⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠷⠶⠖⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⠿⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣶⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⡀⠲⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⣿⣿⣶⣤⣄⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⢤⣴⡆⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀


⠀⠀⠀⠀⣠⡶⠙⠗⠩⢐⡒⠒⠒⠲⠖⠀⠀⠤⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⠷⠁⠂⠠⠒⠈⠀⠀⠠⠀⠀⠠⠀⠁⠁⠀⠈⣩⡀⠂⠉⠈⠂⠙⠛⡣⣒⡤⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠂⠀⠀⢉⣞⣅⡄⠀⠀⠀⠀⢤⢦⣮⡺⣿⣷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣈⣽⡷⠶⣶⣦⣴⣦⣶⣦⣄⣀⣀⣀⠀⠁⠁⠀⠀⠀⠀⠀⠠⠀⢫⣙⣾⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢀⡎⠎⠐⡀⠆⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠿⠷⠷⠶⣶⣶⣶⣤⣡⠀⢠⣤⡢⣿⣟⣢⣄⠀⠀⠀
⠀⠀⢸⡇⠃⠀⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠈⠉⢙⠛⠶⣼⡿⡏⢿⣿⣝⣧⠀⠀
⠀⠀⠀⣳⣤⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡠⠠⠀⠀⢀⣀⣠⣦⣷⣢⣽⣤⡙⠉⣻⡟⣻⠀⠀
⠀⣴⠎⠉⠙⠉⠉⠛⠛⠿⠶⣶⣼⣦⣥⣐⣄⠀⠀⢀⡀⠀⠐⠀⠏⠛⢽⠿⢧⢟⣇⢀⡠⢼⣷⣿⣔⡄
⢸⣻⢀⠀⠀⠈⠀⡠⠄⠀⠀⠀⠀⠉⠙⠛⠻⠿⢷⣷⣤⣤⣅⡀⠘⠓⣛⣭⣿⠞⠙⡻⢿⡟⢿⢿⣿⣧
⠘⣆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢾⣿⠋⣤⣜⢫⣆⢷⡾⠧⣿⡏
