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