Limestone Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡜⣿⠓⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠙⠛⠛⠻⠟⠱⠦⣿⣧⡄⢹⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠒⢈⠚⠰⠿⢿⡿⠟⠉⠚⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡆⠛⢶⡀⢀⣤⣶⡿⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢙⣿
⣿⣿⣿⣿⣿⣆⠀⠁⠘⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⣿⣿⠟⠉⠹⣿⣿⣷⣦⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣴⠀⣨⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⢸⣾⠏⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⣷⣆⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣶⣖⣂⠀⠀⠉⠀⠀⠠⢄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧⣀⠀⠀⠀⠀⠀⠀⠉⠛⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⠉⠙⠲⠭⣿⣦⣀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠐⠢⠤⣶⣄⠀⠀⠀⠀⠀⠀⢀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡏⠀⠀⠀⠀⠀⠀⠈⢷⡇⠁⠀⡖⢲⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⡷⠐⠒⠀⠈⠋⠡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢻⡄⢇⠀⠀⠀⠀⠤⠀⠀⠸⢿⣄⢀⣻⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠇⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠘⢅⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠻⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢾⣦⠀⠀⠀⢸⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⡾⠟⢄⠀⠀⠙⠦⣤⡌⠄⠀⠀⠀⢾⠀⠀⠀⠀⢀⣀⠀⠀⢸⣎⣷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⠇⠁⢸⢶⡦⠖⠉⢿⣿⠓⣀⠀⠀⣼⠀⠀⠀⠠⠎⠀⠀⠀⠘⣿⣿⣆⠀⠀⢀⡤⠞⠦⢦⣄⠀⠀⠀⣀⡘
⠀⠀⠀⠀⠀⠈⢅⣀⣀⠲⣦⡇⠀⠀⠈⠿⢷⣿⣷⣴⣿⣧⠀⠀⠀⠀⠀⠀⣀⠀⠜⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠋⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⡷⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣛⠋⢛⣷⣡⢟⣩⣩⣽⣿⡿⣗⣶⣶⣿⣿⣿⢍⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⣀⡀⣠⡤⢴⣶⢦⢤⣔⣿⣿⡿⣭⢟⣽⣿⣿⣿⡿⢿⠟⢛⣩⣽⢿⣿⣿⣿⣿⣻⣦⣘⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⣿⣷⡿⣷⣾⣯⣿⣿⠗⡻⠟⠶⠛⡛⠉⡨⢶⡿⢀⣠⣾⣿⣯⣿⣿⣿⣿⢟⣾⣵⣿⣻⡷⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡞⣿⣿⣿⣿⣯⡿⣩⣭⡀⠀⠀⠀⢁⠈⠀⠜⢂⡀⢉⡠⣶⠤⣿⣟⣿⣿⣿⣻⣿⣴⣾⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⢿⣿⠛⢿⣿⢿⣯⣷⣉⡡⣛⣅⣟⠷⡞⡛⢿⠟⠊⠤⠐⣵⠶⠒⣀⡸⢾⠟⠋⡬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣮⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣟⣯⣶⣿⡿⣣⢞⡥⠚⡍⠈⠛⠣⡒⠀⠅⠀⠀⠀⣤⡴⠞⢁⣀⣼⡻⢋⢀⠒⣨⣿⣿⣿⣿⣿⣿⣻⣿⣿⣯⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣿⣿⣿⣾⣿⣽⣋⠅⠋⠛⠉⠃⢠⢀⠂⠫⠰⢄⣤⡔⠔⢉⡀⠞⠉⠀⣯⠖⣩⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣭⠟⣡⠔⠻⠛⡻⢛⣉⡵⢔⣴⣶⣾⣫⣀⠚⢊⡀⠐⠊⢁⣴⢿⠽⠂⣮⡿⣟⣽⣿⣿⡿⢻⣿⢻⣿⣟⡽⠁
⠀⠀⠀⠀⠀⣾⡷⣟⣽⣿⣿⣿⣽⡿⢛⣿⡶⢿⠹⣿⢾⣿⣿⣟⣵⣶⡿⣋⣹⠟⠉⠁⠊⣨⢅⣤⠞⣫⡔⣞⡍⢁⣨⣿⣯⣿⣿⣿⠃⢾⣷⣾⣿⡟⠁
⠀⠀⠀⡤⣾⣞⣥⣧⣾⣟⣯⢿⡽⣾⣻⢏⣷⡯⣦⣾⢟⣥⣿⡿⢛⡷⢉⣙⡯⡐⠓⢀⣤⣶⣟⠕⣿⣽⢏⢉⣾⣋⡟⡿⣿⣿⠟⠁⣲⠉⢀⣔⡛
⠀⣠⣞⢟⣛⢛⡋⠉⡫⢾⡿⡯⣜⣿⣿⣿⣾⢟⣳⣾⣾⢛⣩⠟⠡⡞⠓⣯⡷⣠⣤⠾⣫⣬⣷⡿⡋⣵⡵⣟⡿⠿⢡⢾⡟⠃⠈⠈⠁⠀⠸⣿⣿⠇
⠀⣯⠘⠀⠉⠈⠷⠚⡴⠷⣟⣼⣵⣿⢿⣿⣷⣿⠛⢫⠞⠋⠱⣿⣷⠴⢚⣵⡺⢋⣰⣺⣻⣟⠛⢡⣿⡿⠿⣿⡲⢾⣽⣆⣧⠄⠤⠀⠀⠀⠀⠻⣿⣇
⢀⢸⡷⠃⠀⠀⢄⣸⣾⣶⢿⢉⣁⣼⡿⡋⠉⠀⠀⠁⡠⠐⢃⢏⣠⡾⠿⠠⠖⠠⡮⣯⣟⡙⡠⠞⠋⡡⢚⣯⣤⠥⣀⣽⣿⡇⠀⠀⠐⠄⠀⢀⢻⠃
⠀⢾⠋⠀⣫⣶⣯⣻⣿⣷⣿⣿⠿⢋⠯⣠⠜⢀⣤⡆⣤⣪⣿⠟⣋⣤⡜⢫⠾⠛⠻⠟⡉⢽⣿⠗⣢⣷⣿⡿⣟⡌⠏⣟⣿⠿⡄⠄⠀⠀⠀⢀⣾
⠀⠸⣵⣞⣭⣽⣿⣿⣿⡿⡿⠏⣬⠶⢛⡵⣺⣍⣿⣿⣿⣯⣷⣿⣿⣿⣿⢿⣥⡼⢖⣶⢻⣶⣧⣶⣟⡟⣡⣾⣿⠶⣾⣿⣿⠷⣧⡤⠾⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠘⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠺⠿⠷⠾⠶⠒⠉⠁⠀⠀⠉⠙⠛⠿⠿⠶⠾⠷⠟⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣦⣍⣥⣮⣩⣷⣥⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡴⠊⠉⢱⠙⢍⠀⠁⠐⠂⠤⢀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠈⠣⣀⡀⠀⠀⠀⠀⠉⠀⠒⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠠⠄⡀
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠁⠀⠀⠑⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠠⡀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠑⠠⢀
⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠋⠀⠀⠀⢠
⠀⠀⡇⠀⠐⠀⠈⠢⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢰⠉⠒⠃⠀⠀⠀⠈⠢⡀⠀⢨⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠆⠢⠄⠄⠀⠠⡴⡆⣬⡇⠿⣷⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠑⣷⠀⠀⡔⢰⣟⠃⢸⠻⢠⣦⠀⠻⡿⠉⣒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⢨⠃⠀⢈⣢⣻⠏⠀⡘⠀⣸⣿⠖⠀⠀⠀⠈⢻⢦⣑⡒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡠⠀⠀⢰⡽⠀⠀⠀⠀⠀⠹⣿⣄⢀⠀⠀⠀⡄⠀⠛⠟⠀⠀⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠀⠐⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣄⡀⠀⢧⠀⣠⣖⣆⣶⢶⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⡁⠀⢀⡴⠥⣀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠉⠘⢈⣸⡾⠙⠿⠛⠛⣿⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠀⢀⡠⠋⠀⢀⣀⣡⠴⢤⣤⠦⣅⡄⣀⢀⢸⣆⣤⣿⠞⣵⡤⠀⠀⢀⡏⣩⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠔⠋⠀⠀⠈⠁⠀⠀⡈⠀⠔⠙⣈⠄⢈⣡⡬⠋⠉⠀⠀⠀⠀⠀⠀⢸⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡸⠻⣄⠐⠈⢀⠠⠀⠀⠀⢀⠀⠀⠀⢀⡴⢖⡫⠋⣄⠚⣶⡛⠃⢠⠖⠒⠑⠁⢀⠀⡈⡅
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠚⡷⣬⠋⠀⢠⡥⢄⠉⢷⡴⠃⠀⠀⠀⠀⠸⠿⠺
⠀⠀⠀⠀⠀⠀⢰⣷⢀⠀⠀⠀⠀⠀⣷⠀⢀⣤⣛⠦⠆⡤⠁⣠⠀⠀⣿⣴⢶⡏⠀⠀⠀⠀⢀⡀⢠⣀⠸
⠀⠀⠀⠀⠀⠀⠘⣼⠈⣅⠀⠀⠀⢀⣾⣷⣎⣀⣙⣓⣴⢁⡶⣿⣧⠎⠈⠛⠛⠿⠛⠳⣂⢠⠼⣧⣿⣿⠙
⠀⠀⠀⠀⠀⠀⢰⠿⡤⠿⣶⣄⣀⣮⣿⣿⣿⠿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣶⠶⣿⠶⡚⣥⣜⡿⡇⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣆⢰⣿⣿⣶⣿⣷⣿⣿⣿⣿⣯⣵⣾⣦⣯⢻⣿⣾⣧⣴⣶⣿⣿⣞⢿⣿⣧⠅⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠫⠿⠸⠋⠚⠉⠙⠛⠿⢯⣿⣂⣻⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠉⠻⠆⠀⠀⠀⢹⣿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠠⡦⠐⡗⢀⠀⠀⠀⠀⢻⠄⠀⠈⢋⠙⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠙⠳⢦⣤⣀⣄⠀⠈⠁⠉⠀⠀⠘⠃⠀⠀⠀⠀⠉⠀⠀⣤⠾⠟⠛⢷⡆
⠀⠀⠀⠀⠀⠀⣾⠃⠠⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢠⡄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠰⠆⠀⠶⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠐⠃⣀⠀⠘⠃⠀⠀⠐⠛⣷
⠀⠀⠀⠀⠀⠀⢿⡦⠀⠀⢠⡄⠀⠛⠀⠀⠀⠀⢀⣤⠀⠀⠀⠶⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⢻⡄⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⣤⣾⣛⣀⠀⣰⡟⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⠁⣤⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣤⣤⡶⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⢀⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣠⡾⠛⠛⢧⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡞⠋⠀⠀⢀⣈⠛⢷⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠘⣧⠸⣧⣤⣤⡴⠟⠁⣠⡿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸⡇⠀⠹⣦⡈⠳⠶⣤⣤⡾⠋⢀⣴⠏
⠀⠀⠀⠀⠀⢸⡟⢻⡟⣿⠛⠛⠛⢛⠛⠛⠻⢦⣤⣤⣤⣤⣴⣟⣁⣀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⠀⢸⡟⠉⠙⣿⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣿⣦⣤⣤⣤⣼⣧⣤⣤⣼⣧⣤⣴⣿⣤⣤⣿⣤⣼⣷⣤⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣷⠀⠁
⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣿⣀⣀⣀⣸⣇⣀⣀⣀⣿⣀⣀⣀⣸⣇⣀⣀⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠉⢹⡏⣩⠉⠉⢹⡏⠛⠉⠉⣽⠉⠉⠉⡉⢹⡏⠉⠉⠉⣿⠉⠉⢩⡉⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠛⠀⠀⣸⡇⠀⠀⠀⣻⡀⣤⠀⣿⢸⡇⠀⠀⠀⣿⠀⠀⠘⠃⣿
⠀⠀⠀⠀⠀⢘⣛⠛⠛⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⣈⠉⠁⠀⠀⠶⠶⠀⠀⢀⣀⡈⠉⠀⠶⠶⠀⠀⣤⣤⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠴⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡦
⠀⠀⣠⢶⣤⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡠⢶⢂⣿⣽⠟⠃⠒⠆⡀⢰⣶⡄⠀⢀
⠀⠀⣟⣿⣿⠋⠀⢀⡤⠀⢠⡀⠀⠀⡤⢤⣻⣿⡽⠀⠞⢾⠋⠉⡈⢢⡴⠉⠄⠀⠅⠊⠉⣀⣠⣸⡆
⠀⠀⠛⡿⠋⡂⢀⣤⠽⢁⡆⠉⠀⠀⡴⠿⠟⢫⠁⠀⢠⡄⢀⠊⠁⣤⠆⠘⠁⠐⡂⣈⢉⣿⢹⣷⣷
⠀⠀⠹⠀⠰⠀⠈⡟⠄⠾⠁⢠⠁⡀⠐⣇⠀⠀⠂⢀⡨⠀⡬⡄⠠⢵⣦⣾⣷⠨⣷⡟⣼⣿⣌⣬⡿
⠀⠂⠉⠁⠀⠀⠀⡇⢀⠀⡀⠁⠐⠱⢒⢾⣆⡊⠀⢸⢿⣆⢀⣣⣦⣆⢙⣿⣿⣰⡿⡧⣸⣚⢿⣿⣣⣶⣶⣶⣦⣤⠄
⠀⠠⠴⢲⣦⣤⣤⣆⠀⠀⠀⢰⠁⠔⠈⡏⡟⠁⠂⠰⣼⡇⡙⢩⣿⣿⡀⣿⡋⠁⡿⢽⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠉⠻⠛⠛⠧⣀⣀⠀⠀⢀⡜⠁⠀⢠⢰⢸⢇⢦⣿⣿⡿⡀⠈⣟⣾⠃⢸⣰⣞⢈⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠟⠻⢄⣀⠀⡀⠀⢻⣈⣨⣿⣿⣿⠿⠶⠶⣿⡿⠿⠟⠛⠻⠋⡛⠋⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡇⠀⠀⣀⣠⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣶
⠀⠀⠀⢀⢀⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣾⣿⣿⣿⢁⣴⣿⣿⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⢀⣠⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠚⠛⠻⠃⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢣⣤⠼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣾⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⡄⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡷⠃⢰⣿⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠙⠛⠛⠓⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⠶⠾⠿⠿⠿⠿⠿⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠙⠻⠶⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣨⣿⣿⠿⠟⠀⠀⠀⠀⠐⠛⠒⠚⠛⠿⠷⠶⠾⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⢀⡀⣀⣀⠀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣠⠤⠚⠛⠋⢉⣭⣿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠈⠉⠉⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⢀⣨⣤⣢⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣀⣀⣀⣄⣀⣀⣀⣀⣤⣀⣒⣈⣉⣁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠛⠛⠿⡿⠟⢻⠿⠛⠛⠻⠿⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠹⠇⠀⠁⠈⠙⠧⠀⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣦⣦⣘⣿⠏⢠⡄⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⢀⣠⠶⡀⠀⠀⠴⣞⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⡿⠟⠋⠁⠓⠋⠀⠀⠀⣀⣀⣀⠤⠀⠀⠀⠀⠉⠀⠀⢠⣴⣷⣾⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣠⣤⣄⣤⣤⣶⣏⣉⡈⠀⠀⠀⢀⣀⣀⣀⣰⣿⡿⠛⠅⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠙⠛⠛⠛⠛⠛⠛⠛⠿⠳⠾⠮⠭⠴⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠹⣿⣻⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⠀⠀⢀⣀⣽⣟⣿⣏⣸⣿⣿⢶⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠙⣷⣤⣀⠀⠀⣀⣤⡾⠛⠙⣿⣷⣦⣼⠿⠿⠿⠻⣟⢿⣏⠀⠀⣠⡿⠉⠻⡯⣻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⣏⣹⣿⠟⠛⠁⠀⠉⢀⣿⣿⡟⠁⠀⠀⠀⠀⢻⣎⣿⣷⣶⣿⠁⠀⠀⢸⣿⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢤⣄⠀⠙⢦⣤⣄⣀⣼⣿⣿⡿⢧⣤⣀⡀⠀⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠘⣷⡿⢿⣿⣿⣄⣀⣀⢸⣷⣇⢸⠇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠁⠈⠻⣭⣷⣤⡀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠙⠻⢿⣿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣽⣶⣌⣿⣯⣥⠿⠿⠿⠷⠟
⠀⠀⠀⠀⠀⣾⠛⠋⠁⠀⠀⠀⠀⢻⣿⡤⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣴⣶⣿⡀⠀⠀⠀⠀⢸⣟⣽⠟⠉
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢠⠀⣈⣿⣷⣿⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠶⠿⠋⠉
⠀⠀⠀⠀⠀⠈⠛⢦⣤⣶⣶⣶⠿⠟⠛⠋⠉⠉⠉⠀⠀⢀⣤⠶⠛⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⣄⣠⣤⠚⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⡽⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⡶⢶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⠉⠉⡁⢀⡍⠉⢁⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⣋⠥⡐⠀⠀⠁⠁⠀⠉⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠫⠡⠐⠂⢁⣔⣤⡶⠗⠉⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠠⣴⣶⣿⣿⣿⠆⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠾⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⣠⣾⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠋⠁
⠀⠑⢤⣤⣄⠀⠀⠀⠀⣀⣀⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⠟⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⣠⠜⠁⠀⠀⠀⠀⠈⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⣬⣿⡟⠛⡇⠀⠀⠀⠀⠀⢹⣆⣀⣀⣠⠿⠶⢤⣄
⠀⠀⠀⠀⠀⢠⠏⠀⢀⡼⠟⣿⣿⣶⣷⣀⣀⣀⢀⣠⠞⠛⠇⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⣠⣞⣠⠞⠉⣀⣾⣿⠟⢿⡻⠛⢩⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⡿⠃⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡿⢿⣿⠀⣮⡟⠀⢀⡀⣠⣤⢴⠏⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀
⠀⠀⠀⠀⢷⣾⣿⢶⣿⣯⣙⣿⣿⣿⣶⣌⣧⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⡈⠑⢦⡀
⠀⠀⠀⠀⠈⠙⢿⣌⣙⣿⣿⣿⣅⡉⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠫⣿⣿⣦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⠲⣈⣽⣋⣿⣿⣿⣦⣀⡀⠀⠀⣀⣠⠞⠛⢿⣿⣶⣲⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⣴⡀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣄⡖⠀⣫⣹⠿⠄⡄⠠⢫⣖⣠⡄⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⢡⡌⡷⠿⠟⢷⠆⣿⠿⣻⣟⡨⡶⢹⡌⢝⠀⠁⢠⠂⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠺⣿⣤⠚⠧⠼⢝⡒⠠⣶⠒⢥⣹⠰⠼⣧⢧⢽⣎⡈⣈⣭⣹⣎⣍⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣒⠾⣗⠿⣾⡗⣿⣿⣶⢇⣶⢼⣿⢝⣷⢾⡿⣿⢻⠙⠣⣿⢿⣽⣟⣿⣧⣴⢀⣀
⠀⠀⠀⢀⠀⢻⣓⢬⣿⣿⣿⣍⣻⣻⠿⡿⢻⣿⣦⣽⣽⣻⠻⣿⣷⣿⢽⡾⢶⢽⡞⣻⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⣾⣫⡛⣺⡟⣍⠻⣿⣿⣿⣻⣾⣿⣷⣞⣿⣿⣿⣿⡦⣶⣴⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⢠⣷⣿⡧⠿⢯⣤⣿⣿⣿⣿⣿⣿⣟⣶⣿⣿⣿⣿⣿⣿⣭⣧⠴⠶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠈⠉⢉⢰⣾⣿⣿⣿⠛⣛⠋⠉⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣭⣿⣿⠿⠓
⠀⠀⠀⠀⠀⠘⠛⠫⣉⣬⣭⣥⣴⣿⣿⣿⠿⠿⠿⠿⠿⢻⣿⣯⣭⣭⣿⣿⣿⣿⡶⣦⣤⣤⣼⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠤⣶⣢⣤⣤⣴⣾⡿⠿⠿⠟⠿⡿⠶⠚⠛⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⢀⡤⢀⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⢠⣄⠀⠀⠀⣀⠀⠈⠳⢄⠘⠂⠀⠀⠀⡠⡀⠈⠁⢀⣄⠀⠀⠀⡀⠀⠀⠀⢠⣟⣾⠇⡡⠐⣶⠒⠛⠻⢿⣶⡄
⠀⠀⠀⢸⠰⠋⠀⢤⣴⢞⣇⠀⠀⠐⡆⠀⠀⠀⠀⠙⠁⢠⣦⠈⠀⣤⠀⣔⡶⠶⢶⢦⣾⢲⡟⠀⢀⠀⠝⠰⣦⡀⢈⣴⢇
⠀⠀⠀⠸⠀⠁⠀⠨⢻⡿⣿⣠⠀⠀⠡⠀⣰⠋⣉⠙⢦⠀⠀⠀⡄⠁⢈⣿⣷⣦⣾⣼⣧⣿⢇⠄⠐⠢⣄⠁⢋⣀⠀⢊⡩
⠀⠀⠀⠠⠀⡀⠀⠀⡜⣷⣿⣿⠃⣚⣀⡠⠓⠺⣟⠑⣼⠂⠠⠖⠐⠠⣀⣿⣾⡭⠽⢿⡛⠉⣋⢴⣶⠀⢸⡆⢧⣼⠁⠁⠁
⠀⠀⠀⢠⠀⣿⡄⠀⢊⢼⣿⡏⠘⠷⠋⣇⠛⠃⢸⠏⠀⣀⡉⠉⠐⠢⣼⣿⡿⣁⠻⣺⣇⣀⠙⢤⢀⣠⠞⠁⠐⠼⢀⠀⠀⣃
⠀⠀⠀⢈⠀⠙⢷⡔⢠⡀⢿⠄⣔⢃⣤⡈⠉⠉⣩⠀⣴⣍⠀⠀⢐⣾⣿⣿⠕⠈⠉⠉⠉⠁⠎⢀⡀⣀⣠⢤⡦⡄⢯⠄⠀⡏
⠀⠀⠀⠀⣀⠤⢄⣻⡅⠐⠝⣠⡶⠛⠐⡁⠀⠂⠁⠤⠀⠄⢀⢱⣾⣿⣿⡟⠛⣿⣿⡟⣙⢣⠹⠈⠃⢡⠈⡀⡇⣸⢠⠄⣸⣧
