Booklet Emojis & Symbols

⡎⣎⢏⢧⢣⡳⡹⡜⢎⢳⡙⣎⢳⡀⠀⠀⠀⠀⠙⣿⢽⢸⢸⢊⡇⣷⢹⡗⣐⣞⠿⣮⣙⣵⡫⣯⣮⢆
⢹⢸⡜⡎⣧⢣⢱⡜⣎⢧⠹⡌⢧⡙⢦⣄⣀⣀⣠⢮⠗⡏⡞⡼⣰⢃⡞⣰⠏⣼⠏⣮⢉⢡⡻⣿⣵⣻⣧
⣸⢸⡇⣇⢹⠸⡇⣧⢹⡌⣧⠹⣌⠻⠶⠬⢭⣭⣥⣼⢸⡸⣱⢱⢃⡞⣰⢋⡼⢃⡾⢁⡼⠋⣵⠸⠺⡚⡽⢧
⣿⢸⡇⣿⢸⡇⣷⢸⡆⢷⠘⣧⣙⣛⣒⣒⡒⠶⠶⠤⡾⡇⡧⢣⢎⡼⣡⠟⣴⢋⣴⠟⣡⠞⢋⣴⠇⢫⢪⡓⢥
⣿⢸⡇⢻⠈⣇⢸⡆⢿⡘⠷⠬⣭⣭⣭⣭⣍⣙⡛⠶⠶⠇⡇⢏⠞⡴⣣⠞⣡⠟⣡⠞⢁⡴⠛⣡⡶⠟⣁⣷⠜⣥
⣿⢸⡇⢸⡆⣿⠈⣧⣈⣛⣓⣒⣒⣒⣒⠲⢦⣭⣙⡛⠶⠶⠱⢸⢞⡴⣣⠞⣡⠞⣡⡾⢋⣴⠞⢋⣤⡾⠛⣁⣴⠗⣧
⣿⢸⡇⢸⣇⣘⠳⠶⠦⠬⠭⠭⠭⣭⣍⣛⠲⢦⣍⣙⠳⠶⢆⡎⢎⠜⣡⠞⣡⠞⣡⡶⢋⣡⡾⠛⣡⣴⠟⠋⣁⣴⠷⠣
⣿⣈⣛⡶⠮⠭⠭⣭⣭⣙⣛⣛⣒⠲⢦⣍⡛⠶⣬⣉⠛⠶⠞⡘⡜⣞⡵⢛⡥⠞⣡⡶⢛⣡⡴⠟⢉⣤⡶⠟⠋⣡⣤⡾⠇
⣒⣒⠮⠭⠭⣍⣛⣒⣒⠲⠶⢬⣍⠛⢦⣌⡛⠶⣤⣉⠛⠶⠶⡑⣜⢎⠴⣋⡴⢛⣡⠴⠛⣩⣴⠾⢋⣡⣴⡶⠟⠋⣁⣤⣶⠷
⣒⡪⠭⢭⣛⣒⡲⠮⠭⣍⡛⢶⣌⡛⢶⣌⡙⠳⣤⣉⠛⠶⡶⢌⢎⣯⢞⣡⠖⣋⣥⠶⢛⣩⣤⠾⠛⣉⣠⣴⠾⠟⠋⣉⣠⣤⣶⠄
⣲⠭⢭⣓⣒⠮⢭⣙⠳⣌⡙⢶⣌⠛⢶⣌⡙⠷⣤⣉⠻⢶⡼⡣⡫⡳⣫⠴⢚⣩⡴⠞⢛⣩⣤⠶⠟⢋⣉⣤⣴⠾⠟⠛⠉⠁
⠮⢭⣓⠲⣍⠻⣦⡙⢷⣌⠛⢦⣌⠻⢦⣈⠻⢶⣌⡙⠷⣦⡬⡪⣜⢝⡵⠞⣋⣥⠴⠞⢋⣩⣤⠶⠞⠛⠉⠁
⡙⢦⡙⢷⣌⠳⣌⠙⢦⡈⠳⣦⡙⠳⣤⡙⠳⣦⡉⠻⢶⣤⠯⢽⡫⣪⠞⣋⣩⡴⠶⠛⠋⠉
⠳⣌⠻⣦⡙⢦⡈⠳⣄⠙⢷⣌⠻⢦⣌⠻⢦⣌⠙⢷⣤⡽⠔⢟⢝⡵⠋⠉
⣦⡙⢷⡈⢳⣄⠹⣦⡈⠳⣦⡙⢷⣄⠙⢷⣄⡙⠷⣦⣈⣓⣾⡵⠋
⠈⢷⣄⠻⣦⡙⢷⣌⠻⣦⡈⠻⣦⡙⢷⣦⡙⠻⣦⡌⢷⡶⠋
⠀⠀⠙⣧⡈⢷⣄⠙⣧⡈⠻⣦⡈⠻⣦⡈⠻⣦⣈⡥⠊
⠀⠀⠀⠈⢿⣄⠙⣷⡈⠻⣦⡈⠻⣦⡈⠻⣦⡈⠁
⠀⠀⠀⠀⠀⠙⣷⡈⠻⣦⡈⢿⣆⠙⢿⡦⠈
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⢿⣄⠙⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠁

⠀⠀⣿⠋⠉⠉⠉⠉⠙⠛⠛⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⠛⠋⠉⠉⠉⠉⠙⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠛⢻⠋⠉⠉⠉⠉⠛⠛⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠛⠛⠉⠉⠉⠉⢙⡟⠋
⠀⠀⠀⠸⠶⢶⡶⠶⠶⠶⠶⠦⠤⣤⣄⣉⡙⠳⢦⣄⠀⠀⠀⣿⠀⠀⢀⣠⡴⠞⣋⣁⣤⣤⠤⠶⠶⠶⠶⠶⢶⡶⠶⠇
⠀⠀⠀⠀⠀⠘⠗⠒⠒⠚⠛⠒⠒⠒⠲⠿⠿⠿⢷⣾⣿⣶⣄⣿⣠⣶⣿⣷⡾⠿⠿⠿⠖⠒⠒⠒⠚⠓⠒⠒⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉





⣿⣿⣿⣻⣟⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣟⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⢸⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⡇⣷⣤⢀
⠀⠀⠀⠀⠀⢀⣴⠏⡇⡟⣸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣇⢻⢸⠹⣧⡀
⠀⠀⠀⢠⣤⣾⣿⠀⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⣿⣷⣤⡄
⠀⠀⠀⣸⣿⡟⢻⠀⡇⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⢸⠀⣿⢻⣿⡇
⠀⠀⠀⣿⡏⢷⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⡼⢹⣿
⠀⠀⠀⣿⠇⢸⠈⢸⢱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡌⡇⠃⠇⢸⣿
⠀⠀⢠⣿⠸⠀⡇⢸⢸⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⡇⢀⠀⠃⣿⡄
⠀⠀⢸⣿⠀⠀⡇⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢸⠀⠀⣿⡇
⠀⠀⢸⡏⠀⠀⠃⡈⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢁⠘⠀⠀⢹⡇
⠀⠀⣼⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧
⠀⠀⣿⡇⠀⠀⠀⠁⠀⠄⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣂⠥⠀⠘⠀⠀⠀⢸⣿
⠀⠀⣿⠃⠀⠀⠀⠂⠤⠤⣀⣉⠓⠪⢵⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣮⠗⠚⣉⣀⠠⠤⠀⠀⠀⠀⠸⣿
⠀⠀⣿⠀⠀⠄⠀⠀⠐⠐⠀⠤⠭⠕⣲⣭⣛⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣛⣭⣖⠫⠭⠤⠀⠒⠂⠀⠀⠠⡀⠀⣿
⠀⢸⣿⠀⠀⢀⡀⠤⠄⣀⣒⠢⠬⣍⡒⠮⢭⣛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣛⡭⠵⢒⣩⠥⠔⣒⣀⠠⠤⢀⡀⠀⠀⣿⡇
⠀⢸⣿⣌⣒⡲⠤⢭⣑⡒⠤⢍⣓⠲⠬⣙⡒⠬⣙⠢⢍⠻⣷⡄⠀⠀⢀⣾⠟⡩⠔⣋⠥⠒⣋⠥⠖⣚⡩⠴⢒⣊⡭⠤⢒⣂⣩⣿⡇
⠀⠈⠉⠛⠻⠿⢿⣶⣦⣬⣉⣒⠂⠍⠁⠂⠈⠁⠀⠁⠀⠁⠀⠙⢆⡰⠋⠀⠈⠀⠈⠀⠈⠉⠐⠊⠩⠐⣒⣉⣭⣴⣶⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⣀⣴⣾⣷⣦⣀⠀⠀⠀⢀⣀⣠⣤⣶⡾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⡿⠿⠿⠿⠿⢿⣶⠶⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⣀⣀⡈⠉⠑⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠢⢄⠀⠉⠁⠒⠤⢄⡈⠉⠒⠤⣄⡀⠀⠀⠀⡠⠔⠈⣀⠀⠐⢂⣁⠁⠀⠐⠀⠀⠄⠀⠀⠀⢰⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡈⠀⠀⠀⠀⠙⠢⡀⠀⠀⢀⣈⣑⣢⣄⠀⠉⠓⠒⠉⠀⠔⠈⡀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣡⠄⠒⠂⠀⠀⠉⠉⠳⣄⠀⠀⠀⠀⡈⠀⠀⢀⠀⠀⢐⠀⠁⠀⠀⠦⠑⠀⠀⠀⠓⠃⠀⠀⠀⠀⣸⠠⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢐⡀⠭⡁⠒⠂⠤⣀⡀⠀⠈⠣⡀⠀⠰⠀⠀⠀⠆⠀⠀⢐⡈⠤⠒⢐⡨⠙⠇⠀⠈⠭⠝⠋⠀⠀⢀⠃⡼⢱⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢒⡀⠥⠀⠀⡔⣂⠤⠈⠑⠢⢄⡙⣄⠂⠀⠀⡀⠀⠀⠘⠓⢒⠰⠒⠃⠤⣘⠀⠀⣀⡙⠁⠀⠀⠀⡎⣸⣇⡞
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⠳⠶⠄⠀⠠⣨⠱⠊⠀⠀⢕⡢⡉⠉⠀⠀⠀⠁⠀⡐⡠⢀⡡⠤⠒⠈⠁⠀⠀⣀⣀⣀⡀⠀⠀⡸⢠⡿⣼⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠤⢄⡀⠀⠑⠒⠥⠀⠀⣚⠵⠢⠀⠀⠀⠀⡌⠀⠀⢟⡉⠄⢐⡂⠁⠈⠀⠀⢀⡥⠤⣁⠀⠀⢠⠃⣼⢱⠇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠒⣡⠄⣀⠍⣐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢅⠂⠈⡐⠀⠈⠀⠀⠠⢁⠱⡊⠍⠀⢀⠇⣼⠇⡞
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠔⣒⠠⢍⡀⠹⠒⠄⠀⠀⣙⠳⠒⠄⠀⠀⢀⠂⠀⢀⠠⠐⠀⠑⠀⠉⠉⠀⠀⠘⠃⠒⠚⠀⠀⡎⡰⡏⣸⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠐⠈⠑⠂⠀⠀⠀⠥⠚⠄⠀⠀⠀⡀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⡻⢰⠇
⠀⠀⠀⠀⢨⣈⣙⣑⠲⠦⢄⣀⠀⠀⠀⠀⠀⠀⠂⠉⠂⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠒⣒⣒⣒⡒⠲⠶⢡⣳⢣⡏
⠀⠀⠀⠀⠐⠦⠤⠥⠍⠽⠵⢦⢽⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣍⣕⣒⡪⡭⠭⠭⠭⠭⠭⢽⣍⡓⢣⠇⡺⠁
⠀⠀⠀⠀⠀⢸⡶⢶⠒⣒⣲⡶⠒⠶⠿⠿⠿⠿⠷⡦⣤⣀⡐⢰⣤⣾⠿⠿⠇⠉⠁⠒⠒⠶⠶⠖⠶⠤⢤⠤⢤⣄⣈⠉⣼⠃
⠀⠀⠀⠀⠀⠈⠑⠒⠐⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠈⠈⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠃⠐⠒⠒⠲⠬⢥⣆⡐⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠐⠤⢒⠅⢒⠩⠀⢒⠩⠅⢒⠩⠄⢒⠩⠔⠊⠁⢒⠩⠄⢒⠉⠁⡂⠍⢀⠂⠍⢐⠨⠉⢐⠈⠄⢒⠨⠄⠒⠈⠄⠢⢤⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠙⠛⢷⣶⣶⠟⠛⠉⠉⠉⠉⠉⠉⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣀⣀⣀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣀⣀⣀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠛⠛⠛⠛⠛⠻⠇⠀⣿⡇⠀⠿⠛⠛⠛⠛⠛⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠿⠿⠿⠿⠿⠷⠆⠀⣿⡇⠀⠶⠿⠿⠿⠿⠿⠟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠿⠿⠿⠷⢶⣶⡆⠀⣿⡇⠀⣶⣶⠶⠿⠿⠿⠷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣀⣿⣇⣠⣤⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠄⠒⠐⠀⠉⠉⠀⠀⠒⠒⠂⠤⡤⠄⣀⣀⣀
⠀⠀⠀⡴⠒⠪⠭⢄⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⢒⣶⡤⠄
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠬⠭⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⢸⠃
⠀⠀⠀⠱⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⣶⠒⠊⠉⠀⠀⠀⠀⠀⣀⣨⡤⠄⣀⡀
⠀⠀⠈⣿⠯⢤⣀⠉⠉⠑⠒⠢⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⡇⠐⠲⣦⣀⣀⣤⡶⠿⠯⠅⠒⠒⠈⠉⠱
⠀⠀⠀⣸⠀⠀⠀⠉⠓⠢⠤⣀⡀⠀⠀⠈⠉⣉⣛⣲⡦⠤⣤⣌⣧⣤⡶⠿⡿⠁
⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⢠⣶⣾⣉⣛⠷⣦⣄⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠴⣒⣒⣪⣭⠽⣿⠟
⣿⠁⠀⠈⠉⠛⠳⠿⢿⣷⣦⣄⡀⠀⡟⡀⠀⠀⣀⣀⣤⡤⠤⠶⣒⣚⣋⡭⠥⠴⠖⠒⠛⠉⠉⠁⠀⠀⠀⣾
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣿⣯⣭⣽⠶⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣄
⠀⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣀⡀⠀⢠⡀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⠞⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠶⠶⠒⠒⠂⠀⠀⠀⠀⠀⠐⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⡒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣙⠲⠬⣝⡳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠒⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣒⠠⣍⡒⠦⣍⡐⠀⠉⠙⠳⣦⡀⢀⠠⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠬⣔⡢⢍⣒⠨⠅⡂⠈⠁⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣉⣳⣦⣍⡒⠠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣛⣻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠉⠉⠙⠳⠶⣤⣄⣀⣀⣠⣼⣋⣠⣤⡤⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠛⣿⡻⣿⡛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣋⣻⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⢙⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠸⠈⢷⡈⣿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢻⣿⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⢀⣠⣤⠶⠖⠛⠋⣿⣥⠶⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣈⣷⡄
⠀⣠⡾⠋⠉⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠩⣭⣿⣓⣒⣒⣒⣒⣒⣒⣂⣀⠈⠭⠍⠉⠉⠉⢩⣿⣉⠁
⣸⠋⠀⠀⠐⠢⢄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠤⠤⠤⣀⣀⣀⣀⣒⣒⠒⠒⠲⠶⠦⠈⠀⠀⠀⠀⣾⠀⠈⠙⠲⢤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⣀⣀⣈⣉⣉⡉⠙⠛⠛⠒⠒⢒⣖⣶⣶⠶⠭⠭⠭⡉⠉⠉⠉⢀⣀⠀⠀⣿⠀⠀⠀⢀⣀⣉⣿⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⣬⣭⣭⣭⣽⣿⣁⢘⡀⡃⣐⢠⣀⣶⣶⣤⣤⡤⠉⠉⠹⢿⣛⣒⡀⠘⢷⡟⠛⠛⢉⡿⠉
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢄⣀⠉⠉⠉⠉⠉⠉⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠟⠃⠀⠀⢸⡇
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡾⠛⠉⠁⠠⠤⠶⠒⠒⠀⠀⠀⠀⠀⠀⠤⠶⠒⠂⠀⠀⢼⠃
⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣩⣤⣤⠤⠶⠒⣂⣀⣠⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣘⠫⣓⠂⠀⠀⠀⠀⠐⠒⠚⣛⣛⣩⣭⣭⠴⠶⠶⠚⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠉⠙⢲⡤⣤⣀⡀
⠀⠀⠀⠀⢠⠖⠚⠉⠉⠉⠉⠉⠙⠒⠲⠤⢄⣀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡇⢹
⠀⣀⣀⡖⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡀
⡠⢯⠀⢇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢳⠀⡇
⡇⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇
⢳⢸⡀⢸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⢳
⢸⠀⡇⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⡇⢸
⠸⡄⡇⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⡇⢱⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠘
⠀⡇⢸⠀⢸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣇
⠀⢹⠸⡄⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⢸⠀⡇⠀⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣸⠀⢸
⠀⠘⡆⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢸⣀⣠⣤⣖⣪⠭⠭⠥⠤⠤⠤⠤⠬⢭⣀⣸
⠀⠀⡇⢸⠀⢹⠀⠸⣤⢴⣒⣫⠭⠽⠛⠛⠛⠛⠉⠉⠙⠛⠋⠉⠁⡠⠤⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠛⠒
⠀⠀⢧⢸⠀⢸⣤⣒⣫⡭⠤⠴⠖⠒⠒⠛⢉⣉⣉⣙⠦⠤⠤⠴⣚⠵⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛
⠀⠀⢸⡜⣉⣉⣀⣠⠤⠤⠴⠒⠒⠚⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠓⠤⣀⠀⠑⢄⠀⠀⠀⠀⡠⠊⠀⣀⠤⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠤⠤⢀⡀⠑⠢⡈⢆⠀⠀⡔⢁⠔⠊⢀⡀⠤⠤⠀⠒⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢌⢆⡆⣰⡰⡡⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠢⠤⣀⡀⠀⠀⢸⠀⠀⡇⠀⠀⢀⣀⠤⠄⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣸⠀⠀⣇⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⣠⠔
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢀⣿⢶⣤⣀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⢀⠴⠊⠀⢐⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠉⠛⠦⢽⣷⣄⣠⣴⣉⣤⣀⣠⣀⡴⠊⠀⠀⠀⢀⡼⠋
⠀⠀⠀⣾⡷⠶⠤⠤⣄⣀⠀⠀⠁⠀⣀⣤⣴⡶⢊⣁⢹⡇⠀⠀⠀⠀⠀⠀⡨⠛⠛⠻⢭⣙⠒⠤⣄⠀⠀⠀⣠⠖⠁
⠀⠀⠠⠿⠯⠽⠯⠽⠶⠶⠿⠶⣶⡿⠿⠿⠿⠿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠈⠙⠲⢬⣦⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠈⠉⠐⢄⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠳⣦⡙⠢⣄⠀⠀⠀⠀⢀⡠⠴⢚⡁
⠀⠀⠀⢠⠿⣆⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣮⢄⠀⠈⠙⠷⣤⡙⣶⡕⢊⣡⠴⠒⠉⡠⠊⠉⠁⠐⠒⠒⠠⠀⡀
⠀⠀⠀⡈⠧⡀⠑⢄⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⣴⣗⡯⢶⣭⡄⠀⠀⠀⠈⠛⠷⠛⢯⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⣠⢾⡅
⠀⠀⠘⢿⡀⠉⢄⠀⠁⠀⠀⡀⠀⠈⠛⢦⡀⣼⣿⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊
⠀⠀⠀⠀⠙⢦⡀⠑⠄⣈⣠⣴⡶⢶⣾⣿⠿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣑⡒⠲⠤⠤⢀⣀⣀⡔⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠹⣄⢠⣤⡤⠷⠚⠉⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠑⠒⠦⠤⣀⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠠⠤⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣒⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠤⢭⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠉⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠶⠦⣤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⣈⣇⡀
