Bags Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢋⡩⠤⠤⣤⣍⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢁⡞⠁⠀⠀⠀⠀⠻⣷⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢀⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⡄⠀⠀⠀⢀⣀⣀⣀⣼⣇⠄⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠾⠿⠿⢧⢿⠛⠛⠛⠛⠋⠉⢩⠯⠝⣼⠉⠉⠉⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢲⡻⠾⢈⠄⠀⠀⠀⠀⠈⡿⠊⠡⠂⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣈⣇⠀⠀⠀⠀⡏⣂⢁⠀⠀⠀⠀⠀⠀⢿⢒⠈⠁⠀⠀⠀⢸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡄⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡴⠇⠀⠀⠀⠀⢧⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⣼⡟⢿⡇⠀⠀⠀⢹⠶⠷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠆⠀⠀⠀⠀⠸⡆⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢲⡟⠁⠸⣿⠀⠀⠀⠸⣆⡜⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⡄⡄⡄⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢀⠇⡌⠀⠀⢼⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⡴⠁⣀⣀⣸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠃⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡇⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⢹⡇⢰⣿⣿⣿⣿⣿⣾⣿⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠯⠛⠿⠟⠉⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠸⡇⠘⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⡉⢥⠠⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠂⠙⠛⠻⠿⠿⠛⠋⠉⡉⠔⣊⠥⠚⠋⠀⠀⠘⠛⠻⠿⠿⠶⠶⠾⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣀⣀⣀⡠⣀⡽⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠁
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣸⣧⣭⣯⣤⣵⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠟⠉⠁⠈⠉⢪⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡟⡤⢤⡀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⢤⣤⢞⣽⣶⣃⣉⡑⢿⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠉⠉⠉⠉⢻⣿⠒⠒⠛⠲⠶⠤⠤⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣷⡷⡶⠾⠶⢶⣶⣶⣿⣯⣾⣿⣧⣼⣧⣤⣦⣴⣿⣄⣰⣤⡦⣤⠀⣄⡀⣴⠶⢟⢟⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⣋⡿⠃⣈⠀⠈⠫⠟⣿⢁⠀⠀⠀⣠⡤⣄⠉⠉⢹⣏⣿⣟⠛⠛⠛⣿⠿⣿⣧⠀⡫⢊⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏⣻⠀⠀⠸⡭⠀⠀⠈⠸⡐⢔⡹⠀⠀⣜⡿⠩⠃⠐⠀⠘⢿⣢⠞⣿⠀⢐⠵⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡍⣷⠀⠂⢠⣴⠮⠏⢶⣄⠂⣠⡄⢄⠈⠁⠄⠁⠀⠈⠀⠀⢠⡀⢀⡀⠀⠀⢸⣿⡇⠀⡪⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣌⣱⣟⠀⢀⣿⠁⣶⢥⣤⠙⣷⠝⠶⢠⠃⠀⠀⡤⠤⠄⠀⢰⢡⢂⢲⢰⠈⠀⣠⡿⣿⠀⢒⠕⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⢉⣽⡿⠿⢿⣿⡛⠛⠛⠛⠛⠛⠓⠒⠶⠾⠾⠾⣧⡄⠐⡈⠠⢵⠏⠀⠈⠹⡯⢼⡇⠀⡪⠚⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣚⣫⣀⣀⢀⠰⣠⣿⠃⠰⠆⡀⢸⣷⡀⠰⠆⠀⠀⠀⠀⢀⣠⡴⢚⠙⡇⠀⠈⠉⠁⡀⠀⣠⣠⡿⠘⣿⠀⠐⡣⢻⡆
⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠛⠛⠛⠛⢿⡷⠶⠶⠶⠷⢿⣼⣷⣤⣼⣴⣤⣤⣤⣼⣆⠀⠑⡀⣻⠀⡀⠀⠀⠀⠀⡻⣿⣿⠄⢹⡇⠨⢆⠜⣷
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡈⢘⡄⣴⡮⠰⡄⠠⠈⠈⣯⢁⠀⣿⠀⠡⠊⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣔⠅⣿⡹⠳⢗⡟⠀⠀⠀⣿⢄⡀⢹⡆⠈⡀⠍⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠖⠶⣤⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⡀⢿⠈⠉⠉⡄⠀⠀⠀⣿⠔⠠⠈⣇⠀⢤⠴⠘⣇
⠀⠀⠀⠀⠀⢸⡇⡸⢎⠀⠙⣤⠊⠉⠢⣠⠋⠙⢍⡩⠟⠛⡍⢉⠝⠛⡍⢉⠝⢛⡛⣻⣧⠀⠊⢀⠹⣀⡄⡁⠀⢠⠤⢤⣹⡈⠀⠀⢻⠀⠠⠊⠄⢹⡆
⠀⠀⠀⠀⠀⢸⣏⠀⠀⡱⣞⠀⠑⣤⠊⠉⠢⣠⠞⠑⣄⢀⡜⠣⡀⢀⠜⠣⡀⣠⠟⢽⣿⡄⠐⡁⢨⡿⠻⠃⠀⢹⡶⣶⢿⡇⠀⠀⣼⠷⠶⢷⣿⣿⠟
⠀⠀⠀⠀⠀⣸⡏⢣⡜⠀⠈⣢⡞⠀⠳⣠⠖⠉⢦⢀⡼⠣⡀⢀⡜⠣⡀⡠⠚⢦⠀⣸⣿⣧⠈⡪⠀⡧⠀⠀⠀⠀⡈⠡⠘⡇⠀⣸⠟⣨⣶⠿⠋
⠀⠀⠀⠀⠀⣿⡇⠁⠈⣦⠚⠁⠙⣤⠞⠉⠢⣀⡴⠛⢄⢀⠜⢯⠀⡠⠛⢤⠀⡠⠛⢼⣿⣿⡀⠰⠁⢿⣞⣕⡄⠀⠀⣴⠲⣗⣸⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣿⠈⣢⡞⠁⠑⣤⠞⠁⠱⣄⠔⠉⢦⣀⠔⠳⡄⣠⠛⢄⠀⡰⠛⢄⢀⢼⣿⢾⣇⣀⡦⢸⠿⠿⠤⠥⠼⠿⢷⣿⠟⠉
⠀⠀⠀⠀⠀⡿⢎⠀⠈⢢⡔⠉⠣⣀⠔⠉⢢⣀⠔⠓⢄⡠⠞⢣⡀⡠⠻⡀⠀⡴⠻⣼⣯⣿⡏⡛⢻⣿⠄
⠀⠀⠀⠀⠘⠷⠤⣵⣶⣥⣌⣤⣔⣉⣢⣀⡔⠙⢄⡠⠊⠳⡀⡠⠛⢄⠀⡨⠫⡀⢀⢼⣿⡿⣈⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠚⠷⠶⠤⣼⣧⣀⣀⣿⡀⠀⡹⢇⢸⣿⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⣠⣄⣀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡆⠀⡇⠀⠀⠉⠛⠶⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠉⡭⠭⠭⣍⡙⠛⠷⣤⡀⠀⠀⠀⠈⠙⢦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡘⢦⡀⠀⠉⢻⠶⣄⡙⢶⣄⠀⠀⠀⠈⢳⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠀⠳⡄⠹⣦⣍⠳⣄⠀⠀⠈⢷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠲⡄⠙⣦⠹⣏⠳⣌⠳⣦⡀⠈⢧⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣳⣄⠈⢦⠈⠳⣌⠳⣌⢧⡈⢷⡀⠈⢧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠛⠛⠛⠛⠿⠷⠾⣦⣷⣌⣷⣀⣸⣇⣹⣀⣠⣤⣤⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣾⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢀⡠⠺⠣⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡴⠛⠁⠀⠀⠈⠛⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠋⠀⠀⠀⠀⠀⠀⠀⣀⣙⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠷⠶⢾⣿⣶⣶⣶⣶⡶⠿⠿⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⠒⠈⠉⠉⠐⠒⠢⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠓⠓⠒⠀⠤⠤⠤⠤⠤⠤⠼⠼⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠜⡙⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡆⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠞⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⢄⠑⠤⣀⣀⡀⠀⠀⣀⠔⠊⡠⠖⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠉⠒⠾⣭⠭⠭⠥⠔⠂⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠚⠉⠀⠀⠉⠉⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⡆⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⢠⢡⠒⡝⣼
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⡈⠇⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡟⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢱⠸⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⠤⣉⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢸⠀⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢍⣒⡒⠤⢈⣁⡒⠒⠠⠄⣀⣀⡠⠎⢰⠀⠘⠀⠇⠀⠀⣠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠋⠥⠶⠔⣒⡒⠒⠁⠀⠀⢇⠈⠢⢒⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠚⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⡴⠚⣉⡿⣏⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⠛⠉⠙⢦⠀⠈⢻⡀⢸⠃⠀⢸⡀⠀⢳
⠀⠀⠀⠀⠀⣠⠞⢉⠞⠁⠀⠀⢀⣠⣼⠶⠶⠞⠛⠋⠉⠉⠉⠉⠉⠙⠛⠲⢤⣀
⠀⠀⠀⢠⠞⠁⣰⠋⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⣰⠋⢀⡾⠁⠀⠀⣼⡷⠛⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢰⠇⠀⣾⠁⠀⠀⣸⠏⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⡏⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠃⢀⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣾⠀⢠⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡏⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⡇⠀⠈⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠁⠀⠀⠀⠀⣀⡤⠤⡄⠀⠀⠀⢀⡏⢷
⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⠐⡇⠀⠀⢿⠤⠤⠴⠞⠁⠸⡇
⣷⠀⠀⣧⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠦⣤⣠⡤⠖⢷⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻
⢹⠀⠀⢿⠀⠀⡿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠚⠀⣀⣠⣤⠤⢤⡘⡇
⠘⡇⠀⢸⡇⢰⡇⢸⠁⠈⠉⠉⠀⠀⠀⠀⡞⢸⡀⠀⢀⣀⣀⣠⣤⣤⠤⠶⠖⠛⠉⠀⠀⠀⠀⣧⢷
⠀⢷⠀⠈⣇⣼⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢸⡄
⠀⠘⡆⠀⢽⡏⠈⡟⠛⠛⠛⠛⠒⠶⠶⣤⣼⢻⡇⠀⢸⡇⠀⠀⢀⣐⣠⣤⠴⠶⠶⠒⠛⠛⠉⣇⢸⡇
⠀⠀⢹⡀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⣾⢸⡇⠀⣾⠙⢶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠀⠀⠀⢧⣾⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⡇⠀⣿⠀⣸⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠴⠶⢛⡾
⠀⠀⠀⠈⣿⠀⠸⣦⣀⣀⠀⠀⠀⠀⢀⡇⣹⢸⠃⠀⠙⠦⢯⡤⠶⠞⠛⠉⠉⠁⠀⣀⣀⡤⠶⠋
⠀⠀⠀⠀⠹⣄⡀⠀⠉⠉⠙⠛⠛⠒⠚⠓⠋⡾⠀⠀⠀⠀⢀⣀⣠⣤⠴⠖⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣤⣤⣤⣤⣾⡥⠴⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢋⣉⣤⣤⣤⣤⣬⣍⡳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣡⣾⣁⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠶⠶⣤⣄⣸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠃⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠳⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣠⡾⠁⠀⠀⢀⡀⠀⠀⠀⠀⢀⡟⣰⠇⢨⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡄⣿⠋⠁⠀⠀⣼⠋⠉⠛⢻⡶⠀⣼⢱⣯⣀⡾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣂⣤⣤⣤⣿⣀⠀⣿⠀⠀⠀⠼⣧⣄⣀⣠⡾⢁⣼⠇⠿⣽⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠉⠉⠉⠉⢉⣿⣏⡈⣰⣿⠿⣿⣿⣶⣶⣾⣿⣿⡿⣿⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢺⣯⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣠⠟⠐⢣⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢠⣾⠃⢈⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣝⣓⠦⢤⣀⣀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠛⠛⠲⠮⣽⣯⣭⣒⣒⣒⣒⣲⣖⣀⣂⣤⣴⠋⠀⣠⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣀⣀⠀⠀⠉⠉⠉⠉⠉⢭⣭⣤⠴⠞⠋⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠒⠛⠛⠳⢶⣾⣿⢿⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠻⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠙⠛⠶⠦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⡿⢷⣄⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠦⠀⢸⡇⢀⡟⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣸⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡿⠀⠀⣼⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡇⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣤⣀⣀⣇⣀⣸⣇⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠤⠴⠿⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⠶⠤⣝⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⡤⠴⠂⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠿⠡⠄⠸⣷⠠
⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⣀⣀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⡸⠀⢀⣤⠶⠒⠛⢉⠓⣆⠀⠀⠀⠀⠀⠀⡞⠀⢀⣀⢀⣀⡀⣈⣽⣿⣇
⠀⠀⠀⠀⠀⠐⠀⠎⠁⠀⠀⠀⠀⠉⢲⡸⡇⠀⠀⠀⠀⠀⡘⠟⠁⠀⡅⠀⢸⠁⠀⠀⠻⠟⠀⣿⠀⠀⠀⠀⠀⢀⡇⠰⢩⣀⡀⠠⠤⠍⢱⢿⡟
⠀⠀⠀⠀⠾⡟⡞⠂⠒⠒⠒⠒⠀⠀⠀⢳⣷⠀⠀⠀⠀⢠⡇⠀⢤⣸⠆⠀⡿⠀⠀⠠⠠⢒⡒⠛⡄⠀⠀⠀⠀⣤⡇⡇⠸⠃⠀⠀⠀⠀⢸⠀⠛⣧
⠀⠀⠀⡈⠀⣿⠃⠀⡰⠒⠒⠊⠀⠀⠀⠘⣇⣁⠀⠀⠀⢸⣷⢸⠀⠈⠉⡇⠇⢠⠀⠈⠳⡍⠀⣤⢳⠀⠀⠀⠀⡏⡇⣧⢸⡶⠀⠀⠼⠀⢸⠀⡀⣿
⠀⠀⠀⡇⣭⣿⠀⠀⠑⢤⡀⢀⠄⠀⠀⠀⣷⠈⠂⠀⠀⠈⠘⡟⠄⢀⣴⠇⠀⣸⢸⡶⠂⠀⠉⠹⣼⠀⠀⠀⠰⠀⡆⢸⢈⡀⠐⠂⣠⡄⢻⡀⡏⣽
⠀⠀⠀⠑⠙⢧⡀⡀⠀⢀⣈⡁⢀⣀⣀⠴⠛⠂⠀⠀⠀⠀⠀⠳⣆⣀⣸⠀⠀⢻⠋⠀⣈⠀⢀⣀⠼⠀⠀⠀⠀⠳⠗⠚⠘⠁⠁⠀⠀⠒⠛⢉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠓⠛⠂⠈⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⣴⠚⢛⣡⡤⠤⢬⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⠭⠭⢭⣙⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙⢉⡉⢤
⠀⠀⠀⠀⠀⡇⠀⡏⠀⠔⠈⠉⠵⠉⠙⠻⡄⠀⠀⠀⠀⠀⠀⡼⢡⣧⠖⠒⠒⠒⢢⡝⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡆⠐⠀⠠⠤
⠀⠀⠀⠀⠀⡇⢸⠀⣸⢠⠄⠂⠀⣆⠁⠒⢧⠀⠀⠀⠀⠀⢀⠇⢀⡟⠀⠀⠂⠁⠸⠻⡌⢿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡃⠀⠀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⣧⢸⠀⣿⡇⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⢾⣤⡟⠀⠀⠀⠀⠀⠀⠀⢿⠿⡄⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⡇⣿⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣇⣽⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⣿⣸⠀⡇⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡤⠀⠀⠀⠀⠀⢻⠀⢰⢱⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⠏⣿⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠉⠃⡾⢛⢧⣄⡀⠂⠀⠀⢀⠀⢰⡇⠀⠀⠀⣶⠛⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠈⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠀⣿⠀⠔⠊⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⡟⠇⢈⣤⣠⡼⠖⣿⠇⠀⠀⠀⢻⢀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⡿⠀⠀⠀⠀⠀⢹⡚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠘⢦⡀⢀⣯⠀⠀⠀⠈⢉⣫⠵⠛⠉⠀⠀⠀⠀⠈⠿⣦⣤⣤⣄⣀⣤⣤⣤⣤⣤⠶⠂⠀⠀⠀⠀⠀⠠⢾⠧⠀⣀⣀⣴⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣟⣻⣻⣹⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣯⣿⣿⣯⣽⣿⣯⣭⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢚⣩⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠦⣤⡃⡶⠋⠀⠙⣷⠜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⡼⠛⠻⣦⡹⡆⠀⠀⠀⠘⣆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣶⣋⠀⠀⢠⢐⣷⢹⠀⠀⠀⠀⢷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡇⢨⣿⣿⣹⣿⣿⣾⣿⠈⣇⡀⠀⠀⢸⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⠿⣿⢀⣾⣿⣽⡿⢿⣿⣿⣿⠁⣿⣿⡿⣶⣾⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⣩⠟⡇⠀⣽⣿⣟⠿⢿⣿⣿⣿⠀⢽⣿⣿⡿⢿⣽⣿⣧⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠊⣧⠔⠇⣠⡇⡠⠎⠉⣻⠢⢭⣙⠀⣿⣿⣿⣿⣷⣻⣮⢿⡿⣿⣿⠽⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠐⠈⠀⢸⡧⣸⢋⡠⠘⠇⢀⠴⠂⠘⡀⢻⡶⠭⣝⣻⠛⠘⠒⠛⠉⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣀⡰⡆⠁⣹⡟⠉⢡⡠⠞⢧⠀⠠⣘⠧⣄⡠⢶⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⣠⠼⡏⢀⡱⡖⠁⢑⡤⠚⠣⢀⡸⠗⠛⣦⠔⠉⡇⠀⡔⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⢀⡤⠚⢇⢀⠸⢖⠁⢀⣤⠐⠙⣀⠜⢮⠀⡳⡞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠚⢃⡈⢶⠈⢣⣀⠠⠲⡁⢀⠶⠊⠉⢀⡤⠛⣅⡤⠃⠀⢀⠀⠀⠀⠈⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⠈⢠⡠⠶⠉⠁⣤⠔⠛⣄⡀⠕⠆⢈⡡⡔⠉⡱⠆⠀⠞⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡙⣄⣠⡔⠛⣄⢀⣤⠒⠉⢂⡠⠚⢣⣠⠜⢋⡀⢾⠀⠂⠀⠀⠀⢀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠬⣦⠈⢢⡠⠷⡎⢁⡱⡖⠉⣈⢶⠂⡹⢦⡀⠀⡀⠀⠀⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢉⢣⡠⠞⣋⡠⠒⣉⢤⡊⠃⠀⠀⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⢼⣎⣁⠤⠛⠻⡄⣀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⢂⠔⣈⡔⠊⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⣪⢚⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡤⠔⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠰⠑⠁⢼⠀⠀⠀⠀⠀⠀⠂⢀⣶⠶⠞⠺⢿⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠤⠄⢀⠃⠀⠀⢸⠀⠀⣤⡤⠴⢖⠛⠙⡇⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠈⠰⠠⠆⠼⠶⢯⡇⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠤⠖⢈⣀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠉⠀⠀⡘⠀⠀⠐⠗⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠸⠀⢸⡇⠀⠀⢸⠀⢠⣷⠦⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠠⠀⠠⠀⠀⠀⢠⠰⠀⠀⠃⠀⠀⠀⠀⠐⠀⣹⡇⠀⠀⣸⣿⣻⡿⢶⣾⣯⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠂⠐⠨⠀⠀⠠⢰⡈⠋⠉⠙⢹⠀⢹⡗⠀⢨⣿⣿⠿⣧⣾⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠸⡇⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⣖⣊⡇⣁⣾⣷⣸⢶⣷⡿⢛⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡇⠀⠈⠉⠛⡟⡟⠛⠊⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉⠐⠀⠄⡀⡗⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠾⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢷⣶⣿⣿⡿⠿⠟⠋⠁⠀⠀⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡤⣉⠓⢾⣿⣿⠟⠑⠶⠖⠀⠀⠀⠀⠀⢀⡙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣾⠿⠃⢀⣿⣷⣾⣿⣟⣀⣀⣀⣀⠀⠀⣠⣴⣾⣿⡦⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⠁⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠇⠀⠛⠛⠹⠋⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠉⠙⢧⣠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠐⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣤⣤⣤⠖⠀⠀⢠⣿⣿⣿⣀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣇⠀⣀⣈⣭⣥⡴⠶⣸⣿⣿⡟⠹⣿⣿⣦⣤⣄⡀⣀⣀⣀⣀⣀⣀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣿⣿⣿⣧⣀⢿⣧⠼⠃⣠⣿⣿⣿⣧⢸⣿⣿⣿⣿⣯⠻⣿⡿⠿⣿⣿⣿⣆
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⢿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣯⡓⣀⢸⣿⣿⣿⣧
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠃⡇⠘⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⡏⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡟⢿⠃⠀⠉⠟⢰⠃⠀⠛⣿⣿⣿⣿⣿⣿⣿⣯⣠⣿⣷⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠀⠀⠀⠀⢸⣻⡶⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⣬⣿⣿⣿⣿⣿⠝⠉⠉⣿⣻⣟⣽⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣰⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣿⣿⣼⠃⠘⠛⠻⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣿⡿⠿⣿⣾⣿⣿⡇⡰⢈⡇⠀⠀⠀⠀⠀⠀⠀⡼⠻⢻⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢻⣿⣿⣷⡄⠀⢀⣿⣿⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⣧⣴⣾⣿⣿⣿⣄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣏⢀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⡾⣿⣿⣿⣿⡀⣹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⡿⠃⢹⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣰⡀⠀⠀⠀⠀⢸⡿⠟⡇⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⠀⢼⠁⠀⢁⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣿⣧⣾⣿⣿⣷⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⣿⣿⣿⣄⣰⣿⣿⣿⢿⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣤⣠⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣶⣮⠉⠛⣻⠋⠻⢿⠟⣋⣴⣾⡿⠛⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⡝⠁⠀⢰⠿⠟⣤⣦⣾⣿⣿⣿⣿⣿⠟⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⢘⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⣧⢸⣿⣿⣿⣿⡿⠟⠁⢩⡿⠋⠀⣬⣿⣿⣿⡿⢛⣡⣤⠄⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⣽⣿⣷⣶⣶⣾⣧⣄⠀⣸⣻⣿⣿⡿⠋⠁⠀⠀⠀⠛⠁⢀⣿⣿⠟⣿⣿⣿⠿⢛⣥⣶⠟⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠁⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣶⣶⣦⣤⣤⣤⣶⣿⣿⡇⠀⢻⣿⣿⣦⣾⠉⡀⢾⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣌⣙⣿⣿⣿⣿⢻⣿⣿⣿⣷⣿⣿⡇⠀⠹⣶⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⣼⣿⣟⡛⠓⠀⢾⣿⣿⣿⣿⣿⡟⠛⣛⣛⣛⣛⠛⠛⣛⣋⣉⣉⡌⠀⣿⣿⣿⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣦⣈⠋⡿⠟⣛⣿⣇⠹⢛⣿⣿⣿⠾⣿⡿⣿⣏⣀⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣿⣽⣿⣥⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠎⠀⠀⡃⠀⠀⠀⡆⠀⠀⢨⠀⠀⠀⢐⠀⠀⠱⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣠⣮⣤⣤⣤⣮⣤⣤⣤⣧⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤⣼⣤⣤⣤⣵⣤⣤⣤⣵⣄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡸⢇⠀⠀⡨⢇⠀⠀⢨⢇⠀⠀⡸⢅⠀⠀⡸⢇⠀⠀⠨⢇⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⣠⣱⣪⣥⣤⣵⣪⣥⣤⣭⣨⣤⣬⣅⣬⣤⣤⣵⣊⣅⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⡟⠛⠛⠛⢻⡟⠛⠛⠛⢻⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣧⣠⣤⣤⣼⣧⣤⣤⣄⣼⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⡿⠿⠿⠿⢿⡿⠿⠿⠿⢿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⡀⣰⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣆⢀⡀
⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠘⠻⢿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣾⠏⠀⠀⠂⡀⣠⡴⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢦⣠⣾⡿⠋⠀⠀⠀⠀⠘⠟⢀⣠⡾⠁⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢿⣇⡴⠋⠠⣀⠄⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠾⠋⠀⠈⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠛⢸⣿⠄⣰⣿⢿⣷⠀⠀⢀⣴⡿⠿⠇⢸⣿⣾⡇⠸⣷⢻⣿⣻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠿⢶⣌⣿⣿⣿⡁⢸⡿⠀⠀⢸⡏⠀⠀⢀⣼⣿⠹⣧⣴⡿⢿⡿⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⡿⣿⠀⠙⣷⢸⡇⠀⣀⡘⢷⣶⣶⡿⢻⣯⣤⣽⣿⣴⣾⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⡿⡆⠑⠦⠀⠉⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⣀⣶⠆⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡶⢰⣿⣿⣿⣿⣿⣿⣿⣗⠸⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⠿⠃⢸⣿⡿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠸⣿⣇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡧⠀⠀⠀⠀⠿⡿⠃⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠈⠍⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⢰⣦⣤⣽⣿⣿⣷⣭⣻⣿⣿⣿⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠿⠷⠶⢶⣦⣭⣭⣍⣛⣛⣛⣿⡿⠿⣿⣿⠟⠛⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⣐⠒⠲⢤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣠⠴⡖⠒⠮⡕⣄⣠⠼⠶⠶⠊⠉⠛⠛⠀⠉⠁⠀⠘⠛⠋⠩⠿⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠀⢸⡀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠸⢰⠀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣧
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢀⠀⠈⠉⠛⢺⠖⠁⠂⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢠⡮⠊⢩⠋⠀⠀⠈⠉⠙⡶⠊⠉⠉⠀⠉⠉⠉⠓⠲⠤⣔⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻
⠀⠀⠀⠀⢀⢳⠁⢀⠇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⡜⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢲⡆⢀⣱⠇
⠀⠀⠀⠀⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠄⠀⠀⠀⠀⢶⣋⠭⠿⣿⠶⠒⡋⢀⡭⢿
⠀⠀⠀⠀⡄⠆⠀⢠⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⠴⠖⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠈⠲⣴⣲⣆⢀⣤⣤⣀⣰⡯⣷⠝⠊⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠉⠉⠁⠀⠀⠃⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⡀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⣤⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⢡⠀⢣⠀⠀⠀⠀⠀⡇⠀⠀⠈⢣⡤⠄⢁⣀⣀⣴⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⠘⡆⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢦⠀⣀⡠⠴⢺⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⡇⢷⠀⠸⡄⠀⠀⠀⢷⠀⠀⠀⠘⠦⣄⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢰⣸⡀⠀⠘⢆⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠴⠞⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢇⠳⡀⠀⠀⠱⢄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠢⢄⣀⣀⣉⣉⠽⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠊⠉⠉⠁⠈⠉⠓⠦⢤⣄⣀⣀⣀⣀⣠⠼⠓⠒⠒⠒⠒⠋⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠶⠶⠿⣼⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣽⣷⣤⣄⢀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⣷⠢⠎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢬⡆⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢏⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⡇⠀⢠⠂⠀⠀⠀⠀⠀⠀⡏⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⠄⡸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⡔⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢜⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠊⡀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⢎⠎⣽
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠞⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⡐⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠊⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠦⣄⣀⣀⣠⡴⡟⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢥⣾⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢪⢺⡆
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠀⠀⢀⠔⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡱⢣⢾⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⡰⢣⠞⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⡡⠃⣧
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡱⠃⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠱⢁⢺⡇
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠳⠦⢼⠇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠻⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂

⢀⡀⠀⠴⠿⠇⠀⠀⠈⣉⠀⠀⠀⣠⣦⡦⠀⠘⠟⠀⣀⠀⠈⠁⠘⣿⣿⡏⠀⣀⠀⠀⣀⡀⠀⠾⠿⠀⠀⠀⠈⡉⠀⠀⠀⣰⣶⡄⠀⠻⠟⢀⡀⠀⠉
⠙⠋⠀⠀⣀⡀⠀⣴⣿⣿⡇⠀⠸⢿⣿⡇⠀⢀⠀⠀⠋⢰⣶⣦⡀⠙⠋⠀⠈⠉⠀⠀⠛⠃⠀⠀⣀⠀⢠⣾⣿⣿⠂⠀⠿⣿⣿⠀⠀⡀⠀⠘⠃⣴⣶
⠀⣠⣤⡀⠙⠋⠀⠙⠻⠃⠀⣠⡄⠀⠁⠀⢰⣿⣷⠀⢀⣾⣿⣿⡷⠀⠰⣿⣿⣿⡗⠀⠀⣤⣄⠀⠛⠃⠈⠛⠿⠃⠀⣤⡄⠈⠁⠀⢾⣿⡆⠀⢠⣿⣿
⢀⣿⣿⠿⠀⠀⣴⣿⣆⠀⠀⠿⠇⠀⠀⠀⠀⠉⠁⠀⠀⠙⠉⠀⠀⠀⠸⠿⠿⠟⠁⠀⢠⣿⡿⠟⠀⠀⣾⣷⡆⠀⠸⠿⠃⢀⣤⡄⠈⠉⠀⠀⠈⠛⠁
⡌⠉⠁⢀⣴⡄⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠾⠛⠃⠀⠀⢾⣷⣦⣄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠘⠿⠿⠇⠰
⣇⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⢀⣤⣄
⠋⠀⠀⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⡀⠈⠋⠁⠸
⣾⣄⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⣤⣄
⠛⠉⢠⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠿⣿
⣠⣄⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢀⡄
⠉⠉⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⣾⣷
⠀⣤⣦⣀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⢰⣿⣿
⢀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉
⠄⠈⠁⠠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⣾⣷⣦⡄
⡀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⠘⠿⡿⠷
⣧⠀⠀⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠿⠃⢀⣴⡤
⠉⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⣈⠉⠀⠸
⣿⣦⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠇⢀⣶⣦
⠋⠁⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠘⠻⣿
⣠⣄⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢡⡄
⠈⠁⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠈⠀⣾⣿
⠀⣴⣶⣄⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⢸⣿⡿
⢠⣿⡿⠛⠀⠀⢿⣿⣷⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⢀⣀⠀⠉
⠆⠀⠀⢾⣿⠁⠀⠀⠁⢀⠀⠀⠀⣾⣿⣿⠀⠀⠸⣿⣿⣶⠀⠹⠛⠀⣀⡀⠀⣴⣶⡶⠀⠀⠀⢰⣿⠀⠀⣀⣄⣤⠀⠀⠰⣿⣿⡇⠀⠀⢻⣿⣷⡦
⡄⠀⠀⢀⣀⠀⠀⢠⣾⣿⣿⡁⠀⠀⠉⠁⣠⣶⠀⠙⠻⠋⠐⠃⠀⠀⠿⠿⠀⠈⣡⣤⡀⠀⠀⣀⡀⠀⣼⣿⣿⡧⠀⠀⠀⠀⠉⢀⣴⡦⠈⠛⠛⠃⠘
⣧⠀⠀⣿⣿⡟⠀⠈⠁⠉⠉⢀⣾⣿⠀⠸⠿⠋⠀⠀⣴⣶⠀⣠⣶⡆⠀⠀⣶⣿⣿⣿⡦⠀⢸⣿⣿⠇⠀⣀⣠⣄⠀⢰⣾⣇⠀⠸⠟⠁⠀⢠⣶⡦⢀
⣁⠀⠈⠛⠛⠀⠀⢠⣶⡶⠀⢾⣿⣿⠇⠀⢾⣦⣤⠀⠀⠀⠀⠛⠉⠠⠖⠀⠺⠟⠉⢉⡀⠀⠈⠻⠃⠀⠰⣿⡿⠟⠀⣿⣿⣿⠂⠀⣿⣦⡄⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡏⢉⣹⣾⢋⡻⣦⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣧⣴⡿⠻⠶⠟⠛⠷⠾⢿⣔⣶⡿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⣷⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡩⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣔⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏⠆⣿
⠀⠀⠀⠀⠀⠀⢰⡏⠄⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⠏⠀⠰⢹⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠀⠈⢷⣍⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⡾⠁⠀⠀⠀⠜⣧
⠀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠀⠙⠷⣤⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣤⠾⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣄⣂⡀⠄⣶⣿⣿⣷⠤⢀⣂⣨⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣧⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣧⣤⣥⣤⣄⣐⣀⣀⣀⣒⣀⣐⣀⣀⣁⣀⣈⣤⣬⣥⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠍⠲⡲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡭⣿⣝⢟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀⢸⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣛⢿⢿⡧⡇⣿⣿⢸⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⣿⣿⣿⣿⣿⣿⡏⢹⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣷⣿⡇⡏⠀⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿
⠀⠀⠀⢰⣶⣾⣿⣶⣾⣿⣿⢹⣿⣿⣿⣿⣷⠘⡀⠀⠀⠀⢸⣿⣿⣿⡏⠉⠁⠁⠉⠁⠁⠀⡆⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⠀⢣⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⢸⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠿⠿⠿⠿⠧⠞⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣀⣛⣯⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡰⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠡⡐⢌⠳⡀⣷⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢶⣿⡵⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⢌⣿⣠⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠴⠖⠋⠑⢦⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠘⠛⢻⢇⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⣸⣿⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⣀⣟⡟⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⡬⢧⣿⠾⠓⠁⠀⡠⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣲⠆⠘⠁⠈⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⢾⡰⢧⠐⢛⣦⠀⠀⠀⡇⢰⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⢹⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣻⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠰⠰⠀⠀⠚⠸⠂⠀⠀⠀⠀⠀⠀⠀⡟⠐⡏⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⣧⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢠⠀⡀⠀⢀⡀⡄⠀⠀⠀⠀⠈⠀⢰⠇⢸⠇⠀⠀⠀⣠⢞⡵⠒⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣴⡇⠀⠀⠘⡇⢷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⣽⠀⠀⡴⢛⡵⠋⠀⠀⠀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢧⢸⡀⠀⠀⠀⠀⠒⠒⠀⠚⠀⠰⠀⠀⠀⠀⠀⠀⢸⡇⢨⡇⠀⠈⠓⣯⠰⡋⢶⠟⣱⠀⣸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢄⠀⠀⠀⠀⢸⠈⣇⢀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⣼⠀⠀⠀⠀⠸⡄⠈⠠⠞⠁⢀⡏
⠀⠀⠀⠀⢀⡤⡶⠋⠀⠀⠀⢀⣤⣓⠂⠄⡀⠘⢦⢈⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣤⡤⠶⠃⠀⠀⠀⠀⠀⠙⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠠⡏⠈⠂⠤⢤⣴⡶⢍⠫⠤⠴⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢻⡴⡿⠐⣲⡄