⠀⠀⠀⠀⢸⣶⡄⢹⡷⠀⠘⣉⢐⡠⣄⠈⠀⢀⠘⠓⠂⢤⣈⣿⣿⡟⠀⠖⡠⠀⣻⣷⡝⢻⣤⣧⣿⣼⣾⠿⠟⠋⢄⠀⣿⣿⡆
⠀⠀⠀⢐⠁⣤⣤⡾⠃⡀⣤⡀⠘⠦⠞⠀⣠⣊⠁⠀⠲⣴⣿⣷⣿⣒⣦⠘⡤⣀⠠⢿⣿⣿⠟⠋⠫⠰⢠⠤⠴⢥⠀⠀⣿⣿⠃
⠀⠀⠀⡇⠀⢻⡄⠀⠀⡯⢽⣇⠈⠀⠀⣠⠭⢀⣈⠐⣴⣿⣿⡟⡉⠄⠥⡤⠘⣥⢂⠼⢿⣿⣷⡀⠀⢀⠀⢾⣶⠄⣷⢀⣿⣿
⠀⠀⠀⠱⡄⠈⠁⢠⣄⢻⣹⣿⠀⠀⠀⠇⢰⣶⣬⠱⣿⣿⣿⣿⡁⢾⠆⣹⠀⠸⠃⠀⢉⠛⣻⣏⠉⠈⡂⢭⡧⠾⢣⣾⣿⡏
⠀⠀⠀⠀⠀⢢⠀⠈⠛⢹⣖⣿⡇⠀⠀⠈⠢⣍⣁⣠⣿⡯⢿⣿⠟⠗⠚⠁⣀⠀⠈⣬⣽⣿⣿⡟⢤⣴⡇⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠑⠄⢶⣦⠙⠋⠁⣴⣋⡓⣄⠀⠙⠛⠉⣆⣼⣿⢗⠲⡀⠀⣰⡆⡄⠀⠈⠀⠀⠈⠀⠉⣼⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣔⣶⣹⠀⠹⣟⣡⠏⠀⠐⠄⠀⠛⠓⠛⠽⠤⠃⠈⢛⡗⡧⠀⠀⠀⢙⡷⠀⠀⠀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⢲⠶⠶⢤⣤⣤⠼⡥⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⣈⣠⣀⣀⣀⡀⠀⠔⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠛⠐⠒⠒⠆⠀⠉⠒⠶⠶⠒⠛⠛⠛⠛⠻⠿⠿⣭⣿⣿⠟⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⢴⠀⠘⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢦⡀⠘⡆⡌⢀⢞⡰⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠢⡐⡀⢣⠐⢣⠎⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡆⢹⡑⠂⡰⢡⢮⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡈⢻⠀⠷⢣⠊⠠⠺⠂⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠀⡀⠀⠀⠐⠋⠃⡸⢈⡄⢀⠃⢀⠶⠁⢄⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⢅⠠⢄⡼⡀⢴⡘⠃⢎⠀⠸⡼⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠁⠈⠸⡀⠇⢠⡃⠠⢊⠄⠀⠀⠀⠀⠀⣀⡀
⠀⢀⣀⣠⠶⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣤⢄⠀⠀⠀⠠⠀⠱⠀⠄⣰⠗⡦⠋⠀⢡⣨⠃⠀⠀⠀⠄⣠⠚⠃
⣠⠾⢉⠥⠀⢈⠉⣃⠀⠰⡖⠀⣀⠈⢡⣀⠐⡁⡆⠱⡀⡄⠀⣠⠀⠂⠀⠑⠀⡍⠃⡴⠀⠰⢁⠁⠀⠤⠞⠁⢲⡀
⢘⡶⢈⠠⡆⢠⡄⠏⠀⠀⠈⣦⠈⠦⡀⠹⠣⡐⡡⠀⠁⠐⡀⠀⢣⡀⢄⠠⠘⠆⠀⡃⡰⠣⠈⠁⠙⠓⠚⠛⠋⠡⣶
⠘⠒⣎⠀⠂⡰⠖⠀⠀⠀⠀⠀⠙⠢⢌⡓⠀⡇⠀⢿⠄⡄⠁⣈⠀⠃⠈⠂⠘⠉⠀⠋⡎⠀⠀⠀⠀⠀⠀⠀⢠⣟⡏⠀⠀⠀⡤⡄
⠀⣴⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢌⠈⠂⡜⠰⡇⣰⠉⡧⢐⡀⠘⡄⠉⠀⠊⢀⠀⠀⠀⠀⠀⠀⠀⡏⣯⠀⠀⠀⢄⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠈⠀⢀⠀⠀⣫⠀⢇⠀⡋⠑⡎⠀⣿⠀⠀⠀⠀⠀⠀⡼⢱⠇⣠⠖⠱⠄⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣂⠀⢀⡄⢀⠀⢀⡄⠐⠱⠄⠈⢢⠁⠉⡐⢄⠙⠁⠘⠁⢸⡇⢀⡀⠀⠀⢀⡼⢑⡛⠩⠞⠓⠊⠙⠂⠐⠛
⠀⠀⠀⠀⠀⠀⠠⣐⠉⡀⢤⠀⠀⡐⠀⠒⠈⠐⠐⠂⠐⠀⠀⡀⠀⢑⠈⠈⠅⠀⠀⠸⠇⢸⠁⢀⢄⠉⡀⢋⡀
⠀⡀⠀⠀⢿⣤⡀⠀⠠⡖⠀⣥⠀⢏⠀⠀⠘⢉⠳⣦⣀⡀⣠⠀⢦⠀⢡⠄⢠⠀⠀⡷⢀⠎⣰⠃⠀⠀⠙⠊
⠀⠉⠁⢄⡘⠂⢄⠀⠶⣨⣧⡀⠟⠈⡆⠀⠀⠀⠁⠠⢬⠉⠣⢥⠀⠳⢈⠂⠘⡀⡈⠐⢋⡜
⠀⠀⠀⠀⠈⠒⠾⣙⢧⡈⢂⠹⡖⠀⣣⠀⠀⠀⠀⠀⠀⠙⢦⣈⡁⠦⡈⠑⠀⠙⠁⡱⠉
⠀⠀⠀⢀⡄⠀⠠⠈⠶⠀⠈⠳⡈⠀⢉⠢⠄⠀⠀⠀⠈⡷⢄⠈⣈⠧⢀⠰⡐⠒⠋
⠀⢠⡴⠢⠽⠬⣀⣀⢀⣈⡩⠴⠦⠈⠒⠵⠄⠉⠐⠫⠶⠚⠆⠃⠀⠰⠞⠷⠄⠛
⠀⠀⠉⠈⠀⠐⡲⢈⠉⠉⠀⠊⠀⠉⠀⠉⠑⠒⠂⠈⠃⠀⠙⠆⢳⣄⠌⣦⢠⡔
⠀⠀⠀⣀⠄⣨⠆⠀⠀⢘⣠⠀⠲⠄⢐⣐⡂⠬⠤⠋⠁⠐⠦⢌⡁⠥⣈⡐⢦⢹⡄
⠀⠀⠒⠀⠀⠀⠀⠶⠳⠖⠂⠉⠁⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠈⠑⠦⣀⢿⢉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⡙⠃⡀⢄⠃⡔⠶⡤⠄⣀⡈⠸⡇⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠉⠙⠂⠤⠍⠒⡤⠄⠘⢣⠖⠦⠂⡀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠐⣂⣐⣂⡆⠁⠐⠄⢀⡈⡀⠤⠤⠤⡒⢂⡉⠆⢀⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⡈⠭⠅⢀⠶⢄⣤⠞⠀⢠⡙⣋⠁⢌⠉⠀⠀⢄⠆⠨⢊⣍⣠⠤⠍⠲⡭⠟
⠀⠀⠀⠀⠀⢀⡤⠒⣫⠄⠚⠁⡀⠠⠚⠁⠊⡅⢡⠈⠰⢀⡅⠀⠀⠠⠄⠀⠀⠠⠔⠈⠁
⠀⠀⣠⡴⢊⠅⠓⠂⠁⠀⣄⠊⠀⠶⢀⠠⢁⠐⠁⠀⠠⠄⠠⠐⣃⠀⡘⠁
⠀⠀⠙⠃⠀⠀⠀⠀⢀⠔⠁⠀⡀⠀⠃⠐⠋⠀⠀⢀⠔⠂⠄⠀⠙⠊
⠀⠀⠀⠀⠀⠀⢠⠴⠁⠐⡁⣈⠁⡶⠊⠀⠀⠀⠀⠈⠋
⠀⠀⠀⣴⣎⠠⢀⠖⢀⠀⠀⠟⠐
⠀⠀⠀⠀⠀⣠⠛⠀⠁⠀⠀⠃
⠀⠀⠀⠀⣸⢁⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢩⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⢳⣦⡀⠀⠀⢀⣀⣴⠞⠃⠀⡾⣿⣿⣿⣦⣀⣀⡀⠀⣠⠶⠶⢛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⣡⠴⣻⣿⡛⣳⣖⡭⠋⠉⠀⣀⠀⠙⠿⣿⡇⠘⠉⣩⣿⣾⢧⡄⠀⢾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠺⢽⠿⠛⠉⠀⡠⠶⠖⠋⣽⣷⡦⠀⠉⠁⠠⣾⣿⣿⢿⣷⡦⠀⠈⠋⠀⢹⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠛⠷⠍⠀⠀⠀⠀⠀⠈⠉⠑⣿⣀⢤⡐⠄⠀⢺⣿⣿⣿⣿⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡄⠀⠀⠉⠛⠻⢿⡪⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣴⣶⡶⠶⠶⠶⠿⣿⣤⣴⣧⣧⣧⣶⣶⣿⣀⣠⣴⣾⠶⠷⠤⣴⣿⣷⣤⣤⠤⠤⠤⠴⣾⣿⣶⣀⣠⣤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠒⡒⠒⠾⠭⠽⣛⣷⣤⣀⠀⠀⠀⠀⠀⠤⠤⠤⠬⣉⣉⣉⣉⠐⠒⠒⠢⠭⠍⢙⠒⠲⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠭⠍⠉⣙⣒⡲⠭⢍⣓⠂⠤⠀⠀⠠⣤⣤⣄⣀⣶⣦⣬⣭⣭⠓⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠠⠤

⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣼⣿⣧⡀⠀⢀⠀⠀⠀⠀⢀⣼⡏⢻⡗⢄⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢠⣿⣿⣿⣄⢀⣿⣧⠀⠀⣰⠿⠉⢆⠀⢻⡀⠈⠢⢄⣤⣶⣿⡏⢢⡀⠀⠀⢀⣤⣀⣠⡾⠃⠑⣤⣆⢰⣿⡀
⠀⣿⣿⣿⣿⣷⣼⣿⣿⣄⣼⣿⣦⣄⠀⠑⠈⢧⡀⠀⢘⣿⡿⠋⡇⠀⠉⢶⠲⢅⠘⣿⢝⠁⠀⠀⣼⣿⣿⣿⣇
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠉⠓⠀⠀⠀⢁⣴⠿⠋⠀⠀⢻⠀⠀⠀⠑⠀⠑⠘⠀⠀⠀⢀⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠸⠿⣟⣿⣿⣿⣿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⡀⠀⠀⢠⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡀⠀⠀⠀⠰⣄⠀⡰⢒⣶⣤⡀⠀⠀⣀⠠⠰⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣿⣿⣿⠿⠛⣻⣿⣲⣤⣈⠴⠏⣀⢤⣿⡿⠋⠁⢀⣹⣁⠁⠠⣿⣿⠓⠖⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢏⡷⣧⣭⣿⠿⠛⠋⠁⠙⠺⡿⣯⣽⣛⣭⡴⠿⢄⠠⢴⣿⣿⠏⢠⡠⠀⠉⣶⣄⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡠⢿⠭⢤⡿⣿⠇⡔⠁⠀⠀⠀⠀⠀⠀⠈⣎⠉⠙⠿⣵⠾⢥⣦⣤⣭⣶⣿⡿⠿⠿⣲⡝⣿⣿⡟⠿⢤⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠙⢧⡴⣴⡣⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢔⣿⣦⣤⣴⣘⡷⠿⠿⣥⣤⣶⣿⣿⡟⣏⠀⠀⠠⢽⡄
⠀⠀⠀⠀⠀⠀⣼⡠⠀⠀⡀⠀⣀⡈⢷⠊⣴⣿⡷⠀⠀⠀⠈⠱⣄⠀⠄⠀⢰⠀⠀⣀⣈⡋⠹⠍⠉⠀⢠⣦⠀⠀⠀⠀⠀⠀⣈⡧⢤⠠⠃⠈⠒⣤⡀
⠀⠀⠀⠀⠀⣠⣋⣽⣲⡖⠒⠶⠤⠗⠻⠃⠉⠉⠁⠠⠔⠓⠦⣴⡌⠑⠒⢐⡁⠰⣿⣿⣿⣿⠢⠀⠂⠈⠺⢉⠀⠒⠲⢤⡆⠀⠀⡁⢀⠀⠀⢄⢿⡏
⠀⠀⠀⠀⠀⡿⠻⠿⣃⡐⠆⠀⠀⢀⡔⠛⢢⡄⠀⣷⣶⣤⣤⣶⡟⠀⠀⠉⠁⠀⠀⠉⠉⠀⠐⠾⣤⡖⡱⠁⠀⠀⠀⠀⠹⣟⡄⠀⠀⠀⠘⡆⠉⢧
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⢲⡀⢀⠞⠀⠀⠀⢱⠀⠻⠿⠯⠑⠉⡀⠰⠴⡀⠀⠀⠀⠀⠀⠀⠀⠓⡀⢰⠃⠀⠀⠀⠀⠀⡀⡿⡄⠀⠀⠀⡸⠁⠀⢰⣷
⠀⠀⠀⢸⣖⠳⠴⠴⠚⢋⡜⢁⡏⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⢀⣀⠀⢀⠈⠵⡀⠀⠀⢠⠀⣠⠔⠁⢸⠀⠀⠀⠀⠀⠀⡰⠁⠉⠓⢒⠚⢁⡔⣿⣿⣿⡇
⠀⠀⠀⠚⠻⣟⠒⠒⠂⠁⣢⡄⠧⣀⡀⢐⣀⠄⠣⢦⣀⠔⠛⠉⠉⢋⠲⣬⡄⠹⡛⠚⢀⢀⠁⡰⣶⣾⠧⣄⣀⢠⡔⠋⠀⣀⣀⡀⡅⠀⠀⠙⣾⠟⠉
⠀⠸⣄⠀⠀⠸⡀⢰⣿⣿⡿⠃⠐⢾⣶⣶⡆⢠⣀⠰⣇⢀⡀⣄⠀⣾⣿⣾⠃⠀⠈⢁⠣⠤⠾⣷⣿⠋⠀⠀⠀⠄⡈⠀⣰⣿⣿⣿⠙⠢⣀⡜⡹
⠀⠀⢸⣷⠂⠂⢀⡌⠉⠁⠀⠈⠉⢲⡘⠛⠁⠀⢠⣤⣽⢿⣿⣿⣿⣿⠞⢩⠀⠊⠉⠓⠦⣀⠀⠀⡈⠀⠢⢄⣰⠂⠈⢱⠸⢿⣿⠟⠉⠳⣌⠥⡷⣤⡀
⠀⠈⢿⣿⡿⣦⡌⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣿⣿⠆⠤⠭⢤⡀⢰⡇⠀⠀⠀⠀⠀⠈⠙⢾⡇⠀⠀⠀⡛⠦⠤⠚⠀⢠⡏⠀⠀⠀⢻⠒⠃⢸⡇
⠀⠀⠀⠉⠛⢿⠀⠀⠀⠀⠀⠀⠠⢰⠁⣀⡀⠀⠉⠉⠁⣆⠀⠀⣸⠁⠈⢧⡘⠆⠀⠀⠀⠀⠀⢸⠙⠤⠀⢐⣥⡜⠄⠆⣀⠘⢆⡀⣀⣡⣯⠒⠈⢻
⠀⠀⠀⠀⠀⠘⢆⡀⠠⠤⠒⠋⡠⢣⠂⠁⢹⢷⠀⠀⠰⣄⠘⠒⠋⢀⣤⡀⠻⢄⡈⠄⠀⢠⠄⠇⢀⡴⢰⣿⣿⡇⣇⡀⢈⡧⠠⠥⠤⣼⣇⡀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⢿⡛⠋⠁⠀⠈⠒⠒⠁⠀⣞⠀⠀⠀⡓⠀⢰⣿⣿⣿⢆⣬⡙⠖⣦⡴⠊⢀⣿⣷⡎⢩⠩⡄⠀⣠⠊⠀⠀⠀⠀⠈⢿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠀⠀⣤⣶⣶⣢⠀⠀⠘⠓⠀⠈⠠⡄⣈⠙⠛⠁⠈⠉⠀⣴⠈⠁⠈⠓⠢⡁⢀⠈⠫⢄⡤⣇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠛⠿⠿⠿⡇⠀⡀⠀⠀⠀⠀⢀⣗⣄⡀⠈⠑⢦⡖⢀⡨⠷⣄⠀⠀⠰⠋⠙⠦⠀⠠⠁⠈⠒⠤⠤⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣀⣀⠀⠔⠋⠈⠙⠧⣄⡀⠘⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣴⣿⢦⠀⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠟⣷⣿⣿⣿⣿⣿⣿⣦⠂⣴⣿⣿⣿⣯⣴⣿⣿⣀⢀⠠⡀
⠀⠀⠀⠀⠀⢀⣶⠿⠮⣿⣿⡘⣿⣿⡄⣌⠀⣠⣦⣦⢀⠸⠃⠀⠀⠁⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⢢⣍⠛⠘⠻⢾⣿⣿⡏⣌⣧⣥⢤⠄
⠀⠀⠀⠀⢀⣾⡜⢂⣀⣽⣿⣿⣽⣿⡑⣿⣿⣆⡻⣿⣥⣤⡚⢿⣎⡀⠐⢳⢯⡿⢿⣋⣙⣛⣬⡉⢿⣿⡏⢰⣷⡞⢴⣾⢏⡙⢿⣿⣿⣿⣌⣃
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣄⡻⣿⣿⣿⣏⣀⠀⣝⠳⡀⣽⡀⢀⣿⠿⢻⣿⡆⢵⣾⣿⣧⠐⢿⠠⣤⡙⠈⠁⣈⠿⣿⠛⣻⣿⣇
⠀⠀⠀⡼⡿⣿⡿⠛⡩⢚⣿⡏⢹⣧⣾⣯⡿⢿⣾⣿⣿⣿⣧⣼⣮⣿⣿⣌⣓⣨⡿⠘⢿⣿⣿⣦⢿⣿⣿⣿⣿⣧⣹⡿⠇⣀⢼⣦⣡⡚⢻⣿⣿⡄
⠀⠀⢠⣇⠀⠃⣀⣸⣿⣿⣿⣶⣿⣿⣿⣷⣦⣦⣼⣿⣿⣿⣿⣿⣿⣿⡿⢛⣽⣿⣿⣶⣦⣬⣟⣻⣷⣿⢿⣿⣿⣿⣿⣷⣄⣼⣎⣿⣿⣿⣼⣷⣆⣄⡀
⠀⠀⠀⠉⠉⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠯⠭⠍⠉⠉⠉⠹⠟⠉⠉⠈⠛⠛⠟⠿⠿⠉⠉⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠙⠛⠛





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⡛⣉⣙⣛⡟⣫⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠈⠁⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⣿⣿⠀⢀⠐⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⣿⣿⡄⠀⡀⡤⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⣿⣿⣇⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⢈⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠠⠄⢰⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⣦⠀⠀⠀⠈⠀⡂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡀⢘⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⡨⠁⠐⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⡀
⣿⢿⡿⡿⣿⢿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣻⣿⣿⣛⣛⣛⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠒⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⢄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠄⠀⠆⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠈⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠆⠀⠀⠀⠀⢂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⢆⠀⠀⠐
⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣀⠄
⠀⠀⠛⠀⢦⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⢀⢈⠛
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⠶⠶⠶⢶⢶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢿⡤⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠄⠀⡌⠂⢠⠀⡀⠀⠁⠐⠶⠖⠀⠀⠀⡀⠸⡆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⢀⠀⠒⠀⠄⣀⣸⣛⠁⣹⡽⡛⢳⡀⠀⣤⡆⠠⡴⠊⢀⣽⣎⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠴⠃⢄⣉⠁⠀⠰⢖⣫⢞⠛⢻⡞⢻⣿⠷⠾⣶⣷⡉⢀⣲⠀⠐⣠⠟⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⠀⠀⠐⠋⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠇⢻⣿⠀⠒⠛⠛⠁⠐⠀⠑⠛⠀⠀⢒⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣫⣶⣰⣤⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⢯⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢿⡿⠋⡼⣿⡟⢸⠇⠙⠀⠀⠁⠀⠉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⡠⣄⡀⠀⠀⠀⣀⣴⣿⣿⣪⣴⠋⢀⠞⣴⠋⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⣿⣷⡿⠝⠛⢻⣶⣾⢟⡿⣻⣿⠟⠁⡴⠋⣼⠋⠀⢀⣾⠀⠀⠀⠀⠀⠀⣀⣄⠀⢀⠺⣝⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⣹⣏⣿⣷⣷⣤⣶⣿⣿⡿⣫⣾⣿⠁⢁⡜⢀⣾⣇⡆⢸⣿⣿⠀⢄⠀⠀⡆⠀⣿⣏⡆⠈⢦⡈⢻⣿⣦⡀
⠀⠀⠀⠀⠼⠿⣿⣿⣿⣾⣿⡿⢿⡟⠿⢿⣿⣿⣯⣾⠛⠹⠿⣶⡞⢀⠈⢿⣿⠁⣿⣿⣿⠀⢸⣄⣀⢱⢸⣿⠋⢹⢂⣤⣻⣮⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠻⠏⠉⠉⠀⠙⠲⣬⣟⠾⡟⠃⠀⠀⢠⣽⡿⠞⠀⠘⠉⠶⢽⣿⣿⠂⣿⣿⣿⡿⠟⠉⢦⠘⠋⢀⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢲⡟⠓⠲⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡏⡵⠠⣤⡋⠀⠀⣴⠃⠈⠈⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠉⠃⢀⣴⠷⢤⣀⡀⠈⢦⣀⡠⠀⠛⠁⢰⡿⡆
⠀⠀⠀⠀⠀⠀⢀⣤⠞⣁⡠⢴⠖⠊⠉⠀⡸⠀⠀⠈⠉⠒⠦⢤⣁⠀⠀⠀⢸⡇⠹⡄
⠀⠀⠀⠀⠀⡴⠟⠓⠉⠀⠀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠉⠒⠦⢼⡇⠀⢳
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠰⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢷⠀⠀⢧
⠀⠀⠀⡼⠁⢀⠀⡀⠀⠀⡆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠈⡇⠀⠈⣆
⠀⠀⣼⠃⠀⠨⡇⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠁⠀⠀⠀⠸⡀⠀⠸⣆
⠀⠀⣿⠀⠀⠀⠉⠀⢀⡃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠉⠀⠀⠀⣧⠀⠀⢻⡄
⠀⢸⢹⡄⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠈⢇⠀⠀⢿
⠀⢸⢸⡇⠀⠀⠀⠎⠀⢠⠀⠀⠀⠀⢀⡇⠀⠀⠀⡀⠀⢰⠆⠆⠀⠀⠀⠀⠀⣸⠃⠀⠈⢆⠀⠸⣇
⠀⡇⠀⡇⠀⠀⠄⠀⠀⢠⡀⠀⠀⠀⡜⢃⠀⠀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⠇⠀⠄⠀⠘⣄⠀⢻⡄
⠀⡇⠀⣷⠀⠎⠀⠁⠀⠀⠁⠀⢀⡞⠀⠘⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠄⠀⠘⣆⠀⣷
⢸⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢄⠀⠸⡆⢨⡆
⡸⢀⠆⢸⠀⠀⠀⠀⠀⠀⢀⡼⢡⣦⣴⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⢀⡀⠹⡄⢇
⡷⠀⠀⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠇⠀⠙⢦⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣼
⡇⠀⠀⠸⡇⢀⡴⠊⠁⠰⣦⠀⣾⠁⠀⡆⠀⠀⠀⠙⢦⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠹
⡇⠀⠀⠀⠶⠋⠀⠀⠀⠀⢻⠄⣿⠀⠀⡄⠀⠀⠀⠀⠀⠙⢦⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠀⠀⠀⡇⠀⠀⠀⠈⠋⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢤⠀⡄⢶⠀⠀⠀⠀⠂
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⡟⢰⠁⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠰⠇⠸⠂⠀⠀⠀⡄
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠋⡰⢃⣞⠣⣀⠀⠁⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠠⠔⣿⡿⠃⠀⢠⣾⡿⠀⠈⠀⠀⠀⠙⢳⣦⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡇⠀⠀⢀⡀⢀⡤⢞⣉⣤⡖⠘⢟⡉⠀⠀⠀⢦⡀⠀⠀⠀⠻⢦⠀⠐⡶⣶⣿⣿⠉⣆⠀⠀⠢⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢐⣲⣾⡿⠿⠿⠛⠉⠁⠝⠒⢂⣠⣴⣶⣶⣿⣿⣿⣤⡀⠀⠙⠃⠀⠀⠈⠻⣄⡀⠀⠀⠈⠓⠠⠀⠈⠛⠁⠀⠛⠓⠦⢄⡈⠑⠤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⠿⠿⣿⣿⣿⡿⠟⠋⠁⠈⠻⢶⡆⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠄⠀⠀⠀⣤⣤⣤⣼⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⡀
⡠⠒⠀⠐⠒⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠄⠀⡀
⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⣀⠈⠀⠀⠀⢈⡉⠀⠀⠈⠀⠀⠀⡀⠀⠀⠄⠀⠀⡀⠐⠀⠀⠀⠘⠀⢀⠀⠈⡄
⢸⠠⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡉⢀⠈⠁⠀⠃⠀⡉⠀⠄⢀⢀⢀⡀⡐⠂⠀⠀⢀⠐⠂⠀⠀⠁⠀⠀⡀⡀⠀⠈⠀⠀⠀⠄⠀⠠⠃
⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⡄
⣨⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠀⢤⣤⣤⣤⢤⠤⠠⠀⢀⠀⢀⣄⣀⣀⣠⣠⣠⣤⣤⣤⣦⣤⣴⣶⡶⣶⠶⠶⠶⠾⠾⠿⠓⠛⠛⠛⠛⡛⠛⡛⠛⡛⠛⠉⠁
⠇⠀⣀⠸⠉⠉⠉⠉⠍⠙⠉⠩⠂⢀⠉⠁⠈⢉⡉⡛⠛⠋⠠⠈⠥⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠐⠀⠈⠁⠈⢁⠈⠀⠀⠤⠀⠀⠀⠀⠀⠠⠀⠀⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠕⠀⠀⠀⠀⠀⠃⠠⠅⠰⠀⠄⠠⠤⠃⠀⠀⠀⠁⠀⠁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⢰⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠀⠐⠒⠒⠒⠒⠒⢛⡉⢉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠈⠈⠀⠀⠐⠀⠀⠀⠀⠐⠳
⢸⡄⠄⠀⠀⠀⠐⠀⠀⠐⠀⠰⠀⠀⠘⠛⠛⠋⠁⠀⠀⠀⢀⠀⠀⠀⠒⠒⠚⠋⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⢨⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠰⠀⠀⠀⠀⠈⠁⠀⠩⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⣂⣀⣀⣀⡀⠀⠀⠠⠀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡀⢀⠀⢀⣀⡀
⣿⣿⡿⠛⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⢀
⠀⠀⠀⠀⠀⢀⡄⢦⣄⠀⠃⠎⠙⠂⠀⢶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠀⣤⢀⠀⠀⠈⡏⠀⢻⡦⠀⢦⡀
⠀⠀⠀⠀⠀⠀⣁⠀⢉⠰⣄⠀⠀⠀⠐⠦⢙⡀⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠸⠃⠀⠁⠈⠃⠀⠀⠁⠀⠈⠀⠀⠈⠃⠀⣀
⠀⠀⣠⠀⠃⠀⠏⠀⠈⠀⠙⢶⡇⠘⠀⠤⣀⠑⠈⡀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⣤⠔⠀⠀⠀⠀⠀⠐⠀⠇⠀⠆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠸⢻⠷⣤⠀⢀⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠟⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠙⠏⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢢⡀⢠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠆⠈⢡⡙⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢫⠀⢀⡁⠀⠐⣤⣄
