## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

## Step 2: Identify the variables and their corresponding symbolic representations
The variables are 'cherry pies', 'oreos', and 'apples', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## 3: Translate the objective function into algebraic terms
The objective function to minimize is $6x_0^2 + 7x_0x_1 + 9x_0x_2 + 4x_1^2 + x_1x_2 + 6x_2^2 + 8x_1 + x_2$.

## 4: List the constraints in semi-algebraic form
Constraints:
1. $9x_1 + x_2 \geq 27$ (calcium from oreos and apples)
2. $16x_0^2 + 9x_1^2 \geq 29$ (calcium from cherry pies squared and oreos squared)
3. $16x_0 + 9x_1 + x_2 \geq 29$ (calcium from cherry pies, oreos, and apples)
4. $13^2x_0^2 + 8^2x_2^2 \geq 42$ (sourness index from cherry pies squared and apples squared)
5. $2x_1 + 8x_2 \geq 16$ (sourness index from oreos and apples)
6. $13x_0 + 2x_1 + 8x_2 \geq 16$ (sourness index from cherry pies, oreos, and apples)
7. $12x_0 + 5x_1 \geq 16$ (healthiness rating from cherry pies and oreos)
8. $5^2x_1^2 + 2^2x_2^2 \geq 20$ (healthiness rating from oreos squared and apples squared)
9. $12x_0 + 5x_1 + 2x_2 \geq 20$ (healthiness rating from cherry pies, oreos, and apples)
10. $5x_0 + 5x_1 \geq 27$ (cost of cherry pies and oreos)
11. $5^2x_1^2 + 8^2x_2^2 \geq 40$ (cost of oreos squared and apples squared)
12. $5x_0 + 5x_1 + 8x_2 \geq 40$ (cost of cherry pies, oreos, and apples)
13. $x_0 - 9x_2 \geq 0$ (relationship between cherry pies and apples)
14. $9x_1 + x_2 \leq 80$ (calcium limit from oreos and apples)
15. $13^2x_0^2 + 2^2x_1^2 + 8^2x_2^2 \leq 94$ (sourness index limit)

## 5: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'cherry pies'), ('x1', 'oreos'), ('x2', 'apples')],
    'objective_function': '6*x0^2 + 7*x0*x1 + 9*x0*x2 + 4*x1^2 + x1*x2 + 6*x2^2 + 8*x1 + x2',
    'constraints': [
        '9*x1 + x2 >= 27',
        '16*x0^2 + 9*x1^2 >= 29',
        '16*x0 + 9*x1 + x2 >= 29',
        '169*x0^2 + 64*x2^2 >= 42',
        '2*x1 + 8*x2 >= 16',
        '13*x0 + 2*x1 + 8*x2 >= 16',
        '12*x0 + 5*x1 >= 16',
        '25*x1^2 + 4*x2^2 >= 20',
        '12*x0 + 5*x1 + 2*x2 >= 20',
        '5*x0 + 5*x1 >= 27',
        '25*x1^2 + 64*x2^2 >= 40',
        '5*x0 + 5*x1 + 8*x2 >= 40',
        'x0 - 9*x2 >= 0',
        '9*x1 + x2 <= 80',
        '169*x0^2 + 4*x1^2 + 64*x2^2 <= 94'
    ]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="cherry_pies", lb=-gp.GRB.INFINITY)
x1 = m.addVar(name="oreos", lb=-gp.GRB.INFINITY)
x2 = m.addVar(name="apples", lb=-gp.GRB.INFINITY)

# Define the objective function
m.setObjective(6*x0**2 + 7*x0*x1 + 9*x0*x2 + 4*x1**2 + x1*x2 + 6*x2**2 + 8*x1 + x2, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(9*x1 + x2 >= 27)
m.addConstr(16*x0**2 + 9*x1**2 >= 29)
m.addConstr(16*x0 + 9*x1 + x2 >= 29)
m.addConstr(169*x0**2 + 64*x2**2 >= 42)
m.addConstr(2*x1 + 8*x2 >= 16)
m.addConstr(13*x0 + 2*x1 + 8*x2 >= 16)
m.addConstr(12*x0 + 5*x1 >= 16)
m.addConstr(25*x1**2 + 4*x2**2 >= 20)
m.addConstr(12*x0 + 5*x1 + 2*x2 >= 20)
m.addConstr(5*x0 + 5*x1 >= 27)
m.addConstr(25*x1**2 + 64*x2**2 >= 40)
m.addConstr(5*x0 + 5*x1 + 8*x2 >= 40)
m.addConstr(x0 - 9*x2 >= 0)
m.addConstr(9*x1 + x2 <= 80)
m.addConstr(169*x0**2 + 4*x1**2 + 64*x2**2 <= 94)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0 (cherry pies): ", x0.varValue)
    print("x1 (oreos): ", x1.varValue)
    print("x2 (apples): ", x2.varValue)
else:
    print("The model is infeasible")
```