3D Statue Park: U shapes

Question

This is a three-dimensional Statue Park puzzle.¹ The five squares in the below image depict the five levels of a $4\times4\times5$ cuboid. The goal is to fit inside it the four three-dimensional pieces shown in the picture, so that the resulting space satisfies the following conditions:

• The pieces can be rotated or reflected along any axis.
• No piece can be orthogonally adjacent to another piece.
• The cells not occupied by pieces must all be orthogonally connected.
• Cells marked with a black circle must be part of a piece, and cells marked with a white circle cannot be part of a piece.

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_{¹ The Statue Park puzzle type was invented by Palmer Mebane. The original rules can be found on his blog.}

Answer 1

Step 1:

The two black dots on the top layer tell us that two Us must be placed vertically, going downwards into the the puzzle.

Step 2:

The two remaining black dots cannot be part of the same shape. The dot on layer 4 cannot be part of a U that is entirely on layer 4. There cannot be a dot above it. Using the left given white dot on layer 2, the only way for a U to be placed is to be parallel to the two other Us, with the given dot at the very bottom.

Step 3:

This resolves the orientation of the left U-shape on layer 1 (as well as the right U-shape, which could have been resolved earlier using the right white dot on layer 2). The final U-shape then must be placed entirely on layer 5, and there is only one way to place it (since it cannot occupy any of column 2 except for the bottom cell).

Answer 2

Deusovi beat me but here's the solution in Minecraft:

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