Yogurt Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠧⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣴⠟⠛⠛⠛⢳⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠉⠉⣻⡿⠋⠁⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡿⠋⠀⠀⠀⣠⡶⠟⠉⠉⠛⠿⣶⣄
⠀⠀⠀⣾⣿⣿⠋⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⣹⡆
⠀⠀⢰⣿⡟⠛⠿⠿⠿⠿⢶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⠿⠿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢿⣷⠀⠀⠀⠀⠀⣴⡿⠉⠉⠉⠁⠀⠀⠀⠀⣿⡄
⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡿⠀⣠⢶⣄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣿⠃⠀⣿⠀⢹⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠸⣿⣿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠃⠀⠀⠈⠳⣞⣀⣤⣤⣶⣾⣿⡇
⠀⠀⠀⣿⡇⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⣿⠇
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠰⣿⡛⢿⡿⣩⡿⠃⠀⠀⠀⢀⣴⠿⣷⣀⣤⣤⣤⣤⣤⣰⡿⢻⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⢻⡆⢸⣿⣤⡾⡟⢿⣶⡿⣁⠻⣿⡏⣽⠁⣿⠁⣛⣙⠃⣙⣹⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣇⢸⣿⠋⣾⣿⠀⡏⠸⡿⢠⣿⠃⣿⠀⣿⢸⣿⣿⢀⣿⠯⣤⣴⣒⣒⣶⣿⠃
⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸⣿⢸⣿⡈⠿⢟⣴⠿⣳⣤⣍⠻⣆⢛⣠⣏⢸⣿⣿⠋⠀⠀⠀⠀⠹⡅⣼⡿
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠸⣧⣼⠟⠛⠛⣿⠏⣾⡋⢀⣿⢂⣿⠛⠉⠙⠋⢱⠃⠀⠀⠀⠀⠀⢀⣷⣿⠇
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣙⣛⣻⣵⠾⠋⣀⠤⣎⣑⡞⢦⠀⠀⢀⣠⠴⣋⣾⡟
⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠈⠉⠉⠉⠀⠐⣏⡽⠶⢧⣠⣷⣞⣉⣭⣭⡶⠿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣷⠈⠉⠙⠛⠛⠻⠿⠿⠿⢿⣿⣶⣶⣶⣶⣶⡶⠿⠿⠿⠟⠛⠛⠉⠁⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡠⠊⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠀⠐⠀⠒⠒⠂⠀⠀⠀⠀⠀⢒⠎⠀⠀⠀⠀⠀⠀⢠⣎⡀⠀⠀⠀⡠⠊⣠⠖⠁
⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣠⠔⠷⠦⠤⣭⡨⠋⣠⠞⠧⢄⡀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⡠⠋⣠⠞⠉⢛⣶⠀⢹
⠀⠀⠀⠀⠀⠈⢻⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⣀⣠⣔⣊⣁⣀⣀⣀⣀⡤⠮⠤⠾⢑⣒⣉⠥⠴⢺⠃
⠀⠀⠀⠀⠀⠀⠈⡏⠑⠒⠢⠬⠍⣉⣘⢓⣒⠛⠓⠒⠒⠛⠛⠒⠒⠒⠒⠒⠀⠀⢒⣛⡒⣈⣉⡉⡭⠤⠐⠒⢊⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⠒⠒⠦⠄⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠄⢀⡄⢈⣉⠲⢄⡀⠀⠀⠀⠀⡤⣀⠴⣴⣀⢀⣠⡶⣦⢳⣀⣤⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⠀⠓⢺⢷⠛⠾⡇⡠⢌⠓⢤⡀⢫⠗⠻⢒⠻⢼⣺⣳⠖⠛⡻⣺⡷⠆⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠁⠉⡟⢻⠦⢕⡊⠸⣀⡼⣀⠄⠈⠙⠶⢬⣐⠠⢨⡯⣇⠃⡔⠐⢰⣇⣠⠤⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⠁⠀⢀⠼⠓⣿⣍⠹⠤⣎⡤⢣⠖⠀⣹⠟⠓⠛⠚⠛⠋⠉⠁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⠦⠔⠁⠀⠉⠒⠤⣀⡘⠶⠊⣏⡠⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠦⠤⠤⠤⠶⠒⠛⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠴⠶⠶⠒⠒⠒⠚⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠴⠶⠶⠒⠒⠒⠚⠛⢦
⠀⣤⠴⠖⠚⠛⠉⠉⠉⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠖⠚⠛⠉⠉⠉⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⢧
⠀⢹⡄⠀⠀⠀⣠⡴⠞⠋⠉⠀⠰⣿⡍⠉⠙⠻⣍⠛⢶⣄⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣀⡴⠚⠉⠁⠀⠰⢯⡉⠉⠉⠻⡍⠛⠶⣄⠀⠈⣧
⠀⠀⢿⡀⠀⢰⣏⣤⠴⠶⠒⠛⢛⣿⠃⠀⠀⠀⣿⠛⠓⢾⣇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢰⠏⣠⠤⠶⠒⠒⢉⡽⠀⠀⠀⠀⢸⠛⠒⠾⣇⠀⠘⣆
⠀⠀⠸⣧⠀⠘⣿⡀⠀⢀⣠⡶⠟⠁⠀⠀⠀⣰⠏⠀⠀⣴⠇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣯⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⣠⠏⠀⠀⣠⠏⠀⠀⢹⡄
⠀⠀⠀⣿⣆⠀⠈⠙⠶⢿⣭⣀⣀⣀⣀⣴⣞⣡⣤⠶⠛⠁⠀⢀⣀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠈⠙⠶⢭⣁⣀⣀⣀⣀⣤⣚⣁⣤⠴⠚⠁⠀⢀⣀⣠⠇
⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣩⣤⣤⣴⣶⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡄⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣉⣠⣤⠤⠶⠒⠛⠉⢹⠃
⠀⠀⠀⢸⣿⣿⣦⣶⣶⣶⣾⣿⡿⠿⠿⢿⣿⣿⣿⡟⣻⢿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠹⠦⠴⠶⠖⢒⣺⠿⠯⠭⢭⣅⣀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠸⣿⣯⣿⢟⡿⠟⠉⠀⢲⣄⢠⣿⠀⠈⠙⣿⣶⢭⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡠⠚⠉⠀⢲⣄⢀⢿⠀⠈⠙⢦⡀⠀⠀⠀⡟
⠀⠀⠀⠀⣿⢹⣷⡟⠁⠀⣠⣶⣟⣿⣿⣿⣶⣶⠾⠁⠹⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⣠⠖⣛⠋⡫⠈⠂⢐⠞⠁⠙⡄⠀⢰⡇
⠀⠀⠀⠀⢹⡏⣿⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠂⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⢰⡥⡄⠉⠛⢲⣠⠀⡽⡖⠛⠂⢹⠀⣸
⠀⠀⠀⠀⠸⣯⣿⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⡧⠀⠀⣾⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⢌⠈⠀⢀⢼⠋⠀⣑⠲⡧⠀⠀⣸⠀⡏
⠀⠀⠀⠀⠀⢿⡼⣧⡀⠸⣯⣿⣿⣿⣿⣿⣾⠟⠁⠀⣰⣟⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⢧⡀⠸⡮⠅⠈⣉⠀⠈⢊⠜⠀⠀⣰⠃⣸⠃
⠀⠀⠀⠀⠀⠘⢿⣞⡿⣦⣝⡻⠿⠷⠿⠛⠁⣀⣤⣾⣟⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢦⣈⠲⠤⠷⠥⠚⠁⢀⡠⠞⢁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣿⣿⣶⣾⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢬⣍⣓⣒⣒⣚⣩⣥⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠒⠂⠤⠄⠤⠀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠄⠒⠊⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠒⠂⠤⠤⠤⠤⠀⠀⠀⠀⠀⠠⠤⠤⠄⠄⠒⠒⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣀⠀⣀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⡀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣾⠃⣤⣤⢀⣤⣤⣤⢠⡄⡤⣤⠰⡗⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠿⠀⢧⡼⠸⣧⣿⢿⣼⡇⡇⠀⠀⠷⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠀⠀⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠀⠀⠄⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠄⠀⢀⠀⠀⢀⠀⠠⠐⠐⠤⣄⡀⠀⠀⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠌⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠜⠜⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠂⠢⠤⠄⣊⠥⠊⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡠⠊⡠⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⢤⣤⣤⣤⣤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⡀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡠⠊⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡰⠁⠀⢀⡠⠊⡠⠊
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠴⠊⠉⠙⡲⢊⡰⠮⣁⠢⡀
⠀⠀⠀⠀⠳⡠⠄⣀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⣀⡀⠊⢀⠀⠀⣠⡂⠤⠃⢀⡂⠥⡔⠁
⠀⠀⠀⠀⠀⢩⠀⠀⠈⠉⠀⠀⠘⠋⠒⢚⣛⡐⠂⠀⠀⠀⠁⠉⠀⠈⠉⠁⠀⠒⠸
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠐⠂⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠢⣮⡴⢴⢌⢑⠠⡀⢲⠮⠇⠿⣱⣾⠧⠇⠷⡿⠄⠀⠰
⠀⠀⠀⠀⠀⠀⢃⠀⠈⣟⠗⠱⢈⢧⣼⡠⠀⠙⠲⠤⣀⣸⢻⣄⣀⣀⣧⠤⠒⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡊⡹⠓⠌⡛⠂⢹⣁⠼⠃⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠬⠭⠠⠤⠤⠤⠔⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⠄⠀⣀⣀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⢀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠤⠀⠀⢀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⠀⢀⣴⡾⠿⠷⢦⣤⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⠟⠉⠀⠀⠀⠀⣨⡿⠁
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣾⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⠟⠁⠀⠀⠀⣀⣴⠾⣿⣷⣤⣀
⠀⣠⣶⣿⣿⠿⢿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣅⣀⡀⢀⣤⣾⠛⠁⠀⠀⠈⠙⠛⢷⡄
⢰⣿⣿⣯⣀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣠⣿⡄
⣾⣿⡟⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡿⠿⠛⣿⣇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⢿⣷⠀⠀⠀⠀⠀⢠⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢀⣿⡏⠀⣠⣤⡄⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣸⡿⠀⠸⣧⠀⢻⡄⠀⠀⠀⠀⢸⣿
⢻⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣶⡿⠁⠀⠀⠻⣤⡾⠁⠀⣀⣠⣤⣾⣿
⢸⣿⡏⠛⠻⠿⠿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣤⣤⣤⣤⣶⣶⣾⡿⠿⠟⠛⠉⢹⣿
⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⣠⣴⣦⣄⣴⠿⣷⡄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢹⣿⠀⠀⠀⠀⠀⢿⣧⡈⠻⢋⣼⠟⢀⣀⣀⠀⢀⣾⠟⣹⣿⣴⣶⡾⣷⡾⢷⣦⣿⠁⣽⣧⡀⠀⠀⠀⠀⢰⣿⡇
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠈⣿⡆⢻⣧⣶⢟⣛⠛⣿⡿⣣⡌⢹⣿⠁⣾⠀⣿⠀⣩⣥⣍⢠⣭⣿⠇⠀⠀⠀⠀⣸⣿
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢁⣾⣿⡇⣍⠐⠿⢇⣼⣿⠀⣿⠀⣿⠀⣿⣿⡇⣸⡿⠶⢤⣶⣛⣛⣳⣿⡟
⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⣿⡇⢸⣿⠘⠿⢟⣱⡿⢟⣶⣬⡉⠻⡀⢛⣠⣿⢀⣿⣿⡿⠁⠀⠀⠀⠀⠙⣇⢸⣿⠇
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣿⣥⣾⠟⠿⠿⣿⡟⢰⡟⠁⢨⣿⢀⣿⠟⠋⠛⠛⠋⡾⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿
⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢻⣇⠙⠿⠿⢟⣡⡾⠋⢀⣀⡴⠒⣾⣧⡀⠀⠀⠀⢀⣠⠞⣹⣿⠃
⠀⠀⠀⠀⢻⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⢠⠚⢻⣁⣟⠉⢻⣠⠷⠶⢖⣛⣭⣴⣾⣿⡏
⠀⠀⠀⠀⠈⣿⣿⠛⠻⠿⢿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣠⣭⣭⣤⣼⣿⣷⣶⣾⡿⠿⠛⠉⠁⣼⡿
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⡞⠛⠳⣦⡀
⣿⠀⠀⠀⠘⢷⡄
⡇⠀⠀⠀⠀⠈⢿⡄
⢿⡀⠀⠀⠀⠀⠈⢿⡄
⠘⣧⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠹⣦⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠹⣦⣠⣤⣤⣴⣶⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⣄⠀⠀⠀⢻⣍⡾⠁⠀⠀⠈⠙⢯⡉⠁⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠲⣄⡀⠀⠀⠀⠉⠉⠉⠛⠻⠷⠶⠶⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⢠⠾⢧⣤⡤⢤⣿⣄⣀⣤⠤⠀⠀⠀⠉⠓⠶⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠈⠙⠶⠤⠶⠚⠛⠛⠶⠶⢶⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢻⣄⣀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠋⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠈⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠋⠉⠉⠀⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠶⠶⠾⠛⠛⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⡀⠀⣀⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠶⠖⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠶⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢱⠀⠀⠀⠀⠀⣀⠤⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠳⠞⣀⠀⠀⠀⡰⠁⠀⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⢼⡉⢱⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢸⠀⠀⠄⠈⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠑⡄⠀⢉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠐⠉⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠈⠉⠁⠉⠿⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠊⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢂⠲⠄⠘⠃⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠈⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣰⠃⠀⣀⣀⠤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠾⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⢁⡴⠋⡁⠀⠀⠀⠒⠂⠀⠈⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠔⣁⢀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠘⠀⠀⠀⢑⠂⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠣⢀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠡⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠠⠀⠉⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠖⠂⢀⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠁⠂⠀⠄⣀⠈⠉⠀⠐⠒⠀⠀⠒⠒⠀⠉⠀⠀⡀⠤⠐⠈⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠨⡀⢄⠄⢒⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠖⢁⢁⡄⠁⠀⡊⡰⠀⠀⠢⠠⠄⠔⡀⠀⠊⠀⢀⠅⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠦⠀⠸⡀⢁⠼⠹⢰⠄⡐⠀⠀⠀⠀⠀⡌⡁⠀⡀⠀⠀⠆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠤⠔⠀⠉⠀⠁⠀⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣄⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡇⠀⠀⠀⠣⣜⡆⠀⠀⠀⡴⠋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⢺⡄⡗⠢⣀⡴⠲⠤⠔⠛⢢⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⡕⠛⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢣⡈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠳⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠛⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠒⠤⢤⣩⣉⣉⣛⠒⠒⠒⢋⣉⣉⣁⡤⠤⠖⠊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⣀⣀⣀⣀⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠑⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣔⠦⣶⡤⢤⢤⣄⣦⣶⡖⢲⡒⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⣷⠩⣘⡸⣤⣽⡋⣻⣠⣽⣇⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠀⠀⣀⣠⣄⣁⡠⢄⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠃⠲⣠⡤⢤⡏⠉⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣄⣀⣀⠤⣀⣁⣀⣀⣁⣀⣠⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠦⠤⠤⠤⠶⠖⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢒⡶⣦⣤⡀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠿⠊⠁⠀⠙⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢸⡄⠀⣠⣾⣋⣀⣀⡀⢀⡴⠃⢀⡴⠁
⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⣷⡟⠉⠈⠭⣍⣉⣩⠋⠀⡰⠛⠶⠤⣄
⢀⣴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠙⠒⠢⠤⠖⠢⣜⡁⣠⡾⠟⠀⢀⣠⠞
⣏⡀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠚⠛⠒⠒⠒⠛⠉⠉⠁⣀⡤⠖⠋
⠈⠉⠉⢹⡀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠖⠒⢺⠇
⠀⠀⢠⠞⠃⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠈⢳⡄⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡗
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣤⠴⠋

⠀⠀⠆⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⡀
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⡀⠀⠀⠀⠀⠀⡠⠐⢂⣉⣤⣤⠤⠤⠤⠶⠶⠶⠤⠤⢤⣤⣌⣀⡂⠄
⠃⠤⡌⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣉⣉⠉⠉⠉⢡⠀⠘⠀⠀⠀⠀⣎⠀⠘⠯⣤⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣠⣬⠟⠀⡨
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠈⠀⠠⣤⣀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠄⠐⠊
⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠛
⠀⠀⠘⣿⣿⡻⣿⡿⣻⢟⡛⣿⢛⢻⡟⣿⢻⡟⡛⢻⡛⢛⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣿⣿⣿⡎⣴⣿⢸⡇⡏⢽⣿⡇⣿⢸⡇⢋⣼⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⠸⡇⣧⢻⢸⡇⠿⣸⡇⡌⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⡄
⠀⠀⠀⢸⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠀⠤⠄⠠⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠤⠤⠀⠀⢀⡀⠀⠀⠀⠀⢀⠠⠐⠁⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠈⠁⠀⠀⠀⠐⠒⠀⠀⠐⠂⠂⠀⠀⣈⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠛⢷⡆⠀⠐⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠌⠀⠀⠑⢈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠐⠁⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⢄⠀⠀⠀⠑⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠠⠤⢄⠀⠀⠀⠀⠠⠐⠄⣀⡀⢀⡀⠠⠄⠂⡡⠚⠁⠀⠄⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠃⡄⠀⠀⠵⢊⡀⠢⢀⠀⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢎⠀⠀⠀⠀⠉⠀⠀⢆⡈⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠀⠠⠊⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠀⠂⠄⠀⠉⠀⠐⠒⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠂⠁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠈⠁⠒⠒⠀⠀⠀⠤⠤⠀⠀⠒⠂⠈⢀⡠⠐⠁

⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠟⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣦⣀⢹⡆⠀⠀⠀⠀⠀⠀⣴⠟⠉⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⡿⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣨⡇⠀⢸⣦⣄⠀⠀⠀⢀⣤⣄⡀⠀⢀⣤⠶⣤⡀
⠀⠀⢀⣀⣰⣿⣁⠀⠙⣿⠀⣼⠁⠙⠷⣤⡴⠋⠁⠙⠛⠿⠛⠁⠀⠈⠻⣆⣀⣤⠶⠷⣶⢶⣤⣤⣤⣄⡀
⢠⡾⠋⠉⢩⣤⡼⠂⠀⠿⣤⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢻⣄⠀⠉⠻
⣿⠀⠀⢦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⢻⡄⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠁⣀⡾
⠀⠻⣦⣀⠀⠁⠈⠙⠳⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⣶⠶⠒⠚⠛⠛⠉⠁⢀⣠⢴⡟
⠀⠀⠀⣿⡛⠳⠶⢤⣤⣀⣠⣄⡀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⣠⣿⠴⠶⠛⠋⠁⢸⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠈⣿⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠶⠚⢩⡟
⠀⠀⠀⠀⢻⡄⠀⠉⢻⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⡞⢦⣀⡴⡆⢀⣤⡀⣠⣤⡀⢠⣤⠀⣶⣦⡶⠲⣦⠶⠲⢶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⢧⣄⢉⡴⡿⢋⡀⢹⠇⠾⠷⣼⢻⣶⣿⡿⠘⢡⡟⣦⢠⡼⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢹⣿⠘⣇⠸⠃⣼⣄⣶⢸⣻⠄⠉⣼⡇⣶⣄⣹⣿⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⠟⠀⠈⠛⠛⠁⠉⠛⠋⠙⠓⠚⠋⠛⠃⠙⠃⠙⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣦⣄⣀⣀⡤⠴⣤⡀⢀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣯⣉⣁⣀⡀⠀⠉⠁⣀⢰⠟⠻⠿⠗⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣇⣿⠓⠞⢻⣸⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠉⠛⠓⠶⠶⠶⠞⠉⠳⠶⠶⠶⠶⠶⠶⠤⠤⠴⠶⠖⠋⠀⢰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠛⠶⠤⢤⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀
⠀⠀⠀⠀⡼⠉⢳⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⢷⠀⢸⡆⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠳⢼⠁⠀⠀⠀⣰⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⣠⠶⠦⣄⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢷
⣇⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀
⠙⠦⠤⠦⢤⣧⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡤⠞⠉⢨
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⡟⠁⣀⡤⠚⠁⢀⢄⡴⠋
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡠⠊⢀⡴⢋⡤⠚⠁⢀⡤⢚⡵⠋
⠀⠀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠠⣀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥⠞⣩⠖⠉⢀⡤⢚⣡⠔⠋
⠀⡼⠁⠀⣀⠤⠤⡀⠀⡌⠁⠉⢳⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⠒⠃⠢⠤⠠⡄⠠⢾⣿⠴⠋⢀⡠⠒⣡⠴⠋
⠀⡇⠀⠀⠁⠀⠀⠈⠉⠀⠀⠀⠘⠤⠴⠒⢆⠀⠀⣀⡤⢠⡀⠀⠀⠀⢀⠀⠀⡀⠜⠁⡠⠔⣡⠴⠋⠁
⠀⢹⡀⠀⠀⢂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠁⠀⠀⠂⣁⠵⠊⢀⠔⣩⡴⠋⠁
⠀⠀⠱⣄⠀⠀⠑⠤⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠦⡀⣠⠞⢹⠃
⠀⠀⠀⡇⠉⠂⢄⡀⠈⠑⠠⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⢀⠴⠃⢀⡟
⠀⠀⠀⡟⠢⡀⠀⠈⠑⠢⢄⡀⠈⠀⠀⠐⠀⠠⠤⠤⠤⠤⠤⠤⠴⠖⠋⠁⢀⣤⠏
⠀⠀⠀⡇⠀⠈⠒⠄⣀⠀⠀⠈⠁⠒⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⡽⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠒⠠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⢻⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠈⢳⡀⠀⠁⠂⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣉⣀⡎
⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⣀⣠⡤⠴⠚⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⢠⠏⠀⠀⢠⡎⠀⠇⢄⠀⠀⠀⢸⡄
⠀⠀⠀⢀⡶⠛⠓⢦⡀⠀⢧⣄⣀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠜⠁⢀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⢾⠀⠀⠀⠀⠙⣆⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⠐⠁⠉⠁⠀⠀⣠⡞
⢀⡀⠀⠈⠷⠤⠶⠶⠦⣸⡇⠀⠀⢸⣅⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⡠⠴⣞⠁
⢯⠿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡴⠋⠀⠀⠹⣄⡀
⠀⠀⠀⢀⣠⠴⠖⠒⢲⣄⠀⠐⠿⠂⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠎⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⣤⡀
⠀⢀⡴⠋⠁⠀⣠⠞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡞⠀⠀⠀⣼⠃⠀⠀⢀⣀⣤⡤⠶⠶⠖⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠙⠶⣤⠞⠁⢀⣤⠞⠋⠉⠀⣀⡀⠠⠤⠤⠔⠲⠒⠒⠂⠀⠙⠋⠉⠉⠉⠉⠛⠛⠓⠒⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⡴⠋⢀⡤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢀⡎⠀⡔⢉⣀⡤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠙⠉⠐⠀⡀⠀⠈⠋⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⡼⠀⠀⢳⡊⠀⠀⠈⠙⠲⠤⠄⠠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⡀⣀⣀⠴⠚⠋⠉⠙⡇
⠀⠀⠀⠻⣄⠀⠀⠙⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⡀⠀⠀⣠
⠀⠀⠀⠀⢸⠙⠲⢤⣀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⡥⠿⠲⠄⠀⢁⣠⠞⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠈⠉⠓⠲⠤⣀⡀⠀⠉⠉⠛⠓⠲⠶⠤⠤⠤⣄⣀⣀⣀⣠⣄⣤⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⢀⣠⠴⠋⠁⢰⠇
⠀⠀⠀⠀⢸⠀⠈⠀⠠⣀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⠤⠖⠚⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠉⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠚⢉⡇
⠀⠀⠀⠀⠀⡇⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣇⠐⣇⠑⠄⠀⢀⠔⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼
⠀⠀⠀⠀⠀⢿⠀⠘⢆⠀⠛⢁⡠⠞⡴⠚⠛⢦⠀⠀⢀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⢦⠀⠅⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇⠰⠋⠀⡘⠀⡴⠄⠀⡄⠐⣡⠶⠤⠗⠀⠰⠊⠿⠀⡖⠂⠰⠉⡭⠑⡌⠧⢄⠀⢀⠴⠊⠀⠛⠀⢸⠁
⠀⠀⠀⠀⠀⠸⡇⠀⠀⢧⡄⠀⠀⢧⠀⠙⠁⢀⡟⢸⠁⣖⠀⠀⠄⡼⠀⢠⣠⠄⢐⠰⠀⠀⠛⢅⠀⢸⠀⢸⠀⠀⡤⠀⢀⡏
⠀⠀⠀⠀⠀⠀⣷⢄⡀⠀⠁⠀⠀⠈⠓⠤⠖⠁⠐⣌⠢⣼⠁⠖⠂⢁⠀⠀⠀⢀⡌⢀⢀⠏⠲⠞⠀⢸⡀⠼⠀⠀⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠸⣆⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠓⠦⠒⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠈⠀⠒⠠⠤⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠉⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠖⠚⠋⠉
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠉⠛⠛⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⡆⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⡇⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠁⠀⠀⢠⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⡏⠀⠉⠳⡀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠱⣤⡤⠤⣹⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⠔⠊⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣲⠟⢁⡠⠖⠉⢀⢆⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣠⠞⣡⠔⠉⢀⠤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢀⠀⣀⠀⠀⠀⠐⠤⠴⢄⡀⡀⠀⢀⠀⠀⠀⠀⠀⣠⡷⢚⠵⠊⢀⠤⢊⡥⠞⠁
⠀⠀⠀⠀⠀⠀⡏⠀⠠⠒⠀⠢⡀⠂⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠊⠐⠂⠙⡿⠚⠁⠠⢂⡥⠞⠉
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠱⠄⠀⠒⠉⢢⠀⢀⡠⠤⣄⠄⠀⠠⢐⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠶⡁⠀⣢⠖⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⢀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⢈⡼⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⠀⠉⠒⠤⢀⠈⠀⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠴⠒⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠠⠀⠀⠀⠈⠀⠒⠠⠄⠀⣀⣀⣀⣀⣀⠠⠤⠐⢊⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⠒⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠸⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠐⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⢤⣤⣀⣀⣤⡤⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⠵⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⠞⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢦⣄⠀⠀⢀⣴⠟⡩⢚⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠿⠷⠶⢶⣶⣶⣶⣶⠏⠀⠀⠀⠈⢷⣴⢟⡥⢊⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣠⣴⣿⡿⢊⣴⡿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡴⢾⣿⣿⡿⢋⣴⣿⡿⠟⢀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⢁⣴⡿⣿⣫⡴⠟⢋⣩⣴⣾⡟⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣴⣞⣛⣛⣛⣛⣛⣛⣛⣛⣩⣥⣶⠿⠛⠋⢁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡤⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠲⠦⠤⠤⢤⡿⠒⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣠⠔⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢩⡏⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠾⠾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠂⠑⠀⠀⠂⠀⠀⠀⠀⠁⠐⠒⠀⠄⠠⠁⠄⠁⠠⠡⠀⠐⠃⠀⢀⠀⠀⠝⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢎⡀⠆⠇⠀⠀⠀⠶⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⣀⠀⠈⠀⠀⢀⠀⠈⠵⢂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⢀⡀⡠⠀⠀⡀⠀⠀⠠⠂⠂⠁⡀⠀⠀⠰⠀⠀⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⣠⠀⠨⠤⠍⠀⢠⠀⠀⠀⠈⠐⠩⠊⠀⢀⠀⢈⠀⠀⢢⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⢷⠁⡿⠘⠎⠈⡇⠱⠇⠈⠀⠀⠀⠈⡄⠰⠂⣤⠀⢈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣄
⠀⠠⠄⠤⠀⠀⠀⠀⢈⢇⣉⣀⢉⣙⠃⠀⠁⠁⠉⠈⠈⠈⠁⠁⠀⠀⠒⠀⢐⠀⡆⢰⡆⡠⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⠈⠛
⠀⠀⠀⠀⠀⠀⠠⠀⢘⡛⡛⣛⣛⣻⡃⢜⡇⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠗⠀⠇⠸⠃⠈⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠁⢁⠠⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠩⠀⢤⠄⢀⢠⠋⡉⢲⠀⠀⢀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⢠⠄⠀⢀⠠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⠤
⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⡄⠀⠀⠀⠈⠀⡄⠀⠀⠈⠁⠈⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⣀⠠⠀⠀⠀⠀⠀⠀⠀⠽⠃
⠀⠘⠘⢈⠈⠁⠀⠀⠀⠀⣀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠀⢀⣄⠀⠄⠀⠀⢽⡦
⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠉⠈⠀⠀⠉⠈⠁⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⠚⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⢀⡠⠤⢄⡀⡠⠖⠉⠀⠀⠀⠀⠠⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣰⠋⠀⠀⠈⠋⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⢂⠏⠙⠚⠉⠓⣄⣀⡀
⠀⠀⠀⠀⡴⠚⠙⠛⠁⠀⠀⠀⠀⢀⣀⣀⣠⣴⠃⣠⠖⠒⢤⡀⢀⡤⢥⡏⢀⡀⠀⠀⠀⠀⠀⠉⠒⠒⢦
⠀⠀⢀⡴⠃⣀⣠⠴⠦⡄⠀⠀⠰⠉⠀⠉⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠦⠀⢠⠴⠦⣄⣀⠀⠈⠣⡀
⠀⠀⣾⠀⠘⠁⠀⠀⠀⠈⠀⣀⣀⣠⠤⠤⣄⠀⠀⠀⣠⣤⣀⠀⠀⡤⠒⠲⣄⣤⣄⠀⠉⠀⠀⠀⠈⠃⠀⠀⣸
⠀⠀⣿⠑⠤⣀⡀⠀⠀⠀⠞⠁⠈⠁⠀⠀⠈⠀⡠⠞⠁⠀⠈⠷⢤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⡤⠞⢹
⠀⠀⢻⠳⢄⡀⠉⠙⠒⠲⠤⠤⣤⣀⣀⣀⣀⡈⠁⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣠⡤⠤⠴⠒⠚⠉⢁⣠⠔⣻
⠀⠀⠘⡆⠀⠈⠙⠒⠶⠤⢤⣀⣀⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣀⣀⣀⡠⠤⠴⠒⠛⠉⠀⢀⡏
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⢁⡞
