Howdy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢷⡀⠀⠀⣠⣄⣠⠄⣾⠀⠀⡀⠀⢸⡄⢠⠏⠀⣼⡃⠀⠀⡟⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣠⠞⢻⡀⢰⡏⠉⣷⠀⢻⣀⣰⢷⣀⣼⠁⠸⢦⠾⠙⢧⣠⡜⣧⡤⢾⢃⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢷⡀⠀⠻⡌⠧⡤⠏⣀⣤⡉⠁⠀⠉⠁⣀⣤⡀⠀⠀⢀⣠⠤⠶⠒⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢠⠴⣀⢰⡏⠈⠻⣦⣀⣤⡾⠋⠀⢿⡆⢰⠏⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⣔⡆⠘⠑⠃⣼⠀⠀⠀⠀⠉⠛⠛⠃⠀⠸⣿⡸⣄⡀⣀⡴⠋⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⢳⡄⠀⠦⠊⠢⠄⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣿⡇⠈⠉⠉⢀⣴⢟⣛⣷⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢻⣶⣜⣉⠙⢸⣿⣄⣀⣤⡤⠼⠙⠦⣤⣤⣴⣿⣷⢀⣠⣴⣿⡣⠃⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⣿⣯⣙⠛⠯⣬⣭⣁⣹⣇⣄⣺⣭⣥⣬⠵⢞⣫⣿⣿⡁⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⣶⠿⣿⡿⠓⠦⢤⣀⣀⣘⣉⣀⣙⣃⣤⠴⠾⣿⣿⠋⠉⠻⣦⡀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢰⡿⠁⢰⡟⠀⠀⠀⣸⡃⠉⠉⠉⠉⣿⠁⠀⢀⣀⠈⠻⣧⠀⠀⠹⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⡿⠀⠀⢸⠁⠀⡰⢫⢭⣿⣆⠀⠀⠀⠸⠀⣾⣋⣉⣳⣄⠹⡇⠀⠀⢹⡆⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⡇⠀⠀⠸⠀⠀⠻⠃⠀⠛⠛⠀⢀⡿⡄⠀⠉⠀⠀⠀⢀⠠⡁⠀⠀⠸⣷⡟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⡀⠀⣀⣠⣴⡟⠰⠃⠀⠀⠀⡏⠀⣸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⣟⠛⠻⣤⣼⠛⠛⠋⠷⠾⠋⣩⠟⠀⠀⢠⠆⠀⢸⣧⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣶⠈⠳⢦⣄⣴⠲⣄⣾⠛⡶⠞⠁⠀⠀⠀⣾⠀⢀⣾⣿⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠸⣧⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣾⠇⣠⣾⣱⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠲⣤⣤⣴⣿⠷⢄⣀⣸⣄⠀⣳⣀⣠⣿⡴⢾⣿⡿⠟⢋⡿⢛⡿⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⣏⣙⣶⡆⢰⢰⢤⣁⡈⢩⢭⡀⡄⡶⢰⢸⡄⡇⠴⢻⡇⠘⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡯⠉⠘⡇⢸⢸⠀⡇⢹⣼⠀⡿⣿⡃⢸⢸⠳⣇⡠⠛⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡳⠻⢸⡆⣇⢸⣿⠞⠁⡇⢳⠸⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⡁⠛⢸⢸⠀⠀⠗⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡷⡌⣖⣦⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣶⠈⡇⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠃⡃⡇⠀⡀⠀⠀⠀⠀⣼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡼⢁⣤⠘⡆⠀⢠⡏⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣾⢀⡇⣾⢩⡀⠀⠀⠀⡛⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠃⡏⡸⢨⠇⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣠⠚⠁⢿⢰⢁⠧⢸⠅⠀⠀⢰⢠⢈⡇
⠀⣤⡀⠀⠀⠀⡮⢰⡵⢁⠎⠀⢸⠁⠆⠀⠀⠀⣀⠤⠒⡀⠀⠀⣓⢸⡰⢃⠞⢓⠂⢸⡘⢀⠰⠀⠀⢀⠏⡸⢐⠁
⠀⡇⢱⠀⠀⢀⡃⢌⣡⠎⠀⠀⡟⢨⠀⣀⠤⠊⣠⡆⠸⢰⠂⡆⡇⢸⠇⡘⠀⡎⢸⢸⠏⣲⢸⡀⢀⡌⢠⡗⢸
⠀⠹⢬⣉⣙⣩⠀⡞⠁⠀⠀⢠⠅⠿⠚⢁⠴⢆⢻⡇⡅⣘⠀⣧⡇⣾⠀⡇⢰⠃⡆⠜⢠⠻⡀⠦⠊⡴⢻⠇⡎
⠀⠀⠀⠀⠀⢸⢐⡇⠀⠀⡠⢞⠠⡴⡇⠜⠀⡔⢸⡇⣫⡇⠄⢽⢀⣿⠈⠧⠃⡸⢇⢀⡎⢀⣉⡚⡫⠤⠚⠐⢓⢒⣒⣒⣐⣩⠂
⠀⠀⠀⠀⠀⢘⠨⢃⡴⢊⡤⡦⢸⢸⡇⣅⢠⠇⡺⡇⠸⢁⡶⠤⠞⠀⠳⠤⢚⡡⠄⠞⣋⣩⠤⠔⠒⣲⢃⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡯⠘⢃⠔⠉⠀⢟⢸⠀⠳⣈⢋⡼⠁⠙⠒⠋⠀⠀⣀⠤⠒⣈⡡⠔⠒⠉⠀⠀⠀⠀⣰⢃⡜
⠀⠀⠀⢀⠔⠁⢼⠁⠀⠀⠀⠩⢸⠀⠀⠈⠉⠀⠀⠀⠀⣀⠴⢊⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⡴⢃⡞
⠀⠀⠀⠈⠶⡇⡽⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⢀⡤⢊⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣠⠊
⠀⠀⠀⠀⢰⠅⡆⠀⠀⠀⠀⠘⡄⢧⠀⠀⠀⢠⠋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠖⠁
⠀⠀⠀⠀⢸⠠⠇⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣎⣥⣚⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠳⣄⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠐⣦⠄⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣉⣉⣉⣩⡜⠁⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠒⠲⠤⣀⠀⢠⣮⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣠⠤⠤⣴⠿⠶⠶⠤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠠⣀⠀⢀⣼⣋⣺⣿⣿⡽⠿⢟⣛⣿⣭⣽⣶⣾⣟⣿⣭⣭⣿⣯⣭⣿⣻⣿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣌⣉⣉⣉⣉⣥⣤⣴⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠷⣶⣽⣚⣾⠿⢿⣿⣿⢿⡿⣟⣣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠋⠉
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⣴⡿⠿⠿⠷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠀⠈⣷
⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢻⡇
⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⠉⠉⠙⣦⡀
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠈⠻⢷⣤⣤⠶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣾⣶⣾⡿⠛⠩⣿⣍⣙⣿⣿⣦⣤⣤⣄⠀⠀⠉⠛⠷⣦⣄⣀⣀⣀⣀⣠⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡷⠏⠉⠁⠀⠀⣠⠾⣯⠉⠀⣹⢷⡶⢾⡷⠀⣀⣀⣠⣤⣼⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⡞⣩⣭⣛⠿⠁⠀⠀⠰⠛⠚⠿⠀⢰⣷⢿⣧⢸⣟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣶⣤⣤⣄⣀⣳⣤⣈⡿⠀⠀⣀⣤⠶⠛⠛⠛⠳⠦⣍⣉⣹⡧⣼⣧⣀⠀⠀⠀⢀⠀⣸⣤⣶⣤⣄⣄⣠⣴⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣟⣛⣿⣷⡴⠞⠋⠀⠀⢀⣠⣶⣶⣤⣌⢩⣍⠀⠀⠈⠙⠛⠓⣻⠏⢀⠈⠀⠉⠛⠋⠉⠉⠛⠳⠏
⠀⠀⠀⠀⠀⣤⣤⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣧⣤⣴⣶⣶⣿⡟⠋⠉⠬⣭⣧⣠⣿⠿⣶⣶⣶⡶⠛⠁⠐⣿⢧⣤⣠⣤⣴⣢⣤⣠⣦⣤
⠀⠀⣀⣀⣷⣿⣿⣆⢠⣿⣿⣤⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣶⣤⣄⠉⣿⡆⢨⣙⣲⣶⡶⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠙⠈⠛⠋⠈⠛⠋⠃
⠀⢾⣿⣿⣿⣿⣿⣿⠾⣿⣿⣿⣶⣶⣶⣦⡄⠀⠀⠀⠀⢀⣤⣶⣿⣿⣝⡛⣻⣿⣿⣷⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠛⣿⣿⠿⠻⣿⣶⠏⢹⣿⣿⣿⣿⣿⡗⠀⠀⠀⢠⣞⣷⢯⣷⣿⣿⣿⣿⣿⣯⣽⣧⡙⣯⣷⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣦⣤⣤⣤⣤⣤⣤⣤⡤⣦⣤
⠀⣤⣤⣦⣴⣾⡆⣿⣇⠀⢠⣿⠿⠿⠛⠉⠀⠀⠀⢀⡿⡦⣹⠞⢩⣿⣽⣿⣿⣟⣿⣻⡿⣿⣿⣿⡿⢷⣦⣤⣀⡾⢩⡵⢻⣟⣒⢢⣴⠶⠒⠒⠒⠒⠒⠛⠛
⣺⣿⣿⣿⣟⡙⢷⣾⣏⣴⣿⣹⣿⣿⡆⠀⠀⠀⠀⣸⠷⢴⡋⣸⣿⣿⣿⣿⣿⠷⣿⢾⣇⣿⣿⣿⣿⢿⣀⣰⣏⣡⣾⣁⣼⡛⣷⠟⠋
⠀⠘⠉⠿⠋⠻⢶⣿⣿⠋⠉⠉⠟⠟⠃⠀⢀⣾⢍⢁⣴⣤⡇⠻⣿⡟⢹⣿⠛⣿⠛⣯⣩⣿⣿⠿⠿⠀⠉⠉⣽⣿⣟⣲⡟⠙⠉
⠀⠀⠀⠀⠀⠀⢠⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⠀⠙⠁⣼⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⠛⠟⠁⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠘⠛⠿⠿⠒⠒⠿⠷⣿⠆⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⡶⠀⢸⡇⠴⢀⣾⣿⣿⠟⠛⢿⣿⣿⣿⣿⡇⠀⠶⠶⠶⣶⣶⣾⣿⣿⣯⣭⣭⣥⣤⣤⣤⣤⣭⡉⠛⠿⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⠟⠁⠀⠀⠸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⠛⢻⡏⠀⠀⠀⢀⣤⣹⡟⠛⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⣄⣘⣧⡄⠀⠀⠈⠛⠋⢷⡶⢮⣽⣿⠆
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠅⢰⡀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡇⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠒⢦⠀⠀⠀⢀⡞⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢠⡇⡧⢈⡇⢀⡤⡄⠀⠀⠀⠀⠀⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⢴⡄⢘⡆⠀⠀⣼⠁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠓⢼⠃⣿⠀⠜⠀⠀⠀⠀⢸⢉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡏⢰⠇⣸⠁⠀⣸⠇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡴⠋⠁⣿⡁⢼⢀⣇⠀⡃⠀⠀⠀⠀⡧⠠⢀⡃
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢨⢁⡟⢠⠏⠀⢀⡟⢠⠇⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⢸⠌⣱⢀⡞⠀⡴⡆⢠⡁⣎⠸⡀⠀⡇⠀⠀⠀⢸⠃⡐⢠⡇
⠀⡏⢢⡀⠀⠀⠀⣸⠁⠾⠇⢠⠎⠀⠀⢸⠁⡼⠀⠀⠀⢀⡴⠊⢁⠀⡼⢀⣀⡀⢸⠆⣹⡾⢀⡏⠀⡿⠀⣀⣧⠇⡐⠀⡇⠀⠀⢠⡏⢠⡏⢰⡇
⠀⣇⠘⣅⡀⢀⣀⡿⠀⣃⡼⠋⠀⠀⠀⡏⢰⡇⢀⣴⠖⠋⠠⣞⡧⢘⡇⣟⠀⡕⢸⠀⣼⠃⢼⠀⢠⠃⠸⠀⣿⠂⣼⡀⢧⠀⢠⠟⢀⣿⡇⢸⠁
⠀⠙⢦⣌⣉⣉⣠⡆⢸⠃⠀⠀⠀⠀⢸⠆⡸⠖⠙⢀⡴⢺⡁⢻⡇⢨⠰⡏⡀⢣⣸⠀⣿⠀⡎⠀⣸⠁⣖⠠⠋⢠⠟⣄⠙⠒⠋⣠⠏⡼⢀⡻
⠀⠀⠀⠀⠀⠁⢸⡃⢸⠀⠀⠀⠀⣠⠾⠀⣀⣴⠂⣼⠀⠸⠇⣸⡇⠰⣰⠇⠄⢹⠇⢐⣿⠀⣇⣰⠇⣰⣏⠀⢀⡞⠀⠈⠓⣚⣚⣁⣴⡇⢤⠧⠤⠲⠖⠒⠒⠒⡲
⠀⠀⠀⠀⠀⠀⢸⠁⠾⠀⢀⡤⠞⢩⡜⠠⡇⢼⠀⡆⠀⣸⠃⣼⡇⢘⡟⠀⣔⠐⢀⡹⠘⢦⠈⣁⡴⠃⢈⣓⡿⠴⠲⠊⡋⣉⣁⣤⣤⠀⡶⠖⠒⠒⠚⠓⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣸⠀⣯⠔⠋⣠⠜⠉⣏⠐⡇⢻⡀⢳⣤⠏⢰⠏⢣⠀⢁⡴⠋⠉⠉⠁⠀⠀⣉⡡⠐⠊⣋⣨⡤⠶⠚⠊⠉⠁⠀⡼⢃⣸⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠁⣤⠞⠁⠀⠀⣏⠘⡇⠀⠳⣤⣀⡴⠋⠀⠈⠉⠋⠁⠀⠀⣀⡤⠖⢉⣁⡤⠖⠚⠉⠀⠀⠀⠀⠀⢀⣠⣼⣷⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⢀⠼⠉⠤⡟⠀⠀⠀⠀⠀⡧⢈⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡴⠚⢁⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⣿⣯⢿⣿⣿⣿⡯⣟⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢸⣠⡆⢐⡇⠀⠀⠀⠀⠀⢱⠀⣵⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣁⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣯⣳⡾⠻⡾⣿⣿⣿⣿⠛⢿⡟⣟⢿⡿
⠀⠀⠀⠀⠀⢩⡇⢸⠇⠀⠀⠀⠀⠀⢸⠂⢼⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡟⢿⣥⡀⠀⣿⣯⣿⣯⡏⠀⣀⣍⣽⣿⡇
⠀⠀⠀⠀⠀⢸⠃⣼⠀⠀⠀⠀⠀⠀⠀⣇⠰⡇⠀⠀⠀⢠⠏⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⣁⡼⣿⣹⣛⠻⠟⠛⠳⡿⠚⠛⢿⣿⣿⡖⣿⠇
⠀⠀⠀⠀⠀⢸⡁⣾⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⡘⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⣰⠖⠉⠀⢿⣾⣄⣻⠀⠀⠀⠀⠀⠀⢸⣿⣽⡇⣿
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⢉⣠⠖⠋⠀⠀⠀⠀⢸⣷⠉⣿⣷⡀⠀⠀⠀⣰⣿⡿⢹⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠲⠶⠶⠶⠒⠊⣉⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠘⠿⣽⣦⣠⣾⣽⠿⠃⣼⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄⠀⣿⣻⣿⣵⡇⠀⣼⣿⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄⠀⠀⠏⣹⣿⡁⠟⠀⠀⠼⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⣿⣿⡽⠂⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⡀⠀⠀⠀⠀⠁⢀⡠⠤⠒⠊⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠈⠒⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠛⠓⠒⠒⡶⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣴⣶⣶⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣄⣀⠀⠀⣀⣀⣤⣶⠿⠛⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠻⠿⠟⠛⠛⠁
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠿⢿⣿⡿⣿⡾⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢳⠸⡇⠀⠀⠀⠀⠀⠀⢰⠒⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣆⠀⠀⠀⡞⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⡸⢘⡇⣠⢄⠀⠀⠀⠀⣞⠀⡇
⠀⠀⠀⠀⠀⠀⡸⢁⠖⡆⢸⠀⠀⡼⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡏⢸⢀⠇⢸⠀⠀⠀⢀⠧⠀⡇
⠀⠀⠀⠀⠀⢀⡇⠼⢠⠁⡞⠀⢰⡁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⡀⢠⠊⣀⠘⠇⣹⡸⠂⢸⠀⠀⠀⣸⠐⠀⡇
⡶⢄⠀⠀⠀⢸⢀⣷⠃⡼⠁⠀⡟⢸⠁⠀⠀⢀⠤⠒⠉⡆⢀⠀⢸⠐⣷⠋⡼⢉⡅⡁⣧⡍⠂⣸⠀⠀⢠⠃⣸⢈⡇
⣇⢸⡀⠀⣀⡏⠐⣠⠞⠀⠀⢰⠁⡎⢀⡤⠖⢁⣴⡏⣰⢰⡃⡅⣼⢨⡟⢸⠁⢸⠀⠆⡿⢀⣇⢸⠀⢀⠎⣰⣏⢸⠁
⠘⠢⣌⣭⣠⡆⢼⠁⠀⠀⠀⡼⢈⠗⠉⣠⠞⡇⢹⡇⠔⡨⠄⣳⡌⢸⡇⣺⠀⡏⢰⠀⠉⣼⠳⡌⠓⢉⡴⢱⠃⡝
⠀⠀⠀⠀⢰⠅⡊⠀⠀⣀⠴⢋⢰⢺⠁⡎⠀⡗⢸⡇⣳⡗⢂⠸⠃⡼⡆⠹⠜⢡⠿⣄⣠⠃⣀⣈⠭⠥⠔⠺⢐⢃⣊⣉⣙⣉⣉
⠀⠀⠀⠀⢸⠂⣇⣠⠚⣡⠼⡇⣹⢸⠂⡇⢰⢇⡞⡇⠸⢁⡾⠤⠴⠃⠘⠧⢖⣣⠤⠒⢊⣉⡭⠴⠖⠒⣻⢁⠏⠉⠉
⠀⠀⠀⠀⢸⠀⢋⡠⠊⠁⠀⡇⢾⠈⢧⣈⢁⡼⠁⠙⠒⠊⠀⠀⢀⡠⠔⡊⣁⠴⠖⠋⠉⠀⠀⠀⠀⣰⢃⡜
⠀⠀⠀⡰⠋⠰⠏⠀⠀⠀⠀⡇⣹⠀⠀⠈⠁⠀⠀⠀⠀⢀⠤⢚⣩⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡞
⠀⠀⠀⠧⡶⢸⠅⠀⠀⠀⠀⣗⠸⠀⠀⠀⠀⠀⠀⡠⠞⣡⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣡⠊
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢻⠈⡆⠀⠀⠀⢠⠚⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠞⠁
⠀⠀⠀⢠⠃⡽⠀⠀⠀⠀⠀⠈⠲⠇⠀⠀⢀⠇⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣋⣠⠞⠁
⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⢣⡀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠀⢩⣍⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣉⠒⠲⠒⢒⣫⠏⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⠦⠤⣄⡀⠀⠀⠀⣰⢃⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠑⢒⡼⡫⡿⣿⣻⣟⣿⣯⡽⠿⣟⣛⣯⣭⣽⡷⣶⣶⣶⣶⣶⣿⣭⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠲⠶⠶⠿⠾⠛⠛⢋⣉⣭⣴⣖⣛⣻⣯⣵⡾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⢯⣽⣟⡛⣿⣛⣻⣻⣽⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢯⣽⣿⣿⣿⣿⣯⠿⠞⠋

⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛
⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠞⠛⠂
⡆⠠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡆
⡄⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡴⣿⣏⣙⣿⣿⣆⠀⠀⠀⠀⠀⠀⣤⣄⡤⠞
⠄⠀⠨⠁⠀⠀⠀⠀⠀⠉⢹⠿⠉⠀⢉⣶⣤⣤⡀⠀⠀⠀⠀⠀⣰⣿⣯⠶⣿⡧⢬⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⠇
⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⣿⠦⠤⡀⠀⢠⣿⣿⡵⢞⣿⣟⠒⠶⢾⣿⣧⠀⠀⠀⠺⡿⢿⣧
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣾⣿⣏⡴⢛⣿⡯⣍⣑⠒⣿⣿⡆⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⣄⡚⠺⣄⣿⣿⣏⡴⣫⣿⡷⢤⣬⣉⣻⣿⣷⠘⢻⡟⠁⠀⣀⣀⣀⣀
⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠉⠉⠉⠉⠛⠿⣿⣦⣸⣿⣿⣯⠞⣩⣿⣟⣓⢦⣬⣹⣿⣿⣰⣏⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤
⡉⢿⣿⣶⣦⣤⣤⣤⣶⠿⣿⡇⡸⠀⠀⠀⠀⠀⣀⣀⣈⡙⢿⣿⣿⣷⠞⣡⣾⣿⣭⡉⠛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠩⠿⠉⠉⢹⣿⣿⣿⣿
⠄⠀⠙⢿⣷⡈⣹⣿⠇⣸⣿⢳⠃⠀⣠⡴⠞⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⣥⣾⣿⡗⠮⣽⣿⣿⣿⣿⣿⠁⠉⠉⣤⠀⠰⠿⠀⠸⢿⣿⣿⡟
⠀⠀⠀⠈⢿⣿⡿⢋⣼⡿⣣⠏⣠⡾⠋⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣿⣿⡟⢉⣉⡁⢿⠇⢀⠉⠠⣶⠆⠾⠇⣰⣿⡿⠁
⡀⠀⠀⠀⠘⣿⣷⣿⡿⣱⠏⣼⠋⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⣃⣄⠀⠿⠗⢀⣴⣶⣦⣾⣿⡿⠁
⠃⠀⢀⡟⠂⢿⣿⣿⣴⠃⣸⠃⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣫⣤⡴⠗⣸⣿⣿⣿⡿⠟⠁
⠆⠀⠸⠇⠀⠀⠙⣿⣿⣷⣿⡄⠀⢀⣴⣶⡆⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠿⠛⠋⠁
⡋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣆⠙⠟⢻⣷⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠘⣿⣿⣦⡀
⠂⠀⠀⣧⠄⠀⠀⠀⠀⠀⠈⠙⠻⣷⡄⢸⣿⣾⣿⠁⢀⣠⣤⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⢀⡀⢀⣄⠀⢿⡿⣿⣿⡆
⠇⠀⠀⠁⠀⠀⠀⠀⢀⣠⣤⡶⣾⠟⠁⢸⣿⣿⡿⠀⣸⣿⡿⣿⠀⣿⣤⣼⣿⠀⣠⣾⣿⣿⠀⣿⣷⣼⣿⠀⠈⢷⢹⣿⣿
⡆⠀⠰⠋⠀⠀⢠⣾⠟⢋⡥⠚⠁⠀⠀⢸⣿⢿⡇⠀⢹⣿⣀⣿⠀⣿⣿⣿⡿⢠⣿⢁⣿⣿⣄⠹⣿⢿⣿⡇⠀⢸⠃⣿⣿⡄
⠆⠀⠀⠀⠀⣰⡟⠁⣰⠋⠀⠀⠀⠀⢠⣿⠇⢾⣷⠀⠘⢿⣿⠟⠀⣿⠛⠛⠁⠈⠿⠿⠋⠻⠿⠁⠀⠀⣿⡇⢠⡟⠀⣿⣿⡇
⡄⠀⠀⠀⢰⣿⠁⢸⡃⢀⣀⣤⣤⣾⣿⠏⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣜⠀⠀⣿⣿⡇
⣥⡄⠀⠀⣾⣯⡙⢾⣧⡈⠛⠛⠛⠉⠁⠀⠀⠀⠙⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣯⠀⢀⣽⣿⡇
⠙⠃⠀⠀⣿⣿⣿⣶⣌⢻⡶⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⠄⠀⣰⣿⣿⡴⢃⣴⣿⡿⣿⣿⡀
⡄⠀⠀⢠⣿⣧⣍⠛⢿⣷⣽⣦⡀⠀⠈⢷⣀⣠⣤⣶⣶⡶⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠀⣠⣾⣿⣿⣿⣷⣿⣿⡋⣠⣿⣿⣷⡀
⠁⠀⠀⢸⣿⡿⢿⣷⣦⠙⣿⣯⣷⡄⢠⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣯⢹⣿⣿⡿⠻⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣾⣿⣷⣄⠙⢿⣷⡌⢿⣷⢿⣾⣿⣿⣷⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠟⢈⣿⣿⡿⣶⡿⠇⠉⠙⠻⢿⣿⣿⣦⡀
⠀⠀⢠⣿⣿⡙⢿⣷⡌⢿⣿⡘⣿⣿⣿⣿⡿⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⢉⡉⠀⠀⣶⡆⠀⠀⠀⣿⣿⣿⣷⢸⡆⠀⠀⠀⠀⠀⠈⠛⠻⠿
