## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of vitamin E', 'milligrams of vitamin C', 'milligrams of vitamin K']. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_1$ for 'milligrams of vitamin E'
- $x_2$ for 'milligrams of vitamin C'
- $x_3$ for 'milligrams of vitamin K'

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is $5.01x_1 + 5.67x_2 + 6.82x_3$.

## 4: List the constraints in symbolic notation
The constraints are:
1. $2x_1 \leq 25$ (cognitive performance index of $x_1$ has an upper bound, but first, we need to establish its direct relationship)
2. $2x_2 \leq 25$
3. $5x_3 \leq 25$
4. $2x_1 + 2x_2 \geq 7$
5. $2x_1 + 2x_2 + 5x_3 \geq 5$
6. $2x_1 + 2x_2 + 5x_3 \geq 5$ (redundant with constraint 5)
7. $6x_1 - 6x_2 \geq 0$
8. $-4x_2 + x_3 \geq 0$
9. $2x_1 + 5x_3 \leq 21$
10. $2x_2 + x_3 \leq 19$

## 5: Correct and simplify constraints based on the problem description
From the problem description:
- The cognitive performance index for $x_1$ is 2, for $x_2$ is 2, and for $x_3$ is 5.
- Direct constraints:
  - $2x_1 \leq 25$
  - $2x_2 \leq 25$
  - $5x_3 \leq 25$

## 6: Establish the correct set of constraints
1. $2x_1 \leq 25$
2. $2x_2 \leq 25$
3. $5x_3 \leq 25$
4. $2x_1 + 2x_2 \geq 7$
5. $2x_1 + 2x_2 + 5x_3 \geq 5$
6. $6x_1 - 6x_2 \geq 0$
7. $-4x_2 + x_3 \geq 0$
8. $2x_1 + 5x_3 \leq 21$
9. $2x_2 + x_3 \leq 19$

## 7: Write down the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'milligrams of vitamin E'), ('x2', 'milligrams of vitamin C'), ('x3', 'milligrams of vitamin K')],
    'objective_function': '5.01x1 + 5.67x2 + 6.82x3',
    'constraints': [
        '2x1 <= 25',
        '2x2 <= 25',
        '5x3 <= 25',
        '2x1 + 2x2 >= 7',
        '2x1 + 2x2 + 5x3 >= 5',
        '6x1 - 6x2 >= 0',
        '-4x2 + x3 >= 0',
        '2x1 + 5x3 <= 21',
        '2x2 + x3 <= 19'
    ]
}
```

## 8: Implement the optimization problem using Gurobi
```python
import gurobi

def optimize_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="milligrams_of_vitamin_E", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="milligrams_of_vitamin_C", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x3 = model.addVar(name="milligrams_of_vitamin_K", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Objective function
    model.setObjective(5.01 * x1 + 5.67 * x2 + 6.82 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(2 * x1 <= 25)
    model.addConstr(2 * x2 <= 25)
    model.addConstr(5 * x3 <= 25)
    model.addConstr(2 * x1 + 2 * x2 >= 7)
    model.addConstr(2 * x1 + 2 * x2 + 5 * x3 >= 5)
    model.addConstr(6 * x1 - 6 * x2 >= 0)
    model.addConstr(-4 * x2 + x3 >= 0)
    model.addConstr(2 * x1 + 5 * x3 <= 21)
    model.addConstr(2 * x2 + x3 <= 19)

    # Optimize the model
    model.optimize()

    # Print the status
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin E: {x1.varValue}")
        print(f"Milligrams of vitamin C: {x2.varValue}")
        print(f"Milligrams of vitamin K: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```