⠜⡰⢃⢎⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡱⣙⢮
⢩⠒⡍⢦⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢃⡞⡱⢎
⠣⡍⡜⢢⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠁⠋
⠱⣘⠰⡡⠌⡄⠀⠆⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⣁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠓⣌⠣⡑⠎⡴⣉⠒⡨⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⡱⡘⡬⢄⠤⣉⠦⡙⡜⣢⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠍⣆⢣⠱⢩⠔⣌⠓⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⡐⠈⠆⢰⡙⠶⣩⠖⡱⠎⠷⠹⠳⢟⡳⢦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⠒⣌⠱⢊⡜⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠰⡈⠅⣠⠳⡜⡧⠰⣍⠓⢙⣀⣥⢳⣤⣖⡪⣗⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢡⠚⡤⢃⠧⡘⢆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠘⢦⡡⢗⢦⣔⠉⣀⣬⣵⣾⣿⡿⠛⠉⠁⠉⠈⠒⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⡓⢬⡑⢎⡱⢌⠠⣀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⢦⡰⣤⠠⠁⢯⡘⣧⡿⣻⣿⣿⣿⣟⠉⠀⣈⣤⣴⣶⣿⣦⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⢚⡱⡌⢦⡑⢎⠰⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠱⢫⢷⣽⡳⣩⠀⠀⠦⣹⣿⣿⣿⣿⣟⣲⣿⣿⣿⡿⠿⣿⣿⣿⣮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⢍⠶⣉⠦⡙⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡦⠀⠀⠉⠘⠑⠀⠀⠀⠀⠘⠘⢿⣿⣿⣿⣿⣿⠟⠁⢠⣠⣄⡈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⢎⡣⢇⣚⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠠⠭⠿⠿⢿⣇⠀⠀⠀⠙⠻⣿⣆⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢣⢎⠵⣉⢦⠳⡤⢄⡀⠀⡰⢎⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠰⣿⣾⣿⠿⢻⠂⠀⠀⠀⢸⣿⡿⠀⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣧⢎⡮⣥⢎⢧⡙⢮⡱⢫⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠀⠰⣀⡄⣀⣭⣶⣿⣿⣿⣶⣬⡀⢠⣾⣿⣿⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⢮⢧⢳⡱⣎⢷⡹⡶⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡱⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀
⡏⣎⢷⡹⣈⢷⣹⠱⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⠀⠀⠀⠰⢀⡈⠀⠈⠰⠈⠀⠀⠀⠰⠉⠀⠉⠹⢿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠆⡰⢀
⡽⢎⢧⡳⣹⢎⡗⠀⠀⢀⠶⣄⢆⠶⡹⠄⠀⠀⠀⠀⠀⠀⠀⢅⠂⡐⡄⣀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠈⣿⣿⣿⣏⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⢌⡱⣃
⣝⣫⢮⡕⣣⢟⡼⢣⡖⣎⡳⡜⢾⡜⠥⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⣳⣿⣷⡆⠀⠀⠹⢷⣀⠀⠀⠀⣠⣿⣿⣿⣿⡄⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢂⠱⣡
⡞⢦⣳⡚⣥⢻⣜⡳⣝⢶⡳⡹⢳⣆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣿⡿⠟⠁⠀⡀⠀⠀⢈⠦⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠑
⣝⡳⢦⣛⣬⢳⢎⠷⡎⢧⢳⢱⢣⡜⡁⠀⠀⠀⠀⠀⠀⠀⠀⡘⢼⠋⠁⠀⠀⠀⠀⠐⠈⠄⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀
⠒⣿⣘⠳⢎⠻⣌⠳⡜⣣⢓⢮⡱⠘⠀⠈⠄⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⠀⠀⠀⢈⠀⠘⠄⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⠔⣒⢎
⢫⡔⡣⢏⡎⡳⣌⢳⣉⠶⡩⢖⣩⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⢡⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢮⠹⡜⢎
⢣⡜⣱⢃⡞⡱⣌⠧⡜⣲⢩⡚⣤⢋⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⠹⠯⠽⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢣⢓⡜⢣
⢣⡜⡱⢎⡜⡱⡜⢲⢩⠖⣡⠳⡌⡝⢦⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡄⠀⣍⠲⣌⢣⠚
⠁⠎⠑⡌⡐⢣⠱⣩⠖⠁⠈⠑⠈⠈⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⠀⠀⠒⠀⡰⢌⠲⣌⢢⡙
⠀⠀⡡⠐⢌⠂⢧⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠃⠀⡜⣲⣾⣿⣷⣾⣶⣦⣄⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠯⠁⠀⠈⠀⡰⢑⢪⡑⢆⠦⡙
⠀⠐⠠⠉⢆⡘⠢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠰⣘⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠂⠁⠀⠱⢊⢦⡙⣌⡒⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠂⢌⠷⣿⣿⣿⡇⠱⢻⣿⣿⣿⠀⠀⢠⣿⣿⣿⣟⠷⢞⣳⡶⣆⠀⡀⠀⠈⠂⠻⣼⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⠄⠘⣃⣿⢿⣿⡄⢾⣿⣿⣿⠀⠀⣯⣿⣿⣿⢧⡳⣾⣿⣿⣿⣷⣼⣤⣄⡀⠄⠀⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠈⠀⠌⠹⢾⣿⣿⣸⣿⣿⡿⠀⢰⣿⣿⣿⣯⢯⢷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣳⣦⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠸⣹⣿⣿⣿⣿⠁⢀⣿⣿⣿⣿⣯⣛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