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡶⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⢔⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠭⠭⣭⣍⣉⣉⣩⡭⠭⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⣀⡸⠀⠀⠀⠀⠀⠀⠀⢎⡀⡱⠀⢀⣄
⠀⠀⠀⠀⠀⠀⡃⠈⡄⠀⠖⠲⠀⠀⢀⠴⡄⠀⠀⢠⠀⠄⠀⡌⠀⡂
⠀⠀⠀⠀⠀⠀⡭⢭⠀⠀⠀⢸⠀⠀⠨⣀⠸⠀⠀⠐⠀⠀⠀⢨⢭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢰⠀⢦⠀⠀⠸⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠊⠉⠑⢄
⠀⠀⠀⠀⠀⢀⡄⠠⠒⠒⠀⢸⠉⠉⢹⠀⢘⠉⠉⠙⠀⠒⠂⢆⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⠀⠸⠀⠀⠈⠉⠁⠀⢐⠀⡇⠈⠢⡀⠀⠀⢀⣤⣤⣒⡀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠷⢀⠀⠐⠚⠀⠀⠀⠀⠀⠀⢸⠀⣨⠀⠀⠀⠀⠀⠀⠘⠒⠃⢀⠴⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠒⠂⠀⠐⠒⠒⠂⡞⠅
⠀⠀⠀⡂⠀⠁⠒⠀⠤⠤⠀⣀⣀⣀⣈⣉⣁⣀⣀⣀⡀⠠⠤⠄⠐⠊⠁⠀⡇⠀⠀⠰⠆⠀⠀⠀⠤⠤⠄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡆
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐⠒⡏⠄
⠀⠀⠀⣉⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢈⠀⠀⠀⢐⠒⠂⠀⠠⠤⠤⠤⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆
⠀⠀⠀⠈⠉⠒⠠⠤⢀⣈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⢉⣀⠠⠤⠒⠈⠨⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠈⠉⠁⡏
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠠⠤⠄⠄⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆
⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠐⠒⠒⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠒⠒⠒⠒⠒⠒⠂⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣄⠤⢀⡤⠠⢄⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠉⡠⠔⢒⡠⠶⠺⣦⠴⠦⣴⠳⠶⢄⠀⠩⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⠊⡠⠊⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠑⣄⠈⠢⡱⡀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀
⠀⠀⠀⢠⠃⡔⠁⠀⢀⠃⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠈⠆⠀⠑⡘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⢀⠆⡜⠀⠀⠀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠐⡈⠆⠀⠀⠰⡁⠑⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢁⠇
⠀⠀⠎⡜⠀⠀⠀⢠⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠐⡈⠄⠀⠀⠋⢲⡐⠢⠤⠬⣉⣉⠈⠉⠀⠁⣉⣀⣀⡉⣁⠠⠀⢐⡴⠙⠆
⠀⠈⡶⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⡀⠀⠀⠰⡆⠀⠀⠁⠦⠌⣁⠐⠂⠀⠄⠀⠠⠨⠤⠀⠀⠠⠀⠀⠒⣈⣁⣰⠋
⠀⢰⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠀⠀⠀⠈⠒⠤⡉⢈⡁⠀⠐⠒⠀⠀⠀⠈⠉⢉⣀⣁⠤⠒⠁
⠀⠈⠒⠀⠄⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠰⠂⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠂⠉⠂⠤⢀⣀⠠⠔⠊⠁⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠞⣩⣷⣶⡡⠚⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⢊⣵⡶⣮⠔⠣⢤⣀⣀
⠀⠀⠀⠀⠀⠀⡞⠉⠥⠄⣠⡾⢤⢤⣼⡀⠀⠠⣌⠁⠉⢳⡀⠀⠀⠀⢰⠋⠡⠤⢀⡴⠧⠤⢤⣇⠀⠀⢤⠁⠈⠑⡆
⠀⠀⠀⠀⠀⠀⢿⣓⠲⠶⠤⠤⠤⠤⠤⠥⠤⠤⠶⠒⣚⣿⠀⠀⠀⠀⠸⣗⣲⠶⠤⠤⠤⠤⠤⠬⠤⠤⠴⠶⣒⣻⠇
⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⡹⠃⠀⠀⠀⠀⠀⠹⣍⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⢿⣸⢸⠿⡸⣯⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣯⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠘⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠤⠦⠄⠉⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠦⠴⠀⡆⠐⠶⠖⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠋⢀⣼⣿⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣆⠀⠀⣀⠁⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣯⣿⣿⣦⣀⠈⠉⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢿⡏⠀⣀⠨⣃⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠋⠁⠀⠈⠉⠉⠉⠁⠀⠈⠙⢿⣿⣿⣶⣶⣇⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢸⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⣷⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⡾⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢠⡇
⠀⠀⠀⠀⣠⠶⠚⠉⠈⠳⣄⠀⠀⠉⡛⠯⡍⠙⠚⠉⠀⢀⡴⠋⠉⠓⠦⣄
⠀⠀⢠⠎⠀⠀⠀⠓⠲⠦⣤⣉⠓⠺⢬⣍⣥⣤⠤⠴⣚⣉⡤⠤⠖⠒⠂⠈⢳⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁


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

⠀⢐⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢢
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠄⠀⡀
⡠⠿⠿⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠿⠧⡀⠀⠀⠀⠄⠊⣥⠴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠒⠲⠦⣄⠐⠄
⠉⢐⠂⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⡒⠀⠀⠀⠀⠀⠃⠄⠉⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠶⠖⠚⠋⣠⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠤⠄⠙⠃⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠷⢤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠃
⠀⠀⢸⣿⡿⣿⡿⣻⢟⢿⡟⢻⡟⣿⢻⢛⠻⡟⠛⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀
⠀⠀⢸⣿⣿⡎⣴⡿⢸⢸⢹⣿⡇⣿⢸⠘⣡⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⢸⣿⣿⣇⣿⣷⠸⢸⡟⢦⣇⣋⣾⢰⣹⣿⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠠
⠀⠀⠀⡋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⢀⡀⢀⠀⡀⠀⠤⠂
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⢉⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠁⠀⠀⠀⠐⠀⠀⢐⠠⠀⠀⣭⠐⠒⠈⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⢿⠆⡀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⠀⠀⠀⠀⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠔⠠⠐⠁⠑⣄⠠⢀⡀⠣⠀⠒⠂⠐⣂⠨⠔⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⡄⠀⠀⠈⠐⠂⠉⡈⠂⠄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡀⠀⠀⠀⠀⠈⠑⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠄⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠡⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠁⡀⠤⠀⠁

⠁⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉⡽⠻⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡇⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉
⣀⣸⣇⡤⠀⠀⠀⠀⠒⠒⣲⣿⡉⢀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠁
⢨⠟⢯⠀⠀⠀⠀⠐⠂⠘⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡠⠀⠄⠀⠀⠀⠀⠂⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠀⠐⠶⣼⣧⠴
⠀⠀⠀⠀⠀⡄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠠⠞⠉⣇
⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠑⠀⢸⡇⢐⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⡆
⠄⠀⠀⠀⠀⣀⣧⠄⠀⠀⠀⠀⠀⠀⢤⣄⣼⣀⡥⠀⠀⠀⠐⠒⠒⣾⣟⠉⠀⠀⠀⠀⠀⠀⠾⠂
⠀⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠐⠂⢠⠞⢻⡅⠠⠀⠀⠀⠐⠀⠞⠀⠘⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡇⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⡷⠒⠁⠀⠀⠆⠀⠀⠀⠘⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡀⠈⢓
⠳⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠰⠆⠀⠀⠀⠺⠀⠀⠀⠀⠲⢦⣿⡤⠆⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡆⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠋⠹⡀⠐⠀⠀⠀⠀⠘⣺⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣇⣨⠄
⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⢠⠀⠀⠐⠆⠀⠈⠉⢉⡿⣯
⣾⡖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠋⠀⠈⠃⠀⠀⣄
⠁⠙⠈⠀⠀⠀⠀⠹⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣀⣧⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⠋⠁⠀⠀⠀⠀⢀⠜⠻⡄⠄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠁⠀⣿⣠⠤

⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠎⠀⠴⣾⣿⣿⣇⠀⠑⠈⠛⠒⠀⠀⠐⠀⠀⠀⠀⠞⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠤⠔⠚⣱⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⢧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⠏⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰
⠀⢀⡠⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠄⠀⠀⠉⠉⠈⠁⠀⠀⠤⢀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⣾⡿
⣠⠋⠀⠀⠀⠀⠀⠓⢀⠀⠀⠀⠀⠛⠢⢀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠁⠒⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣡⣾⣿⠗
⡄⣠⡤⠤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠉⠉⠉⠉⠀⠋⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠁⡄⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⣀⣢⣤⣤⣶⣶⣦⣀⣶⣿⣿⣿⡥⠐
⠀⠀⠀⠀⢈⠀⠀⠀⢀⠀⢀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠓⢦⡄⠊⠋⠘⠁⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⢿⢿⣿⣿⣿⣯⣤⡤
⣄⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⣤⢦⡀⠀⠀⠀⠀⢀⡀⠀⢀⣩⣿⣿⣏⠻⢿⣿⣿
⠉⠁⢄⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡄⠠⠒⠉⠀⠀⠀⠀⣾⡿⠉⠀⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠑⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⣷⣆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠠⠓⣄⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⢀
⠀⠀⠀⠀⠀⢿⠙⠒⠒⠒⠀⠀⠳⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⣸⡇⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠁⠈⠙⠷⠀⠄⠀⠀⠀⠐⠲⠤⢬⣿⣿⣿⡇⢀⣿⣇⣠⣤⠤⡀⠊⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠒⠂⠀⣠⣤⣿⣿⣿⣷⣸⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⣠⠃⠀⠀⠺⠆
⠀⠰⠀⢠⡀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⡿⠿⣿⡆⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣰⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⡀
⠀⠐⠠⠼⢷⣠⠀⠀⢀⠠⠀⠘⠁⠀⠀⠀⡀⠀⠀⠀⠀⢉⡉⣤⠶⠟⣡⠂⠀⠈⠻⡄⠀⠀⠀⠀⠉⠉⠓⠀⠀⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚
⠀⠀⠀⠀⠈⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠩⠄⡔⠁⠀⠘⠁⠀⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⠀⠀⠀⢀⣠⣰⣶⣤⣄⣇⠀⠀⠀⠀⠀⠄
⢀⡴⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠈⠻⠙⠻⠰⠦⢤⡄⠀⠀⠀⠀⠈
⠉⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⠈⠓⠊⠧⣀⠠⠤⠴⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣈⠀⠀⣤⠀⡀⠀⠀⣠⣮
⠀⠀⠀⠀⢶⠠⠤⡀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⢿⣆⢀⣻⣦⣵⣶⣿⠟⠉
⣀⡀⠀⠀⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠹⣦⣷⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠛⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⡎⠀⠀⠀⢠⣴⣿⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢟⠿⣠⢀⠀⠁⠈⠁
⠀⠀⢀⣠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠊⠀⠀⠐⠦⢄⡀⠀⠀⠀⢽⣿⣿⣦⣄⡇⠀⣰⣿⣿⡿⢁⠤⠀⠂⠀⠀⠀⠐⠂⠀⠞⠀⠈⠀⠘⠳⢌⣀⠒⠂
⠙⠀⢄⠀⠀⠀⡀⠀⠀⠐⠄⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣀⠀⠙⠻⣿⡏⠁⠀⣿⣿⣫⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣿⣿⣿⣶⡄⢤⠚⠒⣨⣿⣯⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠘⠀⠀⠀⠀⠒⠛⠉⠉⠀⠐⠚⠛⠙⠉⣻⡿⢿⠇⣀⣶⠚⣵⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠕⣫⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡾⣋⠔⣡⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣴⠟⠉⠉⠻⣆⣠⡾⣫⢞⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⠃⠀⠀⢀⣤⣿⡫⢚⣵⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⣠⡴⣾⣿⡿⢋⣴⣿⡿⠆⣀⣿⠇
⠀⠀⠀⠀⠀⢼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⣡⣾⣫⣫⡴⠟⣋⣥⣶⣿⠛⠁
⠀⠀⠀⠀⠀⠈⠛⣿⡷⠶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣴⣞⣛⣛⣛⣛⣛⣉⣉⣭⣵⡶⠟⠛⠉⣸⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠒⠒⠢⠤⠤⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣼⢀⣠⠴⠚⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣿⠁⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⠤⠤⣼⠗⠋⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠷⠶⠶⣶⣶⣶⣤⣤⣤⣤⣾⡿⠟⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠀⠐⠆⠀⠀⠀⢐⠁⠀⠈⣡⠕⠀⠀⠀⠸⢏⣙⡆⠀⠀⠀⠀⠀⢠⡆⠤⠀⠀⢤⠀⠀⠀⢀⡠⠜⠿⢦
⠀⠀⣈⠙⠓⣾⣛⣷⠀⢰⣿⡿⢿⣂⣸⡄⠀⠀⡀⠀⠀⠀⣸⣄⠀⠀⠠⣷⣦⣽⣷⢆⣰⡆⠀⠀⠀⣾⡟⠒⠚⠈⢰
⠀⠀⠦⡀⢘⣫⡭⠶⠀⠈⣿⠂⠂⠈⠉⠀⠀⠀⠰⢸⢸⣿⢯⡇⠀⠃⠀⠀⠹⠿⠀⠀⠻⠆⢄⡤⠀⢸⡷⣦⡄⠀⣼
⠀⠀⠁⠀⠬⠷⠚⠉⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠊⠁⠀⠐⠀⠀⢀⣀⡀⠀⢠⣤⣤⡤⠀⢸⣄⣈⡁⠀⣈
⠀⠀⠀⠀⢀⢀⠀⠤⢀⡢⠀⠀⠀⢠⣾⣛⣛⡿⠞⠀⠀⠀⠀⠀⠀⠀⠤⡅⠈⠧⠺⠖⠶⠶⠶⠖⠀⠈⠉⠉⠉
⠀⠀⠀⠀⢸⣿⣟⡶⣶⢤⣌⡀⠀⠀⠐⠀⠤⢍⣙⣷⠀⠀⠠⠔⠀⠀⣤⣶⣢⠀⠀⠀⠀⠀⢀⣚⣁⠚⠻⢶⢂⣰⠄
⠀⠀⠀⠀⠀⠷⣿⣿⣻⣏⣫⡟⠀⠀⠯⣉⡓⠢⢤⣙⠀⠀⠁⠀⠀⠀⡠⣾⡏⠀⠀⠀⣄⡀⣉⣉⠉⢀⢠⠼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠈⠑⠺⠗⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠙⠂⠐
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡀⠀⢹⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⠄⠀⠀⠀⠂⠤⡀⠀⣨⡿⠋⣠⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁⠀⠀⣠⠂⠀⠀⠀⣠⣿⠯⡷⢂⡔⠁⠀⠸⡌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⠀⠀⢀⡠⠔⠲⣶⣿⣷⠛⠉⠉⠻⣅⠀⣠⠇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠑⠚⠉⠁⢀⣠⣭⣿⣿⣊⣄⡀⠀⠈⠉⠁⠀⠀⢸⠋⣉⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠰⠦⣀⡐⠲⣲⢺⠋⣿⡿⠋⠉⠉⢿⡍⢦⣄⣀⣀⣀⡴⢫⡮⣐⣤⡀
⠀⠀⠐⠀⠀⠀⢀⣤⠒⠀⠀⠠⡈⠉⠛⠁⠂⢀⣿⠁⠀⠘⠷⢸⣧⠈⠻⣉⠩⠁⠀⢿⡴⢥⣄
⠀⠀⠀⠑⠝⢋⣠⠔⠶⠲⠤⡀⠈⠲⣤⣤⣤⣮⡿⠀⠀⠀⢀⣈⣿⣧⣄⣀⢀⣠⢾⠏⡰⣁⠈⢀
⠀⠀⠀⠀⠀⠌⠁⠀⠀⣀⠀⠈⠳⢶⠴⣶⡿⢻⣧⣀⣀⣀⣀⣀⣿⢯⠻⡋⠭⠄⠂⢀⡿⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡴⡺⠉⢩⡯⣷⣄⡀⠀⠀⣡⣾⡟⠉⠉⠉⣿⣿⡜⣄⠑⠦⢤⠤⠖⡩⠀⠀⠀⠀⠀⣠⠂
