⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⡷⣽⣯⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣴⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣫⣿⣿⣿⣿⣿⠾⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⢽⡶⣿⡻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣟⣿⣿⣟⣻⣿⢿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡿⣿⣗⣯⣷⣉⣿⣹⣶⣴⡾⣷⡶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣟⣻⡿⣿⣧⣸⢟⡿⣿⣯⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⣿⣿⣿⣻⣿⣧⢿⣶⣼⣿⣷⣇⢿⠿⡿⢷⣻⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣻⣿⣿⣿⡿⢼⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣇⣹⣿⢻⣿⣯⣶⣯⡿⣷⣷⣿⣿⠿⣿⣻⢿⣻⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣧⣾⣬⣝⢿⣻⢟⣼⣨⣭⣿⢾⣟⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣿⣟⣿⣿⣧⣿⣻⣽⣿⣿⣿⣯⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣽⠿⡷⢟⣾⣿⢤⣤⣭⣧⡼⣟⣷⣾⣿⣿⣟⣿⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⣼⣿⣿⣿⣧⣿⣛⣹⣿⢩⣿⣭⣿⣯⣿⣾⣷⣻⣟⣛⡫⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⡇⣸⣿⣿⡿⢗⣵⣫⣣⣮⣿⣿⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⡮⣯⣿⣿⠛⣻⣯⣿⣿⣽⢡⣿⣿⣿⣿⢿⣻⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣸⣿⣶⣾⣧⣿⣿⠻⡋⣽⣯⣟⣮⣯⣝⡁⡙⢚⣻⣿⣤⣁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣟⣯⣿⢻⣿⠛⣲⣼⣿⣿⣿⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣯⣿⣿⣼⣓⣻⣿⣋⣿⣷⣿⣭⢯⣍⠙⢹⣽⣷⣿⣽⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢀⣿⣿⣿⣿⢿⣻⣷⣿⣳⣿⡿⣿⠟⣿⣿⢀⢡⣿⡿⠛⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣽⣿⣟⣻⣿⣶⠷⣿⠀⢸⣗⠚⣿⠒⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣿⣿⣻⣛⣿⣿⣿⣇⣀⡀⢺⣿⣆⢸⣷⢿⡇⡉⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡾⣿⣿⣿⢻⣯⣶⣿⣿⣿⣿⣟⢿⣬⣿⣿⠌⠸⣿⣷⣤⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣻⣿⡯⣗⣿⣿⣿⣿⣯⣟⣿⣿⢿⣤⣤⣿⣴⣧⡄⠀⠀⠀⠀