⣼⣢⣼⣿⡤⠀⣇⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢿⣿⣿⣿⣇⣿⣿⣿⣿⣿⠟⢻⣶⣦⣿⣴⣿⡤⢀⡀⠀⠀⠀⠀⢀⣠⠆⢀⣴⠇
⠀⠀⠀⠀⠀⣰⢻⣿⣽⡿⢿⣿⣿⣶⣟⣭⣥⢶⣻⣿⣛⠛⠋⢓⣦⠤⠶⣶⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⢸⣼⣿⣩⠻⠿⠏⠉⠉⣹⣥⡤⠸⠿⠟⣿⣿⣿⣿⣿⣺⣿⣋⣽⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣧⣤⣄⣀⣨⣿⠀⠀⠀⠰⢛⡁⠈⡽⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡻⣶⣤⣾⡶⣏⢀⣰⣄⣩⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣷⣶⣄⣭⣻⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠉⠉⠀⠉⠉⠻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢿⡄⠀⠀⠀⠀⠀⠀⠈⢯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠞⠁⠈⢹⣤⣿⣿⣿⣿⣶⣂⣤⢴⠶⠿⠾⣿⣿⣟⠛⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡒⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢼⣿⣤⣾⡅⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢛⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⡛⣿⣿⣿⣤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣦⡼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣿⡏⣳⣿⣿⣾⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣇⠹⠻
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣮⣝⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣿⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣀⣤⣿⣟⣿⢟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠞⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⣦⣶⡹⣧
⠀⠀⠀⠀⢲⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⢿⣏⣻⣯⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⡏⠻⠆⢹⡄
⠀⠀⠀⠀⠘⢿⡿⠻⢿⣿⣿⣿⣿⡿⢷⣾⣿⣿⢿⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣱⣬⡀⠀⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⢷⣾⣿⣯⣿⣾⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣻⣽⠟⡿⢿⣷⣿⠟⠃⢀⠐⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠷⣾⣿⣿⣿⣿⣿⣾⣿⡇⢀⣀⣀⣤⠂⣠⡆⣰⡿⠻⣿⠿⠾⠿⢷⣦⣿⣤⣤⡼⣄⠐⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣿⣿⣿⣿⣯⢿⣧⣽⣿⣿⠛⣽⢟⣼⡿⣶⣿⣦⣾⠟⠁⠀⠄⠀⢀⠙⢿⣿⡿⣿⠀⠟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣿⣿⣿⣯⡙⣷⣿⣿⣿⣿⣾⣇⣿⣿⣤⣯⣄⠉⠀⠀⠀⠀⠀⡀⠈⡁⠀⢿⡟⠀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣇⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⢀⠀⠀⠀⠰⠀⠋⠰⠇⠀⣸⡷⠈⠿⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠆⡴⠀⠋⡀⠸⠃⡄⢰⡇⢀⠀⣰⣟⡼⠟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣏⠙⣠⠀⣾⠇⢠⡀⠃⠀⠀⣨⣾⡟⢻⣷⠿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣇⣈⣀⣿⣧⣴⡾⣿⣿⣯⣾⣛⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠿⣿⡆⠀⠩⣙⣟⢿⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣟⣿⣿⣿⣽⣆⣾⠟⢂⣴⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣧⣼⢋⣴⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡇⢠⠈⠻⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⢀⠀⢀⣀⢠⣶⣤⣤⣤⣤⡥⠘⠷⠊⠓⣄⡠⣤⣤⣀⣶⣄⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠑⠋⠑⠖⢠⣬⡿⢻⣷⣤⢠⣌⣙⠛⠛⢿⣶⣭⣿⣿⣿⢿⣷⡟⠉⠣⠦⠘⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠐⠒⠛⢿⣦⠟⢿⡿⠻⠿⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠁⠀⢀⣀⣼⣟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠓⠦⠉⠉⠉⠉⠉⠉⠉⠛⠻⣶⡄
⠀⢀⣠⣤⣤⣤⣤⡶⠶⠦⣶⠾⠛⠲⠶⣶⡶⠶⠶⣶⡶⠲⠶⠿⢛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⢀⣿⠋⠀⠀⠀⠈⠙⠂⠀⠀⠁⠀⢦⡀⠈⠑⠀⠀⠀⠁⠀⣴⠾⠋⠙⠦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠄⣠⡾⠛⠳⣦
⠈⣿⣄⠀⠀⠀⡀⠀⠐⣤⣠⣄⣤⣤⡷⠴⠦⠶⠶⠾⠶⢾⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠁⠀⠀⠀⢹⣧
⠀⠙⢿⣷⣦⠀⠨⣾⡿⣯⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡀
⠀⠀⠀⠉⠻⣧⠀⠸⣿⣳⣿⣽⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⠀⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠟⠉⠈⠙⠳⢦⣄⣀
⠀⠀⠀⠀⠀⢹⣧⠀⠸⣷⢿⣿⣿⡿⣿⣷⣶⣾⣿⣿⣿⣽⣿⣻⠶⣦⣤⣬⣍⣉⣙⣛⣛⣛⣛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢻⡆⠀⢹⣧⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⣾⣭⣽⣭⣭⣭⣭⣿⣛⣟⣻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⢶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢿⡼⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣾⣷⣿⣷⣯⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⣿⢿⣾⣿⣿⣯⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣏⢿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠛⠻⢿⣿⣿⣽⣟⣿⡿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⢤⣤⣌⣁⣈⠉⠙⠛⠛⠛⠿⣷⣾⣯⣿⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⣤⣤⣀⣉⡉⠉⠉⠛⠛⠻⠷⣶⣶⣿⣿⣿⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⢦⣤⣤⣀⣀⡀⠈⠉⠛⠛⠿⠶⣾⣿⣿⣿⣽⣟⣿⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠶⠶⢦⣤⣤⣉⣀⠈⠉⠙⠛⠛⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠶⠶⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⠉⢦⠀⠁⢢⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⢬⠹⠟⠸⠷⠦⠆⠀⠀⠀⠀⠀⠀⢰⠀⣔⣠⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⡉⢠⣤⡄⢲⡆⠀⠀⠀⠀⠀⠀⠀⣻⣹⡿⡌⢉⢸⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡓⠀⣎⣛⢻⣄⣴⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣷⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠺⣿⣹⣤⠙⠯⢷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⣿⣛⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢢⣤⣾⣿⡅⡼⣾⣷⣶⢶⡶⣤⣤⣤⣾⣿⣿⣷⣿⣿⣾⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢟⢺⣤⡟⠃⠁⢠⣾⣿⣗⠺⠆⠘⢙⠯⣹⠧⢮⠫⣿⣭⣿⣥⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡑⢿⣟⣷⣠⣍⣟⠼⣻⣧⠷⠲⠅⢈⡁⠬⠨⣚⡃⢌⣽⣾⡿⣟⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⣽⡎⠻⠯⢬⡞⠀⢙⣣⡼⣇⢴⣆⡹⣏⠣⢈⠘⣏⢵⠊⣭⣮⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⠌⣰⣛⡺⡎⣦⢒⡰⢃⡝⠤⡚⣈⢠⣩⢤⡞⣹⣧⣾⣵⣾⡟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡟⣽⣣⣧⠡⢳⡄⣒⣹⠈⡂⠁⣄⢻⣶⣻⣾⣿⣹⣿⠿⠢⠍⠿⣷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣵⡿⠰⢊⣀⠱⡖⣎⠉⠩⠽⢧⣾⣿⣿⣿⣾⠿⠛⠁⠀⠀⡌⠀⡻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠈⣐⡒⠜⡘⢿⣶⡈⡋⠙⢍⠀⢠⠀⠀⠀⠐⠁⢀⣰⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⢤⣤⣋⣠⡈⠥⠃⠀⠠⠔⠚⡗⢈⣭⣐⡘⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣾⣷⣦⣦⠨⣿⠻⢤⡤⠀⠘⠂⣐⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⠿⢿⣿⣶⣔⣣⣴⡶⢦⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣀⡌⠀⠀⠉⠻⢿⣿⣵⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠈⠀⠀⠀⠀⠀⠀⠨⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⡀⡠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣅⡤⣀⣴⠟⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢉⡙⢶⢏⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡕⠁⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡘⠠⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣏⠀⠀⠈⢘⣦⣤⣀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⢉⡽⠋⠁⠀⠀⡿⠋⠙⢯⠀⠀⠀⠉⠳⣄⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡰⣻⠋⠀⠀⢠⠏⠀⠀⠀⠀⡼⠁⠀⠀⠈⡇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠻⡷⡄
⠀⠀⠀⠀⠀⠀⡞⢱⠃⠀⠀⠀⡏⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢱⣹⡀
⠀⠀⠀⠀⠀⡼⠀⡞⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣇⣇
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⢹
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣼
⠀⠀⠀⠀⣴⣿⣿⣧⡀⢀⣤⣾⣷⣶⣄⢀⠀⣈⣷⣶⣶⣦⣄⠀⣿⣠⣴⣶⣶⣦⣀⣹⡀⣀⣴⣾⣿⡄
⠀⠀⠀⠀⢻⡃⠈⠻⠿⠿⠟⠁⠀⠙⠿⢿⠿⠿⠋⠀⠀⠙⠿⣿⡿⠟⠋⠀⠈⠙⠿⢿⣿⠿⠋⠁⠹⡿
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠉⠐⠢⢄
⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⡠⠊⠁⠀⠀⠀⠀⠉⠢⣄⠀⡁⠀⠀⠀⠐⢀⠀⠀⠀⠀⠐⠂⠈⠉⠉⠉⠐⠢⢄
⡘⠀⠀⠀⠀⣠⡀⠀⠀⠀⠈⠰⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠠⠊⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠁
⠡⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡇⠀⠀⢀⡆
⠀⠑⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡇⠀⠀⠌
⠀⠀⠀⠑⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠡⠀⠰
⠀⡔⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⢬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠑⢞⡡⠇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠀⣠⠔⠶
⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐⠄⠊⠀⠀⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠤⠠⢀⡤⢀⠀⣠⡀⢀⣀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠆⠀⠀⢠⡄⠀⠉⠈⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠄⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⠄⠚⢄⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢐⡀⠀⠈⢑⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠂⠀⠀⡐⠀⠀⢀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁⠢⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡐⠈⠀⡄⠀⠈⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⠄⡠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠜⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠀⠈⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠌⠀⠀⠔⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠐⠀⠰⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠤⠄⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠰⠶⠦⠤⠤⢤⡤⢤⠄⠠⠤⡴⠶⠄⠐⠒⠒⠚⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠔⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠈⡄⢦⠀⠀⠀⠠⡤⠤⠴⠛⢢⠀⠠⡆⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠈⢤⡑⠠⠤⠤⠜⠃⡀⠀⠀⠱⠡⢦⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠉⠀⠑⠒⠊⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠉⠘⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠙⢋⠔⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣲⡀⡀⠀⣴⠁⢰⣃⡈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⡀⠀⠀⠀⢀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣴⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣪⡵⣿⣺⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢗⣈⣰⣿⣿⣷⣥⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠉⠀⢈⠓⣮⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠛⣈⣁⠀⡤⢼⣍⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⢠⢂⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢿⢠⠀⡤⢀⠈⠀⢉⠾⠉⢿⠁
⠀⠀⠀⠀⠀⠀⡠⠌⡢⣠⡥⢟⠎⢊⡱⠰⠎⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⢋⣰⠂⠒⣀⣦⡝⡣⡂⡤⠞⡀
⠀⠀⠀⠀⠀⣒⣊⣱⡶⢛⡊⣠⠴⡟⢔⡂⠡⠊⠁⠙⢉⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⡱⢯⣍⠁⣈⠀⠤⠿⡎⠀⡥⡔⢧⡇
⠀⠀⠀⠀⠀⣍⢤⣿⠥⠐⡾⢓⢍⠞⠃⠀⠠⠀⠀⠀⢀⠁⠡⠿⠛⢷⠲⠶⣦⣤⢤⣦⢾⡿⢿⢳⣾⡑⠒⢌⡁⠤⢠⠶⠀⢘⣀⣂⣴
⠀⠀⠀⠀⠀⢠⠘⡗⢂⣹⣋⣅⡤⢉⠄⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣤⠀⣌⡁⢁⣘⣩⢟⢻⡿⢘⡉⠆⠡⢞⡁⢉⣨⠿⡏⢁⣸
⠀⠀⠀⠀⠀⠀⠆⠘⢉⣻⡄⣤⡠⠀⠀⠀⠀⠄⠂⠀⠀⠠⢀⡀⠈⠀⠀⢈⡷⠋⠅⠠⠁⣭⠢⡀⢈⡀⠀⡀⠒⢀⡁⢞⢳⠺⠛⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⡝⢳⣖⢀⡀⠐⠀⠀⠀⢠⣀⡈⠀⠘⠁⣈⡟⠁⠐⠀⠄⠀⠀⠁⠄⠈⡈⠓⠀⢁⢢⡐⢌⣡⣿⡀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣞⡼⣿⣡⠊⠀⠀⠦⣀⡀⠄⢀⠀⢼⣿⠾⡃⠁⣔⢂⠃⡀⠀⠀⠀⠠⣘⣷⣿⠀⢙⣠⠦⡌⢶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⣋⠐⠒⠀⠀⠂⢄⠸⠰⠠⡾⠁⣿⢠⢋⠜⠑⠧⢖⣂⣀⣲⡶⣟⡣⠴⠆⠘⣧⢔⢁⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⢸⡧⡍⢠⡤⠀⣷⣂⡶⢬⡳⣬⡻⣷⣾⣿⣷⣾⠿⢻⠯⢅⣮⠔⠣⢀⠀⡀⢀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣶⠟⢷⡶⠿⠙⢧⡜⠛⠋⣹⣭⠉⢡⠟⡤⡳⢫⣔⠼⣧⣠⡉⠊⢔⣄⣈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢶⡆⠄⢓⢁⠘⠐⠆⠀⡁⠈⡁⢀⡘⢆⢹⡲⠚⢱⡙⢎⠃⣼⣫⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⣤⡈⠠⠚⠀⢤⠝⠀⠀⢚⠡⠄⠘⠚⣴⢯⣾⣶⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣯⣀⠒⠑⢎⡱⠄⠆⠺⡽⣶⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣯⠐⠛⠶⢏⢄⠳⣐⣶⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣓⠘⣋⢒⣳⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠟⠉⠀⢻⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣀⣀⡀⠀⠀⠀⠀⠰⠋⠹⠲⠿⠷⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⣿⠿⠖⣶⠶⠄⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠏⢇⢠⠖⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⠇⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⡞⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠀⠀⠈⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠨⠀⡄⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠀⠂⠀⢀⠀⢰⠀⢠⠀⠀⣼⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠶⠤⢤⣀⣀⣤⠤⠴⠖⠂⠐⠋⠱⣩⣿⣦⠀⠰⠸⡄⠀⠃⠀⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡤⠞⠀⢇⠿⢿⡁⠀⠘⠁⠙⢆⠀⠀⢰⣾⡧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⠖⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢤⣀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠩⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠔⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠾⠃⠀⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠐⠀⠀⠀⢀⢈⡓⢦⡀⠀⠀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠀⠀⣣⠀⠀⠐⠶⠖⠀⣂⠀⢘⠳⠲⣄⠙⢄⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⢠⠎⠀⡼⠋⠀⠁⠀⠀⣞⡀⠀⠉⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠰⡇⠀⠀⠀⠀⠼⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠹⡆⡞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠀⠀⠀⠀⠀⢐⡅⣠⣠⠆⠀⠀⠈⢇⠀⠀⠀⠀⠀⣀⠴⠻⠈⠃⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠈⠀⡄⠈⠑⠦⠶⠖⠋⡅⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠠⠀⠂⠀⣠⢀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠴⠂⠀⠀⠀⢀⡄⠀⠐⠣⠋⢀⣾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠄⠀⠀⠀⠈⠁⠀⡴⠀⢀⠎⠀⠀⣠⠟⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠔⠂⠸⠤⠀⠈⠰⠖⠋⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠀⠀⠀⣠⠎

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⡴⣾⣽⣝⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣠⣤⣶⣿⣿⣤⡀⠀⢠⣶⣤⣸⣼⣿⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠐⠷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣀⣴⣶
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣀⣴⣾⣏⣹⣿⣿
⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣀⣤⠶⣯⣽⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⠉⠉⠻⠿⠟⠀⢸⣏⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠟⠃⠸⠿⠛⠉⠀⠀⠀⢿⢟⠀⠀⣡⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣄⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢸⣿⣿⣿⣿⣿⠿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⢜⣿⣿⠟⠋⠁⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡶⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⡛⢏⣤⣎⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠱⣿⣦⣿⣴⣿⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠁⠉⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡛⡛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⠁⠀⠀⠠⣴⣿⣷⣤⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⠿⢿⡏⠙⡷⢾⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠂⠀⢠⣼⣿⡿⠯⣽⣛⣲⡯⣽⣿⣛⣿⣿⣿⣭⣙⠛⣿⠚⢓⣿⢾
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⢠⣾⣿⣟⡛⣷⡿⠷⢞⣛⡾⢋⣉⠁⠀⡬⠛⠻⣿⣝⢛⣿⢃⣾
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⣿⡿⠃⠈⣹⣏⣛⣶⣯⠟⣹⣧⣶⣾⡛⢛⠛⢻⢿⣿⣏⣬⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣰⣿⣷⡶⣶⣿⡶⢿⣫⣿⣽⣟⠟⢿⡈⢙⣮⠤⠼⠃⠀⠘⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠈⠀⠀⠀⠈⠛⠛⣻⠿⡿⠯⠉⠀⠉⠁⠠⢄⡀⠈⠁⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⡠⡀⣀⣠⣄⡤⣤⣦⣄⡀⠀⢀⡀⢠⢦⡄⢿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣷⣷⣯⣬⣝⢹⣿⣇⠀⠈⣷⣠⡄⠈⠻⣿⣷⡧⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣼⣿⣿⣿⣿⣿⣿⡿⣿⣆⠀⡘⣿⣿⡁⠀⢻⡿⠿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⣷⣿⠆⠙⠒⠇⠷⠊⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⡽⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢉⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣦⣤⣄⣈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⢁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⢀⡼⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⣄
⠀⠀⠀⠀⣠⡟⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⣧⡀
⠀⠀⠀⣰⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢷⡀
⠀⠀⣰⠏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠠⢄⡐⠒⠒⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢷⡀
⠀⢰⡏⠀⣾⣿⣿⣿⣿⡿⠿⢿⣿⠿⠿⢿⣿⠿⠿⣿⡿⠿⠿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⣷
⠀⣿⠀⣼⣿⠋⠀⠀⠀⠉⠂⠀⠈⠀⠠⡄⠈⠃⠀⠀⠁⢀⡴⠋⠉⠳⠀⠀⠀⠀⠀⠀⣀⣀⡀⢄⣠⠾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⡇
⢸⡇⢠⣿⣿⣆⠀⠀⠀⣀⣀⠤⠴⠶⠶⠞⠚⠒⠒⠛⠛⢻⡁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿
⣾⠁⢸⣿⣿⣿⣷⣧⠀⠙⡍⠑⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹
⣿⠀⣾⣿⣿⣿⣿⣿⣧⠀⠹⡄⠀⠀⠀⠉⠉⠉⠙⠓⠒⠲⠦⠤⣈⣙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸
⣿⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⢳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠐⠒⠒⠒⠤⠤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠭⢿⣿⣿⣿⣿⣿⣿⡄⢸
⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⢸
⢻⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠸⡆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⣼
⠸⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢳⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⡿
⠀⢿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣈⡉⠛⠓⠒⠶⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠁⣸⠇
⠀⠘⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣈⣉⠛⠛⠛⠓⠒⠶⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⢠⡟
⠀⠀⠘⣦⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀⣀⠉⠉⠛⠲⠶⢤⣤⣤⣀⣀⡀⠀⣿⣿⣿⣿⣿⠃⢠⡟
⠀⠀⠀⠘⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣈⡉⠉⠛⢹⣿⣿⡿⠃⢠⡟