⢃⠀⠀⠀⡠⡊⠀⠠⠠⠃⠉⢀⣚⣛⣻⣛⣿⣿⡁⠀⠀⠀⣸⣿⣳⢌⢳⣥⡴⠖⠛⠲⠀⠀⠀⠀⠖
⠀⠀⠀⠁⠀⠁⠀⠑⠄⠀⢀⣴⡿⠟⠛⠛⠉⠁⠈⠀⠘⠉⠑⠁⠉⠙⠉⠉⠙⢦⡀⣀⡲⠆⠀⠉
⠀⠀⠀⠠⠀⠀⠀⠊⠀⡨⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠐⠈
⠀⠀⠘⠀⠀⠀⠀⢀⡼⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠐⠬⡙⢶⡄
⠀⠀⠀⠀⠐⠀⠐⠉⠀⣼⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠢⠁⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡖⠒⠠⣀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣉⢳⡀⠀⠀⠳⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⠹⡀⠀⠀⠀⠀⣀⠠⠤⠄⠀⡔⠱⢄⡾⠻⣗⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠆⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⢀⡼⢈⢁⡞⠁⢀⢹⡌⢦⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸⠀⠀⠀⠀⢸⡏⠀⣠⠤⠤⢤⣞⠐⣤⡟⠀⠀⠀⠀⢷⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⢆⠀⠀⠀⣸⠇⡜⠀⠀⠀⠐⢤⡽⣻⠄⠀⠀⠈⠠⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠂⢠⣞⠥⠀⡇⠀⠀⣀⡑⡖⣵⣋⡀⠀⠀⢀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⢯⡤⢖⡿⠃⠀⠀⠀⡸⠁⢿⠭⠉⡟⠶⣄⠑⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠘⠛⠉⣀⣀⢤⠓⠖⠁⠀⠀⠀⣰⠃⠀⠈⢳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⡀⠀⠀⠀⠀⠚⠠⠀⠀⠀⣠⠞⠁⣀⣠⣤⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡏⠀⠈⠲⢄⠠⣉⠃⠐⠀⠀⠀⢿⠐⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠒⢇⠀⠀⠀⠐⢵⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⢀⠀⠠⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣷⣫⠟⠉⠛⢦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠛⢁⣤⣤⡀⠀⠀⠀⢀⡀⣠⠴⠛⠁⡌⠉⢁⡀⠀⠀⠈⠑⢲⠜⠁⣤⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⠉⠀⠙⢶⡄⠄⡉⣟⠡⠀⠊⢀⣄⠀⣸⣷⠀⠀⢀⣴⠇⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⠀⠉⢶⣄⡤⠀⣠⠄⢈⣿⣀⣿⣿⡖⠊⠀⠁⠀⠠⢇⣼⣿
⠀⠀⠀⢠⣦⣀⡀⢈⡴⠁⠀⠀⠀⠀⠀⢈⣿⠋⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⠄⠀⢀⢡⡶⢋⢉
⠀⠀⠀⠀⠈⠙⠻⣮⣼⠷⢤⣀⠀⠀⣠⣾⠗⠀⣀⠈⠀⢨⣿⣿⣿⣿⣿⣿⣶⣿⢠⣶⣤⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠲⣬⡁⠀⠀⠉⠷⢶⣿⣃⡀⣀⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⢀⣴⠄⠈⠙⢶⡀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣉⣀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠛⠋⠲⠦⣀⡀⠉⠲⢦⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠈⠛⠉⠀⠀⠀⣠⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠷⣶⣤⣽⣿⣿⣿⣿⡿⠿⠟⠿⠛⠋⠀⢀⡄⢀⡀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢁⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠴⠿⠟⠉⠀⠀⠈⠀⠀⠀⠊

⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤⠄⠀⠤⠀⠀⣀⡀
⢀⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣀⠀⠀⠀⠀⠀⠄⢀⣨⣤⠴⠶⠶⠶⠒⠒⠒⠒⠶⠶⠶⠤⣬⣀⠀⢄
⠁⠀⡄⠀⠀⠈⠉⠭⠭⠍⠉⠉⠉⠉⠉⠉⠉⠭⠭⠭⠩⠥⠀⠀⠠⠄⠈⠀⠀⠀⠀⡦⡀⠘⠳⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠞⠁⡠
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠛⠉⠁
⠀⠀⢸⣿⣿⣙⠿⢋⡾⢩⠹⡏⣌⣿⢹⡏⣿⢰⢸⣯⢉⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣿⣿⣿⡆⣿⡇⢿⢆⡇⡿⣿⢸⡇⣿⠈⣼⣿⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⣿⣿⣿⣇⣿⣧⡸⣸⣇⣃⣿⣌⣱⣿⣴⣸⣿⣨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠿⢿⣿⣿⣷⣶⣶⣶⣿⣿⡿⠿⠛⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⢸⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠉⠁⠀⠒⠂⠀⣀⠀⠔⠈⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣤⣄⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠂⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠀⡀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⢀⡀⠄⢊⠄⠀⢀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠄⠀⠔⡈⠨⠈⣦⠔⠄⢄⠀⠂⠄⠁⣀⠉⠁⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢔⠀⠀⠀⠁⠐⠈⠀⡌⠁⠠⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢈⠂⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠐⢈⠀⢂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡈⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠐⠂⠀⢀⠀⠊

⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠰⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠀⠀⠤⠴⣿⣿⡷⠶⠄⠀⠀⠀⠀⠀⠀⠴⡚⠘⣧⠌⠀⠀⠀⠀⠀⢰⣂⠒⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢤⣠⣦⣤⡲⠂
⠀⠀⠀⠀⠀⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠓⠘⠦⠀⠀⠀⠀⠀⠀⢈⣁⣴⢤⣈⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠻⡆⠄
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣀
⠀⠀⠀⠀⠐⠀⠀⡂⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠛⠁⠀⠉⠀⠀⠀⠀⠐⢀⠈⣤⡤⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢷⣂⠐⠛⠃
⠀⠀⠀⠀⠈⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠨⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⢘⡭⠺⠗⡦
⠀⠀⠀⠀⠀⠀⠁⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠄⠀⠀⠰⠆⢀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠰⠄⢰⡇⣄⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠰⢊⣥⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⣅⣄⣀⠀⠀⠀⠀⣤⣼⣿⣤⠄⠀⠀⣠⠞⠀⠀⠀⢀⡀⠀⠻⠀⢤⠄
⠀⠀⠀⠀⠀⢤⠐⡒⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡤⠤⠄⠀⠀⠀⠀⠈⠉⣼⡁⢸⣿⠀⠀⠀⠀⠀⢸⣿⢿⣷⠛⠛⠋⠁⠀⠀⠀⠀⠀⠠⠒⡀⠰⠁
⠀⠀⠀⣜⡤⠠⠉⠃⠸⠿⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠟⠳⢼⡀⠀⠀⠀⠀⠚⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠈⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠆⡐⠂⠀⠠⠒⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⢠⣿⣀⣀⠀⠀⡋⠀⠀⠀⠀⠀⠴⠄⠒⠀⠧⠄⡤⠀⠀⠀⠀⢠⡤⠴⠾⠿⠶⠖⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣾⣿⣿⡿⠋
⠀⠀⠙⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⣛⠛⠛⣭⠄⠀⠀⠀⠀⠁⣓⢻⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⣿⣧
⠀⠀⠐⠋⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠂⠀⠀⡑⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⡄⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠐⢛⣳⣼⡯⠶⠛⠁⠀⠀⠀⠄⠤⠀⠀⠀⢠
⠀⠀⠉⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢨⠠⠐⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⡖⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⣧⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠐⠠⢠⠄⠀⠀⠀⠀⣰⠶⠂⠈⢡⡿⠄
⠀⠀⠠⡀⠊⡡⠊⢁⢀⠀⠀⠀⠀⠀⠀⠉⢺⣉⠀⢫⡴⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⠀⠀⠀⠀⠀⢀⣼⠇⡠⠢⣄⣣⡀
⠀⠀⠁⠀⠭⠀⠈⠙⠉⠠⠌⠀⠀⠀⠀⠀⣤⡴⠛⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠀⠁⠆⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⣠⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⣤⠞⢉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣠⠴⠞⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡴⢞⣫⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⢀⣤⣶⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣫⡥⠶⠶⣦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⣭⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣀⡴⣛⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣰⡟⠁
⠀⠀⠀⣠⢞⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡟⠁
⠀⣠⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣴⠞⣡⠟
⣼⢫⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣁⡴⠋⣡⡾⠃
⣿⠈⠷⣤⣤⣤⣤⠶⠾⠛⠉⣀⡴⠞⠉⣠⡾⠋
⠘⢷⣤⣄⣀⣠⣤⣤⠶⠞⠛⢉⣀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠻⠿⠿⠿⠿⠶⠶⠶⣶⣦⣄⡀
⠀⢀⣠⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⢹⣿⠆
⠀⢾⣅⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠟⠋⠀⢠⣿⣇⡀
⠀⠀⠙⢷⣦⣍⠛⠛⠿⠿⠶⠶⢶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠟⠛⠛⠉⠁⠀⣠⣤⣀⠀⠀⠈⠙⣿⡇
⠀⣴⣾⢛⣋⣡⣤⡶⠀⠀⣀⣤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⢀⣾⠟⠙⠛⢿⣶⡶⠾⠟
⠀⠈⠙⠋⠙⣿⡿⡷⠶⠚⠋⠻⣷⣄⣴⠟⠙⢷⣤⣀⡀⣸⡿⣿⣦⣀⣀⣿⣿⣦⣤⣶⣤⠾⠿⠷⠾⠋⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠛⠋⠁⠀⠉⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⣀⣴⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⢠⠾⠛⢿⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⢀⣴⣶⡄⠻⣧⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⣿⡃⠀⠙⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⢿⣿⠛⠛⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⣤⡄⠀⠀⢷⡄⠀⠈⢿⣦⠙⠃⠀⠀⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣤⠙⢶⣤⡾⠛⠃⠀⠀⣴⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠠⠞⠙⣧⠀⠀⠀⠀⠀⢀⣼⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠁⠀⠈⢧⠀⠀⣴⡶⢶⡌⠻⣦⣀⢀⣼⠀⠀⠀⠀⢠⣾⠃⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠈⢧⣸⣏⠀⣸⡇⠀⠀⠉⠉⠁⠀⠀⠀⣴⡿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⢤⣤⣤⡶⠾⢿⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷⠀⠀⠀⣀⠀⠀⣠⡿⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠘⣿⣦⠀⠀⠉⠛⠛⠋⠁⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣴⣶⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣤⣤⣶⣶⣶⣶⣶⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠛⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⣿⣿⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣿⣿⣿⣷⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣾⢷⣿⣿⣿⣿⣿⣿⡼⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣮⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣵⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⣿⣿⣿⣿⣿⢿⣫⣟⢿⣿⣿⣿⣿⣿⣷⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣿⣿⡿⣻⣵⠿⠋⠛⢿⣮⣟⢿⣿⣿⣿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⣿⡿⣟⣵⡿⠛⠁⠀⠀⠀⠀⠈⠻⢷⣯⣻⢿⣿⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣚⣿⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⡓⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⢯⣭⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⢉⣇⣴⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⢀⣀⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⣉⣠⣤⣤⠶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠉⠉⠻⣿⠉⢠⡟⠁
⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋
⠹⢴⣤⣤⣤⣤⣶⣶⣞⣻⠭⠭⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠛⠋⠉⠙⠛⠛⠛⠉⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⠀⠈⡟⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⠓⢦⣀⠀⠀⢻⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢁⡞⠁⣀⣠⠞⠉⠉⠙⠲⣬⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⢸⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⣀⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣀⡀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⢟⡽⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣶⣿⣷⣦⣤⣤⣤⣤⣤⡤⠶⠞⠛⣡⣮⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⠿⢿⣲⠦⣄⣀⠀⢀⣜⡵⠁⠀⡇
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠓⠾⣽⣿⠏⠀⠀⢀⡇
⠀⡴⠚⠉⠉⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢰⡁⠢⢄⣀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠙⠲⠦⢬⣉⣑⠂⠠⠄⣀⠀⠀⠀⠈⠉⠳⢦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠬⣍⣒⠠⠄⡀⠀⠈⠙⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⢴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢍⡒⠀⠀⠀⠙⠻⣭⣭⠭⠝⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⢉⣩⣭⣷⣶⣤⠤⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣾⠀⠋⠁⠀⠀⠀⠀⠀⠀⠑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢬⣓⠶⢤⣤⣤⣤⣤⢴⣪⠵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠋⠉⠉⠉⠉⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣤⡴⠶⠶⢶⠶⢤⣤⣀⣀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⣼⣡⠖⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠇⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⠀⣀⣠⡿⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⣆⠀⠈⣿⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⣾⣿⡄⠀⢹⡛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠭⡿⢷⣦⣼⣧⠌⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠴⠶⠏⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠙⠉⠙⠛⠲⠶⠶⠶⠶⠶⠒⠚⠛⠋⠉⠉⠉⠀⣀⣀⣤⣤⡴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠉⠒⠒⠒⠒⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠴⠖⠚⠛⠛⠛⠙⠉⠉⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠐⣀⠀⠀⠀⠈⠻⣦⣀⣤⡰⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠒⠒⠂⠤⠤⠤⠔⠒⠁⠀⢠⡤⣼⠻⣏⡷⣺⢤⡄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⢾⡛⣯⣽⣻⣾⢶⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣾⣶⣷⣿⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⢀⣠⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⡀⢀⣠⠤⠊⠁⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠶⠶⠤⠤⠴⠶⠶⠶⠒⠋⠉⠁



