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

Let's denote the number of soccer balls as $x_1$ and the number of basket balls as $x_2$. The profit per soccer ball is $4, and the profit per basket ball is $5. The manufacturing machine time constraint can be represented as $5x_1 + 7x_2 \leq 700$, and the air filling time constraint can be represented as $3x_1 + 4x_2 \leq 500$. The objective is to maximize the profit $4x_1 + 5x_2$.

## Step 2: Write down the symbolic representation in the required format

The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'soccer balls'), ('x2', 'basket balls')],
    'objective_function': '4*x1 + 5*x2',
    'constraints': [
        '5*x1 + 7*x2 <= 700',
        '3*x1 + 4*x2 <= 500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 3: Convert the symbolic representation into Gurobi code

Now, let's convert this symbolic representation into Gurobi code in Python:

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="soccer_balls", lb=0, ub=None)
    x2 = model.addVar(name="basket_balls", lb=0, ub=None)

    # Define the objective function
    model.setObjective(4 * x1 + 5 * x2, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(5 * x1 + 7 * x2 <= 700, name="manufacturing_machine_time")
    model.addConstr(3 * x1 + 4 * x2 <= 500, name="air_filling_time")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: soccer balls = {x1.varValue}, basket balls = {x2.varValue}")
        print(f"Maximum profit: ${4 * x1.varValue + 5 * x2.varValue:.2f}")
    else:
        print("No optimal solution found")

solve_factory_problem()
```