### numDistinctIslands2 
from typing import *
def f_gold(grid: List[List[int]]) -> int:
    def dfs(i, j, shape):
        shape.append([i, j])
        grid[i][j] = 0
        for a, b in [[1, 0], [-1, 0], [0, 1], [0, -1]]:
            x, y = i + a, j + b
            if 0 <= x < m and 0 <= y < n and grid[x][y] == 1:
                dfs(x, y, shape)
    def normalize(shape):
        shapes = [[] for _ in range(8)]
        for i, j in shape:
            shapes[0].append([i, j])
            shapes[1].append([i, -j])
            shapes[2].append([-i, j])
            shapes[3].append([-i, -j])
            shapes[4].append([j, i])
            shapes[5].append([j, -i])
            shapes[6].append([-j, i])
            shapes[7].append([-j, -i])
        for e in shapes:
            e.sort()
            for i in range(len(e) - 1, -1, -1):
                e[i][0] -= e[0][0]
                e[i][1] -= e[0][1]
        shapes.sort()
        return tuple(tuple(e) for e in shapes[0])
    m, n = len(grid), len(grid[0])
    s = set()
    for i in range(m):
        for j in range(n):
            if grid[i][j]:
                shape = []
                dfs(i, j, shape)
                s.add(normalize(shape))
    return len(s)