⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣾⣶⣿⣼⣹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠀⠁⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⢻⣿⣏⣿⡛⢿⡟⢟⣟⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⣽⣿⣿⣶⣷⣼⣿⣾⣿⣿⣿⣿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠐⠲⠤⠤⢀⣀⠀⠀⠀⡇⣾⣿⣿⣿⣷⣖⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣒⠒⢷⣿⣿⣿⣿⣿⠝⠛⠁⠀⣿⣿⣿⣿⡽⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⣀⣴⡚⠉⠀⣉⣽⣿⣿⣿⡟⠹⡇⠁⠀⠀⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⢶⡿⢽⡁⠀⣠⠌⠁⠸⣿⣿⣿⣾⡿⠃⠀⢀⢀⣿⢿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡗⠒⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡠⠄⠒⠛⠩⣄⠔⣫⠄⣠⣷⠞⠓⣤⡤⠽⠿⢿⠿⠿⢿⣞⣀⡈⠉⠐⣿⣿⡷⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⠃⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⡓⠊⣉⢶⠤⠀⠙⡦⡠⠴⡯⠔⠒⠉⠀⡀⢀⢄⠆⣀⡀⠀⢸⣿⠿⡿⠉⠩⠷⢮⣥⠽⢉⡹⣟⢯⣹⣿⣿⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠒⠢⠤⢄⠀⠀⠀⠀⠀⠀⢠⢞⡪⣠⠊⡡⠊⠀⢀⠎⡼⠁⡠⠑⠁⠀⠀⡘⠀⠀⡈⠈⠉⠉⠓⢼⣧⣀⡇⢀⣒⣒⣒⣒⣛⣧⠤⠍⢓⡦⠌⣛⠿⣉⣀⣀⠀⠀⠀⠀⠀⠠⣀⠀⣇⠀⠀⠀⠀
⢂⠘⣧⣷⢹⠀⠀⠀⢸⡆⠹⡰⡳⡒⡄⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⣸⣿⣿⡏⠉⠀⠈⢦⠀⣤⠉⡟⡿⢳⠚⠯⣚⠅⡲⢻⡟⣿⡆⠀⠐⣄⠀⢈⠳⣯⠀⠀⠀⠀⠀⠹⡄
⠈⢳⣄⡇⠈⠐⠀⠀⢾⡱⡀⠙⢷⡙⢮⡢⣱⣆⠘⠦⡆⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠂⣿⠀⡇⠤⠃⠤⡄⠀⡱⠺⣮⣿⡾⡇⠀⠀⢠⢳⡄⠱⡘⣀⣀⣀⠀⠀⠀⢹
⠀⠀⠉⠻⢷⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠙⢮⣢⡅⠰⠊⠀⠀⢁⣴⣿⡛⢧⠀⠀⠀⠀⠀⠀⠉⠀⢿⠀⢠⡞⢡⠞⠀⠀⠘⣿⣷⣷⠀⠀⠈⢸⠈⢶⣳⡹⠀⢧⠙⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠊⠭⠟⠫⠽⣛⢿⣛⠛⠛⠙⢛⡿⡇⠀⠀⣴⣿⠏⠹⡔⢄⠑⢴⠊⠉⠀⠀⠀⠀⡼⠀⠀⡁⠀⠀⠀⠀⠀⣹⣿⡏⢠⠱⢠⢸⠀⠀⡳⣿⡳⠘⠀⠱⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣭⣶⢍⠢⢄⡸⢰⣡⢠⣾⡟⡁⠤⠂⠈⠢⡑⠻⣯⢶⣤⢄⣀⠀⢁⠔⠀⠀⠀⠊⠙⣦⢔⡵⢹⣇⡸⢆⣿⠇⡀⠀⠈⢮⣟⢄⠀⠀⠙⢦
⣀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣻⡎⠃⠀⢱⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠘⢻⠭⠄⠒⠒⠷⠴⢋⠴⡻⠁⠸⣾⠙⣄⡿⣰⠁⠀⠀⠈⢎⡄⢟⢲⡒⠒
⣿⣷⣦⣄⢘⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢦⣸⢀⣿⠇⠀⢀⡠⣄⡀⠀⠀⠤⠤⠐⣒⣤⠄⣀⡀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⣿⡇⣻⠽⣟⠪⠖⠀⠀⠈⡙⠈⢸⣎⢶
⠿⣿⣿⣿⣿⣯⡉⠉⠓⣄⠀⠀⠀⠀⠈⢑⠄⠹⣿⡏⣠⢖⠵⣚⣥⠔⠂⠀⠀⠀⠀⠐⣒⡫⠥⣛⠯⣭⣒⠠⣀⠢⣄⠀⠀⠀⠀⢸⣧⣷⡿⣧⣤⠀⠀⠀⠀⠁⠀⠘⣣⠚
⣄⠈⢛⣿⣿⣿⣶⠾⠿⢛⡇⠀⠀⠀⠀⢬⡑⢼⢺⡇⡝⢑⣯⡪⢅⣒⣫⡥⠤⠤⠤⣌⣓⠚⠹⢶⣭⣀⠉⠛⢦⡑⢄⠡⡀⢀⠀⢸⠏⠀⠙⠒⢭⣀⠀⠀⢠⠤⠒⠊⠁
⣿⡿⠛⠉⢹⣿⣿⣧⣔⠛⡿⠀⠀⠀⠙⠂⠬⣓⣏⣏⣠⣟⡧⡚⠉⠀⠀⠀⡴⠊⠀⠀⢠⠙⢿⡷⠦⢽⡻⡦⢄⠘⢢⡱⡀⠈⠀⡌⢀⡤⠐⣒⣛⣛⣛⣲⣤⣄⡀⢀⡶
