## Step 1: Define the symbolic representation of the problem
Let's denote the number of chickens as $x_1$ and the number of potatoes as $x_2$. The objective is to minimize the cost, which is $6x_1 + 2x_2$. The constraints are:
- Protein: $20x_1 + 2x_2 \geq 80$
- Fat: $4x_1 + 3x_2 \geq 50$
- Carbs: $4x_1 + 7x_2 \geq 100$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

## 2: Convert the problem into a symbolic representation
The symbolic variables are:
- $x_1$ for chickens
- $x_2$ for potatoes

The objective function is: $6x_1 + 2x_2$

The constraints are:
- $20x_1 + 2x_2 \geq 80$
- $4x_1 + 3x_2 \geq 50$
- $4x_1 + 7x_2 \geq 100$
- $x_1 \geq 0$
- $x_2 \geq 0$

## 3: Represent the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'chicken'), ('x2', 'potato')],
    'objective_function': '6*x1 + 2*x2',
    'constraints': [
        '20*x1 + 2*x2 >= 80',
        '4*x1 + 3*x2 >= 50',
        '4*x1 + 7*x2 >= 100',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 4: Implement the problem in Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="chicken", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="potato", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Define the objective function
    model.setObjective(6 * x1 + 2 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(20 * x1 + 2 * x2 >= 80, name="protein_constraint")
    model.addConstr(4 * x1 + 3 * x2 >= 50, name="fat_constraint")
    model.addConstr(4 * x1 + 7 * x2 >= 100, name="carbs_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of chickens: {x1.varValue}")
        print(f"Number of potatoes: {x2.varValue}")
        print(f"Total cost: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```