⠀⠀⠀⠐⣤⣊⠉⠉⠉⠒⠒⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠃⢠⣤⣄⡀⠀⢀⣀⣤⣶⣛⡋⠥⠀⠀⠀⠂⠆
⠀⠀⠀⠀⢹⠀⠀⠉⠒⠦⣤⣀⠀⠀⠀⠉⠉⠛⠒⠶⠦⠤⣄⣀⡀⠀⠈⢸⡄⢀⣀⣽⣷⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣤⡤⠤⠔⠒⠒⠉⠉⠉⠛⠒⠿⠛⠉⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⣲⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⣧⣄
⠀⣾⠟⠷⢶⣬⣉⠓⠦⣤⣀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠄⠒⢒⣈⣩⠤⠤⠶⠒⠛⡿⠁
⢸⡏⠀⠀⠀⠀⠉⠙⠓⠒⠿⢿⣶⣤⣀⠀⡸⢡⠀⣀⣀⣠⣤⡤⠶⠶⢖⣚⣉⣩⡥⠤⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢿⣿⣭⣭⣵⠶⠶⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣠
⠀⠈⠙⠛⠷⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠞⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⣦⣟⣀⣤⣤⠤⠶⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⢋⣉⣤⡶⠾⢛⣩⣿⣷⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠋⣉⣠⣤⡶⠟⠛⣉⣤⣴⡾⠿⠛⠛⣉⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠉⢁⣀⣤⣴⠾⠟⠋⣉⣤⣴⠾⠛⠋⣉⣠⣤⣶⣾⡿⠿⠛⠋⠉⠁⢸
⠀⠀⣀⣤⡶⠟⠛⠉⠁⢀⣠⣤⣶⠾⠛⢋⣉⣤⡶⠞⠛⣉⣡⣤⣶⡾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢰⣿⣇⣀⣤⣴⣶⠿⠛⣋⣩⣴⠶⠛⣛⣩⣤⣶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⠙⠛⠿⣿⣶⣾⣿⣿⠿⢿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⢀⣿⣿⣦⣀⡀⠀⠀⣀⣀⣠⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠉⠛⠿⠿⠿⠿⠿⠟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣦⣀⣀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⣿⣿⠛⠻⠿⢿⣿⣿⣿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠋
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠟⠛⠉
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⠋⠉
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⠋⠉
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⣶⣿⠿⠟⠋⠉
⠈⠻⣷⣄⣀⣀⣀⣀⣤⣤⣾⡿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠂⢌⢕⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣣⢊⠔⠕⣱⠋
⠀⠀⠀⡠⠀⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠞⠕⡃⣢⠞⠁
⠀⠀⠀⡌⢉⣭⣉⣐⣒⣩⡁⣒⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⠤⠤⢠⠤⣀⣀⣀⡀⠀⣴⠕⡡⠋⣮⠞⠁
⠀⠀⠀⢹⠌⠛⠛⠛⠋⠽⠿⠻⣿⣶⣶⣿⣦⣄⠐⣀⣠⢤⣒⡭⣥⢤⢶⡶⠶⣶⠶⢖⣲⣖⠠⠬⡵⢃⢊⢖⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠚⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠁⠀⠛⢚⣠⠖⢅⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠴⠶⠶⠖⢒⣒⣚⣛⣛⣋⣛⢛⣓⣒⣒⣒⣒⣒⣒⣒⠒⠢⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⠟⢉⣉⣉⣉⣉⠙⠛⠛⢛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⠛⡿⢹⣿⠋⣽⠃⣽⠃⣿⡿⢡⣿⢳⡏⠀⠀⠁⠉⢉⣉⣁⡉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠘⣇⠘⢛⠇⠏⡇⠏⡆⠹⠃⣿⣿⢸⡇⠀⠀⠀⠀⠀⠠⠤⠤⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣿⣶⣿⣤⣿⣤⣴⣦⣼⣧⣴⣷⣼⣿⢸⣇⠀⠠⠄⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣼⣿⣿⣙⣛⣛⣛⣿⡿⠿⠿⠿⠿⠿⠿⣿⣮⣿⡀⢈⣉⣉⣉⣭⣭⣭⣭⣭⣭⡷
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠛⠛⠉⠉⠭⠍⠁⢀⣀⣀⣋⣉⠉⠛⠛⠛⠻⠿⠿⠛⠻⣿⠿⠿⠟⠛⢛⣛⣛⣋⡉⠩⠭⠉⠙⣛⣓⣲⣼⣯⡇
⠀⠀⠀⠀⠀⠀⢀⡟⣼⡋⠛⠛⠛⠛⠒⠶⠶⠶⠦⢬⣭⣭⣤⣈⣉⣀⣓⠒⠀⠀⠀⠒⣚⣛⣉⣭⣭⡭⠷⠶⠖⠚⠛⠛⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣇⢿⡤⠍⠀⠀⠉⠉⣀⣐⠒⠒⠀⠀⠀⠀⠨⠭⠭⠉⣉⣹⠟⠛⠉⠉⠉⠙⠒⠒⠂⢀⣀⣀⠀⠀⠀⠀⠈⣡⣀⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣌⣉⣉⣓⠒⠒⠠⠤⠤⢤⣍⣉⣠⣤⡶⠶⠦⣴⣆⡠⡟⠁⠉⠁⠀⠀⠀⣀⣉⠀⠤⠤⠀⠒⢒⣈⣉⣩⣤⣤⣶⡾⠿⣿⣟⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣶⣶⣦⣾⠁⠀⣲⢶⣖⡒⠬⣙⡻⢦⣔⡂⢉⣉⣡⣤⣤⣶⣶⡾⢿⣟⣛⣯⣭⣭⣷⡶⠶⠟⠛⠛⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⢻⡿⢻⡿⣿⣿⣿⠃⠎⠀⠈⠛⠈⢙⠓⠶⣬⣓⠾⣝⡻⠿⣿⣿⣿⣿⡷⠿⠿⠛⠛⠉⠉⠀⠐⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠘⢣⡌⣡⢿⣿⠇⡜⠀⠀⡀⢀⠀⠀⠀⠁⠒⠭⣛⠶⣭⣛⠮⣝⡻⢿⣅⠀⠀⠀⠀⠀⡀⠀⠈⠀⠒⠀⠄⣈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣷⣿⣷⣿⣿⡏⡰⠡⣼⢷⡿⣿⢰⡴⣆⡤⠀⠀⠀⠉⠒⢬⠟⣶⠹⢷⠉⣿⠀⢐⡈⠅⢀⣀⡤⠤⠖⣛⣭⣭⣶⠟⠃
⠀⠀⠀⠀⠀⢀⣠⣤⣴⡾⠛⠛⠻⠿⠿⣿⡟⢠⠁⠀⠈⠁⠀⠀⠀⠈⠁⠷⢂⡀⠀⠀⠀⠊⣼⠃⣀⠎⣼⣧⡼⢷⣚⣯⣭⣶⠶⠾⠛⢋⣉⣤⣿⣶⣤⣤⣀
⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⡿⢀⠃⠀⠀⠀⠷⢿⣴⢂⡤⢀⠀⠛⣵⡃⠀⠀⣰⠇⢸⡟⣸⣿⣷⠿⠟⠛⢋⣉⣤⣤⡶⣾⣿⣽⣿⡷⠿⣟⣿⡿
⣸⣿⣯⣭⣍⣙⡛⠛⠿⠿⣿⣿⣿⣿⣿⠁⡜⡒⠄⠒⠲⠄⢠⣄⡉⠳⢿⠖⠀⠉⠁⠈⢠⡟⠀⡼⣰⣟⣁⣠⣤⣶⣾⣿⣯⣽⣿⡿⠿⣛⣩⣵⠶⠛⢻⡏
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣭⣿⠃⡞⠹⣷⣄⠀⣀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⢀⡟⠀⣸⢡⣿⣿⣿⣿⣿⣿⡿⠟⣛⣭⣵⠶⠟⡋⠉⢀⣀⡭⠾⣛⣷
⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡴⠁⠀⠙⠻⢿⣿⣧⣠⣤⣤⠀⠐⢦⠀⢀⠀⡾⠁⢰⢇⣿⣟⣻⢿⣿⣩⣴⠾⠛⠋⠁⢀⣀⡤⠔⣚⣩⣵⠶⠛⠉
⠀⠀⠀⠀⠈⠉⠉⣻⢿⡿⠿⣿⡏⡸⠁⠀⠀⠂⢤⣀⣸⣯⠻⠿⣿⣤⣤⣀⠇⠄⡼⠁⢠⢃⣾⣿⣶⣬⣿⣿⠅⢀⣀⡤⠔⢚⣩⣤⣶⣿⡋⠁
⠀⠀⠀⢴⣶⣾⣿⣿⣼⣏⣱⡿⢠⠁⠀⠀⢤⡀⠈⠙⠿⢷⢤⣀⠈⠉⠉⠁⡘⣸⠃⢀⡏⣼⣿⣿⣿⣿⣿⣿⣚⣩⣥⣶⡿⠟⠋⠡⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⠁⠇⠀⠀⠀⠰⢌⡘⠲⢤⡀⠀⠈⠁⠀⠀⢠⢱⡏⠀⡼⢱⣿⣙⠛⠛⣻⣿⡿⠿⠛⠉⠁⠄⢚⡀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⡀⠀⠀⠀⠠⣌⡑⠂⠭⠑⠆⠀⠀⢀⢆⡟⠀⢰⢣⣿⣏⠉⠉⣻⣉⠡⠄⠂⢀⠤⠒⣋⠥⠖⣚⣩⣼⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁⢀⡏⣾⡿⠤⣈⣀⢘⡖⢀⣀⠤⠖⢊⣁⣤⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣷⣦⡴⠶⠶⠶⣤⣀⠀⣸⠃⠀⡜⣼⣿⣶⣦⣤⣀⡉⠉⣉⣤⣶⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠶⠛⠛⠋⠉⠀⠀⠀⠀⠀⡀⢀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⢀⣴⠞⠛⠋⠉⠁⠀⠀⠀⢀⡀⣄⠈⡷⠼⠦⠷⢽⠟⠛⠊⠁⠀⠀⠉⠛⠿⣿⣿⣿⠿⠛⠉⠁
⠀⢰⡿⠡⣄⠒⢄⠀⠀⠰⣄⣳⠬⠗⠊⠉⠀⠀⠀⡀⠠⣄⠠⣄⢳⣬⡷⠤⠀⠀⠀⠙⠻⢶⣤⣀
⠀⣾⠇⣤⠈⠑⢦⡀⠀⢄⠀⠀⠀⠀⣀⠀⢦⣘⡶⠼⠗⠛⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄
⠀⢿⡆⠿⢟⣠⡄⡈⠲⢄⠀⠀⡀⠀⠹⠓⠚⠀⠀⠀⠀⠀⡠⡤⢔⡒⢪⡭⠶⠖⠚⠋⠁⠀⠀⠀⠀⠈⡙⠷⣄
⠀⠈⠻⣦⡘⠉⢾⣇⡆⠀⠙⠢⡈⠒⣄⠀⠀⠀⠀⠀⠠⠤⠐⠒⢉⡥⠴⠖⠛⠀⢀⣀⣠⠤⠴⠒⣋⣩⣥⣴⠟
⠀⠀⠀⠈⠻⣶⣄⠙⠃⢰⢀⡄⠈⠓⢤⡀⠢⣀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⢒⣈⣩⣤⠶⠖⠛⠋⡉⢥⣾⠃
⠀⠀⠀⠀⠀⠀⠙⠷⣤⡈⠙⠻⣷⣆⡄⠈⠲⣄⠀⡀⠀⣀⣉⣥⡴⠶⠞⠛⠉⠉⠀⠀⠀⠉⠠⠤⠐⠚⡿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠈⠻⣇⠀⡀⠀⡸⢁⣾⣋⡁⠠⠄⠐⠒⠈⠄⠐⠀⠀⢐⣂⡠⠤⠒⠒⣃⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠛⢠⠇⣼⣯⠄⠰⠂⠀⠀⠀⡄⠤⠖⢚⣋⣡⣤⣶⡾⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⣸⠀⢿⡧⠤⠄⢚⣁⣭⣤⡶⠾⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡴⠾⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⡇⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡶⣶⣿⣿⣿⣿⠿⠋⢸⡏⠻⣿⣿⣿⣳⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⣶⡿⢻⣏⣩⣿⡿⠿⣛⣫⠽⠚⠉⠀⠀⡀⢘⡇⠀⠀⠙⠿⣿⣿⣟⣿⠭⣿⣿⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠛⠋⣩⡥⠴⠒⠋⠉⠀⠀⠀⣠⠴⠊⢡⠐⡇⠀⠀⠀⠀⠀⠉⠛⠯⣭⣓⣲⠯⠭⢽⣟⠛⡟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠀⠀⣿⠀⠀⠀⢀⣠⠤⠒⠉⠀⢀⠂⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠀⢺⠀⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢹⠀⠀⢹⠀⠀⠈⠀⠠⠄⣀⣀⣠⣋⣀⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢺⠀⠀⢸⠀⠀⠸⠀⠀⠀⣀⠜⠁⠀⢀⠜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡄⠀⢸⡇⠀⠀⡆⠀⠈⠁⢀⡤⠒⠁⠀⡀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡇⠀⠸⡇⠀⠀⠱⠐⠒⠉⠁⠀⣀⠴⠊⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⡇⠀⠀⡇⠀⠀⠀⠀⢀⠄⠔⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡇⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠖⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣯⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⢸⠀⠀⠀⠠⠴⠉⠁⠀⠀⠀⡤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⢀⡽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣧⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠸⡇⠀⠀⠀⡠⠴⠒⠃⠀⠀⣠⠟⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⠀⠀⣇⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⢸⠀⢸⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣤⠤⣿⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣸⣤⣼⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠶⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠍⠙⠛⣛⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠁⠒⠒⠒⠒⢶⠄⠀⠀⠈⠉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣌⣁⣈⣁⠶⠂⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠤⢀⣀⣀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⡟⣰
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢏⣾⠏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣰⡟
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⢋⣾⠏⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣰⡟
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⢃⣾⠏⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⡟
⠀⣾⣻⠿⣷⣾⣿⣯⣭⣭⣿⣛⣻⣿⠿⠿⠿⣿⡿⣱⣿⢃⣾⠏⠀⠀⠀⠀⠀⢸⢛⡶⢶⣶⡦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣾⠃⣰⡟
⠀⣿⡿⠈⠀⠐⢚⡉⠉⠉⠙⠛⢛⡻⠿⠿⢿⢷⣶⣿⢃⣾⠏⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡉⠉⠉⠉⠘⠛⡋⠿⠟⠿⠖⢲⣾⠃⣰⡟
⠀⢿⣷⣤⣤⣤⣤⣀⣀⣉⠠⠀⠀⠀⠀⣉⡀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢸⣜⣦⣤⣄⣠⣀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠿⠾⣶⣶⣶⣮⣥⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠒⠿⠯⠴⣶⣔⣖⣂⣡⢼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆⠀⠀⠀⣀⠀⠀⠠⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣍⣘⠲⢶⣁⣟⢃⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣇⡄⠀⡀⠸⢀⣞⢉⣀⠀⠐⠀⢨⣦⠉⢿⣦⣿⣀⣻⠀⢿⡄⢠⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣸⣿⡆⠀⠄⠸⣾⣼⣜⣿⣧⠀⠀⠈⢻⠳⠬⢿⣭⠟⠷⠛⠁⡰⠞⣲⡛⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣽⣿⣿⡄⠈⣦⢽⣟⢿⣿⣃⠀⠀⠀⢸⣷⡛⣇⠀⠓⢰⢭⡄⠀⠀⣛⢲⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡘⠁⠺⠓⣽⣿⣦⡈⠓⠈⠙⠛⠛⠀⣀⠴⠛⠛⠛⢯⡛⠷⣶⢨⡀⢀⠀⢛⡤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠞⣡⠿⠿⠉⢲⠦⠤⠘⠂⠉⠁⠀⠀⠀⠀⠀⠘⠛⠻⣌⠳⡘⡵⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⡷⢶⠞⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠳⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠄⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣦⠀⠀⠀⡄⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣶⠀⠀⣠⠎⠀⠀⠀⡤⣴⡒⢲⣄⠷⠀⠀⠉⠂⠀⢠⡖⣻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠤⣉⡉⠙⠀⠀⠀⠀⠀⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣶⣶⢶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠃⠀⠀⠀⡼⠛⡠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡃⠠⠇⢈⢷⡀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢤⡔⣻⡿⣝⣶⠄⠘⠀⠈⠠⣤⣀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠛⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡁⢨⠌⣯⡙⠒⠠⢶⡄⠀⠈⢻⡁⠰⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⠛⣧⠙⠻⣥⠴⠞⣡⠂⣀⣀⠀⠀⠁⠉⠉⠛⠻⠦⣤⡀⣀⣀⣤⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⢀⠁⠖⠒⠎⠠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡀⢣⠆⣤⡀⠿⠀⠢⠄⠈⠱⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠆⠁⢠⡏⣰⣶⠦⡴⢤⠦⠀⣤⠀⠀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡄⢸⡇⠠⠄⠈⠙⠄⣀⣈⠟⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⣹⠃⢠⡄⠰⠚⢤⠿⣭⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣟⣛⣻⣻⣛⣛⣿