⠀⠀⠀⠀⠈⢷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⣁⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠋⠉⣁⣠⣤⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣲⣶⣶⣶⣶⣶⣶⣶⣶⣚⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⡩⠡⢤⠤⠶⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣢⡀⠐⠉⠀⠀⠀⠀⠀⣀⠳⣆⠀⠀⠀⣄⠀⠀⠀⠀⣰⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣀⣀⣤⣤⣴⡶⣈⣤⣴⣿⣦⣠⡼⠃⠀⠀⠀⠀⣏⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⣿⣿⣿⣿⣿⠿⣧⠛⠁⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠞⡛⠛⠛⣉⠀⠐⠉⠁⠠⣄⣈⡀⠀⠀⠀⠠⠀⢀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢒⠀⣉⣤⣤⣖⣹⣤⣀⠀⢲⠤⠴⠴⠋⠤⠀⠀⠀⠀⡇⠈⠄⢀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣷⣞⠯⣽⣿⣌⢻⣷⣼⣿⣶⣴⣆⣴⣶⣶⣆⠀⡇⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣽⣿⣶⡬⣍⢉⡛⢩⡙⠛⢿⡽⠟⠛⢿⠛⠂⠘⠚⣻⠆⠈⠀⠀⠠⠀⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⠿⣥⣴⣫⡭⢿⣭⣾⣻⣶⣦⣶⣶⣶⣄⠀⠂⣨⠉⠑⠀⠀⢆⣋⣠⣤⣠⣄⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣿⡾⠿⢛⣿⣧⢴⣞⡱⣞⣿⣾⣫⣽⣿⣽⣿⣽⢶⡆⡀⢀⠆⠀⠀⠀⡴⠚⠋⠁⠀⠛⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⡹⢷⣏⡎⣿⣷⢛⣡⠾⣭⠱⢟⣭⠿⢯⡶⢻⣥⠾⣿⣟⠀⠀⢠⠀⠀⠀⠀⢀⡁⠠⡶⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣯⢟⡾⡾⣫⢯⢿⣅⣺⡉⠛⠉⢷⣟⣣⠶⢋⣵⢾⡝⣿⡂⡰⠀⠀⠀⠀⡤⠾⣷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣴⠿⢃⠦⠆⣼⢟⠿⠆⣴⣿⠏⣩⢜⣫⣶⡽⣿⣿⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣟⣭⣼⢋⡔⡛⠀⠀⠀⣿⣻⠡⣺⡗⡭⠛⣿⠁⠀⠀⡀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⢦⠴⢿⡀⠀⠉⢀⠀⢼⣿⠃⠀⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣵⢚⡽⢯⣿⣿⣿⠏⠀⠀⠀⠀⠐⠈⠂⠀⠀⠀⠀⠈⠛⠆⠁⠘⠀⠀⠰⠀⠘⡄
⠀⠀⠀⠀⢀⣤⣶⣾⣿⡶⢖⡦⠽⣿⣷⠞⣸⣧⣿⣽⣿⣦⡀⠀⠀⠀⡀⠀⣰⣀⡀⠀⠀⠀⠀⢠⡠⠃⠂⣄⠀⠣⣠⠃
⠀⠀⠀⣤⠟⣩⠍⢐⣻⠿⠛⠛⢛⣻⣿⣯⡄⠀⢻⠻⣿⢾⣷⣤⣤⣀⢁⠔⠁⠈⠉⠉⠉⠀⠀⠘⠁⠊⢀⠁⠈⡼⠃
⠀⠀⠘⢣⠟⢠⡾⠋⢀⣴⣶⡿⠋⠉⠠⠌⠙⠂⡈⢷⠐⠺⠟⠟⢿⣷⣯⣀⣀⣀⣠⣤⣤⣤⣀⠈⠀⠀⠀⣀⠊
⠀⠀⠀⠾⠀⠀⠀⠄⢚⢠⡯⢀⣔⣊⡠⠀⠉⠀⠀⠈⠀⢀⡀⣬⠱⠹⣿⣿⡻⣿⢿⣿⣿⡇⠀⠀⠈⠄⠠⠉
⠀⠀⠀⣷⠀⢰⠃⠀⢸⣿⠃⣩⡽⠛⠱⠏⠀⠀⣐⣩⣄⣤⣾⣿⣷⣟⣹⣿⡏⣷⣘⢿⣇⣀⠀⣀⠎⠃
⠀⠀⠀⠘⡄⠘⠀⠀⡾⢣⠁⣽⣣⠘⠏⠀⡤⠎⠙⣻⣿⣿⣻⣿⣷⣿⣿⣿⣻⣿⣿⣋⣙⣿⡿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢿⠙⠐⢠⠇⠀⠀⠟⠀⠹⣿⠻⡟⠁⠀⠈⠛⡑⠏⠉⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠐⠀⠀⠀⠀⢠⡄⠀⠈⠛⢰⠀⠀⠀⠀⠀⠘⣶⠄⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⢀⣀⣠⣄⢠⣤⠀⠠⠤⠀⠀⣤⣄⡀⠉⠀⠤⢀⢾⡀⠀⢀⢀⣄⢀⡘⣯⠢⣄⣀⣤⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠛⠚⠒⠒⠂⠀⠲⠚⠻⠟⠿⠛⠿⠓⠆⠒⠲⠛⠻⠟⠷⠾⠷⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠒⠒⠚⠁⠀⠈⢉⡉⠉⠉⣉⣅⡀⠀⠀⣛⠉⠉⠉⠳⠶⠦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠋⣠⠦⣶⣤⠾⠛⠒⠛⠋⠉⠛⠛⢉⡀⠉⠛⠛⠉⠑⠲⠿⠶⠦⣍⠋⠉⠁⠹⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⣴⠶⠞⠋⠄⠐⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠙⠳⣄⣀⠀⠈⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣶⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠈⠙⠻⢄⠀⠀⠈⢳
⠀⠀⠀⠀⣠⣴⣶⡟⠇⢠⣴⠟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡤⠘⢷⡄
⠀⢀⣴⣿⣿⣿⣿⡇⡄⠀⢱⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⢮⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠰⡀⠀⢹
⣰⣿⣿⣿⣿⣿⣿⡇⢿⣦⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠻⣰⠇⡇⣽⢙⡗⣶⢤⣤⣀⣀⠀⠀⠀⠈⠁⣠⠗⠀⡸
⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣀⠀⠹⣄⡀⠁⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠑⠽⣼⠇⡟⢸⠃⡏⢸⠀⡏⢸⠉⡟⢲⠲⣤⠋⠁⠀⠙
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣌⠻⣷⠀⠀⠙⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠌⢿⣗⣺⣤⣇⣸⠀⡇⣼⠀⡇⣾⢀⣯⠦⣶⡏
⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡷⣌⡛⢷⣦⣄⠀⠈⠷⠴⠤⣄⠀⢀⣀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠹⣯⢽⣓⣾⢯⣷⣯⢼⣧⠿⢋⣠⡾⠋
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣌⠙⣶⣭⣛⠷⣶⣶⣄⡀⠀⠉⠀⠈⣉⠷⠤⠒⠒⣦⡠⠤⢤⡀⠐⠦⢼⠯⠔⢮⠭⠔⢚⡩⠄⡘⠟⢉⡠⣾⠇
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣧⡈⢿⣏⠛⠶⢭⣝⣛⠿⠿⣷⣤⣀⣠⣤⣄⣁⣠⣄⣀⣀⣤⣄⣠⣤⣶⣤⣾⡷⠭⠔⣂⣡⠴⠚⢁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣷⣝⣧⡀⠀⠀⠉⠙⠓⢶⠶⣬⣭⣭⣭⣛⣛⣛⣛⣛⣛⣛⣩⣭⡭⠥⠴⠒⠚⠉⣁⣠⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣤⣄⡀⠀⠀⠈⡄⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⠶⠼⣤⣿⣄⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣤⠯⠟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠁⠀⢱
⠀⠀⠀⠀⡐⠉⠈⠑⠃⠀⠀⠐⠀⠐⠀⠘⠂⠢
⠀⠀⠀⠀⢃⠀⠀⠄⠀⡀⠀⠀⠁⠀⠐⠀⠀⠸
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠈⠣⣀⠠⠀⠀⠀⢀⠀⡐⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠖⠉⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⡀⠆⠈⠀⢠⠀⡠⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢀⢀⠠⡙⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠇⠀⠘⢨⠖⠁⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⢆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠢⠀⠀⡄⠂⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠈⠈⢀⠕⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠁⢺⠁
⠀⠀⠀⠀⠀⠀⡀⠔⠃⠀⢀⣀⠀⠀⠀⢀⣀⠀⢀⠇⠈⡡⢊⠒⠤⢀⡀⠀⠀⠀⢀⡀⣤⠒⠈⠀⢀⡀⢸
⠀⠀⠀⢀⡴⠊⠀⠀⠑⠊⠁⠀⠑⠤⠤⠃⠈⠁⠁⢀⠜⠀⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸
⠀⠀⠀⢇⠀⠀⠀⠀⢰⠀⠀⠀⠢⠂⠀⠀⠐⠀⠐⡀⠈⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠹⠂⠀⠀⠈⡄⠐⠀⠀⠀⠐⠀⠀⠀⠀⠉⠒⠂⠉⡇⠀⠀⠀⠀⠈⠠⠀⠀⠂⠆⠀⠀⠀⠰
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠂⣡⠙⠢⠀⠠⠊⠀⠀⠐⠀⡆
⠀⠀⠀⠀⠂⠄⠠⠄⠀⠀⢣⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠱⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⣀⣀⠀⠀⠀⠀⠀⢀⣀⠠⠠⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠉⠀⠈⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡀⠀⠀⠀⠘⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠅⣕⣀⣠⣔⠧⢨⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠀⠘⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠀⣀⣀⣀⣀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡈⣠⣽⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⢹⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⡿⠟⠻⣷⣿⣿⡇⠀⢿⠻⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠉⠁⠀⠀⠀⠀⠈⠙⠻⠿⠧⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣤⣤⡀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣯⡤⢤⣤⣶⣶⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⡿⠃⠀⠈⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⡿⠁⢄⢹⠀⠀⠀⠀⢸⣿⡿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣿⣿⢿⣿⠏⠀⠀⠈⣆⡅⠀⠀⠀⠀⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣷⡿⠃⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢿⣿⣿⡮⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠻⣿⣡⡿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡀⠀⠀⠀⠔⠚⠉⠙⣿⣄⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢸⣿⣯⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⣾⠁⠀⣴⡆⠀⠀⢀⡴⠋⣹⣦⣀⣀⣤⣾⠟⠀⠀⠀⠀⠀⢠⠸⣿⣿⣷⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣄⢸⣿⣤⢶⣾⠏⠀⣠⠞⣉⣀⣼⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣸⠀⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⡿⣻⡿⢃⣼⣷⣾⠟⠉⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣻⣾⠟⣡⣿⡿⠋⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣮⣭⢼⣿⣿⠃⠀⠀⠀⠀⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠏⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢱⣀⣀⡠⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠑⠤⠄⠈⠀⠉⣷⠲⠀⢦⢆⢄⠍⢙⢒⠢⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠲⠤⠄⠀⠀⠀⠀⠒⠛⢲⠒⠲⠤⣌⣀⡃⠫⢪⠖⢈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠼⣅⣀⣀⣀⠤⠤⢄⣀⣀⡠⠤⠞⠀⠀⠀⢀⣀⣉⣹⣶⣰⠋
⠀⠀⠀⠀⢀⣠⠤⠖⠒⠋⣭⣽⣋⣁⣀⣀⣠⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉⠀⠀⡴⡵⠧⢤⣀
⠀⠀⡴⠊⠁⠀⠀⠀⢀⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣞⠀⠀⠀⠀⠙⠢⡄
⠀⠸⡅⠀⠀⢀⣤⣾⣷⢶⣴⣇⣀⣀⣀⣀⣀⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⢾⠏⠀⠙⡄⠀⠀⠀⠀⡿
⠀⠀⠈⠲⢤⣀⠙⠯⣗⡯⠖⠉⠀⣀⡯⠕⠒⠒⠒⠒⠒⠲⠤⢬⣉⡉⠉⠉⠙⠛⠓⢶⡁⣀⠤⠚⠀⢀⣠⠴⠊
⠀⠀⠀⠀⠀⠈⠉⠑⠶⣮⡭⠭⣝⣓⣒⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣶⣶⣤⣤⡴⢯⣤⡶⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠬⠭⢭⣭⣭⣭⣭⣭⡭⠭⠭⠤⠴⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⢶⠿⠿⠆⠶⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡖⠂⠰⠛⠈⠛⢛⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢤⣀⣠⣄⣠⣄⢀⣀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⠀⢀⣠⣴⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠛⠛⠛⠋⠈⠉⠙⠛⠋⠙⠉⠛⠻⠆⣤⡄⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠛⣫⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠉⠠⣦⠀⠀⠀⢀⣴⡄⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡍⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⢠⣤⣶⣤⣀⣴⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⣀⠀⠀⠀⠀⠀⠀⣯
⣈⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠿⢷⣤⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⢀⣿⡆
