## Step 1: Define the symbolic representation of the problem

Let's denote:
- $x_1$ as the number of coats
- $x_2$ as the number of skirts

The objective function to maximize profit is: $12x_1 + 16x_2$

The constraints based on the given information are:
- Designing time constraint: $1.5x_1 + 3x_2 \leq 45$
- Printing time constraint: $2.5x_1 + 3.5x_2 \leq 70$
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Step 2: Convert the problem into a Gurobi-compatible format

The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'coats'), ('x2', 'skirts')],
    'objective_function': '12*x1 + 16*x2',
    'constraints': [
        '1.5*x1 + 3*x2 <= 45',
        '2.5*x1 + 3.5*x2 <= 70',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 3: Write the Gurobi code to solve the optimization problem

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(lb=0, name="coats")
    x2 = model.addVar(lb=0, name="skirts")

    # Set the objective function
    model.setObjective(12 * x1 + 16 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(1.5 * x1 + 3 * x2 <= 45, name="designing_time_constraint")
    model.addConstr(2.5 * x1 + 3.5 * x2 <= 70, name="printing_time_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of coats: {x1.varValue}")
        print(f"Number of skirts: {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```