⠀⠀⠀⢀⢺⡯⣿⣻⣿⣝⣻⣿⣻⣿⣿⡏⠉⠁⢸⡇⢻⣾⣿⣼⡇⠀⣼⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠐⣯⣿⣿⣿⣿⣿⣿⣿⣳⣿⣟⣖⣿⢧⣿⣿⡆⠁⠉⠛⠿⢿⣿⣦⣸⣲⡀⢀⠀⢀⣀⣤⣴⣶⣾⣿⢻⣿⢷⣗⣫⣵⣿⣿⣿⣿⣷⣿⢟⣷⣿⣿⣟⣻⣿⣷⣤⣄⣀⡀⠀⣀⣤⣾⣟⣿⣶⣾⣟⣹⣯⣿⣿⣿⣛⣻⢻⣛⣹⡃⡄⢟⣻⡟⣿⣿⣋⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢨⣿⣿⡽⣿⣿⣟⣿⣻⡿⣿⣟⣻⣿⢖⣿⣿⡄⢷⣿⣄⠐⠀⣸⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⢿⢿⣼⢿⣻⢛⣟⣿⣶⣿⡿⣟⢪⣿⣾⡆⠀⡶⠀⣼⣇⠀⣶⠟⣶⣤⣞⣳⣶⡞⢺⣿⠛⣯⡿⠯⣿⣿⣟⣿⣿⣟⣿⣾⢼⢫⣿⠼⣯⣾⣸⣾⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣰⣻⣯⡏⣿⣿⠻⠿⣾⣷⢟⡿⣿⣿⣉⢻⣿⡡⠈⠻⣿⣿⣿⡿⠏⢸⣞⣾⣿⣯⣻⣶⣞⣫⡿⣷⣿⣊⠺⢫⡿⢿⣿⡿⢳⡿⣿⣿⣷⣝⣕⣿⡟⠛⣟⣰⣯⣽⡄⣝⠛⣷⢯⣷⡿⢛⢧⠼⡻⣦⢟⣯⣿⣱⡿⣧⡿⣻⣿⣿⣿⣿⣿⣿⣟⣷⣳⣿⣾⣞⠂⠀⠀⠀⠀⠀⠀⠀⠀ ⠹⠿⢹⣽⣽⣼⣭⣿⣿⣿⣿⣿⣿⣏⡇⣶⣶⢧⣶⣴⣾⣿⣿⣶⣷⣾⣿⣿⣿⣯⣿⣯⣽⣽⢿⣿⣾⣏⣹⣿⡏⢹⣯⣿⣼⣿⣿⡿⣼⣽⣯⣿⡇⣠⣿⣿⡀⠸⣿⣽⣿⣏⣼⣿⡷⣿⣧⣿⣿⣧⣽⣾⣿⣾⣿⣿⣿⣵⣿⣿⣿⣿⣿⣯⣧⣿⣶⡇⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⣭⣽⡿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣯⣿⣬⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⣿⢿⣿⣿⡟⢿⡾⣧⣼⠻⡏⢿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢸⢿⣿⣿⣿⣿⣻⣿⣿⣿⡟⣿⣿⢁⠁⢉⣷⣿⣿⣿⣆⠉⠉⠇⠈⢉⣿⣿⣿⣿⣆⡉⠰⡁⢸⣿⣷⣿⣿⣷⣀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⢻⢻⣿⣿⣿⣻⣿⣿⣿⠈⣿⣿⡿⠿⢿⡿⠿⣏⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣤⣿⣿⣿⣿⣿⣿⠯⢻⡱⡟⣻⠝⢈⣼⣿⡿⠛⠛⠿⣿⣷⡀⠈⣴⣿⣿⠟⠛⠻⣿⣷⡄⠀⢸⣿⣟⠛⠛⠿⣿⣷⡀⣹⣿⡟⠛⢛⠛⠛⠛⠛⠛⠛⠛⢸⡻⣽⠐⠠⠀⣿⡇⠂⠂⢸⣯⣿⣯⣭⣭⣽⣭⣿⠛⣿⣿⡟⠛⢻⣿⢿⡋⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣯⣿⣷⣿⣼⣿⠉⢻⣾⣿⣿⣙⣉⣿⣿⡟⠀⠀⠀⠀⠋⠉⠁⢠⣿⣿⠁⠀⠀⠀⠘⣿⣿⠀⢺⣿⡗⠀⠀⠀⢹⣿⣗⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣯⣟⣿⣿⣿⣿⣟⠟⣫⣵⣿⣬⡼⣿⣿⣮⣟⣠⣿⣽⣿⣿⣿⣿⢬⢿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢠⣷⣿⣿⣹⢿⣀⣼⢻⡿⣿⣉⣡⣿⣿⡇⠀⠀⣶⣶⣶⣶⡇⢾⣿⣿⠀⠀⠀⠀⠀⣿⣿⢀⢾⣿⡇⠀⠀⠀⢸⣿⣯⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣻⣿⡿⠿⣿⡿⢿⣿⢟⣿⢻⣤⣬⣿⣷⢟⣿⣯⣽⣭⣿⣿⣿⠶⠿⣻⣿⢟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢈⣿⣿⣿⣽⣿⣏⣒⠖⣳⣿⣿⡽⢻⣿⣧⠀⠀⠩⠉⢹⣿⡇⠈⣿⣿⡄⠀⠀⠀⢠⣿⡿⢈⣻⣿⡇⠀⠀⠀⣸⣿⡟⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣽⣾⣿⣿⡇⠂⣿⡇⢙⣿⣙⣿⣫⣦⣬⣍⣿⣿⣿⠟⣻⣯⣿⣽⣿⣿⣿⣟⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