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⢸⣿⣭⣶⠶⠮⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⢀⣼⡟
⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣛⣛⠛⠷⢤⣴⣶⠆⠀⠀⠀⣀⣤⡄⠺⠛⠉⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣉⣉⣭⣤⣶⣶⣶⣧⣤⣴⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟
⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠛⠛⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⢠⣤⣿⠟⠃
⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣯⠤⠆⠀⠀⠀⠀⣰⣿⠉⠉
⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣦⣤⡾⡿⠇
⠀⠀⠀⠉⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠛⠉⠉⠀⠉
⠀⠀⠀⠀⠘⠋⠉⢉⣴⣶⣤⣄⣀⠀⠀⠀⠐⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛⠂⠰⢄⣴⠟⠉⠀⠀⠀⠀⢀⣀⢀⣀⢠⣴⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣀⣀⣴⠟⠻⢮⡉⠛⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡄⠀⢼⣿⠋⠁⠀⠀⠀⠙⢦⡈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠹⣦⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣷⡀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣦⣄⣀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣷⡀⠀⠀⢀⣤⣤⡴⢾⣃⣄⡀⠨⣙⣻⣿⣶⣾⠿⠟⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠻⣄⡀⠘⠛⠛⠚⠉⠉⠉⠉⠓⠶⣯⣭⣛⠾⢯⡛⠋⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠈⠻⣿⣓⣀⣛⣛⣿⣽⠿⠷⠀⠀⠀⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣚⠉⣥⡶⢦⡄⢶⣤⣤⣤⣉⣉⣉⣉⣋⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠀⠈⠻⣆⡙⢦⡀⠀⠀⠉⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡷⣄⣸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⢁⢵⢱⠿⡀
⠀⠀⠀⠀⠀⢀⣎⡿⠷⡷⠦⣄⡀⢝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡻⢻⣶⡄⠈⣦⣇
⠀⠀⠀⠀⠀⠬⣔⣛⣯⡽⢛⣉⣉⣶⣄⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠟⢻⣿⣿⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⢻⠋⣄⣿⡿⠛⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⢨⢶⣿⣿⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠸⣠⣿⣽⣶⣺⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣠⡴⣟⠳⡗⢿⣿⣿⣿⡿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⢿⣎⢙⣻⠿⣷⣤⡤⠤⢾⢫⣽⣧⢱⣷⡛⣾⣿⣿⠛⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢋⣾⣭⣶⠿⣿⣾⣾⠋⠦⠈⠈⠀⠐⠻⢿⢾⢧⡺⣿⠋⢀⠀⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣺⣟⣿⣿⣯⢾⣾⣿⣶⡆⠀⢀⡀⠀⠄⠈⣻⣶⣿⡟⠁⠐⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡟⠁⡢⠛⣿⣿⣿⣷⣟⣿⣿⣿⣾⣿⡿⠝⠁⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⢻⣾⡾⣠⣴⣿⣿⣿⣾⣿⣿⡿⢿⡟⠫⡀⢠⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠰⣰⢦⡔⢂⣄⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣏⢐⣡⣺⣷⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣇⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⢸⣟⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⡧⠒⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⢀⠀⢀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠄⣤⠨⠙⠓⠺⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⣀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢢⣤⣤⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⢉⠟⠖⠖⢂⠀⢲⡌⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡐⠈⠙⠐⠊⠓
⠀⠀⠀⠀⠀⠀⠀⡼⢛⡘⢀⠂⠀⢀⠀⢁⠀⠀⠀⠠⠥⠀⠀⢀⡀⠀⡀⡴⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⢻⡋⣯⠀⢐⣀⡛⢀⠈⢡⠀⢡⡎⠁⠀⠍⠑⠂⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠗⢀⡈⠉⡉⠲⣿⣶⣾⡇⠀⡄⠀⡇⠀⠀⠘⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠸⠛⢬⣽⣏⡀⡐⠇⡠⠀⢀⢴⠄⡆⠀⠃⠀⢀⡈⠀⠀⠤⠀⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⣹⡅⢉⣤⡅⣰⠫⠀⣀⠘⢂⠉⠀⠋⠄⠐⠂⢀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⢈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠺⣿⣿⣿⡧⣶⣿⣎⠉⠁⠀⠀⠉⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⡖⠙⢻⣿⡏⣍⣦⡔⠉⢦⣦⡄⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⡾⢋⠻⣷⣾⣇⢀⡌⠉⡠⠁⠀⠀⠀⠀⠀⣁⠀⠀⣤⠤⠀⠂⠤⠀⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⢈⠷⣁⠀⡔⠉⠉⠋⠳⠾⣶⣦⣶⣦⣭⣥⣾⠿⠶⠟⠁⠀⡀⠁⠐⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠘⢀⣀⡤⠀⠀⠀⠀⠀⠠⠆⣰⠃⢰⡎⠋⡠⠒⠀⠀⡸⠟⠡⠉⣠⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡼⠟⣡⠆⠀⠀⢀⠄⣠⡔⠈⢀⠈⠞⡄⣹⠀⣐⠚⢑⣀⣀⡾⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠀⠠⠟⠃⢁⣀⠀⠁⢰⠋⣀⡴⢐⢿⣍⠑⣚⣥⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡧⢲⣧⣀⣉⣤⣀⣩⣿⣿⢿⣿⡥⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠿⠛⢿⡿⠟⠑⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡚⣄⣉⣶⡀⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠘⠻⣧⣀⣤⣌⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠋⢠⡼⠀⠙⠯⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⣤⣤⣤⡀⢰⡾⠀⠻⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠈⢫⣿⠞⡓⠀⠀⠁⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡏⠀⢀⠃⣬⠀⠘⠷⠽⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠴⢺⡅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠐⡂⡄⠉⠀⠀⣠⠀⠘⡛⠻⢶⣦⣤⣄⣀⣀⠠⠶⠶⠦⢀⡀⡌⠀⡆⢠⡴⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⡓⠊⠉⠁⠀⠀⠀⠴⠆⠔⠁⡀⠀⠈⡁⡀⠈⠁⠀⠀⠄⠈⣄⠀⣠⠀⣄⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠠⠬⠁⠒⠃⠀⢀⡄⡀⠀⠀⣀⠆⠀⣁⠘⠀⣆⠀⠂⠐⠀⡩⠀⠉⢠⡀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠀⠒⠈⠁⠀⠀⠀⠀⠀⢨⠄⠀⠀⣠⠠⠁⠀⡄⠉⠀⢠⡄⢠⠁⠀⠀⢠⠇⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠲⠆⢀⠀⠐⠂⠀⢰⠀⠘⠂⠠⠈⠁⠀⠀⠄⠀⠁⢰⡯⠂⠾⠀⠀⡆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣆⠈⣉⠀⠀⠈⠁⠜⠳⡀⠀⠐⠟⠂⠀⢈⠄⠀⠐⠋⠒⠂⠀⠀⠀⠸⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣛⠠⢀⣀⠈⠉⣡⠀⠀⠁⠰⠀⣀⢀⡚⠃⠈⠉⠀⠀⠀⠀⠀⠸⢶⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠲⠤⡤⠀⠉⠠⠖⠠⠞⠀⠟⡋⠘⠃⠀⠀⠆⢠⠀⢀⠀⠀⠄⠏⡀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣐⠾⠿⢛⣁⡄⢀⡰⠔⢁⣆⠀⣆⡀⣰⠀⠊⡨⠀⠇⣴⠄⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢉⣤⡀⠈⠋⠁⢰⣶⠘⣀⡈⠁⢁⠄⡊⠀⠃⡀⠀⢺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣢⣴⡇⢀⡉⠀⠂⢀⠘⠑⢃⣼⢤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⢾⣷⣌⣷⣤⢥⡿⣿⡾⠿⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⣇⠀⢇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠛⣩⠿⠏⢫⡤⠑⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠁⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠈⣿⣾⡊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠆⠀⢤⣨⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠄⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠄⢐⠁⠀⠀⡐⣿
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⢄⠁⠈⠈⠀⠀⠀⠄⣿
⠀⠀⠀⠀⠀⠈⣙⠯⣝⣷⠀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⠁⠤⠀⠀⠂⠀⠀⠀⠀⠚⢺⡀
⠀⠀⠀⠀⠀⠀⢟⢿⡇⠀⡀⠒⠀⢻⡶⠦⢖⣂⠂⠛⠛⢘⠛⢎⡣⠩⠇⠀⠀⢐⠀⠀⠀⢀⡀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⠄⠖⡈⢋⡄⣀⠀⡀⢚⠁⡄⡀⠂⠑⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠂⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⣛⡟⠄⠉⠠⠃⠀⠐⠂⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⡹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⣹⠅⡂⡀⠀⠀⠀⠀⠙⠂⡈⠁⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠈⣷⡶⢦⠤⠚⢀⡆⠒⣠⢠⠄⠀⠁⠀⠉⠀⠀⠠⠀⠠⠀⠀⠂⡠⢀⡄⠀⠄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠣⣴⡂⢠⡴⠛⣉⣰⣥⣶⣾⣴⢶⣦⣿⡟⠿⡀⠀⠀⠀⠀⢊⢀⠨⡹⠓⢦⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⡻⣷⣾⠿⠿⠁⠡⠄⣉⣃⢺⠉⠓⣼⠂⠀⠀⠀⠀⠀⠈⣬⠁⢐⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣧⣿⡄⠀⠀⠀⠀⢀⠀⢄⡀⠀⠠⠠⠄⠂⠀⠀⡠⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠂⠀⠀⡀⠠⠀⠀⠁⠀⠁⢆⠀⠄⠀⢀⠁⣂⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠀⠄⠀⠶⠚⠐⠄⠋⠠⢤⠄⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣆⡀⢀⣀⢄⢀⢐⢁⣸⣥⣕⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢷⠟⠙⠛⢀⡝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣠⠶⢠⠞⠲⢲⣶⠒⠦⣤⣠⡴⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠰⡾⠉⠘⡏⠉⡂⢋⠀⠀⡀⢀⣐⣁⣀⣐⣋⠀⡀⢀⠿⠉⠉⢛⡦⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡆⢉⠻⣂⣀⣻⣭⣦⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⡏⠈⠀⢀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⣻⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟⠛⠯⠽⢛⡿⣶⠽⠷⣎⢿⣿⣿⠿⣿⣿⣿⣿⣷⣾⣧⡀⠀⠀⡴⢢⣄
⠀⠀⠀⠀⠀⠀⣖⢠⣿⣿⣿⣿⣿⣿⡿⣽⢏⡿⣭⠉⠛⠋⠐⠁⠀⠈⠋⠀⠀⠒⠡⠞⠕⠛⠛⡻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣽⣯⣨⣇
⠀⠀⠀⠀⠀⣼⡿⠉⣿⣿⣿⣿⣿⣿⢿⠃⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠁⠘⢿⡻⣿⣟⣿⣿⣿⣿⣿⣏⢉⣄
⠀⠀⠀⠀⢠⣿⣷⡖⠉⠻⣿⣿⣣⡴⠜⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡠⠢⢶⣔⣿⣿⣿⣿⣿⣿⠃⢨⣿⡀
