⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣌⣀⣼⠄⠄⣠⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠇⠀⠀⠀⢨⠀⢕⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢅⠀⠀⢀⠀⠘⠃⢸⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠶⠿⠘⠀⠐⠤⠬⣩⡚⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⠿⠛⠉⠁⠛⠛⠿⣶⣯⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⠭⢹⠀⠀⠀⠉⠉⠛⡫⣺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡰⡸⣿⠘⠀⠀⠀⠐⠀⣽⢳⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡮⢦⣿⢠⠀⠀⠀⠀⠁⢾⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠫⢺⣿⠀⠐⠀⠀⠀⡐⣸⠇⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⢪⡊⡱⣿⠘⢰⠀⠀⠁⡴⣇⠣⠘⢳⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⣤⡔⠞⡴⢢⠉⠟⢺⠛⡎⠸⠀⠳⠈⢦⣹⢶⡿⠀⣰⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣀⢠⢁⠔⠩⣻⢙⣦⣇⣠⣀⣸⣐⣿⣤⣷⣶⡻⡩⠟⣉⠀⠁⢟⢀⡀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠉⣀⠄⠀⠀⠅⠐⢽⢺⡄⠀⠀⠙⠾⢧⣿⣽⣯⣿⣯⠿⠿⠛⠒⣉⣠⣀⣍⣙⠟⠔⠢⠍⠀⢈⠒⢄⣀⠀⢉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢁⠀⢐⠁⠀⠀⠀⠀⡠⠸⣄⣿⣶⣀⠀⠀⠀⠀⠀⠒⠶⣶⣶⣶⣤⣤⣤⣍⣿⣿⠿⠤⠈⠢⠀⠀⠀⡈⠠⠢⡰⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⠀⢃⢠⠖⠀⡴⠁⠂⠠⠈⠷⣾⣄⠛⠶⣤⣀⠉⠓⠶⣤⣍⣻⣿⣿⠿⠛⠀⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⠀⡁⠸⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⡀⣀⠅⠁⠀⠞⠀⠀⢾⣶⠂⠡⠀⠀⠈⠉⠉⡟⡿⠛⠋⠉⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⡆⠀⠑⢈⠰⠀⠀⠀⢮⣀⠄⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢩⠀⠊⢀⣯⠆⠑⠠⣿⡝⣆⠀⠀⠀⠀⠀⠫⢂⣇⠀⠀⠀⠀⠀⢀⢖⡽⠚⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡃⠀⠩⠋⠲⠝⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡴⠀⡀⢹⠟⠀⠈⠀⠀⣇⡪⢻⡀⠀⠀⢼⠰⠌⣇⠀⠀⠀⣠⢹⡕⣱⠇⠀⠀⠀⠀⠀⠀⠀⢮⣿⣤⠀⠀⠀⠀⢕⢸⢡⠧⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⡮⣂⠘⣾⣇⢄⠀⠀⠀⠀⣯⡅⢨⠿⢤⣾⣮⣶⣞⣤⢤⡃⠰⠨⡶⠂⠀⠀⠀⠀⠀⠰⢢⣜⡿⣽⣷⠠⢰⡰⠀⢩⡄⡈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢸⠰⡄⢀⣿⡕⠑⠀⠀⠀⠀⣻⣷⠟⠁⠀⠀⠀⠀⠀⠉⢯⢔⣲⠀⠀⠀⠀⠀⢁⢬⢌⢧⣱⣿⣿⠁⠌⢈⣬⣸⢠⡇⣧⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⣿⣿⣿⠼⢸⣿⢿⣄⡶⡶⢼⣫⠁⠠⠤⠬⢒⠀⠀⠅⠄⠀⠹⠽⣀⣀⣠⣀⠀⢀⣤⡀⣫⡙⣿⣡⢀⣀⣿⣷⢿⣄⡆⣇⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢫⣯⡿⣿⣷⣿⣽⣇