Brat Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣧⣍⡻⣿⡏⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣤⣽⣟⡻⢿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⣛⣿⡛⠻⠷⢮⣽⡿⠛⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠔⠒⠲⠤⡄⠀⠀⡤⠶⠒⠒⢼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠲⠚⡖⠢⠀⠀⠀⠠⠖⡶⠲⠚⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠂⠀⠀⠀⢐⠆⠀⡐⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠉⠀⠈⠁⢀⠜⠀⢸⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠈⠀⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⢿⣷⠈⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⢻⡆
⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠙⠓⠦⣴⣶⣤⣤⣤⣤⣤⣤⠴⠚⠉⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⢠⡴⠋⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢈⣁⣒⣲⣿⣆⣀⣀⣸⣿⣾⣭⠥⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠙⠳⣄
⠀⠀⡴⠋⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠈⣧⣠⡴⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠛⠛⠶⢦⣤⣸⡇⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢦⣀⠹
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⡾⣇⣀⣤⣀⠀⠀⠀⠀⠈⢻⡷
⣼⠁⣠⠖⠀⠀⠀⠀⢀⣤⣤⢏⣽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠻⣾⡇⠀⠘⢧⠀⢤⡀⠀⠀⢻
⣧⢾⠏⠀⠀⠀⠀⡼⠋⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢦⡀⠈⣇⢸⣳⡄⢀⡼
⠉⡾⠀⠀⣴⠀⣸⠁⢀⡾⠁⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠈⢳⡄⢹⡼⠇⠳⠋
⠸⣇⡀⢰⢿⣀⡿⢀⡾⠁⠀⣾⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣷⠀⠀⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⠹⠶⠖⣶⠚⠛⠉
⠀⠀⠙⠻⠘⠋⠀⠘⠷⣦⣤⡇⠸⣧⣀⣀⣀⣀⣀⣀⣀⣸⠁⣿⣀⣀⣷⢸⣤⣤⣤⡤⠤⠶⠶⠚⠛⠋⠉⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠐⣶⠤⣤⣀⣉⠀⠈⠉⠉⠁⣠⡄⠀⢀⣀⣠⣤⣶⠋⠀⠀⠀⠀⠀⠀⣼⢁⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠉⣿⡟⢛⣟⣉⣉⢹⡿⠉⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠛⣭⡶⢦⡗
⠀⠀⠀⠀⠀⠀⢰⡏⠴⣶⠶⢁⡀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⣈⣴⠋⠉⠉⢙⣇⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⢠⡾⠙⠦⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠧⡤⠶⠚⠉⠙⢦⣀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠁⠀⠘⠓⢟⠓⠒⠒⠒⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⡄⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡉⣹⠟⠲⣦⢤⣤⣤⣀⣀⣠⣤⣤⢴⣶⠿⣏⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢹⡏⠀⠀⠈⠳⠶⠞⠋⡎⠙⠦⠴⠚⠁⠀⠹⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢻⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⠶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠦⠤⠤⣤⠤⠤⠤⠤⣴⠶⠶⠶⢒⣒⣒⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⣖⡒⠒⠛⣿⣉⠉⠉⣩⢵⣺⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠦⣍⡷⠙⠖⣩⠿⣍⡓⠋⠁⠻⠴⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣨⣤⠶⠋⠁⠀⠀⠉⠓⠲⠶⠤⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣀⣀⢀⡀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣼⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⢸⣿⣷⣶⣶⡿⡿⣿⣿⡿⣿⣿⣿⣿⡧⠿⠖⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣧⣾⣴⣿⣿⣿⣿⣿⡟⣿⠓⢚⠛⢻⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡩⢹⠝⡏⣿⢹⠸⣆⣇⣿⣐⣸⠃⢸⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠘⡅⣪⡤⣴⢶⡶⣶⣿⣿⢻⢹⠋⠀⠘⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠉⠇⠻⠘⠃⠇⣿⣙⠈⢈⠃⡄⠀⠇⠀⠄⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢭⠷⠾⠶⠶⠾⠶⠶⠒⠒⠚⢻⠃⠤⠄⠀⠀⠐⠂⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡜⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠰⣯⠀⣏⡖⠀⠚⠒⣆⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⡇⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢰⠀⢸⢦⠠⠤⠤⠾⠇⠀⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⠀⠀⢀⠀⠀⡇⠀⠀⢰⣶⠶⡶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠀⡄⡇⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⣷⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠒⠒⠒⠁⠠⣿⠀⠀⢸⣿⣿⣿⣿⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠘⡋⠉⣋⡉⠉⣉⣵⠀⣿⠀⠀⠀⡉⠉⠉⠉⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⠀⠀⠉⠋⠉⠉⠉⠉⣉⣀⣿⡄⠀⠀⠿⠹⠻⠍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⠀⢸⢴⣶⣭⢭⣭⡭⠭⠭⠽⠗⠺⡇⠀⣀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠛⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⠶⠶⠤⣤⣀⠀⠿⣿⣿⠟⠛⠛⠿⢿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣍⡋⠀⠀⠀⠀⠘⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠰⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠲⠴⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠐⠋⠋⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣅⡀⡘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢱⣆⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠂⠀⢀⣤⣦⣴⣴⠆⠀⠀⠀⣾⣿⣏⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠟⠋⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿
⢀⣴⣿⣄⠀⠀⠀⠈⢉⣀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠈⠻⠟⠁⠠⠀⠀⠀⠉⣀⣀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣠⡤⠶⠚⠉⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣷
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠉⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠘⣿⣿⣿⠿⠇⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣆⣀⡀⠀⠀⠀⠀⢀⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠻⣿⣶⡿⠆⠀⠀⠀⠻⣿⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠀⠀⠀⠀⠀⠀⠀⠀⡔⡇
⠀⠀⠀⠀⣾⠇⠀⠀⠙⠁⢀⣠⣤⣤⣤⣀⠀⠀⠉⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢴⡇⠀⣾⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣀⣉⠛⠿⣿⣿⣿⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢀⣤⠙⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⣄⡠⢠⠀⢤⡤⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠐⠖⠀⡄⠐⠖⠊⢻⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠘⠀⠀⠀⢀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⠀⠀⠀⠃⠀⠀⠀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⣀⠉⠉⢀⣠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣄⠈⠉⢁⣠⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⠉⣻⣿⣿⡷⢾⡿⠿⣿⣿⠿⣿⡶⢾⣿⣟⣿⠛⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠙⣷⣿⢟⡵⠚⠁⣠⣾⣿⣿⣷⣄⠉⠓⢬⡻⣿⣾⠉⡉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠟⣁⣴⣾⣾⠿⠛⠁⠈⠛⢿⣷⣷⣦⡉⠺⣿⣬⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢽⣿⣇⣾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣧⣹⣟⡭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡿⠃⠴⠶⢤⣄⡀⠀⠀⠀⠀⢀⣠⡤⠶⠦⠙⣿⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣠⣄⣀⠈⠙⠂⠀⠀⠐⠋⠁⣀⣠⣀⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⢀⣴⣿⡿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡈⠿⣿⣿⣾⡿⠆⠀⠀⠺⢿⣿⣿⣿⠿⢡⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⣿⣆⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⢀⣴⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡿⣿⣷⣄⡀⠘⢿⣿⣿⡿⠃⢀⣤⣾⣿⢿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢻⢹⣿⣿⣶⣤⣈⣁⣤⣾⣿⣿⡟⡇⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣇⣼⣼⣿⣿⠿⠟⠉⠉⠻⢿⣿⣿⣧⣇⣸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⣧⠀⠀⠙⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⢰⣴⣃⣀⣀⣰⣆⣀⣀⣸⣼⡄⠀⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡄⣾⣿⣿⣟⣿⣻⣟⣿⣽⣿⣿⡇⢸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠋⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡿⠿⣿⣿⣿⢿⣿⣶⡆⠀⠀⢰⣶⣾⣿⣿⣿⣿⡿⠟⣛⣷⣾⣅⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠀⠀⠿⠹⣿⠷⠬⡽⣷⣶⣾⡿⠉⠶⠉⠿⠿⠙⠛⠀⠀⣿⡟⡿⠀⢠⡟⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢃⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⣿⣎⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⣼⣿⣿⢿⣦⣀⣀⣀⣀⣀⣠⣿⠃⠀⠀⠸⣷⣄⣀⠀⢀⣀⣀⣤⡾⠃⢠⠀⣰⣿⣿⣿⣦⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢰⣿⣿⣿⣇⠈⠉⠙⠛⠛⠋⠁⠂⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠁⠀⠀⢨⣼⣿⣿⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠀⡟⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⢸⡇⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⢸⣇⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢸⠷⣿⣿⣿⣿⣷⡀⠀⠀⠘⠶⠀⠒⠲⠖⠒⡀⠒⠀⠀⠀⠀⠀⡠⢺⣿⣿⣿⣿⣿⠋⠀⣾⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠂⠐⠐⠚⠁⠀⠀⠀⠀⡠⠚⠁⢸⣿⣿⣿⣿⡏⡄⢰⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⡄⢹⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠁⠀⠀⠀⢸⣿⣿⣿⣿⢳⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣿⠈⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣼⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⢰⣿⡆⢹⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⣿⣿⣿⢸⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠆⢸⣿⣿⣸⣿⠉⢿⣿⡿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⢸⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣸⠀⣿⡏⢻⣿⣿⡆⢸⣿⡇⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢻⣿⣿⣼⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡏⣼⣿⡇⢸⣿⣿⡇⢸⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣿⠃⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⣿⣿⣿⣠⠆⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⠃⡿⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣼⣿⣿⣿⡟⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢹⣿⣿⣧⠸⠀⢡⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⣰⣿⣿⣿⡿⠁⡀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠈⣿⣿⣿⢀⣄⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⡿⢁⣼⡇⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢀⡸⠇⠀⠸⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠃⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠑⠠⢤⣄⣀⣀⠀⠀⠀⠙⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠟⠋⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣶⣤⣀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠤⠄⠀⠀⠒⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠶⠶⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠈⢟⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⣦⣤⣤⣀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣇⣠⣤⣤⠴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇⠀⠀⠀⠀⠀⣀⠴⠒⠛⢉⣿⡟⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢀⣠⣴⡶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⣠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣴⡿⠛⠁⣠⡾⠀⠹⣿⡄⢠⡾⣿⠀⠀⠀⢀⣴⠶⣤⡶⠀⣸⣿⡇⠀⠀⣰⡟⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⣾⠇⠀⢀⣾⡟⠁⠀⣰⣿⠁⠀⠀⣿⡇⠸⣿⣷⣶⣦⢠⣿⠃⢀⣿⠇⢰⣿⡿⠀⢀⣴⠟⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡿⠀⠀⢠⣿⡏⠀⠀⢰⣿⠁⢀⡾⢱⣿⠏⣾⡏⠀⣸⣿⢠⡿⠹⠿⠶⠟⠋⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢸⣿⣀⡀⣠⡿⠃⢀⡾⢁⣿⡏⣠⣿⣧⣾⠏⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣷⠀⠀⢿⣿⣭⡿⣿⣅⠀⠊⠀⠸⣿⣴⠏⠈⠉⠁⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢸⡟⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⣾⡇⠀⠀⢸⣿⠇⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣿⠁⠀⠀⣾⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠈⣹⡟⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣀⣿⣡⣴⡿⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣤⣄⣀⣀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠉⠉⠉⠐⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢖⣒⡒⠒⠒⠺⠤⠄⣀⡀⠀⠀⠠⠊⠉⠉⠚⠣⠷⣤⣀⠤⠤⢄⣀⠠⠤⠄⠤⠤⡀
⠀⠀⠀⠴⠶⢶⣦⡤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⠄⠀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⡠⠔⢂⣉⡁⠀⠤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢢⡌⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠞⠂⠉⠀⠀⢀⣠⠶⠊⠅⠒⠒⠒⢢⡅⢀⠒⢒⡒⠒⡔⠀⠤⠤⠤⠤⠀⠤⠀⠀⠧⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡆⣐⠡⡴⢥⠤⠟⠐⠛⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠒⠚⠒⠒⠠⠤⠀⠀⠀⠀⠤⠤⠤⣔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢺⡁⠀⡑⢄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠀⠉⠂⢳⡀⠀⠐⣡⠊⠉⠒⠢⡀⠀⠀⠈⣼⡄⠀⠀⠀⠀⢀⠴⠂⠈⠉⠑⠄⠈⠳⢌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠓⠈⠉⠉⠉⠐⢴⠀⠀⠀⢦⠄⠘⣄⣀⣀⡈⠘⠦⣀⠀⡠⠋⢀⡔⠒⠆⠀⢰⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠀⢠⠇⠀⡸⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠠⠤⢀⡀⢀⠠⠂⢀⠀⠀⣀⠀⠀⠐⠆⠹⠃⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⠴⠖⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠁⠀⠀⠀⠀⠀⣀⡄⠤⠂⠉⣄⡤⡿⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠘⣉⠁⠈⡀⠀⣇⢀⢵⠐⢉⠖⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠒⢤⡔⠉⠈⠈⠁⠁⢉⣀⠠⠃⠁⠀⠀⡠⠔⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠈⠁⠀⠉⠀⢀⣀⡤⠖⡩⠇⠀⠀⡉⠀⡀⠴⠉⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠤⠤⠄⠀⠤⠤⠒⠒⠊⠁⠀⠀⡼⠀⠚⠀⠒⠉⠀⠠⢤⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠤⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡁⠀⢠⣀⠀⠀⠀⠀⠀⠐⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⢺⡵⠉⠀⠀⠀⠈⡆⠀⢀⠄⡠⠔⠊⠁⠌⠀⢁⠉⠛⡟⠝⠒⠀⠀⠀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⢀⠂⠀⠀⠀⠀⠀⠀⡿⠁⠁⠀⠀⠀⢀⠌⠀⠀⠜⠀⢠⠁⠀⠓⠀⠀⠀⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠠⡤⠤⡤⢤⡀⡠⢻⢄⣀⡠⠠⡞⠁⠀⢀⠀⠀⠀⡌⣮⠉⠀⠀⠀⢠⠃⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡀⠀⢇⠀⠋⡠⠓⠊⠀⠀⢀⣈⣀⣀⣾⣐⣲⣾⣴⡁⠀⠀⠀⢀⠆⠀⣘⠓⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠉⠀⠈⠀⠀⠀⠀⠀⡤⠃⠀⢈⡥⠍⠁⠉⠙⢦⣀⢀⠶⠒⠠⠤⢏⠋⠉⠐⠢⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠉⡶⠾⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠑⡖⢲⡀⠀⠀⠀⠀⠀⠀⠔⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⢚⠧⢄⡈⠉⠒⠒⠒⠀⠒⠒⠩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠓⢹⣇⣀⣀⣏⣉⣐⣶⢶⡒⡒⠂⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠀⢐⠿⠀⠀⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢣⠑⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠿⠷⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠶⠶⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⢶⠶⠶⠶⠶⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⣰⡟⠁⣠⣤⣶⣾⡟⠛⠉⢉⣠⠴⠎⠉⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠱⢦⣌⠉⠉⠛⢿⣶⣶⣤⣄⠈⢿⡆
⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣇⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⣿⠓⠶⣤⣄⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣸⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢶⣤⡀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠉⢉⣤⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢁⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠸⠏⠉⢿⠟⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠻⡿⠉⠹⠇⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣤⣤⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢈⡷
⠀⠀⠀⠀⢀⣠⠄⣤⠘⢷⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡾⠣⣤⠠⣄⡀
⢀⡴⠞⠉⠙⢧⠀⠈⣶⠉⠐⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠂⠉⣶⠁⠀⡼⠋⠉⠲⢦
⣎⠀⠀⠀⠀⠀⢷⣄⡼⠳⠖⠉⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠈⢙⡶⠚⠉⠉⠉⠉⠓⢾⡋⠁⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠉⠲⠚⢧⣠⡞
⠫⡤⢤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠸⢧⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡼⠇⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠉⠛⠂⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠓⠛⠉⠲⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⢊⠛⣤⡀⢀⡀⠀⠀⠀⠈⢻⡶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢾⠟⠁⠀⠀⠀⢀⡀⢀⣤⠛⡑⣦⣀⡀
⠀⠀⠀⢀⡴⠖⠉⠉⠉⠉⠳⣄⣹⠏⠉⠁⢺⠇⠀⠀⠙⢦⣈⠉⠉⠛⠛⢳⣶⠶⠶⢶⡒⠟⠛⠉⠉⣀⡴⠋⠀⠀⠸⡗⠈⠉⠹⣏⣴⠞⠉⠉⠉⠉⠒⢦⡀
⠀⠀⠀⢹⡀⣤⠔⢶⡆⠀⠀⣬⠃⠀⠀⠀⣾⠀⠀⠀⠀⠀⠉⠓⠢⣤⡴⠋⠀⢠⠂⠀⠙⢦⣤⠶⠊⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⣅⠀⠀⢰⡶⠢⣄⣀⠟
⠀⠀⠀⠀⠘⠧⠞⠁⢟⡀⢸⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⢻⠀⠀⠀⠀⢸⠃⢀⡸⠈⠳⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣀⣀⢀⡀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠙⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⢸⠿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠋⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⡉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⡸⠀⠀⣇⣀⣀⣀⣀⣀⣀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣯⣭⣿⣿⣿⣯⣤⣤⣤⣤⣽⣿⣿⣿⣭⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠀⢀⣠⡴⠶⠟⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠉⠙⠷⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⢸⣧⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣤⡤⠔⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡀⢀⣠⡾⠛⠉⠙⠻⣦⡀
⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⠿⠛⠉⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣴⠟⠉⠀⠈⠉⠛⠷⣶⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⡶⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⠖⠒⠛⠛⠛⠛⠻⠿⠿⠛⠓⠲⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣼⠇
⣿⡀⠀⠀⠀⠀⠀⢀⣠⢴⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠉⠃⠀⠀⠀⢀⣼⠏
⠸⣧⠀⠀⠀⠀⠰⠋⢁⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠙⢷⣄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⣴⠄⠀⠀⣿⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠙⠻⢶⣤⣄⣠⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⡀⢙⠀⠀⠀⠀⢀⡋⢀⡀⠀⢠⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡆⠀⢀⠃⠀⠀⠀⡎⠀⢀⠙⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢧⡀⢈⠇⠀⠀⠀⢣⡀⢈⡗⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⡆⠈⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢲⠿⠋⠉⡱⠶⠲⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠴⠎⠀⠀⠊⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠿⢦⡙⠛⠷⢶⣤⣤⣤⣤⣤⣤⣴⠶⠟⠛⠟⠛⠲⠶⠶⠶⠶⠶⠶⠶⢤⡴⠂⢶⠶⠔⠚⠓⣦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⠉⠀⠀⠀⠈⠛⠛⠰⠦⠤⠤⠤⠼⠋⣀⠙⠶⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢆⠀⠀⠶⠐⠀⣄⣠
⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⢀⣤⡶⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⠚⠳⢦⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠈⢳⡄⢀⡼⠿⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣠⣽⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠁⠀⢙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣎⠙⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠴⢿⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠉⠉⠉⢿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠒⠖⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡿⠻⣿⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠞⠛⣧⠀⡿⠛⠳⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⣯⢀⡿⠀⠀⠀⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡀⠀⠀⢹⣾⡇⠀⠀⡀⠀⠈⠙⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣿⣿⣶⣶⣯⣬⣿⣶⣾⣯⣷⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⡖⠂⠉⠱⣤⣴⣶⣶⣶⣲⣶⣶⣶⡂⠀⠠⣤⣤⣄⣤⣤⠤⢤⡔⠒⠠⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠋⣉⡉⠥⠄⠒⠂⠉⠉⠓⢶⠒⡖⠃⠀⠀⠀⠀⣀⠤⠤⠤⢌⣦⡀⠈⢇⡸⠁⠘⡏⠹⣤⠀⣀⡼⠭⢝⣒⡲⠤⣄⡀
⠀⠀⠀⠀⠀⡔⠁⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠓⠤⢄⣀⠀⠀⠀⠀⠀⠁⠀⢠⠇⠀⠘⢲⠆⠀⠀⠀⠀⠉⠓⢦⡈⠳⡄
⠀⠀⠀⠀⢰⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡔⠒⠚⠋⠀⠀⠀⠀⠀⠀⠚⠂⡲⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸
⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⣇⣀⣀⠀⠄⠀⠐⠒⠛⠓⠒⠀⠠⠄⠐⠂⣀⠀⡠⠐⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠜⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡌
⠀⠀⠀⠀⠀⠸⡀⢰⠀⠀⠀⠀⠠⣃⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡜
⠀⠀⠀⠀⠀⠀⢃⠀⡄⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠆⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼
⠀⠀⠀⠀⠀⠀⠸⡄⢱⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⣠⡴⢺⠁⠀⠀⢀⡠⠒⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡞⠀⠀⠀⠀⠀⠀⠀⡸⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣧⠘⡀⠀⠀⠀⠈⠑⡖⠒⠉⠁⣀⡤⣬⡄⠤⠒⠉⠀⠀⠀⠈⠂⠈⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢰⠃⢠⠃
⠀⠀⠀⠀⠀⠀⣷⣼⠀⣇⠀⠀⠀⠀⠀⠁⡰⠃⡞⠀⠀⠀⠁⠀⠀⠀⠔⠊⠁⠀⠙⢆⠘⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢠⠇⢠⠏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣷⡀⠀⠀⠀⠱⠁⠀⠧⣀⠸⠇⠀⠀⠀⠀⡀⠀⣴⡄⠀⡬⠀⡜⠀⣀⠤⠰⠿⢅⠀⠀⠀⠀⠀⠀⠀⡎⠀⡞
⠀⠀⠀⠀⠀⠀⢹⣿⢣⠘⣿⣷⠀⠀⠀⢢⠐⠲⡀⠸⢉⠃⠀⠀⠀⠀⠘⠢⠤⠤⠊⠀⠀⣄⠀⠖⠂⠀⠀⡸⠀⠀⠀⠀⠀⠀⡸⠀⡸
⠀⠀⠀⠀⠀⠀⢸⡿⠙⡄⢸⣿⠧⠀⠀⠈⣦⡀⠈⠲⡤⢀⣀⡀⣀⣀⡀⠲⠄⠀⠀⠀⠀⢨⡆⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢰⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠛⠁⠰⣇⠸⠏⠀⠧⠀⠀⠀⢙⡶⢤⡀⠒⢰⡾⠋⠀⠀⣀⠠⠤⠐⠒⢀⠜⠁⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡎
⠀⠀⠀⠀⠀⠀⠠⠂⠈⢻⠀⡟⠁⠀⠀⠀⢠⡼⠂⠁⢈⣀⣀⠖⠂⡚⠁⠀⠀⠀⠀⡰⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠣⡇⠀⠀⠀⠹⢧⢄⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⢞⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡄⠙⣦⠀⠀⠀⠀⠸⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⣧⣤⣴⣷⣤⣄⡀⠀⣹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇
⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⣀⣰⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡰⢾⣿⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡞⠀⢰⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢁⡴⠋⢀⡴⠃⣠⠞⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⠀⠐⠒⠒⠊⠁⠀⠒⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣹⢘⡷⢻⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡱⡄⠈⠋⠀⠟⣋⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠑⠲⠋⠸⠎⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⡄⠀⣀⠐⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠝⠂⠀⠀⠰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⠉⠑⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⣏⠴⠋⠢⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡤⠀⢀⠀⠀⠀⢰⣚⠰⣆⡀⠀⠀⠀⢀⠀⠀⢠⢰⠆⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡠⠐⠈⠀⢀⡭⠠⢉⠀⠀⠈⠀⠐⠁⡀⠀⠀⠀⠑⠀⠈⡏⠄⠉⠁⠀⠁⠀⢤⠁⢀⠂⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⢮⣯⡇
⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⣀⣀⠆⠂⣠⠊⠂⢌⠀⠀⠀⠀⡁⢀⠰⠀⠔⠊⠄⠠⠂⠢⠀⠁⠄⡰⠀⠀⠀⠀⠆⠈⠁⠀⠀⠀⠀⢈⡅⢀⠠⠄⠀⠠⠀⡀
⠀⠀⠀⠀⢀⡜⡖⠳⠘⡿⡋⠀⠗⢴⢃⢔⢲⡀⣀⠀⠀⠀⠐⠀⡯⡝⣀⣓⣀⣀⣛⡀⢻⣝⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠂⠄⣠⠀⠀⠀⠠
⠀⢀⣀⡄⠀⠤⠴⢄⣜⠄⠀⠠⠦⠀⠁⠜⢀⢌⠒⠀⠀⠀⠀⢢⣿⡀⠆⢀⢁⡀⡈⡘⢨⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⡌⣐⣆⡷⣠⠠⡦⠀⡤⠈⢀⠐
⠀⠠⠿⢁⢈⠄⢀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠴⡂⠀⠀⠀⠀⠀⠑⠢⠑⠢⠅⠁⠈⠬⠐⠉⠃⠂⠀⠀⠀⠀⠀⠀⠝⠤⡲⢺⠑⡍⠶⠀⠂⠠⢅⠀⢐⠆⠈⡜
⠀⠆⠈⣶⣀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠊⠀⠀⠀⠀⠀⠀⠄⠷⡄⡇⠠⠠⠀⠆⣤⢱⠾⠠⠀⠀⠀⠀⠀⠀⠀⠞⠋⢻⠒⠒⠀⠀⠀⠀⠈⠀⡀⠀⠁⡀
⠀⢀⡀⠈⠋⠶⠀⠂⠠⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⡅⠰⡄⢠⠆⣡⠇⠴⠃
⠀⠀⠁⠀⠛⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⡄⠘⠚⠃⠘⠒⠃⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠠⠂⠐⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠁⠀⠔⠄⢠⠌⡀⠀⠀⠀⠠⠂⠁⠀⠐⢤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠐⢀⣡⠀⠀⠀⠀⠀⡆⠀⢀⡈
⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠈⢂⡀⠁⣀⡱⠃⠀⠀⠰⠀⣀⢃⠈⢁⠸⠂⠈⢉⣰⢒⠀⡀⠀⡠⠀⠀⠀⠀⠠⡄⠈⠡⠒⡄⠴⠠⡄⠈⠇⠀⠒⠀⢰⠄⠀⡀
⠀⣆⠀⢡⡄⠀⠰⠀⠀⠀⢆⣱⠀⠀⠀⠀⡤⠀⠀⠀⠢⣃⢀⠁⣐⣦⢀⠪⠀⡄⢀⣈⠀⠈⠀⣀⠲⠌⠀⠀⠘⡦⠬⡤⠞⠀⠀⠙⡀⢀⡀⠀⠀⠁⠀⣴⠁
⠀⠈⢦⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠂⡤⢀⡄⠀⠀⠀⠀⠁⡈⢉⠋⠁⠀⠀⡶⢆⠜⠛⠒⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠠⡀⠀⢂⢀⠊⠀⠀⠀⢀⠀⠈⠀⣀
⠀⠀⠠⠀⢴⣇⡒⢀⠀⠀⠀⠀⠀⠀⢈⡒⠡⡀⠀⠀⡄⠎⠀⠀⠆⠀⠁⡀⠐⣤⢀⠀⡄⣄⣀⡀⠈⠁⠀⠀⢀⣄⠣⡀⣀⢆⢀⡟⣦⢅⡴⣠⣼⡔⠂⠁
⠀⠀⡀⢘⡂⢸⠠⠆⠠⠤⠖⠖⠀⠚⠀⠐⡦⠠⡄⠀⠀⠀⠀⠀⠉⠀⡀⠀⠈⡟⢼⠀⠩⠊⡧⢌⡀⠀⠀⠀⠀⢀⠕⠂⠀⠵⠎⠸⠇⠁⠢⣘⡂⢤⠶⡄
⠠⠎⠉⢲⡗⢿⡰⠁⠐⡓⡇⠀⠀⠀⠀⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⡄⢰⠀⢰⠁⡠⠀⠀⠀⠠⠈⠈⠀⠀⠀⠡⡤⠄⠀⠀⠀⠐⢄⣦⠒⠠⠫⡝⢂
⠀⠀⠀⠈⠹⢀⣁⠉⠁⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢀⠀⡁⠀⢀⢠⠀⠀⠐⠒⠀⢀⠈⠀⠀⠀⠀⠆⠓⠘⠀⠀⠀⠀⠐⠛⠚⠒⡱⠌⠫⡓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⠻⣷⣿⣿⣿⡿⣿⡿⠛⣓⣱⣢⣔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣲⣿⣷⣿⠙⣿⠓⠀⣠⣤⣾⣿⠿⢟⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠿⠈⠻⠯⠟⠛⠙⠛⢛⣩⡔⠋⠁⣀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠴⠦⠀⠀⠐⢿⡍⠑⠂⣑⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣤⠀⢀⠖⣦⡄⠀⠀⢸⣷⠖⣚⢿⣯⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠟⠀⠀⠈⠫⠄⠀⠀⠀⠉⢸⡇⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠄⠀⠀⢤⠀⠀⠀⠀⠀⠀⠈⡀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⡠⠀⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠡⡐⣀⣰⡦⠊⠇⠀⠀⠀⠰⠤⠇⡸⠃
⠀⠀⠀⢠⠒⢉⠀⠀⠀⠈⠳⣬⣝⠶⡑⠀⢂⢀⠀⠀⠀⠈⠵⠀⠀⠀⠀⠀⠐⠁⠀⠐⠁
⠀⠀⠀⢡⠀⢤⣧⡀⠀⠀⠀⠈⠙⠥⣾⣽⣫⢥⣓⣆⠈⠢⠤⠤⢤⠄⠐⠀⠀⠀⣀⠟⡆
⠀⠀⠀⠀⢣⡀⠙⣍⡖⠀⣀⠀⡀⠀⠈⠉⠛⠿⠾⣽⣿⣦⣢⣠⢚⠀⡀⠤⠒⠈⠀⠀⠑⣄
⠀⠀⠀⠀⠈⡝⠦⣄⠚⠻⡺⠷⢬⠷⣤⣐⡠⡀⠀⠀⠀⠉⠙⣯⠳⡊⠢⡀⠀⠀⠀⢀⣼⣿⠃⢄
⠀⠀⠀⠀⠀⢡⠐⠈⡍⢢⣤⣉⠓⠊⠩⢿⢍⣊⣆⣂⠀⡄⠀⠌⠀⡇⢤⣼⣷⣄⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠢⣇⢁⠈⠇⢹⠉⠳⡶⣤⣬⣁⣀⣈⣉⣁⣂⣄⠈⠑⢄⠉⠛⠿⠿⠿⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠔⠀⠈⠀⠀⠈⠳⣖⠤⣤⠄⠀⠀⠀⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⢠⠤⠤⠤⢤⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠷⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⢤
⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢺
⢸⡇⣀⣂⡒⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⢂⣰⢰⢶⠰⢶⠶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⣹
⢸⡿⠿⣿⣿⣿⣿⣶⣦⣤⣀⠀⢢⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡰⠁⣀⣤⣴⣶⣿⣿⣿⡿⠿⣿
⢸⡇⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣶⣤⣤⠄⠄⣠⠀⢀⡤⠠⠄⣤⣤⣶⣾⣿⣿⣿⣿⠿⠟⠋⠉⣀⠀⠀⣿
⢸⡯⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠈⡆⢸⡀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿
⠘⣿⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⢦⡀⠀⢃⡈⠀⠀⡴⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠀⠀⣿
⠀⣿⠀⠐⠛⠉⠁⠀⠀⠀⠈⠉⠛⠲⠦⠤⠄⠀⣷⠀⢸⡇⠀⣼⠀⠠⠤⠤⠶⠛⠋⠁⠀⠀⠀⠀⠉⠙⠢⠀⣿
⠀⣿⠒⢻⠛⠛⠛⢿⣷⠶⠖⠲⠒⠲⡶⠶⠂⠀⢸⡃⠈⠁⠀⣯⠀⢠⠤⢴⡷⠶⠶⠶⣶⣶⠶⠖⠒⠒⡖⠒⣿
⢸⣿⠀⡈⢧⡀⠀⢬⣽⡷⠀⠀⣀⣴⠇⣰⣿⠂⢸⡃⠀⠀⠀⣗⠀⣾⣇⠘⣧⣀⠀⠀⠬⣽⡿⠀⢀⡼⢁⠀⣿
⢸⡿⠀⢻⡄⠙⠶⣤⣤⣴⠶⠟⠋⠁⢠⡾⣿⣾⢸⠃⠀⠀⠈⣿⡶⣿⠻⣄⠈⠙⠻⠶⣦⣬⣤⡶⠟⢁⡾⠀⣿
⢸⡇⠀⠀⠻⣆⠀⠀⠀⠀⢀⣀⣤⠾⢋⣲⡿⣉⣸⠀⠀⠀⠀⣿⢁⣿⣗⡚⠳⢦⣀⡀⠀⠀⠀⠀⣠⠞⠁⠀⣿
⢸⡿⣧⠀⠀⠈⠉⠀⠘⠛⠉⠉⠀⣠⣾⡿⣁⡟⣸⠁⠀⠀⠀⣿⡈⣏⣿⣿⣄⠀⠈⠉⠛⠃⠀⠈⠁⠀⢀⣠⣿
⢸⡇⠈⠙⠀⡀⡀⠀⠀⠀⣀⣤⣾⠿⠋⠨⠿⠃⣿⠀⠀⠀⠀⣿⡇⢻⡯⠙⠻⣿⣥⣄⡀⠀⠀⠀⢀⡀⠚⠁⣿
⢸⣷⠀⠀⠀⠉⠉⠛⠛⠙⠉⠋⠐⠀⠀⢀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠋⠙⠙⠉⠋⠉⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠥⠊⢀⡀⠀⠀⠀⠀⠀⢀⠈⢂⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠾⠉⠃⠄⠀⡀⠀⠀⠀⠀⠐⠫⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠀⠀⠀⣀⠀⠀⠀⠀⠀⢥⠀⠀⠀⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇⠠⡖⢶⣿⠀⠀⡆⠀⠀⣾⡶⢲⠀⣼⡟⣆⠀⠀⠀⠀⠀⠀⠀⠄⢡⣿
⢸⣿⣇⠄⡀⠀⠀⠀⠀⢀⣠⡼⠋⡘⢿⣦⣤⠄⢻⣆⠀⠁⠀⣠⣿⠤⣤⣴⠟⠅⠈⢷⣄⡀⠠⡐⠄⠃⠘⠘⣿
⢸⣿⡏⡰⠀⠁⠀⠀⠈⠀⢸⠃⠀⠍⠒⡀⠀⠀⠈⢛⠷⠶⠾⠟⠁⠀⠀⠀⠌⠀⠀⢀⣧⠀⠀⢠⠀⠒⠂⠀⣿
⢸⣿⠅⠁⠀⠀⢰⠀⢀⠀⣸⠀⠀⠀⠀⠀⠀⢒⣘⡂⠈⠀⣀⣆⠀⠀⢄⠐⠀⠀⠀⠀⣿⠀⠀⣹⠀⠈⠃⠀⣿
⢸⣿⠀⢰⠀⠀⣸⡆⠀⠀⣿⠀⠀⢀⠔⠀⣰⠟⠉⠹⠷⣤⣤⣬⣭⡑⠀⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇⠈⠁⠀⣿
⠀⣿⠀⢸⡆⠀⢸⡇⠀⠂⣻⠆⠀⠡⠂⢰⡏⠀⠀⠀⡄⢀⡀⠀⠈⣿⡀⠠⡀⠀⠀⠠⣿⠀⠀⢹⡇⣐⣲⢺⣿
⠀⣿⠀⢸⣇⠀⢸⡇⠀⠀⣺⡂⠀⠀⠀⣾⠇⠀⢰⡟⠛⢻⠛⢿⡀⢸⡇⠠⠈⠂⠀⠠⡟⠀⠀⣾⠁⢐⣿⠰⣿
⠀⣿⠀⢸⣷⠀⢸⡧⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⣿⠳⣶⡟⢤⢾⡇⠘⣇⠀⡀⠀⠀⢨⡇⠀⠀⣿⠀⢲⡏⠘⣿
⠀⣿⠀⠈⣿⠀⠀⣿⠀⡀⠈⡇⠀⠂⠘⠃⣠⣾⡿⢿⣿⣷⣾⣾⣯⡀⠙⠀⠀⠀⠀⢸⠃⠀⢸⡇⢀⣹⡇⣘⣿
⠠⣿⡇⠀⢿⣇⠀⢹⡇⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠴⠀⠀⠀⠀⠀⠰⠀⡜⠀⢌⣾⠃⢰⣿⠠⣸⣿
⢀⡿⢩⠀⠘⣿⣔⠈⣷⡀⠁⠘⣅⠀⠀⣠⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠃⠀⣾⠇⢀⣾⠇⠐⣴⣿
⠀⣿⢜⠀⠀⢹⣷⡀⠘⣧⡀⠀⠐⠀⠀⠤⠤⠈⠙⠛⠿⠿⣶⣶⠾⠟⠋⠠⢄⠀⠀⠀⣼⠏⠠⣽⡟⠀⢈⢾⣿
⠀⣿⡲⠬⠀⠀⢿⣆⠀⠈⢷⡐⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⣾⠃⠀⣱⡿⠀⠠⣉⢶⣿
⠀⡿⣙⡰⢂⡀⠈⢿⣦⠀⠀⠱⣄⠀⢀⡏⠀⢰⠀⠀⡖⠀⠀⠀⠀⠈⠀⢸⡇⠀⠜⠁⡀⣼⡟⠀⠀⢄⠩⢧⣼
⠀⡟⡘⠶⡭⢁⠀⠀⠻⣷⡀⠀⠀⠀⠸⣧⣀⠀⠀⢸⠃⡇⠀⠀⠀⠂⠀⢻⡇⠀⢐⣠⣿⠏⠀⡀⠀⣄⢳⣞⣹
⠈⠂⠀⠚⠒⠓⠘⠶⠶⠿⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠻⠿⠿⠛⠛⠞⠛⠛⠓⠚⠛⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠋⠉⠉⠉⠉⠉⠙⠚⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡌⠀⢣⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⡰⠀⠀⠀⠱⣀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠃⠀⠀⢀⠁⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠑⢀⡀⠀⠀⠀⠣⡀⠀⠀⠀⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠈⡄⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⠝⠁⠀⠀⢠⠐⢄⠀⢘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠸⠆⡇⢘⠀⣾⣿⣿⣆⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠀⡇⠈⣆⠇⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⠀⢢⠀⠈⠢⣿⣿⣬⠗⠁⠀⠀⣀⠀⠀⠈⠿⣥⣿⠟⠀⠀⠀⡘⠀⠘⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠄⠁⠈⠧⡘⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠂⢀⠴⠁⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡬⠒⠤⢀⡀⠀⠑⠀⠤⠤⠤⠀⠀⠀⢀⡀⠤⠒⠥⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡼⠁⠀⠀⠀⠀⠁⠙⢌⡀⠀⠀⠀⠀⠉⠉⡠⠁⠀⠀⠀⠈⠳⡄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⢣⡀
⠀⠀⠀⠀⢀⣠⣴⠯⠤⠖⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⠤⢽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢈⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣧
⠋⠉⠉⠉⠉⠉⠉⠉⠙⢇⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⢘⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⢣⡀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠤⠤⢤⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤
⠀⠀⠀⠀⠀⢾⠦⠀⠀⠀⠀⠑⡄⠀⠀⢀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⢺⠓⠀⠀⠀⠀⠀⠈⠢⣠⠁⠜⠓⠄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠏⠂⠄⠚⡲⡛⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⠀⡄⢤⠻⠘⠂
⠀⠀⠀⠀⠈⢻⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣉⠴⠟⡀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⠈⠓⠂
⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁
⠀⠀⠀⠀⠀⣸⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢼⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⢸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⡈⠉⠉⠉⠈⠑⠓⢄
⠀⠀⠀⠀⠀⢺⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢳⠀⠀⠉⠁⠐⠒⡶⠒⠈
⠀⠀⠀⠀⠀⣹⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠤⠀⠀⠀⠀⣤⠤⢤⣤⣤⠄⠠⠤⢤⡤⡤⠀⢀⡀⣀⣀⠤⢼⡀⠀⢧⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣇⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠵⠒⠛⠛⠓⠲⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⠏⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣤⡀⠀⠀⠀⠀⠈⣿⠏⠘⠉⢿⡟⢹⣿⠏⠀⠀⠀⠈⢣⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢧⡀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡿⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠯⠭⠝⠛⠛⠻⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⢀⣀⣠⡤⠤⠖⠒⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣤⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢹⠇⠀⡀⠀⠀⠀⠈⠒⢤⡀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠀⣀⣤⡴⠶⠶⠶⠶⠶⠶⠛⠛⠋⠁⠀⠀⢸⡆⠀⢱⡀⠀⠀⠀⠀⠀⠙⢦⡘⢿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠹⣎⣿⡄
⠀⠀⠀⠀⠀⠀⣼⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⢤⢤⠀⠀⠘⣆⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⠀⠀⡇⠀⠀⢀⠔⠒⠓⠢⡀⠀⠀⢀⡴⠚⠉⠙⠲⡀⠈⠳⣄⡀⠀⠀⠉⠙⠛⠶⣄⡀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⣿⠀⢰⠃⠀⠀⠀⠀⠑⠀⢠⠏⠀⠀⠀⠀⠀⠱⠀⠀⠀⠉⣿⠒⠦⢤⣀⡀⠈⠻⣄⢻⡇
⠀⠀⠀⠀⠀⠀⢹⣷⠀⠈⣧⠀⡿⠀⠈⠀⠀⠀⢿⠏⠉⠀⠘⠉⠉⠉⠹⡿⠀⠀⠀⠀⠀⠀⢹⠀⣀⣀⠈⠻⣦⡀⠸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠈⠙⣷⠀⠸⣷⠀⣿⠁
⠀⠀⠀⠀⠀⣠⡾⠃⢀⡼⠁⣸⡇⠀⠀⠀⠀⠀⠀⢠⠖⠛⠢⡁⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠉⢀⡟⠀⢠⡿⢠⣿
⠀⠀⠀⠀⡼⠋⠀⠀⠎⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣴⠟⠀⢀⡿⠁⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠁⠀⣧⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡇⠀⠀⣾⠁⠀⠀⢻⣷⡀
⠀⠀⠀⠘⣧⠀⠀⠀⠀⢿⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⣇⠀⠀⡇⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⢷⡀⠀⣏⠳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡀⠀⠈⠻⣷⡄⢻⣄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠇⠀⠘⢦⡈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠘⢿⣆⠀⠀⢸⡷⠀⠙⣆⠀⠀⣠⠟
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠈⠀⠀⠀⣸⠃⠀⡿⠉⠓⠶⢤⣤⣤⣤⡤⠴⠚⠁⠻⣤⠀⠀⠀⡿⠀⣰⠋⠀⠀⣠⠏⠀⢠⡏
⠀⠀⠀⠀⠀⠳⠀⠀⠀⠠⡀⠀⠠⡎⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢾⠁⠀⠻⠦⠀⠸⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠛⠛⠉⠉⠉⠉⠙⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡶⢛⢦⡀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⠔⢁⡞⠉⠉⠛⢦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣰⠋⠀⢸⡀⠀⠀⠀⠈⢏⠳⣄⢀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢰⠇⣞⠉⢹⢧⠀⠀⠀⣀⢸⣀⠬⣅⣎⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠈⠁⠀⠀⠳⢄⢀⡔⡉⠀⠀⠀⢱⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠀⢹⠀⠀⠀⠀⠀⠀⢠⣿⠈⠃⠀⠀⠀⠂⠀⠀⠈⢷⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⢀⠸⡆⠀⠀⠀⠀⠀⠀⠙⠦⠤⠠⠀⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡩⠛⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢇⠉⠦⠐⠉⠉⠷⠤⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⡰⠣⡀⠁⠂⠀⠀⠀⠀⠀⡴⠊⠉⠑⠤⣧⠀⢀⠀⠀⠀⠉⣳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠓⢤⠬⠵⠦⢄⠀⠀⠀⠘⡖⠤⢄⡀⠀⠸⡀⢈⠀⠀⠈⠉⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠠⠄⣀⢠⣉⣦⠀⠀⠘⠢⠤⣈⡉⠒⡇⢸⣄⣀⣄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠑⠊⠀⠀⡀⡎⠀⠉⠁⠀⡆⠀⠈⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠤⠄⡄⣀⢀⣀⡀⠴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢰⠶⠶⢶⠶⠶⠶⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣛⣀⣠⠤⠶⠶⠤⢄⣄⡋⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠁⠀⠀⠀⠉⠉⢙⣻⡶⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡾⠁⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⡠⠋⣿⠎⠀⠀⠀⠀⠀⢀⣴⠫⢥⣤⡄
⠀⠀⠀⠀⠀⡼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠉⠠⠶⣭⡀
⠀⠀⠀⠀⣼⠁⠀⠈⠀⠀⠀⠀⠀⣠⠔⠚⠋⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠹⣆⠀⠂⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⢳⡄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠶⠶⢤⣄⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⡇
⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠑⢦⣄⠀⣼⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠙⢲⠄⠀⠀⡿⠤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠱⣏⣀⡤⠤⢴⡎⠀⠀⢀⣀⣄⡀
⠀⠀⢀⣤⠤⠚⠁⠀⠀⠀⠀⠀⠚⠃⠤⣄⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⣤⠾⢧⣀⣤⣾⡤⠴⠊⠉⠀⢀⡿
⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠙⢦⣠⡶⠋⠀⠀⠘⣯⣁⠀⠀⠤⠤⠤⠖⠛⠳⠦⠤⢤⣀⡀
⠀⠀⠙⢦⣀⡄⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣹⣆⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠚⠿⢧⣀⡀
⠀⠀⠀⠀⡘⠀⠀⢫⠳⣤⡀⢀⡴⠞⡿⠋⢁⣤⣤⣤⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢦⣄⠉⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠉⠲⠦⠼⠷⠀⠉⠋⠀⠘⣇⠀⠀⠀⠀⠀⠉⢀⣀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡤⠟⠳⢄⣀⣀⡴⠃⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠒⠒⠚⠓⠉⠉⠀⠀⠀⠀⠉⢉⡿⡶⠲⠦⠤⠤⠤⠤⠤⠴⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⢀⡎⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢟⣷⣤⣄⣀⣠⠟⠀⠀⠀⠀⠀⠀⢠⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⠏⠈⣽⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⠀⠀⣩⠉⣳⠄⠀⠀⠀⠀⠀⠀⡾⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⡿⢷⣴⡆⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⡼⠓⢂⡼
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠓⠛⠙⠒⢶⠖⠶⠶⠖⢲⢶⡞⢋⡟⠉
⠀⠀⠀⣀⢀⣠⡔⠉⠀⠀⢀⡀⠀⠀⡰⢁⡔⠀⢀⡴⠁⠈⠉⠉
⣴⠞⠋⠙⣦⡳⣝⢳⣤⣄⣀⣉⣙⣿⣛⣉⣉⣡⣼⡷⢢⣤⣀⣀
⣷⣀⣀⣸⣇⣛⣋⣭⣞⣋⣤⣟⣯⣻⣍⣉⣙⣷⣯⣞⣻⣄⣀⣈⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠻⡿⡿⠿⠿⠿⠿⠿⢿⡿⢟⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣷⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠⠤⠀⠀⠀⠀⠠⠄⣀⣤⣶⣦⣤⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣤⠤⣄⣀⠀⠀⠤⠼⠿⣿⣿⣿⣿⣆⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠁⠀⠀⠀⣀⠈⠀⠂⢠⠀⠉⠙⠿⣿⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡈⠐⠠⡀⠀⠁⠂⠄⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢸⠈⢧⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢐⠠⠀⠀⠑⠢⢀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠀⠀⠳⣄⡀⠀⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢤⣤⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⡠⠁⠀⠀⠄⠈⠙⠶⣄⣀⠀⠀⠉⠀⠆⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣁⢤⡉⢿⣿⣿⣿
⣿⣿⣿⣿⠟⠀⠀⣠⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠛⠋⠤⠔⠒⠀⠀⠀⠘⠛⠛⣉⡉⢤⠤⠈⠩⠄⠴⣅⠀⠙⠛⠋
⣿⣿⣟⣁⠠⠔⠊⣁⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠊⠉⠉⠈⠉⠉⠁⠈⠀⠒⠀⠀⠤⠠⠒⠀⠀⠀⠀⢀
⣿⣿⣿⣿⠃⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢹⣦⣤⣤⣶⣿