⠀⠀⠀⠀⠀⢿⣿⣃⣤⡤⢿⣿⣿⣫⡄⠢⠂⣀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⣻⣿⣿⣿⣿⣿⡟⢿⣷⣼⣿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⡶⠀⢈⡽⠿⣿⣧⣨⣵⣁⣴⣢⡄⣀⠀⢀⠀⢀⡀⠀⠀⠀⣠⣄⣠⠠⢶⣶⣶⣾⣿⣿⣿⡿⠿⣯⠀⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣀⣴⠾⠉⠉⢻⡿⠿⢿⣯⣿⣷⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠿⢿⠛⣻⣿⡋⠀⠀⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣭⣻⠷⣤⣴⣿⡄⠀⣠⡿⠁⠈⠛⠉⠉⣿⠛⠛⢻⡫⠉⢹⣿⠉⣁⡀⣀⣨⣿⡷⢶⣻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣮⣭⣏⣛⣛⡷⠲⠶⡶⠶⢾⡿⡶⠤⠾⠱⠶⠟⢓⢚⣛⣯⣭⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣛⠗⣲⠖⠒⠒⢲⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠧⠄⠰⠶⠶⠾⠿⠿⠛⠛⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠢⠂⠄⠢⠀⠀⠢⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠛⠛⠋⣠⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣷⣤⣀⣸⣿⣿⣿⣿⣿⣿⡟⠻⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⢸⡟⢿⣿⣿⣿⣿⣧
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⣤⣀⣀⠀⠀⢠⡆⠀⠈⣇⠈⣿⣿⣿⣿⣿⡀
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣤⣬⣯⣛⡛⠋⠉⣿⡇⠀⢠⡟⠀⣿⣿⣿⣿⣿⡇
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣾⣛⣀⣤⣟⡓⠁⣿⣿⣿⣿⣿⡇
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣦⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣭⣴⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⢧⠀⠈⣆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢆⠀⠀⠀⢳⠀⢸⠀⠀⡰⢋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡀⠀⢸⢠⠏⢀⡞⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡇⠀⠾⠁⠀⢸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⠀⠀⠀⠀⠀⠀⠳⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠴⠶⠶⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⢛⠟⣻⠁⠀⣠⠀⠀⢀⡀⠀⠀⣄⠀⠈⢏⢻⡓⠲⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠉⠀⠀⠀⠈⠙⠁⠀⢞⣠⠁⠀⡞⢱⠀⠀⣇⣱⠀⠈⠉⠀⠀⠀⠀⠉⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⡴⠒⠒⠟⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠛⠚⠓
⢳⡀⠀⢀⡀⠀⠈⠉⠙⠒⠒⠦⠤⠤⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠴⠒⠒⠒⠋⠉⠉⠀⢀⣀⠀⠀⣠
⠀⠉⠉⢻⠉⠳⣄⣀⣤⡴⠦⣄⠀⠀⠀⣀⣀⣀⠀⠀⠀⠉⠉⠉⣉⡉⠉⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⣠⠶⡖⠦⣄⣀⡴⠋⣽⠉⠉⠁
⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⠈⠓⠒⠛⠉⢸⠈⠛⠦⠤⠤⠖⠋⢹⠉⠛⠦⢤⡤⠖⠋⠀⡇⠈⠙⠲⠶⠞⠁⠀⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠈⠓⠲⠤⣄⣸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⣁⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⠦⠧⠤⣤⣤⣤⣀⣀⣸⣄⣀⣀⣀⣠⣤⣤⡼⠤⠤⠶⠶⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⡄⢨⡝⠽⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠈⢣⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⣧⣵⡷⢶⣵⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣀⢢⡉⠱⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠂⠀⠀⠀⠹⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠀⠉⢃⢣⠘⡙⠆⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡁⠂⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢆⠀⡌⢧⠂⠸⡄⢀⢺⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⠂⡄⢀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢧⣸⠎⢃⢶⣛⡛⡉⣻⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⢀⡤⣤⣄⡴⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠋⠀⡙⢷⠇⣐⣒⡶⠣⠭⣐⡙⢃⣤⠄⡀⠀⠀⠀⠀⠀⣠⠘⠻⡏⢷⢴⢼⡄⡴⢉⡘⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢤⣘⡃⣘⣇⣀⣴⣟⠨⣝⣐⣖⠺⢿⡀⡌⠛⣻⠝⢿⡆⡘⠗⣞⢷⡈⢞⣂⡵⣇⡈⣹⠉⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢠⣤⠟⢁⣆⣰⠊⢐⣾⣃⣭⣏⠡⣶⠺⠂⣩⠻⣧⠈⣙⣷⢊⠷⠎⠅⣏⡄⡐⠉⢿⠁⣌⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠟⠺⢧⡉⡽⠁⣉⣩⣅⠔⢳⣖⣩⡗⠒⣻⣿⡏⣤⣴⠋⣿⠞⢯⣥⣉⠉⠙⡿⠞⠛⠖⢣⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣏⢈⣡⠚⠚⠏⣹⠐⡦⠴⠦⣦⣤⡊⣩⡿⠒⣹⢛⡉⠡⢧⡜⣹⠛⡩⢈⠼⣆⣷⣟⡇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⣿⢿⣠⢶⡅⢹⡄⢣⢾⣤⣉⡝⠙⢳⣏⡼⠯⢉⣢⣥⢝⢋⡞⠣⣎⡄⢩⢪⣉⣘⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡭⢶⣉⣷⣱⡙⡧⢥⣿⣿⣷⠶⣦⡶⠿⣏⣠⡏⢺⡃⣵⡀⠘⠿⣇⠜⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⣿⣻⠟⣛⡛⢿⣿⣿⣿⣽⣯⣵⣚⣛⣱⣒⡟⣻⣧⣿⣶⣋⣷⣿⣐⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⣾⣏⣭⠤⣾⣫⠙⢋⠛⠿⠿⠿⠯⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⣷⣬⣡⠄⣠⣄⣄⣹⢤⣽⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣱⣯⣏⣉⣀⣩⣽⡟⢏⣉⡉⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣿⣡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⣿⡿⠿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢉⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢅⠓⢀⡘⢀⠄
⠀⡰⠁⠉⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠒⠁⢁⠈⠁
⢀⠷⠄⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣈
⡜⠒⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⠀⠀⠀⠀⢸⡀
⠱⡠⠀⠀⢀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠘⣆⠂⠀⠀⠀⠀⢄⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠈⠩⠋⠀⠐⠂⠀⣁⡀⣄⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⡇
⠀⠀⠀⠀⠀⡂⠀⠀⣸⡅⠤⠄⠈⠑⠤⠀⣀⠤⠤⠤⡤⠤⠤⡤⠲⠅⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⢨⠋⢠⠁⢀⡠⠰⠒⡀⠀⠰⢮⠁⠈⢈⡤⢴⠙⠶⠄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠘
⠀⠀⠀⠀⠀⠀⡁⠀⢉⠡⠀⡀⢎⠀⡠⠐⡠⠜⡡⠁⠰⠠⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠩⣀⠠⠎⠀⠀⠉⡐⠈⠀⠚⠃⠈⠀⠀⠂⢀⠄⢠⡐⠠⠆⣀⠀⠀⠀⠀⠀⠄⠰⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⢠⢀⠐⠀⠪⠐⠐⠀⢀⠀⢆⠐⠙⠒⠠⠅⠀⢠⠄⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠁⠀⠂⠁⠀⠀⠐⢁⣈⠀⡆⠙⠀⠙⠄⠀⠠⠁⠀⠁⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢄⣀⡠⠀⡠⠈⠁⡀⠁⠀⠀⠀⠐⠀⠄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⠆⣀⡁⠀⠐⠅⢠⠀⠀⠀⠀⠀⡀⠀⠀⠄⠠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡈⠓⠇⠀⢀⠀⠀⠀⠱⠀⠈⠁⠂⠈⣠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠠⠀⠢⢤⠠⠤⠀⠐⠋⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠑⢄⠀⠁⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠀⠀⠙⠦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣶⣶⡄
⠀⠀⠀⠀⢀⣠⡴⠶⢿⣿⣿⣿⣯⣿⣿⡻⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣝⢹⣿⣹⣷⣤⣀
⠀⠀⠀⣀⣻⣿⣡⣄⡘⣿⡿⠟⠁⠛⠿⠿⠋⠀⣝⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⠛⠋⣁⣬⡏⠈
⠀⠀⢸⡏⠹⣿⣿⡮⡿⠂⠀⠀⠀⠀⠀⠀⠐⣿⣋⣿⡿⢧⡀⠀⠀⠀⠀⢀⡞⢁⣀⣤⣤⡶⡶⠟⠛⠋⢻
⠀⠀⣾⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣿⠀⠀⠀⢰⣿⠿⠛⠋⠉⠁⠀⢀⣀⡴⠄⠚
⠀⠈⢿⡀⣠⣶⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢳⣤⢸⣿⠀⠀⠀⢸⡥⠤⠴⠒⠛⠉⠉⠁
⠀⠀⢿⡏⢻⣷⡿⠛⠀⠀⣄⠀⠀⠀⠀⠀⠀⠘⠿⠾⣃⣰⠆⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣦⣤⣤⣀⣀⡀
⠀⠀⠁⣙⣯⡀⠀⠀⣶⣞⣿⣯⡿⢠⣞⣩⣷⡆⢀⣞⣽⣷⠀⢠⣦⣶⣶⣿⠋⠛⠻⠛⠁⠘⠛⢛⣉⣭⠿⡧
⠀⠀⠀⠹⣿⡷⠦⢶⣬⡛⠛⠛⠁⠈⠻⢿⡯⣤⣾⣾⡿⠋⠀⣨⣿⠘⣫⣍⡀⠀⠀⠀⠀⠒⠺⢯⣅⣀⡤⠇
⠀⠀⠀⠀⠈⠻⢷⣿⣷⣝⣓⣻⣿⣿⣞⣫⣼⡿⠿⠋⠀⠀⠀⢷⣝⣿⣿⣞⣁⣴⣶⡄⢀⣴⢶⣄⣀⣉⣷⣶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣮⣻⢟⣉⣑⡬⢫⣭⣝⣫⣶⣾⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⢻⢻⢻⣹⣿⣿⣽⢹⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠛⠛⠛⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⠛⣛⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢊⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠰⢿⣴⣧⢴⡮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⠠⡉⢳⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠈⢡⠣⠀⡑⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣄⠢⢳⡀⣰⣠⡼⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⠐⠂⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡥⠘⢾⣧⡄⣑⣒⡰⡚⢁⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡧⣄⣛⡒⢙⡳⠫⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⠇⠸⡄⢓⣺⠄⢯⠯⡠⣾⡇⢐⠶⢆⡰⣖⠐⣷⣴⢧⢰⡽⢙⡭⠑⢷⠺⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣩⡵⢋⡍⠽⠉⣱⣎⣹⡓⢠⡽⡏⢀⠿⡆⠙⣺⡖⡻⠜⠃⣋⠉⠞⠳⠛⣘⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⠸⢧⡉⡝⢈⣉⣍⠔⢷⡎⣽⡒⣺⣶⠏⣠⡝⣹⡖⢿⣤⣉⠉⠳⠞⠓⠚⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⢨⣵⡉⠛⢩⣆⠳⠲⠲⢤⣆⣩⡟⣊⣝⣩⡁⠿⣄⡝⢉⠇⡐⣷⢽⣿⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣣⡞⣷⡈⣆⣱⠳⢦⢼⣄⡙⡳⢚⣇⡥⢶⢪⡠⣏⠚⣴⣠⢳⣤⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣷⣾⣾⣤⣯⡿⣿⠟⣩⡿⢏⣹⠶⡯⢽⣂⠿⣤⡌⣵⣦⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣻⣷⢶⣂⡤⠙⣻⡏⣲⢷⣶⠚⣒⣶⣴⣿⣿⣿⣶⢿⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣵⣿⣭⣿⣣⠬⣷⠲⣭⢽⣤⣾⣽⢿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣾⣿⣷⣟⣷⣿⣵⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⣿⣿⣾⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠐⠶⠶⠶⠶⠶⠒⠒⢲⡤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠋⣠⣤⣠⠶⠤⠴⠒⠲⠖⠛⠓⠶⠖⠲⠤⣤⡬⣙⠛⠛⢓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡴⠶⠊⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢉⠛⢦⣀⠀⠱⠦⣀
⠀⠀⠀⠀⢀⣀⡄⠁⢸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠑⢦⣀⠀⢣⡀
⠀⢀⣴⣾⣿⡟⠀⠘⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢁⠀⠱
⣰⣿⣿⣿⣿⣧⢱⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡍⡏⡟⢲⢢⢤⣄⣀⡀⠁⠀⠁⣨⠆⢀
⣿⣿⣿⣿⣿⣿⣎⢻⣄⠘⠦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣇⡇⡎⢸⢸⢸⢀⠏⡏⡖⣲⠉⢀⡈
⠹⣿⣿⣿⣿⣿⣿⣷⣝⢦⣄⡘⠒⠢⣄⣀⡀⠀⠀⠀⠀⠀⢈⠀⠂⠀⠀⠘⢿⣷⣷⣾⣼⣸⢀⠇⡧⠟⣩⠟
