Positioning a fake mirror for best effect

Question

Challenge

Premise

Consider a mosaic of \$m\times n\$ tiles, in \$k\$ unique colours designated by integers. Example (\$3\times6\$, four colours):

4 1 3 2 4 2
1 2 4 2 1 3
4 3 2 1 4 4

My poor man's mirror is a pane of glass of width \$\sqrt{2}\cdot\min(m,n)\$. I stand it diagonally on the mosaic, like so:

4 1 3 M 4 2
1 2 M 2 1 3
4 M 2 1 4 4

For this example I can pretend it reflects exactly two full tiles:

x 1 x M x x
x 2 M x x x
x M 2 1 x x

No matter what diagonal I choose, this is the greatest number of full tiles I can fake-reflect. Yay.

Task

Input: an integer matrix of \$m\$ rows and \$n\$ columns where \$2\leq m\leq1000,2\leq n\leq1000\$. The number of unique values is \$k\$ where \$3\leq k\ll mn\$.

Output: three integers, in any format. The first and second respectively represent the row coordinate and column coordinate of the matrix element ('mosaic tile') at the left end of the 45-degree diagonal where the fake mirror should be placed for 'best effect', effectiveness being defined as shown above. The third integer is 0 or 1, respectively meaning a rising (bottom left to top right) or falling (top left to bottom right) diagonal.

For clarity's sake, here are some simple test cases.

Example 1

Input:

4 1 3 2 4 2
1 2 4 2 1 3
4 3 2 1 4 4

Output: 3 2 0

Example 2

Input:

3 6
4 7
5 8
1 2
2 1

Output: 4 1 1 or 5 1 0 (not both)

As you can see, a unique solution isn't guaranteed.

Example 3

Input:

2  7  4  10 7  8  9  5  6  4  2  4  10 2  1  7  10 7  2  4  10 10 8  7
6  5  6  2  2  3  6  1  6  9  7  2  10 3  4  7  8  8  3  7  1  8  4  2
3  3  7  6  10 1  7  9  10 10 2  6  4  7  5  6  9  1  1  5  7  6  2  7
7  10 3  9  8  10 9  3  6  1  6  10 3  8  9  6  3  6  2  10 1  2  8  1
7  7  8  1  1  6  4  8  10 3  10 4  9  3  1  9  5  9  10 4  6  7  10 4
1  10 9  7  7  10 3  3  7  8  2  2  4  2  4  7  1  7  7  1  9  9  8  7
5  9  5  3  8  6  5  7  6  7  2  7  9  9  7  10 8  8  7  3  5  9  9  10
9  3  8  2  9  2  1  3  6  3  8  5  7  10 10 9  1  1  10 2  5  1  6  9
8  7  6  2  3  2  9  9  9  7  9  5  8  3  8  2  2  5  2  2  10 10 3  5
7  1  1  2  3  2  10 1  2  10 3  3  2  1  4  2  5  6  10 9  6  5  3  8
8  9  5  2  1  4  10 6  8  6  9  10 10 8  1  6  10 6  4  8  7  9  3  5
8  1  5  7  1  8  7  5  8  6  4  5  10 1  6  1  4  4  10 7  6  3  3  6

Output: 1 10 1

Edit - indexing

The example outputs are 1-indexed, but 0-indexing is allowed.

Remarks

• This is code-golf, so fewest bytes wins.
• Standard rules, I/O rules and loophole rules apply.
• If possible, link an online demo of your code.
• Please explain your code.

Answer 1

Charcoal, 91 bytes

≔⟦⟧θＷＳ⊞θＩ⪪ι ≔Ｌ§θ⁰η≔⊖⌊⟦Ｌθη⟧ζＦ⁻ＬθζＦ⁻ηζＦ²⊞υ⟦Σ⭆⊕ζ⭆⊕ζ⁼§§θ⁺ι⎇λμ⁻ζμ⁺κθ⁺ι⎇λξ⁻ζξ⁺κμ⎇λι⁺ιζκλ⟧Ｉ✂⌈υ¹

Try it online! Link is to verbose version of code. 0-indexed. Explanation:

≔⟦⟧θＷＳ⊞θＩ⪪ι 

Input the mosaic. (These 12 bytes could be avoided by requiring the input to be in JSON format, but I was too lazy to punctuate the example.)

≔Ｌ§θ⁰η

Get the width of the mosaic.

≔⊖⌊⟦Ｌθη⟧ζ

Get the inner size of the mirror, i.e. the distance from the first to the last character of the mirror in terms of diagonal steps.

Ｆ⁻Ｌθζ

Loop over the possible row(s) of the top left corner of the mirror's enclosing square.

Ｆ⁻ηζ

Loop over the possible column(s) of the top left corner of the mirror's square.

Ｆ²

Loop over the possible rotations of the mirror.

⊞υ⟦Σ⭆⊕ζ⭆⊕ζ⁼§§θ⁺ι⎇λμ⁻ζμ⁺κθ⁺ι⎇λξ⁻ζξ⁺κμ⎇λι⁺ιζκλ⟧

Calculate the number of tiles it reflects exactly. Exact matches are counted twice and the diagonal is also counted but this doesn't affect the relative score. Push this number along with the potential solution to the predefined empty list.

Ｉ✂⌈υ¹

Output the solution with the highest number of exactly reflected tiles.