Gada Emojis & Symbols

⠀⠀⠀⠀⢀⡀⠠⠤⠠⣤⣀
⠀⠀⠀⠈⠀⣀⣴⠚⣇⡛⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⢠⣼⠁⡠⠃⠍⣋⠑⠛⢄⡀⠜⠣⠔⠂⠀⣤⣰⠆⠁⠒⠁⠤⠤⣍⡖⠂⠄⠂⠄⡀
⠀⢠⡰⠧⠠⡫⢂⡔⠂⣄⡨⠞⠁⠀⡀⠄⣒⠤⠚⢁⡀⡢⠌⠈⠌⠒⠠⠀⠉⠊⠆⠄⡀⠢⡐⢄
⠀⠈⣟⣡⡟⡰⠃⢱⠔⡋⠀⠀⠀⢘⠰⠂⢁⠀⡡⠀⠠⠉⠀⡂⠄⢀⠀⠁⠢⣁⠢⣀⠪⡢⡈⠠⡑⡀
⠀⠀⠀⠀⠊⢄⠖⠁⠈⠀⠴⠠⢞⡁⠤⠎⢅⣈⠀⠀⠀⠀⠀⠈⠑⠠⡑⠤⡀⠀⠑⠬⡣⡈⠊⡢⠈⢌⢄
⠀⠀⠀⠘⠳⠋⠀⠁⣄⣡⠞⠈⠑⠂⠀⠒⠀⠠⢉⠒⠄⡀⠀⠀⠀⠀⠈⠢⡈⢢⠀⠀⠈⢈⢆⠐⠕⡀⠀⡀
⠀⠀⠀⠙⠁⠀⠄⣠⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⡢⢄⠀⠀⠀⠀⠈⠣⡑⠄⠀⠀⠙⣆⠈⢦⠀⡐
⠀⠀⠀⠈⠀⢈⠘⠁⠠⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⣄⠀⠀⠀⠀⠘⠌⢆⠀⠀⠘⣆⠘⡆⢠⠄
⠀⠀⠀⢤⠤⠊⢐⢥⠀⠀⢁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⠀⠀⠈⢆⠂⠀⠀⢡⠀⢿⢸⠄
⠀⠀⢀⡴⢀⠁⠀⠀⡀⠀⠀⠌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠄⠀⠀⠀⠀⠀⡄⠀⢸⠀⣾⡏
⠀⠀⢩⠤⠾⠀⠀⠀⢠⠀⠀⠈⢂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄⠀⠀⠀⠀⢠⠀⣸⣟⣕⠁
⠀⠀⠌⠀⢠⢀⠀⠀⠂⠂⠀⠀⠀⠡⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠄⠀⠀⠀⢸⡴⠫⣃⢟⠁
⠀⠘⠀⠀⠀⠄⠀⠀⠈⠌⠄⠀⠀⠀⠈⢊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⡀⠠⠃⠜⢰⢏⠅
⠀⢰⠐⠀⠀⠀⢣⠀⠀⠐⠌⢄⠀⠀⠀⠀⠀⠈⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⢚⠱⠥⢲⠋⣼
⠀⠀⡌⡄⠀⠀⠱⡡⠀⠀⠈⢊⠠⡀⠀⠀⠀⠀⠈⠢⢁⠂⠄⡀⠀⠀⠀⢀⡐⣠⠆⡔⠉⡂⡞⢀⡞⡷
⠀⠀⠠⠐⠄⠀⠀⠐⠔⢄⠀⠀⠑⢬⡢⢀⠀⠀⠀⠀⠀⠁⠂⢤⣀⠀⣀⠤⡺⠋⡃⠀⠞⢁⡴⡋⡱⡁
⠀⠀⠀⠑⢌⠢⡀⠀⠈⠳⣀⢀⠀⠀⠈⠐⠅⡐⠠⠀⠀⠀⣠⣞⠀⡀⠔⠂⢑⡲⠞⢉⣴⠋⣆⠼⠀⠟⣄
⠀⠀⠀⠀⠀⠓⢌⠢⢀⠀⠈⠑⠕⡀⢀⠀⠀⠀⢈⣖⠂⢉⢩⠄⠒⢁⡴⡳⠉⢠⠖⠇⣀⡶⢩⡞⢡⡞⡸⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠍⢂⣄⣀⣀⣁⣦⢭⡭⣻⢭⢒⣭⡠⠟⠋⢋⣀⡴⠊⢁⣥⠞⢩⠁⣹⠇⣰⡁⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠋⠙⠂⠈⠒⠎⠹⠆⠈⠛⢭⡨⢀⡀⡀⡧⢀⡑⠠⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠈⠐⢈⠁⡻⠁⡔⠀⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⡈⠘⠑⡈⠀⢈⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣡⡰⠁⠰⣻⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⡟⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿⠇⣨⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠷⠶⠶⠾⠷⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠞⠛⠋⣿⠏⠉⠉⠉⠉⢿⡏⠛⠛⠻⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣽⠟⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠻⣯⡙⠳⣦⣄
⠀⠀⠀⠀⢀⣴⠟⠋⠀⣠⡿⠁⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠈⢿⣄⠀⠙⠻⣦⡀
⠀⠀⠀⣴⠟⠁⠀⠀⣰⡟⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠻⣦
⠀⢀⣾⠋⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠙⣷⡀
⠀⣾⠃⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠘⣷
⣸⠇⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠
⠘⣷⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣾
⠀⠘⣧⡀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢀⣼⠃
⠀⠀⠘⢷⣄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠙⢷⣄⠀⠀⢻⣆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣰⠟⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡙⢷⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢀⡾⢋⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣀⣀⡈⢿⡄⠀⠀⠀⠀⣸⠏⢀⣀⣀⣴⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⠛⠛⠿⠿⠿⠿⠛⠛⢿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⣤⣤⣤⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠻⠾⠷⠟⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⠶⠶⣶⡶⠟

⠀⠀⠠⠁⣠⡾⠛⣣⠼⡺⠖⣳⣤⠤⠤⣤⢄⣀⠀⠠⠄⢄⡠⠤⡔⠽⠤⠀⠠⢆⢠⠤⢀⡀
⠀⠀⢃⣾⠛⠢⢂⠔⣠⠒⠓⢤⡤⢛⠉⠀⢉⣠⢐⡨⠟⠛⡩⠤⠤⠮⢟⣙⠛⠳⢦⡄⡁⠂⢕⠢⡀
⠀⠰⣾⠅⣽⠕⡡⢊⡀⣹⠖⠡⠂⠁⣤⣈⡠⠒⡉⠤⠴⠢⠮⢅⣀⠈⠁⠒⠭⣐⠢⣈⠊⢕⢄⠈⠪⡳⡀
⠀⠀⠛⠚⢋⡎⢣⡠⢋⠔⠁⡠⠂⣠⠗⢉⡤⠶⠰⠉⠈⠉⠉⠑⠢⢍⡒⢄⡀⠀⠑⠢⡑⢤⠑⢝⢆⠈⢎⢂
⠀⠀⠀⠀⡆⣰⠊⡀⠀⡠⢊⡩⡯⢔⣒⣛⣒⠒⠲⢄⡀⠀⠀⠀⠀⠀⠈⠢⣌⠢⡀⠀⠈⠓⣕⢄⠑⡵⡀⠣⢣
⠀⠀⠀⣠⡙⠡⠈⠀⢺⡲⢋⢊⠊⠁⠀⠀⠀⠉⠑⠒⠌⡑⠤⡀⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠈⠪⣣⠈⢞⡄⢡⢃
⠀⠀⠀⠠⡄⢁⣄⡠⢋⢔⡎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡢⣄⠀⠀⠀⠀⠈⠣⡑⣄⠀⠀⠙⣧⠈⢿⠀⢎
⠀⠀⠀⠐⠠⢁⡔⢁⢂⠁⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⠀⠀⠀⠘⡌⢆⠀⠀⠸⣇⠘⡇⢸⠆
⠀⠀⠀⠰⢖⢫⢊⢼⡆⠀⠈⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢦⠀⠀⠀⠀⠈⢆⢆⠀⠀⢻⠀⣿⣺⠂
⠀⠀⣠⠴⢃⡇⠃⠘⢰⠀⠀⠘⡌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡳⡀⠀⠀⠀⠘⡜⡄⠀⢸⣀⣿⡇
⠀⠀⢠⢷⠾⢸⠀⠀⠇⡆⠀⠀⠈⢎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠱⡀⠀⠀⠀⠁⠇⢀⣾⡻⣾⡁
⠀⢀⢣⠁⠀⡼⠀⠀⠸⡰⡀⠀⠀⠀⠢⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢡⠀⠀⠀⠀⣴⠞⠵⣽⡏
⠀⢸⠘⠀⠀⢡⡇⠀⠀⢱⠱⠀⠀⠀⠀⠈⢎⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⣄⣀⢔⡕⡣⠎⣸⣃⠃
⠀⢸⢰⠀⠀⠈⣾⡀⠀⠀⠣⡱⡀⠀⠀⠀⠀⠑⢌⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠃⡡⡫⠺⢤⢟⢁⡎
⠀⠀⡄⠆⠀⠀⠘⢝⡄⠀⠀⠑⢌⢦⡀⠀⠀⠀⠀⠈⠢⢍⠢⠤⣀⣀⠀⠀⢀⣠⢃⡼⣪⠊⣸⣰⠋⣠⠞⡎
⠀⠀⠱⡈⢄⠀⠀⠈⢪⢢⠀⠀⠈⠳⢝⠦⡀⠀⠀⠀⠀⠀⠉⠒⢤⣀⣉⣉⣡⢴⠟⠛⡄⡡⠋⢀⣴⠉⡷⡁

⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡂⠀⠀⠀⢀
⠊⠀⠀⠑⠀⠀⢀⡀⠐⠂⠁⠀⠀⣀⠍⡁⢠⡀⢰⣵⣄
⠀⠀⢀⠄⢈⢖⡴⡀⠀⡠⢠⠂⢀⡘⢰⢀⠎⡃⠤⠈
⠀⠐⢁⠀⠠⠋⠊⠀⠔⡑⠁⣀⡸⣴⢃⠞⠀⢌⠻⡀⢀⣀⡏⠄
⠰⢐⡵⡊⠀⠀⠀⠈⠈⢀⡀⢫⠜⢁⣮⠀⠀⠙⢇⢡⠀⡘⡸⠀⢃
⠃⠉⠊⠀⠀⠀⠀⢀⠀⠸⠀⠊⠀⠡⣉⠠⠄⠀⠀⠀⡀⠰⠁⣴⢹
⠀⡠⢔⠥⠊⠀⡤⡠⠀⠀⠀⠈⠀⠠⠈⡍⠉⠥⠤⡰⡱⠁⣸⢣⢋⡄
⠀⠂⠁⠀⡶⢆⡡⠈⡀⡆⠀⠀⠀⢦⡇⡇⠀⠀⠈⠔⢀⡴⣡⡛⡘⠄
⠦⡀⢋⡡⠔⡉⠔⢉⡀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠎⣰⢃⡐⣁⡄
⠒⣉⠡⠶⡟⠀⢀⣂⢛⠒⠂⠀⠀⠂⠀⠀⠀⠀⡰⢃⠞⢁⠚⡰⠣⢇
⠉⡀⢦⠰⣄⠀⠀⠇⠘⠀⠀⠛⠓⠀⠀⠀⡄⢃⠰⢘⠸⠁⠜⠱⠃⢠⠂
⡀⠹⠸⠁⠂⠌⠀⠀⠀⠀⡂⠄⠀⣠⠼⢉⠔⠉⡔⠃⠀⢠⠖⠲⠄⣠⣅
⠃⠀⢀⣀⡈⠷⣒⡬⠶⠈⣀⡴⢚⡤⠞⢁⠐⢁⠀⣀⠄⠀⠀⠃⠀⢑⠝⠀⡄
⠀⠀⠘⠒⢂⣉⣁⡴⢔⣊⠵⠚⢩⡰⢈⡤⢒⡤⠈⢅⡀⠀⠀⠀⠀⢈⠘⣳⡺
⠀⠀⠀⠀⠀⠁⠙⠊⠉⠂⠀⠀⠀⠐⠏⠂⢁⠀⠀⣠⢔⢀⡠⢀⡀⠉⠇⠵⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠓⠡⢦⣨⢍⡁⣐⠦⠕⠆⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⣁⣀⠰⢔⠢⠄⠂⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠒⠉⠀⠀⠈⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡐⠊⠩⢐⡂⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣈⠁⠀⣠⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣍⠀⣐⠬⠀⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⣀⡾⠆⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣠⠟⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄⠀⣤⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣴⣿⣿⠉⠈⠻⣿⣿⣿⡟⠉⠀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠛⠉⠉⠉⣉⣹⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣤⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠀⠀⠀⠀⣰⣾⣿⣿⠿⠿⠿⠿⠿⣿⣿⣷⣄⠀⠈⠻⣿⣷⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⢀⣀⠀⠀⠈⠋⠁⠀⠀⢀⣠⣶⣿⣿⠿⠿⣿⣷⣄⠀⠈⠻⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⣀⣴⣿⣿⠟⠉⠀⠀⠀⢸⣿⣿⣷⣄⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⣠⣿⣿⢻⣿⣿⣷⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣾⣿⠇⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⢸⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⣼⣿⡏⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⢸⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣿⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⢀⣿⣿⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⢠⣾⣿⠏⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⠉⠻⣿⣿⣾⣿⡟⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠈⠻⣿⣿⣄⠀⣀⣤⣶⣿⡿⠟⠁⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⡿⠟⠉⠀⠀⣀⣠⣴⣾⣿⡿⠛⠁⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠙⢿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⢀⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈⢻⣿⣿⣿⣷⣶⣦⣤⣴⣶⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣤⣼⣿⡟⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⢀⣠⣾⣿⠟⠉⠀⠙⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⠿⠋
⢀⣴⣶⣶⣶⣤⣾⣿⠟⠁⠀⠀⢀⣴⣾⣿⠟⠁
⣾⣿⠏⠉⠻⣿⣿⣇⠀⠀⢀⣴⣿⣿⠟⠁
⢻⣿⣷⣄⠀⠈⠻⣿⣷⣶⣿⡿⠋
⣼⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⡄
⣿⣿⣆⠙⣿⣿⣷⣄⢀⣸⣿⡇
⠘⠿⣿⣿⣿⠿⠿⣿⣿⡿⠟⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣎⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣮⣤⣤⣠⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣥⣅⠀⣘⠙⠷⡾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣭⣀⡺⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠉⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠠⢄⠀⠀⠀⢀⠀⠀⠀⠩⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢠⠂⠀⠀⠀⠀⠇⠄⡰⢀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠔⠛⠁⢐⠏⢀⠫⡔⡐⠀⡙⢏⢀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢵⠃⠀⠀⠀⠀⡤⠙⠊⠴⠀⠀⠙⠈⠀⢡⠃⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢠⠀⠀⠀⠐⠁⠀⠀⠐⠌⠁⠁⢀⠀⠎⡼⢝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠤⡈⠑⢠⠀⠀⠀⡄⠸⠀⠈⠃⢀⡜⡱⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠰⢤⢊⠕⡂⠔⠠⡀⠠⠀⠀⠁⠀⠀⠀⢠⠎⡜⡁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⢘⠁⠀⡅⠁⠉⠠⡀⠀⠀⢀⠔⡡⢋⠆⡑⡅⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠰⢸⠪⠁⠀⠀⠀⠀⠀⠀⢀⠔⡀⠄⠔⠁⢈⠄⡅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⣀⡰⣖⠬⠀⣈⠤⣊⠥⢊⠑⠁⢀⠀⠀⠐⠀⠛⢃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠬⢤⣶⣮⡒⠋⠅⣂⢔⠢⠦⠴⠀⠀⠀⠀⢠⡨⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠧⢐⠀⡢⠀⢀⢡⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⡉⣀⠤⠐⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒⠉⣀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⡀⠔⠂⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡚⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠹⣿⣿⣿⣿⣿⣷⣷⣶⣶⣾⠇⣍⡉⠈⠻⢿⣿⣯⣿⣿⣿⣿⣿⣿⣾⣥⣄⣀⣴⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢳⣶⣬⣭⠛⣿⡟⠛⠛⠛⢒⠘⠳⣄⣑⠆⠈⢹⠿⠛⣉⣭⣵⠶⢂⣤⠶⣿⣿⠗⢶⣤⣄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⣧⣤⣀⠀⠚⠻⢟⡥⠚⣡⣶⣶⣄⣀⣀⣀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢉⣭⣿⢿⠷⣨⣷⡎⢀⣼⢿⡿⣿⣿⣿⣿⣿⠗⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⢿⣯⣠⠕⢿⣿⣷⣌⡁⠀⠀⠀⢠⡾⠟⠁⣼⣿⣿
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠭⠧⠤⠭⠭⢉⣟⣳⣶⣤⣄⣀⣀⡸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⣻⣇⠉⢭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠴⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⢠⡀⠀⣿⣧⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢤⢤⣤⣀⡀
⣭⣿⣿⣿⣿⣶⣶⣤⡤⠀⠀⠀⢠⣴⣲⡴⠺⢯⠃⣒⡀⢚⡐⢾⣻⣛⣶⣄
⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠞⠀⠐⠠⢀⡈⡹⢣⠟⣸⣯⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷⠄⠈⠢⣤⣀⣼⣡⢏⡼⣿⡮⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡬⢲⣦⣼⣿⠛⣱⠟⢹⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡟⣫⣥⣞⣡⣾⣻⣷⣾⣿⣿⣿⣾⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⣽⣿⣏⠤⠒⠛⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣶⡖⠛⠉⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣥⣀⣈⣭⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠘⢀⠇⠀⠘⡃⡠⢂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠀⠉⠠⠏⢠⠶⠈⠛⠳⠀⠀⢠⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡤⠀⠀⠢⡄⠀⠠⣁⡶⢂⡀⠴⢆⡀⠰⣶⡠⡀⠂⣰⠧⡁⠀⡶⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡁⠀⠁⠁⠀⡄⠀⠀⠄⢁⣻⢴⣿⠟⠗⠈⠹⣷⡈⣧⡄⠂⠈⠀⣤⠀⡀⠐⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠉⠁⠂⠤⠔⠊⠰⠆⠒⠀⠀⣿⠟⢣⢃⠠⠃⠰⡿⢷⡾⠀⠀⠓⢠⡄⠑⠤⠠⠔⠂⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⢃⡼⣫⠀⠀⣤⠂⣵⣾⢷⣾⣆⠰⣿⠔⠁⠊⠀⠐⠀⠘⢄⣷⠀⣴⣧⡾⣦⣅⠢⡄⠀⠀⣼⢦⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠰⡎⠑⠀⠀⣼⣷⣤⣿⣷⡷⠉⢈⡀⠀⡔⣀⠀⣤⠀⠁⢀⣄⠀⣀⠉⠙⣰⣽⠿⣥⣿⡆⠀⠰⠀⢳⡄⠹⠄
⠀⠀⠀⠀⠀⠀⣀⠀⡾⠁⣸⠁⠈⠢⠀⠎⣹⡾⣭⣟⣁⠞⠛⣇⣸⣤⠻⡥⢄⡈⢳⠎⣽⣦⣙⠛⢧⣘⣛⣥⢿⡌⠣⣀⠥⠓⠺⡆⠑⡦⠀⡄
⠀⠀⠀⠀⠀⠘⠏⣑⠛⠢⣅⡀⠐⠀⠠⡞⠻⢿⡿⠏⣠⠀⢀⣄⡉⢁⠂⠐⢉⡁⠂⠰⣉⢱⣠⡀⠀⡬⠿⢿⡿⠿⠶⠀⠐⠄⠀⣨⡤⠿⡂⠳
⠀⠀⠀⠀⠀⠉⡀⢾⡕⢃⣎⣽⣶⠛⡆⣨⡙⡶⢶⡤⣤⡠⢄⡙⠆⢾⣃⣼⣧⣙⢂⣹⡶⢤⣁⣠⣤⣰⠒⣦⠶⢲⡈⢱⠾⢤⣟⣁⠚⢾⡷⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠉⢙⠷⠤⣡⢿⣿⣡⣄⣐⠁⠀⢈⣡⢿⣍⣩⣿⢙⡁⣷⣌⡻⢾⣉⡄⢀⠙⣀⣅⡘⣯⣿⣄⣤⠾⠏⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢎⡙⢋⡴⠖⠂⢚⢉⡍⠙⢋⡍⢉⣉⣉⣉⣉⣭⣩⣌⢉⠉⣃⠉⣡⡍⠉⣉⠙⠓⠲⠶⣌⡈⠛⠱⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣢⣔⣉⣔⡲⠿⠉⠟⠳⣿⡷⠃⠀⣴⡖⠶⢴⡂⣄⢪⡶⢲⣬⠀⠛⣶⣶⠿⠏⠁⠷⠖⢤⣉⣨⣆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⢿⣹⡗⣺⠓⣿⣶⡿⣄⡐⣷⠟⣉⣁⠚⠁⢨⠇⣀⣛⠛⡷⢂⣬⡷⣶⡿⠲⣦⠺⣻⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡷⣌⡛⢷⣯⡿⣻⣿⣀⡉⢩⡅⠄⠀⣼⠍⠉⣀⣿⡟⣿⣽⡿⢟⣠⣾⠟⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠀⠉⢀⣈⠉⠛⠟⣟⣛⡂⠠⣔⣠⢀⣚⠛⠿⡿⠛⢉⣁⠀⠉⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⡘⡠⠛⢦⡄⢡⣼⢉⠉⣈⡩⢈⠁⡤⠾⢃⡖⠌⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⡤⠀⠀⢸⣇⡘⠢⠀⠁⠀⣠⡷⠀⠀⢠⡌⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣼⣇⠹⣷⠳⢂⡼⢃⡿⣠⠾⣢⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢡⣮⣖⣀⠹⠀⠤⢊⡐⣪⣲⣇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣻⢿⣿⣶⣃⣚⣷⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣾⣭⣙⢲⡍⠙⠋⠙⠋⣹⣄⣋⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣽⣻⣯⡔⢲⡖⣠⣽⣛⡯⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣭⡉⠷⣴⣦⠿⠉⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⢶⣣⠀⣉⠡⢬⣘⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢁⣄⠀⠑⣴⣿⣇⠄⠁⢀⣄⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⠀⡏⣹⡋⠀⠐⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡟⣭⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡴⠛⠻⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣼⠉⢄⡉⣀⠘⡸⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢇⡀⢻⠀⡈⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡅⣀⡈⣑⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣷⡿⠞⡱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢬⠀⡗⢲⢠⡖⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠀⠃⠘⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣉⣹⣿