⠀⠀⠀⠀⠙⠶⣤⠤⠤⠴⠶⠾⣐⣶⡾⠟⠁⢀⣀⣀⣀⣀⣀⣀⣈⣷⣶⣖⣒⣒⣒⣿⣁⣀⣀⡠⢤⣤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢠⠇⣦⠗⣆⠖⢠⢄⡴⡄⣠⠀⠀⠀⠀⠀⠀⠀⠊⠇⠫⠊⢆⠀⢀⡄⡐⡋⠀⠀⠉⠐⠠⣠⣄⡀
⠀⠀⠀⠀⠀⢠⡀⠠⠂⠀⠀⠀⠀⠀⠀⢠⡏⡜⠒⢋⢛⢖⢼⢸⣷⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠘⠂⠫⠘⣉⣡⣿⣣⣤⣴⣴⡿⠀⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣃⣀⣁⣴⣋⠁⠀⠀⢀⡞⠴⡳⠰⣋⠇⠈⢢⠀⠀⠘⢖⣾⣊⣀⡀⠀⠀⠩⠆⠠⠟⢋⣉⣃⠀⠉⠉⠉⠉⠉⠀⠀⠘⢣⡀⠐⠃
⠀⠀⠀⠀⢺⡭⣤⣤⣤⡤⠤⣿⠀⠀⠈⠓⠒⠲⠶⠶⠒⠒⢺⡄⠀⣴⡟⠤⠔⠒⠒⠒⠒⠒⠚⠻⠛⠋⠉⠃⠀⣀⣤⡤⠞⠏⠃⠀⠀⠀⠈⠻⣉
⠀⠀⠀⠀⠀⣣⡀⠀⠀⠐⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢭⠓⠈⡤⠀⣀⡜⣀⠀⠀⠀⠀⠀⠙⠫⡉
⠀⠀⠀⢰⠏⠁⠭⠖⣛⠻⠭⠆⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⣚⣨⢴⣺⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣷⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⣷⢶⡶⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⢹⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣀⢳⣤⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⣶⣦⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠹⣿⣿⣿⣿⣿⣇⠀⠄
⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⢀⠘⢿⣿⣿⣿⣿⣇⡤
⠀⠀⠀⠀⠀⢈⠈⢻⣾⣿⣿⣿⠿⢿⡿⡙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠸⠀⢀⣻⡿⢯⠯⠁⠀⠠⣇⣧⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣅⣿
⠀⠀⠀⠀⠈⠲⠤⠀⡰⠋⠀⠵⠀⠀⠈⠉⠁⢉⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠⣄⡀⠀⡄⣠⣈⣿⠿⠋⠁⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣼⠷⣿⣆⢠⣿⣿⡇⠀⠀⠀⣠⡴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣽⣻⣿⡿⣧⣴⠖⣿⣿⣷⣾⣿⡇⢰⣤⣀⣀⣠⡄⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡥⣿⣿⡻⢿⣿⡟⣿⢉⣷⢸⡿⣿⠻⣿⣇⢸⣿⣿⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣱⣿⣏⠙⣴⠇⠀⣿⡀⡻⣾⣿⣿⣷⣿⣿⠈⣿⣿⢷⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⣿⣷⡿⠀⢾⣏⢻⣿⣿⣿⣿⣿⣿⣿⣧⢿⠁⡞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⣲⣿⣿⣇⠀⣸⣿⣶⣿⣿⣟⣿⣿⣿⣿⣿⣿⣶⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠿⣿⣿⣿⣦⢿⣿⣿⣿⡿⢡⣾⣿⣥⡿⣿⣷⡙⣿⠇⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣞⠛⠛⠉⡁⣰⠺⣿⣿⢃⡼⣿⡉⣼⣿⠟⣫⣷⣹⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢭⣭⣶⣾⣿⣿⣿⣻⣿⡛⢿⣿⣿⣷⣿⣾⣿⣟⢏⣽⣟⣀⡤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠹⣿⣿⣿⣿⡿⠿⠟⠛⠩⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠘⠿⢿⣽⣿⣗⠀⠀⢢⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠛⠛⠛⠓⠂
⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠈
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠊⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄
⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠴⠋⠀⠀⠐⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢦⢤⣄⣀⠀⣸⠁
⢠⠴⠒⠒⠒⠒⠚⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⢄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂
⡇⠀⠀⠀⡟⠆⠀⠤⠤⠀⠀⠐⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⡇⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠤⠤⢤⣀⠀⢀⠀⠀⢀
⡇⠀⠀⢠⣧⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠀⠀⢸⡟⠦⠤⣒⣒⣨⡥⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠀⠀⠀⠀⠀⠉⠒⠲⠤⠔⠁⠸⣀⠀⠄⠀⠀⠀⡆⢧
⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠀⣠⡤⠶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠈⠉⠒⠒⠲⠾⣅⡼
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠿⠴⠟⢶⠗⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢉⣉⣉⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢤⢄⠀⡀⢢⣀⣠⣿⠤⠤⠤⠒⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢳⣡⠸⡀⠀⠻⠀⠀⠀⠐⠂⠈⠀⠈⡇⠀⠀⠀⢰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠒⠀⠁⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⡙⡵⠛⠻⣿⡙⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢱⠀⠀⠀⢣⢷⠀⠎⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢸⢸⠀⠀⠀⠈⣸⠀⢠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢸⢀⠀⠀⠀⢀⣻⡄⢈⢰⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠴⠒⢀⠈⠉⠉⠉⠉⠉⢹⡼⠈⠉⠁⠀⠀⠀⠈⣨⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠉⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠺⢿⠗⠀⠀⠀⠀⢐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡊⢽⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⡀⠀⠀⠀⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⢄⠔⡠⡀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠪⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠃⢊⢀⠔⠀⠀⠀⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢪⠂⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠐⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠐⢽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡈⡐⡄⣔
⠀⠀⠀⠀⠀⠀⠀⡧⠀⣹⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢂⠎⠈⢀⢌⠔⠌⡴⠣
⠀⠀⠀⠀⠀⠀⠀⡇⠂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⡡⠡⠀⡀⠠⠪⠀⢊⠎⠸⠀⡐⠇⠰
⠀⠀⠀⠀⠀⠀⢰⡇⠨⣺⠀⠀⢀⠄⠀⠀⠀⠀⠀⠊⠌⠜⠐⠁⡠⠔⡑⡑⠑⠡⠂⡊⠠⠀⠌⢀
⠀⠀⠀⠀⠀⠀⢸⢇⠄⣺⠀⢘⠑⡁⡁⡠⠠⢀⠄⡀⡠⠀⠊⢊⢊⢊⠊⠌⡠⡑⡁⠀⠐⡀⠄⢀
⠀⠀⠀⠀⠀⠀⢸⢕⠬⣺⠀⢸⠨⡪⢊⠔⡰⠡⢊⠈⠔⡑⡑⡑⡡⠢⢂⠄⠘⠔⡐⡀⠐⠀⠈⢀⠊
⠀⠀⠀⠀⠀⠀⠸⢵⣾⠿⣆⢸⠐⡔⡡⢊⠔⡁⡡⢂⢊⢌⢜⡔⡑⡡⢪⢊⢎⠌⡄⡰⠀⠄⢂⠠⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠒⠒⠒⠒⠂⠀⠁⠣⠤⠬⠬⠌⠔⠃⠃⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠄⠀⢀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⢂⠄⡠⢐⠁⡀⢄⠈⢀⠅⠠⡈⠀⡁⡊⠔⡡⢂⠄⡨⢂⠔⡀⢀⠔⡠⢂⠔⡠⢀⠁⡀⠪⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠀⠁⠊⠀⠁⠈⠀⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⢶⣶⣾⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣩⣿⣿⣥⣶⠶⠶⠾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡶⢧⡀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⡾⣷⣦⣞⣱⠃⠈⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣟⡿⣿⣟⣿⣿⣶⣇⣀⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⢻⣿⣿⣿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡛⣾⣿⣳⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⡟⢃⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣻⢡⣿⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣛⣛⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠙⠻⣿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠉⣁⣀⣤⣤⣤⣉⡁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⢀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠉⠓⢤⡀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠣⡀
⠀⠀⠀⠀⠀⡰⠃⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠘⢦⡀
⠀⠀⠀⠀⢠⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠙⢦⡀
⠀⠀⠀⠀⢸⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠱⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⢤⣀⡀⠀⢸⠀⡀
⠀⠀⠀⠀⢸⠀⣇⠀⠀⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣇⠀⣸⠀⠇
⠀⠀⠀⠀⠈⡆⢸⠀⡼⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠘⣦⠏⡜
⠀⠀⠀⠀⠀⢸⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠐⡋⢉⠠⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠡⠁⠊⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣪⡿⠟⠉⣉⣉⣉⡉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⣿⠋⢀⠖⠊⠉⠙⠻⣿⣷⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡿⠁⢠⠃⠀⠀⠀⠀⠀⠹⣿⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢿⡄⠀⠀⠀⠀⠀⢰⣿⣇⣠⣠⣤⡴⠶⠊⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡿⣷⣤⣤⠶⠶⢞⣛⣫⣿⡯⣷⣶⣶⣴⢴⣿⡧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⣉⣉⣩⣽⣿⣿⡄⠾⠿⠟⠛⠹⡿⣴⣿⠀⠁⠘⠻⣿⣷⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢿⢻⣿⡟⠻⠉⠓⣿⣉⣿⠀⠀⠀⠀⠀⠱⠹⣷⡇⠀⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣵⣿⣿⠀⠀⠀⠀⢛⢿⣧⠃⠀⠀⠀⠀⠀⠑⣾⡇⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣏⣙⢾⢸⣿⢿⣿⣆⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣼⠋⣼⣿⣿⡄⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠹⢋⠁⠀⠀⠀⠀⠈⣿⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⠰⣼⡿⠋⢛⣿⣿⡟⠀⣥⣶⣿⣿⣿⣧⠀⠀⠀⡘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠙⠻⢿⡍⢠⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣴⣿⣿⣿⠀⠰⣿⢿⢸⠛⣿⣿⡆⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠉⠺⣿⣧
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣼⡾⢸⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣸⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠛⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⣀⣀⠀⠉⠛⢿⣿⣿⣿⣿⠃⢁⣀⣀⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⣿⣷⣄⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠉⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠈⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣤⣤⣤⣶⣶⡆⠀⠀⣀⠀⠤⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢰⣟⠸⡏⢾⣳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣜⠛⠀⠁⠀⠀⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣍⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⢍⣿⣋⢻⠘⣶⡈⢻⡿⣫⡶⠚⢋⣽⡿⣫⠴⣌⠛⢹⠋⡎⠇⡜⣧⢏⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣷⣼⣿⠇⠃⠙⠀⠀⠛⠃⠀⠰⣿⣍⣷⡿⣻⣿⣿⣡⢿⡆⠈⠀⣁⡀⡁⠁⢠⣬⣙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢚⣿⠿⠫⠦⡤⣾⣿⠀⡄⡀⠀⣃⠈⠉⠡⡾⠿⠟⠛⠷⢾⡇⢠⢢⣿⠇⣰⣰⢸⣧⠙⢷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢙⡉⣠⣾⢿⣄⢳⣟⡷⢞⣷⢧⠶⣭⠖⣙⣥⣤⡭⣽⣿⣿⡞⢅⡈⢷⣽⣿⣵⠏⢀⡝⣧⠀⠻⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠋⡡⠶⢶⣦⣙⠛⠛⠁⡉⠶⣀⡾⣿⢃⡵⣫⣝⢆⢹⣿⢀⣷⡀⠉⠉⣁⡠⢾⠀⢹⡆⢀⣬⡓