⠀⠀⠀⠀⣠⠂⢠⠀⡜⠑⣀⠀⢀⠀⠀⠀⠙⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣶⣧⡈⠀⠀⠀⠈⠁⠠⣄⠀⠹⢦⡀
⠀⠀⠀⠀⠁⠂⢘⠀⠀⢀⠉⠀⠜⠀⠀⠀⢼⠁⠘⢿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠛⢿⣿⣦⡀⠐⠁⠀⠀⠀⠑⣄⣀⠉⢲⣤
⠀⠀⡀⠓⢠⠀⠸⣆⠀⠀⡠⠀⠀⠀⠘⢦⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠻⣿⣷⣄⠀⠀⠀⠀⠈⢿⣷⣜⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣿⣧⠀⠀⠙⠛⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣇⣽⣟⠀⠀⠀⠀⠀⠀⠋⠓⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢟⣿⣥⣧⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⡀⢀⣀⠄⣀⣸⣿⣿⣿⣷⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⢁⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⢹⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⠛⠛⠁⢺⣸⣿⣿⣧⣿⣿⣿⣿⣿⡿⣿⣶⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⠋⠀⠀⣶⠀⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀⠛⠿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠋⠀⠀⠀⠀⠀⠸⢰⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠆⣼⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠃⣿⣿⡇⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣼⡇⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡝⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣣⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠷⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⣤⣤⣄⣀⠀⢀⣀⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠸⣍⡇⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠸⠂⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⢀⠠⠀⠀⢠⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢤⡴⡦⠴⠶⠒⡖⠓⠀⠘⠛⠉⠁⠀⠂⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢰⠀⠂⠀⠀⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠔⠀⣸⡄⠠⠆⠨⠀⠀⠀⠀⠀⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⡄⠀⠁
⠀⠀⠀⠀⢠⣔⠉⠻⠖⠀⠀⠀⠀⠀⠀⡄⡀⣠⡾⠁⠐⢠⠀⠀⠀⠀⠀⢸⡄⣀⡀⢀⢀⢀⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠀⠚⣷⡄⣿⠁⠒⠂⣠⠀⠀⠀⠀⠀⠐⠘⠓⠓⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡞⣧⠕⠋⠁⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣼⣿⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⣿⣷⡀⠈⡄⠁⠈⠀⡀

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⢾⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠉⣰⣎⡴⢃⣰⠃⠀⠈⠢⡀⣠⢴⣏⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡒⠁⠁⠁⠊⠈⠀⣠⠏⡸⠀⠀⠀⠀⠘⢏⠋⢸⡆⠈⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣄⢰⡇⠀⠀⠀⠀⠀⠊⠁⠂⠁⠀⠀⠑⠄⠀⠀⠁⠀⠀⠀⢠⠀⣸⡆⢠
⠀⠀⠀⠀⠀⣸⣄⣾⣷⣿⣸⣇⣧⠀⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⡄⠀⣰⠀⣼⣇⣿⠧⠟⠂
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⡀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣧⣠⡿⠖⠋⠁
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⡲⢤⠄⠠⠷⠘⠛⠒⠛⠉⠁
⠀⠀⠀⠀⣰⣿⡆⠀⢰⡀⠀⠀⠀⢀⡀⢄⡴⠏⠚⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠺⠇⠰⢉⣰⣡⡊⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣷⠀⠀⡀⠀⠀⣺⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠏⠙⢄⠢⢄⠀⠄⠰⣿⡇⠀⢸⣿⡄⢠⡇⠀⣠⠿⠿⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⢀⡜⠀⠀⢣⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⡘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠒⠠⢑⠢⣀⣀⡠⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⡸⠁⠀⠑⠠⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⡀⠀⠀⠀⠀⡱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠈⡖⢤⣀⣐⠁⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⠎⡐⠁⠀⠀⠉⠒⢄⡀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⢊⡜⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠉⠉⠉⠉⢩⡟⠒⠦⠤⣀⡀⠀⠀⠑⢌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠤⣀⣀⡀⠀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢈⠷⡀⠀⠀⠀⢥⣤⣭⠑⠂⠠⠤⢀⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⢃⡬⠘⣦⡀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⣰⡗⠉⠀⠀⠘⠿⢦⠀⢹⡏⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠄⠀⠀⠑⢂⠀⠀⠀⠀⠑⠂⠃⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠓⡀⠀⠀⠀⠀⠀⠀⠀⠜⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠐⡀⠀⠀⠀⠀⠀⠌⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⢖⠄⠡⠒⢎⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡼⣀⠀⠀⠀⠀⠄⡃⠀⠀⠀⡂⠀⠀⠀⠀⢀⠴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢱
⠀⠀⠀⠀⠙⡦⣁⠒⠠⠄⣰⣕⠠⠤⢒⣔⠠⠤⠒⢊⠥⠞
⠀⠀⠀⠀⠀⠈⠢⢉⠒⠠⠀⠤⠤⠤⠤⠠⠀⠒⣉⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠠⠄⠀⠒⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢠⣾⣥⠜⠀⠁⠀⣀
⠀⢀⣀⣴⣄⠀⠰⣦⠔⠃⣀⠀⢴⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢠⣿⣷⡶⡈⠀⠀⠘⢿⡀
⠀⠉⠉⣥⠴⣾⠧⠾⠦⠀⠉⠙⠳⠨⠓⠀⠀⠀⠀⠀⡀⢠⡴⣰⠾⢃⣶⣿⣿⡿⠈⡥⠐⠀⠀⠀⠐⠀⣀⣴⡞
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢁⡼⡿⠋⠉⠈⠐⠺⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣞⢀⣾⣧⠆⠈⠁⠀⣀⡤
⠀⠀⠀⠀⠀⠀⣰⠻⢏⣾⣿⡶⢦⠀⠀⠀⠙⢧⠀⠐⡆⠀⢸⣆⣠⣄
⠀⣠⠄⣴⠂⠞⣻⣶⣿⣿⡿⠂⡀⠀⠀⢠⡄⠀⠲⢤⣀⣆⣾⠿⣋⣁⠀⠀⠀⠀⠀⢀⠠⢆
⠀⣀⠀⡄⠀⢸⡿⠋⠀⠚⠀⢀⣿⡀⣶⣼⣧⣾⣾⣤⡾⠿⠷⣶⡿⠋⠀⠀⠀⠀⠠⠁⠀⠀⡀
⢀⣿⣼⣷⡀⢠⣤⣾⣶⣴⣿⣾⡿⠟⠻⠿⠛⠉⠁⠀⠴⠟⠛⠉⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⡆
⠈⠉⠉⠙⠓⠫⢤⣀⣀⠂⣨⠉⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠉⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠒⠼⠿⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣧⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡀⡟⠀⠼⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡄⢈⠁⠀⠠⡘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠴⠁⣿⠀⠀⠀⡁⠸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢠⡀⠀⡿⠀⠀⠀⢁⠀⢿⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠈⠀⢸⡇⠀⠀⠀⠘⠃⠸⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠤⠀⢹⡄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡆⠀⠀⣸⠁⠀⠀⠀⠀⢠⠀⠀⢷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣈⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠂⠀⠘⣇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠠⠍⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠶⠀⠀⢸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣷⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⣶⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢹⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⣷⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⡆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢈⡷⠸⠂⠀⠀⢠⡏⠀⠀⠀⠀⠀⢰⣆⣤⡄⠀⠀⠃⠀⠀⠀⠀⢱⡄⢠⡌⣷
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠘⠷⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡏⠀⠀⠘⠀⠘⠃⠀⠀⠀⠈⠁⢸⡇⠨⣇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠧⠀⠀⠀⠈⣄⠉⠀⠸⣇
⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⢿⡀⠀⠀⠙⣄
⠀⠀⠀⠀⣼⠉⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⢳⡀⠀⠀⢹⡆
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢷⡀⠀⠀⢱
⠀⠀⠀⢰⡏⠀⢹⡀⠀⠀⣰⠆⠈⠀⣤⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠄⠀⠀⢣⡀⠀⠘⣇
⠀⠀⠀⢸⡇⠀⢸⡇⢠⡶⠀⠀⠀⠀⠀⠰⠆⠀⠀⣿⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠠⠄⠀⠈⢷⠀⠂⣿
⠀⠀⢠⣿⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣷⡀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣤⠀⠈⣷⠀⢹⡆
⠀⠀⢸⡏⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠀⠈⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢧⠀⣇
⠀⠀⣾⡀⠊⠀⠈⣇⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⢠⡆⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠈⢶⣹
⠀⠀⣿⠁⠀⠀⠀⣿⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢿
⠀⠀⣿⠀⠀⠀⠀⣿⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣰⡆⠛⣄⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⢠⡄⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢶⡄⢦⡀⣦⠀⢀
⢰⡇⣰⠇⣴⡁⠀⣿⠀⠀⠀⠀⠀⠀⢈⡿⢀⡿⢹⡇⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⡾⢁⡾⠀⣿⠀⠻
⠀⠀⣿⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁⠘⠁⠈⠛⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃
⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣐⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠠⠄⣠⢀⡆⠀⠀⠀⣍⡈⠙⠀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠼⠅⣠⡿⠿⠷⠷⠴⠶⠀⠠⠶⠟⠡⢬⢱⠾⢇⢀⣀⣿⣿⣿⣶⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠺⢇⣶⣾⢬⣼⣿⣿⡿⠟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣶⣬⣿⡿⠀⠁⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡞⣿⣦⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣿⡿⢩⣿⣿⣦⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡃⣼⣿⣿⣿⣦⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⢹⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡃⠿⠿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠦
⠀⠀⠀⠀⠶⠛⣻⠋⠓⢠⣀⠐⢻⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⡆⠀⠀⠛⠘⢿⣿⡛⣿⡟⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠚⠃⠴⢂⣈⣥⣤⡬⠷⣿⡿⠟⠻⠉⠀⠀⢀⡟⠇⠀⠀⠀⠃⠀⠽⣷⡈⢷⡄⠁⠀⢿⠛⠻⠟⠋⠉⠛⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢵⠀⠴⡓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣄⣀⠀⢸⣿⣦⣖⢦⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠿⢿⣶⠿⠿⣿⣿⣦⣿⣿⣿⣿⣿⣷⣺⣶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⣿⣷⣶⣶⣶⣿⣦⣭⠿⢿⣿⣿⣿⣿⣿⣷⣬⡉⠳⢆
⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣟⢻⣿⣿⣿⣿⣿⠟⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡥⡀
⠀⠀⠀⠀⢠⣿⣿⣿⠏⣾⣷⣲⣾⣿⣿⣿⣿⢟⣿⣮⢉⣿⣿⣿⡿⣚⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣻⣆
⠀⠀⠀⠀⡼⢿⣿⣿⣠⣹⣿⢿⣛⣿⣧⡋⠁⣈⣹⡿⣷⡝⠻⡜⡿⣿⣿⣿⡿⢋⣿⣿⣿⣿⣿⣯⠹⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣣⣸⣿⣿⣿⣾⣷⣦⣿⢿⣿⣿⣷⣿⣿⣿⣷⣿⣷⣾⣤⢹⣿⣟⡵⠛⠛⠿⢿⣿⣿⣿⣷⣄⢻⡿⣿⣭⡇
⠀⠀⠀⢸⣿⡟⣿⣿⠟⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⢿⠿⣄⠀⣸⠑⠲⣿⢻⣿⣿⣿⡿⡷⣴⣿⣿⣿⣿⣦⡀⡟⢸⣧⡀
⠀⣀⣤⣿⣿⣿⣿⡏⢱⣿⣿⣿⣿⡿⣿⣯⡟⠉⠊⣤⣄⣿⡈⢃⢇⠀⠹⣯⣿⣿⢿⢧⣟⣿⣿⣿⣌⠈⣿⣿⢛⡿⢿⣧
⠘⣷⢿⣯⣿⣿⣻⣷⣼⣿⣿⡿⢛⣿⡿⠀⡀⠰⢻⣷⣿⣿⣧⠰⣾⣄⠀⢻⣿⣿⣷⣿⣿⣟⣿⣿⣿⡂⣿⣿⣏⣳⡰⠃
⠘⣷⣾⣿⣿⣿⣿⣷⣿⣿⣿⠿⢋⠝⠗⠉⣉⣷⣿⣿⣿⣿⣿⡄⠸⢻⡄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠘⢿⣻⣿⣿⣿⣿⣽⣿⣿⢟⠕⢧⣥⣾⣿⡏⠉⠛⢿⣿⡿⣿⡀⢡⣿⡀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠇
⠀⠀⠀⠛⢿⣿⣿⣿⣿⡿⠇⣀⣴⣿⠛⡋⢘⠎⠳⣤⣶⣿⣷⣿⣷⡀⠻⣿⣀⣾⣿⠿⣿⡿⠛⠋
⠀⠀⠀⠀⠈⢿⣿⣿⠿⡿⠍⢿⡿⢷⣶⣶⣤⣶⣿⠾⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠛⠺⠟⠿⠿⠁⠀⠉⢿⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣴⣤⠂⠀⠀⢀⣿⣿⣷⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠀⠊⠁⠀⠀⠀⠚⠁⣿⣿⣿⣿⣆⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣦⡀⠙⠳⠌⠺⢦⣄
⠀⠀⠀⡀⣀⣀⣀⣄⣀⣀⢀⣀⣀⠀⣀⢀⡀⢀⣀⡀⠀⡀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠉⠁
⠀⠀⣩⣍⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠰⠿⠿⠿⣦⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢴⡁⠀⠀⠀⡀⠀⠃⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠋⠀⠀⠑⠄⢄⠀⠀⠑⢄⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⡀⠀⣄⡸⠀⠀⠀⢠⠞⠑⢤⣤⣶⣿⣶⣶⣷⡤⠤⠟⠀⢀⠀⠒⠒⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠈⠀⠀⠀⠀⡰⠃⠀⠀⠀⠛⠿⠟⢿⣿⣿⢿⣆⠀⢐⣉⡠⢀⣀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠉⠉⠁⠉⠝⢢⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡀⠻⣧⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠒⠀⠰⠀⠐⠊⠉⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣬⢷⡀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢈⠥⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠽⣧⢳⡄⠀⠀⠀⣆⠁⠀⢀⣠⣠⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠻⣄⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠠⠀⠀⠈⢀⠃⡀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⠀⠀⠀⠀⣠⢃⠁⠀⠀⢸⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠘⣦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⣀⡴⠁⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠻⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⣠⡟⠀⠀⠀⠀⢦⠀⠀⠀⠘⢆⠀⡀⠀⠀⠈⣝⡻⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣠⠾⠋⠐⠀⠀⠀⣠⣾⠟⠁⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡀⠈⠀⠱⠈⣿⣯⡓⣬⡳⣄
⠀⠀⠀⠀⠀⠀⡴⠟⢁⣴⠟⠁⢀⡜⠀⠀⣰⣾⠟⠁⠀⠀⠀⣀⣾⡟⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⢹⣦⣀⡀⠀⣸⣿⣿⣾⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠑⢾⣏⣁⣀⣠⡞⠀⣠⡾⠟⠁⠀⠀⠀⣠⣴⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠲⢦⣴⣴⣾⣿⣿⣿⣤⣤⣀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠒⠀⠉⠀⠀⠀⠀⠐⢲⡤⠀⠀⠀⠀⠀⠀⠀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢢⠀⠀⠀⠀⠒⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠲⣦⠀⣷
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡀⣀⡀⠠⠀⠘⠀⢀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢇⡄⡀⠈⣠⠀⠈⢩⡔⠃⣀
⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠆⠀⢀⠀⠀⠀⠈⠉⠀⠃⢉⡠⠏⠘⠉⠠⠴⠚⠁⠞⠃
⠀⠀⠀⠀⠀⢸⢻⣿⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⢠⣶⠀⠀⢸⠀⣿⠀⢸⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠤⡤⣤
