Grid of Sums

In the following, two squares are adjacent if they share a vertex. (This means orthogonal and diagonal adjacencies.)

Consider a \(3 \times 3\) grid beginning with empty squares. You will play 9 turns. In each turn, select an empty square. If this empty square is not adjacent to a filled square, write a 1 to the square. Otherwise, write the sum of all numbers adjacent to it to the square.

Your objective is to maximize the final number that you write. What is this maximum?

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