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠇⢁⡞⣴⣟⡴⠋⠛⠉⣀⣀⣐⠺⠟⢦⡟⢸⣿⢀⣿⣼⠸⡿⣜⠻⢿⡟⢈⣡⢁⣮⢾⡾⣷⠈⣿⣷⢠
⠀⠀⠀⠀⠀⠀⣴⢠⢹⣆⠀⠸⡿⣡⡿⢃⡔⠛⠛⠳⢤⣪⣍⢻⣶⣝⠋⣿⣿⡇⣿⢠⣷⠤⠞⣨⡆⡿⠁⣼⣌⣉⡴⡏⠀⣿⡟⣸⢳
⠀⠀⠀⠀⠀⣸⢇⡎⣠⢻⡆⠀⠻⣿⢱⢁⡶⢛⣿⣶⣄⡙⢷⣞⣁⠏⣷⠈⠳⣏⣻⡎⢡⡰⠆⠘⠃⢸⣇⠻⣽⡧⠛⠀⣼⣿⣧⡏⢸⡇⣀
⠀⠀⠀⠀⠐⣣⡪⣼⢇⣾⡇⠀⠛⢀⠀⠸⣿⡟⠛⠲⣌⢷⡄⠙⠻⠾⠟⢀⡆⢈⠙⠻⠦⠝⠋⢀⣼⠈⠻⣷⣶⠶⠻⢚⠁⣬⠙⢿⣿⢆⠉
⠀⠀⠀⠀⢸⣿⣷⣶⣿⠟⢀⡄⢧⣙⡤⠦⠹⣷⡘⠀⠈⢣⣻⣄⣄⠤⠴⢟⣵⣌⠻⢶⡶⠶⠖⡋⠅⣈⡃⠖⠂⠀⢀⣀⣳⠶⠢⠟⢋⣎⠻⣷
⠀⠀⠀⠀⠀⠉⠉⠉⣁⡤⠾⣶⣤⡔⢲⡾⠶⣬⡻⢦⡀⠀⠳⣽⣯⡳⣿⣿⣵⣮⡓⠦⢭⣭⣥⣴⡴⠋⢀⡥⢖⣻⣯⣭⣙⣿⠀⠀⡆⢻⠾⠈
⠀⠀⠀⠀⠀⠻⢿⡰⣥⣶⢊⣬⣭⣽⣦⠙⢄⣤⣝⠶⣌⡑⠶⢽⣿⣻⣿⡻⡶⠤⠭⠭⡤⢴⣭⠋⠀⣰⡋⡭⢠⣦⣭⣭⡭⢦⣤⣶⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠁⠈⠓⠚⠉⠓⠂⠉⠛⠶⠦⠭⠝⠛⠚⠉⠓⠚⠛⠛⠋⠀⠈⠉⠁⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⠁⠀⠉⢆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠈⠉⠉⠳⣍⠉⠒⣄⠀⠀⢸⠀⡇⠀⠀⠀⠀⢨⠀⠇⠀⠀⢀⠔⠚⣩⢝⡉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠖⠉⠀⠈⠆⠀⠘⡆⠀⢀⣃⣧⣀⣀⣀⣀⡎⡜⠀⢀⡔⠁⠀⡔⠁⠀⠈⢢⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⢸⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⢎⡀⠀⡜⠀⠀⠀⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠊⠀⠀⠀⠀⣠⠚⠁⠀⠀⢀⡠⠤⠐⠒⠒⠂⠤⣀⡀⠀⠀⠀⠉⢲⡁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠎⠀⠀⠀⠀⡔⠁⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⡜⠀⠀⠀⠀⡜⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃
⠀⠀⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⢰⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠸⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⡌⠀⠀⠀⠀⡞⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢻⠀⠀⠀⠀⠀⢡⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡌⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢲⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⢠⠁⠀⠀⣀⠠⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠄⢀⣀⡀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⠴⠾⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠗⠒⡄⠀⡇⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⡜⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠁⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠃⠀⠀⠃⠀⡇⠀⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠄⢰⠀⠀⢰⠁
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠁⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⠀⢸⠀⠇⡘⠀⠀⡸
⠀⠀⠀⠀⠈⡄⠀⠘⡄⡇⠀⠀⠀⢸⠀⠉⠉⠈⠉⠉⠁⠀⠀⠐⡖⠒⠒⢐⡆⠈⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⢸⠀⣷⠁⠀⢠⠃
⠀⠀⠀⠀⠀⢃⠀⠀⠱⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣠⠃⠀⢀⠏
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⣄⡇⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⢃⠀⠀⠀⠉⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⡦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⢀⠀⠀⢶
⠀⠀⠀⢠⠀⠁⡀⠀⠚⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠦⠴⠴⣦⠖
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⡞⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣤⡶⣦⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠂⣤⡤⠤⠀⢀⣶⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠁⠀⠀⢄⢸⠳⣾⣗⠇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠐⠒⠒⠖⠒⠖⠒⢛⢛⢓⠒⢺⣟⠀⠀⠀⠀⠁⠒⠚⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡨⠼⠚⠀⠀⠀⠀⠈⡉⡈⡘⡀⡘⠀⢂⡁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣷
⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠐⠐⠈⠈⡀⣐⣀⣀⣀⣠⣿⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠑⣄⠢⡈⢻⠀⠀⠀⠠⣸⠀⠀⠀⠠⣤⣀⣨⡿
⠀⠀⠈⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⣈⣺⠀⠀⠀⠀⢸⣷⠀⠈⠛⠳⠧⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⣀⠤⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠚⡻
⠀⠀⠀⠀⠀⢄⡠⠤⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⣴⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⡀⠤⠥⠤⠬⣙⡆⠀⠀⠀⠀⠀⠀⣾⡇⢀⠩⠐⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠉⠰⡄
⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⣰⡿⠀⠃⢀⡴⠒⣈⣉⢲⠀⠀⠀⠀⢠⡇⢀⠀⠤⠀⣠⣽⣷
⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⠀⢦⣇⠀⠀⠀⢰⠉⠀⢠⠀⣸⠀⠀⢈⣃⣸⠀⠀⠀⠀⢸⠀⢰⠖⠒⠐⠂⡟⢿⡄
⠀⠀⠀⢩⡖⠀⡠⠠⠄⠀⠀⠈⡟⡀⠀⠀⣿⣀⠈⠛⠂⡇⢠⠀⢠⠂⢀⢴⠀⠀⠀⠛⢠⢀⡄⠀⢠⠀⡅⠀⣿
⠀⠀⣏⣸⡇⠀⠑⠄⡀⠄⠀⠀⣇⠁⠀⠀⠉⠛⠀⠠⡆⠀⣄⡶⠂⠀⠙⢺⠀⠀⠀⠀⢀⢂⡀⠀⠠⠄⣃⣨⡽
⠀⠀⠈⠈⠃⠠⠀⠠⠀⠀⠒⠂⠉⠀⠀⠀⠀⠈⠓⠤⠇⣄⣏⠀⢈⡤⠴⠊⠀⠀⠀⠙⠉⠈⠀⠤⠤⠬⠥⠞⠁

⠀⠀⠀⠀⠀⣀⡤⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣒⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣀⡀
⠀⠀⠀⢠⠏⢩⠖⠒⠲⣿⠿⣦⠀⠀⠀⠀⠀⢠⠎⠕⣒⣒⣒⠫⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⢸⠀⡇⠀⣀⠠⠉⡀⠬⣇⠀⠀⠀⠀⡘⢘⡏⠀⠀⠀⡿⡕⡇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠨⡇⡇⡿⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⢀⡿⠃⠀⠀⢸⢸⢀
⠀⠀⠀⢀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⡀⠀⠀⠀⢠⡈⠺⡄⠀⠀⠀⢸⡇⠀⠀⠀⣸⢼⠈
⠀⠀⠀⠺⢻⡧⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⡼⠂⠐⠇⠀⠀⠀⠈⡇⠀⢱⠀⠀⠀⠸⡇⠀⠀⠀⠉⢸⠀⠠⠔⣎
⠀⠀⠀⠀⠀⢸⠸⠟⢀⣤⣤⢷⣾⠀⠀⠀⠇⡀⠀⠀⠀⠀⠠⠜⠁⠀⠸⠂⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠣⠤⢾⠀⠀⠀⠩⠛⠋⠁⠀⠀⠀⠘⠓⠶⠶⠤⠴⠶⠶⠖⠛⠀⠀⠀⠀⠀⠚⠁⠀⠠⠷⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⠶⠶⠛⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠟⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠀⠈⠉⠉⠉⠉⣉⣁⡀⠀⣀⢀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢽⣿⡇⠀⠀⠀⠀⢰⣹⣿⣸⠀⠻⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣽⣿⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠸⡄⡞⠉⣭⣭⣭⣄⢸⠀⡏⠀⠐⠀⠐⢸⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⡇⣸⣿⣿⣿⣿⠘⡀⡇⠢⠀⢴⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠻⠿⢿⣿⠿⠆⡇⢇⠀⠀⠀⠀⠄⠿⡿⢻⡟⠛⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⢈⣒⣒⣲⠶⠎⢍⣁⣨⠭⠭⠽⠿⠟⠛⢇⠸⡇⠀⠀⠈⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⢠⣀⠤⡀⢸⡇⠀⠐⣄⠐⠆⠀⢸⠀⣷⠀⠀⠰⣶⣿⣿⣿⣿⣤⣤⠄⠀⠀⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⡠⠂⠈⠐⠀⢸⣷⠀⠀⠀⠗⠦⠄⠸⠄⠛⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⢠⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢻⡇⠀⠶⠇⠀⠈⣿⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⣸⣿⣿⣿⡿⢃⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠘⠦⣤⣤⣤⣶⣖⣚⢉⣉⣉⣩⣭⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿⠟⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⢀⡤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠐⠂⠀⠐⠂⢸⢸⠈⠁⢸⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠻⠃⠀⢸⠈⡠⠦⠆⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣅⡘⣒⣒⣒⣒⣊⣉⣀⣓⣀⣭⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣮⡻⢿⠿⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⡟⠉⠛⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠀⠀⠀⠐⠚⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢻⣷⡄
⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣷⣌⡻⣆⠀⣀⠠⠄⣀
⠀⠀⢄⠀⠀⡠⠄⢸⣿⣷⣏⢷⣽⣿⣮⡇⠀⠀⠀⠈⣷⠞⠉⠑⠢⣄⠀⢀⡀
⠀⠀⠀⠑⠄⣈⠀⣸⣿⡟⣟⣸⢿⠇⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⠋⠁⠉⠑⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣶⡿⠄⠛⠠⠤⠄⣀⠼⠄⣄⡀⠀⠀⡏⠀⠀⠀⠀⣸⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠙⠛⡟⠀⠀⠀⠀⠀⠸⠁⠀⠀⠠⠁⠀⢸⠁⠀⠀⠀⢠⡏⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⠀⠲⡀⠀⠀⠀⠀⠰⠧⣀⣴⢇⡜⠀⡟⠀⠀⠀⢀⡞⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠲⣤⣀⣀⣀⣤⣴⠋⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⣴⣷⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⡿⡟⠹⡿⢶⢶⣿⣷⣴⢶⣯⣀⣀⣠⣾⣽⣏⢻⣾⣿⡿⠋⣡⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠞⠁⠃⠀⠀⠈⡇⠛⢿⡀⠀⠉⢍⣍⠙⢿⣯⣻⣫⣾⡿⠀⣮⡈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⢢⠈⠻⣿⣿⣿⡇⢸⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⡄⠀⠀⠀⢡⡀⠙⠹⣿⠀⠘⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⡀⠀⢀⣀⣼⣣⠀⠈⠀⠀⣀⠀⣟⠦⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠙⠛⠛⠛⠋⠥⢤⣤⣤⣤⡼⠋⠦⠖⠚⠙⠓⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠙⠒⠦⠤⠴⠖⢦⡀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠴⠁⠀⠄⠐⢿⣿⣦⡀⠀⠀⠀⣀⠠⣴⣷⣦⣤⣆⠄
⠀⠀⠀⠀⠀⣿⣃⡭⢧⣀⠀⠉⠛⠿⣷⣶⣤⣤⣾⣿⣾⣿⣻⣷⣤⡀
⠀⠀⠀⠀⢸⣿⢋⡀⠀⠈⠳⣦⠀⠀⠐⠛⣿⣿⣿⣿⣯⣿⣿⣿⡿⣿⣿⣶⣤⣤⣀⡀
⠀⠀⠀⠀⢸⣿⣼⣷⣄⠀⠀⠘⢷⣴⣶⣾⣯⣽⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠰⣽⣿⣿⣿⣷⣶⣤⣮⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣙⣛⣻⡿⠿⢿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣙⠛⠻⠿⠿⣿⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣟⣻⡟⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣘⡛⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣽⣿⣓⡶⢮⠭⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠿⠿⠿⠛⠛⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣿⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀⠈⠙⠿⠟⠀⢼⠘⣄⢸⣧
⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⣤⣀⣀⣼⣿⣴⣿⡆⢀⡀⡇⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠀⠀⠂
⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠊⠁⠀⣀⡉⠛⢿⣷⠘⢿⣿⠈⠀⠊⠀⠀⡃⠀⢀⡀⣀⠀⠀⠀⠀⠀⠐
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢋⣴⠋⠀⠀⣼⣿⡯⢡⠀⠀⢹⣦⣸⣿⡄⠀⠀⠀⠰⠛⠛⣿⣿⣿⣁⠀⡀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖⡒⢀⣴⠞⠁⢰⠘⠀⡀⠻⠿⠛⠁⠀⠀⠘⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣷⡀⠄⠀⠀⠀⢀⣸⢆⡀