⠀⠙⢿⣿⣿⣿⣿⣿⣯⠓⣬⣛⠶⣬⣤⡀⠈⠒⠀⠒⣦⠀⠤⢤⡀⠠⣀⠠⢌⣷⠫⢝⠻⠽⠛⠉⠴⠟⡁⣴
⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣌⢿⡝⠲⠭⣝⣛⡿⠷⣤⣤⣤⣬⣤⣤⣀⣤⣤⣤⣴⣦⡾⠶⢁⣀⠤⠒⢉⣼⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣿⣆⠀⠀⠈⠉⢛⠳⡖⠒⠺⠿⠯⠭⠽⠿⠒⠒⠒⠊⣉⣡⣤⣶⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⠦⣤⣌⣄⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣷⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⣿⣿⣧⣾
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣿⣿⣿⣿⣿
⠀⠀⢀⡔⣽⢿⣿⣿⠿⠷⢤⣦⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠡⢰⠆⠲⠆⢿⣿⣿⣿
⠀⢀⣾⣹⡇⠀⣈⣤⡤⠤⠤⣤⣉⢻⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⣧⣀⣠⡞⠛⢛⡉
⠀⣿⣸⠏⢀⣾⡏⠀⠀⢀⡀⠈⠙⢷⣿⣶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢿⣿⣿
⢸⣿⡇⠀⣾⠏⠀⣐⣶⣿⣿⣦⣤⣄⠙⣿⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢺⣿⣿⣇
⣿⡿⡇⢠⡟⢠⣾⢿⣿⡿⠿⠿⠿⣿⣷⡸⣿⣧⣸⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁
⣿⡇⠀⢸⣧⢾⡇⣼⠁⠀⠀⣀⣀⣈⣿⣿⣟⢿⢿⡀⠀⣀⣤⣶⠚⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⢹⡇⠀⠈⣼⠸⢁⡏⣀⡴⠟⣫⣽⡏⣩⠛⣿⡷⡾⠛⠋⠛⠋⠚⠋⠈⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠇⠀⠀⡏⠀⢸⣿⠏⣠⣾⣿⣛⢛⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⢸⠁⣴⡟⠉⣰⡟⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣰⣾⠿⡟⢠⣿⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⢋⠄⠀⣀⠔⠃⠀⣾⣿⣿⣿⣰⣯⡾⠛
⠀⠀⣴⣿⠃⠀⣵⣾⡇⣼⡿⣿⡟⠀⡆⠀⠀⠀⢠⠆⠉⠐⠁⠀⠈⠁⠀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⣸⠏⠀⠀⠀⢸⣿⡇⢸⡇⢿⠃⠀⠀⡀⢇⣠⣤⣊⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⠟⠁
⠀⢻⠀⠀⠀⠀⢸⡿⡇⢈⡇⢸⡇⠀⠸⠁⠐⠉⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠈⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠻⡆⠀⠀⠀⠀⢸⣦⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⡄⠀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣀⣤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⠟⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠋⢸⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⠟⠀⠀⣿⠏⠀⢸⡿⠃⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸⣯⠀⢸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠇⠀⢸⣿⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡛⡂⠀⠀⠀⠀⠀⢸⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠃⠀⠒⠾⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠙⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠉⠱⠓⠒⠀⢈⣿⣦⣀⡀
⠀⠀⠀⠀⠀⣀⠔⠁⠙⠛⠛⠀⠀⣴⡶⠾⠋⠉⠉⠉⠙⠓⠦⣀
⠀⠀⠀⢀⡸⠐⠿⠉⠉⠀⢀⣤⣶⠿⠀⢀⡴⠒⠒⠲⢤⡤⣀⡈⠓⠲⣄
⠀⢀⠞⠷⢿⣟⠓⠂⠀⠀⣾⠃⠀⠀⠀⣸⡄⠀⠘⢯⠑⣿⠀⠉⡦⡀⠉⠓⠦⢄⡀
⠠⢊⣴⣦⣤⠄⠀⢠⠞⠛⠋⠀⠀⠀⢸⠀⠙⠦⢤⣤⡿⠟⢀⡔⣁⡼⠄⠀⠀⠀⠈⠙⠲⢤⣀
⡔⣀⣀⣀⠀⠀⡴⠟⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⣀⣀⣠⠔⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠏⠀⠀⠉⠙⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⣉⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⢤⣽
⠈⠙⠢⠀⡄⠀⢳⣤⡤⠤⠴⠶⠶⠖⠒⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠳⡄⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠒⣒⠩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠦⣷⠙⠦⠘⠘⠦⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠥⢤⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠓⠶⣆⠓⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⡀⠐⠲⠆⢀⡀⠀⠄⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⠻⠷⠾⣀⡰⢿⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠉⠀⠘⠉⠀⣀⣀⣤⡥⠆⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣟⡀⠀⠐⢤⡙⠂⠀⢰⢦⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢻⣍⣁⣈⡀⣠⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⡴⠀⠀⠄⠩⠆⠀⠂⠘⠈⢣⡞⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠋⡝⠉⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡡⢚⠁⣀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡟⣡⡏⠚⠁⠀⠀⠀⠀⠀⠀⣤⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢍⡠⠖⠃⠀⠀⠀⠀⣀⣤⢏⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⠁⠀⠀⢀⠄⠒⢎⣩⣤⡅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣀⡜⠁⠀⠃⠀⠹⢛⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡿⣫⠐⠀⠀⠀⠀⠰⢟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠓⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⣿⠀⠀⠀⠀⠀⠀⠾⢃⠀⣄⡆⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⣀⠀⠀⠰⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⡟⢈⠠⠀⠈⠀⠒⠀⠠⣄⣤⡤⠈⠂⢀⣴⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⢀⠀⠦⠀⠃⡴⢠⣦⣤⡺⢿⠅⠐⠂⠀⣸⠆⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠠⣄⣀⠀⢀⠙⠉⢠⣀⡐⠚⠃⠀⠚⠙⠛⠛⠃⢠⡤⢁⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠃⠈⠛⠀⠈⠃⠀⠀⠀⢤⠀⠉⠡⠄⠀⠀⠀⣀⠁⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠀⠀⣈⡒⡂⠀⠀⠀⠒⠀⠀⠂⠀⠀⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣈⣭⣁⣰⣶⡶⣦⣤⢠⠴⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡁⠈⠀⠀⢍⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⡔⠨⠀⢀⣠⣴⠖⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⣀⣔⣶⡀⢸⡉⢲⣄⡀⠀⠀⠀⠀⣪⣅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢛⡀⠚⡛⠻⠍⢁⠘⠇⠈⠀⣁⠄⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⢟⡥⠈⠠⡆⠀⠀⠀⠊⣠⠄⡰⣍⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠁⠀⠀⠀⢡⡀⠀⢀⣠⠾⠛⠉⠁
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⣩⡶⠾⠋⠁
⡄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⢀⡄⠀⠐⢀
⣉⠀⢀⡀⠈⢱⡄
⣶⣶⣶⣶⣶⣿⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⢛⣛⣻⣛⣙⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣏⣹⣿⣿⣿⣽⣛⣛⣿⣿⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠿⢰⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠂⢠⡑⠎⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠉⠀⠀⠐⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡆⠀⠀⠀⠀⠀⠀⣀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⣉⣿⡃⠀⢴⣶⣶⠁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣾⣿⡇⢈⣁⡄⠀⢿⡀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠟⠀⢻⣿⣿⣇⠈⣿⠆
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡏⢶⠖⢻⣿⠛⣿⡟⢻
⠀⠀⢀⣀⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠾⠧⣁⢀⡐⠃⢿⡧⣄⠐⢉⣶
⠀⠀⠣⢈⡁⠠⠄⠀⠀⠀⡀⠀⠰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⠆⠉⠘⡷⠗⡶⢀⠙⠛⠈⣋⠃
⠀⠀⠀⢰⠇⠀⠠⠶⠄⠀⣭⣀⡀⠙⠶⢅⣛⠂⣐⠰⠂⠀⣤⣤⣀⣀⢀⣠⣄⠀⠀⠰⠀⠀⠀⠇⠀⠣⠀⢀⠁⠘⠧⡤⡤⠀⢠⣄
⠀⠀⠀⠈⠀⠀⠲⠒⠀⠲⣀⡀⠈⠁⣄⠀⡀⠀⠈⢘⢂⡈⢹⣤⣛⢛⣦⠛⠡⣤⢀⠀⣰⠈⠄⢠⠀⡼⠀⠈⠳⠄⠠⠂⠁⠐⣾⡇
⠀⠀⠀⠀⠀⠀⢰⣤⠄⠊⠨⣑⡉⣄⠀⠂⢁⡠⣷⣶⢞⡡⣤⠄⠃⠘⠅⠟⠠⠂⢙⢠⠏⠠⡆⢸⡖⢰⠁⠀⠀⡄⠀⠐⠰⡾⠟⡹⡀
⠀⠀⠀⠀⠀⠀⠳⠃⠀⠐⠒⡎⠱⢤⣧⠔⣯⠁⠮⢠⡏⣘⡿⢶⡀⠈⠚⣹⠃⢉⣉⣁⡸⠀⠿⠾⠠⠈⠄⢐⠈⢡⡴⠀⠰⠙⠂⡃⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣯⡽⠞⣡⡄⣸⣿⣸⡦⢾⠷⢉⣽⠾⡝⠩⣼⠋⡙⠃⠀⢁⡀⠀⢀⠀⠀⠈⡀⣡⠼⠀⢠⡘⢇⣠⣂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠉⠤⠁⠚⣷⣾⠿⣿⠟⢀⡴⢛⣓⠿⢀⡄⠡⠄⣉⠈⠁⠛⢀⣀⡠⡀⣀⠄⠉⠀⠀⠘⠏⣤⠸⡇⠱⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠫⣤⣴⢓⣷⠛⣇⣿⡍⣁⣤⣤⣆⣈⠩⡖⢫⣧⣤⡉⢄⣤⠾⢤⡖⢀⠲⠁⠚⠁⢒⣸⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⣀⣿⡶⠚⠋⠈⠠⠾⡛⠻⢿⣿⣾⣿⡟⣋⠙⠳⣤⣰⢭⣿⡿⢿⠙⠁⠰⠗⢸⠇⣤⣄⣸⢎⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠹⢿⡟⠂⠀⢀⡌⢀⠉⠂⠀⠉⠉⠉⠙⠉⠑⠛⡍⠙⠂⢤⢀⡉⠳⠀⠐⠠⡖⢂⠇⠨⢭⡟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠂⠀⠀⣽⠉⠁⠤⠕⠀⠔⠃⢐⡀⠁⠀⠀⢀⣠⣤⣤⣁⣠⣉⣀⡀⣀⣁⣸⣀⣀⣸⣼⣧⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠉⠀⠙⢁⡄⠀⠈⠅⠂⠘⠦⠶⠉⠁⠀⠁⣉⣉⣉⠉⣍⣹⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠙⢋⣭⣤⡎⠀⠀⠀⡠⠀⠈⠶⡦⢘⣛⡋⣀⣟⡉⡛⢛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⣫⣄⠚⠃⣮⢤⣌⣸⠊⢛⣿⠿⠻⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢀⡿⣿⣵⢋⠸⣿⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⣿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣼⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡇⠀⢰⣯⣉⡁
⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⢻⡁⠸⣇⠀⣿⠈⢉⣹⡟⠛⠲⢦⣀
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢹⠟⠋⠛⣿⡾⠋⠀⠑⣄⠀⠀⠙⢷⣄
⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⢰⡟⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⣠⣤⡏⢠⣾⡁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣷⢀⡤⠞⠋⠙⢳⣼⢥⣿⠞⠛⠓⢶⣤⡀
⢸⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹⡟⠁⠀⠀⣴⠋⠀⠀⠈⢷⠀⠀⠀⠙⣷⡄
⢸⠃⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠘⣿⡀
⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣸⣧⠶⠶⠾⣇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣻⡇
⢸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣟⣀⣤⡴⠞⠛⠋⢁⣀⣤⡴⠚⠋⠙⠲⣼⣄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣿⡇
⠀⢿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠴⠞⠛⠉⢀⣀⣠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⡿⠀⠀⠀⠀⣰⣿⠁
⠀⠘⣷⡀⠀⠀⠀⠘⣿⡀⠀⡴⠋⠁⠀⢀⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣴⠟⠁
