## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are 'rotisserie chickens', 'hot dogs', and 'chicken thighs', which we can denote as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to maximize is $4x_1 + 8x_2 + 7x_3$.

## 3: List the constraints in symbolic notation
The constraints given are:
- $2x_1 + 21x_2 + 17x_3 \leq 283$ (dollar cost constraint)
- $10x_1 + 6x_2 + 13x_3 \leq 368$ (carbohydrates constraint)
- $6x_2 + 13x_3 \geq 41$ (carbohydrates from hot dogs and chicken thighs)
- $10x_1 + 13x_3 \geq 44$ (carbohydrates from rotisserie chickens and chicken thighs)
- $10x_1 + 6x_2 \geq 86$ (carbohydrates from rotisserie chickens and hot dogs)
- $2x_1 + 21x_2 \leq 205$ (cost constraint for rotisserie chickens and hot dogs)
- $21x_2 + 17x_3 \leq 180$ (cost constraint for hot dogs and chicken thighs)
- $2x_1 + 21x_2 + 17x_3 \leq 107$ (total cost constraint, noted but seems to be a repetition or error given the context of a 283 upper bound)
- $6x_2 + 13x_3 \leq 333$ (carbohydrates from hot dogs and chicken thighs upper bound)
- $10x_1 + 6x_2 + 13x_3 \leq 157$ (total carbohydrates constraint, seems to be an error or repetition given the 368 upper bound)
- $x_1$ is an integer (rotisserie chickens must be a whole number)
- $x_2$ and $x_3$ are not restricted to integers (hot dogs and chicken thighs can be fractional)

## 4: Correct and Clarify Constraints
Upon review, there seems to be a mistake in the problem statement regarding the total cost and carbohydrate limits. The constraints should be considered as:
- $2x_1 + 21x_2 + 17x_3 \leq 283$
- $10x_1 + 6x_2 + 13x_3 \leq 368$
- And the specific bounds and requirements as listed.

## 5: Symbolic Representation
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'rotisserie chickens'), ('x2', 'hot dogs'), ('x3', 'chicken thighs')],
'objective_function': '4*x1 + 8*x2 + 7*x3',
'constraints': [
    '2*x1 + 21*x2 + 17*x3 <= 283',
    '10*x1 + 6*x2 + 13*x3 <= 368',
    '6*x2 + 13*x3 >= 41',
    '10*x1 + 13*x3 >= 44',
    '10*x1 + 6*x2 >= 86',
    '2*x1 + 21*x2 <= 205',
    '21*x2 + 17*x3 <= 180',
    '2*x1 + 21*x2 + 17*x3 <= 107',
    '6*x2 + 13*x3 <= 333',
    '10*x1 + 6*x2 + 13*x3 <= 157',
    'x1 >= 0 and x1 is an integer',
    'x2 >= 0',
    'x3 >= 0'
]
}
```

## 6: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="rotisserie_chickens", vtype=gurobi.GRB.INTEGER, lb=0)
    x2 = model.addVar(name="hot_dogs", vtype=gurobi.GRB.CONTINUOUS, lb=0)
    x3 = model.addVar(name="chicken_thighs", vtype=gurobi.GRB.CONTINUOUS, lb=0)

    # Objective function
    model.setObjective(4*x1 + 8*x2 + 7*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(2*x1 + 21*x2 + 17*x3 <= 283)
    model.addConstr(10*x1 + 6*x2 + 13*x3 <= 368)
    model.addConstr(6*x2 + 13*x3 >= 41)
    model.addConstr(10*x1 + 13*x3 >= 44)
    model.addConstr(10*x1 + 6*x2 >= 86)
    model.addConstr(2*x1 + 21*x2 <= 205)
    model.addConstr(21*x2 + 17*x3 <= 180)
    model.addConstr(2*x1 + 21*x2 + 17*x3 <= 107)
    model.addConstr(6*x2 + 13*x3 <= 333)
    model.addConstr(10*x1 + 6*x2 + 13*x3 <= 157)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Rotisserie Chickens: ", x1.varValue)
        print("Hot Dogs: ", x2.varValue)
        print("Chicken Thighs: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```