⠐⠠⠿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⠟⠉⠀⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣾⣿⣠⣀⣀⠀⠀⠀⠗⠙⣿⣿⣿⣿⣿⣏⣀⣤⣾⣿⣿⣿⣦⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡯⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡇⠈⠉⠁⠹⣆⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠣⣂⡀⢸⠳⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣸⡇⠀⠀⠀⠀⠘⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠷⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⣦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⢮⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠋⣿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⡀⠙
⠀⠀⠀⠀⠀⠒⠶⣶⣤⠤⢤⣤⣤⣶⣿⠆⣀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⣰⡇⣸⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢯⣭⣭⣭⣤⣨⣛⣳⢦⣄⣤⣴⡍⢀⠤⠞⠋⠰⠁⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⣩⠟⢠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⠛⠛⣋⡈⣩⣌⣁⢀⣀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣵⡆⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣌⣛⠛⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣄⣸⣦⡀⠀⠀⢸⣿⣿⣿⡿⠿⠛⢛⣉⣀⣨⣭⣭⣴⣶⣶⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⠈⠂⠀⠀⠀⠀⠀⠀⠠⠐⠁⣭⠟⠀⢀⠀⠀⠀⠀⠐⢻⣷⠀⣾⡿⣿⣿⣿⣤⣤⣤⣾⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⡻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠈⣷⠾⠄⠀⠀⠀⠀⠇⢀⡀⠐⠀⠀⠀⠀⠰⠂⠉⠀⠚⠓⠸⢫⣤⡛⡉⠁⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠁⠋⠀⠀⢀⠁⠀⠀⢀⡒⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢡⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢾⣿⣿⡟⢭⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠂⢀⠀⠀⠀⠟⠲⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠞⠃⠿⢿⢦⠹⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢦⣴⣿⣷⣤⣬⠋⠐⢦⣶⠶⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢽⣷⡄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⣀⡀⠼⠅⠀⠉⣦⠟⠓⠤⣺⣟⣻⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠐⡀⠸⣿⠄⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃⠈⠀⠀⠀⠀⠈⠉⣽⠿⠭⢤⣀⠀⠀⠀⠀⠤⣄⡰⠁⠀⠈⠀⣐⣤⣤⣴⣶⠷⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠵⣈⣁⣀⣈⠽⠋⠀⠁⠀⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠤⠄⣐⣒⣒⣒⣒⣒⣒⣠⣤⠤⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠼⠿⠒⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⠒⠲⠬⠭⣒⡲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠊⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⢦
⠀⠀⠀⠀⢦⠣⡈⠲⣄⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⠀⠐⠒⠐⠒⠂⠀⠀⠀⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢈⡇
⠀⠀⠀⠀⢸⠀⠈⣳⣄⡙⠲⣤⡒⢉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠄⡀⠀⣀⠴⢋⢼⠃
⠀⠀⠀⠀⣇⠀⠓⠂⠬⣍⣙⡶⠁⠀⠀⢀⡉⠴⢶⠤⠤⢤⠄⠠⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠄⠄⢚⣩⠤⠚⠁⢸
⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡾⢁⠀⠀⠀⠭⠽⠶⠶⠦⠤⠤⠤⢤⣀⡀⠉⠉⠀⠛⣍⡡⠤⠔⠒⠂⠉⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠚⠛⠉⠉⠛⣶⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠻⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣀⠀⠀⠀⢸⢋⠄⠀⠀⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠑⠢⠤⣄⣀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠚⢦⠀⠀⠀⠀⣶⡒⠠⠿⡆⢸⠼⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠙⡟⠁⢸⠀⠀⣀⡠⠤⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⢦⣿⣟⡶⠲⠦⠤⠤⣤⣄⣀⣀⣀⣀⣤⠤⠤⢴⡧⠀⢸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠂⠤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣰⣭⣜⣦⠀⠀⠀⠀⢀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠻⡀⠰⠠⠀⠀⠀⠀⠐⠊⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠀⡉⢉⠉⠁⠘⠂⠶⡀⢀⠊⠀⠀⠈⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠕⠀⠀⠀⠂⠠⢀⠀⠀⠀⢀⠀⠈⠢⡀⠀⠀⠀⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡄⠀⠘⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠁⠀⠁⢻⠄⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠉⠀⠓⠐⠠⠄⠠⠀⢀⠈⠗⠀⠠⠀⠈⢁⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⡀⠀⠀⠇⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡎⠰⡆⠁⠀⠀⠰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢙⡇⠈⠀⠠⠉⠀⠁⡀⠀⢀⠊⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠁⢸⡀⠈⢀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡐⠂⠊⠀⠀⢠⣖⠀⠀⠀⢸⣷⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠂⠀⣐⢫⡘⢣⠈⠹⡿⡘⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⣀⣀⣀⠀⠀⠔⠀⠀⠀⠀⠘⢫⡰⠂⣐⢓⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⡠⠒⠠⢍⡽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢮⡷⢒⡤⢮⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡠⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⢛⣿⣶⣦⣄⣀⣀⣀⣀⣀⣀⣠⣴⣶⣿⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⣴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠷⣦⣄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣠⡾⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢷⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣰⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢻⣆⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣼⡏⠀⣰⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⣆⠀⢹⣧
⠀⠀⠀⠀⠀⣰⡟⠀⢰⣿⡟⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢻⣿⡆⠀⢻⣆
⠀⠀⠀⠀⢠⣿⠁⢀⣿⣿⡇⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⢸⣿⣿⡀⠈⣿⡄
⠀⠀⠀⠀⣾⠇⠀⣾⠇⣿⡇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢸⣿⠸⣧⠀⠸⣷
⠀⠀⠀⢰⡿⠀⢰⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣿⣇⣨⣿⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡆⠀⢿⡆
⠀⠀⠀⣿⠃⠀⣿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠙⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⣿⠀⠘⣿
⠀⠀⢸⡟⠀⣸⡟⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢻⡇⠀⢻⡇
⠀⠀⣿⠇⠀⣿⠃⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⣿⠀⠸⣿
⠀⢠⣿⠀⢰⣿⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⡇⠀⣿⡄
⠀⢸⣿⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣆⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⡇
⠀⠀⣿⠀⠘⣿⠀⢸⣿⠀⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠀⠀⠀⠀⠀⣿⡇⠀⣿⠃⠀⣿⠁
⠀⠀⢻⣇⠀⢻⣇⢸⡇⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⢸⡇⣸⡟⠀⣸⡟
⠀⠀⠈⢿⣆⠀⠻⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠟⠀⣰⡿⠁
⠀⠀⠀⠈⠻⣷⣄⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣠⣾⠟
⠀⠀⠀⠀⠀⠈⠙⠻⣷⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢀⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⢀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢠⢒⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠕⢿⠓⠚⠑⡀⠀⠀⠠⠂⢉⣥⠛⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣀⠀⠿⠸⠨⠇⢀⡀⢁⠠⠂⠀⡠⠀⠀⠑⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠋⠙⠒⠒⠀⠤⠀⡀⠀⠈⠙⠣⣀⠰⠀⠀⠀⠀⠐⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠁⠀⠀⠈⠁⠂⠤⡀⠀⠀⠠⠀⠀⠀⠱⡀⠀⠀⠀⠀⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣸⣶⡀⠀⠘⡄⠀⠀⠀⠘⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⣀⡀⠐⡀⠈⠈⡷⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠂⠉⠃⠠⠀⡀⠀⣀⢀⠀⠉⠷⠀⠐⠀⠀⠙⠃⠀⠀⢀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠽⠀⠀⠀⠀⠆⠀⠀⠩⠀⠀⠈⡀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠃⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠃⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡤⠀⡀⠀⠀⠀⠈⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⡄⠀⠸⣇⡈⡕⢰⠀⠀⠀⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣀⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠇⠀⡔⠀⠀⣈⢢⠀⠀⡈⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠈⠀⠀⠂⠅⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⡄⠀⠀⠀⢸⣇⢀⠀⣠⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠂⠠⢈⡀⠂⠊⠁⠀⠀⣤⣀⠀⠀⠀⢨⣿⠏⢰⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠆⠀⠀⡗⣇⠔⢖⡖⠲⣶⡆⠖⠐⠦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠑⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠷⡏⢢⠌⠑⠆⣼⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣇⡏⠃⡤⡋⡐⢒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⠳⣌⣀⠈⠁⠟⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠲⢀⡀⠄⢀⠀⠀⠀⣀⠄⠀⠇⠈⠛⣭⣴⣞⣩⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠁⠠⠄⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠯⣷⡄⠀⠀⠀⠀⠀⠀⠀⡞⡾⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠘⡇⠀⠀⢀⣶⣶⣶⣖⠛⡗⣒⣛⣿⣫⡽⠿⣤⠀⣀⣤⡀
⠀⠀⢸⣿⣏⣉⡉⡏⠉⠉⠉⠉⡏⠉⣹⣟⣿⡿⠛⠻⣿⠋⠉⠁⠉⠀⣰⠦⣿⣺⣧⠀⢹
⠀⠀⢸⠉⠉⠉⠉⠋⠉⠉⠉⢩⣯⣭⣭⣽⣿⣤⣤⡤⢼⠤⠤⠤⢿⣯⣽⣰⡇⠈⢹⡗⠾⣄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⡇⠀⠀⠀⣸⣀⣀⣀⣸⣾⡀⠙⠷⢤⣸⠁⠀⢀⣩⣷⣦
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠏⠉⠉⠉⠛⠉⠉⠉⠉⠉⡇⠀⠀⠀⠙⠓⢤⡼⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠢⣄⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠦⣄⣿⠗⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣂⠄⠀⢈⢈⠁⠀⠉⠳⣬⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⣹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡼⠷⢒⣲⣿⠶⢿⣛⠦⣤⣤⣤⣤⣤⠤⠖⠋⠁⠀⠀⡰⠋⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠫⡫⡀⠈⢳⢄⠀⠱⡵⡈⠙⠲⠦⢤⣤⣤⣤⡤⠶⠋⠁⣀⢔⡽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⢘⠜⡦⠀⠡⣦⢄⡞⠧⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⠋⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠴⢞⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠬⠥⠂⠢⠭⠊⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⢃⡜⠁⠀⠀⠈⡇