⣿⢿⣟⠷⣦⣍⣽⣿⣿⣿⣯⣇⠀⠻⣿⣷⣶⣴⣶⣿⣿⠇⠀⠘⢿⣿⣶⣦⣶⣿⡿⠃⠄⢹⣿⣷⣶⣶⣾⣿⠟⠘⣹⣿⠏⠉⠉⣿⣿⠏⡉⢹⣿⣿⢟⣽⣟⣍⣁⣄⣉⣡⣌⣉⣜⣷⣿⣿⣋⡽⣽⢍⣿⠶⢿⣿⡯⠷⢿⣧⣼⠛⡷⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠘⣿⡞⢷⣤⠟⢻⣾⣟⣿⣿⣿⣀⣠⣈⣽⣿⣿⣻⣯⣀⣀⣀⣀⣀⡍⣻⣿⣿⣯⣀⣀⣄⣼⣿⣿⣿⣿⣿⢁⡀⣀⣽⣿⡃⠀⠀⣽⣿⠂⡁⢸⣿⣿⢿⣿⢿⠉⠉⢉⣿⡋⢩⣿⣿⣽⣿⣋⣯⣿⣿⣻⣻⣂⣹⣻⣷⣷⡿⣇⣹⡇⠷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣲⣱⣵⣿⣿⣾⣾⣾⡿⣛⠻⠿⠿⣻⣿⣿⣻⣿⣿⣿⣿⣏⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣟⣻⣿⣽⣤⣔⣿⣿⡅⠀⠀⠺⠿⠃⠀⠸⠿⠿⣼⢝⣫⣴⣾⠟⠻⣷⣶⡿⣷⣶⡞⣷⣷⣿⡷⣽⣿⣿⣿⢟⣯⡿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠈⣉⣽⢉⣉⣭⣉⠁⣿⠀⠁⠀⣿⣉⣉⢹⣿⣉⢉⠉⢹⣿⠉⣼⣶⣸⣟⣿⣫⣿⣯⣏⣿⣻⣿⠻⢶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣨⣽⣿⣿⣯⣤⣤⣤⣴⣶⣤⡮⢽⡽⣿⣷⣿⣷⡳⣾⢩⣯⣿⠉⣻⣿⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠠⣼⡟⠉⣰⣿⢋⢻⣿⣿⠀⠀⢀⣿⢛⠙⢘⣿⠛⠛⠀⢸⣇⠀⢹⠛⢡⣟⣛⣾⡽⣿⡟⢿⡼⣯⢿⣤⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⢿⣿⣿⢿⣿⣿⣿⡏⠉⣿⡏⣿⣿⣿⣛⣻⣟⣟⣳⣛⣿⣙⣒⠛⣙⣖⣫⣛⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠜⠿⣶⡿⠋⠺⠿⠋⣟⣻⣟⣟⠻⣟⣟⢟⡛⣀⣀⣀⢘⣛⡀⠀⠀⣈⣿⡭⣿⡻⡟⣧⣼⠻⢿⣫⡿⣍⠀⠀⠀⠀⠀⣿⣿⠀⡑⢸⣿⣿⢿⣿⠀⠿⠇⣀⣛⣃⡙⢿⣻⡿⠯⠿⣿⣻⣻⣿⢿⣝⣟⣿⠫⣯⣽⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣭⣿⣭⠯⣍⣿⣧⡽⣿⠁⣷⣿⣦⣿⣷⣾⣻⠚⠯⠀⠀⠀⣷⣧⣿⣿⣿⣟⣛⣾⣆⣿⣿⠛⣦⠀⠀⠀⠀⠀⠛⠛⠀⠁⢸⣿⣿⢾⣿⠀⠀⠀⢰⢛⣗⣾⠛⣾⡟⢻⢞⣛⣿⣻⣷⣿⣛⣛⣿⣿⡿⣿⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡶⣿⢽⣛⡶⣽⣻⣿⣝⣿⣯⢟⡘⣻⡟⢯⣽⠃⠀⠀⠀⢀⣿⣿⣷⡳⣿⣿⣤⣺⠹⣿⣹⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⢾⣾⡄⠀⠀⠘⢷⣿⣛⡶⢫⣷⣞⣽⣿⢿⣿⡿⣷⣿⣿⣷⠉⠡⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⢘⣯⣥⣿⣿⣧⣿⣿⣟⡻⢾⣿⣿⣧⠛⠀⠀⠀⠀⢈⣿⣿⣿⣮⣯⣿⣭⣷⡾⡹⢏⢿⡱⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣽⣯⡤⠀⠀⠀⠈⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⡿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣏⣾⣯⠥⣿⡷⣭⣹⠶⣿⣯⣯⠀⠀⠀⠀⠀⢻⣿⣿⠛⢻⣾⣿⠁⢸⡏⢿⣧⠀⣿⠛⠻⠛⠟⠻⠛⠟⠻⠛⢿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠐⢵⣿⢾⣿⣿⣿⢿⣿⣯⡟⡵⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣹⣿⣿⣗⣿⢳⣾⣿⣿⠉⠀⠀⠀⠀⠀⠺⡿⣿⠻⣞