input: Write a Python function that, given n rectangles defined by their bottom-left and top-right coordinates, determines the largest possible square area that can be formed within the intersection of any two regions, or returns 0 if no such intersection exists.
  
  from typing import List
  
  def largest_square_area(rectangles: List[List[int]], corners: List[List[int]]) -> int:
      """
      Given a list of n rectangles in a 2D plane, each represented by their bottom-left and top-right corners, determine the largest possible area of a square that can be formed within the intersection of any two rectangles, if such an intersection optimally exists. If no intersections exist, return 0.
  
      Examples:
      >>> rectangles = [[1,1],[2,2],[3,1]], corners = [[3,3],[4,4],[6,6]]
      1
      >>> rectangles = [[1,1],[2,2],[1,2]], corners = [[3,3],[4,4],[3,4]]
      1
      >>> rectangles = [[1,1],[3,3],[3,1]], corners = [[2,2],[4,4],[4,2]]
      0
  
      Constraints:
      n == rectangles.length == corners.length
      2 <= n <= 10^3
      rectangles[i].length == corners[i].length == 2
      1 <= rectangles[i][0], rectangles[i][1] <= 10^7
      1 <= corners[i][0], corners[i][1] <= 10^7
      rectangles[i][0] < corners[i][0]
      rectangles[i][1] < corners[i][1]
      """
output: def largest_square_area(rectangles: List[List[int]], corners: List[List[int]]) -> int:
      n = len(rectangles)
      max_side = 0
  
      for i in range(n):
          for j in range(i + 1, n):
              x1 = max(rectangles[i][0], rectangles[j][0])
              y1 = max(rectangles[i][1], rectangles[j][1])
              x2 = min(corners[i][0], corners[j][0])
              y2 = min(corners[i][1], corners[j][1])
              
              if x1 < x2 and y1 < y2:
                  max_side = max(max_side, min(x2 - x1, y2 - y1))
                  
      return max_side ** 2