Remainder Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢶⡿⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠗⠘⠃⠈⠛⠻⠇⣹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠙⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠒⠲⠦⢼⣿⠺⠀⠀⠀⣴⠿⠛⢷⣆⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⣏⠀⠀⠚⢿⡄⠠⣤⡶⠦⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠹⣄⢑⠦⣾⣷⠀⠀⠀⠀⠀⢸⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⣤⡀⠀⠀⢀⠀⠀⠀⠀⠘⣷⣄⠀⠐⢟⢻⣆⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣬⣿⣦⡀⢮⡷⠀⠀⠀⠀⠈⠛⠀⠀⠈⠂⢻⣆⠀⠀⢠⡾⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⣷⡍⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⡔⢫⣼⠟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠋⣠⣿⣿⠧⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣒⣲⣉⣈⣴⡿⠏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢩⣾⠛⠛⠉⢿⣿⣇⣀⠤⠤⡄⠀⡀⠀⠠⠀⠀⢸⡿⣟⠛⣟⣛⣁⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡵⠟⠛⠙⢀⠀⢀⢸⣿⣿⣠⣦⣤⡴⠀⠃⠸⡴⠶⠟⠈⣡⡿⠉⠀⢉⠏⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠉⠀⣤⡄⠀⢀⣤⡼⠿⠿⢿⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⢀⣿⠃⢢⣄⢉⡾⠛⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⢠⡼⠃⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⠀⠀⠀⢰⣿⡁⠀⠀⢻⣾⣄⣀⠀⠼⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⣿⠁⢀⣤⡄⠀⠀⠀⠉⣦⡀⠰⠈⠀⢀⣤⡶⠛⠁⠀⢀⣴⠟⠋⣿⡃⠀⠛⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⡇⠀⠘⠖⠲⣤⠀⠀⣀⣤⣿⣦⡤⠐⢡⣿⣁⠀⠀⠀⠈⠁⠀⠀⠸⣷⣤⣤⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡅⠀⠀⠼⢶⣿⡷⠟⠋⠉⠉⠉⠀⠈⠉⠛⢿⣤⠖⠀⢀⠀⠀⠀⣀⣼⠿⣿⠀⢀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⠇⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣷⣶⠆⠀⠀⠛⠉⠀⠀⢸⣇⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢀⣠⣤⣤⡋⢀⡤⠀⢹⣿⣶⣶⣤⡀⠁⠀⠀⣴⠟⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠫⠉⢻⣷⡎⠀⠀⢸⡇⢠⣶⡀⠙⠻⢿⣶⣿⣠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡞⣿⡇⠀⠀⢸⠇⢸⣿⠿⣦⡀⠀⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣠⠃⡠⠙⣿⡎⠀⠀⠀⠀⠀⢸⠀⢰⣿⡇⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⣿⡶⠁⡴⣿⠇⠀⠀⠀⠀⠀⠈⠀⣾⠏⠻⣦⣀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣠⠿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣷⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠁⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡇⠀⠀⠀⠐⣼⠇⠀⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣿⠁⠀⠀⠈⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⣠⡼⠃⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⠈⠁⠀⣀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠐⢂⣂⠤⠤⠤⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⣫⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⠞⢰⣴⠀⠀⠀⠃⢨⣉⠀⠀⠀⠀⠀⠀⠀⣴⣤⣴⡆⠀⠀⠀⡿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⣵⣾⣻⡿⢿⡆⠀⠀⠀⠀⡇⠀⣿⢗⣠⡀⠀⠀⠀⠀⣋⣾⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⡇⠀⡀⠘⠃
⠀⠀⠀⠀⠀⣠⢿⣿⣭⣾⣿⢿⡆⠀⠀⠀⠀⡇⠀⢋⣏⡟⠉⠁⠀⠸⠀⣉⠈⠀⠀⠀⡖⠐⠂⢈⣉⣉⡉⠁⠀⠀⠀⡇⢠⡤⠤⠤⠄
⠀⠀⠀⠀⢀⣿⣿⠟⠉⢳⣶⡜⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠉⢱⣶⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠌⣉⡁⢰⣄
⠀⠀⠀⠀⠀⠀⢀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢬⣼⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠀⠀⠀⢰⡆⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⢧⣀⣀⡴⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣿⡗⠋⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠝⠃⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣤⣤⣦⣄⢀⣤⣤⣴⣷⣖⣶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣞⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣤⣶⣆⣩⣹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡯⢋⣤⣶⣾⣿⣿⣶⠟⠋⠉⠁
⠀⠀⠒⣂⣺⣻⡻⣝⣦⣠⠀⠰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿⣿⡿⢻⣟⣻⣥⣴⣿⣿⡿⠟⠋⠉⢀
⢠⣤⣤⣤⣬⣬⣥⣈⡀⠂⠀⢀⣨⣯⣿⣿⣏⣿⣿⣿⡿⣻⣚⣭⣷⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⢀⣶⣾⣣⣤
⠀⢰⣷⣿⣷⢶⠾⢿⠛⣳⢦⣄⡉⠉⠉⠉⠙⠚⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠊⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿
⠀⢹⣜⣿⣿⣤⣤⣤⣠⡶⠷⠻⣿⠛⢓⣶⣤⣌⡛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣠⣧⣤⣶⣜⣿⣿⡿⠇
⠀⠸⢚⠿⣿⣿⡿⣉⢀⠀⢀⣠⠤⠾⢛⣻⣿⣿⠛⠓⢦⣍⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣾⣽⣿⣀
⠀⠂⢀⢁⠘⠫⡋⠀⡥⠀⠈⠀⠀⠋⢉⣽⡟⠁⣀⣤⣄⣈⣁⣤⣄⠀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠌⢸⡖⢎⣤⢈⣴⣀⠀⠀⢀⣴⣿⣿⡾⠿⢿⣿⣿⣿⣿⣿⣿⣷⣎⣿⣿⣿⣿⣿⣿⣫⣿⣭⣿⣻⣿⣿⣽⡋⠁
⠀⠀⣁⡬⠳⠾⣟⣷⡇⠇⠀⢰⣿⣿⣿⣋⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⠗
⠀⡛⠛⢁⣠⣔⠀⣨⣾⣤⣶⣿⠏⢉⣀⣤⡄⠀⣶⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣷⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠈⠛⢛⣥⣀⣸⣿⡿⣿⡟⣺⢿⣿⡯⠁⠀⠈⠙⠓⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⡿⢦⡈⣛⡿⣿⣿⣷⣾⣦⣀
⠀⠀⢲⠀⠀⣸⣿⡿⣯⣠⣾⣷⡟⠂⠹⠷⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣿⣿⡻⠄
⠀⠀⠎⣣⠔⠻⠉⢀⢮⠟⡏⠛⠃⠀⠀⠀⠀⠀⡀⠁⠀⠄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣙⢿⢿⣿⣿⣷⡤
⠀⠀⠀⠀⠂⠀⠗⠴⠒⢂⠄⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⣿⠉⠙⣏⠋⠙⠷⠮⢍⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⠿⣿⣷⣄⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣴⣿⡿⠿⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⣠⣾⣿⠟⠀⠀⠻⢿⣿⣿⠿⠇⠀⠘⢿⣿⣦⡀⠀⠀⠀⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⢀⣀⣸⣿⣿⣀⣀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣾⣿⣿⣷⣄⣤⣶⣿⣿⠿⠿⠿⠿⠿⠿⣿⣷⣦⣄⣴⣿⣿⣿⣦⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⠟⠁⣨⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣯⡀⠙⢿⣿⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠅⡙⠋⢡⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠚⠉⠇⠐⠦⠀⠀⣊⣡⠌⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁⠀⠀⠀⠀⠃⠀⠀⠀⠉⣿⣆⡙⢓⢀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⢀⣤⣤⡤⣁⠀⣀⠁⠾⠁⠢⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⣿⡄⢻⣿⣿⠀⠠⠐⢤⣶⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡰⠗⠂⠀⠐⠖⠤⣄⡀⠀⢠⠟⠀⠙⣦⡜⠃⠀⠀⢸⣴⣦⣴⣾⣿⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⣀⠟⠠⣄⠀⠀⠀⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠈⠃⠀⠄⠀⠀⠑
⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⠂⣀⠀⠂⢀
⠀⠀⠀⠀⠀⠻⢟⠆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣸⣱⣇⣀⣿⣷⡶⣄⠀⠀⣠
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⣿⣿⡇⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠜⣿⣿⣿⣧⣿⡟⢿⠻⡿⠶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠚⣀⡛⠗⠂⠁⠀⠀⠀⠘⢿⣿⣷⣿⣿⡴⢦⣼⠒⠛⠁⠀⠀⠀⠀⣰⣶⣤⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠋⡀⠀⣠⠶⢼⡂⠐⠒⠚⠻⡅⣿⣿⣆⣸⡿⠋⠀⠀⠀⠰⣾⣹⣿⢿⣏⣓
⠀⠀⠀⣠⣿⠟⠁⠀⣠⡙⠀⣹⡄⠀⠀⠀⠀⠀⠀⢱⡻⢿⣿⡹⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⢿⣻⣦
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠸⢡⣤⡈⠃⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠛⢦⡵⣴⡆
⠀⣸⡿⠃⠀⠀⠀⠠⠀⠤⠀⠃⠀⠀⢀⢀⡀⣀⠀⠸⠘⣿⣿⠂⠀⠀⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀
⢠⡿⠁⠀⠀⠀⠢⠀⠀⠂⣿⠁⠀⠀⠀⠀⠀⠉⠳⡖⠠⢿⣿⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠞⠀⠀⠀⢀⡤⠀⠠⣀⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⠈⠂⢠⡟⢺⡄⠀⠀⢀⡄⠀⠀⠴⠂⠀⠀⠊⠀⠀⢨⣉⣷⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⣼⠋⠙⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠛⠲
⠀⠀⠀⠀⠀⣿⠁⠨⡁⢂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⢶⡤⠀⠀⣀⠀⠘⢿⠔⠂⠐⠶⠆
⠀⠀⠀⠀⠀⠈⡁⠇⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⣾⣇⣀⣬⣽⠷⡄⠓⠀⢀
⠀⠀⠀⠀⠀⠀⠙⠒⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠓⠀⠀⠉⠻⢿⣿⣿⣿⣿⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠠⠠⠀⠈⠛⣿⣿⡅⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⢀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⢳⡌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠧⠀⣠⣿⣿⣭⢦⣦⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢫⣴⠷⣿⣿⣿⡟⠸⢿⡗⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡳⢚⠃⠀⣴⣿⣿⣿⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢁⡜⠀⠘⣿⠃⢋⠸⠃⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠁⠀⢲⡍⢠⡀⢸⡀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠛⢻⡇⣸⠇⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢰⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⡀
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠤⠦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠂⠀⠈⣻⠟⠃⠀⠀⠀⠀⠀⠀⢠⣅⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⠔⡈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣰⣟⢄⡀⠐⠁⠀⠀⠀⠀⠀⠀⠐⣦⡈⠹⣷⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢁⠀⠀⣰⣦⣾⣾⡵⠀⠀⠀⠀⠈⢹⣿⣿⣦⣻⡀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣧⣤⣿⣮⡙⢦⣦
⠀⠀⠀⠀⠀⠀⢀⠀⠨⠁⢀⡼⠀⣾⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠠⠀⠀⠀⠀⢼⣿⢃⣹⣿⣧⣤⣥⢰⣄
⠀⠀⠀⠀⠀⢰⣿⣦⣀⠘⣪⠅⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡤⣸⣿⣿⣿⣿⣿⣿⣧⠀⡀⠀⠀⠀⢲⠀⢸⣶⢼⡿⣿⣿⡿⢧⣷
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⡿⣿⣿⣦⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢀⣠⣏⢠⣾⣿⣿⣿⣛⣭⣾⣿⡘
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣁⡀⠀⢸⠋⠀⠀⢻⣿⡿⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠐⣿⣿⡟⠁⠻⣿⣿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣷⣶⣷⡴⠇⠀⠀⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠤⠝⠋⠀⠀⠀⠈⠁⠋⣛⣿⡿⠟⢁⡄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⣴⠊⠒⠉⠀⠁⠌⠛⠓
⠀⠀⠀⠀⢀⣹⡾⣿⣿⣿⣿⡿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣠⠄⢻⣟⣡⣾⣿⣶⣶⣶⣷⣶⣔⢶
⠀⠀⠀⠀⢨⣿⣿⣷⣮⣯⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⠖⠋⠁⠀⣽⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢸⠿⠛⠉⠉⠉⠉⠉⠁⣿⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⢀⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⡖⠂⠉⠁⠀⠙⣢⣙⠃⠀⠀⠀⠀⠀⢠⠿⡿⣽⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠇⠐⠺⣷⠀⠀⠉⢙⡀⠀⢰⡄⠀⠀⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣃⣾⠄
⠀⠀⠀⠀⠀⢸⡶⣿⣿⡆⠀⠀⢸⣿⡇⠀⢿⣤⣼⣿⣿⣿⣿⡿⢫⣿⡿⠿⢿⣟⣻⠿⠋⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣶⣶⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠉⠛⠻⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⣠⣴⠾⠟⢋⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⡝⠻⠷⣦⣄⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣿⣧⡀⠀⣠⡀⢠⡄⢀⣀⣀⡀⠀⢀⣀⠀⠀⣀⣀⡀⠀⢀⣼⣿⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡏⠻⠏⣿⡆⢹⣷⡿⠁⢸⣿⣭⡅⠀⣼⣿⡄⠀⣿⣽⡿⢰⣿⠹⠟⢹⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢷⣤⣴⠟⠁⢼⠟⠀⠀⠸⣿⣶⠄⠰⡟⠸⡷⠀⢿⠿⡦⠈⠻⣦⣤⡾⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣤⣤⣤⣤⣤⣄⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣤⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣍⣉⣉⣉⣉⣿⡇⢸⣏⣉⣉⣉⣉⣹⡇⢸⣿⣉⣉⣉⣉⣩⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠉⠉⠉⠉⠉⣿⡇⢸⡏⠉⠉⠉⠉⢹⡇⢸⣿⠉⠉⠉⠉⠉⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⣿⡀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⢀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠛⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⠿⠿⠿⠿⠿⣷⡄⢰⡿⠿⠿⠿⠿⢿⡆⢠⣿⠒⠁⠈⠉⢣⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠷⠶⠶⠶⠶⣿⡇⢸⡷⠶⠶⠶⠶⢾⡇⢸⣿⠶⠶⠶⠶⢶⣿⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣶⣶⣶⣶⣶⣿⠆⢸⣷⣶⣶⣶⣶⣾⡇⠸⣿⣶⣶⣶⣶⣶⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠋⡉⣒⠶⡶⠶⠖⠶⠖⠒⢶⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣱⠞⢦⠢⢬⡥⢭⣉⣍⠐⠒⠦⣈⠑⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣾⣟⠙⣿⠟⢷⣴⣧⡀⢨⠉⠻⢾⣄⠈⠛⣄⠹⡇
⠀⠀⠀⠀⠀⢀⣴⣪⣻⣿⣷⣾⣿⣿⡿⣿⣶⣼⣧⠈⢻⠻⣶⣤⡆⠘⢻⣇⡀⣇⡇
⠀⠀⠀⠀⢰⣧⣟⣿⣿⣾⣿⣿⣿⣿⣧⣨⣿⣛⢿⣿⣿⣤⢀⣿⢿⣷⣾⣿⣿⠙⡆
⠀⠀⠀⠀⣇⣿⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠻⠟⠁⠀⠈⠈⠿⣿⣧⢹⡀
⠀⠀⠀⠀⢿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣇⢧
⠀⠀⠀⠀⠈⢫⣻⣤⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠉⢿⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢻⣿⣿⣿⣿⢇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠻⣿⣿⣿⣿⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠘⣿⣿⣿⣷⣞⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠹⠿⣿⣿⣦⣀⡀⣀⡀⠀⠀⠀⢀⣴⠿⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠻⣿⣿⣿⣷⣾⣿⣿⠟⢁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⠉⠈⢭⣉⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠈⠙⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣀⡀⢰⣆⡖⠐⠻⠁⠀⠲⡦⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢸⣿⣿⠀⢈⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠖⢷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣄⡀⢀⣉⡻⣿⣿⣤⡀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⠁⣼⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠉⠁⠙⠿⣿⣿⣷⣦⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠴⠄⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⠉⢉⣙⠻⣿⣇⠀⠀⠀⠈⠻⢿⣿⣇⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⡆⣭⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢋⠀⠀⠿⠿⠃⠈⣿⣷⣄⠀⠀⠀⠀⡑⢤⣤⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣥⡁⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢀⣴⡦⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠘⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡀⢶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⣯⠁⠀⢀⡠⣾⣿⡿⠿⣿⣆⠀⠐⠤⢠⠏⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼⠅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣷⣠⡞⠀⣿⣻⡄⠀⣸⣿⣦⣠⣤⡟⢀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢪⣿⣿⣇⢠⣿⡿⣠⢊⡁⠀⠈⠙⠿⣿⡟⢁⠂⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠛⠉⢉⡙⠻⢿⣿⣦⣄⡀⠀⠈⠀⣿⣿⡦⡀⠈⠻⣿⣻⣦⠀⠀⠀⠀⠒⢶⠀⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣷⣦⣄⠉⠀⠀⠈⠛⠿⣿⣷⣾⣿⣿⣿⡄⠀⠠⠤⠹⣯⣮⠁⠀⠀⠀⣉⣉⠀⣤⣶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⢀⣽⣿⣿⣿⣿⣿⡿⠓⠒⠂⠙⠋⠁⠀⠀⠀⢤⢤⡄⠉⢉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣏⠉⣻⢿⣦⡀⢀⣤⣀⣦⠀⠀⠀⠀⠂⠶⠃⠀⢸⡇
⠀⠀⢀⣠⡴⣶⣶⣶⡤⣤⣄⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡇⡠⢹⣿⣦⣏⣿⣿⣿⣿⡧⠜⠀⠀⠀⠀⠀⣤⣤⡄⣤⣬⣥
⠀⠘⢉⣠⣾⠿⠛⠁⠀⠀⠙⣿⣦⡀⠀⠀⢠⣿⣮⣍⡴⠋⣹⢿⣿⣿⣿⣿⠀⠀⣼⡇⠀⠁⠉⠛⠻⠽⠃⠘⠀⠀⠀⠀⠀⣚⣛⣆⣍⣩⣭
⠀⠀⠛⠋⣙⣛⣋⣻⣿⣧⡾⠟⠁⠀⠀⠀⣼⣿⣿⣿⣥⠢⣿⠀⠉⣿⡿⠁⠀⣼⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⠋⠋⠉⠙
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠀⢸⠂⠀⣿⣷⣶⣾⢿⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡬

⠀⠀⠀⠀⠀⠀⠞⠹⠛⣭⠙⠉⡙⢉⡉⠋⡉⢏⡹⢿⠙⣿⡭⢿⣿⠏⣿⣿⣿⠿⢛⡋⢉⡃
⠀⠀⠀⠀⠀⠀⣻⡁⠀⢸⡿⣮⣄⠂⡤⠌⣰⡋⢹⣇⠻⠏⣵⢦⣴⡜⣿⠉⢛⠂⢻⡓⢛⡇
⠀⠀⠀⠀⠀⠀⢮⡠⠐⢸⣧⢾⢛⣟⣷⢦⣿⢙⣿⠁⢴⣿⣄⢘⣳⣿⣿⠀⣽⠅⢹⡇⢽⡇
⠀⠀⠀⠀⠀⠀⢠⣀⠈⣿⡎⢹⣶⡅⢸⣶⠹⡆⣹⡟⢃⣽⠴⡿⠁⡏⠲⠀⢸⠆⠰⠇⠺⡇
⠀⠀⠀⠀⠀⠀⢨⣀⡴⣇⡽⢾⠀⡧⠼⢈⣿⣿⡤⡾⡍⠁⠀⠁⣸⡇⢙⠃⢚⡃⢘⡃⢚⡇
⠀⠀⠀⠀⠀⠀⢨⣠⠼⣿⢇⣸⠯⢇⢸⢟⡾⠯⠀⢁⡃⠠⠄⠨⠸⡇⢉⡁⢉⡁⢈⡁⢹⡇
⠀⠀⠀⠀⠀⠀⢨⣩⡛⢿⣤⡻⣄⡜⢋⣼⡇⠀⠅⠐⢀⠀⠀⠈⠐⣇⢨⡅⢩⡅⢨⣥⠏⡇
⠀⠀⠀⠀⠀⠀⢀⣩⠁⡛⢋⡻⣿⣷⣬⣿⠃⢀⣀⠐⠀⠀⠀⠀⠁⣿⢸⡄⢠⣥⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣉⡜⠋⣘⠃⢨⡝⢿⡇⠉⠀⡀⠀⠀⠀⠀⣠⣞⣿⣦⠶⠿⠥⣤⣀⡀⡇
⠀⠀⠀⠀⠀⠀⢸⢿⣡⡦⢛⡸⠇⠴⢾⡇⢂⣀⣀⣀⣤⣶⣿⣻⡉⢶⠴⠇⠸⠖⠗⠘⠋⡃
⠀⠀⠀⠀⠀⠀⢸⠂⢹⣦⣽⣃⣾⣧⣾⡿⠿⠯⠿⠶⠶⢟⣛⣛⣛⣶⣥⡾⠶⠿⠾⠒⠛⡇
⠀⠀⠀⠀⠀⠀⢸⠄⠠⡅⡞⢡⡞⣿⣖⣛⣛⣛⣛⣉⣉⣭⣭⣭⣭⢭⠤⡿⣷⠾⠖⠓⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡀⢃⠛⡴⢣⢻⣟⡽⣯⣩⣭⠭⠤⠴⠶⠶⠶⢚⣛⣛⣛⣛⣾⡕⢻⠘⡇
⠀⠀⠀⠀⠀⠀⠸⠤⠦⠦⠵⠼⠯⠦⠴⠤⠿⠿⠶⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⠶⠇

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡴⠶⢶⣴⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⣶⣾⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⣉⣉⣉⣽⣿⡟⠛⠁⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠠⠔⣊⣉⣩⣭⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠶⠛⠉⣉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣋⣠⠴⠒⠋⣉⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣋⣁⣤⣴⡾⢿⣟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⣋⣡⣴⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣵⣶⣿⣿⣿⣛⣉⣡⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⡶⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡙⢏⣿⣿⡛⠖⠛⠋⠀⠀⠀⠀⠀⢀⡶⢛⣡⣾⣿⣿⠃⠐⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡘⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⠋⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣾⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⣶⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢺⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣼⠿⡯⠋⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣤⡀⠀⠀⢠⣾⢛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⢏⠙⠌⠋⠻⣯⡀⠀⠀⠀⠈⠙⢷⣤⣀⠀⣀⠀⡀⢤⣀⠀⠀⠹⣿⣦⡀⠹⣷⠀⠀⣹⡯⠀⠀⢹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡣⠕⠨⠢⠁⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠪⡦⡆⣽⣿⣿⣿⣿⣿⣷⡀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣾⣿⢿⡳⢝⠢⡈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠨⣟⢿⣶⣶⣿⡟⠛⠉⠉⠈⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣙⠦⡉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⡠⢋⣴⣿⠿⣿⡇⠀⠀⠀⠀⣿⣿⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣸⡟⠑⢮⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⠀⠀⣰⣿⣿⠗⣋⣽⡇⠀⠀⠀⠀⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣭⣟⠿⣿⣾⣿⣿⣿⣯⣵⣾⠃⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠉⠀⠈⠛⣿⣿⡿⠟⢻⡇⠀⠀⠀⢠⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⣠⠖⠛⣧⠀⠀⢰⣿⣿⣿⠿⠿⠿⠶⣶⣤⣄⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⠀⠀⠹⡄⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣟⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠏⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⢸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⣚⠚⢒⣪⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣿⡇⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⡭⠛⠉⠛⠭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠤⢀⣀⣀⣈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢓⠁⠀⠀⠀⠉⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⡀⠀⢀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠘⠒⣶⠒⡒⢲⠒⡖⢲⠒⣖⢲⡇⡇⠀⠀⠀⠀⠀⠀⢀⠜⠁⢣⢨⣛⡁⡆⠈⠢⠀⠀⠀⠀⠀⣔⣖⡤⡾⣸
⠀⠀⠀⢱⠆⠲⠐⠆⠰⠀⠆⠰⡀⠇⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢩⠉⡏⠀⠀⠀⡇⠀⣴⡎⠉⠹⠞⠉⠫⠏⠉⢳
⠀⠀⠀⠀⢯⡌⢥⠩⡌⢥⠨⡄⢡⡄⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣘⠀⡃⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠘⡖⠐⠂⠒⠐⠂⠒⠐⡇⠣⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣬⣶⣥⣤⣤⣤⣇⣠⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⢤⢤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⡤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡔⡲⡢⡤⠤⠤⠤⠤⠤⡤⣚⡸⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇⠀⠀⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠮⣒⡣⠋⠉⠉⠉⠉⠉⢇⣣⣋⠇⠀⠀⠀⠰⣍⣉⣉⣉⣉⣉⣉⣉⠇⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣴⣶⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣾⣿⡿⢉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣿⣿⠁⡈⠀⠏⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣤⡀⠀⠀⠀⠸⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⡆⢰⡁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡇⢠⣼⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣦⠠
⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⣷⣤⣀
⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠺⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠘⣿⣿⣿⣿⣿⣿⣍⣳⣦⣄⡀
⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⠿⠛⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠙⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⡳⡀
⠀⠀⠙⢯⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⡀⣀⣤⣴⣿⣿⣿⠏⠀⣰⣮⡟⢋⠝⠫⠟⠉⠛⣋⠀⢂
⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣍⠋⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⠷⠏⠀⠀⢀⡰⣟⠁⠚⢋
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋⠕⠛⠻⢿⣿⣿⢻⣿⣿⣿⢻⠇⢀⣠⠴⣂⡛⠰⠀⠀⠰⣿
⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⢻⡿⢿⠛⠋⠞⢰⣿⠃⣾⣿⠿⢿⣷⣦⡀⣽
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⢻⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣀⣼⣇⡼⠁⠿⠁⠂⠀⠀⠈⢻⣿⠈
⠀⠀⠀⠀⣌⣉⣉⣥⣤⣶⠒⠻⣿⡛⠿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⡴⠂⠀⠈⡐⠀⠀⠀⣸⡏⡄
⠀⠀⠀⠀⣿⡘⣿⣿⣿⣿⣷⡀⠹⣿⠃⠀⢙⡄⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⠿⠋⣀⣀⣀⣀⣴⣤⣶⣶⣿⣾⣷
⠀⠀⠀⠀⣿⡿⣾⣥⣌⠉⠛⢿⡆⢭⣄⠺⣿⣿⡄⠀⠀⠙⣧⣶⣎⢻⣶⣦⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢃⠔⠋⡍⠉⠛⠋⠉⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣧⣴⡤⠀⠀⠉⠁⠛⠓⠋⠓⠋⠉⠁⠀⠀⠀⠈⠻⠟⢸⣿⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⢣⡴⣣⣤⣤⡄⠤⣀⡀⠻⣯⡙⣿⣿
⠀⠀⠀⢰⣿⣿⣯⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⣤⢸⣿⣿⣿⣿⣿⣄⠙⢿⣦⡉⢻⠟⠁⠰⡾⣟⣷⣾⣷⣌⢻
⠀⠀⠀⢸⡿⠛⠛⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢿⣧⠙⣿⣸⣿⣻⣿⣿⢿⣿⣇⡾⣿⣿⣆⡀⢀⡀⢈⢯⣼⡏⢿⣿⣿
⠀⠀⣰⣾⢿⡟⣫⠄⢀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⡧⢈⣿⡏⡽⣿⣤⣈⣿⡿⠀⢸⣿⣿⣷⡈⣷⢸⣷⡽⣇⢸⣿⣿
⠀⢀⣿⣷⣟⡽⠧⢀⡌⢀⣀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠⣟⢻⡟⠰⡄⣿⣿⣿⣿⠃⠀⠀⠻⣿⣿⣷⡀⠘⡇⢻⣽⣎⣿⣿
⠀⢸⣿⣿⣿⠂⣠⣾⣷⣿⡿⠛⠁⠀⣠⣴⣶⣾⣿⣿⢿⣇⣀⠠⣶⣶⣿⣶⣤⣼⣟⣫⣩⣿⣿⣿⣿⡟⠀⣠⣤⣄⣉⣿⣿⣇⣠⣿⡀⢘⣿⣿⡏
⠀⠀⢿⣿⣯⡄⢹⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣉⣁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠃
⠀⠀⠈⢿⣿⣿⡜⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠻⣿⣷⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢛⣿⣿⣿⠿⢻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢿⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣄⠀⣤⣀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣇⢿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⠋⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣦⣴⣤⡼⠿⠟⠛⠛⠀⠈⠛⠀⠀⠀⠀⠀⠐⠐⠀⠠⠄⢠⠀⠀⡠⠀⢀
⠀⠀⠀⠀⣾⣿⣸⣿⣉⠘⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠛⣹⣿⣿⣀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣰⣦⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠙⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣤⣀⠀⠀⠀⡙⠉⠉⠛⠻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠒⢦⠀⣀⣠⣿⣀⣀⣀⡀⠀⢀⠤⠤⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢡⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠻⢿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⢼⣏⡀⠀⠀⠸⠻⠿⠿⢿⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⠺⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣖⠒⠲⠤⣄⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⣙⢦⡀⠀⠙⠪⣧⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠛⠂⠀⠀⠈⠃⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣂⡀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠎⠀⠀⠀⠀⡀⠀⠀⡀⠀⠙⠦⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠣⡀⠀⢠⡠⣄⡠⠴⠃⠀⢠⠖⠉⠀⢀⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠆⠀⠀⠀⠀⠈⠒⢢⠀⠀⣘⡚⢣⠉⠋⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠄⠒⠂⢊⠔⠤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣨⡘⡻⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠒⠀⠀⠂⠬⣐⠒⢒⣛⢆⢳⡎⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢉⠆⠐⠎⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣯⠋⠉⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⡏⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣰⡟⡹⢽⡲⢒⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⣤⣀⡀
⣿⣿⣿⣿⣶⣶⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⠿⠟⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠋⠉⠀⠀⠈⠻⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡏⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣾⡧⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣆⠈⠁⣀⣠⣤⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠄⠀⡀⠈⠀⠀⠀⠐⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⡁⠐⠉⠂⠀⠀⠀⠄⠀⠀⠀⢀⣴⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣆⠀⠠⠀⠀⠀⡈⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠲⡄⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠄⡔⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⡖⠲⠒⠒⠂⠐⠂⠒⠀⠐⢺⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⠀⢰⠀⠃⠀⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⢰⠀⠃⠀⠀⠀⢰⠀⡄⠀⠀⡈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⠈⡀⢰⠀⠀⠀⠀⠀⢁⠀⠀⠀⡇⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⠀⠀⡇⢸⠀⠀⠀⠀⡇⠸⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠧⠤⠔⠃⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⢤⡈⢇
⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠖⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⠒⠒⢲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠄⠛⠉⠉⠀⠀⠀⠀⠘⠛⣿⣦⡌⠙⠶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⣁⠙⣿⣔⡐⠚⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠶⢋⡡⡢⢝⣿⣤⣀⣈⡁⠚⣦⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠁⠀⡀⠐⠂⠉⣀⣀⠀⠁⣒⣒⣂⣭⣉⣷⣾⣿⣭⣤⣤⡄⣀⠈⢳⡄⠀⠀⣠⠔⠁⠠
⠈⠈⠌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠟⠀⠀⠀⠀⠀⢈⡁⢠⣤⣴⣶⣿⠿⠟⠛⠉⠉⠙⠻⢿⣿⣧⠄⠀⠁⠀⠹⣤⠊⠉
⠀⠀⡐⠀⠀⢉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢰⠏⠀⠀⠀⡀⠔⠃⠈⣡⡴⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⢱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠀⠀⠀⠀⣠⠇⢀⠃⠀⠀⢀⠂⡠⢔⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢇
⠀⠠⠠⠀⠄⠀⠄⠀⠀⠡⠠⠀⠌⡼⠉⠉⢣⣠⡏⠀⠀⢀⠀⣰⡗⣩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⢂⠸⡧
⠀⢡⢡⠉⡌⠈⠌⠀⠀⠁⠠⠉⡜⡵⡤⠠⢼⣿⠁⠀⠀⡌⢠⣟⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⢢⠈⡄⣿⠀⠀⠈