⣿⣿⣿⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠿⠿⠿⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠤⠄⠀⢀⣀⣄⡀⠀⠠⠤⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⠷⠄⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠤⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⢀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⠿⠋⠁⠀⠈⠐⠠⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠄⠂⠁⠀⠙⢿⣿⣿⣿
⣯⣠⠄⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣀⢈⣿⣿
⣿⣏⣠⡟⢀⡄⢀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠰⣆⠘⣿⣿⣿
⣿⣿⣿⣤⣾⣤⣾⣿⣿⣶⡶⠒⠀⠀⠠⣤⠄⠀⠀⠀⢀⠀⠀⠀⣤⡄⠀⠀⠒⠂⠰⣯⣴⣿⣿⣄⣵⣄⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠤⣄⣚⣒⣒⣒⡒⠲⠆⢾⠒⢞⠛⣛⣛⣛⣛⣲⡦⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠾⠋⠩⠩⠀⠈⠚⢀⡿⠀⠚⠛⠩⠍⠉⠟⠓⠆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⣀⣀⣀⣀⣀⣀⣀⣩⣶⣌⣐⢀⣀⣀⣀⣀⠐⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠐⠢⡀⠀⠀⠀⡐⠀⠣⠀⠠⡀⣀⠀⠀⠀⢚⠒⠂⣘⡈⠉⠛⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⢀⡠⢂⡊⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⡅⡀⢠⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠉⢀⠀⠀⡀⠀⠀⢀⠄⠂⠙⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣀⠀⣳⠘⠀⠀⠑⠠⢀⣀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡡⣤⠜⢹⠋⢿⠱⣆⣡⡀⠀⠀⢠⣄⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠳⠏⠂⠒⠊⠀⠘⣏⡘⠀⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠤⢂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⣆⠈⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡠⠐⠀⠉⠈⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⡔⡄⣤⡘⣧⠀⠀⠀⠀⠈⢀⠔⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡇⢰⡐⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠉⠭⠤⠤⠀⠱⠼⠆⠠⠀⠀⡤⠁⠀⠀⠀⠀⢣⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⡠⠊⠀⠀⠀⠀⡠⠀⠈⠄⠰⠀⠀⠀⠀⠁⠈⠉⠀⠈⢹⠀⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢄⡀⠃⠰⠀⠀⠀⠀⢀⠴⠞⠅⠀⠀⠀⠀⡴⠀⠀⠀⢹⡀⡃⠀⠀⠀⠐⡂⠀⠀⠐⠺⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⢀⡠⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⣀⠊⠘⡀⠀⢀⣀⠇⠐⡀⢀⠀⢀⡭⠉⠉⠉⠉⠀⠀⢰⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠥⠼⠤⠬⠆⡀⠀⠐⠀⠀⠒⠂⠀⠤⠎⠀⣄⡊
⠀⠀⡀⠄⢠⠀⠐⠂⠐⠃⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠴⠦⠄⠀⠄⢀⠀⠀⠀⠀⣤⠚⠉⠁⠀⠀⠀⠄⠐
⠀⢮⠀⠠⠁⠀⠀⠄⠂⠀⣼⠂⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠉⠉⠁⠀⠀⠀⠈⠉⠈⠙⠉⠉⠀⠀⠀⠁⠈⠁⠀⠀⠈⠀⠐⠂
⢀⠁⠀⢀⠀⠀⠀⠀⠀⡴⠹⠖⠒⡖⡶⠒⠀⠰⠶⠀⠀⠐⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠠⠄⠠⠤⠤⠄
⠈⠸⠀⠀⠑⠂⠐⠢⠐⢁⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⢉⠠⠀⡤⢀⡀⠒⠁⠀⠀⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠠⠀⡃⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠶⠤⠎⠀⢉⠴⠀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠸⠤⠤⠤⠆⢧⣄⠤⠄⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠿⢿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⠟⣡⣾⣿⣷⠸⠄⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣩⣍⡉⠛⠈⠻⣿⣿⣿⣷⣄⢸⡏⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣸⠟⠉⠀⠀⠰⣶⡙⣿⣿⣿⣿⡜⣿⡌⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠿⠟⢛⣡⡀⠀⠀⢀⣴⣷⣝⣿⡌⣿⣿⠿⠟⢈⣃⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣴⣶⣾⣿⡆⠹⣷⡀⣰⣿⣿⡟⣛⣭⣵⣶⣶⣾⠿⠛⢻⡿⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣰⣿⣿⡿⠋⠀⠈⠙⢷⠸⣿⡟⣰⡿⠻⠿⠛⠛⠙⣏⢿⣷⣅⣶⣦⣍⡙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠙⢰⣿⣱⡄⠀⠀⠀⠸⠞⣏⢿⣿⣿⣿⣿⣿⡷⠆⠉⣿⡇
⠀⠀⠀⠀⢀⣴⡿⠋⠰⠺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣽⣷⡽⣆⠀⢀⣀⣤⡼⠆⠁⢠⣍⡉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⣴⡇⠀⠀⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠁⠽⠛⣿⣿⣬⣯⣭⣥⣀⣀⣀⣀⢸⣿⣿⠀⠀⢠⣴⣿⡇
⠀⠀⠀⠀⢹⡇⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢛⣉⣁⡙⠄⣤⣙⣛⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⢀⣠⣴⣿⣿⠟⣩⣿⣿⣿⣿⣿⣦⡈⠛⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠛⣉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⣿⡀⠀⠑⢲⣦⣄⡀⠀⠀⠀⠀⢠⣿⠋⣠⣶⣿⣿⣿⠀⢘⡻⠿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠰⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣆⠀⠉⠉⠙⠿⣷⣤⣀⣰⡿⢃⣼⣿⣿⣿⣿⣿⡇⢺⣿⣷⣶⡍⠉⠀⠀⠀⠀⡀⠢⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⢀⣤⣌⡙⠛⠛⢁⣾⣿⣿⣿⣿⣿⣿⡇⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⢭⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣴⣶⣶⣶⣦⣬⣉⣉⣉⣉⡉⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⠟⣉⣴⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⢿⠿⢋⣤⣾⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣿⣿⣿⡇⠀⢰⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠂⠉⠉⠉⠉⢻⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⡀⠀⠀⢿⡄⠁⠁⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠰⠤⢐⡤⠖⠋⠀⠀⠀⠀⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠋⣉⠟⠃⠈⠶⠴⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡿⡀⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢠⠟⠷⠙⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠛⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⢀⠇⠀⣠⢞⣓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢸⢀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠴⠃
⠀⠀⠀⠀⠸⡘⠤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⢀⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠙⢤⡚⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣰⣁⣀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡾⢻⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⡙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠸⣁⣉⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠛⠯⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣤⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠿⣿⣦⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉

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

⢀⡤⠎⡀⠴⠛⠀⠁⢰⣟⣓⣀⣌⢠⡞⣿⣿⣿⢿⣆⡀⠀⠀⠀⠘⠿⢿⣶⣾⣿⣿⣿⢦⣄
⢉⡴⠟⠡⠂⣰⣶⣾⣯⣿⣿⣙⢟⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠉⢤⡽⢷⣄
⣮⡶⠄⢀⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠚⠂⠀⠛⠂
⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢰⡄⠀⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢻⡄⠈⣷⡄
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢿⡄⠘⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣷⠘⣿⣿
⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠘⢿⡆⠈⢿
⣿⣿⣿⣷⣿⣿⡝⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⠀⠀⠀⢿⡀
⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⣋⣮⣽⣿⣧⠀⠀⠀⠘⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡷⠀⠀⢸⣿⡿⠉⢹⣿⣿⠄⠀⠀⠀⣹⣷
⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠟⠁⠀⠀⠀⠻⣿⣇⢀⣿⣿⠀⠀⠀⠀⠘⢿
⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣬⣿⡏⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠇⠀⠀⠀⠀⢠⣤
⣿⠿⠿⢿⡟⠩⢿⣿⣿⣿⣿⡐⠀⠀⠀⠤⣤⣴⡶⠿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⣾⣿
⠃⢀⣴⠟⠂⢀⣿⣿⣿⡇⡈⢻⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢉⣿
⠀⣼⠋⠀⢀⣾⣿⣿⣿⡏⠁⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣤⣤⣴⣿⠇⠀⠀⣰⡄⣸⡟
⠀⣿⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠈⠲⠶⢛⣯⣿⣿⡿⠁⠀⠀⢠⣿⢠⣿⣧
⠀⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⣠⣴⣿⠏⠀⢀⡀⢤⣿⣿⡿⠁⠀⠀⠀⣾⡏⣼⢧⣿
⠄⢸⣧⡶⠀⢻⣿⣿⣿⣇⠀⠀⠀⢻⣾⣿⣆⠀⠀⣰⣿⣿⣟⣫⣤⣾⠿⣷⢺⣿⠟⠀⠀⠀⠀⣸⡿⣰⢟⣼⣿
⣾⠀⣿⡇⠀⠈⣿⣿⣿⣿⡄⠀⠀⠈⢿⣿⣿⡧⢀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠟⢡⡟⣾⣿⣿
⡩⠀⢿⣧⡀⠀⠿⣿⣿⣿⣷⡀⠈⠀⠈⣻⣿⣿⣾⣷⠈⢻⣧⠛⢿⣿⢿⣿⡗⠀⠀⠀⠀⠀⠀⠀⣾⢷⣿⣿⣿
⣷⡀⠈⢻⣇⠀⠀⠈⠻⣿⣿⣿⡄⠀⠘⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⣶⣿⡇⠀⠀⠀⠀⠀⡀⢠⣇⢸⣿⣿⡟
⣝⢷⡀⠀⠹⣷⣄⠀⠀⠀⠛⠿⣿⣆⠀⠀⠀⢿⣿⣿⣿⣄⠀⠀⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⣇⣸⡿⢧⣿⡿
⣿⣎⠻⣄⠀⠈⠻⣦⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠙⢿⢿⣿⣷⣧⣄⡀⠀⢨⣿⠀⠀⠀⠀⠀⣿⠃⠘⣿⣿⡇
⣿⣿⣧⡘⢧⡀⠀⠀⠓⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣷⣤⣿⠀⠀⠀⠀⠀⣿⠀⢰⣿⣿⡇
⣿⢻⡏⣻⣌⢳⡄⠀⠀⠈⠙⢿⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⢿⠀⣿⣿⣿⣇
⠀⠘⢿⢃⠙⠄⡙⢆⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⢠⣿⣿⣿⣿
⣠⣿⣷⣮⣤⡀⢡⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡀
⠛⠛⠿⣿⠉⠑⠄⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶
⣿⣿⣏⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣯⣿⠀⢻⣿⣿⣿⠛⣛⡻⢷⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡄⠀⣴⣻⡛⣷⠹⣟⡄⣼⣿⣿⣿⢿⣿⢿⡽⣶⣿⡽⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⠴⣼⡏⠟⣢⠼⠛⠋⠀⠀⠀⠀⠀⠉⣉⠓⢶⣔⠩⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡧⣵⢋⣀⣀⣀⠀⠀⠀⠀⣴⣾⠟⠋⠙⠂⢻⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣈⣷⣉⣉⠙⠻⠿⠀⠀⠈⢉⡠⠴⠶⣤⣤⣼⣨⣍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⢸⣼⢿⣿⣿⣿⣳⡄⠀⠀⠀⣺⣶⣷⠿⣿⡟⢛⡟⡿⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⣀⠘⣿⣿⢿⣟⠿⠟⢳⡀⠀⠜⠛⢛⣋⡴⠟⠉⢸⢟⠻⠿⢂⠌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣧⠀⠉⠁⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢈⢪⣉⣋⢶⡿⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠳⡸⡆⠀⠀⣰⣿⣛⣅⣷⡀⠀⠀⠀⠀⠀⢼⣷⣗⣿⣷⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣷⣰⣷⣿⣄⠈⢻⡀⠁⠀⢉⠛⠁⠀⠀⣀⢾⣜⠉⣬⡿⣟⡻⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣮⣿⢈⢻⣈⡳⣄⠙⠛⠛⠃⠀⡀⠠⢊⡁⢮⣾⣧⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢹⡏⣯⡿⣒⠀⠀⠀⠀⠀⠀⠈⣷⡁⠅⢳⠑⢬⠳⣤⣤⠖⠙⠀⣒⢶⣬⣛⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣷⡓⠌⢳⣦⠳⢙⡟⠀⠀⠁⢼⠷⣣⣈⣭⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⡛⠀⠠⠀⢿⡔⣻⡆⠀⠀⠀⣼⠀⢹⣿⣿⢟⡿⡻⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⠋⠙⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣟⡙⠛⢷⣶⡛⠻⣿⣿⣿⣷⣄⣴⣶⣦⣤⣴⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣶⣿⣿⣿⣿⣿⠄⢹⣿⣿⣿⠉⠁⠈
⣿⣿⡿⠉⠹⣿⠟⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡛⣿⣿⣿⣷⣽⣿⡏⠀⠀⠀⠀⠀⠀⠐
⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⡀
⣿⡧⠀⠀⠀⠀⠀⠐⠀⢀⣀⡀⢉⠿⣷⣤⣄⣼⣿⠌⢀⣶⠾⣟⣃⣀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⡶
⣿⣷⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣮⣿⣽⢿⣧⣶⣿⡿⠛⠙⣿⣏⣙⣿⣥⣶⣤⠈⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⠛⠉
⣿⣿⡇⠀⠀⠀⠀⢰⠋⢤⣼⣿⣿⣿⣿⣿⣿⣷⣽⠧⡉⠳⣲⣴⣿⣿⣿⣿⣷⡞⢱⣿⣤⡀⠀⠀⣨⣿⣿⣿⣿⣿⡇
⣿⣟⠀⠀⠀⠀⠀⢁⠴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣟⠢⡙⢄⣻⣿⣿⣿⣿⣿⣿⣟⡿⣿⣌⣱⣆⠀⠉⢻⣿⣿⣿⣿⠃
⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣳⣟⣽⣿⡏⠂⢀⣽⣿⣿⣿⣽⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀
⣿⠋⠀⠀⠀⠀⠊⡩⢤⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⢋⣿⣿⣿⡯⢹⣿⣿⡿⣾⣹⣿⣷⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣤⡼⠁
⣿⡆⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡻⣟⣿⣿⣿⣿⣟⢷⢿⣯⣿⣿⠻⣿⠿⣺⣾⠿⢾⣿⣙⠀⠸⣿⠟⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆
⣿⣇⠀⠀⠀⠀⠘⡡⠾⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣎⣧⣉⢻⣿⣾⣥⣾⣿⣿⣿⣿⠺⣿⡀⠐⠓⠀⠀⠀⣿⣿⣿⠿⠛⠋⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣤⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢿⣧⢹⡏⣿⣿⣿⣿⣿⡞⣿⣧⡈⢹⣿⣿⠓⠌⠟⠋⠁⠉⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧
⣼⣿⣿⡀⠀⡄⣸⡿⠟⢻⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡝⡿⣿⠀⣿⣿⣇⠸⣿⡿⢇⣄⠀⠀⠂⠈⣻⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠁⠁⢀⣤⣾⣿⣿⣿⣿⣿⢿⣿⢹⣿⠟⣿⢀⠛⠀⢻⣿⣿⣷⣶⣶⣿⣏⣀⣤⡤⣾⣏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣄⠀⠀⠀⢠⠟⠛⢙⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣂⣴⣶⢤⢼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠁⠀⠀⣤⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢟⣁⡤⣧⡴⠁⢀⣴⣿⣿⣿⣿⣿⣿⣧⣹⣾⣿⠿⣿⣿⣷⣾⢼⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⠛⠋⠉⣄⣤⣫⠏⠸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷⡘⣿⡋⢁⣿⣿⣿⣿⣿⣷⣄⣠⣤⣦⣄⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠶⠶⠶⠄
⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠦⣈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⣤⣾⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡉
⠀⣤⠙⣿⣿⣆⠐⣶⡄⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢀⢴⣓⠀⡀
⠀⣿⣄⠉⢻⣿⣭⣿⣯⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢺⣯⣬⣶⣡⣾⣿⢣⠈⠄
⠀⠻⣬⡆⠀⠙⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣿⣿⣿⣿⣿⣿⠏⣧⣄⢀⠀⢀⢀⢄
⠀⠀⡝⢿⣦⡀⠈⠻⣿⡟⣿⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⢩⣥⢀⣤⣬⣃⣾⣿⣿⣿⣿⣶⡆
⠀⢠⡧⠈⢻⣷⡄⠀⠘⢧⣿⠻⡇⠀⠀⠀⠠⢀⣀⣀⠀⣠⠏⠀⠀⠀⣀⠴⢷⣿⣼⣿⡯⠀⠀⣀⣬⡟⢉⣿⣿⣿⣿⣿⣿⡡⠂
⠀⢸⣿⣮⠄⠈⢡⣄⠀⠀⢿⡄⢧⣤⣤⣤⣷⣶⣦⣽⣽⣿⣤⣶⣤⣶⣤⣴⣿⣿⣭⣽⣿⣤⣨⣿⣿⣧⣿⣿⣿⣑⣘⣋⠁
⠀⠈⢰⢸⠀⠀⢰⠿⣿⢇⢨⡞⠈⠛⠛⠛⠛⠛⠛⠛⢋⠛⠉⠛⠛⠛⢛⣛⠛⡛⡛⢿⣛⠛⠛⡿⡶⠓⠀⠀⠀⠀⠀⠉⠉⠈⢹⠿⠂
⠀⠐⣠⡀⠀⠀⠀⠀⣟⣀⣩⠁⠀⠀⠀⠀⠈⠉⠉⢉⣉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡏⠉⡉⠉⠈⡹⣽⣟⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⣷⠀⠀⠀⢀⣿⠃⢹⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⣿⠦⠀⠀⢸⢿⣆⣈⣠⣤⣿⣶⣀⣤⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣾⡇⠆⠀⡟⣸⢯⢉⠁⢩⣿⣏⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣷⢸⡀⢸⣇⣛⠸⣾⣾⠚⣿⡏⠀⠐⣖⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉
⠀⠀⣸⣿⣷⣾⣿⣿⣶⣧⣄⡈⢹⣿⡇⠉⠉⠏⠉⠉⠉⠙⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⡀⠀⠀⠈⣿⡿⠿⠿⠟⠋⠙⠃
⠀⠐⠿⠿⠿⠟⠉⠉⠫⣿⣭⣽⣼⣙⠃⠁⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠒⠶⣶⣶⡦⠋⠉⠉⠋⠉⠈⠉
⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⠁⠀⠃⠀⠈⠁⠈⠀⠀⠀⠀⠁⠁⠑⠈⠁⠁⠃⠉⠁⠘
⠀⠀⡀⠀⢀⠀⠠⠀⡄⡀⢀⠠⠀⡀⢀⠀⢄⢀⠀⠠⠀⢀⠠⠀⠀⢀⠀⠄⠀⠀⢀⠀⣀⠀⠀⡠⠂⡄⡀⠠⢀⠀⡀⠀⠀⠄⠐
⠀⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢠⣶⠾⠛⠋⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⢿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢿⣿⣿⣿⣿⣷⠄⢻⣧⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⣤⣤⡀⠀⠈⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣤⣼⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣷⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣠⣿⠟⣿⡿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣋⣹⣿⡆
⠀⠀⠀⠀⠀⢠⡿⠁⣰⡿⠃⣿⡟⠉⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠉
⠀⠀⠀⠀⠀⠸⣧⠀⠛⠳⠀⣿⡇⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⣤⣤⣄⡀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢻⣶⠾⢿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣸⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠾⠿⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡯⠉⠛⠛⠻⣿⡏


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⠶⣦⣄⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣶⣥⣀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀⢻⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢟⡿⠉⢿⡀⢹⡇⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⢋⡞⠀⠀⢸⣧⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣶⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣹⡏⢰⡞⠀⣰⡇⠈⠿⠀⠂⢀⣿⡿⣿⡏⢾⣿⣦⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠙⢻⣿⣿⣿⣯⣿⣿⣿⣾⣿⣿⠃⢸⠀⢰⣿⠁⠀⠀⠀⢀⠞⣿⣿⣟⡹⣿⡄⢀⡙⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠳⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠿⠀⠈⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠟⠉⠛⠟⠙⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⢐⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠓⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣦⡊⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⡧⠀⠀⠀⠀⢻⣿⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⠙⠛⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠘⢿⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣹⣿⣿⣿⠃⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣓⣦⣼⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿⣿⣿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠙⠉⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⡟
⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⠁⠽⡿⣶⠀⢰⡾⡿⠍⠁⡟⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⣄⣀⠀⠀⠀⠨⢳⡀⠀⠀⠉⠀⠈⠁⠀⠀⣠⠫⠀⠀⠀⢀⣀⡴⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠒⠦⠖⠂⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠦⡀⠀⠀⠈⠢⠈⠁⠀⠀⠀⢡⡀⠀⠀⢄⠈⢙⣓⣛⠁⢀⠄⠀⠀⣠⠃⠀⠀⠀⠉⠠⠊⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠈⠐⠢⣶⣤⣍⡀⠀⠀⠈⠙⠢⣄⠀⠁⠀⣉⡁⠀⠁⢀⡤⠚⠉⠀⠀⠀⣈⣥⣴⡦⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣀⠀⠀⠀⠙⢿⡶⢄⡀⣀⠴⣾⠟⠁⠀⠀⢀⣠⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⠠⠙⠂⢉⠴⠚⠁⢀⣤⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡶⠿⠛⠉⠀⠀⠀⠀⠀⠤⠔⢊⡵⠟⠙⠻⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠒⢀⣴⣯⣤⣄⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠤⠐⠀⠈⠉⠉⠉⠀⠀⠀⠔⠅⠀⠀⠈⠉⠉⠉⠀⠐⠠⢤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⣀⡤⠔⠈⠀⠀⠀⠈⠐⠤⣄⣀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⡀⠀⠀⡠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠁⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠈⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⢀⣤⣾⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣷⣦⣄⠀⠈⠛⢛⣋⣠⣤⣴⣶⣾⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⢠⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⣤⣶⣶⣿⡿⠿⠟⠛⠋⢉⣠⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⣰⣿⣿⠋⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠿⠿⠛⠋⣉⣁⣤⡄⢠⣾⡿⣿⣮⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠘⢋⣉⣡⣤⣶⣶⣿⡿⠿⠟⠛⢉⣉⠀⢤⣴⣶⣶⣄⣿⡟⠋⠁⢸⣿⡀⠈⠁⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣈⣤⣴⣶⣾⣿⠿⠿⠛⠛⠉⢁⣤⣤⣶⡆⣾⡟⠻⠷⠸⣿⣄⣸⣿⢸⣿⡿⠿⠀⣿⣧⢀⣶⣷⠀⢹⣿
⠀⢠⣴⣶⣿⡿⠿⠟⠛⠋⣉⣤⣤⡀⣶⣿⠿⣿⡆⣿⣏⣀⣄⠻⣿⣶⣶⣄⢿⣿⠛⠋⠀⣿⣧⣴⣶⡈⠻⠿⠟⢁⣀⣤⣤⣶⣶⣧
⠀⠀⢉⣩⣤⣤⣄⠸⣿⡰⣿⡋⠙⠛⢸⣿⣤⣾⡟⢸⣿⡛⠋⠀⣤⣄⣸⣿⠸⣿⠄⠀⠀⠘⣉⣉⣤⣴⣶⣾⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠈⣿⡏⠙⣿⡇⢿⣇⠙⠿⠿⢿⣦⣿⣏⠹⣷⣄⣿⣷⡾⠿⠙⠛⢛⣁⣠⣤⣴⣶⣿⡿⠿⠟⠛⠋⢉⣀⣤
⠀⠀⠀⢹⣷⠀⢹⣿⠸⣿⡴⣶⣤⣾⡿⠸⠿⠀⠙⣉⣈⣤⣤⣶⣾⣿⠿⠿⠛⠛⠉⠉⢠⣤⣶⠆⠀⢀⣿⣿⠇
⠀⠀⠀⠈⣿⣷⣾⠟⠀⠛⠃⢈⣉⣡⣤⣴⣶⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⢀⣾⣿⡟
⠀⠀⠀⠀⢉⣁⣤⣤⣶⣶⣿⡿⠿⠛⠛⠉⢉⣤⣴⣦⣄⣀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠟⠁⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠈⠿⠛⠋⠉⠁⠀⠰⣶⣷⣦⡀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠰⠖⠀⠀⢻⣆⢸⣿⡄⠀⠀⠀⠀⠀⢰⡟⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⢠⠴⣦⣄
⠀⠀⠀⠀⠀⠀⣠⠴⣲⠆⡀⠘⠓⠋⠀⠀⠀⠀⠀⠻⢾⡟⢿⣾⣧⠀⠀⠀⣿⠁⠘⠟⠁⠀⠀⣞⠛⣉⣉⣩⡿⠓⠾⠟⠀⣤⢦⣤⡀
⠀⠀⠀⢀⣴⣏⣀⠐⠛⢩⠟⣀⣾⣷⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢮⡐⠻⡆
⠀⢠⠞⠉⠀⠀⠈⠻⣦⡞⣥⣿⠉⠈⠀⢩⡟⠀⣀⣀⡀⠀⢀⣠⡤⠶⠶⣖⠒⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠶⠃
⠀⣿⠀⠀⠀⣀⡀⠀⣹⠀⠈⠉⢻⣤⠤⢿⠀⠀⢯⡉⠉⠙⠻⣅⡀⢀⡀⠈⢳⡀⡀⠀⠈⠙⠳⣄⠀⠀⠀⠀⢰⡶⣤⠏⣷⢾⣆
⠀⠹⣄⠀⠸⠋⣠⣠⠏⠀⠀⠀⢸⡆⠀⢸⡇⠀⢠⣿⣶⡄⠀⠈⠋⠀⣙⣦⡀⠉⠁⠰⠋⣴⣀⣈⢷⡀⠀⠀⠈⢳⣄⠀⠀⠹⣿⡄
⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⣷⢰⡯⠤⢴⡶⣠⠀⣈⣭⣭⡉⠁⠀⠀⠀⠀⣀⣠⣀⠀⢿⡀⠀⢀⡾⢧⣄⣴⠚⠉⠁⠀⠀⣠
⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠹⣿⠀⠀⣾⠞⢁⡼⠋⠀⠀⠙⠶⠚⠉⢻⡄⠈⠳⣄⠘⠛⠉⠛⠻⢦⣠⡟⠁⠀⠀⠀⣴⣾⡋
