## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables and the objective function, and list the constraints.

### Variables
- $x_1$ represents the number of small balls thrown.
- $x_2$ represents the number of large balls thrown.

### Objective Function
The objective is to maximize the score, which is given by $5x_1 + 2x_2$.

### Constraints
1. Total balls thrown: $x_1 + x_2 \leq 20$
2. Minimum small balls: $x_1 \geq 6$
3. Minimum large balls: $x_2 \geq 5$
4. Maximum of either type: $x_1 \leq 12$ and $x_2 \leq 12$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'small balls'), ('x2', 'large balls')],
    'objective_function': '5*x1 + 2*x2',
    'constraints': [
        'x1 + x2 <= 20',
        'x1 >= 6',
        'x2 >= 5',
        'x1 <= 12',
        'x2 <= 12'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, ub=12, name="small_balls", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(lb=0, ub=12, name="large_balls", vtype=gurobi.GRB.INTEGER)

    # Objective function: maximize 5*x1 + 2*x2
    model.setObjective(5*x1 + 2*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 + x2 <= 20, name="total_balls")
    model.addConstr(x1 >= 6, name="min_small_balls")
    model.addConstr(x2 >= 5, name="min_large_balls")
    model.addConstr(x1 <= 12, name="max_small_balls")
    model.addConstr(x2 <= 12, name="max_large_balls")

    # Update model
    model.update()

    # Solve the problem
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal score: {model.objVal}")
        print(f"Throw {x1.varValue} small balls and {x2.varValue} large balls.")
    else:
        print("The problem is infeasible.")

solve_problem()
```