## Step 1: Define the symbolic representation of the problem
The problem involves variables 'bananas', 'eggs', and 'apple pies' which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $3.99x_0 + 4.83x_1 + 2.67x_2$.

## 3: List the constraints in symbolic notation
Constraints:
1. $29x_0 + 15x_1 + 1x_2 \leq 160$ (fat)
2. $28x_0 + 28x_1 + 8x_2 \leq 210$ (calcium)
3. $24x_0 + 5x_1 + 25x_2 \leq 365$ (umami index)
4. $8x_0 + 23x_1 + 20x_2 \leq 348$ (carbohydrates)
5. $29x_0 + 1x_2 \geq 52$ (fat from bananas and apple pies)
6. $29x_0 + 15x_1 \geq 35$ (fat from bananas and eggs)
7. $15x_1 + 1x_2 \geq 29$ (fat from eggs and apple pies)
8. $29x_0 + 15x_1 + 1x_2 \geq 44$ (total fat from all)
9. $29x_0 + 15x_1 + 1x_2 \geq 44$ (same as above, redundant)
10. $28x_0 + 8x_2 \geq 33$ (calcium from bananas and apple pies)
11. $28x_0 + 28x_1 \geq 60$ (calcium from bananas and eggs)
12. $28x_0 + 28x_1 + 8x_2 \geq 60$ (calcium from all)
13. $24x_0 + 25x_2 \geq 119$ (umami from bananas and apple pies)
14. $24x_0 + 5x_1 \geq 83$ (umami from bananas and eggs)
15. $24x_0 + 5x_1 + 25x_2 \geq 117$ (umami from all)
16. $8x_0 + 23x_1 \geq 76$ (carbohydrates from bananas and eggs)
17. $23x_1 + 20x_2 \geq 71$ (carbohydrates from eggs and apple pies)
18. $8x_0 + 23x_1 + 20x_2 \geq 71$ (carbohydrates from all)
19. $6x_0 - 4x_1 \geq 0$
20. $6x_1 - 9x_2 \geq 0$
21. $29x_0 + 15x_1 \leq 160$ (fat from bananas and eggs)
22. $29x_0 + 1x_2 \leq 68$ (fat from bananas and apple pies)
23. $28x_0 + 8x_2 \leq 190$ (calcium from bananas and apple pies)
24. $28x_0 + 28x_1 \leq 85$ (calcium from bananas and eggs)
25. $28x_1 + 8x_2 \leq 109$ (calcium from eggs and apple pies)
26. $8x_0 + 23x_1 \leq 150$ (carbohydrates from bananas and eggs)
27. $8x_0 + 23x_1 + 20x_2 \leq 310$ (carbohydrates from all)

## 4: Define the symbolic variables and problem representation
```json
{
    'sym_variables': [
        ('x0', 'bananas'),
        ('x1', 'eggs'),
        ('x2', 'apple pies')
    ],
    'objective_function': '3.99*x0 + 4.83*x1 + 2.67*x2',
    'constraints': [
        '29*x0 + 15*x1 + 1*x2 <= 160',
        '28*x0 + 28*x1 + 8*x2 <= 210',
        '24*x0 + 5*x1 + 25*x2 <= 365',
        '8*x0 + 23*x1 + 20*x2 <= 348',
        '29*x0 + 1*x2 >= 52',
        '29*x0 + 15*x1 >= 35',
        '15*x1 + 1*x2 >= 29',
        '29*x0 + 15*x1 + 1*x2 >= 44',
        '28*x0 + 8*x2 >= 33',
        '28*x0 + 28*x1 >= 60',
        '28*x0 + 28*x1 + 8*x2 >= 60',
        '24*x0 + 25*x2 >= 119',
        '24*x0 + 5*x1 >= 83',
        '24*x0 + 5*x1 + 25*x2 >= 117',
        '8*x0 + 23*x1 >= 76',
        '23*x1 + 20*x2 >= 71',
        '8*x0 + 23*x1 + 20*x2 >= 71',
        '6*x0 - 4*x1 >= 0',
        '6*x1 - 9*x2 >= 0',
        '29*x0 + 15*x1 <= 160',
        '29*x0 + 1*x2 <= 68',
        '28*x0 + 8*x2 <= 190',
        '28*x0 + 28*x1 <= 85',
        '28*x1 + 8*x2 <= 109',
        '8*x0 + 23*x1 <= 150',
        '8*x0 + 23*x1 + 20*x2 <= 310'
    ]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="bananas", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="eggs", vtype=gurobi.GRB.CONTINUOUS)
x2 = m.addVar(name="apple_pies", vtype=gurobi.GRB.CONTINUOUS)

# Objective function
m.setObjective(3.99*x0 + 4.83*x1 + 2.67*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(29*x0 + 15*x1 + x2 <= 160)
m.addConstr(28*x0 + 28*x1 + 8*x2 <= 210)
m.addConstr(24*x0 + 5*x1 + 25*x2 <= 365)
m.addConstr(8*x0 + 23*x1 + 20*x2 <= 348)
m.addConstr(29*x0 + x2 >= 52)
m.addConstr(29*x0 + 15*x1 >= 35)
m.addConstr(15*x1 + x2 >= 29)
m.addConstr(29*x0 + 15*x1 + x2 >= 44)
m.addConstr(28*x0 + 8*x2 >= 33)
m.addConstr(28*x0 + 28*x1 >= 60)
m.addConstr(28*x0 + 28*x1 + 8*x2 >= 60)
m.addConstr(24*x0 + 25*x2 >= 119)
m.addConstr(24*x0 + 5*x1 >= 83)
m.addConstr(24*x0 + 5*x1 + 25*x2 >= 117)
m.addConstr(8*x0 + 23*x1 >= 76)
m.addConstr(23*x1 + 20*x2 >= 71)
m.addConstr(8*x0 + 23*x1 + 20*x2 >= 71)
m.addConstr(6*x0 - 4*x1 >= 0)
m.addConstr(6*x1 - 9*x2 >= 0)
m.addConstr(29*x0 + 15*x1 <= 160)
m.addConstr(29*x0 + x2 <= 68)
m.addConstr(28*x0 + 8*x2 <= 190)
m.addConstr(28*x0 + 28*x1 <= 85)
m.addConstr(28*x1 + 8*x2 <= 109)
m.addConstr(8*x0 + 23*x1 <= 150)
m.addConstr(8*x0 + 23*x1 + 20*x2 <= 310)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Bananas: ", x0.varValue)
    print("Eggs: ", x1.varValue)
    print("Apple Pies: ", x2.varValue)
else:
    print("The model is infeasible")
```