⢿⣗⠲⢿⡿⠋⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣴⣏⣘⣃⠀⢀⣠⠾⠁⡠⢄⢀⡿⠀⣿⠀⠀⣶⠈⠙⢿⣿⣖⣷⣡⢀⣿⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣡⠔⠊⣞⠑⢲⡿⠀⢰⠇⠀⠰⣄⠌⢿⣳⣻⣸⡯⠔⡂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣋⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⠥⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠲⡆⣼⠉⢒⡧⢄⢀⠀⠈⢆⠀⣧⢿⢽⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠉⠛⠿⣿⡿⣿
⠀⠀⠀⠀⠀⢀⠴⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡇⢸⠀⣰⣀⠀⠈⢢⢀⣘⡆⠿⣇⣁⣠⠶⠋⠀⠀⠀⠀⣀⡤⣿⣏⠙⠷⣶⣤⣤⡿⠟
⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠀⠁⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠉⠉⠒⠢⢤⣀⡙⠶⢿⣷⣮⣷⣦⣤⡉⠻⣷⣄
⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⣩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⣽⣦⣝⣿⡿⠿⠛⠿⣦⣼⣿⣿
⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢻⣅⢻⣞⠟⣖⢤⡀⣠⠝⠛⠿
⠁⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠰⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢦⢿⣆⢹⡷⢵⣅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠣⠻⣾⠇⣸⠏⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⢄⠏⢴⡄⢀⢹⡃⣿⡦⣧⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣦⡀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢁⡞⠀⠉⠈⢦⣇⠃⣽⣬⣿⢱⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢮⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⠄⠀⠀⠀⢠⣤⣴⣶⣶⡤⠤⠤⠀⠐⠚⠉⠀⠀⠀⠀⠀⢻⣆⢸⣿⡏⣾⡄
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠙⠓⠼⣯⣓⠐⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠁⣇⡹⢱
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠬⠙⠒⠬⣿⢷⠦⣄⡒⠀⠀⠀⠀⢐⡣⠀⢳⡇⠀
⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢀⣴⣿⣿⣏⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⠟⠳⢮⣀⠍⠳⣿⡀⠀⠀⠉⠻⡝⢾
⠀⠙⢿⡋⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⠴⢋⣼⣿⣿⣿⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠊⠁⠀⠀⠀⠉⠛⢶⣏⢷⣄⠠⣄⠀⠘⡆
⣆⠀⠈⢳⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⡖⢫⣾⣿⣿⣿⠹⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠙⣷⡙⣷⡌⢣⠀⢳
⣿⣷⣆⠈⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢺⣿⣿⡿⢻⠀⡿⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⠀⢀⣨⣤⣤⣤⣤⣄⣀⣀⣘⣿⡈⣿⢎⠳
⢿⣿⣷⡀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠓⠦⢾⣿⣿⢅⣸⡾⠀⠀⠀⠴⠀