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠀⠀⣾⣭⣄⡀⢰⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢃⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡿⣅⢀⡸⡀⣿⣿⠃⣤⠶⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣠⠴⠾⠛⠛⠉⠛⠒⠀⠀⠀⠀⢸⣿⣿⡎⢹⣾⡇
⠀⠌⠌⠀⠡⠡⠡⠀⠀⠌⠌⠤⠡⠡⠀⢸⣷⠀⠀⠀⢱⣿⣿⡀⠀⣠⣴⣦⣤⡢⠀⠀⠀⠀⠀⠀⠀⣠⡶⡶⠿⣷⣦⡀⠀⠀⠀⠀⠸⠹⣿⣷⠈⣿⡉
⠀⡈⡈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⠀⢠⣼⡆⠀⠀⠸⢹⣿⡇⠘⠛⠯⡭⠀⠉⠶⠀⠀⠀⠀⠀⠘⠙⠲⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡀⣿⠟
⠀⠐⠐⠀⠂⠀⠂⠀⠀⠀⠀⠀⢂⠂⠀⠀⣸⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣾⣿⣧⣸⡇
⠠⠡⠡⠈⠄⠄⠌⠀⠀⠠⠠⠀⠌⡄⠄⣀⣸⣇⡿⢀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠹⣿⣷⣶
⠀⢁⢃⠈⠀⠀⠀⠀⠀⠀⠂⣀⣼⣾⣿⣿⣿⣿⡇⠰⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⢰⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠻⣿⣿⣷⣦⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⠂⠀⠀⢸⠀⠀⣿⣿
⠀⠄⠄⠠⠀⠠⠠⠀⢀⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠃⠀⡀⠀⠀⠸⠀⠀⠘⠀⢰⣿⡇
⠀⠈⠈⠀⠁⠀⠁⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢹⣿⣿⣿⣧⡀⠀⠀⠀⠈⠛⠿⠷⠾⠋⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡆⠀⡀⠀⣸⣿⡇⣾
⠀⡀⡐⣀⠀⢀⢂⠀⣿⣿⡇⠀⠀⣄⠘⣆⠀⠀⢻⣿⣿⣇⡈⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠸⢀⣥⣾⣿⣿⣰⡿
⠀⡸⡀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠸⣦⠘⣧⠀⠀⢿⣿⣿⡠⠀⠀⠈⡹⢿⣿⣿⣷⡀⠀⣀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡗⠂⠸⣿⣿⣿⣿⡇
⠀⠽⠡⠀⠄⠄⠀⢸⣿⣿⡇⠀⠀⠀⢹⣧⡹⣷⡀⠸⣿⣿⣧⢂⠀⠀⠈⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡟⣿⡟⡀⠘⠀⢿⣿⣿⣿⣷
⠀⢱⢣⠉⠈⠈⡌⢸⣿⣿⡇⠀⣤⠀⠀⢻⣷⣿⣷⠀⣿⣿⣿⣟⣦⠀⠀⠀⠸⡞⣹⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀⠀⠂⢻⣧⠇⠀⠁⠈⠹⣿⣿⣿
⠀⢃⢂⠀⠀⠐⡀⢸⣿⣿⡇⠀⣿⣿⣄⠈⢿⣿⣿⣧⢸⣿⣿⣿⣿⣧⢆⠐⡀⢻⣣⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣼⣰⠃⠸⣧⠈⠀⠀⠀⠀⠹⣿⣿
⠀⢚⠆⠀⠀⠀⠀⢸⣿⣿⡇⣀⢹⣿⣿⣷⣜⣿⣿⣿⣿⣿⠙⣿⣿⣿⣾⣦⢳⡈⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣼⠀⣿⣆⠐⠀⡄⠀⠀⢿⣿
⠀⠜⠄⠀⠠⠠⠠⢸⣿⣿⡇⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⣿⣿⣿⣿⡎⣷⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡞⣼⣿⣆⢠⣷⠀⡆⢸⣿
⠀⣼⡌⠀⠀⠠⠀⢸⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⢻⣿⢿⣿⣷⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣶⣧⣼⣿
⠐⡸⡐⠂⠂⠂⠂⢺⣿⣿⣧⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠈⠀⠠⠔⠊⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢵⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⡄⠀⠀⡆⠰⠀⢀⣀⠄⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢁⡀⠀⠧⣈⡀⠬⠃⠀⠂⠓⠘⠂⠑⠐⠠⣙⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⡀⢆⠐⠀⠀⠀⡀⠀⡄⠀⢠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠸⡀⠀⢸⠠⡄⢼⡀⠰⠤⣇⠀⡸⢠⠉⠁⠤⠀⠀⡄⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⡀⠀⢧⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠐⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠣⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣉⣉⠁⠐⠒⠈⠉⠉⠁⠀⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣒⣈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⠪⢭⣯⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠤⠤⠤⠬⣥⠤⠔⠒⡶⠚⣁⣴⡠⠴⢠⠀⠀⠀⠀⠀⠈⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠇⠀⠀⢀⡏⠀⠀⠀⠈⠉⠁⠀⠀⢀⠈⣆⠀⠀⠀⠀⠲⣌⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠸⢄⡀⣠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠀⠘⢆⠀⠀⠀⣀⡏⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠉⠑⠦⠔⠀⠀⠀⠀⠀⠀⠀⡤⠴⠉⢹⠳⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠒⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠊⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⢄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠤⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢍⣙⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠞⠉⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠰⡪⠗⡋⠐⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠿⡧⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠠⠼⠄⠐⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠧⢄⣀⣀⣀⡠⠔⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⣉⣤⣴⣀⣀⣀⣀⣀⣀⣀⣐⡂⣄⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠃⡰⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠡⡘⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠿⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢯⣉⠙⠛⠳⠦⣤⡀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠳⣦⡀⠀⠀⠙⠳⣜⢦⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⡓⠶⢽⣦⡀⠀⠀⠈⠙⢧⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠥⠆⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠖⠋⠙⠲⢄⠀⠀⠀⠤⠄⠈⠙⠦⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣰⠁⠀⠀⠀⠀⠈⢳⡤⠚⠑⠲⡀⠀⠀⠈⠑⢦⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⣄⠀⠀⠀⢀⡀⡇⡀⠀⠀⣰⡃⠀⠀⠀⣠⠴⠒⠈⠁⢠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢈⡒⠤⣤⡼⠟⠛⣿⠞⠉⠁⠀⠀⠀⣜⣀⡤⠤⣄⣸⢦⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣄⠀⠠⢀⠂⠀⠀⡔⠊⠙⢢⠀⠀⢀⣈⣉⠙⢧⠀⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠀⠀⣠⠖⢤⠤⠧⠦⠴⢃⡴⢄⡀⠉⠈⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠉⠑⠋⣠⠋⠀⠀⠀⣠⠏⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠑⡄⠀⠀⠀⠀⠀⢹⣀⣀⠀⣧⡀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠙⠁⣄⠈⠀⠘⠲⣄⠀⠀⠀⠀⠀⠈⡆⢣⠘⣆⠀⡼⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⣤⢤⡤⣄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⠘⡴⡍⠋⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣃⣸⡀⣘⡂⠉⣉⠑⠢⢄⣀⠀⢀⣘⣯⠇⢱⢄⡴⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢬⣉⣁⠠⣴⣸⡈⠿⢆⠀⡐⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡷⠐⠚⠷⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢸⠙⠉⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣸⠗⠀⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢺⣤⣤⣧⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠰⢭⡃⡴⠧⠭⡿⣉⣉⡩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠥⠭⠭⠥⠤⠕⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡀⠔⢮⢴⠠⡰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢽⣹⠞⡁⢸⡘⣿⠚⡵⢟⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣟⣓⣇⠈⠸⣦⠀⢰⡄⢤⠈⠀⣍⠉⠭⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡀⠄⠀⡉⠖⠁⠀⠘⠈⠈⢁⠘⣆⠁⠸⣀⣋⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠄⡐⡘⠂⠉⠀⣘⠈⠀⠃⠈⠀⠉⠀⠀⠀⡐⡂⠀⣈⢇⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠌⠠⢀⠆⡥⠑⠀⠁⢲⣽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⣫⠀⡀⠀⠀⠁⠸⠀⠊⠈⠈⠈⣑⢠⡔⣆⠘⣼⢻⣶⡂⠗⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⠌⠉⢀⡠⣤⠠⣴⠞⣰⢿⢶⡾⠲⠎⠀⢩⠟⠝⣧⡑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣞⢂⡇⠧⢪⣀⣾⣾⣼⣪⣷⠟⣦⠀⢩⠆⠲⣆⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠆⡳⣯⡯⣞⢿⣻⠽⡖⠟⢏⠱⢧⣘⣧⣼⡛⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠐⠳⠛⣯⢎⠰⢇⢃⡴⡴⣚⣷⣯⠌⡁⠀⠀⠀⠀⠑⠪⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡍⢁⣏⣈⢡⣞⣿⡵⣶⣮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣭⢭⣽⠙⣝⠌⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⢏⢴⡥⡔⢦⣴⣚⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣚⣻⡯⠄⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⣉⣁⣤⠴⠖⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣁⣤⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠞⠀⠉⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡈
⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡷⠆⠀⠤⠀⠠⠂⠀⠀⠀⠀⡰⠒⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠘
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸⠀⠀⢀⠄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠃
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⢠⠑⠤⢀⠌
⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠁⠂⠀⠀⠄⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡟
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡿⢿⣻⣛⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠿⢿⣿⡿⠿⠿⢟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠠⠶⢤⡀⠀⢀⣀⣀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠓⠂⢻⠀⠤⠤⠀⢁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠀⢀⠰⠆⠠⠀⠀⠀⡌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠒⡂⠤⠈⡠⢊⠵⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠻⣷⡄⠀⠉⠀⢰⣡⢋⠊⣄⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⢻⠀⢩⡛⠓⠒⢂⡿⢿⠾⢻⠟⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⡿⠟⠻⠿⠛⠋⢉⡆⢸⡏⢸⡀⠀⠑⣄⡠⠼⣷⡀⢠⠂⠀⣰⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠀⢧⠀⠀⠀⠀⠀⡼⠀⠈⠁⠀⡇⢀⡴⣿⣿⡄⢻⣿⡀⠀⠀⠈⢆⠀⠈⠉⠙⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣦⠀⢸⡀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢳⠈⣴⣿⣷⡀⢸⣿⣷⠀⠤⢂⡌⠱⡀⠀⠀⡔⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⡐⠦⣄⠀⡇⠀⣌⠉⠉⠉⠛⠃⠀⠀⢸⣴⣿⣿⣿⣇⣿⣿⣟⡃⢀⣾⠀⠀⠑⣄⠘⢀⣴⡇
⠀⠀⠀⠀⠀⠀⣴⠛⠧⣀⡀⠀⠈⢳⣷⡀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣷⠀⠀⠘⡆⠀⠀⠈⢲⡟⣹⣇
⠀⠀⠀⠀⠀⢰⡟⢆⠀⠀⠻⢯⡛⣯⣽⠇⠈⢻⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⠘⢿⣿⣷⠀⠀⠑⠀⠄⠀⠀⢻⡁⠈
⠀⠀⠀⠀⢀⣿⠇⠈⠑⠢⠤⠤⣙⣰⣿⠀⠀⣸⡿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠘⢿⣿⣧⣀⠀⠀⠀⠀⠀⠀⡳⢬⡳⡀
⠀⠀⠀⡠⣫⠵⣚⣛⡛⠛⠟⠞⣿⣿⣿⠀⠀⣿⠁⠹⡄⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠈⢿⣿⣿⢧⡀⠀⠀⠀⠀⢱⠈⠇⣷
⠀⠀⢸⣿⡿⠏⠁⠘⠙⠳⣄⣴⣿⣷⡛⡀⢰⡏⠀⠀⠱⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢈⣿⣘⣧⡙⠂⠀⠀⠀⠀⢇⢀⡏⣠
⠀⠀⢸⠉⡜⠀⢀⡤⠆⠀⠈⡙⢿⡿⠛⢥⢾⢧⡀⠀⠀⠹⡀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣾⣿⣟⠉⢿⣤⡀⠀⠀⠀⠈⠚⣴⠃
⠿⠿⠿⣦⣀⣴⠋⠀⡠⠔⣫⣾⡏⠐⠦⡀⠓⢟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⢾⣯⣛⣷⣄⢀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠙⠿⠷⣶⣦⣼⣄⣏⣻⡅⡙⠶⢁⠐⠠⠜⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣈⡹⠿⠆⠉⠳⣝⣃⡀⠈⠗⠒⠙⠇⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⡀⣀⡤⣤⣀⠠⠴⠶⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠶⠖⠁⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⢀⢀⣀⠤⠴⠞⠓⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⡏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢹⡆
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢰⣱⠀⠀⢀⠤⠒⠉⢹⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠸⠼⢿⠀⠀⠀⢀⡰⡞⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠘⣄⡠⠖⠁⡰⠁⠀⠀⠀⠀⢀⡜⠀⡔⠒⠚⡦⡀⠀⠀⣀⠴⠦⡀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⠢⣀⢠⠃⠀⠀⠀⢀⡠⠊⠀⠀⣧⠀⠠⡇⣈⣖⣉⡖⠒⠒⡎⠲⣄⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠀⠀⠀⣧⠤⢤⡇⡄⠀⢼⡗⠒⠒⡇⠤⠬⢤⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡇⠖⠒⢺⣟⠉⠉⡇⣗⣒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡇⡋⣉⣹⣏⢉⢉⡇⡒⡒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁⠈⠉⠉⠀⠉⠉⠁⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠸⣇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣸⠇

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣄⠀⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠁⠉⠙⠓⢦⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢤⡦⠟⡶⠄⠙⠒⠤⣆⡀⣰⠀⠀⠀⣠
⠸⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣳⣌⠀⠉⠘⡀⠀⠀⠉⠉⠹⠔⡦⢊⡏
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⠴⠶⢄⠈⠁⠀⢀⣄⣤⣾⣄⣄⣀⠀⠀⡰⡇
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡴⠃⡜⠁⢠⠎⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⡏⡷⠤⠚⠉⠀⠀⠈⠉⣻⣦⡄⠸⢤⡄
⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠉⠀⠀⠣⠔⠁⠀⡌⠀⡜⢀⠊⡇⠠⡲⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⡁⠀⠧⣀
⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⡶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⠊⠀⠀⠀⡰⠋⠀⠀⠤⣀⠀⠀⠀⠀⢀⣿⡟⠛⠂⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠰⢦⣀⡈⡙⠀⠀⢀⣸⠋⠁⢰⡚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠸⢮⣁⡬⠃⠀⡴⣋⣉⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣩⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣡⣦⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠈⠀⠠⡀⠀⠀⡠⠤⠴⠃⠁⠈⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢄⠀⠀⠐⡀⠀⡀⠘⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠉⠉⠓⠃⠀⣧⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠉⠓⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣀⠴⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠉⠁⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⢾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡉⠁⠀⠀⡼⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠹⢄⣀⣠⠾⡁⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⢠⠜⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⡄⠀⠀⠀⠆⠀⠈⡀⠀⠀⢼⣷⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠿⠃⠿⠿⠿⠿⣆⠀⠀⠰⡀⠀⠁⠀⠀⣾⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠈⠀⢀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣧⡀⠀⠱⡀⠠⠂⢀⣭⣭⣏⣉⣉⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠀⣰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣦⡤⠊⠀⣴⣿⡉⠀⠙⢿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⢀⣠⠚⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡟⠠⠂⢸⡿⠃⢤⡄⠀⠀⠹⢿⣷⡄⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢰⣏⣉⠀⣩⣭⣭⣭⣭⡁⣩⣭⣭⣭⣭⣭⡸⠀⠀⡈⠀⠀⠠⠄⠀⠀⢰⡀⡘⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⡆⠀⣀⣀⣀⣀⣿⣿⡇⣿⣿⠿⠿⠿⠃⠛⣻⠿⠿⣿⣿⡇⠀⢀⠀⠀⠤⠀⠀⠀⠐⠀⢐⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠃⠉⠁⠀⠀⠀⠉⠉⠙⠓⠢⢄⡀⠀⠀⠀⠉⢬⠀⠘⢁⡿⠙⠁⢆⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠉⠳⡄⠀⠀⠀⠚⠀⠘⢇⣠⣤⣈⣦⣤⠀⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢻⠵⠶⠷⣖⣛⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⢃⠀⣀⣉⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⠄⠀⠀⠀⠀⠀⠑⡅⠐⠴⠀⠀⠉⠙⠝⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠛⢥⢀⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠗⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣣⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⣀⣤⣤⣾⠿⠛⠉⢀⣤⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⣠⣶⣶⣸⣰⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠻⣿⣿⡉⢹⣿⣇⣀⡀
⠀⠀⠀⠀⣠⣴⡄⡆⣰⣦⡀⠀⠀⠀⠀⣀⣤⣶⢨⡍⣁⠀⠀⠀⠀⠀⢸⣿⠟⢻⣿⣿⣿⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⣻⠿⠿⠿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⡿⢻⣿⣾⣾⣿⣷⣀⠀⠀⠀⣼⡿⠀⢸⣿⣿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣭⣼⣿⣦⣴⣿⠂
⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⠀⠀⣸⣿⠁⣾⣿⣿⡿⠈⠛⠿⠿⠗⠀⠈⠁⠀⢸⣿⣿⠁⠀⢸⣿⢠⣤⠀⠀⠀⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⣾⡟⣼⣿⣿⣿⡇⣿⡇⠀⠈⠁⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣷⣶⣿⣿⣷⣶⣿⢂⣀⣿⣿⣿⣿⠛⠉⠻⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠈⠀⣿⣿⢿⣿⡇⠈⠀⠀⠀⠀⢸⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⣀⣰⣿⣿⠋⠉⠻⣿⣿⣼⡿⣿⣿⣿⣷⣀⠀⣠⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⣸⣿⡏⠈⢿⣿⡄⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠉⠻⣿⣷⣤⠀⢴⡿⡿⠋⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣿⣿⠀⣿⣷⠀⠀⠀⠀⢠⣿⡟⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⡿⠿⣷⠆⠀⠀⢹⣿⡿⣿⣿⣿⢿⣿⡁⠈⠁
⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣿⣶⣶⣿⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠦⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠠⢤⣄⣀⣉⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡁⠀⠀⠀⠀⠀⠀⠈⠛⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢀⡤⠶⠀⠀⠰⣒⡒⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣀⡀⠘⠉⣰⣀⡀⠉⠉⢀⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢞⠝⣩⠧⡄⢈⣉⡀⢀⠔⢁⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠁⠋⠀⢀⣹⠀⠶⠀⡨⠛⠉⠁⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⢣⠀⠀⠀⢸⣯⣴⠶⠾⡇⠀⠀⢀⡞⠳⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⢀⠜⠳⢤⣀⡾⢿⡁⠀⣠⠷⡄⢀⡸⢷⡀⠀⠀⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⡶⠛⢷⣤⡀⣩⡃⢀⡿⢯⡀⢀⣽⣅⣠⠾⠓⠶⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣠⡾⠋⠀⠀⠀⠈⢻⠉⢷⡾⢧⣠⡷⠛⠰⡏⠀⠀⠀⠀⠈⢧⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡎⢠⡾⠋⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⡼⢹⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⣄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⡼⣴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠇⠈⣧⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⢿⣶⣧
⠀⠀⠀⠀⠀⡸⠹⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⠀⠀⣿⡄⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠈⢷⠹⣇
⠀⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠀⢀⡤⠞⢹⡆⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠙⢦⣀⠀⠀⠀⠀⠈⠇⢹
⠀⠒⠒⠒⠚⢧⡀⠀⠀⢀⣠⣖⣉⣀⣀⣬⣧⣤⣤⠴⠧⠤⠤⠾⠷⠤⠤⠤⠷⠤⠤⠽⢷⣤⡀⠀⠀⠀⣸⣧⣤
⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣼⣷⡀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⢀⣾⣧⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣿⠈⠛⠟⠘⠟⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⠃⠻⠛⠁⣿⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣛⠉⠀⠀⢠⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⡄⠀⠀⠉⣛⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⡶⠀⠀⠙⠿⠛⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠿⠋⠀⠀⢶⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠀⠀⢀⣀⣀⣀⠀⠀⢀⣤⡶⠟⠛⠛⠻⢶⣤⡀⠀⠀⣀⣀⣀⡀⠀⠀⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⠋⠉⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠉⠛⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠁⢸⣧⠀⣠⣴⣦⣄⣠⣴⣦⣄⠀⣼⠇⠈⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢹⣿⠋⠀⠀⣽⡏⠀⠀⠙⣿⡏⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣿⣧⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣠⡿⠃⠙⢿⣄⠀⠀⠀⠉⢛⡂⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠙⣷⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋⠻⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠻⠛⠃⠈⠛⠟⠁⣽⣿⣿⣿⡇⠀⣠⣶⣤⠀⠨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠁⠀⢀⣀⣀⡀⠀⠈⠻⠿⢿⣿⡆⠀⠻⣿⡿⠀⠠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣴⣾⣿⣷⣆⣠⡀⣀⣰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣷⡄⠀⠀⠉⠉⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣶⣶⡄⢠⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣷⣶⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣶⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣖⠀⢻⣿⠀⢰⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣆⠘⡇⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⣿⣿⣿⣧⣰⣿⣿⣿⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠈⢿⣿⣿⣟⣹⣿⣿⣿⠃⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⡇⠀⠀⠀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢈⣽⣶⣀⣴⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣻⠿⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠹⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⠀⠀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠻⣿⡿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣀⠀⠄⠐⠀⠀⠀⡀⠀⠀⢀⠛⠁⢂⠀⠀⠀⠤⠜⠉⠀⠀⠀⠈⠀⠀⠂⠀⠉⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠞
⡔⠈⠁⢱⠀⠀⢲⠂⠀⠀⠁⡄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣌⡀⠀⠀⣄⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠐⠄⣀⣀⣠⢴⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⢨⠁⠈⠉⠃⠀⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠘⠁⠐⠀⠀⠈⠛⠁⠠⠖⠀⢀⠀⠀⠠⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠃⠈⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠢⡀⠀⢄⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⠈⠁
⠀⠀⠀⡤⠯⠤⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡓⠀⠂⠀⠂⠀⠀⠆⠀⠀⠃⠐⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠰⠀⠀⢀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠶⣶⣒⠲⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠲⠶⠶⠶⠦⠤⣄⣀⣀⡀⠀⠤⣄⣀⡀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⠁⠀⠀⢀⣀⠀⠠⠄⠽⠏⠉⠉⠙⠛⠛⠛⠓⠀⠀⠀⠀⠜⠀⠀⠀⠈⠉⠑⠚⠂⠈⠉⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠰⡊⠀⠀⠀⠀⠀⠀⠛⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡘⢧⠀⠀⠀⠀⣭⠷⠀⠀⠀⠀⣀⠤⠴⠖⠒⠒⠒⠒⠒⠤⢇⢀⣠⠴⠒⠚⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⠈⠳⣄⠘⡷⣌⢷⡀⠀⠀⡞⠁⠀⠴⠄⠀⠀⠀⠀⠀⢀⣸⠏⡄⠀⠀⠀⣠⣾⣁⣠⣤⡤⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈⣯⣷⠀⠀⠙⠒⠶⠶⠤⠤⠤⠄⠒⠚⠉⠙⠲⠶⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠲⢶⣖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣁⠤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠚⠉⠉⠁⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠒⠒⠚⢛⣻⠿⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣠⡤⠴⠚⠋⠁⠀⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⣯⡽⠛⠉⠀⢀⣠⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠤⢴⡖⠠⡤⢤⣤⣀⣀⣈⣊⠥⠤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠲⣀⠀⠇⠀⣰⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⠳⢤⣸⠀⡤⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⢢⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠲⢤⡀⢀⡍⢹⡇⠀⡉⠑⠒⠒⠒⠒⠂⢸⠓⠃⠀⠀⢀⡀⠀⢰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡀⠀⢙⠛⠀⠘⢦⠀⠘⣦⣄⡀⠀⠀⠀⢺⢤⣄⠚⢳⡏⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡁⠀⢸⡄⠀⠀⢸⠀⠀⠸⡄⠉⠛⠒⠤⢼⡿⠘⠦⡼⠁⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠈⡟⠀⠀⢳⠀⠀⠘⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢠⠋⠈⠉⠁⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⢰⠇⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⣧⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⣀⡀⣀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠠⣾⣿⣿⣿⣿⣿⠿⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣎⣡⣌⣠⠥⠀⠀⣼⡀⠀⢀⣀⣤⣴⠇⠀⠀⠙⢿⠿⠋⠁⠀⠀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠓⠒⠲⠶⠶⠶⠖⡾⠉⢻⠋⠉⠁⢠⡏⠀⠀⠀⠀⠀⠱⣄⠀⣠⡔⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣀⣇⣀⣾⣀⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠈⠛⡹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠧⣄⠀⠀⠀⠤⠬⠭⠕⣾⠉⢁⣀⣁⡬⠷⣠⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⠈⠉⢓⡶⠶⠒⠒⠋⠙⠣⠍⢡⡀⠀⢐⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠀⠐⠀⠀⠆⠂⠂⠀⠀⠂⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⠘⠀⠃⠁⠐⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠲⣤⣤⠒⠚⠉⠉⠈⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠛⠦⠤⠤⠤⠤⠤⠮⠽⠯⠥⢤⠭⠽⠤⠤⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡴⠟⡋⠈⠉⠉⠈⠃⠒⠶⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⣿⡇⠀⢀⠴⠦⠀⠀⠀⠀⠤⣠⣭⠉⠙⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢷⣄⣰⢋⡙⠢⠀⠀⠰⢋⣭⢱⡀⡠⡿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠀⠳⣝⣣⠜⡏⠈⠛⢬⠭⠞⡁⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠞⠃⠀⠘⠦⠖⠃⠀⠀⠒⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠠⢤⣄⣀⡤⠔⠂⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠈⠉⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⠉⠉⠓⠒⠉⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡇⠘⢦⡀⠀⠀⡠⠃⣸⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠘⡄⠀⣹⢖⠚⡅⢰⠃⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠦⠀⠀⠀⠀⠀⠘⣤⣯⡙⢻⣧⡎⠀⠀⠀⠠⠆⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢾⣏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡐⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠐⢼⣈⡀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⢦⠈⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡷⢄⠑⣼⡇⠀⠀⠀⠀⢹⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣇⠀⠑⣼⡇⠀⠀⠀⠀⠈⡄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣭⠑⢦⣸⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠶⠶⠶⠶⠦⠾⠦⠶⢤⣤⣤⣷⣤⣤⣤⣤⣤⣿⣵⡶⠿⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠿⠶⠶⠖⠒⠒⠒⠒⠛⠺⠶⣤⡄
⠀⠀⠀⠀⠴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⢰⡶⢶⣦⠀⣀⡀⠀⢀⣀⣀⠀⢀⣀⣶⠀⢀⣀⠀⢀⣀⠀⠀⠀⣴⡶⣶⠄⢀⡀⣀⠀⠀⢀⣶⣶⣶⣶⢰⣶⣀⠀⢠⡶⠀⢀⡀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⣾⣧⡾⠏⣾⠋⣿⠀⣿⠋⣿⢲⡟⢹⡟⢰⣿⣽⡇⢸⡟⠁⠀⢸⡿⢀⣿⠀⣿⠋⣿⠃⠀⠀⢁⣿⠁⠁⣸⡏⣹⠇⣸⡇⠸⣯⣙⠠⠟
⠀⠀⠀⠀⠀⠰⠿⠂⠀⠀⠿⠾⠋⠰⠿⠀⠿⠸⠷⠾⠇⠘⠷⠞⠀⠿⠇⠀⠀⠘⠷⠾⠋⠰⠿⠐⠿⠀⠀⠀⠼⠿⠀⠠⠿⠂⠿⠀⠿⠀⠷⠾⠟⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠿⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣆
⠀⠀⠀⣸⡟⠀⣀⡀⠀⠀⢀⣴⠟⠁⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣼⣏⣴⣾⠏⠀⠀⢀⡾⠃⠀⠀⠸⣇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠰⠿⠿⠋⢹⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣼⡇⣀⠀⣾⠃⠀⠀⠀⣤⠾⠷⠙⢷⣄⠀⠘⢷⣄⠐⠛⢿⣷⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣿⡿⢿⣷⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠻⠶⣦⣽⣿⣶⣾⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⠇⢸⣿⡇⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣷⠟⢻⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣦⣀⠀⠀⠀⠀⠀⢤⠤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⢿⡁⠀⠀⠀⣀⣀⣤⣶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢹⡟⠳⠶⠶⠾⠛⠋⠁⠀⠀⠘⠷⣶⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⠷⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠈⠉⠉⣹⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢀⣼⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣤⣤⣾⡃⠀⠀⢀⣿⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠉⠉⠻⣷⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣻⣶⣶⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠒⠚⠋⠉⠉⠉⠉⠛⠛⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠀⠉⢩⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠈⠁⠀⠀⠀⠉⠉⢛⣳⣒⠀⠀⠀⠀⠀⠀⠐⢶⡚⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠢⣄⠀⠉⠉⡿⠉⢉⡟⠀⣀⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣠⡤⣷⣤⠴⠶⢦⣄⠘⠀⠀⡼⠁⢠⡏⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢉⣿⡋⠀⠀⠀⠀⠸⡆⢀⡾⠁⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠛⠛⠲⠶⢤⣤⣄⣀⠀⠉⠛⠛⠋⠉⡟⠃⠀⠀⠀⠀⢀⣷⠏⠀⠀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠒⢿⣄⠀⠀⠀⢀⡼⠁⠀⠀⣰⠟⣉⣁⢹⡄