⠀⠀⡀⠀⡀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⣀⠀⢀⠀⣀⠀⠀⣀⠀⠀⢀⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠹⡟
⠀⠀⣇⢀⡇⠀⢠⡘⣧⡀⠀⠹⣆⠘⠀⣟⠀⢨⠀⢹⡆⢠⣿⠀⢠⡜⢧⡀⠀⠘⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⣠⣴⠁
⠀⠀⡇⠀⡧⠀⣬⠈⠹⡄⠀⡄⠈⠧⠀⢻⠀⠘⠀⡄⠹⠀⣿⠀⣤⠉⠙⣧⠀⣤⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠅⣠⣼⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠁⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⢹⡆⢳⣰⢁⡍⣷⢸⡏⣿⠘⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠥⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠁⠰⠃⠀⠁⠁⠈⠁⠈⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠶⢤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠐⢲⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣈⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠢⢤⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠓⢒⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣉⣉⣁⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠤⠤⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⠒⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣈⢙⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠦⢀⣩⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠒⠢⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣛⢏⣑⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⣠⡄⠰⣦⡀⠙⠿⣷⣍⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠘⠋⠉⠭⠿⠻⠟⢋⡉⠙⠳⢌⠻⢦⣄⡀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣷⣄⠉⠀⢀⠤⠈⢉⡑⢀⣀⠀⠁⠠⢍⡛⠓
⠀⠀⠀⠀⠒⠀⠀⠀⣠⠊⠙⠻⣷⣄⡉⠀⠀⠀⠰⠃⢈⠙⢷⡄⣀⡈⠳⡖
⠀⠀⠀⠀⡀⠀⠑⠰⠁⠀⠀⡰⠉⠻⢿⣦⣤⡙⠀⠀⠀⠀⡸⠡⡭⠻⠛⡿⣆
⠀⠀⠀⢸⡇⠀⢰⠉⠁⠄⠞⠀⠀⠀⠀⡉⠻⣿⣶⣄⡀⠶⠀⠀⠀⡆⠀⢻⠻⡆
⠀⠀⠀⠘⢿⣄⡄⠀⠀⠔⠊⠁⢀⢀⡞⠤⢀⡀⠙⠻⣿⣷⣤⣀⠐⠇⠀⠀⠀⣿
⠀⠀⠀⠀⢄⠉⢿⣦⡐⠀⠀⠀⠐⠏⠀⠀⡀⠐⢀⠀⠀⠉⠻⣿⣷⣶⣤⣄⣀⢘
⠀⠀⠀⠀⠸⡀⠌⠙⠻⣷⣤⡀⠛⢄⠈⠛⠁⢠⣯⡄⠀⠀⢀⠎⠈⠻⢿⠘⣿⣿
⠀⠀⠀⠀⠀⢱⣄⠀⠂⠈⠛⢹⣷⣤⣀⠀⠞⠁⠀⠀⠛⡰⢇⠀⠀⣠⠃⠀⠀⠏
⠀⠀⠀⠀⠀⠀⡘⢷⣄⣀⠀⠀⠈⠛⠋⣿⣦⣄⡀⡠⠊⠁⠀⢈⠍⠓⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠟⢶⣙⠿⣧⣀⡀⠀⠀⢀⡙⠛⢿⣷⣤⣠⠔⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⡈⠉⠷⢤⣝⡻⢶⣤⣀⣀⡀⠂⠈⣉⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠄⡉⠙⠷⠂⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠴⠦⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠧⠤⠼⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠓⠛⠛⠛⠓⠚⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠒⠒⠓⠓⠒⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⡟⣿⢛⠛⣿⠻⣟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡟⢉⠏⠉⠹⡉⢻⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⡏⢱⣿⠄⠸⣿⡄⢻⡇⢹⡆⠀⠀⠀⠀⠀⠀⡾⠀⣼⠀⢸⠀⡄⠀⡇⠀⢇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⡇⢸⣿⠀⠠⣿⡇⢸⣷⠀⣷⠀⠀⠀⠀⠀⢰⡇⠀⡇⠀⢸⠀⠗⠀⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢸⣿⠀⠀⣿⡇⢸⡟⢰⡇⠀⠀⠀⠀⠀⠀⣇⠀⣧⠀⢸⠀⠀⠀⡇⠀⡼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢿⡄⣿⣐⣀⣿⢀⡿⣡⠟⠀⠀⠀⠀⠀⠀⠀⠘⢦⡘⣆⠘⡄⠀⢰⠇⣰⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢛⣋⣋⣛⣛⣛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣉⣉⣉⣉⣉⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣭⡭⢭⣭⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢭⡥⠤⢭⡭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡏⠀⠀⢀⣀⠀⢀⡀⠀⣀⣀⠀⣀⠀⠀⠀⠘⡇⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⡇⠀⠀⢿⣰⡆⡾⢽⡄⣇⡼⢲⠯⣧⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣙⢛⣟⣛⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣒⣶⣿⣦⣤⣾⣷⣖⣊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⠿⠿⢿⠻⡿⢫⡿⠿⢿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⡿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⣿⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⣿⣯⣥⣴⣦⡄⠀⠀⠐⣄⢀⡴⠀⠀⠀⢤⣶⣦⣭⣿⡧⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣭⠉⣿⡶⠀⠀⢀⠀⢹⣾⠃⢀⠀⠀⠰⣾⣏⢩⣽⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⡿⢋⣾⣿⡄⣘⣶⣾⠁⠈⡟⠀⢸⣶⣞⡁⣼⣿⣆⠻⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⢿⣿⡇⠈⠛⢿⠀⠀⠁⠀⢸⠟⠊⠀⣿⣿⡟⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡇⠀⠈⢻⣿⣆⢠⠃⠀⠀⠀⠀⠀⢣⣀⣼⣿⠏⠀⠀⢿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣴⣦⠀⠀⢻⣿⠇⠀⣤⣤⣠⣤⡄⠀⢻⣿⠏⠀⠀⣶⣼⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢳⡈⣿⡀⠀⠀⠙⡟⠁⠀⠀⣸⡟⣠⠎⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠟⢹⣿⣿⡟⢰⠀⢻⣿⣷⣤⡀⠤⠦⠄⣠⣴⣿⣿⠇⢰⡀⣿⣿⣿⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣵⣿⠀⠘⣿⣿⣿⡀⠀⠀⠀⣨⣿⣿⡟⠀⢸⣷⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣼⣇⢁⠙⣿⣿⣶⣶⣾⣿⡿⠁⠁⣾⣌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⢹⣿⢻⣿⣜⣶⣌⡻⣿⣿⡿⢋⣤⣞⣼⣿⢻⣿⠃⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠻⣿⣿⣟⡻⢿⣿⠿⢛⣿⣿⣿⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⢿⣆⢀⣴⠟⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠈⠒⣦⣤⣤⣤⣀⣀⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⢄⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠉⠢⡈⠙⠳⠒⠚⠉⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⣰⠀⠀⡀⢀⠀⠰⡀⠈⣦⠆⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠀⠀⢸⡇⠀⠇⠀⠰⡡⠊⣤⣄⢳⢀⡏⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⣠⣴⡟⠀⠤⢤⡀⠈⠀⠀⠀⠀⣸⢞⠇⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣿⣿⠀⠐⠒⠦⠄⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⣟⡏⠀⠀⠒⠤⡀⠀⣠⠄⢠⣿⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣿⡟⠒⠤⠤⡀⢠⣾⣧⢄⡸⣿⡆⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⣀⣀⠤⠖⠋⢀⣠⠟⠋⢿⣐⠠⠀⠀⣾⡿⡇⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣼⡀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡇
⠀⠈⠓⠒⠒⠒⠊⠉⠀⠀⠀⣼⠮⢭⣥⣠⣿⡵⢃⣼⣿⣿⣿⣿⡛⢶⡦⢾⣤⣤⣤⣤⡜⠸⣿⠇⠀⠀⠀⠀⠀⠀⡏⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠷⠌⣿⡶⠿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣠⠈⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡄⠀⣼⠁⠀⣿⣿⣿⣿⣿⣿⠙⠿⠿⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢀⡟⠀⠀⢹⣿⣿⠿⠿⢿⡄⠀⠀⡇⠀⠀⠀⢀⣀⣼⣀⣀⣤⣴⠟⠛⠉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠦⢀⡾⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⢹⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠃⢈⣰⠟⠁⠀⠀⠀⠀⢸⣧⠀⠀⢠⠀⠀⠃⠀⠀⢸⣿⡿⠋⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠧⣤⣤⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⢠⠁⠀⠀⢀⣼⠋⠀⠀⠀⢋⠹⣿⡆⠀⠀⠀⡄⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⡀⠀⠀⣾⠃⠀⠀⠀⠀⠸⡄⢿⣧⠀⠀⠀⡇⠀⢰⡏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠤⠀⠈⢷⠈⢀⠄⢻⣶⡄⠀⠀⠀⢀⠁⢸⡿⠀⢀⡀⢱⠀⠈⢇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠀⠀⢨⡇⠀⠀⢸⣿⣿⠀⠀⠀⠈⠓⠚⡇⠀⠠⠄⠀⢧⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⣸⢿⣷⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⣧⣠⣤⣤⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡌⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⢀⣀⠰⠖⠈⠁⠀⠰⡄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⠀⠐⣄⠀⠱⡄⠹⣆⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢈⠿⣆⠀⢈⡦⣀⣿⣴⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣸⣷⣶⣿⣿⣆⣩⣿⣿⣾⣿⣿⡿⠏⠀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣸⣿⡿⠀⠈⠉⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⡿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡿⣱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠸⢱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡼⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠀⢤⣄⣀⣴⣊⡠
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆
⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⣶⣲⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⣠⠞⠟⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣸⡟⠀⣿⡄⠀⠀⢀⠞⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣿⣿⣿⣿⠟⠉⣉⣽⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠘⢶⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠲⢶⣿⣿⣿⣿⣿⣟⣋⣭⣄⣠⣶⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣯⣶⣾⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠉⠛⠿⣿⡿⢿⣿⡟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⣠⠴⠶⠦⣤⡤⠖⠒⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⣁⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡎⠀⠀⡀⠘⠢⣄⠀⠈⠣
⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡐⢄⡂⢄⢢⡘⣦
⠀⠀⠀⣿⣾⢏⡠⠀⣠⠔⠡⢾⣿⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣖⠈⡂⣤⡘⢮⠃
⠀⠀⠀⠸⣯⠞⢠⣞⠁⢊⣴⣿⣿⣯⡻⠛⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⣿⣿⣷⡌⠳⣝⢦⡱⡄⠁
⠀⠀⠀⠀⠿⣶⡿⢑⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠹⣾⣿⣿⣿⣿⢝⢦⡙⣷⡝⡆
⠀⠀⠀⠀⠀⠙⠇⣼⠇⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡡⣵⡈⠿
⠀⠀⠀⠀⠀⠀⠀⠙⢲⣟⣿⣿⣿⣿⣿⠁⣤⠀⠀⣀⡂⠀⠀⡇⠈⣿⣿⡿⣿⣾⣿⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⣼⣿⡟⠐⣿⡀⠀⣒⡆⠀⠀⣤⠄⡘⢿⣿⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⣰⡇⠀⠯⠄⠀⢀⣿⣦⠀⠈⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣀⣤⠤⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢙⠘⠿⣯⠁⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⡄⢠⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⢆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⠈⣷⠀⠀⠀⡀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠼⠛⠉⠹⣟⣷⣰⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣦⡈⠛⠏⠒⢦⢾⣗⢹⣦⠀⠠⠈⠑⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⡙⣀⠈⠃⣄⢆⣈⠏⡿⠇⠀⢰⡄⠀⠈⠉⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠂⠀⠀⠚⠚⣿⠀⢀⠀⢀⣾⠁⣤⠀⣠⣿⠟⠲⠂⠀⠒⠂⠀⠈⠉⠀⠀⠈⠉⠉⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠢⡤⠛⣠⠇⠀⣾⡇⢀⣿⠀⠀⣆⣼⠇⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠸⣦⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠣⣐⣃⣤⠛⠆⣸⣿⠷⠀⠁⠀⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠄⠀⠀⠀⠀⠀⠀⢠⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠉⠳⣂⣴⣿⣿⣇⡗⣤⠀⢀⡶⠉⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢷⣆⡻⠙⣉⠁⠈⠃⡀⣀⣤⠃⢺⡀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⣼⠃⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣼⣍⡀⣦⣸⣿⣿⣏⢀⡀⡉⣉⡉⠀⠀⠄⠐⠒⠁⠀⡄⠀⠀⠀⢸⠉⠛⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⢻⢿⣿⣿⡟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣴⠀⠠⠀⠈⢆⠀⠀⢀⣴⣜⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⡟⠀⠀⠠⣀⠘⢳⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠀⠀⠀⠑
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠂⠀⢠⣀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡤⠋⠀⠀⣔⢂⡀⣨
⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠈⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠉⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠈
⠀⠀⠀⠀⡠⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡼⠀⡿⠾⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢁⡴⢁⠞⣡⠎⠉⠃⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣾⣾⡅⢤⣾⡾⣤⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠛⢿⣿⣿⣿⣿⣿⡦⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⠃⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠇⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠅⠀⠀⠀⠀⠘⡆
⠀⠀⠀⣠⣤⠶⠶⠶⠶⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠒⠉⠉⠈⠉⠉⠙⠲⣄⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⠣⠀⠀⠀⠀⢀⣴⡆⠀⠠⠻⠿⢿⣧⠀⢠⣤⣶⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣤⣤⣤⣴⣿⣿⣯⠂⠀⠀⠀⠀⠈⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣀⣀⠄⣀⣴⣆⡀⠘⣿⡯⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠟⣿⣍⣉⣀⣀⣈⣙⣿⢻⠿⠿⠿⣿⢿⣿⣿⣿⣿⣿⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⢀⡞⠁⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⢀⠤⠔⠒⠚⠛⠃⠒⠢⠤⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠘⠿⢿⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠦⢤⣀
⠀⠀⠀⠁⠈⡃⠀⠀⠀⠀⠀⠀⢙⣀⡀⢰⣄⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡂
⠀⠀⠀⠀⡀⠁⠀⢄⠀⠀⣰⣿⣸⠏⠁⣰⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⡀
⠀⠀⠀⠀⠆⠀⠀⢀⣳⣤⡍⠉⢠⢆⣴⣿⣿⣷⡄⢠⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷
