## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of hours to run the north factory.
- $x_2$ represents the number of hours to run the south factory.

## Step 2: Convert the natural language description into a symbolic representation
The objective is to minimize the cost, which can be represented as $200x_1 + 400x_2$.

## 3: Define the constraints based on the natural language description
The constraints are:
- $20x_1 + 30x_2 \geq 75$ (minimum of 75 long coats)
- $15x_1 + 25x_2 \geq 30$ (minimum of 30 short coats)
- $10x_1 + 30x_2 \geq 40$ (minimum of 40 mini coats)
- $x_1 \geq 0$ and $x_2 \geq 0$ (non-negativity constraints, as the number of hours cannot be negative)

## 4: Express the problem in the required symbolic format
```json
{
'sym_variables': [('x1', 'hours at north factory'), ('x2', 'hours at south factory')],
'objective_function': '200*x1 + 400*x2',
'constraints': [
    '20*x1 + 30*x2 >= 75',
    '15*x1 + 25*x2 >= 30',
    '10*x1 + 30*x2 >= 40',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 5: Translate the symbolic representation into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="x1", lb=0, ub=gurobi.GRB.INFINITY, obj=200)
    x2 = model.addVar(name="x2", lb=0, ub=gurobi.GRB.INFINITY, obj=400)

    # Add constraints
    model.addConstr(20*x1 + 30*x2 >= 75, name="long_coats")
    model.addConstr(15*x1 + 25*x2 >= 30, name="short_coats")
    model.addConstr(10*x1 + 30*x2 >= 40, name="mini_coats")

    # Set the objective function
    model.setObjective(200*x1 + 400*x2, gurobi.GRB.MINIMIZE)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours at north factory: {x1.x}")
        print(f"Hours at south factory: {x2.x}")
        print(f"Total cost: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_production_planning_problem()
```