⣤⣤⣤⣤⣤⣤⣤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠛⠶⠶⠛⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣾⣿⣽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣭⣤⣤⣤⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⣿⣿⣿⣿⣿⣿⣯⣩⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠁⠀⠀⠈⠙⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢰⣿⡏⠀⢀⡔⠑⢤⣀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠘⣿⣇⠀⢈⠀⠀⠀⠙⢠⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠹⣿⣦⣄⣉⣓⣊⣴⣿⠟⠁⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣶⣄⡈⠙⠛⠿⢿⠟⠛⠁⣀⣠⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣶⡆⢸⣶⣶⣿⣿⣿⣿⡿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⡹⠿⠛⠛⠚⠉⠉⠙⢏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣴⡇⠀⠀⠀⣤⣤⣤⣤⣶⡀⠀⠀⠀⠐⢄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⢍⣀⡠⠤⣾⠁⠀⢀⠈⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠿⣿⣶⣄⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠔⠁⠀⣀⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⢀⡾⠿⠿⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⡿⠁⠀⠀⠀⠀⠈⠙⠓⠆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⢁⣴⠾⠟⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣥⠟⠁⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠁⣨⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠐⠿⠜⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⢭⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣠⣾⡫⣦⣽⣾⣿⣧⠴⡖⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣠⡾⣿⣛⣷⣿⠿⢻⡅⠲⠀⠉⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⢀⡾⣫⢻⣾⠿⢫⡅⠛⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⣰⢏⣾⣿⠟⡥⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸⠀⠀⢀⡼⣳⣿⢟⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢸⠀⢀⡾⣱⡟⠱⠈⠀⠀⠀⠀⣠⣤⣴⣶⣤⣤⣀⠀⠀⠀⣾⣭⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣄⣼⣳⢏⠘⠂⠀⠀⠀⢠⣾⠟⠉⢻⣿⡏⠻⣿⣷⣄⣰⣿⣷⢾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⠾⠏⡏⠈⠀⠀⠀⠀⠀⠈⠿⣧⣤⣾⡿⠃⠀⢹⣿⣿⣿⣿⣿⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⠦⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣟⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⢿⣷⠂⢨⣳⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣏⡽⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠟⣿⣿⣯⡳⣯⣻⢿⣷⠲⣾⢿⣿⣿⣿⣿⡿⣿⣹⡷⣫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡉⠛⠾⣭⣿⣒⣻⣾⣷⣚⣳⣯⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⠿⠿⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢠⣿⡅⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⣽⡖⠲⠀⠀⠀⠈⠉⠀⠀⣄⠙⠍⠢⣄⠀⠂⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⣿⣿⣶⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⡄⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣪⣷⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠘⢓⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠂⠠⠂⠀⠠⠀⠀⠀⢈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣹⣿⣿⣿⣿⣿⣿⡟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠿⣿⣿⣿⡇⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠤⠴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡬⠀⠀⠀⠈⠉⠉⢀⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠞⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣧⣤⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣄⡤⠴⡛⠛⠉⠉⠁⠀⠀⠀⠠⣤⢼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣹⡿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣴⣿⣿⣿⠿⢷⠄⠐⠊⠩⠁⠀⢀⣦⠤⠀⠀⠀⠀⠀⡤⠀⠹⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠑⠀⠀⠉⠁⣤⢰⡉⠛⠿⠛⠋⠉⢄⠀⠀⡏⠁⠀⠀⠀⠀⠀⠈⢿⣶⡟⠀⠀⢀⡴⠃⠈⣬⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣦⡰⣧⠀⢠⠉⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠎⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠻⠉⠿⡆⠹⣾⣿⣧⢿⣇⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠐⠀⠄
⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠘⠉⠀⠻⢿⣽⡌⠻⠈⢁⡀⠰⠀⠀⠀⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠃⠤⠀⢤⡄⣠⡔⠂⣤⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠙⢣⡾⠆⣀⠤⣴⣆⠂⠀⠀⠀⠀⠀⢀⡀⠦⠀⢀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⣾⣇⣀⠀⣴⠆⠀⠈⠉⠠⠀⠄⠐⣦⠀⠻⠟⠘⠟⣧⣶⡆⠈⣰⣦⠦⢺⠁⡀⠿⣲⡏
⠀⠀⠀⠀⠀⢀⣾⡿⠋⢰⣿⣿⠟⠡⣴⡿⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠀⡾⠛⠖⠃⠈⢡⣆⠓⠳⠚⡏⠀⠉⠓⠀⠁
⠀⠀⠀⠀⢠⣾⡏⠀⠀⣸⣿⠏⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠘⠉⠀⠀⡀
⠀⠀⠀⢠⣿⡟⠀⠀⢰⣿⣿⠆⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⣼⣿⣵⣣⢀⣿⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢠⣿⣿⣿⣿⢰⣿⣿⣿⡀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠇
⣷⣄⢸⣿⣿⣿⣿⠀⣿⣿⣿⣧⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢀⢉⣿⣦
⠹⣿⣧⢻⣿⣿⣿⣦⣿⣿⣿⣵⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⡀⢀⠀⣀⠀⡀⢀⠀⣀⡀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢸⣻⡿⣿⣸⣿⣄⡟⣾⢺⣉⡁⣇⣾⣷⣸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤⡀⣠⣄⢀⣤⡀⣀⢀⡄⣄⢠⢠⣤⢄⣀⡄⢠⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠼⢤⠜⠧⠤⠻⢤⠟⠻⡼⠇⠟⠿⠀⠇⠸⠿⠟⠿⠻⢼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡍⣴⡌⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠃⠻⠶⣍⡵⠾⠃⠚⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠤⠤⠤⠍⠁⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠦⠶⢐⠀⣛⣃⣈⣈⡁⢀⣤⡴⠻⠁⢸⠷⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⣿⠀⠖⠒⠲⠂⠀⠛⠶⠖⠒⠖⢸⡇⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⢰⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣳⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣃⣀⣀⣀⣀⠀⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣀⠀⠀⢀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⡭⠭⠤⢤⡉⠉⢹⠀⠀⠀⠀⠀⠀⡏⣭⠭⠭⠭⠤⠥⠬⣭⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠌⢛⣛⣛⡛⠁⣠⠞⣿⠳⣄⠀⠀⠀⡇⢿⣀⣘⣛⣀⣀⣃⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣛⡦⠐⣶⣶⠖⠀⣇⠀⠹⣖⣿⠀⠀⠀⡇⠠⠤⠀⠠⠤⠀⠠⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠬⠤⠬⠭⠭⠤⠤⠌⢳⠶⠞⠁⠀⠀⠀⡇⣶⣲⣲⣶⣖⡆⣶⢲⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡛⢛⣛⣛⣛⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⡇⣿⢿⢿⣿⡿⡇⣿⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠐⠖⠖⡀⡀⠶⠖⢸⠀⠀⠀⠀⠀⡀⡇⣩⣭⡍⣩⣭⡅⣿⢻⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣁⣈⣉⣉⣉⣁⣀⣀⣸⣝⠲⠶⠶⠚⠃⣇⣛⣒⣃⣙⣒⣃⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⡦⠄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⡀⠀⠀⠀⠀⠀⠙⠿⠻⢿⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⣰⢷⢭⣣⡄⠀⠀⠻⢆⡀⠉⣁⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠯⠴⠒⢈⣡⠔⣱⢯⣻⣻⣧⣭⠁⠄⣀⠀⠉⡉⠀⠊⢲⡔⢤⡢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠠⠌⠾⢟⣿⢿⢿⠿⣽⢷⣎⠀⡇⢀⢥⣀⣠⠉⠀⠈⢉⡴⠫⡍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⣞⣽⣷⣶⠀⡦⣿⢛⡟⣾⣻⣄⠋⢀⡀⡏⠒⠬⠛⢃⠀⢹⠛⠂⡊⡀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣮⢿⣾⣿⣾⡟⢻⣮⢋⠈⣟⣡⠔⡻⠧⠈⣷⠤⣄⢤⡄⠁⠀⠀⣄⠾⠅⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣌⣟⣽⣾⣧⡗⢹⡯⢆⠴⠏⠐⠂⠀⠣⣀⡎⢒⢫⣉⣿⣠⠀⢆⣹⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⢳⣿⣿⣛⣿⢏⠴⣁⡤⠒⠉⠉⠉⠓⣾⡃⢨⠼⣬⡿⠋⡓⢼⡛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡻⢷⡼⣯⢻⣿⣽⡿⣿⡇⢀⠀⠀⠀⠀⠸⣿⢟⠮⣈⡷⣣⣼⡿⢦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣷⢿⣭⣳⣭⣹⣎⣽⣿⠟⠀⠀⠀⠀⠀⠀⢠⡇⣦⢣⣝⣶⣧⣼⡹⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⡿⠿⣿⢽⣿⠃⠀⠀⠀⠀⠀⠀⣰⡿⠻⡆⢱⣍⡹⡧⠒⠸⣷⡊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣌⡉⠉⠿⡧⣻⡏⠀⠀⠀⠀⠀⠀⠛⠋⢁⣞⢱⣿⣌⣻⡇⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢥⡧⢄⡇⣿⡇⠀⠀⠀⠀⠨⠂⠀⠰⠿⣿⣦⣟⢪⢋⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡹⠛⢧⠀⠀⠀⠐⠀⠀⠀⠀⢸⣾⢿⣿⡧⢐⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀⣯⡶⠥⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⡟⡙⣦⣤⣀⣀⣀⣴⣷⣤⡔⢁⢻⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢾⣦⠬⢿⣤⡋⠤⣾⡓⢛⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⢀⡜⠉⡹⠃⠢⣄⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠬⠡⠔⠛⠦⢠⠨⠾⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⢹⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣬⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣝⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣦⣭⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⣜⣸⣿⣍⣉⣉⣉⣉⣉⣩⣭⣽⣿⣿⣿⣯⣩⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢻⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⢱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠮⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠿⢸⣿⣭⣭⣭⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⢿⣿⡿⠿⠿⣿⣿⠿⠿⣿⣿⡿⠿⢿⣿⡇⣿⣿⣛⣛⣛⣛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠾⠿⠆⣿⡰⠿⠷⢸⡇⠿⠿⢸⣿⠸⠿⠇⣿⡇⣿⣿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⣛⡻⣿⠟⣛⣛⢿⡟⣛⣛⢻⣿⢛⣛⡛⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣛⣛⣃⣿⣜⣛⣛⣸⣇⣛⣛⣸⣿⢸⣿⡇⣿⡇⣿⣿⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣩⣭⡙⣿⢋⣭⣍⢻⡟⣭⣭⢻⣿⢸⣿⡇⣿⡇⠻⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣭⣭⣥⣿⣮⣭⣭⣼⣧⣭⣭⣼⣿⣬⣭⣥⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⡆⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢳⢀⠀⣉⣀⣀⣀⣆⢠⠀⡄
⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢄⣳⣾⣿⣿⣿⣿⣿⣿⣾⡇⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⢁⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣾⣿⣟⢻⣿⡄⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠈⡸⢿⣿⣻⣿⣿⣻⣿⣿⣿⡟⠑⠡⣐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠈⠿⠛⡿⠿⡿⠛⠋⠡⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠢⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⡇⠠⠄⠀⢀⣀⣤⣤⣤⣤⣶⣤⣤⣄⣀⠀⠀⠠⠄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⣠⣶⣿⣿⣿⣿⣿⣿⣽⠿⢿⣿⣿⣿⣶⣄⠠⡴⣺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⢿⢃⣾⣿⣿⣿⣿⡛⢿⣿⣿⠃⢔⣛⡝⣿⣿⣿⣿⣷⡘⣿⡇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢸⣿⣿⣿⣿⣿⣿⣿⡋⠙⠳⣾⣿⣸⣿⣿⣿⣿⣿⣇⢻⢯⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⡀⠀⢸⣿⣿⣿⣿⣿⣜⢿⡿⢠⣤⣿⡿⣫⣿⣿⣿⣿⣿⡿⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣼⣯⣵⣾⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣦⣦⣧⢻⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⢰⢢⣍⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⣿⠿⠿⣦⡙⠟⣉⡐⠲⠄⠀⣀⣉⠁⠀⠠⣍⣛⡿⠟⣰⢏⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⣾⣠⣅⠠⠄⠀⠀⠈⠁⠐⠒⠀⠍⠛⠛⠂⠋⠉⠀⠀⠉⠁⠉⢀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⣇⠘⢻⣿⣿⠲⢦⣇⣐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⢀⣤⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠠⠍⠻⣥⣤⣈⣉⣙⡓⠒⠒⠀⠀⠐⠒⠚⠛⣉⣉⣴⣶⡿⠋⠄⠀⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠂⠀⣠⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⠿⢛⣿⣿⣿⣿⣿⣷⡄⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠸⢿⣿⢿⣿⣿⠷⠔⠲⠢⠬⠋⠐⠂⠀⢿⣿⣿⣿⣿⡿⠇⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣆⠀⠀⢿⠋⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠋⠋⢿⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⢦⣶⣠⣦⠀⡏
