Implement sparse matrix storage, addition, and multiplication

Design a way to store a sparse matrix (most entries are zero) and implement efficient operations.

You are given matrices using their non-zero entries:

• Matrix A has shape (m, n) and is represented as a list of triples (row, col, value) containing only entries where value != 0 .
• Matrix B has shape (n, p) (for multiplication) and/or shape (m, n) (for addition), represented the same way.

Implement the following:

1. Sparse storage : Choose a representation suitable for efficient operations.
2. Addition : Compute A + B (only when shapes match), returning the result in the same sparse triple-list form (do not output explicit zeros).
3. Multiplication : Compute A × B , returning the result in sparse triple-list form.

Constraints

• Dimensions can be large (e.g., up to 1e5 in each direction), but the number of non-zero entries k is much smaller than m*n .
• Input triples may not be sorted.

Output requirements

• Return only non-zero entries.
• You may return triples in any order unless otherwise specified.