⠀⠀⠈⠻⣦⡀⠀⠀⠹⣷⡸⣷⣶⡒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠈⠙⠳⢦⣤⣙⣿⣿⠉⠙⠛⠶⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠶⠦⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠈⠉⠛⢶⣄
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠙⣷
⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠞⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠶⢶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠣⠄⠁⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠒⠦⣄⣁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠈⠀⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⢢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠐⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠠⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠉⠁⠀⢤⢽⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠞⠁⠀⠀⠀⣀⣤⠤⠄⠀⠀⠘⠳⠦⠶⠚⠁⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠄⠐⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⠂⠀⠀⠀⢀⠣⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠔⠚⠳⣄⠀⠀⠀⠀⡠⠊⣀⠀⠀⠀⠀⠘⠁⠀⢠⠇⠀⠌⢀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⠭⠟⠁⠀⠀⠀⠀⠀⠀⣠⠏⠀⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⣤⣂⡀⠀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣀⠄⠀⣠⠾⣃⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠒⠲⠬⣝⠶⣤⣀⣀⠐⠠⠒⠒⠒⠒⢉⣉⡤⠖⢋⣁⣨⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⠩⠭⠭⠥⠬⠭⠽⠖⠚⠋⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠐⠑⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢈⠀⠈⠁⠀⠘⠀⠀⢤⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⠀⢀⣀⡤⠤⠤⢤⣀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣁⠀⢠⢲⣾⣿⠑⢆⣀⠶
⠀⠀⠀⠀⠀⠀⠤⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠙⠁⠀⠀⠀⠀⢸⠶⢋⠀⠀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⣄⠀⠀⠀⢰⠃⠀⠈
⠀⠀⠀⢀⡠⠤⠤⠄⣀⠀⢠⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣰⡗⠀⡄⢈⠁⠒⠚⣧⡀⠀⡰
⠀⠀⠐⠛⠀⠐⠀⠤⡀⠁⠠⡇⠀⡜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡿⣇⡷⠙⠧⡀⠀⣷⡌⠋
⠀⠀⠀⠀⢀⣠⠄⠀⠠⡀⢲⠨⣄⢠⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⣁⢚⢄⣦⣷⣸⣰⣄⣌⣄
⠀⠀⠀⠰⠂⣠⣄⡩⡤⣀⡈⠳⣿⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠌⣡⢾⡟⢣⣛⣻⣈⠌⡏⢦
⠀⠀⢠⢇⣵⡉⠀⢄⣀⠀⠙⠷⣌⢣⡀⠀⠈⡄⠀⠀⠀⠀⠀⢀⣧⣀⡀⢊⡥⠊⠠⢴⡍⠹⡈⢷⠐⡼⣧
⠀⠀⠀⠟⠁⣀⠠⠤⢤⠵⢦⣀⠉⢦⡀⢳⠀⡃⠀⠀⠀⠀⢀⣼⡿⢞⠺⠛⢉⡘⠓⣺⠧⠤⣵⣼⠚⡇⣸⢷
⠀⠀⠀⠀⢊⠄⠶⠒⢉⣑⡦⣈⠛⢆⢻⠈⠀⠇⠀⠀⠀⣠⢊⠉⢀⡌⠀⠈⠁⢆⡾⢁⡔⣋⡉⠙⢶⡘⠉⢿⡇
⠀⠀⠀⠎⠀⢀⡘⠛⢉⣋⡙⠀⠙⢌⣳⠀⣴⠀⠀⢀⡼⠁⢸⡠⠋⠀⠀⢀⡜⣾⡷⠋⠀⠀⠈⠀⢸⠃⠀⠈⣿
⠀⠀⠀⠀⢠⣧⠞⢋⣭⠾⠛⠛⠓⢢⣕⣃⣸⣦⡴⠋⠀⠀⠘⢁⡀⠄⢀⡁⠚⠉⡠⠔⠀⠀⠠⢀⡟⠀⠀⠀⡏
⠀⠀⠀⠀⢾⠁⡜⢡⠃⢀⡤⠒⣩⢟⢿⡟⡻⡟⠀⠀⣀⣴⣁⠠⠄⠀⡀⠀⠀⠉⠀⠀⠀⡠⢡⡞⠀⠲⡄⢠
⠀⠀⠀⠀⠨⢸⠀⢸⣤⠋⠀⣴⠁⢡⠎⡔⡇⢱⠀⢸⡣⠞⠃⢀⠊⠁⢉⠀⠀⠀⠀⡠⢒⣴⠏⠀⠀⠀⣵⠄
⠀⠀⠀⠀⠀⠹⣄⠈⣿⠆⢸⡿⢠⡟⠀⡃⣇⠈⢷⡉⠀⣠⠔⢋⠀⠨⢤⡤⠤⠖⣋⠴⢋⡜⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠈⢺⣌⣷⠸⣇⠀⣧⣜⢶⣌⢻⡿⣦⣬⣭⣿⣷⣶⣖⣾⣽⢁⣠⣞⢁⣠⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣿⠀⣺⡇⠀⠉⠛⠿⠶⠀⠀⠈⠈⠉⠉⠁⠁⠉⠁⠼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢢⡀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣠⠞⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡄⠀⠀⢀⣤⠛⣤⠴⠋⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡶⣿⠁⢸⣏⣀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣴⡿⠃⠀⠀⠈⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡠⠄⣀⣼⡀⢉⣿⡿⢅⣔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠡⣀⠀⣟⣿⣶⡿⣶⢤⣝⡙⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠉⠙⠽⢤⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⣲⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⣿⣡⠇⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠉⠉⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠚⢻⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣀⡞⠀⠀⠀⢀⠀⡺⠁⠀⢄⣀⡀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠠⠊⡠⠘⠂
⠀⠀⠀⠀⠀⠀⣀⡄⢈⡀⠀⢀⠀⡠⠤⠼⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢀⠤⡀⠀⢀⠠⠊⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⡡⠄⠂⠁⠀⠄⢀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡀⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠦⢀⡀⠀⠀⠠⠀⠀⠀⣀⡠⠖⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠐⠒
⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡲⣶⡵⠞⠣⢀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠮⠭⡿⡏⠱⣿⣿⡷⠼⠻⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠲⠓⢚⡩⠶⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣘⣷⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣷⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣯⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⣿⣍⣑⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣟⣿⣟⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠛⠛⠿⣿⣿⣿⣿⡏⠉⠻⣿⡛⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣗⣻⣞⢿⣇⠀⠀⠀⠀⠀⣰⡟⣁⠶⣋⣭⣤⣤⣼⣿⣿⣿⡃⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢾⣯⣻⡟⢿⢿⣿⡀⠀⠀⠀⠀⣿⣴⣷⣶⣶⡒⢒⠀⠉⠛⣿⣿⠇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣧⣟⡾⣷⣿⢿⣧⣄⡀⠀⣀⣿⣿⣿⣿⢸⣹⣌⡛⢦⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⣿⣛⢻⣿⣿⣿⣿⣿⣿⣵⣟⡗⠽⡿⠷⣮⣥⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣇⣶⡿⠟⣿⣿⣿⣿⣿⣿⣿⣳⣿⣷⣶⣑⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣻⣿⣿⣿⣿⣿⣞⣿⣿⡿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⠀⣀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣵⣶⣦⣤⠄⠂⠀⠢⢢⡀
⠀⠀⠀⠻⢂⡿⢂⣿⣷⣶⣎⡀⠐⠂⠀⠙⢿⣿⣽⣿⣿⣿⣿⣿⡿⠷⠿⣿⡿⣿⣟⠡⠀⠐⠆⠙⠻⠷⠦⠤⣤⢶⡧⠄
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⢑⣬⡉⢻⣿⣿⠿⠿⠋⠀⠀⠀⠀⠁⠀⣀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠓⡜⡇⠀⠀⠀⠀⠀⠀⡠⢄⣀⢘⣻⡶⣼⣿⣿⡷⣆⣤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⢀⡀⣄⢰⣞⣲⣶⣶⣶⣞⣿⠿⠿⠿⣿⠿⠻⠏
⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣦⣤⡀⠀⠀⠾⠛⠛⠉⠉⣸⠝⠉⣩⣽⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⢐⠢
⠀⠀⠀⠀⠀⠉⠛⠛⡿⣿⣿⣿⠿⠟⠓⠀⠀⠀⢤⣴⠷⣞⠛⠋⠉⠀⠀⠀⠁⠂⠘⠀⠡⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣿⣿⡾⠶⠖⠒⢄⣉⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠉⠋⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢣⠔⠋⠑⢆
⠀⠀⠀⠀⠀⢀⠤⠤⢄⣸⠀⡠⠀⢀⠀⠀⠀⠀⢸⡀⣀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠀⠁⠀⠀⢦⠊⠑⠀⢀⠀⠈⡆
⠀⠀⠀⠀⠀⢧⠀⠀⠐⠀⠢⡄⠀⠀⠈⠀⠀⠋⠀⠐⣰⠁
⠀⠀⠀⠀⠀⠘⢆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣀⠑⡄⠀⠀⠀⠀⠀⠀⠔⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣂⡤⠒⠈⠉⠙⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠀⡤⡀⢨⡀⠀⢀⠤⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣴⡄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠳⣼⢠⠀⢠⠘⡇⡴⠋⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠗⢠⠐⠒⠒⠄⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⡈⢰⠀⠈⠰⠏⠔⠀⡈⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣦⠀⢰⢦⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⠈⠆⠀⠀⡧⠊⠀⡐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠉⠁⢀⠟⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡥⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠤⢤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴⠚⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠖⠋⢩⣀⣠⠤⠢⣄⠀⠀⠀⡔⠢⣄⠠⠋⠀⣠⠞⡕⠈⠉⠓⠲⠤⠤⠤⠤⠤⠶⠖⢛⠀⠀⠀⠀⠒⠂⢸
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠀⠀⠈⠑⠒⠊⠀⠀⠀⠀⠀⡔⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⡄⠐⠠⠂⠀⡀⢸
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡆⠀⠄⠄⠀⠀⠠⠔⠀⠀⠀⠀⡞⠀⠃⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠈⠀⡘
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠢⠔⠀⠀⠀⠒⠂⠀⡆⠀⠑⠒⠀⡁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⢲⠄⠀⠀⠀⢣⠐⠤⠂⠀⠀⠐⠤⠀⠀⠄⠀⠘⠢⠤⠔⠊⡇⠸⠀⠀⠀⢀⠜⠢⡀⠀⠠⠀⡎⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⡆⠀⠀⠀⠴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠢⢀⠠⠂⡴⢄⠀⠉⢀⡜⠁⠀⢀⡄⠀⡆
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠠⠀⠀⠀⠘⠦⣀⣀⡤⠞⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠉⠃⠂⢹⠀⠀⠀⠹⡄⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⢀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⢈⣦⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠉
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠉⡟⠀⠀⠒⠒⡏⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠇⠀⠀⠀⠀⡇⢰⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣄⠀⠀⠀⣠⠅⢜
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡮⠞⣲⠞⠷⠰⠿⠚⡇⡶⣵⠄