⠉⠉⠉⣩⡿⢟⣡⣤⣴⣄⣀⣤⠄⣿⠀⠀⠀⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⢳⠀⢀⢲⣤⣄⣴⠟⣠⡾⣿⣦⣴⠾⠟⠋
⠀⠀⠀⠿⠟⠛⠉⠁⠛⠛⠉⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⣿⣦⡧⣰⠏⠘⠉⠀⠉⢁⣠⣤⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡤⠸⡇⠀⠀⠹⣄⠀⠀⠘⢆⣀⣀⡤⠴⣄⠀⠀⣀⣀⠀⠀⡼⠁⢀⡿⠁⣰⠇⢀⡤⠖⠛⢻⣋⣤⡴⠏
⠀⠀⣀⣴⠾⢻⣿⣴⠾⣷⠾⠋⠀⠀⢷⠀⠀⠀⠙⣦⡀⠀⠀⠉⠁⠀⠀⠈⠛⠉⠁⠸⠦⠾⠁⢀⡾⠁⢠⠏⢀⡞⠀⡴⠂⠀⠉⠀⠹⡄
⠀⠾⠋⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢠⡟⠀⠸⡇⠸⡇⢰⣷⣄⢠⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠴⢤⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠖⠖⠒⠛⠋⠁⠀⠀⢠⡟⠀⠀⠀⠳⡄⠁⢻⠀⣼⣿⢠⠏
⠀⠀⠀⠀⠀⠀⠻⣤⣼⡀⠀⠀⠀⠀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣤⣄⡀⠈⢳⠏⠀⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠹⣄⠀⠀⣾⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⡷⠀⠈⠓⠛⢡⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠹⣆⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣷⠀⠀⣀⡞⢻⠷⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⣿⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⡀⠀⠀⢻⠀⠀⠀⠀⠀⡇⠀⠘⢧⣤⠀⢠⠶⠞
⠀⢀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣈⣷⣄⣀⣀⡶⠋⠁⠀⠀⠀⠀⠀⠉⠳⣄⣸⣄⣀⣀⣀⣠⣧⣀⣀⣀⣈⣉⣉⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⢛⡏⢀⣸⣧⡀⢠⣠⣀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣬⣥⣭⣤⣤⣅⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣸⠋⠉⢉⡽⠗⠈⢧⣀⠀⠈⢯⠉⠁⢹⡙⢀⡁⠉⣡⢶⡖⠛⠛⢛⡷⠶⣄⠀⠀⠀⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠰⣏⣉⣁⣀⣀⣀⡼⠃⠀⠀⠀⠈⠙⠲⢬⣳⡄⠀⠀⠛⠋⠀⠙⠒⣃⠀⠀⡼⢦⣀⡘⣇⠀⠀⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠖⠚⠋⠉⠙⠛⠁⠀⠈⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⡀⠘⡴⠛⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣃⠀⠸⠁⣰⢻⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⣉⠀⠀⠈⠁⠀⢻⠢⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡇⠛⠛⠉⠀⠀⠀⠀⠠⠇⣦⡀⣠⠶⠄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠽⠖⢲⡀⠀⠂⢀⡀⡠⠊⠈⠀⠀⠈⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⡏⠈⡀⠠⠄⠀⠠⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⣀⠀⠠⠩⠀⣀⠀⣀⢇⣀⡄⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡕⠀⠀⠀⠐⠀⡈⠀⠁⠀⠀⠠⠂⠁⠀⠉⠀⠀⢰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡳⡀⠀⠀⠀⠈⠀⢀⢠⢸⠀⠀⠀⠁⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡌⠇⠀⢠⠀⠀⠀⠉⠈⠁⣀⠀⠀⠀⣀⣴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠘⢀⠀⠰⠚⡄⣤⡐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠔⠀⠀⠀⠂⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⢤⣴⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣿⣿⣿⠟⠛⠀⠘⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠀⠉⠉⢽⣆⠠⢀⡀⠀⠀⠀⠀⡠⠀⠲⣤⠀⠀⠃⠀⠉⠯⠉⠁⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⣎⠁⠀⠀⠀⢠⣄⢀⡶⠾⠿⠀⡀⠉⠀⠀⠀⢀⠅⠀⠀⡘⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠈⠑⠢⠤⢄⣀⡈⠉⠁⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⢰⡀⠀⠀⠑⠁⠀⠀⠈⠈⡙⠀⠀⠀⠀⠀⡠⠘⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡟⠇⠂⠀⠀⠀⠛⠂⠂⠀⠀⠀⠚⠀⠀⣤⣧⠀⠀⠀⠀⢠⠇⡆⠀⣀⠴⠋⠅⢰⡟⠋⠙⣦
⠀⠀⠀⠀⠀⠰⠀⡔⠀⡄⠠⡄⠀⠈⢲⡏⠛⠙⢻⡹⡶⣗⠘⠋⠀⢹⠀⣰⠀⠓⠶⠶⠒⠖⠛⠁⠚⠃⠀⠀⠀⠀⠀⠀⠉⠓⠐
⠀⠀⠀⠀⠀⠀⠸⣷⡄⢸⠂⠓⡿⣾⢻⣿⠀⢰⣦⠁⠤⣭⣢⡄⠀⣼⣀⠀⠀⠀⠀⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢐⣟⠀⣠⣀⣬⡄⠋⢀⡉⠐⠛⠛⠛⠃⠋⢀⠁⠀⢙⣿⡅⠀⠀⠁⠉⢩⠀⡀⡇⠀⠀⠀⠀⠀⣻⣏⡼⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠈⠁⢰⠢⢎⣀⣶⡄⠀⠀⠀⠀⢸⣇⠀⠸⠟⠀⠀⠀⡀⠀⠀⢀⡅⠃⠀⠀⠀⠀⠀⠿⠻⢴⡟⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠘⠢⠀⠀⠀⠀⢸⢈⠀⠀⠀⣤⠠⢤⡁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠈⠉⠀⣀⣈⢎⣿⣶⣤⣀⣂⡄⠲⠄⠀⠀⡄⠀⡧⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢀⣀⢀⡀⣰⣤⡘⣧⣿⣿⣿⣿⡟⠃⡰⠶⠒⠚⣷⣖⠀⠐⢈⣁⠀⠐⠃⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠤⠔⠞⠋⠘⠇⠈⠁⠣⣼⣟⣻⣿⣿⣿⣿⣆⠀⠁⠀⠀⠀⡾⠃⠀⠀⠀⠉⠀⠀⡀⠢⢴
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠞⡻⢁⢀⣤⣄⣤⣘⢦⣀⢠⣤⣴⣿⣿⣹⠉⠈⡟⠿⠘⠀⠀⠀⠀⠀⢀⣺⠁⠀⠀⠀⢠⠌⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠠⠋⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⠀⠀⣶⣿⣳⣿⡆
⠀⠀⠀⠐⢤⠀⠀⢀⡞⣰⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⢸⡿⣿⣿⣧⣀
⠀⠀⠀⠀⠁⠀⢸⣿⣧⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣥⠆⠀⠀⠀⢻⡿⠅⠀⠈⠀⠈⠻⣏
⠀⠀⠀⠀⠀⠀⣰⡟⣽⡇⣸⣿⡇⣿⣿⣿⣿⢿⠿⠛⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠋⠋⠀⠀⠀⠀⠘⢿⣿⡦⠤⢤⡄⣀⠐⠋⣀
⠀⠀⠀⠀⡄⠀⢿⢣⣿⡀⣿⣿⠃⣿⣿⣿⡇⠀⠀⠀⠠⣶⣿⣿⣿⡿⠿⠿⢿⡿⣿⠿⠶⠖⠤⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⣾⣿⠀⠀⢺⣿⢰⣿⣿⢘⣿⣿⣿⣿⠀⠀⠀⠲⠼⢉⠈⡯⠛⠛⠷⠶⣤⣄⡀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⢼⣿⡇⠀⢀⠀⠸⣿⡿⣼⣿⣿⣿⣿⡆⠀⢀⠐⠀⢠⠐⠇⠰⣷⣷⣄⣘⠙⠛⣿⣷⣶⣦⣌⣉⡙⠒
⠀⠀⠀⠸⣿⢇⠀⢐⡤⢳⣿⣿⡿⣿⣿⣿⣿⣷⠀⣾⣿⠀⢸⡆⠀⠀⠸⠁⠈⠙⠻⢶⠀⠀⢀⣀⣉⠛⠻⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠐⢿⣿⢀⡿⢀⣼⣿⡟⢁⡟⠻⣿⣿⠟⢀⣿⣿⡇⢨⡁⠀⠐⠀⠀⠀⠀⠀⠸⠆⠀⠀⢿⠛⠻⢶⣦⣭⡙⣏⠙⠿⣿⣶⣦
⠀⠀⠀⠀⠉⢳⣼⠀⢉⣴⡟⢠⠚⠛⣫⣼⠋⠀⢼⣿⣿⣿⣬⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠃⢹⠀⠙⢰⣶⣤⣆⡉
⠀⠀⠀⠠⠞⠸⢃⣴⣾⣿⠡⠃⣠⣾⡿⠃⠀⢀⣿⣿⣿⡯⠡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⡀⠀⠀⢸⠀⠀⢘⣿⠉⠙⠋
⠀⠀⠀⠀⠀⡟⠀⣶⣿⡅⣡⠾⢋⣼⠃⠀⠀⠀⣿⣿⣿⣿⡲⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⢠⣰
⠀⠀⠀⠀⠘⢀⣼⣿⡏⢀⡴⣾⣿⠃⠀⠀⠀⣀⣽⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣛⠘⣻⠆
⠀⠀⠀⠀⣶⣿⣿⡿⢶⣶⣶⣿⣿⣀⣤⣤⣤⣿⣿⣿⣿⠇⠀⠀⠸⡆⠀⠀⠀⠀⠀⠠⠂⡐⢪⣅⣉⠐⠐⠫⠍⠉⠉⠉⠙⠛⢿⣶⡀
⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠚⠿⠿⠿⠿⠻⠿⠛⠛⠛⠿⠷⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⢌⡙⠻⢤⣘⣂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣒⣭⣁⠀⠀⢀⣄⡀⠀⠀⠀⠈⠿⢷⣦⣌⡙⠳⠦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠈⠉⠀⠘⠃⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠂⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣹⠢⢄⡀
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⠃⠭⣁⠀⢿⡲⢦⣄⣀
⠀⠀⠀⠀⠀⢠⣿⠀⡦⣿⠃⡶⠶⣂⣿⣿⠑⠊⠀⠀⣄⠀⠍⠀⢈⢁⠠⢠⣴⣛⢺⣿⢦⡀
⠀⠀⠀⠀⠀⢸⣟⣠⣏⣿⠄⣠⡄⢹⣻⣿⡇⠀⢔⣦⣟⡥⠀⠀⢸⣙⠺⢶⣚⡋⣸⣿⡇⢩⣭⡄
⠀⠀⠀⠀⠀⢸⡏⠿⢿⣿⡄⢽⠇⢾⣿⣿⣇⣰⣶⣖⣤⡥⠀⣶⣿⣶⣾⣤⣤⣥⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣷⢲⣾⡇⣒⣀⣤⣼⡿⣿⡿⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠸⣿⣄⣿⠟⠛⠉⠁⠀⠀⣿⡇⠀⠀⠀⣤⡿⣿⣿⣿⣻⠿⣛⠋⠉⠉⠛⡋⢹⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣼⡿⢱⣹⣿⡿⢿⡟⠁⠀⠀⠘⣆⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠟⠁⡾⠿⣿⢃⡟⢀⣠⡤⠶⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⣿⣿⠇⡼⠱⣿⣥⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠟⢸⣷⠶⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠻⢻⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠉⡄⠃⢸⣿⡿⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁⣇⠀⢾⣿⡏⠉⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⢸⣿⠀⠸⣿⠄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠶⠏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠀⣤⠾⠻⢶⣠⢟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡾⠋⠉⠳⠎⠁⠀⠀⢠⣯⠟⠶⣶⠀⠀⣠⡆
⠀⠀⠀⠀⢀⣤⡶⡖⠷⠚⠛⢀⡤⢾⠁⠀⢴⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⢔⣫⠾⠃
⠀⠀⠀⠀⢼⡏⡌⣶⡒⠚⠋⠉⠀⣸⠀⠀⠀⢨⡟⠋⠀⠀⣷⠿⣄⣀⠀⠀⠀⠰⠏⠲⣏
⠀⠀⠀⠀⠈⠻⠇⢹⠻⡄⠀⠀⢀⡟⠀⠀⠀⠘⢷⠄⡠⠚⠉⠠⡌⠛⠒⣆⣀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⣆⠀⣼⢃⠀⠀⠀⠀⢠⡞⠀⠟⠰⠇⠀⠀⠀⠘⠟⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⢧⣿⣸⠀⠀⠀⠀⢸⢠⣄⣑⣠⣄⠀⠀⠠⣄⣰⠃⢀⣹⠆
⠀⠀⠀⠀⠀⡀⢀⡿⠀⠀⠀⢸⣯⡼⢷⡀⠀⠀⠘⢦⡀⠀⠈⠉⠀⠶⣤⡾⠃⣤⠈⢷⡀
⠀⠀⠀⠀⣼⡏⠘⣧⠀⠀⢀⣏⡇⢹⡈⠳⣄⣀⠀⣤⡍⠳⠀⡤⢶⡞⠉⠲⠛⡉⢣⡠⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢸⣿⢷⡛⡿⣇⠀⣼⠷⠤⣄⠙⠲⠤⣭⣉⡁⢹⡤⠖⠋⠉⠉⠙⠓⠶⡌⢆⡙⢌⢾⣧⡀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣀⣤⣀⠙⢦⡀
⠀⠀⠀⠘⠟⠛⠛⠓⣿⣰⠓⠲⡄⠙⣦⠀⣀⣠⠞⠁⠀⠀⠀⣰⠏⣉⣉⣹⡆⢹⠈⠣⠈⠛⠋⢥⣄⠀⠀⠀⠀⢰⢋⡤⣠⡌⠋⢛⡛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⠶⠷⢄⡈⠳⣯⠀⠀⠀⠀⠀⣴⣫⠎⠁⢀⢸⠏⣿⠙⢦⡄⠀⣿⣿⣿⠿⣷⣶⡀⢸⣄⣒⣈⣀⠀⠈⣻⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣫⠞⠁⣠⣀⠀⠉⠳⠘⢷⣄⣀⣠⠞⣱⠃⠀⢈⣹⣆⣀⣧⢀⣾⠇⠀⣿⣿⡿⢿⣧⣾⣷⡀⢳⡒⠛⠉⠀⠙⣽⠋
⠀⠀⠀⠀⣿⡀⡶⠋⢀⡴⢚⡁⠉⠛⠶⣤⣄⣀⡭⠟⢁⡼⠁⠀⠀⠈⠁⠋⠃⢁⣾⠟⠀⠸⣿⣻⣤⣿⢿⣎⠈⢿⣆⠉⢓⣦⠴⠦⠼⢦
⠀⠀⠀⠀⣿⣷⣳⠞⣩⡴⠁⣁⣂⣀⣀⣀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠉⠉⣿⠻⣷⣿⣷⠶⠛⣷⠋⠀⣀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠻⠟⠱⠮⢋⡶⠛⠉⠉⠀⠉⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠸⠀⣇⠹⡿⢋⡴⠚⣉⡷⢾⣇⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡶⢦⣤⣄⡀⣠⣴⣶⠶⠶⢶⣶⣶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦⠙⢁⣠⠞⠁⢀⣤⠟⠒⠀⣤⣤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠉⠙⢋⣾⠃⠀⠀⠀⢹⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣠⠾⠋⣠⠖⠀⠀⠀⣀⡴⣋⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣇⣠⠞⠁⠀⣴⠚⠋⠉⣻⠳⡌⠧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠘⠀⠅⢠⡿⠡⠌⠡⠄⠀⢤⠄⢻⠆⣿⠀⠠⠦⠀⠃⠠⠤⠀⠀⠘⠃⠰⠆⠑⠎⠠⠙⣧⣠⣤⣤⣤⣤⣀⣀⣿⣤⣝⠒⣲⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠿⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣭⡤⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣿⣿⣿⣿⣯⣴⠶⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣾⣿⣿⣿⣿⣟⣢⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣩⣭⣴⣿⣿⣿⣿⣿⣿⠛⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠽⢋⣉⣽⣿⣿⣿⡄⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣿⣿⣿⣿⣦⡙⣧⠀⠀⠀⠀⠀⢀⠖⣉⡽⢿⣿⣿⠇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣿⣿⣯⠻⠿⣼⣿⠀⠀⠀⠀⢠⣿⣾⣿⡷⢫⣿⣿⣦⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠉⠘⠃⠀⠀⠀⠟⠉⠉⠐⠚⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠶⠶⠶⠦⣄⠤⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣤⣤⣴⠏⢀⣠⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣟⣉⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢹⡟⠛⠋⠉⠀⠉⠛⣿⡿⣿⣿⣧⣾⣿⣿⡿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢡⢎⣴⣿⣿⣿⣿⣿⠿⠛⠉⠀⢀⣠⣤⣤⣴⡿⣧⡀⠀⠀⠀⠀⣼⣿⠁⠈⣿⣿⣿⣿⣿⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⠁⢄⣠⣴⣾⣿⣿⣿⣿⣿⠀⠹⣟⢦⣠⡴⠛⠉⣿⡆⢀⣿⣿⣿⣿⣿⡇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⡄⠹⣧⣻⣿⣿⣿⣿⠀⣼⠁⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢏⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⢿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⡇⢠⣇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⡿⢟⣓⢮⡻⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⢋⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⣿⠋⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⣶⣾⣿⣿⢸⣿⣿⣾⣿⣿⣿⣿⣿⠿⠟⠻⣿⣷⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⡿⠋⣿⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠄⠀⠀⠀⣸⢀⡀⠀⢠⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⣰⣤⠚⣀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣷⣧⣾⣥⣶⣷⣾⣷⣼⣯⡴⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⢿⣿⢿⠏⣻⣷⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣷⣿⣿⡿⠿⣿⠟⠻⢿⣿⣿⣶⣭⣍⣙⠻⠿⠿⠿⣿⣿⣿⣿⣿⣶⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡀⣾⠛⣿⣿⣿⠟⠁⡘⢀⡄⠀⠀⡿⠿⠟⠛⢻⣿⣿⣾⣷⠂⠀⠉⠛⠿⣿⣿⣿⣮⣄⡿⠋
⠀⠀⠀⠀⠀⠠⣝⢿⣴⡏⠸⣷⡆⣠⢁⣾⣷⡇⢾⣅⢀⠆⢀⠎⠹⣿⠟⠁⠀⠀⠀⠀⠀⠀⢩⣭⡍⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠠⠌⢻⡉⣧⢱⣼⣷⣿⣸⣿⣿⣷⣽⣿⣿⣶⣮⣼⣹⣯⣀⣤⡀⣀⣤⣤⣄⠀⢸⣿⣃⣤⣤⡀⠀⢀⣠⣤⣤⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⠁⣿⡟⠉⣹⣿⣋⣉⣿⣷⢸⣿⠋⠁⠹⣿⡄⣾⣏⣉⣹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⡏⠉⠀⠀⠀⣿⡇⠀⢹⣿⣍⣉⣭⡍⢸⣿⣆⣀⣠⣿⠁⢿⣏⣉⣩⣭⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⣿⣿⣿⢿⣿⣿⣿⣿⡀⠂⠀⠀⠀⠀⠀⠙⠃⠀⠀⠙⠛⠛⠉⠀⠘⠛⠛⠛⠋⠁⠀⠈⠙⠛⠛⠁⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⠻⣿⣿⡋⠓⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣟⣶⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡆⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣿⣿⣿⣿⣿⢿⣾⣻⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⡀⣄
⠀⠀⠀⠀⠀⢠⡾⣿⣷⣿⣿⣻⢼⣿⣿⣿⣿⣿⣽⣷⡀⠀⠀⠀⢀⣿⣿⣿⣿⣏⢾⣟⣽⣏⣷⡠
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣾⣵⣯⣶⡄⣰⣽⣿⣿⣿⣿⣹⢿⣿⣷⣿⣳⣿⡆
⠀⠀⠀⠠⣼⡿⢿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣠⠠
⠀⠀⠀⢠⣿⣿⣾⣿⣟⣿⢿⣿⣿⣿⣿⣿⢿⣇⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣯⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⡿⠾⠛
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣿⣿⣿⣿⣿⡷⣿⣿⣿⣼⡿⡿⣿⠿⢇
⠀⠀⠀⠀⠁⣹⣿⣿⣿⣾⡿⠟⠻⣻⣿⣿⣿⣿⣿⣯⡞⡾⣿⣿⣿⣿⣝⣿⣿⣿⣿⣶⣦⣿⣤⣠⠤
⠀⠀⠀⠀⠀⠿⠛⠛⢿⡅⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠛⢿⣿⣿⣻⣿⣿⣿⣿⣟⡹⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⢟⣿⣿⡿⣿⡿⠿⠟⠁⠀⢀⣴⣳⣿⣏⣿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡾⠿⠛⠉⠁⠀⠀⠀⢀⣴⣿⣿⠽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠈⠉⠉⠁⠀⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢀⡀⠀⠀⠀⠀⢀⡉⠃⠀⠀⠀⠀⠑⠄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠋⠩⠉⣀⠀⠀⠂⠠⠀⡀⠈⠁⠒⠠⢀⠀⠐⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢀⠀⠁⠂⠄⡀⠀⠈⠐⠆⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠈⢻⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢀⡈⠑⠄⡀⠀⠈⠐⠄⡀⠀⠡
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠀⠳⡀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢑⠠⢀⠀⠈⠐⢄⢃
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠄⠀⠀⠘⢦⡀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠤⣤⣅⢀
⠀⠀⠀⠀⡐⠀⠀⠀⢀⠌⠀⠀⠀⡔⠀⠈⠓⢦⠄⢀⠀⠀⠈⠐⠂⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠔⣨⢀⠴⡀⢂
⠀⠀⢀⠔⠀⠀⠀⡠⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⠐⠨⠄⢐⠂⠠⠄⠀⠀⠀⠨⠍⠑⠉⣉⢡⡄⠒⠋⢁⠀⠴⢁⠀⠑⠀⠐⠒
⢀⠔⠁⠀⡠⠔⠁⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⠄⠁⠈⠁⠀⠀⠈⠁⠁⠉⠒⠀⠤⠀⠀⡠⢤⠀⠊⠀⠀⠀⠀⢀
⠀⠀⠛⠆⠰⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠦⢀⢀⠀⡠⠔⠁
⠀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠐⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠠⠀⠂⠁

⡡⣼⡃⠀⠀⣰⡿⢰⣼⣿⣿⣿⣷⣄⠸⠃⠀⠘⡅⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿
⢈⣩⡁⠀⠀⣿⣧⢸⣿⣻⣿⣿⣿⣈⡿⠀⢀⡿⠃⢀⣶⣿⣿⣿⣿⣿⣿⢭⣻⠿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣇⡀⢠⣧⠀⢀⣶⣿⣿⣿⣿⣿⣷⣿⣏⡄⠀⣼⣿⣿⣿⢿⣬⡉⢣⡀⢻⣶⣹⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣯⣬⠟⣠⣾⣿⣿⣿⣿⣾⣿⣿⣷⠀⠀⣽⣿⣿⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟
⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠏⢻⡿⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣠
⣾⣿⠏⠁⣈⣉⣉⣽⡟⠛⠁⢀⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣼⣷⣴⣾⡿⢀⡸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡋⣠⣼⣿
⣿⣿⣿⣿⠿⠛⠛⢉⣿⡆⠀⠛⠻⢶⣄⠈⠹⣿⣿⣿⣟⡥⠶⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣿⣿⣇⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠙⣿⣦⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣿⣿⣿⣿⣿⣿⢠⣿⣿⠿⣿⣿⠿⠛⠁⠉⠛⣿
⣿⣿⡿⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡤⣄⣀⠀⠀⠀⢠⣶⣾⣩⣿⣶⢾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠿⠿⠀⠀⠰⠃⣸⢃⣀⣶⣤⡀⠀⢸⠟⢻⡿⣿⣿⣿⣿⣷⣶⣿⣿⡄⠀⢾⣿⣿⣿⣿⠟⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⣤⡤⠖
⢤⣄⠀⢀⣴⣿⣿⡿⠿⢿⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠛⠿⠟⠀⠀⠀⠀⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⡄⠀⢠⣿⣿⣿⣿⣷⠂
⣄⣀⣤⣺⣿⣿⠉⠉⠀⠉⠀⠀⠀⠀⠀⣠⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠈⠁⠙⠛⠉⠁
⣿⠟⢹⣿⣿⣿⠀⠀⠀⡀⠀⠀⠠⠒⠉⠉⢀⣿⣿⣿⣿⣿⣷⣀⠀⢿⣿⣿⡟⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠
⢋⣾⣿⡗⢻⣟⠧⣄⣀⣉⣳⣄⠀⠀⢉⣀⣾⢻⣿⣿⣿⣿⣿⣿⣷⣄⣙⠋⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⠂⠀⠀⣠⣴⣿
⣹⣿⣿⣿⠈⠉⠀⠹⣿⣿⡿⠃⠉⠹⣿⣿⣿⣌⠻⠛⣿⣿⣿⣿⣿⣿⣿⠇⠈⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⣉⠁⠀⠀⢻⣿⣿⣿
⡿⢻⣿⡇⠀⠀⣿⣷⣽⠿⠁⠀⠀⠀⠙⠻⢿⣿⡆⢴⣿⣿⣿⣿⣿⠿⠿⠗⠂⠃⠀⠀⠀⠀⡼⠻⣿⣿⣿⣿⣻⣿⣿⡟⠀⠀⠀⠀⠉⠛⠿
⠿⠀⠘⣿⣆⠀⠙⠿⠀⠀⠀⣄⡀⠀⠀⠀⣬⢿⣿⣿⡟⠋⣿⣿⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⢳⣄⠹⣿⣿⣿⣿⠟⠁
⣄⡀⠀⣿⢯⠀⠀⠀⠀⠀⣈⠀⠀⠀⢿⣿⡯⠃⣹⣿⣧⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⢹⣿⡿⠁
⠿⠣⣸⣿⣤⣄⠀⠀⠀⠐⠉⠓⠀⠀⠀⣨⣵⣾⣿⣿⣿⣷⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⣿⠃⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⣠⡦⠈⠻⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⡇⠙⠁⠀⠀⡀⠀⠀⠈⣧⠀⠀⢠⡇⠀⠀⠀⠻⠽⠏⢴⣶⡇⢠⣦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⡀⠀⡼⠀⣀⢀⡠⣄⠀⠀⠀⠀⠀⠘⠷⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠃⠀⠃⠀⣇⠀⠀⢘⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠴⠄⡄⠀⠀⢀⣤⣤⣤⡀⠀⠉⠲⠤⠜⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠁⠀⢠⣿⣿⣿⣿⣿⡆⠀⣀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡆⠀⠀⠀⠘⣿⡉⠙⣿⣿⠃⠀⠉⠁⠛⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣾⣿⣿⣷⣾⣯⣷⣿⣿⣶⣤⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⢉⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠙⠣⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣽⣿⣿⡇⣿⣿⣿⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢀⣄⢀⣀⢀⣀⡆⢀⢀⣀⢀⣀⠀⣀⡆⣀⢀⣀⠀⣀⠀⣀⡰⢀⡄⢀⠀⡀⢀⠀⢠⡀⠀⡄⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠇⢠⣬⢥⣺⣽⠀⡟⡈⢹⠀⠯⣸⠀⣇⡿⣀⢇⣸⠀⣑⠸⣊⠵⢇⡸⢅⠇⠐⠮⠂⢁⡭⡇⡇⣗⡉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠁⠀⠈⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠙⠻⢿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣷⣦⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⢀⡾⠉⠉⢉⣉⡟⣿⣿⣿⣿⣿⣟⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠚⠓⢒⣛⠛⠷⠒⠒⠒⠒⠛⠛⠉⠉⣽⣿⣯⢹⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣙⣛⣁⣠⣶⣾⣿⣿⣶⣦⣤⣤⣭⣭⣥⣼⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢛⢿⣿⣿⢟⡽⢶⣾⣝⢿⣿⣿⠉⠉⣿⣿⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⣀⣼⣿⣏⣿⣦⣾⣿⣿⡏⣿⣿⣦⣶⣿⣿⣧⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⢝⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⠿⠿⠿⠿⠿⠟⠛⣿⣀⣠⣼⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⢶⣶⣿⣿⣿⠟⢻⣿⢿⠿⠿⠿⠿⠟⠛⠉⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⣿⣟⣿⡷⢶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⢼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣏⣥⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣟⣡⣤⣤⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠑⠁⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣤⣴⡶⠶⠿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣉⡤⠔⠒⢂⠉⠉⢳⣶⣉⡩⠭⠿⣷⣿⣿⡻⣦⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠊⣁⠔⣊⠥⣒⡨⢭⣿⣯⣽⣷⢖⣻⣦⡙⣿⣿⡜⣧⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⠃⢠⢞⡵⡪⠖⢉⡠⠔⠁⡜⢹⠢⢌⡑⠿⣿⣧⢸⣿⣿⡼⣇⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⡎⢀⠃⢠⡥⣫⢊⡠⠒⠁⢀⡠⠊⡠⠃⠀⠀⠈⠳⣮⡻⣮⣿⣿⣷⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡘⠀⡸⡱⣡⢋⡠⠔⣒⡡⠔⠋⠀⠀⠀⠀⠀⠀⠈⢻⡌⣿⣿⡸⣿⡄⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠃⡇⢠⢣⡗⡩⠔⠊⠉⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣟⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣼⠏⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢈⣻⣦⣀⣿⢰⣿⡇⢸⡇⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⠀⠀⠑⠀⢿⡿⠀⠀⠀⠀⠀⠀⠿⠿⣯⠟⠉⢿⣸⡏⣿⠸⣿⠀⠀⠀⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡬⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⣿⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠸⣄⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⢠⡾⠃⢿⣹⡆⣿⡀⠀⠀⠀⠀⢸⡄