⠀⠘⣿⠀⠀⢸⢠⣿⠀⠌⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣞⢷⡀
⠀⠀⣿⡄⠀⠘⢸⡿⢊⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⢄⢸⢿⣧⣷⡀⠀⠀⣀⠀⣀
⠀⠀⢿⣧⠀⡞⢸⣿⠟⠁⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⡶⣋⡄⠀⣀⣼⠇⣾⣿⣛⢯⣿⣷⣾⣯⢁⠈⠢⣄
⠀⠀⠘⣿⣦⡗⢸⣿⠀⠀⢸⣼⡇⠀⣀⠀⠀⢀⠠⠀⢀⠎⠀⣼⡿⠋⠀⠀⡿⠋⢸⣧⡈⠛⢿⣿⡿⣿⣷⣿⣷⣀⣉⢻⡶⣦⣀
⠀⠀⠀⠘⢿⣇⢾⡏⢀⣄⢸⣿⠃⠀⠸⣷⡀⢉⡤⠚⠉⢀⡾⠋⠀⣰⣧⠞⠀⢼⠙⢿⣿⣦⡺⣿⣿⣮⣿⣿⣿⡿⠋⠻⣧⡿⣿⣿⣆
⠀⠀⢀⣶⠀⣿⢸⡇⢸⣿⢸⣿⡆⢀⣤⢿⡗⠉⠐⣠⡾⠋⠁⠀⣰⣿⢏⠄⡆⢠⠸⣄⠈⢿⣷⣿⣿⡿⠛⠉⠁⠀⠀⠀⣽⠿⣿⣄⡈
⠀⠀⠘⣿⣆⣿⣿⠁⣸⣿⣼⣿⡇⣿⣿⣿⣿⣶⠟⣫⣤⣶⠶⠋⠉⣠⠎⢀⡁⢸⡆⠨⣦⡈⠿⠟⠉⢰⢸⡇⠈⠁⠀⣰⣿⡀⠀⠉⠃⠑⠂
⠀⠰⣾⣿⣿⢿⣿⡇⠈⡻⣾⣿⠃⣤⠙⠿⠿⠗⠛⠛⢻⣃⣤⠈⣰⣟⣀⣾⡇⠘⠛⠀⠉⠀⠀⠀⣶⠸⢿⠁⣾⢠⣿⣿⡛⢻⣶⢤⡀
⠀⠀⠀⠀⣈⣿⣿⡇⠀⠙⢻⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣥⣾⣿⣷⠿⠋⠀⠀⠀⠀⢠⢠⣄⠀⢹⣄⣿⣰⠇⠈⠙⢻⣿⣿⣿⣿⣧
⠀⠀⢀⣈⣠⣿⣿⣿⡓⠶⣿⣿⣿⡿⠀⠀⠀⢀⣀⣀⡀⠀⠸⠿⠏⢠⡀⠀⠀⠀⠀⠀⠸⣇⣿⠀⠀⠙⡿⠋⠀⠀⠀⠀⠈⠙⠛⠛⠻⠟
⢰⠂⠉⠉⠻⣛⡙⣉⣙⠓⠿⢿⣅⠀⠀⠀⠀⠨⠽⠟⠋⠀⠀⠀⠀⠈⣿⣦⣤⣀⣀⣤⣄⣸⣿⣤⡤⠼⢿⣶⣦⠙⢟⠗⣶⢷⣤⣤⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⠿⠒⣘⠛⠿⠿⢷⣶⣴⣦⣤⡀⠀⢀⣀⠀⠀⠙⠿⠟⠛⠀⡈⢙⣿⠋⠀⠀⠸⣿⡿⣷⡆⠀⢀⣀⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠁⠀⠉⣙⣯⠁⠀⠈⠛⢯⣤⣤⡤⠀⠀⠀⠀⠹⠿⡿⢦⣄⣤⢀⣿⠷⠛⠋⠙⠛⠿⠿⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠖⠙⠒⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡤⠒⠚⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⢣⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⢻⣿⣿
⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⣇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⡋⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⣿⣿⣿⡏⠁⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠚⠛⠋⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠑⠢⠤⠤⠖⠚⠛⠉⣿⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠙⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡎⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡼⢸⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⢣⠀⠀⠀⠀⢰⠁⡜⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⠁⣾⣇⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡇⢰⠁⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢀⡇⣸⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢇⠀⠀⢰⠁⡇⠀⠀⣇⠀⢀⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⢀⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣆⠀⡸⠛⠀⠀⠀⠘⢄⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⣖⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⣁⣴⣿⣿⣿⣿⠛⠛⢻⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠟⣵⣧⣤⣶⣿⣿⣛⠋⠙⠺⣧⠀⣿⡟⠃⢸⣦⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣄⣤⠀⣠⡶⡿⠀⣼⣿⣿⣿⠟⣁⣽⣿⣶⣤⠀⠀⠙⣿⡇⠀⢸⣿⢲⣲⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠛⠛⠉⠈⠐⣿⣿⠟⠛⠀⣰⣿⣿⣿⣿⡟⢷⠀⠀⢯⣷⠀⠀⠻⢶⣿⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣦⡀⢀⠉⠁⢀⣴⣦⣿⡿⠃⠀⠀⣠⣿⣿⢏⣵⡎⡟⣻⡆⠀⢸⡿⣷⣤⡀⠀⠁⠀⣀⣬⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⣻⠕⠀⠀⣰⣿⣿⣏⣾⣿⡇⣿⢿⣇⠀⠈⡇⠐⠛⠿⠿⠿⠿⣿⡧⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⢄⠐⡯⠍⠀⠟⠈⠀⣀⣤⣴⣟⣉⣑⣫⡽⢯⡰⡼⣞⣿⣦⡀⠈⠒⠒⠐⠒⢖⣺⣭⣠⢀⡇⣠⡆
⠀⠀⠀⢀⣦⣴⣶⡄⢀⠀⠙⠂⡀⠤⣴⡇⢀⣀⣰⣴⣾⣿⣿⣽⣯⣾⣿⣿⠁⠈⢿⣿⣼⣟⣿⣿⣷⣦⣀⣀⣰⣿⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⢀⡾⠿⢿⡿⣿⣿⣦⣼⣧⣤⣴⣾⣿⣿⡿⣿⣟⣛⣿⣿⣿⣿⣿⠟⣿⣇⣀⠘⢿⣿⣟⣿⣟⠿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣷⣤⣾⣥⡾⠽⣿⣿⣿⣿⣿⣿⣿⣶⣭⣏⣭⣶⡶⠿⢋⣿⣿⠾⢟⠿⠻⣷⣀⠻⣿⣿⣿⣷⣽⣿⣛⣻⣟⡽⢿⣿⣾⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣿⣉⣽⣿⣿⣿⣿⣿⣿⣿⣥⣤⣾⣿⠿⠖⠒⠿⠳⣶⣝⠿⣷⣍⢻⠿⠯⠽⠋⠉⢿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣍⣿⣿⣿⢿⣿⣿⣿⠿⠿⠟⢛⣉⣤⣦⣤⣤⣤⣀⣤⣤⣭⣾⢿⣿⣦⡂⠀⠐⢒⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⢸⣷⣿⣿⣿⣿⣾⣟⣿⣿⣿⣿⣾⣿⣿⠿⡷⠞⠉⠈⠉⠉⠉⠉⠙⠛⠛⠛⠿⣿⣿⣥⣽⣻⣿⡶⢖⠻⠷⢋⣭⣫⣿⣿⡿⠋⠁
⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣤⣤⣴⡾⠟⠛⠉⠉

⣷⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠐⣄⣤⣀⡜⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠀⠀⣏⠀⢈⡇⠀⣀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠗⠨⠍⢉⡛⣋⠡⠅⠚⢭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⡇⠀⠀⠙⠀⠀⠀⠘⠀⠀⠐⣗⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠁⠐⠀⢙⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⡿⠉⠁⠁⠀⠈⡹⢩⠁⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⢞⠁⠘⠀⠀⠀⠠⠁⡄⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠋⠉⠉⠉⠃⠀⠙⠀⢄⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠑⢆⠀⣦⠄⠀⠀⠀⠀⠠⡄⠀⠀⠈⢆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⣷⡀⢀⣠⠄⡸⢣⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣧⡉⠀⢀⡁⢼⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣤⣤⣍⣳⣴⠉⠀⢸⣯⣀⣄⣤⣤⣀⣠⣇⣠⣄⣠⠇⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠤⠼⠤⠾⠤⠤⠤⠤⠤⢤⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⢴⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢽⠀⠀⠀⡠⠐⠂⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠡⡈⠑⠤⡀⢴⠁⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠭⠟⠦⠄⢿⡼⣿⣿⢵⣏⣉⣉⣉⣉⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣤⡀⠙⡤⠐⢁⠆⠀⠀⠀⠀⠀⡠⠶⠏⠚⢿⡦⠵⠶⠶⠯⣾⡙⣉⣉⣙⣉⣷⣲⣛⣲⣿⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠛⢄⡠⢎⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⢰⡿⠭⡭⠭⠭⣿⢿⣒⣛⣓⠫⣟⣲⣒⣾⣿⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⡔⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿⠽⠭⠽⠯⣿⡿⣷⣺⣷⢬⣷⢶⣶⠯⣿⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠭⠭⠭⠭⣿⣯⠯⠿⠯⣽⡋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣶⠒⠒⣶⠶⠶⣶⠶⠶⣶⠶⣶⠶⡶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⣶⠶⠶⣶⣶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠾⢛⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⢀⠀⣤⣦⡀⠀⢀⡞⠀⣼⠇⠢⣽⣿⡟⠛⢷⣦⣤⠀⠀⠀⡶⠀⠀⠀⠰⢄⡀⠀⠀⠀⢀⠀⠴
⠀⠀⠀⠀⢰⠀⠈⢀⣿⣿⣿⣯⣉⡠⠂⠸⡁⠀⠈⠻⣧⠀⠈⢻⣿⣶⣶⣿⣧⣤⣄⣀⣀⣾⡿⣿⣷⠀⡸⠠⠖
⠀⠀⠀⠀⠀⠀⣿⣬⣿⢿⣿⠋⠻⣿⡀⠀⢈⣦⣤⣤⣼⠃⠱⣄⣿⡟⠳⢿⠿⣿⠋⢉⡽⠟⠛⠿⢿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⡛⢿⣤⣹⣿⣶⣿⣿⢿⣿⣿⡃⠀⠈⢻⣷⠀⠀⠀⠈⠙⠛⠀⠀⠠⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠈⣡⠀⠙⢷⣤⣌⡹⢿⣿⣿⣿⡛⢿⢿⣿⣦⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⡶⠘⣿⣄⠀⣹⠿⣇⠀⢈⣾⢿⣿⣿⣆⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⠃⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣬⣽⣿⠋⠀⢿⣷⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣷⣤⣴⣾⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⠟⠻⠿⠻⣧⣄⠀⠻⣿⣿⠙⠿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⢸⡁⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠿⡟⠉⠀⠀⠀⠻⠿⣿⣧⣄⢹⣿⡀⠀⢉⠏⣹⣿⠀⢀⣀⣀⣀⠀⠼⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣷⣶⣿⣆⠈⣿⣴⠟⠉⣬⣁⡿⠶⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣄⣰⣿⣷⡶⠂⠈⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⢶⣶⣄⠀⠀⠀⠀⠀⠀⠙⡶⠼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣤⣤⡶⠋
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠌⠀⢠⣴⣿⣿⠿⣷⣿⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⢁⣼⠟⠋⠁⠐⠛⢻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠙⠻⣿⣿⣤⡀⠐⠿⠋⠀⠀⠀⠀⠀⣼⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠤⠤⣉⡐⠁⠀⠀⣀⣀⣤⣤⣠⣴⠟⠉⠀⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣾⣆⣿⣦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁
⠿⠿⠿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆
⠀⠀⠀⠀⢹⣿⡀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂
⡄⠀⢠⠄⠈⠁⢠⠀⠙⠧⠀⢀⡀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠽⠞⠂⠀⠀⠀⠀⠀⢀⠀⠘⠓⢆⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼⣄⣦
⣷⡀⠈⣰⠂⣠⣾⠇⠀⠀⠀⣾⡇⢠⡧⠀⣠⣠⠀⢠⡄⣀⠀⠀⠀⠀⢀⡄⠀⠐⠛⠓⠆⠀⡀⠀⠀⢀⠀⠴⠋⠀⠀⠀⠈⠷⠀⣄⠀⠀⠿⣿⣿
⠿⠧⠀⠃⢠⣿⡿⠯⠀⠠⠀⠛⠙⣿⣷⠀⢻⡆⢰⣏⠁⠉⠀⠠⡆⠀⠈⠁⠃⠀⠀⢀⣀⠀⠱⠦⠠⡞⠒⠼⢁⠑⡈⠀⣦⢠⠀⠁⠀⠀⠀⠉⠋
⠀⠠⠂⠀⣼⣿⡧⠀⠀⠀⠀⠈⠀⠉⢉⡁⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠐⠶⠿⠇⠶⠾⠀⠁⣸⣿⠀⠈⢰⣇⢠⡀⢰⡄
⡸⠃⢠⡆⠛⠛⠓⠀⢤⣄⣀⡀⠙⠂⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣧⠈⢗
⠁⢀⣿⣓⠀⠻⡄⠀⠀⠉⠁⢀⠁⠀⠀⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄⡀⠀⠀⢀⡀⠀⠀⠀⠥⠀⠀⠈⠉⠉
⠀⣰⣧⡈⠁⠀⠀⠀⠈⠀⠀⠈⢿⡀⠀⠀⠀⣈⠁⠀⢰⣆⡀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠖⠀⢒⣢⡀⡙⣠⣤⣤⣼⠇⠀⠀⣀⣀
⠀⠽⠻⠋⠁⠀⠀⢀⣀⣄⢀⣀⣬⣿⡤⠥⠤⠾⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠼⠷⢦⣄⣀⡙⠃⠤
⠀⠀⡀⠀⠀⠈⢀⣀⣤⡾⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠶
⣀⣠⣤⡴⠶⠾⠛⠛⠛⠋
⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠐⠚⢛⠛⠛⠛⢃⡀⢀⠀⢂⡂⠒⠛⠓⠆⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⠗⠆⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⣸⢀
⠌⠀⠀⠐⠀⠀⠈⠂⠀⠀⠠⣤⢀⡄⠈⠀⠀⢀⣉⠠⡤⠅⠦⢤⢴⡅⡈⣞⠛⣣⣔⡐⢀⡛⢁⡋⠘⠀⠂⠋⠛⠠⠞⠠⠁⠋⠀⢸⣶⠂⢰⡿⠈
⠇⠀⠄⠠⠨⠄⢄⣀⡀⢀⡙⠈⠋⢉⡉⢈⣂⣀⣂⠐⠒⠀⠆⠒⠐⠀⠶⠂⠴⡀⠠⠁⣤⠄⠈⢁⣁⠁⠋⠐⠀⠒⠖⢲⣴⠀⠀⢸⠉⠀⣿⠃⠀⠀⢀
⠀⠰⠀⠷⠂⠛⠙⠳⠂⠒⠖⠂⠐⠢⠨⡄⠠⠁⠬⠄⡁⣋⠁⣀⢘⠀⠂⠐⠰⠤⠄⢀⠀⢠⡄⣈⠁⢈⡚⠀⠘⠁⠞⡔⠈⣧⠀⠀⠀⣘⣿⠀⠀⢰⠇
⠄⠰⠀⠏⠀⠖⠚⠋⠙⠁⠉⠃⠉⠃⠘⠉⠂⠐⠂⠶⠀⠆⠆⡤⢠⢈⢉⣄⢂⣐⣂⠆⠤⠬⠀⡃⣘⢘⣹⡀⠉⢀⡤⢘⡇⠀⠀⠀⠀⣿⠏⠀⢀⡟
⠀⠆⠰⠟⠀⠞⠘⠃⠀⠂⠞⠉⠙⠇⠀⠸⠃⠻⠆⠶⠚⠂⠰⠖⠰⠆⠬⢩⡀⣀⠈⠂⠘⠃⠐⠀⠯⠨⠝⠛⢻⣅⠁⠾⠀⠀⡠⠂⣸⠏⠀⠀⣼⠁
⠀⢰⡴⢶⡀⢲⠀⠸⠀⠺⠀⠐⣿⠧⠐⠀⡆⣰⡆⡰⠂⠀⡖⢰⠆⢴⠆⢠⢀⠁⢀⠙⠀⠂⠣⠂⠆⣠⣄⠘⠛⠋⠀⠀⢀⡾⠁⠀⡟⠀⠀⠀⣽
⠁⠀⣁⠀⡁⡀⡄⢠⣁⠀⢀⠀⣤⠀⢀⣀⠀⢀⡁⠀⡀⠀⢁⠀⠀⠈⠀⡁⠁⠀⠋⠘⠜⠑⠛⠂⠀⠀⠀⠀⠀⠀⣠⣴⣾⡁⠀⣰⠇⠀⠀⠀⣧⢰
⣤⠀⠹⠆⠁⠀⠣⠀⠘⠀⠛⠀⠉⠀⠀⠙⠀⠸⠁⠀⠿⠀⠏⠰⠀⠀⠘⠃⡇⠀⡾⠁⡀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠚⠻⠟⠻⠶⠿⣤⣀⣈⣘⡇
⠀⠀⠀⢠⠄⠀⠀⠀⠐⠀⣠⠀⢰⡄⠀⢠⢤⣠⡄⠀⣤⡄⣠⠀⢀⣀⠀⠀⢀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⠀⠀⡀⠉⠓⠒⠶
⡶⠈⢀⠰⡆⠀⡘⠈⠇⠀⠸⠇⠀⠁⠀⠛⠀⠀⠀⠰⠇⠁⠋⠀⠈⠁⠀⠀⡞⠀⡿⠀⢀⡴⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠛⠓⠀⠀⠀⠀⠒⠂⠀⠀⠰⠶⠦⠄⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⣀⣇⠀⠀⣤⠀⠀⠀⠀⠀⠢
⠀⠀⠀⡀⢠⡀⠀⣄⠀⠀⣠⠄⠀⣄⠀⠀⢠⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⢰⠇⠀⠀⠀⡐⠏⠅⠀⡀⠁⠀⠀⠀⠀⠀⠙⠃
⠘⠇⠀⢀⠀⠁⠁⣬⣄⣀⠀⠴⣦⠀⠀⠀⠀⢀⣈⣤⡁⠀⠙⢳⡄⠀⠀⠀⠠⢀⣼⠋⠀⡼⠀⠀⢵⠰⠂⡀⣀⠀⠀⠀⠀⠶
⡃⠐⠻⠿⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁⠀⠀⣈⠀⠿⠴⠄⢀⠈⠁⣠⡼⠃⠀⠀⡀⠀⠂⠀⠈
⠷⠀⠀⠀⠠⣶⠶⣾⣦⠀⠀⠀⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠈⢉⠶⢖⠒⠀⠀⠀⠀⠩⣦⠀⢠⡈⠁⠀⠀⠀⠀⢤
⣤⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⡀⣄⠀⠀⠉⠋⠱⠄⠠⠄⠸⣷⡙⠂⠈⠰⠆⠉⠀⠃⠀⠀⠒⠈⠁
⠀⠘⠓⠀⠀⠀⠀⠀⠽⠀⠀⠈⠠⠀⠈⠓⠀⢀⡀⠤⠄⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠶⢀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⣡⡞⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⣀⠐⠆⠀⣤⢀⣂⠀⠀⠀⠀⢈⠁⠀⠘⠓⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠲⠀⣠⣬⡿⠿⠿⠷⠤⠤⠶⠄⠀⠠⣤⣦⠉⣤⢀⡉⣤⣼⡁⠀⠀⠀⣤⣤⣶⣦⣀⣴⣤⡟⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢠⡮⢐⠉⣈⠠⠆⢶⣶⣾⣿⣿⣿⠏⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘⢡⡼⠇⣅⠾⣶⣽⣟⣿⡻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠡⢂⣀⣵⣿⡟⠀⠀⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠟⢿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡈⠃⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠂⣿⣶⣀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢴⣶⡶⠃⢶⣿⣳⡀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢱⣿⡇⠃⠀⣿⣿⣿⣿⡀⠸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢛⡟⣱⣥⡟⣿⠀⠘⠻⡿⣿⣟⣿⡆⢹⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⣼⠟⢚⣻⠞⣐⣿⣿⣷⣿⠀⢀⢰⣏⢹⣿⣿⣧⡌⢿⣿⣿⣝⣿⣶⣄⡀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⠁⠁⠀⢠⣚⠧⠼⠩⢈⢋⣵⣶⣿⣿⣿⣿⡇⠀⠈⣾⣿⡆⣿⣿⣝⡄⠸⣿⣯⣿⣧⣍⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡴⣺⢶⡛⠃⡀⢈⡅⣵⣦⣶⣼⣯⣿⡟⠁⣶⠏⣸⠟⢳⣷⡘⢿⣿⣿⣦⠈⢿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⣀⣤⣴⡶⠀⢃⡀⠉⠊⢉⣀⠀⠌⠁⢸⣧⣿⣿⣿⣡⡿⣿⣥⣾⢃⠀⠋⠀⢸⡎⠹⢶⣝⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⡛⠛⠿⠶
⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠲⠤⠀⠠⠺⡿⢛⢡⣖⡥⣪⣿⣏⠐⠙⢫⢋⡆⠀⠈⠀⠀⠁⠀⠺⠻⣧⠹⣿⡝⠳⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⡻⣍
⠀⠀⠀⠀⠈⠙⠀⠐⠀⢀⣀⢠⣤⣤⢄⠠⣽⣫⡽⠿⠛⠍⠀⠀⠀⠀⢸⢠⠇⠀⠠⠀⠈⠇⠀⠐⣿⣆⠘⢷⡄⠈⠃⠈⡻⠿⠿⣿⣿⡟⠻⠿⠮⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠖⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡄⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠀⠀⠀⢀⠀⠀⢰⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⡀⠀⣀⠤⠄⠉⠁⠀⢀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⣿⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠋⢁⣴⣿⣿⣿⢿⣻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⣦⡀⠀⠀⠀⢀⣤⣷⣾⣷⠆⣼⣿⣿⡿⠋⢣⣯⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⣠⣿⣿⢆⡀⠀⠙⠛⠂⠀⠛⣨⡿⣯⡟⣡⣞⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢿⣠⡤⡤⣤⡤⠤⠤⠤⡜⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠙⣦⡀⡆⠀⠊⠻⢛⣿⣿⢟⣵⡿⣹⠿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⢁⡤⠀⠀⠀⠀⠀⣀⣀⣤⣤⣦⣿⣷⡅⢸⠆⠰⢏⡱⣆⠞⣩⣺⣫⣦⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢠⣾⣿⠀⣤⡖⡼⡻⢻⡏⣻⠻⢻⣿⣧⣃⣸⣤⣶⡯⠖⠃⠀⠀⠀⠀⣠⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠁⠈⢛⠻⣻⣿⣿⠋⢘⡿⠁⠀⠀⠡⢶⣾⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⡰⠞⣫⠞⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣉⣢⣴⣾⣟⣥⣿⣿⣿⣤⣦⣤⢠⣤⣦⡶⡶⠿⢿⣿⣿⣿⣷⣔⡄⠐⣱⣴⣷⣽⣞⠁⢐⢰⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠈⣷⢾⣿⡍⠉⠙⢛⠉⠉⠀⠀⣲⠦⣤⣬⣭⣍⣉⡁⢄⠠⠄⢈⡿⠿⠷⠶⠛⡋⠉⠁⣀⣀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠄⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠐⠀⠀⠑⠀⠂⠀⠊⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡌⠐⠄⠀⠀⠀⠀⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠠⠀⠨⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠐⠻⠇
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠰⠁
⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠉⠈⠀⠈⠡⡀⠈⠈⠨⠈⠉⠩⠡⠀⠀⠀⠀⠂⠀⠀⡠⠀⣠⣀⠀⡀⠀⢠⠄⡍⠉⢉⠩⢤⠀⣠⢠⢠⠄⢠⣆⡀⢠⡇⠆
⠅⠀⠀⠀⡀⡀⠀⠀⠀⠈⠶⠃⠄⠤⠄⠀⠉⡯⢀⠉⡋⡉⢀⡙⡐⠚⠚⢀⠂⠐⠆⠆⠠⠄⠄⡈⡥⣄⡀⠀⢸⠟⠀⣿⠁
⠁⠠⢠⡤⠱⠘⠯⠠⠭⠤⠠⢈⢃⠈⠙⣙⡂⠒⠖⠂⠀⠄⠤⠠⢡⢀⠁⡈⢁⡀⠘⠃⠢⠂⠰⠂⡤⡈⢣⠀⠈⠀⢸⡏⠀⠀⡔
⠀⠶⠰⠂⠐⠔⠃⠒⠀⠒⠐⠂⠒⠒⠀⠆⠤⠁⠍⠬⠅⡍⠀⣀⢀⣀⠤⠡⠌⠁⠃⡊⢛⡇⠘⢀⡄⢰⠊⠀⠀⠀⣿⠃⠀⣼⠁
