3D Space Exploration

Farmer John's cows have finally blasted off from Earth and are now drifting through space in their Moocraft. To reach their kin on Io, a moon of Jupiter, they must first cross a dangerous asteroid belt.

Bessie is piloting the craft through an $N \times N \times N$ ($1 \le N \le 100$) sector of space. Every asteroid in the sector is a group of $1 \times 1 \times 1$ blocks of space-rock joined face to face. Two blocks that touch only along an edge, or only at a single corner, belong to different asteroids.

Count the number of distinct asteroids in the sector.

The sector is given as $N$ square slices stacked one behind another (front to back). Each slice is an $N \times N$ grid in which * marks a block of space-rock and . marks empty space. A block at slice $s$, row $r$, column $c$ shares a face with the blocks at $(s\pm1, r, c)$, $(s, r\pm1, c)$, and $(s, r, c\pm1)$, and blocks joined this way belong to the same asteroid.

For example, consider a $3 \times 3 \times 3$ sector. Here M marks the Moocraft's start at $(1,1,1)$ and D marks the destination at $(3,3,3)$; these markers only help orient the map and are not needed to solve the problem.

   Close slice    Middle slice     Far Slice     Assembled with overlaps
     +---+            +---+          +---+                  +---+ Far
     |M..|            |..*|          |...|                  |...|
     |.*.|            |.*.|          |.*.|                +---+.|
     |...|            |*..|          |..D|                |..*|D|
     +---+            +---+          +---+              +---+.|-+
                                                        |M..|.|
                                                        |.*.|-+
                                                        |...|
                                                  Close +---+

Visual inspection shows three asteroids, including a long one running straight through the middle of the sector. The next figure labels each block with the number of the asteroid it belongs to:

                                 +---+ Far
                                 /|...|
                                / |.1.|
                               /  |...|
                              /   +---+
                             +---+   /
                            /|..3|  /
                           / |.1.| /
                          /  |2..|/
                         /   +---+
                        +---+   /
                        |...|  /
                        |.1.| /
                        |...|/
                  Close +---+

• The first line contains a single integer $N$ ($1 \le N \le 100$).
• The next $N^2$ lines describe the $N$ slices in order. Each slice is given as $N$ lines of exactly $N$ characters, where each character is * (a block of space-rock) or . (empty space).
• Counting after the first line, lines $s \cdot N + 1$ through $s \cdot N + N$ (for $s = 0, 1, \dots, N-1$) form slice $s$. Within a slice the lines are its rows and the characters within a line are its columns, so the same row and column of two adjacent slices are aligned and share a face.

• Print a single integer: the number of distinct asteroids in the sector.