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠫⡓⠶⠶⠛⠁⠀⠀⠀⠁⠸⠽⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡁⠐⠒⠠⠤⠀⠄⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⠶⠶⠶⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⣚⡉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠟⡱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⣉⣙⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠀⢀⡾⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⢀⣉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⠟⠀⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡌⠉⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡼⠁⠀⠙⢦⡀⠒⠦⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣠⡞⠁⠀⠀⠀⢤⡀⠀⠀⠀⣸⣿⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⢀⣤⠤⠤⢶⡟⠳⣄⠀⠀⡴⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠿⣏⢀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⠓⢺⣿⡆⠀⠺⠞⠉⣾⢕⡾⠃⠈⠑⢦⣀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⢾⣿⠇⠀⠀⣀⡤⣿⠘⣇⣾⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠤⠖⠋⠁⠀⣿⠀⡇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣶⠖⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠴⠶⣶⣶⠶⣶⣒⣂⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⡦⠬⣀⣸⣦⣀⣀⣀⣀⣀⣀⣤⣤⣴⡖⣻⢛⡙⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠷⠾⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡀⠤⠤⢠⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡠⠀⢄⠸⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡅⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⠀⠁⠸⠀⠀⠀⢠⠄⠀⠢⡀⠀⠀⢧⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⡶⠒⢲⠶⡶⠒⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠠⠁⠀⠀⠸⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠛⠛⠛⠒⠛⠚⠟⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠴⠴⣿⡩⢽⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽⠀⠀⠀⠠⠤⠄⠀⢰⠀⠀⠀⢈⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣏⢼⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠒⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⠀⢀⡀⣀⣀⠠⠔⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⡏⠈⠈⠉⡉⡉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣩⣩⡩⡍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣓⣓⣂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡼⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣰⡻⢤⣀⣀⣀⢀⣀⣀⣀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢹⡇
⠀⠀⢀⣀⣀⣀⣤⣤⣼⣧⣿⣯⣼⣯⣥⣴⣮⣽⣯⣽⣫⣋⣝⣩⣉⣙⣽⣫⣪⣗⣡⣠⣟⣕⣽⣟⢙⢁⡽⡭⣫⠿⡿⣩⠞⠕⣽⠁
⠀⠀⠸⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠂
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠚⠛⠛⠓⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠚⠚⠓⠚⠛⠛⠒⠛⠛⠓⠛⠛⠓⠚⠒⠒⠒⠲⠶⠶⠴⠶⠴⠶⠋

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠨⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣠⣀⣤
⠀⠀⠀⠲⠀⠁⠀⠀⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢣⠐⠠⠤⠄⠠⠰⡿⡿⢿⠿⡿⠿⠿⢿⡟⠀⠀⠀⠸⠳⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡂⠀⠀⠀⠁⠁⠉⡉⢉⢉⠩⠉⠉⢹⣇⠀⠀⠀⠀⠁⠁⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠠⠄⠤⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠀⠘⡧⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢖⠑⠛⣿⠀⠀⠀⣸⠀⠀⠀⣀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣐⣾⠀⠀⠀⢸⡆⠀⠚⠺⢯⡿
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠠⠐⠃⠀⠉⠀⠀⠀⠀⠀⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⢚⡇
⠀⠀⠀⠀⠀⠒⠀⠁

⠀⠀⠀⢀⡶⢛⣻⡟⠶⣷⢦⡀
⠀⠀⠀⣿⢀⣾⣿⣿⣧⣼⣷⢿⣦⡀
⠀⠀⠀⣿⢻⣿⣩⡿⣿⡿⣿⣾⡏⣷
⠀⠀⢰⡟⢿⡿⢷⣾⣿⣥⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠈⣷⠘⢿⣛⣒⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠉⠉⢉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠈⠳⠶⠼⠿⣿⠋⠉⠉⠀⠀⠀⢀⡴⠟⠉⠀⣀⣤⣶⡿⠿⠛⣁⣬⢓⣴⡗⣀⠈⡁⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⣴⠏⠀⣀⣶⣾⢟⠩⣀⠒⣢⠬⢐⣴⣿⢟⣐⠆⣼⣯⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠄⣶⡿⡋⢊⠀⠵⠔⣂⣥⣾⢟⡫⡀⠇⢆⣾⡟⡹⣿⣧⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⢪⣾⠏⡐⣎⣠⣴⣶⠿⢟⡋⠁⢈⠢⢈⣴⠟⢹⣦⡒⢊⢻⣧⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⢢⡿⢃⣵⡾⢟⣋⠁⠒⠔⣀⣁⣠⣴⠾⠛⠁⠀⠀⠙⢿⣮⠀⢻⣿⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠲⠋⣡⣿⢋⡰⣌⣥⡶⠿⠛⠛⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠹⣷⡠⢻⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠋⣰⡿⠅⣣⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⠀⠀⠹⣷⡘⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⢰⡿⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣱⠟⠀⠀⡦⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⡆⠀⠀⠘⣿⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢸⡏⠀⠀⠀⢿⣀⣠⡿⠀⠀⢀⣴⠆⠀⠀⠀⠹⣄⣀⣼⠃⠀⠀⠀⣿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⡇⠀⠀⠀⠀⠉⠉⢀⡀⠀⠿⢧⡀⠀⠀⠀⣀⡈⠉⠁⠀⠀⠀⠀⣿⣀⣴⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⢀⣀⡀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⣿⣿⣶⣌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠐⡇⠀⠀⢠⡾⠋⠉⠙⢳⡄⠀⠀⢸⠇⠀⠀⠀⣰⣟⠰⣀⠽⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡼⣟⡻⠿⠾⣆⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠀⠈⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠟⠛⠛⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⣛⠛⠛⠁⠀⠀⠉⠙⢋⡟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⡿⠀⠔⠁⠙⠷⣦⡀⢀⣀⣴⡾⠋⠁⠒⡀⢹⡆⠀⠀⠀⢀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⠆⠀⠉⢷⡀⠀⢸⡇⢘⣀⠀⠀⠀⣿⠉⠉⠉⣿⡇⠀⠀⠈⣕⠀⣷⠀⠀⣰⠏⠀⠀⠲⢾⣟⡀
⠀⠀⠀⠀⠀⣠⠞⠋⠁⣀⣤⣤⠶⠿⣧⠀⢸⡇⣸⡟⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠈⣿⡀⢻⠀⢠⣿⠷⢦⣤⣄⡀⠈⠛⢦⡀
⠀⠀⠀⠀⢸⣇⣤⠶⠛⠉⠀⠸⣆⠱⠹⣧⢸⡇⣿⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⢀⣾⢓⠀⣼⠁⠈⠙⠛⠶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢃⢻⣿⡇⢻⣷⠀⠀⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⣾⢇⠂⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⡆⢻⣇⢸⣿⠀⠀⠀⣿⠁⠀⠀⣼⡇⠀⠀⠀⣿⢃⣾⠃⡌⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠈⢿⢾⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⢠⣸⣿⣿⠏⡳⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠉⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡰⠀⡴⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⡇⡗⠒⠂⠤⠀⠀⠀⠐⢈⣹⢪⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠇⡿⡿⢻⣿⠦⠀⠀⠀⠘⠻⠟⠐⢄⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠊⠉⠉⠑⢄⠈⠇⠰⡀⠀⠀⠀⠐⢾⣿⠗⠀⢀⠊⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⡜⠀⠀⠀⠀⠀⠘⡄⡆⠀⡇⠀⠐⢄⡀⠀⠀⡠⠀⠘⡄⠀⠀⠀⠀⢱⡦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡇⠀⣠⡇⠀⠀⠀⠀⢉⠏⠀⠀⠡⠈⠲⢄⠀⠀⢸⡟⣿⠤⣶⡄
⠀⠀⠀⡜⠁⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣇⡴⣹⠀⠀⠀⠀⠀⡎⠀⠀⠐⠢⠤⢄⡀⢱⠢⣼⡇⣿⠀⢹⡇
⠀⠀⠸⠀⠀⠀⠀⠀⡴⠁⠀⣀⠀⠀⠀⠀⠀⠀⡟⠐⠁⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠈⠙⠚⢿⣼⣿⢦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰⠇
⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡠⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⢠⡤⠮⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡆⠈⠛⢿⠟⠡⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣇⠀⢹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢺⣿⡆⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡆⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢿⣿⣿⣿⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⣀⣀⣀⣀⣀⣀⣙⣛⣁⣀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠤⠠⠄⠤⠤⠤⠤⠤⠤⠤⠀⠀⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠖⠂⠀⠀⠀⠀⠐⠒⠂⠒⠀⠀⠀⢇⠆⡀⢠⠀⠀⠀⠀⠀⠀⢡⠀⢀⡀⠀⠈⠀⠀⡡⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⢸⠃⠀⠀⠀⢤⣬⣀⡀⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢄⠀⢾⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠑⠐⠂⡤⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣠⣄⠀⠈⠳⣅⠈⠀⠀⠀⠀⠀⠀⠀⠰⠃⠐⡋⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠁⠠⠀⠨⡒⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠈⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⣉⡁⢋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢁⠀⠀⠀⠀⡀
⠀⠀⠀⡬⠹⡄⠀⠀⠀⠀⠀⠀⠂⢀⡡⠀⠀⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣃⡈⠂⠐⠓⠠⠀⠂⢠⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢱⡀⠀⡠⠂⠐
⠀⠀⠈⠀⠀⠈⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣶⣾⢋⣈⡯⠥⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠁⢀⣀⠤⠟⠛⠫⢀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠋⠀⢀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡄⠚⠁⠀⠀⠀⠀⠐⠑⠀⡂⠀⠀⠀⡅⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠄⠁⠀⢤⡀⢘⣁⣀⠈⡅⠂⠐⠋⠀⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠐⠒⡺⠋⠀⠀⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⣴⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠍⠿⠿
⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠛⢻⡇⠀⠀⠀⠀⠀⣀⣀⡀⣠⣤⣀
⠀⠀⠀⢠⣾⣿⣿⣿⣦⠤⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⢺⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠶⠿⠗⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣀⣀⣴⣶⣾⣿⡆
⠀⠀⠀⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠟⠈⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡏⠚⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⡻⠿⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⢿⣿⡟⠁⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⢀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠂⠑⠛⠘⠀⠃⠀⠃⠂⠃⠋⠞⠁⠐⠃⠀⠋⠘⠁⠐⠑⠃⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⡀⠀⠀⣢⠠⢄⡀⠀⠀⠠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠠⢠⣀⠀⢀⣤⢠⡄⠆⢄⡀⡨⠃⢠⡰⠌⠖⠆⢀⠀⢀⡃⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⡀⠀⠈⠐⠀⡠⢊⣭⡭⣒⠏⣤⣤⢓⣬⡆⣷⡂⣦⠄⡠⣸⠤⠼⡂⠈⠈⢊⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣀⣦⣯⡷⡱⣧⠾⣭⣓⡻⠎⣓⣛⡵⣧⢞⡕⣹⢫⡞⣅⢢⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠙⠛⠛⠙⠓⠚⠛⠲⠻⠻⠙⠛⠚⠛⠳⠿⣯⣶⠿⠿⠟⠓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠂⢀⠁⠀⢄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠐
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⡆⠀⠈⠁⠀⠀⢀⠎⠀⠀⠀⡀⠊⠁
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡅⠦⠠⡀⢠⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠒⠶⠒⢊⡁⠴⡀⠀⠀⠙⠂⠦⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠀⣤⠄⠐⠠⠊⠀⠸⠴⠦⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⠐⠂⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⡴⠚⠋⠀⠀⠀⠀⠛⠠⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠖⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⡀⢀⡀⠀⠀⠀⠀⢀⠀⠐⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⡿⠿⠿⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⡙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶⣤⣤⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠁⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡉⢻⣿⣿⠏⠀⠀⠀⢀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣷⡄⢹⡿⠀⠀⠀⠀⠈⣀⡀⠉⠛⠋⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠈⠁⠀⠀⠀⢠⠟⠻⠟⠳⣦⣦⡄⠀⠀⣰⡿⠉⠉⠉⠛⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⣿⠻⠿⠟⢳⡄⠀⣨⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠂⢀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣇⣀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⠞⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⡴⠶⠶⠶⠶⢦⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣤⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣨⠛⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⠑⠶⠶⠶⠶⢤⣶⡿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⣀⡠⠴⠞⠋⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠁⠀⢻⡀⠀⠀⠀⠀⢉⣷⣶⣶⣿⡇⠉⠁⠀⠀⠀⠀⢀⡿⠉⢻⣄⡀