⣳⣮⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣓⠣⢜⣺⡯⢝⣽⣫⣽⣿⡷⡿⣿⣾⣿⣿⣿⣯⣯⣾⢛⡐⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡳⣲⢾⡿⣿⣿⡿⠄⡠⠀⠷⢷⣜⡄⠀⠀⠀⢠⠉⠁⣀⠀⠀⢀⢵⣮⢼⡷⠓⠶⢮⠞⢉⣯⣿⣿⣿⣿⣿⣿⣿⠿⠷⡳⢒⣒⡧⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣟⣻⣻⣿⣗⣷⣗⣗⢪⢀⠂⣀⣿⣹⣦⠀⠀⠀⠈⠛⠀⠀⣠⢏⣕⡅⠀⠀⡀⡢⡀⣸⡕⢻⣿⣻⣞⣿⣿⣿⡿⣯⢶⣁⣝⣹⣻⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣟⣿⣿⣿⣽⣿⢱⢱⠎⢠⢣⣞⣫⡷⢻⣷⢶⡤⢶⣗⣫⣚⡇⣛⡮⣧⠐⠁⢅⡱⠣⢣⡼⣻⣿⣻⣿⣿⣿⣻⣿⠸⡿⣻⣹⣿⡴⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣻⣿⣿⣿⣿⡿⣎⣼⣾⣻⢟⡱⡮⡷⠋⢻⣝⡜⡎⢺⠀⠘⣮⡼⢺⣵⣏⣯⠇⠂⡐⢂⣻⣳⣿⢼⣿⣿⣿⣿⣿⣧⣯⣽⣾⣹⣗⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠟⢻⣿⣿⣿⣿⡇⢏⣸⡧⡯⣯⡾⠀⠀⠀⠘⣝⢻⡧⢾⠀⠀⠀⠉⡻⣝⢗⣯⣟⣧⡈⡑⢺⢻⣿⡗⡇⠹⣿⣿⣿⣷⣿⢿⠋⡏⠆⢦⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡷⡀⠠⠀⡸⣿⣽⠀⣾⡟⣷⣮⠧⠖⣆⠅⢀⠀⠘⣜⣣⣬⠅⠨⠐⡀⢅⢉⢛⠛⢮⡳⣿⣿⡪⣿⣿⡿⠇⠀⢹⣿⣿⡿⠒⢄⢀⣄⢅⢼⡜⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢳⡃⢭⣚⠯⣿⡅⡿⢿⢽⢨⡰⢆⠉⡻⢠⡄⠀⠙⣧⡽⡷⠀⠈⡈⢎⠃⠸⠨⢄⡫⡿⢫⢝⣿⣿⣯⠀⠀⠸⣯⣰⡯⣚⡎⠲⣲⣽⣿⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢑⡪⢌⣒⡎⢾⡮⣿⣏⣾⣽⣦⡿⣷⣚⢌⣱⣬⠄⡀⠄⠹⡮⡝⠖⠃⣉⠑⡑⣦⡓⡳⡚⡬⢟⣫⣿⣿⡇⠀⠀⠀⢹⣟⣮⢘⡃⠤⠝⠨⠫⡼⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡎⣙⠮⢠⡥⣻⣿⣿⣻⣿⣿⣟⣿⣧⢆⡱⣿⣏⡱⢫⡸⣇⢺⡀⡞⠫⢃⣲⣮⢽⣷⣽⣽⢷⣿⣿⣿⡇⠀⠀⠀⠈⡗⣟⡎⢇⣸⡰⡐⣚⡗⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆⢊⠏⠟⣯⣾⣿⣦⣙⣿⣿⡽⣿⣿⣯⣼⣻⣿⣿⣭⡣⡟⣷⢿⣝⡔⣽⣞⣿⣿⣷⣝⡫⣿⣿⠏⣿⣿⠀⠀⠀⠀⢻⣾⣳⠿⣒⠰⡨⣩⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣑⡒⠇⢽⣻⣿⠟⣽⣿⣿⣿⣿⣧⣿⣿⣿⡁⢿⣿⣿⣿⣷⣷⣿⣯⣷⣿⣿⣿⣿⡻⣇⣿⣿⡻⣾⣿⡿⠀⠀⠀⠀⠀⣿⡿⢗⠊⣐⡰⢿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠶⣧⡵⢡⢟⢻⣿⠀⣫⣻⣿⣿⣻⢿⣿⣿⣿⣿⣷⣶⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⡅⢠⣍⠴⠞⣛⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣔⢛⣻⠇⠀⠚⣷⣗⣟⡟⢹⣿⡟⢿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⢡⠢⠘⠉⠡⢸⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡂⡐⢀⡕⣮⣯⠀⠀⢸⣧⣿⠟⢮⣓⠬⣷⣎⢿⡧⢐⣕⡮⣧⣽⣯⣭⡻⢿⡻⣛⣯⡽⢟⣇⣽⣻⣿⣯⠇⠀⠀⠀⠀⠀⠀⢹⡀⡐⡀⠀⠡⢘⢼⠀
⠀⠀⠀⠀⠀⠀⠀⢐⡅⠐⣒⢧⣼⠃⠀⡘⢿⣝⣿⣿⢿⣾⡽⠶⣬⣚⠭⡧⠿⠿⢛⣛⢺⠶⠿⠿⠾⠾⣶⣟⣛⣽⢿⢾⣿⣿⣧⢄⠀⠀⠀⠀⠀⠀⡇⣕⡂⠀⠐⠨⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡲⠇⠏⠀⢀⢠⡿⠘⢷⣺⠻⣾⣿⢷⢛⣦⣫⣏⢲⣜⡳⢄⢛⢷⣿⢘⢿⣟⠊⢟⣿⡏⠚⣧⠐⣿⣿⣷⣓⠀⠀⠀⠀⠀⢷⠻⠀⠀⠁⢈⡇⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⢻⠀⠀⢡⣯⢑⣻⠿⡷⢯⡷⣭⡛⠿⢿⣞⢿⣿⡬⣽⡏⡿⡾⣽⣤⠿⣿⠀⠱⠏⡧⠀⢯⠀⢻⣿⢟⣯⠇⠀⠀⠀⠀⡼⠉⠉⠉⠀⠀⠸⠀⠀