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A nine-cell region is the smallest subset of the plane that can contain all twelve free pentominoes, as illustrated below. (A free polyomino is one that can be rotated and flipped.)

A twelve-cell region is the smallest subset of the plane the can contain all $35$ free hexominoes.

What is the smallest region of the plane that can contain all $108$ free heptominoes (shown below)? All $369$ free octominoes?

Also, is there an existing OEIS sequence for this? If not, I'll add one once there is a bit more data. (The sequence begins $1, 2, 4, 6, 9, 12, \cdots$.)