⠀⠀⠀⠀⠀⠛⠻⠟⠟⠋⠀⠀⠉⠀⠀⠀⠀⠘⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠶⠾⠯⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠃⠀⠀⠀⠀⣰⣾⣄⣀⠀⠀⢀⣴⣶⣿⣧⠴⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⣴⣿⣿⣾⠿⣾⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠉⠉⠉⠛⠛⠋⠉⠉⠙⠻⣶⡀⣤⡀⢸⠅⢱⡀⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡏⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢹⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠻⠀⣠⣶⡴⠃⡀⠀⣤⣦⣤⢤⡤⠀⠀⠁⠀⠀⠶⠶⢦⣉⣡⣠⠼⡿⢿⠿⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣤⣇⣢⣼⠀⠀⠀⠀⠀⣰⡟⠛⠛⠉⠙⣷
⠀⠀⠀⠀⠀⡀⣠⠞⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⠡⣤⠀⣤⡀⢿
⠀⠀⠀⢀⠜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⠀⠈⠁⡾
⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⡀⠀⠀⢳⡀⠂⠈⡠⢧
⠀⠀⢸⣇⣀⡤⢿⡿⠋⠀⠀⠈⢻⠁⠀⠈⠳⢤⢾⣿⠶⠖⠁⠈⠓⠒⠢⢄⣀⣀
⠀⠀⠘⢿⣷⣤⣦⣄⠀⠀⠀⢀⣈⣷⡀⠀⠀⠈⠉⠙⠂⠚⠚⠋⠁⢠⣤⡄⠀⠀⠙⣦
⠀⠀⠀⠀⠈⠉⠲⢌⣳⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠰⡄⠀⠀⠰⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣯⠹⣿⣧⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⣸⣷⣦⣤⠎⣥⣀⣀⣀⣤⣾⣿⣿⣿⣷⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⢉⡈⠀⠠⣄⠉⠉⠁⠀⠈⠉⣩⠋⠈⢹⠟⠋⣡⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⠷⠉⠀⢠⣄⠀⠀⠀⠀⢠⠎⠀⠀⡠⠋⠀⢈⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣤⡀⢠⣄⡀⠀⠀⡠⢇⣀⠴⢊⣠⣴⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⢷⣄⠀⠀⠈⡀⣠⣾⡁⠀⣠⠴⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣄⣀⣀⣈⣸⣿⣿⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⣓⣿⠭⠭⠅⠀⠀⠀⠉⠻

⠀⠀⢄⠀⠀⠳⣄⢰⠀⠀⠲⡄⠀⢦⠀⢰⠀⠀⢀⢶⢰⠀⠀⠐⡄⠀⡄⠀⠀⠰⡄⠀⠀⠀⠀⠀⢰⡄
⠀⠀⣎⢧⡀⢤⣌⣻⠄⠀⠀⠹⡄⠈⢧⠘⣆⠀⢸⠀⢯⠃⠰⡀⢹⠀⠹⡄⠀⠀⠹⡄⢸⡆⠀⠒⠁⢱
⠀⠀⡇⠑⠹⠆⠱⠈⠑⠄⠀⠀⡽⡄⢠⡆⠘⣆⠸⠀⠀⠱⡀⠳⡌⠠⠳⡙⠄⠀⢀⠔⠋⠸⡀⠀⠀⢸
⢡⠀⠰⡀⠀⢀⠐⠀⠀⠀⠀⢸⢇⠱⣼⠹⣄⠘⠀⠀⠀⠀⠡⢀⠘⡄⠀⠱⢀⠀⠁⠀⠀⠀⠁⠀⠀⠘
⠈⢧⠀⠁⠀⢸⢇⠀⠀⠀⠀⠘⠈⠂⣿⠀⠈⠣⡀⠀⠀⠀⠀⢈⡇⠰⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢀⠀⠳⠄⣷⡈⠄⠉⠒⠂⠀⡿⠄⠀⣿⠀⠀⠀⠈⠳⢄⠀⠀⠊⠸⠸⣆⠀⠀⠻⠀⠀⠀⣀⡠⣶⠀⢠⠎
⠈⢣⡀⠀⠰⡈⠢⣀⠀⠀⠀⢣⠀⠀⢿⠀⠀⡀⠀⠀⠀⠳⡀⠀⠁⠀⡿⠀⠀⠃⠀⠀⠀⠀⡠⢃⡴⠋
⠀⠀⠳⣄⠀⠁⡀⠀⠁⢀⠀⠰⡇⠠⢸⠀⠀⠁⠀⠆⠀⠀⢣⠀⠀⡠⠃⠀⠄⡀⠀⠀⠀⠪⠔⠋⠀⣠⠆
⠀⠑⢤⡈⠓⠄⠙⠿⢈⡁⠀⠀⠳⡀⠀⢆⠄⠀⠀⠀⠀⠀⠘⠀⠈⠀⢀⣀⣀⠬⠟⠁⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠂⡴⠁⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⢦⡀⣉⡁⠀⠀⣀⣀⠄
⠀⠉⠑⠒⠲⠶⠒⠊⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠙⠒⠶⠒⠂
⠐⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⡄⠑⠲⠤⠤⠤
⠀⠀⠀⠀⠀⡴⢰⡄⠀⠄⠀⢤⣴⣦⡄⠀⠀⡀⠀⠈⠃⠀⢀⠀⠀⢠⣴⣤⡤⠀⠐⠀⠀⡇⣄
⠀⢀⣠⠴⠊⠀⢈⢧⠀⠀⠀⢈⠻⠿⢾⡂⠈⠀⠀⠀⠀⠀⠀⠁⢸⡷⠿⠛⠀⠀⠀⠀⣸⡇⠈⠲⣄⡀
⠀⠁⠀⠀⠀⢠⠇⣸⠀⠀⠀⠀⠁⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⡀⣏⢱⡀⠀⠀⠈⠁
⠀⠀⠀⣀⠴⠃⢠⠇⠈⣆⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡼⠁⢹⡀⠳⣄⡀
⠀⠂⠉⠀⠀⣠⠋⠀⣧⠘⡆⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠈⠀⠀⣸⠁⡼⠀⠳⡀⠀⠉⠒
⠀⠀⢀⡠⠚⠁⣰⠃⠘⣆⠀⠀⠊⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠠⠀⠀⠀⣸⠃⠸⡄⠙⠦⡀
⠀⠀⠁⠀⣠⠞⡥⠂⠀⠘⢴⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡶⠃⠀⠐⢜⢦⡀⠈
⠀⡀⠴⠊⢁⡞⠀⠀⡆⠀⢸⡀⢤⠂⠁⠈⠓⡛⢶⣦⣴⡶⢛⠞⠁⠠⢚⡤⢠⡇⠀⢰⠀⠈⢳⠈⠓⠤
⠀⣠⠖⠁⣸⠀⣠⠆⣗⡃⢸⣧⠈⠭⠐⠀⠀⠈⠒⢿⡿⠀⠁⠀⠀⡂⠡⢀⣽⠇⢨⣽⠠⡀⠈⡇⠈⠲⣄
⠘⠁⠀⣠⡧⠞⢁⡴⣻⠀⢸⠚⣧⡀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⢀⣾⠙⡇⠘⡟⢦⡙⠦⣝⡄⠀⠈
⠀⠀⠉⠁⣠⠔⠋⣰⠃⠀⡼⢠⢇⠹⡗⠒⠛⠉⠁⠈⠁⠉⠉⠛⠒⣺⠋⢸⡀⣧⠀⠙⣄⠙⠢⣄⠉⠁⠂
⠀⢀⠴⠋⠀⠀⡰⠁⠀⢰⢇⠞⡾⡆⢘⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⢸⢿⢳⡸⡄⠀⠘⣆⠀⠀⠙⠢⡀
⠀⠁⠀⠀⠀⠠⠁⠀⠀⠸⠘⠸⠇⠇⠸⠨⢙⠦⣄⣀⣀⣠⠴⠋⡆⠇⠘⢸⠀⠇⠇⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⢄⡀⠀⠀⠀⢃⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠁⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡠⠠⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠀⠢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠱⠀⠀⠀⠀⠀⠈⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠠⠀⠀⠀⠌⠀⠀⠀⣤⠲⣦⠀⠀⠀⠀⠀⣠⡶⢢⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⡐⠀⠀⠀⠀⣿⣿⣿⠂⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡐⠀⠀⠰⠀⢃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠊⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⢰⣄⠀⠀⡀⠀⠈⢰⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⢀⠔⠃⠠⠄⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡀⠀⢀⠄⠈⠉⢀⠀⠀⠀⠂⠈⠐⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⠀⢔⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠃⠉⠄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠋⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠑⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠁⠸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠢⠘⠀⠑⠀⠒⠀⠀⠀⠈⢀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢠⡇⠀⠂⠀⢀⠂⠒⠀⠚⠇⠈⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⢣⡀⠀⠀⠀⠀⣀⠀⠈⠆⠀⠀⠊⠀⠀⡄⠀⢀⠄⠀⠤⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⠿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠓⠲⢤⡀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠒⠒⠲⠤⣄⡀⠀⠀⢀⣠⡤⠤⠶⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠒⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⣠⡀⠀⠀⠀⠀⠙⠦⣄⠀⠀⢈⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠺⡁⠙⠦⣄⣀⠀⠀⠀⣀⣩⡗⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠳⣄
⠀⠙⢆⠀⠈⠉⠉⠙⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⠤⢤⣀⣀⣀⣀⡤⠔⠋⠁⠀⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⣀⣀⣀⡤⠤⠖⠚⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠄⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢻⣸⡇⠀⠀⠀⠀⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠉⠑⠀⠈⠁⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⠀⠀⢠⠀⢀⠈⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⣶⢤⣿⡂⠈⡄⢸⡀⣹⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡼⠁⡟⠀⠙⠚⠓⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣀⣀⣀⣀⣀⠀⢀⣴⡾⠟⠋⠉⠁⠀⠀⠈⠉⠻⠿⠿⠛⠋
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠉⠁⠀⠉⠉⠙⠻⢧⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⣦
⠀⠀⠀⠀⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⣤⣤⣤⡤⠶⠞⠛⠒⠄⠈⠳⣦⣀
⠀⠀⠀⠀⠸⣦⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠙⣷⣤
⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣄⣀⣀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣷⡀⠀⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣤⣤⣴⣶⣶⠾⠟⠛⠉⠈⢻⣷⡄⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣸⣿⣿⠙⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡴⠶⠶⠛⠛⠉⠁⠀⠀⢀⣰⠖⢦⡀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠋⠁⢸⠀⠙⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠶⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⢶⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⡍⠁⠀⠀⡐⠀⠀⠀⠀⢰⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠰⠁⠀⠀⠀⢀⣾⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠁⠄⠘⠀⢀⢺⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠸⡶⠶⠶⠲⠒⠒⠺⠒⠒⣻⠚⠓⠒⠒⠶⠾⠿⠟⢛⡟⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣆⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣎⢃⠂⡔⡰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆⠀⠰⢁⠀⡀⣿⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠘⢧⡀⠀⡴⠀⢻⠀⠀⠀⠀⣠⡞⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠻⣶⣁⠀⢻⠀⠀⢀⡴⠋⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠙⠷⣾⣤⣴⠟⠁⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠉⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣤⡶⠟⠋⠁

⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀⠀⢀⡠⠔⣒⠶⢤
⠻⡉⢑⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠛⠒⠈⠛⠋⢀⢔⡛⠉⣉⡭⠶⠶⠒⢒⠖
⠀⠘⢦⣂⣀⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣶⠴⠤⣀⣈⠀⢂⣍⡉⣳⠯⢅⡐⠀⢀⣀⠸⠦⣤⣀
⠀⠀⠀⠀⠀⠙⡄⠱⡄⠀⠀⠀⠀⠀⢀⣴⢯⡄⣇⠈⢱⠀⠃⠀⠀⠀⠙⣱⠊⡦⢹⠀⠐⠢⢍⡲⢤⡼⠷⢄
⠀⠀⠀⠀⠀⠀⡽⠀⣹⠀⠀⠀⠀⠀⠀⣠⠏⠀⢈⠟⣥⡀⠀⡴⣶⣦⠀⠘⠦⠀⠋⠰⡀⠠⣄⠈⠁⠈⠳⡄
⠀⠀⠀⢀⡤⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠟⢲⠊⢈⣄⣛⣁⠀⠛⠛⠃⠀⠀⡄⣄⠀⠢⡑⠤⣈⠓⠤⣈⢷⡻⠄
⠀⠀⢀⡞⠁⢠⠏⠀⠀⢀⣠⠶⠞⠛⠓⢲⠃⡤⢼⠿⣿⠟⠃⠀⠐⣧⢀⣀⡏⢸⡆⠀⡇⠀⠈⠃⠀⠈⠣⢳⡀
⠀⠀⢸⡁⠀⣏⠀⠀⣴⠟⠁⠀⠀⠀⠀⣨⢾⡀⡈⣳⠛⠲⠦⠶⠚⣡⣿⢻⠇⡸⠀⡞⠀⠀⢆⠀⠀⢦⠀⣬⣧
⠀⠀⠈⢧⡀⠹⠶⣾⠁⠀⠀⠀⠀⠀⠀⠉⢠⠿⣄⢏⡰⠀⣠⢸⠏⢰⠃⡏⡜⠁⡀⠳⠀⠀⠈⠳⡄⢸⡆⢹⠈
⠀⠀⠀⠀⠙⠲⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡈⠈⠁⢉⠈⠁⡀⡏⠀⢿⠁⠀⢇⠀⠀⡄⠀⠀⣷⢸⣷⡾
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡀⠀⠀⠱⡼⢧⠀⠠⠎⠀⢠⠁⠡⠀⠘⠆⠀⠈⣧⠀⣷⠀⠀⣿⢸⠋⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣦⣤⣀⡁⡸⢇⠀⡆⢰⡇⠀⠀⠀⢣⠀⣄⠀⠃⢀⣏⠀⡼⢻⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠈⢺⠳⡀⢣⡀⠀⢀⠞⠀⣼⠀⢀⡞⠹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡼⠛⠁⠀⠀⣴⠛⡇⠀⠀⠙⢾⢱⢤⣯⣴⣾⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⢠⡞⠁⠀⠀⢀⡼⠃⠀⡇⠀⠀⠀⢀⡏⣾⠛⠻⠿⢛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣇⠀⠀⣰⠋⠀⠀⠀⣇⠀⠀⠀⢸⠀⣹⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⢸⡀⠀⡇⠀⠀⠀⠀⢻⡀⠀⠀⢸⡀⡇⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⡇⢸⠁⠀⣇⠀⣀⠀⠀⠘⡇⠀⠀⠈⣧⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠙⠚⢧⠈⠉⡉⠉⠹⡄⠀⣇⣀⡀⢠⣿⠀⠄⠼⠧⠶⠦⡄
⠀⠀⢀⠀⠀⠀⠘⣧⣤⣤⣄⣤⠾⢦⠤⠧⠞⠴⠃⣰⠋⠀⠀⠀⠈⠳⣄⡀⢀⡄⣠⡿
⠀⠀⠊⠃⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠴⣤⢦⡴⠴⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢨⠆
⠀⠀⠀⠀⠀⠠⠂⣨⠛⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢠⠛⠙⠒⠁
⠀⠀⠀⠀⡐⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡏
⠀⠀⠀⠐⠀⠀⢸⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⡇⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⡁⠀⠀⠋⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠁⠀⠀⣀⠤⠀⡀
⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠤⠤⣄⠐⠦⣤⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⣀⣠⣀⡈⠢⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⢊⣀⣤⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⠀⣰⠃
⠀⠀⠀⠀⠀⡐⠀⡼⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⣰⠉⠈⠉
⠀⠀⠀⢀⠌⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⣸⠧⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢰⠧⠀⠀⠀⠚
⠀⠀⠀⡀⠀⠈⠁⠀⠀⠒⠒⠂⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠚⠁⠀⠐⠦⠤⠖⠙⣄
⠀⠀⠀⠈⠓⠦⢤⣤⣤⣀⣤⣤⠀⠈⠳⠦⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣴⣒⣡⣤⣤⠀⠈⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡎⢠⠊⡠⢀⡄⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⣧⣞⠀⣸⣷⣿⣄⡄⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠻⣿⣷⣦⣼⣿⣿⣿⣿⣧⢔⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⣿⣿⣯⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡝⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⡄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠒⠲⠤⣄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣀⣀⣤⠴⠖⠒⠒⠂⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣶⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⣧⠀⠀⠀⠀⠀⢰⣿⣇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣷⣿⣒⢿⣦⡀⠀⠀⢠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⠀⣀⣀⣼⣿⣿⣿⣿⣮⡛⢿⣆⡀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣿⣿⣽⣿⣿⣿⡗⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣮⡛⣿⠿⢿⣿⠿⠿⠋⠀⣽⣿⣿⣿⠽⣟⣭⢠⡏
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⣹⣿⣿⡿⠿⠛⠿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠈⠀⠀⣀⣶⣤⣨⣿⡻⣻⣾⣿⣿⠏
⠀⠀⠀⠀⣰⣿⢿⣿⡿⠋⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣤⣀⣠⣴⣾⣿⣿⣿⠿⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠁⠁⠘⠛⠁⠀⠀⠀⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⣀⣀⣠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠐⠟⠠⠌⣹⣻⣿⡯⡾⠻⠄⠉⠁⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⣖⣶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠁⠠⣾⣿⠃⢹⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⢨⡄⠀⡐⠢⠀⢨⠀⢛⣀⣠⣾⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠈⠁⣒⠒⢠⣆⠀⠀⣾⠋⣽⣿⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⡧⠁
⠀⠀⠀⠀⠀⢀⡡⠈⢉⢿⡽⠀⢀⣿⣶⣿⠈⠀⠀⠀⠀⠀⠀⠈⠂⢀⣠⡶⠊⠀⠰⣆
⠀⠀⠀⠀⠤⠥⠐⠲⢤⣷⡧⠀⣸⣿⣟⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⠀⠀⢢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠰⡒⣉⢐⠿⠟⠁⣠⠟⠉⠀⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠁⠀⠀⠀⠀⢠⣾⠟⣆
⠙⠒⠀⠁⠈⠙⡓⢶⡶⠒⠈⠀⠀⠀⠀⠐⣶⣾⣾⣿⣿⣿⣿⡿⡌⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⠀⠀⠀⠀⠀⢀⠉⢃⠀⠀⠀⠀⠀⠀⣹⡏⠀⢠
⠀⠀⠀⠀⠀⠰⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⡉⠃⣻⣿⣿⣷⣿⠻⠿⡀⠀⠀⠀⠀⢠⠀⠈⡗⠄⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢁⣄⠋⣿⣿⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⢹⣂⠀⠀⠀⠀⡄⠃⢇⠀⠀⠀⠀⢀⠔⡞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⣿⣹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠈⢻⣿⣿⡀⢄⠀⠀⠀⡼⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠀⢢⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣻⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⠈⠀⠀⡼⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡏⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢠⠂⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⡠