⣶⣿⣿⣶⣿⣷⣾⣾⣷⣿⣶⣶⣿⣶⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣿⣶⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⡟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠿⠁⠀⠀⠀⣠⣤⠀⠀⠀⠈⠀⠀⠀⠛⠃⠀⠀⠀⠿⠿⠁⠀⠀⠀⣠⣤⠀⠀⠀⠉⠀⠀⠀⠛⠃
⠀⠀⣀⡀⠀⠈⠙⠋⠀⠀⠀⣶⣷⠀⠀⠀⣀⣄⡀⠀⠀⠀⣀⡀⠀⠈⠙⠋⠀⠀⠀⣶⣷⠀⠀⠀⣀⣄⡀
⠀⠠⠿⡿⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠙⠛⠀⠀⠀⠠⠿⡿⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠹⠛
⠄⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢲⣿⡖⠀⠀⠀⢲⣶⠄⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢲⣿⡖⠀⠀⠀⢰
⠀⠀⠀⢲⣶⡄⠀⠀⠀⣠⣄⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⢲⣶⡄⠀⠀⠀⣠⣄⠀⠀⠈⠁⠀⠀⠀⠈
⣶⠄⠀⠈⠉⠀⠀⠀⠀⠛⠛⠀⠀⠀⢰⣾⡄⠀⠀⢰⣶⡄⠀⠈⠉⠀⠀⠀⠀⠛⠛⠀⠀⠀⢰⣾⡄
⠈⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⢀⠀⠀⠋⠉⠀⠀⠀⠈⠈⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⢀⠀⠀⠋⠉
⠀⢴⣷⡄⠀⠀⠈⠀⠀⠀⢰⣿⡷⠀⠀⠀⣤⣶⠀⠀⠀⢴⣷⡄⠀⠀⠈⠀⠀⠀⢰⣿⡶⠀⠀⠀⣤⣶
⠀⠘⠉⠁⠀⣤⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠘⠉⠁⠀⣤⣦⠄⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠁
⣀⣄⠀⠀⠀⠉⠙⠀⠀⠰⣾⡖⠀⠀⠀⣠⣀⠀⠀⠀⣀⣤⠀⠀⠀⠉⠙⠀⠀⠰⣾⡖⠀⠀⠀⣠⣀
⠿⠿⠁⠀⠀⠀⣠⣤⠀⠀⠀⠉⠀⠀⠀⠛⠃⠀⠀⠀⠿⠿⠁⠀⠀⠀⣠⣤⠀⠀⠀⠉⠀⠀⠀⠛⠃
⠀⠀⣀⡀⠀⠈⠙⠋⠀⠀⠀⣶⣷⠀⠀⠀⣀⣄⡀⠀⠀⠀⣀⡀⠀⠈⠙⠋⠀⠀⠀⣶⣷⠀⠀⠀⣀⣄⡀
⠀⠠⠿⡿⠂⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠙⠛⠀⠀⠀⠠⠿⡿⠂⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠹⠛
⠄⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢲⣿⡖⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢲⣿⡖⠀⠀⠀⢰
⠀⠀⠀⢲⣶⠄⠀⠀⠀⣠⣄⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⢰⣶⡄⠀⠀⠀⣠⣄⠀⠀⠈⠁⠀⠀⠀⠈
⣶⠄⠀⠈⠉⠀⠀⠀⠀⠛⠛⠀⠀⠀⢰⣾⡄⠀⠀⢰⣶⠄⠀⠈⠉⠀⠀⠀⠀⠛⠛⠀⠀⠀⢰⣾⡄
⠈⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⢀⠀⠀⠋⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣷⣧⠀⠀⠀⢀⠀⠀⠋⠉
⠀⢴⣷⡄⠀⠀⠈⠀⠀⠀⢰⣿⡶⠀⠀⠀⣤⣶⠀⠀⠀⢴⣷⡄⠀⠀⠈⠀⠀⠀⢰⣿⡶⠀⠀⠀⣤⣶
⠀⠘⠉⠁⠀⣤⣦⡄⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠘⠉⠁⠀⣤⣦⡄⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠁
⣀⣄⠀⠀⠀⠉⠉⠀⠀⠰⣾⡖⠀⠀⠀⣠⣀⠀⠀⠀⣀⣤⠀⠀⠀⠉⠉⠀⠀⠰⣾⡖⠀⠀⠀⣠⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠒⠈⠉⠁⢀⣈⣢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡶⠿⠋⣉⣭⠶⠛⣋⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠈⠉⠛⢛⠟⠛⠭⢠⡄
⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⣢⣀⠀⠀⣾⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠓⠒⠦⢤⣄⡀
⠀⠀⢀⠴⠊⠉⠀⢔⣮⡅⠻⢮⣿⣿⣿⡶⢤⣤⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣌⣬⣵⢦⠄⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⣶⣾⣿⣭⠵⠶⠛⠋⣉⣉⣿⠿⠿⢿⣯⣯⣷⣒⣶⣧⣄⣀⣀⣿⠛⠉⠁
⠀⠀⠈⠢⢄⡀⠀⠈⠐⠀⠀⢴⣿⠿⠖⠒⠛⣩⣽⣛⡥⠴⠟⠛⠙⠛⠓⠒⠋⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⣀⣀⣀⣀⣀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠁⠈⠁⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠃⠀⠀⢀⠴⠚⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠠⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⠄⣠⢼⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠂
⠀⠀⠀⠀⠀⢀⡀⢁⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣤⠼⠴⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⣯⣿⡇
⠀⠀⠀⠀⠀⢘⠦⠸⠏⠀⠀⠀⠀⠀⠬⠿⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⠁
⠀⠀⠀⠀⠀⢨⣤⡤⠬⣥⠀⠀⠀⠀⢤⢤⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠈⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣼⡄⠐⠚⣆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡄⠀⠀⠀⠀⣠⣽⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠒⠲⢾⣿⣿
⠀⠀⠀⠀⠀⡧⠀⠀⠀⠉⠉⠀⠀⠀⢀⣤⣤⣾⡏⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⣿⡿⢆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠉⠉⠿⢦⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡐⠂⢄⣀⡀⣀⣁⡤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢰
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡼⠀⠀⠀⠀⠀⣤⠗⠀⠀⠀⠈⠉⠙⠻⢿⣿⡿⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢨
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠴⠖⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⡀⠀⠀⠀⠀⠒⠄⢀⣀⣀⣀⣀⣀⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⣠⣴⣶⡿⠿⠛
⠀⠂

⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⡒⠒⢄
⠀⠀⠀⢠⠒⡉⢀⠠⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⡃⠀⠀⠠⠀⠈⠀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣰⢦
⠀⠀⠀⠰⠀⠂⠀⠀⠐⠀⠄⢠⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⢇⠀⠉⢀⣇
⠀⠀⠀⠀⢣⠀⢰⠉⠂⠚⠀⠀⠃⠔⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠉⠁⠈⠣⡀
⠀⠀⠀⠀⠀⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⡴⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢔⠞⠝⠀⠣⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠠⠃⡌⠀⠀⠣⡈⠁⠒⢽⢄⡠⡀⣀⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣀⣤⠁⠀⠃⠀⡇⠀⠀⠀⠘⡄⠀⠀⠣⢕⠁⡇⠓⢖⠊⡦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠈⠩⠀⡸⠆⣀⢃⠀⠐⣄⠀⠀⢀⣄⣀⠴⣗⢤⠭⠑⢆⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠬⠉⡾⠥⢓⣄⡠⠢⣉⠔⠫⣉⠜⢅⠔⢀⣀⢴⡤⠵⢤⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⢹⠒⠆⠨⠇⠀⢠⠀⠀⡄⠀⠰⠀⢠⡡⠔⠈⠉⠀⠀⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡆⠘⡀⠸⠀⠈⡀⠀⠄⠀⡄⢀⠊⠀⢀⠀⢠⠦⣔⡑⠤⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣘⠀⢀⡀⢸⡤⠓⠠⣇⠀⠃⠀⠀⠀⠇⡆⠠⠀⡀⠱⡋⠀⠀⠀⡈⠃⠀⢀⡀⢐
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠙⠀⠈⠂⠀⠁⠀⠀⢸⡀⠐⠀⠀⡐⠀⣧⠲⠠⢄⠀⠈⠙⠉⠁⠀⠀⡠⠁⢈⠇
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⠇⣀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠓⠊⠑⠤⠊⠉⠈⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠄⠒⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣖⣒⣲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⠿⠟⢋⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡿⠛⢿⢦⡉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣐⡒⠂⢈⣥⣶⠟⠋⠉⣉⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠈⠉⠉⠉⠛⣉⠱⠄⠂⠈⠰⣠⣾⡀
⠀⠀⠀⡀⠄⣒⣯⣵⠶⠚⣉⣤⣶⣾⠟⠋⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢈⣁⡀
⠀⠀⡈⢰⢯⣿⢋⠁⠀⠈⠉⠛⠋⠁⠀⠰⠾⠿⠿⠿⣿⣿⣭⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠆⠀⠀⢠⠏⠀⢻⣿⣿
⠀⠀⢿⣈⡸⢏⠘⠃⠠⣤⣤⣭⢤⣄⣀⣀⡀⠀⠀⠐⡿⠋⣀⠀⠐⡴⣾⣶⣄⠀⠀⠀⠀⢰⡿⠋⠈⣱⠀⠀⠀⢀⡐⠁⠀⡜⢸⣿⡿
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣈⢙⠛⠺⠿⠽⠧⠃⠂⠀⠀⢀⠰⣼⠀⠀⣷⣶⠆⡟⠁⡤⠀⠀⠎⠀⠀⣀⡟⢀⣤⣶⣿⣿⣿⣾⣧⡾⠋
⠀⣠⣤⣄⣀⣶⣌⠙⠣⠭⠽⠿⢿⣿⣿⣿⣷⣶⣦⣤⣬⣍⣀⡸⣦⣭⣴⢞⠁⠀⠁⠀⠄⠀⠀⡄⣿⣵⣿⣿⣿⣿⣿⣿⡿⠋
⣾⣿⣿⣿⣿⣿⣿⣷⣦⣅⣀⡀⠒⢶⣦⣤⣤⠭⠍⠉⠉⠑⠚⠀⠈⠁⠀⠀⠀⠀⢀⠌⠀⠀⠔⡿⢹⣿⣿⣿⣿⣿⣿⠟⠁
⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⣀⠀⠀⣀⣠⣴⣀⠀⡄⢀⡼⠇⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣸⠀⣸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠁⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠐⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢀⡄⢠⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⣆⠀⠀⠀⠀⠀⠂⠀⠀⣀⠄⠀⠀⠀⠀⢑⣶⠄⣀
⠀⠀⠀⠉⠓⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠔⢊⠍⡖⠀⠀⠀⠀⡀⠐⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠠⢠⠀⠄⠘⠐⠀⠀⣩⡁⠀⠆⠑⠂⠖⠩⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⢀⠀⢀⠀⠒⠀⠴⢀⠂⠁⠨⡶⠤⡴⠒⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡿⢎⠽⣇⢵⣾⠔⡄⡠⠀⠀⠈⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠈⠐⡄⠀⡀⠀⣠⠀⠀⠁
⠀⠀⠀⠀⠂⢄⠂⠀⠀⠀⠈⠉⠑⣄⠀⠀⠀⠈⠰⢥⠿⠀⠀⠡
⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀⠀⠀⢀⣬⠆⠠⡆⠀⡼⢨⠄⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⠋⢺⢻⡇⠀⠉⠘⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣴⡤⣚⠀⡘⢘⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢤⠥⠇⠐⢺⡌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⡤⣀⠀⠀⠀⢼⡇⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠔⠀⢲⡄⣜⡤⠈⠐⠉⠙⠘⡷⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⣏⣸⠆⠑⠂⢃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠆⠀⠀⡇⡧⠄⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡷⣷⡦⠀⢀⣀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⡆⣇⡀⢉⠉⠂⡀⠟⠉⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠂⠠⠄⠀⠀⠀⢀⣀⣄⠀⣿⣯⣁⣨⣋⡖⣭⡤⠂⣀⡆⠀⠀⠀⣠⡠⡘⢓⣁⢀⠒⣆
⠀⠀⠀⠘⠁⣶⠀⠀⠀⣠⢘⡈⠉⠅⠘⠻⠿⠿⠟⠓⠛⠀⠿⠿⠃⠀⢀⠄⠁⠐⠀⠇⠺⠊⠙⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⡖⣲⡶⢶⣶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢫⣍⠙⠻⠶⣦⣀
⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⢀⣀⡀⠛⢻⡏⠀⠀⠈⠙⠻⣦⣄
⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠖⠛⠉⣯⠀⠀⠀⠀⣄⠀⢶⠀⠸⣯⠁⠀⠀⢛⡀⠀⠀⠀⠀⠀⠙⢷⣄
⣠⡾⠋⠀⠀⠐⠒⠒⠒⠉⣩⠉⠀⠀⠀⠀⢀⣤⠀⠀⠘⣿⠛⣦⠀⠙⠓⠛⠀⢀⣠⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢻⣄⠀⠀⠀⠀⠙⠳⢶⣼⠃⠀⢴⡛⣷⠀⠻⢦⢶⡄⠀⠉⠀⢀⣀⣠⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠈⠻⢦⣄⡀⠀⠀⠀⠙⠂⠀⠈⠛⠋⠀⠀⠰⠶⠃⠀⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⢸⡟⠳⣦⣄⡀⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⢸⡇
⠀⠀⠀⠀⠘⣷⠀⠀⠉⠛⠻⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠋⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠶⠶⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠈⣿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢺⡏
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠚⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⠉⠛⠒⠠⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠐⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢦⡀⠉⠉⢛⡒⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⢷⣠⡴⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⣀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠇⠀⣠⣄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠠⣄⠉⠀⠈⠷⠾⠁⢼⣛⡃⠀⣼⡷⣦⠀⠀⣶⠀⣾⠀⠀⢹⡏⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠙⠲⢤⣀⡀⠀⠀⢹⡇⠀⠉⠀⠘⠀⠀⠙⠛⠋⠀⠀⠀⢀⣠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠁⠀⠛⠋⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠉⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠋⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⣠⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠁⢀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⢀⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠉⠀⠀⢀⣠⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⢀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⢟⣻⣯⣭⣍⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⣠⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣩⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⢀⠠⠶⢛⣩⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⣴⣶⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠠
⠀⠀⠦⢄⢀⠘⠂⠀⠀⠐⠀⣈⡭⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⢐⣢⡄⠀⠀⠀⠐⠀⢠
⢀⠀⠀⠀⣙⡀⠈⠀⠀⠀⠂⠀⠀⠀⠃⠀⠈⠀⠁⠀⠀⠀⠀⢚⡩⣥⡶⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⣒⡀
⠙⠛⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠐⠀⠀⢀⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣉⠀⠐⠤⢐⡀⢠⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡿⠀⠀⠀⠀⠀⠂⡤⠀⠀⠀⠀⠉⠛⠓⠲⠾⠭⠦⠀⠀⠘⠤⢄⣀⣀⡇
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⢀
⠀⠀⠘⠀⠀⢈⡁⠀⠩⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣟⣿⣿⡂⠀⢦
⠀⠀⠁⠀⠸⠋⠙⢛⠛⢤⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠘⠛⠉⠁⠀⠀⠀⢸
⠀⠀⠀⡠⢤⣮⣆⣠⠤⡎⠀⢀⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⢃
⠀⠄⠀⡇⠀⠀⠀⠈⢉⢲⠄⠀⠒⠠⠰⠖⠁⠀⠀⠀⠀⣠⠀⡀⠀⠄⠀⢀⠀⠀⠀⠐⡀⢸⠀⠀⠒⠀⠈⠆
⠀⠀⠀⣷⣀⣀⣀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⢌⣉⠙⠰⠂⠦⢤⡀⠀⠀⠉⢺⠀⠀⠀⠀⠀⠀⣀
⠀⠠⣀⡍⠠⠄⣀⣉⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣂⠱⠂⠀⠉⢕⣒⡒⠡⢬⠀⠀⠀⠀⠱⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠴⠬⣄⣉⣡⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠙⣿⣿⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⣿⠏⢠⡀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠟⠛⠛⠛⠛⠛⠛⠛⠋⢀⣾⣇⠈⠿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⡀⠒⢶⣶⣶⣶⣶⣿⣿⣿⣦⣤⣤⣤⣤⡀⠀⠉⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄⡈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⢋⣠⣶⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⠀⣸⣿⠿⢿⣿⠁⢠⣾⣿⠿⠋⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⣰⡿⢁⣠⡀⠻⡇⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⢀⣶⡿⠻⣿⣄⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⣠⣾⡿⠋⠁⠀⠘⢿⣷⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⣶⣶⣿⣿⠟⢾⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣰⣿⡟⣿⡟⠁⠀⠆⢹⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢻⡿⠁⠓⠤⠀⠀⢂⡁⢸⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⠈⠰⠀⠄⠀⠢⢸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣀⣣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⣹⠀⠸⠿⢋⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⡆⠀⠁⠁⠀⣒⠀⠉⠹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠹⠟⠈⠀⠀⣀⣥⣶⣦⢤⣑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠛⠋⣁⣠⣤⣾⣶⣾⣿⣿⣿⠿⠻⢷⡟⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⣸⣣⡇⡘⣿⡄
