## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are ['chicken drumsticks', 'bowls of pasta', 'corn cobs', 'black beans'], which we can denote as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to maximize is $7x_0 + 5x_1 + 7x_2 + 4x_3$.

## 3: List the constraints in symbolic notation
The constraints are:
- $19x_0 \leq 423$
- $12x_0 \leq 556$
- $4x_1 \leq 423$
- $10x_1 \leq 556$
- $4x_2 \leq 423$
- $22x_2 \leq 556$
- $16x_3 \leq 423$
- $4x_3 \leq 556$
- $19x_0 + 16x_3 \geq 56$
- $19x_0 + 4x_2 \geq 38$
- $4x_1 + 4x_2 \geq 36$
- $19x_0 + 4x_1 + 4x_2 \geq 53$
- $19x_0 + 4x_2 + 16x_3 \geq 53$
- $4x_1 + 4x_2 + 16x_3 \geq 53$
- $19x_0 + 4x_1 + 4x_2 \geq 66$
- $19x_0 + 4x_2 + 16x_3 \geq 66$
- $4x_1 + 4x_2 + 16x_3 \geq 66$
- $19x_0 + 4x_1 + 4x_2 \geq 62$
- $19x_0 + 4x_2 + 16x_3 \geq 62$
- $4x_1 + 4x_2 + 16x_3 \geq 62$
- $12x_0 + 10x_1 \geq 58$
- $12x_0 + 10x_1 + 4x_3 \geq 86$
- $12x_0 + 22x_2 + 4x_3 \geq 86$
- $12x_0 + 10x_1 + 4x_3 \geq 98$
- $12x_0 + 22x_2 + 4x_3 \geq 98$
- $19x_0 + 4x_1 \leq 159$
- $4x_1 + 16x_3 \leq 176$
- $4x_1 + 4x_2 \leq 159$
- $19x_0 + 4x_1 + 4x_2 \leq 302$
- $4x_1 + 4x_2 + 16x_3 \leq 141$
- $19x_0 + 4x_1 + 16x_3 \leq 328$
- $19x_0 + 4x_1 + 4x_2 + 16x_3 \leq 328$
- $12x_0 + 22x_2 \leq 328$
- $10x_1 + 4x_3 \leq 435$
- $12x_0 + 10x_1 + 22x_2 + 4x_3 \leq 435$

## 4: Define the symbolic variables and constraints for the JSON output
```json
{
    'sym_variables': [
        ('x0', 'chicken drumsticks'),
        ('x1', 'bowls of pasta'),
        ('x2', 'corn cobs'),
        ('x3', 'black beans')
    ],
    'objective_function': '7*x0 + 5*x1 + 7*x2 + 4*x3',
    'constraints': [
        '19*x0 <= 423',
        '12*x0 <= 556',
        '4*x1 <= 423',
        '10*x1 <= 556',
        '4*x2 <= 423',
        '22*x2 <= 556',
        '16*x3 <= 423',
        '4*x3 <= 556',
        '19*x0 + 16*x3 >= 56',
        '19*x0 + 4*x2 >= 38',
        '4*x1 + 4*x2 >= 36',
        '19*x0 + 4*x1 + 4*x2 >= 53',
        '19*x0 + 4*x2 + 16*x3 >= 53',
        '4*x1 + 4*x2 + 16*x3 >= 53',
        '19*x0 + 4*x1 + 4*x2 >= 66',
        '19*x0 + 4*x2 + 16*x3 >= 66',
        '4*x1 + 4*x2 + 16*x3 >= 66',
        '19*x0 + 4*x1 + 4*x2 >= 62',
        '19*x0 + 4*x2 + 16*x3 >= 62',
        '4*x1 + 4*x2 + 16*x3 >= 62',
        '12*x0 + 10*x1 >= 58',
        '12*x0 + 10*x1 + 4*x3 >= 86',
        '12*x0 + 22*x2 + 4*x3 >= 86',
        '12*x0 + 10*x1 + 4*x3 >= 98',
        '12*x0 + 22*x2 + 4*x3 >= 98',
        '19*x0 + 4*x1 <= 159',
        '4*x1 + 16*x3 <= 176',
        '4*x1 + 4*x2 <= 159',
        '19*x0 + 4*x1 + 4*x2 <= 302',
