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

Let's denote:
- $x_1$ as the number of ducks shot
- $x_2$ as the number of geese shot

The objective is to maximize points, with each duck shot worth 5 points and each goose shot worth 6 points. Therefore, the objective function can be represented as $5x_1 + 6x_2$.

## Step 2: List the constraints based on the problem description

1. Shoot at least 5 ducks: $x_1 \geq 5$
2. Shoot at least 3 geese: $x_2 \geq 3$
3. Shoot at most 12 ducks: $x_1 \leq 12$
4. Shoot at most 7 geese: $x_2 \leq 7$
5. Total animals shot is at most 15: $x_1 + x_2 \leq 15$

## 3: Symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'ducks'), ('x2', 'geese')], 
    'objective_function': '5*x1 + 6*x2', 
    'constraints': [
        'x1 >= 5', 
        'x2 >= 3', 
        'x1 <= 12', 
        'x2 <= 7', 
        'x1 + x2 <= 15'
    ]
}
```

## 4: Convert the problem into Gurobi code

To solve this linear programming problem using Gurobi, we will use the Gurobi Python API.

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="ducks")
    x2 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="geese")

    # Set objective function
    model.setObjective(5*x1 + 6*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 >= 5, name="min_ducks")
    model.addConstr(x2 >= 3, name="min_geese")
    model.addConstr(x1 <= 12, name="max_ducks")
    model.addConstr(x2 <= 7, name="max_geese")
    model.addConstr(x1 + x2 <= 15, name="total_animals")

    # Update model
    model.update()

    # Solve model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution:")
        print(f"Ducks: {x1.varValue}")
        print(f"Geese: {x2.varValue}")
        print(f"Max points: {5*x1.varValue + 6*x2.varValue}")
    else:
        print("No optimal solution found")

solve_arcade_shooter_problem()
```