⠀⠀⢀⣠⠖⠋⠁⠀⠀⠈⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠀⠀⠀⠀⠀⡇
⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢈⡆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠼⠴⠧⠴⠶⠒⠒⠒⠓
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡜⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠴⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⢨⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣀⣤⡤⠴⢶⠖⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉
⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠋⠉⠉⠀⠀⠀⠀⢸⡒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⡀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⠿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠲⠏⣀⡸⠀⠀⢸⠀⠀⠀⢝⠁⠐⠀⠀⠀⠀⠀⠀⡀⠩⠀⠉⠉⠁⡩⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠢⡀⠀⣠⠆⠀⠀⠈⢧⢀⡀⠁⠀⠀⠀⡰⠀⠀⠀⠈⠁⠀⠦⢀⠖⠈⠄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠑⠂⠥⠤⠘⠂⠀⠀⠀⠀⠀⠀⠠⠀⡜⢀⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡔⡎⠀⡖⠁⠆⠀⠀⠤⠒⠒⠒⡆⠀⠀⠀⠀⠀⣠⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄⠔⠈⣆⢈⡿⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠳⡄⠀⠀⠀⠀⢀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⢸⡖⠂⠁⡄⠀⠀⠀⠀⠀⠀⠁⠀⢢⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠌⣏
⠀⠀⠀⠀⠀⠀⢸⡁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢤⣄⣀⡀⠔⠀⠀⠀⠀⡆⠈⢀⣤⠀⠀⠸⣼
⠀⠀⠀⠀⠀⠀⢸⠷⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠁⡌⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠖⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⠀⠀⠀⠀⠀⡀⠀⠈⠀⠓⠚⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠘⠆⠀⠆⠠⠀⠀⠀⠀⠀⠆⠣⣄⡁⠀⠀⢀⡔⠋⠉⠁⠉⠑⢄⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡜⢠⠔⠠⡀⠀⠀⠀⠈⣠⠔⠚⠁⠀⠀⡌⠀⠈⢢⣆⡠⠠⠀⡇⠀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠁⣇⢀⠆⠀⣐⣤⣤⠀⠘⠶⠀⠀⠀⠀⢇⡱⠀⢨⠏⠂⠀⣰⠇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠈⠿⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡶⠎⠦⢤⡾⠊⠀⠘⠶⠄

⠀⠀⠀⠀⡧⡷⢗⢛⠡⡠⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠐⠂⠈⠁
⠀⠀⠀⠀⢱⠏⣛⣁⠬⠔⠒⠸⠀⢂⠠⠄⠀⠒⠈⠈
⠀⠀⠀⠀⢘⠉⠁⠤⠐⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣻⡿⣿⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠠⣤⠠⠠⠀⠀⣿⣛⣷⡿⠫⣿⣿⣼⠀⠀⠀⠀⠐⠒⠓⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⣰⡂⠀⠀⠀⣯⣭⣭⣤⣬⡍⢙⡙⠀⠀⠀⠀⢰⡂⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡤⡄⠀⡷⢶⡟⢻⣿⠖⣾⡿⠀⠀⠉⠁⠉⠉⠉⠉⠉⠁⠀⠀⢰⠆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠉⠁⠀⠀⠀⠐⡆⠀⡀⠀⣿⣿⡟⠉⢹⣸⣿⡆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣃⠀⠁⠀⣿⣿⡇⠀⠸⣿⣿⣷⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⢅⡀⠀⠀⠋⠙⠛⠀⠀⠻⠿⠿⠲⠶⡤⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⡀⠀⠀⠀⠀⠀⠀⠏⠀⠙⠀⣐⣶⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⡀⠀⠊⡽⢿⣝⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡘⠳⠶⠾⠋⠈⠉⣀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠙⠛⠛⠻⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠓⠒⠶⠾⠿⠃⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣇⣀⡀⠀⠀⠲⠶⠶⠶⠶⠟⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣦⣴⠟⠛⠉⢙⣧⠀⢤⣤⣤⣤⣤⣤⠐⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠲⠶⠶⠶⠶⠶⠾⢿⣿⡇⣄⣀⣀⣀⣀⡄⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠐⠲⠶⠶⠖⠃⢘⣿⡃⣀⣈⣉⣉⣉⣥⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠙⠛⠛⣛⠁⢸⣿⠃⢠⣀⣉⣉⣉⣤⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣉⣉⣉⣉⡀⢸⣿⡀⢀⡉⠉⠉⠉⡁⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣉⣉⣩⡀⢸⣿⡇⠈⠉⠛⠛⠛⢋⣘⣿⣦⣤⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣀⣤⣄⣈⣿⣧⠴⠶⣶⣚⠛⠉⠉⠀⠀⣀⠤⠄⢀⣉⡛⠳⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣃⣤⠤⠴⠶⣾⣛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠙⠶⣶⣬⣀⡀⠉⠛⠛⠑⠠⠄⣀⣌⣭⡙⠻⠷⣶⣤⣄⡀
⠀⠀⠀⠒⠒⠋⠉⣉⠶⢤⣐⣪⠭⠴⠛⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣀⠐⣀⠼⠶⠶⠞⣀⣀⡤⠤⠀⣥
⢸⡀⠐⠦⡀⠀⠳⣍⣹⣷⡷⢄⡀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⣿⡴⠖⢚⣋⣩⠭⠖⣒⣋⢉⣿
⠸⣧⣀⠀⠈⠓⢤⡀⠈⠉⠓⢲⠞⣍⡛⢶⣤⠤⠒⣊⠡⠤⠉⠂⢀⣀⣠⡤⠶⠖⠛⠉⠉⠀⢸⡇⠀⠀⠄⠒⠊⣉⣡⡤⢼⣿
⠀⣇⠈⠳⢤⡀⠀⠈⠓⢦⣀⠀⠳⣬⣿⣧⡿⠴⠚⣁⣤⣴⡶⠟⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠠⠴⠒⠋⠉⣁⣠⣤⣾⣟
⠀⣇⠙⠦⣄⠙⠳⣄⡀⠀⠈⠙⠲⠄⣀⣤⡴⠾⣛⡉⠁⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠤⠶⠒⠛⢋⣉⣥⡤⣾⣟
⠀⠋⠙⢦⣈⠑⠦⡀⠙⠦⣄⠀⠀⠰⡿⢖⣚⣩⡤⠴⢒⣋⣭⠤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⠶⣖⡻⠭⣙⣠⣤⣽⣿⠿
⠀⠀⠳⣦⡈⠓⠀⠈⠐⠀⠈⠙⠢⢤⣿⣉⣭⡴⠶⠛⠉⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣭⣽⣶⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⣀⣠⣤⡶⠂⢸⡇⠀⣀⣀⣤⣴⣶⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠈⣿⡤⠴⣒⠿⢛⡉⠁⣀⣤⣼⣷⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⢸⣿⣠⡤⠶⣚⣭⣶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣏⡥⠖⠛⠉⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⣰⡿⠿⡛⠛⡍⠙⠒⠒⠒⠒⠶⠶⠶⠲⠶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠺⠓⢦⡀
⠀⠀⠀⠀⢸⣏⣅⠐⡁⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⠤⢄⡀⢯⡉⠀⠀⠀⠙⢦
⠀⠀⠀⠀⣰⢿⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠷⠀⠀⠂⠉⠻⠀⠀⠀⠂⠀⠀⠱⣄
⠀⠀⠀⠀⡏⡀⠀⠀⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡷⠄⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠛⣿⠀⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠙⠦⣀⠀⠀⠀⠀⠪⣡⡀⠀⠀⠀⠀⠃⠈⠳⡄
⠀⠀⠀⠀⠀⣿⠀⠀⡄⠀⠀⠐⠀⠖⠀⠳⡀⠀⠀⠀⡀⠀⠀⣆⠙⢦⡀⠀⠀⠈⠑⢤⡀⠀⠁⠂⠉⢦⠀⠀⠠⣢⠀⢀⡽⠂
⠀⠀⠀⠀⠀⢻⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠡⣀⠙⠶⡄⠀⠀⠀⠈⠳⣄⠀⠀⠈⢳⡀⠀⣁⠗⠋
⠀⠀⠀⠀⠀⠘⠷⣤⣬⣦⣤⣤⡄⢠⡤⠤⠤⠶⢶⣄⣀⠐⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠓⢄⣰⠋⠳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣰⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣇⡔⠓⠒⣦⠀⡠⢤⣀⡀⡟⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⠏⠀⢀⠀⣿⠞⠀⠀⠈⢻⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⢠⠂⠀⠉⠀⢀⡄⠀⠈⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣀⣡⢿⡀⠁⠀⢰⢿⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠁⠘⠦⣀⣀⠞

⣀⣀⣄⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⣭⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣀⣀⣤⣁⠚⠛⠛⠛⠳⠾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠉⠉⠉⠉⠉⠉⠱⠂⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠢⢄⡀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⠀⣲⠂⠈⠉⠉⠛⣮⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣡⠃⠀⠀⠀⠀⠀⡏⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠒⠀⠤⠤⠤⠶⠗⠇⠀⠀⠀⠀⢀⣼⠃⠀⢀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣥⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠦⠤⢤⡤⠤⠖⠒⢲⠖⠒⠒⠒⠒⡶⠒⠒⠒⠒⡶⠶⠶⢴⡦⠦⠴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡼⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢳⠀⠀⠀⢧⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⢣⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⡆⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠶⠺⠒⠒⠒⠒⠒⠒⠺⠒⠲⠶⠶⠶⢿⠶⠶⠒⢠⠛⠒⠒⡒⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⣼⡇⠀⢀⣶⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⣀⠀⠀⡸⢇⣀⣀⡼⣄⣀⣀⣷⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠋⣿⠋⠉⠀⠉⠉⣹⣉⣉⣉⣀⣉⣉⣉⣀⣈⣒⣚⣁⣈⣉⣉⣀⣸⡁⠀⠀⠀⠈⣷⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⡿⠤⣀⠀⠀⢸⠉⠏⠉⠈⠀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠉⣉⠙⠉⠉⡇⠀⠀⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠉⣉⣄⢀⣄⣸⡄⠀⢸⠐⡮⠀⣇⣀⣎⡳⢳⠞⠦⠃⢿⡦⠸⠻⠌⢓⣦⠀⢠⠇⣰⠚⠋⠛⢿⢤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣵⢰⣾⠁⠈⠉⢈⠿⠼⠀⢸⣄⡥⣤⡭⠤⠤⠴⠶⠲⠶⠶⠖⠒⠒⠒⠒⠚⠒⠒⠾⢘⣏⢠⣤⡴⣄⣤⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠸⣄⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢰⠛⡄⠀⠀⣸⡇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⢁⣀⠀⢠⢦⡀⠀⠀⠀⢀⣠⣤⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⢀⢶⡀⠸⣼⠀⢻⢲⡖⠋⢇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠛⢊⣙⠒⠓⣤⠿⡴⢲⣀⡀⡧⠴⣏⢀⡤⣀⠹⣄⣀⣀⡴⠶⢋⣦⢷⠴⣄⣴⢺⠜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠑⡦⢣⡠⠇⠐⢇⡤⢤⠈⢁⣽⣅⢖⡌⣻⢶⡚⢀⡏⢀⡟⢥⠀⣬⣹⣄⠀⠁⠸⡄⠀⠡⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠙⠣⠤⣶⠦⠴⠋⣛⠉⣆⢈⣿⠮⣉⡴⣇⣠⠟⡟⣉⠙⠧⣏⠛⠁⠀⣉⢧⣤⡖⠳⠼⠶⣏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡸⠽⢀⠈⢳⠀⠸⠭⢇⠈⣻⠣⢴⠡⡏⣝⠹⣆⠙⠋⢀⡶⣏⣰⣀⣶⡼⠀⠀⡳⢀⠀⡘⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⣇⠀⣸⡟⢛⣶⠞⣆⡸⠉⢳⡠⠾⣆⡑⢚⣡⠋⡵⢚⡛⣿⠀⠀⢠⠿⢶⠀⡀⣱⠊⠒⠉⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡁⣽⠋⠁⠻⢤⠃⢸⡃⢠⣙⣇⢯⡀⣤⠉⢻⣆⠑⠔⠋⡸⠛⣤⣾⣋⡈⠋⠉⣯⣤⠀⡀⢀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⣸⢯⠿⣤⠜⠳⠔⠓⠶⠋⠈⠒⠬⠤⠠⠜⠀⠉⠉⠉⠀⠀⠀⠀⠙⠦⠚⠓⠁⠘⠖⢹⠇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⢀⢠⡤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⢤⡤⠤⠤⠤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⠀⠀⠸⡆⠀⠀⢡⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⡄⢀