⠀⠿⠦⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡀⠀⠀⠀⠐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢧⠄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠊⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢧⠁⠀⢤⣱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠉⠉⠉⠀⠀⠁⠀⠀⠀⢻⡀⠂⠀⠀⠉⠁⠠⠀⠀⠀⠀⣡⠣⠒⠀⠀⠀⠀⠀⠀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠈⠉⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠉⠁⠀⠀⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠘⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⢡⡀⠈⢧⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢠⡀⠀⠀⠀⢈⣧⠀⠀⣧⢀⣾⣀⣼⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢻⡶⠖⠛⠙⠳⠶⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣰⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠈⠁⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣷⣶⣶⣤⣶⣶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⣶⣶⣿⣿⣷⣶⣶⣤⣀⠀⣠⣶⣿⠿⠛⠉⠀⠀⠀⠀⠠⠾⠛⠉⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣦⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⣤⣤⣤⣤⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣀⡀⢀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣴⣾⠿⠛⠉⠀⠀⠹⢿⣿⣶⣤⣤⣀⣀⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠋⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣶⣽⣥⣮⣬⣿⣥⣿⣻⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠹⠟⠋⠉⠉⠙⠒⠒⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣴⣴⣶⣯⡥⠤⠶⣦⣄⣀⠀⠒⠺⠿⣝⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡁⠀⢠⣼⣿⡹⣴⣦⠄⢸⣿⣍⠙⠓⠂⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠁⠛⠛⠀⣤⣇⢻⡃⢿⣁⣴⣿⡿⣯⡛⢦⠀⠈⠛⢷⡶⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠁⠀⢠⣄⣀⣿⡏⢧⡳⡈⠿⣯⡙⠳⣌⠑⠈⠳⡀⠀⠀⠹⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢻⢺⡟⣿⡿⣷⠈⢣⡈⠀⠈⠻⣄⠈⢧⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡼⢸⡇⠹⡇⠹⡄⠀⠑⠄⠀⠀⠹⡆⠀⠀⠀⠀⠀⢢⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣼⡇⠀⠘⠀⠹⣦⡀⠀⠀⠀⠀⠙⢄⡀⠀⠀⣀⠀⢹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⡀⠀⠀⢀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣸⠳⡄⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡏⠋⠀⢹⡄⠀⠀⣰⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⣧⠀⣰⠃⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠠⣀⠄⠐⠀⠈⠉⠐⠂⠤⡀
⠀⠠⠆⠀⠀⠀⠀⠀⠀⢄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠂⠐⠐⠂⠠⠤
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⡂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢛⠋
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠂⠔⠀⠀⠀⠀⠀⠀⢠⢤⡁⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠃⠂⢀⡀⠀⠊⡁⠀⠀⠀⠀⢠⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠀⠂⠀⠀⠀⢀⡏⠀⢠⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡄⠀⠀⠀⠈⠐⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⣀⠄⠀⠂⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠠⠄⠰⡎⠀⠀⠀⠀⡘⠀⠀⡂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠀⢀⠀⠀⠘⡀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⢠⠀⡆⠀⣀⢀⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠘⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⡆⠀⡆⢦⠶⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⢠⡈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠦⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠈⢿⣿⣦⡀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢰⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠋⠀⠀⣠⣾⣷⣄⠀⠀⠙⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⡿⠟⠋⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠙⠻⢿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿
⢠⣿⡏⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣀⣤⣾⡿⠿⠿⠛⠛⠛⠛⠿⠿⣿⣷⣦⣀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢹⣿
⢸⣿⠇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿
⣸⣿⠀⠀⣼⠋⠉⠉⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⠉⠉⠙⣧⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿
⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⢹⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⢸⣿⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣿
⠘⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⢿⣷⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡿
⠀⠸⣿⡆⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣿⠇
⠀⠀⢿⣷⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣾⡿
⠀⠀⠘⣿⣇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣸⣿⠃
⠀⠀⠀⢹⣿⡄⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⣿⡏
⠀⠀⠀⠀⢻⣿⡄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⣿⡟
⠀⠀⠀⠀⠀⢻⣿⡄⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀⠀⠉⠛⢶⣤⣀⠀⠀⣀⣤⡶⠛⠉⠀⢀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣀⠀⠈⠙⠛⠛⠋⠁⠀⣀⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⣠⣤⣶⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⢉⣿⢿⡗⠄⢾⣟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⢀⠀⠃⢀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣯⡠⠇⠀⠯⣠⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣇⠘⡟⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣟⠿⣿⣍⣉⣿⣯⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠸⣿⣿⣿⣻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣲⣦⣄⡀⠀⠀⠀⠀⠘⢷⣼⠃⠀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠃⠀⠀⡠⠉⠘⣿⡇⠈⡇⠐⠒⠒⠲⠾⣿⡐⠀⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⢹⠃⢀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠘⣿⣿⢿⡇
⠀⠀⠀⠿⠟⠁⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⢀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠘⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⡀⠀⠀⠑⠘⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⣿⣿⢛⢿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠿⠛⠉⠀⣶⣭⣛⠿⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣾⣭⣭⣭⣭⣭⣿⣿⣿⡛⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⢻⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟
⠀⠀⠀⠀⠀⠀⢸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⡻⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣭⣻⠿⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⡄⠚⠀⠀⠀⠀⠀⠀⠰⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⢦⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣷⡄⠀⠀⠀⠀⠀⣠⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⣾⠈⣷⣤⠀⠀⠀⠀⢋⢸⣿⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢹⢧⡁⢸⣿⣷⣿⠏⠀⠀⠀⠘⢿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣿⣿⣆⠀⠀⠀⢄⣻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡷⢾⣿⣿⣿⠁⠀⠀⠀⠀⣸⡘⢿⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠄⣟⣰⣿⠒⠀⠀⠀⠀⢹⣷⣼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠰⡧⢾⣷⡧⠀⠀⠀⢀⣼⡗⢰⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⢴⣾⣿⣗⡀⠀⣀⢠⣿⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⣮⡋⣿⣏⠶⠀⠈⢸⣿⣿⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢰⠟⡝⠛⣿⠀⠀⠀⣾⣽⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣦⡧⠀⣿⠰⡟⠀⣿⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡇⢸⣿⡇⠀⣿⠄⣦⢠⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣇⠀⣿⠋⠀⢹⠀⣍⢸⡇⣺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣀⡇⠀⠺⠁⠀⠸⠀⠀⡀⠸⢦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⣰⡇⣤⣴⣂⣀⡄⢠⢸⣷⣦⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠁⠀⢨⠅⠀⠀⠈⠶⠿⠯⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡀⠀⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡋⠉⠉⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡉⠉⠁⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠷⣾⣷⣾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣶⣾⠛⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠏⠀⠀⠈⠉⠙⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢀⡀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢺⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣄⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣶⣶⣤⠀⠀⠀⢰⡗⠀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠉⢁⣀⣀⠀⢀⣀⡀⠈⠙⠛⢻⣿⣿⣿⣿⣦⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢸⣿⣇⠀⠀⣴⣦⣄⠸⣿⣿⣷⡾⠿⠿⢷⡆⣰⠟⠉⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠊⠉⠻⣿⣿⣿⠟⠛⠀⠀⠘⠿⢃⡀⠀⠀⣸⣿⣶⣶⣶⣶⣿⠟⢁⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠛⠃⣀⣼⣿⣷⠀⠘⠿⣿⠟⢸⣿⣿⡄⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⠀⠀⠀⠛⠋⠀⠀⣀⣤⡄⠉⢻⣿⠁⠀⠀⠀⠀⠀⠸⣿⡿⠁⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠶⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⢠⣿⣿⠉⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⣼⣿⣶⣤⠿⣿⣿⣇⠀⠀⠀⠀⠘⠋⠁⠀⠀⢀⣠⣾⡿⠛⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣡⣤⡄⠀⠈⠻⣿⣿⣇⠀⠈⠻⣿⣄⣀⡀⠀⠀⣿⣧⡠⢿⣿⣏⡀⠀⠈⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠈⢻⣿⡇⠀⠀⠀⣿⣿⣿⣦⣴⣿⠿⠇⠀⠀⠙⠛⠀⠀⣀⣀⡀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠀⠁⠀⠀⠀⠈⠻⣷⣤⡀⠀⠈⢻⣿⣿⠟⠁⠀⠀⠿⠇⠀⣠⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢙⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣸⣿⣷⣶⣿⠗⠰⣿⣿⡿⠋⠉⢀⣀⣤⣀⣤⣶⣶⣶⡾⠛⠛⠉⠉⠙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠂⠰⠾⠿⠟⢀⣠⣤⣄⡀⠿⠟⠀⠈⠉⠁⠀⠀⢹⡿⠗⢾⣿⣿⣿⠿⢿⣿⣿⡉⠀⠀⠀⠀⠀⢾⣿⠁
⠀⠀⠀⠀⠀⠰⠄⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣷⣤⣤⠿⠃⢠⣤⣶⠖⠋⠀⠀⠀⣿⣿⠃⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⣤⣤⣤⠀⠀⠈⠻⠿⣀⣤⣴⣿⣦⣤⣿⠿⠿⣿⡇⠀⠸⣿⣿⣄⣀⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⡀⠀⠘⠿⠏
⠀⠀⠀⠻⠿⠏⠀⢀⣤⣴⣾⡿⠛⠁⠀⠙⠛⣁⣀⠀⠀⠻⠄⠀⠀⠉⢿⣿⣏⠁⠀⢀⣾⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣀⣰⣿⣿⡷⠀⠀⠀⢠⣤⣴⡾⢿⣧⠀⠀⢸⠿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠸⣿⣷⠄⠀⢴⣷⠄⢀⣿⣿⣿⡟⠋⠀⠀⢀⠀⢸⣿⡟⠀⢸⣿⡇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢾⣿⡟⠋⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⣿⣿⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⣤⣼⠿⠁⢀⣤⣶⣿⡿⠛⠉⠀⠀⢀⣤⣿⡿⠃⠀⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠙⠋⠀⠀⠘⠿⠟⠁⠀⠀⢀⣀⣀⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⢰⣶⣦⣀
⠀⠀⠀⠀⠀⣤⣤⠀⠈⠙⠋⢠⣤⣄⠀⠀⣀⠀⠀⣠⣴⣶⣿⣿⠋⠀⠈⠙⠛⠛⢠⣤⣤⡄⠀⠈⣿⣿⡿⠃⠀⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠟⠉⠁⠀⠀⠀⠈⠋⠛⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠙⣿⡿⠀⠰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠈⠁⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⣿⣿⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠃⣀⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠋⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠉⠙⣷⠄⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡄⠉⠀⠀⠹⠿⠀⠀⠀⠀⠀⠁⠀⠀⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠹⣿⣷⡄⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠙⠛⢻⣶⣄⠀⠹⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣀⣼⡏⠀⢸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠃⠀⠈⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⢸⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠸⣿⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠺⣦⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣠⡄⠈⠉⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣭⣭⣭⣛⡻⠷⣦⣄
⠀⠀⠰⣄⡀⠀⣀⡀⠠⠤⠤⣴⣶⣾⣿⣿⣿⣿⣯⣭⣿⣛⣿⢿⣻⢷⣤⣽⣿⣷⣄
⠀⠀⠀⠈⠛⢷⣶⣬⣭⣽⣿⣿⣿⣿⣿⣿⣷⣺⣦⡴⠶⠦⢌⣁⣈⠙⢮⡛⣿⣿⣿⡳⣄⡀
⠀⠀⠀⠀⠙⢦⣌⣙⠛⠿⠿⠿⠿⠿⠛⠛⣉⡭⠶⠒⣛⣛⣳⣿⣿⣿⡷⠿⣿⣿⣿⣷⣹⣿
⠀⠀⠀⠀⠀⠐⠾⠿⣿⠿⠿⣟⣻⣽⡾⢛⣡⡴⢞⣫⣭⡭⠠⣤⡰⣬⣭⣅⣙⣦⡉⠻⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣻⣿⣿⡿⢋⣴⣿⣯⣼⣭⣿⣿⣿⣷⣬⡛⠶⣭⣭⠛⠛⠛⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⣻⣿⢏⣴⣿⣿⠟⢉⣥⡾⠟⢋⣭⡽⠟⠛⠿⠿⠃⠀⠀⠰⢤⠾⣷⠒⠻⠦⡀
⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⢯⣾⣿⠟⠁⣰⣿⠟⠀⣴⣿⡍⠀⣴⠞⠂⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⣿⣿⠏⠀⣼⣿⡟⠀⢸⣿⡿⠀⣼⠃⣠⣤⡀⠀⠀⠀⠀⠀⢿⣁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠛⢰⣿⣿⡟⠀⣸⣿⣿⡇⠀⢸⣿⣧⠀⣏⣾⣿⣿⣿⡄⣠⢄⣠⡴⠿⠿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣱⠀⣿⢸⣿⡇⢰⡌⣿⣿⣆⢿⣿⣿⣿⣿⣿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⣿⠀⡇⢸⣿⣇⢸⣷⣹⣿⣿⣾⣿⠋⠀⠀⠙⠇⣿⣏⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠇⢸⣿⢻⡌⣿⣷⣿⣧⠹⣿⡄⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠈⢿⡎⢿⣽⣿⢿⣿⣧⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢷⡈⢿⣿⣆⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠻⣿⣧⡈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡀⠀⠉⠛⠳⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⣠⡀⠉⠓⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠒⡄⠀⠀⡴⠁⠀⠀⣠⠔⠠⠚⠁⠈⠓⠤⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⣰⠊⠉⢈⡽⠅⠴⠃⡠⠊⠀⠶⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠈⠑⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⡏⠀⠀⠳⣄⣀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢷⠀⠂⠀⠀⠀⠐⠀⠀⠀⢀⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠜⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠈⠢⣄⣀⣀⣀⣀⣀⣀⠔⢁⣴⢁⡤⠤⢼⠀⠀⠀⠀⣠⠞⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⣀⣠⣔⡩⠖⠉⠈⠁⠀⠀⠀⠣⠤⠒⠯⢤⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣥⠀⠀⠀⢰⠁⠈⠉⠒⠒⠤⠤⠤⢤⣤⣤⠤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⡎⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠣⡀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⢱⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢤⣀⠤⠜⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡎⠸⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠁⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⢀⠤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⢀⣀⡀⡜⠀⠀⡧⢄⣠⣄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠈⠉⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠚⠉⢀⣠⠖⠋⠙⠲⣄⡀⠙⠓⠦⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⣠⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣀⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠙⠦⠴⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣠⠖⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢿⣓⠲⠄⢂⠀⠹⢷⡟⠁⢀⠀⠴⢒⣻⠃⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣓⡖⠂⠀⠉⠉⠐⠄⠳⣄⠈⠁⡴⠋⠔⠋⠉⠀⠀⠒⣞⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠍⡉⠁⠀⠀⠀⠀⠀⠀⠸⣆⡾⠁⠀⠀⠀⠀⠀⠀⠉⡉⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠐⠿⣿⣿⣷⣶⣦⣄⠠⣀⡀⢻⠃⣀⡠⢀⣤⣶⣾⣿⣿⣿⠗⠀⠐⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣛⡉⠁⠀⠠⣹⣿⣿⠋⣁⣒⠶⣬⡳⣽⣷⣼⢰⣿⡵⣫⡴⠖⣂⠉⢻⣿⣿⡁⠀⠀⠀⣉⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠁⠀⣀⠄⠀⣁⡿⣡⣾⣿⠿⠓⠀⠁⠈⠀⠀⠀⠈⠀⠁⠀⠻⢿⡿⢦⡹⣦⡁⠀⢄⡀⠀⠉⠻⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⢠⣄⣨⡥⢊⣵⡿⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣷⣍⠢⣭⣁⣤⠈⠙⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⢀⠀⠀⠀⠀⢲⡿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣗⠂⠀⠀⠀⢄⠀⠙⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠀⠀⠀⢋⠞⣯⠀⠀⠠⣶⣦⡀⡄⠀⠀⠀⠀⠔⣰⣶⠦⠀⠀⢸⡟⢎⠃⠀⠀⠈⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢀⣀⣀⣠⣤⠖⠋⣰⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⢷⡌⠓⢦⣤⣀⣀⣀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠛⣉⡍⠀⠀⢀⡾⠁⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠙⣆⠀⠀⠀⢉⡛⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡾⠁⠀⠀⠀⠈⠀⣠⣿⡿⠁⢀⠂⠀⣀⢀⡀⠀⠀⣀⢀⡄⠈⢂⠀⠹⣿⣷⡀⠈⠃⠀⠀⠀⠹⣦⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣠⡟⠀⢀⡴⣺⠎⢀⣴⠋⡼⠁⣸⡎⠀⠀⠈⠛⠻⣦⡾⠛⠋⠀⠀⠘⣧⡀⢹⡌⠳⣄⠈⢿⡲⣄⠀⠘⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡎⡾⠁⠴⠋⠁⣰⠃⣰⡿⢻⡀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⣴⠿⣷⠀⢳⠀⠈⠳⠄⢻⡌⣇⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠄⠀⠀⠐⢁⣴⡿⠁⠘⣻⠀⠤⠶⠚⠉⠙⠲⠤⠄⢸⡟⠄⠹⣷⣄⠑⡀⠀⠀⡄⣷⢿⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢀⠀⡜⢠⡞⠁⠇⠀⢠⡿⣷⣄⣀⠀⠀⠀⠀⣀⣴⡿⣧⠀⠀⠃⠙⣦⠈⡀⢀⢠⢩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠁⣾⠁⠀⠀⠀⣼⠁⢸⠿⣿⣷⣦⣾⣿⡿⢻⠃⢸⡆⠀⠀⠀⢸⡆⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⢿⡄⢠⡆⠀⠛⠀⢸⡀⢹⡌⢻⠋⣸⠃⣸⠀⠘⠃⠀⣠⠀⢸⢇⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠁⠈⢧⠀⠀⢸⡄⢷⡌⠿⠈⠘⠏⣰⠏⣴⠀⠀⢀⡏⠀⠡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠻⢦⣟⠀⠀⠀⠀⣿⠼⠋⢀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠋


⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⣤⣼⣿⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣠⣴⣾⣿⠿⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⠛⠿⣿⣷⣦⣾⡿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣉⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣬⡿⠀⠀⠀⠀⠷⠐⠟⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⣠⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣄
⠀⣸⣿⠋⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⢹
⠀⢻⣷⣀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣤⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⡿
⠀⠈⠻⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⡀⠀⠀⠀⢰⡟⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⢿⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠻⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣅⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⢿⣷⣦⣤⣄⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡿⠟⠋⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⡿⠛⠛⠛⠛⠁⠀⠀⠀⠻⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⡀⠀⠀⠀⠀⠀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠶⠆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠉⠉⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⡀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠋⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠗⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠶⠺⠟⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠃⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣴⠤⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠛⣀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⠇⠀⠀⠀⠋⠠⠾⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⢀⣀⣀⣀⠈⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⡏⢙⣿⡆⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠘⠻⠟⠋⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣤⣤⣤⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⢀⣤⣄⠈⠙⠿⣶⣤⣄⡀
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠛⠉⢀⣀⡤⠞⠋⢹⠉⠛⠦⣄⣀⠈⠙⠛⠻⠿⠶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⣿⠇⠀⢠⣤⣤⣤⣤⣤⠤⠶⠖⠚⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠉⠛⠒⠶⠦⢤⣤⣤⣤⣤⣤⠀⠀⢿⡇
⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⡇
⠀⠀⠀⣿⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⡇
⠀⠀⠀⢿⡇⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿⠀⠀⣿⠃
⠀⠀⠀⢸⣷⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣿
⠀⠀⠀⠈⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⡇
⠀⠀⠀⠀⠸⣷⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠹⣷⡄⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠉⠛⠶⢤⣄⣀⠀⠀⢸⠀⠀⢀⣀⣤⠴⠞⠋⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢶⣶⣤⣤⣤⣴⣶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⡆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠉⠀⠀⠈⠓⠦⠤⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢉⡭⠼⢤⡀⠀⠀⠀⢀⣠⣀⠀⠰⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡴⠁⠀⣄⠀⢸⣠⣄⣰⠁⣀⠀⠹⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢣⡀⠀⠈⢡⣤⣤⢤⣤⡀⠁⠀⣰⠃⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢁⠆⠰⢈⡻⢌⠻⠃⢠⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡤⡆⠀⢠⡏⠀⠀⢼⣁⣈⡆⠀⠀⢧⠀⠀⠀⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⣆⠀⢄⠈⢛⡋⠀⡀⠀⡼⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢶⠀⠀⠈⠓⢬⡉⠙⠋⣩⠵⠊⠀⠀⠀⠀⢠⠇⡰⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣰⣀⠀⠀⠈⠢⠜⠀⠀⠀⢀⡜⣆⡴⠋⠈⠳⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠈⠓⠦⣄⡀⢀⣠⠤⠖⠋⡇⠁⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠤⣳⡀⠀⠀⠀⠙⠋⠀⠀⠀⣰⢃⣠⠤⣄⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠈⢳⠀⠀⠀⢹⠉⠀⠀⢰⠏⠀⠀⠀⠀⠙⡆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠲⣄⠈⡇⠀⠀⢸⠀⠀⠀⡞⠀⣠⠔⠀⠀⢠⠇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠳⣇⡀⠀⢸⠀⣀⣀⣃⠞⠁⠀⠀⡠⠏⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠉⠉⠁⢠⠞⠉⠉⠉⣻⠋⠁⠀⠉⢦⠀⠚⠛⠓⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣅⠀⣅⣀⠸⡀⡄⠀⡀⢻⡀⣀⢀⡆⡸⠀⠀⡆⢈⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠉⠀⠀⠀⠀⠈⠉⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣉⠔⠈⠢⡀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠈⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠁⠉⡧⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠔⠊⠉⠁⠐⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⠴⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠀⠒⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠊⠉⠉

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠒⠶⠀⠒⠉⠉⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠰⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠊⢹⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢀⠀⢻⡀⠜⢀⣴⠇⣠⣾⣧⣀⣼⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠓⠤⣤⡤⠦⠶⠞⠛⠛⠉⢛⣿⡀⠀⠀⠀⠀⠠⣄⣤⣤⣄⣠⡞⠁⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⠀⠀⠀⠀⣿⢿⢿⣿⣯⠀⠀⠀⠀⠀⢀⣾⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⢼⠁⠀⠈⠉⠛⠳⠀⠀⢀⣤⣿⣿⣧⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠉⠝⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠉⠚⢆⠇⠀⠀⠀⣴⠋⠀⠀⠀⣀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠛⠛⠙⠛⠓⠈⠁⠙⠙⠛⠛⠀⠘⠘⠛⠛⠛⠀⠐⠛⠛⠙⠓⠛⠛⠛⠙⠛⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⡄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠇⠘⣿⣿⠀⠀⠀⠀⠈⠛⢿⣿⡛⠿⣿⣷⡀⠈⠛⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⢈⣿⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠠⣶⡿⠟⠀⠀⠀⠀⠿⠿⠟⠁⠀⠶⠿⠟

⢺⣽⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣯
⠀⠻⣿⡿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⢿⣿⠟
⠀⠀⠈⠛⢮⣙⠾⣟⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣻⠷⣋⡵⠛⠁
⠀⠀⠀⠀⠀⠈⠓⢭⡛⠮⣝⡿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢿⣫⡵⢋⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣻⢾⣽⡛⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⢛⣯⡷⣏⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡹⠳⣮⡻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢟⣵⠞⢏⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠳⣭⡻⢿⣷⣤⣀⠀⠀⢀⣤⣾⡿⢏⣭⠞⣋⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠳⣭⠛⣋⣥⣾⡿⢟⣭⠞⣋⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣴⣾⠛⢛⡥⠞⢋⡤⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⢛⣥⠞⣋⡴⠒⣥⡘⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⣋⣵⠞⣋⡴⠞⠁⠈⠓⢦⣙⠳⣮⣛⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡄⠀⣠⣴⡿⠻⣉⣵⠞⢋⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠳⣮⣉⠿⢷⣦⣀⠀⢠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠘⠛⣁⣵⠞⣋⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⠳⣦⣈⠛⠃⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱⣄⠀⢈⠫⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠙⠃⠀⣠⢟⡄
⠀⠀⠀⠀⠀⣀⣴⣷⣿⡷⠞⢃⡘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢀⡘⠳⢾⣿⣶⣦⣀
⠀⠀⢀⣤⣾⣿⠿⠋⠁⠀⠀⠀⠁⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠀⠀⠀⠈⠙⠿⣿⣷⣤⡀⠠
⢠⣶⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣶
⣀⣙⣽⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣯⣋
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⢚⣉⠉⠲⣄
⣼⢰⠋⠉⠣⠀⠈⠳⣤⣀⠀⢀⣠⣤⠶⠶⠶⠲⠶⠶⠶⣤⣄⣀
⡇⡎⠀⠀⠀⠳⢄⣀⣨⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⢀⣠⡤⠖⠚⠛⠉⠉⠙⠛⠶⣤⡀
⣇⠇⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠟⠒⢋⣩⡤⠴⠶⠒⠒⠛⠓⠲⣤⡀⠈⢿⡄
⢻⢠⠀⠀⢠⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣄⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣿
⢸⣼⡆⠀⡟⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣸⣀⡀
⠀⡟⣷⣀⡇⡜⢉⡇⠀⠀⢀⣶⠀⠀⠀⠀⢀⠖⠢⢄⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠏⠉⠛⢷⣄