⠀⠀⠀⠀⠀⢀⣤⠄⣠⣶⣿⠿⠻⡻⢿⣿⣥⣾⣿⣿⣿⣟⣿⠿⢋⠉⠀⠈⠙⣿⣶⣅⠘⢷⣄
⠀⠀⠀⠀⠀⠘⠿⢫⣵⣿⡥⠒⠁⠀⠈⠻⡿⢟⣿⣿⣿⣿⣿⣿⣿⡿⣕⠢⢤⣾⡟⠁⣦⡀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⢿⡏⠰⣿⡿⠷⣨⣷⣼⡟⠆⠸⣿⣷⡜⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣾⢏⡿⠃⣸⠀⠀⠀⣀⣅⣀⣿⠛⣧⡄⠀⣿⡎⣿⡎⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢃⣾⣷⡶⣿⠀⠀⠈⠁⣟⣽⣿⢀⣿⣷⠆⣿⠃⣿⡿⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣾⣿⣿⢂⣿⣷⡀⣀⣰⣯⣽⣿⡇⢿⣄⢀⣿⣰⣿⡇⣸⡇⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⣾⣿⠿⠋⣼⠿⢿⣿⠴⢾⢿⠿⠿⠿⢘⣿⠟⢣⠟⣁⣼⣿⣿⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢼⣿⡿⠏⠀⣤⣤⡎⡾⢾⠗⠈⣟⠴⢦⣴⣶⣾⣧⠔⠡⣺⣽⣿⣽⣿⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⠟⠋⠀⠀⠀⠀⡜⣠⡟⠹⠷⠶⠿⡏⠉⣿⣿⠟⠁⠐⡈⢹⣿⣿⡿⠈⡀⠁
⠀⠀⠀⠀⠀⣘⠉⠉⠉⠛⠿⣧⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⣛⣽⠿⡷⠿⠿⠆⣶⠇⢸⡿⢾⡻⠋⣠⡼⠁
⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⡴⠋⠱⣦⣀⢀⣾⣿⣵⣾⣷⣾⣿⣶⣶⣷⣾⡆⠀⢹⣦⣶⡂⠀⡠⢞⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠀⠀⠀⠸⣿⠋⠡⠿⠛⠛⠋⠭⠭⠭⠿⠿⠿⢷⠶⠖⢿⣿⣿⣿⣟⣒⠑⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⣿⣥⠽⠬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠻⢶⣴⠿⢿⣄⠀⠀⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣶⣦⣶⡞⠁⠀⠀⢈⣷⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠖⠋⠉⠻⢿⣧⣬⣁⣠⣼⣿⠁⢀⣀⣸⡃
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠴⠉⠩⠭⢌⢻⣿⣿⡿⢿⡉⢉⣿⣃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠒⠖⠃⠀⠀⠀⠀⠀⠁⣗⣀⣶⡆⢛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣸⠂⠀⠀⠀⠀⠀⠠⣿⠅⣿⠇⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠤⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣿⣆⠀⠀⠀⠀⠀⢀⣽⣸⡟⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⠴⢒⣾⡷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⡙⣿⣿⣤⣤⣀⣴⣿⣿⣿⣵⠿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠋⠁⠒⠻⡿⠿⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⣿⣿⣿⣿⣿⣿⠿⠛⠑⠉⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⣄⡀⠀⣾⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⡗⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⢶⡖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣴⣷⡄⢛⣿⣿⣿⣖⣦⣼⣀⡀⠀⠀⢹⠀⠀⠀⣥⣾⡗⠆⠀⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠻⠿⢟⠛⠉⣉⣤⣶⣛⣻⡿⣿⠿⢿⣿⣿⣶⣦⣤⡏⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠎⣸⢀⣆⠠⣤⣀⡀⠀⠀⠀⠀⠀⠈⠒⠰⠤⢄⣀⣀⠙⠉⠁⠀⠐⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⣴⣯⣀⢨⣦⢈⣁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢿⣖⠲⣶⡤⣀⡈⠛⠛⣩⣟⣛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠺⠾⢇⣀⣿⣶⡙⢦⠛⣿⡶⢆⠉⠉⠁⠀⡈⠳⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠁⠀⠀⠀⠀⠀⠀⠙⢀⣬⣏⣆⣀⡠⠥⣦⠷⣦⡄⢠⠀⠀⢽⡇⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣁⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣨⣟⣁⡙⣿⣳⢶⣿⢻⠘⠀⠀⢸⡇⠠⡅⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⣭⣿⠛⠿⢾⣿⣦⣭⡿⢀⡁⣰⣿⣿⣿⣧⣌⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣂⣾⣿⣿⡶⠶⣞⣿⠷⠉⢙⣲⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠈⠀⠸⠿⣓⡲⠶⠶⠾⠟⠻⠉⠉⢠⣶⡿⠿⠷⠞⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠄⣀⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠒⠒⠂⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⢠⡶⢶⡀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⠀⠀⠈⠳⣤⣷⠀⠀⠀⠀⢠⡞⠁⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣸⡀⢸⣤⡀⠀⠀⣠⣄⡀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠼⠯⡀⢙⡇⢸⡀⠙⠶⠞⠁⠈⠙⠋⠁⠀⠈⢷⡤⠖⠛⠖⢶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣄⡞⠛⠁⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠉⠛⠒⠶⠶⠦⢤⣤⣤⣤⣤⠤⠤⠶⠶⠖⠛⢉⠉⢉⣀⡴⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠙⠒⠲⠷⠦⠤⠤⠤⢤⡤⠤⠤⠤⠤⠴⠶⠒⠚⠋⠉⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡤⠶⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠳⠎⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢠⣄⠀⣀⠀⠀⠀⣀⠀⢀⠀⣀⣀⣄⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠻⣌⢋⣽⢞⡉⡞⠹⢧⡿⣧⣿⡏⢢⡿⡄⣼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⣯⠰⣜⣡⢧⣶⣸⣇⣈⣿⣰⢦⣽⣇⡇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠁⠀⠈⠀⠀⣠⡀⠉⠁⢀⣁⠈⠀⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣞⣛⣉⡉⠉⠋⡁⡼⠷⠾⠆⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠉⠸⠥⠟⠚⠻⠇⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⠦⣄⣀⣀⡴⢦⣀⣀⣀⣀⣀⣀⣀⣀⡤⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣤⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠘⣧⣈⣷⠀⠀⠀⠀⠀⣰⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣇⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡛⠙⢿⠀⣼⠛⢶⣄⡴⠛⠛⠶⠶⠋⠉⠙⣦⣀⣠⠴⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢉⡥⠿⠀⠺⠦⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢣⡀⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠛⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠶⠚⠁⣀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⠤⣤⣀⣠⣈⡉⠉⠙⠋⠋⠉⠉⠉⠉⠉⢀⣁⣀⣠⡦⠴⠒⠋⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠦⣄⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠶⠞⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡞⢦⡠⢆⣠⣤⣠⣤⡄⢤⡄⣶⣴⠒⢦⠖⠲⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠓⢦⡴⡟⢡⡄⡏⠘⢳⣞⠿⢿⣇⡁⢿⡇⠼⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⠆⠙⠦⠔⠻⠿⠾⠦⠤⠾⠶⠻⠞⠧⠖⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣀⣀⣴⣤⣀⣀⡤⠤⣄⢀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢯⣥⣤⠄⣀⢀⣤⢾⠛⠛⠃⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠘⠳⠋⠉⠘⠟⠀⠀⠀⢀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠲⠤⢤⣤⠞⠢⣤⣤⣤⣤⡤⠤⠤⠤⠴⠋⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⢿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣿⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠋⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢱⡤⠀⠒⠂⡛⠛⠛⠰⠄⠀⠀⠛⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⢣⠀⢰⠄⢀⣤⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠒⠾⠖⢾⠗⠉⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣀⠶⢘⡿⠀⠀⣼⠞⢷⣴⣶⣴⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⢠⠛⠀⠀⠀⣧⠀⣸⠏⠀⣀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⢰⠇⠀⠀⠀⠀⡿⢀⣦⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⢦
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣇⡀
⣶⠉⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠋⢉⣦
⠛⣅⠒⠒⠀⢉⣭⡅⠐⠒⠒⠒⠂⠀⣀⣀⣀⣀⣤⣤⣤⣩⣤⣤⣤⣀⣀⣀⡀⠀⠒⠒⠒⠂⠀⢨⣭⠁⠀⠒⠂⣩⠛
⠀⡟⠛⠉⠀⠠⠶⠄⠀⢀⣉⣉⣉⣉⣉⣉⠁⠈⣁⠀⠀⠁⠀⠀⠀⠈⣉⣉⣉⣉⣉⣉⣉⡀⠀⠠⠶⠶⠀⠉⠛⣿
⠀⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣼⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣷
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢿⡀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⠁
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠠⠆⠀⠀⠀⠀⠀⠘⣶⣿
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣎
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣿⣿⣇⠀⣤⠀⢠⣄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢠⡾⣿⣯⣿⣷⠆⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠈⢓⣤⣴⠟⠁⠿⠙⠋⠁
⠀⠀⠀⠀⢸⡃⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠘⢿⣿⣤⣼
⠀⠀⠀⠀⠀⠉⠒⠂⣉⠁⠔⠒⠒⠂⠀⠀⠀⢀⣀⣀⣤⣄⣤⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠖⢳⣄⣀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠙⠿⢿⣿⣿⣿⠶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣶⣶⣤⣤⣴⡶⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠊⡉⠙⠢⣄
⠀⠀⠀⠀⠀⢀⠔⠚⠉⠁⢀⠀⠂⢈⠀⢠⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠀⠀⠀⠀⢰⠁⠀⠆⠀⠂⠀⠠⠄⠈⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠃
⠀⠀⠀⠀⢇⠀⠄⠀⠠⠆⠀⠄⠀⠰⠂⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣩⣯⡢⡀
⠀⠀⠀⠀⠘⡆⠀⠆⠀⠀⠀⠠⠄⢀⡤⢤⡀⠀⢸⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠊⣹⡇⠀⠙⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡀⠠⠀⡠⠤⣀⢀⡼⠀⠀⠱⡀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠛⠦⠔⠤⠎⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠁⠠⠄⠈⢀⠁⠀⠂⠀⠘⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⣀⣴⢶⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣠⢴⠞⡽⠁⣿⡀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠊⡴⠃⡜⠀⠀⢷⠘⠦⢼⡀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡞⠀⠸⠁⠀⠀⠈⠢⡀⠀⠈⠑⢦⡷⣤⠖⡆⢀⣤⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢱⣈⡱⠃⢸⠤⠷⣆⠒⠛⣣⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⠤⠴⡞⠉⠉⣿⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⢅⠑⠂⠺⣄⢀⣈⣳⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⡄⢠⠃⠑⠦⠤⠧⣀⠀⢘⣦⣀⠀⣀⡤⠛⠒⠒⠋⢹⠎⠳⣏⠉⠉⠳⢄⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡪⠵⠐⠸⠿⠭⢍⡲⢄⣠⠖⠲⡄⡨⠛⢦⣀⡤⠒⢦⡰⠒⢄⣀⠴⣦⡤⠶⠳⣖⢢⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠂⡴⠚⢳⠤⣤⡈⣈⣣⠀⠀⢨⠀⠀⠀⡍⠀⠀⢠⠀⠀⢸⠁⣀⠷⠛⠊⠉⢉⠉⡉⠉⢑⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⡀⠀⡆⠀⢹⡄⠀⠘⡄⠀⠀⡇⠀⠀⣾⠀⠀⡾⠊⢁⠀⠁⠂⠀⠀⢀⡤⡀⠀⠀⠠⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢇⠀⢹⠀⠀⢧⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡞⠁⠤⠀⠄⣀⣀⣎⣓⡋⢀⣉⡒⣄⠐⠀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡞⠀⠀⠀⣀⣘⣄⡤⠓⠢⢜⡄⠀⢸⠀⠀⠀⠀⢀⠃⡸⠀⡄⠠⠀⠀⢧⠊⠁⠀⠀⠀⠀⢉⠇⢘⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⠹⠷⠉⠉⠻⡀⠀⠙⠂⠀⠀⠈⣷⠀⠀⡆⠀⠠⠀⢸⠀⣇⢀⡀⠀⣈⠀⠆⡙⠒⡦⠐⠒⠚⡁⠐⠀⢠⠏⠙⢲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠗⠒⠲⠶⠶⠶⠒⠒⠒⠚⢳⡘⠊⠈⢦⠀⢀⣀⡀⠂⠈⣠⣀⣀⠖⠊⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢀⡴⢥⣀⡀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠑⠉⠀⠙⠢⠞⠁⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣄⣀⣀⣀⡠⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢰⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⡛⠂⠈⠙⣷⡀⠰⣦⡴⢄⢀⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⣠⣶⣶⣿⣿⣿⣿⣏⠂⢄⠀⠈⠁⠀⠈⢿⣼⠋⠙⠙⢢⣀⣀⣼⣿⢷⣤⣀⠀⢀⡾⠿⢧⡄
⠀⠀⠈⠻⠻⠌⠻⠿⢉⣽⣿⣿⡄⠀⠀⠀⢀⣀⣤⣤⢀⡀⠁⠀⠀⢀⣁⣈⠉⠀⢀⣥⣾⣟⠣⠀⠀⠀⠐⠳⡤⠴⠤⢦⣀
⠀⠀⠀⠀⠠⡶⠾⣆⠈⠻⢿⠁⠀⠀⢀⣴⣾⣿⣿⣿⣇⠙⠳⠀⠀⠀⠉⠉⠁⠀⠁⠀⢀⣿⠀⠀⠂⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠿⣿⣿⣿⢿⡟⡩⠛⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠙⠯⠀⠰⠁⠀⣫⠿⠛⠙⠁⠀⠀⠀⠀⠲⣿⣿⡌⠀⠀⠀⢠⣿⣿⣇⣀⡄⠸⠁
⠀⠀⢸⣿⣷⣤⢤⣀⡀⠀⠀⠁⠀⢀⡀⠀⠀⠁⠤⠐⠁⢀⣶⣶⣼⣦⡀⠀⠀⠀⠀⠱⠿⢾⠂⠀⢸⣿⣿⣿⣋
⠀⠀⠀⠛⠉⠉⣻⣿⣿⣾⣷⣀⣄⢸⣿⡄⣠⡀⠀⠀⠉⠚⠿⣿⡝⠋⠙⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡀⣠⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣀⠀⢠⣶⣀⡄⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣦⣦⣀⣤⡀
⠀⠀⠀⠉⠙⠛⠛⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣠⡄⠀⣠⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣟⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠓⠉⠙⠋⠋⠉⠛⠛⠛⠿⠶⠿⠇⠀⠀⢠⣦⣼⡿⠟⠛⠛⠻⢿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡫⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡵⣫⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣠⠶⠶⢦⡀⢀⣴⣟⠕⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⠁⠀⠀⣀⣿⣿⢟⣵⣿⣋
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣠⣴⣾⡿⢋⣵⣿⡯⢉⣹⡇
⠀⠀⠀⠀⢰⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠟⢋⣴⣿⣏⡴⢟⣋⣥⣶⡟⠉
⠀⠀⠀⠀⠀⠉⢻⡿⠶⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣭⣭⣭⣭⡷⠾⠛⠉⢁⣿
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⢀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠂⠠⠤⠤⠤⠤⢤⣀⢀⣸⡧⠒⠊⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣟⡠⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⢹⡏⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣾⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣠⠶⠈⠛⠛⠛⠋⠱⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⣤⡄⠀⣴⡆
⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣇⠀⣿⣷⠀⣿⣿⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢹⣿⠀⣻⣿⡄⡏⢹⡇
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣿⡆⢿⢿⡇⠀⠸⣿
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⢸⢸⡇⠀⠈⣧⠀⠀⣿⡆
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⣼⢸⣧⠀⡸⣿⢀⡄⢹⡇