⠀⠀⣾⣿⠛⢿⣦⡙⣿⣼⣿⣿⠿⡟⠁⢸⣿⢰⣶⣶⠀⠀⣠⣤⡀⠀⠀⣿⣿⣧⠀⠈⠀⢸⣿⡆⠈⢹⣿⣿⡄⠛⢾⠟
⠀⣸⣿⣋⣠⣤⣿⣿⡿⠿⠋⣡⠔⠛⠚⠈⣿⣆⠙⠋⠀⠀⠙⠛⠁⠀⠀⠙⠛⠋⢀⡀⠀⠀⠉⢡⣶⣼⣿⣿⠁⠀⢸⣇⣀⣴⡦
⣴⠿⠿⠿⠛⠋⠉⠀⠀⢀⣼⣧⢠⡦⠀⠀⠘⢿⣦⣀⣠⣶⣤⡀⠀⣴⣶⡀⠀⠀⢿⣿⡆⠀⠀⢀⣉⣹⣿⠏⠀⠀⠀⠉⢻⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠈⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇⠀⠈⠋⠀⠀⠀⠀⠉⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⢧⣴⣿⠆
⠀⠀⠀⠀⠀⠀⢀⡼⢷⣬⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣾⣿⣿⣷⡄⠀⣤⠀⢙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⣴⣋⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣧⡀⠛⢁⣾⣿⡟⠀⠀⠀⠀⠤⢤⣤⣄⡀⠸⣧⣸⣿⠄
⠀⠀⠀⠀⡠⠛⠣⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣶⣼⣿⡟⠀⠀⠀⠆⠀⠀⠀⠀⠀⠱⣤⡈⣿⡇
⠀⠀⠀⢸⣧⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⡶
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡴⠋⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⠲⡾⠀⠀⠀⡈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠚⠁⠀⠀⡾⠁⠀⠈⢷⡀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠈⠓⠀⠀⢠⡞⠉⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠞⢁⣀⣠⡤⠄⠐⢦⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⡴⢶⣶⣿⠟⠛⢿⡉⠉⢧⡀⠀⡀⠛⣆⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠒⢉⣠⢴⣿⠟⠁⠀⠀⠀⢻⡄⢾⡄⢀⡙⣯⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣿⠶⠛⣩⡴⠋⣿⢀⡀⠀⠀⠀⠘⡇⢨⣥⡟⠉⠛⠀⣠⡴⠋
⠀⢀⡴⠒⠚⠻⠧⣍⢹⡦⠤⠖⠚⠋⢉⣠⡴⠚⠉⠀⠀⣿⠺⠇⠀⠀⠀⢰⡇⢸⡿⣇⣀⡴⠞⢉
⡾⠋⣉⠍⠙⣧⠤⠟⠁⣀⣀⣤⠴⠚⠛⣧⠀⠀⠀⠰⠏⣿⡀⠀⠀⠀⠀⣼⠃⠈⣯⣿⡿⠟⣩⣍⡉⠻⡆
⠹⣆⠛⠖⠻⡛⢛⣿⣏⡉⠁⠀⠀⠀⠀⢹⣦⣤⣤⠴⠞⠉⠳⣤⡴⠶⠚⠛⠀⠀⡿⠋⠀⠞⠉⠉⠙⣠⠃
⠀⠈⠳⣄⠀⠁⣾⠁⣈⣉⣉⣽⠿⠖⠚⠉⣁⣀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢤⡤⣤⣤⠴⠚⠁
⠀⠀⠀⣹⠞⠛⠋⠉⠉⠉⠁⠀⠀⠀⠈⢉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠑⣶⠄⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⢁⣠⠟
⠀⠀⠀⡿⢦⣄⣀⣁⣠⣭⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢦⣴⠃⠀⠀⠀⢻⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⣴⣋⣀⡀
⠀⠀⢠⡇⠀⣾⠉⠁⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠻⣦⡀⣰⠇⠀⠀⠀⠀⠘⡄⠀⢀⡾⣡⣾⠶⠞⠛⠉⠉⢉⣩⠿
⠀⠀⢸⠇⠀⡗⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠶⣏⠉⠙⢦⣀⠀⣽⠃⠀⠀⠀⠀⠀⠀⣷⣶⢿⣾⡍⠂⣉⣀⣠⡴⠖⠋⠁
⠀⠀⣼⠀⢰⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢉⡿⠁⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⠋⠈⠉⠉⠉
⠀⢀⡿⠀⠸⢷⡆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠋⠀⠀⠀⠀⠀⠀⣠⡴⠋⠊⣰⠃
⠀⠈⠷⣄⡀⠀⠙⠳⣄⡀⠀⠀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⢀⡞⢳⡄
⠀⠀⠀⠀⠙⠲⠤⣀⠈⠙⢦⡾⢋⡹⡆⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠋⠁⠀⠀⢀⡴⠋⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣯⣀⣙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⢀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⡉⣼⠁⠀⠀⠈⣆⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠉⠁⡇⠀⠀⠀⠀⣸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣄⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠿⠀⠁⠀⠀⢀⣤⡟⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠾⠙⢦⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⡤⠶⢄⡰⢟⠁⢹⡶⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀⣈⠓⣤⠟⣤⣦⣤⣀⣀⣀⣀⡾⢁⡄⠈⠁⢀⣿⠋⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⣿⣯⣉⠛⢻⣿⡏⠉⠉⢉⡏⣿⣿⡟⠁⠀⠀⠀⢉⡷⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠉⠹⣎⣓⡶⢶⠞⢳⡏⠀⠀⠀⠀⠀⢠⡿⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠉⠛⠻⣤⡤⡶⠈⢧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣧⠀⢳⠀⠀⠀⠀⠀⠀⠙⠛⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠤⠤⠴⣿⣤⣾⣷⠶⠦⠤⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠐⠂⠉⠁⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⣀⣸⠧⢄⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠙⢧⡀⠀⣠⣼⢫⡍⢨⠏⠉⢻⡛⣟⢻⠛⠿⣥⣀⡀⢀⡾⠋⠉⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣷⠞⠁⠛⠿⠶⠏⠀⠀⠘⠷⠿⠛⠀⠀⠀⠉⠙⠻⢦⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⡇⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄⠀⣰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇⣶⡆⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢹⣿⠁⢠⣿⠃⠀⠀⠀⠀⠀⠀⢠⣄⠀⣤⣟⢿⣿⢸⣿⡇⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⣿⣷⣿⠋⠀⣿⡏⠀⠀⣠⣴⣶⠄⡀⢸⣿⣾⡿⢉⣿⣿⣼⣿⡇⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣼⣿⠟⠁⠀⢸⣿⣁⣴⣾⣿⣿⣿⣸⣿⢸⣿⣿⠇⣸⣿⣿⣿⣿⣇⣠⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢹⣿⠀⠀⢀⣼⣿⣿⡿⠁⣷⣿⣷⣿⣿⣿⣿⣿⣠⣿⣿⣿⡟⠘⠻⣟⣁⣼⣿⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣠⣶⡿⣿⡇⢾⣇⢠⣿⢿⣿⣿⠿⠿⠃⠳⠿⣋⣤⣽⣷⡿⠿⠛⣻⣿⠋⠉⠉⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⣿⡇⠘⢿⡿⠋⠈⠛⠁⠀⣀⣴⣾⡿⠟⠋⠉⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠀⢀⡄⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⢸⣿⡀⢀⣴⣾⠟⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⠛⠂⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣀⣀⣀⣀⣤⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⠸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁⠀⠘⣿⣿⣍⣩⡁⢈⣽⡟⢛⣭⣭⣍⠙⢻⠟⣁⡴⠞⣹⢆⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡷⢋⠝⠅⢠⠝⠛⡍⠃⠈⠰⠿⠴⢶⣫⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣰⢋⠉⠙⢷⣧⣿⡇⡇⣼⣷⠇⠀⣀⣠⠤⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⢻⡸⡇⣾⡆⢳⠈⠀⠙⠿⠉⠀⣈⣀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠟⠁
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠘⡷⢤⣈⣁⡼⠀⠀⠀⢀⣀⣤⣾⣿⠀⠀⠀⠀⠀⢸⡿⠟⠋⠁
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⣤⠹⣆⠈⠉⠉⠉⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣠⣿⣴⣶⣶⣾⠏
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⢷⣄⠐⣔⠻⠿⠿⠿⠟⠋⠀⠀⢀⣠⣾⣿⣿⣿⣿⡟⠋
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢹⣷⣦⣄⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣷⣄
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠁⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⢿⣿⣿⣓⣒⣺⠿⠿⠿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⠒⡒⠒⢒⠛⠒⣒⡛⠛⡛⡛⠛⠛⠛⢛⢛⢛⠛⣛⡛⠒⡒⡒⢒⢒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢹⡏⠀⣾⠋⢹⣆⢻⡇⢠⣿⡄⣸⠃⠈⣿⠉⢻⡆⠹⣧⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⡟⠘⣿⠃⢸⣿⠘⣿⣼⢹⣧⡟⠀⠘⣿⠃⢼⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠆⠼⠷⠀⠙⠦⠼⠃⠀⠿⠧⠠⠿⠇⠀⠰⠿⠤⠿⠃⠀⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢻⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⣜⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣣⣾⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣯⡾⠳⣸⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⣶⣾⡤
⠀⠀⠀⠀⠀⠀⢠⡞⣉⣵⣶⣶⣶⣰⣶⣾⣿⣷⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠟⠿⠓
⠀⠀⠀⠀⠀⠀⠘⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠭⣝⣛⣛⣻⣿⣿⣿⡿⠟⠛⠉⠀⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⠀⠀⢀⣴⣆⠀⢰⡏⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⡿⠁⣾⠁⢸⣇⣿⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⡿⠀⠀⠰⣟⣼⠇⠀⢸⡟⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⢹⣼⡇⠀⠀⢘⣯⠋⠀⢠⣼⡇⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢰⡏⠀⠀⠘⣿⠃⠀⢀⣾⠿⡴⡾⠛⢘⡧⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣷⣴⠻⡇⠀⠀⢠⣿⣄⣠⡼⠃⠀⠀⣠⡴⠟⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠉⣷⠀⠀⢿⣄⢀⡾⠀⠈⠁⠀⠀⣰⠾⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠃⣠⡟⠀⠀⢀⣀⡀⠀⠀⠀⠀⣤⣤⣄⠀⠀⢼⠛⠁⣸⠿⡇⠀⠀⣿⠀⠀⠈⠉⠋⠁⠀⠀⠀⢠⡾⠉⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠠⣯⡾⠃⠀⠀⢰⡟⢹⡧⠀⠀⢠⡾⢿⣀⣹⡷⠟⣻⠄⠀⠸⣷⡇⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⣾⠃
⠀⠀⠀⣀⣴⢿⡏⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⡿⠀⠀⠉⢩⡧⠀⠸⣇⠀⣠⡟⠳⠲⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠏
⠀⣤⡾⠋⠁⠸⣗⠀⠀⠀⠐⣿⡾⠋⠀⠀⢸⡇⠀⠀⠀⣸⡏⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⡀⠀⠀⠸⢧⣄⣀⣠⡶⠋
⠀⠁⠀⠀⠀⢈⡧⠀⢀⣠⢾⡏⠀⠀⠀⠀⣸⣷⡀⢀⣠⡟⠀⠀⠀⠀⠀⣆⠉⠉⠀⠀⢠⡤⠊⠀⠀⠲⣤⣀⣀⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢐⣯⣴⠟⠁⢸⡇⠀⠀⠀⢠⡿⠈⠋⠛⠁⠀⠀⢀⡤⡤⣄⠙⢅⠒⠂⠘⠛⠒⠒⠢⢄
⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠸⣇⠀⠀⢀⡿⠁⠀⠀⠀⠠⢀⡨⢹⡆⡇⡯⢌⡈⢦⡀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠃
⠀⠀⠀⣴⠟⢹⡇⠀⠀⠀⠀⢻⣄⣤⡟⠁⠀⠀⠀⢠⠔⠁⠀⢨⡧⣧⠁⠀⠈⠙⡏⢦⣄⡀⠀⠀⠀⠀⠃⠀⣼
⠀⠀⣼⠏⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡖⠁⠀⠀⠀⠀⠈⡇⠈⠢⢄⠀⠀⠸⣶⠉⠑
⠀⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠑⢄⠀⠟
⠀⠀⠙⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⠀⡇⠀⠀⠀⠀⠀⠳⣌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⢀⠀⠀⢀⡇⠀⠀⠀⡇⠀⢀⠇⠀⠀⠀⡤⣄⠀⠀⠉⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠸⡄⠀⠈⣷⠀⠀⢀⡇⠀⠎⡳⠀⠀⣸⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡒⠁⠀⠀⠀⠉⠁⠒⢻⡗⠈⠉⠸⠀⢹⡟⠒⠒⠁⠀⠀⣀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡈⠉⠀⠣⠙⠀⠀⠀⢀⣤⠴⠖⠚⠃⠂⡒⠁⠀⡘⠓⢠⡀⠀⠀⠀⠁⠉⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⣄⣀⣤⠞⠋⠀⣬⣄⣀⣀⠄⠀⠀⠀⢡⠀⠘⡏⢱⠤⣀⣀⢀⣠⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣯⣷⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⢀⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣻⢀⣄⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢻⢿⠇⠀⡾⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡇⣼⢽⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣇⣾⡿⠀⢸⣽⠃⠀⠀⠀⣀⡀⠀⠀⢸⢳⢀⣞⡷⣾⣷⣇⣿⣿⠀⠀⢀⢯⣿⡇
⠀⠀⠀⠀⣿⡆⠀⠀⢸⢽⢿⡟⠁⢀⣯⡟⠀⣀⡴⣮⡿⡞⣠⣄⢸⢺⣟⡞⢀⣷⣿⣿⣿⣻⠀⠀⡾⣿⣿⠇
⠀⠀⠀⠀⠹⣿⣒⣖⣿⣺⠋⠀⠀⢸⣿⣧⣞⡽⢾⣿⣟⡇⣿⣾⣼⣾⣿⡇⣸⢿⢿⢯⡿⣟⣦⣞⡿⣻⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣞⣾⢾⣺⠁⣸⢾⡗⣧⢿⣾⣧⣿⡽⣧⢯⡿⣟⡿⢁⣉⣚⣭⠴⣿⣗⣖⣖⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⣗⣇⡴⣯⠟⣻⡿⢸⣾⣀⣯⡟⣗⢿⡽⠣⠟⠁⠙⢒⣫⢴⣲⣿⠽⠟⠚⢋⣟⡿⠁⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡷⣽⠞⠁⠀⣿⣯⠈⠣⠯⠞⠀⠈⠉⠀⢀⣠⢴⣻⠿⠚⣭⡁⠀⠀⠀⢀⣾⡿⠁⠀⢀⣤⣤⣤⣤⡀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣣⣻⠁⠀⠀⠀⣿⣺⠀⠀⠀⠀⠀⢀⡠⣾⡭⠞⠉⠀⠀⢸⣿⣷⡄⠀⣠⢞⡞⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⢹⢹⠀⠀⠀⠀⢸⣾⡀⠀⠀⢀⣴⣫⠟⠉⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠘⠻⠇⠀⠀⣼⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⡀⠀⠀⠀⢀⣀⡤⣾⡽⠞⢻⣿⣿⣿⣿⣿⣽⡾⠀⠸⣿⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣿⣿⡯⠿⠗⠋⠁⠀⠀⠀⠻⣿⣿⡟⠩⢊⣹⠀⠋⠀⢫⠋⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠉⠙⠛⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠉⠃⠚⠁⠈⣄⣿⡎⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⡎⠳⢄⣠⣤⡀⠀⡈⠉⠀⠀⠀⢠⣿⡿⡫⠄⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡇⠈⠢⡄⣉⣉⣉⡀⠄⢚⡃⠀⠛⠛⢰⡡⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠱⣄⠀⠐⢲⣤⣴⣶⣾⠟⠀⠀⢀⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⡌⠳⡐⠻⠿⠿⠛⠁⠀⢀⡴⠿⣿⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡴⠋⠀⠀⠹⣏⣁⣀⡴⠂⣠⣿⣄⣠⣤⣤⣤⣤⠴
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣠⣴⣿⡏⣀⣴⣿⣿⣿⣿⣿⡟⠋⠁
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣦⣀
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠉⠉⠃
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⣀⣀⣀⣠⡤⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣁⣀⡀⢁⣀⣀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠈⣈⡀⢀⣁⡀⢀⣀⣀⣀⡀⢀⣈⣁⠀⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⣿⡇⠀⣴⡏⠀⣿⡆⠸⣿⡀⢠⣿⡇⢠⡿⠀⠀⣿⡏⠈⣿⡆⠹⣿⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⠾⣿⡇⠐⣿⡿⠀⣿⣿⠀⢿⣇⣾⢻⣷⣼⠇⠀⠐⣿⡷⠀⣿⡿⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣿⡇⠀⠻⣇⠀⣿⠇⠀⢘⣿⡏⠈⣿⣟⠀⠀⠀⣿⣇⢀⣿⠇⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⡈⠉⢉⠀⠀⠈⠉⠁⠀⠀⠈⡉⠉⠈⠉⢉⠀⡀⠈⠉⠉⠉⠁⢀⠀⠉⡉⡁⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠠⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡌⠙⢦⣤⡞⠁⣄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣰⡷⠞⠀⠀⠀⠙⠿⠀⠙⠻⣷⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣀⡀⠀⣼⡟⠒⢿⣧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⣯⠭⣝⢦
⠀⠀⢸⠇⢿⢸⡿⠀⠀⠈⠿⣿⠦⣄⣀⡿⢦⣄⣀⣀⣀⣀⣀⣀⣤⡴⢿⣠⠔⣿⠏⠁⠀⠈⠂⡇
⠀⠀⢸⠘⢸⢸⣷⠀⠀⠀⣠⣿⣷⣄⠙⠻⠶⣦⣭⣉⣉⣋⣉⣩⣥⠶⠛⢁⣼⣿⣦⡀⠀⠀⢃⡇
⠀⢠⡧⢤⠸⡼⣧⠀⢠⡾⠋⠀⢠⠟⠓⠢⣄⣀⡀⠈⠉⠉⠉⢁⣀⣤⠶⣟⠛⢷⡈⠻⣆⠀⣽⠃⡀
⠀⠈⢀⡈⠛⠃⢻⡀⣾⠁⠀⠀⢸⠀⠀⢀⢊⡉⠉⢉⢉⡉⡉⠻⢁⢀⡀⢈⠂⠀⢇⠀⢹⡾⢃⡀⢠⡀
⠀⠀⠈⠻⣶⡖⠈⢻⡇⠀⠀⠀⠀⠸⠇⢨⠂⡟⠀⠍⡎⢡⠸⠠⠋⡎⠀⠛⠃⠀⠈⠀⠀⣷⠀⠀⣀⡿
⠀⠀⠀⠀⠉⠙⠻⢸⡇⠀⠀⡶⠒⣆⣶⠀⡇⢲⡀⢰⡆⡶⠒⣦⣶⡠⠒⢸⡆⣶⠀⡗⠀⣿⠀⣀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠀⢿⡀⢀⡗⠖⠃⢯⣀⠏⣏⢧⡿⣇⡗⠴⠋⢸⠳⣄⢸⡱⡟⣧⠇⠀⡿⠀⠀⢠⡄
⠀⢀⣄⣨⣿⣶⠟⠃⠈⢷⡀⠁⢰⡀⠀⠁⢠⠀⠀⠀⠁⠀⠀⡔⠈⠀⡈⠀⣠⠁⠀⢀⡼⠃⢤⣧⠄
⠀⠈⠋⠉⠉⠃⠀⢀⠂⢄⠙⠳⠾⠷⣄⠀⠀⣧⡀⠀⠀⢀⣼⠃⢀⡜⢀⡴⢿⣤⠴⠛⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠛⠲⠶⠛⠙⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢷⣄⠀⠀⣀⡶⠋⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⠻⠿⣿⡦⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⣠⠞⠳⣦⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⣶⢶⣄⠀⠀⠀⡀⢠⣤⣠⣤
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣷⣄⠀⠀⠀⠀⣿⡀⠀⠀⠀⣀⡾⣇⣀⡀⠀⣸⣿⡇⠀⠀⣠⣾⢏⠖⠑⠻⣆⠀⠘⠛⠛⠻⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⠀⡏⠀⠀⠀⠀⠘⢿⣿⡶⢤⣴⣿⠟⠶⠚⠻⣦⠀⠈⡽⠛⠛⠿⣛⣷⢖⣿⣿⠛⠃⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠟⠶
⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⣿⣿⣷⣤⣉⠙⠛⠓⠚⡷⠞⠲⢿⠛⠛⣉⣭⣶⣿⠿⠿⣶⡄⠀⠀⠀⠀⣷⠀⠀⠀⡀⢀⣤
⠀⠀⠀⠀⠀⠐⠚⠢⠈⠓⠂⠀⣷⠀⠀⠀⠀⣰⡿⠋⣽⠟⠁⠈⠙⡷⠲⠶⠶⠶⢶⡶⠚⠋⠉⠉⠻⣿⡀⠀⠈⢿⡆⠀⠀⢠⡟⠀⣀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠶⠀⠀⣠⡀⠰⡇⠹⣆⠀⠀⣰⡏⠀⢠⡟⠀⠀⣀⡤⣥⡀⠀⠀⠀⠈⣷⠀⣠⡤⢤⡀⠘⢷⡀⠀⠈⣿⡀⠀⣼⠇⠐⠛⠋⠁⠁⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇⠀⠁⠀⠹⣦⠀⡿⠀⠀⢸⡇⠀⢰⣡⠚⣿⣻⡄⠀⢀⡀⠁⠸⠧⠴⠶⠽⠆⠘⠇⠀⠀⢸⣇⣼⢏⢀⣀⠀⠀⠀⠀⢾⣧⣶
⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢰⡆⠀⢰⣮⣻⡇⠀⠀⠀⠁⠀⠈⠉⠀⠈⠉⠀⠀⣾⣽⠀⠀⠀⠀⠀⢀⡄⢙⠆⠀⠀⠈⣿⣷⣏⢸⣿⡄⠀⢰⣷⢸⣿⢟⠀⢀⡄
⠀⠀⡀⣶⣄⡀⠘⠷⣿⡇⢸⡇⢴⣽⣿⡟⡇⠀⠀⠀⠀⠀⠠⣤⣀⣀⡀⠀⣀⣀⣀⣤⠀⢠⣤⢶⡿⠁⠈⠀⠀⠀⢠⣟⣿⣿⣾⣿⣧⡀⠈⣿⢸⣿⡼⢠⣸⣷⡆
⠀⠀⣿⢸⣿⣇⡀⠀⣿⣧⡾⢃⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠻⣌⠉⠳⠶⠋⠉⠀⣈⡛⠉⣠⡞⠁⠀⠀⣸⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣄⠙⢻⣿⠀⢸⣿⡟
⠀⣠⡘⢿⣿⠟⢁⣠⣿⣧⣴⣿⣟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠸⣇⠈⠛⠶⠴⠏⠳⠼⠏⠹⠛⠁⠀⠀⠀⣠⡟⠀⣰⠿⢿⣿⣿⣿⣿⡿⢿⣟⣦⡼⢿⡷⠤⠼⠿⠦
⠚⢛⠭⠭⠭⠭⠍⠁⠈⠗⠀⠠⠠⠬⠭⠍⠡⠌⢧⡀⠀⠀⠀⢻⣦⠀⠀⠀⡆⠀⠀⡀⠀⠀⡼⠁⣠⣼⣿⣡⡾⠋⠀⠖⠶⠀⠀⠐⠒⢒⣒⡂⠀⣀⠒⠒⡂⣀
⠀⠀⠀⠀⠀⠂⠈⠉⠉⠙⠂⠀⠉⠁⠀⠀⠈⠉⠀⠉⠛⠶⠶⠿⠟⠛⠢⠤⠿⢦⣤⡿⠤⠾⠿⠛⠛⠛⠛⠉⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⣀⠀⢀⣀⠀⣀⣄⡀⢀⣀⠀⣀⡀⢀⣀⢀⣀⣀⣀⢀⣀⠀⢀⣀⠀⠀⠀⢀⣀⣀⣀⢀⣀⠀⣀⡀⣀⡀⠀⢀⣀⢀⣀⣀⣀⢀⣀⠀⢀⡀⣀⣀⣀⠀⢀⣀
⠀⠀⣿⣀⣀⡏⣸⠃⠀⠹⡎⣿⢀⡟⣇⢸⠏⢸⡇⠀⠙⡆⠹⣄⡼⠁⢀⣄⠀⢸⡇⠀⣹⢸⡇⠀⢹⠁⣿⣧⠀⣸⣿⠈⡇⠀⢹⡏⣏⣰⠋⠀⢹⠀⡿⣇⠈⡏
⠀⠀⣿⠉⠉⡇⢻⡀⠀⢠⡇⢸⣼⠃⢻⣼⠀⢸⡇⠀⢠⡇⠀⢹⠁⠀⠘⠛⠀⢸⡟⠛⠉⠸⡇⠀⢸⠀⣿⠸⣤⠇⣿⠀⡟⠛⠋⠀⡟⠙⣦⠀⢸⠀⡇⠘⣧⡇
⠀⠀⠛⠀⠐⠛⠈⠛⠒⠋⠀⠘⠛⠀⠘⠛⠀⠘⠓⠒⠋⠀⠀⠛⠃⠀⠀⠀⠀⠘⠓⠀⠀⠀⠙⠒⠋⠀⠛⠂⠛⠀⠛⠘⠛⠀⠀⠘⠛⠀⠘⠃⠛⠛⠛⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡿⠟⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠛⠁⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⣶⣦⡀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠓⠶⣤⣄⣀⠀⠀⢰⣄⠀⠀⠀⠀⠘⣿⣿⣿⠢⣀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⢠⣼⣿⡄⠀⠀⠀⣀⢻⣿⣿⠀⠈⠻⠒⣿⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣹⡇⠀⠀⠀⠈⠀⢹⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢃⣀⡤⠀⠠⣦⣌⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⡇⠀⣄⡰⢇⡀
⠀⠀⠀⠀⠀⢀⠐⣀⢀⠀⠀⠀⠀⠀⣤⡤⠈⠀⠁⠀⠀⠐⢿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠱⠄⠰⠃⠀⠈⣇⣼⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⡼⠋⠀⢰⣤⣤⣔⣤⡄⠈⢿⡀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠝
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⢀⣼⡿⠛⠙⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⡛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣀⣀⠀⠀⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠺⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡈⣿⡧⣶⣶⣾⠿⠛⠋⠚⠒⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣶⠿⠿⠿⠟⠛⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣥⣴⡶⠿⠿⠛⠋⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⡃⣴⡄⠀⢀⣭⣾⡶⠿⠈⠀⡈
⠀⠀⠀⠀⠳⠤⠤⠔⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣴⣶⠟⠛⠉⡀⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠏⠉⠂⠄⠀⠁
⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣠⣤⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣮⣛⣛⣯⣽⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣧⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⣿⣿⣿⣿⣦⠹⣿⣷⣌⡙⠻⠿⣿⣿⣷⣍⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⢿⣻⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣦⣤⣭⣭⣍⣉⣙⣛⣛⣛⣛⠛⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣭⣽⣿⣿⣿⣿⣿⢏⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢫⣭⠭⠭⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣻⣿⣿⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠒⠲⢤⣤⣤⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣿⣿⣶⣤⡉⠙⠛⠁⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣚⣿⣿⡏⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣄⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⡤⠒⠉⠁⠀⠀⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⣿⣿⣦⣤⣤⣼⠉⠉⠉⠉⠉⠉⠉⠉⣉⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⢂⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠻⢿⣯⣭⣽⣿⣿⣭⣽⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⢻⣿⣧⣤⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠏⠘⢹⣿⣻⠿⠟⠃⠁⣹⣿⣿⢫⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⡀⢻⡇⠀⠀⠀⠐⠛⠛⢁⣉⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⢸⡿⡇⠀⠀⠀⠀⠀⠀⢹⣷⡶⠶⠏⠀⠙⠿⠞⠁⠀⠀⠀⣸⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡶⡽⠆⠀⠀⠀⠀⠀⠀⠀⢸⣟⡓⠂⢶⡄⠀⠀⠀⠀⠀⣀⣰⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣿⢸⡇⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠦⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⣠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⣾⣧⣠⣆⡀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣄⠀⠀⣀⣤⣴⠇⠀⠀⢀⠔⡌⠀⠀⢠⠄
⠀⠀⠀⠀⠀⡀⠀⡄⠀⢨⣤⣶⣿⣿⣿⣿⡿⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣷⣏⠴⠃⡸⠀⠀⠀⠘⡀
⠀⠀⠀⡆⠀⡇⠠⡗⢧⣈⡉⠈⢹⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⡰⠁⠀⠀⠀⠀⠈⠢
⠀⠀⣸⣿⣦⣷⣴⣿⣿⣿⠇⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢋⣠⡶⠀⠞⠀⠀⠀⠀⣠⠁
⠀⠈⠋⣿⣿⣿⣿⠟⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠶⢟⣁⡀⠀⠀⠀⠀⠀⡴⠃
⠀⠈⡄⠈⠿⡟⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡁⣰⣶⠂⢸⠉⢆⠀⠀⣠⠊
⠀⠀⢱⡀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠿⢋⣿⠟⠹⠁⠀⠘⠀⠀⢣⡼⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⠬⠿⡿⠁⠀⣴⢿⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢛⣿⠃⠀⡼⢣⣿⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢈⡴⠉⣿⡆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣁⠤⠀⠒⠉⠀⠀⠸⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣀⣀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠰⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠆
⠀⠀⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟
⠀⠀⠀⢻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⡄⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⢠⣴⣾⣿⣿⣿⣿⣿⣻⠟
⠀⠀⠀⠀⠀⠘⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣭⣝⣛⣛⣛⠿⠿⢛⣛⣛⣫⣭⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⢦⣤⠤⣄⠀⠀⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⠲⠶⢤⣀⠀⠀⠈⢧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⠀⢀⣷⠀⢀⡏⠀⠀⠀⠀⠉⠳⣄⠀⠀⢳⠀⢻⡄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣾⠀⣠⠏⣿⣠⠟⠀⠀⠛⠓⠀⠀⠀⠈⢧⢀⣸⠂⠀⢳⡀
⠀⠀⠀⣠⠞⢹⡇⠀⠀⠀⠀⣼⢿⡾⠁⠀⠈⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠈⣿⠋⣧⠀⠀⢳⡀
⠀⠀⢠⡇⠀⠘⡇⠀⠀⠀⣰⠃⠀⠀⠀⢰⣶⠀⠀⠄⠀⠉⠁⠀⠀⠀⠀⠀⣿⡴⠷⢤⣄⣀⣙⣶⡤⠄
⠀⠀⢸⠀⠀⠀⠙⣆⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⢷⠀⠀⠀⠀⡀⣀⣏
⠀⠀⠸⡇⠀⠀⠀⣼⠯⣷⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣼⠀⠀⠀⣸⣿⣿⣿⡄
⠀⠀⠀⢿⠀⠀⠀⡿⠦⣬⣷⡶⡶⣦⢶⢀⡀⠀⠈⠓⠚⠁⠀⠀⢸⡟⠏⠁⢸⠁
⠀⠀⠀⢸⠀⠀⢠⠇⠀⠀⠀⣇⢷⢹⠘⡟⢧⣀⣀⣀⣀⣤⣤⢶⠏⠀⠀⢀⡟⢦⡀
⠀⠀⠀⣸⠀⣠⠏⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠉⠳⡍⠛⡿⢧⣼⣄⣀⡤⢾⡀⠀⠙⢦
⠀⠠⠴⠗⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⢲⡤⠤⠤⣿⠋⠁⠀⠀⢸⡅⠀⠀⠙⢦⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠳⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢰⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠳⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡟⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⡴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⢟⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⡏⣿⡉⠠⠀⠀⡄⠀⠀⢀⣴⢆⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢌⠑⡿⠀⠀⠀⠀⢿⣦⣶⣾⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠏⠀⠀⠀⠁⠀⠀⠀⠤⣾⣿⣿⡗⠂⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠒⠶⠆⠖⠀⠀⠠⢤⣤⣴⣿⣿⣿⣿⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠙⠢⣄⡀⠠⠀⣿⣿⣿⣿⣿⣿⡀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⢰⣆⣉⠳⢤⣿⣿⣿⣿⣿⡏⢹⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣉⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⠀⠈⣋⣝⠓⡿⣽⣿⣿⣿⣿⣿⣿⣿⣯⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢹⣿⣀⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠞⠁⢳⡀⠀⣸⣿⣿⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⡀⠀⠀⠞⢀⠀⠸⡇⣀⣿⣿⣿⣾⣎⡿⣯⣽⣟⣿⣻⣿⣿⣿⡿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣾⣥⡟⠀⠁⠀⣾⡁⠀⠐⠚⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠐⣯⠱⠄⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⢐⣿⣿⣿⣿⠛⣿⣾⠀⢸⡄⠀⢱⢌⣹⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣷
⠀⠀⠀⠐⠀⠚⠹⢿⠁⠀⠹⣿⡷⠺⢻⣕⠁⢸⣿⣎⣾⣿⣉⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠙⣿⡀⠀⠿⢿⣿⣿⣿⣧⣿⣉⢩⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠟⠻⣿⣷⠊⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⣀⣴⣸⣿⣿⣿⣿⠁⢹⡏⡉⠻⢿⠫⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣷⣟⠁⠀⠀⠈⠓⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡆⠀⠀⡀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⣀⡀⣀⡄⠀⣀⣠⠴⡿⢁⠐⢯⡥⠆⠐⣧⡶⣋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠐⠒⠆⠐⠐⠠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠀⠀⣤⣤⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣾⣿⣿⠿⠟⠛⢻⡏⠋⢘⣼⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⣠⣶⣾⣿⣿⠿⠛⠛⠛⠉⠁⠀⠀⠀⢰⣿⣶⣾⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⣀⠀⠀⠹⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠉⠽⠿⠭⠁
⠀⠀⠀⠀⠀⠀⣀⣄⡠⠖⢋⣁⣤⣶⣶⣶⣤⣤⣀⣀⢀⡀⠀⠀⠀⣠⣾⢻⡏⣷⡄⠀⢐⣳⣦⡀
⠀⠀⢀⣠⣶⣾⡿⢋⣰⣾⣿⠿⣿⣿⡲⣿⣿⠉⠀⠀⠀⠀⣴⣾⣿⣿⡇⠈⣸⡿⣿⣄⠈⠙⠋⠁
⢀⣾⣿⣿⣿⡏⣴⣯⣿⣿⣿⡞⡋⠁⠀⠀⠸⠅⠀⠀⠀⠀⠀⠻⣿⢿⣇⣰⣿⠃⠈⢿⣦⡀
⠘⢱⣿⣿⣿⠇⣾⡿⠟⣹⡏⠉⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣿⡟⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣀⣀⡀
⢠⣽⡿⠟⠁⠀⠀⠀⣨⣿⣧⡀⠈⣿⢛⠓⠒⠁⠀⠀⠀⠀⣠⠌⠣⠁⠀⠻⣿⡠⣄⠀⠀⠀⠙⢿⣶⣄⣠⡾⠟⠉⠁
⡿⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣈⣸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠂⣶⡆⡖⠀⢈⣛⠈⠁⢀
⠃⢀⣴⣿⣷⣤⣴⣾⠁⠀⠈⠉⠛⠛⢿⣿⣶⣦⡀⠀⠤⡄⠀⠀⠀⢠⣴⣢⡄⠀⠀⠀⠈⠳⣄⠠⣿⢟⡀⠀⣿⡆
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣯⣻⣿⡄⡼⠀⠀⠀⠀⢺⣯⠔⠌⢀⠀⠀⠀⠀⠈⠛⣙⠘⢷⡀⠙⠿
⠀⢀⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⡀⠘⢿⣿⢹⣿⠁⠀⠀⠀⠔⠈⣿⣟⠁⣾⠀⠀⠀⠀⠀⠀⠖⠁⢄⠳⣤⣄
⠀⠸⠱⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣧⠀⠀⢻⣷⠙⠀⠀⠀⠀⠀⢀⣿⡟⠘⣿⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⠄⠀⠀⠹⠈⣇⠀⠀⠀⠸⣷⣿⣿⠀⢻⡇⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡿⢀⣠⠄⠀⢸⣏⣿⠀⠀⠻⣦⣤⣤⠟
⠀⠀⠀⠀⠈⠛⢿⣿⣇⣷⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⢿⠉⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡌⣇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⠀⠀⠀⠀⢸⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠳⡄⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠿⢿⠿⠿⡿⠿⢿⡿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡶⣷⣶⣾⣶⣴⣤⣄⣀⠀⢀⣤⣶⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡟⠀⠀⠠⠴⣶⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⢸⢹⣧⠀⠀⠀⠈⠙⣿⣿⡟⠀⣿⣿⠿⢿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢧⠀⠹⣿⣦⡀⠀⠀⠀⠈⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⡟⠛⣿⡄⠀⠀⠴⠣⢤⣀⣨⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣷⠀⣴⣿⣿⣾⣿⣿⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠘⢦⣀⣿⣾⡯⠀⢰⣤⣤⡟⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢹⣿⡀⠈⢿⣿⣿⣷⣤⣈⠉⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠸⣿⡇⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡟⠃⠀⢿⣷⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠃⠀⣠⣾⣿⣆⠀⠀⢷⣾⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⢀⣴⣿⡿⢿⣛⣫⣽⣿⣿⣿⠀⢰⣿⣿⣿⣿⣦⠀⠈⠋⢠⣾⣿⡿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣠⡿⠟⠉⢀⠈⢿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣷⡀⠀⣩⣿⠟⢠⢟⣾⣿⣿⣿⣿⣿⣿⠏⣠⣿⡿⢷⣄
⢰⣿⣷⡄⠀⣼⣆⠈⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠏⢐⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡄
⣾⡿⣿⢻⠀⣿⣿⣷⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⣧
⣿⢂⢿⡌⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣧
⡏⠈⣸⡇⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡘⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠁⣷⠀⣿⡿⠿⠿⣿⣿⣿⡿⠿⠿⣿⠿⣿⡇⢻⣷⣿⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠀⢻⠗⠤⢍⣛⡛⠿⠟⠛⣋⣩⣴⣿⡇⢸⣿⣿⢸⣿⣿⣿⣬⣛⠻⢿⣿⣿⣿⣿⣿⠭⣿⣿⣿⣿⢿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣛⢿⣿⡟⣿⡇⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣶⣮⣤⡬⢷⣶⣾⣿⣿⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠃⠿⣧⢸⣿⡇⣉⢹⣿⠿⠛⠿⠿⠛⠫⠀⠀⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢼⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⠄⣰⠷⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⡦⣤⡄⠀⢀⡠⣶⢾⠻⣇⡑⠶⠒⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⡷⣿⢇⢴⣆⠀⠀⠛⢧⣹⣷⣷⣾⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⡍⠉⢿⣻⣧⡈⡻⡇⣩⣳⡈⢀⠰⢩⡿⢿⣿⡿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣀⣀⣯⣍⣷⡘⢿⣟⣿⡗⢸⣷⠎⠀⠈⣿⠃⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⠀⠀⠀⢰⡏⠀⡜⠻⠟⠙⠛⣿⡻⣿⠃⠀⢹⣦⣾⣷⡏⠀⠀⠀⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣎⠳⣤⣿⣆⠢⠀⢠⣿⣶⣾⣿⣷⣿⠃⠀⢸⣿⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⣿⣿⡻⠆⢠⣴⠟⠉⠉⠛⠛⠿⢟⡂⠀⣸⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⢟⣿⠃⣴⠟⠁⠀⠀⠀⠀⠀⢀⣆⣡⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢅⣿⣯⡾⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠸⣿⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠰⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠈⢻⡀⠀⠀⠀⠀⠀⠀⣄⢇⡀⢀⠂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠟⠋⠁⠀⢈⣧⡜⠃⡦⠑⠷⠦⣿⠶⠽⠟⠳⠋⠻⠷⠶⠦⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢷⣴⣷⣿⣿⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣴⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⣿⣿⣿⣿⣿⢟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠛⠛⠋⠉⠉⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠋⢉⠉⠉⠙⠲⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⣠⣤⡴⠖⠉⢀⣤⣶⣶⣶⣶⣶⣶⣤⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠸⠿⠟⠉⠀⠀⣴⣿⣿⡿⠛⠛⠉⠙⠛⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⡴⠚⣉⣠⣤⣤⣬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⣰⠋⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀
⢠⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠙⠒⠦⢤⣀⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠀⠀⢀⣀⣠⣤⣤⣴⣉⣹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⣶⢿⣿⣯⣴⠾⠿⢛⣫⠭⠗⠋⠉
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣩⣭⠶⠟⠛⠉⣁⡤⠖⠊⠉
⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠋⣉⣠⠤⠖⠊⠉
⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣉⠤⠴⠒⠋⠉⠁
⠀⠀⠀⠙⠢⣅⣂⠤⠤⠀⠀⣀⣀⡶⠼⠒⠋⠉⠁
⣶⣶⠶⠶⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⢦⣤⣤⠶⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣾⣿⣋⣉⠉⠉⠉⠉⠒⢒⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣍⡲⠤⣄⠀⠀⢾⣿⠟⠛⠛⠉⠉⠀⠀⢀⣠⠴⣚⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⣒⠋⠉⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⢿⣿⣯⣯⠁⢡⣶⣾⣿⣍⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠄⠉⠈⠁⠽⠃⠀⠙⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⠠⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡆⢰⠿⠆⠻⠻⠶⠶⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⣶⣧⠸⣶⣄⠐⠂⣀⠈⠉⠈⠀⢀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠦⣄⡤⠐⣧⡀⠀⠀⠀⠀⠀⢀⣷⣿⡱⠉⠀⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣴⠇⠘⠓⢶⣤⡀⠀⢠⠞⣿⣄⣀⣠⣄⡠⠜⠁⣿⣇⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⣾⡄⢀⢀⣘⣇⠀⢀⣠⣴⣿⣿⣿⡋⠙⠀⠀⢰⣿⢻⣶⣎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢄⠀⣿⡇⠀⠀⠉⣿⠁⠀⢀⣸⣿⣿⣿⢿⡶⠤⣠⣾⠇⣸⣿⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠙⢻⡄⠈⠙⡟⠓⠉⣩⢿⣿⣿⢿⣦⡀⠀⠀⠀⣴⣿⠋⣿⠁⠀⠀⠀⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⢸⣷⣤⡤⠾⠒⠊⠁⠈⢻⣿⣮⣻⣿⣆⣼⣼⣿⣧⠈⠁⠀⠀⠈⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡙⠇⣿⢿⣿⡟⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣤⣿⣿⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣽⡄⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⡿⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠰⣴⡄⠀⠸⡿⣿⢿⡀⣰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⡴⣫⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠄⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠌⠀⠜
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠌⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⢀⠖⠀⡰⠆
⢀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠘⢄⠃⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠊
⠢⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⡠⣃⠔⠁
⠀⠀⠑⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠬⠊
⠀⠀⠀⠀⠈⠂⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠲⢤⡀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠱⣄⠀⠀⠀⠀⣠⠔⠁⠀⣀⣤⣴⣦⡀⠀⠀⠀⠴⠿⢿⣿⣦⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⠜⠁⠀⢀⣾⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠼⠁⣠⠀⠒⠒⠤⡀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡃⠀⠀⡴⣦⠀⠀⢸⠀⠀⣴⣶⠀⠀⠘⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡜⠉⠉⠉⢢⡤⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠉⠀⠀⡘⠀⠀⠈⠉⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠛⠁⠀⠈⢄⠀⠀⠀⠀⠠⠉⢄⠀⠀⠀⠀⠐⠁⠀⠲⠶⠄⠀⠀⢸⠀⠀⠀⠘⡅⠀⠑⠤⠐⢇⢀⡰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡃⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠁⠒⠂⠈⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠉⠣⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣤⠆⠀⠀⠀⡏⠀⠀⠀⠀⡰⢁⠞⠢⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡟⠀⠀⠀⣼⣁⣀⣀⠴⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⡀⠀⢿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠁⠀⢀⣼⣿⣄⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠟⠳⣄⠈⢻⡿⠋⠉⠉⠉⠛⠉⠁⠈⠉⠙⢿⣿⣿⡿⠁⠀⡠⠊⠀⠈⠉⠉
⠀⠀⠀⡀⠠⠀⣀⡜⠁⡴⠋⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⢠⠎
⠀⠰⠉⣀⡀⠀⠈⠁⠎⢀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⢀⠗⠉⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠎⠀⠀⠀⣠⡖⠀⡄⠀⠰⡀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠘⠦⠤⠾⠇⠆⠀⠱⣄⠀⣈⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠒⠀⠀⠀⠀⠀⣀⡀⠀⠬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣀⠀⠤⠄⠀⠒⠂⠀⠈⠉⣀⣀⠠⠤⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠤⠤⠀⠒⠒⠀⠉⠉⣀⣀⡀⠤⠤⠤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠆⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⡴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠶⢄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣶⣶⣾⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢀⠄⠀⠀⣀⠐⠈⠀⠀⠀⠀⠀⠀⠁⠒⣅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣦⣤⣄⣠⣶⣶⣶⣄
⠀⠀⠀⠐⠁⠀⠀⠀⠨⠆⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠌⢂⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⡇
⠀⠀⠀⡇⠀⠀⠚⠀⠀⠀⠀⠀⠀⠜⠂⠒⠂⠒⠒⠀⠤⢈⡔⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⢹⣿⠃
⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⢄⢘⠀⠀⠀⠀⣿⣿⡏⣿⣿⣿⣿⣿⣿⠉⠁⢀⠀⠀⠀⠀⠐⢡⠘⠁
⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⠈⢙⠀⠀⠀⠀⠘⠿⣇⢿⣿⣿⣿⡿⠃⠀⠠⣿⡆⠀⠀⠘⠿⢾
⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⠀⠀⠀⠈⠀⠁⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⡀⠀⠀⢀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⡀⠀⠀⠀⠀⠐⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⣈⠣⡀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣒⠖⠲⡒⠒⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡀⠀⠀⠀⠀⡈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣿⣦⡀⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣿⣿⠣⢌⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠘⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢿⠀⠒⠤⢄
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⡤⢲⠆⢰⣿⠀⠀⠀⠀⢿⠀⠐⣇⠀⠀⢸⡄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢠⠏⠀⡞⠀⡼⢹⡄⠀⠀⠀⢸⣇⠀⣿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡏⠀⢸⡃⣰⠃⠈⢧⠀⠀⠀⣸⠛⢦⠟⠙⠦⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠁⠀⠀⠀⠀⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⢀⡤⠶⠶⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠉⠉⠉⠉⠉⠓⠄⠀⠀⠈⢧⠀⠀⠀⢸
⠀⠀⠀⠀⢃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡜
⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⠀⠠⠖⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠔⠋⠉⠉⠓⠦⡀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠔⠒⠲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣸⠁
⠀⠀⠀⠀⢸⠀⠠⡀⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⡿⠀⡀⢹⡄
⠀⠀⠀⠀⠘⣇⠀⣽⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⣧⠊⠀⢸⠃
⠀⠀⠀⠀⠀⠙⣆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⢀⣀⣀⠠⠤⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⠀⣐⣰⣆⢀⠂⢲⠀⠀⠀⠖⠀⢀⠀⠀⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠰⠀⢬⣿⢮⡙⠀⣸⠁⡆⡄⣹⠀⣯⠀⠀⠀⠀⠁⠉⠉⠉⠀⠈⠉⠉⠋⠉⠋⠋⠉⠘⠉⠛⠊⠹⠏⠿⣻⣿⣿⣿⣿
⠸⡆⢸⠛⡾⣴⡀⢹⡆⣣⡄⣟⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣾⣿
⢶⡀⢸⣷⡇⢻⡟⠿⡇⣹⡂⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣿
⣸⡁⢸⣿⡏⢸⡓⠀⣇⡿⠇⠉⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⢀⡀⣦⢴⢂⠀⠀⠼⠷
⢸⡇⢸⣿⡧⣘⡇⠠⢾⡇⡀⣵⠰⠀⡄⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢤⣼⣦⡿⣥⣿⣿⣾⡆⠀⠀⢲
⢸⠀⢠⡿⠓⢿⣿⠀⢺⡇⡇⢿⣀⠰⡇⣬⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠿⠏⠙⠟⠀⠀⣉
⢸⡀⠲⣿⠀⠈⠈⠂⢼⡏⠓⠾⣿⢴⣿⣿⠀⣟⢦⣶⣿⠟⠁⠈⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡘⢏
⢸⡄⠀⡟⠀⢀⠀⠀⢸⡇⠀⠘⠀⠾⡇⠀⠈⠙⠾⠗⠐⠆⢀⣠⣶⡄⠀⠀⢀⡠⣄⣢⣄⡠⠾⠸⠂⠀⠁⢸⡿⠿⠿
⠸⠇⠾⣇⠀⠀⠀⠀⢨⡇⢀⠀⠀⠲⣷⠀⠠⠀⠀⠀⠀⠈⠻⣟⠁⠀⠀⠀⠈⠹⠿⣋⣉⢀⡀⢤⡀⢶⣤⡞⣷⡢⡷
⠀⠀⠠⢿⠀⠀⠀⠦⡀⡇⠀⠀⠀⠀⠷⠀⠀⠀⢤⡀⢠⠀⠀⠸⣿⣮⠀⠀⢀⠀⠀⠀⠉⠙⠛⠛⠓⠛⠿⠿⠶⢤⠯
⠀⠀⠀⢺⡄⠀⠀⠀⢰⣷⠀⠀⠀⠀⠻⠀⠀⡠⠀⢄⠈⠁⠀⠀⠈⠙⠓⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣀⣀⠀⡀⠀⠀⠄⠀⠀⡀⠐⢾⣦⡁⢁⠀⡀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉
⢀⢠⠀⠀⠀⠀⠀⠛⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠻⠰⡄⠀⠀⠈⡁⢸⣦⡴⡈
⢈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⣠⣿⠄⠀⠀⠠
⢰⡄⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⢠⣾⣻⠀⢷⡇⠀⠀⣿⣿⣿⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⢸⡇⠀⠀⠀⠀⠀⠄⠀⢸⣧⠀⠀⠀⠀⢹⣛⣿⠀⠈⡧⣆⢀⣾⣿⠃⠁⠀⠘⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⣾⡟⠁⠀⠀⣷⣿⣿⣿⣿⣤⣆⣤⡄⠹⠟
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢩⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣀
⠀⠙⢿⠀⠀⠀⡄⠀⣠⣄⠐⠂⠀⣂⢰⠀⠀⠀⠂⠈⠠⠄⢹⣿⣿⣿⣿⣿⣿⠋⠿⣿⣿⡆
⠀⠀⢿⡅⠞⠃⠀⠀⠀⠀⠁⠀⠴⡈⣿⠁⠀⢦⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⡗⢿⣿⣿⣆
⠀⠀⣼⡇⠀⠀⠀⠚⠁⠀⠀⠐⠤⣬⡽⠀⠀⣸⠶⢷⣤⠴⠛⢉⣿⣿⠟⠻⣿⣿⣴⣶⣿⣿⡽
⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡇⠈⠀⡾⠉⠀⠒⠀⣿⣿⣿⢾⣿⣿⡇
⠀⠐⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡼⣿⣿⠘⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⢰⣿⡷⠀⠀⠉⠀⠘⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⡰⣹⠇⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠠⠄⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⢀⡤⠒⠚⠉⠉⠙⠒⠲⢤⡀
⠀⢀⡜⠁⠀⠀⠀⢀⡤⢤⡀⠀⠐⠈⢣⡀
⠀⢸⠀⠀⠀⠁⣰⠃⠀⠀⠙⢦⠀⠈⠆⢻⡀
⠀⢸⠀⠀⠓⠒⠉⠙⡄⠀⠀⠘⡆⠀⠀⠀⢧
⠀⠘⢆⠀⠀⠀⠀⣀⠟⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠈⠙⠊⠉⠉⠀⠀⠀⠀⢀⡇⠀⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡂⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⡠⠖⠋⠓⢦⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⠀⠀⠀⠈⠧⠐⠒⠒⠉⠋⠀⠈⠉⠉⢙⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠫⠭⠍⠉⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠀⠀⠀⠀⢀⠀⠈⠈⠓⢤
⠀⠀⠀⠠⠖⠒⠊⠀⠀⠁⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠱⣄
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠔⠣
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⡀
⠀⠀⣀⠞⠀⠀⠀⠠⠠⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡
⠀⠀⠣⠶⣈⣩⡅⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢤⡚⢢⡀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⡖⠚⢀⠀⠀⠙⢯⠁⠒⠒⠒⠒⠒⠒⠒⠹⠁⠀⠱⠀⢀⠼⠀⠙⣆⠀⠀⠘⡆⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⣀⣤⡴⢥⣤⠀⠀⣔⠁⠀⠀⢀⢸⡆⠀⠃⠘⣉⣁⣀⣀⣀⣀⠈⠑⠀⠄⠠⣄⣀
⡠⣀⣠⠖⠉⠒⠤⠤⠔⠀⠂⠤⠇⠄⠈⠀⠠⠞⠒⠒⠒⠒⠶⠶⠒⠋⢀⣠⣴⣆⠀⢀⠀⡥⢀⣈⣉⡡⠤⠻⠀⠀⠠⠇⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⣀⡀⠉
⢧⠄⠒⠒⠉⠁⠒⠒⠒⠀⠒⠀⢖⣦⣦⠤⠜⠈⠁⠂⠉⠑⢲⠄⡠⠚⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡤⢂⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡰⠉⢀⣀⡀⠈⠄⠀⠀⠀⠀⠘⠀⢀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠁⢀⠔⢀⡆⠀⠀⠀⢀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⡄⠈⠉⠁⠀⠀⢀⡀⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠈⠑⠂⠀⠀⠂⠁⠀⢀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⣄⣀⣀⡦⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣒⣦⣤⣤⣶⣶⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠟⠛⠛⠛⠛⣽⣿⣿⣿⣿⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣇⣀⡀
⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⠿⢿⣿⣶⣦⣄
⠀⡴⠞⣉⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣤⣄⡉⠙⠿⣷⣄
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠉⠉⠓⠄⠈⢿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⠃⣹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣾⠿⠗
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣼⡸⣯⣿⣿⠀⢱⣾⣿⣿⣿⣿⣿⣿⠸⣿⡟⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣕⣻⡌⢿⢰⡇⠙⢿⣿⣿⣿⣿⣿⡇⠹⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⠉⢱⠘⠇⣿⠀⠀⢹⣿⣿⣿⣿⣯⣄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣿⠀⢸⣆⠀⠉⠀⠀⢞⣽⣿⣿⣿⠻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⢼⣿⣿⢷⣆⢸⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⣷⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⣙⠻⣷⣿⡻⣿⣿⣿⣿⣷⣦⣘⠛⠛⣛⣛⡸⠁
⠀⠀⠀⠀⢀⣠⠿⠀⣰⣿⣶⣌⠳⣌⠻⣿⣾⣝⢿⣿⣿⣿⣿⣿⢏⠉⠉
⠀⠀⣠⡴⢛⣡⣴⣿⣿⣿⣿⣿⣷⣽⣷⣮⡻⢿⣷⣝⢿⣿⣿⡟⣼⣆⣀⢤⡀
