## Step 1: Define the symbolic representation of the variables
The variables are ['CBRN platoons', 'artillery batteries', 'water purification units', 'medical platoons']. Let's denote them as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The symbolic representation of the variables is:
- $x_0$ : CBRN platoons
- $x_1$ : artillery batteries
- $x_2$ : water purification units
- $x_3$ : medical platoons

## Step 3: Formulate the objective function
The objective function to minimize is: $4x_0^2 + 7x_0x_1 + 9x_0x_2 + 7x_0x_3 + 3x_1^2 + 2x_1x_2 + 8x_1x_3 + 5x_2x_3 + 5x_3^2 + 3x_0 + x_2$

## 4: List the constraints
The constraints are:
- $3x_0 \leq 117$ (logistics footprint of CBRN platoons)
- $6x_0 \leq 107$ (fun factor of CBRN platoons)
- $2x_1 \leq 117$ (logistics footprint of artillery batteries)
- $1x_1 \leq 107$ (fun factor of artillery batteries)
- $5x_2 \leq 117$ (logistics footprint of water purification units)
- $6x_2 \leq 107$ (fun factor of water purification units)
- $7x_3 \leq 117$ (logistics footprint of medical platoons)
- $6x_3 \leq 107$ (fun factor of medical platoons)
- $5x_2 + 7x_3 \geq 17$ (total logistics footprint of water purification units and medical platoons)
- $3x_0^2 + 5x_2^2 \geq 23$ (minimum logistics footprint of CBRN platoons squared and water purification units squared)
- $3x_0 + 2x_1 \geq 11$ (total logistics footprint of CBRN platoons and artillery batteries)
- $2x_1 + 5x_2 \geq 19$ (total logistics footprint of artillery batteries and water purification units)
- $3x_0 + 2x_1 + 5x_2 + 7x_3 \geq 19$ (total logistics footprint of all units)
- $6x_0 + 1x_1 \geq 14$ (total combined fun factor from CBRN platoons and artillery batteries)
- $6x_2 + 6x_3 \geq 20$ (total combined fun factor from water purification units and medical platoons)
- $6x_0 + 1x_1 + 6x_2 + 6x_3 \geq 20$ (total combined fun factor from all units)
- $3x_0 + 2x_1 \leq 106$ (maximum logistics footprint from CBRN platoons and artillery batteries)
- $2x_1^2 + 7x_3^2 \leq 97$ (maximum logistics footprint from artillery batteries squared and medical platoons squared)
- $2x_1^2 + 5x_2^2 \leq 59$ (maximum logistics footprint from artillery batteries squared and water purification units squared)
- $3x_0 + 5x_2 + 7x_3 \leq 101$ (maximum logistics footprint from CBRN platoons, water purification units, and medical platoons)
- $x_0, x_1, x_2, x_3$ are integers

## 5: Convert the problem into Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="CBRN_platoons", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="artillery_batteries", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="water_purification_units", vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(name="medical_platoons", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(4*x0**2 + 7*x0*x1 + 9*x0*x2 + 7*x0*x3 + 3*x1**2 + 2*x1*x2 + 8*x1*x3 + 5*x2*x3 + 5*x3**2 + 3*x0 + x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(3*x0 <= 117)
m.addConstr(6*x0 <= 107)
m.addConstr(2*x1 <= 117)
m.addConstr(x1 <= 107)
m.addConstr(5*x2 <= 117)
m.addConstr(6*x2 <= 107)
m.addConstr(7*x3 <= 117)
m.addConstr(6*x3 <= 107)
m.addConstr(5*x2 + 7*x3 >= 17)
m.addConstr(3*x0**2 + 5*x2**2 >= 23)
m.addConstr(3*x0 + 2*x1 >= 11)
m.addConstr(2*x1 + 5*x2 >= 19)
m.addConstr(3*x0 + 2*x1 + 5*x2 + 7*x3 >= 19)
m.addConstr(6*x0 + x1 >= 14)
m.addConstr(6*x2 + 6*x3 >= 20)
m.addConstr(6*x0 + x1 + 6*x2 + 6*x3 >= 20)
m.addConstr(3*x0 + 2*x1 <= 106)
m.addConstr(2*x1**2 + 7*x3**2 <= 97)
m.addConstr(2*x1**2 + 5*x2**2 <= 59)
m.addConstr(3*x0 + 5*x2 + 7*x3 <= 101)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("CBRN platoons: ", x0.varValue)
    print("artillery batteries: ", x1.varValue)
    print("water purification units: ", x2.varValue)
    print("medical platoons: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'CBRN platoons'),
        ('x1', 'artillery batteries'),
        ('x2', 'water purification units'),
        ('x3', 'medical platoons')
    ],
    'objective_function': '4*x0^2 + 7*x0*x1 + 9*x0*x2 + 7*x0*x3 + 3*x1^2 + 2*x1*x2 + 8*x1*x3 + 5*x2*x3 + 5*x3^2 + 3*x0 + x2',
    'constraints': [
        '3*x0 <= 117',
        '6*x0 <= 107',
        '2*x1 <= 117',
        'x1 <= 107',
        '5*x2 <= 117',
        '6*x2 <= 107',
        '7*x3 <= 117',
        '6*x3 <= 107',
        '5*x2 + 7*x3 >= 17',
        '3*x0^2 + 5*x2^2 >= 23',
        '3*x0 + 2*x1 >= 11',
        '2*x1 + 5*x2 >= 19',
        '3*x0 + 2*x1 + 5*x2 + 7*x3 >= 19',
        '6*x0 + x1 >= 14',
        '6*x2 + 6*x3 >= 20',
        '6*x0 + x1 + 6*x2 + 6*x3 >= 20',
        '3*x0 + 2*x1 <= 106',
        '2*x1^2 + 7*x3^2 <= 97',
        '2*x1^2 + 5*x2^2 <= 59',
        '3*x0 + 5*x2 + 7*x3 <= 101'
    ]
}
```