⠀⡇⢃⡹⠀⣇⢿⡇⠀⠀⢸⢻⠀⠀⠀⠀⡌⢠⣘⣾⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⣷⣀
⠀⠹⡌⠣⣀⢈⣟⣀⡈⠁⢸⢸⡄⠀⠀⠀⠙⠬⠿⠛⠓⠒⠀⠀⠀⢀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢿⢷⡄
⠀⠀⢻⢦⣌⡟⠁⡀⠉⠀⠘⡄⠳⣄⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠈⢷⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⡄
⠀⠀⠀⠑⣼⠉⠈⠉⠃⠀⠀⠙⣆⠈⣳⣤⣠⣤⠶⠋⠁⠘⠀⠀⢀⣴⠛⣧⠀⠈⠳⠶⠒⢉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠹⡄
⠀⠀⠀⠀⣿⠒⠒⠲⠀⠀⠀⡴⠋⠉⠉⠉⠀⠀⣀⣤⣦⡤⠴⠚⠉⠀⠀⠙⠷⣦⣤⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠳⢬⣶⡤⢤⢤
⠀⠀⠀⠀⣿⠤⠤⢄⠀⠀⣼⠧⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡈
⠀⠀⠀⠀⢿⣀⣀⣀⠀⢰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⡇⣀⡈⠀⢸⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣿⠉⠉⠁⢸⠀⠘⣧⡀⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢸⡉⠉⠀⢸⡀⠀⠈⣷⣤⡈⠳⣄⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠈⡏⠉⠂⠘⡇⠀⠀⢹⡄⠉⠓⠛⠓⠶⠤⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠒⣿⠉⢧⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣏⠈⠁⠀⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢠⣏⣉⡁⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠓⠂⠄⠀⡼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⡗⠀⠀⠀⠀⠀⠀⠀⣟⣁⡐⢂⣀⡼⠃⡾⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠻⠖⠻⠦⠼⠃⠀⠀⠀⠀⡿⠄⡠⡀⣀⠀⠀⢻⠀⡟⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⠀⠀⣧⠛⢂⠄⠄⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠦⢴⣤⠵⠖⠋⢠⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣃⠴⡄⠀⠀⠀⣀⡼⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠴⣯⣤⡶⠿⠉⢱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠽⢶⣤⣽⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠼⣧⣄⡠⠾⠻⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⠟⠷⢄⣠⣽⠦⠄⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⡤⠂⢀⠟⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠻⡀⠸⢤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⢀⡾⡟⠉⠀⠀⣤⡀⠀⣀⠀⠀⡀⠀⢀⣤⠀⠀⠉⠻⢷⡀⠀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣾⡇⠈⣧⣥⠀⠀⠀⢸⡇⠀⢾⡁⣀⡷⢀⣸⠇⠀⠀⠀⣈⣼⠁⢸⣷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢆⣸⣅⠷⣄⣀⣌⢉⣅⣈⣉⣯⣥⣼⣯⣱⣀⣠⡾⣨⣇⡰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠀⣿⣧⡤⠞⠁⣽⣿⠈⠣⣤⣬⣿⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⡗⢀⡸⢃⡈⠤⢘⡇⠁⢩⣛⡏⣿⣻⡯⢿⣏⡇⠤⢁⡘⡇⡀⢚⣿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⡈⠠⣼⣌⣰⠤⠛⣥⣶⠿⠀⣀⣄⠀⠀⣿⢈⣐⣸⡇⠿⣶⣼⠛⠴⣆⣡⣧⠄⢁⣱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢻⠞⠋⣩⠿⢠⡀⢉⣿⡛⠀⠐⠁⠀⠀⣿⠀⠈⢺⡇⢚⣿⡈⢀⡄⠿⣍⠙⠷⡟⠀⣀
⠀⠀⠀⠀⠀⠀⣀⠄⣉⠛⠉⣠⣾⠗⣦⣼⠟⣋⠛⠣⢌⣁⣀⠄⠀⢻⠀⣂⣨⡭⠜⠛⣙⠻⣧⣴⠺⣷⣄⠉⠛⣉⠠⣀
⠀⠀⠀⠀⠀⡾⢿⠀⠻⣀⣠⣮⡍⢽⡟⠡⠞⠁⠉⠉⠈⠉⠉⠙⠳⠞⠋⠉⠉⠁⠉⠈⠈⠳⠌⢻⡯⠉⣵⣄⣀⠟⠀⡿⢷
⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⠇⠀⣠⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣄⠀⠸⣿⣿⣷⡀⠀⠈
⠀⠀⠀⠀⠀⠐⣊⣯⣭⡁⠀⠀⣰⡿⢛⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⢿⣆⠀⠀⢀⣭⣽⣛⠂
⠀⠀⠀⠀⢀⡜⠛⢉⣽⡿⢃⡾⠁⣵⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣮⠈⢷⡘⢿⣯⡉⠛⢣⡀
⠀⠀⠀⠚⢩⣧⣴⡉⠛⠉⡏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢹⠈⠛⢉⡦⣼⡍⠓
⠀⠀⢲⡴⡟⠁⢘⠏⢠⡶⠶⢻⡗⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢼⠛⠶⢶⡄⠹⡓⠈⢻⢦⡞
⠀⠀⠘⣆⣀⣶⠇⠐⠊⠠⠰⣤⡗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣤⠆⠄⢑⠂⠸⣶⣀⣼⠃
⠀⠀⠀⠈⠻⠿⡶⢴⣶⣤⢠⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⡄⣤⣶⡦⢶⠿⠟⠁
⠀⠀⠀⠀⠀⠠⣦⢀⣿⣿⣮⡁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢨⣵⣿⣿⡀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠍⠹⢁⡻⣿⡆⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢰⣿⢟⡉⠏⠩
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⢿⣷⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⡿⠾⡇⠀⠀⢀
⠀⠀⢀⡄⠈⢀⣿⣄⢰⣤⡆⠀⢻⡿⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣾⠿⡟⠀⢰⣤⡆⣠⣿⡀⠁⠠⡀
⠀⠀⠘⠟⠛⠙⢿⣿⢮⠀⢤⣀⢀⣀⣸⣦⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⣶⣇⣀⣀⣀⡦⠀⠴⣿⡿⠋⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠸⠙⠇⠀⣏⠈⢿⠟⠛⣿⣦⡈⠢⣄⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⣠⠔⢁⣴⣿⠛⠻⡿⠉⣹⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⣨⣿⡿⠴⡿⠋⠻⡿⣄⠈⢻⣦⣤⣀⡀⢀⣈⣤⣴⡟⠁⣠⢿⠟⠙⣿⠦⢿⣿⣅⠀⠙⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠸⠆⢰⠗⠀⣀⠀⣿⣶⠿⣈⣿⣀⣥⣤⣤⣿⣁⠻⣶⣿⡀⣀⠀⠺⡆⠰⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠽⠾⠿⠟⠃⠙⠛⠉⠉⠉⠙⠛⠋⠘⠻⠿⠥⠯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿⣆⣀⡀⠀⠀⠀⢀⣠⡴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣤⣤⣿⣿⣿⣿⣿⣷⣶⣏⠁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢤⣄⣤⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠈⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠚⠧⢤⣤⣤⣟⣿⣿⣿⣿⣿⣿⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠀⠈⠉⠉⠉⢉⣿⠉⠻⢷⣌⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡜⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡜⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡝⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣽⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⠉⠙⠁⠉⢀⡴⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠄⠀⠰⢀⠀⠁⢭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⢤⡤⢠⠀⢰⠂⠸⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢳⡀⠙⠒⠂⣴⣿⠋⠀⢀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⣧⠀⠀⠘⢯⠁⠀⠠⠈⠀⠀⢇⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠳⡴⣄⡞⠓⢃⠐⣐⡇⠀⣸⣷⠆
⠀⠀⠀⠀⢠⣀⠀⠐⠤⠤⣴⣶⣄⠀⠀⠘⠳⠆⠀⠸⠇⠀⠀⠀⠀⠉⠉⠙⠛⠛⣟⠈⠑⠀⠄⣀
⠀⠀⠀⠀⢈⠉⠻⠖⣒⣋⠡⠙⠠⠁⠠⠀⠀⠀⠰⠄⠒⠚⠓⠊⠀⠀⠂⠰⡄⠀⣀⠠⢴⡦⡄⠀⠌⠧⠄
⠀⠀⢀⣶⡷⠔⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⡿⠿⠿⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⡛⠉⢩⠀⠠⠄⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⢠⠀⢠⠀⠀⣤⣀⣴⠆⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣧⠀⢻⠀⢸⡀⠀⠿⢻⢯⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢜⠦⠼⠧⠬⠧⠤⠖⠁⠈⢿⣆⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠠⢤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠸⣆⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⢿⡆⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣶⣾⣿⣿⣿⣿⣶⣦⣤⣀⣸⣿⣿⠀⠀⠀⠀⠀⠸⡇⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠇⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠆⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠶⣦⣤⣤⣤⣄⣀⣠⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠈⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠉


⠀⠉⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⢯⡤⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⢤⡽
⠀⠀⠀⠀⠀⠀⠀⢷⠡⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠌⡾
⠀⠀⠀⠀⠀⠀⠀⠘⡗⣃⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣈⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡰⡩⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢍⢂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣻⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣟⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠚⣋⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣙⠓⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣿⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣿⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠻⣷⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢳⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⡞⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⣿⣆⠀⠀⣰⣿⠟⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠹⣿⣦⣼⣿⠋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣸⣿⡿⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⡠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡠⠊⠳⣌⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢋⡠⠊⠀⠀⠀⠈⠳⢌⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⣩⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣉⠻⢿⣶⣤⡀
⠀⠀⢀⣀⣤⣤⣶⠿⢟⣋⠥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢭⣛⠻⢷⣶⣤⣤⣀⣀
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠛⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⢾⣦⣰⢄⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡠⡆⣤⠆
⠀⠀⠐⣶⣿⣯⡘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⣽⣿⡶
⠀⠀⠘⠋⠙⢿⣷⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⡿⠋⠙⠂
⠀⠀⠀⠀⠀⠈⠙⢿⣷⣌⠳⣄⠀⠀⠀⠀⠀⢀⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠀⠀⣠⠞⣡⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⢱⣦⡔⣪⡷⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡶⢮⡓⢤⣴⡊⣡⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⢞⣾⣥⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⣽⣮⡳⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣴⣿⡟⡩⣽⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣯⣔⢷⣿⣦⡛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠱⠻⠉⠉⠾⣻⣮⡻⣮⡢⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⢔⡵⢋⣴⣿⠶⠉⠹⠟⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⢇⣃⠀⠀⠀⠀⠈⠺⡻⣮⡻⣮⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⡵⢋⣵⢿⠟⠁⠀⠀⠀⠀⣄⡘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠈⠪⣻⣮⡹⢮⡢⡀⠀⠀⡄⠀⢀⠔⡵⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣦⡙⠌⠢⡀⢀⠔⠁⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡻⣦⡀⠈⠣⣤⣾⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠾⡻⣦⡀⠈⠳⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⣠⣿⣮⡻⣦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⣴⣟⠛⠁⠈⠺⠿⣦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠈⠫⣻⣦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⣾⡿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⣦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣠⣾⢟⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣷⣄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣷⣄⠈⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠞⠀⣠⣾⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⣿⣷⣄⠀⠳⣤⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣠⣾⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣷⣄⠈⠳⢄
⠀⠀⠀⠀⠀⠀⣼⡄⢠⣾⢿⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢪⡻⣷⡄⢠⣧
⠀⠀⠀⠀⢀⠂⠈⢰⡯⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢽⡄⠑⡐
⠀⠀⠀⠀⢂⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⣠⢀⠀⣀⡀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠸⠛⠀⡀⣀⣉⠋⠀⠀⠀⠀⣝⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠠⠊⢀⠀⠀⠀⠀⡸⡀⠈⠀⠀⢀⡀⠀⠀⠀⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠖⠀⠦⡄⢀⢔⠁⢄⠘⠃⠀⠊⡀⠀⠑⠄⠀⠀⢈⡄⠑⡀
⠀⠀⠀⠀⠀⠨⠔⡀⠀⠀⢦⠀⠠⠀⠀⠀⠀⠀⠠⡰⠀⢨⠄⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠉⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢠⡀⠀⡀⠀⠀⠀⢀⠀⠀⡠⠐⠂⠀⠀⠸⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢤⣤⠄⠠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠈⠾⢤⣤⣤⣤⡴⠢⡐⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠟⠇⠀⠠⠄⠸⠀⠀⠈⣏⠁⠀⠀⡁⠀⠈⢈⠃⠐⠂⠀⠀⠦⡀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠄⠘⢠⠄⠀⠀⠑⢶⠊⠀⢲⠔⠊⠀⠀⠔⠁⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠌⠈
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠰⡄⠀⠈⠂⠠⠤⠠⠐⠀⠀⠀⢀⢠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠂⠀⠀⠀⠊⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⣶⠘⠶⠀⠀⠀⠀⠀⡀⠀⢀⠞⠈⠁⠀⡐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠛⣀⠒⠁⠁⣴⠔⠁⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣦⠈⠓⠀⠁⡀⠒⠊⡰⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢄⡀⠉⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠐⠞⠈⠀⠢⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠒⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠢⡀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠳⠀⠀⢄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠀⠀⠄⠀⠀⠔⡨⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⡀⠀⠀⢀⠒⠁⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠠⠄⠊⠀⠀⠀⠐⠂⠤⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣅⢸⠀⠀⠄⠺⡀⢠⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣤⣤⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠻⠋⠉⠉⠉⠉⠿⠿⢿⣿⣿⠟⠋⠉⠉⠉⡉⠉⠛⢿⣿⣿⠿⠿⠋⠉⠉⠉⠙⠻⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⣀⣀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣄⣀⡀⠈⠉⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣠⡴⢛⣿⣷⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣠⣾⣿⡟⠷⣄⠀⠀⠉⠻⣷⡄