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠉⠉⠈⢠⠇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠐⠒⠒⠒⠒⠂⣶⡶⠊⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠂⢀⡀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⣄⠀⢦⠀⠀⡤⡄⢀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠂⠒⠒⠒⠒⠒⠂⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠀⠀⠀⠀⡀
⠸⣼⠁⠸⡇⠀⢰⠆⠈⠃⠂⠀⠆⡄⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡆⣠⡾⠂⠀⠁⠀⠰⠁⣼⡿⠁⠄⢉⣼⠟
⠀⢿⠀⠀⠰⠦⠀⠀⠆⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⡟⣡⣤⡄⠀⡐⠀⠀⡿⠀⢠⠚⣿⠏
⠀⢀⣠⠞⠀⠀⠀⡄⣠⠞⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠙⠊⠁⡐⠁⣠⢀⠁⠀⠸⠛⣉⡴⡆
⢀⡏⠀⠀⠀⠀⠀⡯⠁⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠰⣀⡀⠀⠻⣄⣼
⠈⠁⠀⠀⠀⠀⡘⠁⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠦⣤⠆⣈⡱⠀⠶⠘⠉⠀⠀⠈⠀⠢⠀⢿⠿
⠀⢦⣄⣄⣀⣤⣤⡶⠾⠋⠁⠀⠂⠒⠀⠈⠈⣈⣴⡀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠈⣉⣭⣍⠀⠀⠀⠀⠠⠠⠇⠀⠀⢀⠈⢿⣀⡇⠀⢀⠀⠤⠄⠂⠂⠁⠀⠈⠃⠀⠀⠀⠀⠲⠶⠶⠶⠆⢠⡀⢆⠀⠀⠁
⢠⣶⣿⣿⡝⣿⢦⡀⠡⣤⠠⠄⢐⣈⠀⠐⠟⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⠀⠀⠨⢻⡄⠀⡀
⠀⢿⣷⣻⣿⡈⢻⣿⣖⠬⣑⣚⠿⠛⠀⠀⠀⠀⣷⣶⠀⠀⣀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁
⠀⠈⣿⣷⢿⣧⠀⢙⠋⢲⠈⠀⠀⢠⣦⠀⠀⠀⠉⠀⠀⠀⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⠟⠊⠩⣂⣥⣾⠈⠀⠀⠀⣞⣘⡇⢀⡀⢀⡄⢀⣤⡀⡀⢠⣠⣠⡔⢻⢳⢠⣠⡄⣠⢦⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⠰
⠀⠀⡀⣀⣴⣾⣿⣿⣿⢸⠀⠀⠸⠋⠉⠹⠘⠴⠛⠶⠘⠶⠋⠳⠞⠘⠇⠻⠸⠘⠺⠃⠟⠳⣾⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⡄⣿⣿⣿⣿⣿⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠵⠃⠀⠀⠀⣀⡀⠀⠐⣤⠌⡞
⠀⠀⠀⣿⣿⠟⠋⠀⠀⠉⡱⠂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠦⠤⠃⠀⢀⡔⠀⠈⢂⠀⠀⠊⠀⠀⡴⠁⠀⠐⣈⠤⠲⡇
⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢀⡀⠀⡀⢢⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠀⠀⢘⡀⠀⠀⠀⡀⠀⠀⢠⠘⢀⡠⠔⣁⢀
⠀⠀⠁⠬⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣊⡡⢦⡼⢻⣥⣀⡀⠀⠈⠀⠚⠋⠀⠈⠓⠤⠔⠚⠁⠀⠀⠀⠣⠀⢐⠊⢠⡀⡀⠀⢀⣴
⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⡤⠖⢉⠈⠁⠀⠀⠁⢾⣿⣿⡏⠊⠓⠒⢲⡆⠀⢠⡀⠀⠀⠀⡿⠈⠀⠀⠀⡄⠈⠉⠉⡀⠰⣾⣿⣿
⠀⠠⠆⠘⠀⢀⠀⠄⠚⠉⠀⠀⠀⠁⠀⢶⣄⠀⠸⣿⣷⣗⢄⣔⢲⣾⣟⡀⣼⡟⠀⢀⣼⣿⠀⠀⠀⠀⠔⠀⠀⠂⠀⢠⣿⠛⠿
⠀⠀⠀⣠⡴⠏⠃⠀⠀⠀⠈⠀⠄⠢⠀⠀⢹⠷⡄⣿⣿⣮⢿⣻⣿⡫⣞⢳⣾⡇⠀⢮⣨⠏⠐⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠋⠁⣺⡶⠆⠀⠀⠀⠀⠀⠈⢻⣦⡀⠓⠃⢹⡿⡁⡀⠀⠉⠙⠋⡛⠇⠑⠀⠘⠁⢠⠀⠀⠀⠀⣀⠀⠀⠀⡿⠋⣤⡤⣶
⠀⠀⠀⠀⠸⣉⣧⢤⣄⠀⠀⠀⠀⢀⣀⣀⣉⡁⣀⢘⢙⡂⡑⣀⣈⣉⣉⡀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⣤⡞
⠀⠀⠀⠀⠀⣤⣹⡤⠋⠀⠀⠀⠄⢸⠐⡇⢸⠀⡇⢰⠂⡟⢰⡆⢸⡇⠀⡇⢸⣿⡆⠀⠀⠸⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⠿⡿
⠀⠀⠀⠀⠀⠈⠨⠁⠀⠀⠐⠀⠀⠸⠶⠷⠞⠐⠳⠼⠦⠗⠘⠧⠘⠦⠴⠇⠈⠉⠀⠀⠐⠀⠀⠀⠰⠂⠀⠀⠀⠀⠠⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⣄⡄⠀⠀⡀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠤⠁⠀⠠⠁⠀⠀⣤⠀⠈⠃⠀⠀⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠰⠀⠀⠃⠒⠀⠀⠀⠀⠀⢀⡽⠂⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢧
⠀⠀⠀⠀⠀⣴⠀⠀⠀⢰⡁⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⣠⣧⡀⠠⠀⠀⣴⠀⠀⠁⢘⡇⠀⠀⢀⡀
⠀⠀⠀⢧⣀⡄⢠⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣀⠄⠀⠀⠀⠀⠀⠘⢨⣿⠀⠀⢸⡉
⠀⠀⠀⠙⠛⡙⢿⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠎⠇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡼⡀⠀⠈⠁⣠⠀⠀⠀⠀⠀⠀⠀⣸⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢃⣰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠫⠄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡠
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⢀⠀⠀⠀⠀⠀⠀⠠⣠⡇⠡⠀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⡀⠄⠀⠀⠀⠀⠈⠛⡏⢛⠀⠄⠁⡞⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⠁⠀⣀⠁⠀⠸⢆⠀⠀⠀⠀⠀⣀⢹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣄⠐⠔⠇⠀⣀⠀⢀⠘⡄⠀⠀⠀⠀⠘⠃⠡⡠⠠⢀⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠙⠰⠀⠁⠀⠈⠶⣌⢣⡱⡀⠀⠀⠀⠙⠂⠀⠳⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠸⠀⠀⢈⠀⢶⣤⠈⠉⠛⢧⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡧⢂⠓⠂⠙⠟⠂⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣴⣿⡀⢀⡖⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣴⣾⡿⣽⣿⣿⣿⣿⣿⣿⡿⣷⡞⠫⡏⠉⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⣇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠥⠀⡀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⠶⠚⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠠⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⢀⣀⣤⣴⡖⡀⠀⠀⠀⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣤⣤⣶⣶⣿⣿⣿⣿⠟⢩⣗⠠⠀⢰⣷⠈⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢠⣀⣿⠀⠀⠋⢀⣤⠀⣀⣍⠴⣍⡛⢿⣿⣿⡿⡟⢿⠿⢿⣿⠛⠍⠀⣼⣿⡀⠒⣸⣿⠀⣤⠆⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣼⡿⠞⠀⢈⠗⠲⣂⠄⢈⡅⠉⢁⡀⠆⣉⡩⠛⠉⠁⢀⠐⠂⡄⠑⠐⠭⣢⡀⡀⢨⠑⠣⠛⠻⢇⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠊⠀⠀⠠⠖⠧⢀⡄⠀⠀⡀⠀⠁⠘⠋⠈⠁⠀⡀⠄⠂⠙⡀⢰⣷⠀⠠⡀⠀⠀⠈⠻⠗⡤⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⢀⠂⠠⢸⣇⣾⠀⠈⠁⠀⠀⠀⢠⡆⠈⢀⠀⠂⢠⣇⣸⣿⡀⣀⣀⠄⢀⠅⠀⠁⢠⡀⠀⢀
⠀⠀⡆⠀⠐⠀⠀⠀⠤⣀⣀⠀⢀⣠⣤⡤⣬⣍⠀⠀⠴⠄⠀⣠⡄⣺⣇⢒⠶⠖⠠⡚⠛⠙⠉⠉⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠁⠨⠑⠠⠄⠉⠀⢀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠆⠨⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠘⢿⣿⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡰⢠⢤⣄⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⣤⣸⣿⣯⣴⣿⣧⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡄⢒⡀⣀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⡿⠿⠿⠛⠛⠛⠷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⡟⠂⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⡇⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⣛⡃⠀⠀⠤⠄⣹⣿⣏⣹⣿⣷⣦⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⣷⠀⠀⠰⢾⡗⠿⠛⢶⣶⢂⣄⡀⠀⠀⢠⠈⡄⠀⢻⣿⠀⠀⠀⠙⠛⠻⣿⠧⠜⠿⠃⠘⢻⣽⣯⠈⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠀⣿⡀⠀⠀⠀⠀⠀⣀⡀⠉⠑⠈⠁⠀⠀⢨⠈⡁⠀⢸⣇⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠠⠂⠈⠻⠓⠿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣆⢸⣷⣶⣆⠒⠀⢈⣭⣿⡁⠀⠀⠀⠀⠀⠸⡔⢰⡀⠐⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣅⣶⣶⡄⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠹⣾⣿⣿⣷⣷⡿⢯⣉⠁⠒⠀⠀⠀⠀⠀⢘⡚⠾⣇⠀⣿⣿⣿⣿⢶⡚⢼⠗⠀⠀⣀⣠⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠹⣿⣿⣿⣿⣷⣤⣿⡁⠀⠀⠀⠀⠀⠀⠰⣂⠏⢻⣶⣾⣿⣿⣿⣿⣇⣯⣤⣶⣾⠿⠿⠿⠄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣜⢿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠠⢾⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣷⣶⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⣄⠀⠘⣿⣆⢻⣿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢾⣆⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣆⣨⣽⡟⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠉⠻⣧⢿⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⠹⣿⣿⣌⢻⣿⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣧⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⡿⣿⣷⡆⢻⣿⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⡂⠀⠀⠈⠽⢿⣿⡄⢻⣿⠀⠀⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠺⣧⡈⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠇⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠡⡠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⣄⢳⡀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠂⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣦⠀⡄⣿⡆⠀⠀⢀⡀⣀⣀⠀⣀⡴⠞⠉
⠤⠤⢭⣷⣶⣶⣶⣤⣀⠀⠀⠀⠹⣿⣯⣷⣺⠄⠘⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇⢣⣿⡇⠀⠀⠈⣻⢏⡩⠟⢉⠴⠂
⠀⠀⠀⠈⠉⠙⠻⢿⣾⣿⣦⣄⠀⠘⢿⣿⣿⢣⠀⣇⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣿⣸⣿⡃⣠⣤⢟⡵⢟⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⡈⢿⣿⣿⣧⣸⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⣸⣿⣿⣷⡿⣻⢟⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⡸⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⡔⠁⠀⠀⠀⠀⠀⢸⣿⣿⣯⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣿⡇⠀⠠⢆⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⡠⢚⣭⠀⠀⣯⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢱⠁⢸⣿⠀⠀⣿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⢸⡀⠸⠿⠀⠀⠿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⢞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⢈⣵⢟⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣛⣛⣛⣛⣛⣛⣛⣛⢻⠀⠀⠀⠈⠁⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣛⣛⣛⣛⣛⣓⣒⣛⣸⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣭⡍⣭⡍⢭⡍⢩⣯⢹⠀⣠⣴⢶⣤⡀⠀⠀⠙⢿⣿⣿⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽⡅⣼⣧⣼⣧⢸⣧⠘⣼⣿⠰⠶⢿⢿⣀⣀⡀⠀⠙⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣶⡆⣶⡦⣴⣦⢺⣿⢰⠹⣿⣿⡋⣰⣿⢋⣙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⠶⠶⠶⠶⣶⣶⣾⠀⠈⠉⠁⣿⣿⠾⠯⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠀⠀⠀⠀⠈⠻⠷⠾⠟⠁




⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿

⠀⢠⡤⠤⠤⢤⣤⠠⠄⠠⠤⠤⠤⠤⠠⠤⠤⠠⠤⠄⣤⡤⡤⡤⠤⠤⠄⠤⠤⠤⡤⡤⢤⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⢠⣤⢤⠄⠠⣤
⠀⢸⠁⠀⣰⠞⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠄⠁⠁⠂⡀⢀⡀⠠⠀⠐⠀⠀⠑⢄⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢧⡀⠀⢿
⠀⢸⠀⠀⢣⠀⢀⠏⠙⠍⠐⠢⠤⣤⣤⠴⠶⠛⠉⠀⠀⢐⡁⠀⠀⠉⠒⠐⠁⠀⠀⠐⣔⠀⠀⠑⠪⠵⣤⣤⣄⣤⡤⠔⠺⠉⠀⠀⢨⠀⠀⢸
⠀⢸⡇⠀⠸⣧⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣄⣾⠀⠀⢸
⠀⢸⣇⡴⠃⠇⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⠔⢁⡤⡠⣏⡒⠄⠀⠀⠀⠀⠀⠀⠀⠰⣉⠇⣶⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡀⠈⡄⢣⣀⣿
⠀⢸⣉⡱⠞⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡾⠁⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠀⠈⢿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣏⣹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠖⠉⣰⠏⠀⠀⢠⠿⠁⠀⡀⢀⡀⡀⠀⠀⠀⠰⡀⠀⠀⠻⣌⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⢠⡾⠁⠀⠀⣠⢟⠇⠀⢀⣀⣸⣇⡇⠀⠀⠀⠀⢣⠀⠀⠀⠙⣷⡀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⢀⣴⠋⠀⠀⠀⣰⠏⣾⠴⠚⠌⠁⠀⠀⠘⡄⠀⠀⠀⠀⢇⠀⠀⠀⠈⢻⣆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⣰⢏⡜⠁⠀⠀⣀⡤⠄⠒⠒⠛⠆⠠⡄⠀⠈⣇⠀⠀⠀⠀⠙⣿⣄⠀⠈⠙⢦⣄⠀⠀⠀⢸
⠀⢸⠀⣠⣴⠟⠉⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⣴⢃⠎⠐⠀⡠⢚⡡⠀⠀⠀⠀⠀⠐⢆⡸⠀⠀⠘⣆⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠉⠻⢦⣄⣸
⠀⢸⠟⠋⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⢀⡾⠃⢾⠀⠀⢊⡜⠁⠂⠀⠐⠶⠞⠂⠰⡾⠁⠀⠀⠀⠘⣆⢀⠀⠀⠀⠀⠙⢿⡳⣄⠀⠀⠀⠉⢻
⠀⢸⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡴⢁⡾⠁⠀⠘⢄⢸⣈⠘⠿⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⣆⢧⡀⠀⠀⠀⠀⠱⣌⢳⣄⠀⠀⢸
⠀⢸⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⣠⡟⠁⠀⠀⠀⢀⣿⠹⣄⠁⠀⠀⠐⠆⣄⠾⣿⣧⡆⠀⠀⠀⠀⠀⠘⠈⢷⡀⠀⠀⠀⠀⠈⠢⡙⠳⣄⢸
⠀⢸⣄⣴⡟⠁⠀⠀⠀⢀⡼⠃⣰⠟⠀⠀⠀⠀⠀⣸⠇⠀⠘⡏⠀⢄⡀⠈⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠈⠀⢈⣻
⠀⢸⡅⠀⠈⠉⠒⠆⠴⠿⣄⣾⣏⡀⠀⠀⠀⠀⢠⡿⠀⠀⣠⠏⠠⡀⠈⠙⠋⠐⣖⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣆⡀⠤⠄⠒⠚⠉⠁⢹
⠀⢸⠉⠒⠂⠠⠽⠥⣀⡀⢀⣤⠀⠈⠉⢋⠉⠤⠛⠓⣶⠞⠁⡄⠀⠀⠁⠠⡀⣠⠊⣣⣄⠄⠲⢖⠖⠒⡂⠉⠉⠛⣤⠁⠈⣂⢀⡀⠿⠤⠒⢺