⠐⠄⠾⠃⠘⠈⠀⠀⠀⠉⠙⠃⠀⠛⠈⠃⠐⠚⠀⠲⠂⠆⠠⠩⢤⡄⢙⠀⠃⠐⠐⠇⠍⠛⣛⣏⠁⠏⠀⡀⠀⢸⠃⠀⢠⡇
⡀⢰⠚⠆⠘⠀⠛⡀⠃⠀⠻⠓⠈⠰⠤⠿⠰⠃⠀⠇⠗⠰⠖⠠⡠⢀⡄⡃⡃⡓⠐⠠⠾⠄⠉⠁⠀⠀⣼⠁⠀⡟⠀⠀⢸⠁
⠀⠀⣦⢠⠀⡆⠲⣦⢀⡆⠴⠀⠠⡶⠀⡆⠀⣆⢀⡄⡀⠀⠀⠄⡀⠀⣨⠉⠁⠉⠀⠀⠠⠀⡀⣀⣲⣿⣧⣄⣼⠃⠀⠀⣸
⠋⠀⠉⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⠀⠁⠀⡉⡈⠀⠁⠀⠀⠀⠃⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠉⠉⠑⠒⠲⠾⣀⣀
⡄⠀⠀⣅⠀⠠⠰⠄⠀⡧⠀⠇⠀⠞⠉⠋⠀⡏⠃⠇⠀⠛⠀⠂⡼⢠⡇⠀⣠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠
⠁⠀⠀⠈⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⠤⠀⠀⠠⢤⠄⠀⠀⠀⠁⡼⠃⢠⠃⠀⠀⢀⣹⠀⠀⣠⠀⠀⠀⠀⠠
⠀⠀⢀⠀⡄⠄⡄⠀⠀⠤⠀⢤⠀⠀⠠⠀⡀⡀⠀⡀⠀⠀⠀⠐⠀⠀⠀⡟⠀⠀⠀⡘⠉⠀⠀⠁⠀⠀⠀⠀⠉⠃
⠈⠀⣀⠀⠈⠀⠼⢤⠀⠐⠶⠀⠀⠀⠀⠤⢤⡄⠀⠉⠳⢀⡀⠀⠐⣠⠟⠀⡰⠁⠀⠖⠂⠀⢀⠀⠀⠀⠐⠂
⡅⠈⠙⠛⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢤⢄⢙⣛⠃⠀⠀⢰⡾⠁⠀⠠⠀⠈
⠉⠀⠀⠀⠙⠋⠛⠃⠀⠀⠀⠀⠀⢴⠶⡀⠀⠀⢠⡀⣌⠉⢄⡀⠄⠠⠀⠙⣀⠐⠆⠀⠀⠀⠀⠉
⠉⠴⠄⠁⢀⠀⠐⣆⠀⠐⢘⡂⠰⠄⠀⠁⠀⡀⠌⠛⠀⠀⠈⠉⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠛⠳⢤⣀⡀⣀⣠⡴⠞⠛⠉⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠚⠛⠙⠛⠶⢤⣤⣤⣤⡤⠶⠛⠉⠉⠉⠉⠳⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠈⣧⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⡾⠀⠀⠀⠀⠀⡞⠁⠀⠉⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⣀⣤⣤⣬⣧⠀⠀⠀⠀⠀⢸⡇⢀⣀⠤⢤⣀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣧⣤⣤⣀
⣀⣤⠶⠚⠋⠁⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠈⠳⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠙⠓⠶⣤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣼⠁⠀⠀⢀⡀⠀⠀⠹⡄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⢠⠞⠉⠉⠳⡄⠀⢻⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⢀⡏⠀⠀⠀⠀⢻⠀⠘⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⣧⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡟⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣧⠀⣸⠀⠀⠀⠀⠀⠈⣇⠀⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣿⣰⠃⠀⠀⠀⠀⠀⠀⠹⣄⣿⣰⠏
⣿⡿⠛⠟⢿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣾⣷⣷⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠠⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⢀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⢰
⠖⠒⠲⠶⠶⠶⠶⢶⢤⡔⠒⠶⠒⠂⢀⣠⡠⠰⠴⠤⠶⠖⢒⠒⠒⠒⠚⠛⠛⢛⠙⠛⠻⠋⠉⠉⠉⠉⠉⠉⠩⠉⢠
⠤⠀⠀⠀⠀⠀⠁⠐⠀⠂⠐⠀⠀⠠⠄⠄⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠐⠄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠒⢒⣒⣉⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠘⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠄⡀⣄⠐⡌⠉⠉⠑⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠐⠆⠀⠄⠈⠑⠈⠀⠈⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠤⠤⠔⠒⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠿⣦⡀⢠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠎⠈⠁⠀⠀⠂⡀⣀⠀⠀⢀⢻⠀⠀⠱⠈⢏⡳⠤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢴⠀⡰⠃⠀⢁⠊⠕⠁⠀⢠⠋⠀⠆⠀⠀⠐⠀⠀⠷⢀⠄⢡⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⣁⢀⠔⠇⠐⠂⠀⠀⡸⠀⠀⠃⠀⠀⣴⣄⠢⠠⠤⠐⠤⣠⡔⠢
⠀⠀⠀⢀⠐⠁⠀⠀⠀⣠⠋⠀⡁⠀⡘⣀⠀⠀⢠⠄⢸⠀⠀⠐⠂⡰⠓⣌⢢⠐⠶⠀⠀⠢⠀⣀⡤⣈⠶⢤⡀
⠀⠀⠀⠀⠃⠀⢀⠐⡈⢀⡰⣊⠀⠈⠉⠉⢀⡴⠁⠂⠘⠀⠀⠀⠀⠤⠀⠊⢸⠀⠰⠄⠀⠀⠀⠈⠁⠀⠀⠈⣙⢦⡁
⠀⠀⢠⠀⠀⠀⠀⢀⠡⠂⠀⠁⠀⠀⠀⣠⠎⠐⠆⠀⢑⠀⠀⠀⠀⠀⠀⠀⢬⠄⠀⠀⠀⣠⠀⠀⠀⠈⠓⠀⠀⡀⠍
⠀⠀⠂⡠⣤⣀⠈⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⣸⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠼⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⣆⡠⣄⢠⠀⠀⠅⢠⡄⠀⡀⠐⢀⠎⠁⠓⠤⣀⠂⢀⡰⠚⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠄⢀⡠⠤⠤⠥⠀⠂⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⡴⠊⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡈⠀⢉⠀⠀⠀⣰⠀⠀⠀⠀⠀⠞⢂⣶⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠢⡀
⢀⠎⠀⣁⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢚⠠⠒⠀⠘⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⠈⠘⢤⠈⡃⢳⠢⡌⢆
⠘⢤⡀⠋⠀⠀⠐⢰⡤⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⢀⠆⠀⠀⠀⢰⠀⠀⠀⠀⡜⡄⠀⠀⢃⠀⠀⠆⠀⠀⡅⡄⠃⠀⡀⠐⠈
⠀⠀⠈⠒⢄⡰⠐⠉⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠄⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠰⢰⠀⢀⠀⠀⢀⠀⠈⠄⠀⠁⢦⠀⠊⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⢠⢁⡆⠀⠇⠀⠀⢀⡄⠀⠀⡤⠀⠀⠀⢠
⠀⠶⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠚⠈⡆⡖⠁⠂⠈⠂⠄⡆⠨⠀⠀⠀⠀⠀⠀⠀⠎⠂⣀⠜
⠠⠂⠀⠀⢤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠈⢆⠀⠀⠀⠀⠘⠀⠀⠀⢠⡇⠘⡄⠈⠀⠀⠀⢁⠃⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠑⠆⡀⠁⠀⢀⡀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠂⠃⠀⠀⡘⠀⠀⠘⡀⠆
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠱⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⢸⠄⠀⡀⠀⠃⢸⢰⠀⠀⠀⠀⠹⡞⠀⠀⠀⠀⠀⠠⠷⠀⠉⢩⠈⠉⢆
⠀⠀⣡⠆⠹⢂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢠⡆⢸⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠙⠀⠀⠈⡄
⢀⠄⠁⡆⠰⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠀⠀⡅⠀⢈⠁⢸⠀⠀⠃⢠⠤⠤⢀⡄⠀⠀⠀⠀⡌⢂⠀⠀⠀⠀⡜⡅
⠀⠀⠀⠁⠐⢂⠀⠃⠀⠀⠀⠀⠀⡆⠀⠀⠠⡔⠀⡆⠀⠀⢃⠀⢸⠀⡘⠀⡌⡀⣸⠤⠒⠀⠃⢡⠀⠀⠀⠃⠈⢇⠀⠀⡘⢀
⠀⠀⡠⠂⠀⠀⠢⠘⠀⠀⠀⠀⠀⢆⠀⠀⢀⠀⢠⠃⠀⠀⠈⠀⢸⢀⠃⠀⠀⡀⠀⠀⡄⠀⠇⡄⠀⠀⢠⠀⠀⠈⠆⡐⠀⠸
⠀⢃⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠈⠄⠀⠈⢀⠂⠀⠀⠀⠀⠂⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠰⡆⢈⠐⠇⠅
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠠⢠⠃⠀⠘⣇⠀⡠⠀⡀⢀⡀⢐⠀⠀⠀⢂⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⢤⡀⣆⡀⠔⠊⠀⠀⠀⠈⠒⠠⠦⠚
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⠄⣶⡿⠂⠀⠀⠀⠀⢀⢿⣿⣿⣷⣄⠀⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠘⠁⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣷⣄⢻⣷⣄⠀⠀⣀
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢰⠇⠀⡆⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⣿⣿⣿⣦⡀⠉⠿⣦⣈⢳⣤⡀
⠀⠀⠀⠀⢸⣗⠀⠀⡄⠀⠁⠀⢸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠻⢧⡙⣿⣶⣄⡀⠀⢀⣀⣤⣤⣄
⠀⠀⡄⠀⣹⣿⡀⢠⣷⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⢻⣿⡿⠿⠿⠟⠛⠛⠛⠛⠻⠶⠀⠉⠀⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⣸⣷⠀⣿⣿⠅⢾⣿⡆⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⡖⣿⣿⢖⣿⣿⣗⠀⢠⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣦⣤⡀⠀⢀⣤⣴⡿⠂⠉⢻
⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣤⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⣿⣿⡟⢊⠥⠤⠎⢘⣿⣇
⣿⣿⣯⣝⣻⡿⢻⣿⣿⣿⣶⣾⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠁⠀⣀⣤⣤⣿⣿⣶⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⡶⠦⠤⠀⠀⠀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⡤⠀⠀⠀⠀⠀⠀⡀⠀⢠
⡄⠀⠀⠀⠈⠀⠀⠀⣅⣀⣀⡠⢀⠤⡤⠵⢤⠶⠄⢶⢒⡴⡄⣐⢂⡂⡂⡀⠐⠀⠆⠀⠋⠀⣧⠂⣾⠈
⠃⡀⣀⢐⡴⣦⢂⣂⠈⣐⠆⠘⠬⠄⠩⠬⠈⠈⢈⢁⣂⠐⠐⠒⠀⠰⠤⡥⢁⡌⡉⠻⡀⠘⠁⢸⠇⠀⣠
⠀⡄⠔⠠⠴⠒⠆⠐⠐⠂⠘⠂⠠⠡⠅⠥⢴⢀⠒⣀⢀⣤⠌⢈⠁⠃⠒⣒⠐⢀⡈⡶⠁⠀⢀⣿⠁⢰⠇
⠐⠐⠛⠐⠈⢃⢀⠈⢉⡃⢀⠃⠙⠐⡛⠀⠒⠰⠐⠫⢤⠈⢁⢈⡐⠘⠉⠋⣹⡯⠘⠃⣠⠂⣼⠃⠀⣾
⠄⠛⠙⠀⠃⢁⠈⠁⢘⠉⠀⠘⠚⠀⠁⠘⠘⠁⠋⠐⠀⠰⢠⡄⠬⠌⠘⠃⠀⠀⣀⣴⠇⢀⡇⠀⠀⡇⡄
⡤⠘⠇⠃⠳⠉⠃⠸⠈⠁⠈⠃⠸⠀⠻⠰⠱⠀⠀⠃⠆⡰⠂⠀⠀⠀⠀⠀⠒⠒⠛⠿⠳⠾⢥⣀⣐⡇
⡀⠀⢐⠀⢀⢠⠁⢰⠀⠆⠀⡖⠶⠀⡖⠦⠆⠰⠆⠄⡤⢸⠁⢀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠐⠀⠠⠀⡉⠙
⠁⠀⠈⠃⠀⠀⠀⠤⠴⠄⠀⠀⠠⠀⠁⠠⣤⠀⠀⠀⠁⡞⢀⡎⠀⠀⣈⡇⠀⣀⠀⠀⠀⠠
⢀⠀⠠⢤⠀⠄⠀⠰⠂⠢⠀⠀⠄⠄⠀⢀⣀⠀⠀⠐⠀⢀⡸⠁⡀⢠⠛⠀⠀⠀⠀⠀⠀⢈⠁
⠌⣠⣴⠄⠀⠚⠒⠀⠙⠂⠀⠀⠐⠒⠂⠀⠈⢣⣤⡀⠑⠞⢁⡜⠀⠈⠋⠀⠐⠂⠀⠀⠉
⠇⠀⠀⠠⠶⣶⠄⠀⠀⠀⠀⣀⡁⠀⠀⢀⢛⠶⡖⠃⡀⠀⠻⠄⢠⡈⠀⠀⠀⠤
⠂⣠⠐⠀⠀⠠⡄⠀⠔⠀⢨⡉⠘⠀⠀⠘⠮⠁⠀⠃⠀⠀⠄⠈⠁
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠂⠀⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠙⣷⡎⠀⠈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⠀⠀⠀⡠⠤⠽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣶⣾⣄⡀⢀⣸⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠶⠚⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⢀⣴⣦⢀⡞⠀⠠⠤⡀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠄⢀⡴⢋⡀⣿⣼⠃⣴⣄⠀⢈⡀⠄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣆⠀⢘⣚⡭⠚⠻⠄⣙⠋⡘⠉⢻⣷⣿⠿⠶⠤⢀⡀⠹⠷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠀⢠⡿⠛⣿⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⡇⠀⠀⠀⠀⠈⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⢀⠜⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷
⠀⠀⠀⠓⠒⠛⠉⠳⣤⣴⣿⣦⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣀⡀⢀⣠⣤⣴⣶⣿⣷⣦⣴⠿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡤⡄
⠀⠀⠀⡠⠖⠁⠀⢀⣀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠇⠐⢌⠘⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⢷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⢀⠉⠐⢄
⠀⢀⠎⠁⢄⠀⠀⠜⠸⠀⠀⠈⣢⠀⠀⠀⠀⠀⠀⠀⢨⠀⢘⡀⠀⠱⡀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢰⠛⢆⠀⠀⠀⠀⠀⠀⡔⢇⠀⠀⢦⢈⣄⡀⠱⣄
⢠⠎⠀⢀⣦⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠐⠉⠁⠠⡀⠃⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⡰⡉⠀⡄⢆⠀⠀⠀⢀⠌⠀⠀⢱⠈⢠⠀⠸⠉⢆⠈⡄
⠈⠲⢄⠈⠉⠁⠀⢀⠋⣴⣶⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠀⠀⢈⠾⠀⠀⠘⠀⠈⡇⠀⠀⠀⠀⢰⢁⠇⠀⢀⠈⢆⠀⠀⢩⠀⠀⠀⡆⢰⠘⠀⠀⡀⠀⠂⠸
⠀⠀⠀⠙⠢⣄⢀⡪⠚⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣀⠄⠂⠁⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠆⡜⠀⢀⠀⠀⠘⡄⠀⠀⢃⠀⠸⠁⢰⢄⠔⠊⠍⠑⢄⠠
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡆⠀⠀⡞⢠⠃⠀⡎⠁⡀⠀⠘⡀⠀⠀⢆⣀⠠⠀⠀⠀⢐⠀⠀⠀⠙
⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⢰⡆⢳⠀⣜⡠⢺⠐⠤⣀⠀⢣⠀⠰⠃⠀⠀⠈⠇⠀⠀⠀⢀⠜⡀⠀⠀⡴
⠀⠛⠁⠀⠠⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢏⠀⠈⡀⠀⠀⠑⢺⠂⡌⠀⠀⠀⠀⠈⠲⠀⠀⠈⠂⠈⠱⠎
⠐⡎⠀⠀⠀⠶⠀⠄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⡖⠂⠌⠱⡀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢸⠀⠈⢇⠀⢡⠀⠀⠀⠆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠃
⠀⠘⠦⣒⠀⠁⠀⠀⠀⣀⠰⠂⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠡⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠀⠀⠈⢆⠈⡄⠀⢠⠠⠁
⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢡⠀⠀⠸⢠⠀⠀⠀⠈⢆⠂⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢼⠀⠀⢢⠀⠀⠀⠀⠈⡄⠀⡇⢸⠀⠀⠀⠀⠈⢶⢰⠇⠀⠀⠀⠀⠀⠀⢠⢄⠀⡤⠶⠤⠷⠧⡄
⠀⠀⢾⠁⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡄⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢡⢨⠃⣿⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠘⠊⠌⠀⠠⣿⠀⠀⠘⡄
⠀⠀⢀⠔⠁⢸⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⢱⠀⠀⢸⠀⠀⠸⡾⠀⠿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠁⠀⠀⠀⣸
⢀⡤⠁⠰⡄⠀⡆⠘⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⢀⠂⡆⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠛⠀⣤⠤⠤⠀⣤⠀⠀⠀⠀⠀⠀⡏⢧⠀⠀⠀⠀⠀⣰⢻
⢸⠀⠀⠘⠀⢀⢇⠀⠈⡆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢡⢀⠂⠀⠃⠀⠀⠸⠀⠀⢰⠃⠀⡇⠀⡔⠉⠀⣿⢀⠥⡀⢹⠀⢆⠀⠀⠀⢠⠁⠈⢇⠀⠀⠀⢠⠃
⠘⠀⠀⢀⠔⠁⠈⢢⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⠇⠀⢸⠀⢸⠀⠀⡏⠈⠐⠊⠀⠀⠀⢱⠈⠉⠀⠀⠀⢸⠀⠀⠈⠆⠀⢠⠃⠀⠇
⠀⣆⠔⠁⠀⠀⠀⠀⠣⣉⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢰⠀⠀⠞⠀⠀⠀⠀⠘⠀⢸⠀⡇⠀⠀⠃⠴⠀⠀⠀⠇⠀⠘⢰⠀⠀⠀⠀⡜⠀⠀⡀⠈⢄⠂⡄⢠
⠀⠙⡀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢰⠀⡌⠀⠀⠀⠀⠀⠀⠁⢨⢠⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢯⠀⠀⠧⠀⠬⠘⠃⡜
⠀⠀⠘⠄⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡀⡸⠀⠀⠀⠀⠀⠀⠀⢠⢐⡜⠀⠀⠸⡄⠀⢀⡄⣀⣀⣀⣀⠀⢐⠀⠀⠀⠈⡆⠀⠀⠀⡆⠀⡜
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⠳⢎⡀⠀⣥⠀⢈⣢⠎⠀⠀⠀⠀⠘⠤⢄⣠⣁⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣆⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠿⠀⠙⠉⠻⠶⠟⢳⣤⡀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠟⠀⢀⠀⠀⠀⠀⣠⣦⠀⠸⢧⣤⠶⠟⣻⢄⣴⡄⣠⠾⣆⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⡟⠙⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⢼⠿⠞⠁⠛⠃⠀⠙⠉⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣁⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠶⠂⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠤⢤⢤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠀⠀⠤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢳⣤⣷⣤⣤⣴⠋⢀⣤⣀⠙⣧⣤⣤⣤⣤⡾⠉⣠⣤⡈⢻⣦⣤⣤⣾⣴⠞⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⣀⣀⣠⣿⠀⢿⣤⡿⠀⣿⠀⣀⣀⣸⡇⠸⣧⣼⠇⢠⡟⠀⣀⣀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠁⠉⠙⢷⣤⣀⣤⡾⠋⠀⠀⠁⠉⠻⢦⣄⣀⣤⠞⠁⠀⠈⠁⠉⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⡤⢤⠀⢠⡀⡇⠀⡿⡄⡰⡇⣤⢠⠤⡀⡅⡤⢤⢠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠂⠳⠚⠘⠝⠃⠃⠀⠃⠙⠃⠃⠛⠘⠀⠃⠃⠃⠘⢸⢛⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡠⢔⡟⠉⠓⠲⠤⣀⡀⠀⢀⣠⡴⠶⠖⠒⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⢀⠤⡄⢀⡴⠛⠃⠉⠢⠤⠔⠉⠀⠀⠀⠀⣀⣠⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠠⡙⡜⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⡀
⠀⠀⠀⠀⠆⢱⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠰⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⡜⠀⠁⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⠀⠀⢀⣀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⣀⢱⡜⠁⣀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠱⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇⠘⢿⡶⣺⡀⢸⣿⣭⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡠⠃⣠⠤⠎⢷⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠉⠣⣀⣉⣡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⢠⠎⣠⠊⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇
⡰⠃⣞⣁⣀⣀⠤⠔⠚⣻⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡇
