## Step 1: Define the symbolic representation of the variables
The variables are 'blueberry pies', 'cherry pies', and 'peanutbutter sandwiches'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_0$ for 'blueberry pies'
- $x_1$ for 'cherry pies'
- $x_2$ for 'peanutbutter sandwiches'

## 3: Define the objective function in symbolic notation
The objective function to maximize is $4.12x_0 + 5.17x_1 + 3.44x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
1. $4.51x_0 \leq 214$ (dollar cost constraint for blueberry pies, but we have a general budget constraint later)
2. $14.6x_0 \leq 314$ (calcium constraint for blueberry pies)
3. $4.11x_1 \leq 214$ (dollar cost constraint for cherry pies, but we have a specific budget constraint for $x_1$ and $x_2$)
4. $8.51x_1 \leq 314$ (calcium constraint for cherry pies)
5. $13.38x_2 \leq 214$ (dollar cost constraint for peanutbutter sandwiches)
6. $7.36x_2 \leq 314$ (calcium constraint for peanutbutter sandwiches)
7. $8.51x_1 + 7.36x_2 \geq 77$ (calcium from cherry pies and peanutbutter sandwiches)
8. $-5x_0 + 7x_1 \geq 0$ (relationship between blueberry pies and cherry pies)
9. $4.11x_1 + 13.38x_2 \leq 135$ (budget for cherry pies and peanutbutter sandwiches)
10. $4.51x_0 + 4.11x_1 + 13.38x_2 \leq 135$ (total budget constraint, but it seems there is an upper bound of 214 for total cost, we will use that)
11. $14.6x_0 + 7.36x_2 \leq 256$ (calcium from blueberry pies and peanutbutter sandwiches)
12. $8.51x_1 + 7.36x_2 \leq 282$ (calcium from cherry pies and peanutbutter sandwiches)
13. $14.6x_0 + 8.51x_1 + 7.36x_2 \leq 282$ (total calcium constraint)

## 5: Consider variable bounds and types
- $x_0$ can be fractional
- $x_1$ must be an integer
- $x_2$ must be an integer

## 6: Define the symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'blueberry pies'), ('x1', 'cherry pies'), ('x2', 'peanutbutter sandwiches')],
'objective_function': '4.12*x0 + 5.17*x1 + 3.44*x2',
'constraints': [
    '4.51*x0 <= 214',
    '14.6*x0 <= 314',
    '8.51*x1 + 7.36*x2 >= 77',
    '-5*x0 + 7*x1 >= 0',
    '4.11*x1 + 13.38*x2 <= 135',
    '4.51*x0 + 4.11*x1 + 13.38*x2 <= 214',
    '14.6*x0 + 7.36*x2 <= 256',
    '8.51*x1 + 7.36*x2 <= 282',
    '14.6*x0 + 8.51*x1 + 7.36*x2 <= 282'
]
}
```

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

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

    # Define variables
    x0 = model.addVar(name="blueberry_pies", lb=0, ub=None)  # Can be fractional
    x1 = model.addVar(name="cherry_pies", lb=0, integer=True)  # Must be an integer
    x2 = model.addVar(name="peanutbutter_sandwiches", lb=0, integer=True)  # Must be an integer

    # Objective function
    model.setObjective(4.12 * x0 + 5.17 * x1 + 3.44 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(4.51 * x0 <= 214)  # Dollar cost constraint for blueberry pies
    model.addConstr(14.6 * x0 <= 314)  # Calcium constraint for blueberry pies
    model.addConstr(8.51 * x1 + 7.36 * x2 >= 77)  # Calcium from cherry pies and peanutbutter sandwiches
    model.addConstr(-5 * x0 + 7 * x1 >= 0)  # Relationship between blueberry pies and cherry pies
    model.addConstr(4.11 * x1 + 13.38 * x2 <= 135)  # Budget for cherry pies and peanutbutter sandwiches
    model.addConstr(4.51 * x0 + 4.11 * x1 + 13.38 * x2 <= 214)  # Total budget constraint
    model.addConstr(14.6 * x0 + 7.36 * x2 <= 256)  # Calcium from blueberry pies and peanutbutter sandwiches
    model.addConstr(8.51 * x1 + 7.36 * x2 <= 282)  # Calcium from cherry pies and peanutbutter sandwiches
    model.addConstr(14.6 * x0 + 8.51 * x1 + 7.36 * x2 <= 282)  # Total calcium constraint

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Blueberry pies: ", x0.varValue)
        print("Cherry pies: ", x1.varValue)
        print("Peanutbutter sandwiches: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```