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠛⠀⣈⠹⠴⠁⠀⠀⠚⡀⢉⠀⠰⡏⢡⠀⠀⡷⣤⠤⠽⠤⠄⠗⠐⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠐⠀⢠⠋⠀⠀⠸⣿⠀⠶⠛⢳⠆⠤⢳⠈⠂⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠱⢄⠀⢀⠌⠀⡌⠀⠀⠀⠀⠘⣆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠓⣠⠆⠰⠁⠘⠀⠀⢠⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⡀⠀⠀⠃⠀⠀⠈⡖⠀⠀⠀⡄⠀⠀⣄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⢿⠠⠀⠀⠀⢱⠀⠀⠸⠀⠀⢠⠄⠀⠀⠀⡇⠀⠀⡗⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⢼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢆⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢀⠟⣇⠀⠀⠀⠀⠀⠀⠈⠢⢘⠀⠀⠀⠀⠀⠀⡇⢰⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠⣠⡟⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⢀⡷⢸⡇⠀⠀⣼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⡇⢸⠃⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠹⡄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⡇⢸⠂⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢳⡼⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⠀⠀⠀⠀⣗⠈⡆⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣆⠀⠀⠀⠘⡄⢹⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⠸⢆⠀⠀⠀⠀⣠⠎⠀⠀⢀⠄⠉⢃⠀⠀⠀⢡⡞⡄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠂⠀⠀⠈⢿⡒⠊⠉⠀⠈⠙⢦⡌⠀⠀⠈⡄⡠⠞⠁⢻⡐⡇⠀⠀⠀⡀⢳⡀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠈⠀⠀⠀⠈⢇⣻⠀⠀⠀⢀⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⠠⠄⢷⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡀⠀⠀⠀⠈⢿⡀⢚⡀⠰⠄⠈⡅⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠲⣼⣴⣄⠤⠤⠀⠀⠀⠀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣵⣿⡿⣟⢯⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢠⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠰⠠⡀⡀⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⢰⢏⠠⡼⠳⣄⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⣿⣿⣷⣤⡠⠄⠸⣿⣿⣿⣿⣶⣦⡀⢸⣿⣯⣡⣆⣼⡯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠿⠁⠀⠈⡹⣿⣿⣿⣿⣿⡗⠈⢻⣿⣿⣿⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣦⠀⠈⠻⣿⣿⣿⣿⡿⢁⡤⠈⠻⠛⠃⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⢐⣸⣿⣿⡇⠀⠀⠀⢨⣉⠡⠄⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠟⠋⠙⠛⠁⠛⠃⠀⣀⣤⣤⣙⣧⡀⠀⢀⣠⣴⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢀⡀⢀⢀⢀⠻⣿⡿⣿⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠈⠈⣈⢁⣀⡠⠋



⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢒⣒⣒⣒⣒⣒⣒⣒⣒⠂⠀⠀⠀⠈⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣩⣭⣭⣭⣭⣭⣭⣭⣭⡁⢰⠶⡗⠒⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⢈⣹⣏⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⢛⣛⣛⣛⣛⣛⠛⣷⠀⠀⠈⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡉⠉⠉⠉⢹⡇⣿⠐⠒⠒⠒⠒⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢈⣉⠉⣉⠉⣉⡁⣿⢈⣉⣉⣉⣉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠤⠀⠤⠀⠤⠄⣿⠠⠤⠤⠤⠤⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠐⡒⠀⡒⠀⢒⠂⣿⠐⡒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣬⣭⣤⣭⣤⣭⣥⡿⢈⣭⣭⣭⣍⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠢⠤⠤⢆⡀⠂⠀⠀⠀⠀⠂⠐⠢⠤⠤⠤⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠤⣄⣀⣀⠀⠀⠢⢀⠀⠀⠀⠁⢀⣀⣠⢤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠊⠀⠀⠀⠀⠢⡈⢂⠄⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢰⠃⠀⠀⢠⣸⣷⡄⢸⡎⢀⣾⡟⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢘⣇⠀⠀⠈⠻⠟⠋⠰⠆⠸⠛⠟⠃⠀⠀⢠⡇⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⢤⣀⣀⡠⠖⣇⣠⠲⢤⣀⣈⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠤⣄⠀⠰⠀⠄⠠⠈⠀⠁⠁⠐⠀⠀⠀⠐⢀⠠⠊⠑⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠼⠜⠚⠄⠠⠠⡀⠠⠄⡀⠀⡄⠀⠄⡤⠀⠠⠓⠢⠧⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⡐⠀⡄⣄⢀⡄⠑⡄⠍⢠⠀⠁⠀⠄⠨⠉⢀⠬⣰⣀⠀⢠⣀⢖⠙
⠀⠀⠀⠀⠀⠀⢠⣀⢀⠁⠁⠔⠎⢀⠠⠄⢠⡀⠄⢀⠀⠀⠀⠄⠠⢠⢠⠱⠀⠈⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⢀⠀⠀⣀⢀⠔⠀⢀⡀⠡⠄⠁⢀⡆⠀⢆⢸⠀⠀⠀⠾⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣆⠀⠂⠈⠀⠁⢀⡀⠃⠀⠀⠐⠀⢀⠄⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠃⠀⠐⡀⢐⢀⡙⠀⠑⠢⠘⠚⢑⠚⠁⠒⢠⠀⠀⠘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⡘⢠⡇⠈⡀⠝⢀⡌⢀⡀⢀⣀⢭⢀⠣⢼⣆⠃⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠨⢁⠂⣡⡘⠆⢡⠂⣤⣄⢐⠀⣤⠑⡄⠅⢀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠉⠳⠿⣽⣶⣤⣄⣺⣷⠶⠝⠉⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢲⢹⠙⡇⠆⠉⠉⠰⠈⡏⡟⠖⢠⡀⠀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠸⠼⠴⠃⠁⠀⠀⠈⠐⠧⠇⠆⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠀⠀⠀⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣴⡏⠀⡠⡄⠙⡆⠀⠀⣠⣾⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠉⢧⡜⠈⡇⠀⠳⠤⠴⠷⢶⡾⠁⠈⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⡀⠀⠀⢀⣧⠀⠹⡄⠀⠀⠀⣰⠛⣆⠀⠀⣸⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠃⠀⠀⠈⡟⣆⠀⠙⢦⠀⢼⡁⠀⠈⠙⢲⡇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠹⡌⢧⡀⠈⠙⢦⣉⠓⠦⣤⠞⠀⠀⣀⣤⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠐⠀⠘⢦⣉⠳⢤⣀⡈⠉⠙⠓⠛⢛⣉⣡⠤⠖⢻⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢬⣉⣉⣛⣛⡭⠭⠷⠖⠲⠖⠉⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢰⠞⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡤⠴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣴⢦⠀⠀⠀⠈⠁⠀⠈⠻⣆⠀⠀⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣀⠔⠒⠋⠙⠛⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣠⣼⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣠⡴⠞⠁⢸⠀⢰⠇⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣇⡇⠀⠀⢸⠀⢸⡉⡭⣍⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢷⠀⠀⢸⡄⠈⠳⠶⠛⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠳⢦⣴⣇⡀⠀⠈⢧⡀⠀⢳⡀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠳⣄⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠒⠒⠛⣩⠥⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠓⠦⠤⠤⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⡃⢋⡷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠉⠉⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⣀⡀⠀⠀⠀⣀⣠⣞⠳⢤⣤⣀⣀⣀⣠⡤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠒⠒⠒⠛⠉⢸⡏⠉⠛⣛⠉⠁⠀⠀⠈⠉⡿⡄⠈⠳⢤⣀⡀⠉⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡼⠁⠀⠀⢸⡀⠀⠀⠀⠀⢠⠇⠙⢦⣀⠀⠈⠉⠉⠉⠉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁⠀⠀⠀⠈⠳⣄⣀⣀⡴⠟⠀⠀⠀⠉⠓⠲⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣾⣿⠏⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⡿⠋⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⡿⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢾⣶⠈⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣧⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⣺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⡿⣏⢿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣹⡝⣿⣏⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣄⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡛⣿⣿⣶⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⣿⡹⠿⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣬⣽⣿⡽⠽⠿⠿⠿⠾⠿⠷⠤⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣽⣷⣦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡃⠂⠀⠀⠉⠀⠉⠀⠀⠁⡀⠀⠈⣀⣽⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢁⣰⡿⠟⠋⠁⠀⠀⠀⠄⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⣄⡞⠉⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⠶⣄⡴⠖⠛⠛⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢸⣧⣤⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣴⠋⠀⠀⠈⢷⠒⠺⢧⡀
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⣸⣇⣀⡀⣀⣤⠤⣯⣀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠿⣄⠀⠀⣀⡾⠀⠀⠀⢳
⠀⠀⠀⠀⢀⡼⠗⠲⠤⣤⡞⠉⠀⠀⠈⠙⢧⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠈⠙⢿⡉⠀⠀⠀⢀⡾
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋⠀⠹⣦⣀⣀⣀⣠⠟⠲⠶⠖⠋
⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡟⠦⣤⣠⡤⣞⠁⠀⠀⠀⠀⣠⠏⠀⠀⣠⠟⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠙⢦⣄⣀⣠⣾⠀⠀⠀⠀⠀⠈⢻⡶⣶⠒⠋⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⣤⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣀⡾⠁⠘⢧⡀⠀⠀⢰⠏⠀⠀⠀⣀⡴⠞⠉⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣛⣛⠛⠉⠀⠀⠀⠈⢷⡀⠀⡿⠀⠀⣠⠞⠉⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢠⣶⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠈⣧⢸⡇⢀⡞⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠈⠳⣤⣸⣾⣱⠏⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⣹⣿⠃⠀⢀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠿⠋⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣀⡀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⢢⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢣⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⢳⠀⠠⠂⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡘⢆⡀⠀⠀⢮⠀⠀⠀⣀⣀⠤⠤⠤⠬⠷⠤⣤⣀⣀⣀⡂⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠱⣀⠈⠉⣢⡀⢳⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⢤⣄⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠖⠓⢌⡶⠊⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠋⢀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠗⠀⠀⠛⠋⠁⠀⢠⣶⠀⠀⠀⢀⠊⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠟⠁⠀⠠⠐⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡷⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡼⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⠀⣸⢳⣄⡐⣏⢣⡀⠀⠀⠀⠀⢹⣌⠪⣄⣀
⠀⠀⠀⠀⢀⡠⠄⠒⠒⠒⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠙⡆⢳⠒⠒⢞⠯⡁⠈⠙⢲⡌⡇⠀⠀⡴⢳
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣢⠴⠒⠃⠀⠀⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠧⠞⠀⠀⠈⣗⡼⠀⠀⠀⠉⠀⠀⠸⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠤⣀⠀⢀⠜⠉