⠳⠤⠤⠤⠶⠒⠒⠉⠉⢸⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢉⣁⠀⠀⠙⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⢣⠀⠀⠀⠀⠀⠀⣾⠁⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⢠⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠀⠘⡇⠀⣸⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣴⠞⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⢠⡇⠀⢹⠀⠀⠀⠱⡜⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠟⠁⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠞⠋⠀⠈⠙⢷⡀⠀⢀⡾⠀⠀⢸⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⠶⠛⠀⠀⠀⠀⠧⠤⣀⣀⠤⣤⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣶⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⠖⠛⡟⠻⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣖⣒⣚⣓⠚⣲⢈⣲⣓⢀⣘⣆⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠁⠀⢤⠀⠀⠋⠤⠀⡄⠀⠀⠿⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⡉⠉⢉⣉⠉⠉⠉⣉⠋⠉⠉⠋⢉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠀⢠⡄⠀⠀⠤⠀⠀⠀⠠⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢹⠉⠉⠉⠉⡹⠉⠉⠉⠉⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣉⣱⣋⣉⣀⡀⣄⣉⣉⣉⣩⣏⡉⢉⣉⣸⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⣄⠡⠤⢧⣀⠀⡀⡠⠴⠄⠄⠤⠤⠦⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⢥⡤⠴⠤⠤⠄⠀⠤⠧⠄⠤⠄⠀⠐⠐⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⣲⠒⠚⠒⠒⡖⠒⠚⠂⠀⢲⠒⠐⠓⠒⢲⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢰⣋⡉⠁⠉⠁⡆⠒⠒⠓⠒⣀⡆⢀⡉⡉⣉⢉⡁
⠀⢀⣀⡠⣤⠶⠶⠛⡲⢄⡞⢲⢤⠖⡲⢾⢀⡭⣠⠗⣰⠞⢓⠎⡰⢖⠄
⣠⠜⡣⢂⠀⢀⠔⢊⡴⢋⡝⢕⢅⡪⢒⢕⡡⢊⡡⠊⡠⢔⠁⢊⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣭⣬⣯⣴⣥⣭⣴⣥⣬⣤⣭⣤⣬⣴⣥⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠀⠀⠉⠁⠀⠂⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠙
⠀⢀⢀⠠⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠄⠀⠠⠁
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⡊⢉⣹⣿⠛⢉⣉⡁⠈⢉⡉⠉⠉⣡⣿⠟⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣉⣷⠟⠉⠀⠀⠈⠙⢿⣄⠈⢈⣱⡾⠋⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣉⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣋⠵⡀⠀⠀⠀⠀⠀⠢⡀⠀⠀⡰⢆⣉⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⣰⠦⣾⣽⡤⣶⠀⠀⠀⠀⠀⠀⠈⢄⢐⡦⣼⣽⡦⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⢀⣏⠉⠛⡦⢤⣶⠤⡞⠃⠀⠀⠀⠀⠀⠀⠀⢻⡧⢤⣧⠤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⢻⣸⠶⡶⢿⣖⡒⣾⣤⣤⣄⣠⣄⣀⣀⠀⠀⠀⣫⡖⡒⣎⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠰⡀⡸⠁⣠⠠⠄⠤⢳⠨⡠⢋⠛⠉⢉⣁⣀⣷⠋⠄⠠⢡⡀⠘
⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠄⠇⠠⠧⠴⠀⠀⠀⠀⠀⠹⠀⡔⢰⠐⠒⠒⢺⡀⠁⠆⠀⠀⠀⢿⠇⢿⡆⠒⠒⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣨⣤⣀⣀⣼⠀⠂⠀⠀⠀⠀⢸⡇⣸⣇⣠⣠⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⠉⠁⠉⠉⠉⡀⠀⠀⠀⠀⠀⣼⡖⡍⠈⠉⠉⠉⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⢆⠀⠠⠀⠀⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡜⠀⠀⠀⠀⠀⣀⠼⡆⡈⡀⠀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠼⣈⠂⢠⠀⠀⠀⠀⢀⠈⢀⡇⢊⡹⠜⣁⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣮⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⢸⠉⠈⢹⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠠⠅⢒⠚⠢⠆⠚⢒⠂⠵⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠀⠰⠄⠀⠀⠀⠆⠀⠀⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡤⠤⠤⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⠀⠀⠀⠀⡴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⢀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⢰⠂⠀⠀⠀⠀⢣⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡴⠋⠀⢸⣷⡀⣠⠴⠂⡈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠁⠀⣀⠚⢿⣧⡀⠀⢠⠄⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠛⠦⣤⣤⢬⣷⡴⠀⠀⢠⡿⠄⡄⠠⠄⢠⡀⠠⠀⠠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠬⠤⠤⠥⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⢠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠁⠀⡠⠐⠀⠀⠴⠁⠀⠀⠀⣀⡀⣀⠀⡐⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢀⣄⠘⠢⠄⠀⠀⠀⢰⡁⠀⠀⠀⢰⠁⠀⠁⠛⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⢁⠂⠸⢦⠤⢤⡇⣀⠀⠀⢀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠈⠂⠌⠀⠂⠀⠀⠀⠀⣂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠰⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠒⠒⠲⢄⡀
⠀⠀⠀⠀⠄⠊⠀⠀⠀⣀⡽⠓⢝⣧⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡆⢤⡘⡧⣤⠀⠙⡄
⠀⠀⠀⠁⠀⠀⢰⡴⠚⡁⠀⠺⠓⣇⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢦⣍⢦⡑⡜⡰⠚⠀⠓⣄
⠀⠀⠀⠀⣠⠤⠚⢠⠊⠱⣤⣴⢀⡋⠠⠍⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠀⠲⠮⠲⣄⣟⡴⢏⡰⢢⡏
⠀⠀⠀⣰⠁⣀⡠⣣⠋⠳⠤⠬⠦⣏⡠⠳⣶⠲⠂⠀⠀⠀⠀⠀⠀⠀⠈⠦⡘⢆⣢⣔⢆⡿⠂⢜⡼⢃⠃
⠀⠀⢠⡧⢞⡡⢞⡕⠋⠂⠀⢀⣀⣏⡴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡵⠲⠭⣣⡷⢊⠟⠶⠛
⠀⠀⠀⠺⠛⢉⣉⣀⣀⢀⡤⠊⠀⡟⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣄⠀⠈⡟⠁
⠀⠀⠀⠀⠻⢿⣿⣿⢿⠋⣀⣤⣾⠱⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣟⡳⢤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠑⠛⠛⠛⠋⠩⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠀⠈⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠤⠟⣻⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡾⠟⣷⠽⢥⣌⡽⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⡤⡷⠁⣈⢃⡾⣮⣭⣿⣍⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠠⠥⠀⠘
⠀⠀⠀⠀⢠⡟⡸⡡⠀⣏⣾⡃⢠⣤⠬⢍⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⢠⣮⠞⠕⢀⣔⠝⣿⢋⣅⠢⡀⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠑⢬
⠀⠀⠀⠿⠃⠀⠴⠋⠀⠔⢿⣏⠔⠙⠲⠷⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⣜⢁⠜⠹⠘⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣀⡵⠥⠀⠀⠒⠠⢤⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠿⠾⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠖⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢠⣦⡴⣦⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠈⣧⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠸⡇⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣤⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠸⣇
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢹⡆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣹⠈⠃⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢿⡄
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⠔⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣷⡀
⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⣧
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠹⣇
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢻⡄
⠈⢿⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡿⣄⠀⠀⠈⢿⡀
⠀⠘⣧⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⠀⣸⠁⠘⢷⡀⠀⠘⣷
⠀⠀⠸⣆⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠙⣦⡀⠸⣇
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠳⣄⢹⣆
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⡄
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⡟⠀⢀⣠⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠙⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣧⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⣦⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠀⠀⠈⠳⣄⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⠶⣄⠀⠀⣧⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠈⠛⢦⣽⣄⠴⠞⠋⠁⠀⠀⢻⡆⠛⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠈⢧⡀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣠⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠳⡄⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⡈⠐⣤⡀⠀⠈⡒⠒⠛⠛⢠⠴⠶⠤⢤⣤⣤⠤⠤⡤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠔⠛⠛⠃⠴⢬⡍⠀⠤⣤⠀⠈⠸⠗⠀⢈⠀⠀⠤⣵⣄⣀⣆⠀⠀⠀⢨⣁⠷
⠀⠀⠀⢠⡖⠚⠉⠉⣑⡿⣽⣷⠀⠀⠀⠈⠛⠷⠄⠀⠤⣤⣤⡄⠈⠋⠙⣀⠀⠘⠛⠋⢃⣛⣆⠬⠟⠿⣇⣠⣆
⠀⠀⠀⠘⣟⣶⣤⡀⠉⡉⣉⣩⢴⣢⣄⣀⣉⣉⠙⡀⠀⣈⣁⣀⣁⣀⣼⣉⣰⣯⣠⣦⠼⢿⣟⣀⣴⡶⢻⡋⠀⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣟⢹⣿⠋⠿⣿⠻⡟⣻⣿⣽⣿⣿⣿⣩⣭⣙⣣⣽⣿⣿⡄⣿⣠⣶⣾⣿⢉⣡⠖⠻⡖⠋⡄
⠀⠀⠀⣼⣿⣿⣿⡿⠿⠿⣿⣿⠿⢟⡻⠟⣟⣛⣛⣻⣉⣛⣻⣛⣛⣻⣛⣛⣛⣻⣿⢟⢁⠼⣿⣹⡥⠴⢂⣱⣡⡶⠄
⠀⠀⢠⡿⣿⣿⣿⢿⢿⡷⠿⢾⠿⠿⠿⠿⢯⡭⢿⡟⠛⢟⣚⠛⢻⠿⠻⡽⣿⣿⣿⣯⢀⠀⣷⣋⣠⣾⡟⣉⣡⣴⣶
⠀⠀⢿⣿⣿⣿⣷⣶⣚⣓⣒⣚⣓⣛⣛⣟⣻⣿⣭⣯⣿⣯⣭⣭⣭⣭⣿⣽⣿⣯⣿⣿⡾⠟⠉⣿⣭⣶⣿⣿⠃⠫⡁⠀⡀
⠀⠀⠈⢿⣿⣿⠿⡿⠟⠟⠻⠛⢛⡛⣋⣩⣩⣭⣍⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣯⣿⣷⣼⣿⠿⢿⡋⢳⡈⠂⣠⠟⠃⣹⡶
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠿⠿⢿⡿⢿⡿⠛⠛⠿⠿⠛⠻⠛⠻⣿⣿⣛⣿⣿⣿⣿⡤⣺⢟⣛⡋⠭⣄⣴⣿⡿⠃
⠀⠀⠀⢠⣿⣿⣧⢤⣤⣼⣯⣤⣿⣷⡾⠶⢶⡶⠾⢿⠓⠛⢛⡻⠟⢿⣿⣛⡟⣯⣙⣽⣯⣽⣿⣷⣤⣾⣿⣿⣿⡿⢿⡻⠁
⠀⠀⠀⠘⣿⣿⣶⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠭⠑⢀⣴⡞
⠀⠀⠀⠀⠘⣿⣿⣿⠛⡏⡍⠉⢋⠉⣩⠉⠙⠉⠉⢏⣩⡉⠹⣭⠉⡉⠉⠄⠀⠐⢿⣿⣿⣿⣧⣤⣿⣿⡶⢟⡟⣡⠃
⠀⠀⠀⠀⠀⠈⣿⡷⠶⠷⠾⠶⠾⠿⠿⠶⠶⠗⣿⠶⠳⠷⢾⠶⢛⣻⠒⠺⠟⠛⠛⣿⣿⢿⣧⠀⣤⣭⣴⣾⣾⡟
⠀⠀⠀⠀⠀⠀⣽⣿⣶⣴⣧⣭⣭⣽⣶⣤⣶⣬⣥⣤⣶⣶⣾⣷⣶⣶⣦⣶⣶⣶⣶⣾⣿⣿⣿⣿⠟⠛⠟⣋⡼⠁
⠀⠀⠀⠀⠀⠀⢾⣿⡿⢿⣿⣿⣿⣟⣛⣻⣿⠿⠿⡿⠿⠿⠿⠿⠟⠟⠿⡛⠟⠛⣛⣿⡻⠿⠗⠓⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠙⠛⠛⠓⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣰⢷⡔⢮⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃⡼⠃⠀⠉⠈⠻⣿⣿⣿⣦⣀⠀⢀⣤⠾⠫⠞⠉⠁⣙⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠞⠀⠈⠀⠀⠀⠀⠀⠀⠙⠏⢿⡙⢿⡿⣟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡿⣶⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠉⠙⠳⠦⣀
⠀⠀⠀⠀⠐⠁⠀⢠⣇⣸⣧⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣀⣀⠀⠀⣾⠀⣀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⢀⡠⠴⣚⣫⡭⠥⠶⠶⢒⣲⣶⡶⠶⠶⠶⠦⠤⣤⣀⡀⠈⠉⠙⠛⠻⠿⢷⣿⣄⣀⣼⣿⣼⡇⠀⠀⣸⡄⣾⡆⢀⣀⠤⠂
⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠀⠀⠀⠠⢄⣀⡉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠋⠉

⠀⠀⠀⠀⣀⠤⠤⠤⠶⠶⠶⠶⠶⠦⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⢰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣙⣛⠒⠶⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠈⠛⡮⢭⡵⣖⣒⣒⣒⣶⣶⣶⣒⣒⡯⡽⢶⠒⠒⠫⠭⠝⣛⣒⠶⠦⠭⣭⣍⣛⣛⣒⠲⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠖⠚⢛⣧
⠀⠀⠀⣀⢀⠗⠘⠇⣰⣧⡆⠉⠉⠙⣦⠀⣗⠉⠙⢺⣤⠴⠖⠒⠶⠦⣄⡉⠉⠙⠒⠒⠢⠭⠭⢭⣉⣛⣛⡛⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⢉⡟
⠀⢠⠋⠀⠸⣄⠀⡼⠁⢀⠀⢀⠀⢰⣿⠀⠈⡀⠒⢬⡇⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣤⣤⡤⠶⠋
⠀⢧⡀⠀⠀⠀⣿⡇⠀⠀⢸⡁⠀⠀⠉⠃⠈⢵⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣸⠿⣶⣤⣤⡼⠿⣆⣀⣀⠽⢶⠀⠀⠀⢀⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⢀⣠⠤⠊⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣷⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠘⠧⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢲⣶⣤⣤⣀⣀⢀⣀⣀⡴⠀⢀⣽⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠂⢼⣿⣿⠽⣿⣍⠁⠀⠛⢻⣶⣿⣁⡸⠿⣿⣇⣸⣷⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⡤⢤⣤⣂⠈⣿⡴⠞⢻⢿⡋⣠⣷⠀⣀⣤⣤⣴⢿⣿⡙⠁⣉⣻⣿⣃⡀⢀⠈⠉⠉⠙⠿⠷⣶⣤⣤
⠀⠀⠀⣀⡤⢧⠀⠹⣏⡿⠋⠀⠀⠀⠈⠠⠟⢛⣿⠟⠋⠁⣸⡇⢸⣿⣯⣿⢛⣿⡋⣗⣮⡀⠀⠀⠀⠈⢐⣸⣿⣿⡇
⠀⠀⠘⣷⣄⣼⣄⠀⢈⠣⣤⣤⣀⣀⣠⣤⡎⠉⠘⢹⣦⣐⣻⣿⠛⠉⠙⠻⣄⠹⣿⢿⣀⣟⣦⡖⠾⣯⡿⠦⡌⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⣼⡇⢸⢭⡿⣧⡀⠀⠀⠀⢸⡟⣛⣿⣄⣀⣀⣤⣾⡿⠿⣶⡾⠻⢾⡄⢀⢨⣇⠀⠀⠛⢿⡇
⠀⠀⢠⣿⣧⣍⣻⣿⡶⠚⠮⣶⡎⣿⠀⠈⢿⣦⢤⡄⠊⣿⠟⢿⣿⣅⡷⠛⠁⠀⣸⣟⣷⡈⠱⠌⠓⢾⣽⠀⠀⢠⣿⡇
⠀⠀⠉⠉⠹⣿⠿⢧⣦⠠⢤⣌⣹⣿⣦⣀⡸⠟⠚⠁⡐⠉⣠⣸⠛⢧⣄⠀⠀⢠⣧⣄⣩⣿⣤⣤⡶⠾⢿⠀⢰⠷⠿⠿⣶⡀
⠀⢠⣴⡞⠉⠛⢧⣞⣹⡄⣠⡉⣿⠛⠮⡟⠀⠀⠀⠀⠈⠀⣹⣅⠂⢀⢹⣃⣠⠋⠉⣾⡁⠉⠀⠉⡗⣤⣺⣤⢿⢙⣦⣠⠞
⠀⠀⠹⣧⠀⠀⠀⢀⣩⢷⡛⠛⣦⠑⢾⣥⡀⡤⣤⠤⡴⢻⣿⡏⢹⣹⡿⠉⠙⣿⡷⠛⠙⣦⡀⣼⡉⠁⣀⣞⢸⣿⣟⠁
⠀⠀⣾⢭⣆⢀⢴⡹⠏⠈⡹⡶⡳⣤⣤⡻⣷⡁⠘⣷⠗⠛⢻⠇⠂⢻⣳⣄⣡⣬⣆⡂⣀⣿⡽⠷⠿⠽⠛⠚⢛⣿⣋⣈⣭⣦⡄
⠀⠀⢃⠀⢈⣸⣼⣷⠄⣰⣿⣂⣷⡎⠀⠙⣿⡙⢦⣾⣤⡔⢾⣤⣴⣿⣯⣄⠉⠉⣹⣷⠿⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠙⠛⠛⣽⣿⡋⠉⠁⠈⠋⢻⢄⣀⣽⣼⠿⠛⠛⡙⢻⣇⡀⠀⢻⠀⠀⣰⣿⣾⠈⠀⠧⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⣐⣏⣿⣿⣷⣶⣤⣤⣾⣿⣿⡁⠀⠀⢀⣴⣾⣿⣯⣿⣶⣿⣧⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠁
⠀⠀⠀⠀⠀⢀⡈⠙⠛⠿⠿⢻⣯⣥⠽⠿⠛⠒⠒⠛⠛⠛⠻⠟⠻⠿⠛⠛⠛⠛⣛⠟⢻⠿⠽⠿⢿⣟⣿⢿⣛⡛⠋⢭⠜⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⡏⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣸⡇⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⠃⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⣿⠀⠀⠀⢸⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⣾⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢹⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢹⣇
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢿⡇⠺⡧⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢶⡆⢶⡄⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⢹⣇
⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠸⣧⡶⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠈⢻⡆
⠀⠀⠀⢀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢻⣆⠀⠀⠈⢿⡆
⠀⠀⠀⣸⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⣇⠀⠀⠈⢿⡄
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣧⠀⠀⠈⣿⡄
⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢸⣧
⠀⠀⣿⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠸⣿⡀
⠀⢰⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢹⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡸⣷
⢀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⢸⡏⠀⠀⠀⢸⡇⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⢸⡇⠀⠀⠀⠈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡷⠈⣿⠈⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢻⡗⢹⡟⢹⡇⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠏⠘⠷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠾⠃⠾⠀⠸⠇⠀⠀⠀⠀⠀⠀⢸
