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```#include <mpi.h>
#include <vector>

// A function to perform local matrix multiplication.
void local_matrix_multiply(const std::vector<int>& A, const std::vector<int>& B, std::vector<int>& C, int block_size) {
// Implement your matrix multiplication logic here.
// For example, a simple triple nested loop for matrix multiplication.
}

int main(int argc, char* argv[]) {
MPI_Init(&argc, &argv);

// Assuming we have Px * Px * Pz processors and square matrices of size N
const int Px = 2; // Number of processors in each row/column in a 2D layer
const int Pz = 2; // Number of layers
const int N = 4; // Matrix size (N x N)

int rank, size;
MPI_Comm_rank(MPI_COMM_WORLD, &rank);
MPI_Comm_size(MPI_COMM_WORLD, &size);

// Validate our assumptions
if(size != Px * Px * Pz) {
if(rank == 0) {
std::cerr << "The number of processors does not match Px * Px * Pz." << std::endl;
}
MPI_Finalize();
return -1;
}

// Create 3D Cartesian topology for processors
int dims[3] = {Px, Px, Pz};
int periods[3] = {1, 1, 1}; // Enable wrapping for circular shifts
MPI_Comm grid_comm;
MPI_Cart_create(MPI_COMM_WORLD, 3, dims, periods, 1, &grid_comm);

// Get new rank and coordinates in the grid
int grid_rank, coords[3];
MPI_Comm_rank(grid_comm, &grid_rank);
MPI_Cart_coords(grid_comm, grid_rank, 3, coords);

// Calculate block size for each processor
int block_size = N / (Px * Pz); // Assuming N is evenly divisible by Px * Pz

// Allocate memory for local blocks of A, B, and C
std::vector<int> A_block(block_size * block_size, 0);
std::vector<int> B_block(block_size * block_size, 0);
std::vector<int> C_block(block_size * block_size, 0);

// Distribute initial blocks of A and B to processors (omitted for brevity)
// ...

// Perform initial local matrix multiplication
local_matrix_multiply(A_block, B_block, C_block, block_size);

// Get rank of right and downward neighbors for A and B shifting
int right_src, right_dst, down_src, down_dst;
MPI_Cart_shift(grid_comm, 1, 1, &right_src, &right_dst); // Shift along the second dimension (j)
MPI_Cart_shift(grid_comm, 0, 1, &down_src, &down_dst); // Shift along the first dimension (i)

// Circular shift and local multiplication for Pz steps
for(int step = 0; step < Pz; ++step) {
// Circularly shift A rightwards and B downwards
std::vector<int> A_recv(block_size * block_size), B_recv(block_size * block_size);
MPI_Status status;

// Shift A rightwards
MPI_Sendrecv_replace(A_block.data(), block_size * block_size, MPI_INT, right_dst, 0, right_src, 0, grid_comm, &status);

// Shift B downwards
MPI_Sendrecv_replace(B_block.data(), block_size * block_size, MPI_INT, down_dst, 1, down_src, 1, grid_comm, &status);

// Perform local matrix multiplication with the received blocks
local_matrix_multiply(A_block, B_block, C_block, block_size);
}

// Gather the local C blocks into the global C matrix (omitted for brevity)
// ...

// Clean up
MPI_Comm_free(&grid_comm);
MPI_Finalize();

return 0;
}
```