⠀⠀⠂⠀⠀⠀⠀⠀⠘⣄⡆⠋⠳⣄⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⣠⣴⣿⡹⣆⠘⣯⣷⢸⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⢸⢠⢉⡟⠒⠦⠤⢤⡤⢤⠤⠤⠒⠚⣿⣾⣹⣇⠹⣆⠹⣾⣿⣇⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠄⢸⣸⠁⢀⡤⢄⣀⠃⢸⠀⠠⠤⣀⠈⢿⣿⡿⣦⠹⣆⠛⡿⡿⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⢸⣾⡿⢰⠁⠀⠀⠉⠉⠉⠀⠀⠀⠈⢻⣄⢿⡇⠜⠷⣝⢦⡸⣿⣆⠀⠀⠀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⡈⠀⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢻⡎⠁⠀⠁⠈⠻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢨⡾⣠⠃⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠛⠁⠀⠀⠀⠀⠀⠁⠉⠛⠿⢿⣷⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⢠⡿⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠒⠉⠀⠀⠈⠑⠒⠤⠤⡤⣀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⡶⣶⡶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⡟⣁⡄⠶⠶⠶⠂⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⣤⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⠖⠁⠀⠀⠱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠓⠶⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡾⢄⣀⣀⠀⠀⠀⠀⠈⠉⠁⠒⠖⠶⡶⠖⠤⣀⣀⡀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⠀⣸⠛⠳⣤⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣴⣫⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⠃⠻⠀⠀⠈⢷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣰⠋⠀⢻⠁⠀⠈⠉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠾⠿⠰⡇⠀⠀⠀⠈⠂⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠸⠛⠀⠀⠀⠁⠀⠐⢄⠻⠿⠋⡄⠀⠀⠀⠀⠀⠀⠀⠁⠞⠁⠀⣇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡏⠉⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠈⣇⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡀⢈⣱⠆⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡾⠁⠀⠀⠀⠻⡀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⢦⡀⠀⠀⣸⢇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠻⣍⠐⠖⠂⠀⣉⣤⡍⠳⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣄⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢆⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠓⠀⠀⢀⣤⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣴⠟⠑⠦⠀⠀⠀⠀⠂⠉⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢡⠀⣸⠋
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠈⡋⠐⠂⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣰⡏⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣄⠀⠀⠀⠀⡇⠀⠀⠀⢠⡟⣀⣤⡀⠀⠀⠀⣼⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠀⢿⡄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⡇⠀⠀⣤⠋⠔⠁⠀⠹⣆⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⢿⡀⠀⠀⠀⣠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡷⠒⠛⢀⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢀⡞⠁⠈⢿⡀⠀⠀⢸⣇⠀⠀⠀⠀⢸⡅⠀⠀⣃⠶⠂⠉⠉⠉⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠈⠻⢦⣄⢸⠿⢶⣄⣀⠖⢨⡅⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⡝⠁⠠⢦⠀⣰⡆⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠄⠀⠀⠠⠘⠋⠁⠀⠀⠀⣠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⢇⡀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡀⠀⠈⠁⠀⠀⠀⠐⣥⡄⠀⠀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠀⠀⠁⠀⢀⣼⠀⠀⠀⠀⣤⠘⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡋⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡼⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⠀⠀⠀⠀⠀⠀⢠⡤⠊⢁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣂⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⠃⠀⠀⠀⠀⠀⠀⡲⠟⠂⠈⠉⠉⠉⠋⢥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡞⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠈⠙⠛⠛⠒⢲⡟⠁⠄⢤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠩⠿⠛⠓⢦⣤⣐⣲⣤⣤⣤⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⠉⠙⠛⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠂⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠋⠉⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠒⠂⠠⡀⠀⢀⢴⡲⢦⣶⣂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠶⢀⣩⣤⣤⠀⠈⢳⣎⡂⠐⢡⡄⠀⠒⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡐⠡⠒⠉⣁⡠⠤⢤⣁⣠⠞⢉⡉⠷⡤⠤⢈⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⣠⠞⠁⣠⠔⣋⣁⠵⠚⠋⠀⠶⠇⠀⠀⠈⠓⠤⣀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⠃⢀⡔⢁⣼⡿⡡⠀⢀⠀⠀⠀⠀⢣⠀⠠⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢠⠎⢀⢶⣿⡱⠁⠠⢎⠃⡠⠀⡄⠀⢀⠀⢡⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣧⠉⠀⡘⡜⢣⠁⢀⠓⡌⠀⡏⠀⡇⠀⠀⡄⠀⢷⡀⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡇⠀⠀⣷⠁⠈⠀⡌⢀⠃⢰⡇⠀⡇⠀⠀⣷⠀⠘⣧⠀⢆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⣿⠆⢿⠙⢷⣿⣄⣸⣇⢀⡇⠀⠀⣿⠀⣀⣿⣄⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠸⠀⠀⡿⡀⠨⡆⠻⢓⣶⣶⣽⡅⠈⠉⠁⠀⣤⣬⡥⢦⡾⢣⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⡄⠀⢃⡀⢱⠀⠈⠛⠿⠧⠜⠀⠀⠀⣼⣿⣿⠦⠜⠇⠻⢀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠳⠤⠠⠊⠀⢰⣿⡇⠀⠀⢦⠀⢃⠀⠀⠀⠀⣠⠐⠂⠠⡀⠀⠀⠀⡘⢢⡀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠴⠋⡉⢷⠀⠰⠀⢣⠀⢳⡀⠀⠸⠅⠲⠲⠤⡜⠀⣠⠞⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⡤⠈⠀⠀⢤⠀⢆⠀⢻⢦⣄⣀⠦⠀⣊⣠⠞⠁⠘⡀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⢁⡴⡇⠀⠈⠀⠈⠂⠈⣸⣈⣉⠋⡟⣩⠧⣤⠀⠀⢡⠀⠐⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠛⠈⠁⡠⠋⠀⡇⡄⠀⠀⢰⠖⠋⠁⢉⠉⠙⢿⣉⡄⢦⡈⠢⠀⢣⡀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠀⡴⠁⠀⢀⠇⡇⢀⠔⢁⣤⠒⠉⠓⢄⠀⠀⠀⡆⢰⠉⠹⠂⢀⠉⠀⠀⠀⣬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠾⠀⢀⡰⠋⣀⡇⠈⠠⡈⠉⠢⢄⡀⠀⠱⣔⡸⠏⡇⣠⣊⢠⠋⠐⢦⠀⠀⢸⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠉⢸⠀⠀⡀⠀⢶⣤⡌⢣⠔⣷⣟⠁⠀⢰⣯⡜⠋⡇⠀⠀⢇⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠁⢸⠀⢣⠈⢠⡞⣯⢘⡏⠱⢛⣽⣃⠜⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣀⠞⠉⠣⣸⡀⠀⢦⡖⠀⢱⡟⠁⢠⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡌⠀⠀⢸⣇⠸⠃⣼⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⢀⣼⡇⠀⢸⡏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢟⠞⣿⠃⢸⡇⢀⣿⢸⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠀⠈⠺⣏⡀⠸⣷⣼⠏⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠙⠀⠀⠀⡇⠈⠑⠂⠻⠿⠐⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠁⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⢭⢿⣟⠁⠀⠀⠀⠀⢣⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠀⠀⠜⠁⠀⢸⣄⡀⢀⡀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⡠⠂⠁⠈⠳⣄⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⣠⠖⠁⠀⡹⠋⠁⠀⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣠⠔⠉⢀⣠⣤⡤⢤⣄⠙⠇⠀⠸⠀⠀⠀⠀⠀⡞⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠑⠒⠒⠉⢀⢴⡾⠋⠉⠀⠀⠀⠈⠱⡆⠀⠀⠀⠀⠀⠰⡙⠀⠀⠀⣌⣀⠀⠀⠀⢀⡀⢠⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⣠⠆⠡⠋⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠀⠀⠀⠀⠀⢀⡔⠉⠙⠛⠛⠛⠛⠉⠁⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠒⠊⠉⠚⡵⣓⣤⣤⣀⣒⣦⡤⢤⠴⢶⡌⠀⢈⠷⠶⠦⣤⣴⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣿⡆⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⢹⠀⠸⣀⣀⣀⠼⠟⠳⣀⣀⣀⣀⣀⡀⠠⠤⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⣀⠀⠤⠤⠤⠐⠒⢒⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣮⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣿⣿⣿⣿⣯⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⡄⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⢿⣿⣾⣿⣿⣯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⣷⣜⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣣⣴⣾⣿⣿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣾⣿⣻⡟⣿⣿⣿⣿⣽⣿⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢀⢼⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠼⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠘⠃⣸⣿⣿⣻⣿⣿⣿⣿⣽⣿⣏⣹⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡿⠁⠀⠀⣿⣽⣿⣯⣿⣿⢿⣿⣽⣏⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⣿⣿⣟⣿⣿⣽⠅
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠻⣟⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣈⠴⡞⠋⠃⡰⡏⡇⠀⠈⠳⠦⢌⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡠⠊⠀⠀⠁⠀⠀⠀⠀⠁⢠⠂⠀⠂⠀⠙⢦⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣼⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⣣⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⠏⣠⣿⠀⢀⠀⡆⠀⡄⡀⠀⠀⠀⠀⡄⠀⢸⢠⠸⣿⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⣾⣻⢟⣿⠸⡀⢧⠰⡇⠹⡄⢀⠄⡄⡇⢰⠈⢺⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠓⡏⠣⠻⠀⠳⡁⠁⠹⡀⠙⠎⠀⠏⠀⡹⠀⠘⡽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠀⠀⢀⠀⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠤⠤⠀⠀⠤⠤⠤⠤⠤⢤⣸⣫⣡⠀⠀⠐⠲⠶⠐⠀⠘⠀⠀⠀⠂⠲⠶⠒⠀⠀⢰⢹⣣⣀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠬⠍⣹⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⢠⠀⢀
⠀⠀⠀⠠⡀⢠⠀⠀⡆⠀⠘⠀⡘⠀⠀⠀⠸⢼⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡯⠄⠀⠀⠀⠸⡀⢸⠀⢀⠀⠀⢠⢀⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⠚⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠑⠘⠂⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠠⣤⠀⠀⣤⠄⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠳⡄⣠⠃⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡮⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣻⣿⣿⡇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣼⢃⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⡀⠹⡇⠀⠀⠀⠀⢀⣾⢁⣼⣿⣿⣿⣯⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⠄⠵⠾⣶⣤⣄⣸⢃⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⡿⠟⠁⠀⣀⢀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣩⡄⠀⠤⢶⣃⠅⢚⠃⣸⣿⣿⣿⣿⣿⠿⡿⠟⠁
⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠴⣮⠥⢲⢖⣿⣿⣾⣶⣾⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⣠⣤⠵⠞⠹⣿⣿⡿⢻⣿⣿⣿⣿⡟⠻⣯⡀
⠀⠀⠘⢿⣷⣾⣿⣿⣙⣿⣿⡿⣿⣿⢷⣾⣧⣦⣴⣶⣴⣿⢉⡁⣈⣻⣿⠿⠏⠀⣀⠈⣙⢳⣦⡤
⠀⠀⠀⠈⠻⣿⣯⣿⣿⣟⣿⣿⡿⣿⡾⣻⣿⡿⢋⣿⡿⣿⠟⣫⡿⣛⣿⣾⣿⣷⣎⣙⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢯⣿⣿⣿⣿⣾⣿⣥⣾⣿⣿⣤⣶⠞⢋⣾⡿⣫⣿⠟⣿⣿⢻⣿⣿⡿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢸⣯⣍⣩⣤⣾⠿⠻⠾⠟⠁⠈⠛⠋⠘⠋⠻⣷⠈⣿⣧⠹⣿⡛⠿⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⢹⡯⣽⡏⠽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢤⣴⣾⠋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠿⡟⠟⠛⠋⠹⠛⠛⠟⠙⠛⠿⢻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣟⡯⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢽⡻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡧⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⠹⡟⠛⠛⠿⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⢻⠏⣸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣆⠣⠀⠀⠀⠀⠘⠿⠿⠷⣄⠀⠀⠀⠀⡠⠚⠁⠀⠀⣙⢛⡛⠁⠞⢰⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠗⠘⠻⠟⠓⠲⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠐⠤⠀⠀⠤⠃⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣵⢻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣏⣇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣿⡏⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⠿⠉⡻⢿⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⢿⠋⠿⠿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⢻⡿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡿⠀⣄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢨⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣜⡥⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠭⠥⠤
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⢠⠞⢁⣠⠾⠭⢄⣠⠤⠤⢄⣀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⢀⣠⠟⠊⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⢀⠇⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⣆⠀⠀⡸⠀⠀⠀⠀⣠⠔⠁
⠀⡼⠁⠈⠑⢦⡀⢠⠞⠁⢀⠀⠀⠀⠀⠀⠈⣶⠰⠧⢤⣀⣠⠞⠁
⢰⠁⠀⠈⣻⡶⠄⠀⠠⠾⣛⠀⠀⠀⠀⠀⠀⢸⣧⡀⡠⠊⠉⠓⢤⡀
⡎⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢫⠀⠀⠀⠀⠀⠉⢢⡀
⢀⠀⠀⠀⠀⢀⡐⠂⠁⠲⣄⠀⠀⠀⠀⠀⠀⢸⡧⣄⣇⠀⠀⠀⠀⠀⠀⠙⢄
⢘⣄⡤⠖⠛⠉⣠⠄⠀⠀⣸⠀⠀⠀⠀⠀⢠⣿⡤⣸⡉⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠁⠹⣄⣠⠔⢋⡴⠂⠒⠊⠁⠀⠀⠀⢀⡴⠋⢸⡃⠘⡳⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠐⠋⢓⡴⣏⣀⣀⣀⣀⣀⣠⡤⠖⠋⠀⠀⢐⡂⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⠃⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠸⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣀⠤⠔⢒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠧⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣦⡀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠛⠻⠙⡌⠉⠉⠙⡄⠀⣀⡠⣤⠶⡖⠒⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢤⣤⣄⣀⣀⣤⣄⠤⢞⠗⠋⠁⠀⠈⠀⠉⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠆⠤⠤⠤⠔⠶⠖⠒⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡶⣾⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠀⠀⠀⠒⠀⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢹⣿⠀⠀⠐⠟⠀⠀⠙⡉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠙⠓⠀⠀⠀⠀⠈⣀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣌⡁⣀⠀⠀⠀⠀⠙⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟⠿⣻⣿⡇⠀⠀⠠⠤⠤⣤⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣿⡿⢶⣿⣿⡷⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⠤⣶⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⡿⠁⡀⠀⠀⡀⠀⢘⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⡐⠂⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢀⠀⠀⠀⠈⠉⠀⠀⢱⣤⣤⣑⣤⣼⡆⠈⠁⠀⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⣿⣀⡭⠖⠁⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⣦⡄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠛⠻⣿⡀⢰⡯⠀⠀⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣼⡇⡜⠁⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⡴⠿⠛⠛⠁⠀⡇⢰⠀⠀⠀⠀⠀⢹⣏⣟⣢⠊⣿⣇⠈⠁⠀⠀⣷⢠⢦⡀⠀⠀⠀⠀⠀⠀⢠⠇⢠⡿⢰⠇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⢠⠞⠀⠐⠒⠀⠀⣰⠃⡾⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⣿⡄⠀⠀⠀⢿⣸⣎⡟⢂⠀⠀⠀⠀⢀⠞⠀⣸⡇⡞⢀⣤⠴⣆⣤⣶⣿⣷
⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⡜⣿⠀⡇⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⡿⣾⠁⠀⠓⢄⡀⠀⢠⠃⠀⣿⡋⠹⠿⠿⠟⣋⣉⣭⣭⣭
⠀⠀⡠⠁⠀⠀⠀⠀⡴⠋⠀⢸⡄⡇⠀⠀⡞⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⡄⣟⣶⡞⡄⠀⣿⢃⣠⣴⣾⣿⣿⣿⣿⣿⣿
⢀⡼⠁⢠⣤⣄⣀⠞⠁⠀⠀⠀⢷⣧⠀⡼⠀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⠟⢤⡀⠀⠀⢃⠙⣿⠇⡇⠀⢿⣏⣛⣿⣿⣿⣿⣿⣿⣿⡿
⡾⠀⠀⠒⠶⡖⢛⣂⠀⠀⠀⠀⠸⣿⡼⢁⡜⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡿⠀⠀⠙⢦⡀⠀⠀⠳⠀⠀⠰⠈⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠑⢦⡀⠀⠀⠈⠉⠘⢦⡀⠀⠀⠀⡛⠁⡼⠀⣾⣿⡿⠿⠿⢻⣿⣿⣿⣿⢀⠀⣼⠁⠀⠀⠀⠀⠈⠳⢤⣀⢀⡄⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠑⢄⣀⠀⠀⠀⠀⠈⠓⠤⣜⠀⢠⠇⠸⠟⠁⢀⢛⢛⡛⡛⠛⠛⠻⢨⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠙⠻⣟⡒⠒⠂⠀⣀⠍⠑⠋⡀⠀⠀⠀⠀⣒⣂⡨⠁⠀⠀⠒⠢⡄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠓⡤⢉⡀⠀⠀⢠⠿⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠃⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠓⠒⡽⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄
⣶⣶⡶⠶⠶⠶⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⠰⠶⠖⠒⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣾⣿⣧⣶⣤⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⡀⣀⡀⠠⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⣤⣤⣀⣉⣀⣘⡇⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⡇⠀⠀⠀⢸⡿⣿⣛⣛⣉⣩⣭⣽⣿⣯⣿⣯⣭⣿⣿⣿⣿⣿⣿⡀⢺⣇⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⡟⢻⠻⣿⡟⢿⠛⣿⣿⣿⣿⣿⣿⡏⠉⢻⠉⠉⠉⠉⠉⢉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⣾⣴⣏⣵⣭⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠛⠛⠛⠛⣛⠉⠉⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠂⠀⢻⡟⣛⠛⡟⢻⣿⢿⡟⢿⠉⣿⢹⡟⢹⣯⢽⠁⣿⠀⢸⣿⠀⡏⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⡠⠘⣷⠏⠀⡏⠨⠿⠸⠇⠘⣀⣿⣼⣿⣼⢿⣴⣖⡏⠀⠸⣿⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢱⠄⣒⣶⡶⣶⡟⣿⡛⣿⢻⣿⣻⡘⣯⢹⡉⠀⠀⠀⢿⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠉⢹⡇⢻⡅⣿⠂⣿⢸⡏⠹⠁⠇⠀⠇⢀⠀⡀⢸⠀⠀⡀⠀⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠦⠬⡬⠥⢄⣤⣄⣤⣀⣀⣓⣒⣒⣂⣲⣤⣾⠃⠀⠀⠀⡀⣁⢀⢀⡄⠠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡿⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⡏⠀⠛⠛⠁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠷⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣿⠛⠉⡿⢫⡍⠉⢭⣥⣤⡄⢠⠁⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢈⡟⠚⠀⡷⠺⠀⠀⠀⠀⠀⠻⢾⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁⠐⠀⠀⠀⠀⠀⠀⢸⡆⠶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢃⡆⠀⢸⢰⡄⢀⣀⣀⣀⣾⣿⡄⠀⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⠀⠈⠂⠁⠈⠤⠌⠉⡁⠈⠠⠀⠁⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⣿⣿⣟⣛⣟⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⢸⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⢸⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡟⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⢸⡇⡇⠀⠈⠁⢉⡉⠉⠉⠉⠉⠉⠀⢸⠀⣷⠀⠀⢠⢼⣿⣿⣿⣿⣿⣟⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢰⡇⠁⠀⡿⠛⠛⠟⠙⠛⠛⠛⢛⣱⠀⢠⡿⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⣤⣤⡶⠷⠶⠶⠶⠾⠿⠄⢸⣧⠀⠀⠀⠀⢷⡴⣭⣥⡬⠤⠭⠭⠭⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡿⢹⡇⠀⠀⠀⠞⠂⠘⢂⠓⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠄⠀⣧⣼⣵⣭⣭⣭⣭⣭⣭⣭⣥⣤⠶⠶⠶⠺⡇⠀⠀⡘⠀⠀⣈⣂⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡤⠴⠗⠛⠒⠚⠓⠒⠂⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠒⠛⠀⢀⡟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⣉⣥⣤⣶⣤⣍⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣴⣾⣿⡿⢿⣟⠻⢿⣿⣿⣿⣿⣿⣿⣦⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣾⣿⣿⠏⠀⠈⢿⣷⣦⣉⣻⣿⣿⣿⣿⣿⣿⣄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⣿⣿⡏⠀⠀⠀⠀⠈⠉⠛⠉⠭⣿⢻⣿⢿⣿⣿⡆⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⣿⣿⣧⡴⠶⢤⣤⡀⠀⠀⢠⣤⣴⡤⠯⢶⣿⣿⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⣿⣿⠋⢀⣤⠀⣀⣙⢧⠀⠀⠀⣠⣤⡄⣤⣄⠉⣿⣿⡆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⡒⣿⣿⡿⣭⡿⠈⠁⠀⠀⠻⢿⡯⣿⡿⣗⣸⣿⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⣿⣿⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⣿⣿⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⡼⢻⣟⣷⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣧⢠⡿⠀⠀⠀⠀⠀⣠⣤⡤⢄⠀⠀⠀⠀⠀⠅⢾⢻⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣴⣇⠀⢀⠀⠀⠈⠠⠤⠀⠀⠀⠀⠀⢀⣶⣾⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⣿⣿⣿⣿⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⡀⠠⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⣰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠛⢿⣿⣿⣿⠿⠁⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠙⠟⠛⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣀⡤⠤⢤⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣹⠞⢁⣠⢾⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠁⠀⣠⣖⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⢀⡞⠁⠀⢱⠀⠀⠀⠀⠀⠀⠠⠄⠀⢰⣇⠀⠀⠘⣦⠎⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⠉⠉⠷⣀⠀⢣⡀⠀⣠⠴⠒⠚⠛⠛⠛⠓⠲⣤⠴⠳⠦⠔⠚⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣰⠇⢀⠀⠀⠀⠀⠙⢦⡀⠈⠲⣅⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⣀⡤⢲⣮⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⣾⣤⣶⠭⢛⣒⣒⠺⢆⠀⠈⠛⠒⠒⠒⠒⠒⢉⣡⣴⠺⠯⢴⣤⠏
