## Step 1: Define the symbolic representation of the variables
The variables are 'chicken thighs', 'chicken breasts', 'hot dogs', and 'blueberry pies', which can be represented symbolically as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $6x_0x_1 + 7x_0x_3 + 3x_2^2 + 5x_2x_3 + 9x_0 + x_1 + 5x_2 + 3x_3$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $5x_0 + 2x_1 + 5x_2 + 9x_3 \leq 84$ (total protein)
- $4x_0 + 6x_1 + x_2 + 4x_3 \leq 58$ (total healthiness rating)
- $2x_0 + 5x_1 + 8x_2 + x_3 \leq 79$ (total fat)
- $5x_0 + 5x_2 \geq 8$ (protein from chicken thighs and hot dogs)
- $5x_0 + 2x_1 \geq 19$ (protein from chicken thighs and breasts)
- $2x_1 + 9x_3 \geq 18$ (protein from chicken breasts and blueberry pies)
- $2x_0^2 + 8x_2^2 \geq 10$ (fat from chicken thighs and hot dogs)
- $2x_1 + 9x_3 \leq 54$ (protein from chicken breasts and blueberry pies)
- $5x_0^2 + 5x_2^2 \leq 48$ (protein from chicken thighs and hot dogs squared)
- $5x_2 + 9x_3 \leq 47$ (protein from hot dogs and blueberry pies)
- $5x_0 + 2x_1 + 5x_2 \leq 45$ (protein from chicken thighs, breasts, and hot dogs)
- $5x_0 + 5x_2 + 9x_3 \leq 49$ (protein from chicken thighs, hot dogs, and blueberry pies)
- $5x_0 + 2x_1 + 5x_2 + 9x_3 \leq 49$ (protein from all)
- $6x_1 + 4x_3 \leq 24$ (healthiness rating from chicken breasts and blueberry pies)
- $x_2^2 + 4x_3^2 \leq 54$ (healthiness rating from hot dogs and blueberry pies squared)
- $4x_0 + 6x_1 + x_2 + 4x_3 \leq 54$ (total healthiness rating)
- $2x_0 + x_3 \leq 72$ (fat from chicken thighs and blueberry pies)
- $2x_0 + 8x_2 \leq 21$ (fat from chicken thighs and hot dogs)
- $5x_1 + x_3 \leq 69$ (fat from chicken breasts and blueberry pies)
- $5x_1 + 8x_2 \leq 70$ (fat from chicken breasts and hot dogs)
- $2x_0 + 5x_1 + 8x_2 + x_3 \leq 70$ (total fat)

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'chicken thighs'),
        ('x1', 'chicken breasts'),
        ('x2', 'hot dogs'),
        ('x3', 'blueberry pies')
    ],
    'objective_function': '6*x0*x1 + 7*x0*x3 + 3*x2^2 + 5*x2*x3 + 9*x0 + x1 + 5*x2 + 3*x3',
    'constraints': [
        '5*x0 + 2*x1 + 5*x2 + 9*x3 <= 84',
        '4*x0 + 6*x1 + x2 + 4*x3 <= 58',
        '2*x0 + 5*x1 + 8*x2 + x3 <= 79',
        '5*x0 + 5*x2 >= 8',
        '5*x0 + 2*x1 >= 19',
        '2*x1 + 9*x3 >= 18',
        '2*x0^2 + 8*x2^2 >= 10',
        '2*x1 + 9*x3 <= 54',
        '5*x0^2 + 5*x2^2 <= 48',
        '5*x2 + 9*x3 <= 47',
        '5*x0 + 2*x1 + 5*x2 <= 45',
        '5*x0 + 5*x2 + 9*x3 <= 49',
        '5*x0 + 2*x1 + 5*x2 + 9*x3 <= 49',
        '6*x1 + 4*x3 <= 24',
        'x2^2 + 4*x3^2 <= 54',
        '4*x0 + 6*x1 + x2 + 4*x3 <= 54',
        '2*x0 + x3 <= 72',
        '2*x0 + 8*x2 <= 21',
        '5*x1 + x3 <= 69',
        '5*x1 + 8*x2 <= 70',
        '2*x0 + 5*x1 + 8*x2 + x3 <= 70',
        'x0 % 1 == 0',
        'x1 % 1 == 0',
        'x3 % 1 == 0'
    ]
}
```

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

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="chicken_thighs", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="chicken_breasts", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hot_dogs")
x3 = m.addVar(name="blueberry_pies", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(6*x0*x1 + 7*x0*x3 + 3*x2**2 + 5*x2*x3 + 9*x0 + x1 + 5*x2 + 3*x3, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(5*x0 + 2*x1 + 5*x2 + 9*x3 <= 84)
m.addConstr(4*x0 + 6*x1 + x2 + 4*x3 <= 58)
m.addConstr(2*x0 + 5*x1 + 8*x2 + x3 <= 79)
m.addConstr(5*x0 + 5*x2 >= 8)
m.addConstr(5*x0 + 2*x1 >= 19)
m.addConstr(2*x1 + 9*x3 >= 18)
m.addConstr(2*x0**2 + 8*x2**2 >= 10)
m.addConstr(2*x1 + 9*x3 <= 54)
m.addConstr(5*x0**2 + 5*x2**2 <= 48)
m.addConstr(5*x2 + 9*x3 <= 47)
m.addConstr(5*x0 + 2*x1 + 5*x2 <= 45)
m.addConstr(5*x0 + 5*x2 + 9*x3 <= 49)
m.addConstr(5*x0 + 2*x1 + 5*x2 + 9*x3 <= 49)
m.addConstr(6*x1 + 4*x3 <= 24)
m.addConstr(x2**2 + 4*x3**2 <= 54)
m.addConstr(4*x0 + 6*x1 + x2 + 4*x3 <= 54)
m.addConstr(2*x0 + x3 <= 72)
m.addConstr(2*x0 + 8*x2 <= 21)
m.addConstr(5*x1 + x3 <= 69)
m.addConstr(5*x1 + 8*x2 <= 70)
m.addConstr(2*x0 + 5*x1 + 8*x2 + x3 <= 70)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Chicken thighs: ", x0.varValue)
    print("Chicken breasts: ", x1.varValue)
    print("Hot dogs: ", x2.varValue)
    print("Blueberry pies: ", x3.varValue)
else:
    print("The model is infeasible")
```