## Step 1: Define the symbolic representation of the problem
The problem involves three variables: 'milligrams of vitamin B7', 'milligrams of vitamin B12', and 'milligrams of calcium'. 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 $1 \cdot x_1 + 7 \cdot x_2 + 1 \cdot x_3$.

## 3: List the constraints in symbolic notation
The constraints given are:
1. $11x_1 \geq 82$ is not correct, it is $11x_1 + 7x_3 \geq 82$
2. $11x_1 + 7x_2 \geq 77$
3. $11x_1 + 7x_2 + 7x_3 \geq 77$
4. $7x_1 + 22x_2 \geq 42$
5. $7x_1 + 22x_2 + 18x_3 \geq 42$
6. $20x_2 + 13x_3 \geq 58$
7. $3x_1 + 13x_3 \geq 50$
8. $3x_1 + 20x_2 + 13x_3 \geq 50$
9. $6x_1 - 2x_2 \geq 0$
10. $11x_1 + 7x_3 \leq 161$
11. $11x_1 + 7x_2 + 7x_3 \leq 196$
12. $7x_1 + 18x_3 \leq 221$
13. $3x_1 + 20x_2 + 13x_3 \leq 93$

## 4: Define the symbolic representation of the problem in JSON format
```json
{
    'sym_variables': [('x1', 'milligrams of vitamin B7'), ('x2', 'milligrams of vitamin B12'), ('x3', 'milligrams of calcium')],
    'objective_function': '1*x1 + 7*x2 + 1*x3',
    'constraints': [
        '11*x1 + 7*x3 >= 82',
        '11*x1 + 7*x2 >= 77',
        '11*x1 + 7*x2 + 7*x3 >= 77',
        '7*x1 + 22*x2 >= 42',
        '7*x1 + 22*x2 + 18*x3 >= 42',
        '20*x2 + 13*x3 >= 58',
        '3*x1 + 13*x3 >= 50',
        '3*x1 + 20*x2 + 13*x3 >= 50',
        '6*x1 - 2*x2 >= 0',
        '11*x1 + 7*x3 <= 161',
        '11*x1 + 7*x2 + 7*x3 <= 196',
        '7*x1 + 18*x3 <= 221',
        '3*x1 + 20*x2 + 13*x3 <= 93'
    ]
}
```

## Step 5: Write the Gurobi code for the optimization problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="x1", lb=0)  # milligrams of vitamin B7
    x2 = model.addVar(name="x2", lb=0)  # milligrams of vitamin B12
    x3 = model.addVar(name="x3", lb=0)  # milligrams of calcium

    # Objective function: Minimize 1*x1 + 7*x2 + 1*x3
    model.setObjective(1 * x1 + 7 * x2 + 1 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(11 * x1 + 7 * x3 >= 82)
    model.addConstr(11 * x1 + 7 * x2 >= 77)
    model.addConstr(11 * x1 + 7 * x2 + 7 * x3 >= 77)
    model.addConstr(7 * x1 + 22 * x2 >= 42)
    model.addConstr(7 * x1 + 22 * x2 + 18 * x3 >= 42)
    model.addConstr(20 * x2 + 13 * x3 >= 58)
    model.addConstr(3 * x1 + 13 * x3 >= 50)
    model.addConstr(3 * x1 + 20 * x2 + 13 * x3 >= 50)
    model.addConstr(6 * x1 - 2 * x2 >= 0)
    model.addConstr(11 * x1 + 7 * x3 <= 161)
    model.addConstr(11 * x1 + 7 * x2 + 7 * x3 <= 196)
    model.addConstr(7 * x1 + 18 * x3 <= 221)
    model.addConstr(3 * x1 + 20 * x2 + 13 * x3 <= 93)

    # Optimize the model
    model.optimize()

    # Print the 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()
```