⠀⠀⠀⢀⣤⠖⠀⠁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠁⠉⠻⠶⣤⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢠⡟⠛⢿⣿⣿⣿⠟⢿⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠃⠀⣸⣿⣿⣿⠀⠀⠠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣚⡛⠙⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⢆⠊⣉⠷⠶⣟⢊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡈⠉⠛⠶⢄⠈⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⠠⠂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠒⠦⢤⣀⠀⠀⠀⢠⠖⠁⠐⠦⣠⠖⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⠄⠀⠀⠀⠀⠈⠙⠲⢴⢻⠀⠀⠀⠀⠈⠀⠀⠰⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠴⢾⣞⡁⢙⠤⠤⠤⣀⣀⠀⢀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⢔⢥⠀⠀⣡⣠⣌⠚⢤⢀⡂⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⡘⢢⠀⠀⠀⠘⢆
⠀⠀⢴⡞⠋⠀⠀⣌⠈⠙⣦⠀⠷⡕⢄⠀⠇⠀⠀⢀⡤⠈⠓⠄⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠙⢦⠀⠀⠈⠢⡀
⠀⠀⠀⣿⢶⣄⠀⠈⠁⠂⠁⠙⢄⠈⢢⣑⣄⡠⢚⡡⠃⢀⡤⠞⢋⡜⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠑⢄⠀⠀⠈⢆
⠀⠀⠀⢸⠀⠈⠳⢤⡀⠠⢠⠀⠈⠢⣀⠭⠐⠉⣁⡴⠞⠛⠋⠉⢉⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡱⠜⢆
⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡈⠃⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⣠⠖⠁⠀⠀⠈⠙⠂⠤⠄⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡦⡀⠑⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣦⣤⠞⢻⠁⠀⠀⢈⣦⡴⠊⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣈⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀⡏⠀⠀⠀⢀⢀⡠⣤⠶⠭⠤⡤⣤⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠭⠹⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠈⠘⡅⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⢰⠻⡀⠀⠀⠀⠸⡀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣴⡞⢻⠛⡟⠉⡟⢙⡟⣦
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠹⡄⠀⠀⠀⢣⣿⠀⠀⠀⠀⣿⠀⠈⣧⠀⠀⠀⠀⠀⢿⣿⡿⡟⠻⡞⢳⡞⣿⡏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⣤⠞⠀⠀⠹⡄⠀⠀⠈⣇⠀⠀⠀⠀⠻⠀⠀⣿⠀⠀⠀⠀⠀⠸⡟⠓⠷⠶⠷⠚⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣿⠛⠉⠀⣀⣀⣸⠀⠀⠀⠀⠹⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠹⣄⣀⠤⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠿⠿⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠿⠛⠉⠀⠀⠀⠀⣿⢸⡀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⢦⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡴⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡞⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠤⠴⠄⠀⠀⢀⣀⣤⣤⣤⣤⣴⡛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡜⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠋⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠁⣀⡴⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⠀⠀⠘⠃⠀⢀⣀⣠⣴⣾⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠶⢦⠠⠀⣤⠀⣤⣤⢤⢤⢤⣤⡄⣠⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠆
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢨⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⡀⢀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⡞⠁⠀⢹
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠘⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⢠⠀⣸⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠀⠐⠂⠀⠀⢀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⠂⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠟⠃
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠵⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⡂⠀⠀⠭⢖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠅⠀⠀⡭⢵⡂⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣅⠀⠀⣈⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⣒⡫⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣖⣚⠁⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠍⠀⠀⠭⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⠀⠀⠉⠑⠂⠀⠄
⠀⠀⠀⠂⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⣭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⣘⢐⡂⡂⠐
⡀⠐⠀⡁⠐⠀⠐⠀⡁⠀⠀⠀⢰⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⠡⠀⠄⢈⣉⡁⡀⣂⢀⣀⢀⣀⣠⠨⠅⠄⠆⣐⢀⡀⡆⢀⣀⡀⣀⣄⢀⣀⣀
⠀⢀⠀⠀⠈⡁⢀⠀⠄⠄⡂⠭⠨⠐⠀⠠⠀⠀⠀⠀⠀⠀⠨⠀⠄⣒⢀⠀⡃⢒⠈⠈⠁⠛⠙⠁⠈⢤⣺⡥⡬⠅⠦⢽⠸⠇⠯⢿⣙⡆⣿⢿⣧⣼⣭
⠀⠠⠀⠂⠀⡀⢀⠀⠀⡀⠂⠅⠨⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⢐⠀⡂⠸⠨⠀⠃⠒⠀⠀⠀⠀⠙⠚⠃⠀⠉⠉⠠⠄⡭⣸⢻⡇⣿⣧⣌⣹⡿
⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠅⠁⠈⠨⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠅⠁⠨⠀⠅⠠⠘⠈⡇⠤⠂⠀⠐⠀⠐⠂⠀⡄⠛⢽⠠⡄⠛⠚⠻⠃⠛⠓⠞⠛⠓
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄⠄⣀⢘⠈⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠁⢠⠀⠁⠈⢀⠀⡀⠄⠀⠀⠀⠀⠀⠐⠂⡀⠂⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠂⠀⠀⠁⡁⠂⣀⠐⢀⠈⠠⠀⠀⠀⠀⠀⠀⠐⠂⡀⠂⠀⠀⡀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⢀⠐⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣲⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⢦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⣴⣿⣿⣿⣿⡏⠙⠿⡇⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣅⣀⣰⣿⣿⣿⡟⠛⠁⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⣠⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠲⠋⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡉⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⠎⠉⠉⠉⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⡰⠂⠀⠀⣠⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⣼⣿⡿⠋⠉⣻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⡟⠀⠀⣰⣿⡿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣆⠰⠛⠋
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠒⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡰⠠⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠘⡆⠀⠀⠀⠀⠲⡀⢸⡰⠃⡠⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠈⠉⠉⢳⠁⠞⠁⠔⢀⡤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⠔⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢲⣥⣤⠤⠖⠊⠀⠀⠀⠀⢈⢀⡀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢀⢀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠑⡄⠀⠀⠀⠀⠁⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠫⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢰⠁⠀⠀⣀⠄⠈⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠗⠊⠁⡀⠀⠊
⠀⠙⣍⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠈⠀⠀⠀⡀
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠐⢀⡖⠱⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠧⡀
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣼⠀⠀⡀⠀⠱⣴⣥⠀⠢⣀⣠⠴⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠰⢎⣉⠉⠑⡆⢿⠄⡞⠀⠀⢺⠝⠊⠀⠀⠈⣡⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠊⢡⠤⠂⠀⠃⠀⡜⠀⠀⠀⠁⠀⠀⠀⠀⠈⣓⡄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠒⠀⠠⣤⠄⠨⣭⠀⢀⣒⣀⣐⡀⣀⣉⣩⠤⠘⣆⣠⠀⠀⠀⠄⠐⠂⠈⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠚⠀⠀⢳⠀⠀⠀⠀⣻⠀⡎⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠚⢻⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠼⣤⣀⣀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠻⠭⠤⠬⠿⠯⠭⠭⠭⠿⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠹⣿⣉⣉⣉⣹⡿⠁⠀⠀⠀⠀⠘⢿⣍⣉⣉⣩⡿⠃⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣉⣀⣀⣉⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠉⠙⠛⠷⢿⠆⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣄⣀⣀⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⡠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣶⠆⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⣿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠛⠁⣀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡠⢀⠐⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠟⠂⠀⠰⠟⠶⠲⠾⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠠⡀⢄
⠀⠀⠀⠀⠀⠜⡠⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠣
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠
⠀⠀⠀⠄⡐⠀⠀⢀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢂
⠀⠀⠀⠐⠀⠀⠀⢸⡇⠀⣠⣟⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠘⣿⠀⠀⠀⠂⠀⡀
⠀⡀⠀⠁⠀⠀⠀⢸⡇⠀⢸⡿⢿⣿⣎⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠈⠿⠄⠙⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣤⣄⡀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⢠⡄⠐⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣀⠀⢈⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⡇⠀⠀⠀⣿⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣿⣷⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠇
⡇⠀⠀⠀⢈⡝⠂⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠻⣿⣿⠟⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⣶
⠇⠀⠀⠀⠈⠃⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠄⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⠼⠋
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠃⠀⡀⠀⠀⠀⢿⣦⣀⡘⠀⠀⠁⠂⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠠⠀⠀⢰⠃⠀⠈⢻⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⣿⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠃⠀⠈⠓⠀⠀⠛⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⡈
⠀⠀⠀⠀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰
⠀⠀⠀⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣼⡟⣴⠶⠶⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡔
⠀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⠀⢸⣿⠃⢀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢦⣄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠢⡀⠀⢼⡿⠀⢈⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⢈⡉⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⢀⣽⠀⠀⠀⠀⠀⠐⢁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⠀⠀⢠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠛⠐⠋⠁⠀⠀⡠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢂⡁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣥⣬⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣛⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⣛⣛⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠾⠭⠭⣭⣽⠛⠛⣉⣉⣥⣭⣉⣡⣴⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣯⣍⣡⣄⣀⠀⠀⢚⡔⢩⣿⣿⣿⣿⣹⣗⢜⢿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣷⣯⣐⠈⣟⠿⣿⣿⣶⣄⠀⡞⢀⣿⣿⣿⣿⣿⣿⣿⣦⣩⠉⢱⡛⠙⠋⢿⣿⡿⣛⢿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡿⡿⡇⣧⣌⣿⣶⡄⠈⠁⠀⡇⢼⣛⣿⣿⣿⣿⣿⣿⡿⣌⠒⣫⣶⣿⣿⣿⣿⢹⣿⣿⣿⡇
⠀⠀⠀⢸⠿⣿⣿⣿⣿⣷⣧⣿⣿⣽⣿⣿⣾⣷⠇⡼⢿⣿⣿⣿⣿⣿⡷⠆⢺⡾⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣶⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢽⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠴⣇⣹⣆⡿⣥⠿⠶⠁⠃⢲⠀⠀⠀⠀⠀⠀⣀⢀⡄⠈⠤⣶⠀⠄⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇
⠀⠀⠀⢠⠁⠀⠀⠀⠈⠙⢉⣥⠈⠒⠀⠀⠀⠸⡆⠀⠀⠀⢈⠀⠀⠀⠀⠀⢠⠇⠄⢐⠀⠈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠂⠀⠀⠀⣀⠦⠈⢿⡃⠀⠀⠀⢈⠁⠌⠀⠀⠄⠙⠀⢁⠁⠀⠀⡾⠀⠀⠀⠌⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢐⠘⠁
⠀⠀⠀⢠⠐⠀⢉⡀⢻⡄⠩⠄⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢈⣀⣀⠀⠀⣼⠁⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠐⠁⠏⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠊⠀⢀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠆⠀⠀⠠⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢠⠆
⠀⠀⢀⠀⠀⠠⡄⠀⡄⣄⠀⣤⡴⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⢔⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠀⠀⠀⢀⠐⠤⢠⠈⠀⣠⠋
⠀⠀⢸⠀⠀⠀⢳⠀⠄⣸⠀⠾⢀⡀⡿⢻⡆⠀⠀⠀⠀⠀⠬⠬⠿⠧⠳⠚⠒⠈⠉⠑⠚⠙⠃⠀⠀⠀⠀⠀⠀⠀⢄⡘⠆⠠⠀⢀⡴⠁
⠀⠀⠘⣧⠀⠀⠉⠀⠈⠁⠀⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠮⡤⠄⣠⠎
⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡴⠃
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⢤⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⣠⢄⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠈⠁⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠵⠄⠠⡾⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣄⠘⢂⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠠⡷⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡔⠂⣈⡊⢷⡀⠹⠦⠀⢈⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡷⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⠀⠈⠂⠠
⠀⠀⠀⠀⠀⠀⢠⡀⠸⡏⠁⠰⡷⠶⠠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠱⠦⢀⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⢁⡾⠁⣀⣒⣲⠆⠀⡱⠯⠀⠀⠐⠀⠈⠀⣈⡀
⠀⠀⠀⠀⠀⡄⠒⢤⠀⠀⠀⣀⣀⡤⠴⠒⣲⠟⠉⠉⢡⣿⣥⠄⠀⠈⠉⠉⠙⠒⠒⠠⠎⠀⢨⠆
