To solve this optimization problem using Gurobi, we need to translate the given objective function and constraints into a mathematical formulation that can be represented in code. 

The objective is to maximize: $1 \times \text{chicken breasts} + 6 \times \text{sashimi} + 3 \times \text{chicken drumsticks} + 1 \times \text{hamburgers}$.

Constraints:
- Healthiness ratings are given for each food item.
- Multiple constraints regarding the total combined healthiness rating of various combinations of these items, including minimum and maximum requirements.

Given variables:
- $x_0$: chicken breasts
- $x_1$: sashimi
- $x_2$: chicken drumsticks
- $x_3$: hamburgers

Healthiness ratings:
- Chicken breasts: 1.17
- Sashimi: 1.68
- Chicken drumsticks: 1.79
- Hamburgers: 3.04

We will define our variables and constraints according to the problem statement, ensuring that we respect the integer requirements for $x_1$, $x_2$, and $x_3$ (sashimi, chicken drumsticks, and hamburgers) but allow for non-integer values of $x_0$ (chicken breasts).

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="chicken_breasts")
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="sashimi")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="chicken_drumsticks")
x3 = m.addVar(lb=0, vtype=GRB.INTEGER, name="hamburgers")

# Objective function
m.setObjective(1*x0 + 6*x1 + 3*x2 + 1*x3, GRB.MAXIMIZE)

# Constraints
m.addConstr(1.17*x0 + 1.79*x2 + 3.04*x3 >= 23)
m.addConstr(1.68*x1 + 1.79*x2 + 3.04*x3 >= 23)
m.addConstr(1.17*x0 + 1.68*x1 + 1.79*x2 >= 23)
m.addConstr(1.17*x0 + 1.79*x2 + 3.04*x3 >= 16)
m.addConstr(1.68*x1 + 1.79*x2 + 3.04*x3 >= 16)
m.addConstr(1.17*x0 + 1.68*x1 + 1.79*x2 >= 16)
m.addConstr(1.17*x0 + 1.79*x2 + 3.04*x3 >= 26)
m.addConstr(1.68*x1 + 1.79*x2 + 3.04*x3 >= 26)
m.addConstr(1.17*x0 + 1.68*x1 + 1.79*x2 >= 26)
m.addConstr(1.17*x0 + 1.68*x1 <= 34)
m.addConstr(1.79*x2 + 3.04*x3 <= 101)
m.addConstr(1.17*x0 + 3.04*x3 <= 122)
m.addConstr(1.68*x1 + 3.04*x3 <= 106)
m.addConstr(1.68*x1 + 1.79*x2 + 3.04*x3 <= 41)
m.addConstr(1.17*x0 + 1.79*x2 + 3.04*x3 <= 67)
m.addConstr(1.17*x0 + 1.68*x1 + 3.04*x3 <= 111)
m.addConstr(1.17*x0 + 1.68*x1 + 1.79*x2 + 3.04*x3 <= 111)

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    for v in m.getVars():
        print("%s = %g" % (v.varName, v.x))
else:
    print("No optimal solution found")
```