⠀⠀⠀⠀⠀⠀⡖⢱⡀⠀⠀⠀⠀⠙⠃⢸⡀⠈⠉⠀⠁⣑⠀⠀⠀⠀⠀⠀⣀⣔⣊⣵⣋⡠⣀
⠀⠀⠀⠀⠀⠀⢹⠆⠉⠁⢲⣀⣀⣀⣀⡤⠭⠥⢒⣋⡉⣏⠀⠀⠀⣠⠴⠋⠓⡶⠒⠦⣄⠁⠀⢣
⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⢠⠒⠒⢲⡶⠃⣀⣀⣀⡀⠈⢻⠀⠀⢎⡀⠀⢀⡜⠀⠀⠀⠈⠑⠒⠋
⢀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠜⠀⣀⡨⠟⠉⢛⡉⠀⠀⠀⠸⡀⠀⠀⠈⠉⠉⠉⠛⠛⢓⣂⠴⠂
⠀⣢⠞⠙⢿⣿⡻⢿⣦⡴⠒⠉⠁⠀⠀⡠⠊⠁⡊⢂⢄⡠⢧⣀⣀⣀⠤⠤⠲⢶⣉⠉⠀⢀⣀⣀⣀
⠀⠹⠤⢴⠞⠁⠀⠁⠀⠙⠦⣄⠀⠀⠂⢅⠟⠤⣨⠔⠋⠀⠀⠀⠀⠙⢄⣠⣮⢝⣓⡀⠐⠣⣄⡰⠧⡝
⠀⠀⠀⠙⠒⠂⠐⠂⠤⠤⢤⣴⣻⡒⠲⠤⠭⠾⠧⠤⡄⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠉⠹⣤⣟⢒⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⣀⣠⡤⠤⠤⣗⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣠⡔⠊⠉⠉⠉⠉⠁⠒⠂⠤⠤⢴⠛⢗⠬⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢊⠯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠱⡈⢦⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠱⠰⠃⢀⣆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢷⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠉⠀⢠⠾⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠸⡄⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠶⠁⡴⠀⠳⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⠀⠀⢸⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⠁⠀⠀⠀⠘⠀⠀⠀⠀⢠⡞⠋⠙⠛⢶⡷⠀⡇⠀⠀⡾⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢻⠀⠀⠀⠈⠀⠀⠀⠀⣐⠊⠉⠀⠀⠀⠈⠀⠀⠇⠀⡼⠁⠀⡏
⠀⠀⠀⠀⠀⠀⡰⠉⢩⡀⣀⠔⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠁⠀⢸⠁
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⡀⠐⠸⠀⠈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠘⣆⠸⡄⠀⠀⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠋⠁⠀⢇⠀⠘⠉⠙⣦
⠀⠀⠀⠀⢀⣠⠽⢦⣤⣤⠤⠚⠀⠀⠀⠀⠀⠀⠈⠳⢌⣉⣁⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠴⠂⢸
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢴⢶⠌⠉⠉⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠎⠀⠀⠀⠀⠠⠦⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⠋
⠀⠀⡇⠀⠀⠠⠀⢄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡈⠀⢀⠠⠐⠀⠀⠀⠸⠆⠀⠐⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⣧
⠀⠀⢷⡀⠀⠀⠀⣰⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡀⠀⢃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠈⠳⡞⠋⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠌⠱⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠴⣂⡸⠀⠀⢀⡞⠀⠀⡀⡀⠀⠀⣰⠋⠀⡼⠀⠀⠀⠀⠐⠀⠀⡧⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠐⢢⡀
⠀⠀⠐⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⠀⠀⠀⢀⠎⠈⠀⠀⠀⠀⠀⢣⡐⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠑⠀⠈⡇
⠀⠀⠀⠈⠳⡆⠀⠀⠘⡄⠀⠀⡀⠀⠀⡜⠠⠃⠀⠙⠲⣤⡴⠶⠚⢫⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠁⠉⠀⠀⠘⠱⠀⠀⠀⠀⠀⠠⠤⠐⠈⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠖⠋⢆⠀⠁⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠓⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀⣨⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠛⣮⣄⣀⣀⣴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⣀⣠⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⡉⠓⠚⣉⠀⠀⠘⠻⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡀⠂⠀⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠐⠀⠀⠀⢀⡄⠀⠁⠀⠈⠐⠢⣄⡀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠔⣉⠀⠈⠳⡄⠀⠀⠀⠀⠀⢉⠢⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡞⢀⠀⠀⠈⠀⠝⣿⡦⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠜⠁⠘⣀⠠⠀⠀⠀⠈⠁⠡⠀⠀⠀⣀⠠⢤⡌⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠆⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⠀⠀⠀⠙⠁⠀⢺⠀⠠⠈
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠄⠀⢀⠀⠀⠀⢠⠂⢀⡗⠒⠊
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠑⢦⢦⣄⢂⡠⠔⠒⠁⠀⣼
⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⢄⠀⢸⠈⠄⠀⠀⠀⠀⠀⢿⡈⠠⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠈⢄⠀⣆⠀⠀⠀⠀⢄⠀⠀⠉⣠⠇
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⢄⠀⠺⠀⠑⠄⠀⠈⠠⡷⠲⠚⠁
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠈⡦⠄⡀⠀⠀⠄⣈⠵⢦⡀⢀⠀⢄⡀
⠀⠀⠀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⣤⢀⣠⠤⠘⠁⠀⠀⠀⠀⠈⡰⠀⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⣀⠀⢠⠊⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⡀⠔⠊⢀⣀⣀⣀⣀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⣀⣀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢯⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢉⣠⠒⠁⠂⢤⡟⠀⠀⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢤⡀⠀⠶⡶⠛⠚⠛⠙⡞⠀⠀⠀⡴⠃⠂⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠜⠁⠀⠀⢀⡜⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣄⡂⠰⢤⣀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠼⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⢄⠀⠀⠀⠀⠉⠙⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⢧⠀⠀⠀⠀⠀⡴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠿⠿⠿⠿⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣶⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠿⣷⣶⣤⣄⡘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠶⢶⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣥⣤⣤⣤⣄⣀⠀⠀⠀⠀⠈⠉⢛⡿
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣶⡶⠟⣡⢶⡀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⡇
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⡇⢸
⣠⠶⣤⣠⠀⠀⠀⢀⣼⠇⢀⣾⠁⠀⢀⡀⣠⠞⠛⠛⠷⠀⠀⠀⠀⠀⠀⣠⠞⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣠⠾⢿
⣿⠀⠘⢿⣆⠀⢠⣿⡟⠀⣼⠃⠀⠀⠙⠿⣿⣶⣶⠦⠀⠀⠀⠀⠀⠀⠀⠻⠿⣶⣦⣬⣤⠶⠖⠀⠀⠀⠀⠀⣿⡟⢁⣤⣾
⠘⢷⣄⡀⠙⢀⡿⣿⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡴⠋⢠⡟
⠀⠀⠉⠛⠆⣼⢳⡏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣠⠴⠋
⠀⠀⠀⠀⢠⡏⣾⠁⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⣸⢱⡟⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⡏⣼⠃⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠈⢻⡆⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠃⣿⠀⠀⡿⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⠘⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⢠⡏⠀⠀⡇⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠃⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⠶⣶⣤⣴⡶⢾⡿⠉
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣄⣀⣤⣤⣀⣸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⣿⡄⠀⢸⡿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡤⠤⡒⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡡⠖⣩⠴⠞⠛⢻⣧⠀⠀⠀⢀⣶⣶⡄⠉⠓⢦⣀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣣⠖⣭⣦⠀⢹⡀⠀⢀⣾⡏⠀⠀⠐⠋⡿⠀⢀⣴⣦⡀⠙⢷⡀
⠀⠀⠀⠠⡞⢁⠀⢀⡴⠊⠀⢠⡟⣿⡆⠈⣧⣴⣿⡟⠀⠀⠀⠀⠀⠇⠀⠸⣧⡼⠃⠀⢸⣿⡄
⠀⠄⡰⢙⣀⡬⢟⡵⣆⣠⣄⢸⡟⠹⠇⠀⢛⣻⡯⠄⠒⠂⠙⠛⠻⠿⣷⣶⣦⣄⠀⠀⢠⣯⡇⠇
⢠⠠⡇⠀⠀⠀⣮⣼⠟⣡⠏⠘⢃⣤⠶⣿⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⣾⣿⠇⠺⠆
⠀⢻⣧⠀⠀⠀⠈⣿⣶⠋⣠⢾⣟⠁⣸⠁⠀⠀⠠⡞⠙⠀⠀⠀⠀⢀⠀⠀⠘⣿⣿⣿⣿⠋⠠
⠀⠀⢻⣷⣄⠀⠀⠈⢛⣾⣿⠏⠈⠤⠇⠠⠀⠀⠰⠉⠀⠐⠒⢦⠀⠀⠃⢠⠀⢣⣿⣿⣃⢀⡤⡀
⠀⠠⠀⠻⣿⣿⣦⣤⣾⠟⠁⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⢤⡆⢸⣇⠀⠈⠀⠀⠘⣾⣿⣇⠸⣠⠆⣤
⠠⠀⠀⠀⠈⠙⠻⣿⣯⣷⣶⣶⡀⠀⢀⢞⣿⣿⣿⠿⣿⡀⠃⢸⣿⡀⠰⣤⣏⠀⣿⡿⠎⠛⠛⢿⣷⢸
⢡⠄⢠⠀⡏⠙⡆⠸⢇⣿⣿⡋⣳⠀⢈⠸⣿⣿⣿⣀⣸⡇⠀⢸⣟⣿⢦⡿⠙⣷⠈⠛⠤⢠⣾⣾⡟⣾
⠇⠀⠀⢀⠙⠒⠁⠀⠀⣿⣿⣿⡿⣌⠸⣆⢻⣿⣿⣿⣿⢿⠀⠈⠻⠿⠋⠀⢒⡟⠀⣠⣾⡿⣿⠿⠇
⠀⡄⠀⠀⠀⠀⠀⠀⡸⠾⠿⢛⣽⣿⣆⢺⣷⢿⣿⣿⡿⣫⣴⢶⣏⠀⢀⡆⢎⠀⣾⢯⣫⠔⠂⠀⠀⠀⡆
⠀⠰⣀⠀⢀⣀⠀⢀⡇⠀⠀⣼⡿⠻⠿⣧⠙⠻⠿⠷⠋⣿⡏⠀⠈⢳⠀⠀⢈⠇⠻⣮⢿⠀⠀⠀⠀⠊⠐
⠀⠀⠀⠉⡛⠻⠓⠀⢿⣷⣶⡄⠀⠉⠉⠉⠀⠀⠀⣴⣄⢙⣿⠀⠀⣼⡀⢠⠏⡀⣴⠟⡘⠀⠀⣤⣤⡤⠖
⠀⠀⠀⠀⠈⠓⡀⠀⠀⠙⢻⠃⠀⠀⢀⠀⢰⣶⣤⣼⣿⣿⡿⢀⣼⣟⡤⢘⣄⣿⣧⣤⣿⣦⣴⣧⣀⠈
⠀⠀⠀⡀⠀⠰⢿⡆⠀⠀⣼⣦⣿⣤⣼⣷⣾⣿⠟⠛⡏⢋⣁⣌⣉⣭⣀⣉⣩⣼⣿⠿⠿⠿⠿⣿⣿⣄
⠀⠀⠀⡨⠇⠀⠈⠀⠀⢸⣿⣿⣿⠛⢿⠀⠸⣿⠃⠀⢠⢤⣼⣿⣿⡿⠟⠛⠉⠁⠀⢦⣾⢷⡄⠈⠙⢟⢧
⠀⠀⠊⠀⠠⡀⠀⢀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣯⣿⡿⣻⠛⣷⣤⢤⣰⠛⣶⢸⡟⢸⣇⣴⢶⡌⢾
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢻⣿⣿⣿⣿⣦⣼⣶⣿⡿⠛⠁⠀⠙⣾⠉⣠⡾⠉⠙⠁⠀⠁⠺⢻⡟⠛⠃⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢫⡙⠛⠛⠿⣟⠟⡁⠰⣋⡷⡀⠀⠸⣟⣭⠶⢛⣏⠉⠛⠳⣄⠘⣧⣄⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠱⡄⢠⣤⡌⢿⡍⠳⡍⠛⢁⣴⠞⠉⠀⠆⠀⠁⠀⠀⠀⣽⠇⢀⡴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢇⠀⢳⠀⠀⠛⣠⣦⡾⡿⠁⠀⢉⠇⠀⠀⢀⣠⣴⣟⡥⠖⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢰⡘⢶⣬⣴⣶⣾⣟⡿⠏⠀⠀⠀⠋⠐⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣍⡙⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⣉⠭⢟⣿⡿⣻⡟⠉⠹⣿⡻⣯⣛⠭⢍⣉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠧⠤⢔⣺⢝⡫⢊⠞⠀⠀⠀⠙⢌⠪⡻⢕⡒⠤⠤⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣋⡭⠵⣚⡩⠖⢁⠀⠀⠀⠀⢀⠀⠑⠮⣑⡪⠭⣍⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢳⢶⠚⠉⠀⠀⢺⢟⡛⢟⠀⢹⢟⡛⢷⠂⠀⠉⠙⢶⢶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡟⠀⠀⠀⠀⠘⠬⠥⠎⠀⠘⠮⠭⠞⠀⠀⠀⠀⢸⡶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠒⠒⠒⢄⠀⠀⠀⠘⠁⠀⠀⠀⠀⣠⠔⠒⠒⢾⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⡇⠀⢶⣀⢀⣠⡇⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⡆⠀⠀⢠⠃⠀⠈⠢⠥⠚⠀⠀⠀⢇⠀⠀⢀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⢹⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⡾⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⠀⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣼⣰⣧⣟⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣞⣇⣧⣸⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡤⠴⠶⢦⣤⣤⣤⣤⠤⣤⣤⣤⣤⣴⠶⠶⠤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠒⠒⣋⢥⡾⡫⢺⠃⠀⠈⢿⢮⡻⡪⣍⡒⠒⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣉⡩⠭⣒⡭⢊⠕⣁⠀⠀⠀⣌⠳⣝⠪⣗⡪⢭⣉⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣗⣒⣚⡭⠕⠊⣡⠔⠋⡀⠀⣠⡈⠳⠤⠙⠲⠭⣕⣒⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢹⣹⠁⠀⠀⢠⣤⣾⡿⣇⠀⡏⢧⠦⡦⠀⠀⠀⠀⢱⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡇⢀⣀⡀⠀⠳⣤⡴⢃⡄⠙⠶⠾⠃⠀⢀⣀⣀⢸⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠁⠀⠈⠱⡄⠀⠀⢀⡁⠀⠀⠀⠀⡔⠁⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⡇⠀⠔⠁⠈⠓⢄⠀⠀⡇⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢩⢧⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠑⠀⠱⢄⡀⣰⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢸⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⢸⠀⡏⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⡇⢓⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠚⠓⠓⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠓⠛⠚⠒⠛⠛

⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿
⠀⠀⠀⠀⣿⡷⢀⡀⢶⣩⣾⣦⠀⠀⠀⠀⠀⢠⣤⡄⠀⢠⣿⣿⡞⠁⠀⠀⠀⠀⠀⠍⡅⢨⡁⠺⣿⣿⣿⣆⡀
⠀⠀⠀⢠⣶⠞⠉⠀⠀⢹⣿⣿⡇⠀⠀⠀⢀⣼⣟⣡⣄⣼⣿⣿⣿⠀⠀⠀⢀⣰⣿⣷⣧⣿⣷⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⣿⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⡟⠉⠁⠀⠙⣻⣿⡿⠀⠀⣰⢻⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠂⠀⠀⠀⠈⣿⡏⠉⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿⠁⣿⠀⠀⠀⣻⡿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠉⠀⠀⠛⠋⠁⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣤⡈⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠟⠠⠉⠀⠀⠀⠀⠀⣐⠀⠄
⠀⠀⠀⠀⠀⢙⣏⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢏⣹⠺⡿⠇
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠘⠁⣿⠋⠀⠀⠀⠀⠀⠀⢛⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠙⣿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠾⣿⡟
⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠃⠀⢁⡖⠀⠀⠀⣸⣿⣄
⠀⠀⠀⠀⢸⣿⡟⣳⣀⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⢸⣿⡏⠁⠉⢿⢛⡟⢹⡇⠀⠀⠀⠀⢰⢾⣿⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢸⡿⣇⠀⠀⠈⠈⠁⠈⠁⠀⠀⠀⠀⠀⣻⣿⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⣠⣤⣤⣾⡷
⠀⠀⠀⠀⣿⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⡿⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠿⡿⠃
⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⢰⣿⣶⣄⣦⡘⠦⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⢿⣿⠀⡜⣽⡟⢻⣴⡖⢠⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣤⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⡰⣿⣿⡇⠈⠈⠀⠰⢿⣆⡀⣻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢽⣿⣿⡇⠀⠀⠀⣰⡄⠀⠀⠠⣌⠀⠈⠿⣤⡄⠀⠀⠀⢀⣩⡻⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠹⣿⡇⠀⠆⣾⣿⠇⢀⣤⣼⣿⡽⠦⠀⡿⠇⣰⣦⡀⢸⣿⣟⣿
⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣙⣿⣿⡀⠀⣿⡌⢀⠹⠀⠀⢹⠏⠀⠔⣹⠃⣿⡿⠀⣤⣶⣦⡀⠻⡿⠋⠀⢿⣿⣿
⢀⡀⠀⠀⠀⠀⠐⢿⣿⣷⡙⣿⣿⣏⢀⣿⣿⣆⡁⣤⣄⣿⠀⠀⢠⣤⠀⠟⠁⢸⣿⣿⢿⣶⣶⣦⣄⣀⠀⠫⠛
⡜⣹⣦⠀⠀⠀⢷⣦⠙⢿⣷⣿⣿⣷⣿⣿⣁⣿⣿⣛⠉⠉⡀⠔⠘⠁⠀⠀⠀⣰⣿⣓⣮⣻⣿⣿⣿⣿⣷⣦⡀
⣿⢿⣿⡆⠀⠀⠈⠟⠂⠘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⠀⢧⠀⠀⣴⠟⣷⡄⣿⣿⣿⣯⡉⠛⣿⣽⣿⣿⣿⣽
⠸⠈⠧⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇⠀⢀⠀⠀⠈⣿⠃⠻⣿⣿⠻⣿⣟⣯⣿⣷⣿⣿⣿⣿⣿⣿⢻
⠀⠀⠀⢠⡄⠀⠲⠀⠀⠀⠀⠐⠉⠁⢘⠽⢿⠀⠀⣼⣧⠀⠀⣯⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠉⠉⠉⢹⠋
⠀⠀⢤⣄⠀⠀⠀⠀⢀⣠⠀⠀⠀⠠⣯⠎⢸⣤⣴⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠄⠀⠀⣠⣾⣀
⠀⠀⠀⠀⠻⠆⠷⢤⠋⠙⠀⠀⠀⠀⠀⠀⣿⣿⡏⡻⠛⢻⣿⣿⠐⢀⠴⠛⠉⠸⣿⣿⣿⣿⠦⠄⡼⣿⣿⣿⣧
⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⡀⣸⣿⣿⣰⣶⣶⣄⠀⠀⣿⣿⣿⣿⣶⣾⠁⢙⣿⣿⢿
⠀⠀⠙⠇⢿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣰⡇⢸⡉⠀⣤⣷⣻⣿⣿⣸⣿⣽⡿⠀⠀⣿⣿⣿⣿⣿⣿⣤⣾⣿⡿⠃
⠀⠀⠀⠘⠒⠛⠻⠾⠶⠶⠄⠤⠀⠈⠙⠛⠟⠛⠚⠻⠿⠿⢾⣿⣿⣯⢀⠀⠀⣾⣿⣿⣿⣿⡻⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⡟⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⣀⠠⠐⠉⠀⠈⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡤⢶⣀⡉⠁⠀⠀⠀⠐⠊⠑⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠀⠀⠹⠏⠂⠀⠀⢀⣄⠒⠉⠛⢢⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⠰⢀⠂⠂⠠⡄⠀⠀⣀⠞⠁⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠈⠇⠎⠀⠙⠓⠒⡴⠒⠯⠔⠒⠀⢀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡋⠂⠀⠀⠀⡐⠁⡀⠀⠀⠀⠐⠂⠤⡁⠢⠌⣧
⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⢄⠐⢄⠔⢀⠜⠙⠆⠀⠀⠀⠑⢢⣀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠸⡄⠀⠀⠈⠢⡀⠠⠊⠀⠀⠀⠈⠆⣀⠀⠊⠀⡠⢷⡀⠀⠀⡀⠀⠠⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠖⠈⠁⠀⠀⡇⠀⠀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠄⠀⢀⣴⢀⠴⠃⠀⢠⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡤⡤⠤⠚⠁⠉⠀⠀⠀⢀⠃⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡈⢀⠇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢠⠌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠎⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⡀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠛⠀⠤⡀⠀⠀⢰⠁⠀⠀⢀⠠⠠⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢳⣤⠇⠐⠀⠀⠀⣀⠄⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⡤⠤⠶⣟⠀⣀⣠⣴⠞⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠓⠲⠶⠤⠤⠶⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣠⡤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠛⢻⣿⣿⣦⠟⠁⠈⠩⢭⣝⣿⣷⣶⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠛⢿⣿⡿⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⣴⠖⠒⠦⣄⡹⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠃⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠇⠀⠀⠀⠀⠝⠓⠶⢶⣄⠀⠀⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡀⢹⣧⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⣹⠃⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙⢷⣶⣶⣶⣶⣯⣭⣥⣾⣿⣦⣄⣠⣴⣿⣷⣮⣭⣭⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠠⠚⡟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠉⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣛⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠀⠀⠀⠀⠀⠀⠸


⡟⠛⢻⠟⠿⠿⠿⠿⡟⠿⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⣿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣔⢠⣀⡶⣤⣾⣼⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⣿⣷⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⣧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠁⠙⠚⠓⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⢀⡀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⢠⣶⣿⠟⠀⠀⣀⣴⠟⢁⣀⣴⣷⣄⠀⠀⠀⠀⠀⢀⡰⣋⡥⠄⠀⠐⢾⡿⣿⣧⡐⠀⠀⠀⠀⠀⠐
⠀⢠⣿⡿⠋⣠⡴⠛⠉⠤⠀⠁⠀⢼⡟⠁⠰⢾⡟⣩⡴⠜⠋⡡⠔⣀⠤⣠⢔⡿⠟⣉⢩⣶⡜⠆
⣰⣿⠋⢀⡼⠋⠀⠄⠀⠀⠀⠀⠀⠀⣤⠶⣻⠟⣂⣤⠶⠂⡼⣠⡞⣠⢎⣵⢋⠴⣫⠞⣩⣴⢾⠀⣄
⡿⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡵⠛⢡⣚⡭⢒⣡⡞⢀⡞⡼⢇⡼⢋⡆⢰⢁⣾⠟⠋⠊⣿⣎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⣾⡇⠀⡀⠀⠀⠀⠀⠀⠀⢸⡟⠁⡵⣫⢿⡗⡖⣶⠀⡞⠙⢛⢸⢃⣟⠀⠈⠘⢀⠔⠁⠘⣩⣗⡆⡀
⠀⢸⣿⠀⠀⣧⡀⠀⠀⠀⠀⢠⠎⠠⠈⣼⣣⢏⡔⣼⠃⡸⠀⠀⡾⠈⡼⣾⣿⡖⠸⢠⠂⠀⢘⢡⣷⡽⡇
⢀⣿⠏⠀⢠⣿⣷⢦⡀⠀⠘⢁⣔⡁⠻⠣⠁⠼⠽⢃⡾⡱⢣⠜⡵⢠⣿⢿⢧⡇⠀⣿⣤⠀⣟⠣⠛⡃⢉
⣸⠏⠀⢀⣾⣿⢃⣼⡿⠦⢤⡄⠞⠁⠚⠀⡋⡀⡼⡿⡘⠁⠀⣸⠃⠘⣞⣈⠏⠀⠀⣿⡏⢠⠌⢘⠟⠁
⡟⠀⣠⣿⡟⢁⣾⠟⠀⣤⣾⠇⡀⡄⣵⢰⢿⢣⡇⢂⡕⡎⡀⣡⠇⡤⡉⠠⢲⣧⣀⣪⢼⡉
⡇⣰⣿⣿⢀⣾⠏⠀⣾⡿⠃⠀⠉⠀⣿⢀⠋⠼⢡⠏⡼⠼⡡⢁⠞⠀⣴⣷⢶⠟⠃⠁⠈⠁
⠈⣿⣿⣿⡀⠈⢠⣼⠏⡀⠀⠀⠀⠀⠀⣭⠀⠀⡟⠘⡴⠈⠀⢨⢀⣸⡇⣼⠈⡜
⠀⠙⢿⣿⣇⠀⢸⡟⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢧⢳⡃⡎⠊⣿⠤⠂⢸⠀⠀⠂
⣧⡀⠀⠻⢿⣿⣶⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡏⡷⠐⢠⡼⢶⠃⠈⠀⠀⠈⠉
⣿⣿⣄⠀⠀⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠃⠘⠀⠂
⡘⣿⣿⣷⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣷⣌⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠄
⠟⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣶⣄⠀⠀⡝⡍⡏⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⣤⡖⠺⠧⠑⣄⠹⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠛⢿⠷⣏⠱⢁⠀⠈⠆⠀⠀⠀⠀⠀⢀⣀⠀⠸⡃⢱⢄⡄⠮⠀⠈⠛⡇
⠀⠀⠀⠀⠀⣿⣿⠏⢥⡤⡘⣂⡀⢠⡏⢆⡀⠃⣀⡆⣀⣀⣀⠤⠤⠬⠥⠄⠀⢇⠀⣌⠄⢄⠂⢀⠎⠁
⠀⠀⠀⠀⠀⢍⠁⠿⡴⠒⠚⠁⣸⣸⡿⠆⢉⠁⠀⢃⠀⠀⠀⠀⠒⢒⠉⠀⠀⠈⠢⠜⠶⡮⠔⡏⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠈⣿⣷⣼⣶⠮⠥⢾⡿⠡⠀⡸⠀⠀⠸⡀⠀⠀⠀⡀⠉⠀⡠⢖⢄⣇⠀⠈⡧⠔⠁⠀⠀⠤⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡩⠝⠉⠙⠤⣴⣿⣷⠀⠀⠀⠀⠀⢀⠅⠀⠀⠚⢀⡴⠊⠔⠁⡸⠈⠉⣻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠎⠀⠀⣀⠤⠴⡆⠀⢹⠀⢰⠤⠤⠐⠊⠀⠀⠀⠐⠁⠀⠀⠀⠀⢇⣀⣠⠃⢃⠀⠀⠀⡠⠒⠚⠂⡤⠃
⠀⠀⠀⠀⠸⠀⠀⠉⠀⠀⠀⢡⡠⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠷⠷⠧⠊