⠀⠀⠀⠀⠀⠣⡀⠀⠓⠒⠉⠁⠀⠀⣠⣞⣡⣤⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠌⢢⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡜⠀⠀⢈⠔⢀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢧⠀⠀⢀⠠⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡀⠀⠈⠄⠀⠁⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢀⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠢⣀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠋⠈⠉⠓⠒⠒⠶⠶⠖⠒⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠤⠄⡀⠀⠀⡀⣀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦⣄⠀⠸⣿⣿⣿⣄⡀
⠀⣰⠏⠀⠀⠈⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠂⠀⠈⢻⣷⣄⠉⠿⣿⣿⣿⣆
⢠⡏⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⡀
⣾⠀⠠⢄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠈⠞⢑⠀⠸⡇⠙⠿⠟⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣲⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⡀
⣿⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⠀⠀⢀⠠⠄⠀⢠⡇⠀⠀⣠⣾⣿⣿⣿⡇
⢸⡆⠀⠓⠒⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⣸⠁⣰⣿⣿⣿⠿⣿⣿⣷
⠀⢻⡄⠀⠀⠀⠠⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣰⠇⠸⣿⣿⠋⠀⠀⠹⣿⣿⣆
⠀⠀⠹⣄⠀⠘⠓⠲⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠁⠈⠡⢀⡼⠃⠀⠀⢻⣿⣇⠀⠀⠀⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠨⠬⠑⠀⠀⠐⠪⠍⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⡀⠀⢀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣦⣤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⠉⠀⠀⠀⠈⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡀⢹⡀⠀⠀⠀⠀⠀⣠⠀⠙⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠉⠉⢻⣿⣎⣳⡈⠃⠀⡔⠋⠁⠀⢠⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢌⣙⣿⣿⣶⣬⣟⢠⡀⠀⠂⠆⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠿⣽⣯⣹⣾⠟⠁⠀⢰⣆⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⣰⠃⣽⠣⠀⠀⠀⠙⢻⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⣂⣤⡀⠀⠀⠁⠀⡇⠀⢀⣦⢒⣤⠋⠀⣯⡀⠀⠀⠁⠐⠀⠀⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣛⣁⣒⣈⢉⣛⡿⠷⠦⠀⠀⣈⣤⣾⣿⣿⣯⣤⢤⣾⣿⣦⣤⣤⣤⣴⣶⣼⣿⣶⣦⣤⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⢀⠁⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡇
⠀⠀⠀⠀⠀⠈⠀⡌⢞⠀⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⡴⠀⠀⠀⣀⢉⣈⣹⣉⣉⣹⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⡿⠧⠼⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠈⠲⠊⡉⠁⠈⠐⠦⢼⣄⣠⣤⣴⣾⠋⠀⠀⠀⠈⣨⡟⡉⣹⣿⣿⣻⣿⣩⣿⣿⣿⣩⣿⠉⠉
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠁⠀⠛⠒⡶⠤⠙⠛⡛⣿⡏⠀⠀⠀⣠⣾⣿⠂⠀⣿⣿⣷⣿⣷⡋⣽⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠈⠉⠰⠠⠤⠤⣤⣄⣀⠀⠀⣀⢴⣿⣿⡿⠁⠀⣠⣼⠿⠿⠟⡡⣾⣿⡭⣿⣟⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⣿⣿⣿⠋⠁⠀⠸⢿⣧⣴⣴⣾⣧⣿⣭⣠⡬⠻⣿⠏⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⣧⠿⡝⠿⢶⠤⣤⣌⣉⠿⠿⢿⣿⣿⡷⠀⠉⠁⠰⣿⣄⣀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠈⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣲⡖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⣿⣂⠘⠁⡰⠷⣤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠟⠊⠀⠀⠀⠀⠀⠀⠻⣐⠒⢐⣗⠠⠿⠅⢰⡇⠀⠀⠀⠀⠀⠀⡔⠉⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣡⣒⣁⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠅⠒⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠥⢬⠁⠀⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠈⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⣀⠀⢀⠀⠀⠀⠀⣀⠦⢒⣂⣠⣤⣤⣀⣒⠠⢇⡀⠀⠀⠀⠀⠀⠀⡯⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⠖⠋⠉⠀⠀⢰⠀⠀⠈⠉⠓⢮⡑⢄⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⠞⠁⠀⠙⠀⠀⠀⠘⠀⠀⠀⠘⠁⠀⠈⢢⡱⡀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠐⠂⡜⣴⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠄⢱⡜⡄⠒⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⢰⠃⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⢠⠀⠀⠀⠀⠀⢀⠔⡣⠎⣴⣉⠙⡦⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢸⠀⣀⡀⠀⠀⠀⠀⠈⠓⢤⠖⠁⠀⠀⠀⠀⠀⣀⡀⢸⠌⠀⠀⠀⠀⠀⡒⢰⠥⣰⡀⢸⠉⠓⢲
⠀⠀⠀⣸⣷⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠦⢼⢀⡇⢀⠋⢐⠗⡼
⠀⠀⢠⣉⣀⣀⣶⣶⣶⣶⣎⣁⣉⡆⠀⠀⠀⡈⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡎⡈⠀⠀⠀⠀⠀⠈⠲⣤⣁⣍⡱⣓⠔⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠒⠉⠑⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡐⠉⠑⠂⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡳⣄⠀⠖⠀⠀⠀⢰⠀⠀⠀⠐⠆⢀⡴⡫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠢⢝⠲⠤⣤⣀⣈⣀⣀⡤⠴⣚⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠓⡽⡂⠀⠀⠀⠀⢀⠋⠁⠒⠂⠀⠀⠒⠂⠉⢃⠀⠀⠀⠀⠀⠀⠀⡠⡺⡉⢱⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣎⠵⣳⣚⠳⡲⠽⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠅⠸⣗⠀⠇⢣⣬⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣄⠀⡈⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⣖⠒⢒⣶⣒⠀⠀⠀⠀⠀⠀⠀⢀⢖⡭⠚⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠣⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡆⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠃⠐⣶⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠶⠶⠴⠭⠯⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠉⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢈⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠥⠀⠀⠠⠍⠀⠈⠿⠛⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠄⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠀⠀⠀⠀⠀⣀⡴⠀⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠋⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠻⠿⠁⠀⠭⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⠍⠀⠈⠡⠀⠀⠹⠟⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠤⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⢀⣠⠆⠘⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⠿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡟⢦⡈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢶⣿⣿⣯⣿⣿⣿⣿⣿⣜⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣼⣿⣿⣦⣿⡿⢿⡟⢿⣿⣿⣿⣿⠿⣌⠙⠋⡿⠘⣿⣿⣿⢹⡿⢿⠏⠀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠻⣿⣿⣏⡉⢃⠀⢿⠀⠉⠙⠟⠙⠂⠙⠃⠘⠀⠀⠘⠿⠇⠸⠃⠀⣿⣿⡿⣿⣀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣾⣿⣤⡤⢄⣀⢙⣧⣄⣹⣶⣾⣷⠶⣶⡶⣴⣷⣶⣶⣾⣶⣤⣆⣴⠄⢿⡏⠀⠘⢿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⣥⣥⣤⢀⣉⣾⣿⣿⣿⠃⠈⠿⡷⠿⣷⣹⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠿⣿⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣩⢷⡆⠰⠂⠈⠛⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠙⠿⠻⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⢷⡀
⠀⠀⠀⣿⣿⣿⣿⡟⢛⣿⣩⡾⠟⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣷
⠀⠀⠀⣿⣿⣿⣿⣷⡿⢿⡁⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣠⠼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣫⡅⠘⣉⡴⠋⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠸⣿⡿⣿⣿⣿⣷⣿⣿⣿⠟⠉⣰⢞⣵⠂⠀⠀⠀⡰⠚⠀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣾⡿⠛⢻⡇
⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⡿⣷⣾⣽⠟⠁⠀⠀⠀⠀⢀⣴⡾⠿⣿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣉⡀⠀⠻⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢛⣿⣿⣤⣤⣀⠀⠀⠀⣴⠟⣻⠞⠻⣿⣿⡇⢰⡆
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⣠⣶⣿⣿⣿⡍⠛⠻⠉⢻⡆⠀⠀⠀⠈⣁⠀⠀⠉⠁⠀⠘⡇
⠀⠀⢠⡶⠊⠉⠉⠛⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣿⣁⡀⠹⠛⠃⠀⠀⢀⠂⠈⠀⠀⠀⠀⡀⠉⠓⠶⠀⠀⠀⠀⢹
⠀⢠⠈⣠⣶⣶⣤⣀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣀⠀⠀⠄⠁⠀⠀⠀⠀⢠⣄⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠸
⠀⠸⠇⠿⢿⡀⠉⠛⢿⡀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠐⣦⠈⠇⠀⢼⣿⡇⠀⠀⠘⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠱⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠆⣴⣀⠀⠀⢿⣦⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠁⠀⠀⠀⠀⣨⣦⡄⠀⠀⣸
⠀⠀⠀⠀⠈⢧⠛⢷⣄⠀⠈⠁⠰⣀⠰⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⢿⣤⣴⠶⠶⢶⣤⣀⣴⠋⠀⠹⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠓⠤⣍⡀⠀⠀⠀⢻⡄⠰⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣿⣿⣿⡄⠀⠀⠀⠈⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠉⠉⢠⣀⣠⡤⠀⠀⢤⣤⣄⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⠋⠀⠀⠀⠀⠉⠛⠦⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣰⣷⣤⣤⣤⡄⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⠃⠘⢿⡉⠙⢿⣿⣿⣷⣦⣤⣄⠀⠀⣠⡾⠋⠀⠀⠰⠶⠶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠉⡆⠹
⠀⠀⠀⣀⣀⣀⣠⣾⣿⠟⠁⠺⣿⣿⣿⡆⠤⠀⠈⠳⣤⡙⠛⢿⣿⣿⣿⣤⣰⣿⠃⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣷⣀
⢀⡴⠚⠛⠛⣿⣿⠛⠁⠀⠀⠀⠙⠻⡟⢳⠄⠀⠀⠀⠀⠙⢶⣄⠈⠻⣿⣿⣿⡇⢀⣶⠆⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠁⠘⠛⠿⠿⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡦⠀⠀⠀⠀⠀⠀⠈⠻⠦⣈⣿⣿⣧⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢄⡀⠀⠀⠀⣴⣿⣿⣿⡟⢿⠇⠀⣠⠁⠀⠀⠀⠀⠀⠉⠛⠛⠳⢤⣤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡄⠀⠘⣷⣀⣼⣿⣿⣿⣿⡏⠀⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠀⠀⠀⠀⠙⢶⣄⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⢧⣤⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠿⢶⠶⠶⠛⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡿⠧⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠁⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢣⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣿⢱⣿⣀⣴⠏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠿⣿⣿⠛⣿⠇⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠚⠋⢿⣶⡇⣿⣿⣸⡿⣴⠋⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣼⠀⣿⡅⣿⢻⡿⠸⡿⠀⡎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠘⣤⠿⣿⡇⠘⠷⠀⠁⠄⢡⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⢈⣿⣿⡿⠛⠁⣠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⡾⠀⣸⣿⣿⣦⣤⣾⣿⡇⠀⢻⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠟⠁⣠⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠻⣿⡿⠀⠘⠛⢉⣠⣼⣿⣿⣿⣧⠀⠈⠁⣠⣾⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⡟⢀⣼⣿⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣀⣉⣉⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⠀⠉⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆⠀⠀⠉⠛⠛⠋⠁⠀⠈⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠓⠒⠂⠠⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⣠⣶⣿⣿⣧⣽⣿⢿⣦⣠⣄⡀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⣶⢤⣤⣾⣿⣿⡯⠛⠁⠀⠀⠈⢉⣙⡛⣿⡌⠉⠤⠤⠤⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡔⠁⠀⣼⣿⣿⣿⡿⡏⠀⠀⠀⠢⣄⠀⠀⠀⠻⣿⢿⣇⣀⠤⠔⠂
⠀⠀⢀⠤⠖⠲⣶⣦⣶⣉⠀⠀⠈⠳⣼⣿⣿⣿⣿⢿⠛⠃⠀⠀⠙⠄⠈⠀⠀⠀⢀⡿⠛⠉
⠀⠀⠀⣴⣂⡤⠈⠙⠿⣿⡷⣤⡀⠀⢸⡙⠿⠟⠛⢫⠀⠀⠀⠀⠀⠀⠐⣅⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠙⢿⣷⣶⡀⠉⠻⣿⠿⣦⡌⡇⠀⠄⠀⠘⢆⠀⠀⢿⡦⠘⢀⣈⠐⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠠⣤⣀⡀⠉⠻⣿⣮⡀⠈⠛⢾⣿⣅⠀⠇⠀⠀⣈⣧⡄⠠⠴⠊⠵⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠈⠻⢷⣧⢀⠀⠉⠻⣾⡤⠀⠉⠙⠳⢄⡐⠿⠟⠉⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠷⣄⣀⠀⠀⠉⠑⠲⣤⣤⡀⠉⠒⠦⣴⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠤⣂⡀⠀⠀⠹⣿⣾⣆⠤⠀⠻⣦⣀⠉⠓⠦⣄⠀⠀⠀⢀⣴⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⠤⠀⠈⠻⢿⣦⠐⠂⠘⢿⣦⡀⠀⠀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢬⣿⡛⠳⢿⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠛⠀⠈⠁⠀⠀⠀⠀⠻⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⣀⠤⠔⠒⠒⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⣴⡏⠤⠄⠒⠂⠀⠂⠤⠄⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⢁⣤⠴⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠙⠶⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣀⠈⠳⡀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣤⣼⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⣿⣿⣄⡄⢳⠀⠀⠀⠀⢰⡇⢰⣾⣿⣇⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢻⣄⡀
⠀⠀⣼⠁⢸⡇⠀⠀⠘⠉⠉⠛⠻⢤⡀⠀⠀⠻⣿⣿⡿⠃⣸⠀⠀⠀⠀⢸⡇⠹⣿⣿⣿⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣺⠉⣷
⠀⠀⢹⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢄⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠻⣄⣀⣉⣀⣠⣤⠖⠋⠉⠉⠛⠒⠀⠀⠀⣿⢀⣿
⠀⠀⠀⠙⠶⣿⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣄⣰⠟⠉⠱⡄⠀⠀⠉⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣴⣟⣻⣄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⣼⠐⢻⣶⣈⢉⢩⡏⢹⡆⢀⣠⡤⠖⢳⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣷⣶⣿⣼⣼⣼⣿⣾⣼⣗⣺⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢀⣤⠶⠚⠋⣉⣀⠤⠤⠤⠤⠤⢄⣈⡉⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠾⠋⠁⠀⠊⠉⠀⠀⠀⠂⠀⠀⠀⢀⠆⠈⠉⠒⠄⠈⠛⢦⡀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡀⠀⠀⠀⠀⠲⣄⠙⢶⡾⢧⣄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠚⠈⢢⡈⠻⣤⠿⠻⣷⣿⡿⣆