⠀⠀⠙⠶⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⢃⠔⠠⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣝⡷⡪⣯⡯⣿⣗⣷⠟⠋⠀
⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢐⠙⣛⢿⣷⣿⣷⠟⣳⣿⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠶⠤⣆⣀⡀⠀⠀⠀⠀⠀⠚⠚⢫⡾⣺⠿⠞⠛⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠢⢤⣤⡀⣀⣘⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀


⠀⡔⠉⠀⠀⠀⠀⣀⣠⣴⣾⢿⡿⣿⢿⡿⣿⢿⡿⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠠⡠⣬⣟⣉⣩⣭⣽⣦⣤⣠⣭⣦⣀⠉⠲⡀
⢸⠀⠀⠀⣠⣴⣿⢿⣻⢯⣿⣛⡾⣽⢯⣿⡽⣯⣟⣷⠻⠾⠏⠀⢀⣀⣀⡀⠀⠀⠀⠀⣼⡄⡠⠊⠀⣿⣿⣿⣿⢺⣫⢟⣟⣟⡟⣿⣾⠀⠀⡇
⠘⢀⣴⡿⣯⢷⣻⣯⢿⣻⢾⡽⠿⠙⠛⠉⣉⣁⣤⣤⣶⣶⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⢛⣴⠁⠀⠀⢿⢿⡿⣿⣼⡿⢿⣿⣷⢽⣿⠾⠀⠀⠁
⢨⣾⣯⢿⡽⣯⢷⠿⠛⠉⢠⣴⡶⣿⣿⣿⣿⣟⡟⣏⣾⣷⣾⠾⠞⢛⣁⠀⣠⢤⣤⠀⠀⢕⠷⣄⡰⣿⣻⣿⣿⢹⠺⣯⡵⠴⡱⣮⢹⡀⠀⠀
⢸⣟⡾⣯⢿⠉⠁⠀⠀⠀⠈⣿⣿⣳⠷⠿⠟⣛⣩⠏⢡⡴⠦⣄⡼⠋⢹⣿⠃⣀⡤⠀⠀⠀⠑⡶⠍⣷⣿⣿⣿⣐⣯⣾⣻⣜⣵⣝⣹⠆⡁⠀
⢸⣟⣿⣻⠇⠀⠀⠀⠀⠀⠀⢃⡀⣠⠼⣧⠛⣿⠀⠀⠀⡇⠀⣿⡇⠀⣸⠸⣇⣠⡇⠀⠀⠀⢀⡀⠤⠻⠿⠧⡜⢟⠣⠜⠃⡠⠀⢟⣧⣼⡿⡇
⢸⣟⡾⣏⠀⠀⠀⠀⠀⣾⠀⢸⣷⠃⠀⣿⠀⡿⠀⠀⢈⣇⠼⠁⢙⣊⣥⠤⠔⠂⢁⡠⠒⠉⠀⠀⠀⠀⠠⡄⠡⢭⣩⣭⣽⠷⢿⡿⠛⢿⡇⠐
⢸⣯⢿⣽⡄⠀⠀⠀⢀⡿⠊⣽⠙⠦⠴⠃⠐⣃⣀⡤⠼⢲⠚⠉⢹⣀⣠⠀⠀⠀⠀⠀⠀⢀⠄⡂⠤⢁⣣⣴⣷⢋⠉⣿⡏⢠⠀⡀⣿⡜⡅⡌
⢸⣯⣟⡾⣽⣦⣄⡀⠘⠃⢀⡡⠤⠔⢒⣻⡏⢉⡀⡤⣴⠾⠿⠿⠛⠩⠝⠁⠀⠀⢀⠰⣌⣦⣾⠖⠻⡟⠋⠛⠇⣸⣧⠈⢃⣸⣷⣼⣿⣿⡧⢗
⢘⢷⣯⣟⡷⣯⣟⡿⣷⣦⣄⣀⠀⠀⠘⠗⠃⠈⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⣠⣥⡞⢩⠉⢻⡇⢠⠀⠣⢠⡓⣤⣹⣿⣿⣶⣿⠿⣛⢯⢳⡁⢰
⢸⠀⠙⠾⣽⣳⣯⢿⡷⣿⡽⣯⢿⡷⣶⣤⣄⣀⠀⠀⠀⠀⣰⠁⠀⣠⡴⡟⣭⢲⠃⣿⣧⡐⠇⡠⣿⡡⢎⣿⣶⣿⢿⡛⡵⣊⠶⣙⣾⢣⡇⢸