        '4*x1 + 4*x2 + 16*x3 <= 141',
        '19*x0 + 4*x1 + 16*x3 <= 328',
        '19*x0 + 4*x1 + 4*x2 + 16*x3 <= 328',
        '12*x0 + 22*x2 <= 328',
        '10*x1 + 4*x3 <= 435',
        '12*x0 + 10*x1 + 22*x2 + 4*x3 <= 435'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="chicken_drumsticks", lb=0)
    x1 = model.addVar(name="bowls_of_pasta", lb=0)
    x2 = model.addVar(name="corn_cobs", lb=0, integrality=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="black_beans", lb=0)

    # Objective function
    model.setObjective(7 * x0 + 5 * x1 + 7 * x2 + 4 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(19 * x0 <= 423)
    model.addConstr(12 * x0 <= 556)
    model.addConstr(4 * x1 <= 423)
    model.addConstr(10 * x1 <= 556)
    model.addConstr(4 * x2 <= 423)
    model.addConstr(22 * x2 <= 556)
    model.addConstr(16 * x3 <= 423)
    model.addConstr(4 * x3 <= 556)
    model.addConstr(19 * x0 + 16 * x3 >= 56)
    model.addConstr(19 * x0 + 4 * x2 >= 38)
    model.addConstr(4 * x1 + 4 * x2 >= 36)
    model.addConstr(19 * x0 + 4 * x1 + 4 * x2 >= 53)
    model.addConstr(19 * x0 + 4 * x2 + 16 * x3 >= 53)
    model.addConstr(4 * x1 + 4 * x2 + 16 * x3 >= 53)
    model.addConstr(19 * x0 + 4 * x1 + 4 * x2 >= 66)
    model.addConstr(19 * x0 + 4 * x2 + 16 * x3 >= 66)
    model.addConstr(4 * x1 + 4 * x2 + 16 * x3 >= 66)
    model.addConstr(19 * x0 + 4 * x1 + 4 * x2 >= 62)
    model.addConstr(19 * x0 + 4 * x2 + 16 * x3 >= 62)
    model.addConstr(4 * x1 + 4 * x2 + 16 * x3 >= 62)
    model.addConstr(12 * x0 + 10 * x1 >= 58)
    model.addConstr(12 * x0 + 10 * x1 + 4 * x3 >= 86)
    model.addConstr(12 * x0 + 22 * x2 + 4 * x3 >= 86)
    model.addConstr(12 * x0 + 10 * x1 + 4 * x3 >= 98)
    model.addConstr(12 * x0 + 22 * x2 + 4 * x3 >= 98)
    model.addConstr(19 * x0 + 4 * x1 <= 159)
    model.addConstr(4 * x1 + 16 * x3 <= 176)
    model.addConstr(4 * x1 + 4 * x2 <= 159)
    model.addConstr(19 * x0 + 4 * x1 + 4 * x2 <= 302)
    model.addConstr(4 * x1 + 4 * x2 + 16 * x3 <= 141)
    model.addConstr(19 * x0 + 4 * x1 + 16 * x3 <= 328)
    model.addConstr(19 * x0 + 4 * x1 + 4 * x2 + 16 * x3 <= 328)
    model.addConstr(12 * x0 + 22 * x2 <= 328)
    model.addConstr(10 * x1 + 4 * x3 <= 435)
    model.addConstr(12 * x0 + 10 * x1 + 22 * x2 + 4 * x3 <= 435)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Chicken drumsticks: ", x0.varValue)
        print("Bowls of pasta: ", x1.varValue)
        print("Corn cobs: ", x2.varValue)
        print("Black beans: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```