⠀⠀⠀⠀⠀⠑⠤⠄⠤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠙⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡲⣋⠔⠒⠀⠐⠒⠢⢄⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠥⠀⢰⣋⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡠⠊⠁⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⡜⠛⠲⠦⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⡴⠊⢀⣤⣷⡶⠉⢻⣦⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⠒⠀⠀⠀⠀⠀⠀⢰⠋⣿⠆⡧⠀⠀⠙⣎⠂⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⡿⣤⡶⢶⠢⣄⠀⠀⠀⠀⠈⠐⡀⠞⠁⠀⠘⠊⠙⠆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⡃⠈⠳⠈⠊⠛⢓⡀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠃⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣠⠞⠀⠀⠀⡆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠳⣄⡀⠀⢮⠀⠂⢠⠆⢀⡴⠁⠀⠀⠀⠀⡇⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠈⣙⣲⢭⣬⣥⠴⠛⠖⠊⠑⠢⣄⡀⠇⠀⠀⠀⠈⠢⣄⣀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⡐⢧⣀⡴⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠱⣤⡖⠀⠀⠀⠀⠀⠀⠀⣤⡟⠁⠱⡀⠀⠀⠀⠀⠀⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠖⠊⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⢀⠜⠉⠙⡄⠀⢣⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣠⢃⠆⠀⠀⠀⠀⠀⡴⠃⠀⡠⠖⠉⠀⣈⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠴⣋⠔⠁⠀⠀⠀⠠⡀⡜⡠⠔⢊⣀⠤⠒⠋⠁⠀⠀⢡⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⠀⢀⠔⢉⡽⣢⣄⡀⠠⠶⢊⣵⣄⠃⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠙⡰⡇⠀⠃⠈⢻⠞⠁⠀⠉⠉⠉⠉⡳⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢳⠠⡄⠀⠦⢠⡀⠀⣀⠀⠀⠀⠈⠓⢢⡚⠁⠀⠀⠀⠀⠀⡆⠀⣠⡾⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡷⢵⡦⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠜⢠⠼⢫⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⣠⡴⢒⣛⢣⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⢰⡖⣦⣶⠒⣶⠂⡆⣴⡆⣶⣶⢰⡖
⠀⠀⣰⢛⠃⡫⣬⢼⢇⣘⣧⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠧⠸⣧⡿⣿⡅⣿⠀⡇⡿⣷⣿⣿⢸⡇
⠀⠀⣿⢛⡋⣿⠼⠯⠀⣷⣸
⠀⠀⠸⣞⠣⠨⠦⠺⠒⠂⠇⠀⢀⣤⣤⣄
⠀⠀⠀⠀⢠⣶⣿⣷⣦⡄⠀⠀⢸⠟⠛⠙⠃⠀⠀⣠⣶⣶⣶⡄⠀⠀⣠⣤⣤⡀⠀⠀⠀⠐⠛⠉⣠
⠀⠀⠀⠀⢹⣯⣀⣀⣹⡇⠀⠀⠸⠶⠄⠶⠆⠀⠀⠿⣿⣟⣿⣿⠀⢰⣏⣥⢤⠹⠀⠀⠀⠲⠶⣿⣿⡇
⠀⠀⠀⠀⢰⡀⠁⠈⢁⠁⠀⠀⠀⣀⣤⠀⡀⠀⠀⠈⠛⢣⣟⠉⠀⠀⠉⣡⣤⡀⠀⠀⠀⠀⠰⣦⣟⠁
⠀⠀⠀⠀⠈⣃⡀⠀⣏⣀⠀⣴⣾⣷⣶⣾⣿⣿⣤⣴⣆⠙⠉⢶⣤⡄⣠⡏⠋⠉⣷⣤⣴⣾⣦⣺⣿⣶⣤
⠀⠀⠀⢰⣿⣿⣳⣾⣽⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠀⣠⣼⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠿⠿⠿⠶⠿⠿⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠄⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠁⠀⠀⠀⠀⠈⠑⢄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⢀⣀⣀⣀⠀⠀⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠏⠀⠊⠷⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡀⣀⠀⠀⠀⠀⠀⠀⡴⢅⠀⠀⠀⡠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⡀⠐⡉⠀⡠⠚⠄⠠⡤⠐⠁⣠⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡀⠪⠂⠈⢁⣀⠈⠢⠊⠌⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⣤⠶⠂⠀⠁⠘⣳⠀⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠀⠚⠛⠐⠠⡀⢺⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⠀⠀⢴⣷⣴⣶⡟⠷⠶⡀⠈⠚⠀⠀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⣦⠉⢸⠉⠒⢬⣉⠑⠛⠿⠿⠛⡇⠠⠵⠬⠦⠠⠤⠤⢡⠄⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠹⠲⣿⣿⠣⠄⠀⠀⠀⠀⠀⠀⠒⠐⠂⢁⠤⠋⠀⠀⠠⠄⢂⡠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠅⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⣦⠀⠀⠀⠀⠀⡰⠁⢀⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠊⠳⠤⣤⣄⣀⣀⠀⡠⠁⣰⠇⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⠤⠴⠒⠒⠈⠉⠉⢁⠊⠁⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⡀⠀⠀⠀⠀⠀⡠⠂⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡇⠀⠀⠀⠀⡁⠀⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠀⠈⠀⠠⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⠀⠀⠀⠀⠀⠀⠀⠈⢂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⠎⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠉⣉⠤⠔⠒⠚⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⡤⠄⠀⠀⠠⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠓⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⣀⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣠⡾⠀⠀⠀⠉⣛⠿⣆⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠋⠉⠉⢢⡤⡞⠁⠀⠀⢳⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣧⠀⢠⣤⣸⠃⢹⣄⣛⣫⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢰⠉⠑⠒⠒⠁⠈⠉⢹⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠐⠚⠋⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠹⡀⠀⠀⠀⠀⢀⠔⠋⠙⠃⠀⠀⠀⢸⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢥⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠶⣦⣹⠀⠀⠀⠀⠐⠴⠶⠒⢛⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⢷⢶⣶⣤⣤⣤⣴⣶⢻⢳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣽⣿⡽⠼⠽⢏⠁⣩⣿⢿⢼⣼⣼⢿⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⣭⢽⣿⣽⡯⣿⠽⢽⢽⢼⣽⣿⢟⡛⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⠉⣯⣯⡭⠽⠐⢻⣻⣇⣀⣈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⡶⡟⣿⣿⡷⣿⣿⣿⣿⣻⣿⣯⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡗⡷⣷⣧⣷⣿⣟⣟⣛⣺⣼⣾⣛⣁⣿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢻⠞⠟⠳⠟⠟⠛⢻⣿⣿⣷⣿⣿⠿⠾⠾⠿⠿⠟⠛⠛⠛⠫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣿⣭⣿⣿⣏⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣇⢀⢀⠀⠤⢤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠟⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠈⢷⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⣀⠀⠀⠀⣤⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢰⠄⣿⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠜⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡆⠀⣿⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠇⢁⠀⠀⠄⣀⡤⠔⠁⠀⠉⢿⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠦⠈⢦⠀⠀⠀⠀⠀⠀⡀⢈⠿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠢⣈⡃⠈⢠⣠⠴⠶⠫⠙⠳⢁⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⣟⢆⠀⠁⠀⠀⠀⠋⠀⠠⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠷⣄⡀⡀⠀⠀⠔⠰⢀⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⣁⠒⢓⠀⠨⠦⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⢀⣀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⡤⠖⠋⠉⠳⣄⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⣴⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠁⠘⠦⠴⠊⢇⠀⣀⡄⠀⠀⢠⡇⠀⠀⡠⠲⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⣀⠀⡀⠰⣢⠀⠉⠀⢈⢶⠾⠁⠑⠒⠋⠀⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⠀⠈⠛⢿⠄⠀⠀⠀⠀⠘⠁⢀⡄⢀⠀⠀⠀⢠⣇⣀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠘⠃⢰⣲⣾⠷⠄⠀⠀⠀⠻⠟⠛⠋⠀⠀⠀⠀⢀⣂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣫⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠢⣀⠜⠉⠁⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢆⠀⢸⠀⠀⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⣤⠜⠀⣬⣤⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠴⡻⠤⠞⠀⠘⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠁⠀⠀⠀⠀⠘⢾⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠀⣼⢸⡟⢉⣹⢳⣤⡀
⣿⣿⠟⠉⠙⠛⠛⠛⠛⠻⣿⣿⣿⡿⢦⣤⣤⣮⣷⡀⢀⡴⠚⠓⠒⠲⢧⡀⠀⣠⣏⠘⠛⢯⢴⣾⣷⣹
⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠁⠀⠀⠀⢀⣩⡻⣿⣠⠖⢒⣒⢦⡀⠹⡴⢹⡟⠛⠓⠛⠿⣷⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡀⠀⠀⠀⠀⣀⣤⡘⢷⣝⢿⣍⠛⢿⣿⠀⡇⠀⢹⣾⠃⠀⠀⠀⣴⡟⠀⣀⣀
⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣶⣾⣽⠁⠀⠙⠻⣿⣷⣮⣓⠈⢧⡼⠁⢀⡼⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠘⠓⠾⠿⣿⣿⠿⠿⠛⠛⠻⢿⣻⡄⠀⠀⠀⠈⠻⣝⠪⣝⣾⣶⣾⠟⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠙⣿⠀⠀⠀⠀⠀⠀⣹⣿⣿⠟⠁⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⣿⣿
⣆⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢸⠃⠀⢀⣠⡖⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⠉⠙
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢾⠀⠀⠈⢁⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣀
⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠀⠀⡠⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦⣀
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣦
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠗⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⡏⢙⠲⠦⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣾
⣿⣿⣿⣃⡏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠒⠒⠶⢤⣴⣶⣶⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣶⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣙⣿⣿⣿⣟⡘⠘⠿⢿⡿⣿⣿⣿⡻⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⢹⣿⡿⠁⠉⠛⠓⠂⠀⠶⢾⡳⢬⡑⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣾⣿⡇⠘⠒⠒⠀⠠⠀⠦⠤⠘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠛⣿⣷⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣽⣿⣦⠀⠀⠐⠀⠀⣀⠜⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷⣄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⣃⣀⣀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠋⠡⠐⡉⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⢀⠔⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣿⣿⡿⢡⡎⠀⠀⠀⠀⠀⠀⠨⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⢣⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣤⡤⠶⢤⡀⠀⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣴⡿⠁⠀⠀⢀⣀⠀⠑⣄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⡇⠀⢠⣾⡿⠟⢿⡆⠼⡣⠭⢦⡴⠶⠶⠤⣄⣤⣄⣀⠄⢠⣆⢁⣐⣐⡢⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⣿⣿⡄⠟⢙⠀⠀⣨⣆⠢⣥⠔⠚⠀⠀⠀⠀⠘⢻⡄⠹⣆⣀⡼⠟⣉⡉⢻⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠻⣿⡄⠈⢦⣴⠟⠛⠁⠀⠀⢀⡀⣀⠁⠀⠀⠈⢻⡀⠈⠳⣁⡤⠛⡟⢸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠘⣿⡈⢪⣿⡄⠀⠀⠀⢾⣼⡷⣿⠛⠳⠞⠉⠿⣥⡀⡀⠀⠡⠀⢣⡟⡰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⡇⠀⣿⡇⢢⠀⣴⢛⣃⡀⠀⠀⢀⡜⠀⠀⡞⠃⠽⣾⣧⢠⡿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠇⠀⡿⣰⠒⣾⠇⣦⣄⠙⠳⣄⠘⠆⠀⠀⠰⠋⠀⣻⡿⢺⡇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⣿⠀⣸⡅⣧⢀⣿⠀⢻⣿⣿⣦⣌⠃⠀⠀⠀⣠⣤⠤⣼⠇⣸⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⣿⠇⡼⠁⣿⡉⠁⣿⣇⠀⠙⠻⠿⢹⠀⠀⢀⣤⣴⣶⣾⡟⣴⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣾⡟⢸⠁⢸⠁⠻⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣸⡏⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣿⠁⣾⠀⢸⠀⠀⢸⣿⡟⠀⠀⠀⣀⠄⠠⠤⡀⠀⠀⢻⣿⣿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⠀⠸⡀⠀⠳⢤⣿⡿⠁⣦⡀⠀⠸⣴⠀⡰⢃⠰⢠⣾⡇⢿⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠻⣄⠀⠳⡄⠀⢸⡿⠁⠀⣶⣬⣦⣤⣬⠁⢰⠀⣴⡿⢩⠃⠘⣿⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⣦⠀⢻⠀⠸⣇⠠⠊⠈⠉⠋⠙⠉⠐⠒⣿⢯⡼⠃⠀⠀⢹⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡏⠀⣸⠀⠊⠳⡈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠈⡷⠀⠑⠀⠀⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⠃⣰⠃⠀⠀⠀⠀⡷⠖⠒⠛⢛⣛⠛⠛⠻⠀⠀⠑⢄⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⣿⡀⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠐⠀⠀⠀⠈⡄⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⠷⣜⠶⠤⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣰⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠠⠔⢒⣼⠏⠉⢳⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡫⠊⠁⠀⢠⡔⠋⠁⠀⠀⡀⢹⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⠁⣀⠀⡀⠚⠁⠀⣀⣀⣈⣖⣽⣿⣿⡛⣹⡆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢠⣧⠀⠀⠀⠁⠀⣤⠞⢡⣿⣿⣿⣿⣿⣿⣧⣌⠓⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠊⣿⣶⣀⡠⠶⣖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡼⡛⠫⠕⠠⠈⠅⠀⠛⠛⠿⢛⣛⣿⣿⣿⣭⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⢻⣇⡀⠀⠀⠀⠀⠐⠠⣶⣚⣿⣿⡿⢿⣿⣿⣿⣿⣷⣊⣳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡛⣿⣿⣷⣤⣄⣤⠀⠀⢠⣽⣿⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡇⠻⠿⣯⣁⡈⠁⢲⠾⣿⣿⠟⠟⠁⢀⣩⣿⠿⠟⣹⡞⠛⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⣄⣾⣿⣿⣿⣷⣶⣴⣿⣿⣦⣶⣿⣿⣿⣿⣷⣰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆⠙⣯⡈⠛⣿⢿⣿⢹⣇⣿⡿⣿⠟⢁⣼⠏⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣶⣶⣌⣀⠀⠂⣴⣾⣿⣦⡀⠀⢀⣩⣶⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣿⣿⠘⢿⡏⢻⢃⠀⣻⠇⠘⣟⠁⠘⡟⢹⠿⠙⣿⣿⣦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⢸⣿⣿⣿⣿⣦⡀⢹⡸⠈⣠⣠⣜⣡⣀⣀⠀⠗⡉⢀⣼⣿⣿⣿⣿⠇⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠩⣦⡀⠻⣿⣿⣿⣿⣿⣮⡑⢤⣶⠦⣬⣤⠴⣶⡤⢊⣵⣿⣿⣿⣿⣿⠟⢀⣴⠝⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⣾⣿⣤⣈⠻⣿⣿⡎⠀⢳⣄⠉⠀⠈⠁⠀⠈⣠⡾⠁⢑⣿⣿⠟⣡⣤⣿⣷⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⣿⣿⣮⡻⠷⠆⢤⣿⡗⢾⣿⣿⣶⢾⣿⠧⠴⠾⢿⣵⣿⢿⡿⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣿⡄⠠⡀⣈⣿⡀⢹⡟⠁⣾⣁⠀⡄⢀⣿⠟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⣡⣤⣶⣶⣿⣶⣶⠖⣿⣶⣶⣤⣀⠻⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⣴⠾⣟⣿⣿⣿⣾⠸⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣼⠙⣿⣿⣿⣿⣿⠿⢊⣀⡀⠹⢻⣿⡿⣿⣿⠁⢨⣴⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⢘⠻⠿⠿⠛⣡⢋⡀⡉⣄⠐⠢⠍⠿⠋⡄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⢸⣷⢶⡶⠋⠁⢈⣀⣀⠈⠙⠳⠶⢞⣧⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣧⣿⣆⣠⣄⠈⢋⡉⠝⠁⠀⡠⢂⡼⠿⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢠⠀⠀⡈⣴⠎⣀⢈⡆⠐⣨⣴⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣾⣿⣿⣿⣿⣷⣿⣮⣶⠞⠉⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠑⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⠀⠈⢻⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢷⣜
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⢀⣀⡀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⣀⠀⠘⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢄⡀⢷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⡙⠧⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⡀⠸⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢀⠔⠁⠈⢆⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⡇⢀⠞⠈⠁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⢄⠀⠀⣱⡄⠀⠈⣀⠤⣀⠀⠀⠀⢠⠃⠇⠀⢦⠸⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢎⢆⠞⡡⠃⠀⢰⡁⢐⡾⠤⠐⢶⠁⢸⣤⠉⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠒⠊⠁⠀⠀⢠⢿⠁⠀⢀⡴⠋⠀⣾⡷⣊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣠⠞⠁⠀⠀⣀⡠⠄⢚⡶⠒⠊⠁⠀⠀⠀⠘⠹⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡄⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⡴⠋⣀⠤⠒⠋⠉⠀⢀⡴⠃⠠⠀⠀⠀⠀⠀⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⡴⠁⠀⢀⠔⠁⠀⠰⡁⡀⠀⣀⡤⠖⠉⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⢣
⢀⣠⡄⣢⠄⠀⠀⠀⢠⠃⠀⠀⠀⠀⡇⠀⠀⢀⡞⠀⢀⠔⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢧⢀⡠⣒⡆
⠋⠈⠑⠪⢕⢕⢲⣠⠇⠀⠀⠀⠀⠀⡇⠀⠀⠎⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠳⢯⣏⠒⠀⠈⢇⣠⢿
⠀⠀⠀⠀⠀⠑⢵⣟⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⠛⠀⢀⠜⢁⡎
⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⡀⠀⣠⠔⠁⢠⠂⠣⣀
⠀⠀⠀⠀⠀⠀⠀⢨⠬⢷⡄⠀⠰⠶⠓⠐⠐⠒⠒⠒⠒⠒⠒⠒⢂⠖⠀⠀⠀⢀⠤⡤⠀⢄⠀⠀⠀⠀⢀⡀⠥⡞⣳⠎⠈⠑⠦⠃⠰⡄⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠃⠀⠀⠀⠀⠣⢔⢻⠿⢸⠀⣀⣤⣯⠁⢀⡴⠁⠛⠁⠀⠀⠀⠀⠀⣷⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢎⠜⠫⡀⡭⠐⠈⠉⠹⢔⡝⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⠋⠙⢧⠀⡠⣫⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢰⡆⠀⠀⣶⠆⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡵⠁⠀⠀⠀⠀⢪⡀⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡡⠛⢷⡬⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⢩⠋⠁⠒⠒⠂⢠⠤⠃⠀⠀⠀⠀⢎⣸⠐⢈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⢻⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠖⠳⠧⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢭⢸⠖⠛⠃⠀⠉⠓⣮⢦⣵⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣱⣄⠸⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠡⠃⠀⠀⠀⠀⠀⠀⠻⠞⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⣛⣛⣷⠶⠖⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠰⠒⣿⣿⣶⡼⠚⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⣁⣿⣿⣯⡉⠄⠀⢠⣸⠻⠿⢓⣀⣀⣄⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣰⣾⣿⣿⡄⣈⠙⠍⠀⠀⠀⠀⠉⠢⢌⠉⠊⠫⡀⠀⠈⠐⠢⠤⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡜⠻⠛⠛⠑⡎⣼⡵⢿⣦⠀⠀⠀⠀⠀⠁⠒⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠉⡗⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⢿⢴⣾⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣆⠀⠀⠀⠈⠀⠀⠘⢁⣆⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣖⣺⢶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣴⣆⠀⠀⠀⠀⠀⠈⣿⣷⣦⡸⣿⣿⡿⡿⢻⣿⣿⡟⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣧⠀⠀⠀⣀⠄⢻⢿⣿⣧⠘⣏⠀⠀⠠⠤⠍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢸⣸⡟⢿⢇⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣯⣿⣿⣿⣿⣿⡀⡀⠀⠀⢀⡼⢃⣿⢸⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣰⣿⣿⣿⣿⣿⣿⡇⠐⠀⠒⠉⠀⠱⣿⢸⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡿⣿⣿⣿⣿⣻⠃⠀⠀⠀⠀⠀⣿⣿⢸⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣿⣿⣹⣿⡏
⠀⠀⠀⡠⠒⠈⢀⣀⡀⠀⡀⠀⢘⣿⣿⡇⣏⣿⣿⣿⣿⣄⣤⠀⠀⠀⢰⣿⢽⣿⡿
⠀⠀⢸⣷⣴⣾⣿⣿⣿⣷⢃⡠⠾⣿⣿⢇⡿⣽⣿⣿⡟⠙⠊⠀⠀⠀⢹⡇⠈⢿⡗
⠀⠀⠀⠚⠿⣿⡿⣿⣯⠇⠀⢘⠀⠽⢿⡀⠻⣿⣿⣿⠀⠀⠤⠄⠀⠀⠈⠡⠀⠰⡀
⠀⠀⠀⠀⠀⠀⢻⠻⣇⠢⠞⠋⠀⠀⠀⠳⡄⠘⣿⢻⠷⣶⢤⡀⠀⠀⣔⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠉⡉⣷⡤⣄⠀⠀⠀⠀⠀⠐⣦⡈⢊⠠⣿⢿⣖⠂⠘⢿⣿⣷⣄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠘⠋⣽⡱⡼⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⡻⠁⠀⠀⠀⠈⠻⣿⣶⡄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡌⣀⠀⡄⠀⠀⠀⠀⠀⠈⣿⣿⣾⣥⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣢⣀⣣⠀⠀⠀⠀⠀⢸⣷⣯⣛⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣲⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣰⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⡏⠀⠹⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣇⠀⢰⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣤⣿⠿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⣿⣿⣿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣻⣿⣿⣿⡆⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣿⠉⣿⣿⡀⠀⠐⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⣿⣿⡟⣿⣿⡄⠀⠀⢀⣀⣬⣦⣶⣶⠋⠁
⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⣿⣿⠆⠹⣿⣷⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⣁⣸⣿⣿⣶⣿⣿⠿⠿⠻⣏
⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣿⣿⣤⣀⣽⣿⣆⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣤⣶⣿⣿⡿⠿⠻⣿⣿⡁⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⡿⠟⠛⠛⢿⣿⠿⠿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⢻⣿⣿⠀⠀⠀⢹⣿⣷⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⣿⣿⠀⠀⠀⠈⣿⣷⣀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣇⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣿⣿⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣤⣀⣀⢈⣿⣿⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣶⣿⠿⠿⠇⠀⠀⠀⣀⣠⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠻⠿⠿⠿⠿⠿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⡞⠋⣹⠟⣿⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡿⠀⢹⡆⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⣿⠟⠿⠾⣿⡶⠶⠿⡿⠓⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡿⠀⠀⠀⠸⣧⠀⣸⠇⠀⠀⢀⡤⠀⣀⣤⣿⣷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠶⠶⠶⠶⠿⠟⠉⠀⠀⣰⠟⡰⠋⠉⢸⡿⣿⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠁⠀⠀⢸⡇⢹⡇⠀⠀⠀⣀⣹⣷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣿⠟⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⣁⣨⣿⡶⠶⠛⠋⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⢠⡟⠀⠹⣆⠀⠀⢘⣷⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⣤⡶⢾⡟⠋⠉⢹⣧⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣼⣧⣤⣤⣾⣷⣶⠶⣿⣿⠟⠛⠻⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠉⠁⠀⣼⠇⠀⠀⠀⢿⡄⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠀⠀⠀⠈⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢠⡟⠀⠀⠀⠀⠘⣧⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠈⢷⣄⢸⡏⠀⠀⠀⠀⠸⣿⣀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣾⣅⣀⠀⠀⠀⣀⣹⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣀⣀⣀⣤⣽⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠠⡀⣀⣀⣀⠀⡄⢠⠀⠀⣤⣶⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠐⣴⡿⢯⣵⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠟⢿⣦⣆
⠀⠀⠀⠀⠈⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⡿⢧⣾⣿⣿⣿⣿⠟⠛⢿⣟⢿⣿⣧⠀⠀⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣰⡷⠀⠈⣿⣿⣿⣿⡀⠀⣠⣿⣦⡽⡟⠀⠀⠠⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⡇⠀⠈⣹⢿⡖⠂⠘⣿⣿⣿⠿⡇⠛⠛⢻⣧⡥⠀⠀⠤⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠘⣷⡀⢰⣿⣿⣷⣠⣿⣤⠄⠀⠘⢧⠐⢤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢈⡿⣷⣮⣽⢻⣿⣿⣍⡤⠀⠀⠈⣷⣿⠋⠓
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠨⠛⢷⣾⣿⣿⡋⠀⢱⣴⠿⠋⠀⠃⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠿⣷⣯⠉⠂⠀⠀⠀⠀⠀⠰⠀⠀⠰⠋⡿⢿⣉⣲⣿⠋⠑⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⠋⢁⣅⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠉⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠄⠀⠀⢀⣤⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣿⣿⡟⠉⠙⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⢦⣀⠀⣾⣿⣿⣿⣶⡄⣴⣿⣿⣿⣷⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢹⣿⣿⣿⣿⡉⢉⣇⠁⢸⣿⣿⣿⣾⡏
⠀⠀⠀⠀⠁⠀⢺⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⡐⠿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⢀⣀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣽⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢦⡌⠉⠛⢿⣿⣿⠉⠀⠀⣿⣿⠟⠛⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⠻⢿⣿⣿⣟⣿⣿⣿⡿⠛⡏⠀⠀⠀⠀⠀⠀⠀⠙⠛⢳⡼⠟⠛⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠟⠉⠉⢹⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁⢠⡄⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣢⣤⣤⣤⣤⣤⣤⣴⣻⣤⣄⡀⠀⠀⣀⡜⢣⡘⠋⣹⣶⣶⣶⣶⣿⣦⣐⡶⠃⣈⡶⠂
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡹⡟⢤⣸⣾⣿⣿⡿⢿⣿⣿⣿⣿⣷⡖⠹⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣰⣁⠃⠈⣿⣿⣿⣿⣥⣸⣿⣿⣿⣿⣿⣧⠀⢰⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡻⠉⣋⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⢶⠟⠂
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⠇⠙⠻⣍⢻⣿⣿⣿⣶⣿⣿⣶⣾⣿⣿⣿⣿⠷⡧
⠀⠀⠀⠀⡀⠈⠻⡿⢟⠀⠙⢿⣿⣿⣿⡿⠏⠘⠿⠿⠋⠀⠀⣠⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣧⣀⣿⣿⣿⣐⣴⢻⠏⠀⠀⢀⠀⠀⠀⠛⣷⡊⢦⡝⠛⣿⠿⡷⠉⣳⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣋⠁⡯⠐⢸⡎⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣏⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠙⣷⠊⠀⠀⠀⠈⠁⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠿⠋⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⢢⢰⡀⠀⡧⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠉⢉⣠⢴⠖⠂⠒⠁⠀⠀⠀⠀⠀⠀⢸⡁⢀⣼⢠⣠⡏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢠⠔⡀⠇⡘⡆⠀⠘⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡀⡀
⠀⠀⢀⠠⠄⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢈⡲⠀⠀⠀⠀⠀⠀⣡⠇⠸⠀⡏⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠼⠓⠒⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠖⠉
⠀⠀⠀⠀⡴⠊⠁
⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣫⣛⣭⣋⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠖⠒⠒⠒⠒⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣤⣀⡀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣽⣭⣉⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣠⡾⠟⣩⣥⡤⢤⣤⣀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠉⣴⠟⠁⠀⠀⠀⠀⠉⢳⣆⣠⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣆⠀⠒⠶⠾⠿⠛⠀⣼⠏⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠰⢶⣶⣶⠿⢀⡾⠁⠀⠀⣀⣙⠷⢦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢶⣤⣄⣀⣠⣤⡴⠟⠁⠠⠶⠛⠛⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⡄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠉⠙⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢘⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣷⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡉⠁⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡝⢷⣦⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡘⠧⠀⢄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⠞⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠿⢷⣄⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠾⠟⠛⠋⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢸⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣛⣹⣏⣛⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡖⣿⠏⢉⡇⠀⣿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⣟⠀⢸⡇⠀⣿⠉⣻⡙⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠟⢹⠋⢹⠏⣿⡏⢙⡏⠉⣿⠉⣹⣿⣿⠋⠁⠀⠀⠈⢹⣿⡟⠓⡿⠋⢹⡏⠉⣿⠀⣿⣁⣹⣇⣹⣋⣻⣿⣦
⠀⠀⠀⠀⠀⠀⢀⡴⢿⣋⣿⣿⣿⣿⣿⣿⣾⣴⣿⣟⣻⠛⠛⡿⠛⣿⠉⢻⣦⡀⠀⢀⣴⠿⠿⡷⠞⢿⠛⢻⣿⣿⣿⡿⡿⠛⢻⡏⢹⠉⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⣿⠚⣿⠛⢿⠛⢿⢿⣿⡿⠿⣷⣶⡷⠤⣿⠤⢾⠿⠿⣶⠿⣿⠦⢴⡷⠴⣾⠦⣾⣯⣿⣿⣶⣿⡒⣻⠓⣻⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⠴⣿⢤⣴⠤⣾⠤⣼⡿⣦⡘⣿⡷⠶⣿⠦⣾⠦⢤⣷⠤⣾⠤⢤⣧⣤⣧⣼⣿⡿⢧⣄⡉⠉⠛⣿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣷⠶⣷⣤⣿⠿⢋⣴⠟⢿⣿⡧⢤⣧⣤⣼⡤⣤⣧⠀⣤⣀⣠⣇⣀⣯⡾⠃⠀⠀⠈⠙⠳⢶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣥⡾⠛⠁⠀⠈⢻⣄⣀⣷⢀⣼⠀⢐⣇⠀⣹⠀⢨⡏⢉⣿⣷⠀⢀⠀⠀⠀⠀⠈⢉⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠉⠀⠀⠀⠀⠀⢘⣯⣩⣯⣈⣿⡉⢹⠋⠀⣿⢀⣹⡿⢿⣯⡽⢷⠿⣷⣶⣶⣴⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡿⣿⣷⠀⢀⣤⣴⢶⣿⠛⠻⡿⠛⣿⢿⣿⣷⣶⣿⠋⢿⡷⢾⡿⠟⢿⠟⠛⡟⠻⡏⠉⡟⠉⢻⠉⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣿⠶⢾⡿⢿⡿⠻⣿⠶⢾⠶⠶⡿⠒⢿⠒⢿⠟⢛⣿⡿⢿⠟⠻⡿⠓⢾⠷⠾⡗⠺⡿⢲⣿⠶⠿⠺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠦⢾⠦⢼⡧⢠⣿⠤⣴⠤⠴⣷⣶⣾⣶⣾⠿⠿⣿⠶⢾⡶⠶⡷⠄⠿⠖⣺⣷⠶⡟⢻⠗⠒⣿⣾⡿⠁
⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⡉⠹⣿⣦⣿⠦⢼⡧⢴⣿⣶⣿⣶⣾⣿⣏⣿⣱⣾⣀⣀⣿⣀⣸⣇⣠⣧⣴⣿⣋⣹⣇⣠⣇⣰⣄⣴⡿⠏⠀⠀⠀⣀⡀⡀
⢀⣤⣾⣿⣥⠖⣠⣾⣥⣤⣾⣣⣄⣙⣿⣿⣶⣾⣧⣼⣿⠿⣿⣿⣿⣿⣿⣏⣀⣹⣧⣴⣿⡛⢻⠉⢨⣇⣀⣿⣤⣼⣷⣿⣿⣿⣿⣷⣾⣿⣿⣿⢟⣿⣿⣵⠖
⠈⠹⠿⠟⢋⣼⠟⠉⠾⠛⠙⠛⠿⠟⠁⠾⠉⠉⠉⠉⢻⡆⣿⣿⣿⣿⣿⣿⣯⣽⣉⣿⣿⣴⣿⣿⡿⠿⠛⠛⠻⢿⣿⡿⠟⠹⠟⠛⠋⠁⠈⠀⠺⠛⠉⢠⢶
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠋⠈⠻⠛⠛⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠉⠉⠁⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠡⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠉⠈⠉⠐⠀⠀⠀⠂⢴⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡠⠊⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣤⡀⢀⣀⣁⣀⣤⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠆⠀⠀⠈⠛⠛⠀⡀⠚⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠙⢿⠿⠿⢿⡿⠃⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠃⠠⠃⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡉⠈⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢴⣾⠿⠿⣷⣦⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠂⠈⠑⠁⠐⠋⠈⡀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠉⠉⠈⠑⢄⠀⠀⠀⠀⠀⢉⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠸⠀⠀⠀⢀⠀⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⢤⠄⠒⠄⣀⠠⠖⠃⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠤⠖⠀⠀⠉⠀⢀⡠⣄⡀⠄⠊⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⢠⣤⡇⠀⠉⢁⡀⠀⠄⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠈⠳⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⡀⠳⢤⡀⠀⠀⠀⠀⠀⢠⠄⠤⠠⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⡟⠂⠀⠀⠀⠀⠀⠀⠀⠐⠩
⠀⣉⣉⠀⠉⠀⠁⠀⠸⡀⠀⠀⠀⢠⠃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⡅
⣠⠤⠶⠂⠀⠀⠀⠀⠀⣠⢤⣔⡒⢼⣦⡶⠉⠢⠀⣀⣀⣀⣀⣀⡀⠄⠮⣥⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠁⠑⠊⠀⠈⠫⡁⣝⠀⠉⠁⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢜⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠌⠳⠤⠤⣠⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠢⡀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠢⢤⠤⡤⠤⠤⠤⠬⠆⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠂⠀⡀⠀⠀⢠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢸⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⢈⠄⠀⠀⢅⠂⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠘⡂⠀⠀⠈⠑⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⢿⣿⠛⠻⠷⠶⢿⣿⣿⣿⢿⡛⠙⡋⠀⠀⠤⣤⣄⡘⣧⡃⠀⠀⠀⠴⠀⣸⡯⠐⣯⣍⡀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⠿⢄⣻⡅⠘⠃⠀⢸⣿⣿⣿⡟⠀⠈⡁⠀⠠⢠⡗⢸⠗⠆⠱⣄⠀⠂⢀⣼⡇⠀⣿⢸⣽⠃⠀⣿⣼⠿⣟
⣿⣿⣿⣾⠂⢈⣉⠀⠀⠂⠀⢸⠀⢸⣿⣿⠀⠀⣇⣀⠀⣀⠁⠀⠀⠀⠀⠈⢿⣀⡾⠀⠁⠀⢡⣶⣦⠘⢠⡿⠾⠷⠋
⣻⡇⠙⠛⠀⠈⠉⠀⠀⠀⠀⠸⠀⠘⣩⣿⣿⠷⠄⠀⠀⠘⠄⢦⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⢀⣊⣄⡀⠩⣥⣶⣦
⠙⠃⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠤⠀⠀⠀⠀⠂⠈⠀⠐⠂⠰⢿⡇⠀⣠⠔⠛⠉⠀⡃⠐⡇⠮⠙
⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢹⣿⣿⠟⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠞⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠙⠂⠀⠐⠒⠻⢷⡤⢤⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠑⢾⠀⠀⠀⠀⢠⣴⠠⣿⣿⣧⣄⠀⠀⢀⣠⡾⠀⠀⠀⠉⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠉⠀⣸⣿⣿⠏⠀⠀⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠈⠙⠉⠁⠈⠉⣤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⠃⠁⠀⠑⠀⠀⠀⣁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢀⠀⠀⠀⠠
⠀⠀⣀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠈⢸⣿⠇⠉⢠⡿⢰⣾⡆⣀⠇⢠
⠀⣴⣾⣿⣿⢿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠀⠀⠘⠛⠀⠀⠀⠀⠈⠿⠁⠉
⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⡀⢀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣁⣠⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠄⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠲
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣽⣥⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣶⣾⣿⣿⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠙⠛⢒⣦⡄
⠀⠀⠀⠀⠀⠀⣠⢄⡞⠁⠀⠀⠀⠀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⢠⠞⣹⠏⠀⠀⠀⠀⡴⠿⣳⠶⠆
⠀⠀⠀⢠⠏⠀⠋⠀⠀⠀⠀⣀⡤⠿⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠳⡄⠀⠠⠤⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦
⠀⠀⢸⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢹⠶⠶⠤⣄⣀⠀⠀⠀⢠⠏⠀⠀⢸⠆
⠀⠀⠘⠢⣄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠙⠦⣄⡟⠀⠀⠀⣸
⠀⠀⠀⣀⣨⠟⠀⠀⠛⠉⣽⡖⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⣳⡖⠋⣹⢃⣠⡤⠖⠲⡆
⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠈⠢⣄⣠⠔⠋⠁⢻⠛⠋⠉⣀⣀⡠⠼⢥⣀⣀⡀
⠀⠘⠦⣄⡄⢠⡀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣆⠀⠀⠀⠀⠈⠛⠦⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⣀⣤⠾⠿⢄⡀
⠀⠀⠀⢼⠀⠘⣍⠢⣄⣠⠞⣽⠉⠠⠶⠦⣌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠓⢦⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠻⣄⣀⣀⣀⡤⠴⠚⠓⠲⠤⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠋⠉⠓⠶⠖⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⣠⠆⠀⠀⠀⠀⢀⡞⡟⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⣀⣀⡴⠃⠀⠀⠀⠀⢀⡞⣸⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠹⠁⢽⣁⠈⠁⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣸⣶⢀⡀⠀⣰⠬⠗⠀⠀⠀⠀⠀⣼⡛⣦
⠀⠀⠀⠀⠀⠀⣽⠙⠻⣧⠾⢥⣤⣀⣀⣀⣀⣤⡼⢧⠶⠃
⠀⠀⠀⢀⣀⡔⠃⠀⢀⠀⠀⣠⢊⠜⠀⣠⠏⠙⠒⠋
⡴⠚⠉⣯⡹⣝⡶⣤⣌⣉⣻⣟⣉⣩⣤⡷⣲⠤⢤⡀
⢷⣄⣀⣧⣭⠵⣚⡥⠖⠙⢮⡳⢤⣉⠛⠊⢹⠀⠀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠿⠋⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣦⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⣼⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⣿⣴⣿⡿⠃⠀⠀⠀⠀⠀⡘⠘⠃⢻⡇⠉⠃⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡇⠀⠀⠀⠀⠀⠀⣤⣄⣰⣶⣾⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠏⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⣠⠆⠀⠀⠁⠀⠀⡜⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢢⠀⠀⠀⠀⢂⠨⠏⠥⢐⡠⡀⠀⠄⠼⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠜
⠀⠀⠀⠀⠈⠉⠂⠀⠀⠐⠀⠀⠀⢤⣿⡜⡡⡂⠶⣖⠤⡈⢺⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠂⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣦⣜⡟⣼⣯⣤⣴⣶⠷⣱⠈⠩⣀⡀⢦⠠⣦⣦⣴⠀⠐⡆⠒⢴⡖⠤⣀⠀⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣠⣴⣾⣿⣯⣿⣻⣿⣿⣿⣿⠿⠛⠉⣀⣨⠗⣶⣾⡸⠧⠖⠚⠚⠋⠉⠉⠉⠉⠉⠉⠈⠓⠃⠚
⠀⠀⣴⡿⣿⢿⣾⣷⣿⣿⣿⣿⣿⣽⣧⢰⡆⡧⠟⠛⢋⠉⠀⣀⣠⡄⣰⣠⠍⠠⠬⢵⣧⠅⣉⠀⠐⣶⡀⠀⠀⠀⠀⢠⣄
⠀⠀⠘⣿⢿⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⠉⡀⠄⣲⠦⠿⠛⠚⠛⠉⡁⠀⢠⠄⠀⡆⡤⠀⠤⠀⠀⣀⠀⡁⠀⠀⠀⠀⠀⠁
⠀⠀⣀⣹⣿⣷⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⠗⢉⡠⣤⣼⡏⡽⠽⠓⠂⠉⠉⣉⣀⣀⣉⢈⢁⠁⠒⠓⠠⠆⠂⠄⠠⠈⠀⠀⠈⣠
⠀⠀⢀⣨⣿⣻⣿⣿⡾⣿⣿⣿⠿⣿⡾⣿⣖⠽⠛⢁⡠⣖⣂⢀⣐⠼⠟⠒⠓⠒⠒⠓⠂⠰⠯⠅⢥⣶⡆⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⢿⣟⣿⡿⠟⠋⢡⣀⡤⢟⠛⠛⢁⣰⠜⠁⣁⣴⠠⠀⡀⣀⠈⠵⠿⠿⠛⠰⠾⠷⢲⣢⠀⣄⡇⠀⢀⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠸⠟⠁⣴⠼⠋⠁⣄⡠⢾⠌⠒⢈⠠⣐⠥⠔⢻⡈⠈⢁⠠⠔⢐⣂⣰⠤⠭⠤⣂⡂⠄⡈⠙⡇⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠰⠾⠷⠂⠁⢀⠗⢊⢠⣔⡾⠥⠽⠒⡊⣔⡀⠼⡟⠀⢀⡦⠼⠑⠂⣉⠤⠶⠶⠛⠋⠉⢲⠣⢤⣇⠀⠠⣀⣀
⠀⠀⠀⠀⠀⣶⠼⠛⡁⢀⡵⠚⠉⣹⣾⣯⠗⠉⠤⠄⠒⠉⢈⣀⠤⢂⢐⠛⠁⠀⡠⠔⠋⠉⠉⡻⠖⢊⡡⠴⠞⡓⢀⣐⡆
⠀⠀⠀⠀⢠⣅⣦⠕⠚⠉⠀⠀⣉⣁⡠⠀⠀⠠⠶⠃⠉⠉⢉⢁⠀⢀⡀⠄⠋⠁⠀⡀⠄⠂⣁⠤⠴⠃⠂⢀⠖⠉⠀⠀⠱⡀
⠀⠀⠀⠀⢈⠕⢰⠁⠀⣠⠃⠉⠀⠀⡀⠤⠠⠄⠀⠀⠀⠀⠐⠂⠀⠁⣀⠠⠐⠀⣁⣤⠴⢁⠀⠀⢂⠔⠊⠀⠀⠀⠀⠀⠀⡅
⠌⠀⠤⠤⠀⠀⠀⠠⠀⢆⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⠀⠤⠴⠊⠁⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠀⠀⠔⢀⠜
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠒⠰⡁⠀⠀⠀⠀⠀⠀⢀⡢⠔⠒⠋⠀⡀⠠⠴⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠂⠄⠤⠤⠤⠀⠎⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉
⠀⠀⠀⠐⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠰⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠇⠀⠀⠀⠁⠁⠀⠀⠀⠀⠑⢄⣀⣀⠀⠠⠤⠒⠊⠁⠀⠀⢀⠀⡀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠉⠁⠂⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠈⢣⠴⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠐⡀⠀⠀⠀⠑⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⣄⠀⠀⠀⠈⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⠑⢄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡄⠊⠐⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢶⡆⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠾⣴⠶⢦⣀⠀⠀⠀⠀⠀⢀⠊⠻⠟⠀⠀⠀⠈⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠸⠄⠚⠿⠣⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠂⠀⠀⣠⡴⢾⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢀⠀⡒⠚⠂⠐⠚⠁⢠⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣄⠁⡒⠁⠀⠀⠀⠀⠀⢀⠂⠀⠁⠀⠀⠀⠀⢠⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠁⠄⡀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡀⠀⠀⠀⠀⠀⠀⠍⠂⠀⠀⠃⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠻⢶⣤⣄⡀⠀⠀⠰⠄⠂⠤⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠉⠛⠁⠰⠀⣗⣶⣆⣀⡄⠀⠀⠀⠀⠠⠄⠀⠀⢀⠇⠀⠀⠀⠀⠀⡼⠉⠾⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⡄⢀⠉⠋⢏⡿⣧⣄⣤⣠⠤⠄⢚⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠐⠄⣂⠙⣳⣰⣾⠵⠚⣛⣚⣛⡷⠉⠀⢀⠀⠀⢣⡀⠀⠀⠀⠸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡴⠖⠁⠀⠈⠀⠀⠀⠀⠉⠁⠁⠀⠀⢀⠾⣀⠀⠀⠙⠳⣤⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠐⠤⠀⢀⡀⠀⠄⠊⠀⠀⠉⠉⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡞⠁⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠀⠈⠒⢄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠁⠂⢄⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣇⣀⡀⣀⠀⣀⠠⢠⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⣈⠉⠉⢉⡁⠠⡞⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠈⡝⠂⠠⣄⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠤⠤⠤⠤⠀⠀⡌⠀⠀⠀⠡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠦⠤⠤⠤⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⢧⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣤⠂⠠⢶⣶⣄⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡔⣠⡿⠋⡁⠀⠀⠀⠈⢻⣧⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⡀⣼⠃⠀⢐⠁⠀⠀⠀⠰⠀⠙⣄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⣼⠃⢀⣀⣀⣀⠀⠀⢀⣀⡀⠀⠈⢡⣀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢰⡟⠈⠉⠉⠉⠉⠀⠀⠈⠛⠛⠛⠢⢸⣿⣷⠦⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⡇⠰⡛⣿⠃⠀⠀⠀⠀⠘⣿⣣⠄⢸⡿⠙⣤⡈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢣⠘⣿⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⣨⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢀⠈⢿⣿⡆⠀⠰⣶⣶⠶⠶⣶⣶⠆⠀⢰⣿⡏⣿⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⡙⣝⠇⠀⠀⠙⠻⠿⠿⠟⠁⠀⠀⢰⣿⣷⣳⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠞⠳⢤⠀⠤⢄⣠⣄⡠⠤⠠⡤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢈⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇⢀⡤⠒⠉⠹⠆⠂⠀⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⠖⠁⠀⠀⠀⠀⠊⡌⠀⣀⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⡖⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⠀⢀⣠⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡟⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