To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables for the quantities of coats and skirts, formulating the objective function that represents the total profit, and listing the constraints based on the available designing and printing time.

Let's denote:
- \(x_1\) as the number of coats produced,
- \(x_2\) as the number of skirts produced.

The objective function, which is to maximize the total profit, can be represented as:
\[12x_1 + 16x_2\]

Given the constraints:
1. Designing time: \(1.5x_1 + 3x_2 \leq 45\)
2. Printing time: \(2.5x_1 + 3.5x_2 \leq 70\)

Non-negativity constraints (since we cannot produce a negative number of items):
- \(x_1 \geq 0\)
- \(x_2 \geq 0\)

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of coats'), ('x2', 'number of 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']
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("X-Luxury_Cloth")

# Add variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="coats", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="skirts", lb=0)

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

# Add constraints
m.addConstr(1.5*x1 + 3*x2 <= 45, "designing_time")
m.addConstr(2.5*x1 + 3.5*x2 <= 70, "printing_time")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of coats: {x1.x}")
    print(f"Number of skirts: {x2.x}")
    print(f"Total profit: {m.objVal}")
else:
    print("No optimal solution found")
```