<p>You are given an <code>m x n</code> integer matrix <code>grid</code>​​​, where <code>m</code> and <code>n</code> are both <strong>even</strong> integers, and an integer <code>k</code>.</p>

<p>The matrix is composed of several layers, which is shown in the below image, where each color is its own layer:</p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2021/06/10/ringofgrid.png" style="width: 231px; height: 258px;" /></p>

<p>A cyclic rotation of the matrix is done by cyclically rotating <strong>each layer</strong> in the matrix. To cyclically rotate a layer once, each element in the layer will take the place of the adjacent element in the <strong>counter-clockwise</strong> direction. An example rotation is shown below:</p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/22/explanation_grid.jpg" style="width: 500px; height: 268px;" />
<p>Return <em>the matrix after applying </em><code>k</code> <em>cyclic rotations to it</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/19/rod2.png" style="width: 421px; height: 191px;" />
<pre>
<strong>Input:</strong> grid = [[40,10],[30,20]], k = 1
<strong>Output:</strong> [[10,20],[40,30]]
<strong>Explanation:</strong> The figures above represent the grid at every state.
</pre>

<p><strong class="example">Example 2:</strong></p>
<strong><img alt="" src="https://assets.leetcode.com/uploads/2021/06/10/ringofgrid5.png" style="width: 231px; height: 262px;" /></strong> <strong><img alt="" src="https://assets.leetcode.com/uploads/2021/06/10/ringofgrid6.png" style="width: 231px; height: 262px;" /></strong> <strong><img alt="" src="https://assets.leetcode.com/uploads/2021/06/10/ringofgrid7.png" style="width: 231px; height: 262px;" /></strong>

<pre>
<strong>Input:</strong> grid = [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]], k = 2
<strong>Output:</strong> [[3,4,8,12],[2,11,10,16],[1,7,6,15],[5,9,13,14]]
<strong>Explanation:</strong> The figures above represent the grid at every state.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>2 &lt;= m, n &lt;= 50</code></li>
	<li>Both <code>m</code> and <code>n</code> are <strong>even</strong> integers.</li>
	<li><code>1 &lt;= grid[i][j] &lt;=<sup> </sup>5000</code></li>
	<li><code>1 &lt;= k &lt;= 10<sup>9</sup></code></li>
</ul>
class Solution:
    def rotateGrid(self, grid: List[List[int]], k: int) -> List[List[int]]:
        # mod as the number of real rotations
        row = len(grid)
        col = len(grid[0])
        steps = min(row, col) // 2
        res = [[0 for _ in range(col)] for _ in range(row)]
        def nxt_move(up_r, down_r, left_c, right_c, cur_r, cur_c, direction):
            if direction == "down":
                return [cur_r + 1, left_c, direction] if cur_r + 1 <= down_r else [down_r, left_c + 1, "right"]
            if direction == "right":
                return [down_r, cur_c + 1, direction] if cur_c + 1 <= right_c else [down_r - 1, right_c, "up"]
            if direction == "up":
                return [cur_r - 1, right_c, direction] if cur_r - 1 >= up_r else [up_r, right_c - 1, "left"]
            if direction == "left":
                return [up_r, cur_c - 1, direction] if cur_c - 1 >= left_c else [up_r + 1, left_c, "down"]

        def helper(up_r, down_r, left_c, right_c):
            cur_k = k % (2* (down_r - up_r + right_c - left_c))
            cur_r = cur_c = up_r
            direction = "down"
            # Go to the point we want to take actions
            for _ in range(cur_k):
                cur_r, cur_c, direction = nxt_move(up_r, down_r, left_c, right_c, cur_r, cur_c, direction)
            start_r = start_c = up_r
            start_direction = "down"
            for _ in range(2* (down_r - up_r + right_c - left_c)):
                # res[start_r][start_c] = grid[cur_r][cur_c]
                res[cur_r][cur_c] = grid[start_r][start_c]
                cur_r, cur_c, direction = nxt_move(up_r, down_r, left_c, right_c, cur_r, cur_c, direction)
                start_r, start_c, start_direction = nxt_move(up_r, down_r, left_c, right_c, start_r, start_c, start_direction)
        for i in range(steps):
            helper(i, row - 1 - i, i, col - 1 - i)
        return res
128