## Step 1: Define the variables and the objective function
The variables are ['milligrams of vitamin B7', 'grams of fat', 'milligrams of potassium', 'grams of protein']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. The objective function to maximize is $5x_0 + 2x_1 + 2x_2 + 6x_3$.

## Step 2: Define the constraints
The constraints given can be categorized into two types: equality and inequality. We will express them using the variables $x_0$, $x_1$, $x_2$, and $x_3$.

### Energy Stability Index Constraints
1. $13.69x_0 \geq 80$
2. $6.32x_1 + 2.58x_3 \geq 55$
3. $6.32x_1 + 12.48x_2 \geq 54$
4. $13.69x_0 + 6.32x_1 + 12.48x_2 \geq 69$
5. $13.69x_0 + 12.48x_2 + 2.58x_3 \geq 69$
6. $6.32x_1 + 12.48x_2 + 2.58x_3 \geq 69$
7. $13.69x_0 + 6.32x_1 + 12.48x_2 \geq 67$
8. $13.69x_0 + 12.48x_2 + 2.58x_3 \geq 67$
9. $6.32x_1 + 12.48x_2 + 2.58x_3 \geq 67$
10. $13.69x_0 + 6.32x_1 + 12.48x_2 \geq 70$
11. $13.69x_0 + 12.48x_2 + 2.58x_3 \geq 70$
12. $6.32x_1 + 12.48x_2 + 2.58x_3 \geq 70$

### Cognitive Performance Index Constraints
13. $20.29x_0 + 13.15x_2 \geq 60$
14. $20.29x_0 + 19.67x_1 \geq 43$
15. $13.15x_2 + 24.51x_3 \geq 103$
16. $19.67x_1 + 24.51x_3 \geq 78$
17. $19.67x_1 + 13.15x_2 + 24.51x_3 \geq 109$

### Linear Constraints
18. $5x_0 - x_2 + 8x_3 \geq 0$
19. $13.69x_0 + 2.58x_3 \leq 226$
20. $13.69x_0 + 6.32x_1 \leq 270$
21. $6.32x_1 + 2.58x_3 \leq 104$
22. $6.32x_1 + 12.48x_2 \leq 165$
23. $13.69x_0 + 6.32x_1 + 12.48x_2 + 2.58x_3 \leq 165$
24. $20.29x_0 + 13.15x_2 \leq 198$
25. $19.67x_1 + 13.15x_2 \leq 222$
26. $13.15x_2 + 24.51x_3 \leq 120$
27. $20.29x_0 + 19.67x_1 \leq 317$
28. $20.29x_0 + 19.67x_1 + 13.15x_2 + 24.51x_3 \leq 317$

## Step 3: Implement the problem in Gurobi
We will use Gurobi's Python interface to model and solve this problem.

```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="milligrams_of_vitamin_B7", lb=0)
    x1 = model.addVar(name="grams_of_fat", lb=0, integrality=1)
    x2 = model.addVar(name="milligrams_of_potassium", lb=0)
    x3 = model.addVar(name="grams_of_protein", lb=0)

    # Objective function
    model.setObjective(5*x0 + 2*x1 + 2*x2 + 6*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    # Energy Stability Index Constraints
    model.addConstr(13.69*x0 >= 80)
    model.addConstr(6.32*x1 + 2.58*x3 >= 55)
    model.addConstr(6.32*x1 + 12.48*x2 >= 54)
    model.addConstr(13.69*x0 + 6.32*x1 + 12.48*x2 >= 69)
    model.addConstr(13.69*x0 + 12.48*x2 + 2.58*x3 >= 69)
    model.addConstr(6.32*x1 + 12.48*x2 + 2.58*x3 >= 69)
    model.addConstr(13.69*x0 + 6.32*x1 + 12.48*x2 >= 67)
    model.addConstr(13.69*x0 + 12.48*x2 + 2.58*x3 >= 67)
    model.addConstr(6.32*x1 + 12.48*x2 + 2.58*x3 >= 67)
    model.addConstr(13.69*x0 + 6.32*x1 + 12.48*x2 >= 70)
    model.addConstr(13.69*x0 + 12.48*x2 + 2.58*x3 >= 70)
    model.addConstr(6.32*x1 + 12.48*x2 + 2.58*x3 >= 70)

    # Cognitive Performance Index Constraints
    model.addConstr(20.29*x0 + 13.15*x2 >= 60)
    model.addConstr(20.29*x0 + 19.67*x1 >= 43)
    model.addConstr(13.15*x2 + 24.51*x3 >= 103)
    model.addConstr(19.67*x1 + 24.51*x3 >= 78)
    model.addConstr(19.67*x1 + 13.15*x2 + 24.51*x3 >= 109)

    # Linear Constraints
    model.addConstr(5*x0 - x2 + 8*x3 >= 0)
    model.addConstr(13.69*x0 + 2.58*x3 <= 226)
    model.addConstr(13.69*x0 + 6.32*x1 <= 270)
    model.addConstr(6.32*x1 + 2.58*x3 <= 104)
    model.addConstr(6.32*x1 + 12.48*x2 <= 165)
    model.addConstr(13.69*x0 + 6.32*x1 + 12.48*x2 + 2.58*x3 <= 165)
    model.addConstr(20.29*x0 + 13.15*x2 <= 198)
    model.addConstr(19.67*x1 + 13.15*x2 <= 222)
    model.addConstr(13.15*x2 + 24.51*x3 <= 120)
    model.addConstr(20.29*x0 + 19.67*x1 <= 317)
    model.addConstr(20.29*x0 + 19.67*x1 + 13.15*x2 + 24.51*x3 <= 317)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("milligrams of vitamin B7: ", x0.varValue)
        print("grams of fat: ", x1.varValue)
        print("milligrams of potassium: ", x2.varValue)
        print("grams of protein: ", x3.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```