⣿⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⢀⣀⣀⠀⢀⣼⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠘⢃⡀⠀⣀⠀⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⢀⣀⣀⡀⣀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠨⠁⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠀⠀⢁⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⣀⠀⢀⢸⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⣁⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣚⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣹⢉⡧⡁⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠐⠁⣸⡇⠀⢄⠀⢈⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⡠⠒⠀⣰⡀⡇⠀⠀⠑⠀⡽⣗⣬⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠚⡁⢈⣤⢂⡠⠂⠀⡀⠉⣇⠘⢂⡀⡈⠛⢷⣤⡀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⢀⣤⣒⣚⣯⣄⡉⣒⣁⣸⣿⣿⣃⣢⣖⣜⣀⡩⠵⠟⠋⠁⢀⣀⢍⡙⣝⣢⣤⣀⣀⠈⣙⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⣠⣤⣤⣴⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠂⠀⢀⡰⠖⠂⠐⠂⠐⢪⠟⡿⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠟⠛⠉⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠋⠁⠐⠈⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠋⠁⠀⠀⣀⣠⣤⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢉⡀⠀⣀⣤⣴⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠸⣿⣄⣀⣀⣠⣴⡾⠟⠋⠁⢀⣀⣤⣵⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡟⠿⠻⣿⡋⣁⣠⣤⣶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣷⡀⢠⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡷⢾⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣌⢿⣆⢀⣠⣶⠾⣿⠿⠿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣦⣿⣿⣋⣀⡴⠃⠰⠆⣀⣿⣷⣶⡿⠟⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣼⡿⡛⠻⠟⢻⡿⣾⣶⣾⠛⠉⠀⣩⡴⣾⠋⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣸⠏⠙⢧⠀⢿⣌⠳⣤⣉⠋⢀⣀⣴⣋⣀⣿⠀⠀⠁⠸⣿
⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⢻⡟⠃⣿⡶⠀⠈⠃⠠⢶⡆⢈⣿⠿⠛⠛⠋⠙⠛⠻⣧⣆⣄⣠⣿⡄
⠀⠀⠀⠀⠀⣼⣟⣿⣼⣿⣤⣽⣶⣇⢶⣤⡀⡀⠀⣴⡶⠿⢿⣦⣤⡀⠀⠀⠀⠀⠉⢻⡏⡀⠙⠻⣷⣶⣆
⠀⠀⠀⠀⠀⠈⠿⢷⣾⣿⡙⢻⣭⣿⣦⣬⣴⣷⣾⣿⣧⣀⣸⠟⢹⣿⣶⣆⢺⣂⣀⣸⣇⠛⠂⣀⣿⣾⠟⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠘⠓⠆⠹⡻⣿⣿⣠⣤⣬⣿⢉⡷⠛⠛⠻⢦⡞⠋⠁⢸⣿⢿⡿⠛⠙⢿⣿⣶⡿⠁⣠⡾⠷⣦⡙
⠀⠀⠀⠀⠀⠀⢠⣴⡿⢷⣄⠈⠉⣩⣭⣉⣿⣾⣇⣀⠻⠷⣾⠀⣤⣾⣼⡿⠸⠷⢶⣶⠿⢋⡀⠀⠀⠻⣷⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠿⠷⠾⠟⠀⡀⢿⣿⣙⣿⠟⡛⢻⠿⠿⠿⠛⢛⡉⣁⠀⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠘⠛⠋⠀⠁⠉⠙⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢀⡴⠊⠀⠀⠀⠀⠀⢠⡦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠊⠀⠀⠀⠀⠀⠀⠀⠘⠃⠤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢡⣤⣄⣀⣀⠀⠀⠀⣴⡞⠲⠤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠁⠀⣈⡶⠉⠁⠀⠀⣸⣿⡿⠛⠶⣤⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣹⣷⣶⣶⠖⠀⣠⡄⢀⠀⠀⠀⠀⢀⣄⡉⠁⠀⠀⠐⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⣾⣽⡷⠋⢀⣶⣶⣤⣤⣿⣷⡤⠠⣤⣄⠀⡿⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⡄⠀⠀⠀⠊⢻⠁⠀⠸⡿⠃⠀⠘⠛⠋⠀⠀⣼⣿⣿⡇⣰⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⣿⣿⣿⣿⣿⡟⠉⠉⠀⠰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⢠⡦⠶⠆⠰⣿⡿
⠀⠀⠀⠀⠀⠰⠀⠰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⣀⣼⢿⡴⠃⠀⡤⣄⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⢀⣤⣿⣿⡏⣠⡾⠋⠁⣼⣧⣤⣤⣦⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⡀⠀⠀⣠⡴⠀⠀⠀⠀⠀⣈⠉⠀⠐⢿⣿⡟⣿⠃⠀⠀⠀⠈⠯⣽⣽⡿⣽⣋⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠈⠉⣿⣧⠀⠀⠀⢀⣼⠌⠀⠀⠀⠘⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⡵
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⠀⡄⠀⠀⢹⠻⡦⠀⠐⣾⣿⠀⡀⠀⠀⠀⣿⢹⠇⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣻⢸⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣠⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣤⣠⣤⣿⢸⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⢰⡀⠀⠀⣸⡄⠀⢻⣾⣿⣿⡘⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⠏⠀⠀⠀⢹⣷⠀⠐⢨⣬⡽⠃⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠉⠀⢦⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠂⢧⣀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠁⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⠾⣥⣵⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣿⣿⣞⣿⣷⣿⡿⠿⡿⣻⠲⣖⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣩⣿⡗⣿⣿⣷⣶⣿⡟⠁⣶⣾⡧⣻⣄⢵⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢿⡿⣿⠿⣾⣿⣷⣜⣿⣷⢽⣶⣾⣿⣾⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣸⣿⣿⣷⢮⡉⠿⣿⠍⠿⠆⠾⠟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢸⣿⠉⢗⣿⣕⢽⣦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠀⡀⠀⠀⠀⣸⣻⢻⠀⠘⣧⢿⣤⡈⢫⣮⣍⣿⣦⣀⡀
⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⣀⡾⠀⢰⠇⣴⡿⢱⠀⠀⠀⡟⣾⣿⣕⢶⣮⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠛⠚⠛⠓⠋⠀⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣿⣿⢠⠀⠿⣿⡟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⣿⣇⡸⣿⣟⠀⠀⠀⠉⠃⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣾⣿⡁⠀⣼⣿⣿⣷⣿⣿⣷⠀⠀⠊⠀⠢⠀⠀⠀⢶⣾⣧⠀⠀⡤⢠⡶⣿⠉⠀⠃
⠀⠀⠀⠀⣀⣠⣾⠁⠆⢀⣠⣾⡇⠀⠻⢾⣿⣿⣿⢀⣾⣿⣿⢻⣿⡷⣿⣿⣷⣦⣀⡀⠠⠀⠀⠠⢾⣷⠁⠀⣨⠀⠀⣿⠁
⠀⠀⠀⠀⣯⣿⣿⡀⡀⣿⢿⣿⡁⡠⠀⡀⠿⡜⢿⡇⠹⠟⠋⡌⠀⠈⠙⠉⠹⠿⣿⢷⣄⡀⠐⠄⡸⢁⠰⢳⡏⠀⣄⠀⣀⠸⣧
⠀⠀⠀⠀⢋⢉⣨⢠⣯⣺⣼⣸⣡⣧⣆⣿⣀⣷⡌⣥⣿⡄⠱⣿⡄⣂⠀⠀⠀⠀⠀⠙⢋⡻⣿⠠⠀⣾⡇⢺⣷⠀⣿⣇⣿⣽⣿⣶⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣤⣷⡀⠀⠤⣤⣶⣶⣾⣷⣿⣾⣿⣿⢷⣿⢿⣿⣿⣿⣿⣿⡟⣿⠃
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡌⠉⠙⠛⢿⣿⣿⣼⡟⡟⡯⣿⣯⣷⣿⣿⣿
⠀⠀⠀⠀⠘⢿⣏⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⢁⡀⠀⡀⢰⢠⢟⠯⣯⣁⣷⢹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣇⣿⢿⡿⣿⣻⣽⣿⣿⡿⠿⠛⠋⠉⠁⢠⡀⠀⠀⠀⠀⣰⡀⣼⣷⣤⣾⣿⣿⣿⡤⠊⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⢹⣺⡟⠉⣿⣿⣿⣿⡄⠀⣾⡆⠀⠀⢽⡅⠀⠀⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢼⠁⠷⣾⣿⣿⣿⡤⠆⣿⣫⡄⠸⣿⣿⡅⠀⠀⠀⠀⠈⢈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⣿⣿⣿⣶⣨⣿⣟⣡⡔⣿⣷⠞⣀⠀⠀⠀⢀⣸⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⡷⢂⠀⢀⣈⣿⡷⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣀⣈⣻⣿⣿⣿⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡅⢠⣽⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢠⠁⠢⠤⣀⣀⣀⣀⡀⡀⠀⠀⠀⢀⡀⠀⡀⠀⠤⠤⠤⠤⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠸⡀⠀⣀
⠀⠀⠀⠀⠀⠀⢳⣄⣀⣤⡶⢶⣶⣶⠐⢶⣶⣤⢠⣤⣤⠦⢶⣶⣶⠄⢶⣿⣴⢶⣶⣤⣦⣆
⠀⠀⠀⠀⠀⠀⢸⡏⢿⠉⠁⠘⣿⠃⠀⢸⡏⠁⠀⢿⡟⠀⠀⢿⣿⠀⠘⣿⠀⠙⠋⣩⢱⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣤⣄⣀⣏⡀⠀⠈⠇⠀⠀⠸⠃⠀⠀⠘⡏⠀⠀⣻⡀⢀⣤⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⢻⣻⡿⣿⢿⠿⠻⡿⠛⠿⡿⠿⠿⡿⠿⠿⡿⠿⢿⠟⠻⣿⣿⢿⢻⡆⣧
⠀⠀⠀⠀⠀⠀⢸⡿⠯⣇⣻⢸⡄⠀⣧⠀⢀⣇⠀⠀⡇⠀⢸⡇⠀⣸⠀⢀⡇⣿⣼⠞⣫⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠻⢧⣸⢿⡀⣸⣿⡀⣸⣗⢀⣸⣷⢀⡿⠄⠼⠛⠛⠛⠛⠉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠓⠖⠛⠓⢦⣠⠤⠤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠑⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠦⢤⠀⠀⠀⠀⠀⢀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠀⠁⠀⠀⠈⠓⠒⠦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⣀⡠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠶⠶⠶⠴⠒⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣀⠀⣄⣤⢀⣄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡟⢸⣿⠸⣿⠈⠙⢄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢰⣷⢻⣿⠀⢻⡳⣄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣿⠃⠈⠁⠀⠀⠹⡜⡆⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⠓⠃⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⣼⠁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣙⣦
⠈⠉⠉⠉⠉⠁⠻⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⣀⣀⣤⠤⣤⣀⣀⡀⣸⠋⠉⠉⠉
⣦⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣦⣬⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣽⣥⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣴⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡊⠁⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠙⢦⡀⠀⣀⡤⠶⠚⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢹⠋⠁⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠢⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⣄⠀⠀⠀⠺⠀⠀⠀⠀⠀⣠⠀⠀⢉⠀⠀⡝⠉⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢀⡀⠐⠚⠋⠀⠀⠀⠀⠀⠀⠛⣶⢺⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⣄⠀⠀⠀⠀⠀⠰⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣀⣤⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢈⠳⣄⡀⠀⠀⠠⡇⠀⠀⠀⠠⠀⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠻⢦⡀⢀⡇⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠙⢶⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠟⠋⠁

⠀⠀⣀⣀⣠⣀⣄⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠈⣽⢍⣉⣉⣰⡀⠠⠤⠄⠤⠠⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⣀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠢⣤⡄⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⢄⠘⠑⢀⡀
⠀⠀⡇⠀⠉⠀⢨⠿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠳⡀⠀⠁⠀⠀⠀⢸⠀⠀⠠⠐⣄⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⣧⣶⢀⣠⡇⠀⢀⠂⡈⠀⢴⠀⠀⠃⠀⠀⠀⠈⠇⠀⠀⠀⠹⡀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠈⢳⠈⢠⠈⠃⠠⠀⡈
⠀⠀⣏⡃⢨⣏⣀⣀⡉⠀⠀⠀⠀⠀⢀⣀⠈⠂⠀⠀⠀⣨⠀⠀⠀⣇⠀⠒⢿⠀⢸⠀⠀⠀⠀⠀⠀⠓⠈⠀⠀⢐⠢⠁⠀⠀⡀
⠀⠀⡇⠁⡤⣯⠥⠬⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠒⢐⣒⠂⠅⠀⠀⢸⠀⠀⠀⠀⠀⠀⡐⠺⢄⠀⠀⠀⠈⠆
⠀⠀⡇⠀⠀⢛⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠈⠉
⠀⠀⡇⠀⣀⣀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⡂⠡⣈⠃⡀⠁⡄⠐⢤⡀
⠀⠀⡇⠀⠀⠀⠀⢻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠛⠃⠈⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣉⢱⣾⣿⣏⢁⠸⠃⣀⣤⠠⡀⠈⠉⡆
⠀⠀⡇⠀⠀⠀⠀⣄⣀⣉⠲⠄⣀⠤⠀⠀⠠⠠⠤⡤⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢀⠰⠿⣷⡺⣿⠳⣶⠿⠯⠻⠏⢸⠆⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠛⠂⢈⠙⠅⠣⡐⢙⡶⢄⠑⠆⠈⠄⢸
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠈⠁⠈⠈⠉⠈⠉⠈⠉⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠀⠀⠈⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⡄
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄

⠀⠀⠀⠀⠀⡘⢰⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣄
⠀⠀⠀⠀⠀⠘⡜⡃⠸⠀⡇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⢖⣾⢋⡽⠋⠙⠟⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠹⣇⠀⣇⢷⣱⠘⢆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⣬⡗⠋⠉⠉⠴⣿⣿⡛⢿⡿⡟
⠀⠀⠀⠀⠐⠐⠀⣟⡆⢹⢞⣡⠲⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⠡⠀⠀⠐⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⡁⢸⡀⡇⠀⠀⢠⣤⠘⠲⠋⠀⠀⢃
⠀⠀⠀⠀⠀⠂⢃⠻⣴⡄⠘⡔⡁⠸⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠻⣇⣀⡀⠸⣾⡄⠀⢠⣶⣶⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠢⣈⡉⠤⠛⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⢋⡛⠓⠢⠒⠒⠒⠠⢠⡄
⠀⠀⠀⠀⠀⡀⠀⠒⠚⠛⠛⠓⡣⠖⠚⣻⠷⢆⣠⣶⡿⠃
⠀⡆⠈⠀⣸⣶⣶⣾⣿⣶⣞⣟⣿⣭⣿⣬⣡⣿⣿⣿⣿⡀
⠀⠠⢀⣾⣏⣉⣹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣶⣦⣤⣤⡤⢴⠖⠲⠤⣄⡀
⣠⠟⢆⣙⣿⣿⣿⣿⣿⣿⣿⣽⣒⠄⠛⢿⡏⣁⣤⣾⣿⣿⣟⣻⣿⣿⡗⣿⣠⣄⡀⢺⣇
⢳⣶⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣻⢿⣾⣿⣿⣷⣄⠨⠆⠁⠄⠉⠀⠀⢄⣀⣠⣤⣤⣤⡀
⠈⢹⣿⠿⠀⠀⣠⣻⣿⣿⣿⣿⡟⠄⠀⣋⣻⣜⣿⣿⣿⡉⡮⠛⣿⣿⣯⡍⣀⡈⡻⠶⣦⡶⡶⠟⠛⠛⠛⠻⣿⡏
⢀⣾⣿⣶⣶⣶⣿⣿⣿⣿⣿⡟⡁⣡⠞⣿⣇⢹⣿⣿⠛⠉⠐⠂⠈⠻⣿⣿⣿⡿⣽⣧⠆⢊⠁⠀⢀⣤⣴⣾⣿⣆
⢸⠾⠷⠄⢘⣠⣤⣿⣉⡹⣿⢀⢁⣠⣴⣴⣤⣿⣿⣃⣂⣀⣠⣤⣤⣤⣝⡪⠉⣂⣀⣀⠐⢃⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠻⣾⠿⢟⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⣻⠈⣿⣿⣿⢻
⠀⠀⠀⡀⣶⣾⣿⣿⣿⣿⣿⣿⣟⠺⢍⠐⠀⠀⠂⠠⠈⠙⠻⠿⠿⢿⠿⠈⠿⢿⣿⣿⡷⠃⠐⠂⠀⠀⢸⣿⣿⣯⣽
⠀⠀⡴⣤⣿⣿⣶⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣷⣶⣾⢠⣆⣠⣀⣢⢀⣤⣤⣄⠀⠀⠀⢀⡤⢦⣄⠀⠀⠀⠙⢻⣿⠋
⢀⡿⠿⠿⢿⢿⣿⣿⠿⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣽⣿⣭⣤⣤⣴⣶⣤⣴⣷⣾⣿⣿⣿⣿⠃
⣾⡹⡈⡀⠀⢀⣀⣧⣀⠀⠀⠀⢸⣿⡟⠿⠿⠿⠟⠛⣿⡟⠛⢻⡟⠙⠛⠿⠿⠿⠿⣿⠛⢛⣿⣿⣿⣿⣿⣿⠁
⣏⣵⢾⣡⡾⡿⠿⣧⣬⠟⠀⢢⣨⠹⣇⣴⣶⣿⠀⢰⣿⠳⠀⣾⡇⠂⡠⠀⠀⠀⢺⣿⡆⢸⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣶⣶⣤⣄⣉⠠⠀⢊⣠⣥⣾⣭⣍⣁⠈⢘⣧⣤⣤⣿⣇⠄⠀⢀⣺⣤⣼⣿⣖⣸⣿⣿⣿⡟
⠙⠉⠉⠉⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠻⢿⣿⡿⠿⠟⠛⠛⠿⠿⣿⡿⠟⠛⠛⠛⠻⠿⣿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣐⡠⠔⠒⠉⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠘⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⢀⣴⠎⠁⠀⠲⣋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢆⠡⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⡀⠋⠀⠀⠀⢀⠀⠀⠀⠀⠈⢇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⡄⠣⠄⠐⠄⠒⣀⠀⢀⠀⠀⠀⠀⠈⢧⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣤⠀⠀⢼⡏⢸⢳⡀⠘⢧⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠤⠟⠀⢻⠁⠈⡎⠑⡄⢘⣆⠥⠈⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠃⢀⡀⠦⠄⠠⠈⠁⠀⡡⠚⡋⠀⠐⡤⡗⡀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⠀⠴⠞⠁⡠⠏⠀⠀⠀⡤⢎⠀⠀⠂⠀⠐⠀⡴⠳⠀⡀⡀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠉⠀⠀⠀⠀⣐⠀⠀⠉⠇⠀⠀⠀⠛⠣⢄⢇⠠⠈⠢⣄⠀⠀⠀⠀⠀⠈⠾⠤⠂
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠁⢠⠄⠀⠀⢣⠀⢀⡄⠀⠀⠀⠀⠀⠼⡃⠀⠙⡄⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡃⠀⠠⠀⠁⠀⠀⠀⠀⠀⢸⠄⢸⠀⠀⠠⢀⡀⢔⠀⠀⠀⠀⠡⠀⠈⢠⡑⠄⡀
⠀⠀⠀⠈⢩⡀⠈⠁⡅⠇⠀⠀⡄⠀⠀⢃⠀⠀⠀⠀⡧⢠⠀⠀⠀⠀⢹⠈⡆⠀⠘⠀⠘⣮⡊⡀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠈⠒⢤⣡⡀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠂⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠳⡈⢦⡀⠀⠀⡀⠇⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠄⡀⠀⠀⠀⠨⡄⣀⣀⠀⡠⠤⣅⣀⣀⣘⣤⣥⣤⣇⠤⠤⠤⠿⢛⡈⠉⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠒⠊⠁⠀⠀⠀⠨⠴⠀⡀⠀⠱⠄⠘⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠒⠒⢆⣀⣠⡄⢀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠶⠿⠀⣐⡋⣀⣀⠀⠠⠆⠁⠈⠉⣹⡍⢉⡉⠀⠀⠆⣦
