## Step 1: Define the symbolic representation of the variables
The variables are 'ham sandwiches', 'bowls of cereal', and 'black beans', which we will denote as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $7.18x_1x_2 + 6.24x_2^2 + 6.71x_2x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $8x_1 + 10x_2 \geq 26$
2. $8x_1 + 13x_3 \geq 26$
3. $10x_2 + 13x_3 \geq 34$
4. $8x_1 + 10x_2 + 13x_3 \geq 34$
5. $4x_1 - 3x_3 \geq 0$
6. $10x_2 + 13x_3 \leq 97$
7. $8x_1 + 10x_2 \leq 78$
8. $x_1$ is an integer
9. $x_2$ is an integer
10. $x_3$ is an integer

## 4: Create the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'ham sandwiches'), ('x2', 'bowls of cereal'), ('x3', 'black beans')],
'objective_function': '7.18*x1*x2 + 6.24*x2^2 + 6.71*x2*x3',
'constraints': [
    '8*x1 + 10*x2 >= 26',
    '8*x1 + 13*x3 >= 26',
    '10*x2 + 13*x3 >= 34',
    '8*x1 + 10*x2 + 13*x3 >= 34',
    '4*x1 - 3*x3 >= 0',
    '10*x2 + 13*x3 <= 97',
    '8*x1 + 10*x2 <= 78',
    'x1 is an integer',
    'x2 is an integer',
    'x3 is an integer'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name="ham_sandwiches", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="bowls_of_cereal", vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(name="black_beans", vtype=gurobi.GRB.INTEGER)

# Set the objective function
m.setObjective(7.18*x1*x2 + 6.24*x2**2 + 6.71*x2*x3, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(8*x1 + 10*x2 >= 26)
m.addConstr(8*x1 + 13*x3 >= 26)
m.addConstr(10*x2 + 13*x3 >= 34)
m.addConstr(8*x1 + 10*x2 + 13*x3 >= 34)
m.addConstr(4*x1 - 3*x3 >= 0)
m.addConstr(10*x2 + 13*x3 <= 97)
m.addConstr(8*x1 + 10*x2 <= 78)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Ham sandwiches: ", x1.varValue)
    print("Bowls of cereal: ", x2.varValue)
    print("Black beans: ", x3.varValue)
else:
    print("The model is infeasible")
```