⠀⠀⠀⢀⣀⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠐⠒⠤⣀⠀⠀⠀⠀⢰⠢⢄⡀⠀⠀⠳⣀⠈⣳⠟⠒⠒⢢⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠈⠉⠙⠂⢲⠉⠉⠁⠁⠒⠤⣀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⢇⠀⠈⠓⢤⡀⠀⡹⠉⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡟⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠈⡆⠀⠀⠈⢦⠀⠀⠀⠙⡼⠓⠀⠈⠁⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡜⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣰⠋⠓⠤⣀⠀⠙⠒⠒⡺⠱⢄⣀⢀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⢧⠀⠀⠀⠙⠢⡀⢰⡁⠤⠤⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⢤⣄⡀⢠⠤⠔⠊⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⠄⠀⠀⠀⠸⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢙⣿⣿⣿⣿⣛⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠘⠟⠙⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣀⣀⠀⠙⠛⠛⠛⠛⠉⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⣟⣯⣽⢿⣿⣷⣶⣤⡀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣯⠯⣿⣷⠛⡝⢯⠉⢫⡻⣶⠊⠀⠀⠠⠂⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⡯⠉⢿⣠⣿⣿⡆⡶⡸⣆⢀⢷⢻⣿⠛⣷⠄⢐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡷⠀⡿⣧⣾⣿⣿⣦⣘⣿⡾⠞⣚⣻⣶⡿⠟⠋⡦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⣠⣿⣴⡧⢿⣿⣋⣿⣯⣿⡾⣷⣿⣿⣿⣿⢱⠰⠠⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣽⣿⣞⠻⣿⡿⣿⣿⣿⣹⣽⣿⣿⠙⡌⣿⡄⡆⡃⠛⡆⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⠘⢿⣷⣦⣷⣿⣿⣿⡿⠛⠛⡇⠀⢳⢹⣷⣷⣠⣸⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡇⠘⡏⠉⢹⣄⣹⣿⣷⣶⣶⠿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣵⣤⡷⣾⠻⣿⣋⣿⣯⣽⡿⠶⠿⠟⠁⣿⣋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠻⠿⠿⠟⠛⣯⣤⣤⣾⠀⠀⠀⠀⠀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣹⣆⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧⡀⠀⠿⠿⣿⣿⣯⡟⠀⠀⠀⢠⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠉⠉⠀⠀⠀⢠⣾⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣀⣀⣀⣀⣠⣴⣿⣿⡟⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣴⡉⣰⡋⠃⣞⢳⢼⣀⡗⡷⣼⢹⡏⡇⡷⣼⢴⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠉⠁⠈⠁⠀⠉⠀⠁⠈⠀⠁⠁⠁⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠚⠙⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠦⠀⠀⠀⠐⠂⠀⠘⠀⠀⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⢀⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⡠⣀⡀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⣶⢸⠁⠀⠈⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣡⡀⠀⠀⢤⣤⣤⣤⣤⡤⠀⠀⠀⢀⠠⣀
⠀⠀⠀⠀⠛⠼⠀⠀⠀⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠟⠀⠀⢀⢀⢀⡀⠀⡀⠀⠠⠁⢇⠀⣨⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⠀⡄⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⢰⣦⣰⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⡀⠀⠀⠉⠉⠉⢉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡄⢀⣇⣀⣀
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠘⢧⢀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠴⠲⠄⠀⠀⠀⠱⡁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠈⠉⠉⢀⡀⠀⠀⠀⢀⠀⢠⣠⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠘⠷⠿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠀⠈⠉⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠶⠿⠷⡀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⢀⠀⠠⠀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠄⠚⢃⠀⠀⠀⠀⢀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠉⠉⠝⠀⠀⠀⠀⡀⢀⣤⡀⠀⠀⠀⢀⠀⢀⡠⠀⠀⠀⠀⠀⢡⡶⣶⡀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠿⠾⠾⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠃⠘⠉⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠘⠷⠿⠁⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠠⡀⠠⠀⠀⡠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢊
⠀⠀⠀⠀⢀⠈⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡊⠀⠀⢅⠠⠀⠀⠀⢀⡒⠁⢠⠐⠈
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⠀⠐⠀⠀⠀⠥⣀⠀⠀⠀⢝⠀⠁⠀⢀⡡⠤⠄⡌⡠⡁⢀⠌
⠀⠀⠀⠠⣢⢖⠔⠀⠀⣀⠀⢀⠀⢀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠤⢀⡀⠀⠑⡂⠁
⠀⠀⠐⠀⠁⠀⠀⠀⠀⠋⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠈⠈⠀⠂
⠀⠀⠀⠀⠂⡊⠀⠀⠀⠀⠠⠄⢰⠀⢂⠀⢁⠀⠁⠀⢀⢀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠂⠴⠀⠁
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⠈⡅⡐⢀⠠⠀⠁⠀⡀⡀
⠀⠀⠀⠀⠀⢠⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠤⠀⠀⠀⠀⡤⣂⡂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠰⠆⠤⠤⢀⡄⠀⠉⠁⠠⠀⠠⡄
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⡄⠐⢆⠡⡨⠆⠂⠠⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠂⢴⠋⠀⣠⠄⠛⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⢀⠰⠆⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⡀⠀⠐⣎⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠘
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡗⠁⡨⠉⠉⠓⢄⠐⠈⣈⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⡘⡔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣵⣶⣶⡌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠿⠿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠉⠙⠛⠛⠉⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣟⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣀⠀⠀⠀⠀⢠⣀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⡿⠀⠀⠀⠀⠀⣿⡇⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⠀⠀⠀⠀⢀⠻⣿⠆⢀⠉⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠻⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⠐⠒⠒⠚⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣴⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⡿⢉⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢀⣀⣀⣐⡂⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⡤⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠆⣠⣾⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣀⠛⠀⠤⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢿⣿⠿⢛⣥⣾⣿⣿⣿⣿⡙⠿⡧⠴⠾⠿⠟⠉⠉⠟⠛⠛⠉⠉⠀⠀⠀⠀⠩⠿⣏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠉⢿⣿⢿⣿⣿⠀⢺⣧⣿⣿⣿⡟⠉⠀⢀⠤⠤⠀⣀⡀⠀⠀⠀⢠⣶⣿⣿⣆⣠⣶⡀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⣠⣴⣶⣦⣄⢻⣄⠈⠻⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⣶⣾⣿⣿⠀⠹⣿⡦⠀⢐⣶⣴⣿⣿⣿⣿⣷⣶⣶⣤⣄⠀⠀⠀⠀⠉⠻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆⠀⠸⣿⠿⠛⠁⠀⢴⠿⢃⣠⠘⡛⢹⣿⣿⣻⣿⣿⣺⣿⣿⣿⠃⠀⠀⠀⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⣿⠀⣀⠀⢀⣠⣾⣿⣏⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡏⢸⣿⢦⣿⣿⣿⠂⠈⣘⣀⣼⣿⣿⠿⡿⠿⠸⣿⣿⠇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣯⠀⠙⢿⣾⣿⡿⠟⠛⡛⢛⠻⣟⠛⠻⠶⣶⣿⣧⣄⡀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠋⢠⡄⠈⣿⡟⡀⢀⢑⡀⠈⠉⠈⠀⠀⢀⡠⠙⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⣧⠀⢿⣠⣾⠟⠀⡄⠀⠈⠛⠀⢂⠁⠄⠀⠀⢰⠃⢼⣦⣉⠉⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢿⠂⣲⠛⡃⠀⠠⠉⠠⠀⠀⠠⠌⠀⠀⠀⠀⡀⠀⡈⢁⡜⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⣗⣶⠊⠏⠐⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠲⠆⠀⠀⠀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⡆⠀⢾⣶⣆⠀⠀⠀⠁⠀⣀⣀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠐⠆⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠠⣿⡇⠀⢸⣿⡿⠑⢄⠀⠀⠐⠈⠛⠷⢦⣤⣤⠄⠲⣾⣿⢶⡀⠀⠀⠙⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡇⠀⢸⣿⣧⢰⣄⢁⡀⠆⠀⠀⠀⠀⠀⠄⢠⣤⣼⣿⡘⠳⠀⢠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣏⡀⠁⣸⣿⣟⣁⣙⣾⣷⡿⠟⢢⣀⣀⠀⣀⣨⡟⠛⣿⡇⡀⢠⠃⣜⢸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡯⣿⢾⢿⣿⣿⣿⣿⣿⣿⠷⠄⠬⠿⠟⢻⣿⣿⣿⣾⣿⣿⡇⢠⡈⣿⢠⠶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣹⠿⣥⣿⣾⣌⣿⣿⠿⣿⣿⣿⣿⣿⡷⠶⠶⣮⣭⣝⣿⣿⣿⣥⣇⠀⢖⡋⣨⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠉⠙⠉⠁⠀⠈⡿⠧⠿⡿⠤⠀⠃⢀⣾⣿⣝⣿⣿⣿⢹⣿⠀⡐⠀⢒⡉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠀⢹⣟⣛⣿⠂⢰⣇⢠⣙⠦⠒⠀⣀⣀⠀⠀⠀⠈⠉⠉⠀⠿⢀⡱⣤⣤⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⠉⠉⠉⡠⠋⠙⠃⠉⠉⠀⠀⠉⠉⢹⡆⠀⠀⠈⠀⠀⠀⣩⡇⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣶⣶⡖⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣺⣶⣤⣠⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡿⠷⠂⠀⠀⠀⠐⠼⢿⣽⣿⠿⠿⡿⢏⡁⠀⢀⡀⠀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⢿⣿⣿⣿⠿⠃⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣵⡟⣭⣷⣹⣿⠁⣴⠛⠛⠛⠛⠛⠛⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣿⣧⡻⣟⣣⡿⣹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⡄⣿⣿⣿⠀⣿⣿⣿⣶⣶⣾⣿⣿⠀⣿⠟⠻⡟⠛⡿⠛⢿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⣶⣦⢸⣿⣿⡇⣿⣿⣿⠀⣿⣷⣒⣒⣒⣒⣲⣿⠀⣿⠦⠴⡧⠤⡷⠤⢾⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⣿⣯⣭⣭⣭⣭⣭⣿⠀⣿⡤⢤⡧⠤⡧⠤⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠸⠿⠿⠇⠿⠿⠿⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⢿⣤⣤⣧⣤⣧⣤⣼⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢀⣀⣀⣀⣀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⠀⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣼⣿⣿⣿⣧⣀⣀⢀⡀⣄⡀⢀⣀⣀⣠⣤⣤⣾⣿⣿⣿⣦⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⢿⡿⠋⠉⠉⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣤⢀⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠏⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣫⣤⣶⣶⣶⣶⣶⠆⠀⠀⣠⣀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣰⣿⣿⣿⣿⣿⣶⣤⣀⠀⠘⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⠄⠀⠀⠈⢿⣿⣿⣿⡟⠀⢸⣿⣿⣷⣄⣻⣿⣿⣿⢫⣾⠟⣿⣿⣿⡧⠸⠿⠍⠉⠻⣿⣿⢰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢃⠊⠈⠂⠰⣄⢿⣿⠋⠀⣀⡾⢟⣿⣿⣿⣿⣿⣿⣯⣾⣫⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠳⣷⠹⣿⣿⣯⡉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣾⣿⡎⠁⠀⠀⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠃⠉⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿
⣰⣶⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡀⠀⠀⢿⢿⣿⣿⡿⢿⣶⣶⠇⠙⢿⣿⣿⣿
⣿⣙⠻⢿⣿⣿⠇⠀⠀⢠⣾⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⠀⠂⠀⠀⠀⠀⠀⢠⣿⣀⡼⠃⠀⠀⢀⣾⣿⣿⣿⣷⡾⠋⠀⠀⠀⠹⣿⣿
⣿⣿⣿⣶⣦⡀⠀⠀⠺⣿⠘⢿⣿⠟⠁⠈⠋⠉⠝⠛⢿⣿⣿⣦⣄⣀⠀⠀⢀⣀⠀⠙⠋⠀⠀⠀⣠⢮⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢻⣿
⣿⠿⣿⠿⠿⠿⣿⠶⣿⡏⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⢿⣿⣿⣀⣀⣀⢀⠀⠀⣀⠀⠻
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠮⡉⠙⠙⠋⠛⠑⠐⠒
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣩⣷⣀⣘⣆⣆⣀⣤⡤⠤⠶⠶⠞⠟⠁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠜⠃⡠⠂⠀⠀⠀⠀⠀⠀⢀⣴⣁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣤⣶⣶⡶⠒⠶⣿⡿⣲⣿⣷⢸⠁⡤⠊⠀⢀⣠⣴⣞⣛⣿⡇⣀⡼⠀⢀⡆⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣗⠂⢰⠟⠓⣿⣏⣀⡼⠿⣿⣿⠋⣉⣶⢇⣴⡟⠻⠿⠿⠿⠿⠿⢷⣿⣾⣴⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠛⠹⠿⣿⠿⠷⡿⠿⢷⢿⣉⣠⡖⣁⣰⣏⡡⠾⣽⣓
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⠖⣻⡶⣟⢻⣿⡉⣸⣟⡛⡓⡠⢶⡾⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠛⠿⠛⠋⠉⠀⠋⣻⣷⣾⣯⡙⠉⠉⠹⢿⡿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢡⡀⣤⣶⣿⣯⣤⣴⠶⠾⠿⠻⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣮⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⢻⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡟⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⢤⣤⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠾⠟⢩⡈⠀⠀⠀⠈⠉⠁⠀⠀⠖⠲⣦⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠁⠀⠀⠀⠀⢷⡀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠙⣿⣆
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⣸⣿⠀⠀⠀⠀⢸⡟⢀⡴⠃⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢀⣿⠇⣾⡇⠀⠀⠀⣿⣧⡶⠶⠶⠶⣤⣶⣾⠛⠋⠉⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⣿⣇⣠⡶⠟⢿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣤⣀⣀⣾⡿⠋⠙⢻
⠀⠀⢠⣿⠁⠀⢀⣠⣤⣤⣽⣇⠀⠀⠀⠀⠀⠀⢃⣀⣠⣤⡴⠶⠶⠃⢻⣿⣿⡀⣤⡀⠹⣦⠀⠀⠀⠀⢀⡴⢁⡠⣀⡼⢩⣿⠋