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣇⣿⢸⣿⠀⡇⣿⢸⡇⠘⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⡏⠘⣿⠀⡇⣿⣾⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⡇⡀⣿⡖⡇⢹⣿⠃⠀⢿
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡇⡇⢻⡇⡇⢸⣿⢰⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⢸⣿⡇⠘⣿⢸⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣷⡇⠸⣿⡆⠀⠿⢻⠀⢸
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⠾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣰⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣰⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⣀⣠⠞⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢸⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡛⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣴⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣇⠎⠠⡀⢃⡄⠀⠈⠀⠀⢃⠈⠅⠀⠀⠠⡋⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠟⠦⠬⠤⠿⠧⠖⠁⠁⡀⡏⠦⠤⠦⣤⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠁⠀⢠⣶⣾⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠘⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠋⢰⠁⠀⠈⣿⡂⣤⢃⠶⠯⠄⠀⣿⣇⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡙⠛⠉⠻⢿⣿⣿⡿⢿⠀⠀⠀⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠂⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠡⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠷⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣄⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣏⣿⣟⣃⡒⠠⢸⡇⠀⠉⠉⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⠓⢺⡒⠦⠀⠾⠎⠉⠁⠈⠀⣄⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢫⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠃⠀⠉⢘⠛⠛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣽⠿⠟⢻⣷⣾⠧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡀⠀⠀⢨⡾⠤⣀⡀⠀⠀⢀⣀⣨⡇⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠰⠀⣀⣆⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠀⣉⣀⣀⡹⠿⠛⢿⣿⣿⣿⣿⣿⡾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣶⣤⠤⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⠀⢀⡾⠵⠥⠤⠴⠲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣶⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠂⠠⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠭⠭⠉⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠓⣀⣀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠣⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⠀⠀⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠤⢃⡜
⠖⠀⠀⠀⠀⠄⠀⢀⣀⣐⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠔⠊
⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠁⠁⠀⠀⠀⢃
⠀⠀⠈⢦⠀⠀⠀⠀⠀⣀⣀⣀⡴⢣⠖⣂⡠⠀⠑⢄⠀⠀⠀⠀⠀⢀⣀⣠⡶⠀⠡⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠒⠀⠀⠀⠻⠛⠛⠒⠁⣤⣾⡡⣇⠁⠀⠀⠀⠈⠛⠒⠒⠒⠚⠛⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠾⣹⠒⠙⣡⡟⢟⡙⠐⠤⢀⣀⡠⣤⣠⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠭⠗⢲⣞⣻⣏⢙⡬⠹⠁⠒⠒⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠽⠙⠒⠊⠀⠀⠀⠀⠈⠢⠄⣀⠀⢀⡀⢱⡀⠄⠐⠊
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠛⠛⠉⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠴⠶⠒⠒⠒⠛⠛⠉⠉⠉⢳⡀
⠀⣞⠋⠉⠉⠀⢀⣠⡤⠶⠶⣶⠶⠶⣶⡦⣤⣀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣞⠋⠉⠉⠀⢀⣠⠤⠶⠒⣲⠶⠶⢶⡦⢤⣀⠀⠀⢳⡀
⠀⠸⡆⠀⠀⡴⠋⣁⣀⣤⣤⢿⡇⠀⠀⠙⣦⣭⣷⡄⠀⢳⡀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡴⠋⢀⣀⣀⠤⢽⠇⠀⠀⠙⡦⢬⣳⣄⠀⢳
⠀⠀⢻⡀⠘⣷⠋⠉⠀⣀⣴⠟⠀⠀⠀⣰⡇⠀⢀⡿⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⣷⠋⠁⠀⢀⡠⠊⠀⠀⠀⢠⠃⠀⢀⡿⠀⠈⢧
⠀⠀⢸⣧⠀⠈⠳⢦⣟⣋⡀⠀⠀⣠⣾⣋⣤⠴⠋⠀⠀⣀⣸⠆⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⠳⢤⣚⣉⠀⠀⠀⣠⠔⣁⣠⠴⠊⠀⠀⢀⣸
⠀⠀⠘⣿⣧⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣤⣴⣶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⡏⣇⠀⠀⠀⠀⠉⠉⠉⢉⣉⣉⣡⡤⠴⠖⠒⠋⢹
⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⠿⠿⠿⢿⠿⣿⣿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠶⠖⠒⢒⣻⠿⠿⠿⢥⠤⣀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢻⣿⡼⣿⠟⠉⠀⣘⣧⣤⣿⠀⠀⣹⢿⣿⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠖⠉⠀⣈⣦⡀⢋⠀⠀⣙⢦⡀⠀⢀⡇
⠀⠀⠀⢸⣿⣿⠏⠀⢠⣾⣿⣿⣿⣿⣿⣟⣡⣀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠏⠀⢠⠊⠽⣇⢐⠀⠁⢜⣁⡀⢳⠀⢸⠃
⠀⠀⠀⠀⣿⣿⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⡁⠀⢸⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡞⠛⠀⢀⠴⠈⠄⣼⡀⠀⢸⠆⡾
⠀⠀⠀⠀⢹⡾⣇⠀⢿⣿⣿⣿⣿⣿⣽⡿⠁⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢆⠀⢣⣄⠀⠛⠀⠐⠅⡜⠁⠀⡜⢠⠇
⠀⠀⠀⠀⠈⢷⣛⢷⣌⡻⢿⣿⡿⠟⠋⣀⣤⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠳⣌⠣⠤⠨⠧⠔⠋⢀⡠⠊⢀⠞
⠀⠀⠀⠀⠀⠀⠉⠛⠾⢿⣿⣶⣶⣾⣿⡿⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⢭⣕⣒⣒⣒⣯⡭⠴⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⣋⣁⣤⣀⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣾⡭⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣽⠟⢹⠃
⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢠⡟
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⠼⠖⠋⠀⠀⠀⣠⠟
⠀⠀⣇⠠⢠⣤⠤⠤⠴⠒⠛⠉⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠙⠷⣦⣄⣀⣀⣀⡀⠀⠀⢀⣀⣀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⣄⠰⠾⡂⢤⠐⠢⡀
⠀⠀⠀⠀⣀⣠⡖⢯⢤⡜⣼⠳⣤⣄⣀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠼⠁⣤⠇⠘⢁⡤⠃
⠀⣴⠚⠉⠀⡠⢿⠠⠀⠀⠀⠀⣥⠀⠀⠉⠓⢄⠀⠀⠀⠀⠈⢳⣴⠋⠀⠱⢴⠋
⠀⠳⣄⠀⠀⠑⠀⠱⠤⠄⠒⠈⠉⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⢸
⠀⠀⠀⠉⠓⠒⠲⠆⠤⠤⠤⠴⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠚⠘⠂⠀⠚⠘⠓⠀⠀⢀⣤⡖⠖⠚⠙⠛⠒⠒⢒⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡋⠀⠀⠀⠠⠀⢀⡀⣀⣀⡹⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠯⢑⠦⠈⢩⠮⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡻⢦⣀⣀⠀⠀⠀⠀⢀⣀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣼⣶⢊⡉⠓⣷⡘⣄⡸⢧⠀⠀⠀⠀⠀⠀⠀⠘⢦⡁⠪⠸⣀⠍⠉⢸⣉⡩⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⡼⢋⣍⣰⣷⣊⡈⡇⠘⠤⣿⠁⠀⠀⠀⢠⡀⠀⠀⠀⠈⠛⠒⠦⠤⠤⠤⠤⠖⠚⠁
⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⡀⠀⠹⣄⠈⠁⠙⠈⠁⠋⠀⠀⣠⠃⠀⠀⠀⠰⠁⢈⡢⠄⠐⢒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢹⠀⠀⠈⠳⣤⣠⣤⣤⣴⡶⠞⠁⠀⠀⠀⣀⠥⠄⣌⠀⣀⡠⠞⠚
⠀⠀⡔⠒⠀⠀⠀⢀⣀⣀⣸⣦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠁⠀⠐⣄⢢
⣠⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⢠⡀⠀⠀⠀⠀⠀⣀⣿⠀⠓⢄
⠈⠀⠄⠠⠀⢀⣀⣀⣀⠤⠬⠭⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠑⠃⠀⠀⠀⠀⠈⠉⠉⠡⠀⣸⠂
⠀⠀⠀⠀⢀⠴⠛⠁⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⡀⣀⢀⣀⠴⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡋⣹⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣩⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣬⣿⣟⣻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⢽⣿⣿⣟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣶⣟⢿⣿⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣻⢶⡿⠉⠸⣿⠟⢻⣻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡞⣿⣶⣲⣿⣿⣿⢿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⠇
⠀⠀⠀⠀⡆⠀⠀⠀⢀⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢰⣷⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠏⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡟⣾
⠀⠀⠀⠀⠐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⡤⢰⣶⣿⣿⣿⢿⣿⡟⠻⠛⠟⣉⣹⠀⠀⢼⣿⣿⣷⡘⣤⠀⠀⠀⠀⠀⣤⣄⢠⣤⡦
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡤⡁⣸⣿⣿⣿⣇⡸⠁⢲⠆⠉⠀⠸⠀⠻⡀⣾⣿⣿⣿⠇⢻⡆⠀⠀⠀⠰⣿⡏⢺⡿⠁
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠹⣿⡿⣿⣿⣄⣀⠀⠚⠛⠀⢄⣀⠸⣿⡟⣽⣿⡜⢫⠈⣗⠀⠀⠀⠀⢹⡟⠼⠁
⠀⠀⠀⠀⡄⡇⠘⣇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡁⢿⣇⣿⡿⠋⠀⢳⠐⠂⠀⠀⠀⠁⣬⣇⣿⢻⠂⢺⠀⣻⡆⠀⠀⠀⢸⣋⠂
⠀⠀⠀⠀⠈⣿⠀⢨⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢠⡟⣸⡿⢃⣄⠀⠀⠀⢻⣶⡆⠀⠀⠀⠉⢿⣿⠈⢳⣾⠀⠉⠀⠀⠀⠀⢘⢈
⠀⠀⠀⠀⠢⣅⡶⠈⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠛⠈⣿⣶⣀⠀⢸⣿⣿⡄⠀⣴⡇⢸⣿⢀⠈⣿⠀⠀⠀⠀⠀⣀⢈⡻⠄⠀⠀⣶⠁
⠀⠀⠀⠀⠀⢻⡅⠸⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⡄⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⠘⣧⠘⢠⠿⡀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⣇⣴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠡⠘⣾⠀⠃⠀⠀⠀⠀⠀⠀⠙⣮⡁⣰⠏
⠀⠀⠀⢣⡀⠐⣧⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⡇⣹⠀⡦⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿
⠀⠀⠀⠀⢳⣤⡄⠁⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⣼⠇⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡇⢸⡄⢡⡀⠀⠀⠀⢠⠀⢀⠀⢻⣧⠿⡆
⠀⠀⠀⠀⢸⣿⡽⢡⡟⠀⠀⠀⠀⠀⢀⠘⠀⠀⢠⣦⡿⢡⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⡼⣧⣿⡇⠀⠀⠀⣾⣴⠃⠀⢸⣧⣿⣷
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⡈⠃⠀⠀⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⠀⠸⡇⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠸⣿⣿⣿⠛⣧⡄⠀⠀⠀⠙⢶⡴⢠⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⣿⣶⠟⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠸⣷⣀⣘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣏⢿⡶⣄⠀⠀⣼⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢉⠋⢿⠜⠋⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⢟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣹⣿⣿⣤⡀⠈⠄⣿⢏⠻⣿⢻⣿⣿⣿⣿⣿⠷⣶⠀⣻⡇⠐⣊⡾⠁⣿⣿⣿⣷⣿⣿⡇⠀⠀⣦⢰⡏⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⠿⠿⣶⣀⣀⣛⣿⢿⣿⢥⣿⣿⣿⣿⣿⡿⣿⣸⢿⣆⡈⣧⠰⣌⣿⣿⣿⣟⣽⣿⠇⠀⢀⣿⣿⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⡿⢟⣷⣼⣿⣿⣿⣍⡹⣾⡏⠻⢿⡿⣧⠐⣿⢿⣿⣿⣿⣿⣿⡆⢠⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠿⠛⠿⠿⠿⣿⣷⣿⣿⣏⣻⣹⡾⣿⣿⡦⣥⠘⣿⡼⢿⣿⣫⣭⣶⣿⣟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠟⠈⠉⢉⣹⣿⣿⣿⣿⣧⠆⣾⠷⠛⠛⠉⠉⠉⠉⠙⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠛⠛⠻⢿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⢿⣿⠀⠀⠀⠀⢠⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠸⣿⣧⣀⢀⣴⣾⡿⠉⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠈⠻⣿⣿⠟⠋⠀⠀⣠⣶⡦⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡀⠀⠀⠀⠀⢻⣿⡄⠀⢶⣿⡿⠛⠁⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⢠⣿⡿⠋⠉⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⡦⠀⠀⠀⠀⠘⢿⣿⣄
⠀⠀⢸⣿⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀
⣰⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣼⣿
⠈⠛⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⢠⣾⡿⠿⠟⠋
⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠉⠉⠉⠉⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣀⣀⣀⣀⣤⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠛⠛⠛⠛⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣠⣤⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠉⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣤⣤⣤⣴⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠶⠶⠶⠶⠒⠒⠲⠶⠶⠶⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠠⠄⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠄⠒⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠢⠤⠄⣀⣀⢀⣈⡉⠉⠉⠉⠁⠉⣉⡀⣀⣀⡠⠤⠔⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻⣴⠏⢀⡀⠀⠀⢀⢀⠀⡀⢀⣀⠀⣰⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⣟⢹⢴⡏⣿⢸⡇⣇⣿⠉⠁⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⠀⠉⠋⢈⣟⣿⠈⠋⠉⠙⠀⠀⠈⠁⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⠀⠀⠈⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠐⠀⠀⠀⠀⠀⡤⠂⠀⠁⠀⠉⠳⢆⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡔⡜⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠱⡀⠀⠀⠀⡠⢈⠜⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⠴⠁⣀⣂⣂⠡⠔⠁⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠔⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠂⠈⠠⣄⠀⡄⠀⠀⡚⠁⠀⠐⠀⡨⠀⠀⠀⠀⢸⡟⠙⡆⠀⠀⠀⠀⠀⠀⢀⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣠⢶⣶⣤
⠀⠀⠀⠒⡤⢤⣴⡼⣦⠀⢀⣴⣷⣶⡤⠀⢘⡀⠀⠀⠀⠀⠀⠈⠉⢡⣄⠀⠀⠀⢰⣦⣌⣛⣦⣤⠀⣤⠀⠀⠀⠀⢠⣷⡦⠤⠤⠗⢠
⠀⠀⡙⠂⠀⣴⠿⢞⣛⠀⠀⢿⡟⠉⠉⠓⠚⠃⠀⠀⠘⡄⢠⢠⡔⡾⣿⠃⢰⡀⠈⠛⢻⣿⠟⠉⠺⣯⡂⠀⠀⠀⢘⣿⣀⡀⠀⠀⢸
⠀⠀⢈⠑⠀⠶⢟⣋⡭⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣷⣇⡏⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠑⠊⠀⠘⡟⠙⠿⠀⠀⠿
⠀⠀⠀⠀⠀⠛⠉⠀⢀⠀⣀⠀⠁⠀⠀⠀⢀⣀⣴⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢠⢴⢢⡄⠀⡒⠖⠒⠒⡁⠰⢿⡤⢤⣄⠀⠴
⠀⠀⠀⠀⠀⣄⣁⠀⠰⠀⠄⢂⣀⠀⠀⠰⠾⠯⠬⣝⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠖⠆⠀⠓⠊⠛⠙⠓⠛⠛⢙⡁
⠀⠀⠀⠀⠀⢻⣽⣟⡿⢶⠖⣦⣤⠀⠀⠀⠒⠂⠠⠤⢀⣹⣇⠀⠀⢀⢔⠂⢐⢢⢲⣾⣵⠀⠀⠀⠀⠀⠀⣀⣙⣁⡛⠛⠛⣛⣀⣼⠆