⢸⠀⠀⠀⠀⠉⠛⠯⣿⣷⣻⡽⣯⢿⡽⣯⢿⣻⣟⡶⣤⡀⣿⣿⣰⣿⡏⣺⡴⣿⠀⣿⣿⣿⣾⣵⣿⡿⢟⡯⣛⡼⢧⣛⠶⣭⢞⡽⣞⣯⡧⢸
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠹⠯⣟⣿⣻⣽⣞⡿⣵⢿⣿⣿⣿⣿⣿⣳⢿⣿⣤⣿⡿⠿⡛⣭⠱⣎⡷⣳⣭⣿⢯⣟⡿⣼⣻⣾⣿⣿⣿⢚
⢈⠀⠀⠀⠀⠀⠀⢀⣤⠴⠆⠀⠀⢀⣠⡈⠙⠳⢯⣟⢾⣻⡾⣿⣿⣿⣿⡯⠟⠛⣩⢳⣜⣣⠿⣴⣫⢷⣻⢷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣹
⠠⠀⠀⠀⠀⠀⠀⠸⣤⣀⢐⣲⣴⢺⣷⣟⣶⡆⠀⠙⣯⡷⣿⣻⣿⣏⣄⠰⣩⢓⣬⣷⣾⣽⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⡿⢋⢷⡇
⠐⠀⠀⠀⣀⡀⠀⠠⢤⣬⠿⠼⠿⠼⠿⠿⣽⡇⠀⠀⣽⣟⡷⣿⣽⣯⠿⣟⠷⢿⣾⡿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⡿⣟⢏⠶⣘⠷⡁⢿⡇
⢨⠀⠀⣼⣟⡿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⢯⣿⣳⣿⣭⢻⣌⢫⢯⠙⣻⣿⣻⣞⣟⡛⣟⠻⡏⢙⣿⣯⢿⡜⣎⠳⣭⣛⠄⣻⠆
⠸⢀⣰⡿⣾⣽⣳⣯⣟⣿⢯⣟⡿⣶⣶⣶⣶⣿⣟⡿⣼⣻⢾⣽⡿⣞⢧⡚⣤⢛⠀⣽⣷⣻⣞⠶⣙⢬⡓⢧⢈⣿⣯⣟⢾⣡⢛⡴⣫⠄⣻⠆
⠀⢸⡿⣽⢳⣯⣷⣟⣾⢯⣟⡾⣽⣳⣟⣾⣽⡾⣽⢯⣳⣯⣿⣿⣻⡽⢮⡱⢆⡫⠀⣿⣷⣻⢾⡹⡜⢦⡝⠦⢘⣿⣷⣻⢞⡴⢫⣜⡳⠂⣽⠂
⠀⢸⣟⡮⣗⠾⣜⢯⡻⣟⠫⢿⣳⣟⣾⣻⢾⡽⣯⢛⡉⢼⣿⣞⡷⢯⣳⡙⢦⢓⠀⣿⡾⣽⣏⢷⡙⢦⣹⢃⠸⣿⡾⣽⢞⡜⣣⢞⡭⠃⣽⠂
⠀⢸⡿⡼⣭⣛⢮⡟⣵⢻⡀⢻⣷⢯⡾⣽⡳⣝⢦⢻⠁⢼⣿⢾⡽⣯⢳⡹⢜⡊⠀⣿⣟⣷⣻⢮⡙⢦⣓⠃⢸⣿⡿⣽⢞⡼⡱⢮⡝⠀⣿⠀
⠀⢸⡟⢳⣧⡏⡟⣾⣭⢳⡇⢹⣿⡞⣽⢳⣯⡝⣮⠉⠀⣾⣿⣯⡟⣷⢫⡜⢣⡜⢠⣿⣿⡞⣷⢫⡜⢣⣮⡅⢸⣿⣿⣽⢺⣴⢩⣷⢩⠐⣿⠀
⠀⢸⣟⡯⣞⡽⣳⢯⡽⣒⡃⣹⣷⣻⡽⢯⣞⡳⣍⠆⠀⣾⣿⢾⣽⡳⣏⡳⣍⠰⠠⣿⣷⣻⡽⣎⡳⣍⠶⡁⢺⣿⡷⣯⡳⢎⡳⣜⢣⠐⣿⠀
⠀⢨⣟⡷⣝⡾⣽⡳⣏⡗⡆⢽⣷⢯⣟⡿⡼⣝⢮⠄⠀⣿⣿⣻⢾⣝⣧⢳⣌⢳⠐⣿⣯⢷⣻⢵⡳⣬⠳⡅⢺⣿⣟⡷⣏⡟⣴⢫⠖⢈⣿⡃
⢠⠐⣯⠿⣽⣛⣷⣻⡝⣞⡁⢻⣿⣻⣞⣷⢻⡝⣮⠄⠘⣿⣿⣽⣻⢾⡼⣳⢮⡝⢨⣿⣟⣯⣟⡾⣵⢫⣷⠃⣼⠿⣏⣿⢳⢿⣜⣯⡚⢠⣿⡇
⢸⠀⠹⣟⣳⢟⡾⣳⡿⣡⢞⠇⠻⣧⣟⠾⣯⣽⠶⢋⡴⡏⠹⣾⡭⣟⣳⡽⢫⣴⡳⠃⠹⣾⡼⣹⡞⡟⣥⣞⠞⠀⠙⢾⡝⡾⢧⣏⡵⢦⠏⡇
⠀⠢⡀⠈⠛⠾⣵⣏⣾⠷⠋⠀⠀⠙⠺⢿⣵⣫⣞⠧⠋⠀⠀⠈⠛⠾⠭⠶⠟⠊⠀⠀⠀⠈⠑⠓⠽⠚⠋⠁⠀⠀⠀⠀⠉⠛⠚⠓⠋⢁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