⠀⠀⠀⠀⣠⡾⢋⣾⠁⡰⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⣆⠀⢹⣿⢿⣿⡄
⠀⠀⣠⠞⠁⠀⣼⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡱⠀⢿⣄⣠⡿⠈⠉⠙⢦⡀
⢀⡾⠃⣴⠛⠛⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣿⠷⢶⣄⠀⠀⠀⠻
⣾⠁⠘⣃⣀⣴⡋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⢹
⠹⢦⣀⢠⡟⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣴⠟⠛⣿⠋⠀⠀⣠⡴
⠀⠀⠉⠛⢻⣤⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣀⣀⣼⠷⠞⠛⠉
⠀⠀⠀⠀⢿⡀⠀⣿⠰⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼⠋⠙⣧
⠀⠀⠀⠀⠈⠛⠛⢿⡀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠠⢄⢠⠃⣰⠿⠦⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡰⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠊⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠑⠢⣄⣀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⡤⠒⠁⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣄⣀⡙⠀⠒⠂⠶⠆⠳⠀⠚⢉⣀⣤⣶⠛⠁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠉⢻⡟⠓⠶⠶⠶⠶⠛⠛⠋⠉⠀⣿⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⠳⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢅⠠⠟⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠲⠂⠀⠀⠘⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⡄⠀⣤⠀⠡⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⡀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠴⣒⡶⠑⢄⠀⠀⠁⢀⣀⠠⢞⢷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠀⠀⠀⠹⣀⠀⠀⢙⠋⠉⠀⢀⣴⣷⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠹⣷⡀⠀⠀⠈⠉⠐⢸⠀⠀⢸⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⡟⢷⡄⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⡇⠈⢻⣆⠀⠀⠀⢸⠀⠀⠀⠘⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠲⠤⣀⠴⢛⠳⢤⡀⢸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠸⡀⠀⠀⠀⠀⢀⢐⠢⠭⣾⡇⠘⡄⠀⠀⠀⡸
⠀⠀⠀⡎⣉⠭⠛⠉⠉⠉⠀⠀⠉⠉⠙⠻⣒⠒⠒⠒⠚⠪⠽⠿⠋⠻⠉⠉⠙⠒⢞⡓⠒⠐⠒⠒⠀⠀⠂⠀⠀⠀⠲⢄⡀
⠀⠀⢸⠈⠁⠒⠢⠤⠐⠒⠈⠉⠉⠀⠒⠒⠢⠷⠤⠤⠐⠒⠒⠒⠂⠀⠉⠉⠉⠉⠒⠚⠓⠀⠀⠀⣀⡠⢤⡢⠀⠀⠀⠀⠙⠢⣀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⠭⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣢⣄
⠀⠫⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠶⠶⣦⣤⡶⠿⠿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡖⠁⢀⣼⣿⣧⣀
⠀⠀⠀⠀⠀⠀⢀⡛⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣀⣴⠿⠛⠁⠀⠉⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠛⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠘⣧
⠀⠀⠀⢸⡏⢠⡾⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⢠⣤⣀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠘⠞⠉⢹⡆
⠀⠀⠀⢸⡇⣶⡀⠀⢰⡾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣀⣸⣷⠀⢀⣼⠃
⠀⠀⠀⢸⡇⠈⠛⠾⠛⠁⠀⠀⢀⣴⠶⠺⠷⢶⡶⠀⠀⠀⠀⠉⢰⠟⠿⣇⠹⣆⠺⣇
⠀⠀⠀⠈⣿⠀⠀⣀⣠⡄⠀⠀⠻⠁⠀⠀⣠⣤⣄⡀⠀⣀⠀⣷⠘⠃⠘⠃⠀⠘⣧⠙⣦⡀
⠀⠀⠀⠀⢿⡄⠀⠙⢿⡿⠃⠀⠀⠀⢴⡾⢉⣹⡟⠃⠀⠈⠁⠘⢷⣄⡀⠀⠀⠀⠘⣧⠈⣷
⠀⠀⠀⠀⠸⣧⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⣿⠀⣿⡀
⠀⠀⠀⠀⠀⣿⠀⣹⠟⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⣿⠀⢻⣆
⠀⠀⠀⠀⠀⢻⡆⢹⣆⢺⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣷⠀⠀⣿⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢸⣧⠀⠹⢦⣍⣁⣴⣄⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠸⢿⣿⣿⡿⠀⠀⢿⡀⠀⠀⠀⠹⣷
⠀⠀⣠⠾⠿⠿⣿⠀⠀⠀⠉⠙⠉⠹⣦⡀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿
⠀⣼⡏⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⣿⡶⣦⣄⣼⠀⣿⡀⠀⠀⠀⠀⣀⣀⣤⣾⣏⡄⠀⠀⠀⢀⣿
⠀⣿⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠉⢹⡀⢸⡟⠛⠛⢛⡿⠛⢋⣉⡠⠮⠀⠀⠀⠀⣼⠋
⠀⠹⣧⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇⢸⣇⠀⣴⠟⢁⣼⣋⣥⡄⠀⠀⢀⣤⣾⠃
⠀⠀⠙⠳⣦⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⣤⠀⣿⡀⠸⣇⠀⠻⠟⠃⣴⡿⠛⢉⣠⡶⠶⣤⣬⣭⡿⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣄⠀⣴⣦⣤⡟⢀⣿⡇⠀⢻⡄⠀⠀⠀⠀⠀⠰⠟⢉⡴⠶⢦⣿⠶⠶⠶⠶⠞⠛⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣏⣀⣽⠟⠛⢻⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⢰⡟⢀⡴⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡼⠟⠛⢿⡄⠀⠘⢿⣀⡀⠘⢇⠀⠀⠀⠀⠀⠀⡇⠈⠀⢸⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣀⣤⣤⣴⣶⣿⣿⣏⠁⠀⠀⠀⠘⣷⠀⠀⠀⠉⠛⠛⠿⡆⠀⠀⠀⠀⠀⢰⣄⣴⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣴⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⠿⢻⡿⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣴⠋⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠛⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉
⢸⣿⣿⣿⣿⡿⠿⠿⠛⠿⠿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢸⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⢀⣾⣿
⠀⠙⠻⣧⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠠⣶⣤⡶⣶⣶⡶⠶⠛⠁⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠀⣿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣀⠴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠶⠟⠛⠛⠛
⠀⠀⠀⣿⡀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣀⣠⣤⡤⢶⣤⠶⠶⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠋⠉⠀⠘⢧⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠾⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⣴⣾⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠖⣶⡟⠁⠀⠀⠀⠀⣠⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣀⣴⡞⣋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣀⣴⣾⡛⣿⠛⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠘⠿⠁⢾⣅⣷⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣴⡟⠿⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠈⠛⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠁⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣁⠀⠀⠀⡀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣾⣿⣿⣶⣄⠀⠀⣀⣀⣀⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠒⠚⠉⠀⠙⠋⠻⢿⣿⠿⠿⠿⣿⣿⣿⣿⣿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⣾⡀⠀⣀⣰⡏⠁⠀⠀⡘⣿⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣼⣿⣿⣇⠀⡄⠀⠙⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⠕⠶⠶⠂⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠨⠀⠀⠀⢹⣿⣩⣔⠒⠶⠶⠄⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢰⣦⣀⢰⡀⢛⣴⡀⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠶⣶⣤⣤⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠀⠀⠈⠙⣿⣿⣷⣿⣌⣿⠀⣅⣽⡜⢷⣤⣼⠿⣿⣿⣆⠀⠀⠀⠉⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⣴⡆⠀⢉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢻⣎⣿⣿⠀⠈⠻⣿⣶⠀⠀⣼⠋⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⢻⣿⠗⠊⠙⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠷⣈⠛⠲⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⣤⠄⠈⠁⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠙⢶⣷⣄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣰⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠙⢿⣿⣿⣿⣷⣶⣠⣤⣾⣿⣿⣿⠟⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⡿⠿⢿⡿⡁⢀⣀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⢿⣿⡿⠟⠉⠀⠀⢈⠁⣠⡄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⢸⣴⠀⠀⠀⠀⠀⠀⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⣀⣀⠀⣧⣴⣿⡏⢠⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢄⢰⡆⠀⠀⠀⠀⣙⣿⣿⣿⣾⣶⣶⣶⣆⡀⠀⡀⠠⠦⠀⢠⡇⠀⠻⣿⣿⣿⣿⣿⣱⣮⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢀⡀⢀⣦⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠿⣞⠛⣧⣴⣿⣿⣵⣤⣤⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⠀⠀⠀⠘⢷⣽⣿⣿⠿⢛⠛⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⣯⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⠁⢀⣦⠐⢦⣈⣾⣿⣴⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣰⣿⣿⣧⣿⠿⠛⣡⣿⣿⣿⣿⣿⣿⣿⡿⠇⠠⣿⠏⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⠻⣿⣾⣿⣧⣄⠀⠀⢼⣿⣿⣿⣿⣿⡿⢀⣶⣾⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⣿⠀⠙⠋⠁⠀⠚⢿⣿⣿⣿⣿⣷⡏⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⢯⡀⠀⠀⣸⣤⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⡤⠀⠀⠀⢉⡥⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣄⠀⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣠⣤⡜⠻⠟⣻⣿⣶⠶⠀⠀⢹⣿⡟⢿⣿⣿⣿⣿⣿⣀⡈⠙⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠁⠈⠄⠀⠈⠛⠛⠀⠀⠀⠀⢈⠟⢡⣦⣨⣿⣿⣿⣿⣿⣿⣶⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢤⣼⡃⠁⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⡗⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣿⡧⠀⠀⠠⣶⡄⠀⢀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣟⣷⠙⢷⣄⣀⡀
⠀⠀⠀⠀⢠⡖⠶⠞⠾⠟⠻⠝⠁⠤⠀⠛⠿⠿⡤⠶⠀⠀⠀⠀⠀⢀⠙⢦⣄⣀⠀⠀⠀⠚⣿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⢸⣦⡕⠠⠀⣀⡀⠄⢤⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣄⠉⠻⣶⣷⣼⠿⠋⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶
⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣤⣤⣬⣝⣛⣙⣿⡿⣿⣷⡆⢀⣸⣟⣿⣿⠛⢿⣦⣤⣤⡤⢶⣦⣶⣶⣶⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣾⣼⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠂⠠⠀⡀⠀⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⣘⡿⠃⠀⠠⠀⠉⣀⢀⣤⣄⠠
⠀⠀⠀⠀⠀⠀⠄⢰⠾⣷⠀⠀⠀⠠⠀⠀⣸⡋⡿⢀⣿⠂⠂⠸⣉⠛⠁⡧⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠀⠈⠂⢌⣉⠁⠿⡿⠀⠘⠄⠈⠛⠞⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠢⠄⡀⠀⠽⠄⢀⣦⣄⡀⣠⠺⠏⠀⠀⠀⠂⠀⠘⡀⡀⢤⣤⠂⠈⠀⠀⠀⡀⠤⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠀⠉⢱⡶⠛⠁⠀⠀⠢⡀⠀⠀⠀⠀⠈⠩⠀⠀⠰⠀⠀⠀⠰⠆⠉⣆⡡
⠀⠠⠄⠀⠀⠀⠂⠢⠀⠀⠬⠀⠀⠀⠈⠦⡂⠤⡀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢘⡠⠴⡏⠉⠓⡢⠤⢀⣀
⠀⠀⠀⢸⣶⣧⣤⡄⣤⢦⣘⠶⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠜⡴⠷⡀⠀⢀⠀⠰⠿⡀⠀⠀⣀⠀⠐⢀⣄⢡⠀⢠⠁
⠀⠀⠀⠘⢿⣿⣿⠀⠋⠁⠀⠀⢂⣣⢀⡀⣠⠊⡀⠀⠀⠀⠀⡀⠇⠀⠀⠐⢠⣦⡀⠀⠉⠉⠀⡌⠀⠀⣦⡴⠊⠀⠄
⠀⠀⠰⠀⠀⠀⠀⠒⠁⠀⠀⡀⢈⡴⠆⢀⠁⣜⡛⡽⣩⣤⣭⡝⢎⠭⢜⡶⡀⢰⢡⣀⠀⠀⠐⠄⣀⡀⠛⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠇⢈⠁⠊⠘⡀⢏⣱⣌⡁⣴⣃⠾⠷⠴⢋⠙⢀⠏⡘⠉⠀⡂⠄⠀⡸⠋⠀⠀⠀⠁
⠀⠀⡀⠔⠈⠉⢀⢀⡄⡈⠉⡐⢶⡶⣦⠲⣴⠦⣭⣥⠃⠈⠀⡿⣖⣘⣿⣶⡵⢾⢃⠒⠀⠈⠉⠁⢠⠀⠐⡒⠂⠀⠉⠉⠂
⢔⠁⠀⡀⠠⡀⠀⢻⠁⠀⠀⠈⠐⠍⣚⣀⢃⠴⠂⣡⢃⠤⣐⣡⡌⢒⠉⠙⢟⢉⣬⢊⣤⠀⢀⣠⣾⠀⠶⠀⠇⣠⠂⣠⠄
⠀⢂⠈⠀⠀⠀⠀⠀⠣⠀⠀⢀⡀⢀⣾⣻⡗⠻⡷⣡⣾⣷⣬⠻⣷⣾⠳⣿⣽⢀⣿⣯⡉⢒⠀⠀⠀⡀⠀⠀⢮⠐⠓
⠀⠀⢀⠀⡀⠄⠂⠁⠀⠀⠀⡈⠉⠒⠩⠙⡛⠻⠶⠧⢿⣥⣾⣴⣿⣼⡧⠿⠧⠾⠛⠋⠀⡈⠀⠀⠀⢇⠀⠰⠾⠤⠄⠀⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢠⠆⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢕⡄⠃⡷⠊⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠁⠀⠀⡠⠦⣀⣬⢵⢄⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣠⣄⠈⠀⣧⣚⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢰⠋⠀⠀⠀⠑⠌⡉⠑⡀⠔⠁⢀⠀⠀⡄⠘⠡⡉⢛⡛⢈⠝⠀⠀⡇⠘⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠜⣇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠺⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠃⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⢤⠶⠞⠥⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠂⣁⣢⠔⠂⠒⢄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠯⢥⣤⠤⣠⣀⣉⣁⣚⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠒⠒⠚⠚⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠈⠐⠤⠤⡤⠒⠀⠀⠀⠀⠀⢂⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠴⠑⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⢄⣀⠀⠀⠀⠀⡠⠠⢀⡾⠁⠄⠐⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠒⠒⠉⠀⠀⠀⢀⢜⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢
⠀⠀⠀⠀⡀⢠⡄⢀⠄⡠⢥⣤⣠⢄⣤⣤⣤⡤⣠⢤⡤⣤⣤⣤⡤⣤⢠⣤⡤⣤⣤⣤⡤⡤⣤⣷⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠭⣭⠉⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⠽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠤⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢹⡧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡩⠖⠛⡰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⢶⢹⡾⣤⣇⡇
⠀⠀⠀⠀⠀⠰⢫⠖⠂⢠⡃⠀⠀⠘⣄⢘⠥⢤⡀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⣉⠤⠤⡄⠁
⠀⠀⠀⠀⠀⢰⡧⡆⢀⠀⠙⠢⣄⠀⠈⠺⡄⠀⠈⣳⡀⣠⠖⠊⠉⠉⡹⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⢼⠀⡟⣄⡴⠲⢜⣧⠀⠀⠹⣄⠀⠀⠹⣍⣀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠈⢁⡀⠤⠿⣥⠀⠀⠈⠳⢤⣀⣽⢩⠑⠒⠒⠒⠒⠶⠦⠤⢄⠀⠀⠀⣠⢽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠚⠛⠓⠈⠁⠀⠀⠀⠀⠒⠁⠈⠑⠦⣄⣀⣀⣀⣀⡠⠞⠀⠀⠀⠳⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⢸⡛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⠛⠲⠤⣄⣀⠀⠀⠀⠈⠀⠉⠳⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠦⢄⡀⠈⠉⠑⠒⠤⢄⡀⠀⠀⠀⣸⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣙⠒⠤⣄⡀⠀⠀⠉⢱⠿⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠃⢀⡆⠀⠀⢿⠤⠀⠀⠉⠓⠦⣀⡏⠀⢸⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠼⢤⣤⣤⡼⠀⠀⠀⠀⠀⠀⠨⠥⢤⣸⢇⡡⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠒⠤⣤⣀⡿⡇⠀⠀⠀⠀⠀⠀⠒⠒⠒⠋⣻⠉⣠⠞⡡⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣀⠀⢱⡏⠙⢒⡒⠒⠒⠒⠒⠒⢲⢶⠞⠙⡷⣮⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢦⣵⢋⣈⠮⠤⠋⠀⠀⠀⠀⡠⢫⡎⣀⣴⡧⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠠⣖⣚⡷⣄⣈⠓⠚⣟⣩⠆⣀⣀⡤⠴⠖⢺⠟⠛⠛⠛⣆⡛⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠓⠒⠢⠤⠤⠤⠤⠤⠤⠛⠒⠒⠺⠯⠯⠭⠗