⠀⠀⠀⠀⠀⠨⠷⣷⣿⣻⣏⡹⣭⠇⠀⠰⢯⣉⡙⠒⠤⣌⣙⠀⠀⠈⠀⠀⠀⠐⡶⣿⡏⠀⠀⠀⢠⣀⡀⢈⣉⠉⣀⣀⣤⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠋⠁⠀⠀⠀⠀⠉⠐⠲⠭⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠙⠛⠥⠬⠥⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠔⣒⣚⣉⣛⠒⠒⠒⠋⠁⢀⣠⠖⠋⠁
⠀⠀⠀⢀⠄⣊⡵⠚⠉⠀⠀⠀⠀⠙⢆⠀⢀⡴⠋
⠀⢀⠔⣡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟
⢀⠊⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡜
⡇⣸⣿⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣡⠎
⢧⠘⢿⣿⣶⣤⣤⣴⡶⠟⣋⠵⠋
⣀⣑⣦⣭⣭⣭⣭⣶⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⠋⠀⠰⣶⡞⢠⡠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣤⣾⣵⣤⠟⠀⠀⠀⠝⣷⣼⡇⠀⠀⢳⠂
⠀⠀⠀⠀⣀⡠⠚⢷⡀⠀⠀⠀⠉⠛⢿⢶⡀⠀⠀⠈⠀⣽⣇⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⢤⣾⣿⠀⠀⠀⠀⠲⢦⣆⠀⠀⠀⠉⠓⠀⠀⠀⠀⣸⢻⢣⣀⠀⠀⠀⠀⣤⠀⠛⠂⢀⣾⠃
⠀⠀⠀⠀⠙⢧⣀⡀⠀⠀⠀⠙⠛⠶⡄⠀⠀⠀⠀⢰⠆⠀⠈⡽⢿⣳⠄⠀⠀⠈⠛⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠳⠆⠀⠀⠁⠀⠀⠀⠀⠠⣤⡴⠋⠀⠀⢠⠀⠼⢃⣽⡇⠀⠑⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣴⠀⠀⢠⠂⠀⠀⠀⠀⢠⣿⡇⠀⠀⣤⠀⠀⢀⡆⠀⠈⠋⢁
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⡺⠋⠠⠵⠋⠀⠀⠀⠀⠀⠼⠿⣇⣀⠀⣿⡆⠀⢸⣿⠀⢀⡀⠈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢇⣤⣼⣿⠀⢸⣧⢠⣿
⠀⠀⠀⠀⣀⣀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣄⣿⡿⣺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢀⣉⡅⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠔⠂⠈⠀⠀⠘⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣶⠀⠀⠀⠀⠀⠀⠆⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣇⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠂⠐⢠⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠋⠀⠀⠀⠀⠈⠓⣄⠀⢀⠔⠉⠙⠦⠶⠂⠤⣄⣀⡇⣀⠤⢄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⡀⠀⢠⣏⣀⡼⠒⠉⠉⠉⠲⣄⣠⢤⡚⡿⠗⠒⠢⢭⡉⢦
⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢹⡏⠉⠳⡀⠀⠀⠀⢀⡤⠟⠀⠀⠀⠀⠀⠸⣎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡾⣅⣀⣀⣉⣑⣦⣀⣏⣀⣀⠀⠀⠀⣀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠭⠭⣷⠀⠀⢀⠇
⠀⠀⠀⠀⢀⡗⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢼⣁⡄⠀⢠⡄⠀⢠⡄⠀⠀⣤⠀⢠⡇⠀⠀⢸
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⢸⡗⢄⠀⢸
⠀⠀⠀⠀⠀⠈⡷⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⡇⠀⢸⡇⠀⢸⡇⠀⠀⢻⠀⢸⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢳⣀⣠⠞⢳⣀⡠⢄⡀⣀⠀⠀⣀⡴⡇⡇⠀⡇⠀⢸⠀⠀⢸⡇⠀⠀⢸⠀⢸⡇⠸⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠁⣀⡀⠙⢹⢂⡏⡝⠀⣇⢷⠀⡇⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠁⡇⣧⠊
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⢀⠀⠀⠂⡾⢸⡼⠁⠀⢸⡘⡄⢹⠀⠈⡇⠀⠀⠇⠀⠀⡞⠀⡜⢰⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⣀⣼⣁⣼⠃⠀⠀⠀⢧⠹⡀⢇⠀⢃⠀⠀⠀⠀⠀⠃⠰⢡⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠂⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⢣⡹⣄⠂⠀⠀⠀⠀⠀⠀⠐⡡⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡑⢀⡉⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠙⢮⡓⠤⣀⣀⣀⣠⠴⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠤⠄⢂⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡦⠤⠤⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠒⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡁⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠁⠀⠴⠁⠀⠈⢆⠀⠀⠈⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠠⠤⠤⠐⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⢈⣁⣈⣁⣐⡲⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢒⣊⠥⠐⠂⠉⠀⠀⢀⣀⠴⠁⠀⣼⠟⠭⢁⡒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢁⡤⠂⠁⠀⢀⣀⠤⠒⢂⢖⡡⠒⠒⠒⠲⡌⢢⠀⠀⠈⠑⢦⡀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢨⠀⢀⠔⠂⠉⠀⠤⠐⠣⠮⠄⠒⠢⢄⣠⣃⣎⣐⠒⠤⢀⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠭⣒⠤⠄⣉⠚⠤⣀⡀⠀⠀⠀⠉⠉⠁⠀⠀⠠⠤⠀⠀⠀⠁⣠⠔⠋⣀⠠⢔⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠒⠭⠅⣒⠬⠍⣐⠂⠤⠤⠀⠀⠀⠀⠤⠤⠐⢊⡩⠤⣒⠭⠑⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠁⠒⠨⢥⣒⡂⠤⠤⠤⠐⣂⠭⠔⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡠⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⠒⠢⠤⢀⣀⢀⣀⣀⢀⣀⠤⠐⠂⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣀⣈⣢⣔⢒⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢴⡢⢤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠙⡄⠀⠀⠀⠀⠀⠀⠠⠈⣤⠮⠄⠛⣨⡁⡖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠁⠋⢣⠖⠉⠑⡄⠀⠀⠀⠀⣩⠡⡄⠚⠠⢀⢻⠅⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⣀⠈⠑⠒⠁⠑⠤⠚⠀⠀⠀⠀⠀⢘⢁⡄⠮⠁⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡠⠔⠒⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠈⠁⠒⠂⠤⢀⣀⣀⡀⠠⠄⠒⠊⠁⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⢀⡀⠀⠀⣀⠤⠐⠊⠉

⠀⡤⣄⣀
⠀⠙⠒⠮⣝⣒⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠉⠙⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣭⣍⣙⣓⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⣟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣧⣀⣠⠤⢤⡔⠉⠁⠈⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⢀⡎⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣽⣦⡤⠶⠶⠦⣤⡀⢀⡴⠖⠒⠒⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⠀⢀⠟⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢻⡏⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢲⡀⠀⠢⠜⠀⠀⠀⠁⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⢀⣰⠏⠉⠁⠀⠸⣆⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡏⠓⠒⠶⢄⠀⢀⣀⡴⠧⠤⠤⠤⠤⠤⠤⠶⠖⢶⣀⡼⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⠀⠘⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⢀⢠⣶⣤⣤⣄⣀⣀⣀⣤⣤⠴⠞⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠉⢻⡇⠘⡶⠶⠶⠶⠶⠶⠖⠒⠚⠛⢹⡇⠀⠀⠀⠀⠀⢻⣳⢦⣄⣀⡤⣶⡵⠋⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣀⣛⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⡁⠀⠀⠀⠠⠤⠤⣤⣤⠤⠤⠶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢹⠀⠈⡇⠀⠀⡏⠉⡇⠀⠀⡇⢰⠇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠸⡆⠀⡇⠀⠀⡇⠀⡇⠀⢀⠇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⢻⠀⠀⡇⠀⡇⠀⢸⠀⣸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⡆⠘⡄⠀⡇⠀⡇⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⡇⠀⡇⠀⡇⠀⡟⢀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⣇⠀⡇⠀⡇⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⠃⠀⠓⠖⠃⠀⠳⠋⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣀⣀⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠛⠓⠶⠶⠾⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁
⣿⠿⠿⠿⣿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣛⣻⣿⣟⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⣼⢻⡆⣼⢿⠀⣿⢷⠀⠀⠀⢠⡾⡋⠉⠉⠙⢶⡀⠀⠀⠀⠀⠀⠀⢀⡟⣷⢀⡟⡇⢸⡛⡆⠀⠀⠀⣴⢟⠉⠀⠉⠻⣦
⠀⢀⡏⢸⡇⣿⢸⡀⣿⢸⡇⠀⢰⢯⡞⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣸⠁⣿⢸⡇⣷⢸⡇⢿⠀⠀⡼⡱⠃⠀⠀⠀⠀⠈⢧
⠀⢸⡇⢸⠇⡟⢸⡇⣿⠀⣷⢀⡟⡿⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡿⠀⣿⢸⠁⣿⢸⡇⢸⡆⣸⢳⠃⠀⠀⠀⠀⠀⠀⠘⣇
⠀⣸⠀⢸⣰⡇⠘⣇⣿⠀⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⡇⠀⣇⣾⠀⢸⣸⡇⠘⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⡄⢿
⠀⢿⡀⠈⠉⠀⠀⠉⠁⠀⣿⢸⡏⣧⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠘⡇⠀⠉⠁⠀⠈⠉⠀⠀⡇⣷⢸⡀⠀⠀⠀⠀⠀⢀⡇⣾
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡿⠈⣧⠈⠷⣤⣀⣠⣤⠞⠁⣼⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⠇⢹⡄⠑⢤⣀⣀⣀⡤⠋⢠⡏
⠀⠀⠻⣦⣀⡀⠀⢀⣠⡾⠁⠀⠘⢧⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⣀⣤⠏⠀⠀⠻⣄⡀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⢹⠃⠀⢹⡅⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡏
⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠸⡇
⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣇
⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⢀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹
⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡀
⠀⠀⠀⠈⢷⣄⣀⣤⠟⠀⠀⠀⠀⠀⠸⣧⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣴⠃⠀⠀⠀⠀⠀⢷⣄⣀⣠⡾
⠀⠀⢀⠀⢀⡉⡉⠁⠀⠀⣀⣀⣀⣀⣀⢀⣉⣉⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡀⣀⣀⣀⠀⢀⠀⠀⢈⠉⠁⠀⠀⠀⢀
⣿⣻⣿⣻⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠁⠀⣀⣀⣀⡀⠀⢀⣴⣦⡀⠀⢀⣴⡾⣿⣷⠆⣀⣀⣀⣤⣤⣤⣄⡀⠀⠈⠉⠻⣦
⠀⠀⠀⠀⢠⣿⠋⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠋⠙⠻⠿⠿⠋⠀⠘⠻⠿⠟⢛⣋⡉⠉⠉⠙⢿⣶⣶⣤⣠⣿⡇
⠀⠀⠀⠀⠘⣿⣷⣶⣿⣟⡁⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⡄⠀⠀⠀⢠⣴⣾⠿⠛⢿⣿⣤⠄⠀⣀⣠⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣡⣤⣤⣤⣤⣶⣶⣾⣿⣿⣻⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⣽⣿⣁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣴⣶⠿⣿⡶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢀⣿⠁⢿⣄⣴⡿⣣⣶⣶⡀⢠⣶⣦⡀⣦⠀⢠⡤⣿⠛⣿⡍⢹⡇⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⠀⠈⢿⡏⣼⡟⠀⢻⣇⣿⠃⠈⠁⣿⡆⢸⡇⣿⣦⣾⠇⠘⣿⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⡀⠀⣿⡇⣿⣇⢀⣿⡿⣿⠘⢿⣷⣿⠃⣸⡇⣿⡟⣷⡀⠀⣿⠄⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣻⣧⠀⠙⠃⠈⠛⠛⠋⠀⠻⠿⠿⠃⠻⠿⠿⠁⠛⠃⠈⠁⠀⠈⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⠀⠙⠻⣶⣦⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⡶⠟⠋⢁⣠⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⣶⣶⣬⣭⣽⣿⣿⣿⣿⣿⣭⣭⣴⣶⡾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠉⡉⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠤⠒⠊⠉⠑⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣶⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⢀⠤⠤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⢀⡤⠤⣥⠈⡀⡀⠀⠀⠀⠀⢩⠉⠙⠶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⠋⢀⣀⣈⣉⡉⠉⠉⠓⠲⠤⣼⣀⠔⠋⠓⢄⣀⣹
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⣿⣀⣀⠟⠋⠁⠀⠀⠀⠙⢦⣠⠤⠴⣖⣻⠟⠚⠛⠒⠲⢤⡈⢳
⠀⠀⢀⡴⠋⠉⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢻⡋⠉⠙⣆⠀⠀⠀⠀⠀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⢹⡞
⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⠒⠢⣄⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲
⠀⢀⠜⠂⠀⠀⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠂⠀⠈⠋⠉⠛⣶⠒⠒⠒⠒⠚⠒⠛⠲⠒⠲⢤⡀⣀⠴⠶⠦⡀
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⢘⠇
⣰⠃⠀⠀⠠⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⢾⡁⣤⠀⠀⢠⡄⠀⠀⢠⡄⠀⠀⢀⣤⠀⢀⣼⠀⠀⠀⢰
⣇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠓⣻⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⢧⡀⠀⠘
⠈⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⠀⢹⡆
⠀⠀⣯⠀⠉⠉⣡⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⢰⡇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠹⣦⣠⠞⠁⡙⣦⠴⠖⠶⣄⣤⡄⠀⣀⣤⡞⢻⠀⡇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⠀⡎⡧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂⣀⠄⠈⠁⡿⢰⠋⡾⠀⠸⡇⢷⠀⢻⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⢀⠇⣧⠞
⠀⠀⠀⠈⡇⠀⠈⠲⡀⠁⠀⠀⠤⢰⡇⣾⡼⠁⠀⠀⢷⠸⡄⠸⡇⠀⠘⡇⠀⠀⢸⡇⠀⠀⠀⡟⠀⢸⠃⡸⢰⡏
⠀⠀⠀⠀⢷⠀⠀⠀⠘⠢⠀⠀⢠⡟⢀⣿⠃⠀⠀⠀⠘⡆⢳⡀⢳⠀⠀⢣⠀⠀⢸⠀⠀⠀⠐⠃⠀⠎⢠⢃⡿
⠀⠀⠀⠀⢸⡀⠀⢠⡀⠠⠄⠒⠛⠶⢺⠇⠀⠀⠀⠀⠀⠹⣌⢳⡀⠣⠀⠈⠆⠀⠀⠀⠀⠀⠄⢀⠔⣠⢃⣾⠃
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠙⣆⠹⣆⠁⠀⠀⠀⠀⠀⠀⠈⠀⢀⡴⢡⡾⠁
⠀⠀⠀⠀⠀⣷⠀⠉⢀⣀⡀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⠦⣄⣀⣀⣀⣀⡤⠖⢉⡴⠋
⠀⠀⠀⠀⠀⢹⠀⠁⢠⣀⡀⠄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣌⣉⣁⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠸⡆⠡⢤⡀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⣇⠐⠤⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⢻⠰⠤⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢠⠄⠀⢠⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠉⠀⠀⡰⠋⠀⠀⠀⢣⠀⠀⠀⠈⠙⢲
⠀⠀⠀⠀⠀⠀⠈⢧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣤⣀⣀⣀⣀⣀⣤⡤⠤⠒⠊⠁
⣿⣿⡿⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⢰⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣾⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠁⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣸⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠒⠒⠒⠒⠲⠆
⠀⠀⠀⠀⠀⡀⠈⠢⠄⣀⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣯⣟⣿⠿⠿⢿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣅⠈⠲⢤⣄⣀⣀⣀⣀⡀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠢⠄⣀⣀⣀⡀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠈⠑⠲⠤⣤⣄⣀⣀⣤⣤⡤⠴⡇
⠀⠀⠀⠀⠈⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠋⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣭⣭⣭⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⢰⠀⡀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⣿⣿⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢣⣸⣀⣃⣸⣀⣇⣨⣁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠷⠿⠸⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