⠀⠀⠀⠒⠒⠉⠁⠉⠙⠁⠓⠒⠚⠓⠦⠤⠼⠆⠒⠒⠚⠓⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠛⠒⠒⠒⠚⠓⠂⠉⠙⠹⠦⠤⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⢟⣻⣛⢳⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠻⠿⣿⣿⣿⢿⠿⠟⣳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠠⢤⣄⡀⠀⠀⣀⣀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿⠀⠶⠀⢷⠀⢨⣍⡉⢃⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣷⠀⠀⢠⡆⢀⡀⠀⠀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠊⠤⢄⣀⠐⢄⡴⢣⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣆⠀⠀⠓⠀⢀⠎⠔⡱⢈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⢻⠀⠟⢷⣤⣤⡤⣼⣾⢸⣧⣿⠇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣶⣿⣿⡿⠿⠟⢛⡝⢻⡟⢸⠀⠈⢢⡀⠀⣠⣿⠈⠁⡌⠁⢀⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢩⡀⠀⠈⠀⠀⠀⡼⠀⠸⠁⠸⡄⠀⣠⣿⣦⠀⢿⣷⠈⠃⠀⠘⣉⠉⠛⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⡇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣇⠰⢉⣿⣏⠃⢰⣿⣧⡀⠀⣀⢘⢦⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣛⠃⠀⢱⠀⠀⠠⠤⠥⠶⣤⡄⠀⠀⢻⢠⣿⣿⣿⡞⣼⣿⣿⡥⠈⢠⠏⠀⠳⡀⠀⠜⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠃⠈⠙⢦⣸⡀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣷⠂⠐⠻⣀⠀⠀⠙⢌⣠⡾⣿
⠀⠀⠀⠀⠀⠀⣾⡅⠉⠐⢦⡄⣠⣤⠿⣷⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢻⣿⣿⣆⠀⠀⢱⠁⠀⠀⠀⢻⡴⠛⠆
⠀⠀⠀⠀⠀⢰⣿⠐⢄⡀⠈⠙⠳⡜⣻⡿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠹⣿⣿⣆⠀⠀⠁⠈⠀⠀⠀⢷⡀⢨⡀
⠀⠀⠀⠀⣠⠿⣋⣠⣤⣭⣝⡛⣧⣴⣿⡇⠀⠀⣿⠙⡆⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠹⣿⣿⣶⣄⠀⠀⠀⠀⠀⢨⡙⢦⢱⡀
⠀⠀⢀⣌⣵⣪⠽⢿⡧⣄⡈⠉⣸⣿⢿⡇⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠹⣿⡿⣟⢧⡀⠀⠀⠀⠀⢇⠀⢠⡇⡀
⠀⠀⢸⣿⢟⠝⠀⠀⠀⠉⠻⣼⣿⣿⣷⡡⣀⣾⠁⠀⠀⠹⡄⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⣿⣇⡿⣦⡁⠀⠀⠀⠀⠈⡆⣼⢠⠇
⣀⣀⣰⡀⠎⠀⣠⠔⠋⠀⢀⡠⣹⡿⠃⠈⢞⢛⣲⣀⣀⣀⣙⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣠⣿⣿⣯⣄⡹⣷⣦⣀⠀⠀⠀⠀⣱⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠙⢷⣤⣴⠃⠀⢠⠊⢡⢾⣿⢀⡙⢣⠈⠙⠧⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠻⠿⣟⠉⠙⠛⠿⢮⣽⣿⣄⣤⠾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠈⠙⠛⠿⢿⣶⣷⡸⣨⣧⢠⣙⠇⠄⣁⠒⠸⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⠟⠛⠀⠉⠳⠭⠤⠀⠐⠉⠁⠉⠂⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⡀⢤⣄⣒⣦⡠⠤⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⡠⠼⠷⠞⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⣀⣀⡠⠤⠖⠚⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠶⢒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠲⠤⣀
⠀⠀⠀⢸⠦⣄⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡤⢾
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠛⠓⠒⠲⠖⠲⠶⠶⠶⠦⠴⠶⠖⠖⠲⠒⠒⠲⠛⠋⠋⠉⠁⠀⢸
⠀⠀⠀⢸⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⢸⠛⢦⡀
⠀⠀⠀⠈⠑⢲⠦⢤⣄⣤⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⡆⠀⢀⣀⣀⣈⣀⣀⣤⣤⣴⠶⠿⠖⢻⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠉⠙⠉⠉⠛⠉⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣄⣸⢧⣠⠴⢻⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢸⠿⢿⣖⣲⠾⠦⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⠥⠤⠴⠶⠶⠚⢒⣋⣩⡴⠋⠀⠀⢸⡆⠉⠠⡙⢆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠙⠉⠛⠒⠖⠒⠲⠒⠒⠶⠒⠒⠖⠒⠒⡖⠚⠛⠋⠉⢩⠁⠀⣇⠀⠔⠈⠀⢻⡐⡀⠈⠢⠳⣄
⠀⠀⠀⣠⠴⠒⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢈⠀⠀⠙⡄⠀⠀⠠⠀⢳⡐⡀⠀⠀⠌⢦⡀
⠀⠀⠀⡷⠤⣌⣀⡉⠒⠶⠤⣤⣤⣄⣀⣀⣃⣀⣘⣀⡀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣄⣤⡴⣤⠖⠛⠻⡖⠁⠀⠀⠀⢳⡀⠀⠀⠀⠢⠱⣄
⠀⠀⠀⡇⠀⠀⠉⠉⠉⡋⠛⠒⠒⠒⠒⠖⠛⠻⠟⠛⠻⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⣿⠀⠀⠀⠹⣄⡠⠒⠁⠀⢹⡄⠄⠀⠀⠐⠘⢦
⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⡀⠀⠀⠹⡄⢀⠄⠊⠁⠘⣆⢂⠀⠀⠈⠈⢣⡀
⠀⠀⠀⠈⢳⠶⠦⣤⣤⣇⣀⣀⣀⡀⢀⡀⠀⢀⡀⠀⢘⠀⠀⣘⣀⣀⣠⣠⣤⣦⣶⣚⣯⡥⠞⡇⠀⠀⠀⠘⣇⠀⢀⠠⠐⠉⢧⡠⡀⠀⠀⠂⠳⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠙⡋⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⢷⡁⠀⠀⠠⠈⠳⡄⢀⠀⠀⠠⠹⡄
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⠀⠀⠳⡔⠁⠀⠀⡠⠘⢦⠠⡀⠀⠡⠹⡄
⠀⠀⠀⠀⠀⡿⢵⣒⡲⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⣒⣋⡭⣗⠀⠀⠀⠀⠀⠀⠀⠙⢦⠐⠀⠀⠀⡠⠳⣄⢀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠙⠒⠒⠒⠶⠶⠭⠭⠭⠭⠭⠭⠭⠭⠤⠶⠒⠒⠖⠚⠙⠋⠉⠁⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠔⠁⠀⠀⡈⠷⣕⠠⠀⢧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⠉⠀⡠⠘⠳⣤⡼
⠀⠀⠀⠀⠀⠉⠒⠶⠤⣤⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣠⣤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠌⠀⢸⡿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢈⠀⢠⣿⠥⠊⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠠⣿⠑⢄⣶⣿⣯⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⢀⣀⣾⣿⣿⣿⣇⡿⣿⠟⢁⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢷⡜⣿⣯⣯⣉⡉⠙⠛⠛⠛⠛⠻⠿⠿⠿⢶⣶⣶⣶⣆⢸⣾⡛⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣻⡻⠗⠀⠀⠀⠀⠀⠉⠀⢴⣦⣼⣿⣟⣛⣋⣀⢿⠗⠒⠲⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣭⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠿⠟⣫⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣤⣤⣀⠀⢫⠀⠘⠛⠛⣷⣾⠟⣉⣮⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠛⠛⠛⠉⠀⠀⠀⠀⠰⣿⡷⢯⣵⣾⠝⣸⣆⠀⢨⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣹⣿⣿⣷⣶⡄⠀⠀⠀⠀⠘⡄⠀⢄⣀⢛⣵⣾⡽⢫⡀⢸⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠈⣄⠛⠋⣀⠹⢛⣵⣏⣻⣾⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢾⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠘⢢⠘⠛⠋⠈⠙⠛⣵⣦⣜⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣽⣟⠻⠿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠸⠷⠀⠈⠙⠇⢾⣿⣯⠛⠲⠄⠀⠀⣀⣠⣤⣤⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⣿⠄⠀⠈⠓⢀⡀⠀⠀⠀⠀⠀⠀⠉⢳⣶⣄⡀⠀⠀⠀⠉⢫⣤⡔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣷⣇⡀⠀⡼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣥⠄⠀⠀⠀⣿⣄⣀⣹⣿⣷⣾⠟⠛⢍⣿⠏⠠⡀⠀⢩⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢐⠳⣶⢆⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣍⣉⣉⣩⣥⣤⣤⣤⣤⣭⣭⣭⣭⣛⣛⡻⠷⢷⣮⣃⠀⠀⠀⠀⠒⠖⠒⠒⠀⠉⠁⠂⢀⠀⣠⣴⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣶⣶⣭⣍⣛⠒⠠⠤⠀⠀⠀⠀⠀⠀⠁⢒⣈⣭⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣤⣤⣤⣬⣙⠻⢿⣿⣿⣿⣿⣷⣶⢶⣶⣶⣶⡿⣿⣿⣿⡿⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⡟⠛⠛⠻⠿⠒⠀⠈⠦⠤⠿⣟⠛⠻⠿⠾⠿⠿⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⣉⠉⠁⠀⠀⢀⡏⢝⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣩⠘⡯⢫⢟⠱⡲⢶⢤⡤⣀⡀⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⢸⡮⢓⠭⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠘⠂⡯⢔⠥⡪⢖⢕⡠⠪⡪⢙⢝⢗⢲⡇⠀⠀⠀⠀⠀⠀⢸⡪⢐⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢗⠕⣪⣆⠵⠵⠒⠚⠚⠉⢹⢟⡿⢦⣀⠀⠀⠀⠀⠀⣿⡪⠕⣺⠁⠀⠀⣀⠤⠤⠀⠀⠒⣲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣀⡇⠀⠀⠀⠀⡖⡉⠉⠄⠀⠐⠀⠀⠀⠀⠀⣾⠕⡊⢔⠮⣷⢄⠀⠀⠀⡭⣒⠅⡞⠀⠀⡜⢁⣠⡤⢴⣴⣼⣧⠾⠏
⠀⠀⠀⠀⠀⢠⠔⠱⠌⣗⢔⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢍⡪⢔⢺⣇⢕⠀⠀⠀⡯⢒⢭⠇⠀⡔⢁⡞
⠀⠀⠀⠀⠀⠀⢣⠀⠀⢻⣂⣺⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠿⠥⠮⠷⠽⠷⠵⠤⠤⠼⠯⠶⠼⠤⠔⢁⡎
⠀⠀⠀⠀⢠⠊⣉⣍⢭⣍⣩⡉⢉⡥⠤⠤⡤⠤⡄⢀⡴⠖⠒⠲⠀⡖⠒⡶⢖⣶⠀⣶⠚⢛⢛⠀⣰⠞⡺⣶⡲⢀⡞
⠀⠀⠀⠀⠈⠆⠳⣡⠀⠀⢻⢻⡈⣇⠁⠀⡇⠁⠁⢸⣥⠄⠀⢸⠀⡇⢰⡯⣐⡏⢸⣿⢌⠢⡸⠀⣿⠅⣪⡿⢀⡜
⠀⠀⠀⠀⠀⠘⡄⢹⣆⢠⡘⣟⣷⠸⡆⠀⡇⠀⢸⠘⡾⠀⠀⢸⠀⡏⢸⣏⣾⠀⣼⣿⣵⣭⠇⠰⣇⣅⡾⠁⡜
⠀⠀⠀⠀⠀⠀⠘⡄⠙⠛⠓⠛⠛⠁⠙⠒⠓⣒⡁⠀⢉⣉⣉⡁⠀⣉⣭⣭⣁⢀⣤⡤⢤⢦⢀⣤⡶⢦⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣯⣟⣿⢣⢸⡍⡏⠀⢸⠀⡏⠀⠀⢸⢸⡇⢸⡳⡟⢸⢕⡏⡢⡝⢸⢻⢢⡏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⡄⢙⢘⡆⣷⠇⠀⠌⠀⡇⠀⠄⡈⢈⠆⣿⣲⠇⣸⣹⣗⣼⠁⣏⣾⡞⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠒⠂⠃⠘⠓⠒⠚⠀⢈⣀⣉⣁⣈⣙⣉⣉⣄⠬⠭⠥⠤⠤⢤⣤⠠⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣂⣀⣀⢀⣀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⡉⠉⢉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣛⣏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠹⠛⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣥⣕⡼⣷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⡽⣦⡆
⠀⠀⠀⠀⠀⢤⡠⢀⠱⣿⣿⣋⢴⠠⢄⠠⣀⠠⢀⠠⢄⠠⢄⠠⠀⣀⠀⣀⠀⡀⢄⡠⢄⠠⡀⠄⡀⢄⡠⣏⡻⠿⢗⡵⡠⣀⠤⣄⣤⡠⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠁⠈⠀⠈⠀⠈⠀⠈⠀⠈⠀⠈⠐⠀⠁⠊⠁⠂⠁⠊⠁⠊⠑⠉⠐⠁⠊⠁⠈⠀⠉⠉⠉⠁⠉⠀⠉⠈⠁⠉⠑⠉⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣦⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡁⠄⠀⠨⣙⠳⣁⠀⠀⠈⠉⠙⢫⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡌⠀⠀⠀⣼⢸⠑⠄⠑⡀⠀⠀⠀⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠀⢒⡀⠛⢧⠀⠈⠂⢑⠀⠀⠀⠀⠈⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠢⡤⡀⠄⠀⣰⣄⢀⠘⠀⠂⠠⠄⠠⠀⣘⡘
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⡀⢿⣧⢠⠿⢤⡉⢅⢀⡀⢰⡄⠀⠀⠀⢰⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢐⡄⠁⠿⣟⣲⡾⠦⣗⣠⠄⣴⣴⣀⣀⢀⣯⡟⠉⠉⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡟⠠⣆⡏⢠⠀⠀⠀⣨⠀⠓⣾⠃⠀⠐⠢⡈⢳⣤⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⡏⠒⠤⢉⢀⠲⡦⣤⡘⢀⣣⣘⣦⠀⡤⠤⠴⠂⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠷⣀⡏⠀⠠⠠⠤⠀⣀⠀⠊⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠊⠀⠀⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⠀⠀⠄⡠⠀⠁⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠐⠀⠄⠀⡀⠠⠀⠀⠐⠡⠀⠀⠀⠐⠀⡀⠀⠄⠂⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡂⠈⠄⠂⠀⠀⠀⠁⠀⠀⠈⠀⣀⣈⠀⠐⠒⠊⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠒⠒⠒⠐⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⠆⢚⡛⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⢛⡓⠰⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣴⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣷⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡘⠿⠿⢋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠿⠿⠋⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