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢀⠀⠀⣽⠛⠓⡤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣈⣀⣰⡇⠀⠀⠠⠃⠀⠉⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣄⣠⠀⠀⠐⢐⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⣿⡿⠳⡸⣩⡆⣿⠈⢿⣻⣿⣿⣿⣷⣿⣯⣄⢠⣿⣯⡏⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⣋⡴⠱⠇⣴⠂⣝⡟⠛⡀⣠⢿⣟⣹⣻⣿⡿⢿⠿⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣟⡉⢁⢀⡄⡰⢃⣠⣿⣟⡷⠿⠮⢞⣻⣛⣬⣭⣤⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠾⠌⣿⣋⣘⣋⣭⣭⣽⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣴⣿⣯⣭⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠻⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠿⠻⠛⠛⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆
⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠀⢀⣠⣖
⠀⢰⣿⣿⣿⣆⠀⠀⢀⣴⣾⣿⣿⣭⡉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⣠⣾⣿⣿⣣
⠀⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⢰⣾⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⠃⠙⡿⣿⠀⠀⠒⠓⠶⠶⠾⢶⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠈⢻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠀⢀⣶⡄⠀⠀⠀⣸⣿⣿⣿⣦⡀⠀⠘⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢻⡙⢻⡅⠀⠀⠀⠀⠀⠀⣒⣼⣿⣿⣦⣄⠀⡆⠙⣿⣿⣿⣿⣷⠀⠀⠀⠉⢸⣿⡈
⠀⠀⠀⠀⠞⠂⠸⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡦⠃⢹⠟⠛⠛⡿⣷⠀⠀⢀⣸⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠟⢻⣿⣿⡏⢠⠇⠀⠀⠀⠁⠙⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠠⢖⣺⣶⣿⣶⣦⣄⠀⠀⠀⠀⠀⣠⣶⣴⣶⡄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡆
⠀⠀⡜⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢰⣿⣶⣿⣥⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠈⠛⠿⠛⠿⣿⣿⣿⣿⠟⠉⠙⠟⠉⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⣻⠏⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣼⣶⣷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣯⡉⠬⢉⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠝⣱⣦⣴⣌⣳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⠰⣿⣿⣿⣶⣿⠋
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⢤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⠀⠹⣎⣩⠞
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣤⣙⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠂
⣼⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡎⠀⠁
⢻⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣥⣶⢶⣣⣿⣹⣿⢋⣭⡃⢀⡀⠀⠀⠀⡀⠠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠸⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢅⣽⣿⣟⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⠻⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⡿⣿⣿⡏⠀⠐⠈⠠⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠾⠢⢄⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⡞⢀⣴⠀⠀⠀⢠⣾⠀⢀⠀⣤⠒⡞⡠⣠⡄⠀⠀⢠⡀⠆⠉⠞⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⠿⣇⣸⣽⣿⣿⣤⠸⣏⣻⠸⣿⣟⣸⣥⠾⡷⢫⣥⠘⣺⡇
⠀⠀⠀⠀⠀⠹⣿⠛⠙⢻⠛⠁⢻⣿⣿⣿⣿⣿⡿⢧⢀⣴⣾⣾⡿⠿⠈⣷⡄⠻⣿⣿⣿⣿⣡⣘⠣⠚⠋⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⢧⠀⠸⠋⠁⢿⣿⣿⣣⣏⣼⣿⣿⣿⣏⠅⠠⠀⡟⠻⣷⣬⣟⣉⣉⡁⣤⠐⣋⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠙⡿⣿⣿⣿⣿⣿⠟⢂⣴⢞⣡⠤⠀⣈⣿⢻⡋⢁⡈⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⢿⣟⠭⠶⣛⣥⣵⣾⣿⣿⣿⣿⢣⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⠓⠉⣻⣦⠳⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡄⠀⠓⢶⡃⢀⠀⢏⣿⣾⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣠⣴⣿⠥⣤⣂⣀⣽⡃⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠈⠣⣱⡄⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⢤⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⡟⢻⡛⢲⣄⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠙⠋⠉⠙⢿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣸⠃⣨⠁⢸⡇⠑⡄⠀⠀⠉⠓⠢⣄⡀⠀⠀⠀⠀⠀⢸⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣏⠀⣾⠃⢠⠇⠀⠈⢧⠀⡸⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣄⣀⡼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠙⢤⣿⣆⣈⣳⣄⣀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠛⠻⢿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠖⠒⠛⠫⠉⠁⠀⠀⣄⠠⠄⣸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠶⠖⠖⢒⡛⠛⣉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢆⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⠁⠉⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠀⠠⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⡤⣣⠟⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠬⠥⠤⠤⠴⠤⠴⠦⠤⠤⠤⠀⠀⠀⠀⠴⠤⠤⠤⠤⠤⠴⠴⠶⠶⠶⠒⣚⣋⣁⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡼⠁⡤⢀⡀⢠⡀⣀⢠⡤⢰⣇⣠⠀⣿⡠⣤⠀⣤⠀⡄⢠⡄⠀⠀⠀⠀⣾⣠⡄⠀⡤⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⢸⡇⢸⠃⣸⡦⣿⢸⡇⢸⡇⢹⠃⣿⢴⡋⣠⡇⢠⣇⢾⠁⠀⠀⠀⠀⣿⠁⡿⢸⡇⢰⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠓⠈⠃⠹⠁⠛⠈⠛⢹⡗⠋⠀⠛⠈⠉⠁⠉⠙⠏⠘⠋⠀⠀⠀⠀⣿⠚⠁⠀⠉⠈⠳⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢠⡗⢐⡖⣛⣑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠼⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠛⢿⣏⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠄⠶⠀⠆⠈⢆⠈⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢆⠀⠐⡌⠀⢦⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡘⣦⢑⠸⠄⠼⠆⠶⠻⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠑⢦⡉⡀⣀⡀⣠⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⠈⠻⢦⡎⠭⢥⡜⢻⡁⠴⠆⢀⡀⠀⠀⠀⠀⠀⠀⣀⠖⣿⢳⢌⣡⣁⣠⠛⠡⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠆⢸⡆⣉⣭⠒⣓⣚⢭⠆⢿⡀⣀⠳⣦⠔⣦⠌⠻⢴⡛⡄⣵⡆⣁⢇⡘⢛⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣥⠟⢋⣄⣹⠋⣩⣖⣩⣝⠉⢴⢸⠇⢠⠻⣆⠉⣻⡗⡸⠧⠻⢘⡉⠑⠊⢛⠁⣌⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠙⠺⢧⡉⡍⠀⣉⣉⡡⠺⣶⣉⣿⠒⣻⣿⠏⣤⡼⢙⣷⠻⣯⣈⣘⠙⠿⠚⠛⠒⣡⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣯⢈⣵⠛⠻⠋⣅⡲⠶⠶⠶⣤⣆⣩⡷⣊⣽⣻⡉⠼⢤⣨⡟⣹⢘⠴⣧⢻⣛⣇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣟⢣⡴⢷⡊⣷⢚⡜⣤⣌⢝⡉⠹⠧⠾⣗⣉⢤⡄⡋⣭⠁⠾⣔⣠⢣⣀⣩⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣞⣿⣴⣦⣍⣹⣓⣿⣾⣿⣛⣿⡟⢛⣥⣼⢑⡏⢸⣧⢘⠙⣣⣨⣱⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢟⠿⣿⣩⡭⠛⡙⠿⣿⠲⣭⣦⣭⣥⣬⣥⣉⣾⣾⣷⣇⣼⣿⣭⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣟⣽⣶⣗⣻⠴⢞⣿⠂⣚⣭⣽⣒⣒⣛⣩⠿⣻⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣽⣷⣯⣵⣟⣴⢟⣽⣿⣿⣿⣭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣞⣾⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠙⣿⣿⣿⣗⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠁⠈


⣿⢿⠿⠿⠿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣻⣿⣿⣿⣟⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠉⣀⣄⠉⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠦⡦⣴⣾⣛⡍⠂⠀⢈⣉⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠂⠘⠃⠀⠀⣙⢟⠃⢀⠀⡹⠛⠧⠀⠈⠀⠀⠀⢹⠗⢆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡅⢎⠯⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠲⣤⢣⢠⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠀⠀⠀⠀⢠⡌⠐
⠀⢀⠒⠀⠀⠀⠀⠀⢹⠶⣴⠋⠁⢀⢉⡁⠀⠀⠀⠀⠀⠀⠐⢤⢤⡄⠀⠀⣴⣶⠠⠂⠀⣠⠆⠈⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⢋⣠⡄⠪⠆⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣁⠘⠃⠉⠀⠀⠴⠂⠀⠀⠀⠙⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠒⡂⠀⢂⣰⠹⠁⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠐⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⠉⣈⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢇⠠⢤⣀⡄⠠⠀⡀⠀⠀⠀⠀⠀⠀⡈⠰⠄⢩⣦⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⢀⡀⠁⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠠⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠭⠁⠀⠀⠈⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠛⠜⠑⠉⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠄⢈⡁⠀⠀⢀⡀⠀⠀⠀⢀⣤⣚⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡷⡾⠳⢌⡀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⢀⡓⠀⠶⠀⢱⠖⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⢦⡜⢿⠃⠀⡈⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣈⠅⠀⠀⢀⡌⠀⠀⠀⢰⣆⡀⠀⠀⠀⠀⠀⢠⣀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠹⠊⠉⠀⠀⠀⢦⡀⠈⠙⠺⠶⣮⣠⣆⢐⣀⣀⣶⣤⠗⠉⡃⠀⠐⡿⠓⠂⠠⣔⠉⠀⠁⠀⢀⣀⠀⠀⠀⡂⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⠀⠈⠙⢿⣟⠛⠉⠉⠀⠀⢠⠀⠀⣤⠅⠀⡤⠂⠁⠀⢀⡀⠀⠀⣩⠿⡄⢢⠇⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠁⠀⠀⠈⠁⠂⠀⠀⠀⠀⢀⡴⠅⠀⢼⡅⢠⠤⠀⠼⢿⣗⠄⠹⠤⣽⠄⠀⠀⣠⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠢⡀⣰⡦⣄⣀⣀⣄⣀⡀⣠⣤⡾⣄⡀⣤⡀⢠⣀⣀⣀⢈⣇⣨⢳⣧⡍⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⡀⢹⡟⠛⠉⠉⠉⠁⢘⡛⠛⠂⠻⡥⢿⢀⣾⡯⣹⣷⣾⣿⣿⡿⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡶⠤⣀⡀⠀⠀⢈⡁⡠⠴⠞⠓⠚⠛⠿⠛⠀⠶⠛⠉⣾⠉⣧⡿⣧⢺⣿⡿⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠸⠿⠿⠖⠒⠂⠰⠞⠂⠀⠀⠀⠒⠂⠀⠈⢿⠟⠀⠻⡟⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢤⡀⠀⠀⠠⠆⠀⠀⠀⠀⢸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