⠁⣿⠀⢸⡇⢾⡛⣧⣿⠈⠀⠀⠈⠀⠈⠀⠀⠁⢸⣿⣿⢳⣿⣧⠀⠀⠀⠀⠀⠚⣿⣻⣻⣟⣻⣿⠉⣿⣿⣹⣟⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⢹⣿⣯⣭⣿⠻⣿⣯⣭⠀⠀⠀⠀⠀⠀⣿⣿⠿⠀⠹⠷⣸⠷⠾⡃⠸⠇⠘⠿⣴⣶⣶⣶⣶⣶⣮⣶⢾⣾⣿⣗⣟⢿⣭⠀⠀⠀⠀⠀⠸⣿⣿⣻⡟⢿⡿⢶⠟⡇⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⢿⣢⣴⡵⢯⣾⣷⡂⠀⠀⠀⠀⣬⣿⣿⣽⣿⣟⢿⣼⣶⣯⣮⣟⡛⠛⢻⣷⡒⠛⠙⣿⣳⠞⢹⣿⡋⠛⠛⢳⣽⣛⡄⠀⠀⠀⠀⢘⣿⣾⣿⣿⣿⣻⡁⠺⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡁⠄⠵⢿⣿⡿⣾⣟⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣛⣻⣿⡟⠛⢛⢻⣿⡆⢾⣿⡇⠀⠀⣸⣿⠀⢸⣿⣿⡄⠀⢸⣿⣶⡆⠀⠀⠀⠀⠐⢿⣛⣿⣻⣿⡏⠁⡯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠈⠿⣿⢿⣻⠟⠀⠀⠀⠀⢀⣻⣿⣿⣏⣉⣉⣿⣿⣧⣥⣤⣾⣿⠇⢸⣿⡇⠠⠀⢹⣿⠀⢘⣿⡿⣿⡄⢸⣿⣑⠂⠀⠀⠀⠀⢐⣿⢻⣿⣿⠛⠀⢀⣫⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⡿⠿⠁⠀⠀⠀⢘⣿⣷⣺⣿⣿⣿⣿⣿⡟⠛⣿⣯⡁⠀⢸⣿⡇⠀⠀⢸⣿⠀⢸⣿⡇⢻⣿⣼⣿⣯⣷⠀⠀⠀⠀⢰⡿⣯⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠠⣿⣿⣿⢭⣁⠻⢿⣿⣿⡏⠀⠘⣿⣷⡄⠘⣿⣷⣄⣠⣿⡿⠀⢸⣿⡇⠀⢻⣿⣿⣿⠧⠀⠀⠀⠀⢈⣽⡿⣻⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣷⣶⠀⠀⢰⣿⠿⠷⠲⠉⠉⠙⠎⠛⠃⠀⠐⠈⠛⠛⠀⠈⠛⠛⠛⠋⠁⣀⡘⣛⣃⣀⣀⣹⣻⣽⣗⡀⠀⠀⢀⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣽⣦⠀⢀⣯⡽⣶⣶⣶⣶⠂⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣄⠀⠠⠀⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣛⠐⠀⢀⡴⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣝⣿⡄⣠⣷⣷⣿⣿⣿⣿⡁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⡗⢀⣼⣿⣿⣿⣶⣄⠀⣿⣭⠀⠀⢰⣺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣽⣿⣿⣿⣿⣿⣿⣿⢿⣯⢻⣿⡾⢿⣾⢿⣿⡽⣿⣯⣷⠿⠯⣿⣿⣿⣷⣿⣿⣝⢿⣿⡿⣟⣾⡿⣿⢻⣭⣿⡿⣟⣩⡽⣼⠙⣭⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⣿⣿⣿⣇⣿⣿⣷⣿⣿⣿⣿⣯⣿⢿⣿⣿⣯⣽⣭⣿⣿⣿⣿⣿⣿⣡⡽⡮⣍⣿⣧⣿⣿⣿⣏⢨⣿⣿⣽⣯⣻⣟⣞⣾⣻⣹⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⡧⢾⡇⣻⢿⡯⢽⡟⠬⡶⣾⣿⣶⣶⣶⣿⢷⣷⣾⣽⠿⢗⣒⠻⣻⣛⢓⡵⣵⣿⣯⠷⣸⡷⡾⣧⡽⣿⢻⡷⢲⣾⣿⣿⣏⣿⡿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣯⣿⡿⠫⢷⣮⣵⣿⣿⣿⣿⣿⣿⣶⣄⢀⠀⣿⣿⠀⠀⠂⢸⣾⡇⠀⢸⣿⣷⡀⠀⠂⢸⣾⡇⣿⣿⡏⢾⣯⣿⣿⠍⣿⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣽⡟⣿⣛⣿⣿⡏⠉⠉⠙⣿⣿⡆⢜⣿⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣷⡀⠀⢸⣿⣧⠶⣿⣿⣿⡻⢯⠧⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣷⣻⣟⣿⣿⣧⣤⣤⣴⣿⡿⠁⠸⣿⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿⡿⣿⣷⡀⣺⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⣞⣿⣿⡿⠿⣿⣿⡋⠁⡀⠐⣿⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠸⣿⣷⣽⣿⣟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢛⣥⣿⣿⡇⠂⠘⣿⣿⣆⠀⠀⣿⣿⡀⠀⠀⣸⣿⡇⡁⢸⣿⡇⠀⠘⣿⣿⣿⣷⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇⣿⣿⡇⠀⠀⠘⢿⣿⣆⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⢸⣿⡇⢀⠀⠘⣿⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⣶⣶⣦⣤⣦⣾⡶⢼⢤⣤⠴⣮⢿⡿⣖⣦⣤⣤⣾⣷⣷⣦⣤⠶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠋⠙⠋⠉⠈⠈⠀⠁⠀⠁⠀⠉⠙⠛⠉⠉⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣤⣤⣠⣀⣀⣀⣀⣤⣠⣄⣀⣄⣀⣀⣀⣀⣀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢉⣉⠉⣉⡉⢉⣛⡉⣉⣉⠉⡉⠉⢉⡉⢉⣉⠉⣉⣉⣉

⣉⣉⣉⠉⣉⢉⡉⢉⠉⡉⠉⣉⣉⡉⣛⢉⡉⣉⠉⣉⢉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣀⣀⣀⣀⣀⣄⣀⣄⣠⣤⣀⣀⣀⣀⣠⣤⣤⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠉⠉⠛⠙⠉⠀⠁⠀⠁⠀⠈⠈⠉⠋⠙⠋⠛⠙⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⣤⣦⣷⣷⣾⣤⣤⣦⣖⡿⢿⣮⠴⣤⢤⢼⡶⣾⣦⣤⣦⣶⣶⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⣿⣿⠘⠀⢀⡇⣿⢸⠀⠁⠟⣿⣿⣿⣿⢿⠙⠀⣆⣿⢿⠘⠀⠀⡇⣿⣿⣇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣷⣿⣿⣿⠘⠀⡇⣿⢸⡁⡇⣿⣸⠀⠀⡀⣿⣿⠀⠀⣆⣿⣿⠘⠂⡇⣿⣿⣥⢛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣟⣿⣽⣷⣿⠸⡇⣿⢸⠀⡇⣿⢸⠀⠀⠀⣿⣿⠐⡀⠁⡋⣿⣿⠿⡿⣿⣿⣞⣿⢟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⡿⣿⣿⣿⣿⣺⡀⣷⣿⡿⣿⢸⠀⡇⣿⢸⠀⠀⠀⣿⣿⠸⠁⡿⣿⣴⣤⣤⣧⣿⣿⣟⣻⣷⣻⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠧⢯⡻⣿⣿⣿⠶⣧⣿⢸⠀⡀⣷⣿⣿⢸⠀⡇⣿⢸⠀⠀⠀⣿⣿⢜⡆⣿⣿⠙⠉⠉⡏⣿⣿⣛⣿⡟⣽⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⢹⣿⠍⣿⣿⣯⢾⡏⣿⣿⡇⣾⢸⠂⠀⡀⣷⣿⢸⠀⡇⣾⢸⠂⠀⠀⣿⣿⠀⢀⣄⣶⣿⣿⣿⣿⣿⣿⣵⣮⢷⠫⡿⣿⣯⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡿⣿⣏⣿⣿⣾⢲⡷⢻⣿