⠀⠀⠀⠀⠀⠀⠀⠠⡔⠈⠉⣥⣶⣶⠀⠀⠀⢙⣂⠀⠤⣤⠄⠈⠉⡂⠀⠙⠒⣩⣥⠔⠛⣧⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣦⡤⣤⣶⣴⣶⡦⢤⣦⣤⣤⡦⠶⢶⣾⣶⣎⢲⠛⣉⣿⠴⢟⣩⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⣿⢷⣾⣷⠶⣷⠶⢾⣿⡿⣛⣻⣿⡿⡿⣿⠿⢾⡿⠋⣿⣖⡁⠄⣡⣠⠄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⢻⣺⣻⣛⣛⣛⣻⣛⣏⣙⣯⣝⣙⣛⣿⣿⣿⣿⢀⣸⣭⡴⣿⣩⣵⡾
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠾⠶⠿⠿⠿⠿⣾⣿⢿⣿⣿⣿⣏⣥⣿⣾⢿⠫⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣶⣶⣶⣶⠾⠿⠿⠿⣿⠿⠿⠿⠛⠟⠻⣿⣿⣿⣿⣿⡅⡻⣚⡗⢈⣡⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠴⢶⣾⢶⣿⣶⣖⣛⣛⣻⣛⣛⣛⣛⣿⣙⣻⣭⣽⣿⣿⣷⣾⣿⠿⠿⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⡿⠿⠿⠿⠿⠿⠟⠟⠟⠻⠟⠟⠛⠛⠛⢻⣿⣿⣿⢉⣿⣇⣤⡶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠶⠶⠦⡴⠾⠧⠶⢶⠶⠿⠶⡷⢾⡖⠲⠾⠛⢿⡿⣿⠉⣭⣥⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⣉⣋⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠛⠛⠻⠿⠿⠿⠽⠛⠛⠭⠬⠧⠐⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣺⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠠⠹⢹⡀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⣠⠃⣸⠀⢳⡀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠀⠈⣧⠀⠀⠀⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⣿⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠀⠀⣠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠠⠄⢠⡏⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣄
⠀⡰⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⠀⠀⠀⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⠄⠊⠁⠸⡣⡀
⡐⠁⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⢿⡀⢀⡠⠐⠉⠀⠀⠀⠀⠀⢣⠙⢄
⠱⠢⣀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⣼⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣠⠼⣿⠁⠀⠀⠀⠀⠀⣀⠤⠒⡍⠱⡈⠣⡀
⠀⠡⠀⠑⠢⡀⠀⠀⠀⠀⠀⠈⠒⢄⠘⡝⠂⡀⠀⠀⢈⡇⣀⡤⠖⣋⠤⠂⡟⢧⠀⠀⡠⠔⠊⠀⠀⢀⠃⠀⠈⢆⢡
⠀⠀⠑⠦⢀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠐⣤⠼⣟⣁⠴⠊⠁⠀⢸⣀⠤⠇⠁⠀⠀⢀⠤⠒⠉⢢⠀⠀⠀⠱
⠀⠀⠀⠈⠢⡑⠢⡀⠈⠒⢄⡀⠀⠀⠀⠀⢣⢹⡏⠀⠹⡏⠀⠀⢀⡤⢴⠏⢁⠂⠀⡠⠔⠊⠁⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠘⢄⠈⠑⢤⡀⠈⠢⢄⠀⠀⠀⢿⠁⠀⠀⣇⡤⠞⠉⠀⡾⣀⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣠
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠒⢄⠀⠁⠢⣀⡘⣤⠶⠋⢇⠀⢀⣠⢴⠋⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠈⠢⢄⠀⠇⠘⠄⣀⣿⠒⠉⠀⡞⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠑⠺⡔⠻⡅⢸⣄⡀⣼⡽⠁⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠘⢄⠹⡀⡟⢰⣿⠁⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠣⡹⣧⣿⠃⠀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣿⡧⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⣿⣿⣟⢦⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⣾⡆⢀
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣾⣷⣿⡿⠛⠻⣿⣗⣤⡀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⢀⣠⣾⣿⣿⣿⣿⣿⣿⣦⣠⣿⣇⣸
⠀⠀⠀⠀⣴⣸⣷⢠⣤⣞⡟⡿⡟⣿⣿⣦⣄⢀⢩⣈⣿⣶⣤⣿⣷⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣾⣽⣿⣛⠛⠻⠿⢿⣿⠟⡏⣿⣿⣿⣧⣇
⠀⠀⠀⢀⣿⣿⣿⣿⣷⣿⠿⠟⠛⠋⠉⠉⠙⠛⠿⡿⣿⣿⡟⣿⣿⣿⣧⣀⠀⠀⠀⠀⠠⠶⠾⣿⣿⣿⣿⣿⣿⣥⣀⣉⠉⠉⡁⢀⣀⣤⣴⣶⣿⣿⣿⣿⣯⡀
⠀⠉⠁⠉⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠈⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⣰⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡴⢦⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣦⣤⣾⣿⣿⣷⣝⣿⣿⣿⣦⣀⡀⣀
⠀⠀⢀⣴⣦⣀⣴⣿⣿⣷⣶⣶⣿⣿⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⣾⣄⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠈⠉⢹⡿⠯⠧⠉⠉⠉⠉⠉⠉⠁⠛⠛⠉⠉⠉⠁⠉⣈⡿⠿⢿⠿⠷⠶⠀⠀⠀⢠⠾⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⢷⣄
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⢀⡠⢚⡘⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡼⠉⢦⡴⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⣠⣾⣷⣾⣿⣿⣿⣿⣻⡛⣦⡀⠀⠀⠀⡠⠞⢉⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠉⠋⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⠀⠐⠋⠀⠂⠀⠀⠀⡄
⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠄

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣀⠄⠀⢀⡾⠀⠀⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠉⠁⠈⠶⠀⠀⠀⢰⠦⡴⢶⢿⠷⣦⠾⠿⢶⢶⡶⠿⠷⣦⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠻⣶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠈⢳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠋⠁⠀⠀⢚⡶⠘⠂⠀⠀⠈⢷⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠙⢦⡀⠀⠀⠺⠃⠀⠀⠀⣀⡴⠞⠻⠏⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠙⢦⡀⠀⢀⡀⠤⠛⠁⠀⠀⠀⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⣦⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢧
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⣤⢤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣇
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡆⠶⠄⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⡄
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠘⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢻⡀
⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣧
⠀⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠙⣆
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠻⣀⠀⠀⢿⡀
⠀⢠⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠻⡄⠀⠘⡇
⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢹⡄⠀⢸⡆
⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠹⡀⠈⣧
⢀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⣿⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠛⡄⠘⣄
⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠟⣀⢹⣷⢸⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠻
⠘⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⣴⣏⡀⠉⢀⡀⠉⠁⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠇
⣷⠀⠀⠀⠈⣇⠀⠀⠀⣠⠞⠁⠀⠉⠉⠉⠉⠉⠙⠒⠐⠚⢶⣄⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤
⣯⠀⠀⠀⠀⣿⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡟⠀⠀⠀⠀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⣸⠇
⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣄⢀⣀⢠⡀
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡿⢠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡿⢰⡏⢠⣇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠉
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠈⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠤⣠⡉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⡇⠀⠀⠀⠘⠙⢿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠿⠋⠉⠀⣿⡖⠀⠀⠀⠀⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⡀⢸⡇⠀⠸⡇⠘⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠃⠀⠁⠀⠇⠀⢀⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠇⠀⣀⣤⠀⠃⠀⠀⠀⢀⣇⣀⣿⠆⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢤⠈⠿⣻⣽⣿⠀⠐⢾⣿⣿⠋⢀⠀⣀⡼⠇⠀⠾⠂⠀⠠⠀⠀⠈⠃⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠘⠯⠧⠄⠀⠀⠻⣁⣠⢾⣧⣄⠀⣀⡀⠀⣀⣀⣤⠂⠀⠀⠀⢀⣼⠃



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

⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⠀⠀⣀⣤⣾⣶⣤⡀⠀⠀⠀⠀⠀⢰⣛⡄⠀⠉⠳⣤
⠀⢀⣴⣴⣿⣿⣷⣴⣄⠀⠀⠀⠀⢸⡇⢸⢛⠛⢻⠁⣿⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⢦⡠⠋⠈⠳
⢠⣾⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠀⢸⡇⣸⠆⠀⢸⡀⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠊⣹⣾⣷⡄
⠀⠀⠀⠘⡆⠑⠲⠄⣄⡀⠀⠀⠀⠘⠟⠁⠀⠐⠀⠙⠛⠀⠀⠀⠀⠀⠘⠛⠉⠙⠟⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⣀⣯⣹⢀⠛⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠉⢹⣿⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⡶⠶⠆⠀⠀⠀⠀⠀⠀⢠⣶⣶⣠⣤⣶⠄
⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⡖⡖⢖⣀⣿⣇⡀⠀⠀⠀⠀⢀⣯⣽⣻⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠉⣾⣀
⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣛⣻⠧⠀⠀⠀⠀⠸⣿⣿⠏⠉⢿⣿⡿⠀⠀⠀⠀⠈⢻⣯⣯⣯⣯⣷⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠴⠦⣟⠀⠀⣾⣿⠀⠀⠀⠀⢠⡾⣅⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡃
⠁⠀⠀⠀⣹⡆⠀⣿⣿⠀⠀⠀⠀⢸⣧⠈⠉⠀⣬⣤⡀⠀⠀⠀⠀⢀⡾⣷⠉⠃⢠⣤⣤⡀⠀⠀⠀⠀⣻⣿⠉⠃⢼⣥⣾⣷⡄
⠀⠀⠀⠴⠣⠇⠠⠿⠿⠀⠀⠀⠰⠋⠘⠃⠀⠛⠻⠿⠛⠂⠀⠀⠀⠋⠀⠛⠀⠀⠛⠻⠟⠛⠂⠀⠀⠰⠃⠸⠀⠀⠘⠿⠛⠟⠁


⠀⠀⠀⠀⡇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠋⣽⣿⡍⠻⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃
⠈⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⡄⣀⡀
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢿⡅
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⢠⡄⠀⠀⠒⠂⠀⠀⠁⣋⠀⠠⡀
⠀⠀⢠⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢻⠟⠋⠉⠀⠀⠀⠀⢀⡠⠤⢤⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠒⠒⠒
⠀⠁⠁⣴⠁⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠱⠘⠷⠶⠆⠀⠀⣶⣄⠀⣀⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠉⠛⢻⣧⠄⠀⠘⠚⠀⠘⢋⢀⠀⢀
⠀⠀⠀⠀⠑⠬⠑⠚⠉⠉⠀⠀⣠⣽⣷⠖⠀⣰⣦⣀⠈⠁⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢶⠛⠋⠉⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⢶⡟⠁⢀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢷⡄⡆
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣿⣧⣄⡀
⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⡿⣞⢧⡙⢦⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣾⡤⢳⡜⣷⣟⢦⡑⢄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⡻⣿⣿⣷⠸⡇⢹⡜⣆⠙⣮⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠃⠀⠐⣰⣿⣿⣿⣿⢸⣷⣾⣄⠐⣌⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠑⣤⣶⣺⣼⣿⣿⣿⣽⡟⢻⣧⠙⢷⣿⠷⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡿⠉⠀⢀⣀⣤⢿⣛⣿⣿⠿⣿⣿⣿⢿⡀⢻⣆⠀⠹⣄⠈⠳⢄⠛⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⡾⢿⣍⣀⣼⣿⣿⠁⠙⣿⢿⣿⣾⣇⠈⡌⢷⣦⣽⣆⠀⠀⠑⢄⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠋⣡⣽⡿⠟⠛⣻⢇⠀⢀⣿⣷⣿⣿⣿⡗⠛⢦⡈⠙⠻⢷⣤⣀⣀⠁⢀⡀⠙⠻⣦⡀
⠀⠀⢀⣀⣴⠿⢻⣿⣿⠟⡇⢠⣤⣾⣿⣿⢀⣴⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠙⠶⣄⠀⠹⣿⣿⠻⠶⠿⢦⣀⢻⣿⣶⣤⣀
⣀⣴⠟⠛⣇⣤⣿⣿⣟⣤⣾⡿⠛⣼⣿⠏⣾⡏⠑⠛⣔⣦⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠳⣦⡈⠙⢻⣦⣄⡀⠉⢧⣻⣿⣿⢻⣿
⣈⣤⣶⠾⣿⣿⣿⣿⠿⠛⢩⣶⣾⡿⠃⢠⣿⣿⠄⠀⠘⣿⣿⣯⣿⣿⣿⣿⡏⢿⣿⣷⡀⠀⠀⠙⣏⠀⠀⣿⡌⠙⠒⢤⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠉⠑⠀⠛⢦⠴⣟⠳⠆⠀⠀⠀⠀⠀⠀⢻⡶⠤⠖⠠⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⢹⠿⠶⠓⠄⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡄⢺⡤⠀⠁
⠰⠿⠀⡰⠶⢊⣲⡿⢦⣤⣼⡀⢀⠀⠀⣶⣶⠳⠖⠋⠙⠛⠒⠂⣀⠀⠀⠀⠀⠀⠀⠷⣶⠀⠨⣄⣴⡤⠈⠉⠙⢺⣷⡆⣀⡀⠈⢪⣿⠀⣿⣿⠷⣆⣀
⠀⠀⠀⠀⠀⢛⣟⣡⣯⣠⡗⠀⣠⣷⣯⣾⠛⠀⠀⠀⠀⠀⠀⠘⠿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠲⡈⢉⣤⡀⠀⠀⠈⠋⠀⠸⠄⠀⠻⠀⠀⠀⠛⠀⠉⠃
⠀⠀⠀⠐⣭⢧⡭⠿⢗⢻⠀⣡⠄⠱⠓⠦⣬⡝⠂⠀⠀⠀⠀⠠⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⢱⠾⠋⠙⢄⡀⢀⡀⡀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠃⠀⠁⠀⠀⠁⠀⠉⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣰⣷⣤⢦⣤⣀⠀⠀⠀⠀⠀⠐⢿⠏⠁⠀⠀⢿⣿⣿⡅
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣀⠂⠀⠀⠑⠚⢤⠀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠚⠋⢹⣄⣀⠀⠀⠀⠀⡀
⣀⣀⣈⢯⡀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣴⠈⠻⠿⣶⣶⡄⠀⠀⣘⠓⣿⠆⣀⡄⠀⠀⠀⠀⠀⣼⡥⠛⠛⢳⢄⣓⡉⢓⠦⠤⠘⣿⢲⣤⠤⣀⡶
⢟⣉⣨⠋⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢻⣷⡾⣟⠀⠀⠐⢇⠀⠑⠤⠏⠀⣦⣀⢀⣰⣿⢿⠁⠀⢠⢆⣨⡎⡿⠏⠀⠀⠀⠾⠛⠿⢖⣿⣧
⠄⠙⠉⠀⠘⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⡿⣇⢸⣿⠄⠘⠛⠦⢤⠶⠶⡤⡶⠊⠉⢙⣧⣀⠱⣄⠀⠆⠈⢻⣟⠷⣦⡄⠀⠆⠀⠀⢸⠟⠛
⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠸⠿⡏⠙⠋⠀⠀⠀⡰⠾⠄⠙⠀⠼⣦⠶⢾⣿⡛⢓⣿⣦⠐⠀⠀⢣⣄⠀⢤⣷⣿⡚⡲⠋⣀
⣤⣼⠀⠀⢷⣀⠀⠀⡐⠿⣣⠦⠀⠀⢘⣿⡿⠵⠃⠒⣶⡽⠢⢤⣄⣼⠉⣶⠀⠀⠀⠀⠀⡝⠓⢻⣹⡅⠈⢳⠺⢶⡾⠓⠈⠓⠀⠻⣽⢉⠁⠉⣻⠉⢤
⡾⠃⠂⠀⠀⠀⢸⡿⣅⡘⢡⠤⠏⠙⠞⠹⠿⠇⣀⢀⣧⣴⡀⢸⠋⠙⠶⠛⠓⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⠼⢶⡞⠃⡀⠀⠀⠀⠀⠈⡏⢠⡿⠛⢢⡄
⣧⣰⡶⣤⣴⡦⣰⣿⣶⡦⠻⠗⠀⠀⠀⠀⠀⠲⠛⣦⢽⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣉⠀⢠⣾⣿⣦⣿⠀⠀⠀⠠⣿⣭⡀⣀⠈⠐
⢈⡿⠟⠁⠈⠁⢸⡟⣏⠁⠀⠀⠀⠀⠀⠀⠀⠈⡥⠛⢺⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣫⣴⢾⠃⠀⠀⠉⠘⢦⠀⢰⡿⡟⠛⠻⠗
⠈⠁⠀⠀⠀⠀⠠⠿⠃⣄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣤⣬⣵⠒⠂⣀⣄⠀⠀⠀⠿⣷⡖⠀⠀⠂
⠀⠀⠀⣌⠙⠛⢊⠀⠀⢸⣦⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⣥⡈⠉⠁⠀⠀⠀⠀⠀⠈⠐⢀⣤
⠀⠀⢠⣟⠀⠀⢹⠀⠀⢺⣽⡤⠀⠀⢀⣷⣸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⡾⠃⢸⣧⣀⢠⠀⠀⠀⠀⠀⠀⠀⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⢈⡛⠁⣹⣋⣀⣼⣿⣭⡇⢼⣤⣸⠛⠓⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠘⠛⢻⣄⢘⣃⣉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⣿⡝⠃⠁⠈⠉⠉⠹⠉⢸⣾⡛⢁⡀⠀⠀⠀⠀⠀⠰⣤⡁⣀⣽⣣⣀⠀⠀⡷⠘⠟⠙⢾⠗⢦⣴⣀⡷⠀⢀⡤⠤⣾⠽⠇⠀⠰⠄⠀⢠⡀⢀⣀
⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⢴⣿⣿⣁⠻⣯⠘⢻⡇⠀⠀⠀⠀⠀⣲⠆⢈⣿⢿⡗⣀⣸⣤⡆⠻⠛⠂⠀⠀⠄⠃⠀⠙⢿⣳
⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠉⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢣⡀⠘⠁⡈⠿⢻⠿⣿⠷⠀⠀⠀⠀⢀⡀⣤⡶⣦⢀
⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢸⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠃⠀⠀⠀⠀⢠⡹⣃⣀⣾⡂⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠨⣿⣿⡻⠷
⡽⠃⠙⠛⠸⡦⣠⡄⡀⡀⠀⢀⡠⢀⣀⠀⡆⠀⠀⠀⠀⠀⠉⢩⡆⠀⡀⠰⣦⣤⡀⢀⠀⣤⡾⣷⣽⣽⢛⠧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠈⠀⠉
⡷⣾⠗⠀⠀⠉⢙⣇⣡⣻⠘⠛⢣⣾⠧⢀⣿⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⠁⠀⠂⠀⠋⠀⢸⠿⠏⠉⣶⣟⣷⠀⠀⠀⠀⠀⠀⠀⠉
⢿⣻⣿⠰⠀⠀⠀⠀⠀⠞⢀⣇⢘⣿⡆⠀⣹⠄⠀⢀⠀⠀⠀⠀⠹⡿⠟⠀⠀⠀⠀⣸⣇⣤⠐⠉⠱⠠⡖⢻⡿⠛⠁⠀⢀⡀
⠉⢛⣇⢀⡀⠺⣿⣰⣦⣿⢻⣿⡏⢉⣛⣀⠛⠀⠀⠉⢀⣀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠈⠁⠀⡀⠀⢠⣌⠀⣙⡦⠀⠀⠟⢉⡶⠓⣀⢠⡤⣔⠓⠴⠟
⠀⠹⣏⢉⣡⣦⣽⣧⡘⢻⣎⣩⣡⣀⣼⡇⠀⠀⠀⠀⢀⡌⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣄⠀⠉⠉⠉⡁⠀⢠⣔⡭⠇⠀⠙⠛⢳⣾⣦⠴
⠀⠀⢈⡟⠙⠋⠉⡑⠉⠁⠀⡄⠈⢉⣿⡀⠀⠀⠀⠀⠀⡑⢦⠀⠀⣤⣤⣬⣭⠀⠀⢀⣿⣼⠁⠀⠉⣧⠀⠐⠋⢋⡀⣶⡟⣧⠀⠀⠀⣤⣰⢩⣧⡀
⣿⠿⠿⠿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