## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'SOC operators' and 'network administrators', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $9x_1^2 + 6x_2^2$.

## 3: Translate the constraints into symbolic notation
The constraints given are:
- $0.86x_1 + 0.09x_2 \geq 20$ (network latency constraint, but note the problem states "at least 20 milliseconds", so we consider it as $\geq 20$)
- $0.64^2x_1^2 + 0.79^2x_2^2 \geq 13$ (data accessibility impact squared constraint)
- $0.64x_1 + 0.79x_2 \geq 13$ (data accessibility impact constraint)
- $0.34x_1 + 0.12x_2 \geq 23$ (cost constraint)
- $7x_1^2 - 6x_2^2 \geq 0$ (specific quadratic constraint)
- $(0.86x_1)^2 + (0.09x_2)^2 \leq 76$ (network latency squared constraint)
- $0.64^2x_1^2 + 0.79^2x_2^2 \leq 32$ (data accessibility impact squared constraint, upper bound)
- $0.34^2x_1^2 + 0.12^2x_2^2 \leq 68$ (cost squared constraint)
- $x_1$ and $x_2$ must be integers.

## 4: Create a symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'SOC operators'), ('x2', 'network administrators')],
'objective_function': '9*x1^2 + 6*x2^2',
'constraints': [
    '0.86*x1 + 0.09*x2 >= 20',
    '0.64^2*x1^2 + 0.79^2*x2^2 >= 13',
    '0.64*x1 + 0.79*x2 >= 13',
    '0.34*x1 + 0.12*x2 >= 23',
    '7*x1^2 - 6*x2^2 >= 0',
    '(0.86*x1)^2 + (0.09*x2)^2 <= 76',
    '0.64^2*x1^2 + 0.79^2*x2^2 <= 32',
    '0.34^2*x1^2 + 0.12^2*x2^2 <= 68',
    'x1 % 1 == 0',
    'x2 % 1 == 0'
]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name="SOC_operators", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="network_administrators", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(9 * x1**2 + 6 * x2**2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(0.86 * x1 + 0.09 * x2 >= 20)
m.addConstr(0.64**2 * x1**2 + 0.79**2 * x2**2 >= 13)
m.addConstr(0.64 * x1 + 0.79 * x2 >= 13)
m.addConstr(0.34 * x1 + 0.12 * x2 >= 23)
m.addConstr(7 * x1**2 - 6 * x2**2 >= 0)
m.addConstr((0.86 * x1)**2 + (0.09 * x2)**2 <= 76)
m.addConstr(0.64**2 * x1**2 + 0.79**2 * x2**2 <= 32)
m.addConstr(0.34**2 * x1**2 + 0.12**2 * x2**2 <= 68)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("SOC operators: ", x1.varValue)
    print("Network administrators: ", x2.varValue)
else:
    print("The model is infeasible")
```