To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of tea and coffee tins, formulating the objective function that represents the profit, and listing the constraints related to the time available for filling and labeling.

Let's denote:
- \(x_1\) as the number of tea tins,
- \(x_2\) as the number of coffee tins.

The profit per tea tin is $11, and per coffee tin is $13. Therefore, the objective function to maximize profit can be written as:
\[ \text{Maximize:} \quad 11x_1 + 13x_2 \]

The constraints are based on the time available for filling and labeling:
- Each tea tin takes 4 minutes to fill, and each coffee tin takes 3 minutes to fill. The total time available for filling is 500 minutes. This gives us the first constraint: \(4x_1 + 3x_2 \leq 500\).
- Each tea tin takes 1 minute to label, and each coffee tin takes 2 minutes to label. The total time available for labeling is 600 minutes. This gives us the second constraint: \(x_1 + 2x_2 \leq 600\).

Additionally, we have non-negativity constraints since the number of tins cannot be negative:
- \(x_1 \geq 0\),
- \(x_2 \geq 0\).

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of tea tins'), ('x2', 'number of coffee tins')],
    'objective_function': '11*x1 + 13*x2',
    'constraints': ['4*x1 + 3*x2 <= 500', 'x1 + 2*x2 <= 600', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Tea_Coffee_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="tea_tins", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="coffee_tins", lb=0)

# Set the objective function
m.setObjective(11*x1 + 13*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*x1 + 3*x2 <= 500, "filling_time")
m.addConstr(x1 + 2*x2 <= 600, "labeling_time")

# Optimize the model
m.optimize()

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