⡽⣧⡾⡷⣸⠷⣯⣿⣵⡵⢓⣛⣻⠻⣒⢗⠿⣽⣾⣷⢷⣿⣶⣶⣶⣿⣾⡶⠬⡟⢽⡯⢿⣻⡇⢾⡧⣿⣻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣧⣹⣻⣾⣞⣟⣻⣯⣽⣿⣿⢨⣏⣿⣿⣿⣧⣿⣍⡮⡽⣡⣿⣿⣿⣿⣿⣿⣭⣽⣯⣿⣿⢿⣿⣯⣿⣿⣿⣿⣷⣿⣿⣇⣿⣿⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣭⠙⣼⡽⣩⣟⡿⣿⣭⢻⣿⡿⣾⣟⡿⣿⢿⣝⣿⣿⣷⣿⣿⣿⠯⠿⣷⣯⣿⡽⣿⢿⣾⢿⡾⣿⢻⣯⢿⣿⣿⣿⣿⣿⣿⣿⣽⡛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡛⣯⣿⢻⡟⣿⣟⣟⣿⣿⢿⣿⣿⣿⣿⡶⣶⣶⣰⣵⣿⣿⣿⣿⣷⣜⡶⢦⡔⣢⣢⣒⣻⣻⢻⣷⣞⣟⢾⣶⣳⣶⣷⣿⣿⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⣽⣷⡽⣿⣿⠩⠟⡿⣶⣶⣶⠿⠹⣧⠀⠀⠁⠟⠿⣿⣿⣿⣿⣿⠿⠛⠈⠠⠀⠀⠆⠿⣿⠿⢻⠟⣴⣦⢿⠘⣿⣺⣿⣿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣷⣯⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⡇⣿⡽⠀⠏⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠙⠀⠀⠄⣿⣿⣿⣿⣦⢿⠻⡙⢠⣿⡏⣷⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣻⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣾⢀⠀⠀⠀⡀⣷⣿⣿⣿⠘⠁⠀⣿⣿⣿⣿⠿⠻⣋⣿⢸⣿⣳⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠿⣿⣿⠁⠛⠿⡿⣿⡿⠻⠙⣧⡇⣿⣿⣿⠘⠀⠀⠀⠀⠀⡀⣿⣿⣿⢾⡄⠁⣿⣿⣿⣿⡋⣿⣿⣿⢻⣿⣾⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⡷⣾⣠⣀⢀⣀⣆⣿⡿⡀⣿⣿⣿⣿⢿⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣭⣿⣿⣦⣿⠛⠉⠩⠋⡟⣾⣧⠀⡄⣷⣿⣿⣿⣿⣿⣿⢿⠿⠙⠀⠀⠀⠀⠄⣿⣿⣿⣿⠾⣾⡿⣛⡺⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣶⠀⡀⣄⣴⣯⣭⣉⠀⡃⠀⠀⠀⣀⣤⣶⣷⣿⣿⣿⣿⣿⢿⠘⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣶⠷⡇⣹⣇⡿⣷⣷⣻⣹⣂⣻⣻⣿⣿⣯⣋⣿⣽⣿⣿⢩⡋⣿⢉⠉⠉⢿⣿⢿⣿⣿⢸⡁⠂⣿⣽⠀⠀⡃⣿⣽⣀⡀⢁⣿⣿⣿⣿⣿⣼⣄⣀⣀⣯⣿⣿⣻⡍⣀⣀⣀⣀⣀⣯⣻⣿⣿⣽⣈⣠⣀⣿⣿⣿⣟⣾⢻⠟⣤⢷⡞⣿⠘⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠿⡷⠛⣼⣧⢿⠷⡯⣿⢿⠶⣿⢍⣽⡽⣋⣿⣿⣷⣜⣉⣌⣡⣉⣄⣁⣍⣟⣽⢟⣿⣿⢹⡉⠏⣿⣿⠉⠉⠏⣿⣹⠘⠟⣿⣾⣶⣶⣷⣿⢹⠄⠃⡿⣿⣶⣦⣶⣿⢿⠘⠀⠇⣿⣿⣶⣴⣶⣷⣿⠻⠀⣇⣯⣿⣿⣿⣽⣍⣦⠷⣟⢿⣿⠈⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⣿⣿⣟⣿⣿⣿⣽⣿⣯⣻⠟⣿⣿⣿⣍⣬⣦⣫⣿⣙⣿⢙⡇⣿⠂⡇⣿⣿⣾⣽⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⡟⣿⣸⠀⠀⠀⡇⣿⣻⢈⡿⣿⢠⠀⠀⠀⡄⣿⣿⠈⡇⣿⢹⠉⠩⠀⠀⣧⣿⢻⡽⣿⣿⣳⠖⣒⣏⣿⣽⣿⣿⣿⢈⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⢟⣿⣻⠿⠶⣿⣿⣿⣭⣽⣯⣿⢟⣷⣿⣬⣤⢻⣿⢟⣿⢿⡿⣿⠿⡿⣿⣻⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠿⡿⣯⣿⢸⠀⠀⠀⡇⣿⢾⢀⣿⣿⠀⠀⠀⠀⠀⣿⣿⢾⡇⣶⣶⣶⣶⠀⠀⡇⣿⣿⣡⣉⣿⡿⢻⣼⣀⢿⣹⣿⣿⣷⢠⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣿⣿⢿⢬⣿⣿⣿⣿⣽⣿⣠⣟⣮⣿⣿⡼⣬⣿⣵⣫⠟⣟⣿⣿⣿⣿⣟⣯⣽⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣼⣗⣿⢹⠀⠀⠀⡗⣿⢺⠀⣿⣿⠘⠀⠀⠀⠁⣿⣿⢠⠁⠉⠋⠀⠀⠀⠀⡟⣿⣿⣉⣙⣿⣿⣾⢻⠉⣿⣼⣿⣷⣿⣯⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡋⢿⣿⢻⠛⡟⣿⣿⠛⣿⣭⣽⣭⣭⣯⣿⣯⢸⠂⠂⡇⣿⠀⠠⠐⣽⡻⢸⠛⠛⠛⠛⠛⠛⠛⢛⠛⡟⣿⣹⡀⣷⣿⠿⠛⠛⣟⣿⢸⠀⡄⣷⣿⠻⠛⠟⣿⣿⣴⠈⡀⣷⣿⠿⠛⠛⡿⣿⣼⢈⠝⣻⡟⡱⢻⠯⣿⣿⣿⣿⣿⣿⣤⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣏⠿⡿⢿⠿⡿⣿⣿⠈⣿⣿⣿⣻⣿⣿⣿⢻⢻⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠀⠀⣀⣷⣿⣿⣷⣿⢸⡁⠰⡉⣆⣿⣿⣿⣿⢉⠈⠇⠉⠉⣆⣿⣿⣿⣷⢉⠁⢁⣿⣿⡟⣿⣿⣿⣻⣿⣿⣿⣿⢿⢸⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⢿⡏⠻⣼⣧⡾⢿⡟⣿⣿⢿⣿⡏⣿⡿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣯⣿⣾⣿⡿⣿⣿⣿⣿⣿⡿⣽⣭⣿⣿⣿⣿⣿⢸⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡇⣶⣿⣧⣯⣿⣿⣿⣿⣿⣵⣿⣿⣿⣾⣿⣾⣽⣧⣿⣿⣧⣿⡷⣿⣼⣏⣿⣽⣿⠸⡀⣿⣿⣠⡇⣿⣯⣽⣼⡿⣿⣿⣼⣿⣯⢹⡏⣿⣹⣏⣾⣿⢿⣽⣽⣯⣿⣯⣿⣿⣿⣾⣷⣶⣿⣿⣾⣴⣶⢧⣶⣶⡇⣏⣿⣿⣿⣿⣿⣿⣭⣼⣽⣽⢹⠿⠹ ⠀⠀⠀⠀⠀⠀⠀⠀⠂⣞⣾⣿⣳⣷⣟⣿⣿⣿⣿⣿⣿⣻⡿⣧⡿⣱⣿⣯⢟⣦⡻⠼⢧⢛⡿⣷⢯⣷⠛⣝⡄⣽⣯⣰⣟⠛⡟⣿⣕⣝⣷⣿⣿⡿⢳⡿⣿⢿⡿⢫⠺⣊⣿⣷⡿⣫⣞⣶⣻⣯⣿⣾⣞⢸⠏⡿⣿⣿⣿⠻⠈⡡⣿⢻⣉⣿⣿⡿⢟⣷⣾⠿⠻⣿⣿⡏⣯⣻⣰ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣸⣾⣯⠼⣿⢫⢼⣾⣿⣟⣿⣿⣟⣿⣿⠯⡿⣯⠛⣿⢺⡞⣶⣳⣞⣤⣶⠟⣶⠀⣇⣼⠀⡶⠀⡆⣾⣿⢪⣟⡿⣿⣶⣿⣟⢛⣻⢿⣼⢿⢿⣿⣿⣿⣿⠛⠛⡟⣿⣿⣿⣿⣿⣿⣸⠀⠐⣄⣿⢷⡄⣿⣿⢖⣿⣻⣟⣿⡿⣻⣿⣟⣿⣿⡽⣿⣿⢨ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣤⣋⣿⣿⡟⣻⢟⡄⡃⣹⣛⢻⣻⣛⣿⣿⣿⣯⣹⣟⣾⣶⣿⣟⣾⣤⣀⠀⡀⣀⣄⣤⣷⣿⣻⣟⣿⣿⣷⢟⣿⣷⣿⣿⣿⣿⣵⣫⣗⢷⣿⢻⣿⣾⣶⣴⣤⣀⢀⠀⢀⡀⣲⣸⣦⣿⢿⠿⠛⠉⠁⡆⣿⣿⢧⣿⣖⣟⣿⣳⣿⣿⣿⣿⣿⣿⣿⣯⠐ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢧⣼⠀⡇⣼⣿⣾⢻⡇⢸⠁⠉⡏⣿⣿⣻⣿⣻⣝⣿⣻⣿⡯⢺⢀⠀⠀⠀⠀⠀⠀⠀⡄⣧⣴⣿⣤⣤⢿⣿⣿⣟⣯⣿⣿⣿⣿⣗⡯⣿⣻⣰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⣀⣤⣷⣿⠸⠌⣿⣿⣬⢿⣟⣿⣿⣿⣿⣶⣯⢻⣿⣿⣿⡾⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⡉⡇⢿⣷⢸⣆⣿⢺⡀⣀⣇⣿⣿⣿⣛⣻⣿⣿⣻⣟⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠒⣿⠚⣗⢸⠀⣿⠷⣶⣿⣻⣟⣿⣽⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠻⠛⡿⣿⢡⢀⣿⣿⠟⣿⡿⣿⣳⣿⣷⣻⢿⣿⣿⣿⣿⢀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣽⣿⣷⣽⢹⠙⣍⢯⣭⣿⣷⣿⣋⣿⣻⣓⣼⣿⣿⣯⡷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⣿⣿⣿⣿⣼⣲⠛⣿⢻⣿⣯⣟⣿⣿⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣁⣤⣿⣻⢚⡙⡁⣝⣯⣮⣟⣯⣽⡋⠻⣿⣿⣧⣾⣶⣿⣸⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣧⣿⣿⣻⢿⣿⣿⣿⣿⢡⣽⣿⣿⣯⣻⠛⣿⣿⣯⡮⣾⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣽⣿⣿⣮⣣⣫⣵⢗⡿⣿⣿⣸⡇⣾⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣯⡫⣛⣟⣻⣷⣾⣿⣯⣿⣭⣿⢩⣿⣹⣛⣿⣧⣿⣿⣿⣼⢀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣴⣿⣿⣟⣿⣿⣾⣷⣟⡼⣧⣭⣤⢤⣿⣾⢟⡷⠿⣽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣧⣯⣿⣿⣿⣽⣻⣿⣧⣿⣿⣟⣿⣱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣿⣿⣟⢾⣿⣭⣨⣼⢟⣻⢿⣝⣬⣾⣧⣯⣿⣿⣿⣿⣼⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣿⣻⢿⣻⣿⠿⣿⣿⣷⣷⡿⣯⣶⣯⣿⢻⣿⣹⣇⡾⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣿⣿⣿⣿⢼⡿⣿⣿⣿⣻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⣿⣻⢷⡿⠿⢿⣇⣷⣿⣼⣶⢿⣧⣿⣻⣿⣿⣿⣠⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣿⣿⣿⣿⣿⣿⣯⣿⡿⢟⣸⣧⣿⡿⣻⣟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡶⣷⡾⣴⣶⣹⣿⣉⣷⣯⣗⣿⡿⣿⣿⣿⣿⣶⢀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣿⣿⣿⣿⣿⢿⣿⣻⣟⣿⣿⣟⣿⡿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣾⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⡻⣿⡶⢽⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣾⣠⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣧⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣷⣿⣿⣿⣿⣿⣿⣴⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣦⣷⠾⣿⣿⣿⣿⣿⣫⣿⣿⣿⣶⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣧⣿⣿⣿⣿⣴⣴⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⣯⣽⡷⣾⣶⣶⣤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