⠀⠀⣼⣿⠀⠀⠉⠀⠀⠉⠙⠿⣷⣦⣴⣶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⢸⡇⢸⣿⡀⢸⣷⠀⣠⡶⢋⡴⢫⡾⢋⣴⡿⠁⠀⠀⠀⢀⡾
⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⡿⠿⠷⢶⣤⡀⢀⣠⡤⠶⣦⣄⠀⠈⠁⠀⠛⠷⠾⠟⠈⢉⡴⠋⠀⣈⣴⣿⠟⠀⠀⠀⠀⣠⡿⠁
⠀⠀⣀⣽⣿⠿⢿⣷⣶⣶⣶⣾⣿⠁⢰⡇⠀⠀⠀⢮⣻⣮⠁⢠⡶⢈⡙⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⢷⡄⠀⢀⣼⡟
⢤⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣬⡿⣦⣼⣿⣿⣿⣿⠀⣀⡀⠛⠃⢶⠀⢙⣿⣿⣟⠶⣄⡀⠀⠀⣿⡁⠸⣷⡀⢸⡇⢠⣾⠏
⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠠⠿⠃⢠⣶⠄⢛⣥⠀⣀⣴⠾⠛⠉⠀⣙⡿⢦⣉⠲⢤⣙⣿⣾⠿⠣⠟⣡⣿⠋
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠈⣷⠈⠳⣄⠐⠋⣠⡀⠀⠈⠉⠀⠉⠀⠀⢀⣴⣟⣉⠀⠀⠙⣷⡄⠈⠛⠻⠶⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠿⣦⡀⠈⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠹⣇⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠳⣄⠀⠀⠀⢀⣠⡶⠟⠻⣶⣄⣠⣿⡀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⠀⠀⠘⢿⣦⡀⠈⢷⣤⡾⠋⠁⠀⣠⡴⠟⠋⠉⠈⣷⠈⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠙⢿⣦⣄⣿⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⢻⣧⣀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠙⢿⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⠿⠿⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢠⣿⡅⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⣽⡖⠲⠀⠀⠀⠈⠉⠀⠀⣄⠙⠍⠢⣄⠀⠂⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⣿⣿⣶⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⡄⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣪⣷⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠘⢓⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠂⠠⠂⠀⠠⠀⠀⠀⢈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣹⣿⣿⣿⣿⣿⣿⡟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠿⣿⣿⣿⡇⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠤⠴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡬⠀⠀⠀⠈⠉⠉⢀⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠞⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣧⣤⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣄⡤⠴⡛⠛⠉⠉⠁⠀⠀⠀⠠⣤⢼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣹⡿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣴⣿⣿⣿⠿⢷⠄⠐⠊⠩⠁⠀⢀⣦⠤⠀⠀⠀⠀⠀⡤⠀⠹⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠑⠀⠀⠉⠁⣤⢰⡉⠛⠿⠛⠋⠉⢄⠀⠀⡏⠁⠀⠀⠀⠀⠀⠈⢿⣶⡟⠀⠀⢀⡴⠃⠈⣬⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣦⡰⣧⠀⢠⠉⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠎⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠻⠉⠿⡆⠹⣾⣿⣧⢿⣇⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠐⠀⠄
⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠘⠉⠀⠻⢿⣽⡌⠻⠈⢁⡀⠰⠀⠀⠀⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠃⠤⠀⢤⡄⣠⡔⠂⣤⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠙⢣⡾⠆⣀⠤⣴⣆⠂⠀⠀⠀⠀⠀⢀⡀⠦⠀⢀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⣾⣇⣀⠀⣴⠆⠀⠈⠉⠠⠀⠄⠐⣦⠀⠻⠟⠘⠟⣧⣶⡆⠈⣰⣦⠦⢺⠁⡀⠿⣲⡏
⠀⠀⠀⠀⠀⢀⣾⡿⠋⢰⣿⣿⠟⠡⣴⡿⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠀⡾⠛⠖⠃⠈⢡⣆⠓⠳⠚⡏⠀⠉⠓⠀⠁
⠀⠀⠀⠀⢠⣾⡏⠀⠀⣸⣿⠏⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠘⠉⠀⠀⡀
⠀⠀⠀⢠⣿⡟⠀⠀⢰⣿⣿⠆⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⣼⣿⣵⣣⢀⣿⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢠⣿⣿⣿⣿⢰⣿⣿⣿⡀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠇
⣷⣄⢸⣿⣿⣿⣿⠀⣿⣿⣿⣧⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢀⢉⣿⣦
⠹⣿⣧⢻⣿⣿⣿⣦⣿⣿⣿⣵⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠀⣿⠁⣿⢧⣿⢀⣞⣧⠀⡿⣦⡇⣼⠋⠉⠈⡏⠀⣾⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠛⠃⠛⠈⠋⠘⠉⠙⠃⠃⠈⠃⠈⠛⠋⠘⠛⠘⠉⠙⠃⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⡤⠦⢀⡤⠦⢀⡤⢦⡄⣤⠀⣤⢠⣤⢠⡄⠤⡤⠄⢤⡄⢠⣄⢠⠀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢷⠸⣧⣠⠸⣧⣠⠘⢧⣠⠇⢻⣄⡿⠸⡏⢿⡇⠀⡇⠀⣼⡇⢸⠙⢿⠘⢧⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢸⣿⣅⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢺⢸⣿⢨⣗⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⣿⢘⡏⡷⡶⣤⣄⠀⠠⠶⢶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⣿⢪⡇⡇⡇⡇⡏⡇⠀⠀⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠀⢸⣸⣟⣙⣗⣇⣇⣇⣇⡇⢐⣖⡲⠀⢨⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⢸⣄⣀⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣉⣁⣀⣨⡇⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣰⠃⠀⠀⠀⠀⢙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⣀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣓⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⠞⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠦⠀⢐⠖⣒⣒⠀⢹⠀⠀⠀⢠⣦⠀⠀⣒⣒⣂⠰⢀⣀⣀⠀⡼⠁⠀⠀⠰⢒⣂⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠨⡽⠀⠨⠶⠯⠿⠀⢸⠀⡴⠛⠋⠀⠀⠀⠉⠉⢩⡿⣏⠁⢸⢳⠖⠦⣄⠀⢀⠼⠯⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠤⠤⠤⠤⠬⠭⠉⠀⡤⠾⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⢙⡦⣄⢸⢸⠀⠀⢨⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠭⠭⠥⠬⠤⠭⠀⠀⡇⣟⡉⠉⠉⣹⢸⡇⠀⠀⠀⠀⣾⠀⠈⠻⣼⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣞⣺⢲⣓⣖⣶⢺⡆⡇⠉⠉⠉⡉⣩⢸⡇⠀⠀⣠⠆⢹⡀⢀⡴⠋⠀⠀⠀⠀⣼⡹⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡯⣿⢽⡯⡯⣿⢽⡇⡇⠂⠂⠀⠂⠒⢸⡇⢀⡞⠁⠀⠀⣟⠉⠀⠀⠀⠀⢀⡴⣣⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⢽⡇⡇⠁⠁⠁⠁⢩⢸⡅⠋⠀⠀⠀⠀⠈⠻⢶⣶⣶⣚⡭⠖⢁⣀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠷⠾⠼⠶⠧⠾⠼⠇⠓⣲⠒⠒⠒⠒⠚⠁⠀⠄⠠⠄⠤⠤⠄⠀⠀⠀⣤⠀⠀⠒⠒⠒⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⡶⠶⠶⠶⣶⡶⢶⠶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠐⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⢀⠀⠀⣀⠵⠒⠂⠂⠒⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⢀⡀⠀⠀⠀⠀⠀⠔⠈⠀⠀⡀⠀⠘⠁
⠀⠀⠀⠀⠀⠘⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠆⠐⠤⢈⢢⠀⠀⡔⠀⠀⠄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⢀⠃⡌⠀⠀⠀⠀⠀⠀⣠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⢠⠌⢰⠀⠀⠀⠀⡠⠔⠈⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠊⠁
⠀⠀⠀⠁⢀⠃⠀⢀⠔⠊⠀⠀⠀⠠⠊⠀⠀⠀⠙⣄⠀⠀⠀⠀⡰⠃⠉⠁⠀⠀⠉⠐⠢⡀
⠀⠀⠀⠀⡜⠀⢠⠂⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠈⠉⢆⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠐⢌⠦⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠑⣌⡄⠀⠀⠀⠀⠀⠀⠨⡤⠓⠊⠉⠊⠀⠀⢱⡉⡀⠀⢀⡭⠚⠲⢤⣄⡀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⡀⡈⠃⠂⢢⠀⠄⠀⠒⢨⠃⠀⠀⠀⠀⠀⠀⠈⠈⠘⠳⣾⠤⢄⡀⠈⡀⠈⠳⡀⠀⠀⠇
⠀⠀⠀⠀⠆⠈⠁⠀⠀⠋⠀⠉⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠒⠂⠀⠀⠀⡟⠩⠤⠀⠁⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠀⠀⠔⠒⠄⠀⠀⠀⢠⠀⠀⠘⠀⠀⠀⠀⠠⠀⢀⡀⣤⡌⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠀⠀⠙⠀⠀⠜⠀⠢⣀⢄⠀⠠⠒⠤⠊⠑⡄⠀⠄⠀⠀⠀⠀⠈⠈⠑
⠀⠀⠀⠀⠀⠀⢔⠐⠐⠀⠀⠀⠀⠀⡒⠒⠉⠀⠀⠀⠁⠒⠒⠶⠀⠀⠀⠀⠐⠈⠄⢀
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⢀⣄⠐⣦⠀⠠⢦⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⣠⡄⠀⠀⠀⡀
⢸⣼⠀⠸⡇⠀⡶⠀⠉⠑⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⢂⣴⠟⠀⠀⠀⠀⠀⢰⡷⠉⠀⢈⣼⠟
⠀⠿⠀⠀⠐⠂⠀⢰⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⡴⣦⠀⠀⠀⠀⢻⠁⢠⡞⣻⠏
⠀⢠⡤⠂⠀⠀⢠⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠛⠁⠀⠀⠠⢆⡀⠀⠘⠋⣡⡾⢇
⢠⠇⠀⠀⠀⠀⢸⠇⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣀⠠⠃⠀⡀⡈⠐⢦⡄⠀⠘⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠾⠠⢬⠇⠘⠃⠁⠀⠀⠀⠀⠘⠂⢹⣛
⠀⠳⢶⣶⡶⠶⠾⠛⠋⠁⠀⠈⠀⠁⠀⠀⣠⠾⡆⠀⠀⠀⠀⠀⠀⢀⡀⠘⢄⠀⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠸⠟
⠀⠀⣠⣤⣶⣆⠀⠀⠐⠀⠀⠈⠀⠀⠂⣤⠹⣦⣾⠀⠰⠀⢂⠁⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠛⠛⠉⠀⣦⠀⠄
⠸⣿⣿⣿⡎⢿⣷⡄⡐⢦⣘⣠⣤⡄⠀⠉⠀⢻⣛⠀⠀⠀⠀⠀⠀⢰⡄⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⢧⡀⢆
⠀⢹⣿⣽⣿⡄⠙⣿⡏⡓⠦⠒⠉⠁⠀⠀⠀⠼⠟⠀⠀⢴⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⢻⣿⠻⠗⡠⢀⡀⠇⠀⠀⢀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⢐⣩⣶⣿⡇⠀⠀⠀⣼⠶⢿⣰⣖⣰⣗⣰⣟⡦⣆⣰⢲⡞⡆⢿⢲⣶⢺⣶⣟⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠁⣾⣿⣿⣿⣿⡧⡀⠀⠀⠁⠀⠈⠁⠉⠁⠉⠀⠉⠀⠈⠁⠀⠀⠁⠈⠀⠀⠀⢀⡼⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⣿⣿⣿⠟⠉⠐⠦⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⢐⡄⠀⠀⠀⠀⢠⡀⠉⡁⠀⠀⢀⡴⠃⠀⠄⢊⣁⡇
⠀⠀⠀⠟⠋⠀⠀⢀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⢀⣀⡆⠉⡀⠇
⠀⠀⡂⣀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠠⠀⡠⣰⡿⣞⠀⠀⠀⠐⠀⠤⠖⠀⠹⢄⡀⠠⠶⠀⠀⠀⠸⠀⠀⠤⢀⠈⠉⠀⠀⣠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⠔⠛⠒⠉⠀⠋⢰⣿⣿⡷⠲⠦⠤⢤⠀⠀⢀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠂⠐⠁⢠⣤⣾⣿
⠀⢀⠄⠐⠀⠈⠀⡀⠆⠉⠀⠀⠐⠀⢠⣄⠀⠘⣿⣯⣇⠄⡀⢠⣼⣷⠀⣾⡗⠀⢀⣼⡇⠀⠀⠀⡀⠄⠀⠄⠀⢀⣿⡿⢿
⠀⠀⠀⣀⣴⣺⠆⠀⠀⠀⠀⢀⠀⠠⠀⠹⡷⡄⢿⣿⣿⣷⣻⣿⢣⣏⢷⣿⡃⠠⣎⣹⠇⠒⠀⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠈⠛⠁⣾⣤⠀⠀⠀⠀⠀⠀⠲⣦⡀⠓⠺⢸⣿⠋⠀⠁⠙⠛⠛⡻⡎⠃⠀⠙⠁⡀⠀⠀⠀⢀⠀⠀⠀⢸⠟⢠⣤⣤
⠀⠀⠀⠀⢺⣹⣤⣤⡀⠀⠀⠀⠀⣀⣈⣉⣃⣀⣘⣋⣀⣁⣀⣀⣀⣈⠁⠀⠀⠂⠀⠂⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣅⡼
⠀⠀⠀⠀⢀⣄⣧⡼⠁⠀⠀⠀⢈⡇⢺⢘⡇⣖⢰⡆⣿⠰⡆⢸⠃⢸⡇⢸⣿⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⢀⡿⡿⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠷⠾⠼⠧⠿⠜⠧⠿⠀⠧⠸⠤⠼⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠁⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠠⠀⠤⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠆⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠉⠀⢱⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠶⠖⠒⢇⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⠄⠈⡇⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⣤⣀⠀⠐⠈⡜⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠉⡇⠀⢀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡀⢀⢁⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠄⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡈⠀⠀⠠⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⢢⣤⣀⡀⠀⠗⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⢿⠀⠀⢤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣹⠀⠀⠘⡄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠉⠹⠁⠀⠀⠀⠈⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣭⣥⣤⣤⣴⣤⣶⣶⣤⣤⣤⣤⣶⣴⣤⡍⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣭⣿⡿⠿⡏⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠇⠀⠀⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠂⠀⠸⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠐⠢⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠠⠁⠀⠀⠀⠀⢀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢀⠀⠸⠀⠀⢀⠅⠀⠁⡉⠄⠒⠂⠁⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠆⣤⡐⡂⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠋⡿⡁⠀⢰⠁⢰⡀⢸⠁⣑⡀⠀⠒⠀⠂⠀⠀⠀⠀⠘⠀⠀⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣈⠀⠐⡅⢄⠀⢢⡜⠱⠀⢰⣧⠱⣀⠁⠀⠡⠀⡄⠀⠐⠀⢡⣀⡀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣻⠍⠂⠀⠹⣌⢣⠀⢻⣶⣶⣤⣻⣧⡘⠛⣄⠀⡀⢀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⣀⡷⣿⡀⠀⠀⠈⠙⠳⣄⢳⡌⠀⠉⠙⠻⠦⠛⠓⠧⠾⠤⣠⣤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⣼⠹⡄⠘⣟⠶⣄⡀⠀⠀⠈⠓⢧⣿⡖⠒⠐⠀⠤⠄⠄⠴⠶⠿⠟⠀⣀⡀⠃⠀⢰⡆⠀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡖⡿⣿⠃⠀⢹⣧⢈⡙⠲⠞⢧⡀⠀⠉⠓⠶⣄⡁⠒⠀⠀⠀⣤⣴⣶⣾⠽⠁⠀⢠⠟⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢩⡿⠩⣿⡅⠀⠀⠻⣾⡇⠀⠀⡌⠙⠓⠚⠦⣄⣀⡉⠑⢲⢤⣤⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡿⠃⠀⢉⣷⣄⠀⠀⠈⠳⣤⣾⣷⣶⣄⣄⡀⡀⡀⢀⠈⠯⠿⣎⣻⣉⠥⠀⠀⠀⠐⠀⠀⠀⡾⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠫⣤⡄⡀⠀⠿⢻⡷⣦⣤⣤⣈⠑⠶⣖⡂⠄⡉⠉⠉⠉⠉⠁⠉⠁⠀⠀⠀⠀⣀⣀⡾⠆⣠⢨⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢿⣷⣆⡀⢨⠀⡟⣸⣿⣿⣷⣦⣬⣝⢳⡀⠈⡁⠒⠲⠦⣤⣤⣤⣄⣐⣫⣾⠟⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣄⣀⠀⢹⡿⠛⠚⠷⣷⡃⣹⣿⣿⣿⣿⣿⠈⣀⠀⠛⣼⠈⣼⣾⣿⠿⠟⠋⠑⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⠚⡟⣷⣤⣑⡀⠈⢀⠜⣧⡿⣽⢿⢨⢏⣿⡀⢿⢀⣰⠿⢞⢋⣉⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡯⢻⣵⣿⣿⣎⠱⣄⠀⠀⣿⣿⣟⡁⢿⣣⣻⣧⠈⣿⠡⠒⠒⠉⢠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⡿⢀⠡⡀⠐⣿⣿⣠⠴⠚⠉⠁⢀⣤⣾⣿⢦⣤⠶⡿⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣷⣘⠛⣯⣿⠙⢟⠁⠀⠀⠀⢀⣾⡿⢾⣽⣮⣿⡿⠁
