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