## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of iron', 'grams of fiber', 'milligrams of vitamin A']. 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 iron'
- $x_2$ for 'grams of fiber'
- $x_3$ for 'milligrams of vitamin A'

## 3: Define the objective function in symbolic notation
The objective function to maximize is: $1 \cdot x_1^2 + 3 \cdot x_1 \cdot x_2 + 4 \cdot x_1 \cdot x_3 + 9 \cdot x_2 \cdot x_3 + 7 \cdot x_3^2 + 3 \cdot x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $27x_1 \leq 234$
- $14x_2 \leq 234$
- $1x_3 \leq 234$
- $27x_1 + 1x_3 \geq 58$
- $27x_1 + 14x_2 + 1x_3 \geq 62$
- $27^2x_1^2 + 1^2x_3^2 \leq 177$
- $27x_1 + 14x_2 + 1x_3 \leq 177$
- $x_2$ is an integer
- $x_1, x_3$ can be fractional

## 5: Express the problem in a JSON format for clarity
```json
{
'sym_variables': [('x1', 'milligrams of iron'), ('x2', 'grams of fiber'), ('x3', 'milligrams of vitamin A')],
'objective_function': '1*x1^2 + 3*x1*x2 + 4*x1*x3 + 9*x2*x3 + 7*x3^2 + 3*x2',
'constraints': [
'27*x1 <= 234',
'14*x2 <= 234',
'1*x3 <= 234',
'27*x1 + 1*x3 >= 58',
'27*x1 + 14*x2 + 1*x3 >= 62',
'(27*x1)^2 + (1*x3)^2 <= 177',
'27*x1 + 14*x2 + 1*x3 <= 177',
'x2 is an integer'
]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi as gp
from gurobi import GRB

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_iron")  # Can be fractional
x2 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="grams_of_fiber", vtype=GRB.INTEGER)  # Must be an integer
x3 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_vitamin_A")  # Can be fractional

# Objective function
m.setObjective(1*x1**2 + 3*x1*x2 + 4*x1*x3 + 9*x2*x3 + 7*x3**2 + 3*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(27*x1 <= 234, name="cognitive_performance_index_iron")
m.addConstr(14*x2 <= 234, name="cognitive_performance_index_fiber")
m.addConstr(x3 <= 234, name="cognitive_performance_index_vitamin_A")
m.addConstr(27*x1 + x3 >= 58, name="combined_cognitive_performance_index_iron_vitamin_A")
m.addConstr(27*x1 + 14*x2 + x3 >= 62, name="combined_cognitive_performance_index_all")
m.addConstr((27*x1)**2 + x3**2 <= 177, name="combined_cognitive_performance_index_squared")
m.addConstr(27*x1 + 14*x2 + x3 <= 177, name="combined_cognitive_performance_index_all_leq")

# Solve the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Milligrams of iron: {x1.varValue}")
    print(f"Grams of fiber: {x2.varValue}")
    print(f"Milligrams of vitamin A: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

The final answer is: 
```python
import gurobi as gp
from gurobi import GRB

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_iron")  
x2 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="grams_of_fiber", vtype=GRB.INTEGER)  
x3 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, name="milligrams_of_vitamin_A")  

# Objective function
m.setObjective(1*x1**2 + 3*x1*x2 + 4*x1*x3 + 9*x2*x3 + 7*x3**2 + 3*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(27*x1 <= 234, name="cognitive_performance_index_iron")
m.addConstr(14*x2 <= 234, name="cognitive_performance_index_fiber")
m.addConstr(x3 <= 234, name="cognitive_performance_index_vitamin_A")
m.addConstr(27*x1 + x3 >= 58, name="combined_cognitive_performance_index_iron_vitamin_A")
m.addConstr(27*x1 + 14*x2 + x3 >= 62, name="combined_cognitive_performance_index_all")
m.addConstr((27*x1)**2 + x3**2 <= 177, name="combined_cognitive_performance_index_squared")
m.addConstr(27*x1 + 14*x2 + x3 <= 177, name="combined_cognitive_performance_index_all_leq")

# Solve the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Milligrams of iron: {x1.varValue}")
    print(f"Grams of fiber: {x2.varValue}")
    print(f"Milligrams of vitamin A: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```