⠀⠀⢀⣠⣤⠤⠀⠀⠴⣶⣛⣿⣻⠽⠟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣆⣈⡀⢸⣧⣤⣼⣿⣧⢣⣺⣿⠟
⠀⠀⠀⠀⠀⠁⠈⠉⠀⠈⠉⠀⠀⠀⣠⣾⣿⠿⢻⡿⠛⣿⠿⠿⣯⣍⣿⣿⣁⣩⣮⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠴⣾⡿⠶⠶⠞⠛⠋⠉⠁⠀⢠⡟⠁⢀⠏⠀⠀⠀⠀⠉⠉⠙⢿⡙⣏⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣎⢿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣮⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂
⠀⠀⠀⣀⣤⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⣴⣦⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠑
⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠖⠀⠀⣀⣤⣾⣿⣿⠇⡿⣄⠈⢄
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⣦⡼⠿⣋⣭⡿⣿⡿⢹⡿⡏⠀⢷⠀⠀⠈⠑⢤⣄⡀
⣿⣿⣿⠟⠁⠀⢠⡴⣤⡤⠔⣚⢉⣻⡾⢟⠍⣁⣾⡿⠡⠿⢱⣧⠱⠈⠗⢄⠀⠀⠀⡙⢏⠃⠐⢤⣀⠀⠀⡀
⡿⠛⠁⠀⠀⢠⣾⣿⡆⠞⠉⡠⠆⠁⣀⣴⣾⡟⡀⠀⢀⣰⣿⣿⡄⢂⣀⠀⢹⡆⡀⠙⠆⠁⡀⠀⢉⣛⠲⣇⣦
⠀⠀⠀⠀⢰⣿⣿⣿⣯⠶⣨⣵⠶⢊⣱⣾⡟⢸⠃⡼⠟⠋⢀⡟⢷⡀⠉⢀⣄⡙⠀⠀⠀⣆⠙⢰⣼⢸⣁⣏⣧
⠀⠀⠀⠀⣼⣿⣿⡛⠁⠛⣩⣴⠶⢿⡿⠏⠰⢃⠼⠃⠷⠀⢏⠀⢈⠓⠀⠀⠈⠇⠀⡧⠐⠟⣀⣿⣿⣾⣿⣿⣿⣧
⠀⠀⢀⣾⡿⢋⣀⣀⣨⣭⢉⣠⣶⡉⠀⠀⢴⢟⠀⢰⡄⢀⣿⢰⠈⢠⠈⠂⣼⢀⣶⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠉⠛⠛⠋⠉⠉⠙⠋⠉⠙⣷⠀⡆⢀⠈⢠⣿⡇⣸⣿⣼⣤⣸⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠨⡀⠇⢸⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡆⠀⢸⡇⡌⠇⢸⣾⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠏⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⡇⣷⣴⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣟⣿⡿⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠇⠀⠀⣀⣀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠘⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠆⠤⢄⣤⠈⠉⠉⠉⠙⠛⠒⠒⠶⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⡇⠀⠀⠈⠁⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠘⢀⡈⠀⠁⢠⠀⣀⡀⠀⠀⠠⠈⣉⠁⠒⠒⠂⠠⠠⡄⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠒⢂⡀⠀⠡⠀⠀⡀⠑⢀⡀⠄⠀⠀⠒⠀⠀⠌⢡⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠻⠶⢤⣤⣄⣀⣀⣀⠀⠀⠐⠀⠠⠄⠁⠰⠐⡀⠄⠐⡀⢠⠃⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠢⠤⠄⠀⠀⠀⠁⠉⠉⠉⠛⠛⠓⠶⠶⣦⣤⣤⣄⣀⣉⣀⡘⠀⢰⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣀⡀⠀⠈⠀⠂⠁⠐⠒⠀⠤⢄⣀⡀⠀⠀⠉⠉⠉⠙⠋⠀⢿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⢡⡀⠸⠉⢹⠛⠲⠶⢤⢠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣒⣀⠀⠈⢘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡤⠁⠀⢈⡀⠓⠂⠧⠤⠃⠊⠀⠀⢠⠂⢐⠆⢠⣤⢤⣀⡀⠘⠿⠇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡘⠁⡴⠋⣸⠃⡞⣡⠖⢀⡄⠀⡀⠀⠈⠑⠚⠠⠾⠅⡤⠀⠄⠀⠀⡸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠇⠒⠲⢠⡏⡄⡘⠘⠀⣸⠃⡜⠐⣫⠃⣼⠀⡴⠀⡂⠀⣄⠀⠀⢠⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠂⡐⡂⢠⠀⡀⠉⠉⠑⠚⠇⠸⢶⠬⡤⣤⠏⣐⠐⠋⠀⢰⠇⠀⠀⡞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⡁⠈⠳⢁⡾⡐⠡⢎⠃⢸⠆⢠⠒⡀⠀⣄⠉⠀⠉⠁⠒⠰⠏⠀⠀⣸⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡈⠋⠙⠋⠾⠁⣆⣘⡫⢀⡟⢠⠃⢏⠣⢠⡟⢀⠃⣠⠀⢰⠆⠀⠀⣰⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠜⠁⡨⠀⣶⢀⡄⢀⠀⢈⠉⠁⠘⠙⠂⠆⡾⠀⣆⡀⠀⢀⡾⠀⠀⢀⣿⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⣀⠀⠁⣰⠃⠎⢠⡜⠀⣾⠀⡔⢉⡤⠀⣦⠀⡀⠀⠀⢉⠙⠁⠀⢀⣎⢯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢠⠄⠀⠈⡁⠛⠘⠿⠔⢖⣶⠃⣜⢀⠒⠃⣸⠃⡘⠁⠀⠀⣸⠃⠀⠀⣞⢸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠠⠋⣔⣹⢀⡏⡰⠁⠀⠐⣦⠀⢀⢈⠉⠉⠒⠋⠀⠷⠶⣒⣠⡏⠀⠀⠸⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠻⠦⢮⣀⡾⢰⣁⠀⠃⢰⡏⠠⠃⢀⡀⢰⡇⢠⠂⠀⠂⢠⠈⠀⠀⢀⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣍⣉⣐⣒⠀⠀⠀⠈⠁⠉⠓⠟⠀⠧⣄⣠⢠⡟⢀⡀⠐⠃⢀⡿⠀⠀⠀⡌⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡯⠿⢍⡉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠉⠁⠈⠃⠉⠉⠾⠁⠀⠀⢰⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⢦⣤⣀⣀⣀⠀⠒⠀⠀⠀⠉⠉⠙⠒⠒⠦⠤⣤⣬⣉⣐⣂⣿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠢⠤⠤⢤⣤⣀⣀⣤⣥⣚⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⢀
⠀⠀⠀⠘⠒⠒⠒⠂⠐⠠⠀⠠⠤⡄⣀⣀⢀⡀⠀⠈⠁⠀⠙⠓⠒⠂⠀⢠⠤⠤⠄⣀⣠⡀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⢠⡀⠀⠈⠉⠀⠐⠀⠲⠦⠤⠤⠤⣤⣤⣀⡀⢨⣁⢈⠈⠒⠀⠤⣀⠀⡄
⠀⠀⠀⠑⠪⢢⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠋⡰⠤⠤⡤⡄⠤⣤⢀⣀⣀⣀⠀⣀⣀⢈⣉⠉⠈⠀⠶⣤⣲⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠒⠒⠲⠦⡄⢠⡄⢀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣉⡘⣋⢓⠒⢒⣒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⢤⣌⣃⣘⠶⢯⠈⡛⠻⣿⠚⠓⡓⣴⠦⠀⣐⠂⢆⠆⢿⡳⠎⣿⠋⡼⡏⠈⡀⠉⠈⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣟⠿⢿⣧⢆⠺⡥⢿⣠⠶⢽⡟⢻⣛⡲⣔⣋⢖⠚⠊⢉⣴⣥⣷⣾⣽⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘⠳⡴⠆⠠⠾⠐⠐⣿⠻⢖⢜⣓⠴⣶⢦⣅⢆⣡⡞⠉⠨⢻⡺⢔⣓⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⢷⠆⠀⠀⠙⢶⡘⢻⠯⡍⢦⢜⡺⡮⠀⠀⢀⠂⠙⠀⠉⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⠀⠑⠉⠃⢳⣼⣙⠖⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⡓⠛⣒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢓⠀⠄⠀⠀⠋⠁⠀⠄⢠⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢠⣤⢦⢊⣹⣢⡀⠀⠐⠀⠀⠀⠀⠀⠘⠀⠡⠐⠈⠁⠄⠀⠀⠀⠀⠀⠄⠌⢙⠂⡀
⠀⠀⠀⡀⠀⡦⠴⢂⣙⡨⡶⣒⣯⣓⣭⠉⠁⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠖⠤⠈⢐⣤⡐⠢⢠⡀
⠀⠒⠀⠐⡥⡴⢲⡄⣇⢆⡻⡭⣿⡷⡶⠀⠀⠀⠏⡹⢶⣭⣤⢤⢀⣀⣀⡀⠀⠡⠀⠀⠀⠀⠀⠈⠐⢂⠀⠀⡆⠸⢷⣗⣼⠱⠰⣤⣤⣀
⠀⢂⢰⡵⠚⢀⡉⢟⡋⠉⠾⣯⡋⡽⠄⠀⣄⢹⢾⢰⡌⡄⢽⣠⡟⣿⣭⣠⢀⡀⢶⢀⠀⠀⠠⠀⠂⠌⠂⠲⣴⢈⢌⢇⡙⢪⡜⣪⣙⣇⢴⣤⡀
⠀⠠⠈⠂⡘⠀⢠⠀⠘⣺⣾⠡⡔⢛⠀⠀⣹⠎⠠⠠⠙⠷⠽⠍⠍⢁⢴⣀⡈⢥⠊⠘⠃⣀⠠⠯⡓⠤⠀⠀⠈⠫⡘⠌⠃⠝⢈⡀⠨⣽⡕⢂⡀
⠀⠐⠚⠚⠒⠊⠒⠒⠁⠛⠀⠛⠩⠀⠀⠤⠐⠂⠐⠹⠰⠤⠠⠘⠳⠈⠷⠴⠗⠈⠣⠀⠧⠀⠠⠀⠉⠲⢨⠄⠀⠁⠃⠠⠁⠐⠠⠬⠘⠶⠄⠉⠲

⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠸⡈⠢⢄⣀⣀⣀⣠⠤⠒⠋⠉⠉⠀⠀⠀⠈⠉⠙⠒⠦⢄⣀⣀⣀⣀⠴⠊⡸
⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠁⠀⠀⠈⠀⢀⣀⣀⣀⣀⠀⠀⠀⢐⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⣠⠖⠉⠀⠀⠀⠈⠑⢄⠀⠄⡠⠊⠁⠀⠀⠀⠈⠑⢦⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡸⠁⠀⠀⣠⠤⠤⣤⡀⠀⢣⡼⠁⠀⢀⣠⠤⠤⣄⠀⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃⠀⠀⡾⢡⣶⣶⣦⠹⡄⠈⡇⠀⢠⡏⣰⣶⣶⣌⢳⠀⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⡄⠀⠀⢧⠸⣿⣿⡾⢠⢇⠤⠥⡀⠸⣆⠻⣿⣷⠟⣸⠀⠀⡇⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⡀⠀⠈⠓⠦⠤⠖⠋⣯⣄⣠⢼⠀⠈⠲⠤⠴⠞⠁⠀⡰⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠑⢤⡀⠀⠀⠀⢀⡠⠞⢦⣰⠟⢤⣀⠀⠀⠀⢀⡠⠞⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠋⠀⠀⠈⠉⠉⠉⠁⠀⢀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⢀⡜⠁⠀⢀⣠⠄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣀⡀⠀⠙⣆
⠀⠀⢀⡼⢿⣁⠀⠼⣥⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢤⡽⠆⢀⡹⠿⡄
⠀⠀⠾⣴⠞⡉⠀⠀⠓⡶⠀⡼⡸⠀⠀⠀⠸⡀⢀⠀⠀⠀⠀⠠⣀⣀⠞⠀⠀⠘⡜⢄⠙⡖⠃⠀⠨⡛⣶⠿
⠀⠀⣰⠃⣸⣶⠞⠁⢀⢓⡟⢠⠃⠀⠀⠀⠀⠈⠁⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡈⢻⢋⡀⠙⢶⣞⠈⣇
⠀⠀⠿⠚⢱⠇⡅⣠⢾⡞⣰⠇⢀⠀⠀⠣⠤⠂⠀⠘⠦⠔⠁⠀⠀⠢⣄⠜⠀⠀⡰⢹⣄⣻⢧⣄⡇⢯⠙⠛
⠀⠀⠀⠀⢸⣤⠟⣽⠈⢋⣿⢠⢸⢆⠀⠀⠀⠀⣄⣠⠀⠀⠰⣀⣠⠀⠀⠀⣠⡌⢇⢸⣿⣍⠈⣏⠳⡾
⠀⠀⠀⠀⠀⠀⠀⢹⡴⠋⠸⣌⡽⠉⠑⠦⣄⡀⠀⠀⣄⣀⠄⠀⠁⣀⡤⠚⠁⠈⢯⣁⡟⠘⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣇⡪⠛⠒⠒⠒⠒⠒⠻⣕⢼⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣊⢉⡀⣉⠑⡄⠀⠀⠀⠰⢋⡉⣈⣁⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠟⠉⠊⠀⠀⠀⠀⠘⠚⠈⠞⠉⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣤⣤⣄⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⣟⣿⠸⠿⣿⣻⠉⢡⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣿⡿⢾⣿⣶⣤⡉⠉⣠⣾⣿⣿⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⠛⢿⣿⣿⣿⣿⣿⣿⣏⡉⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣆⠀⠙⠻⠿⡿⢿⡿⣷⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡏⠀⡀⠿⢿⣿⣷⣿⣦⣶⣿⣇⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠿⣿⣟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⢸⠟⢛⣿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠾⣿⣄⣐⣶⡆⠋⠁⠀⠀⠀⠰⣦⣤
⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⠄⢰⣿⣿⣝⣛⣷⣬⣶⣦⣄⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠸⠿⣿⣿⠿⠿⠓⠁⠀⠋⠂⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣶⠜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⣿⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⣶⠄⠀⣐⠂⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢸⣿⣿⡒⠛⠁⠙⢳⡤⠂⠈⣻⣾⣿⣧⠀⡀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠻⠿⠿⠿⠿⠶⠤⠈⠁⢽⢤⣿⣿⣿⣿⣷⡰⣄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣤⣀⢀⣾⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⢈⠓⠀⠀⠀⢘⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⠠⣤⣤⡼⠀⠀⠀⠑⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⠈⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠐⠤⠈⠁⠀⢀⢂⡴⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠃⢓⠒⠢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠡⠀⠀⠀⢀⠦⣀⡘⠋⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠡⠄⡄⢏⠈⠪⡄⠀⠀⠀⠀⠀⠀⠸⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢨⠻⣅⣀⠀⠀⠀⠂⠽
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠀⠀⠀⠐⢄⠀⡐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠀⠈⡀⢀
⠀⢠⠊⠁⠑⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⣦⡧⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⢸⠀⠀⢸⠸⡆⠀⠀⡇⠀⠀⠀⠀⠐⡜⠀⠀⠀⠀⠈⣃⠒⠠⠀⠑⣤⣀⡀⢦
⠀⠘⡄⡄⠀⡇⣧⠀⠀⠇⢹⡄⠀⠀⠀⠹⣠⠄⠀⠀⠀⠙⠳⢄⠀⠀⡀⠀⠉⠚⢧⡤⣠⢀⢀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡇⠇⠀⢱⢸⠀⠀⢰⠀⠻⣆⠀⠀⠀⠱⡄⠈⢀⠞⡠⠐⠀⠙⠒⠥⠤⢄⣠⣌⣷⡠⠀⢀⣀⠀⠈⣳⢦⢀⡀⣊⣀⢀⡂
⠀⠀⢣⢀⢠⢸⡸⡆⠀⢸⠀⠀⠈⢢⠀⠀⠀⢘⣶⡇⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⢀⡙⠆⠀⠀⠀⠀⠀⠀⣈⡔⠀⢉⠩⢉⡉⠉⢁⠄
⠀⠀⠸⡄⠈⠀⡇⡇⠀⣸⠒⠠⠤⠤⠱⢄⠀⢯⠈⠀⠉⠉⠉⠉⡉⠀⣁⣀⡠⠤⠠⠚⠀⠈⠀⠀⢀⠀⠀⢀⠀⣁⡠⠖⠊⠁⠀⠀⡀
⠀⠀⠀⢇⠀⠀⣇⢳⢠⠇⠀⠀⠁⠀⠀⠈⠳⣆⣀⣀⣴⠥⠔⠲⠷⠒⠁⠠⠀⠀⠈⠒⠄⡠⢔⡮⠒⣋⡤⠒⠋⠁⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠸⡄⠀⠁⢸⡎⢀⠖⠀⠀⠀⠀⠀⠀⡏⢀⡀⠀⠀⠀⠀⠀⠀⠉⠑⢒⠠⠄⢀⢈⣈⡭⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⡀⠀⠈⡇⠂⠀⡠⠔⠒⢒⡢⣄⠁⠀⠈⠉⠉⠓⠲⠦⠤⣤⣀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⠈
⠀⠀⠀⠀⢻⢹⠚⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠉⠑⠒⠤⢤⠄⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢸⢸⢀⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠀⠀⠀⢸⢸⠋⣡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇
⠀⠀⠀⠀⢸⢸⠀⢻⢸⣦⣤⣍⣑⣒⠒⠒⠒⠂⠀⠀⢤⣀⠀⠀⣀⠀⠀⠀⢀⡤⠊⡰⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡏⡼⢲⠄⢘⠁⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⠗⠊⠙⡢⠈⢉⡤⠚⠁⢀⠜⠁⠀⡇⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠃⠀⡸⠠⠞⠓⠒⠢⠤⠤⢄⣀⣀⣀⣀⣀⣤⡤⠔⢙⠋⠁⠀⢀⠔⠁⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⠀⠀⠀⠁⠀⠀⠀⢰⠀⡇⢸⠀⣀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢦⣄⣀⣄⣀⣀⣠⡾⠂⡇⠘⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