## Step 1: Define the symbolic representation of the problem
The problem involves three variables: 'milligrams of vitamin B2', 'milligrams of vitamin D', and 'milligrams of magnesium'. Let's denote these as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $1 \times x_1 + 3 \times x_2 + 6 \times x_3$.

## 3: List the constraints in symbolic notation
The constraints given are:
- $1x_1 = 1$
- $14x_1 = 14$
- $12x_1 = 12$
- $3x_2 = 3$
- $4x_2 = 4$
- $6x_2 = 6$
- $13x_3 = 13$
- $1x_3 = 1$
- $9x_3 = 9$
- $1x_1 + 3x_2 \geq 20$
- $1x_1 + 3x_2 + 13x_3 \geq 20$
- $14x_1 + 1x_3 \geq 19$
- $14x_1 + 4x_2 \geq 16$
- $14x_1 + 4x_2 + 1x_3 \geq 12$
- $14x_1 + 1x_3 \geq 24$
- $14x_1 + 6x_2 \geq 10$
- $14x_1 + 6x_2 + 9x_3 \geq 10$
- $-9x_2 + 6x_3 \geq 0$
- $4x_2 + 1x_3 \leq 47$
- $14x_1 + 6x_2 \leq 69$
- $6x_2 + 9x_3 \leq 40$
- $x_1$ is an integer
- $x_2$ is a continuous variable
- $x_3$ is a continuous variable

## 4: Correct the constraints based on the problem description
Upon reviewing, notice that some constraints are directly stated as equalities or inequalities without coefficients, implying their direct translation:
- The muscle growth index for $x_1$ is 1: $1x_1 = 1$
- The digestive support index for $x_1$ is 14: $14x_1 = 14$
- The immune support index for $x_1$ is 12: $12x_1 = 12$
- Similarly, for $x_2$ and $x_3$, their indices are given but seem to directly translate to specific values rather than coefficients in constraints.

## 5: Formulate the problem in Gurobi code
Given the complexity and the specific request for format, let's directly formulate the problem.

```json
{
    'sym_variables': [('x1', 'milligrams of vitamin B2'), ('x2', 'milligrams of vitamin D'), ('x3', 'milligrams of magnesium')],
    'objective_function': '1*x1 + 3*x2 + 6*x3',
    'constraints': [
        'x1 = 1',
        '14*x1 = 14',
        '12*x1 = 12',
        '3*x2 = 3',
        '4*x2 = 4',
        '6*x2 = 6',
        '13*x3 = 13',
        '1*x3 = 1',
        '9*x3 = 9',
        'x1 + 3*x2 >= 20',
        'x1 + 3*x2 + 13*x3 >= 20',
        '14*x1 + x3 >= 19',
        '14*x1 + 4*x2 >= 16',
        '14*x1 + 4*x2 + x3 >= 12',
        '14*x1 + x3 >= 24',
        '14*x1 + 6*x2 >= 10',
        '14*x1 + 6*x2 + 9*x3 >= 10',
        '-9*x2 + 6*x3 >= 0',
        '4*x2 + x3 <= 47',
        '14*x1 + 6*x2 <= 69',
        '6*x2 + 9*x3 <= 40'
    ]
}
```

## 6: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # milligrams of vitamin B2
    x2 = model.addVar(name="x2")  # milligrams of vitamin D
    x3 = model.addVar(name="x3")  # milligrams of magnesium

    # Objective function
    model.setObjective(1*x1 + 3*x2 + 6*x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 == 1)
    model.addConstr(14*x1 == 14)
    model.addConstr(12*x1 == 12)
    model.addConstr(3*x2 == 3)
    model.addConstr(4*x2 == 4)
    model.addConstr(6*x2 == 6)
    model.addConstr(13*x3 == 13)
    model.addConstr(x3 == 1)
    model.addConstr(9*x3 == 9)
    model.addConstr(x1 + 3*x2 >= 20)
    model.addConstr(x1 + 3*x2 + 13*x3 >= 20)
    model.addConstr(14*x1 + x3 >= 19)
    model.addConstr(14*x1 + 4*x2 >= 16)
    model.addConstr(14*x1 + 4*x2 + x3 >= 12)
    model.addConstr(14*x1 + x3 >= 24)
    model.addConstr(14*x1 + 6*x2 >= 10)
    model.addConstr(14*x1 + 6*x2 + 9*x3 >= 10)
    model.addConstr(-9*x2 + 6*x3 >= 0)
    model.addConstr(4*x2 + x3 <= 47)
    model.addConstr(14*x1 + 6*x2 <= 69)
    model.addConstr(6*x2 + 9*x3 <= 40)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"x1: {x1.varValue}")
        print(f"x2: {x2.varValue}")
        print(f"x3: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```