⠀⠀⠀⣠⣴⡿⢋⡄⠀⠀⠈⠁⠠⠚⣻⣿⣿⣷⣿⣤⣤⡴⡠⢃⠄⠀⠀⠀⠀⠀⠀⠀⠈⠢⠰⣤⣤⣼⣾⣿⣿⣿⠛⠢⡈⠁⠀⠀⠢⣌⠻⣦⡀
⣠⣶⣿⣿⡭⠞⠉⠀⠀⠀⠀⠀⢀⠴⢋⣿⣿⣿⣿⣿⠟⠁⠀⠁⣀⡤⠀⠀⠀⠀⠠⣄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣍⠳⢄⠀⠀⠀⠀⠀⠈⠓⢮⣿⣷⣦
⣿⡟⢉⡡⠄⢀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣼⣿⣿⡟⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢹⣿⣿⣧⡈⠣⡀⠀⠀⠀⠀⠀⠠⡀⠐⢬⣉⠛
⠸⣿⣉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡⣿⣿⣿⣤⣄⠀⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⢀⣤⣼⣿⣿⣍⠙⠄⠀⠀⠀⠀⠀⠀⡀⠈⠳⢦⣈⣹
⠀⢻⣿⠁⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⣿⣿⣷⡙⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⣴⠟⣷⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⢹⡟
⠀⢸⣿⡟⠁⠀⡠⠀⠀⠀⠀⠀⠀⢀⣴⣿⢻⣿⣿⣷⣉⢻⡟⠡⠊⠀⠀⠀⠀⠀⠀⠉⠂⠙⣿⢊⣽⣿⣿⡟⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣿⡇
⠀⠀⣿⣆⣠⠎⠀⠀⢀⠆⠀⠀⣰⠟⣼⠃⣸⣿⣿⡃⠨⣿⠁⠀⠐⠀⠀⠀⠀⠀⠀⡐⠀⠀⢿⡭⠈⣿⣿⣷⠀⠹⡙⣧⡀⠀⠐⢄⠀⠀⠙⢦⣀⣿⠃
⠀⠀⠘⣿⣧⡀⠀⣰⠃⢀⠀⣼⠏⢰⡏⢠⠿⣿⣿⣷⣤⣿⢠⡶⠂⠀⠀⠉⠉⠁⠀⠈⢲⣄⢾⣄⣼⣿⣿⡿⠇⠀⣧⠘⣷⡀⢀⠈⢧⡀⠀⣠⡿⠃
⠀⠀⠀⠈⠻⢿⣿⣧⣴⣃⣾⡏⢀⡿⠀⠀⣼⢿⣿⣿⣿⠇⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄⣿⣿⣿⣿⢷⠀⠀⢸⡆⠘⣷⣌⣷⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡿⠀⢸⡇⠀⠀⠀⢸⢿⣿⡿⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢾⣿⡿⢿⠀⠀⠀⢨⣷⠀⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣾⠃⠀⠀⠀⠈⠘⠛⣷⣾⣿⠆⣀⣤⡤⠶⠶⠦⠀⠀⠀⣿⣷⣼⡟⠇⠈⠀⠀⠀⠀⣿⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⣶⣦⡀⢀⠀⠀⢠⡿⠟⣿⡟⢉⣀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢿⡇⠀⠀⢀⠀⣸⣴⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠈⠙⠻⢿⣷⣦⣼⠃⠀⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⢿⣴⣶⡿⠟⠋⠀⣿⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⠀⠀⠀⢀⣼⠏⠀⠀⣸⡟⠋⠉⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠈⢿⣦⠀⠀⠀⢀⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣷⡀⠀⣠⡿⠃⠀⠀⣰⣿⠷⠒⠒⠀⠀⠀⠀⠀⣰⡿⣿⣧⠀⠀⠈⠻⣷⣄⣠⣾⡟⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠙⠿⢿⠯⠄⠀⠀⣰⣿⣇⠤⠀⡀⠀⠀⠀⠀⣰⡿⠁⣸⣿⣧⡀⠀⠀⠬⣿⠿⠟⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣁⢀⡀⠀⠀⠀⠀⣰⡿⢁⡾⠉⣿⡿⣿⣆⡀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣤⣤⣴⡾⠟⠁⢸⡏⣀⠀⠀⠀⠀⠀⣰⣿⣤⣾⢃⠀⣿⣇⠈⠙⠿⣦⣤⣤⣤⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣾⡏⠀⠈⠀⠀⠀⣰⡿⣿⠡⣴⣋⢀⡟⢻⣆⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⣿⠘⢿⣤⣈⣉⡾⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠀⠀⠀⢸⣿⣇⠀⣿⣿⡯⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠁⠀⠀⠸⣇⠻⣶⣿⣿⠑⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣁⡀⠀⠀⠹⣦⡉⠀⠸⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠈⠙⠒⠄⠒⠋⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣀⣀⠀⡀⠀⣀⣀⣴⡾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⣉⣻⣋⣙⢛⣟⢋⣿⢟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡆⢀⢠⣦⡀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⣿⣇⠀⠀⠀⠀⠀⠠⠃⠈⣿⣿⡿⠟⢛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣫⡷⢟⣛⣛⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⣿⡇⢰⣿⡿⠀⠀⣀⡀⠖⣃⣀⣴⣿⣯⣶⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣀⣿⣿⣿⣿⣾⣿⣿⠟⢃⠄⣷⣄⢀⢸⣿⣶⠀⣶⣿⠇⠀⠀⢉⠛⠛⠉⠉⣸⣿⣛⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣷⣿⣿⡯⢀⣤⣶⣿⣼⡿⢙⣃⣿⡏⠀⣿⣿⢀⡂⣲⣿⣿⣶⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣋⠏⣰⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣟⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⣿⣇⢸⣿⣿⣿⡟⡿⢿⣿⣦⣾⣿⣿⢶⡿⣷⣶⣾⣷⠛⡟⣿⣿⣿⣿⣆⢳⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⣴⣿⡿⣿⡇⢀⣾⣿⣿⣿⣿⣯⢸⡇⡟⠿⣿⣿⡆⠀⣿⣿⣿⣷⡏⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢩⣿⢿⣷⣿⡿⢿⠿⣇⣘⣿⣯⣿⣿⣷⣿⣹⣷⣿⣴⣬⣿⣥⣄⣿⡿⣿⣿⠷⣏⡈⠻⣦⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣏⣿⡟⠻⠿⠿⠟⠛⠛⠉⠈⠙⠛⠿⠷⠿⠿⠿⠟⠋⠁⠉⠙⠛⠻⠿⠿⠟⢻⣿⠆⠈⡀
⢀⡀⠀⠀⠀⢰⡇⠀⠀⠀⠂⢿⣿⣞⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣳⣆⠀⠁⠀⠀⢸⡄⠀⠀⠀⢀
⠀⠙⢶⡄⠀⣿⣿⡄⠀⠀⠀⠘⢿⣿⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⡆⠀⠀⢠⣿⣿⠀⣠⡾⠋
⠀⠀⠀⠈⠳⣿⣿⣿⣄⡠⠊⠡⡘⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠙⢄⣠⣿⣿⣿⠞⠁
⠀⣀⡀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠈⠻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠈⣿⣿⡟⠁⠀⠀⢀⣀
⠀⠀⠙⠛⠶⠤⣈⣿⠏⠲⣄⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣠⠖⠹⣿⣁⠤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠢⢷⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠔⠚⠉⢻⡀
⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⢾⣿⣿⣦⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣤⣾⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣛⡀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⢀⣛⣩⣅
⠀⠀⠀⠀⠐⢤⣤⣤⡞⠉⠻⢿⣷⠋⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠹⣾⣿⠟⠋⠰⣦⣀⡤⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣶⣤⣀⡀⠈⠐⠂⠄⠀⠀⠀⠀⠀⠀⠁⣀⣠⣤⣾⡿⠟⠓⣴⠂⠉⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠀⠀⠒⠀⠒⠠⠄⠘⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣲⣦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡼⠛⠉⠁⠀⠀⠘⢿⡯⠉⠐⠒⠒⠈⠓⠀⠢⠤⠴⠖⠒⠒⠒⠊⢹⡿⠃⠀⠀⠈⠉⠛⢆⠆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⢛⣡⣍⠛⠿⣿⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠛⢛⣋⣭⣴⣾⣿⣿⣿⣿⣶⣦⣍⣙⡛⠻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣙⠻⠿⣿⣿⠿⠛⣩⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣦⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⢠⣿⣿⣿⣿⣿⣷⣤⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣦⣸⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣶⣶⣶⣶⣤⣶⣦⣄⡀⢀⣠⣴⣶⣦⣴⣶⣶⣶⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣭⣙⣿⣿⣿⣿⣛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣿⣿⣿⣿⣛⣭⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢿⣿⣿⣿⣿⣿⠿⠿⢿⣿⡟⠛⠛⠓⠆⠀⠉⠻⠟⠋⠀⠠⠞⠛⠛⢛⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⣿⣷⣶⣦⣌⡀⠀⣶⣦⣤⣀⠀⠀⣀⣠⣴⣶⠆⠀⣈⣤⣶⣾⣿⣿⣿⣿⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣶⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⠉⠉⠉⠉⠉⠃⠈⠻⠟⠁⠘⠉⠉⠉⠉⠉⣥⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⣤⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣤⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣫⣿⣿⣿⣿⠟⢹⣿⣶⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣶⣿⡏⠻⣿⣿⣿⣿⣝⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣿⣿⣿⠟⠁⠀⣿⣿⡏⠈⠻⣶⠲⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⢶⠟⠁⢸⣿⣷⠀⠈⢻⣿⣿⣿⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⠀⠀⣿⣿⣧⠀⠀⠈⠲⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⣽⠦⠖⠋⠀⠀⣼⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⣀⠀⢰⡄⢻⣿⢿⡆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⣿⡏⢠⡄⠀⡀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⡿⢰⡇⠀⠈⣿⣾⣿⡇⠹⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠏⢸⣿⣷⣿⠁⠀⢹⡄⣿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣧⣿⡇⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠃⠀⠀⢸⣿⣼⣿⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀⠀⡀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁⢀⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⢹⡄⠀⠻⣿⣿⣿⣄⠀⠐⢤⣶⣶⣶⣶⣶⣶⡴⠂⠀⣰⣿⣿⣿⠏⠀⣠⠇⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⣿⣿⣿⣿⡆⠀⣿⡄⠀⢻⣿⣿⠈⠀⠀⠀⠈⠙⣿⣿⠟⠁⠀⠀⠀⠁⣿⣿⡏⠀⢰⣿⠀⢸⣿⣿⣿⣿⣧⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢀⣿⣿⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⣿⠁⢀⣿⣿⡀⢸⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣼⣿⣿⡇⢠⣿⣿⣿⣄⣀⣀⣀⡴⠿⠿⠦⣀⣀⣀⣠⣿⣿⣿⠀⢸⣿⣿⣧⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣿⣿⠀⠈⢹⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡏⠁⠈⣿⣿⣿⣷⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡟⢀⠀⠈⣿⣿⣿⣇⢀⡀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⣽⣿⣇⣾⠀⠀⢻⣿⣿⣿⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣿⡏⠀⠀⣷⣸⣿⣯⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣿⣿⣿⣿⣆⢠⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡄⣸⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⢦⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⣴⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⣿⣦⡈⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⣁⣼⣿⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠙⠿⣿⣿⣿⣃⡀⠀⠀⢀⣺⣿⣿⣿⠿⠋⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⣶⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣤⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠙⢻⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡟⢁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⢀⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣚⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⢀⡴⠶⢖⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡲⠶⢦
⠀⠀⡘⢦⣔⣭⣥⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣪⣝⣢⡼⠃
⠀⠀⠘⢯⠟⢿⣿⣍⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⣿⡿⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢯⣿⡾⣲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⣷⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡷⣭⡶⣤⣀⠄⡸⢍⠉⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣄⢉⡙⢇⢠⣀⣤⢾⣭⢿⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣾⣁⡽⢣⣶⠾⠋⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠹⠷⣦⡜⢏⣋⣷⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⢏⢔⣽⣿⢖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡲⣿⣯⡢⠵⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢆⣾⠛⣏⣕⡽⢿⣗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⡿⣮⡪⣿⠻⣧⡰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢫⡾⠁⠀⠈⢻⣿⣮⡙⢷⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⡾⣫⣾⣟⠟⠁⠀⠈⢷⡙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠃⠀⠀⠀⠀⠀⠙⢿⢿⣦⡙⢷⡕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡿⣫⣾⣿⠝⠁⠀⠀⠀⠀⠀⠱⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣿⣦⡙⣿⡱⣄⠀⠀⠀⠀⠀⠀⣠⢞⡾⣫⣾⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⡕⠄⠱⣄⠀⠀⣠⠎⠠⣫⣾⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠈⠳⣎⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀⠈⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⣽⢿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣾⣿⠗⢝⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣾⣿⠟⠁⠀⠀⠙⢝⢿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠑⢝⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⣠⣾⢟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣠⣾⣿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡠⢞⣥⣦⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡰⣬⡳⢄
⠀⠀⠀⠀⠀⢀⣜⡾⢟⡵⣟⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⡻⣮⡻⣷⣣⡀
⠀⠀⠀⠀⢀⣮⠏⣠⣯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢌⠳⠷⡀
⠀⠀⠀⢠⣯⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣽⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠲⠿⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠉⠀⠤⢬⣏⡱⣮⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⣄⠀⢰⣆⣶⡬⣿⡽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⣿⡗⠒⠛⢿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡿⠛⢹⡇
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⠒⠃⢽⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠐⣺⠃⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣴⣿⣷⣿⠛⠀⠀⠀⠈⠙⢲⣤⣤⣴⡞⠉⠉⠛⠿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⡞⠀⠀⠀⣠⣴⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠀⠀⠀⠀⠀⠛⠿⣿⣶⣾⣿⣧⡄
⠀⠀⠀⠀⠳⣴⣶⣦⣬⣽⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣠⣨⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠈⠙⢿⣟⠛⢛⣿⣿⣿⣿⣷⣦⣄⣠⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡁⠉⠛⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⣤⣘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢻⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⠳⠹⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣾⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌


⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠛⠛⠛⠿⠿⣿⣷⣶⣮⣝⣷⣦⣕⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣩⣭⣍⣛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⠏⠉⠁⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢟⣫⣽⣾⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⠋⠀⠀⠀⣀⣀⣀⠉⠢
⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠉⠙⠛⢷⣀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠈⢉⣤⣶⣿⡿⢟⣿⣿⡿⢟⣛⣭⣵⣶⣾⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⣴⣿⡿⠟⢁⣴⠟⣫⣴⣾⣿⣿⣿⣿⣿⡿⠿⢛⣛⣩⣭⣵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⣼⣿⠟⠀⠀⢊⣵⣿⣿⣿⣿⣿⡿⠟⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣠⣴⣶⣦⡄⠀⠀⠀⢈⠁
⠀⠀⠀⢸⣿⠃⠀⠀⢠⣿⣿⣿⣿⣿⠟⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣤⣀⢀⣠⣴⠟
⠀⠀⠀⢸⡏⠀⠀⢠⣿⣿⣿⣿⡟⠁⣰⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠉⠉⠁⠉
⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⡘⣿⣿⣦⣀⡀⠀⠻⣿⣿⣿⣿⣿⣧⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠈⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣮⡙⢿⣿⣿⣶⣤⡀⠙⢿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠹⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣷⠀⠈⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⠀⠀⠈⠻⣿⣿⣧⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⡇⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣤⡀⠀⠙⢿⣿⣿⣿⣿⠃⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠃⠠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⢻⣶⣴⣾⡟⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣶⠿⣿⠤⠒⠋⠀⠙⠓⠦⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⢠⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⢻
⠀⠀⠀⠀⠀⠀⠀⡶⠦⣄⡀⠀⢀⣀⣗⠈⣄⡀⠀⢀⣠⡤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡦⡀⠉⣾⣿⣿⣿⣿⣿⣿⣗⡿⠋⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⢟⣼⣟⠉⠛⠿⣷⡻⣤⣴⡿⢲⣦⣠⣶⡖⠒⣱⣦⢾⣷⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⡿
⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣷⣄⣀⣘⣿⣿⣿⣿⣬⣟⠈⠛⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣦⡼⠁
⠀⠀⠀⠀⠐⠫⠤⣤⣼⣿⣿⣿⣿⣿⣿⣍⣻⣿⣿⣿⣿⣯⣉⡩⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣶⡏⠙⠹⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡋⢑⣿⠏⠙⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⢉⣷⣽⣿⣿⣿⣿⣏⡾⠙⢯⣧⡀⠀⠀⠀⠀⠀⠀⢸⡒⠦⣠⣤⡶⢶⣾⠿⠗⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡼⢀⣴⠿⠛⠿⣿⡿⢿⣿⠿⠟⠦⣄⠙⢧⠀⠀⠀⠀⠀⠀⠀⣹⠶⡧⠤⠙⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⣠⡿⢦⡙⢱⡖⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⣦⣼⠟⠋⠙⠦⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣵⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡿⠋

⠀⠀⠀⠀⢐⡏⠙⢢⡀⠀⠀⢠⣶⣶⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⡞⠉⢶⣄⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⢀⣴⣶⣤
⠀⠀⠀⢠⣾⣷⡄⠀⠱⠦⣄⠀⠛⠛⠀⠀⠀⠀⡎⠀⡠⠾⠀⠀⠳⡄⠀⠙⠳⣄⠀⠈⡇⠀⢀⣀⠀⠀⠘⣿⣿⡿⠂
⠀⠀⠀⠈⠻⠛⠃⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⣸⣔⣊⣀⣀⣠⠀⠀⠈⣆⠀⠀⠈⠃⠀⢿⠀⠻⠟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⠀⢠⣴⣯⡄⠀⠀⠀⠀⢸⡆⠀⢀⣀⠀⡄⠀⠟
⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠁⠀⠀⡠⠊⠁⠀⢀⣤⣦⣤⡀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠈⡇⠀⠈⠋⠀⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠉⠀⠈⠀⠀⠀⡀⣾⣿⣿⣿⡇⠀⠀⣴⣷⠀⠀⡿⠀⠀⣀⠀⠛⠀⣀⡀⠀⠃⠀⣠⡀
⠀⠀⠀⠘⠛⠛⠛⠛⠃⠚⠛⠛⠃⠘⠀⠀⠀⠈⠛⠛⠛⠁⠀⡼⠁⣿⠀⠀⡇⠀⢸⣿⠀⠀⢸⣿⣷⠀⠀⠺⣿⡟⠂
⠀⠀⠀⠀⡀⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⢿⠀⠀⡇⠀⠀⠃⠀⠀⠈⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⣸⣿⣿⣏⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣾⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⡏⠁⠀⠀⡇⠀⠀⢀⣤⡀⠀⠀⠀⠀⠋⠀⠀⠀⠉⣽⠀⠀⡇⠀⠀⠀⢀⣠⣤⣉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠃⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⡀⠀⢀⠢⠀⠀⠛⠿⠟⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣶⣥⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡀⠲⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠸⡄⠀⢹⠀⢱⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡵⠞⠁⠀⠀⡇⠀⢀⠃⢀⣧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣀⣠⣼⣷⣶⣧⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣠⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣀⣠⣤⣶⣾⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣀
⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤
⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠈⠛⠿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⠿⠛⠉
⣤⣀⣀⣄⣀⣀⣤⣤⣀⣠⣬⣉⣉⣭⣉⣉⣉⣉⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣉⣉⣀⣀⣀⣀⣠⣀⣄⣄⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⠔⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⡴⠋⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠞⠁⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡾⠋⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣴⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣠⠞⠁⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⢀⡜⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⡠⠋⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡄⢀⠂⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⠞⠐⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⡣⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⢹⡧⠀⠀⣰⠟⠁⠀⣰⢏⠞⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣾⣁⣤⣾⡃⠀⣠⢞⡵⠃⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠙⢧⣌⠙⠻⢷⣏⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣦⣀⠈⠛⢶⣄⡈⠙⣿⡁⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⢳⡙⠷⣤⡀⠈⠛⢶⣯⡴⠋⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣅⣈⣧⢳⣴⠏⠙⢶⣄⡀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠘⢧⡈⣹⡿⠃⠀⠀⠀⠈⠻⢶⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⣷⣾⡿⠋
⠀⠀⠀⠀⠀⠀⢀⡾⠻⣌⡳⣤⣴⠟
⠀⠀⠀⠀⢀⣴⣟⣓⣒⠾⣿⡟⠁
⠀⣠⡶⣶⢿⣻⢯⣉⢉⡿⠋
⣴⠏⠀⠻⣦⣉⣓⣿⠟⠁
⣿⡀⠀⠀⠀⠙⢛⣿⠂
⠈⠛⢷⣄⡀⣠⡾⠃
⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢠⣄⠀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠻⣟⠓⠂⠻⣾⠑⠚⣻⠟⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢿⣿⡟⠛⠷⠿⣇⠀⠀⠏⠀⣸⠿⠾⠛⢻⣿⡿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡵⠆⠀⠀⠉⠀⠀⢠⡘⣷⣀⣔⡄⠀⠀⠉⠀⠀⠰⢮⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠚⣩⣽⣿⡭⢦⡄⠀⠀⠀⠀⢀⢨⡇⠀⡿⢸⡇⡀⠀⠀⠀⠀⢠⡴⢭⣿⣯⣍⠓
⠀⠀⠀⠀⠀⣴⣯⣿⠁⠀⠀⠙⠀⣄⣄⣴⡼⠿⠃⠀⠀⠈⠟⢧⣦⣠⣠⠀⠋⠀⠀⠈⣿⣽⣦
⠀⠀⠀⠀⠸⠛⣿⡿⢄⠀⠀⢀⣀⡏⠉⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⢸⣀⡀⠀⠀⡠⢿⣿⡛⠇
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡄⠀⠰⣿⠁⠰⠾⣶⡜⠋⠀⠀⠙⢣⣶⠷⠆⠈⣿⠦⠀⢠⡀⢸⣿⡇
⠀⠀⠀⠀⢀⣿⣟⣴⣿⠁⠀⠰⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡆⠀⠈⣿⣦⣻⣿⡀
⠀⠀⠀⢀⣾⠿⠋⣼⠇⠀⠀⠀⠸⢾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡷⠇⠀⠀⠀⠸⣧⠙⠿⣷⡀
⠀⠀⠀⠈⠀⠀⢰⣿⣶⣾⡟⠀⠀⠀⠿⡄⠀⠒⠶⣶⣶⠶⠂⠀⢡⠿⠀⠀⠀⢻⣷⣶⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡿⠋⣿⠁⠀⠀⠀⠀⢿⠀⠀⠀⢈⡁⠀⠀⠀⡖⠀⠀⠀⠀⠈⣿⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠸⠁⢸⣿⠀⠀⠀⠀⠀⢈⡷⠦⠖⠛⠛⠲⠴⢾⠁⠀⠀⠀⠀⠀⣿⡆⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣾⡇⠀⠀⠀⠻⣶⠀⠀⠀⠀⣤⠛⠀⠀⠀⢸⣶⣦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⢸⡇⠀⠀⣀⠀⠘⢾⠳⠞⡷⠃⠀⣀⠀⠀⢸⡇⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣾⡇⠀⠀⣤⡀⢀⣤⠀⠀⢸⣷⣄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠙⣷⠀⢸⠏⠁⠈⠹⡇⠀⣾⠋⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢷⣼⡆⠀⠀⢰⣧⡾⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠛⠉
⣿⣟⣛⣟⣛⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢿⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾
⠈⠻⣮⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟
⠀⠀⠈⠻⣮⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠈⠻⣮⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣮⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣌⠻⣦⡀⠀⠀⠀⠀⢀⡴⠏⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⢦⡀⢀⡴⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡿⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣾⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡾⠋⠙⢷⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡾⠋⠀⠀⠀⠀⠙⢷⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠲⣶⠶⠦⣤⣠⡾⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢦⣀⣤⠶⠶⣶⠖
⠀⠀⠀⠀⠀⠀⠙⢷⡀⢼⡋⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⢙⡧⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢀⡼⣷⠈⠛⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠛⠁⡾⢧⡀
⠀⠀⠀⠀⢠⡾⠋⢀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⡀⠙⢦⡀
⠀⠀⢀⡴⠋⢀⣴⠛⢸⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡏⠻⣦⡀⠙⣦⡀
⣠⣿⡻⣧⣴⠋⠀⢀⣼⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠘⣧⡀⠈⠙⣦⣼⢟⣿
⢿⣌⣿⢶⣝⣛⠛⢋⣁⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⡙⠛⣛⣩⡶⣿⣡
⣶⣾⡶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣷
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠠
⠀⠀⠀⠲⠄⠒⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡈⠀⠀⠀⠀⢀⣀⣀⢀⡔⠀⠀⠀⠀⢀⠖⠁⠀⢰⠃⠀⠀⡹⠀⠀⢀
⠀⠀⠀⠀⡌⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⣇⠀⣠⠤⠴⠁⠀⠀⠉⠂⠀⠀⠀⣰⠋⠀⠀⠠⠶⠶⠦⢤⡴⠒⠶⠇
⠀⠀⠀⢸⠀⠀⠀⡸⠒⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⢠⠁⠀⠀⢸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⢄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠈⣨⠛⣆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠒⣖⠀⠀⠀⠀⠄⠀⠀⠸⣀⣀⣀⣀⣀⡤⠴⠊⠹⡀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢶⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠈⠆
⠀⠀⠀⠀⡎⠀⡜⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠋
⠀⠀⠀⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠈⡄
⠀⠀⠀⠐⡇⠀⠀⢸⣄⣠⣤⣄⣀⣠⠀⣀⡠⠀⠀⠀⠀⡴⠋⠀⠀⠀⡿⠒⠒⠦⣤⠖⠒⠛⠀⠀⠀⠀⢀⠀⠀⠀⢰⣠⠤⠤⠤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⠘⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠈⡷⢦⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠐⢒⣾⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠉⠳⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠆⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⢲⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⡶⠖⠋⠉⠉⠙⢢⣤⣀
⠀⠀⠀⢀⣠⠤⢤⣤⡠⠊⠁⠀⠀⢬⣭⣿⣿⣦⣆⡀⠀⠀⠀⠙⠌⠢⣀
⠀⠐⠉⠁⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠟⡥⢤⣤⡀⠀⠀⠀⠈⢳⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢰⣽⣄⠀⠀⠀⠀⠈⡀⡀⠀⠀⠀⠀⢣⠀⠀⠊
⠀⠀⠀⠀⠀⢳⡀⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⡀⠀⡈⠻⣿⣷⣷⣦⣀⡠⠀⠑⠁⠀⠀⠀⡎
⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣘⣀⣀⠉⠉⠉⣉⣡⠔⠁⠀⠀⠀⠐⡐⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣷⡶⣿⣄⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡗⠀⠀⠀⣠⣿⣿⠋⠀⠀⠈⠁⠀⠀⢠⣿⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠅⠀⠀⠀⠀⢐⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣸⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠀⠙⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠁⠀⠀⣠⣾⣷⣄⠀⠀⠉⠻⢿⣶⣤⣀
⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⡿⠿⠛⠉⠀⠀⣀⣴⣾⠿⠃⠙⠿⣷⣦⣀⠀⠀⠉⠛⠿⢿⣶⣶⣦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⣀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⣿
⢻⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣿
⢸⣿⠀⠀⢸⣿⡿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⠿⢿⣿⡇⠀⠀⣿
⢸⣿⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⣿
⢸⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢸⣿
⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿
⠀⣿⣷⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣿⡿
⠀⢸⣿⡀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⣿⡇
⠀⠸⣿⡇⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⣸⣿⠃
⠀⠀⢻⣿⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⣿⡟
⠀⠀⠸⣿⡆⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⢸⣿⠃
⠀⠀⠀⢿⣷⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣿⡟
⠀⠀⠀⠘⣿⣇⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⣸⣿⠃
⠀⠀⠀⠀⢹⣿⡄⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⢻⣿⡄⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠈⠛⢿⣷⣤⣀⣀⣤⣾⡿⠛⠁⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⣠⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⠀⠀⣀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠟⠛⠉
⣿⣿⡟⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