⠀⠚⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⣷⣝⢿⣧⣿⣿⣿⣿⣻⣷⡄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣷⡌⠻⣧⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠘⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠙⢿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣷⡑⠀⠀⠹⣿⢹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠶⠆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⡠⠤⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⢒⡾⠘⠧⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⠫⠑⠀⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣠⣤⠶⠞⠒⠛⠛⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠍⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠦⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣴⠟⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣇⠈⠉⠒⠒⠒⠒⠒⠒⠒⠋⠉⠁⣠⠞⠁⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⣀⣀⡀⠤⣤⡾⠋⠀⠀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣫⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⡄⢀⣀⣀⣀⣀⣤⣾⠿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧⠀⢰⣏⣷⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣦⣤⡄⠀⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣤⠤⠶⠖⠁⢀⣈⣳⠶⣤⣤⠜⠱⢤⣴⣶⠖⢒⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⡀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢰⠋⣀⡀⠹⣞⢳⠀⠀⢰⣿⡿⠶⢦⣌⣿⠛⠛⢿⣿⣿⣿⣿⣿⣀⣠⣿⣿⢿⣿⣿⣿⠉⣉⣉⣿⣿⣶⣼⣿⡀
⠀⠀⠀⠀⣀⡰⢦⣀⠀⢸⠀⢸⠀⢿⣻⠀⣿⡟⢻⣿⠙⣿⡇⣶⣆⢹⣿⠀⣿⡄⢿⡇⣤⡌⢿⡏⢹⣿⢡⣶⠈⣿⡇⢠⣿⣿⣿⠇⣼⣿⣿⣿⣦⣤⣤⠄
⠀⠀⠀⠀⢹⣄⣀⡞⠁⣧⣤⠈⠳⣈⠛⠿⣿⢿⢸⣿⡀⣿⡇⢹⡟⢸⣿⠀⣿⠇⣼⡇⣿⠇⣼⡇⢸⣿⡈⢿⣤⣿⡇⢀⣉⣹⣿⠀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⣸⠀⢻⠛⣷⣆⠈⣿⠘⣿⡇⣿⡇⢰⡀⣿⣻⡇⣠⣴⣿⠀⣄⢸⣿⡇⢸⣿⢻⣦⠘⣿⠇⢸⣿⣿⣿⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡛⠁⠀⢸⡄⠙⠚⢀⣿⣆⠛⢃⣾⣇⣸⣧⣘⣿⣇⣻⣿⣿⣀⣿⣈⣿⣧⣼⣿⡈⠛⣠⣿⣀⣈⡉⣹⡉⣹⣿⣿⡿⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣀⣄⣸⡟⠶⠶⠿⠿⠬⣿⣿⣛⣙⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠛⠁⠛⣻⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⣤⣼⣷⣶⣶⣦⣤⣤⣤⣀⣈⠉⠛⠻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢶⣦⣤⣉⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠭⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⣿⡿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⡿⢿⡿⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠿⠿⠟⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣾⣿⣤⣈⣁⣪⡍⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⠿⠿⠿⠷⢦⣤⣭⣉⣙⡛⠶⣦⣄⡀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢶⣭⡻⢶⣄⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢷⣦⡀⣿⠆
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡉⢿⣿⡀
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠈⠻⣦⡙⢷⣄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠈⢿⣆⠙⢷⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠞⠛⠛⠀⠹⣦⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⠓⠶⠶⢦⣄⣙⠛⠛⠛⠛⠛⠛⠛⢦⣙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣋
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠈⠙⠻⢷⣶⡀⠒⠀⠒⠚⠛⠳⣮⣙⠳⠶⣦⣤⣤⡶⠾⠋⣠⡾⠋
⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⢶⣦⣤⣄⣀⣀⣀⣉⣻⣶⣶⣶⠶⠶⠚⢛⣻⡏
⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡄⠀⠈⠉⢻⣧⣀⣀⣀⣀⣀⠀⠀⠀⠈⣉⣉⣉⣩⣉⠁⢠⣿
⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣠⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠷⣼⣧⣤⣭⣭⣭⣭⣭⣭⣍⣉⣉⣉⡒⠶⠖⢀⣿⠃
⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣠⢠⣟⠛⠲⠶⠖⠶⠶⠶⠶⠖⠒⠒⠉⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣬⣤⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢸⡟⣼⡋⠉⠒⠒⠒⠲⠶⠶⠶⠶⠒⠒⠉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⠀⠀⣀⣠⣞⠉⠀⣤⣀⣀⣠⣤⡴⠋⠁⠀⠀⣀⠐⠻⣦⠀⢹⣏⠙⠛⠓⠒⠒⠒⠒⠒⠞⠒⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢤⣀⡉⠉⠙⣛⣛⣉⣉⣻⣄⠀⠀⢹⠉⢁⡸⢧⣄⠀⠀⠰⣿⣷⣴⠟⠀⢸⡟⢛⠛⠒⢒⣠⣶⣶⣶⠾⠛⠉
⠀⠀⢀⣠⠴⠛⠉⠛⠃⢠⣤⠼⠟⣻⡆⢹⣿⣩⣿⠉⠙⠻⣆⠘⣿⣉⠀⣀⠽⠛⢦⣤⣤⣤⣴⠆⠀⣼⣷⣿⣿⠿⠿⠛⠋⠉
⠰⣿⣿⡇⠰⠟⠀⠴⠶⢀⣁⣤⣾⡟⠛⢾⣿⣿⠿⣷⠙⠃⢿⡄⠈⠛⠻⠧⠴⠴⠛⠋⣁⣈⣥⠴⠛⠉
⠀⠈⠉⠻⣿⣾⣿⡿⠿⠛⢋⣥⣼⠀⣠⣿⠿⣿⡆⢸⡧⣶⡌⢻⣦⣤⣀⣀⣀⣀⡴⠟⠋⠉
⠀⠀⠀⠀⡿⢛⣥⡀⠾⠆⠀⢉⣽⣶⠿⣿⠶⣿⢇⠈⣷⠀⠀⣠⡍⠻⣿⡟⠙⢿⡆
⠀⠀⠀⠰⣧⣠⡠⠶⠒⠈⢙⠟⢻⡷⣾⣿⠾⣿⡆⠀⢹⣶⣿⡀⠁⠀⠠⠿⢦⡾⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣾⣟⢛⣿⣿⣿⡴⠟⠀⠀⣸⢿⣿⡇⠘⠿⠃⢠⣌⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⢙⣟⢻⠟⠁⠀⠀⠀⠀⠀⣿⢶⣿⣿⡿⠇⣤⣦⠀⣠⡿
⠀⠀⠀⠀⠀⢘⡿⢉⣹⡿⠿⠟⣿⡆⠀⡀⠀⠀⠀⣸⣿⣼⣧⣼⣷⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⢈⡿⠋⢀⡀⢠⠞⠻⣷⣌⠛⠀⠀⣰⣿⣷⣾⣷⣾⣽⣿⣋⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣆⠉⠀⢀⣠⣤⣿⠙⢷⣤⡾⠿⣿⣛⠛⠛⠛⣿⣿⣿⠋⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⠟⠋⠉⠙⢆⠀⠈⠀⠀⠀⠙⢿⡛⠛⣹⣏⢩⣥⣤⡈⢙⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠙⣶⣏⠁⠀⠀⢠⡞⠋⠁⢈⣻⡄
⠀⠀⠀⠀⠀⠀⠘⢃⣀⣀⣀⣀⡤⢤⡾⠀⠀⢀⣀⣤⣤⣤⣀⡈⢻⣅⠙⠛⢁⠙⠲⠶⠿⠛⠻⣦
⠀⠀⠀⠀⣀⣄⣐⣀⡙⠁⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠛⣦⠹⣇⡉⣉⡁⠀⣤⠀⠀⠀⠘⡇
⠀⠀⠀⢰⡏⢉⣿⣧⣹⠂⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⡰⠀⠠⠤⢿⡆⢻⡅⠤⠄⠀⠈⠳⣄⣀⣀⡜
⠀⠀⠀⠘⠿⠿⢄⣈⡁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⢿⣼⡄⠀⠸⠇⠈⣷⠆⠀⠀⠀⠀⠀⠉⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢀⡛⠿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣤⣴⣿⣿⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢀⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⣞⠉⣹⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣼⣧⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⢹⣖⣉⣉⡤⠲⠆⠀⣠⣤⣤⣤⣤⣤⣤⣴⣦⡀⠀⣀⣴⠟⢻⣿⣿⣿⣷⣶⣤⣤⣴⠊⠁⠀⠀⠀⣀
⠀⠀⠈⠉⠉⠁⠀⠀⠄⠀⠹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠿⠿⠏⠹⣿⣿⣿⣿⣷⣄⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⢿⣿⣿⣿⣿⠄⠀⠀⢀⣀⣐⡖⠀⠘⠿⠛⠛⠿⠏⠀⠀⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣻⣿⣿⣿⣿⣋⡁⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢞⠘⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠿⣿⡿⠿⠿⠟⠛⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⡷⣾⣟⣻⣻⣿⣿⡽⣿⢶⣄⠀⣀⠔⠋⠀⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⣶⠗⢿⣿⡏⠡⠎⢳⡀⠈⢦⠹⣿⡀⢀⣀⠀⠚⠁⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣻⠅⠀⢿⣀⣼⣿⣿⠰⡖⡈⢿⡑⡈⣧⢻⣿⠉⢻⣷⠀⢘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠤⢸⣹⣇⣾⣿⣿⣷⣆⣘⣿⣷⣷⠿⢺⠻⣦⣿⠿⡚⡟⣧⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣿⣖⣸⣆⣿⣿⡿⢿⣿⣏⣿⣭⣿⣴⡾⣿⣿⣿⠛⡄⣃⢡⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢷⣻⣏⣿⣧⣼⣿⣷⣾⣿⣿⢿⡟⣾⣿⣿⢿⡛⣿⡇⢡⠸⠈⡜⡗⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⠻⣿⡧⣈⠛⣿⢹⣿⣿⣿⣶⣿⣿⣿⠛⠀⣇⢻⣷⠈⡆⠆⢡⢻⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⡇⠉⢿⣿⣷⣿⡿⢿⣿⣾⡟⠉⠁⢸⣆⣀⣸⣨⣿⣿⣿⣼⡬⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⠀⠸⡄⠉⠀⣷⣀⣽⣷⣳⣶⣿⡿⠿⣻⢿⣿⣮⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣏⣦⣤⣷⣾⠿⢿⡟⠛⣿⣏⣽⣿⣶⣾⣿⠟⠁⢸⣿⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣽⣿⣷⣿⣷⠾⢿⠛⣉⣉⣥⡇⠀⠀⠀⠀⠀⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣆⠀⠉⠉⠀⠀⠈⣻⣿⣿⣿⣷⣦⣤⣀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠛⠟⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠘⠛⠻⢯⣽⣷⡾⠁⠀⠀⠀⢀⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣾⣿⡇⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⢀⡀⠀⣀⣀⣴⣿⣿⣿⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⡿⠀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢠⠳⡀⢴⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣄⡀⢠⣦
⠀⠀⠀⠀⠰⣖⠒⠁⡠⠵⣲⡖⠊⢒⣶⣖⡛⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⡿⢤⣤⣈⣀⣄
⠀⠀⠀⣠⣄⠀⡇⢸⠰⣷⢸⣾⢶⡞⣿⢩⡝⣿⢰⡌⣿⢩⠻⡗⣾⢋⡍⣿⡇⣶⣾⡏⣿⣿⣦⣄⡀
⠀⠀⠈⠧⠼⠁⠷⡄⣧⣍⠻⣷⢸⡇⣿⠸⢣⣿⠸⢇⣿⠸⣱⡇⣿⣎⠳⣿⠀⣬⣿⢡⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢤⠐⠁⢃⠹⠇⣸⡸⢇⣿⣸⣌⣿⣰⣿⣇⣷⣹⣧⣿⡘⢃⣿⣀⠛⣟⣻⣿⡿⠻⠇
⠀⠀⠀⠀⠀⠀⡧⠤⢼⠓⠚⠛⠛⠿⠭⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣆⣠⣧⣤⣤⣉⣉⣙⠛⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠙⠛⠓⠶⢬⣛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡆⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠉⠁⠀⠀⠉⠉⠉⠙⠛⠛⠿⠿⡶⢤⣀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠛⠋⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣿⠛⠛⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠈⣿⡏⠛⠲⢤⣀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣺⠋⣁⠀⢸⡇⠀⠈⢷⢍⠛⣦⣄⠀⠀⠘⢿⣿⣷⣾⣿⡿⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢗⣟⡉⠀⣼⠃⠐⣦⡘⡌⢣⡝⢌⢿⡦⣄⠈⠳⣮⣿⣿⡄⠀⠀⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⠥⠀⣿⠠⡤⠼⡏⢿⠈⠃⠘⣾⢦⡽⡷⣦⣈⠻⣿⣷⣄⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢞⠁⠀⢀⡿⠀⠘⢆⡀⠀⠀⠀⠀⡿⢄⠛⠸⠇⣿⠓⠮⣿⣿⣷⣄⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣻⣇⠀⣸⡇⠀⣠⡤⠛⠒⠀⠀⢀⣇⡬⠷⣄⠀⡟⠀⠀⠀⠉⠙⠛⠿⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣽⣏⢀⡿⠀⢠⣞⣉⠀⠀⠀⣇⣼⢷⡞⠃⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡍⣼⠁⠀⢠⠞⠁⠀⣰⡸⠙⣹⠋⢀⣤⡘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢻⣄⡿⠀⢰⠃⢀⠜⣣⠻⢃⣴⣿⠖⠋⣨⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢹⡇⠀⣧⠖⠁⠘⢣⡶⠛⠁⣧⠀⣿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⢸⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣾⠟⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠻⣿⣄⣀⣀⣀⣼⡿⠁⠀⠀⠀⢻⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡏⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⠉⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠦⢤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠤⣄⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠓⠺⢭⣛⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⣤⣤⣤⣀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⡀⠈⢹⣶⣍⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠒⣚⣋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣾⣿⣿⠷⣌⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⢻⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠓⠦⣄⣉⠛⠛⠓⠒⠶⠖⠒⠒⠛⠛⠛⢉⣉⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠚⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⣿⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⡀⠀⠀⠀⢰⣿⡿⢋⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣥⣂⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣦⡄⢙⡿⣷⠟⠻⣷⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠙⢻⣿⢟⡭⢻⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⣻⡟⠁⠉⠟⠛⠛⠁⠀⠘⠁⠺⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆⣙⣒⠊⠁⠢⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢿⠙⠋⠙⠛⡦⠀⠀⠀⣠⠏⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣿⡁⠀⠀⠀⠈⠁⢀⣤⡾⢃⣼⢏⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡀⢦⢠⣤⣴⣼⣋⣴⣿⣷⢟⣠⣴⣾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣬⣽⣿⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⡏⢉⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣿⣿⣿⣿⣿⠀⢀⣹⣏⠤⣸⡟⠛⠋⣹⣿⣿⣾⣿⡿⣻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣏⣴⣿⣿⡟⢋⠀⠙⢦⣰⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⡟⠉⣵⣿⡟⠁⢸⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠊⡼⠋⠁⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠚⡁⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢀⣬⣼⠏⠀⠀⠀⠀⠀⣿⣅⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⣾⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⢸⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⠛⠀⠀⠈⠀⡞⠙⠛⠟⠙⠳⠛⠻⢿⣿⣿⡿⠿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⣄⣀⣈⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠚⠃⠉⠉⠉⠉⠙⠳⢆⣤⡠⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡄⠀⡦⢘⡇⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⡩⠒⠙⢦⡀⠘⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡇⠀⡷⣾⠀⠀⠀⠀⢠⠃⢀⣤⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⣎⣀⣤⡄⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢄⠀⢧⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠈⡇⠀⠀⢸⠀⠀⠀⢰⣶⡀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠁⠀⢻⡀⠀⢸⠀⠀⠀⠸⠿⠃⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠍⠁⠀⠈⣧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠀⠀⠀⠀⠀⢠⠃⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⣦⠀⠀⠀⣀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⡒⢶⣶⣤⡄⢹⡄⠀⠈⠓⢦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣶⣿⣿⣿⡶⠚⣷⡗⠂⣄⠀⠀⠀⠀⠈⠙⠲⠦⢤⠤⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠜⠛⠿⠿⠿⠧⠤⠿⠛⠀⠈⢧⡀⠶⠶⠶⠶⠶⠶⣾⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡧⠀⠀⠀⠀⠀⠀⠗⠀⠈⠛⠲⠤⣄⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⣆⣀⣀⣠⠤⠖⠊⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢀⡤⣶⠯⠉⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣄⣀
⠀⠀⠀⢀⠔⡹⠊⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣙⣶⣤⣀
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣶⡿⠿⠛⠉⠀⠀⠀⠀⠊⠭⢓⠦⣄
⠀⣰⢃⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡇⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⢸⠃⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠓⠦⠬⣍⣉⣉⣉⣉⣉⡽⢋⠔⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠒⠒⠛⠛⠛⠛⠛⠋⠁
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡫⠚⠁⠀⠀⣀⡤⠶⠛⠉
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠕⠋⢀⣠⠤⠖⠋⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⣀⠠⢖⣋⡬⠴⠒⠋⠉
⠀⠀⠀⠀⠈⠙⠓⠲⠖⠛⠛⠉⠉
⠀⠄⠠
⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⣒⣤⡀⠀⡠⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠐⠋⠉⠉⠉⠛⠓⣴⡚⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣼⡇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣄⠀⠀⢻⡀⠀⠀⢠⡀⠀⠀⣿⡋⢻⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣬⣷⣤⣌⣳⣤⣀⣈⣻⣦⣄⣹⣿⡾⠟⠙⠶⣄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠉⢛⣿⣛⣯⣉⠉⠉⠉⠉⣉⣉⡈⢳⡀⠀⠀⠘⣆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠀⠀⢀⣾⡿⠋⠁⠈⢳⡄⢀⡾⠉⠉⠻⣆⣷⠀⠀⠀⢻⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣾⠋⠀⠀⢀⡀⠀⣧⣾⠀⣀⠀⠀⠹⣿⡇⠀⠀⣸⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡾⢻⡀⠀⠀⣿⣿⢂⡿⢻⣼⣿⡇⠀⠀⣿⣿⢀⡴⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠹⠇⢬⣳⣤⣀⣬⣥⠞⠁⠈⢿⣥⣀⣠⡴⢛⣧⠤⣾⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣾⠛⣧⣀⠉⠉⠉⠉⠁⠀⠀⠀⢀⠈⠙⠛⠁⣸⡿⣾⡏⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠇⢠⡟⢹⡇⠀⠀⠀⠹⣶⣶⣶⡿⠀⠀⣴⢻⣿⠀⢻⣧⠀⠀⠀⠀⠙⠶⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢀⡟⠀⣼⠷⣦⠀⠀⠀⠈⠉⠉⠀⠀⣠⣇⠀⢿⠀⢸⢻⡆⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡾⠀⣼⠃⢀⣿⠀⠀⠀⢀⣀⠀⠀⢸⡏⢿⡄⠸⡇⢸⡎⣧⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡸⠁⣼⠃⠀⣼⠇⠀⠀⢰⣿⣿⡆⠀⠘⣷⠈⣧⠀⣿⠀⠁⣿⣶⡶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⣸⡏⠀⠀⠀⣾⣿⣿⣷⠀⠀⣿⠀⣿⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⢀⣿⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⢸⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢈⣿⣦⡀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⣰⠛⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⡴⠋⠀⠉⠁⠀⠀⣿⣿⣶⣦⣤⣤⣤⣴⡾⣿⡇⠀⠀⠰⢿⡄⠈⠳⡄
⠀⠀⠀⢀⣴⣿⣴⡶⠞⠁⠀⠀⠀⠀⠀⢀⣾⣋⣠⣿⡍⠉⢉⣿⣡⣄⠸⣷⡀⠀⠀⠀⢻⣄⠀⠹⣄
⠀⠀⠀⣸⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠟⠉⠹⣿⣴⣿⠟⠀⠹⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣿⠄
⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠙⢯⣿⡀
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠈⠻⣷⣄
⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠘⢿⣆
⠘⣆⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠈⠛⠛⠻⠿⠛⠋⠁⠈⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠳⣄⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⠳⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⣶⣦⣤⣤⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣦⣤⣤⠴⠾⠦⠀⢀⡀⠠⡄⠀⠠
⠀⢀⡀⠂⠂⠀⠂⠀⠀⠀⠀⣤⣾⣷⡀⠀⠀⠀⣀⠴⠻⠴⠇⠀⠀⠀⠀⠀⠀⠀⠰⠄⡷⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠉⠁⠀⠀⠚⠁⠀⠀⠘⠺⠁⠀⠀⠉⠀⠀⡛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠲⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠒⠀⠁⢀⡛⠿⠃⠈⢡⡄⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠘⠟⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢢⡈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠈⢀⠀⢀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠸⠂⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠃⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠛⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⢀⣀⣠⣤⣤⠤⠄⣀⣀⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠶⠖⠲⣤⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣶⣿⣦⡀⠀⠈⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⢠⣴⣶⡄⠈⠻⣿⣿⣽⡿⡣⠄⢄⢠⠜⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠘⠛⠉⠁⠀⠀⠚⢛⡭⠻⠦⣤⣴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢚⣓⠒⠄⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣼⠟⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⢿⠿⠿⠛⠋⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠾⠷⣶⣻⣿⢿⣿⣿⠀⠀⠀⠀⠀⢀⣼⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡄⠘⣿⣿⣸⣿⣿⡇⢁⣠⣠⣴⣟⠁⠀⡈⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠙⠉⠈⣿⣿⣷⣿⣿⣿⣿⠟⢣⡞⠁⣀⣀⣁⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠉⠀⠀⠀⠀⣴⠞⠉⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢻⣴⠂⠤⢀⡀⢀⣀⣴⣾⣿⣿⡁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣬⡗⠂⠀⣀⠀⢠⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⣿⢆⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⠀⠀⣀⠀⠀⣀⣿⠇⠀⠀⠉⢀⣿⣿⡿⠀⣿⣿⠛⠐⠦⠤⠤⠔⣿⡎⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢤⣤⣡⡴⢿⣧⠀⠀⠀⢀⣾⣿⠿⠃⣾⡿⠋⠀⠀⠀⠀⠀⠐⢻⣧⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⡏⠀⢸⣷⡐⠒⠀⠸⠋⠀⠀⠀⠟⡁⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⠀⠀⣾⠋⠑⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠈⢰⠾⣿⠆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠈⠀⠠⣾⡏⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⣿⡒⢦⡀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡶⠋⠉⠀⠀⠈⠀⠀⣂⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡧⠂⠀⠀⠉⠃⠙⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠘⢿⣶⡄⠛⠜⡀⠀⣠⢤⡀⣼⣶⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⠣⠀⣘⣁⠂⠈⣸⠷⠋⢁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⣀⣀⣠⣴⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⢳⣶⣄⣤⣤⣤⣤⣤⠈⠀⠀⠀⠀⠘⢀⣀⣠⣤⣶⣶⡶⣿⣛⣉⣭⣿⣷⣿⣶⣿⣿⠀⣱⣶⣶⠄
⠀⠀⠀⢀⣺⣿⣶⣶⣶⣶⣴⣿⣿⣇⠠⠘⢦⡙⣿⣿⣿⣿⣿⠋⠀⠈⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣙⣧⡼⣿⣿⣿⠃⠐⣟⠛⠓⠂
⠀⠘⠀⠀⠻⠿⣿⡿⠿⠿⠿⠿⠿⠿⣷⠒⠀⠟⠛⠛⠛⠛⠃⠠⠀⠀⢀⠀⢼⡉⠉⠉⠉⠉⣿⠉⠉⠁⢠⡏⢸⡇⠀⠀⠀⡆⠀⠀⡃
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣤⣠⣆⠀⠀⡀⠀⠀⠀⠀⠀⠉⠓⠤⠤⢤⣿⠀⠀⠀⣸⠁⣾⣧⠀⠀⠀⢰⠀⠀⢥
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⢨⣿⡿⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢸⣿⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠆⠸⣯⢠⠀⠀⠀⠀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠣⠀⣿⡈⠀⠀⠀⠀⡇⠀⠀⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢽⡅⠀⠀⠀⠀⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣇⡐⠀⠀⠀⠇⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠠⠀⢠⡄⠀⠀⠂⠀⠂⠀⠀⠄⠀⢁⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠇⠀⠱⠀⠀⠛⠀⣦⠀⢀⠐⠛⠀⠐⢆⢨⠃⠀⠀⠒⠀⠀⠈⠁⢄⡄⠐⠀⠠⠾⣤⡃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⡀⠀⠀⠐⠂⠀⠱⡀⠢⠃⣀⠀⠀⡀⠀⠀⡤⣨⠀⡀⠰⠀⠀⣀⠀⠀⠀⡀⠀⠀⡀⠔⢀⠀⠄⢀⠎⢀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠀⠀⠁⡀⠰⡏⠠⡀⠰⠊⠃⡈⣄⠀⠉⠀⠄⢠⠉⠀⠋⠀⠠⠁⠀⠖⠰⠉⡀⢴⠈⠁⠨⠦⢔⠁⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠴⠀⠀⠀⠢⡠⠆⠀⢀⠀⢃⠐⠃⠀⡁⠉⠀⠶⠗⠈⡄⠀⠔⠃⠀⠀⢤⡞⢑⠃⠀⢄⡤⠘⢀⠠⢀⡀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠰⠑⢀⠐⣎⢹⡀⠂⠀⠀⣴⠀⠆⠠⠀⠀⢉⢀⠨⠋⡄⢤⠌⡅⠀⠀⠡⣀⡀⠀⠈⠀⠀⠀⠛⣀⡖⢶⠀⢃
⠀⠀⠀⠀⠀⠀⡀⢄⠃⠘⢷⡆⠀⠀⠑⡁⠀⠀⠀⠐⡊⠀⣀⠀⠀⠀⠀⠄⢀⣄⠁⠂⠁⡀⢰⢈⡄⠈⠆⠎⠀⠞⢀⠠⠉⠈⠉⡉⠉⢀⠔⠀⡀
⠀⠀⠀⠀⠀⠐⢯⠀⠑⠀⠀⡅⠘⠁⢀⡈⠀⢦⡼⠋⠐⢀⣁⣴⠀⠒⠈⠁⠀⢠⡄⠒⠈⠉⢀⠡⠀⠀⠰⢐⠓⠈⡁⠀⠰⠥⠂⡡⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⡀⠀⢠⡀⢂⠐⡠⠃⠄⠸⠅⠘⠢⠀⢤⠀⠠⠄⠀⠃⠀⣖⠄⠀⠓⠠⠁⢀⠈⢁⠐⠚⠠⠊⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠣⡀⠀⠀⢀⡆⠘⠈⢀⢁⠀⠄⠠⠇⠀⠀⢠⠄⠀⡀⠈⠀⡤⠀⡰⠀⡀⠈⠀⠀⠄⢠⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⢌⠩⠃⡁⡀⡤⠀⢂⠰⠆⠙⠀⠖⢃⠀⢈⡛⠐⢆⡀⠄⠠⠀⣀⠀⠷⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣁⣈⠤⠤⠤⢶⡖⣢⡸⠄⠂⠶⡒⠶⠲⡗⢣⠠⠄⠤⢏⡀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠍⠉⠉⠉⠉⠉⠉⠓⠒⠺⡀⠀⠀⠀⣒⣄⣇⣨⣇⣠⠔⠳⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣇⠠⠐⣺⡅⠒⣨⣧⣬⣖⠾⣿⣏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠀⠙⢹⠏⣽⠇⢹⡈⢢⡜⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢄⠀⠀⠀⠆⣳⠇⡅⢞⡢⠠⠊⠈⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢀⠽⡀⠀⠀⠀⠠⠂⡆⠈⣇⠀⠔⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠠⠊⠐⠅⡇⠀⠀⢀⠀⡠⢓⠀⢵⠀⠕⠑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡀⠀⠀⠀⠀⠐⠄⡪⡀⠐⢽⠀⠀⠀⡀⠲⢺⠀⢻⢀⠢⠔⠊⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠉⠉⠚⠚⠓⠛⣐⢀⣀⠧⠬⠚⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠄⠀⣀⠀⠠⠀⠄⡀⢀⠀⠀⠀⠀⠠⠀⠄⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⠐⠁⠀⠊⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡶⠶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋⠀⠀⠀⠈⠉⠉⠻⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⠿⠿⢿⣿⣿⣷⣶⣿⣿⣿⣇⠀⠀⠀⣶⣀⣀⣴⣶⣶⣿⠟⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠈⠉⠓⠮⣙⢿⣦⡀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢝⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠑⣽⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣹⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢁⠈⣿⡿⢷⣤⡀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⠄⢸⡇⠀⠙⢿⣷⣤⡀⠀⠈⠿⣿⣿⣶⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠂⣸⡇⠐⢶⢶⡟⢿⣿⣷⣄⡀⠙⣿⣿⣿⡟⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⣿⠃⠶⢬⡇⠻⠀⢻⣿⣿⢿⣶⣌⡳⣿⣱⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣢⢠⣿⠀⠈⢢⣁⣀⠀⢸⣿⣟⠺⣿⣿⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⡍⣼⡇⠠⢞⡻⠆⠀⢠⣸⡟⠛⠂⣿⡿⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢠⡿⠀⠀⣛⡽⡃⢰⣾⣿⢧⢰⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣼⡇⠀⡴⢁⣼⡟⠀⣼⣿⠸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠀⠰⠗⠋⠀⣴⣾⠿⢻⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣿⠀⠀⠀⠀⠀⣼⡗⠀⢸⡿⡿⠁⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⢿⣷⣤⣀⣤⣾⡿⠁⠀⠀⢿⡀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠋⠉⠉⠉⠉⠀⠀⠀⠀⠈⠳⣄⠀⠀⢻⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣤⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣦⣀⠉⠿⠉⢸⣿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣫⢷⣄⡀⠀⠀⠘⠋⠻⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣝⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣦⣤⣄⣀⣀⣈⠙⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣌⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢹⣷⣤⣶⣿⣛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠶⣍⡓⠦⢤⣀⣀⡀⠀⢀⣀⣀⣠⣤⣤⠶⣿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠭⢭⣭⡭⠤⠶⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠓⠶⠤⣤⣤⣤⣤⣤⠤⠶⡚⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡰⠂⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⣆⠀⠀⠙⠳⠤⠤⠤⠖⠋⠀⠀⡜⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣷
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⡞⢯⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⡴⠞⢉⡟
⠹⡈⢧⠀⠈⡳⣤⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⣴⣾⠩⠀⠀⡞⡸
⠀⢳⡘⢧⠀⠀⠈⠉⠉⠉⠉⢹⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢻⠉⠉⠉⠁⠀⠀⠀⠀⡼⣱⠃
⠀⠀⠳⡌⢧⠀⠀⠀⠀⠀⠀⠘⠳⣄⡉⠓⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠚⠉⣀⣤⠞⠃⠀⠀⢀⠀⠀⠀⡜⡵⠃
⠀⠀⠀⠹⣌⢧⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⠲⠦⣤⣈⣉⡉⠉⠛⠛⠛⠛⠛⠉⠉⣉⣁⣠⡤⠴⠖⠛⠉⠀⠀⠀⠀⡴⠃⠀⢀⣞⡼⠁
⠀⠀⠀⠀⠘⢦⡳⡄⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⣫⠞
⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⣠⢴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⠦⣄⡀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⠖⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣍⣓⠲⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠴⢒⣋⣥⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁⠀⠀⢷⣤⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠋⠀⠀⢘⠒⠒⠐⠦
⠀⠀⠀⢀⡞⠋⠀⠀⠀⠈⠉⣇⢀⣀⠀⠀⠀⠀⢰⣟⡆⢸⣻⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠤⣤⢼⣿⣤⣥⣤⣤⢤⣤⣯⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠧⠀⠐⠸⠀⠂⠒⠠⠄⠘⠰⠸
⠀⠈⡞⠀⠗⠃⠘⢺⠖⠀⠘⢿⠋⠙⠍⠀⠐⡖⠖⠀⣖⣒⣒⠒⠂⠀⢰⠀⠀⠀⡰⠀⠐⠀⢹⠈⠈⢸⡏⠉⠉⠸⡇⠁⠈⠹
⠀⠀⡇⣺⢓⣷⢰⡟⠛⡇⢰⣟⣰⠀⠀⠀⠀⡅⠀⢸⡏⠉⢽⠤⠀⠀⠘⠀⢀⠜⠄⠀⠀⠀⣿⣍⣉⣿⢺⠁⠈⠀⡇⣍⣭⣘
⠀⠀⡇⠈⢹⠉⠸⡇⠀⡇⠀⢹⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⢸⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⢺⠀⠀⢸⡐⠀⠀⠀⡇⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠓⠛⠛⠛⠛⠓⠒⢒⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠐⡖⠀⠀⠎⠁⠀⠐⠋⠛⠒⢲⠒⠒⠒
⠀⠀⠘⠛⠋⠙⢻⠛⠛⠛⢻⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠛⠋⠉⠀⠐⠃⠁⠀⠘⠛⠚⠙⠃
⠀⠀⠰⡶⠶⠶⢶⡶⠶⠶⡾⢦⡴⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⢁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀
⠀⠀⠀⢠⡤⣤⠸⠁⣀⣀⡈⢈⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣤⣤⣤⣤⠀⠀⠀⠐⠀⠀⣤⢤⣤⡄⠓⡏
⠀⠀⠀⠨⣇⣸⠨⠨⡇⠈⡇⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠿⠤⠤⠇⠀⡇
⠀⠀⠀⠀⠀⠉⠨⠠⡇⠀⡇⠀⠉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⣠⣤⣤⣤⣤
⠀⠀⠀⠋⠃⠇⠟⠛⠂⠀⠀⣀⣀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠀⠀⠀⠀⢠⠀⠀⠆⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠘⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⡰⢠⠀⠴⡀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⣀⣀⣀⣀⡄⠀⠀⣤⣤⡂⡆⢴⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠁⡠⠊⠀⠀⠴⠀⠀⠀⠀⠀⡇⠀⠀⠤
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠈⠁⠉⠈⠀⠀⠀⠐⠂⠀⠀⠐⠀⠁⠀⠀⠈⠳⠬⠀⠐⠀⠀⠀⠈⠀⠀⠃⠀⠀⠒⠀⠐

⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡆
⠀⠀⠀⠀⡇⠀⡄⢠⠀⠤⠠⡄⢠⠀⡄⠠⠄⢠⠀⡄⠠⠄⢠⠀⡄⠠⠄⢠⠀⡄⢠⠄⠤⠀⡄⠠⠄⢤⠀⡄⢠⠀⡤⠀⡄⢠⠀⠤⠀⡄⠀⡇
⠀⠀⠀⠀⡇⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇
⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⡇
⠀⠀⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⡇
⠀⠀⠀⠀⡇⠀⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⡇⠀⡆⠀⠀⡀⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⢀⠃⠀⢀⠀⠶⠀⠂⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⡇
⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⢸⠉⠉⠃⠞⢱⠃⡟⣸⠸⢃⠇⠏⠐⡃⠀⠀⠀⠀⡟⠀⡇⠀⠘⠀⡆⢸⠀⠄⠊⠝⡰⢹⠇⡎⠐⠀⠀⠀⠖⠀⡇
⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⠃⠁⣞⠈⠀⠀⢸⠁⠀⠀⠀⠀⠋⠐⠁⠁⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠓⠀⡇
⠀⠀⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⡇
⠀⢀⡆⠀⡇⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡇⠀⣄
⢠⣾⣿⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠆⠿⠂⠿⠇⠺⠇⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡇⢸⣿⣦
⢾⣿⣯⣴⡿⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠸⣷⣬⣿⣿
⡌⣾⡿⠟⣧⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⡟⠿⣿⡎
⡏⠀⣴⣾⣿⢋⠃⢀⣤⣾⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⣦⡄⠀⣉⢻⣿⣶⡄⠈
⢀⠘⠛⢛⡷⡿⡗⢁⡀⠈⠛⠛⢿⠀⠀⠀⡀⠀⣴⡄⢠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⠀⣴⡄⠀⠀⠀⠀⢰⠟⠓⠋⠀⣀⠁⡾⣷⣟⠛⠛⢀
⠁⠀⠀⣁⣀⢠⣴⡾⠟⢦⡀⠀⢈⣀⡀⣴⣿⢰⣿⣷⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣷⢸⣷⡄⣀⣈⠀⠀⣠⠞⠿⣶⣤⢀⣀⡁
⣴⢋⡏⠁⢘⣴⣿⠀⢀⣠⣿⣶⠈⢿⡇⣿⣿⠀⠛⠃⢟⣡⣤⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣙⠃⠻⠃⢸⣿⡇⣿⠏⢐⣾⣧⣀⠀⢸⢷⣜⠀⠉⣏⠲
⠙⠻⣇⣠⡾⣯⢛⡷⠫⣽⣿⣘⢆⠀⠀⠈⠛⢀⡼⢂⣻⣟⡛⠉⠉⠀⠀⠀⠀⣄⡈⢉⣿⣿⣿⣃⠺⣄⠘⠋⠀⠁⢀⠞⣼⣿⡩⠳⣟⢫⡿⣦⣀⡿⠛
⡿⠃⠻⣯⣉⠖⣿⢋⠑⢾⡏⠀⠀⢧⠀⠀⣤⢮⣾⡿⠻⢿⠿⠲⡄⠀⠀⠀⣀⢻⣷⣼⣟⣿⠿⢿⠿⠊⢦⡄⠀⢠⠇⢄⠈⣿⢔⢋⢺⡗⢆⣹⡿⠃⠻
⣿⡄⠀⢈⡿⠛⣿⣶⣕⣸⣧⠴⢦⣌⣦⣘⣿⣶⠛⠀⠀⣸⣀⢀⡸⣄⡀⠀⢻⣧⢹⢟⣛⡁⠀⠀⠀⢲⣾⣟⣠⡶⣩⡶⢤⣿⣐⣥⣾⡟⠻⣟⠀⠀⣼
⡛⢿⡖⢹⣇⣠⣤⠋⠀⠉⢱⣶⠴⠛⢿⣍⢫⣭⣽⣿⡆⠘⡝⢿⣔⡮⠜⣦⡈⢻⠴⠿⠋⠀⢶⣶⣚⡏⡩⢋⣽⠟⠳⢴⣶⠉⠁⠈⢣⣤⣀⣿⠑⣾⠟
⠀⣠⣷⣿⠻⣿⣠⣧⣤⣶⣿⣷⣧⣤⣘⣿⣷⡿⣿⣿⠷⠀⣅⣁⣙⣷⣬⠟⠓⢖⠻⣏⡙⣲⠾⢿⣯⡟⠷⣧⣏⠀⢰⣷⣿⣷⣦⣴⣧⣸⡿⢻⡷⣧⡀
⣶⡯⡾⡡⡲⢾⠁⠀⣭⣿⣯⣿⣿⣻⣿⣿⣿⣥⣏⠁⢀⡀⠨⣉⡟⢡⣧⣠⣤⣾⠋⠙⠧⣵⣷⠿⠟⠊⠰⣬⣧⣀⣸⣿⣯⣿⣯⡅⠀⣹⠦⡢⣹⡮⣷
⣸⣿⣫⣬⣵⣿⣀⢀⣈⡽⣿⠿⠂⢙⣴⣿⢿⣶⣶⡎⢉⣿⣶⠿⣏⠀⢼⣿⣟⣿⡿⣀⡴⣿⣿⠏⠀⠀⠀⠻⡅⠘⠁⠺⢿⡿⣍⣀⢀⣸⣷⣭⣮⣹⣿
⠙⠛⠿⣫⣳⣿⣿⣍⣹⡁⠁⡀⣀⡼⠉⠁⡾⠛⠉⠀⢠⣎⣁⣠⣽⣆⣀⣹⠹⠿⠛⠉⢳⣟⠁⠀⠀⣤⡀⠀⠈⠹⣄⡀⠈⠁⢹⣉⣽⣿⣷⣹⡽⠟⠛
⡿⢿⠿⡿⠿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟⣛⣿⣿⣻⣿⣟⣿⣟⣻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠻⣿⣿⣵⣿⣿⣧⣤⣍⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣦⡄⠀⢽⣿⣿⣿⣿⣿⣤⢨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣆⡹⣿⣿⣿⣧⣦⠲⢶⠍⠿⠟⠚⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡿⢡⣾⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⠛⣽⣿⠃⣠⠖⠒⠶⣄⠀⠀⠀⢠⡾⠋⠘⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⣾⣶⣿⡏⠀⠁⠀⢀⡀⠀⠀⠀⠀⠠⣀⣠⣄⢹⣁⣼⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⣿⣿⢣⡹⠁⠀⠐⢯⢉⠏⠀⠀⠀⠀⡌⠹⡠⠟⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣼⣿⢻⡌⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣿⡟⣶⢰⠁⠀⠀⠀⠀⠀⡀⢠⡄
⠀⠀⠀⠄⠄⢄⠸⣿⠿⣿⣿⣿⣿⣶⡊⠓⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠰⣿⣿⣿⣿⡿⠁⠖⠠⢀⡀⠔⢈⠐⡡⠀⣒
⠀⠤⢀⠐⠈⢤⠲⣿⣦⣴⣿⠛⣿⣽⣷⣇⠀⠀⠀⠀⣀⡀⠀⠀⢀⡠⠀⠀⣾⣿⣯⡻⠋⠀⠐⠀⠒⣉⡠⠞⠏⣀⠤⣊⠄
⠀⠀⢂⠱⣄⠀⠥⢌⠛⢿⣿⡄⠙⢿⣢⣽⣆⠀⠀⠀⠘⠛⠓⠛⡏⠀⠀⣼⣿⣿⡟⠀⠀⣠⡇⠴⠀⠻⡦⠔⢊⡡⠐⠁
⠀⠀⠀⠡⡈⠳⢄⠀⢀⡀⠈⣑⠀⠘⢻⣿⣿⣦⡀⠀⠀⠡⡀⠀⣠⠀⢸⣿⣻⣿⡆⠀⢸⠿⠒⠀⠀⠀⠁⠒⣁⣀⠩⠑⠃
⠀⠃⠌⠁⠘⠂⠀⠰⠟⠴⠇⠻⠂⠀⠀⠟⠁⢻⡿⢦⣄⠀⠈⠉⠁⢠⣿⣿⠹⣿⢣⠠⡁⠀⢀⣠⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⠀⠀⠀⣠⡀⠠⠀⠀⢿⡄⠙⠳⢶⣴⣶⣿⠿⣿⡇⠻⢸⠀⠉⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⣶⠉⢇⢳⠀⠀⠈⠅⠒⠦⠤⣌⡉⠁⣴⢻⡇⠀⠈⠰⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠀⠀⠀⣧⣄⣀⠘⠄⠀⠀⠀⠃⠀⢁⣾⠀⢸⢄⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠚⢋⡁⣇⡈⠀⠐⠀⠀⠂⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣻⣷⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣙⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⣿⢿⣿⠾⠟⠋⢿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣶⣶⣙⣻⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣉⠿⠟⣛⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⡿⢿⠿⣿⣿⢿⣉⢛⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⡆⣿⣿⣷⣾⣿⣿⣻⣿⠿⣿⣿⣷⣯⡴⠀⢠⡽⣿⣿⣿⣦⣹⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣿⡿⠙⣩⠷⠀⠉⣽⣿⡏⠙⢿⣿⣿⡟⠁⠀⣿⣿⡈⠙⠷⣾⡉⠀⠀⠉⢿⣿⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⣿⣿⠿⠛⠛⠋⢙⣋⣉⣠⣤⢁⣀⣀⣀⡛⠿⠿⣄⠀⣿⣿⡀⠀⠀⢛⣋⣡⣤⣤⡤⠤⢤⣤⣄⣀⠉⠉⣙⣻⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣷⢸⣿⣿⣿⣿⠋⣰⣿⣿⣿⣿⣿⣿⣦⣄⣀⣉⣉⣀⡠⠞⣡⣾⣿⣿⣿⣿⡀⠀⣰⣿⣿⣿⡜⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⡇⢼⣿⣿⣿⠿⠿⣿⣿⣿⣿⡟⢻⣿⣧⣿⣿⣿⣿⣿⡇⢸⣿⣯⣀⣼⣯⢻⡟⠀⢸⣿⣿⠟⠃⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣼⣿⣿⣿⣿⣯⡾⠃⠀⣰⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡸⣿⣿⣿⣂⠀⠘⠿⣿⣿⡿⠿⠛⣩⣟⠁⠀⠀⠄⠹⣧⠉⠉⠛⠋⠉⠁⠀⣠⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⡹⣿⣿⣶⣤⡄⠀⠀⠀⣠⡼⠻⣇⣤⡄⣀⢠⣤⡼⠷⣄⡀⠀⠀⠀⢾⣿⣿⠿⣡⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡷⠹⣾⣭⣭⣤⡤⠶⠒⠋⠁⠀⠀⠀⠙⢧⣿⣼⠃⠀⠀⠀⠙⠛⢶⣶⣾⣷⣶⣋⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⢿⣿⣿⣿⡆⠀⠒⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⠍⣠⣿⣿⣿⡿⣟⣾⣿⣾⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣷⣂⣱⡶⠊⠁⠀⠀⢠⣿⣧⡀⠀⠀⠀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⣿⣿⣿⡿⠿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠁⠈⠙⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠻⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣷⣦⣤⣤⣴⣦⣤⣶⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠛⣿⣿⡿⠛⠿⠋⢈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣷⢶⣦⣭⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⢟⣿⠟⠋⠋⠄⠁⠘⠉⠉⠛⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⣴⣤⣤⣆⡀⢠⣠⠀⢸⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄⢀⢀⣾⣿⣿⣿⣧⣿⣼⡟⣠⡟⣻⡏⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡏⡎⣸⢻⣿⣿⣿⣿⣿⡼⡹⣹⢣⠾⡷⠦⠸⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⣹⢁⡟⡿⡿⠟⠉⠈⠈⠀⠀⠁⠀⠀⠠⣤⢀⠀⢿⡇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⡟⠉⠸⠧⠃⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡇⢀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡓
⠀⠀⠀⠀⠀⠀⣼⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⡍⠀⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⡶⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⢠⠇⠀⣇⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡅⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⢀⡟⡀⣸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⣼⢱⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠈⠪⡛
⠀⠀⠀⠀⣇⣿⣿⡿⠟⠛⠉⡜⠀⠀⠀⠀⠐⠁⠂⠄⡉⠐⠒⠒⠒⠁⠈⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠰⣽⠛⠁⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡩⠃⠲⢆⠀⠀⢢⠀⠀⠀⠀⠀⠈⣌⡳
⠀⠀⠀⠸⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠰⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠳⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣖⣒⣒⢲⣶⢶⣶⡶⠶⠶⠶⠶⠶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⡷⠀⠀⣶⣶
⠀⠀⣀⡀⠀⠤⠤⠀⠐⠒⠒⠂⠄⠠⠐⠀⡀⣿⣿⣿⣿⣿⡿⠿⠷⡀⠀⠀⠠⠀⠤⠤⠤⠤⠄⠐⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⡠⠘⠦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠔⢈⣆⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠘⣿⠀⠀⠀⡀⠠⠐⠊⠉⠂⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣤⡤⡀⠀⠀⠀⠂⢀⣠⣾⣿⡄⠀⠠⣤⡄⠀⠀⠀⠀⢸
⠀⠀⠐⠟⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡇⢹⠀⠔⠀⠀⠀⣿⡄⠀⠀⠀⠀⢿⡀⢀⣘⠿⠷⠾⠿⠿⠿⠿⢓⣀⠀⣼⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⢱⠀⠀⠀⢿⡸⣄⠀⠀⠀⠈⠳⢜⡿⣿⣶⣤⣤⣤⣴⣾⠿⣫⠞⠁⠀⠀⠀⢀⡞⣾
⠀⠀⠀⢰⡟⢸⡟⢸⡏⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⡿⢠⠀⠀⠀⠘⣷⣌⠳⣄⣀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⣼⠏
⠀⠀⠀⠠⡧⠸⡧⢸⡆⠀⠀⠀⣴⠀⠀⠐⠊⠉⠉⢷⣾⣀⣀⡄⠀⠘⢿⣷⣦⣭⣍⣉⢻⢿⠖⠢⠴⣿⠢⠄⠒⠞⡟⣉⣉⣩⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⠀⢰⠛⢷⠀⠀⡀⣹⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠠⣄⣠⠀⣂⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⡄⠀⠀⠀⠀⠈⣾⡷⠀⠀⠈⡠⡈⠀⠀⢞⣷⠅⠀⠀⠀⠠⣾⢟
⠀⠀⠀⠀⠀⠀⠀⢸⠿⡃⠠⠤⢼⣿⡷⣺⣷⣷⣶⣿⣷⠶⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠁⠀⠀⢈⣿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣀⠴⠚⢻⣿⡿⠿⡿⠟⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣿⡇⠀⠈⢹⡖⠀⠀⠀⡰⣿⠁⠀⠀⠀⠀⣤⡄⠀⢰⣶⢷
⠀⠀⠀⣰⠁⢠⠞⠉⠣⡑⡠⠃⠀⠀⠀⠐⢎⠀⠐⢎⠀⠒⢎⠀⠀⠸⣝⠀⠀⠀⢸⢹⠀⠀⠀⠱⣧⣀⣸⡇⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠁
⠀⠀⣴⠃⡀⢸⠀⠀⠀⢀⠤⢤⡀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⢡⡸⢄⣀⣠⠞⡸⠀⠀⠀⢰⣹⣿⣿⡰⠀⢀⡀⠾⠀⢀⠀⠀⣀⣀
⠀⠈⠛⠉⠉⠉⠁⠀⠀⡏⠀⠀⠉⠩⠍⠁⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⠻⠊⠁⠀⠀⠀⠀⠛⠛⠛⠃⠀⠈⢛⣈⣈⠋⠀⣹⣿⣿
⠀⠀⣠⣤⣀⠀⠀⠀⡸⠀⠀⠤⠂⠀⠀⢲⣾⣿⣿⡄⠀⠀⠀⣠⣤⡀⠀⣠⣴⣤⣀⣀⣀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠀⠈⠀⠀⣽⠿⣿
⠀⠀⣿⡏⠉⠑⢤⣀⣧⣤⣀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣀⠴⠋⠉⢻⣿⢰⡿⣿⣿⣿⣿⢧⣴⡄⠀⠀⠀⡸⣿⣿⣇⡀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⢻⣧⠀⢀⣤⣈⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⢋⣡⣤⠀⢀⣾⠇⠀⠿⢻⣻⠟⣁⡈⠓⠁⠀⠀⠈⢉⣿⣿⡿⠟⠀⠀⠀⠐⠃⠀⠀⠃
⠀⠀⠀⠙⢷⡘⢿⣿⠉⠲⠤⣤⣄⣀⣀⣤⣠⠤⠒⢹⣿⠟⣠⡿⠃⠀⠀⠀⠘⠋⠀⠙⠃⢰⠀⠀⠀⢰⣌⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠉⠳⠬⣦⣅⡤⠂⢄⠀⠀⠤⠀⣄⣼⣾⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠁⠀⠀⢹⣿⣿⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⣟⣶⣤⠀⢀⣴⣺⣷⡎⡇⠀⠀⠀⠀⠀⠀⠀⠋⠰⣀⠀⠀⠀⠀⠀⠀⠀⢐⠓⣿⣿⣿⡷⠀⠀⢸⠿⠿⠿⠯⠿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠙⣤⣉⠛⠛⡅⡈⠛⠋⣉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣠⠰⠀⠀⠠⣘⣿⣿⣧⡄⠀⠀⢹⣶⣦⣤⣤⣶⣿
⠀⠀⣀⠴⣯⣬⡷⢄⠀⢸⡉⠳⠐⠀⠐⠀⠛⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠂⠐⠀⡀⠀⠀⢡⣿⣿⡟⠋⠀⠀⢸⣿⣯⣿⣿⣿⣿
⠀⠀⠘⡾⣭⣭⠷⡚⠀⠈⠳⣤⠁⢤⠄⢡⡤⠞⠀⠀⠀⢰⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⠃⠀⠠⠮⣿⣿⣿⡆⠀⢠⠟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠐⠚⠓⠒⠀⠀⠀⠀⠈⢧⣀⣀⠞⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣄⣀⣀⣠⣴⠞⠛⠋⠁⠀⠀⠈⠰⠿⠿⠏⠀⠀⠀⢀⣤⣤⣴⣴⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣿⣾⡿⣿⣿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣘⢿⣿⣿⣿⣽⣧⣿⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠻⠶⣿⣿⠻⡿⣿⣿⣷⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⠛⠟⠁⢸⠁⠀⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠱⠂⠀⠄⢀⡀⠀⠀⠀⠀⠀⠰⢠⠀⠀⡿⠀⠀⠀⠀⠀⣀⣠⣤⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡁⢠⣶⠘⠛⠏⣀⠀⡀⠀⠀⠙⡿⢦⠀⡇⠀⢀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠘⢈⡇⣺⣿⣯⠉⣴⠉⠧⠀⠀⠀⠀⠀⠀⠀⣷⣾⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠷⣌⠹⠄⠐⠃⠀⠀⠀⠀⠀⠀⣿⡷⠎⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠊⢀⠰⠋⠀⡀⠀⠀⠀⣤⠀⠀⢠⣄⣿⠂⠀⡀⠸⠿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠂⠀⠀⠀⠀⠀⣿⣧⣄⠈⢿⣿⢠⠞⢀⣀⠀⠈⠙⢻⠛⠉⠛⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠠⢀⣀⣀⢉⡙⢷⣶⠀⡿⠂⠀⢿⣿⡿⣄⠀⣸⠀⠀⠀⠀⠀⣿⣛⣧
⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⢸⡇⠀⠰⡆⠻⣝⣹⣿⠃⠈⠛⠄⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⢀⠜⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇⠀⠂⠉⠊⠟⠳⠛⠀⠀⢀⡀⡇⠀⡀⢀⣠⡀⣸⢶⠃⠘⠃⢀⣼⣶⣵⣤⡀
⠀⠀⠀⠀⣷⡄⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⢁⠀⠈⡿⠃⡇⠀⡃⢠⣽⣿⡏⣤⡀⠀⢀⢸⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⣻⣿⣧⣿⣷⡀⠸⣅⠀⠠⢾⠀⢼⠀⣀⢁⣤⡤⢀⠏⠀⠄⠸⢃⠺⣤⣼⡧⢠⠇⢀⡑⢻⠾⣯⣿⠇
⠀⠀⠀⠀⠨⡻⢿⣜⠿⣃⠘⠇⠀⠀⠀⢀⠠⠋⢇⣼⣿⣯⣷⠄⠀⠀⠀⠘⠀⢸⠅⢰⡎⢶⠺⠄⠈⠛⠋⠁
⠀⠀⠀⠀⠀⢌⣵⣬⣧⣽⣦⢷⡄⢀⣾⣿⡓⢿⡿⣿⣿⣷⠟⠀⢠⠀⠀⣿⠀⣼⠀⠸⢠⠃⢠⠀⣠
⠀⠀⠐⠀⢰⣟⣻⣿⣿⣶⣟⣱⣿⢻⣿⠟⢀⡿⠷⣿⣿⣿⡇⣠⣾⡇⠀⢻⠀⠀⣼⣿⣓⣦⣸⣿⣇⣴⢃⣤⡄
⠀⠀⠀⣇⠺⣿⡿⢿⣿⣷⣾⠿⠾⣻⣿⠄⢠⣷⣾⣄⢡⠻⢿⣿⣿⡇⠀⢸⣀⣾⠿⣷⡄⠻⣿⡿⢯⣿⣏⣞⣡⡤
⠀⠀⠀⠈⠟⠴⡿⢻⣄⡹⣸⡟⠙⠟⠁⠐⣿⣿⣿⡿⠘⡀⠀⡿⢿⣅⣠⢸⠉⡇⢰⠿⣿⡄⣠⣴⣟⣻⣽⣯⣥⡤⠄
⠀⠀⠀⠀⠀⠉⠑⠋⠘⡁⠉⣿⠀⠀⣄⠀⠈⠙⣻⣀⣀⣧⢤⣷⣼⣤⣽⠾⠶⢷⣮⠰⢒⡿⣿⣿⣤⣧⡠⡉⠐
⠀⡓⠐⢦⣜⣿⣷⣤⣄⡇⠀⣏⣾⠀⢻⣷⡀⣼⣿⢟⣽⣾⣿⣿⡏⡿⣿⣿⣛⣿⣿⣷⣻⠇⠋⡏⢻⠙⠛⠃
⠀⡇⠀⠀⠈⠉⠛⠛⠛⠛⠀⣿⠇⠀⠾⣿⣧⠟⣧⣿⡿⢿⠁⢸⣴⡇⡿⠿⣯⣭⢿⡟⠎⠀⢀⣴⣾⡿⣿⠿⣷⠆
⠀⠃

⠀⠂⠀⠀⠀⠀⠀⠃⠐⠃⠘⠃⠘⠁⠘⠀⠀⠉⠐⠇⠐⠃⠀⠀⠒⠃⠻⠀⠓⠀⠡⠀⠀⠇⠀⠓⠀⠃
⠀⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣦⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⣿⣛⣋⣉⠉⠙⠛⠛⠿⠿⣿⣛⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣄⣀⣈⣉⣛⣠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢻⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢷⠄⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣿⡀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠈⡿⢿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠐⠂⢻⡇
⠀⠀⠀⠀⠀⠀⣼⡟⢉⣉⡉⠙⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⣤⡾⠃
⠀⠀⠀⠀⠀⢠⣿⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢻⣦⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡅⠀⣿⠁⠹⣷⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣷⠀⠹⡆⠀⣿⡄⠀⣠⡾⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠙⠀⠀⠋⠀⢸⣷⣾⠋⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠟⠛⠁
⠀⠈⠛⠛⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠛⠻⠶⣦⣤⣤⣤⣤⣬⣿⠀⠀⠀⠀⠀⢀⣾⠃⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣿⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⡀⠀⠀⠀⢰⡟⠁⠀⣠⣴⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠹⣧⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠻⣶⡟⠉⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⣻⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠈⠻⣄⠀⠀⠀⢿⡆
⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⡾⠋⠀⠀⠀⠀⣠⡾⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⣼⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣴⡿⠋⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠹⣧⣄⣀⣀⣀⣀⣀⠀⠀⠀⢾⣏⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣠⣴⠿⠁
⠀⠀⠀⠘⣿⡉⠛⢛⣿⣿⣧⠀⠀⠀⣿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏
⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠹⢷⣤⡾⠟⠀⠀⠀⢿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠻⣿⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠶⠶⠶⠶⠶⣿⠀⣿⡶⠶⠶⠶⠶⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢠⣿⠀⣿⡆⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣸⣿⠀⣿⣇⣀⣀⣀⣀⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣷⡈⣣⣌⡟⠛⠛⠛⠻⣿⠀⣿⡟⠛⠛⠛⢻⣁⣼⠡⣼⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⣀⣽⣉⣉⣀⣀⣀⣀⣀⣀⣿⢀⣿⣀⣀⣀⣀⣀⣀⣉⣩⣏⣀⣀⣈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⠘⣿⣭⣭⣭⣭⣭⣭⣉⣭⣭⣭⣭⣭⣽⠇
⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⡯⠁⠈⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣶⣾⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠠⠀⠒⠒⠂⠠⢄⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠚⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠠⠄
⠀⠀⠀⠀⠀⡄⠀⠠⡁⠚⠀⠆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⣄⣠⢠⣤⡄⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠘⠦⡀⠈⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⡜⠀⠀⠀⠀⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠀⡠⠈⢂⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠱⡄⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢨⠒⠆⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈⠀⠀⠈⠢⡀⠀⠀⣜⢉⢉⢉⣘⣻⣻⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⢂⠄⠈⠂⢤⣌⣉⣉⣩⣩⣽⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡂⢀⠘⠳⢛⢗⠿⡯⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠔⠁⡡⠊⠔⡡⡪⢐⠔⠀⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡊⢔⠁⠠⢊⠔⠁⡠⢄⠀⠔⠄⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⡡⢂⢄⠁⣠⠞⡇⠀⠈⠒⠢⢄⠐⢀⣤⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠊⠄⠨⢊⠔⡁⠞⠡⠀⡇⠀⠀⠀⢀⠜⡡⠎⠅⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⢊⠔⡡⡪⠢⢪⢌⢄⢧⠀⡠⠊⠔⡼⠉⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠔⡡⡢⢊⢔⠔⠅⠕⠡⣸⠏⠀⠈⢊⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢊⢌⠔⡽⠣⠢⢊⠈⠘⠡⠀⢡⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⡡⠠⢲⢘⡁⡁⡁⠀⠑⣀⠀⠥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⡀⢉⣬⠇⠄⡀⠀⡄⣭⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠎⠔⠁⠹⣇⠌⠔⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠈⠊⢂⢻⠎⡐⠀⢂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡡⡁⠘⡆⠈⠀⠣⠀⢰⡤⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠈⠈⠄⠀⣴⠀⠀⠠⠡⠘⡇⠀⢀⣀⣀⣀⣀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢅⣀⣤⠤⠤⠤⠊⠙⠆⠀⠡⡠⡁⢙⠒⠒⠂⢀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⠟⠒⠲⠶⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣀⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣆⡄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠴⠀⠀⠀⠀⣴⣿⡶⠆⠀⠀⠀⠀⠀⢈⣽⠿⠋⠀⠀⠀⠀⠀⠺⣿⣟⠷⠃
⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⡀⠀⠀⠀⣠⡒⠉⡀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⡀⠀⠀⣰⣯⡉⢛⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢠⡀⢠⣿⣧⢨⠀⠀⣀⠀⠀⠀⣿⣿⡄⠍⠃⠀⣹⡀⢰⣿⣧⠀⡾⠀⠌⣶⡀
⠀⠀⠀⠐⠒⠒⠛⣿⣷⣞⠓⠚⠳⡞⠛⢳⣶⣶⡄⠛⠋⠒⠒⠚⠻⣿⣾⣷⠖⠻⠗⠚⠛⢻⣾⣇⣠⡐⠛⠁
⠀⠀⠀⠀⢀⣀⣀⣿⣿⣏⣤⣀⣀⣇⣀⣿⣿⣻⣹⡄⢤⣀⠀⣀⣰⣿⣿⣧⣇⢀⡁⠀⠀⣾⣷⣿⢻⠄
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢹⣿⡏⠉⡏⢹⣿⣿⠉⠉⠛⣺⣿⡅⠀⢹⣿⡟⢻⡇⠈⠁⠀⣸⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠤⢤⣿⣿⠧⠬⣿⡇⠴⡧⢼⣿⣿⠤⠤⣴⣿⣿⣷⠤⢼⣿⡧⢼⣏⣴⡦⣴⣿⣿⠇⠠⢹⣇
⠀⠀⠀⠀⠀⣸⣿⣿⡄⢀⣿⣗⠀⠁⠸⣿⣿⡄⣸⣿⡇⠻⠋⠀⢸⠿⠀⠈⣿⣿⡀⣿⣿⣿⠀⠀⢸⡏
⠀⠀⠀⠉⢉⣿⣿⣿⠉⢹⣿⣿⠈⡏⢹⣿⣿⡋⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⣿⣿⡋⣿⣿⡏⠉⠉⠉⡇
⠀⠀⠀⠠⠼⠿⡿⠛⠤⠼⠿⡿⠶⡧⠼⠯⢿⠻⠤⠤⠤⠠⠤⠤⢿⡇⠤⠤⠿⠿⠷⢿⡿⠅⠀⠤⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⣀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠉⠁⠉⠀⠈⠁⠈⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠌⠛⠛⠛⠻⠿⠿⠷⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⣀⡀⠒⠂⣠⣤⠿⠿⠿⣷⣄⠀⠀⠀⠀⠣⡀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠴⠉⠀⢀⣶⠏⠉⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠉⣄⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣶⡀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠛⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠔⠛⠻⠟⠉⠙⠛⢻⡆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠻⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣠⣤⣾⣤⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠉⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⠀⠀⢀⠈⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠐⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⣀⠀⠰⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠁⢠⣤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⠁⠀⢰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣘⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣀⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠉
⠀⠀⠀⠀⠀⠀⢀⡿⢷⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠛⠠⣤⡀⠀⠀⠀⣤⣤⣶⡆⠀⢤⣤⣤⣤⠀⣠⡾⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠿⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠘⠀⠀⠀⠀⠈⠁⢰⣿⣿⣷⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠀⣀⠀⠀⠀⣿
⠀⠀⠀⢴⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠉⠉⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⢉⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⢀⡀⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠁
⠀⠀⠀⠀⠙⢿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠃⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⡿⠃
⠀⠀⠀⠀⢰⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠛⠛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠳⣤⣄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠗⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣼⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⡶⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣴⠟⠁⠀⢠⡴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣾⠁⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣦⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⣿⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀
⠀⣴⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠙⣤⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠉⢷⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠙⢳⣤⡀⠀⠈⠛⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠡⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⠐⠛⠓⠚⠛⠛⢤
⠀⠀⠀⠀⠀⣀⠾⠃⠈⠁⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢾⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⢠⡌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠘⠇⣀⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣴⣦⣤⣀⣠⡟⠛⠀⠛⠛⠁⠀⠘⢻⣤⣤⣤⣤⣤⣤⣤⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡤⠤⠤⠤⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢀⡴⠀⢠⣶⣵⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⣈⡑⠢⡀⠀⠀⠀⠀⢸⠀⠀⠰⠈⠁⢸⡪⢿⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢔⢭⠩⡇⠈⢢⠈⡄⠀⠀⠀⠀⡇⠀⠀⠉⡂⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⡄⢐⠄⡪⣧⠀⠈⡆⡇⠀⠀⠀⠀⢹⠀⠀⠐⠀⣼⢀⠇⠀⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⡂⠠⠤⢄⣱⡀⠢⡪⠘⢦⡀⢹⣹⠀⠀⠀⠀⠀⠀⢠⠅⣰⣣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⡄⠀⢖⠄⡹⡦⣀⠐⠕⠱⣤⣧⣣⠤⠤⢀⡜⠠⢂⣴⠷⠷⠒⠉⠉⠈⡩⠖⠀⠀⠠
⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⠀⠈⠬⠌⠩⠭⠓⣲⣤⡽⠁⢒⣶⢂⣽⣾⡯⠶⠒⠒⠒⠠⠤⠊
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠒⠒⠒⠒⠒⠲⢒⣖⢶⠒⠒⠒⡎⠉⢩⠉⠉⡟⠉⡀⠂⠀⠒⠂⠒⣖⠒⠒⠲⣒⡄⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠫⠀⢀⡀⠀⠀⠚⣁⠠⣺⠗⠡⢪⠇⠠⢕⣨⡆⢠⠌⡥⡂⢔⠠⠰⢪⠄⢦⢰⠍⡉⡇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢉⣏⣀⠀⠀⠀⠡⠪⠔⠡⠪⢰⡄⢀⠫⢑⠨⢹⢀⠛⡳⠶⠀⠕⠕⠅⡎⠹⠨⠂⡇
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠙⠟⠙⡩⠣⢃⢒⢖⢶⣶⡶⡲⢺⠇⠐⠨⢒⠨⢘⡲⡞⡗⡲⠲⢒⠖⣶⣶⡶⣶⣞⠁
⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠈⠃⠀⠀⠀⠀⠐⠬⣒⠥⠊⣏⠀⢀⢶⠀⠀⠀⠘⢉⠈⡀⢹⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢏⠐⠃⠸⡀⠀⠀⠀⠰⡖⠒⠂⠀⠨⡒⠭⡂⣿⡁⠁⠸⠤⣤⠀⠴⡡⡪⣶⢼⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠴⠬⠤⠸⡠⠄⠛⠂⠀⠀⠈⠇⡄⡄⠨⡂⠍⡂⣿⡡⢦⢲⠀⠀⠀⠀⢠⢄⠄⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠲⠁⠀⠀⠠⠆⠀⠀⠋⡅⠀⠠⡂⠅⡂⡿⡀⠀⠻⠀⠀⠀⠔⡔⡠⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠐⡖⠒⠁⠰⠤⣤⠀⠃⠀⠔⡂⠅⡂⣿⠀⠀⠀⠳⡀⠈⠀⠓⡺⠎⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢔⠅⡠⣜⡠⣝⢠⠊⠀⠀⢸⠀⠠⡐⠅⣒⣽⡠⢊⢔⢤⣋⣬⠀⡮⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠑⡡⡪⢊⢟⢔⠝⣾⢀⢀⠀⠀⡆⠨⢒⠩⠄⣪⢔⢕⠑⡫⡪⢊⢹⢶⠇⣕⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠀⠁⠁⠈⠈⠀⣀⠀⠀⠀⢧⣀⣅⣶⣏⣻⠥⠥⠬⠤⠥⠥⠥⠯⠦⠽⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢤⠦⣐⠠⡔⢄⠢⣰⢖⠢⢐⠨⡐⠅⣂⠀⡲⢐⠨⠰⣵⠦⡂⢄⠤⡢⠴⠔⠤⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠿⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠠⠆⠈⠁⠐⠀⠀⠐⠖⢄⡀⠀⢻⣷⣿⣿⣿⣿⣿⣿⣦⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣿⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⡄⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠡⠀⠀⠀⠈⠙⠿⢿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣿⣿⣿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡻⠻⢿⣿⣿⠀⠀⢀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⠀⠀⣀⣀⣉⣰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠂⠀⣀⣀⡀⡤⠤⠒⠚⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⣿⣿⣿⣛⣿⣿⣿⣇⠀⠠⢶⠒⠋⣁⠀⠀⠀⠀⠀⡀⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠁⠈⠛⢋⡩⡟⠿⠿⠛⠋⡉⠁⡞⠂⠀⠀⠀⠐⣿⣴⠆⠀⢀⡼⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢷⢀⣄⢁⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠀⠹
⠀⠀⠀⠠⣤⠴⠆⠀⠀⠉⠈⠿⠈⣿⣿⡝⣧⠦⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⢁⠈⣀⢃⡠⢀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠀⠛⠼⠇⣀⠠⣯⡀⠀⠀⠀⠀⢀⡤⢀⡀⣀⠄⠁
⠀⠀⠀⠀⢀⣄⡠⢀⣼⣧⠄⣸⡇⠀⠈⠀⠀⠀⠹⡄⢙⣥⣉⡷⠿⠀⣾⣞⣨⠃⢌⠻⠇
⠀⠀⠀⢀⣾⠏⠀⣾⡿⠃⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠛⠀⠁⠈
⠀⠀⠀⣿⠏⠀⢰⣿⡇⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⢸⣿⣾⡄⣿⣿⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⣄⠀⣿⣿⣿⡇⣿⣿⣇⠀⢿⣇⠀⠀⠀⠀⠀⢸⣆⠘⠓⢤⣄
⢻⣷⣹⣿⣿⣧⣿⣿⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠙⣷⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀
⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⡀⠀⠀⠼⠁⠀⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⢐
⠀⠀⠀⠁⠀⠀⡀⣤⣀⡀⠀⣡⣄⣀⠐⠀⠀⠀⠀⠈⠀⠠⠘⠛⠈⠋⠀⠀⠀⠀⠀⠂⠐⢀⠂⠐⠶⠢⠶⠀⠄⡠⠀⠤⡀⠀⠀⠘
⠀⠀⠀⡆⠀⠁⢈⠀⢈⠀⠀⡀⠀⣸⠐⠂⠐⡆⠀⠲⠖⠶⠤⢠⢀⡤⠴⣀⡀⠀⠀⠀⢀⡀⢋⣉⠛⠋⠛⣛⡛⠀⠀⠀⠈⠂⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠦⢼⠀⠀⡇⠴⢸⣿⡷⢸⠉⡇⠸⠑⢀⠀⣼⠈⢹⠈⠯⠝⠀⠀⠀⠛⠛⠀⠾⠀⠉⠁⠺⠄⠀⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠃⠀⠘⠀⠀⠈⠀⠁⠀⠀⠘⠀⠙⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⡤⠐⠈⠉⠉⠉⠁⠐⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⠔⢈⠤⢁⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⠒⢄⠀⠀⠃⠀⠁⠀⠀⠀⠢⡁⠀⠃⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠰⠊⠉⡀⠀⢠⣄⠀⠀⠀⠴⢀⠁⢸⣆⠀⢸⡦⢰⠄⢤⢤⡖⠉⠛⠆⠄⢀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠛⠀⠈⠛⠀⠀⠀⠀⠸⠀⠀⠁⠀⠈⠑⠀⠘⡈⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡆⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⠀⠔⣠⡶⠀⢠⠀⢣⠀⠠⣤⡤⠴⠂⠀⠃⠱⡀⠲⣤⣴⠞⠜⠁⢀⠀⠀⠀⠀⠐⢀⠴⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠿⡏⠀⠃⢀⡼⠀⠀⢑⢤⡀⠀⠀⠀⢀⡀⠀⠈⠒⢤⡀⣀⠀⡀⢰⠀⠀⢀⠔⠁⠈⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢠⠄⠀⠀⠐⠁⠁⠠⠀⡉⡄⠈⠁⢈⠋⢩⡇⡀⡀⠀⠀⡀⠈⠘⠇⠠⠄⠐⠃⣠⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠂⠀⣀⠀⡴⠀⠀⢱⡇⠀⠠⠂⠀⠀⠀⠀⢰⠀⠀⠈⠖⠀⠀⠀⠀⠀⠐⡁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠤⠀⠑⢠⡄⠉⠂⠛⡀⢀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡏⠀⣀⠀⠀⠀⠀⠠
⠀⠀⢀⠀⠀⠀⢄⠄⠀⢀⣀⡀⠀⠑⠘⠀⠀⠀⢎⠀⠇⠀⠀⢡⠜⠀⠄⠐⢀⠀⠀⠠⠀⢀⠆⠀⡄⠉⠈⠀⢿⣴⠦⠠⠀⠀⡄
⠀⠀⠈⠀⠀⠁⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠘⡁⠀⠀⠅⠉⠙⠺⠀⠖⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠜⠀⢀⠄⠀⠄⠀⠀⠀⠀⠂⠀⠀⠬⠑⠀⠠⠄⠀⠂⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⣀⠀⠀⠀⠀⠀⣀⠠⡔⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠇⠂⡉⠀⠐⡀⠀⠤⠀⠐⠒⠲⠀⠈⠀⠉⠀⠒⠒⠒⠈⠀⠀⠀⠀⠁
⠁⠂⠼⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠁⠀⡀⠀⠀⠀⢸⠀⠀⡀⠀⢨⠁⠈⠁⠀⠀⠋⠀⠒⠒⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⢸⠀⠀⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢸⠠⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠉⠀⠀⡇⠀⠀⠀⡸⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠁⠀⠀⣀⠠⠀⣀⠄⢠⠀⠀⠠⠆⠀⢀⠀⠀⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠉⠀⠐⠀⠋⠀⠔⠀⢀⠀⠀⠀⠈⠀⡘⠃⠀⠂⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⣿⠃⢀⣤⣤⣤⡄⢠⣄⣤⣤⡀⢠⣄⣠⣤⡄⢀⣤⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⢻⣿⢠⣿⠁⠀⣿⠁⣼⠯⣦⣸⡇⣸⡟⣤⢸⡿⢸⡇⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠇⠈⠿⠶⠞⠿⢀⣿⠶⠶⠟⠀⣿⠷⠶⠟⣁⠙⠷⠞⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⢶⣆⠀⠿⠂⠀⠀⠀⠀⢠⡄⠀⢀⣿⠸⠏⠀⠀⠀⠐⠟⣼⡇⠀⠙⠻⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣾⠋⢰⡷⢀⣿⡶⠛⠘⣿⠟⠃⢸⣷⠞⢿⡆⢀⣴⠟⠻⣿⠀⣴⠞⠛⣿⡇⣸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⣀⣿⠇⣼⡇⢸⡟⠀⠀⢰⣿⣀⠀⣾⠇⠀⣼⡇⢸⣇⣀⣼⡟⢸⣿⣀⣠⣿⠀⣿⣅⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠁⠈⠁⠀⠀⠀⠉⠋⠁⠉⠀⠀⠉⠀⠈⠉⠋⠉⠁⠀⠉⠛⠉⠉⢀⣈⣉⣱⡟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣠⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